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Physics Reports 430 (2006) 211 – 276 www.elsevier.com/locate/physrep Multidimensional quantum dynamics and infrared spectroscopy of hydrogen bonds K. Giese a , M. Petkovi´ c a, b , H. Naundorf a , O. Kühn a , a Institut für Chemie und Biochemie, Freie Universität Berlin,Takustr. 3, D-14195 Berlin, Germany b Faculty of Physical Chemistry, University of Belgrade, Studentski trg 12, 11 000 Belgrade, Serbia Accepted 21 April 2006 editor: S. Peyerimhoff Abstract Hydrogen bonds are of outstanding importance for many processes in Chemistry, Biology, and Physics. From the theoretical perspective the small mass of the proton in a hydrogen bond makes it the primary quantum nucleus and the phenomena one expects to surface in a particular clear way are, for instance, zero-point energy effects, quantum tunneling, or coherent wave packet dynamics. While this is well established in the limit of one-dimensional motion, the details of the multidimensional aspects of the dynamics of hydrogen bonds are just becoming accessible to experiments and numerical simulations. In this review we discuss the theoretical treatment of multidimensional quantum dynamics of hydrogen-bonded systems in the context of infrared spectroscopy. Here, the multidimensionality is reflected in the complex shape of linear infrared absorption spectra which is related to combination transitions and resonances, but also to mode-selective tunneling splittings. The dynamics underlying these spectra can be unravelled by means of time-resolved nonlinear infrared spectroscopy. As a fundamental theoretical ingredient we outline the generation of potential energy surfaces for gas and condensed phase nonreactive and reactive systems. For nonreactive anharmonic vibrational dynamics in the vicinity of a minimum geometry, expansions in terms of normal mode coordinates often provide a reasonable description. For reactive dynamics one can resort to reaction surface ideas, that is, a combination of large amplitude motion of the reactive coordinates and orthogonal harmonic motion of the remaining coordinates. For isolated systems, dynamics and spectroscopy follow from the time-dependent Schrödinger equation. Here, the multiconfiguration time-dependent Hartree method is shown to allow for describing the correlated dynamics of many degrees of freedom. Classical trajectory based methods are also discussed as an alternative to quantum dynamics. Their merits and shortcomings are scrutinized in the context of incorporating tunneling effects in the calculation of spectra. For the condensed phase, reduced density operator based approaches such as the quantum master equation are introduced to properly account for the energy and phase relaxation processes due to the interaction of the hydrogen bond with its surroundings. © 2006 Published by Elsevier B.V. PACS: 33.20.Ea; 03.65.Sq; 03.65.Xp; 03.65.Yz; 82.20.w; 82.30.Rs; 82.53.k Keywords: Quantum dynamics; Intramolecular energy redistribution; Tunneling; Energy and phase relaxation Corresponding author. E-mail address: [email protected] (O. Kühn). 0370-1573/$ - see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.physrep.2006.04.005

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Page 1: Multidimensional quantum dynamics and infrared ...xbeams.chem.yale.edu › ~batista › v572 › rsurf_kuhn.pdf · Hydrogen bonds are of outstanding importance for many processes

Physics Reports 430 (2006) 211–276www.elsevier.com/locate/physrep

Multidimensional quantum dynamics and infrared spectroscopy ofhydrogen bonds

K. Giesea, M. Petkovica,b, H. Naundorfa, O. Kühna,∗aInstitut für Chemie und Biochemie, Freie Universität Berlin, Takustr. 3, D-14195 Berlin, GermanybFaculty of Physical Chemistry, University of Belgrade, Studentski trg 12, 11 000 Belgrade, Serbia

Accepted 21 April 2006

editor: S. Peyerimhoff

Abstract

Hydrogen bonds are of outstanding importance for many processes in Chemistry, Biology, and Physics. From the theoreticalperspective the small mass of the proton in a hydrogen bond makes it the primary quantum nucleus and the phenomena one expectsto surface in a particular clear way are, for instance, zero-point energy effects, quantum tunneling, or coherent wave packet dynamics.While this is well established in the limit of one-dimensional motion, the details of the multidimensional aspects of the dynamics ofhydrogen bonds are just becoming accessible to experiments and numerical simulations.

In this review we discuss the theoretical treatment of multidimensional quantum dynamics of hydrogen-bonded systems in thecontext of infrared spectroscopy. Here, the multidimensionality is reflected in the complex shape of linear infrared absorption spectrawhich is related to combination transitions and resonances, but also to mode-selective tunneling splittings. The dynamics underlyingthese spectra can be unravelled by means of time-resolved nonlinear infrared spectroscopy. As a fundamental theoretical ingredientwe outline the generation of potential energy surfaces for gas and condensed phase nonreactive and reactive systems. For nonreactiveanharmonic vibrational dynamics in the vicinity of a minimum geometry, expansions in terms of normal mode coordinates oftenprovide a reasonable description. For reactive dynamics one can resort to reaction surface ideas, that is, a combination of largeamplitude motion of the reactive coordinates and orthogonal harmonic motion of the remaining coordinates. For isolated systems,dynamics and spectroscopy follow from the time-dependent Schrödinger equation. Here, the multiconfiguration time-dependentHartree method is shown to allow for describing the correlated dynamics of many degrees of freedom. Classical trajectory basedmethods are also discussed as an alternative to quantum dynamics. Their merits and shortcomings are scrutinized in the context ofincorporating tunneling effects in the calculation of spectra. For the condensed phase, reduced density operator based approachessuch as the quantum master equation are introduced to properly account for the energy and phase relaxation processes due to theinteraction of the hydrogen bond with its surroundings.© 2006 Published by Elsevier B.V.

PACS: 33.20.Ea; 03.65.Sq; 03.65.Xp; 03.65.Yz; 82.20.−w; 82.30.Rs; 82.53.−k

Keywords: Quantum dynamics; Intramolecular energy redistribution; Tunneling; Energy and phase relaxation

∗ Corresponding author.E-mail address: [email protected] (O. Kühn).

0370-1573/$ - see front matter © 2006 Published by Elsevier B.V.doi:10.1016/j.physrep.2006.04.005

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212 K. Giese et al. / Physics Reports 430 (2006) 211 –276

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2122. Gas phase hydrogen bond Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

2.1. Anharmonic coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2152.2. Model potentials for hydrogen transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2192.3. Reaction surface Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

2.3.1. Reaction coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212.3.2. A full-dimensional all-Cartesian Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2242.3.3. Fixed reference case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2262.3.4. Flexible reference case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2262.3.5. Selection of relevant modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2262.3.6. Atomic reaction coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2272.3.7. Collective reaction plane coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2282.3.8. Reduced normal modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

3. Coherent quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2293.1. Mean-field vs. multiconfiguration methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2293.2. Laser-driven dynamics and intramolecular vibrational energy redistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

3.2.1. The case of a single minimum potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2313.2.2. The case of an asymmetric double minimum potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

3.3. Mode-specific tunneling splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2374. Semiclassical and classical trajectory-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

4.1. Semiclassical approximation for the calculation of spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2434.2. Classical trajectory approach for tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

4.2.1. The Makri–Miller model (MM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2454.2.2. The extended Makri–Miller model (EMM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2474.2.3. Application of the MM and EMM methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

5. Condensed phase Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2536. Dissipative quantum dynamics of hydrogen bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

6.1. Lineshape theories and nonlinear response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2556.2. Quantum master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2576.3. Vibrational energy cascading in an intramolecular hydrogen bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2596.4. HOD in D2O and H2O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2616.5. HAT and PT reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2636.6. Multidimensional IR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

7. Sundry topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2657.1. Geometric isotope effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2657.2. Excited state PT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2667.3. Laser control of hydrogen transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Appendix A. Determination of the SMC Hamiltonian parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Appendix B. Calculation of multidimensional eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Appendix C. Parities of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

1. Introduction

Hydrogen bonds (HBs) are of pivotal relevance for the understanding of the equilibrium and nonequilibrium propertiesof many molecular systems. The scope of HB research is really cross-disciplinary encompassing physics, chemistry,and biology [1–12]. The first published record of the concept of a HB as a new type of weak bond goes back to apaper by Latimer and Rodebush in 1920 [13,14]. But it was only in the 1930s that the interest revived, most notablyin the context of proton and hydroxyl ion mobility in water [15]. Their unusually high mobility had been knownsince the work of Grotthus in the early 19th century, but the first attempt for a microscopic understanding in termsof Born–Oppenheimer potential energy surfaces (PES) for the nuclear motion was provided by Huggins in 1936 only[16]. The breakthrough in recognition was probably the discourse upon HBs in Pauling’s “ The Nature of the ChemicalBond” [17]. It was also in the 1930s that the signatures of the formation of HBs in stationary infrared (IR) absorption

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 213

spectra had been realized (see, e.g., Ref. [18]). Soon thereafter hydrogen-bonding found its central place in biologywith its role in protein structure [19] and in particular for understanding the DNA [20]. There is no space to cover themany facets of the development in the years that followed. The state-of-affairs as of 1976 had been summarized inRef. [1] and an up-to-date compilation is forthcoming [12].

A new perspective on HBs emerged with the development of ultrafast time-resolved spectroscopies that started inthe 1980s. Having a time-resolution in the pico- and femtosecond domain it became possible to unravel the dynamicalprocesses underlying broad and sometimes structured stationary absorption spectra in particular in the condensed phase.First, dictated by experimental constrains, it has been proton transfer taking place after electronic excitation which wason the agenda [21,22]. During the last years, however, laser sources with the appropriate time resolution in the rangebetween 500 to 4000 cm−1 became available to nonlinear IR spectroscopy, opening the road to the observation ofreal-time HB dynamics in the electronic ground state and thus to the differentiation between the various mechanismscontributing to the lineshape [23].

It goes without saying that this development on the experimental side has triggered substantial theoretical efforts.Apart form the general complexity of condensed phase systems, theory is challenged by two specific features of HBswhich are already relevant in the gas phase: their quantum nature which often comes along with multidimensionality,that is, the coupled motion of many degrees of freedom (DOF). Interestingly, an early account on the multidimensionalnature of HB dynamics comes from the 1936 paper by Huggins [16].

Fig. 1 shows the (empirical) potential energy curve developed in Ref. [16] for the motion of a proton between twooxygen atoms as a function of the distance between the latter. Apparently, the shape of this potential changes from

Fig. 1. Empirical potential energy curves for the motion of a proton between two oxygen atoms in a HB (the proton coordinate is defined with respectto the center of the O–O distance vector). Upon decreasing the O–O distance the shape of the potential changes from a double to a single minimum(reprinted with permission from Ref. [16]. Copyright 1936 American Chemical Society).

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214 K. Giese et al. / Physics Reports 430 (2006) 211 –276

a double minimum for long distances to a single minimum for short ones. This has, of course, dramatic consequencesfor the proton wave function and the associated properties (for a respective recent simulation, see also Ref. [24]). Now,let us assume a situation where an O · · · O motion has been initiated, for instance, by some laser excitation [23] or isdriven by the fluctuations of the environment [25]. For the proton wave function this gives rise to a time-dependentpotential such that, e.g., the respective energy levels are modulated in time. The actual effect on the IR spectrummight range from a broadening to the appearance of some substructure, the details will depend on the time scale ofthe modulation as compared to the time scale of the O–H vibrational motion (∼ 10–20 fs) as well as on the type andstrength of anharmonic coupling.

Thus the presence of a HB, A.H · · · B, is reflected in the change of a number of properties as compared with the caseof a free A–H bond. This fact lends itself to the classification into weak, medium-strong, and strong HBs [4]. Takingthe perspective of IR spectroscopy, one observes a red-shift, a broadening, and the development of an often complexsubstructure of the band associated with the vibration of the bridging H atom. Although the actual numbers differ inliterature (compare, e.g., Refs. [4,26]) weak HBs can be characterized by a HB distance exceeding 2.8–3.2 A with theA–H stretching band being in the range between 3300 and 3500 cm−1. Medium strong or moderate bonds have a HBlength of about 2.5–3.2 A and an absorption in the range of 2600–3300 cm−1, while in strong HBs the bond lengthis smaller than 2.5 A and the absorption is found in the range from 2100 cm−1down to about 1000 cm−1. Specificexamples will be discussed in the following.

The strength of the HB comes along with a typical shape of the PES for the motion of the H atom. For weak HBshaving a relatively large HB distance one often finds a symmetric (A· · ·A) or asymmetric (A · · · B) double minimumpotential as shown for the symmetric case in Fig. 2a. In particular in the symmetric case one has besides the localvibrational motion to account for the tunneling between the different wells. Viewed from the time-domain perspectivethis gives a contribution to the H transfer probability as indicated, e.g., by the deviation of the temperature dependenceof the rate coefficients from the Arrhenius behavior [27]. In the frequency domain, tunneling gives rise to a doubling ofvibrational states and shows up, e.g., in corresponding microwave [28], infrared [29], or fluorescence excitation [30]spectra. The tunneling splitting � is related to the tunneling period T via � = h/T .

Medium strong HBs have a much lower barrier and therefore a larger tunneling splitting is to be expected. Thecase of an asymmetric bond is shown in Fig. 2b. Here, the dynamics is mostly that of a rather anharmonic vibration.However, due to the coupling between the A–H and A · · · B motions, this dynamics will be multidimensional as well.For strong HBs the barrier becomes small or negligible and the zero-point energy (ZPE) of the anharmonic vibrationalmotion may exceed the barrier height as shown in Fig. 2c.

In general one distinguishes hydride, H atom (HAT), and proton (PT) transfer depending on whether two, one orno electron charge is transferred along with the positively charged nucleus [31]. The former two cases can be viewedas proton-coupled electron transfer reactions and, indeed, often the distinction is not very clear and one has to beaware that, for instance, the transfer of the proton can be coupled to the simultaneous transfer of the respective negativecharge along a different pathway, e.g., in small organic molecules through the conjugated system [32]. This mechanisticfeature has, of course, drastic consequences for the actual reaction especially in a polar environment [25]. Leavingaside these electrostatic aspects and returning to the multidimensional picture of nuclear dynamics one can envisage

ener

gy

ener

gy

ener

gy

reaction coordinate reaction coordinate reaction coordinate(a) (b) (c)

Fig. 2. Potential energy curves typical for weak symmetric (a), moderate asymmetric (b), and strong symmetric (c) HBs. Vibrational energy levelsin of the potentials are also sketched.

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 215

two fundamental types of couplings between the HAT or PT reaction coordinate and motions of the surrounding [33].First, there will be promoting modes which modulate the potential along the reaction coordinate such as to reduce thebarrier (cf. Fig. 1). The second type of modes are those whose motion is required to change the molecular structurefrom the reactant to the product configuration, e.g., accounting for the rearrangement of single and double bonds. Theseare also called reorganization modes [33]. The effect of these couplings can be observed as mode-specific tunnelingsplitting, for instance, in electronic excitation spectra of tropolone which has an intramolecular HB [34]. It has beenassociated with the promotion of H atom tunneling along ammonia chains [35] and in a broader sense it also plays arole, e.g., in hydride transfer in enzyme catalysis [36,37].

The multidimensionality of HB quantum dynamics and its spectroscopic signatures is the central theme of thisreview. First, we will consider the problem of defining and calculating multidimensional PES in the gas phase inSection 2. This section is divided into the cases of nonreactive (Section 2.1) and reactive (Sections 2.2 and 2.3) PES.Special emphasis is put on the detailed derivation of a Cartesian reaction surface (CRS) Hamiltonian, which provides astraightforward road to a full-dimensional simulation of about 100–200 DOF by ab initio quantum chemical methods.Having defined the Hamiltonian, the solution of the time-dependent multidimensional nuclear Schrödinger equationby means of the multiconfiguration time-dependent Hartree (MCTDH) method is briefly presented in Section 3.1. Itis emphasized that by construction the CRS Hamiltonian is ideally suited for the combination with MCTDH wavepacket propagations. First, this is illustrated by exemplary calculations of ultrafast IR laser-driven dynamics initiatingintramolecular vibrational energy redistribution (IVR) in a single and a double minimum HB system in Section 3.2.Next, the issue of vibrationally mode-specific tunneling is addressed in Section 3.3. Using the case of intramolecularHAT in tropolone as an example, the determination of an appropriate CRS Hamiltonian is exercised in some detail.

The favorable scaling of classical molecular dynamics with the number of DOF keeps triggering efforts aimedto incorporate approximate quantum effects into such simulations. In Section 4 we first briefly review semiclassicalapproaches which are built on the WKB formalism. This sets the stage for the subsequent discussion of different typesof tunneling trajectories whose performance is tested for the case of double HAT in carboxylic acid dimers.

Condensed phase processes are addressed in Sections 5 and 6. First, the system–bath approach to condensed phaseproblems is introduced in Section 5. Next, we summarize different line shape models of HB systems as well as theresponse function formalism, vital for the analysis of ultrafast nonlinear IR spectroscopies (Section 6.1). The responsefunctions can be simulated by means of a quantum master equation (QME) which is discussed in Section 6.2. As anapplication we give results for the ultrafast relaxation dynamics of an intramolecular HB system which highlights itsmultidimensional nature by virtue of a solvent-assisted energy cascading after OH-stretch excitation (Section 6.3).Subsequently, we give a brief summary of the vibrational dynamics of HOD in D2O and H2O which has been ratherextensively studied by means of nonlinear IR spectroscopy in recent years. After a short account on reactive PT andHAT dynamics in the condensed phase has been given in Section 6.5 we focus on multidimensional IR spectroscopyas a novel means to unravel complex spectra in the condensed phase. In particular it is shown how 2D-IR spectroscopyapplied to a generic dissipative HAT model can be used to scrutinize details of the system–bath coupling.

While this review is focussed on ground state dynamics and IR spectroscopy we have summarized closely relatedtopics in Section 7. First, we address the properties of the vibrational ground state wave function itself and discusshow multidimensionality appears as the interdependence of primary and secondary geometric isotope effects (Section7.1). Second, we briefly quote some results on electronic excited state dynamics where extremely fast PT is initiatedby photoexcitation (Section 7.2). Third, we give a perspective on laser control of PT and HAT in Section 7.3. Finally,we provide a synopsis in Section 8.

2. Gas phase hydrogen bond Hamiltonian

2.1. Anharmonic coupling

In the following we will focus on the properties of adiabatic PES for the electronic ground state assuming the validityof the Born–Oppenheimer approximation for the decoupling of electronic and nuclear DOF [38]. In passing we note thatthere are recent promising attempts to treat electrons and protons on the same footing by separating off the coordinatesof the heavy atoms only (for an overview, see Ref. [39]). The vibrational properties of molecules in the vicinity ofa stationary point on the Born–Oppenheimer PES are often quite accurately described by means of normal modecoordinates. Denoting the Cartesian coordinate vector for the Nat atoms as x = (x1, . . . , xNat ) with xi = (xi,x, xi,y, xi,z)

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216 K. Giese et al. / Physics Reports 430 (2006) 211 –276

and the mass-weighted Cartesian vector as X with Xi,� = xi,�√

mi (� = x, y, z), the normal mode vector Q is definedas a linear combination of Cartesian displacements from some reference geometry Xref by

Q = L(X − Xref), (1)

where L is the unitary transformation matrix containing the eigenvectors of the matrix of the second derivatives of thepotential (Hessian) at the reference geometry. At a minimum of the PES, one has 3Nat − 6 eigenvectors describingvibrational motion and, respectively, three eigenvectors for infinitesimal rotation and translation. The correlation be-tween the HB length and the shape of the potential for the H motion expressed in Fig. 1 strongly suggests that thedynamics of HBs and in particular their IR spectra can only be understood in terms of multidimensional PES for thecoupled DOFs. In other words, the harmonic approximation for the motion of the set of normal mode coordinates,{Qi}, will not be adequate for the present purpose. Nevertheless, it provides a starting point for developing a principalunderstanding in terms of simple models. Early qualitative ideas have been developed in the 1940s [40] and werereviewed by Sheppard [41] as well as Bratos and Hadži [42]. The first quantitative assessment was given by Witkowskiand Marechal for the IR spectra of carboxylic acid dimers [43,44].

The standard anharmonic model Hamiltonian [26,45] for discussing the IR spectra of weak HBs consists of an A–Hstretching oscillator, Q�, coupled to an A · · · B oscillator, Q�, via a cubic anharmonicity K���:

H = T� + T� + 12�2

�Q2� + 1

2�2�Q

2� + K���Q

2�Q�, (2)

where T� and T� denote the respective uncoupled kinetic energies containing the conjugate momenta {Pj }. Notice thatthis treatment neglects any effect of HAT or PT. Next, one assumes that the dynamics of Q� is much faster than that ofQ�. As a consequence it is reasonable to rewrite Eq. (2) by introducing an effective frequency for the fast coordinate:

�eff(Q�) = ��

√1 + 2K���

�2�

Q�. (3)

To simplify matters this expression is considered in a Taylor expansion up to first order in Q� only which gives

�eff(Q�) ≈ �� + K���

��Q�. (4)

Making use of the time scale separation between the two modes one can invoke the so-called second Born–Oppenheimerseparation. This amounts to introducing the adiabatic quantum states, �(adiab)

�AH (Q�), of the fast mode Q� which dependparametrically on the slow mode coordinate Q�. For the respective adiabatic energies one has

E(adiab)�AH

(Q�) = (�AH + 12 )�eff(Q�). (5)

This in turn gives the following shifted oscillator PES for the motion of the slow mode

V�AH(Q�) = 1

2�2

�Q2� +

(�AH + 1

2

)(�� + K���

��Q�

). (6)

Notice that it is to be expected that the HB contracts upon excitation since the elongation of the A–H bond in the excitedstate of the anharmonic PES will lead to a strengthening of the HB. This will lead to a shortening of the A · · · B distancewhich is reflected in the minima of the oscillator PES in Eq. (6) which are located at Q

(0)

� =−(�AH + 1/2)K���/���2�.

The situation is sketched in Fig. 3 where we also introduced the quantum states of the slow mode. From this figure itbecomes clear that the problem has been reduced to a model which is well-known from electron-vibrational spectroscopy[38]. Therefore, it is not surprising that the understanding of the related IR spectra heavily borrows from respectivetheories for electronic transitions. For instance, based on Fig. 3 one would anticipate that the IR spectrum shows aFranck–Condon type progression of lines with the intensity pattern depending on the relative shift of the PES. Thisholds in particular if it is assumed that the dipole moment function depends on the fast coordinate only, i.e.,

�(Q�, Q�) ≈ �0 + ��

�Q�Q�. (7)

Upon increasing the HB strength the IR spectrum may become broader and more structured, which signals that additionalanharmonicities have to be included. Besides the anharmonicity of theA–H stretching vibration, it is the so-called Fermi

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 217

νAH = 0

νAH =1

ener

gy

A .... B

Fig. 3. The adiabatic separation between the fast A–H oscillator coordinate and the slow A · · · B coordinate allows for the definition of shiftedoscillator PES for each quantum state of the fast coordinate.

resonance coupling between the A–H stretching fundamental and the A–H bending first overtone transition which isimportant. Denoting the bending coordinate as Q� we have the modified Hamiltonian

H = T� + T� + T� + 12�2

�Q2� + K���Q

3� + 1

2�2�Q

2� + 1

2�2�Q

2� + K���Q

2�Q� + K���Q�Q

2�. (8)

Now there are two fast modes and the adiabatic separation scheme needs to be modified. This is most conveniently doneby introducing appropriate zero-order states for the uncoupled stretching and bending motions. Denoting the latter by|�AH〉 and |�AH〉, respectively, and expanding the adiabatic wave function in terms of the product basis {|�AH〉|�AH〉}yields a Hamiltonian matrix for the fast mode where the zero-order states are coupled via matrix elements of the type〈�AH|K���Q�Q

2�|�AH〉. Diagonalization of this matrix will yield eigenstates where the stretching fundamental and

the bending overtone zero-order states are mixed. This usually leads to oscillator strength borrowing of the bendingovertone from the stretching fundamental which gives a typical double peak structure in the spectrum. The regionbetween the two peaks is commonly called Evans window. A second quantization description of this approach has beengiven in Refs. [46,47] (cf. the review in Ref. [48]).

In principle this strategy can be extended to encompass strong HBs as well, e.g., by incorporating additional modesor by including higher-order terms in the PES and dipole moment surface [26]. In any case, one should be aware that thepicture given so far may fail due to oversimplification when it comes to a quantitative comparison with the experimentbased on ab initio quantum chemical anharmonic force constants. This does not mean that force constants treated asempirical parameters would not have given a reasonable fit to some experimental data. The importance of higher-orderterms has been nicely illustrated for the moderate HB in the acetic acid dimer in Ref. [49], where force constants up to6th order have been necessary to explain the fine structure of the spectrum in the OH-stretching region.

The expansions of the PES in terms of normal mode coordinates discussed so far are in fact only certain approxi-mations of a more general PES V (Q1, . . . , Q3Nat−6). Expressed in normal mode coordinates the general form of the

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218 K. Giese et al. / Physics Reports 430 (2006) 211 –276

Fig. 4. IR stick spectrum of malonaldehyde as calculated from a 4D model PES including the OH-stretching as well as the O · · · O mode shown inthe upper part of the figure. The PES does not include the possibility of HAT. (For more details, see also Ref. [52].)

Hamiltonian has been given by Watson [50]. It reads

H = 1

2

∑�,=x,y,z

(J� − �)I�(J − ) − h2

8

∑�=x,y,z

I�� + 1

2

3Nat−6∑i=1

P 2i + V (Q1, . . . , Q3Nat−6). (9)

Here, J� are the rotational operators, the 3 × 3 matrix I is the inverse of the moments of inertia tensor, and � =∑ij �

�ijQiPj is the so-called vibrational angular momentum. It contains the Coriolis coupling constants ��

ij=ε��∑Nat

k=1×Lk,iLk�,j with ε�� being the unit antisymmetric tensor and L has been given in Eq. (1) (note that in the present casethe composite index k (k = 1, . . . , Nat, = x, y, z) is used).

Being interested in intramolecular HB vibrations of larger molecules one can restrict the discussion to the case ofno rotation, J� = 0. Furthermore, since the first two terms of Eq. (9) are proportional to the inverse of the moments ofinertia they are usually assumed to be negligible. Notice, however that especially for small molecules this will not bethe case (see, e.g., Ref. [51]).

In principle the PES entering Eq. (9) can be calculated on a numerical grid which circumvents the calculation ofhigher-order derivatives. An example is given in Fig. 4 where we show the IR absorption spectrum of malonaldehydecalculated on the basis of a full 4D PES containing the OH-stretching, OH-in- and out-of-plane bending as well as alow-frequency O · · · O mode [52]. Most interesting in terms of H-bonding is the region between 2700 and 3100 cm−1.Here, one finds three almost equally spaced lines which would suggest a Franck–Condon progression in the OH-stretching vibration with respect to the O · · · O mode. However, the spacing is not fitting the O · · · O frequency and the

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 219

fact that the two spacings are equal is a coincidence. Analysis of the respective eigenstates reveals the strong mixing inparticular with the in-plane OH-bending vibration. This situation resembles Eq. (8), although the PES is substantiallymore anharmonic [52].

With increasing number of DOFs, the effort to calculate the full-dimensional PES increases exponentially. Neverthe-less, since the importance of correlations between modes is likely to decrease with the number of participating modesit is reasonable to expand the PES with respect to such correlations [53]:

V (Q1, . . . , Q3Nat−6) =∑

i

V (1)(Qi) +∑i<j

V (2)(Qi, Qj ) +∑

i<j<k

V (3)(Qi, Qj , Qk) + . . . , (10)

where the so-called n-mode correlation potentials, V (n), have been introduced and the summations run over distinctpairs, triples, etc. In passing we note that any force field PES based on a Taylor expansion as discussed above hasthis form.

In principle, normal mode coordinates can also be used for reactive problems. For this purpose, it is advantageousto use the transition state geometry as the reference for the Taylor expansion. This strategy has been applied to e.g. thecalculation of the infrared spectrum of H3O−

2 in Ref. [54]. It is clear, however, that normal mode coordinates are notlikely to give a very compact representation of the PES away from the reference point. Here, proper large amplitudecoordinates usually will perform much better as will be discussed below. Finally, we remark on the possibility ofextracting normal modes and associated frequencies from harmonically driven classical molecular dynamics whichavoids the calculation of second derivatives altogether [55].

2.2. Model potentials for hydrogen transfer

There is a widely used set of two-dimensional model PES which captures different coupling types between large andsmall amplitude motions (see, e.g., Refs. [56–59]). The dynamics and the associated spectra depend, of course, on thesymmetry of the PES. In the following we will consider symmetric potentials appropriate for A–H · · · A situations. Thiswill provide the background for the discussion of the tunneling splittings in Sections 3.3 and 4. The asymmetry requiredto model the case A–H · · · B can be easily incorporated by adding, e.g., a term linear in the reaction coordinate. Startingpoint is the most simple polynomial expression of a symmetric double-well potential given by the square-quarticHamiltonian,

V (1)(X) = − 12aX2 + 1

4cX4, (11)

where only the large amplitude coordinate X is considered; a and c are constants that determine the two minima atXmin = ±√

a/c and the barrier height �E� = a2/4c. The saddle point is at X = 0. A small amplitude vibration alongthe mass-weighted normal coordinate Q is modeled by the harmonic oscillator PES,

V (1)(Q) = 12 2Q2, (12)

where is the frequency of the oscillator. If the two coordinates are coupled, the dynamics is governed by theHamiltonian

H = P 2X

2+ P 2

Q

2+ V (X, Q) (13)

with the PES having the form, Eq. (10), i.e.

V (X, Q) = V (1)(X) + V (1)(Q) + V (2)(X, Q). (14)

The large amplitude coordinate is anti-symmetric with respect to the molecular symmetry transformation T (permutationand subsequent rotation) while the small amplitude coordinate Q can either be symmetric or anti-symmetric withrespect to T. Thus, if there is a minimum at (X0, Q0) there must be an equivalent minimum either at (−X0, Q0) or at(−X0, −Q0).

The case V (2)(X, Q) = �X2 Q corresponds to the symmetrical mode coupling (SMC) with � denoting thecoupling constant and Q is assumed to be symmetric with respect to the molecular symmetry transformation T.

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220 K. Giese et al. / Physics Reports 430 (2006) 211 –276

The potential, VSMC, can be written as a displaced harmonic oscillator coupled to a double-well,

VSMC = − 12aX2 + 1

4 cX4 + 12 2

(Q + �

2 X2)2

, (15)

with a new constant c = c − 2�. The displacement of the oscillator Q(0) =−�X2/ 2 is proportional to the square of X.The saddle point and minima are at (0, 0) and (±√a/c, −�a/c 2), respectively. The barrier height is �E� = a2/4c.Fig. 5 (top) shows an SMC potential with typical parameters. Two characteristic paths are also given: the intrinsicreaction path (IRP), i.e. the path of steepest descent (in mass-weighted coordinates) from the barrier top and a straightline path connecting the minima.

The SMC Hamiltonian depends on four parameters, a, c, , and �. The set of parameters can be reduced withoutrestrictions by introducing dimensionless positions (x, q) by [56]

X = xmx, (16)

Q = xmq, (17)

where xm =√a/c is the minimum position of x, and by dividing the SMC Hamiltonian by 8�E� = 2a2/c. The new

SMC Hamiltonian HSMC reads

HSMC = −g2

2

(�2

�x2 + �2

�q2

)+ 1

8(x + 1)2(x − 1)2 + �2

2

[q + �

�2 (x2 − 1)]2

, (18)

where the constant 1/8 is added and the oscillator DOF is shifted by �/�2. The new parameters are defined by

g = hc/a√

2a = h x/8�E�, (19)

� = /√

2a = / x , (20)

� = �/2√

ac, (21)

where x =√2a is the harmonic frequency corresponding to the minima of the original SMC Hamiltonian. It depends

on three parameters only. Formally, the parameter g corresponds to the reduced Planck constant of the new SMCHamiltonian. Moreover, it is related to the dimensionless mass m by m=g−1/2 where h=1. The remaining parameters,� and �, have the same meaning as the original and �, respectively. In Ref. [60] a simple method was proposed todetermine the SMC parameters. It will be addressed in Appendix A (cf. Section 2.3.1) and an application is given inSection 4.2.3.

The case

V (2)(X, Q) = �XQ (22)

corresponds to the anti-symmetric mode coupling where the small amplitude coordinate Q is assumed to be anti-

symmetric with respect to T. The saddle point is at (0, 0); the minima are at Xmin = ±√

a/c + �2/c 2 and Qmin =−�Xmin/ 2. A typical PES is shown in Fig. 5 (middle).

The case V (2)(X, Q)=�X2 Q2 is called squeezed coupling. The small amplitude coordinate may either be symmetricor antisymmetric. The saddle point and minima are at (0, 0) and (±√

a/c, 0), respectively. A typical instance of thePES is shown in Fig. 5 (bottom). The PES can be written as an oscillator with X-dependent frequency �(X),

VSQZ = V (1)(X) + 12�2(X)Q2, (23)

�(X) =√

2 + 2�X2, (24)

where for � > 0 the mode is weakened upon approaching the saddle point. The squeezed coupling case is typicallyrealized for out-of-plane modes in, e.g., tropolone [61].

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 221

-6.0

-4.0

-2.0

0.0

2.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

X

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

QQ

Q

Fig. 5. Contour plots of instances of PES showing three coupling types (see text). The contour line spacing is 1/5 of the respective barrier height. TheIRP (thick black line; schematic) and straight line paths (dashed) are indicated. Top: SMC (Eq. (15), a = c = 1, = 0.25, and � = 3/32). Middle:Eq. (22), a = c = 1, = 0.25, and � = 0.0884. Bottom: Eq. (23), a = c = 1, = 0.25, and � = 0.2. The straight line path coincides with the IRP.

2.3. Reaction surface Hamiltonian

2.3.1. Reaction coordinatesThe generic Hamiltonians discussed in the previous section describe a large amplitude reaction coordinate coupled

to an orthogonal small amplitude harmonic vibration. While the parameters of such types of Hamiltonians can be easilyobtained from quantum chemistry calculations as indicated, it is desirable to have a more rigorous method for obtaining

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222 K. Giese et al. / Physics Reports 430 (2006) 211 –276

a Hamiltonian which does not make too many a priori assumptions concerning the dimensionality or functional formof the coupling between large and small amplitude coordinates. There are two general concepts in this respect. First thedescription of vibrations and in particular of IR spectra in terms of normal mode vibrations [62]. Second, the modelingof chemical reactions by means of a reaction path [63]. Both concepts have been blended in the Miller–Handy–Adamsreaction path Hamiltonian [64]:

H = 1

2

(ps −∑klQkPlBkl(s))

2

(1 +∑kQkBk,3Nat−6(s))

2 + V0(s) + 1

2

∑k

(P 2k + �2

k(s)Q2k). (25)

Here, s is the arc length coordinate along the IRP and Qk (k =1, . . . , 3Nat −7) are the orthogonal harmonic DOF. (Theps and Pk are the respective conjugate momenta.) The potential along the reaction coordinate s is given by V0(s) and thekinetic coupling between the different DOF can be calculated from the knowledge of the normal mode transformationvector (cf. Eq. (1)) for mode k, Lk(s), as well as its derivative with respect to s, L′

k(s), via Bkl(s) = L′k(s) · Lk(s) (here

Lk denotes the kth column vector of L, see Eq. (1)).

For small curvature, �(s) =(∑

kB2k,3Nat−6(s)

)1/2, a vibrationally adiabatic approximation is reasonable. It was

shown that the effect of small IRP curvature can be accounted for by an effective s-dependent mass �eff(s) in theadiabatic Hamiltonian [65],

H(ps, s) = 1

2�eff(s)p2

s + V0(s) + E0(s), (26)

where E0(s) is the ZPE of the harmonic modes [64]. If the reaction path curvature is too large then the reaction swathhas to be taken into account. Moreover, the orthogonal DOF Q may become nonunique for distances away from thereaction path that exceed the radius of curvature. For the case of large reaction path curvature a straight line reactionpath Hamiltonian has also been suggested in Ref. [66]. There have been a number of applications of the reaction pathconcept to the determination of vibrational spectra (e.g. Refs. [67–70]), even including anharmonic corrections for theorthogonal modes [71], or rate coefficient calculations (e.g. Ref. [72]).

A large reaction path curvature indicates large couplings among the reactive and the orthogonal DOF. This isreminiscent of the behavior of general heavy-light-heavy reactions [73]. To cope with this situation it has been suggestedto use more than one reaction coordinate. In the reaction surface Hamiltonian approach (and variations thereof), insteadof the intrinsic reaction coordinate, few internal coordinates (i.e., bond lengths r) serve as large amplitude DOF whilethe remaining coordinates (Q) are typically optimized such as to minimize the energy for a given point on the reactionsurface. Their motion is then again accounted for within harmonic approximation [74–77].

The kinetic energy operator of the aforementioned approach has a rather complicated form making a numericaltreatment difficult. Using Wilson’s G-matrix notation one has [75,76]

T = 1

2(pr , PQ)

(Grr GrQ

GQr GQQ

)(pr

PQ

), (27)

where pr and PQ are the vectors of momenta for the large and small amplitude coordinate, respectively. An assessmentof various approximations to Eq. (27) in terms of the vibrational spectrum has been given recently in Ref. [78].

A much simpler structure of the kinetic energy operator is obtained by adopting a formulation based on Cartesiancoordinates such as the CRS Hamiltonian by Ruf and Miller [79]. The CRS Hamiltonian relies on the selection offew atomic Cartesian coordinates that perform large amplitude displacements during the reaction. For instance, forplanar HAT systems the in-plane coordinates of the reacting hydrogen are a reasonable choice. The remaining CartesianDOF are considered to perform only small amplitude displacements and the full PES is approximated by a secondorder Taylor expansion with respect to the small amplitude DOF. However, there may be significant couplings betweensmall amplitude collective coordinates and large amplitude atomic coordinates just because a complete separationbetween these two does not necessarily yield the most compact representation of the PES. For the intramolecular HATin tropolone, the situation is illustrated in Fig. 6. Panel (a) shows an overlay of the minimum geometry (Cs symmetry)and saddle point geometry (C2v symmetry). A significant motion of the heavy atoms upon approaching the saddlepoint geometry is visible; especially the oxygen-oxygen distance is shortened. The chemical bonding has significantlychanged between the two considered geometries, i.e., the PES seen by the two oxygens differs also quite strongly. Thus,

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 223

C C

O O

H

C C

O O

H

(a) (b) (c)

C C

O O

H

Z

Y

Fig. 6. The OCCO–H fragment of tropolone. (a) Overlay of minimum XR (filled circles) and saddle point XTS (open circles) geometry. Coordinateorigin is the center of mass. (b) Atom displacement corresponding to direction w1 (anti-symmetric). The displacement of the remaining atoms issmall. (c) Same for direction w2 (symmetric).

the choice of the atomic (y, z) coordinates leads to a strong coupling of large amplitude (H-atom) and small amplitudecoordinates.

All geometries X (mass-weighted) of the molecule span a N-dimensional vector space, where N is the total numberof DOF. Thus, instead of Cartesian coordinates of individual atoms one may alternatively choose any orthonormal setof N-dimensional vectors to describe large amplitude changes of the molecular geometry. A suitable choice of suchvectors has been proposed by Takada and Nakamura [60]. Let XR be the 3 · Nat dimensional mass-weighted vectorcorresponding to the geometry of one minimum (denoted as right). The center of mass is at the origin. Likewise, letXL and XTS correspond to the other minimum (denoted as left) and the saddle point geometry, respectively. The left(XL) and right (XR) minimum is only unique up to an arbitrary rotation. This arbitrariness is removed by rotating XLaround XR in order to minimize the distance |XR −XL| [60]. Correspondingly, the saddle point geometry XTS is rotatedaround the center geometry, XC = (XR + XL)/2, in order to minimize |XC − XTS|. Then, the two vectors,

w1 = XR − XL

|XR − XL| , (28)

w2 = XC − XTS

|XC − XTS| , (29)

span the so-called reaction plane [80]. The vector w1 corresponds to the direct tunneling direction, i.e., the straightline connecting both minima and the center geometry; the vector w2 points from the saddle point towards the centergeometry. Thus, for a symmetric double well system the displacement vector XR − XTS is partitioned into the twosymmetry components, i.e. w1 and w2 for the anti-symmetric and symmetric motion, respectively. The two directionsw1 and w2 are orthogonal, w1 · w2 = 0, because they transform according to different irreducible representations.

In panels (b) and (c) of Fig. 6 we show the vectors w1 and w2 defined in Eqs. (28) and (29) for the case of tropolone.Notice, that w1 [panel (b)] is similar to the y coordinate, while w2 [panel (c)] describes a concerted motion of thehydrogen together with the oxygens. Notice that the reaction plane concept was used previously by Yagi et al. [80] asa guide for the relevant region of configuration space in their full-dimensional treatment of malonaldehyde. Moreover,the idea was implicitly used by other authors in the context of an anharmonic expansion of the PES for the samemolecule [81].

The relevance of the reaction plane definition can be appreciated in the context of reaction paths proposed byGarrett and Truhlar [82]. They considered a family of paths parameterized by a parameter � (0���1), which forthe case of two equivalent minima coincides with the two turning points on the IRP, ±s0. The reaction path is thengiven by

X(s; �) = (1 − �)X(s) + �[X(s0) + (s − s0)(X(s0) − X(−s0))/2s0], (30)

where X(s) is the coordinate representation of the IRP. For � = 0 this path coincides with the IRP and correspondsto the tunneling path in the small-curvature limit; for � = 1 it is a straight line connecting the two turning pointsand corresponds to the large-curvature limit. It was shown that the least action principle applied to this path yields

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224 K. Giese et al. / Physics Reports 430 (2006) 211 –276

reasonable semiclassical tunneling splittings for molecules like malonaldehyde [83,84], tropolone [84], and variouscarboxylic acid dimers [85]. In passing we emphasize, however, that the family of Garrett–Truhlar paths covers a curvedbut two-dimensional submanifold of the full-dimensional configuration space only.

The direction w1 is essentially the straight line part of the family of Garrett–Truhlar paths. Based on these observations,the physical relevance of the reaction plane can be established, when it is possible to show that the IRP (the otherextremum of the family of Garrett–Truhlar paths) lies approximately in the reaction plane. To be specific, consider theprojection of the IRP onto the reaction plane,

X(s) = [(X(s) − XTS) · w1]w1 + [(X(s) − XTS) · w2]w2 + XTS. (31)

The difference between the IRP and its projection can be expressed by the root mean squared atomic displacement,

�(s) = 1√Nat

√(x(s) − x(s))2, (32)

where lower case letters refer to non-mass-weighted coordinates. The smaller �(s) the closer is the IRP to the reactionplane.An example for a comparison between a full-dimensional and a reaction plane IRP is given in Fig. 7 for tropolone.Apparently, �(s) is rather small indicating the relevance of the reaction plane definition for this case.

2.3.2. A full-dimensional all-Cartesian HamiltonianIn the following we will give a derivation of a full-dimensional reaction surface Hamiltonian in terms of generic

reaction coordinates v1 and v2 with corresponding vectors v1 and v2. As discussed in Section 2.3.1 there are at leasttwo reasonable choices for reaction coordinates (cf. Fig. 6). First, one may choose (v1, v2) to equal certain atomiccoordinates, for instance, the position (y, z) of the reactive hydrogen atom in tropolone. This is identical to the choiceof Ruf and Miller [79]. Second, one may choose (v1, v2) to equal the reaction plane coordinates (w1, w2) discussed inSection 2.3.1 [86]. Note, that in the following there is no inherent restriction as far as the number of reactive DOFs isconcerned.

The reaction coordinates (v1, v2) are assumed to be orthogonal to the 3 infinitesimal translational vectors as well asto the three infinitesimal rotational vectors corresponding to the saddle point geometry XTS. For two generic reactioncoordinates there is a (N − 2)-dimensional subspace of the space of all geometries X, where N is the number of DOF,i.e., N =3 Nat for a molecule with Nat atoms. A basis for this subspace can be obtained, e.g., by considering a projectionof the full-dimensional Hessian at the saddle point geometry onto the subspace. Let U(X) be the full-dimensional PES,then the full-dimensional Hessian at the saddle point is given by

K(f)ij (XTS) = �2U

�Xi�Xj

∣∣∣∣XTS

. (33)

The projector onto the subspace is given by I − P, where I is the identity matrix and P is the projector onto the spacespanned by v1 and v2,

P = v1vT1 + v2vT

2 . (34)

A diagonalization of the projected matrix,

(I − P)K(f)(XTS)(I − P), (35)

yields N eigenvectors; the set of eigenvectors is denoted as {v1, v2, e1, . . . , eN−2}. There are eight eigenvectors withvanishing eigenvalue: two correspond to the reaction coordinate vectors v1 and v2, the others correspond to the 3 infinites-imal translations, and to the three infinitesimal rotations with respect to the saddle point geometry. For convenience, theset of eigenvectors is ordered such, that the 6 translational/rotational vectors are eN−7, . . . , eN−2. The remaining N −8vectors ej are the normal modes of the saddle point with respect to the subspace. Coordinates corresponding to theeigenvectors ej including rotation and translation are denoted as Qj . For systems with two symmetrically equivalentminima, the eigenvectors ej and the associated coordinates Qj are either symmetric or anti-symmetric with respect tothe molecular symmetry transformation T.

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 225

Fig. 7. Comparison between a reaction plane and a full-dimensional IRP for the intramolecular HAT in tropolone using a DFT/B3LYP (6−31+G(d))

PES. (a) Contour plot of the potential cut along the reaction plane spanned by w1 and w2 (in a0 amu1/2). The contour line spacing is 500 cm−1andthe maximum contour line is at 6000 cm−1. The thick black line is the projection of the IRP onto the reaction plane X(s), and the barrier height is2161 cm−1. (b) Root mean squared atomic displacements Eq. (32) for the full-dimensional IRP geometries on the approximate PES (solid line) andfor the quantum chemically determined PES (filled squares). For more details, see Ref. [86].

Any geometry X can be expanded in terms of the set of eigenvectors discussed so far,

X = XTS + v1v1 + v2v2 +N−2∑j=1

Qj ej , (36)

where the saddle point geometry serves as a reference, i.e., the coordinates (v1, v2, Q1, . . . , QN−2) describe displace-ments from the saddle point geometry.

In the CRS framework small displacements �X = X − X0 with respect to a so-called reaction surface X0 areconsidered,

�X =N−2∑j=1

(Qj − Q(ref)j )ej . (37)

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226 K. Giese et al. / Physics Reports 430 (2006) 211 –276

The displacements are orthogonal to the space spanned by v1 and v2 and the reaction surface is parameterized by thereaction coordinates, i.e., Q

(ref)j = Q

(ref)j (v1, v2). The Taylor expansion of the PES with respect to the reaction surface

reads,

U(X) = U(X0) + G(f)(X0) · �X + 12�XTK(f)(X0)�X + · · · , (38)

where G(f)i = �U/�Xi is the full dimensional gradient and the Hessian is given in Eq. (33). For small displacements

of the Q a truncation of the Taylor series after the harmonic term is reasonable. The truncated PES is denoted V in thefollowing. The three translational DOF can be discarded without loss of generality; the three rotational DOF can bediscarded under the assumption, that the infinitesimal rotational vectors do only slightly change on the plane spanned bythe reaction coordinates. The PES is then expressed as, V =V (v1, v2, Q1, . . . , QN−8). The potential values, gradients,and the Hessians on the reaction surface can be obtained by standard quantum chemistry calculations.

2.3.3. Fixed reference caseThe choice Q

(ref)j ≡ constj corresponds to a reaction surface that is independent of (v1, v2) and was termed the fixed

reference case [79]. (Note that this kind of reference has to be distinguished from the choice of XTS as reference inEq. (36).) Formally, due to this choice, the reaction surface becomes a plane. The special case Q

(ref)j ≡ 0 is important

for the formulation of the Cartesian reaction plane Hamiltonian; in this case, the PES reads

V (v1, v2, Q) = U(v1, v2) −∑

i

Fi(v1, v2)Qi + 1

2

∑ij

Kij (v1, v2)QiQj , (39)

where U(v1, v2) ≡ U(X0), and the force Fi acting on mode Qi and the Hessian Kij are related to the full-dimensionalquantities by, respectively,

Fi(v1, v2) = −G(f)(X0) · ei , (40)

Kij (v1, v2) = eTi K(f)(X0)ej . (41)

2.3.4. Flexible reference caseWith a flexible reference, Q

(ref)j = Q

(ref)j (v1, v2), X0 describes a generally non-planar surface. This facilitates the

selection of the region of the configuration space that is relevant for the reaction. It could be obtained, for instance,by partial geometry optimization for a each given point (v1, v2). Inserting Eq. (37) into Eq. (38) yields the PES of theCRS Hamiltonian with respect to the flexible reference,

V (v1, v2, Q) = U (v1, v2) −∑

i

Fi (v1, v2)Qi + 1

2

∑ij

Kij (v1, v2)QiQj , (42)

with abbreviations,

U = U +∑j

FjQ(ref)j + 1

2

∑ij

KijQ(ref)i Q

(ref)j , (43)

Fi = Fi +∑j

KijQ(ref)j , (44)

where all quantities are considered to be (v1, v2)-dependent and evaluated with respect to X0.

2.3.5. Selection of relevant modesThe PES of the CRS Hamiltonian of the previous Section and the exact PES agree only up to second order terms

with respect to Q, but the CRS Hamiltonian is full-dimensional. For an efficient numerical treatment, it is necessaryto select certain modes out of the set {Qj } that are especially important and to introduce a further approximation for

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 227

the remaining modes. The selection procedure depends on the choice that was made for the reaction coordinates. Thegoal is to formulate a reduced dimensional CRS Hamiltonian (h = 1),

HCRS = −1

2

�2

�v21

− 1

2

�2

�v22

−n∑

k=1

1

2

�2

�q2k

+ V (v1, v2, q), (45)

where q = (q1, . . . , qn) is a set of relevant coordinates.The choice of Q-modes is formally arbitrary, i.e., one may switch to another -equivalent- set of modes by a linear

transformation,

Q′ = AQ, (46)

where A is an orthogonal transformation matrix that does not mix modes with different symmetry. It is assumed in thefollowing, that the vector Q′ can be expressed as

Q′ =(

qS

), (47)

where the vector S = (S1, . . . , Sn′) with N = n + n′ + 8 accounts for the set of irrelevant modes. The set of reactioncoordinates plus the relevant modes, {v1, v2, q1, . . . , qn}, is called model coordinates; the irrelevant modes are denotedas spectator modes.

2.3.6. Atomic reaction coordinatesThe simplest choice for reaction coordinates are atomic positions as discussed in Section 2.3.2. One may, for instance,

identify (v1, v2) with the position of the reactive hydrogen atom in the plane of the molecule [79]. For atomic reactioncoordinates the choice of a flexible reference is preferred since otherwise for large displacement from the referencegeometry one would obtain large forces indicating a substantial deviation from some IRP. Thus, the starting point isEq. (42) which can be written in a more convenient form as follows: let Q(0) =Q(0)(v1, v2) be such that V (v1, v2, Q(0))

is minimal among all Q for any fixed value of the reaction coordinates, i.e., Q(0) satisfies,

Fi =∑j

KijQ(0)j . (48)

With this definition the linear and harmonic term of the PES can be joined in a displaced harmonic term,

V (v1, v2, Q) = U (v1, v2) − EQ(v1, v2) + 1

2

∑i,j

Kij (x, y)(Qi − Q(0)i )(Qj − Q

(0)j ), (49)

where the so-called reorganization energy,

EQ(v1, v2) = 1

2

∑i,j

Kij (v1, v2)Q(0)i Q

(0)j , (50)

is introduced.In analogy to Eq. (47), the vector Q(0) can be divided according to

Q(0) =(

q(0)

S(0)

). (51)

Formally, the coupling between model coordinates and spectator modes is assumed to be negligible. Then, the numberof actual DOF of the PES can be reduced by setting,

S ≡ S(0)(v1, v2). (52)

This choice is preferred among all other possible choices, because even when there is a small coupling that is not strictlynegligible, the energetics of the reduced dimensional PES is still equivalent to the full-dimensional one, i.e., the barrierheight is the same.

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228 K. Giese et al. / Physics Reports 430 (2006) 211 –276

2.3.7. Collective reaction plane coordinatesThe minima and the transition state geometries are points on the reaction plane. Therefore, the choice of a fixed

reference, Q(ref) ≡ 0, is reasonable. This implies, X0(w1, w2) = XTS + w1w1 + w2w2. Let Q denote the projectionoperator onto the n-dimensional subspace spanned by the relevant modes q. Diagonalization of the matrix,

(I − P − Q)K(f)|X0(I − P − Q), (53)

for each fixed value of (w1, w2) yields n′ nonvanishing eigenvalues �2i (w1, w2) for the spectator modes as a function

of the reaction plane coordinates. The PES in terms of the model coordinates is given by Eq. (39) with S ≡ 0(cf. Eq. (47))

V (w1, w2, q) = U(w1, w2) −n∑i

Fi(w1, w2)qi + 1

2

n∑ij

Kij (w1, w2)qiqj + 1

2

n′∑i

h�i (w1, w2). (54)

The last term of this expression corresponds to the ZPE of the spectator modes. The ad hoc inclusion of this terminto the reduced-dimensional PES is motivated by intuition; the change of frequencies of many modes may lead to acontribution to the potential, that would have been neglected otherwise. More rigorously, such a term appears withinan adiabatic separation of the relevant coordinates and spectator modes. Here the spectator modes are relaxed to theirequilibrium positions which depends on the relevant coordinates. This introduces not only the ZPE into the potentialbut it also gives a contribution to the kinetic energy [87,88]. A recent systematic study in the context of tunnelingsplittings in the formic acid dimer can be found in Ref. [89].

For all three extremal points on the reduced PES, q = 0 holds. Furthermore, neglecting the ZPE term, the energeticsof the reduced PES, e.g., the barrier height, is identical to the full PES by definition of the reaction plane. This is a veryimportant feature of the present formulation, which distinguishes it from the choice of atomic reaction coordinatesaccording to Ref. [79], because one does not need to include relaxed spectator modes in order to yield the same PESenergetics as in the full dimensional case.

Previously, Yagi et al. [80] used the reaction plane as a starting manifold for the generation of points for a full-dimensional treatment of malonaldehyde. In their approach, the global PES is spanned by a modified Sheppard scheme.The main advantage of the present approximate method, however, is the fact that the PES can be written as a sum ofproducts of function of the reaction coordinates (w1, w2) times the remaining coordinates q. This form makes its use innumerical propagation schemes for the solution of the Schrödinger equation which are based on a product representationof the wave function very efficient (see Section 3.1).

Finally, we note that the dipole moment surface of the reduced model can be approximately expressed in a similarmanner. For numerical convenience, only the first derivative with respect to the reaction plane can be employed, whilethe dipole function is treated numerically exact on the reaction plane. The approximation reads

��(w1, w2, q) = �(full)� (w1, w2, q)|q=0 +

∑k

��(full)�

�qk

∣∣∣∣∣q=0

qk , (55)

where �(full)� is the full-dimensional dipole function and � = X, Y, Z is the Cartesian component of the dipole moment

vector.

2.3.8. Reduced normal modesIn this section generic reaction coordinates (including atomic and reaction plane coordinates) are considered. The

model coordinates—except v1 and v2—are arbitrary in the sense, that any linear combination of them will yield exactlythe same results. There are three sets of coordinates, however, that are unique, these are—as in the full-dimensionalsystem—the normal modes of the reduced (n + 2)-dimensional system at the two symmetrically related minima andat the transition state. These normal modes are called reduced normal modes. By neglecting changes of the zero-pointenergy, which are usually small if all important coordinates are included into the relevant part of the Hamiltonian, onecan compute, e.g., the reduced normal modes of the right minimum geometry by diagonalizing the Hessian

RK(f)(XR)R, (56)

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 229

where R is the projector onto the space spanned by the model coordinates (v1, v2, q1, . . . , qn). The corresponding n+2reduced normal modes are denoted Y

([n+2]D)k with k = 1, . . . , n + 2 in the following. Recall that the vectors v1 and v2

are both N-dimensional. Likewise, the vectors ej are N-dimensional according to their definition by diagonalization ofEq. (35). Thus, the (n+2)-dimensional model coordinates define a (n+2)-dimensional reduced space that is embeddedin the full N-dimensional space.

While for setting up the Hamiltonian it is most convenient to use model coordinates, all physically relevant quan-tities should be related to the unique reduced normal modes. Additionally, to analyze the connection of the reduceddimensional model with the full-dimensional system, one has to investigate overlaps of reduced normal modes withfull normal modes. This task can be achieved by defining a N × (n + 2)-matrix B that transforms from the reducedspace (v1, v2, q1, . . . , qn) back into the full N-dimensional Cartesian space:

B = (v1v2q1 . . . qn), (57)

where the vectors v1, etc., i.e., those vectors that span the so-called reduced space, constitute the columns of thematrix. This matrix always exists by definition and the property, BTB = 1, holds because the constituting vectors areorthonormal.

Since molecular IR spectra are normally analyzed in terms of harmonic transition frequencies corresponding tonormal modes it is necessary to establish a link between the modes obtained in harmonic approximation and forthe reduced dimensional model. In other words, a reasonable reduced model should be characterized by overlapswith certain full normal modes being close to one. Let Y(f)

j with j = 1, . . . , N − 6 be the full normal modes cor-responding to the right configuration XR, and let Yk be the reduced normal modes corresponding to the right mini-mum configuration (v

(min)1 , v

(min)2 , q = 0). Then the projection of reduced normal mode k onto full normal mode j is

given by

pjk = (Y(f)j )TBYk . (58)

An application of this procedure will be discussed in Section 3.3.

3. Coherent quantum dynamics

3.1. Mean-field vs. multiconfiguration methods

In the following we will address the question how the time-dependent nuclear Schrödinger equation,

ih�

�t�(q; t) = H�(q; t). (59)

can be solved efficiently for a multidimensional HB system with general coordinates q. The exact basis set representationof a multidimensional wave function reads

�(q; t) =n1∑

�1=1

· · ·nN∑

�N=1

C�1,...,�N(t)�(1)

�1(q1) · · · �(N)

�N(qN), (60)

where C� is a time-dependent coefficient matrix and �(k)�k

(qk) are time-independent basis functions. The coefficientmatrix scales like O(nN ) with respect to the number of DOF N, where n is a typical number of basis functions for onedimension. Unfortunately, this exponential scaling hampers any attempt of a direct propagation of a multidimensional(say, with N > 4) wave function.

In order to motivate a solution to avoid this dimensionality bottleneck consider a reaction coordinate X coupled toa harmonic vibrational mode Q where the interaction is written in the form of Eq. (14), i.e. q = (X, Q). Suppose thetotal wave function is of Hartree-product form

�(X, Q; t) = �(X; t)�(Q; t) (61)

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230 K. Giese et al. / Physics Reports 430 (2006) 211 –276

with the single particle functions �(X; t) and �(Q; t). Equations of motion for the single particle functions can bederived by the Dirac–Frenkel variational principle [90]

〈��|H − ih�

�t|�〉 = 0. (62)

One obtains, neglecting unimportant phase factors,

ih�

�t�(X; t) = [TX + VSCF(X; t)]�(X; t) (63)

and

ih�

�t�(Q; t) = [TQ + VSCF(Q; t)]�(Q; t). (64)

The coupling between the two coordinates in this time-dependent self-consistent field (TDSCF) approach is realizedby the so-called mean-field potentials, e.g.,

VSCF(X; t) = V (1)(X) +∫

dQ�∗(Q; t)V (2)(X, Q)�(Q; t) (65)

and similar for VSCF(Q; t). While this approach works well for many molecular applications [91], it is likely to performrather poorly for the case of HAT as first pointed out by Makri and Miller [92]. The reason for a possible failure isobvious: If the single particle functions are delocalized along the respective coordinates all details of the interactionpotential are washed out by the integration in Eq. (65). Recently, this has been nicely illustrated in a model simulationof PT along a water chain [93]. However, even in cases where one deals with a single minimum potential only, thelimited flexibility of the Hartree ansatz may surface in the improper description of IVR (see, Section 3.2).

To solve this issue several multiconfiguration schemes have been proposed [92,94], but the breakthrough especiallyin terms of general applicability came only with MCTDH approach developed by Meyer and coworkers [95–97]. Herethe ansatz for the wave function reads

�(q; t) =n1∑

�1=1

· · ·nN∑

�N=1

A�1,...,�N(t)��1,...,�N

(q; t), (66)

where a single configuration is given by a Hartree product of N time-dependent single particle functions,

��1,...,�N(q; t) = �(1)

�1(q1; t) · · · �(N)

�N(qN ; t). (67)

The integers nj in Eq. (66) refer to the number of single particle functions corresponding to a certain DOF qj . Equations

of motion for the coefficient matrix A�1,...,�N(t) and the single particle functions �(j)

�j(qj ; t) can also be derived by the

Dirac–Frenkel variational principle, Eq. (62) [95]. For the numerical integration of the equations of motion the singleparticle function are often expressed in a discrete variable representation grid [98,99]. For more details in particularconcerning the implementation within the Heidelberg MCTDH code [100], see Ref. [96].

The time-dependent basis (i.e., a set of time-dependent single particle functions) can follow the wave packet duringthe propagation, making the ansatz Eqs. (66)–(67) more efficient than using the same number of time-independent basisfunctions (cf. Fig. 8). The efficiency can be further increased by combining certain modes [101], which in fact hadbeen suggested earlier to incorporate correlations in the framework of a Hartree product ansatz [102]. Let q(ck) with1�k�r denote a subset of the full coordinate vector q. There are r �N mutually disjoined subsets in total. A modifiedsingle configuration is now given by a product of functions corresponding to the individual subsets of coordinates,

��1,...,�r (q; t) = �(1)�1

(q(c1); t) · · · �(r)�r

(q(cr ); t). (68)

The full MCTDH ansatz is obtained by replacing � → � and N → r in Eq. (66). For convenience the basis functionsappearing in Eq. (68) are also denoted single particle functions.

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 231

ϕ (1)

ϕ (2)

ϕ (1)

ϕ (2)

Ψ

t = 0

q1

q2

Ψ

t > 0

evolution

time

Fig. 8. Illustration of the MCTDH method. The initial wave packet �(t = 0) moves and spreads during the time evolution. The time-dependentsingle particle functions �(k) can follow the motion of � (only one single particle function per DOF is shown).

Provided that a separation between slow and fast coordinates has been performed and that the fast coordinates aredescribed by a set of quantum states {|�〉}, the MCTDH wave function can be written as [103]

��(Q1, . . . , Qf , t) =n

(�)1∑

j(�)1 =1

. . .

n(�)f∑

j(�)f =1

A(�)

j(�)1 ,...,j

(�)f

(t)

f∏�=1

�(�,�)j��

(Q�, t), (69)

where the Q� are the slow coordinates. Eq. (69) corresponds to the so-called multi-set formulation, where differentsets of single particle functions have been defined for each quantum state.

There are several important extensions of the MCTDH approach outlined so far. For instance, in the multilayerformulation each single particle function itself is treated recursively as a MCTDH wave function [104]. On the otherhand, it has been shown that one can reduce the effort for representing the single particle functions considerably byusing parameterized functions such as Gaussians [105].

The MCTDH approach is also capable of diagonalizing a multidimensional Hamiltonian [97]. First, the Hamiltonian isdiagonalized within the basis of certain initial single particle functions. This yields expansion coefficients correspondingto each eigenstate. The coefficients corresponding to the desired eigenstate are used to build the quantities appearingin the equations of motion for the single particle functions. Then, a small integration step is performed for imaginarytime (t → i�) and a new optimized set of single particle function is obtained. The procedure, which is called improvedrelaxation, is repeated until convergence is achieved [97].

It is important to notice that an efficient implementation of this MCTDH scheme requires to have at hand a Hamiltonianwhich is of product form in the different coordinates. This way mean-field integrals like Eq. (65) factorize and can bestraightforwardly calculated. There is a scheme for transferring general potentials into product form [106], but this isnot feasible for multidimensional situations where also the very task of calculating the PES becomes impossible. Onthe other hand, the CRS Hamiltonian, Eq. (45), is of product form with respect to most coordinates which makes itparticularly well-suited for studying multidimensional HB quantum dynamics using the MCTDH approach.

3.2. Laser-driven dynamics and intramolecular vibrational energy redistribution

3.2.1. The case of a single minimum potentialAs a first example let us consider the HB dynamics triggered by excitation of either the OH or the OD bond in phthalic

acid monomethylester (PMME) shown in Fig. 9. PMME has been the first case for which coherent vibrational dynamicsof a HB on a subpicosecond time scale could be observed using ultrafast 130 fs IR pump–probe spectroscopy [107,108].Monitoring the decay of an initially excited OD stretching vibrational state an oscillatory component corresponding toa 100 cm−1 vibration was found in the signal which survived for about 1.5 ps (decay time about 500 fs). The excited

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232 K. Giese et al. / Physics Reports 430 (2006) 211 –276

Fig. 9. Upper panel: the equilibrium structure of deuterated PMME as obtained using DFT/B3LYP with a 6-31 + G(d, p) basis set [114]. Lowerpanel: the potential along the O1-D coordinate (with respect to the equilibrium value) is shown together with the lowest eigenfunctions.

state itself decayed with a time constant of T1 = 400 fs, whereas the overall vibrational cooling time was about 20 ps.Subsequently, a similar behavior was found for excitation of the OH vibration in the undeuterated species where merelythe T1 time decreased to 200 fs [109]. Furthermore, photon echo measurements revealed a homogeneous dephasingtime shorter than 100 fs [110]. The rather short T1 time seems to be a more general feature of this type of HBs, e.g.,a 200 fs decay was observed for deuterated 2-(2′-hydroxophenyl)benzothiazole in Ref. [111]. In order to unravel thepathways for vibrational energy flow two-color pump–probe experiments have been performed for PMME as wellwhich will be discussed in Section 6.3 [112,113].

The HB is nonlinear (see, Fig. 9) with the O–O distance being 2.56 A, i.e. it should be classified as a medium-strongHB. Apart from an isoenergetic enantiomer [115] there is only a single minimum PES which is rather anharmonicas can be seen from Fig. 9 where the potential is shown as a function of the OD bond distance for the deuteratedspecies. The anharmonic shift for the �OD = 1 → 2 transition amounts to −180 cm−1 in this one-dimensional PES cut.Apart from this shift there is a strong Fermi-resonance with the OD bending overtone [116]. For the normal species,on the other hand, one observes a more significant substructure and broadening in the IR spectrum as seen in Fig. 31below. Here, the �OH = 0 → 1 transition is calculated at 3036 cm−1and the anharmonic shift for the �OH = 1 → 2transition is −430 cm−1 [112]. Notice, however, that the assignment of the overtone transition must be taken with care,because these zero-order states are usually rather diluted among several eigenstates. Overall, these stationary spectraare indicating the underlying multidimensional HB dynamics.

A global view on the dynamics can be obtained by combining a CRS description with a TDSCF propagation. Givena dipole moment surface �, the interaction with the external laser field can be included via the Hamiltonian:

Hfield(t) = −�E(t) cos(�t), (70)

where � is the carrier frequency of the pulse and E(t) is its envelope usually assumed to be of Gaussian

E(t) = E0 exp(−2 ln(2)t2/�2) (71)

or sin2 shape (0� t ��)

E(t) = E0sin2(t/�), (72)

where E0 is the amplitude and � characterizes the pulse duration. Notice that for simplicity we assumed that the fieldis directed along the dipole moment vector.

Using the OH or OD-bond length as the reaction coordinate an essentially full-dimensional description of the couplingto the molecular scaffold has been given in Ref. [114]. There it was shown that indeed the periodic modulations of the

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 233

Fig. 10. PMME pump–probe signal (in arbitrary units) calculated on the basis of a full-dimensional CRS-Hamiltonian and a TDSCF propagation.The pump pulse excites the OH-stretch transition and the probe pulse detects the excited state absorption (�pump =3150 cm−1, �probe =2750 cm−1,� = 330 fs (Eq. (72))). The oscillations are due to a low-frequency HB mode which modulates the O · · · O distance. For more details see Ref. [114].

Fig. 11. MCTDH coordinate expectation value with respect to the equilibrium configuration for a 19D-CRS model of the HB dynamics in PMME-D:(a) OD-bond distance, (b) Low-frequency HB mode (in units of aB

√a.m.u.). (For more details, see Ref. [117].)

pump–probe signal [108] can be traced back to the coupling between the OH-stretching vibration and a low-frequencyvibration influencing the strength of the HB (cf. Fig. 3). Taking a simplified definition of the pump–probe signal asbeing equal to the energy absorbed by a weak probe pulse starting from the state prepared by the pump pulse oneobtains the signal shown in Fig. 10 [114].

The oscillatory signal reflects one aspect of the experimental results, i.e. the periodic modulation of the HB strengthby coherent excitation of an anharmonically coupled low-frequency mode. However, in contrast to the experiment itdoes not decay. In principle this observation could have at least two reasons: First, the TDSCF wave function doesnot provide enough flexibility for intramolecular energy flow. Second, the interaction with the solvent is a key forthe understanding of the relaxation process. In order to scrutinize the first point a reduced dimensional model hasbeen defined in Refs. [115,117] by considering only the most important substrate modes. The selection was based onthe reorganization energies (cf. Section 2.3) as well as on the TDSCF dynamics [114]. For this model wave packetpropagations for the reaction coordinate and 8 and 10 harmonic modes being treated within the MCTDH and theTDSCF scheme, respectively, have been performed in Refs. [115,117]. In Fig. 11 we show the expectation value of theOD bond distance as well as of the low-frequency HB coordinate for an excitation with a � = 300 fs sin2-shaped pulse,Eq. (72). The laser-driving at the resonant frequency is reflected in the fast oscillations in panel (a). Superimposed onenotices in particular a slow modulation with a period of about 500 fs. This is a direct consequence of the anharmoniccoupling to the low-frequency HB mode as can be seen from panel (b). The maxima of 〈Qlow〉 coincide with the minima

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234 K. Giese et al. / Physics Reports 430 (2006) 211 –276

Fig. 12. Expectation value of the uncoupled energy of the OD bond, 〈EOD〉/hc, for the 19D model of PMME-D (see, Fig. 11). (a) MCTDHpropagation, (b) TDSCF approximation. (For more details, see Ref. [117].)

Fig. 13. Enol-keto tautomerism of 2-hydroxybenzaldimine compounds.

of the envelope in panel (a). This implies that the elongation along Qlow leads to a compression of the HB and thus asmaller amplitude of the OD vibration.

The energy exchange between the OD bond and the substrate modes can be monitored by calculating the expectationvalue of the reaction coordinate part of the Hamiltonian neglecting any coupling to the orthogonal normal modes. InFig. 12 we show 〈EOD〉 for the MCTDH (panel (a)) and the TDSCF (panel (b)) cases. Apart from the rather similarrapid oscillations we notice that in the MCTDH case 〈EOD〉 is (on average) slightly decreasing after the initial rise untilabout 0.25 ps while in the TDSCF case it stays constant. This can be interpreted as the onset of energy randomization,i.e. the irreversible (at least in the considered time interval) energy flow from the OD-vibration into the rest of themolecule. This effect is only captured by the MCTDH approach. This finding clearly highlights the importance ofincorporating multiconfiguration effects even for single minimum systems at moderate energies. From the decay of〈EOD〉 after about 0.25 ps (averaging over the rapid oscillations) we can estimate the associated time constant which isof the order of about 20 ps. We will return to this example in Section 6.3 where the role of the solvent for understandingthe experimentally observed [108] much more rapid decay will be addressed.

3.2.2. The case of an asymmetric double minimum potentialThe anharmonicity of single minimum systems such as PMME is still moderate since no HAT takes place which

would require a change in the conjugation pattern of the molecule as shown, for instance, for the enol–keto tautomerismin 2-hydroxybenzaldimine compounds in Fig. 13. Here, not only the incorporation of large amplitude motion but alsoaccounting for multiconfiguration effects in the quantum dynamics is mandatory. Let us consider salicylaldimine (SA)as the simplest example (R&H) for which a number of quantum chemical investigations exist [32,118–121]. At theDFT/B3LYP(6-31+G(d,p)) level the barrier height is 1940 cm−1and the enol–keto asymmetry is 1308 cm−1 [121].

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 235

Fig. 14. PES for the 2D planar HAT in salicylaldimine, where x and y give the position of the H atom with respect to the center of mass and themoments of inertia axes of the enol configuration as well as a relaxed molecular scaffold (cf. Section 2.3.4). Also shown are the probability densitiesfor the first three states (a–c) and for the first state which is localized in the keto minimum (d). (For more details, see Ref. [122].)

Fig. 15. Displacement vectors of the normal modes which enter the 7D model of HAT in SA–H/D [121,122]. The harmonic frequencies are: �4 =336,�6 = 452, �14 = 861, �26 = 1333, �30 = 1513 cm−1.

Restricting ourselves to a planar model, a possible choice for the large amplitude reaction coordinates are the in-planex and y positions of the moving H atom with respect to the center of mass and the moments of inertia axes of the enolconfiguration (cf. Section 2.3.6). The 2D PES for the bare H motion shown in Fig. 14a apparently is rather anharmonic.Moreover, the reorganization energy for the harmonic modes (cf. Eq. (50)) of 3573 and 3112 cm−1 at the transition stateand keto configuration, respectively, points to a substantial coupling. In Refs. [121,122] a 7D model has been suggestedfor SA–H and SA–D, respectively, which includes the five strongest coupled modes shown in Fig. 15. Notice thatthese modes can be roughly classified as being either of the symmetric (�4, �14), i.e. promoting, or the antisymmetric(�6, �26, �30), i.e. reorganization, coupling type.

Fig. 14 also shows some of the lowest eigenstates of the reaction coordinates for normal mode coordinates fixedat the enol minimum. They have been obtained from the solution of the Schrödinger equation for the wave functions��(x = (x, y)) ≡ 〈x|�〉,

[Tx + U(x)]��(x) = E���(x). (73)

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236 K. Giese et al. / Physics Reports 430 (2006) 211 –276

Fig. 16. Absorption spectrum, Eq. (77), of the 7D model of SA–H (black) and SA–D (gray) calculated using the MCTDH approach. The dashedlines show the pulse spectra used for the laser-driven dynamics in Fig. 17. (For more details, see also Ref. [123].)

These states form a crude diabatic basis, {|�〉}, [38] which can be used for a diabatic representation of the CRSHamiltonian:

H diab = 12 P2 +

∑�,

[H��(Q)�� + H�(Q)(1 − ��)]|�〉〈|. (74)

Here, we introduced the diabatic PES

H��(Q) = E� − F��Q + 12 QK��Q, (75)

and the coupling between the diabatic states

H�(Q) = −F�Q + 12 QK�Q. (76)

This representation has at least two advantages: First, it offers a convenient means to analyze the spectrum and associatedwave packet dynamics in terms of the (zero-order) diabatic states. For instance, Fig. 14 shows the ground state as wellas states dominated by bending and stretching fundamental excitations. Second, since the diabatic states are ratherlocalized, the grid on which the first and second derivatives of the PES have to be known is limited.

An overall characterization of the model can be obtained by calculating the linear IR absorption spectrum which isconveniently done via the Fourier transformation of the dipole–dipole autocorrelation function [38]:

I (�) = 4�nmol

3hc

∑�=x,y,z

Re∫ ∞

0dtW(t)ei�t 〈�0|[��(t), ��(0)]|�0〉, (77)

where |�0〉 is the ground state, nmol is the number density of molecules, ��(t) is the dipole operator in the Heisenbergpicture with spatial direction �, and W(t) is a certain window function to account for the limited time interval on whichthe correlation function is known.

In Fig. 16 we show the IR absorption spectrum for SA–H and the deuterated SA–D which has been evaluatedaccording to Eq. (77) using the MCTDH method [123] in the multi-set formulation, Eq. (69). SA–H shows a broadand structured band covering the range from 2200–2800 cm−1. Upon deuteration (SA–D) the substructure almostdisappears and the band shifts into the range from 1800–2050 cm−1. Experimental results are available only forN -phenylsalicylaldimine (R–Ph) in CCl4 solution [124]. Apart from the solvent-induced broadening one observes twobands around 2600 and 2750 cm−1what could be taken as a confirmation of the magnitude of the calculated red-shift.The difference between SA–H and SA–D in Fig. 16 can be traced to the reduced resonant couplings between the reaction

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 237

Fig. 17. Energy change for the 7D model of SA–H (left) and SA–D (right) after ultrafast excitation of the OH and OD dominated absorption band(see pulse spectra in Fig. 16). The data correspond to (from top at t = 750 fs): Exy (gray), EQ26 , EQ14 , EQ30 , EQ6 , and EQ4 (left, OH), and EQ26 ,EQ14 , Exy (gray), EQ30 , EQ4 , and EQ6 (right, OD). (For more details, see also Ref. [123].)

Fig. 18. Tautomerism of tropolone.

coordinates and the substrate modes in the deuterated case. This effect is nicely reflected in the dynamics upon OHand OD stretching excitation. In Fig. 17 we compare the energy expectation values of the reaction coordinate and theuncoupled harmonic modes for SA–H and SA–D after excitation with a 260 fs sin2-shaped pulse. The IVR dynamics ofSA–H is characterized by energy randomization on a time scale of about 700 fs, a number which correlates reasonablywith available experimental data [124]. Furthermore, none of the harmonic modes seems to play a particularly importantrole. For the deuterated species no such fast decay is observed. Instead there is a pronounced energy exchange withmodes �14 and �26 which persists at least up to 5 ps [122]. Hence we can conclude that for the case of a rather anharmonicdouble minimum potential, i.e. for typical HAT systems, a subpicosecond decay of the OH-stretching vibration can becaused solely by intramolecular anharmonic interactions.

3.3. Mode-specific tunneling splittings

An important example for multidimensional HB dynamics is the dependence of tunneling splittings on the excitationof particular vibrational modes in the molecule. This feature has been observed spectroscopically for electronic ground[125] and excited [30,34] states. As discussed in the introduction mode-specific tunneling splittings are just anothermanifestation of, e.g., the promoting role environmental motions can have in HAT and PT reactions.

Before addressing approximate methods for calculating tunneling splittings in Section 4 we will discuss the quan-tum route to IR spectra and related tunneling splittings for the exemplary tautomerization of tropolone, Fig. 18.Tropolone is among the most intensively studied examples for mode-specific tunneling splitting although this effect

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238 K. Giese et al. / Physics Reports 430 (2006) 211 –276

has been observed in other system such as, e.g., the formic acid dimer [29]. Studying the vibrational progressionsin vibronic spectra of tropolone, Alves and Hollas [126,127] as well as the groups of Mikami [34] and Sekiya [30]found a pronounced mode-specificity in the S1-state tunneling. For the electronic ground state Tanaka et al. mea-sured the lowest tunneling doublet to be separated by �0 = 0.974 cm−1 [128]. Redington and coworkers [125,129]attributed a doublet at 741 and 751 cm−1to the fundamental transition of what was called the nascent skeletal tun-neling coordinate [130]. In the region of the OH-stretch fundamental transition a doublet is observed at 3102 and3121 cm−1 [129], i.e. the tunnel splitting is as small as �1 ≈ 20 cm−1. A similar conclusion concerning the magni-tude of �1 was reached in Ref. [131], although based on a different assignment. The controversial assignments canbe taken as an indication that the identification of tunneling doublets in a rather structured spectrum is not straight-forward. Given the complex line shape of the OH-stretching band one may also ask whether it is reasonable toassign a transition to a particular normal mode whose definition is meaningful in the vicinity of the minima on thePES only.

General aspects of mode selective tunneling were addressed theoretically by using two-dimensional model Hamil-tonians adjusted to mimic tropolone [60,61]. On the other hand, Redington put forward a two-dimensional adiabaticmodel which included the OH-stretch and the skeletal tunneling coordinate corresponding to a vibration at around750 cm−1 [130]. The remaining modes contributed only their ZPE. This model has been rather illuminating and afterproperly choosing a tunneling path it could even reproduce the experimental splittings quantitatively. In contrast, inRef. [86] using a full-dimensional analysis of the PES in the vicinity of the minimum we have shown that the intrinsicreaction path approaches the PES minimum not along this skeletal tunneling coordinate but along a combination ofthe two lowest frequency in-plane modes (∼ 360 cm−1). Overall, in view of the highly structured IR spectrum in theOH-stretch region one may argue whether a two-dimensional description is rigorously justified, not to mention theadiabatic approximation.

The CRS approach offers a way to investigate the nature of the IR spectrum based on a multidimensional quantumtreatment [86]. The reaction plane coordinates have already been discussed in Fig. 6. In principle, all remainingorthogonal DOF could be incorporated in harmonic approximation into a 39-dimensional model. In practice, thenumerical effort for the solution of the full-dimensional nuclear Schrödinger equation can be considerable and areduction of the dimensionality along the lines discussed in Section 2.3.5 is advisable. Here, one is guided by theprocess which one intends to model. If one puts the focus on a proper description of the low-frequency part of the IRspectrum associated with the reaction coordinate as well as on the OH-stretching region, overlap and resonance criteriasuggest an overall twelve-dimensional model [132].

In Fig. 19 we show the reduced normal modes of the right configuration for the 12D model and in Table 1 wegive the overlap between the reduced and the full normal modes according to Eq. (58) as well as with the reactionplane coordinates. These overlaps convey some information concerning the target to which the model coordinates havebeen adapted. There are three modes, Y(12D)

1 , Y(12D)9 , Y(12D)

12 , with a comparatively large overlap with the reaction

plane. The reduced mode Y(12D)1 almost coincides with the OH stretch full mode Y

(f)1 . The reduced mode Y

(12D)9 has

a significant overlap with the full mode Y(f)15 , which has OH bend character and shows the largest IR intensity. Finally,

the reduced mode Y(12D)12 has lowest frequency and shows significant overlaps with the full modes Y(f)

33 , Y(f)36 , and Y

(f)37 ,

i.e. it represents those modes associated with the approach of the minimum along the IRP as discussed above. Thusone could call Y(12D)

12 the reaction mode of the 12D model. This is in contrast to the assignment of Ref. [130] which

would correspond to mode Y(12D)10 , i.e. a mode which has a perfect overlap with the full mode Y

(f)28 , but at the same

time has only a very small overlap with the reaction plane coordinates.The ground state tunneling splitting for this model is obtained as �0 = 2.7 cm−1, what is about three times the

experimental value of �(exp)

0 = 0.974 cm−1 [128], which indicates most likely an underestimation of the barrier bydensity functional theory [86] (cf. Fig. 7). An overview of the IR spectrum, Eq. (77), is given in Fig. 20. There are threedistinct regions in the spectrum, marked A, B, and C. The 12D model Hamiltonian has been designed to account forregions A and C only. First, let us focus on region A which contains the information about the reaction mode. Here, itturns out that the resolution which can be reasonably obtained from a time propagation of the autocorrelation function isnot sufficient for a detailed analysis of the spectrum. Low-lying eigenstates can also be obtained by using the improvedrelaxation method as discussed in Section 3.1. There are also other MCTDH based diagonalization schemes as used,for instance, by Coutinho-Neto et al. [133] to determine the ground state tunneling splitting of malonaldehyde withina full-dimensional model potential.

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 239

Fig. 19. The reduced normal modes of the 12D model of tropolone. Frequencies are indicated.

Table 1Comparison of normal modes of the full-dimensional system with the reduced normal modes of the 12D model for tropolone

Full 12D

No. �(f)k

/2c Inten. Overlap Mode �(12D)k

/2c w1 w2

1 3329 133.7 0.97 Y(12D)1 3227 −0.64 0.31

7 1674 202.4 0.87 Y(12D)2 1662 −0.01 0.00

8 1663 2.82 0.89 Y(12D)3 1655 −0.15 0.04

9 1617 104.6 0.91 Y(12D)4 1593 0.06 −0.01

10 1539 124.5 0.94 Y(12D)5 1535 0.04 0.00

11 1522 141.9 0.85 Y(12D)6 1500 −0.26 0.07

12 1480 212.0 0.83 Y(12D)7 1461 0.17 −0.07

13 1454 27.0 0.59 Y(12D)8 1400 −0.04 0.00

14 1349 57.7 0.65

15 1327 227.5 0.72 Y(12D)9 1271 0.60 −0.11

28 752 14.5 1.00 Y(12D)10 753 −0.11 0.03

30 694 7.2 1.00 Y(12D)11 695 −0.03 0.00

33 449 15.2 0.41

36 375 5.3 0.70 Y(12D)12 392 0.32 0.94

37 364 5.2 0.56

Frequencies �k (in cm−1), IR intensities (in km/mol), overlaps of the reduced normal modes with full normal modes [cf. Eq. (58)], and overlaps

of the reduced normal modes with the reaction plane directions, |w1 · Y(12D)k

| (“w1”) and |w2 · Y(12D)k

| (“w2”), are given.

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240 K. Giese et al. / Physics Reports 430 (2006) 211 –276

0.00

0.50

1.00

1.50

2.00

2.50

3.00

250 750 1250 1750 2250 2750

A

B

CInte

nsity

(ar

b. u

nits

)

� / 2c (cm-1)

Fig. 20. Overview of the computed IR spectra of the 12D model of Trn(OH) at T = 5 K. Important regions are marked by A–C. A 3ps MCTDHpropagation has been used together with a Gaussian window function W(t) ∼ exp[−t2/�2], � = 1 ps.

Table 2Excitation energies (εj ) and IR intensities for the 12D [“H-12”] and a 4D [“H-4”] model of tropolone [86]

Final state �f IR intensity (km/mol)

εj /hc (cm−1) �+0 → �f �−

0 → �f

H-12 H-4 H-12 H-4 H-12 H-4

121+ (41+) 357 354 7.1 6.7 14.0 10.6

121− (41−) 371 372 8.8 5.2 6.2 6.4

122+ (42+) 697 690 0.6 0.6 0.7 0.9

122− (42−) 737 744 0.0 0.1 0.5 0.3

101+ (31+) 752 752 0.7 1.7 14.6 20.7

101− (31−) 750 749 15.6 21.5 0.6 1.1

Parenthesized labeling refers to the 4D model.

The results (i.e., eigenenergies and IR intensities) of a still another diagonalization procedure outlined in AppendixB are shown in Table 2 (indicated as “H-12”). The symmetry of the states is either gerade (“+”) or ungerade (“−”).The labeling of states is nk±, where n is the reduced normal number and k is the number of quanta in that mode. Theeigenenergies are given with respect to the gerade ground state level �+

0 .A pronounced mode selectivity is found for the

states 12k corresponding to mode Y(12D)12 : the splitting increases to 14 cm−1 for the fundamental (121) and to 41 cm−1

for the first overtone (122). This is not surprising since according to Table 1 this mode most closely resembles thereaction mode, i.e., that mode, which vibrates in the direction of the IRP. On the other hand, excitation of mode Y(12D)

10which corresponds to the skeletal tunneling coordinate of Ref. [130] influences the tunneling splitting only marginally.

The origin of the mode-selectivity can also be scrutinized by inspecting the respective wave functions. In Ref. [86]a four-dimensional model of this reaction had been studied which focussed particularly on the low-frequency region(compare the respective energies and IR intensities in Table 2). In Fig. 21 we show 2D representations of the 4Ddensities for states corresponding to the excitation of modes Y

(12D)12 and Y

(12D)10 in the 12D model. Apparently, the

mode-selectivity can be traced to the symmetry of the mode, i.e. there is an enhancement of the tunneling splitting forthe symmetric promoting-type mode Y

(12D)12 and no effect for the asymmetric coupling mode Y

(12D)10 .

Next, we focus on the OH-stretching region C whose separate contributions from the gerade and ungerade initialground state are shown enlarged in Fig. 22. The major components of the spectrum can be used to assign a tunnelingsplitting of �1 ≈ (20±8) cm−1.Although the absolute position of the OH-stretch band is shifted by about 250 cm−1withrespect to the experimental value [129,131], the order of magnitude of the splitting is rather well reproduced [86].

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 241

Fig. 21. Densities |�|2 of selected states of the 4D model of tropolone [86]. (The other DOF in this 2D representation have been integrated out.)

The excitation of a symmetric normal mode (approx. Y(12D)12 ) has substantial structure in the reaction plane. In the right panel qas is that reduced

model mode which correspond to an asymmetric normal mode (approx. Y(12D)10 ).

0.00

0.10

0.20

0.30

0.40

0.50

2600 2700 2800 2900 3000 3100

∆1

Ψ+0

Ψ -0

I (�

) (a

rb.u

nits

)

� / 2c (cm-1)

Fig. 22. The OH stretch region computed for the 12D model of tropolone. Transitions with gerade and ungerade initial state are shown separately.Lines are drawn to guide the eye. The OH stretch tunneling splitting �1 is indicated.

It should be noted, that the OH-stretching excitation is energetically above the reaction barrier. Thus, this suggests thatthe splitting is due to dynamical tunneling [134]. A similar conclusion was drawn for the HAT in 3,7-dichlorotropolonein Ref. [135].

The complex structure of the IR spectrum in Fig. 22 deserves a more detailed study to pinpoint the contribution ofspecific modes of the 12D model. This can be done by studying the dynamics underlying this broad band. One conceptwhich is frequently used in the theory of IVR is that of zero-order states [136]. In the present case, the states �(0,R)

n

(�(0,L)n ) corresponding to the harmonic approximation to the right (left) minimum form a complete basis (cf. Appendix

B). Moreover, in the absence of anharmonic couplings, these states would be the eigenstates of the Hamiltonian which isthe motivation for their common use in the interpretation of IR spectra. These states are eigenstates of the right/left-handnormal mode Hamiltonian

H(R/L)0 =

∑j

h�j

(n

(R/L)j + 1

2

), (78)

where n(R/L)j is the number operator of harmonic mode j located on the right/left-hand side. The expectation value of

H(R)0 is defined by the real part of the half-space integral [137]

〈H(R)0 〉R = Re

∫x>0

dr �∗H(R)0 �. (79)

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242 K. Giese et al. / Physics Reports 430 (2006) 211 –276

Fig. 23. IVR dynamics for the 12D model of tropolone. The initial wave packet is the harmonic oscillator state corresponding to a singly excited OHstretch. (a) Energy flow for specific modes (indicated) as well as sums of the harmonic ER+L(t) and anharmonic EA(t) contributions. (b) Same asin (a), but for remaining modes (from bottom to top): mode no. 11, 2, 8, 10, 4, 3, 6, 5, and 7. The curves are shifted in steps of 50 cm−1 (except forcurves corresponding to mode no. 11, 2, and 8); the energy is E = 0 at t = 0 for all curves.

Equivalently, 〈H(L)0 〉L can be defined for the left half-space. The expectation value of the full Hamiltonian H can be

divided according to,

〈H 〉 = 〈H(R)0 〉R + 〈H(L)

0 〉L + 〈HA〉, (80)

where the Hamiltonian HA contains the anharmonic part of the full Hamiltonian. In Eq. (78) together with Eq. (79)one can identify the energy expectation value without ZPE of right mode j, as

〈E(R)j 〉R = h�j 〈n(R)

j 〉R (81)

and analogous for the left mode energy 〈E(L)j 〉L.

In Fig. 23 we show the dynamics after initial preparation of the single excited local OH stretch mode localized atthe right-hand side: �(0,R)

11 . The time evolution of this state with respect to the full Hamiltonian H leads to preferentialenergy flow into those modes, that are coupled to the local OH stretch mode. Thus, the energy flow pattern yieldsinformation about the coupling among modes. From Fig. 23a we notice that during the first 100 fs the energy quicklyredistributes from the OH-stretch to the OH-bend and to the low-frequency reaction mode. Furthermore, there is somepartitioning between energy in the harmonic and anharmonic part of the Hamiltonian which changes only slowly afterthe first 100 fs. Notice that there is only a 250 cm−1 energy contribution flowing from the anharmonic part into the

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 243

harmonic part. Compared to the initial harmonic energy this is less than 8%. This observation justifies the use ofharmonic modes for the present analysis. As far as the total 12D model is concerned, Fig. 23b demonstrates that theenergy is rapidly redistributed over all modes during the first 500 fs. Closer analysis of the initial decay rates in Fig. 23allows the conclusion that there are essentially two decay channels, first via the OH bending vibration and second viaall remaining modes. In other words, the detailed form of spectral pattern as well as the rather small tunneling splittingin Fig. 22 is a consequence of the coupling between all modes of the 12D model.

A similar fast IVR dynamics has been observed in Ref. [137] for a 14D CRS-model of 3,7-dichlorotropolone.Together with the results for the asymmetric double minimum potential this suggest, that rapid subpicoscond IVRafter OH-stretching excitation could be a common feature of medium–strong intramolecular HBs having a doubleminimum PES.

4. Semiclassical and classical trajectory-based methods

4.1. Semiclassical approximation for the calculation of spectra

Traditionally, semiclassical spectra and related tunneling splittings are addressed by stationary WKB theory [138]whose multidimensional formulation [56] starts with the N-dimensional wavefunction �(q) expressed as

�(q) = exp

{i

hS(q)

}, (82)

and the complex-valued function S(q) is expanded in a power series with respect to h,

S(q) = S0(q) + h

iS1(q) +

(h

i

)2

S2(q) + · · · . (83)

Inserting this ansatz into the stationary Schrödinger equation and keeping only the lowest order terms in h, one obtainsthe Hamilton–Jacobi equation for the action function S0 and the transport equation for the amplitude S1. Notice thatsince S0 is in general a complex-valued function the configuration space can be conveniently divided into regions whereS0 is real- (R), imaginary- (I ), or complex-valued (C) [56].

For the calculation of tunneling splittings Herring’s formula [139]

�(sc) = h2∫

X=0dN−1q

{�(L) �

�X�(R) − �(R) �

�X�(L)

}, (84)

has been widely used [56,57,140] where �(R/L) are the right- and left-localized wavefunctions and X is the reactioncoordinate. In the following we will give a trajectory-based picture of the tunneling splitting. In this respect, it isimportant to note that according to Eq. (84) a priori the tunneling splitting is not related to a tunneling path, but to theproperties of the wave function along a symmetry surface (or line in 2D) �. Moreover, as was pointed out by Benderskiiet al. [57], if one interprets tunneling in terms of classical trajectories by means of semiclassical theory, such trajectoriesneed to be followed only up to �. Especially, these trajectories may never reach the opposite well, but, nevertheless,their contribution can be significant [141].

Let us focus on tunneling in a two-dimensional SMC-potential. Takada and Nakamura [56] showed that there aretwo extreme cases of tunneling in a SMC potential: (a) tunneling proceeds through the C region or (b) tunnelingproceeds through the I region. Case (a) and (b) correspond to small and large coupling, respectively, between thereaction and harmonic coordinate. The situation is sketched in Fig. 24. In case (a) the saddle point falls into the Cregion. No trajectories can be defined in that region and the action function is complex; especially, it is complex alongthe symmetry line.

In the I region the semiclassical wave function can be expressed as

�(X, Q) = exp

{− 1

hSI (X, Q) − S(X, Q)

}, (85)

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244 K. Giese et al. / Physics Reports 430 (2006) 211 –276

#

X

Q

IC

CIRP

X

Q

instanton

LSLAIRP

#

HU

C

C

I rT

(a)

(b)

Fig. 24. Tunneling mechanisms (schematic) for the SMC-PES. The saddle point (�) and the IRP (dash–dotted) is indicated. Thick solid and dashedlines correspond to caustics of the allowed and forbidden regions, respectively. (a) Small coupling, and (b) intermediate coupling. The instantontrajectories (dotted) and the trajectories for the locally separable linear approximation (thin solid) are indicated.

where S0 ≡ iSI and S1 ≡ −S. SI satisfies the Hamilton–Jacobi equation for the upside-down PES, E − V (r) andp(r) = ∇SI (r) is the momentum of a classical trajectory on the upside-down PES traversing position r. Here, vectorsr denote a position in the 2D space (X, Q).

It is assumed that there is a single point rs = (0, Qs) on the symmetry line � where the exponent is stationary,

�SI (X, Q)

�Q

∣∣∣∣rs

= 0, (86)

i.e., there is a particular trajectory with momentum perpendicular to the symmetry line at rs . In the vicinity of rs on�, the action SI can be expanded into a second-order Taylor series with respect to Q, SI (X = 0, Q) ≈ SI (0, Qs) +0.5 S′′

I (0, Qs)(Q−Qs)2, where S′′

I is the second derivative with respect to Q. The tunneling splitting can be determinedby inserting Eq. (85) into Herring’s formula for the SMC case, evaluating the derivative with respect to X, inserting thetruncated Taylor expansion for �, and performing the Gaussian integral analytically in the semiclassical limit h → 0[56],

� = B exp{−2SI (0, Qs)/h}, (87)

with the prefactor

B = h

√h

S′′I (0, Qs)

e−2S(0,qs )2|px(0, Qs)|, (88)

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 245

where px = �SI /�X is the momentum in X-direction. Thus, in this approach, the tunneling splitting is dominated bya particular trajectory. This trajectory is defined by the condition Eq. (86) and the boundary conditions of the field oftrajectories corresponding to the I region.

Takada and Nakamura [56] derived analytical expressions for the tunneling splitting by making simplifying as-sumptions: Separability and linearity of the PES in between the hyperbolic umbillic point (denoted HU in Fig. 24)and the take-off point rT. Then, the tunneling splitting according to this locally separable linear approximation isgiven by Eq. (87), where SI (0, Qs) is the action along the path denoted LSLA in Fig. 24. The locally separable linearapproximation can be applied for ground and excited states as long as the invariant torus exists. An application tomultidimensional systems with N > 2 seems not to be feasible because it involves a search for the take-off point inmultidimensional space.

Under the assumption that the harmonic approximation is valid for the wavefunction in the allowed region one canextend the I region to the minima such that R and C regions vanish and the PES is inverted, −V (X, Q). Consider theright minimum: a field of trajectories that emanate from the right minimum in the infinite past can be defined [59].Only an extreme tunneling trajectory is considered, that satisfies Eq. (86) at certain point Qs . Such trajectory starts-offfrom the symmetry line with momentum perpendicular to the symmetry line and approaches the hill along the weakestmode. For a SMC-type PES, Benderskii et al. [59] found a splitting formula similar to Eq. (87), where SI ≡ Sinst isthe action along half of the extreme tunneling trajectory. The extreme tunneling trajectory is called the instanton. Thisapproach was also applied to other types of potentials [57,58] as well as to a full-dimensional model of malonaldehyde[142]. The instanton theory was also generalized to treat low vibrationally excited states [59,143].

The crucial point in multidimensional instanton calculations are the determination of the instanton trajectory as well asthe calculation of the prefactor. For the latter, an adiabatic separation of different DOFs is often done [144–147].A relatedapproximate approach was introduced by Tautermann et al. [83–85] which combined the search for a Garrett–Truhlartype tunneling path (cf. Eq. (30)) with a prefactor coming from a one-dimensional theory by Garg [148]. Recently,Mil’nikov and Nakamura [149,150] gave a rigorous numerical implementation of instanton theory that is applicableto multidimensional systems. The performance of the new method was demonstrated, e.g., for malonaldehyde [151]and the formic acid dimer [152]. In particular it was shown in Ref. [152] that approximate instanton theories maypredict tunneling splittings for vibrationally excited states even in cases where the applicability of instanton theory isnot justified.

Real-time propagations of the semiclassical wave function have been intensively discussed as an alternative tothe stationary WKB approach. In particular the initial value representation of the Herman–Kluk [153] semiclassicalpropagator [154] attracted considerable interest (for applications in the present context, see, e.g. [155–160]). Using thefilter diagonalization method for the spectral analysis of semiclassical correlation functions (see Ref. [161] for a recentreview), spectra including tunneling splittings could be obtained for simple SMC-type model PES [156,160]. However,the anharmonicity of ab initio PES seems to render this approach inapplicable as far as high-resolution information,such as tunneling splittings, are concerned. This was demonstrated in Ref. [160] for the HAT in 3,7-dichlorotroploneusing a CRS-Hamiltonian model.

4.2. Classical trajectory approach for tunneling

4.2.1. The Makri–Miller model (MM)The dynamics of large systems is often successfully described by classical mechanics. Focussing on HB dynamics

and in particular on HAT or PT the immediate question arises how quantum effects can be incorporated into a classicaltrajectory-based method. Tunneling paths are widely discussed in the context of generalized transition state theories[63,162,163]. Usually adiabatic separability of the intrinsic reaction coordinate and the remaining orthogonal DOFis assumed. In Section 2.3.1 it was already mentioned that the effect of small IRP curvature can be accounted for bymeans of an effective mass. Avoiding the introduction of the effective mass leads to tunneling paths that lie on theconcave side of the IRP and cross the symmetry surface at a position displaced from the saddle point. This effect wasalready investigated by Marcus and Coltrin and it was termed corner cutting [164]. The same effect can be found, e.g.,for the instanton path (cf. Section 4.1).

For large IRP curvature the small curvature tunneling approximation breaks down and straight-line paths are favoredleading to the large curvature tunneling or sudden approximation. The straight-line paths go from a turning point on one

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246 K. Giese et al. / Physics Reports 430 (2006) 211 –276

side of the IRP to the other side, i.e., they cross a nonadiabatic region, the so-called reaction swath. However, alreadyMarcus and Coltrin [164] pointed out that the “true” tunneling path should satisfy the least action principle of classicalmechanics. The formal justification is given by the multidimensional WKB theory (cf. Section 4.1): the solution ofthe Hamilton-Jacobi equations in the I region is given by classical trajectories propagated on the inverted PES. Stillanother family of paths was suggested by Garrett and Truhlar and has been discussed already in Section 2.3.1. Finally,tunneling paths (maximum probability paths) have been introduced on the basis of the vibrational ground state density[165]. For a general discussion of various approximations in the context of a two-dimensional model potential, see alsoRefs. [166,167].

To account for tunneling processes in multidimensional trajectory based simulations, Makri and Miller [168] proposeda simple method that is based on the one-dimensional WKB result for the tunneling splitting � in a symmetric double-well potential [138],

� = h�

exp

{− 1

h�

}, (89)

where � is the classical oscillation frequency of the particle in each of the wells and � is the classical action integralfrom the left turning point −q0 to the right turning point q0 > 0 (as before q denotes general coordinates),

� =∫ q0

−q0

|p(q)|dq. (90)

It was noted [168] that according to Eq. (89) one-dimensional tunneling can be interpreted in an intuitive way: a classicalparticle oscillates back and forth in one well; each time the turning point q0 is encountered, there is a certain (typicallysmall) probability, exp{−�/h}, for tunneling into the other well, with the rate of such events being proportional to theoscillation frequency �. For small tunneling probability, the cumulated tunneling probability for the time-interval 0 tot is the sum of the individual probabilities,

�(t) =∑tj � t

exp{−�/h}, (91)

where tj are the times at which the turning point q0 is encountered. The function �(t) is a staircase function withstep-size 2/�. Averaging over an ensemble of trajectories with different phases yields a straight line, the slope ofwhich is proportional to the tunneling splitting,

� = 2hd

dt〈�(t)〉. (92)

This equation is the basis for a generalization to the multidimensional case. To this end, it is assumed that tunnelingproceeds along straight lines given by a tunneling direction d. A turning point is defined by the condition that the sign ofthe momentum p projected onto direction d (i.e., p · d) changes from plus to minus. Each time tj when such a classical

turning point q(cl)j is encountered, the tunneling probability is computed by [168],

�j =∫ �max

0d�√

2[V (q(cl)j + �d) − V (q(cl)

j )], (93)

where �max is the maximum length of the straight-line path for which the square root is real. The cumulated tunnelingprobability is then given by Eq. (91) with the replacement � → �j . The multidimensional �(t) is still a staircase butwith varying step-size and step-height. However, due to the exponential factor only the smallest �j contribute to thesum. The tunneling splitting is obtained from Eq. (92). The ensemble of trajectories is the invariant torus correspondingto a certain eigenstate, i.e., a numerical implementation may use adiabatic switching or normal mode sampling [169].The quasi-periodicity of the dynamics on the invariant torus ensures �(t) to be linear also in the multidimensionalcase. However, when the initial conditions are generated by an approximate method like normal mode sampling, thesampling error leads to deviations from linearity [169].

For a multidimensional system the choice of the tunneling direction d is nonunique. In the original paper, Makriand Miller proposed to use the shortest straight-line connection between the caustics of the left and right invariant

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 247

(a) (b) (c)

Fig. 25. Tunneling directions as proposed by Makri and Miller in Ref. [168]. Tunneling is assumed to occur along straight lines parallel to the givenarrows. Thick black lines correspond to caustics of invariant tori. (a) SMC case as well as asymmetric coupling case with large (b) and small (c)displacement.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 5 10 15 20 25 30 35 40 45

(t)>

t / fs

Fig. 26. Cumulated tunneling probability 〈�(t)〉 for an ensemble of 2000 trajectories corresponding to the first-excited OH stretch 12D model oftropolone (cf. Section 3.3). The maximum propagation time was Tmax = 48 fs. The deviation from linearity is due to the approximate nature of thenormal mode sampling method.

tori. Their original proposal is illustrated in Fig. 25 for the SMC and asymmetric coupling case. For the SMC casethe tunneling direction always coincides with the large amplitude coordinate. For the asymmetric coupling case thetunneling direction not only depends on the coupling strength, but also on the eigenstate of interest. Alternative choiceswere discussed in the literature [169–173].

It was shown that the Makri–Miller model can be used to estimate tunneling splittings for multidimensional systems.For instance, calculations using 2D model [174] and full-dimensional PES [80] or on-the-fly molecular dynamics [175]were carried out. As an example, we show the cumulated tunneling probability for the first-excited OH stretch of the12D model of tropolone discussed in Section 3.3. From the slope a tunneling splitting of 32 cm−1 can be determined,in good agreement with the quantum value of about 20 cm−1(cf. Section 3.3) (Fig. 26).

Finally, it was shown that the model can be used to incorporate tunneling into classical molecular dynamics simu-lations [172]. To this end, a particle that hits a turning point was physically displaced along the straight-line tunnelingdirection with probability equal to exp{−�/h}.

4.2.2. The extended Makri–Miller model (EMM)From the discussion in Section 4.1 it is clear that the straight-line approximation of the Makri–Miller model may

break down in a situation sketched in Fig. 24b. Considering an SMC Hamiltonian, for any set of fixed parameters a, c,�, and such a situation is met for large enough �, since the minima are at (±√a/c, −�a/c 2) and the saddle pointis at (0, 0). In order to deal with such as situation an extended Makri–Miller (EMM) model has been developed inRef. [176] that can also approximately account for tunneling through the I region. The following discussion addresses theSMC case, because only in this case a pronounced mode-selectivity (spanning orders of magnitude) can be anticipated.The asymmetric coupling case is complicated by the dependence of the tunneling splitting upon the phase of the wavefunction along the symmetry line [60]. The squeezed case is not interesting for the present purpose, because straightline paths are well suited, especially the instanton is a straight line in this case (cf. Fig. 5).

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248 K. Giese et al. / Physics Reports 430 (2006) 211 –276

In a one-dimensional system turning points q0 are uniquely defined by the energy E and PES V (q) via conditionV (q0)=E. In a multidimensional system classically allowed and forbidden regions of configuration space are dividedby caustics, that is, a surface (or line in 2D) of points on which trajectories cross from one branch of the momentumfunction to a neighboring branch. This ensemble is mathematically described by the Lagrange manifold (e.g., invarianttorus) on which the branches of the momentum function are defined. For nonseparable potentials one cannot uniquelydefine the kinetic energy for a particular coordinate, thus it is impossible to obtain the turning points [177]. Therefore,in Ref. [176] the following practical but approximate definition for multidimensional turning points was suggested. Let{dj } be the set of N normal mode vectors corresponding to the right minimum of a double-well system. For a harmonicsystem, turning points are given by the condition

dj · p = 0, (94)

where p is the momentum vector. In Ref. [176] it was shown that this condition is also reasonable for a modestlyanharmonic case.

With this prerequisite the algorithm of the extended Makri–Miller model involves the following steps: Initial con-ditions for an ensemble of trajectories are selected either by adiabatic switching or by normal mode sampling. All thetrajectories are initially located either in the right or left well. A trajectory of the ensemble is propagated in the R regionfor a certain amount of time T. During this time the trajectory will hit classical turning points q(cl)

n . These turning pointsare determined by the condition dj · pt = 0 [Eq. (94)].

Each time the trajectory hits a classical turning point q(cl)n , the so-called parity of motion �j of the corresponding

direction dj is flipped from +1 to −1. The use of parities of motion allows to introduce real-valued momenta evenin the classically forbidden region [178]. More details are given in Appendix C. The trajectory is propagated in theforbidden region according to the equations of motion (C.3–C.4). The condition Eq. (94) is also used to determinenonclassical turning points outside the R-region: When the turning point condition is fulfilled for a trajectory in theforbidden region then another parity �k of the corresponding direction is flipped from +1 to −1. Both trajectories in theforbidden region, the one with �j = −1 and �k = +1 as well as the one with �j = −1 and �k = −1, will be integratedin this case. When a trajectory, that is propagated in the forbidden region, crosses the symmetry line �, the complexaction of that trajectory, Eq. (C.5), is computed along the trajectory from q(cl)

n to the crossing point on the symmetryline. There may be several trajectories that emanate from one classical turning point q(cl)

n . Only that action is kept, thathas the smallest imaginary part whose absolute value is denoted Wn. The contributions of all nonclassical trajectorieswith smallest action that have emanated from classical turning points q(cl)

n at time tn are summed up according to

�(t) =∑tn � t

exp

{− 2

hWn

}. (95)

If the initial conditions correspond to an invariant torus then the ensemble average, 〈�(t)〉, is a straight line and thetunneling splitting is given by Eq. (92).

For the case of a SMC PES the algorithm is illustrated in Fig. 27. Vectors of the 2D space (X, Q) are denotedas r. One may distinguish two important cases: In case (i) a classical trajectory (solid) propagated in the R-region hitsa turning point r(cl) “(1)” at the C2-R caustic. The emanating trajectory (dashed) resides solely in the C2 region andhits the symmetry line. In case (ii) a turning point “(2)” of the C1-R caustic is touched giving rise to a trajectory in theC1 region. This trajectory hits several nonclassical turning points at the caustics of the C1 and neighboring I regions.For instance, the third non-classical turning point rT of that trajectory is located at the I − C1 caustic. A second parityis flipped at that point giving rise to an additional trajectory that is propagated in the I region. This trajectory thencrosses the symmetry line. By this procedure, many trajectories are generated for a single classical turning point r(cl)

and several of them will cross the symmetry line. Only the trajectory with smallest imaginary action is kept which ismotivated by the least action principle. It is interesting to note that in Ref. [173] it was demonstrated that for the caseof HAT in HO−

2 trajectories with one and two parity flips are contributing equally to the tunneling splitting.In the case of small �, when the saddle point falls into the C2 region, the parity-based trajectories in that region are

expected to have smaller action than those that propagate through the C1 and I region, and vice versa for the case oflarger �, when the saddle point falls into the I region. The situation in this case is reminiscent of the locally separablelinear approximation. Compared to the locally separable linear approximation there are two important differences:

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 249

Q

I

X

C1

(1)

(2)

C2

case (ii)

case (i)

rT

Fig. 27. Right well of Fig. 24 for an unspecified saddle point. Tunneling trajectories emanating from two classical turning points, (1) and (2), areshown. Cases (i) and (ii) correspond to tunneling through the C and I region, respectively.

(i) the present formulation assumes separability in the C1 region, and (ii) the C1-I boundary is determined by theturning point condition Eq. (94), i.e., the turning point condition is used for the classical forbidden and allowed region.For the locally separable linear approximation the C1-I boundary was approximated by a straight-line continuation ofthe exact C2-R boundary. The two approximations, have been justified by numerical tests in Refs. [173,176] (see alsonext section).

4.2.3. Application of the MM and EMM methodsThe performance of MM and EMM methods has been scrutinized in Ref. [176] on the basis of a generic SMC

model potential which allowed for a wide variation of parameters and related tunneling regimes. The findings can besummarized by plotting the tunneling probability

Ptunnel(Q) = exp{−2Wn/h} (96)

vs. position on the symmetry line Q (where X = 0 on �) for the semiclassical methods, EMM and MM, as comparedwith the density |�(0, Q)|2 of the quantum mechanical symmetric ground state along the symmetry line.

In Fig. 28 results for an SMC Hamiltonian with parameters g=0.04 and =0.64 and for three values of �, i.e. couplingstrengths, are shown. The maxima of the EMM and quantum data are normalized to one, i.e., the pre-exponential factoris assumed to vary only slowly along Q compared to the exponential part. The data of the MM is normalized by thenormalization constant obtained for the EMM. This is justified, because both methods only yield the exponential partof the wavefunction.

In the upper panel (� = 0.16) the saddle point is located at Q = 0.25. Due to corner cutting the maximum of |�|2 atQ=0.16 does not coincide with the saddle point where the energy is lowest along the symmetry line. The scatter plot ofthe EMM appears to be shifted systematically by about 0.025 to the right, but it agrees quite well with the numericallyexact results for Q > 0. For Q < 0 there are two branches of the scatter plot. This can be attributed, respectively, totrajectories that tunnel from the C2 into the I region and to trajectories that tunnel from the C1 into the I region. Thescatter plot of the MM shows a single branch that rises to about the maximum of the QM curve. Note that MM andEMM are normalized by the same constant. The maximum of the MM data is only about 4% lower than the maximumof the EMM data, but the MM sampling deviates strongly from the QM results. This is not surprising, because thereare no trajectories in the I region. However, for the tunneling splittings only the magnitudes of exp{−2Wn/h} matter,therefore the magnitude of the MM tunneling splitting agrees with the exact result [176].

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250 K. Giese et al. / Physics Reports 430 (2006) 211 –276

Fig. 28. SMC Hamiltonian with g = 0.04, � = 0.8 and three different values of �: The QM density |�(0, Q)|2 (dimensionless) along the symmetryline � (Q dimensionless) is compared with scatter plots of the tunneling probability Ptunnel(Q) [cf. Eq. (96)] computed with, respectively the EMMand MM model. The position of the saddle point is indicated by arrows.

In the middle panel of Fig. 28 (� = 0.32) the saddle point is located at Q = 0.5; the maximum of the QM curve isat Q = 0.38. The EMM scatter plot agrees with QM result (systematic shift of 0.025) quite well; the second branch atabout Q ≈ 0 almost vanishes. Conversely, the MM results show a strong deviation. Especially, the maximum of theMM is 84% smaller than the EMM maximum. Upon a further increase of � the MM scatter values become very smallas compared to the EMM values. Exemplary, this is shown in the lower panel of Fig. 28 (� = 0.64). Here, the saddlepoint is at Q = 1.0 and the QM maximum is at Q = 0.76, respectively. The EMM scatter plot agrees quite well with

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 251

Fig. 29. Geometries of FAD (schematic): (a) minimum (C2h); and (b) transition state (D2h) geometry (adapted from Ref. [179]).

the QM results (systematic shift of 0.01). Clearly, in this case only the EMM yields the correct order of magnitude ofthe tunneling splitting [176,132].

To sum up, since the MM approach does not account for tunneling through the I-region, one observes its breakdownfor �= 0.32 and 0.64. On the other hand, the approximate EMM method performs reasonably well in such cases [176].

In the following we will focus on a concrete example, that is the double HAT in carboxylic acid dimers. The formicacid dimer (FAD) is the most simple carboxylic acid dimer. The minimum (C2h) and transition state (D2h) geometryof gaseous FAD is shown in Fig. 29 [179].

FAD was thoroughly investigated by experimental and theoretical techniques. Recently, the ground state tunnelingsplitting of FAD (more precisely (DCOOH)2) was measured by Madeja and Havenith [29] to be 0.00286 cm−1. In anearly theoretical gas phase study, the structure of FAD was found to keep approximately C2h symmetry along the IRP[180] indicating a concerted transfer as suggested by the transition state geometry. Shida et al. [77] investigated a 3Dmodel of FAD; their tunneling splitting value of 0.004 cm−1 corresponds rather well with the experimental finding. Thebarrier height was 49.4 kJ/mol and the step wise transfer mechanism was found to contribute about 20% to the tunnelingsplitting. More recently, Luckhaus obtained a tunneling splitting of 0.0015 cm−1using a 6D model [89]. Approximate[85,147] as well as a rigorous implementation [152] of instanton theory have been applied to the FAD. The most reliableinstanton-based value for the ground state tunneling splitting to date is 0.0038 cm−1[152]. For elevated temperatures(or excitation energies) the reaction mechanism may change [181,182] or not [183]; a clarification considering also thelimit T → 0 was not yet achieved.

For the present purpose a concerted HAT for FAD is assumed, i.e., the transition state has D2h symmetry. Thus, thetunneling in FAD is very similar to that of tropolone and related molecules and the main characteristics of the PEScan be covered by a SMC-PES. Along the IRP first the monomers approach each other and then the hydrogen transfertakes place [181]. This means, one may vary the IRP curvature by substituting the two hydrogens, H5 and H′

5 by, e.g.,fluorines or phenyl rests. In terms of SMC parameters, the coupling strength � can be varied by choosing differentsubstituents. The SMC parameters itself can be determined by the method of Takada and Nakamura [60] as outlined inAppendix A.

Various parameters of FAD are reported in Table 3. Results for a number of different levels of quantum chemistry werealso reported in Ref. [180]. Among them, barrier heights �E� deviate by several kJ/mol and no rigorous convergencewas achieved. The value in Table 3, �E� = 47.6 kJ/mol, is assumed to be a reasonable estimate, although a veryrecent high-level quantum chemistry calculation [DFT/aug-cc-pVTZ with CCSD(T) corrections] points towards alower barrier of 33.1 kJ/mol [152].

The most prominent geometrical feature is the decrease by 0.33 A of the rOO and rCC distances (definition: seecaption of Table 3) for the saddle point geometry compared to the minimum geometry. Conversely, the rCR distancesdo not change, and the two monomers approach each other as a whole.

There are parameters for two derivatives of FAD in Table 3, too: the fluoro formic acid dimer (FFAD) and thebenzoic acid dimer (BAD). In these compounds the two hydrogens, H5 and H′

5 (cf. Fig. 29) are substituted by eithertwo fluorine atoms (F) or two phenyl rests (C6H5). Both molecules, FFAD and BAD, are planar. The barrier heightof FFAD is only slightly increased by 1.5 kJ/mol as compared to FAD, while for BAD it is 7.4 kJ/mol lower. Thevariation of the geometrical parameters of the carboxylic ring is moderate among the species. Most prominent is thedecreased inter-monomer distance rCC for FFAD (3.80 A) as compared to FAD (3.91 A) and BAD (3.89 A). In all threecompounds the monomers move as a whole when approaching the saddle point, which can be deduced from the almostunchanged rCR bond length.

The dimensionless geometrical parameter � for FFAD (1.19) is only slightly smaller than that for FAD (1.28). ForBAD (1.82) it is significantly larger, because the difference between mass-weighted minimum geometry XR and center

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252 K. Giese et al. / Physics Reports 430 (2006) 211 –276

Table 3Parameters for carboxylic acid dimers [MP2/6 − 31 + G(d)]Molecule FAD FFAD BADa

�E� (kJ/mol) 47.6 49.1 40.2�Ezpe (kJ/mol) 1.2 1.5 1.5

rOO (A) 2.77 (2.44) 2.75 (2.43) 2.69 (2.42)rOH (A) 1.00 (1.22) 1.00 (1.21) 1.00 (1.21)rCC (A) 3.91 (3.58) 3.80 (3.50) 3.89 (3.63)rCR (A) 1.09 (1.09) 1.34 (1.33) 1.49 (1.49)

�b 1.28 1.19 1.82

�as/2c (cm−1) 3298 3350 3107�m/2c (cm−1) 203 (497) 173 (382) 119 (199) 1/2c (cm−1) 1480 1181 1236 2/2c (cm−1) 471 439 377

g = 1/8�E� 0.0477 0.0371 0.0477� = 2/ 1 0.318 0.372 0.305� = ��2 0.129 0.164 0.169

Parenthesized values correspond to the saddle point geometry. rOO is the O′2.O4 distance, rCC is the C′

3.C3 distance, rCR is the C3–H5 distance,and rOH is the O2–H1 distance.

aB3LYP/6-31 + G(d) geometry.bcf. Eq. (A.1).

geometry XC is primarily the displaced hydrogens, while the difference between saddle point geometry XTS and XC isthe displacement of the two monomers. Obviously, the monomers are considerably heavier for BAD leading to largermass-weighted distances than for FAD and FFAD.

The effective frequencies, 1 and 2, were obtained by the projection of normal mode frequencies onto w1 and w2,respectively. Direction w2 corresponds to an effective inter-monomer vibration. The frequencies 2 decrease by goingfrom FAD to BAD; this behavior is similar to that of the minimum normal modes of the inter-monomer vibration �m

and it is mainly a mass effect. The frequencies 1 correspond to direction w1 which includes mainly the hydrogenmotion. The effective frequency 1 of FFAD and BAD is rather different from the parent compound. No such differencecan be anticipated from inspecting the anti-symmetric OH stretch frequencies �as.

The dimensionless SMC parameters are also given in Table 3 for the three species. The two incorporated DOF arethe antisymmetric stretch of the two OH bonds (�as) and the inter-monomer vibration (�m), respectively. They wereexcluded from the ZPE difference �EZPE, which is less than 2 kJ/mol in all cases.

The tunneling splitting �E obtained for the three methods QM (quantum), MM, and EMM, respectively, is givenin Table 4. For FAD, the numerical exact result (3.69 × 10−3 cm−1) corresponds well to the experimental value(2.86 × 10−3 cm−1), indicating that the effective barrier height is reasonable. The semiclassical methods, MM andEMM, yield the same order of magnitude as the QM method which is satisfactory in view of exponential accuracy. Thesignificant deviation of the experimental BAD value from the theoretical predictions was explained by the fact that theexperiment was performed for doped crystals of BAD [85].

Recently, Tautermann et al. [85] determined tunneling splittings of various carboxylic acid dimers using an ap-proximate version of the instanton method. The values are given in the last column of Table 4 and marked as “GTpath” (cf. also Section 2.3.1). The quantum chemistry was performed by a hybrid approach using a combination ofB3LYP/6-31+G(d) and G2(MP2) theory. The barriers for FAD, FFAD, and BAD are, respectively, 37.0, 36.6, and32.0 kJ/mol. The present SMC model and the Garrett–Truhlar path results predict that the tunneling splitting of FFADdrops significantly as compared to FAD and BAD. Concerning the SMC model, inspection of Table 4 unveils that thereason is the decrease of the projected frequency 1; this leads to smaller g, and larger � and � as compared to FAD.The effect is compensated for in BAD by the substantial decrease of 2, which relates to an effective inter-monomervibration.

To summarize, the parameters of FAD (and derivatives) do not lead to predominately tunneling through the I region,and the MM and EMM methods yield a reasonable ground state tunneling splitting. Recently, we have investigated

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 253

Table 4Tunneling splittings for carboxylic acid dimers as calculated by semiclassical and quantum methods for a SMC model Hamiltonian

Molecule �E (cm−1)

QM MM EMM Exp. GT patha

FAD 3.69 (−3) 3.20 (−3) 5.45 (−3) b2.86 (−3) 2.2 (−3)

FFAD 3.55 (−5) 2.05 (−5) 4.55 (−5) 5.3 (−4)

BAD 1.20 (−3) 0.64 (−3) 1.41 (−3) c0.22 (0) 2.2 (−3)

aRef. [85].bRef. [29].cRef. [184] (seleno-indigo doped BAD crystals).

the mode specific tunneling splitting for the HAT in the hydroperoxyl-anion (HO−2 ) which is probably the simplest

example for multidimensional vibrationally assisted HAT [185]. Here one finds substantial deviations between MMand EMM [173].

5. Condensed phase Hamiltonian

The standard model of condensed phase theory contains a reaction coordinate (s) coupled to a harmonic bath(coordinates {Z�}, frequencies { �}) as expressed via the Hamiltonian [38]

H = Ts + V (s) + 1

2

∑�

⎛⎝P 2

� + 2�

(Z� − F�(s)

2�

)2⎞⎠ . (97)

Here F�(s) is the coupling between both subsystems. Comparing this expression with the CRS Hamiltonian inEq. (39) we notice that provided the coordinate dependence of the Hessian is neglected the CRS Hamiltonian ac-tually has the form of the generic system–bath Hamiltonian. While Eq. (97) is only linear in the bath coordinates itcan easily be extended to include quadratic couplings as well. In other words the CRS Hamiltonian approach offersthe possibility to calculate a system–bath Hamiltonian at least in principle. In practice this approach is restricted tosituations where the number of bath DOF is finite such that the Hessian can be calculated. This approach has been takenin Ref. [186] to determine the relaxation rates for the HB dynamics in 3-chlorotropolone (neglecting any solvent) andin particular their microscopic origin in terms of specific anharmonic couplings.

In order to derive an alternative more flexible description let us start by considering the scenario of an intramolecularHB in a medium-sized molecule. In most general terms the total system can be separated into an intramolecular anda solvent part. Focussing on the molecule itself the experimental conditions usually will suggest a separation betweenrelevant (i.e. active) and passive DOF. For instance, femtosecond IR spectroscopy ofA–H vibrations highlights processesinvolving the strongly coupled modes such as the HB vibration on a short time scale only. All other modes, whosecoupling to the A–H vibration is weak, will be less important. This does not imply that their effect is negligible. Indeed,together with the solvent DOF they provide a reservoir (or heat bath) for energy release out of the active modes.Furthermore, their fluctuation may cause a modulation of the active DOF’s transition frequencies and by that causecoherence dephasing.

The circumstances described so far are also characteristic for other system–bath situations, e.g., electron transferreactions [38]. The only specification for the situation of HBs so far has been the introduction of two types of reservoirs,i.e. an intra- and an intermolecular one. The motivation for this subdivision also derives from the desire to be ableto have a clear assignment of the relaxation processes. Moreover, in particular for the relaxation of higher-frequencymodes specific intramolecular reservoir modes might become important and identifying them may suggest to includethem into the active system. Notice that for smaller system such as HOD dissolved in D2O [187,188] it is, of course,possible to include all intramolecular DOF into the relevant system.

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254 K. Giese et al. / Physics Reports 430 (2006) 211 –276

S Q

Z

V (s,Q,Z)

Fig. 30. Separation of the DOFs of an intramolecular HB system in a solvent, into an active (s), a intramolecular reservoir (Q), and an intermolecular(solvent) reservoir (Z).

Thus the total Hamiltonian can be written as [189]

H = Hs + Hq + HZ + V (s, Q, Z), (98)

where s, Q, and Z stand for the active, the intra- and intermolecular reservoir modes, respectively (see. Fig. 30). Itshould be emphasized that no assumption has been made concerning the type of coordinates so far. If, for instance,normal mode coordinates are used for the molecule, the Hamiltonian will be of the Watson form, Eq. (9). This impliesthat one has to take care not only of the translational motion but also of the vibration–rotation coupling. These termsare usually taken as part of the bath or the interaction Hamiltonian [187,188].

The system–bath interaction Hamiltonian can be expressed in the standard form [38]

Hs.b ≡ V (s, Q, Z) =∑u

K(u)(s)�(u)(Q, Z) (99)

with the operators K(u) and �(u) belonging to the system and the bath, respectively. Explicit expressions for theseoperators can be obtained from a Taylor expansion of the interaction potential V (s, Q, Z).As noted before this approachoffers a convenient means for characterizing the interaction in terms of transitions associated with specific DOF. Thesimplest term is of bilinear form (LL coupling):

H(LL)s.b =

∑i

si∑�

h �g(LL)i (�)Z� (100)

where we assumed that the solvent bath is harmonic with frequencies � and g(LL)i (�) is the dimensionless coupling

strength between the ith system coordinate and the�th solvent coordinate. Of course, there is a similar bilinear interactionbetween si and the intramolecular bath modes {Qj }. While being important for large amplitude system motion, thisLL coupling will vanish if the dynamics takes place in the vicinity of a potential minimum where si corresponds to anormal mode vibration. In passing we note that Eq. (100) assumes that the coordinates are dimensionless as well. Thiscan be always achieved by defining an appropriate length scale for a given coordinate [116]. Eq. (100) gives rise tosimultaneous one quantum transitions in the system and solvent bath. One of the next terms in the Taylor expansion ofEq. (99) would involve the interaction Hamiltonian of quadratic-linear (QL) form

H(QL)s.b =

∑i

s2i

∑�

h �g(QL)i (�)Z� (101)

which gives rise to two-quantum transitions in the system and additionally is the leading term responsible for puredephasing. Again, an interaction ∝ s2

i Qj is possible as well, even in the vicinity of a potential minimum. The lowestorder term which involves transitions in all three subsystems is given by

H(LLL)s.b =

∑i

si∑�

∑j

h �g(LLL)ij (�)QjZ�. (102)

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 255

By virtue of this term solvent-assisted transitions become possible. Such processes may be important if, for instance,the deexcitation of the system coordinate is not resonant with the excitation of some intramolecular mode. Of course,other terms of the Taylor expansion are possible and the actual choice will depend on the system at hand and shouldbe guided by experimental observations (an example will be given in Section 6.3).

Since the relevant system Hamiltonian will usually be low-dimensional it can be diagonalized, i.e.

Hs|a〉 = Ea|a〉. (103)

In principle the eigenstates |a〉 will depend on all bath coordinates. In order to make the perturbation of the system statesdue to the interaction with the bath small, it has been proposed to include the bath-averaged interaction Hamiltonianinto the definition of the relevant system Hamiltonian, i.e. [186,188]

Hs → Hs + 〈V (s, Q, Z)〉B. (104)

This way, only the fluctuations of the system–bath interaction are entering the model

Hs.b → Hs.b − 〈V (s, Q, Z)〉B. (105)

But there is also a bath perspective: for anharmonic systems 〈a|s|a〉 is likely to be different for different states whichchanges the system–bath interaction. The interdependence of system and bath is often neglected or even does not occurby virtue of the model Hamiltonian, e.g., assuming a harmonic bath the bath-averaged couplings, Eqs. (100)–(102)would vanish. There are, however, situations where such a treatment is of relevance, e.g. if the system–bath interaction istreated not by a Taylor expansion but within a classical molecular dynamics calculation of the quantum system embeddedin a classical bath. Here, it has been proposed to account for the above mentioned interdependence self-consistentlywithin an iteration procedure [188].

The strategy outlined so far can be used to describe nonreactive and reactive dynamics. The separation of the totalHamiltonian, Eq. (98), is tailored for application of perturbation theory with respect to the system–bath interaction.However, often the influence of, e.g., charge fluctuations, of the surrounding solvent or protein is substantial and anonperturbative treatment is necessary. This holds in particular for PT reactions which come along with a pronouncedcharge translocation. A nonperturbative approach which does not resort to such a partitioning of the Hamiltoniancomes at the expense of neglecting quantum effects in the nuclear motion. In this case the forces driving classicalmotion can be generated “on the fly”. For small systems they can even be calculated using quantum chemical methods[181,183,190,191]. This has the advantage that the bond-breaking and making processes are correctly accounted forat the given level of quantum chemistry. For large systems such as proteins, however, the majority of the DOF canbe described using classical force fields within a hybrid quantum mechanics/molecular mechanics scheme, e.g., byusing Car–Parinello molecular dynamics [192–194]. As an alternative to this density functional type of approach,empirical valence bond theory [195] enjoys great popularity as well [196–199]. Here, it is assumed that the groundstate PES of the reactive PT system can be obtained from the diagonalization of a nuclear coordinate dependentHamiltonian matrix whose matrix elements are chosen such as to reproduce, e.g., the energetics of suitable referencereactions.

Notice that these hybrid approaches are not restricted to the limit of classical nuclei. Under the assumption thatthe motion of the proton is much faster than that of the other nuclei, PES for the proton motion can be definedfor each instantaneous configuration of the heavy nuclei along a classical trajectory. This corresponds to the secondBorn–Oppenheimer separation discussed in Section (2.1) [200–204]. Besides the proton motion also a HB coordinatehas been incorporated within this strategy in Ref. [205].

6. Dissipative quantum dynamics of hydrogen bonds

6.1. Lineshape theories and nonlinear response functions

The models of anharmonic coupling introduced in Section 2.1 have also been used to discuss the broadeningmechanism of IR spectra in the condensed phase. For the simple two-mode model, Eq. (2), two approaches canbe distinguished: Using Kubo’s lineshape theory, Rösch and Ratner [206] put forward a model where the dephasing ofthe �AH fundamental transition is a consequence of the direct coupling of the associated dipole moment to the fluctuatingfield of the surroundings. On the other hand, the dephasing has been modeled by assuming an indirect coupling via

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256 K. Giese et al. / Physics Reports 430 (2006) 211 –276

the damping of the low-frequency mode described either quantum mechanically [207,208] or via a stochastic process[209]. For an increased HB strength additional modes such as the A–H bending vibration have to be taken intoaccount. The interplay between the dephasing coupling and the arising Fermi resonance involving the A–H stretchingfundamental and its bending overtone transition has been extensively studied by Henri–Rousseau and coworkers whosupplemented the above mentioned two mode quantum model by a Fermi resonance coupling to a bending overtonetransition while taking into account the direct dephasing of the bending and stretching modes [46,47]. This way theyextended the earlier relaxation-free approach by Witkowski and Wójcik [210] as well as the stochastic description givenby Bratos and coworker for the static modulation limit [211,212]. An overview of these developments can be found inRefs. [45,48].

By now it is well-established that the processes contributing to the lineshape can be unravelled by means of time-resolved IR spectroscopy [23]. A unified description of linear and nonlinear IR spectroscopy can be given within theresponse function formalism in Liouville space [213] (see also Refs. [214,215]). Given the molecule–field interactionHamiltonian, Eq. (70), the linear polarization can be expressed as [213]

P (1)(t) =∫ ∞

0dt1R

(1)(t1)E(t − t1) (106)

with the response function being defined as

R(1)(t1) = i

htr{�G(t1)[�, Weq]}. (107)

Here, G(t) is the Liouville space Green’s function, i.e. G(t) • =U(t) • U+(t), and Weq the equilibrium statisticaloperator of the total system. Eq. (107) is essentially a dipole–dipole correlation function (cf. Eq. (77)) and contains allinformation on the linear IR absorption spectrum. In third order with respect to E(t) the polarization is given by [213]

P (3)(t) =∫ ∞

0dt3 dt2 dt1R

(3)(t3, t2, t1)E(t − t3)E(t − t3 − t2)E(t − t3 − t2 − t1) (108)

with the response function now being defined as a function of the three time intervals between the interactions

R(3)(t3, t2, t1) =(

i

h

)3

tr{�G(t3)[�, G(t2)[�, G(t1)[�, Weq]]]}. (109)

This response function contains all information on third-order nonlinear IR experiments such as pump–probe or photon-echo methods. Eq. (109) can be read in the following way: the first interaction with the laser field via � promotesthe diagonal equilibrium (population) density matrix into an off-diagonal one (coherence). The latter evolves duringthe interval t1 up to the second interaction which turns the coherences into a nonequilibrium population subsequentlypropagating during t2.After the third interaction the system evolves in a coherence during t3. R(1)(t1) and R(3)(t3, t2, t1)

can be simplified if the discussion is restricted to a single transition, say �AH =0 → 1. In this case and after performinga cumulant expansion to second order one obtains for the dipole–dipole correlation function [213]

tr{�(t)�(0)Weq} ≈ |�10|2 exp{−i�10t − g(t)} (110)

with

g(t) =∫ t

0dt ′∫ t ′

0dt ′′C(t ′′), (111)

where we introduced the correlation function of the transition frequency fluctuations ��10(t)

C(t) = tr{��10(t)��10(0)W 0eq}. (112)

Here, the trace is defined with respect to the remaining DOF and W 0eq is the equilibrium density operator for the state

�AH = 0. This correlation function can be calculated analytically, e.g., for the shifted harmonic Brownian oscillatormodel [216]. However, it should be emphasized that this expression is adapted from electronic spectroscopy where theimplied Condon approximation is usually fulfilled. This must not be the case for HB, where the coordinate dependenceof the dipole moment can be of importance (for a recent study of non-Condon effects see Ref. [217]).

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 257

Thus for the understanding of linear and nonlinear IR spectra energy gap correlation functions, Eq. (112), are required.They contain information about dephasing processes, i.e. pure and energy relaxation type dephasing [38]. In a seminalpaper, Oxtoby and coworkers determined dephasing rates from classical molecular dynamics correlation functions andstudied the interplay between anharmonicity of the PES and dephasing [218]. Notice that gap fluctuations may alsoplay a role for the determination of relaxation rates [219].

An alternative description is provided by making use of the fact that it is usually only a small subset of relevantDOF in the sense of Section 5 which will interact directly with the external field while the majority of DOF form abath. Tracing out those bath coordinates and assuming a weak system–bath coupling in the Markovian limit togetherwith an initial factorization of the equilibrium statistical operator into a bath and a relevant system part one obtains,e.g. by using the projection operator formalism, an equation of motion for the reduced density operator �(t) = trB(W)

involving the following Green’s function [38,213]:

Gs(t) = �(t) exp{−iLst − Rt}. (113)

Here, Ls is the Liouville superoperator for the relevant system defined as

Ls • = 1

h[Hs, •] (114)

and R is the relaxation superoperator which will be discussed in the following section. In order to introduce thisapproach into Eq. (109) one has to make the additional assumption that the trace over the bath DOF can be performedfor each time interval separately [213]. This implies a phase randomization between the different propagation intervalswhich allows us to replace G(t) → Gs(t). An early application of this approach in the context of linear IR spectra ofHBs has been given in Ref. [220]. Nonlinear IR spectra of a HAT system have been modeled along these lines, forinstance, in Ref. [221].

6.2. Quantum master equation

Besides providing a means to calculate the nonlinear IR response functions the Green’s function (113) can also beused to study the dynamics of the reduced density matrix, e.g., triggered by arbitrary laser driving. To this end oneneeds to solve the following QME [38]:

��

�t= −i[Ls + Lfield(t)]� − R�, (115)

where the Liouville superoperator for the coherent evolution has been introduced in Eq. (114) and Lfield(t) • =[Hfield(t), •]/h. The relaxation and dephasing due to the system–bath interaction is contained in the superoperator R.Using the explicit form of the system–bath interaction, Eq. (99), it can be expressed in the energy representation givenby {|a〉} (cf. Eq. (103)) as [38]

Rab,cd = �ac

∑e

�be,ed(�de) + �bd

∑e

�ae,ec(�ce) − �ca,bd(�db) − �db,ac(�ca). (116)

Here we introduced the damping matrix

�ab,cd(�) =∑uu′

K(u)ab K

(u′)cd Cuu′(�) (117)

being defined in terms of the Fourier transform of the bath correlation function as

Cuu′(�) = 1

h2 Re∫ ∞

0dtei�t 〈�(u)(t)�(u′)(0)〉B, (118)

where 〈•〉B denotes the equilibrium expectation value of the bath. The energy representation of the QME is sometimesalso called Redfield equation with Rabcd being the Redfield tensor [38,222,223]. The various approximations containedin Eq. (115) have already been mentioned in the previous section. (Notice that Eq. (116) neglects in addition the energy

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258 K. Giese et al. / Physics Reports 430 (2006) 211 –276

renormalization of the spectrum.) There are several approaches which are aimed to go beyond the present limitations,e.g., by including initial correlations or non-Markovian dynamics [224–227] (for an overview, see also Ref. [38]).

The actual form of the damping matrix depends on the system–bath interaction operator at hand. For a bilinear (LL)coupling, Eq. (100), one obtains

�(LL)ab,cd(�) =

∑ij

〈a|si |b〉〈c|sj |d〉C(LL)ij (�) (119)

with

C(LL)ij (�) = �2(1 + n(�))[J (LL)

ij (�) − J(LL)ij (−�)], (120)

where we defined the spectral density

J(LL)ij (�) =

∑�

g(LL)i (�)g

(LL)j (�)�(� − �). (121)

Within the QME approach the spectral density is the central quantity determining the strength of system–bath interactionat a certain frequency. Eq. (121) has been obtained under the assumption of a harmonic bath which is likely to be oflimited use for real solvents. However, within linear response theory the actual fully anharmonic solvent can be mappedonto an effective harmonic solvent. This essentially assumes that there is a uniform distribution of coupling strengthsover a large number of solvent modes. The effective spectral density for the coordinate si then reads [228] (see alsoRef. [229])

J(LL)ii (�) → J (eff)(�) = 2

h2 tanh

(h�

2kBT

)∫ ∞

0dt cos(�t)C(cl)(t). (122)

Here C(cl)(t) is the classical correlation function of the force, −�Hs.b/�si acting on some system coordinate si whosevalue is fixed. Note that the use of (classical) force–force correlation functions has a long history in this respect[230–233].

The vibrational relaxation rate between a pair of states can be expressed as [38]

kab = 2�ab,ba(�ab) (123)

from which the T1 vibrational life-time say of state |a〉 follows as

1

T1(a)=∑b �=a

kab. (124)

In other words, this approach which is sometimes also called Landau–Teller method, allows to calculate vibrationallife-times microscopically from classical molecular dynamics simulations. In practice, the treatment is not restrictedto the LL coupling case, that is, Eqs. (117) and (118) are more general (see below). The use of these expressionsoften comes along with the replacement 〈�(u)(t)�(u′)(0)〉B → 〈�(u)(t)�(u′)(0)〉class [187,188]. Notice, however, thatthis introduces some ambiguity since classical and quantum correlation functions have different properties, e.g., theclassical function will not lead to detailed balance [38]. This is usually accounted for by introducing certain (problemspecific) correction factors [234,235].

Alternatively to a classical description of the spectral density for vibrational energy relaxation one can use modelfunctions. Suppose that the coupling strength can be expressed in terms of the single parameter g

(LL)i and a new spectral

density j (LL)(�) is introduced as being equal for all system coordinates, i.e. J (LL)ij = g

(LL)i g

(LL)j j (LL). A common form

of the spectral density is [38]

�2j (LL)(�) = �(�)� exp(−�/�(LL)c ), (125)

where �(LL)c is some cutoff parameter. As shown in Ref. [236] Eq. (125) can rather accurately reproduce calculated

spectral densities for HB systems in simple systems.

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 259

In terms of the PES expansion of Section 5, accounting for pure dephasing requires to incorporate at least thequadratic-linear (QL) coupling which gives

�(QL)ab,cd(�) =

∑ij

〈a|s2i |b〉〈c|s2

j |d〉C(QL)ij (�). (126)

Within the QME, (115), the total dephasing rate between two levels is then given by [38]

�ab = Rab,ab = 1

2

∑c

(kac + kbc) −∑uu′

K(u)aa K

(u′)bb C

(QL)

uu′ (� = 0). (127)

Here, pure dephasing is accounted for by the last term for which one can use, e.g., the approximate form [116,237,238]

lim�→0

C(QL)ij (�) ≈ g

(II)i g

(II)j

4kBT

hlim�→0

�(j (QL)(�) − j (QL)(−�))

≈ g(QL)i g

(QL)j �pd, (128)

where �pd is a (temperature dependent) parameter. Higher order system–bath couplings such as Eq. (102) can besystematically included yielding expressions for the damping matrix in terms of spectral densities [38,116] and thusadditional contributions to relaxation and dephasing rates. Of course, this approach will generate even more spectraldensities which requires further approximations and lead to additional (usually unknown) parameters. One possibilityto obtain at least the intramolecular part of the Redfield tensor has been given in Ref. [186] for chlorotropolone on thebasis of an ab initio CRS Hamiltonian.

So far we have only discussed those contributions to the Redfield tensor, Eq. (116), which have a straightforwardinterpretation in terms of energy and phase relaxation rates. This restriction is possible in the Bloch limit wherepopulations and coherences within the density matrix are completely decoupled such that∑

cd

Rab,cd�cd → (1 − �ab)Rab,ab�ab + �ab

∑c

Raa,cc�cc. (129)

The validity of the Bloch limit can be appreciated by looking at the secular approximation to the equations of motion,(115). Here, in the spirit of a rotating wave approximation, one retains only those terms on the right-hand side whichobey |�ab − �c′d ′ | = 0, i.e.∑

cd

Rab,cd�cd →∑c′d ′

Rab,c′d ′�c′d ′ . (130)

The conditions for the secular approximation are apparently fulfilled by the Bloch limit terms. However, the right-handside of Eq. (130) also mixes different coherence matrix elements, i.e. a bath-induced coherence transfer, �ab → �c′d ′ , ispossible [236,239,240]. Beyond the secular approximation the bath-induced conversion of coherences into populations,�ab → �cc, and vice versa becomes possible as well [241,242].

6.3. Vibrational energy cascading in an intramolecular hydrogen bond

In Section 3.2.1 we have considered the intramolecular HB dynamics of PMME triggered by ultrafast laser-driving.It turned out that without accounting for the influence of the surrounding solvent the experimental observation of ratherrapid population relaxation of the OH/OD stretching vibration could not be explained. This issue was reinvestigatedusing a system–bath model in Refs. [116,236] and supplemented by further experimental data and simulations in Refs.[112,113]. In the following we summarize the conclusions which have been reached on the basis of a QME study(Section 6.2) combined with an ab initio PES including up to four-mode correlations in a normal mode representationof the relevant system (Section 2.1) as well as two-color IR pump–probe experiments.

In Fig. 31 we show the IR absorption spectrum of PMME-H dissolved in CCl4 together with the spectra of thedifferent pulses which were chosen such as to probe in the �OH-bending fundamental region and to excite in the �OH-stretching, �OH-bending as well as in the C–O stretching region. An additional probe pulse around 2200 cm −1 served

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Fig. 31. Stationary IR absorption spectrum of PMME-H in CCl4 solution. In the lower part of the figure the pulse spectra of the pump and probepulses are shown. (For more details, see Ref. [112]).

Fig. 32. Two-color IR signals as a function of the delay time of probing the �OH = 0 → 1 transition after excitation in the fundamental transitionregions of the �OH, �OH, and C–O stretching vibration, cf. Fig. 31. (For more details, see Ref. [112]).

to monitor the excited state absorption of the �OH-stretching band. From the latter the decay time of about 220 fs of the�OH = 1 state has been inferred. In Fig. 32 the transient signals probed in the �OH = 0 → 1 range are shown for thevarious excitation conditions. For direct excitation of this transition a fast component (800 fs) is observed which canbe attributed to the decay of the bending fundamental state. The same fast component is seen upon excitation of the�OH = 0 → 1 stretching fundamental transition, whereas for excitation of the C–O transition only a slow componentis found, which is present in all data and due to the overall vibrational cooling process.

The initial ultrafast dynamics can be well described within a 5D dissipative quantum model. The five normal modesspanning the PES are depicted in Fig. 33 together with a schematic level scheme showing the fundamental, overtone, andcombination transitions of the 4 fast coordinates, i.e. �OH, �OH, and two modes having out-of-plane bending character,�OH1 and �OH2. Each of these states is dressed by an anharmonic potential energy curve corresponding to the motionalong the low-frequency HB mode �HB.

The coupling between these 5 DOF and the remaining intra– and intermolecular DOF has been treated by includinginto the system–bath Hamiltonian a LL term, Eq. (100), which is responsible for energy dissipation out of �HB into thesolvent as well as a third order term, Eq. (102), which serves to transfer energy from the modes �OH1 and �OH2 intointramolecular bath modes. The latter process is assisted by the solvent and therefore could not be observed in the gasphase simulations discussed in Section 3.2.

An overall picture of the relaxation process can be obtained from Fig. 34 where we show the population dynamics ofthe diabatic states defined for the fast coordinates as a function of time and energy. The cascading nature of the dynamics

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Fig. 33. Normal mode displacement vectors of the 5D dissipative model of the PMME-H relaxation dynamics developed in Ref. [112]. In the rightpart of the figure possible transitions between the four fast mode states, (�OH, �OH, �OH1, �OH2), after excitation of the �OH = 0 → 1 transitionare shown.

is clearly visible and the associated time scales for the �OH and �OH fundamental transitions are well reproduced. Noticethat the modeling includes essentially a single parameter, i.e. the strength of the third-order coupling. The relaxationtime for the low-frequency HB mode was determined from a classical simulation to be about 1.7 ps [236].

The importance of the third-order coupling can be appreciated by simply switching it off. Respective results areshown in Fig. 35. In this case the only means for energy release into the environment is the LL coupling via thelow-frequency mode. Apparently, in the present model this causes a much slower decay of the initially excited statemanifold.

Finally, we note that the low-frequency modulations of the signal after �OH-stretching excitation in Fig. 32 are dueto quasi-coherent wave packet motion of the HB mode (cf. discussion in Section 3.2.1). The nature of this wave packetmotion has been investigated, e.g., for a 3D dissipative model of PMME-D in Ref. [236] where it was shown that it isdue to coherences in the diabatic ground state potential which are excited by the laser field, but may also be triggeredby bath-induced coherence transfer.

The example of PMME shows that the combination of time-resolved IR spectroscopy and dissipative quantum theoryenables one to obtain a detailed understanding of the dynamics underlying complex stationary IR spectra as given inFig. 31. It should be noted that similar observations (ultrafast dynamics, coherent wave packet motion) have also beenreported for the intermolecular HB in the acetic acid dimer [243–245].

6.4. HOD in D2O and H2O

The dynamics of water as the most important solvent has been scrutinized in some detail during the last years [23,246].In particular the microscopic details of the longstanding problem of unusual proton and hydroxide ion mobility becameaccessible to computer simulations [247,248]. Being interested in the fundamental building blocks of these systems,

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Fig. 34. Population dynamics of the fast mode diabatic states (shown at their respective energies) after ultrafast (100 fs) excitation of the �OH =0 → 1region. The dissipative model includes a LL coupling to the solvent as well as third-order solvent assisted coupling between the modes �OH1 and �OH2and two intramolecular bath modes of comparable frequency. Notice that the population of each state is calculated by summing up all vibrationallevels of the low-frequency HB mode for that state. The ground state population is not shown.

Fig. 35. Same as Fig. 34 but without the third-order coupling.

protonated water clusters have triggered quite some attention [249] with the Zundel cation, H5O+2 , emerging as the

prototype system for a strong HB whose IR spectrum is still providing a challenge for interpretation [250–252].These clusters can also be viewed as a means to obtain a deeper insight into the dynamics of proton translocationalong “proton wires” in biological water networks such as bacteriorhodopsin, the prototypical light-driven protonpump [193,253] .

Returning to bulk water, there is a host of ultrafast spectroscopic investigations which usually focus on HOD inD2O or H2O because it allows for a spectral separation between the primarily excited vibrations and the modes of thesurrounding. The central questions are concerned with the understanding of the relaxation and dephasing processeswhich lead, for instance, to the rather broad absorption band. While early time-resolved studies seemed to be limitedby their time-resolution [254], more recent IR pump–probe experiments point to a vibrational relaxation time of theexcited OH-stretching mode of about 0.7 ps [255,256] to 1.3 ps [257]. Considerable mechanistic information comes

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from IR-pump plus Raman-probe experiments which besides an OH-relaxation time of 1 ps allowed to assign theprimary relaxation channel to involve the intramolecular HOD bending and the solvent DOD bending vibrations [258].First theoretical investigations of vibrational relaxation have been presented by Rey and Hynes using a Landau–Tellertype approach which combines a quantum description of the solute with a classical solvent [187]. Their calculatedrelaxation time of 8 ps is too long and indeed the more recent extensive work by Skinner and coworkers gave animproved value of about 2.3 ps [235]. Both agree, however, on the relaxation pathway starting with the HOD bendingovertone. A rather detailed comparison has been given in Ref. [259]. It should be noted that these theoretical approachesassume “intact” intramolecular coordinates for HOD, i.e. the possibility for HB breaking by virtue of energy releaseinto the HB mode is not included. It has, however, been observed in fully classical simulations of OH-frequency gapcorrelation functions [260] and at least in Ref. [256] it has been argued from the experimental point of view, that theuptake of energy by the HB mode contributes to the OH stretching relaxation. For a more detailed discussion of thispartly unresolved issue, see Refs. [23,259].

The OH-vibration of HOD in D2O has also been investigated using IR photon echo spectroscopy. Two-pulse photonechoes probing the evolution during t1 and t3 while t2 is zero in Eq. (109) gave evidence for a rather fast pure dephasingtime of about 90 fs within a Bloch model [261]. This has been assigned within Oxtoby’s model [218] (see Section6.2) to be a consequence of the large anharmonicity of the OH-mode which allows for dephasing already by a linearcoupling model, i.e., Hs.b = sOHf (t), where f (t) is the fluctuating force. Notice that such a term would vanish for aharmonic oscillator coordinate and only the quadratic coupling ∝ s2

OHf (t) would lead to pure dephasing (see Section6.1). A more detailed theoretical investigation in Ref. [262] points to the relevance of electric field fluctuations onlength scales which originate from the local environment during the first 200 fs and from more collective long-rangedmotions at longer times.

Spectral diffusion, that is, the randomization of an initially prepared distribution by the environmental fluctuations,has been probed using 3-pulse photon echo spectroscopy, i.e., the echo peak shift is monitored as a function of the delaytime t2. It is well established that this gives direct information on the gap correlation function, Eq. (112). For the presentsystem it was found that the decay of C(t) has picosecond components indicating structural correlations for times wellbeyond the vibrational relaxation time [263]. Subsequently, even signatures of low-frequency underdamped HB motionhave been observed [264] in accord with the simulations of gap-correlation functions in Refs. [260,262,265].

6.5. HAT and PT reactions

In the foregoing two sections we have considered nonreactive vibrational dynamics of HBs. On the other hand, thekinetics and dynamics of reactive HAT and PT in the condensed phase is at the heart of many chemical and biochem-ical processes (for a general review see, e.g., Ref. [33]). In principle one can distinguish adiabatic and nonadiabatictransfer processes depending on whether or not transitions between the quantum states of the proton are important[38]. Within the QME approach of Section 6.2 there is no need for this distinction since the full information on thequantum states and respective transitions is part of the model and naturally accounted for by solving the equations ofmotion [266–271].

In the context of the hybrid quantum–classical simulations discussed in Section 5, however, the distinction betweenadiabatic and nonadiabatic is essential. Adiabatic PT including the quantum character of the proton wave function hasbeen reported, for instance, for model HB complexes in polar solution [272], excess proton motion in water [273], andenzyme-catalyzed hydrolysis [202]. In Ref. [202] it was also shown that the quantum-classical approach is a limitingcase of a TDSCF ansatz for the total wave function, i.e. it is essentially a mean-field approach.

Nonadiabatic transitions between the adiabatic PES can be accounted for along a classical trajectory of the heavynuclei by means of the surface hopping procedure [274,275]. Combined with a multiconfiguration description ofthe proton wave function this approach has been termed MC-MDQT (multiconfigurational molecular dynamics withquantum transitions) [203]. MC-MDQT has been applied to various processes ranging from excess PT in water wires[276] to hydride transfer in enzymes [277–279].

6.6. Multidimensional IR spectroscopy

Inspired by multidimensional NMR spectroscopy, considerable efforts have been directed towards carrying some ofthe respective ideas into IR spectroscopy. The advantages are obvious: additional information is obtained on congested

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Fig. 36. Homodyne-detected signal, Eq. (131), of a dissipative double minimum system with LL and QL coupling. Left and right column correspondto the Bloch limit and the full Redfield tensor simulation, respectively. In the upper panels t1 = 0 whereas for the lower panels we have t1 = 1000 fs.For more details see Ref. [221].

spectra with the benefit—as compared to NMR spectroscopy—of having a high time-resolution [216,280–282]. Withinthe context of HBs the fluctuations, e.g. of the OH/OD-stretching transition in HOD diluted in D2O/H2O have beena target for 2D-IR studies [283,284]. But also processes in other systems such as the HB breaking in methanol-ODoligomers [285,286] have been investigated. Rather promising are the prospects for unravelling HAT or PT reactiondynamics as shown in theoretical calculations for malonaldehyde [81] and generic dissipative model systems [221,287].Tanimura and coworker, for instance, have demonstrated a novel technique for obtaining chemical reaction rates directlyfrom 2D-IR spectra [287].

2D-IR spectroscopy is rather sensitive to anharmonic couplings of the PES. This feature has been used in Ref.[221] to investigate the dependence of the dissipative dynamics of a reaction coordinate on the type of system–bathcoupling. In particular it was demonstrated that the cross-terms appearing for simultaneous LL and QL coupling, i.e.Rabcd ∝ 〈a|si |b〉〈c|s2

i |d〉, can lead to a breakdown of the Bloch approximation to the QME. Specifically, this termintroduces an efficient bath-induced coherence transfer where the frequency mismatch in the Redfield tensor is of theorder of the ground state tunneling splitting. In other words, although the latter can be small and the related dynamicswell in the picosecond range, there is an influence on the ultrafast dynamics due to coherence transfer. As an examplewe show in Fig. 36 the Fourier-transformed homodyne-detected signal

Shom( 3, t2, t1) =∣∣∣∣∫ ∞

0dt3ei 3t3R(3)(t3, t2, t1)

∣∣∣∣2

(131)

for different coherence times t1. The model system has two pairs of states below the barrier, (|0〉, |1〉) and (|2〉, |3〉). Inthe figure one sees peaks at 3 =�43 =1048, �21 =1630, and �30 =1936 cm−1. Initially the system has been preparedin a nonequilibrium superposition state (|2〉 + |3〉)/√2. Panels (a) and (b) correspond to the Bloch limit for t1 = 0and 1000 fs, respectively. The decay of the different peaks is in the subpicosecond range due to population relaxation(t2-axis). In addition, coherence dephasing during t1 leads to an overall reduction of the amplitude. A similar behavioris seen when the full QME is solved, Fig. 36c and d. However, there is an important difference for the resonanceat 3 = �21, for which the dephasing and relaxation times are apparently longer as compared to the Bloch case.In Ref. [221] it was shown that this is a consequence of coherence transfer of the type �21 → �20 → �21.

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 265

7. Sundry topics

7.1. Geometric isotope effects

The anharmonicity of the PES together with the wave nature of hydrogen motion leads to a measurable quantumeffect already for the vibrational ground state. Consider the case of an asymmetric double minimum potential. Theexpectation value of the bond length 〈RA.H〉quant for an A.H · · · B bond will be larger than the respective classicalvalue, RA.H. Upon deuteration the lower ZPE leads to a reduction of the deviation between quantum and classicalresults such that 〈RA.H〉quant > 〈RA.D〉quant > RA.H. This is called primary geometric isotope effect. For symmetricsingle minimum potentials of strong HBs, A · · · H · · · A, on the other hand, the expectation value of the H or D positionis always midway between the donor and the acceptor.

So far we have not yet considered the coupling between the A–H motion and the hydrogen bond A · · · B motion. Infact, this may cause a simultaneous change of the expectation value of the A · · · B distance upon deuteration. This iscalled secondary geometric isotope effect. It was found in the 1930s in molecular crystals by Ubbelohde (Ubbelohdeeffect) [288] (for an early review, see also Ref. [289]). Generally speaking, for the asymmetric A.H · · · B case thereduction of the A–H distance upon deuteration will cause a weakening of the HB, i.e. an increase of the A · · · Bdistance. The converse will occur for the single minimum A · · · H · · · A case. Upon increasing the A · · · A distance,e.g. for different crystals, the magnitude of the secondary geometric isotope effect will change. Ichikawa compiled acorrelation diagram based on available crystal structures which showed that a HB expansion occurs for A · · · A distancesin between 2.43 and 2.65 A. The maximum expansion is about 0.03 A [290,291]. In fact, much of this behavior hasbeen rationalized by using a simple empirical chain model including the effect of quantum fluctuation of the H/D atoms[292]. In passing we note that solid state [293] as well as low-temperature liquid state [294] NMR offer an alternativeto X-ray or neutron scattering to study geometric isotope effects.

An interesting case occurs when the ZPEs are just below or above the barrier such that there is a substantial differencebetween the H and D wave functions. While the expectation value of the H/D position will be the same, the wave functionwill have a single maximum in the former case while it has two local maxima in the latter case. Consequently, the HBwill contract in the former case and expand in the latter one. Recently, this effect has been studied in the context of anempirical valence bond model by Limbach et al. [295]. The limiting case where the ZPE for H is just above and thatfor D is just below the barrier seems to be realized in the H3O−

2 ion which has been studied using ab initio path integralmolecular dynamics [296].

The calculation of the geometric isotope effect requires knowledge of the respective vibrational ground state wavefunction. This becomes particularly straightforward in cases where this function is rather localized either in a singleminimum potential [299] or in a double minimum potential where the tunneling coupling can be neglected [297,298].Here one can use a PES spanned by the local normal mode coordinates as given in Eq. (10). In Fig. 37 we showan example of a four-dimensional model for the double HB in porphycene. From the two-dimensional cut which

Fig. 37. Geometric isotope effect in porphycene: Two normal mode displacement vectors for which the PES cut in the right panel is shown togetherwith the ground state vibrational probability density (for more details, see Refs. [297,298]).

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covers the symmetric NH-stretching as well as a low-frequency collective ring deformation mode one notices thatthe maximum of the vibrational probability density is not at the coordinate origin (classical limit). Since the doublydeuterated species has a lower ZPE the probability density will cover a less anharmonic part of the PES and its maximumwill be closer to the classical one. Thus, the simultaneous change of NH-distances and HB geometry mediated by theanharmonic coupling leads to the primary and secondary geometric isotope effects.

Finally, we note that multicomponent quantum chemistry [39] offers in principle a rather well-suited alternative tothe construction of a Born–Oppenheimer PES. Here one has the advantage that a geometry optimization of the classicalcoordinates yields automatically information on the geometric isotope effects. However, this does not yet include theeffect of quantum mechanical correlations between the proton and the heavy atom motion, which needs to be takeninto account in a subsequent heavy atom wave function calculation on the electron–proton PES. Unfortunately, for amolecule of the size of porphycene, for instance, this approach is only feasible upon making further approximationsconcerning the quantum chemical model [297].

7.2. Excited state PT

Inter- and intramolecular PT after electronic excitation has attracted considerable attention, e.g., in the context ofindustrial photostabilizers [21,300] and biological photoprotection [301]. In contrast to the IR driven dynamics inthe ground state, the forces experienced by the ground state wave packet upon promotion to the excited state areconsiderable, leading to rather rapid PT reactions. It is also important to note that the PT will be triggered by thelaser interaction at a specific instant of time, opening the way to a better understanding of such reactions in variousenvironments. For instance, using femtosecond pump–probe spectroscopy it was found that the photostabilizer Tinuvincompletes a reaction cycle—excitation, PT, internal conversion, PT—in about 1 ps, where the excited state PT itselftakes about 150 fs [21]. This suggest an essentially barrierless process which is, however, part of a multidimensionalwave packet motion involving several low-frequency DOFs. Oscillatory spectra signaling coherently excited nuclearwave packets where first found in Ref. [302] (see, also Ref. [303]). An even better resolution of the coherent wavepacket motion was achieved in Ref. [304] where a benzothiazole compound has been investigated which has a muchlonger S1 lifetime (for a review see also Ref. [305]). The excitation of four vibrational modes could be identified andthe 60 fs excited state PT was termed to occur ballistically, i.e. without much wave packet dispersion. A theoreticalstudy in terms of normal modes projected on the minimum energy path was given in Ref. [306].

Condensed phase excited state intermolecular PT is of particular importance in the context of photoacid chemistry.Upon electronic excitation photoacids, such as certain aromatic dye molecules increase their acidity substantially andin the presence of a base PT occurs. From the kinetics point of view, this problem has traditionally been handled as abimolecular reaction whose rate is governed by diffusional motion and PT in the encounter complex (for a review seeRef. [307]). Recently, it was demonstrated that the anharmonic couplings between the reaction coordinate for PT andthe vibrations of the proton donor and acceptor can be used to follow the reaction on its intrinsic time scale. To this end,the optical excitation pulse is combined with a mid-IR pulse which observes the transient changes of specific markermodes (for a review, see Ref. [308]). This strategy had been introduced into excited state PT research in Ref. [309]about 20 years ago. It allowed to unravel, for instance, the ultrafast PT steps in the complex between the photoacid8-hydroxy-1,3,6-trisulfonate-pyrene and acetate in deuterated water [310], including the possibility of proton hoppingalong water wires in so-called loose complexes [311]. Although much theoretical work has been focussed on aspectsof PT in acid–base chemistry [25], a microscopic modeling of the transient spectra has not yet been accomplished.

There are also a number of double PT systems which have been studied, putting emphasis on the issue of step-wise versus concerted transfer. For instance, for BP(OH)2 ([2, 2′-bipyridyl]-3, 3′-diol) it was found that step-wise andconcerted transfer occur at the same time, both with a 100 fs time scale [312]. Here, the concerted pathway showeda pronounced dependence on the excitation wave length. Wave packet motion of skeleton modes associated with thetwo reaction pathways have been investigated in Ref. [305]. Excited state double PT has also been investigated forthe 7-azaindole dimer as a model mimicking DNA base pairs. The issue of the transfer mechanism appeared to becontroversial for a long time [313,314]. The most recent time-resolved resonance-enhanced multiphoton ionization(REMPI) study of the gas phase dimer gives evidence for a concerted mechanism [315]. Other model base pairs, likethe 2-aminopyridine dimer, have been studied with regard to the photostability. Here, conical intersections reachedalong the PT coordinate are testifying the breakdown of the Born–Oppenheimer approximation and are the key forefficient deactivation after electronic excitation [316].

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7.3. Laser control of hydrogen transfer

Coherent laser control is emerging as a powerful tool to influence and study nonreactive and reactive moleculardynamics [317–320]. An aspect crucial for the realization of control is the coherence of the evolving wave packet. Withthe demonstration of coherent wave packet dynamics of HBs [108] in solution, the question can be raised whetherelectronic ground state PT or HAT can be triggered by adapted IR pulses [321]. The case where the dynamics isrestricted to the two lowest vibrational states in a double well potential has been extensively investigated (for a review, seeRef. [322]). Taking into account higher excited states, the pump–dump scheme [323] is perhaps the most straightforwardcontrol strategy. For an asymmetric PES the initial state which may be localized in the left minimum is first excited toa state which is energetically above the barrier (pump). A second pulse then de-excited (dumps) the system to a statelocalized in the right minimum. This strategy has been applied to a generic two-dimensional model in Ref. [324] and toa two-dimensional model of thioacetylacetone in Ref. [269]. While parameterized pulse sequences have the advantagethat the optimized pulse has a straightforward interpretation, optimal control theory which optimizes the laser fieldsuch as to achieve, e.g., a maximum overlap between the wave packet and some target state at a given time turned outto be much more powerful in predicting, e.g., new reaction pathways. For instance, in Ref. [269] it was demonstratedthat upon increasing the penalty for high pulse intensities during the optimization, the HAT dynamics changes froman above-the-barrier pump–dump like scheme to that of a through-the-barrier tunneling. The latter mechanism wastermed “Hydrogen Subway” in Ref. [325]. This mechanism which does not involve highly excited vibrational stateswas later shown to particularly useful in condensed phase situation where phase and energy relaxation competes withcoherent laser driving [270,271,326,327]. (For a general account on the influence of pure dephasing on HAT control,see also Ref. [328].)

Given the success of low-dimensional model studies one might ask whether the concepts can be applied to mul-tidimensional systems. In this case making a guess for an analytical pulse sequence would be a real challenge andmuch hope lies on optimal control theory. However, the implementation of the obvious combination of optimal controltheory and MCTDH has not yet been reported. In Fig. 38 we show an example indicating the complications one facesin multidimensional HAT control [122]. For illustration we consider the 7D model of deuterated salicylaldimine dis-cussed in Section 3.2.2 [121]. The two dimensions corresponding to the planar HAT are used to define diabatic states(cf. discussion in Section 3.2.2). Each diabatic state is dressed by a 5D harmonic PES describing the modes shownin Fig. 13. The coordinate dependent diabatic state couplings are included but their effect on the PES is not shown inthe figure. The initial state is the ground state wave packet in the potential curve corresponding to the diabatic statelocalized in the keto well (cf. Fig. 14 d) (thick line). For the generation of the initial wave packet the state couplingsare switched-off. This could be a choice of a product target state in a control scheme. Due to the anharmonic couplings

Fig. 38. Left: diabatic (with respect to the D-motion) potential energy curves along mode �14 for a 7D model of deuterated salicylaldimine(cf. Fig. 13). The thick solid curves represent the ground state (� = 1) and the initial state (� = 10) which is populated at time zero in the dynamicssimulation shown in the right panel. Right: Population dynamics of the ten lowest diabatic states after initial population of the HAT product likestate � = 10 (curves are shifted by (� − 1) × 0.25). For details, see also Ref. [122].

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268 K. Giese et al. / Physics Reports 430 (2006) 211 –276

between the harmonic modes and between the diabatic states the decay of the initial state back to the overall groundstate is rather rapid as seen from the diabatic state populations in Fig. 38. However, there appears to remain somesmall population in the initial state during the first picosecond. Moreover, a detailed analysis shows that the energyis almost equally redistributed over all five harmonic modes. In other words, at least for this system, the dynamics istruly multidimensional and the choice of a reaction path or even of a sufficiently stable target state for laser driving isnot obvious.

8. Summary

The theoretical description of hydrogen bond dynamics continues to provide a challenge since it comprises multi-dimensionality and pronounced quantum effects. This setting would be interesting on its own because it can be arguedthat methods developed in its context are rather likely to be transferable to other systems with less quantum behavior.But, the intriguing aspect of hydrogen bond research is its broad scope in terms of possible application rangingfrom the properties of water via biological light-triggered proton pumps to enzyme catalysis. The last years havewitnessed an enormous increase of the microscopic insight into these processes due to experimental and theoreticalprogress.

In this review we have focussed on recent theoretical efforts to understand the stationary and time-resolved infraredspectroscopy of hydrogen-bonded systems starting from a microscopic multidimensional Hamiltonian. Here, the strate-gies range from anharmonic normal mode potentials for nonreactive vibrational dynamics up to reaction surfaces forreactive processes. Considerable space has been devoted to the presentation of the CRS Hamiltonian approach, be-cause it is not only a reasonable compromise between accuracy and the desire for a truly multidimensional ab initiopotential, but it also provides the interface to commonly used generic system–bath Hamiltonians for condensed phasereactions.

The availability of a multidimensional potential is only useful if that the respective dynamical equations can besolved. Here, the CRS Hamiltonian turns out to be ideally suited for the combination with the MCTDH methodfor propagating wave packets. This statement has been underlined by three applications in Section 3. First, we haveshown how the anharmonic coupling between the OH stretching vibration and some low-frequency hydrogen-bondmode in a medium strong single minimum system can lead to coherent wave packet motions of the low-frequencymode after laser excitation of the OH-mode. While this example did not show any ultrafast intramolecular vibrationalenergy redistribution (IVR), increased anharmonicity as present in a double minimum system was shown to lead tosubpicosecond energy redistribution even in this gas phase situation. In the final example we examined the issue ofmode-specific tunneling in a symmetric double minimum hydrogen bond, which provides just a specific realizationof the general theme of the influence of promoting and reorganization motions in proton or hydrogen atom transferreactions.

Stepping from the gas to the condensed phase the question can be raised whether classical molecular dynamics withits favorable scaling can be supplemented by quantum effects at moderate costs. We have looked at this issue in termsof the ability of trajectory-based methods to account for quantum tunneling. Thereby, we outlined an extension tothe well-known Makri–Miller method which is based on trajectories in the classically forbidden region. Although weshowed that this new method is capable of improving the description, for instance, of tunneling splittings, it must bestated that this comes along with substantial numerical effort.

We have also discussed the more traditional approach to condensed phase dynamics in terms of a quantum masterequation. This approach has been illustrated for the description of energy cascading in an intramolecular hydrogen bond,where its combination with a nontrivial system–bath Hamiltonian provided a microscopic picture for the time-scalesobserved in ultrafast pump–probe spectroscopy. We further presented some results for the two-dimensional infraredspectroscopy of a generic proton transfer system which showed how this rather promising technique can give detailedinformation on the anharmonic couplings.

Nonlinear time-resolved infrared spectroscopy is a valuable means to unravel even complex dynamics which isotherwise buried in broad stationary spectra. On the other hand, the availability of the appropriate laser source willtrigger the desire to not only analyze hydrogen bond dynamics, but to control it by virtue of specifically shaped pulses.Some results along these lines have been discussed at the end of this review together with the challenges posed in thisrespect by the multidimensional nature of hydrogen bond dynamics.

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K. Giese et al. / Physics Reports 430 (2006) 211 –276 269

Acknowledgment

This work has been financially supported by the Deutsche Forschungsgemeinschaft, partly through the Sfb 450. Theauthors thank N. Došlic, J. Dreyer, T. Elsaesser, K. Heyne, H.-H. Limbach, J. Manz, H.-D. Meyer, E. T. J. Nibbering,M. F. Shibl, J. Stenger, K. Takatsuka, Y. Tanimura, and H. Ushiyama for many stimulating discussions during thelast years.

Appendix A. Determination of the SMC Hamiltonian parameters

In Ref. [60] a simple method was proposed to determine the SMC parameters entering Eq. (18). It makes use of thereaction plane idea which is introduced in Section 2.3.1. The saddle point of the SMC-PES is at (0, �) where � = �/�2

is the ratio of the geometry displacements,

� ≡ |XC − XTS||XR − XC| . (A.1)

The frequency parameter � is given by �= 2/ 1. The frequencies 1 and 2 correspond to the reaction plane vectorsw1 and w2, respectively, and are determined by weighting the contribution of each normal mode with a component intothe w1 and w2 direction, respectively,

2k =

∑j

(Yj · wk)2�2

j , k = 1, 2, (A.2)

where Yj with j = 1, . . . , N − 6 are the N = 3 · Nat dimensional normal mode vectors of the (right or left) minimumand �j are the corresponding normal mode frequencies. The SMC parameter g is given by g= 1/8�E�, where �E�is the ZPE corrected barrier height. This procedure has been applied in Ref. [60] to tropolone and in Section 4.2.3 tovarious carboxylic acid dimers.

Appendix B. Calculation of multidimensional eigenstates

In the following we briefly outline how low lying eigenstates of a multidimensional double minimum PES can bedetermined approximately by a combination of improved MCTDH relaxation together with solution of a generalizedeigenvalue problem. For this purpose the states �(0,R)

n (�(0,L)n ) corresponding to the harmonic approximation to the

right (left) minimum serve as initial states for MCTDH improved relaxations (cf. Section 3.1) which gives the relaxedstates �(rel,R)

n and �(rel,L)n . These states form an optimized basis for the diagonalization of the Hamiltonian. The left

and right states are not orthogonal, thus eigenstates and eigenvectors of the Hamiltonian can be obtained by solvingthe generalized eigenvalue problem,∑

n′

∑�′=R,L

(H�,�′n,n′ − εl S

�,�′n,n′ ) · c�′

n′,l = 0, (B.1)

where � = R or L, H�,�′n,n′ = 〈�(rel,�)

n |H |�(rel,�′)n′ 〉 is the Hamiltonian matrix, S

�,�′n,n′ = 〈�(rel,�)

n |�(rel,�′)n′ 〉 is the overlap

matrix, and c�′n′,l are the expansion coefficients for the lth eigenstate,

�l =∑

n

∑�=R,L

c�n,l�

(rel,�)n , (B.2)

with eigenenergy εl . The present procedure requires less single particle functions than the attempt to compute the geradeand ungerade states directly by improved relaxation.Appropriate symmetry adapted gerade and ungerade superpositionscan be formed according to

�(rel,±)n ∝ �(rel,R)

n ± �(rel,L)n . (B.3)

An application is given in Section 3.3 (cf. Table 2).

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270 K. Giese et al. / Physics Reports 430 (2006) 211 –276

Appendix C. Parities of motion

The assignment of parities of motion, �j = ±1, to each DOF has been suggested by Takatsuka an coworkers [178].It insures that real-valued momenta can always be defined as

pj = pj/√

�j . (C.1)

For �j = +1 and −1 the motion is classically allowed and forbidden, respectively. Of course, the transformationEq. (C.1) is noncanonical if �j = −1. Inserted into the usual Hamiltonian it yields

H (p, q; �) =∑j

�j

2p2

j + V (q), (C.2)

where �=(�1, . . . , �N) is the vector of parities of motion. Trajectories in the (p, q) space can be generated by modifiedHamilton’s equation of motion,

˙pj = − �H

�qj

= − �V

�qj

, (C.3)

qj = �H

�pj

= �j pj . (C.4)

The first equation, Eq. (C.3), is the unchanged Newton’s equation of motion. The second equation, Eq. (C.4), determinesthat for �j =−1 the velocity qj and momentum pj have opposite directions. Notice that Eqs. (C.3)–(C.4) are equivalentto the formulation in Ref. [178] and have a canonically invariant form. This implies that in principle the method ofcharacteristics can be used to construct an action function based on a field of trajectories [329]. However, for the presentpurpose we defined the action along a trajectory on a �-sheet (for constant energy) by

S(p0, q0, �; t) =∑j

∫ t

0

√�j pj (�)qj (�) d�, (C.5)

where (p0, q0) are the initial conditions of the trajectory.

References

[1] P. Schuster, G. Zundel, C. Sandorfi (Eds.), The Hydrogen Bond Theory, North-Holland, Amsterdam, 1976.[2] T. Bountis (Ed.), Proton Transfer in Hydrogen-Bonded Systems, Plenum Press, New York, 1992.[3] D.A. Smith (Ed.), Modeling the Hydrogen Bond, ACS Symposium Series, vol. 569, ACS, Washington, 1994.[4] G.A. Jeffrey, An Introduction to Hydrogen Bonding, Oxford, New York, 1997.[5] S. Scheiner, Hydrogen Bonding, Oxford, New York, 1997.[6] D. Hadži (Ed.), Theoretical Treatments of Hydrogen Bonding, Wiley, Chichester, 1997.[7] H.-H. Limbach, J. Manz, Ber. Bunsenges. Phys. Chem. 102 (1998) 289.[8] G.R. Desiraju, T. Steiner, The Weak Hydrogen Bond, Oxford University Press, Oxford, 1999.[9] P. Schuster, P. Wolschann, Chemical Monthly 130 (1999) 947.

[10] T. Elsaesser, H.J. Bakker (Eds.), Ultrafast Hydrogen Bonding Dynamics and Proton Transfer Processes in the Condensed Phase, KluwerAcademic Publishers, Dordrecht, 2002.

[11] A. Kohen, H.-H. Limbach (Eds.), Isotope Effects in Chemistry and Biology, Taylor and Francis, Boca Raton, 2005.[12] R.L. Schowen (Ed.), Handbook of Hydrogen Transfer, Wiley-VCH, Weinheim, 2006.[13] W.M. Latimer, W.H. Rodebush, J. Am. Chem. Soc. 42 (1920) 1419.[14] M.L. Huggins, Angew. Chem. Int. Ed. 10 (1971) 147.[15] J.D. Bernal, R.H. Fowler, J. Chem. Phys. 1 (1933) 515.[16] M.L. Huggins, J. Phys. Chem. 40 (1936) 723.[17] L. Pauling, The Nature of the Chemical Bond, Cornell University Press, Ithaka, 1939.[18] R.M. Badger, S.H. Bauer, J. Chem. Phys. 5 (1936) 839.[19] L. Pauling, R.B. Corey, H.R. Branson, Proc. Nat. Acad. Sci. USA 37 (1951) 205.[20] J.D. Watson, F.H.C. Crick, Nature 171 (1953) 737.[21] T. Elsaesser, in: J. Manz, L. Wöste (Eds.), Femtosecond Chemistry, vol. 2, Verlag Chemie, Weinheim, 1994, p. 563.[22] A. Douhal, F. Lahmani, A.H. Zewail, Chem. Phys. 207 (1996) 477.

Page 61: Multidimensional quantum dynamics and infrared ...xbeams.chem.yale.edu › ~batista › v572 › rsurf_kuhn.pdf · Hydrogen bonds are of outstanding importance for many processes

K. Giese et al. / Physics Reports 430 (2006) 211 –276 271

[23] E.T.J. Nibbering, T. Elsaesser, Chem. Rev. 104 (2004) 1887.[24] M. Benoit, D. Marx, Chem. Phys. Chem. 6 (2005) 1738.[25] P.M. Kiefer, J.T. Hynes, in: T. Elsaesser, H.J. Bakker (Eds.), Ultrafast Hydrogen Bonding Dynamics and Proton Transfer Processes in the

Condensed Phase, Kluwer Academic Publishers, Dordrecht, 2002, p. 73.[26] S. Bratos, J.-C. Leicknam, G. Gallot, H. Ratajczak, in: T. Elsaesser, H.J. Bakker (Eds.), Ultrafast Hydrogen Bond Dynamics and Proton

Transfer Processes in the Condensed Phase, Kluwer Academic, Dordrecht, 2002, p. 5.[27] R.P. Bell, The Tunnel Effect in Chemistry, Chapman & Hall, London, 1980.[28] T. Baba, T. Tanaka, I. Morino, K.M.T. Yamada, K. Tanaka, J. Chem. Phys. 110 (1999) 4131.[29] F. Madeja, M. Havenith, J. Chem. Phys. 117 (2002) 7162.[30] H. Sekiya, Y. Nagashima, Y. Nishimura, J. Chem. Phys. 92 (1990) 5761.[31] S. Hammes-Schiffer, Chem. Phys. Chem. 3 (2002) 33.[32] L. Rodriguez-Santiago, M. Sodupe, A. Oliva, J. Bertran, J. Am. Chem. Soc. 121 (1999) 8882.[33] M.V. Basilevsky, M.V. Vener, Russ. Chem. Rev. 72 (2003) 3.[34] Y. Tomioka, M. Ito, N. Mikami, J. Phys. Chem. 87 (1983) 4401.[35] C. Manca, C. Tanner, S. Coussan, A. Bach, S. Leutwyler, J. Chem. Phys. 121 (2004) 2578.[36] P.K. Agarwal, S.R. Billeter, P.T.R. Rajagopalan, S. Benkovic, S. Hammes-Schiffer, Proc. Nat. Acad. Sci. USA 99 (2002) 2794.[37] R.S. Sikorski, L. Wang, K.A. Markham, P.T.R. Rajagopalan, S.J. Benkovic, A. Kohen, J. Am. Chem. Soc. 126 (2004) 4778.[38] V. May, O. Kühn, Charge and Energy Transfer Dynamics in Molecular Systems, 2nd ed., Wiley-VCH, Weinheim, 2004.[39] M.V. Pak, C. Swalina, S.P. Webb, S. Hammes-Schiffer, Chem. Phys. 304 (2004) 227.[40] B.I. Stepanov, Nature 157 (1946) 808.[41] N. Sheppard, in: D. Hadži (Ed.), Hydrogen Bonding, Pergamon Press, New York, 1957, p. 85.[42] S. Bratos, D. Hadži, J. Chem. Phys. 27 (1957) 991.[43] A. Witkowski, J. Chem. Phys. 47 (1967) 3645.[44] Y. Marechal, A. Witkowski, J. Chem. Phys. 48 (1968) 3697.[45] O. Henri-Rousseau, P. Blaise, Adv. Chem. Phys. 103 (1998) 1.[46] O. Henri-Rousseau, D. Chamma, Chem. Phys. 229 (1998) 37.[47] D. Chamma, O. Henri-Rousseau, Chem. Phys. 229 (1998) 51.[48] O. Henri-Rousseau, P. Blaise, D. Chamma, Adv. Chem. Phys. 121 (2002) 241.[49] J. Dreyer, J. Chem. Phys. 122 (2005) 184306.[50] J.K.G. Watson, Mol. Phys. 15 (1968) 479.[51] S. Carter, J.M. Bowman, J. Chem. Phys. 108 (1998) 4397.[52] N. Došlic, O. Kühn, Z. Phys. Chem. 217 (2003) 1507.[53] S. Carter, S.J. Culik, J.M. Bowman, J. Chem. Phys. 107 (1997) 10458.[54] X. Huang, B.J. Braams, S. Carter, J.M. Bowman, J. Am. Chem. Soc. 126 (2004) 5042.[55] J.M. Bowman, X. Zhang, A. Brown, J. Chem. Phys. 119 (2003) 646.[56] S. Takada, H. Nakamura, J. Chem. Phys. 100 (1994) 98.[57] V.A. Benderskii, S.Y. Grebenshchikov, G.V. Mil’nikov, E.V. Vetoshkin, Chem. Phys. 188 (1994) 19.[58] V.A. Benderskii, S.Y. Grebenshchikov, G.V. Mil’nikov, Chem. Phys. 198 (1995) 281.[59] V.A. Benderskii, S.Y. Grebenshchikov, G.V. Mil’nikov, Chem. Phys. 194 (1995) 1.[60] S. Takada, H. Nakamura, J. Chem. Phys. 102 (1995) 3977.[61] M.J. Wojcik, H. Nakamura, S. Iwata, W. Tatara, J. Chem. Phys. 112 (2000) 6322.[62] E.B. Wilson, J.C. Decius, P.C. Cross, Molecular Vibrations, Dover, New York, 1955.[63] D. Heidrich (Ed.), The Reaction Path in Chemistry: Current Approaches and Perspectives, Kluwer Academic, Dordrecht, 1995.[64] W.H. Miller, N.C. Handy, J.E. Adams, J. Chem. Phys. 72 (1980) 99.[65] Y.-P. Liu, G.C. Lynch, T.N. Truong, D.-H. Lu, D.G. Truhlar, B.C. Garrett, J. Am. Chem. Soc. 115 (1993) 2408.[66] W.H. Miller, B.A. Ruf, Y.-T. Chang, J. Chem. Phys. 89 (1988) 6298.[67] B. Fehrensen, D. Luckhaus, M. Quack, Z. Phys. Chem. 209 (1999) 1.[68] S. Carter, N.C. Handy, J. Chem. Phys. 113 (2000) 987.[69] D.P. Tew, N.C. Handy, S. Carter, Mol. Phys. 99 (2001) 393.[70] R. Meyer, T.-K. Ha, Mol. Phys. 101 (2003) 3263.[71] D.P. Tew, N.C. Handy, S. Carter, Phys. Chem. Chem. Phys. 3 (2001) 1958.[72] S. Schweiger, B. Hartke, G. Rauhut, Phys. Chem. Chem. Phys. 6 (2004) 3341.[73] J. Manz, J. Römelt, Chem. Phys. Lett. 81 (1981) 179.[74] T. Carrington, W.H. Miller, J. Chem. Phys. 81 (1984) 3942.[75] T. Carrington, W.H. Miller, J. Chem. Phys. 84 (1986) 4364.[76] N. Shida, P.F. Barbara, J.E. Almlöf, J. Chem. Phys. 91 (1989) 4061.[77] N. Shida, P.F. Barbara, J.E. Almlöf, J. Chem. Phys. 94 (1991) 3633.[78] I. Matanovic, N. Došlic, J. Phys. Chem. A 109 (2005) 4185.[79] B.A. Ruf, W.H. Miller, J. Chem. Soc. Faraday Trans. 2 84 (1988) 1523.[80] K. Yagi, T. Taketsugu, K. Hirao, J. Chem. Phys. 115 (2001) 10647.[81] T. Hayashi, S. Mukamel, J. Phys. Chem. A 107 (2003) 9113.[82] B.C. Garrett, D.G. Truhlar, J. Chem. Phys. 79 (1983) 4931.

Page 62: Multidimensional quantum dynamics and infrared ...xbeams.chem.yale.edu › ~batista › v572 › rsurf_kuhn.pdf · Hydrogen bonds are of outstanding importance for many processes

272 K. Giese et al. / Physics Reports 430 (2006) 211 –276

[83] C.S. Tautermann, A.F. Voegele, T. Loerting, K.R. Liedl, J. Chem. Phys. 117 (2002) 1962.[84] C.S. Tautermann, A.F. Voegele, T. Loerting, K.R. Liedl, J. Chem. Phys. 117 (2002) 1967.[85] C.S. Tautermann, A.F. Voegele, K.R. Liedl, J. Chem. Phys. 120 (2004) 631.[86] K. Giese, O. Kühn, J. Chem. Phys. 123 (2005) 054315.[87] A. Nauts, X. Chapusiat, Chem. Phys. Lett. 136 (1987) 164.[88] F. Gatti, Y. Justum, M. Menou, A. Nauts, X. Chapusiat, J. Mol. Spectr. 181 (1997) 403.[89] D. Luckhaus, J. Phys. Chem. A 110 (2005) 3151.[90] P.A.M. Dirac, Proc. Cambridge Philos. Soc. 26 (1930) 376.[91] R.B. Gerber, V. Buch, M.A. Ratner, J. Chem. Phys. 77 (1982) 3022.[92] N. Makri, W.H. Miller, J. Chem. Phys. 87 (1987) 5781.[93] O. Vendrell, H.-D. Meyer, J. Chem. Phys. 122 (2005) 104505.[94] A.D. Hammerich, R. Kosloff, M.A. Ratner, Chem. Phys. Lett. 171 (1990) 97.[95] H.-D. Meyer, U. Manthe, L.S. Cederbaum, Chem. Phys. Lett. 165 (1990) 73.[96] M.H. Beck, A. Jäckle, G.A. Worth, H.-D. Meyer, Phys. Rep. 324 (2000) 1.[97] H. Meyer, G.A. Worth, Theor. Chem. Acc. 109 (2003) 251.[98] J.C. Light, I.P. Hamilton, J. Chem. Phys. 82 (1985) 1400.[99] D.T. Colbert, W.H. Miller, J. Chem. Phys. 96 (1992) 1982.

[100] G. Worth, M. Beck, A. Jäckle, H.-D. Meyer, The MCTDH package, Version 8.3, University of Heidelberg, Heidelberg, 2005.[101] G.A. Worth, H.-D. Meyer, L.S. Cederbaum, J. Chem. Phys. 109 (1998) 3518.[102] N. Makri, Chem. Phys. Lett. 169 (1990) 541.[103] G.A. Worth, H.-D. Meyer, L.S. Cederbaum, J. Chem. Phys. 105 (1996) 4412.[104] H. Wang, M. Thoss, J. Chem. Phys. 119 (2003) 1289.[105] G.A. Worth, I. Burghardt, H.-D. Meyer, Chem. Phys. Lett. 368 (2003) 502.[106] A. Jäckle, H.-D. Meyer, J. Chem. Phys. 104 (1996) 7974.[107] J. Stenger, D. Madsen, J. Dreyer, E.T.J. Nibbering, P. Hamm, T. Elsaesser, in: T. Elsaesser, S. Mukamel, M. Murnane, N. Scherer (Eds.),

Ultrafast Phenomena XII, Springer Series in Chemical Physics, Springer, New York, 2000, p. 542.[108] J. Stenger, D. Madsen, J. Dreyer, E.T.J. Nibbering, P. Hamm, T. Elsaesser, J. Phys. Chem. A 105 (2001) 2929.[109] D. Madsen, J. Stenger, J. Dreyer, P. Hamm, E.T.J. Nibbering, T. Elsaesser, Bull. Chem. Soc. Japan 75 (2002) 909.[110] J. Stenger, D. Madsen, J. Dreyer, P. Hamm, E.T.J. Nibbering, T. Elsaesser, Chem. Phys. Lett. 354 (2002) 256.[111] D. Madsen, J. Stenger, J. Dreyer, E.T.J. Nibbering, P. Hamm, T. Elsaesser, Chem. Phys. Lett. 341 (2001) 56.[112] K. Heyne, E.T.J. Nibbering, T. Elsaesser, M. Petkovic, O. Kühn, J. Phys. Chem. A 108 (2004) 6083.[113] K. Heyne, E.T.J. Nibbering, T. Elsaesser, M. Petkovic, O. Kühn, in: T. Kobayashi, T. Okada, T. Kobayashi, K.N. Nelson, S. DeSilvestri (Eds.),

Ultrafast Phenomena XIV, Springer, 2004, p. 389.[114] G.K. Paramonov, H. Naundorf, O. Kühn, Eur. J. Phys. D 14 (2001) 205.[115] H. Naundorf, G.A. Worth, H.-D. Meyer, O. Kühn, J. Phys. Chem. A 106 (2002) 719.[116] O. Kühn, J. Phys. Chem. A 106 (2002) 7671.[117] H. Naundorf, O. Kühn, in: A. Douhal, J. Santamaria (Eds.), Femtochemistry and Femtobiology, World Scientific, Singapore, 2002, p. 438.[118] M. Fores, M. Duran, M. Sola, Chem. Phys. 260 (2000) 53.[119] M. Fores, M. Duran, M. Sola, Chem. Phys. 234 (1998) 1.[120] A. Simperler, W. Mikenda, Chem. Monthly 128 (1997) 969.[121] M. Petkovic, O. Kühn, J. Phys. Chem. A 107 (2003) 8458.[122] M. Petkovic, O. Kühn, Chem. Phys. 304 (2004) 91.[123] M. Petkovic, O. Kühn, in: M.M. Martin, J.T. Hynes (Eds.), Ultrafast Molecular Events in Chemistry and Biology, Elsevier, Amsterdam, 2004,

p. 181.[124] K. Heyne, J. Stenger, J. Dreyer, E. Nibbering, T. Elsaesser, unpublished.[125] R.L. Redington, R.L. Sams, J. Phys. Chem. A 106 (2002) 7494.[126] A.C.P. Alves, J.M. Hollas, Mol. Phys. 23 (1972) 927.[127] A.C.P. Alves, J.M. Hollas, Mol. Phys. 25 (1973) 1305.[128] K. Tanaka, H. Honjo, T. Tanaka, H. Kohguchi, Y. Ohshima, Y. Endo, J. Chem. Phys. 110 (1999) 1969.[129] R.L. Redington, T.E. Redington, J.M. Montgomery, J. Chem. Phys. 113 (2000) 2304.[130] R.L. Redington, J. Chem. Phys. 113 (2000) 2319.[131] R.K. Frost, F. Hagemeister, D. Schleppenbach, G. Laurence, T.S. Zwier, J. Phys. Chem. B 100 (1996) 16835.[132] K. Giese, Multidimensional tunneling in hydrogen transfer reactions, Ph.D. Thesis, Freie Universität Berlin, Berlin, 2005.[133] M.D. Coutinho-Neto, A. Viel, U. Manthe, J. Chem. Phys. 121 (2004) 9207.[134] E.J. Heller, J. Phys. Chem. 99 (1995) 2625.[135] K. Giese, H. Ushiyama, K. Takatsuka, O. Kühn, J. Chem. Phys. 122 (2005) 124307.[136] M. Gruebele, Adv. Chem. Phys. 114 (2000) 193.[137] K. Giese, D. Lahav, O. Kühn, J. Theor. Comp. Chem. 3 (2004) 567.[138] M. Razavy, Quantum Theory of Tunneling, World Scientific, Hoboken NJ, 2003.[139] C. Herring, Rev. Mod. Phys. 34 (1962) 631.[140] M. Wilkinson, Physica D 21 (1986) 341.[141] V.A. Benderskii, S.Y. Grebenshchikov, E.V. Vetoshkin, G.V. Mil’nikov, D.E. Makarov, Chem. Phys. 98 (1994) 3309.

Page 63: Multidimensional quantum dynamics and infrared ...xbeams.chem.yale.edu › ~batista › v572 › rsurf_kuhn.pdf · Hydrogen bonds are of outstanding importance for many processes

K. Giese et al. / Physics Reports 430 (2006) 211 –276 273

[142] V.A. Benderskii, E. Vetoshkin, I. Irgibaeva, H. Trommsdorff, Chem. Phys. 262 (2000) 393.[143] E. Vetoshkin, V.A. Benderskii, S.Y. Grebenshchikov, L. von Laue, H.P. Trommsdorff, Chem. Phys. 219 (1997) 119.[144] Z. Smedarchina, W. Siebrand, M.Z. Zgierski, J. Chem. Phys. 103 (1995) 5326.[145] Z. Smedarchina, W. Siebrand, M.Z. Zgierski, J. Chem. Phys. 104 (1996) 1203.[146] Z. Smedarchina, A. Fernandez-Ramos, W. Siebrand, J. Comput. Chem. 22 (2001) 787.[147] Z. Smedarchina, A. Fernandez-Ramos, W. Siebrand, Chem. Phys. Lett. 395 (2004) 339.[148] A. Garg, Am. J. Phys. 68 (2000) 430.[149] G. Mil’nikov, H. Nakamura, J. Chem. Phys. 115 (2001) 6881.[150] G.V. Mil’nikov, H. Nakamura, J. Chem. Phys. 122 (2005) 124311.[151] G.V. Mil’nikov, K. Yagi, T. Taketsugu, H. Nakamura, K. Hirao, J. Chem. Phys. 119 (2003) 10.[152] G.V. Mil’nikov, O. Kühn, H. Nakamura, J. Chem. Phys. 123 (2005) 074308.[153] M.F. Herman, E. Kluk, Chem. Phys. 91 (1984) 27.[154] W.H. Miller, J. Phys. Chem. A 105 (2001) 2942.[155] S. Garashchuk, D. Tannor, Chem. Phys. Lett. 262 (1996) 477.[156] V.A. Mandelshtam, M. Ovchinikov, J. Chem. Phys. 108 (1998) 9206.[157] V. Guallar, V.S. Batista, W.H. Miller, J. Chem. Phys. 110 (1999) 9922.[158] V. Guallar, V.S. Batista, W.H. Miller, J. Chem. Phys. 113 (2000) 9510.[159] T. Yamamoto, W.H. Miller, J. Chem. Phys. 118 (2003) 2135.[160] K. Giese, O. Kühn, J. Chem. Phys. 120 (2004) 4207.[161] V.A. Mandelshtam, Progr. Nucl. Magn. Res. 38 (2001) 159.[162] D.G. Truhlar, A.D. Isaacson, B.C. Garrett, in: M. Baer (Ed.), Theory of Chemical Reaction Dynamics, vol. IV, CRC, Boca Raton, 1985, p. 65.[163] T.C. Allison, D.G. Truhlar, in: D.L. Thompson (Ed.), Modern Methods for Multidimensional Dynamics Computations in Chemistry, World

Scientific, Singapore, 1998, p. 618.[164] R.A. Marcus, M.E. Coltrin, J. Chem. Phys. 67 (1977) 2609.[165] N. Shida, J. Almlöf, P.F. Barbara, J. Phys. Chem. 95 (1991) 10457.[166] V.A. Benderskii, D.E. Makarov, P.G. Grinevich, Chem. Phys. 170 (1993) 275.[167] V.A. Benderskii, D.E. Makarov, C.A. Wright, Adv. Chem. Phys. 88 (1994) 1.[168] N. Makri, W.H. Miller, J. Chem. Phys. 91 (1989) 4026.[169] Y. Guo, S. Li, D.L. Thompson, J. Chem. Phys. 107 (1997) 2853.[170] T.D. Sewell, Y. Guo, D.L. Thompson, J. Chem. Phys. 103 (1995) 8557.[171] Y. Guo, D.L. Thompson, J. Phys. Chem. A 106 (2002) 8374.[172] V. Guallar, B.F. Gherman, W.H. Miller, S.J. Lippard, R.A. Friesner, J. Am. Chem. Soc. 124 (2002) 3377.[173] K. Giese, O. Kühn, J. Chem. Theor. Comp. 2 (2006) 717.[174] E. Bosch, M. Moreno, J.M. Lluch, Chem. Phys. 159 (1992) 99.[175] M. Ben-Nun, T.J. Martinez, J. Phys. Chem. A 103 (1999) 6055.[176] K. Giese, H. Ushiyama, O. Kühn, Chem. Phys. Lett. 371 (2003) 681.[177] M. Razavy, A. Pimpale, Phys. Rep. 168 (1988) 305.[178] K. Takatsuka, H. Ushiyama, A. Inoue-Ushiyama, Phys. Rep. 322 (1999) 347.[179] J.-H. Lim, E.K. Lee, Y. Kim, J. Phys. Chem. A 101 (1997) 2233.[180] Y. Kim, J. Am. Chem. Soc. 118 (1996) 1522.[181] S. Miura, M.E. Tuckerman, M.L. Klein, J. Chem. Phys. 109 (1998) 5290.[182] H. Ushiyama, K. Takatsuka, J. Chem. Phys. 115 (2001) 5903.[183] K. Wolf, A. Simperler, W. Mikenda, Chem. Monthly 130 (1999) 1031.[184] C. Rambaud, H.P. Trommsdorff, Chem. Phys. Lett. 306 (1999) 124.[185] W.-T. Chan, I.P. Hamilton, Chem. Phys. Lett. 292 (1998) 57.[186] R. Xu, Y.J. Yan, O. Kühn, Eur. Phys. J. D 19 (2002) 293.[187] R. Rey, J.T. Hynes, J. Chem. Phys. 104 (1996) 2356.[188] C.P. Lawrence, J.L. Skinner, J. Chem. Phys. 117 (2002) 5827.[189] D. Borgis, J.T. Hynes, Chem. Phys. 170 (1993) 315.[190] M. Tuckerman, K. Laasonen, M. Sprik, M. Parrinello, J. Chem. Phys. 103 (1995) 150.[191] M.V. Vener, O. Kühn, J. Sauer, J. Chem. Phys. 114 (2001) 240.[192] A. Laio, J. VandeVondele, U. Röthlisberger, J. Chem. Phys. 116 (2002) 6941.[193] R. Rousseau, V. Kleinschmidt, U.W. Schmitt, D. Marx, Phys. Chem. Chem. Phys. 6 (2004) 1848.[194] R. Rousseau, V. Kleinschmidt, U.W. Schmitt, D. Marx, Angew. Chem. Int. Ed. 43 (2004) 4804.[195] J. Aaqvist, A. Warshel, Chem. Rev. 93 (1993) 2523.[196] J. Lobaugh, G.A. Voth, J. Chem. Phys. 104 (1996) 2056.[197] U.W. Schmitt, G.A. Voth, J. Phys. Chem. B 102 (1998) 5547.[198] R. Vuilleumier, D. Borgis, in: B.J. Berne, G. Ciccotti, D.F. Coker (Eds.), Classical and Quantum Dynamics in Condensed Phase Simulations,

World Scientific, Singapore, 1998, p. 721.[199] S. Braun-Sand, M.H.M. Olsson, J. Mavri, A. Warshel, in: J.T. Hynes, H.-H. Limbach (Eds.), Handbook of Hydrogen Transfer, vol. 1: Physical

and Chemical Aspects of Hydrogen Transfer, Wiley-VCH, Weinheim, 2006.[200] H.J.C. Berendsen, J. Mavri, J. Phys. Chem. 97 (1993) 13464.

Page 64: Multidimensional quantum dynamics and infrared ...xbeams.chem.yale.edu › ~batista › v572 › rsurf_kuhn.pdf · Hydrogen bonds are of outstanding importance for many processes

274 K. Giese et al. / Physics Reports 430 (2006) 211 –276

[201] A. Staib, D. Borgis, J.T. Hynes, J. Chem. Phys. 102 (1995) 2487.[202] P. Bala, P. Grochowski, B. Lesyng, J.A. McCammon, J. Phys. Chem. 100 (1996) 2535.[203] S. Hammes-Schiffer, J. Chem. Phys. 105 (1996) 2236.[204] S.R. Billeter, W.F. van Gunsteren, Comp. Phys. Commun. 107 (1997) 61.[205] S.Y. Kim, S. Hammes-Schiffer, J. Chem. Phys. 119 (2003) 4389.[206] N. Rösch, M.A. Ratner, J. Chem. Phys. 61 (1974) 3344.[207] B. Boulil, O. Henri-Rousseau, P. Blaise, Chem. Phys. 126 (1988) 263.[208] B. Boulil, J.-L. Dejardin, N.E. Ghandour, O. Henri-Rousseau, J. Mol. Struct. (Theochem) 314 (1994) 83.[209] G.N. Robertson, J. Yarwood, Chem. Phys. 32 (1978) 267.[210] A. Witkowski, M. Wójcik, Chem. Phys. 1 (1973) 9.[211] S. Bratos, J. Chem. Phys. 63 (1975) 3499.[212] S. Bratos, H. Ratajcak, J. Chem. Phys. 76 (1982) 77.[213] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, Oxford, New York, 1995.[214] S. Bratos, J.C. Leicknam, J. Chem. Phys. 101 (1994) 4536.[215] S. Bratos, A. Laubereau, in: D. Hadži (Ed.), Theoretical Treatments of Hydrogen Bonding, Wiley, Chichester, 1997, p. 187.[216] Y. Tanimura, S. Mukamel, J. Chem. Phys. 99 (1993) 9496.[217] J.R. Schmidt, S.A. Corcelli, J.L. Skinner, J. Chem. Phys. 123 (2005) 044513.[218] D.W. Oxtoby, D. Levesque, J.-J. Weis, J. Chem. Phys. 68 (1978) 5528.[219] H.J. Bakker, J. Chem. Phys. 121 (2004) 10088.[220] C. Scheurer, P. Saalfrank, J. Chem. Phys. 104 (1996) 2869.[221] O. Kühn, Y. Tanimura, J. Chem. Phys. 119 (2003) 2155.[222] A.G. Redfield, Adv. Magn. Reson. 1 (1965) 1.[223] W.T. Pollard, A.K. Felts, R.A. Friesner, Adv. Chem. Phys. 93 (1996) 77.[224] N. Makri, J. Phys. Chem. A 102 (1998) 4144.[225] C. Meier, D.J. Tannor, J. Chem. Phys. 111 (1999) 3365.[226] R. Xu, Y. Yan, J. Chem. Phys. 116 (2002) 9196.[227] Q. Shi, E. Geva, J. Chem. Phys. 119 (2003) 12063.[228] N. Makri, J. Phys. Chem. B 103 (1999) 2823.[229] D. Xu, K. Schulten, Chem. Phys. 182 (1994) 91.[230] R. Zwanzig, J. Chem. Phys. 34 (1961) 1931.[231] R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, New York, 2001.[232] D.W. Oxtoby, Adv. Chem. Phys. 47 (1981) 487.[233] J.C. Owrutsky, D. Raftery, R.M. Hochstrasser, Annu. Rev. Phys. Chem. 45 (1994) 519.[234] S.A. Egorov, K.F. Everitt, J.L. Skinner, J. Phys. Chem. A 103 (1999) 9494.[235] C.P. Lawrence, J.L. Skinner, J. Chem. Phys. 119 (2003) 1623.[236] O. Kühn, H. Naundorf, Phys. Chem. Chem. Phys. 5 (2003) 79.[237] D. Reichman, R.J. Silbey, A. Suarez, J. Chem. Phys. 105 (1996) 10500.[238] Y. Yan, F. Shuang, R. Xu, J. Cheng, X.-Q. Li, C. Yang, H. Zhang, J. Chem. Phys. 113 (2000) 2068.[239] Y. Ohtsuki, Y. Fujimura, J. Chem. Phys. 91 (1989) 3903.[240] J.M. Jean, G.R. Fleming, J. Chem. Phys. 103 (1995) 2092.[241] A.M. Walsh, R.D. Coalson, Chem. Phys. Lett. 198 (1992) 293.[242] O. Kühn, V. Sundström, T. Pullerits, Chem. Phys. 275 (2002) 15.[243] K. Heyne, N. Huse, E.T.J. Nibbering, T. Elsaesser, Chem. Phys. Lett. 369 (2003) 591.[244] N. Huse, K. Heyne, J. Dreyer, E.T.J. Nibbering, T. Elsaesser, Phys. Rev. Lett. 91 (2003) 197401.[245] K. Heyne, N. Huse, J. Dreyer, E.T.J. Nibbering, T. Elsaesser, S. Mukamel, J. Chem. Phys. 121 (2004) 902.[246] B. Bagchi, Chem. Rev. 105 (2005) 3197.[247] D. Marx, M.E. Tuckerman, J. Hutter, M. Parrinello, Nature 397 (1999) 601.[248] M.E. Tuckerman, D. Marx, M. Parinello, Nature 417 (2002) 925.[249] R. Ludwig, Chem. Phys. Chem. 5 (2004) 1495.[250] K.R. Asmis, N.L. Pivonka, G. Santambrogio, M. Brümmer, C. Kaposta, D.M. Neumark, L. Wöste, Science 299 (2003) 1375.[251] N.I. Hammer, E.G. Diken, J.R. Roscioli, M.A. Johnson, E.M. Myshakin, K.D. Jordan, A.B. McCoy, X. Huang, J.M. Bowman, S. Carter,

J. Chem. Phys. 122 (2005) 244301.[252] J. Sauer, J. Döbler, Chem. Phys. Chem. 6 (2005) 1706.[253] W. Kühlbrandt, Nature 406 (2000) 569.[254] H. Graener, G. Seifert, A. Laubereau, Phys. Rev. Lett. 66 (1991) 2092.[255] S. Woutersen, U. Emmerichs, H.J. Bakker, Nature 278 (1997) 658.[256] H.-K. Nienhuys, S. Woutersen, R.A. van Santen, H.J. Bakker, J. Chem. Phys. 111 (1999) 1494.[257] G.M. Gale, G. Gallot, F. Hache, N. Lascoux, S. Bratos, J.-C. Leicknam, Phys. Rev. Lett. 82 (1999) 1068.[258] J.C. Deak, S.T. Rhea, L.K. Iwaki, D.D. Dlott, J. Phys. Chem. A 104 (2000) 4866.[259] R. Rey, K.B. Moller, J.T. Hynes, Chem. Rev. 104 (2004) 1915.[260] R. Rey, K.B. Moller, J.T. Hynes, J. Phys. Chem. A 106 (2002) 11993.[261] J. Stenger, D. Madsen, P. Hamm, E.T.J. Nibbering, T. Elsaesser, Phys. Rev. Lett. 87 (2001) 027401.

Page 65: Multidimensional quantum dynamics and infrared ...xbeams.chem.yale.edu › ~batista › v572 › rsurf_kuhn.pdf · Hydrogen bonds are of outstanding importance for many processes

K. Giese et al. / Physics Reports 430 (2006) 211 –276 275

[262] J.D. Eaves, A. Tokmakoff, P.L. Geissler, J. Phys. Chem. A 109 (2005) 9424.[263] J. Stenger, D. Madsen, P. Hamm, E.T.J. Nibbering, T. Elsaesser, J. Phys. Chem. A 106 (2002) 2341.[264] C.J. Fecko, J.D. Eaves, J.J. Loparo, A. Tokmakoff, P.L. Geissler, Science 301 (2003) 1698.[265] C.P. Lawrence, J.L. Skinner, J. Chem. Phys. 118 (2003) 264.[266] R. Meyer, R.R. Ernst, J. Chem. Phys. 93 (1990) 5518.[267] O. Brackhagen, O. Kühn, J. Manz, V. May, R. Meyer, J. Chem. Phys. 100 (1994) 9007.[268] O. Brackhagen, C. Scheurer, R. Meyer, H.-H. Limbach, Ber. Bunsenges. Phys. Chem. 102 (1998) 303.[269] N. Došlic, K. Sundermann, L. González, O. Mo, J. Giraud-Girard, O. Kühn, Phys. Chem. Chem. Phys. 1 (1999) 1249.[270] N. Došlic, O. Kühn, Chem. Phys. 255 (2000) 247.[271] H. Naundorf, K. Sundermann, O. Kühn, Chem. Phys. 240 (1999) 163.[272] D. Borgis, G. Tarjus, H. Azzouz, J. Phys. Chem. 96 (1992) 3188.[273] U.W. Schmitt, G.A. Voth, J. Chem. Phys. 111 (1999) 9361.[274] S. Hammes-Schiffer, J.C. Tully, J. Chem. Phys. 101 (1994) 4657.[275] J.-Y. Fang, S. Hammes-Schiffer, J. Chem. Phys. 110 (1999) 11166.[276] H. Decornez, K. Drukker, S. Hammes-Schiffer, J. Phys. Chem. A 103 (1999) 2891.[277] S.P. Webb, P.K. Agarwal, S. Hammes-Schiffer, J. Phys. Chem. B 104 (2000) 8884.[278] S.R. Billeter, S.P. Webb, T. Iordanov, P.K. Agarwal, S. Hammes-Schiffer, J. Chem. Phys. 114 (2001) 6925.[279] P.K. Agarwal, S.R. Billeter, S. Hammes-Schiffer, J. Phys. Chem. B 106 (2002) 3283.[280] S. Mukamel, Annu. Rev. Phys. Chem. 51 (2000) 691.[281] M. Cho, Phys. Chem. Comm. 5 (2002) 40.[282] M. Khalil, N. Demirdöven, A. Tokmakoff, J. Phys. Chem. A 107 (2003) 5258.[283] J.B. Asbury, T. Steinel, K. Kwak, S.A. Corcelli, C.P. Lawrence, J.L. Skinner, M.D. Fayer, J. Chem. Phys. 121 (2004) 12431.[284] T. la Cour Jansen, T. Hayashi, W. Zhuang, S. Mukamel, J. Chem. Phys. 123 (2005) 114504.[285] J.B. Asbury, T. Steinel, C. Stromberg, K.J. Gaffney, I.R. Piletic, A. Goun, M.D. Fayer, Chem. Phys. Lett. 374 (2003) 362.[286] J.B. Asbury, T. Steinel, C. Stromberg, K.J. Gaffney, I.R. Piletic, M.D. Fayer, J. Chem. Phys. 119 (2003) 12981.[287] A. Ishizaki, Y. Tanimura, J. Chem. Phys. 123 (1) (2005) 014503.[288] J.M. Robertson, A.R. Ubbelohde, Proc. Roy. Soc. A 170 (1939) 222.[289] A.R. Ubbelohde, K.J. Gallagher, Acta Crystallogr. 8 (1955) 71.[290] M. Ichikawa, Act. Crystallogr. B 34 (1978) 2074.[291] M. Ichikawa, J. Mol. Struct. 552 (2000) 63.[292] S. Tanaka, Phys. Rev. B 42 (1990) 10488.[293] H. Benedict, H.-H. Limbach, M. Wehlan, W.-P. Fehlhammer, N.S. Golubev, R. Janoschek, J. Am. Chem. Soc. 120 (1998) 2939.[294] I.G. Shenderovich, H.-H. Limbach, S.N. Smirnov, P.M. Tolstoy, G.S. Denisov, N.S. Golubev, Phys. Chem. Chem. Phys. 4 (2002) 5488.[295] H.-H. Limbach, M. Pietrzak, H. Benedict, P.M. Tolstoy, N.S. Golubev, G.S. Denisov, J. Mol. Struct. 706 (2004) 115.[296] M. Tachikawa, M. Shiga, J. Am. Chem. Soc. 127 (2005) 11908.[297] M.F. Shibl, M. Tachikawa, O. Kühn, Phys. Chem. Chem. Phys. 7 (2005) 1368.[298] M.F. Shibl, M. Pietrzak, H.-H. Limbach, O. Kühn, 2006, submitted.[299] J.E. Almlöf, Chem. Phys. Lett. 17 (1972) 49.[300] T. Elsaesser, in: T. Elsaesser, H.J. Bakker (Eds.), Ultrafast Hydrogen Bonding Dynamics and Proton Transfer Processes in the Condensed

Phase, Kluwer Academic Publishers, Dordrecht, 2002, p. 119.[301] C. Crespo-Hernandez, P.M.H.B. Cohen, B. Kohler, Chem. Rev. 104 (2004) 1977.[302] C. Chudoba, E. Riedle, M. Pfeiffer, T. Elsaesser, Chem. Phys. Lett. 263 (1996) 622.[303] A. Douhal, Science 276 (1997) 221.[304] S. Lochbrunner, A.J. Wurzer, E. Riedle, J. Chem. Phys. 112 (2000) 10699.[305] S. Lochbrunner, E. Riedle, in Recent Research Developments in Chemical Physics, vol. 4, Transworld Research Network, Trivandrum, Kerala,

2003, p. 31.[306] R. deVivie Riedle, V. DeWaele, L. Kurtz, E. Riedle, J. Phys. Chem. A 107 (2003) 10591.[307] N. Agmon, J. Phys. Chem. A 109 (2005) 13.[308] E.T.J. Nibbering, H. Fidder, E. Pines, Annu. Rev. Phys. Chem. 56 (2005) 337.[309] T. Elsaesser, W. Kaiser, Chem. Phys. Lett. 128 (1986) 231.[310] M. Rini, B.-Z. Magnes, E. Pines, E.T.J. Nibbering, Science 301 (2003) 349.[311] O.F. Mohammed, D. Pines, J. Dreyer, E. Pines, E.T.J. Nibbering, Science 310 (2005) 83.[312] M. Glasbeek, Isr. J. Chem. 39 (1999) 301.[313] A. Douhal, S.K. Kim, A.H. Zewail, Nature 378 (1995) 260.[314] S. Takeuchi, T. Tahara, J. Phys. Chem. A 102 (1998) 7740.[315] K. Sakota, C. Okabe, N. Nishi, H. Sekiya, J. Phys. Chem. A 109 (2005) 5245.[316] T. Schultz, E. Samoylova, W. Radloff, I.V. Hertel, A.L. Sobolewski, W. Domcke, Nature 306 (2005) 1765.[317] S. Rice, M. Zhao, Optimal Control of Molecular Dynamics, Wiley, New York, 2001.[318] M. Shapiro, P. Brumer, Principles of the Quantum Control of Molecular Processes, Wiley, Hoboken, 2003.[319] T. Brixner, G. Gerber, Chem. Phys. Chem. 4 (2003) 418.[320] I. Walmsley, H. Rabitz, Phys. Today August (2003) 43.[321] O. Kühn, L. González, in: J.T. Hynes, H.-H. Limbach (Eds.), Handbook of Hydrogen Transfer, vol. 1: Physical and Chemical Aspects of

Hydrogen Transfer, Wiley-VCH, Weinheim, 2006.

Page 66: Multidimensional quantum dynamics and infrared ...xbeams.chem.yale.edu › ~batista › v572 › rsurf_kuhn.pdf · Hydrogen bonds are of outstanding importance for many processes

276 K. Giese et al. / Physics Reports 430 (2006) 211 –276

[322] M. Grifoni, P. Hänggi, Phys. Rep. 304 (1998) 229.[323] G.K. Paramonov, M.V. Korolkov, J. Manz, Adv. Chem. Phys. 101 (1997) 327.[324] N. Došlic, O. Kühn, J. Manz, Ber. Bunsenges. Phys. Chem. 102 (1998) 292.[325] N. Došlic, O. Kühn, J. Manz, K. Sundermann, J. Phys. Chem. A 102 (1998) 292.[326] O. Kühn, Eur. Phys. J. D 6 (1999) 49.[327] O. Kühn, Y. Zhao, F. Shuang, Y.-J. Yan, J. Chem. Phys. 112 (2000) 6104.[328] E. Geva, J. Chem. Phys. 116 (2002) 1629.[329] V.P. Maslov, M.V. Fedoriuk, Semiclassical Approximations in Quantum Mechanics, D. Reidel, Dordrecht, 1981.