excited states methods: reviewiopenshell.usc.edu/chem545/material/lecture_exstates_tddft.pdf ·...

Excited states methods: Review

Can be derived by considering linear response of time-dependent approximate ground-state w.f.:

HF->TD-HF -> RR-TD-HF (RPA/CIS)CCSD -> LR-CCSD(same as EOM-CCSD) CCSDT -> LR-CCSDT(same as EOM-CCSDT)

Additional approximations:EOM-CCSD > CIS(D) or CIS(2) or CC2EOM-CCSDT > CC3 .....

Similar approach can be applied to DFT:DFT-> (real-time) TDDFT -> (linear-response)TDDFT (RPA or TDA)

Tuesday, November 27, 2012


In all approaches:1. w.f. is described as linear combination of excited determinants;2. Amplitudes are found from diagonalizing some matrix related to the Hamiltonain

Example: EOM-CCSD



Methods applicable to large molecules:

W.F.1. CIS (RPA): N5, singles only. Diagonaize H in the basis of singly excited det-s.

2. CIS(D): N6, CIS + perturbative account of doubles. SOS-CIS(D): N5, very fast!

DFT:1. TD-DFT: N5. Just like CIS, only that Kohn-Sham Hamiltonian is used.

Recommended reading: Dreuw, Head-Gordon; Chem. Rev. 105 4009 (2005)

density matrix (section 4.3), and attachment/detach-ment density plots (section 4.4). In the last section5, two illustrative examples of typical theoreticalstudies of large molecules employing TDDFT aregiven, where special emphasis is put onto the ap-plicability, advantages, and limitations of TDDFT.

2. Wave-Function-Based Methods

2.1. Configuration Interaction Singles2.1.1. Derivation of the CIS Equations

Configuration interaction singles (CIS) is the com-putationally as well as conceptually simplest wave-function-based ab initio method for the calculationof electronic excitation energies and excited-stateproperties. The starting point of the derivation of theCIS equations is the Hartree-Fock (HF) groundstate, !0(r), which corresponds to the best singleSlater determinant describing the electronic groundstate of the system. It reads

For simplicity, we assume a closed-shell ground-stateelectronic configuration, and thus, the !i(r) cor-respond to doubly occupied spatial orbitals and n )N/2 (N is the number of electrons). !0(r) is obtainedby solving the time-independent Hartree-Fock equa-tion, which is given by

with

and

In this equation, h(r) contains the kinetic energy ofthe ith electron and its electron-nuclei attraction,while the Coulomb operator Ji(r) and exchangeoperator Ki(r) describe the averaged electron-elec-tron interactions. They are defined as

Solution of the Hartree-Fock equations for theground-state Slater determinant (eq 1) within a givenbasis set of size K yields n occupied molecularorbitals, !i(r), and v ) K - n virtual orbitals, !a(r).Here and in the following, we use the indices i, j, k,etc. for occupied orbitals, a, b, c, etc. for virtual ones,and p, q, r, etc. for general orbitals. In configurationinteraction, the electronic wave function is then

constructed as a linear combination of the groundstate Slater determinant and so-called “excited”determinants, which are obtained by replacing oc-cupied orbitals of the ground state with virtual ones.If one replaces only one occupied orbital i by onevirtual orbital a and one includes only these “singlyexcited” Slater determinants, !i

a(r), in the CI wavefunction expansion, one obtains the CIS wave func-tion, "CIS, which thus reads

The summation runs over index pairs ia and has thedimension n ! v. This ansatz for the many-body wavefunction is substituted into the exact time-indepen-dent electronic Schrodinger equation,

where T has the usual meaning of the kinetic energyoperator

and Vel-nuc corresponds to the electron-nuclei attrac-tion

where the index i runs over all electrons and K overall nuclei. ZK is the charge of nucleus K. Theelectron-electron interaction, Vel-el(r) is given as

Projection onto the space of singly excited determi-nants, that is, multiplication of eq 8 from the left with!!j

b|, yields

and with

one readily obtains an expression for the excitationenergies !CIS ) ECIS - E0

"a and "i are the orbital energies of the single-electronorbitals !a and !i, respectively, and (ia||jb) corre-

!0(r) ) |!1(r)!2(r)...!n(r)| (1)

F(r)!0(r) ) E0!0(r) (2)

F(r) )"i

n

fi(r) (3)

fi(r) ) hi(r) + Ji(r) - Ki(r) (4)

Ji(r)!i(r) ) ["j

N

"dr#!j/(r#)!j(r#)

|r - r#| ]!i(r) (5)

Ki(r)!i(r) ) ["j

N

"dr#!j/(r#)!i(r#)

|r - r#| ]!j(r) (6)

"CIS )"ia

cia!i

a(r) (7)

H(r)"(r) ) [T(r) + Vel-nuc(r) + Vel-el(r)]"(r) )E"(r) (8)

T(r) ) -"i

1

2#2

i (9)

Vel-nuc(r) ) -"i"K

ZK

|ri - rK|(10)

Vel-el )"i

N

"j>i

N 1

|ri - rj|(11)

"ia!!j

b|H|!ia$ci

a ) ECIS"ia

cia"ij"ab (12)

!!jb|H|!i

a$ ) (E0 + "a - "i)"ij"ab + (ia||jb) (13)

"ia{("a - "i)"ij"ab + (ia||jb)}ci

a ) !CIS"ia

cia"ij"ab (14)

4012 Chemical Reviews, 2005, Vol. 105, No. 11 Dreuw and Head-Gordon

sponds to the antisymmetrized two-electron integrals,which are defined as

Equation 14 can be nicely written in matrix nota-tion as an eigenvalue equation

in which we use the unusual symbol A for the matrixrepresentation of the Hamiltonian in the space of thesingly excited determinants to make the connectionto later occurring equations more clear. ! is thediagonal matrix of the excitation energies, and X isthe matrix of the CIS expansion coefficients. Thematrix elements of A are given as

The excitation energies are finally obtained by solv-ing the following secular equation

that is, by diagonalization of the matrix A. Theobtained eigenvalues correspond to the excitationenergies of the excited electronic states, and itseigenvectors to the expansion coefficients accordingto eq 7.

2.1.2. Properties and LimitationsIn the previous section, the derivation of the CIS

working equations has been outlined, and it has beenshown how excitation energies and excited-statewave functions are obtained. Here, we want tocompile some useful properties of CIS:

(1) Since the CIS wave function is determined byvariation of the expansion coefficients of the ansatz(eq 7), its total energy corresponds to an upper boundof the true ground-state energy by virtue of theRaleigh-Ritz principle. Also all excited-state totalenergies are true upper bounds to their exact values.(2) Owing to Brillouin’s theorem, which states that“singly excited” determinants !i

a(r) do not couple tothe ground state, that is,

the excited state wave functions are Hamiltonianorthogonal to the ground state. (3) In contrast toother truncated configuration interaction methods,for instance, CI with single and double excitations(CISD), CIS is size-consistent, that is, the totalground-state energy of two noninteracting systemsis independent of whether they are computed to-gether in one calculation or are computed indepen-dently. This is immediately plausible having Bril-louin’s theorem in mind, owing to which the ground-state energy within CIS is nothing else but theHartree-Fock energy. And since Hartree-Fock issize-consistent,1 so is CIS. (4) Another useful propertyof CIS is that it is possible to obtain pure singlet and

triplet states for closed-shell molecules by allowingpositive and negative combination of R and " excita-tions from one doubly occupied orbital. Due to itsconceptually simple ansatz (eq 7) and the listedproperties, the CIS excited-state wave functions arewell defined. Hence, their wave functions and corre-sponding energies are directly comparable, which isa particularly necessary prerequisite if one is inter-ested in transitions between excited states.

An analytic expression for the total energy ofexcited states can be obtained from eq 14 by addingE0 and multiplying from the left with the correspond-ing CIS vector. It reads

As a consequence, ECIS is analytically differentiablewith respect to external parameters, for example,nuclear displacements and external fields, whichmakes the application of analytic gradient techniquesfor the calculation of excited-state properties such asequilibrium geometries and vibrational frequenciespossible. Analytic first derivatives for CIS excitedstates have been published,45,46 and also secondderivatives are available.46 They have been imple-mented in various computer codes, for instance,Q-Chem, CADPAC, or TURBOMOLE.47-49

In general, excitation energies computed with theCIS method are usually overestimated, that is, theyare usually too large by about 0.5-2 eV comparedwith their experimental values (see, for instance, refs45, 50, and 51). This is on one hand because the“singly excited” determinants derived from the Har-tree-Fock ground state are only very poor first-orderestimates for the true excitation energies, since thevirtual orbital energies !a are calculated for the (N+ 1)-electron system instead of for the N-electronsystem.1 Consequently, the orbital energy difference(!a - !i), which is the leading term in eq 17, is notrelated to an excitation energy, if the canonicalHartree-Fock orbitals are used as reference. In otherwords, the canonical HF orbitals are not a particu-larly good basis for the expansion of the correlatedwave function, which then needs a high flexibility tocompensate for this disadvantage, that is, doubly andhigher excited determinants are demanded in thewave function. On the other hand, since electroncorrelation is generally neglected within the CISmethod, the error will be the differential correlationenergy, which must be at least on the order of thecorrelation energy of one and sometimes severalelectron pairs. Such energies are typically on theorder of 1 eV per electron pair, and hence, one shouldexpect errors of this magnitude.

Furthermore, CIS does not obey the Thomas-Reiche-Kuhn dipole sum rule, which states that thesum of transition dipole moments must be equal tothe number of electrons.52-54 Thus, transition mo-ments cannot be expected to be more than qualita-tively accurate. While a full presentation of thecomputational algorithms used to evaluate the CISenergy is beyond our present scope, it is still impor-tant to understand the dependence of computationalcost on molecule size. The computational cost for CIS

(ia||jb) ) !!dr dr!

["i(r)"a(r)"j(r!)"b(r!) - "i(r)"j(r)"a(r!)"b(r!)|r - r!| ] (15)

AX ) !X (16)

Aia,jb ) (!a - !i)#ij#ab + (ia||jb) (17)

(A - !)X ) 0 (18)

"!ia(r)|H|!0(r)# ) 0 (19)

ECIS ) EHF +$ia

(cia)2(!a - !i) + $

ia,jbci

a cjb(ia||jb) (20)

Calculation of Excited States of Large Molecules Chemical Reviews, 2005, Vol. 105, No. 11 4013

sponds to the antisymmetrized two-electron integrals,which are defined as

Equation 14 can be nicely written in matrix nota-tion as an eigenvalue equation

in which we use the unusual symbol A for the matrixrepresentation of the Hamiltonian in the space of thesingly excited determinants to make the connectionto later occurring equations more clear. ! is thediagonal matrix of the excitation energies, and X isthe matrix of the CIS expansion coefficients. Thematrix elements of A are given as

The excitation energies are finally obtained by solv-ing the following secular equation

that is, by diagonalization of the matrix A. Theobtained eigenvalues correspond to the excitationenergies of the excited electronic states, and itseigenvectors to the expansion coefficients accordingto eq 7.

2.1.2. Properties and LimitationsIn the previous section, the derivation of the CIS

working equations has been outlined, and it has beenshown how excitation energies and excited-statewave functions are obtained. Here, we want tocompile some useful properties of CIS:

(1) Since the CIS wave function is determined byvariation of the expansion coefficients of the ansatz(eq 7), its total energy corresponds to an upper boundof the true ground-state energy by virtue of theRaleigh-Ritz principle. Also all excited-state totalenergies are true upper bounds to their exact values.(2) Owing to Brillouin’s theorem, which states that“singly excited” determinants !i

a(r) do not couple tothe ground state, that is,

the excited state wave functions are Hamiltonianorthogonal to the ground state. (3) In contrast toother truncated configuration interaction methods,for instance, CI with single and double excitations(CISD), CIS is size-consistent, that is, the totalground-state energy of two noninteracting systemsis independent of whether they are computed to-gether in one calculation or are computed indepen-dently. This is immediately plausible having Bril-louin’s theorem in mind, owing to which the ground-state energy within CIS is nothing else but theHartree-Fock energy. And since Hartree-Fock issize-consistent,1 so is CIS. (4) Another useful propertyof CIS is that it is possible to obtain pure singlet and

triplet states for closed-shell molecules by allowingpositive and negative combination of R and " excita-tions from one doubly occupied orbital. Due to itsconceptually simple ansatz (eq 7) and the listedproperties, the CIS excited-state wave functions arewell defined. Hence, their wave functions and corre-sponding energies are directly comparable, which isa particularly necessary prerequisite if one is inter-ested in transitions between excited states.

An analytic expression for the total energy ofexcited states can be obtained from eq 14 by addingE0 and multiplying from the left with the correspond-ing CIS vector. It reads

As a consequence, ECIS is analytically differentiablewith respect to external parameters, for example,nuclear displacements and external fields, whichmakes the application of analytic gradient techniquesfor the calculation of excited-state properties such asequilibrium geometries and vibrational frequenciespossible. Analytic first derivatives for CIS excitedstates have been published,45,46 and also secondderivatives are available.46 They have been imple-mented in various computer codes, for instance,Q-Chem, CADPAC, or TURBOMOLE.47-49

In general, excitation energies computed with theCIS method are usually overestimated, that is, theyare usually too large by about 0.5-2 eV comparedwith their experimental values (see, for instance, refs45, 50, and 51). This is on one hand because the“singly excited” determinants derived from the Har-tree-Fock ground state are only very poor first-orderestimates for the true excitation energies, since thevirtual orbital energies !a are calculated for the (N+ 1)-electron system instead of for the N-electronsystem.1 Consequently, the orbital energy difference(!a - !i), which is the leading term in eq 17, is notrelated to an excitation energy, if the canonicalHartree-Fock orbitals are used as reference. In otherwords, the canonical HF orbitals are not a particu-larly good basis for the expansion of the correlatedwave function, which then needs a high flexibility tocompensate for this disadvantage, that is, doubly andhigher excited determinants are demanded in thewave function. On the other hand, since electroncorrelation is generally neglected within the CISmethod, the error will be the differential correlationenergy, which must be at least on the order of thecorrelation energy of one and sometimes severalelectron pairs. Such energies are typically on theorder of 1 eV per electron pair, and hence, one shouldexpect errors of this magnitude.

Furthermore, CIS does not obey the Thomas-Reiche-Kuhn dipole sum rule, which states that thesum of transition dipole moments must be equal tothe number of electrons.52-54 Thus, transition mo-ments cannot be expected to be more than qualita-tively accurate. While a full presentation of thecomputational algorithms used to evaluate the CISenergy is beyond our present scope, it is still impor-tant to understand the dependence of computationalcost on molecule size. The computational cost for CIS

(ia||jb) ) !!dr dr!

["i(r)"a(r)"j(r!)"b(r!) - "i(r)"j(r)"a(r!)"b(r!)|r - r!| ] (15)

AX ) !X (16)

Aia,jb ) (!a - !i)#ij#ab + (ia||jb) (17)

(A - !)X ) 0 (18)

"!ia(r)|H|!0(r)# ) 0 (19)

ECIS ) EHF +$ia

(cia)2(!a - !i) + $

ia,jbci

a cjb(ia||jb) (20)

Calculation of Excited States of Large Molecules Chemical Reviews, 2005, Vol. 105, No. 11 4013


Excited states methods: Different flavors of LR


TD-DFT: Success stories and limitations

1. Remarkable improvement over CIS (for valence states, typical errors are 0.2 eV).

2. Inexpensive!

3. Different meaning of amplitudes (we do not have w.f., we have change in density).

4. Failures: Rydberg states, charge-transfer states. Also, artificial states often appear.

5. No doubly-excited states; problems with bond-breaking and conical intrsections


Different types of excited states: Valence, Rydberg, and charge-transfer


Origin of artifactual states in TD-DFT


KS orbital energies (Koopmans) versus deltaE


KS orbital energies (Koopmans) vs. deltaE

A1 T1

A2

A1

A2

A1

T2

T1

1.01

1.01

1.01

0.72

-0.01

-0.01

-0.01

0.28

0.43

0.46

0.47

0.48

0.57

0.54

0.53

0.52

0.54

0.53

0.48

0.51

0.51

0.46

-0.02

-0.03

0.04

-0.02

-0.02

0.02

IP-CISD1

!B97X-D/6-311+G(df,pd)M06-2X/6-311+G(df,pd)B3LYP/6-311+G(df,pd)


TD-DFT: Range-separated functionals

Range-separated (aka long-range corrected aka Coulomb-attenuated) functionalsreduce SIE, improve Koopmans IEs, and, consequently, improve excited states.

energy curves.It is also worthwhile to note another peculiarity of

CT states, namely, that the B matrix (eq 100)vanishes for a long-range CT state in the TDDFT,as well as TDHF, case. Since the neglect of the Bmatrix is equivalent to applying the Tamm-Dancoffapproximation, TDHF and CIS, as well as TDDFTand TDDFT/TDA, yield identical results for theexcitation energies of long-range CT states, which isthe difference of the corresponding orbital energies.This fact can be exploited in TDDFT as a firstdiagnostic for whether one deals with a problematicCT state.

However, if the exact local Kohn-Sham xc poten-tial would be used, which unfortunately is not known,the correct 1/R long-range behavior would be found.This is due to the derivative discontinuities of theexact exchange-correlation energy with respect toparticle number.112,113 In the case of an electrontransfer from orbital i on molecule A to orbital a onmolecule B, the xc potential jumps discontinuouslyby the constant IPA - EAB, leading to a singularityin the derivative of the xc potential with respect tothe density, that is, the xc kernel fxc used in theTDDFT calculation (eq 96). This singularity thencompensates the vanishing overlap between theorbitals i and a when the molecules A and B are beingseparated, and thus, the fourth term of eq 96 does infact contribute to a CT state when the exact xcpotential would be employed, and the correct 1/Rasymptote along the separation coordinate wouldthen be obtained for a CT state. Furthermore, thisclearly explains why standard approximate xc func-tionals in TDDFT fail in describing long-range CTstates correctly since they do not contain the deriva-tive discontinuities.

At present, several different pathways are startingto emerge to address this substantial failure ofTDDFT for CT states and to correct for it. The mostobvious way is to improve the xc functional byincluding exact exchange in the unperturbed Hamil-tonian, either in the form of nonlocal Hartree-Fockexchange or of the exact local Kohn-Sham exchangepotential. The latter are known as local Hartree-Fock (LHF) or also optimized effective potentials(OEPs), which have recently been introduced byGorling106 and Ivanov et al.107 These potentials maybe able to yield the correct asymptotic 1/R depen-dence of excited CT states, since they possess singu-larities as soon as the overlap between the electron-donating and electron-accepting orbitals i and a,respectively, approaches zero. This, however, remainsto be explored in detail, and it seems numericallydifficult to compensate the vanishing overlap withthe divergence of the xc kernel fxc such that the fourthterm of the A matrix (eq 96) cancels the electron-transfer self-interaction error correctly and recoversthe correct 1/R asymptote. Also, satisfactory match-ing correlation functionals remain to be developed.

Inclusion of nonlocal Hartree-Fock exchange hasbeen realized in a few schemes so far.114-116 In allthese schemes, the Coulomb operator of the Hamil-tonian is split into two parts, a short-range and along-range part, as, for example, in ref 114,

where r12 ) |r1 - r2|. The first term of the right-handside (rhs) corresponds to the short-range part and isevaluated using the xc potential from DFT, while thesecond term, the long-range part, is calculated withexact Hartree-Fock exchange. This idea is fairly oldand had originally been suggested by Stoll and Savinalready in 1985.117-120 The scheme (eq 101) has beenapplied in combination with various xc functionalsyielding, for instance, LC-BLYP, which indeed cor-rects the failures of TDDFT for CT excited states.114

A major drawback of this approach, however, is thatthe standard xc functionals require a refitting pro-cedure. Similar in spirit is the approach of Baer andNeuhauser, who also include long-range Hartree-Fock exchange but who employ a different partitionfunction for the Coulomb operator.116 An extensionof this approach has been presented by Yanai et al.,who combine B3LYP80 at short range with an in-creasing amount of exact HF exchange at long rangeresulting in a functional called CAM-B3LYP,115 whichgives excellent CT excitation energies in comparisonwith benchmark calculations. But since they use atlong range at most 60% HF exchange, the long-rangeasymptotic behavior of the CT states is not fullycorrected.115 A slightly different route is taken byGritsenko and Baerends who suggest a new long-range corrected xc kernel that shifts the orbitalenergy of the acceptor orbital to a value related tothe electron affinity. The wrong asymptotic behaviorof the CT states is corrected by a distance-dependentCoulomb term correcting for electron-transfer self-interaction.95

A completely different ansatz to the solution of theCT problem may be represented by time-dependentcurrent density functional theory (TDCDFT), whichhas recently been implemented in the Amsterdamdensity functional (ADF) package.121,122 We havepreviously seen that a correct description of charge-transfer excited states requires the nonlocal HFexchange potential, and current functionals are non-local. However, at present no test calculation has yetbeen performed to show whether TDCDFT would inprinciple yield correct CT excited states. Also, dueto its high computational cost the question remainswhether this method will be applicable to largemolecules soon.

4. Analysis of Electronic TransitionsWhen a molecule is excited from the electronic

ground state to an energetically higher lying excitedstate, the electronic many-body wave function changes.To gain insight in the nature of the correspondingelectronic transition, this change in the wave functionneeds to be analyzed. In principle, this is possible bydirect analysis of the excited state wave function orits electron density. For this objective, the techniquesdeveloped for the analysis of the wave function orelectron density of the ground state are simplyapplied to the excited state of interest. However, herewe want to focus on approaches that are dedicated

1r12

)1 - erf(µr12)

r12+

erf(µr12)r12

(101)

4026 Chemical Reviews, 2005, Vol. 105, No. 11 Dreuw and Head-Gordon


excited states methods: reviewiopenshell.usc.edu/chem545/material/lecture_exstates_tddft.pdf ·...

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