Dissertation for the Degree of Doctor of Philosophy in Physics presented atUppsala University in 2003AbstractBurmeister, F. 2003. Photoelectron spectroscopy on HCl and DCl: Synchrotronradiation based studies of dissociation dynamics. Acta Universitatis Upsaliensis.Comprehensive Summaries of Uppsala Dissertations from the Faculty of Scienceand Technology 805. 61 pp. Uppsala. ISBN 91-554-5531-XThe dissociation dynamics of the ionized molecules hydrogen chloride (HCl) anddeuterium chloride (DCl) have been studied in gas-phase using synchrotron basedphotoelectron spectroscopy (PES).The main inner-valence photoionization band for DCl and HCl was recorded usingmaximum resolution in order to probe an interference pattern between a disso-ciative and a bound electronic state. For HCl+, distorted Fano-type peaks wereobserved even for modest resolution, whereas for DCl+, the pattern was hardlydiscernible. The observation in HCl+ has been explained by a coupling betweentwo adiabatic electronic states, where the bound state was populated throughnon-adiabatic curve-crossing. The nuclear motion of HCl+ is too fast for theBorn-Oppenheimer approximation to be fully valid in this case, whereas for DCl+,with larger reduced mass and therefore slower nuclear motion, the non-adiabaticcoupling is less pronounced, and the vibrational progression almost vanishes.A comparative study between PES and threshold photoelectron spectra (TPES)of the inner-valence bands of HCl and DCl has been performed, showing differ-ences in intensities and shapes of the vibrational bands. These differences wereattributed to the fact that the photoelectron can be regarded as isolated from thecation for PES, but not so in the case of TPES.A resonant Auger electron spectroscopy study of HCl and DCl has been performed,which shows an interference pattern between the atomic and the molecular Auger-and photoelectron channels. The atomic features are associated with ultra-fastdissociation of the molecules, on the same time scale asf the Auger decay. Theobservation shows that the excited molecular system has to be regarded as a su-perposition of fragmented and molecular states.A study of the X-state of HCl+, populated via a core-excited state, shows a se-lective population of the final state. The explanation is that that the magneticorientation of the core-hole is transferred to the final state of the molecule.A setup for the data acquisition of Photo-Electron Photo-Ion Photo-Ion COin-cidence (PEPIPICO) measurements using a Time-Of-Flight (TOF) spectrometerhas been developed. A Time-to-Digital Converter (TDC) card has been linkedtogether with the data treatment program Igor Pro as a user interface. Further-more, the PEPIPICO spectrometer has been characterized to provide a solid basisfor the analysis of future experimental data.Florian Burmeister, Department of Physics, Uppsala University, Box 530SE-751 21 Uppsala, Swedenc©Florian Burmeister 2003ISSN 1104-232XISBN 91-554-5531-X Printed in Sweden by Kopieringshuset AB, Uppsala 2003

Till Emma♥

Sa mjuk, sa varm, sa fin!

Jorge Ben (oversattning Mats Hallgren)

4

List of papers

This Thesis is based on the following Papers, which will be referred to in the textby their Roman numerals.

I. Nonadiabatic effects in photoelectron spectra of HCl and DCl.I. ExperimentF. Burmeister, S. L. Sorensen, O. Bjorneholm, A. Naves de Brito, R. F. Fink,R. Feifel, I. Hjelte, K. Wiesner, A. Giertz, M. Bassler, C. Miron, H. Wang,M. N. Piancastelli, L. Karlsson, and S. SvenssonPhys. Rev. A 65, 012704 (2001).

II. Nonadiabatic effects in the photoelectron spectra of HCl and DCl.II. TheoryL. M. Andersson, F. Burmeister, H. O. Karlsson, and O. GoscinskiPhys. Rev. A 65, 012705 (2001).

III. Confirmation of non-adiabatic vibrational progression in the maininner-valence photoionization band of DCl and HClF. Burmeister, L. M. Andersson, G. Ohrwall, A. J. Yencha, T. Richter,P. Zimmermann, K. Godehusen, M. Martins, H. O. Karlsson, S. L. Sorensen,O. Bjorneholm, R. Feifel, K. Wiesner, O. Goscinski, L. Karlsson, andS. SvenssonSubmitted to Phys. Rev. A

IV. PES/TPES Comparative Study of the Inner Valence IonizationRegion in HCl and DClF. Burmeister, G. Ohrwall, L. Karlsson, O. Bjorneholm, S. Svensson, andA. J. YenchaIn manuscript.

V. Observation of a Continuum-Continuum Interference Hole inUltrafast Dissociating Core-Excited MoleculesR. Feifel, F. Burmeister, P. Salek, M. N. Piancastelli, M. Bassler,S. L. Sorensen, C. Miron, H. Wang, I. Hjelte, O. Bjorneholm, A. Naves de Brito,F. Kh. Gel’mukhanov, H. Agren, and S. SvenssonPhys. Rev. Lett 85, 3133 (2000).

VI. Spin-orbit selectivity observed for the HCl+ (X 2Π) state usingresonant photoemissionR. F. Fink, F. Burmeister, R. Feifel, M. Bassler, O. Bjorneholm, L. Karlsson,C. Miron, M.-N. Piancastelli, S. L. Sorensen, H. Wang, K. Wiesner, andS. SvenssonPhys. Rev. A, 65, 034705 (2002)

5

VII. Using Igor Pro as the user interface together with a FAST TDCcard for PEPIPICO data acquistion: manual and overviewF. Burmeister, M. Gisselbrecht, J. R. Piton, E. S. Cardoso, S. L. Sorensen,and A. Naves de BritoLNLS-CT, 06, 01 (2002)

VIII. Description and performance of an electron-ion coincidence TOFspectrometer used at the Brazilian synchrotron facility LNLSF. Burmeister, L. H. Coutinho, R. R. T. Marinho, K. Wiesner,M. A. A. de Morais, A. Mocellin, O. Bjorneholm, S. L. Sorensen,P. de Tarso Fonseca, J. G. Pacheco, and A. Naves de BritoIn manuscript.

The following is a list of papers to which I have contributed but that are not in-cluded in this Thesis.

Development of a four-element conical electron lens dedicated to highresolution Auger electron-ion(s) coincidence experimentsK. Le Guen, D. Ceolin, R. Guillemin, C. Miron, N. Leclercq, M. Bougeard, M. Si-mon, P. Morin, A. Mocellin, F. Burmeister, A. Naves de Brito, and S. L. SorensenRev. Sci. Instrum. 73, 3885 (2002)

Filtering core excitation spectra: vibrationally resolved constant ionicstate studies of N 1s → π core-excited NOH. Wang, R. F. Fink, M.-N. Piancastelli, I. Hjelte, K. Wiesner, M. Bassler,R. Feifel, O. Bjorneholm, C. Miron, A. Giertz, F. Burmeister, S. L. Sorensen,and S. SvenssonJ. Phys. B 34, 4417 (2001)

The dynamic Auger-Doppler effect in HF and DF: control of fragmentvelocities in femtosecond dissociation through photon energy detuningK. Wiesner, A. Naves de Brito, S. L. Sorensen, F. Burmeister, M. Gisselbrecht,S. Svensson, and O. BjorneholmChem. Phys. Lett. 354, 382 (2002)

Experimental study of photoionization of ozone in the12 to 21 eV regionA. Mocellin, K. Wiesner, F. Burmeister, O. Bjorneholm, and A. Naves de BritoJ. Chem. Phys. 115, 5041 (2001)

A vibrationally resolved experimental study of the sulfur L-shellphotoelectron spectrum of the CS2 moleculeH. Wang, M. Bassler, I. Hjelte, F. Burmeister, and L. KarlssonJ. Phys. B 34, 1745 (2001)

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7

Contents

1 Introduction 9

2 Experimental 112.1 Synchrotron radiation . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Beamline I411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Experiment endstation . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Why BESSY? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Molecular physics 193.1 Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Atoms, molecules and their electronic structure . . . . . . . . . . . 21

3.2.1 The electronic structure of atoms . . . . . . . . . . . . . . . 213.2.2 Electronic structure of HCl . . . . . . . . . . . . . . . . . . 22

3.3 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 The Schrodinger equation . . . . . . . . . . . . . . . . . . . 253.3.2 The diabatic framework . . . . . . . . . . . . . . . . . . . . 273.3.3 The adiabatic framework . . . . . . . . . . . . . . . . . . . 313.3.4 The Landau-Zener formula . . . . . . . . . . . . . . . . . . 333.3.5 Adiabatic/diabatic framework: short repetition . . . . . . . 35

3.4 Franck-Condon projections . . . . . . . . . . . . . . . . . . . . . . 373.5 Fano resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.6 Hydrogen chloride . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.7 Summary of Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.7.1 Papers I, II and III . . . . . . . . . . . . . . . . . . . . . . . 453.7.2 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.7.3 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.7.4 Paper VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.7.5 Papers VII and VIII . . . . . . . . . . . . . . . . . . . . . . 53

Comments on my own participation 55

Why fundamental science? 55

Acknowledgments 56

Bibliography 61

8

9

Chapter 1

Introduction

The name photoelectron spectroscopy reveals the basic fundament of the researchmethod, of which the work presented here is based upon. A photon interacts witha system, in this Thesis the hydrogen chloride (HCl), or deuterium chloride (DCl)molecule in gas-phase, and one or several electrons are emitted and analyzed. Thekinetic energy of the electron(s) is analyzed with a spectrometer. The kineticenergy is related to the binding energy of the electrons in the system according to

hν = Ekin + Ebin, (1.1)

where hν is the photon energy, Ekin is the kinetic energy of the electron andEbin is the binding energy of the electron. Eq. (1.1) is a modified version ofthe photoelectric effect formula, which was proposed by Albert Einstein in 1905.The emitted electrons can have different origins. Fig. 1.1 shows the two cases of

Cor

eel

ectro

nsVa

lenc

eel

ectro

ns

Photo-electron

Auger-electron

Resonant channelDirect channel

Figure 1.1: A sketch of photoelectron and resonant Auger electron channels. Solid

(dashed) lines represent occupied (unoccupied) orbitals.

importance for the work presented in this Thesis:

1. The direct channel: One photon interacts with the sample and causes theemission of one electron from the molecule. The extracted electron is called

10

a photoelectron. The direct channel is non-resonant, i. e. the process occurswhenever the photon energy exceeds the Ebin on the photon energy, seeEq. (1.1). The direct photoelectron channel is shown in the left-hand side ofFig. 1.1.

2. The resonant channel: Here, the photon excites the molecule. A core-electronis put into a previously unfilled orbital. When the excited molecule decays,the excess of energy is released via the ejection of a photon or an electron.The electron decay channel is shown on the right side of Fig. 1.1. Theprobability for an electron to carry the excess energy is orders of magnitudelarger than for a photon, except for deep core levels in heavy atoms (muchheavier than in the here discussed light diatomic cases HCl, CO, etc), wherethe photon channel becomes significant. The discoverer of the decay processin 1925, Pierre Auger, has given the name to the Auger process and the Augerelectron. The resonant channel is highly sensitive to the photon energy sinceit must match the excitation energy.

Note that both cases in Fig. 1.1 result in the same final electronic state.Papers I, II, III and IV deal with the direct channel process. Paper V deals

with an interference phenomenon between direct and resonant channels. Paper VIdeals with a purely resonant process. The outline of this Thesis is as follows:Chapter 2 gives a brief overview of the experimental setup needed for photoelectronspectroscopy using synchrotron radiation. Chapter 3 gives a background to and anextension of the physics that lie behind the results presented in the Papers in thisThesis. The HCl molecule is one of the most studied molecules in the literature.Nevertheless, as this Thesis shows, there is more to find out. The dissociationdynamics can be probed and analyzed in photoelectron spectra, revealing intricatedetails.

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Chapter 2

Experimental

The understanding of physics in nature is based upon experimental studies. Notheory, no matter how beautiful it might appear, can stand the falsification ofan experiment. And theory can be strengthened (but strictly never proven) byexperiments. Over the centuries, experimental setups have grown in size andcomplexity. The laboratory where experiments similar to those discussed here isone of the largest in Sweden, the National Synchrotron Radiation Facility MAX-lab in Lund. This Chapter gives a general overview of the laboratory employedfor the present investigations.

2.1 Synchrotron radiation

Synchrotron radiation has its origin from accelerated charged particles. The par-ticles accelerated can be electrons, positrons (the positively charged twins of elec-trons), protons, etc. The acceleration is achieved by electric and magnetic fields.

There are natural sources for synchrotron light in universe, for example neutronstars quickly spinning around their axis. Here the accelerated charges are electronsaccelerated by the magnetic field around the neutron stars. Charged particlesfrom the sun which are deflected by the magnetic field of the earth is anothersynchrotron source.

On earth, man-made sources of synchrotron light are particle accelerators.The first time synchrotron light was seen was in 1947 at the General Electriclaboratories in New York, where emitted light was observed from an electronbeam [1].

The principle of the synchrotron is to synchronously change the generatorfrequency of the accelerating stages together with the magnetic field in such a waythat the electrons, whose orbital frequencies and momenta are increasing as a resultof the acceleration, always feel an accelerating force and are simultaneously kepton their assigned orbits inside the vacuum tube. In 1949, J. Schwinger publisheda paper characterizing synchrotron light theoretically [2]. There it was shown thatthe energy lost in one turn is proportional to the fourth power of the electronenergy, i. e. doubling the particle beam energy means sixteen times more photons.

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MAX II MAX I

BL I411Undulator

Figure 2.1: An overview of MAX-lab.

Electric fields are used for acceleration where the electrons gain kinetic energy,and magnetic fields for guiding and controlling the electron beam. The magnetsused are primarily dipoles and quadrupoles. Quadrupoles are used for focussingthe electron beam, in analogy with optical systems, whereas dipoles (called bend-ing magnets) are used for making the electrons turn around in the ring. Sincethe magnetic fields are not changed with time, see below, the name synchrotronlight source has lost its original meaning. A more appropriate name used for theaccelerator device today is electron storage ring. As the electrons are acceleratedby the centripetal magnetic force, they emit photons in the forward direction,synchrotron light. Between the magnets there are magnet-free sections. In one ofthese sections, the electron bunches are pushed to keep the kinetic energy each turnby a oscillating electric field. The device is called a radio frequency cavity, due tothe high frequency of the field (50-1500 MHz). The 1st generation of synchrotronlight research was ”parasitic” activity on behalf of the nuclear physicists acceler-ators. The 2nd generation of light sources, starting up in the 70’s, were electronstorage rings dedicated to synchrotron light research. One of these 2nd genera-tion light sources is MAX I in Lund, Sweden, which was inaugurated in 1987.An overview of MAX-lab is shown in Fig. 2.1. MAX stands for Multi-purposeAccelerator for X-rays. A microtron is used for pre-acceleration of the electronsbefore they are ejected into the storage ring. The microtron at MAX-lab is placedunder MAX I. MAX II is a 3rd generation synchrotron light source, which hasbeen operational since 1998. Whereas for 2nd generation light sources one mainlyuses the synchrotron light from the dipole bending magnets, more sophisticatedmagnetic devices have been developed for 3rd generation light sources. Arrays ofdipole magnets force the electrons to proceed in a sinusoidal trajectory, emittingmore photons than the one-turn bending magnets. These arrays are called wig-glers or undulators. Wigglers are used when high photon energies (keV range)are required. The higher the magnetic field, the higher the photon energy. Some-times super-conducting electromagnets are used to produce as high magnetic fieldas possible. The wiggler photon intensity versus photon energy characteristics is

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Table 2.1: MAX II, and beamline I411 parameters

Parameter ValueMAX II:Electron energy 1.5 GeVElectron speed 0.999999%Storage ring circumference 90 mRadio Frequency of Cavity 500 MHzElectron bunch length 20 psElectron bunch periodicity 2 nsBeam lifetime > 10 hMax. Current >200 mAUndulator at BL I411:Period between magnets 60 mmNumber of magnet poles 87Total undulator length 2.65mMagnet gap 22 - 300 mmPeak field 0.65 T

continuous over a long energy range. Undulators typically have a larger number ofshorter dipole magnets in one array. Unlike wigglers, the undulator photon inten-sity versus photon energy characteristics is non-continuous: for certain energies,the photon intensity increases by orders of magnitude. This is due to constructiveinterference of the superimposed photons emitted on the undulator. The wave-length for which the constructive interference occurs depends on the deviation ofthe electrons from a straight path, which in its turn depends on the strength ofthe magnetic field. By changing the size of the undulator magnet gap, and therebychanging the strength of the magnetic field, the photon energy corresponding topositive interference can be changed.

With the development of wigglers and undulators, the demands on the elec-tron beam quality has increased. Source size, emittance and beam stability areparameters that are important this aspect. Some basic parameters of MAX IIand undulator ”Finnen” of beamline insertion device 411 (BL I411) are shown inTable 2.1 [3, 4].

2.2 Beamline I411

In the previous Section, the creation of synchrotron light has been discussed. Thelight coming from the undulator cannot be directly used for the experiments.Among other things, the experiment needs a more well-defined photon energy,than the undulator can provide. A beamline is needed to meet this requirement.An overview of BL I411 is shown in Fig. 2.2. More detailed descriptions of BL I411can be found in Refs. [5, 6, 7]. MAX II, and the undulator lie outside Fig. 2.2,on the left, behind a thick wall of concrete and lead which protects from radiation

14

Und

ulat

orso

urce Cyl

indr

ical

mirr

or

Wat

erco

oled

baffl

es SX

700

Mon

ochr

omat

or

BL I411

Ref

ocus

ing

mirr

or

One

-met

erse

ctio

n

2m

End

stat

ion

with

Sci

enta

anal

yzer

Exi

tslit

Figure 2.2: An overview of beamline I411 at MAX-lab.

from the electron storage ring. All optics elements (mirrors, gratings) work atgrazing incidence, due to the fact that no material has appreciable reflectivity atnormal incidence at photon energies of 60-1000 eV, where BL I411 is operational.First, the photon beam reaches the cylindrical mirror, which focuses the beam inhorizontal direction. Baffles are used for reducing the size of the photon beam tothe order of 0.1 mm2, and cutting the edges of the white light beam.

The heart of the beamline is the monochromator, which selects one photonenergy out of the undulator spectrum of light (mono=one; chromos=colour). Themonochromator of SX700-type at BL I411 consists of three optical components:a large plane mirror, a plane diffraction grating, and an ellipsoidal mirror. Thediffraction grating disperses the photons at different angles, depending on thephoton energy, in analogy with a prism. By changing the angle of the grating, theoutgoing photon energy passing through the fixed exit angle can be varied. Theexit slit, mounted behind the monochromator, has a variable aperture. Dependingon the aperture size (1-800µm), the photon energy band width can be chosen tobe narrow for high-resolution experiments, or broad, for higher flux and betterstatistics. After one last refocusing mirror, the photon beam enters the part of thebeamline dedicated to the experiments. The baffles, the first cylindrical mirror,and the diffraction grating in the monochromator are water cooled, because theyare heated up by the light.

The experiments can be performed at either the endstation with the station-ary Scienta electron analyzer, see next Section, or at the one-meter section. Atthe one-meter section, the scientific groups can use their portable equipment. Forinstance, the collaboration with the Stacey Sorensen group in Lund has used theone-meter section for measurements with a time-of-flight spectrometer for coin-cidence measurements. This spectrometer has the same overall characteristics asthe spectrometer described in Paper VIII.

In the BL I411 case, the beamline meets a second requirement, besides thefunction of the monochromator: the ultra-high vacuum (≈ 10−10 mbar) in theelectron storage ring has to be protected from the relatively high pressure at theexperiment (≈ 10−3 mbar). Different kinds of vacuum pumps are installed alongthe beamline. They make it possible to maintain the pressure difference betweenthe experiment and the monochromator and the electron storage ring. Capillarytubes are used between the beamline sections, to make the photons come through,but to reduce the gas flow from the high-pressure to the low-pressure sections.

15

2.3 Experiment endstation

The experimental research presented in Papers I, V and VI in this Thesis, wasperformed on the end station of BL I411, containing a Scienta SES-200 analyzer.A schematic overview is shown in Fig. 2.3. The synchrotron light interacts with

Photons

Gas cell

5- elementlens system

Magneticshields

Analyserspheres

MCP DetectorPhosphor screenCCD camera

Figure 2.3: The principal parts of a Scienta electron spectrometer.

the HCl/DCl gas-phase molecules in a gas-cell with differential pumping. Thepressure inside the gas cell during the measurements has been estimated to be inthe 10−3 mbar range, whereas outside the gas cell, the pressure is in the 10−5 mbarrange. The emitted photo- and Auger electrons are detected by the electron spec-trometer. The setup works in a crossed-beam configuration where the gas source,the axis of the Scienta electron spectrometer lens and the direction of propagationof the synchrotron light form an orthogonal set. The electron spectrometer can berotated from 0 to 90 with respect to the linear polarization of the synchrotronlight. The measurements at BL I411 were done at the magic angle, 54.7, at whichthe electron intensity is proportional to the case where electrons are collected overall angles. The electron spectrometer consists of two main parts, the electron lensand the analyzer. The electron lens is a complex five-element system, made tocollect electrons, transporting and focussing them on the entrance slit of the ana-lyzer, and also to accelerate/retard the electrons to a pre-set kinetic energy beforeentering the analyzer, the pass energy. The analyzer is composed of one innerand one outer metal hemisphere, with 200 mm mean-radius, hereby the name SES-200, where SES stands for Scienta Electron Spectrometer. Electrons with kineticenergy equal to the pass energy at the entrance fulfill the half-turn between thehemispheres, and hit the centre of a multi-channel plate (MCP) detector on theother side of the analyzer. Electrons with less or more kinetic energy are detectedon a smaller or larger radius relative to the centre of the MCP. The electronsenter the hemisphere at different angles. Electrons with identical kinetic energybut different entrance angles are focused onto the detection plane (the MCP). The

16

MCP detector amplifies the impact of the photoelectron to a measurable pulse. Aphosphor screen on the back-side of the MCP detector transforms the MCP pulseto a visible dot. The position of the phosphor screen dot is monitored by a charge-coupled-device (CCD) camera, and the corresponding kinetic energy is calculatedby a computer. The computer also controls the potentials of the electron lens andthe analyzer. The experimental end station is described in more detail in Ref. [8].

My contribution for future measurements at BL I411 has not been on theScienta endstation, but for a Photo-Electron Photo-Ion Photo-Ion COincidence,PEPIPICO spectrometer, which can be used at the one-meter section on BL I411,as well as an equivalent spectrometer used at the LNLS synchrotron in Campinas,Brazil. A new data acquisition setup has been developed, joining the FAST-COMTec company Time-to-Digital Converter (TDC) card with Igor Pro as userinterface. More detailed information about the data acquisition setup is presentedin Paper VII. Furthermore, the characterization of the PEPIPICO spectrometerhas been made and is available in Paper VIII. This Paper provides a solid basisfor the understanding of the experimental results.

2.4 Why BESSY?

There is a state-of-the-art setup for high-resolution photoelectron spectroscopyusing synchrotron light at MAX-lab, as shown by Papers I, V and VI. Why doesone then choose to go to the Berlin synchrotron BESSY II for the DCl measurementpublished in Paper III?

After Paper I had been written, contact was established with the theory groupof Quantum Chemistry. One of the results of the collaboration with theory wasthe statement ”From the simulations we observe that two additional peaks in theexperimental DCl should appear if the resolution were to be enhanced to around10 meV” in the last sentence in the abstract in Paper II. For the experimentaland theoretical discussion about the Physics of related Papers, see further in Sec-tion 3.7.1. Is a photoelectron spectrum of the inner-valence band of HCl/DCl atBL I411 at MAX-lab of 10 meV within reach, or are the characteristics of beamlineBUS-SGM U125 at BESSY II significantly favourable?

1. The so called Doppler broadening has to be considered, which is an ef-fect which is particularly important for light molecules such as HCL/DCl.The initial kinetic energy of the photoelectrons varies, since the atoms ormolecules from which they are ejected have a thermal motion. If the motionis totally random, this gives rise to an energy spread that has a Gaussiandistribution with a full width at half maximum (FWHM):

∆E = 0.722

√EkinT

M, (2.1)

where ∆E is given in meV, if the kinetic energy Ek of the electron is in eV,the temperature T is temperature in K, and the mass M of the molecule is inatomic units, a. u.[9]. Hereby, the Doppler broadening for the experimentsin Paper I with Ekin = hν − Ebin =(64-26) eV=38 eV, T=293 K, and

17

M=37 a. u. will be 13 meV. This is enough to obliterate the resolutionrequired for the expected outcome of the experiment. With hν=40 eV,the Doppler broadening is reduced to 7.6 meV. The main advantage withbeamline U125/2-SGM at BESSY [10] is the access to lower photon energies,down to 30 eV, whereas beamline I411 at MAX-lab provides photon energydown to 60 eV only, which would give a Doppler broadening of 12 meV.

2. The photon energy bandwidth, which is in the order of 8 meV at MAX-lab,is significantly better at BESSY II, 1 meV.

3. The Scienta software is updated at BESSY II, where a new feature is in-cluded, the Drift Region option. The DCl experiment in Paper III will betaken as an example. The region of interest is the inner-valence band ofDCl, which has a low cross section, and the band can not be seen on theCCD screen. Only after hours of recording and adding swept spectra, thestructure of the band is visible to the eye. By experience, the kinetic energyof an electron for a fixed binding energy is known to drift with time. Thisis due to drifting potentials in the interaction region, the surrounding sur-faces might not be perfectly grounded, etc. This problem can be solved bysweeping a second region (called Drift Region in the software), of an intensepeak, in the present case the peak corresponding to X-state (2Π3/2) v′ = 0,in the outer valence band for HCl/DCl. The cross section for ionizationof this state is orders of magnitude larger than for the inner-valence bandstates and can be seen immediately on the CCD screen. The kinetic energyof the intense peak is chosen, here (40-12.74) eV=27.26 eV. The two regionsare swept consecutively, and the software corrects for possible drifting of thekinetic energy of both regions automatically. In the present experiment adrift of 20 meV over a few hours was noticed.

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Chapter 3

Molecular physics

In Chapter 2 the importance of experiments for a validation of theory was men-tioned. However, it works in the other direction as well. An experimental obser-vation is sterile without a proper theoretical model, which puts the observationin a more general context. The scientific projects presented in this Thesis weredeveloped in close collaboration with different theoretical groups.

This Chapter is disposed as follows:

• An introduction of light can be found in Section 3.1,

• An overview of the electronic structure of atoms is given in Section 3.2.1,

• the electronic structure of diatomic molecules in general and HCl in partic-ular is presentedd in Section 3.2.2,

• the core of this Thesis, which is an overview of the diabatic/adiatic frame-works, which lead to the Born-Oppenheimer approximation is given in Sec-tion 3.3,

• a fundamental tool for the understanding of electron spectra of molecules isthe Franck-Condon projection concept, presented in Section 3.4,

• When electronic states interfere, they give rise to Fano-profiles, which willbe discussed briefly in Section 3.5,

• an overall view of the literature concerning the HCl molecule is presented inSection 3.6.

3.1 Light

Light, when described in the classical (i. e. pre-quantum mechanical) sense ofJ. Clerk Maxwell, can be considered to be an electromagnetic (EM) wave. En-ergy is transferred continuously. In contrast to this classical electrodynamicalview, Quantum Electrodynamics describes electromagnetic interactions in terms ofmassless elementary ”particles” known as photons. Both views have raison d′etre,

19

20

depending on which experimental observation one discusses. For geometrical op-tics, such as the understanding of phenomena interference and diffraction of lightthrough macroscopic slits, the classical view is sufficient. This is due to the factthat the energy transported by a large number of photons is, on the average,equivalent to the energy transferred by a classical EM wave.

The photoelectric effect however, which A. Einstein explained in 1905, as wellas other numerous experimental observations, can not be explained by the classicalwave view. Instead the interpretation of the observations require a quantizationof the EM field for the interaction between light and matter, and the quanta arecalled photons. Photons are stable, chargeless, massless elementary particles thatexist only at the speed of light, c. Moreover, photons are bosons, they are spin-1particles. An arbitrary large number of bosons can be in the same state, contraryto fermions, see Section 3.2.

Each photon has an energy given by the product of Planck’s Constant and thefrequency of the radiation field:

E = hν. (3.1)

When many photon progress with the same state, with the same momentum and

L k, p-state

L k, p-state

Figure 3.1: Angular momentum, L, linear momentum, p, and wave vector, k, of aphoton.

direction, they form a monochromatic plane wave. According to the quantum-mechanical description, a wave transfers energy in quantized packets, i. e. photonssuch that E = hν, and spin angular momentum L of a photon which is either −1or +1. The signs indicate right- (L-state), and left-handedness (R-state), respec-tively. In Fig. 3.1, the two alternatives are shown. The angular momentum of aphoton is completely independent of its energy and linear momentum. Whenevera charged particle emits or absorbs electromagnetic radiation, along with changesin its energy and linear momentum, it will undergo a change of ±1 in its angularmomentum.

If all photons of an EM wave are right- or left-handed, then the light is purelyright- or left-circularly polarized. A purely left-circularly polarized plane wave

21

will impart angular momentum to the target as if all the constituent photonsin the beam had their spins aligned in the direction of propagation. Now, howcan linearly polarized light be explained, which is the case for BL I411 [5], seeSection 2.2? Here each individual photon exists in either spin state with equallikelihood [11].

3.2 Atoms, molecules and their electronic struc-

ture

This Section aims at giving an understanding of the electronic structure of HClneeded for the discussion about the results presented in the Papers and in thelatter part of this Thesis. How can a complicated system composed of two nucleiand 18 electrons be treated in a comprehensive way? First, the electronic struc-ture of a simpler system, namely an atom, is discussed. The results will then beimplemented on the diatomic system HCl.

3.2.1 The electronic structure of atoms

The simplest atom is the hydrogen atom with one electron orbiting around oneproton. The characteristics of the electron can be described by a set of quantumnumbers. n=1, 2,... defines the shell. Within the shell, there are n subshells withquantum numbers l=0, 1, 2, (n-1). l is the orbital angular momentum quantumnumber and gives the orbital angular momentum when multiplied by h = h/2π.When writing the quanta of l, one uses the order s, p, d, f for increasing angularmomentum. ml is the quantization of the orbital angular momentum projectedon to a reference axis (by convention the z-axis), and is restricted to the values 0,±1, ±2, ...,±l. A way to show the shape of the electronic orbitals is to show theboundary surface, which is defined as 90% of the probability is within the surface.Typical examples of boundary surfaces are shown in Fig. 3.2. The p-state for agiven n-value is triply degenerate, with three values for ml (0,±1), where ml=0 isassociated with the pz-orbital,while a linear combination of ml=±1 are associatedwith the px and py orbitals.

Besides orbital angular momentum, the electron exhibits a second property offundamental importance for understanding the energy levels of electrons, namelythe spin angular momentum. The spin momentum quantum number for an elec-tron is 1

2 and the projection of the electron spin along the magnetic field directionis ms = ± 1

2 . The sum of the orbit and spin momenta is called j, l + s = j, andthe projection of the electron of the sum of orbit and spin moments is mj.

So far, only the one-electron atom has been considered. When filling up theshells in a multielectron atom, certain restrictions have to be considered. ThePauli principle restricts the system to allow only one electron for a given set ofquantum numbers. Hund’s rules describe the building-up order for a given sub-shell for minimization of the energy of the total system. For a given many-electronatom, these rules will determine which atomic orbitals are populated.

When considering multielectron atoms, l, s and j are replaced by the vectorsum of all the individual electron moments and designated L, S and J . The

22

s

px py pz

xy

z

x

y

z

xy

z z

yx

Figure 3.2: Boundary surfaces for s- and p-orbitals.

corresponding atomic state with L=0, 1, 2, 3, is designated a S, P , D, F ,...in analogy with the individual angular momenta of the electrons. The state ofa many-electron atom is expressed by a term symbol by using these quantities:2S+1LJ . The projections of the orbital- and spin-components are denoted ML,MS and MJ .

3.2.2 Electronic structure of HCl

Atoms form matter ranging from simple diatomic molecules to macroscopic com-plex units. This Thesis deals with the understanding of the diatomic moleculehydrogen chloride (HCl) and different inherent degrees of freedom which are re-lated to the electronic structure in this system.

In Fig. 3.3, the atomic orbitals (AO) are presented for the chlorine (Cl) andhydrogen (H) atoms. The orbital scheme in Fig. 3.3 is simplified, since only thequantum numbers n (1, 2, and 3 for Cl; 1 for H) and l (0:s, 1:p) are considered forthe atoms. The spin will be considered later in this Section. When considering adiatomic molecule such as HCl, the atomic orbitals for the free atoms (H and Cl)can be used as a basis set for a linear combination, which gives the molecular or-bitals. This approach is called the linear combination of atomic orbitals-molecularorbitals (LCAO-MO). It has been shown to be powerful as a tool for calculationsof molecular orbitals. A schematic view of the energy levels, and how they changebetween H and Cl atoms and the HCl molecule is shown in Fig. 3.3.

The spherical symmetry of an atom is broken in a diatomic molecule. The

23

electrostatic force field is cylindrically symmetric with respect to the internuclearaxis. For one electron in a diatomic molecule, a precession of l takes place aboutthe axis with constant component ml(h), where ml can take only the values

ml = l, l − 1, l − 2, ...,−l. (3.2)

l is projected onto the internuclear axis, the projection ml is denoted λ. Thecorresponding molecular states to λ=0, 1, 2, 3, ..., are designated σ, π, δ, φ,... states, analogous to the mode of designation for atoms. Electron orbitalswith rotational symmetry around the internuclear axis (z-axis by convention) aredenoted σ-orbitals. s-orbitals for atoms form σ-orbitals for diatomic molecules,and so do pz-orbitals. Hereby, the degeneracy of the p-orbitals for atomic caseis removed for the molecular case. The px,y orbitals are perpendicular to each

Cl HCl H

2840

1320

826

1727

7

3p 2π

3s

2s

1s

2p

5σ

4σ

3σ,1π

2σ

6σ*

0

1s

1σ

Cl3pz+H1s(LUMO)

Cl3px,y(HOMO)

Cl3pz+H1s

Cl3s+H1s

Cl2p

Cl2s

Cl1s

AO MO AOcomposition

Vale

nce

elec

trons

Cor

eel

ectro

ns

Ebi

n(eV

)

Figure 3.3: Schematic presentation of atomic orbitals, AO, and their linear combinationsforming molecular orbitals, MOs in the HCl molecule. The composition of the MOs ofAOs is also indicated. The solid (dashed) lines represent occupied (unoccupied) electronicorbitals. The coupling between the orbital and spin momenta of the electrons is notincluded in this description.

other and the z-axis, forming propeller-like probability densities. These electronorbitals are denoted π-orbitals. An example is the highest occupied molecularorbital (HOMO) in HCl, 2π. For HCl, the 2π electron orbitals are non-bondingsince they are localized around the Cl atom, and no orbitals of the same symmetry

24

exist around the H atom for any molecular electronic orbital formation. Electronorbitals around an atom without any bonding/antibonding character to neighboursare denoted lone-pair orbitals.

The HCl molecule has a (1σ)2(2σ)2(3σ)2(1π)4(4σ)2(5σ)2(2π)4 electronic con-figuration in the neutral ground state. An electronic configuration is the setup ofoccupied orbitals of the molecule. The superscript numbers give the number ofelectrons in each molecular orbital. When adding the numbers together one gets18 electrons for the HCl molecule, as expected.

Λ=1

Ω=3/2

Σ=+1/2

(a)

Ω=1/2 Σ=-1/2

2Π3/2

2Π1/2

2Π3/22Π1/2

(b)

Ebi

n

Λ=1

Figure 3.4: (a) Vector diagrams and (b) Energy Level diagram for a 2Π State (Λ = 1,S = 1

2). Σ is obtained by projecting S onto the intermolecular axis. The total electronic

angular momentum about the internuclear axis, denoted Ω, is obtained by adding Λand Σ. In (b), to the left the term is drawn without taking the interaction of Λ and Σinto account; to the right, taking account of it. The X-state in HCl+ is composed by adoublet which can be described in these terms.

When the electronic configuration of an atom/molecule is different from theground state, then one often uses the convention to write only the orbitals whichare changed. E. g. if one electron is missing in the HOMO, it is sufficient to write(2π)−1. When two electrons are missing in the 5σ orbital, and one electron isadded to the LUMO, then one writes (5σ)−2(6σ∗)1, where the ∗ marks the anti-bonding character of the orbital. The electrons in molecules with larger bindingenergies have orbitals which are localized around one atom, and the energy levelsresemble the case of the electronic orbitals in atoms. These electrons are calledcore-electrons. In Fig. 3.3, the lowest lying MO orbitals 1σ, 2σ, 3σ and 1π arecore-electrons, and the binding energy levels remain approximately the same asfor the Cl atom.

So far only the orbital angular momenta of the electrons have been consideredfor the HCl molecule. Now the intrinsic spin of the electron will be regarded. Forone electron, the projection of the spin resultant s onto the internuclear axis, witha constant component ms(h) = ± 1

2 , is denoted σ. (this quantum number mustnot be confused with the symbol σ for terms with λ = 0). The total electronicangular momentum about the internuclear axis, denoted ω, (or mj , which will beused in this Thesis), is obtained by adding λ and σ, just as the total electronic

25

angular momentum j for atoms is obtained by adding l and s. Whereas, however, avector addition has to be carried out for atoms, for molecules an algebraic additionis sufficient, since the vectors λ and σ both lie along the line joining the nuclei.Equivalent annotations are given for a multielectron diatomic atom, where thelower-case letters above are substituted by capital letters. An example is given inFig. 3.4, where the (2π)−1 X-state is described. Just as for atoms, the multiplicity2S+1 is added to the term symbol as a left superscript. Furthermore the valueof Λ+Σ is added as a subscript (similar to J for atoms). The example in Fig. 3.4deals with a 2Π term whose components are designated 2Π3/2 and 2Π1/2.

3.3 Molecular dynamics

This Section aims at giving a better understanding of molecular dynamics. I havechosen a two-level system in the inner-valence region of HCl+ as a showcase. Twoelectronic configurations (4σ)−1 and (5σ)−2(6σ∗)1 are associated with diabaticpotential energy curves, see below. The molecular symmetry is the same for theelectronic states, and the degeneracy of states of the same symmetry is prohibitedfor any internuclear distance. The electronic states are coupled, i. e. they aredependent of each other. Mixing of electronic configurations is possible, whichresults in configuration interaction (CI). For the coupled system, the resultingpotential energy curves repel each other, which results in an avoided crossing. Theconsequence is adiabatic potential curves. To obtain overview, I have neglectedother electronic configurations in the discussion, and the matter is certainly morecomplicated than in this showcase indicates. In Table 3.1 (see Section 3.3.5) thedefinitions for the adiabatic and diabatic frameworks are provided for guidance.The main reference for this Section is Ref. [12]. The two-level treatment wasimplemented on electronic configurations and adiabatic energy curves which stemfrom Ref. [13], since they play a central role for the understanding of the resultspresented in Papers I, II and III. Further reading can be found in Refs. [14, 15, 16].More recent theoretical work done in the field can be found in Refs. [17, 18, 19,20, 21].

3.3.1 The Schrodinger equation

The Schrodinger equationHψi = Eiψi, (3.3)

describes a quantum system, in this case an HCl+ cation. The total energy Ei ofthe system is calculated by applying the Hamilton operator H to the wave functionψi. The HCl+ molecular ion is composed of two atoms having atomic numbersZ1=1 and Z2=17. The Cartesian coordinates and conjugate momenta for the Nel

electrons are denoted ra and pa, respectively. For the nuclei, R1,2 and P1,2 areused. When disregarding the electron spin, the Hamilton operator has the generalform

Hmol = Tnuc + Vnuc−nuc + Tel + Vel−nuc + Vel−el. (3.4)

26

here the kinetic energy of the nuclei is

T ≡ Tnuc =P 2

1

2M1+

P 22

2M2(3.5)

with M1,2 being the masses of the first and second nuclei. The kinetic energy ofthe electrons is given by

Tel =Nel∑a=1

p2a

2mel, (3.6)

where mel is the electron mass and p = ih∇. Since both kinds of particles arecharged they interact via Coulomb forces. The repulsive Coulomb pair interactionbetween the electrons is

Vel−el =12

∑a=b

e2

|ra − rb| (3.7)

(note the factor 12 , which compensates for double counting) and between the nuclei

(note V ≡ Vnuc−nuc, which will be used in the following)

V ≡ Vnuc−nuc =Z1Z2e

2

|R1 − R2| . (3.8)

The attractive interaction between electrons and nuclei is given by

Vel−nuc = −(∑

a

Z1e2

|ra − R1| +∑

a

Z2e2

|ra − R2|

). (3.9)

All quantum mechanical information about the stationary properties of the molec-ular system defined so far is contained in the solutions of the time-independentnonrelativistic Schrodinger equation

Hmolψi(r, R) = Eiψi(r, R). (3.10)

Here and in the following, the set of electronic Cartesian coordinates are combinedin the multi-indices r = (r1, r2, ..., rNel

). A similar notation is introduced for thenuclear Cartesian coordinates R = (R1, R2).

As it stands, Eq. (3.10) does not tell much about what one is aiming at, namelyelectronic excitation spectra, equilibrium geometries etc. However, some generalpoints can be made immediately:

1. The solution of Eq. (3.10) will provide an energy spectrum Ei and corre-sponding eigenfunctions ψi(r, R). The energetically lowest state E0 is de-noted the ground state. In the following, the more formal notation wherethe eigenstates of the molecular Hamiltonian are denoted by the state vector|ψi〉 will also be used. The wave function is obtained by switching to the(r, R) representation: 〈r, R|ψi〉.

2. The probability distribution |ψi(r, R)|2 contains information on the distri-bution of electrons as well as on the arrangement of the nuclei.

27

3. In the Hamilton operator in Eq. (3.4), the intrinsic spin of the electronsand nuclei have been neglected, since they are not needed for the discussionconcerning the molecular dynamics in the following sections and related Pa-pers I, II and III. However, in many cases, a Hamilton operator with theelectron and even nuclear spin included is necessary for a description of themolecular system. Paper VI is an obvious case, where the consideration ofelectron spin is central for the understanding of the experimental results.

3.3.2 The diabatic framework

In the following the molecular system HCl+, with two electronic configurations(4σ)−1 and (5σ)−2(6σ∗)1, (see Section 3.2), will be discussed. How would onetreat that system in the context of the Schrodinger equation? I will start with adiabatic, static approach, based on a single configuration model.

Only the electronic configuration (4σ)−1 is considered initially, and the nucleiare fixed at some point R(0). In Fig. 3.5, R(0) has been chosen at the minimumof the diabatic curve Y1. (In principle, any other R(0) can be chosen.) Thissimplifies the Schrodinger equation considerably, since the nuclear dynamics in theHamiltonian in Eq. (3.4) can be neglected. One has to consider only the electronicSchrodinger equation, from which the eigenstate ϕ1(r; R(0)) and eigenvalue ε1 canbe calculated:

HUel (R

(0))ϕ1(r; R(0)) = ε1ϕ1(r; R(0)), (3.11)

where HUel (R

(0)) is the electronic Hamiltonian for the uncoupled diabatic electronicstate at R = R(0). The superscript U in HU

el (R(0)) stands for Uncoupled, and

the reason for the notation will be explained below. The notation in ϕ1(r; R(0))reflects the parametric dependency of the wavefunction on the choice of R(0), dueto the operator HU

el (R(0)). This is the meaning the notation throughout the text.

The same treatment can be done for the electronic configuration (5σ)−2(6σ∗)1, forwhich the energy ε2 and the eigenfunction ϕ2(r; R(0)) are obtained. The electronicstates ϕi(r; R(0)) are only eigenfunctions of HU

el at R(0). A second fundamentalrestriction is that the electronic states have to be independent of each other: Thesymmetries of the states have to be different. If one wants to define ψ(r; R) atother internuclear distances than R(0), and the symmetry of the states is the same,the following ansatz can be made for the Hamiltonian:

Hel(R) = HUel (R

(0)) + V (R, R(0)) (3.12)

where V (which will result in the coupling of the diabatic states) is:

V (R, R(0)) = Hel(R) − HUel (R

(0)). (3.13)

Two eigenstates ϕ1(r; R(0)) and ϕ2(r; R(0)) are known for HUel . These states form

a two-dimensional orthonormal basis for R = R(0) for independent, uncoupledstates. They correspond to energies ε1 and ε2 at R(0). The states of the coupledsystem differ only slightly from those of the uncoupled system represented by theuncoupled diabatic electronic states ϕi, and one can hope to solve the equationfor the complete molecular system:

Hmolψ(r, R) = Eψ(r, R) (3.14)

28

Ene

rgy

Eavoidedcrossing

R(0) RC RV12

Θ12

U1

U2

Y1

Y2

2VC

Figure 3.5: Schematic figure of an avoided crossing for a diatomic molecule with a two-level electronic state system. The diabatic potential energy curves Yi corresponding toa fixed electronic configuration described by wave function ϕi(r; R

(0)) have degeneratevalues for a given internuclear distance RC . R(0) is an internuclear distance where theuncoupled diabatic electronic states have well-defined eigenenergies εi. If the symmetryof the states is the same, a static coupling Vij between the states will make coefficientsai(R) of the molecular wave function ψ(r, R) = a1(R)ϕ1(r; R

(0))+a2(R)ϕ2(r; R(0)) switch

around RC . The adiabatic potential energy curves Ui correspond to electronic eigenstatesφi(r; R) for all R if R can be assumed to change so slow that the electrons always canadapt adiabatically to the nuclear configuration changes. However, if this adiabatic ap-proximation is broken, a dynamic coupling term Θ12 allows transitions between φ1(r;R)and φ2(r; R).

in terms of ϕi(r; R(0)) by writing

ψ(r, R) = a1(R)ϕ1(r; R(0)) + a2(R)ϕ2(r; R(0)) (3.15)

where a1(R) and a2(R) are expansion coefficients, corresponding to the nuclearwave functions. This is reasonable, since the complete system ψ(r, R) is composedof the electronic and nuclear wavefunctions ϕi(r; R(0)) and ai(R). Eq. (3.15) isthe origin of the well-known configuration interaction (CI) model.

Now one can determine the coefficients ai(R). Let the molecular Hamiltonian

Hmol = T + Vnuc + Hel(R)= T + Vnuc + HU

el (R(0)) + V (R, R(0)), (3.16)

where T ≡ Tnuc and the definition of the electronic Hamiltonian in Eq. (3.12)is used, act on the diabatic two-level system in Eq. (3.15). Multiplication of the

29

resulting Schrodinger equation by 〈ϕi| from the left and integration over the twoelectronic coordinates yields the following equation for the expansion coefficientsai (using the orthogonality of the uncoupled diabatic basis and ket notation)

〈ϕi(r; R(0))|Hmol|a1(R)ϕ1(r; R(0)) + a2(R)ϕ2(r; R(0))〉 =[Ti + εi + Vi]ai(R) + 〈ϕi|V (R, R(0))|ϕ1〉a1(R) + 〈ϕi|V (R, R(0))|ϕ2〉a2(R)= Eai(R). (3.17)

whereTi = 〈ϕi|T |ϕi〉 (3.18)

andVi = 〈ϕi|Vnuc|ϕi〉 (3.19)

N. b.: The index i for the kinetic and potential energy matrix elements for thenuclei Ti and Vi (see Eq. 3.5) is related to the index of the electronic state ϕi, notthe nuclei with coordinates Ra. Note that the off-diagonal terms

〈ϕi|HUel (R

(0))|ϕj〉 = HUel (R

(0))〈ϕi|ϕj〉 ≡ 0〈ϕi|T |ϕj〉 = T 〈ϕi|ϕj〉 ≡ 0

〈ϕi|Vnuc|ϕj〉 = Vnuc〈ϕi|ϕj〉 ≡ 0 (3.20)(3.21)

due to the orthogonality of the states |ϕi〉 Thus, a matrix equation for the coeffi-cients ai=1,2(R) is obtained:[(

T1 00 T2

)+(

ε1 00 ε2

)+(

V1 00 V2

)+

+(

V11 V12

V21 V22

)](a1(R)a2(R)

)= E

(a1(R)a2(R)

), (3.22)

with the terms:Vij = Vij(R, R(0)) = 〈ϕi|V (R, R(0))|ϕj〉 (3.23)

Note the difference between Vi and Vij : Vi is the potential energy for the nuclei(see Eq. (3.19)) for electronic state i, whereas Vij stems from Eq. (3.13). If oneintroduces the Hamiltonian for the motion of the nuclei in the electronic state |ϕi〉as

Hϕi = Ti + Yi(R) (3.24)

one getsYi = εi(R(0)) + Vi(R) + Vii(R, R(0)) (3.25)

which are the diabatic potential curves Yi in Fig. 3.5. The εi are the diabaticelectronic energies related to HU

el (R(0)) in Eq. (3.11). Eq. (3.22) then simplifies to:[(

T1 00 T2

)+(

Y1 00 Y2

)+(

0 V12

V21 0

)](a1(R)a2(R)

)= E

(a1(R)a2(R)

).

(3.26)

30

Now the result of the matrix elements Vij can be identified:

• Vii gives a contribution to the diabatic potential energy curve Yi, see Eq. (3.25).

• Vij ; i = j couples the diabatic electronic states. This is denoted the staticcoupling, since it stems from the potential energy operator in Eq. (3.13).

What is the relation between the diabatic electronic energies Yi and the eigenvaluesE in Eq. (3.26)? The condition for the existence of non-trivial solutions of thispair of equations is that the determinant of the coefficients of the constants a1(R)and a2(R) should disappear:∣∣∣∣ T1 + Y1 − E V12

V21 T2 + Y2 − E

∣∣∣∣ = 0 (3.27)

This condition is satisfied for the following values of E:

E± =12[(T1 + Y1) + (T2 + Y2)] ± 1

2

√[(T1 + Y1) − (T2 + Y2)]2 + V12V21 (3.28)

If one assumes fixed nuclei: T1 = T2 = 0, the only energy E, which remains inHmol is the potential energy, which for the coupled system is defined as U , i. e.U = E. Then Eq. (3.28) simplifies to

U1 = 12 (Y1 + Y2) + 1

2

√(Y1 − Y2)2 + 4V12V21

U2 = 12 (Y1 + Y2) − 1

2

√(Y1 − Y2)2 + 4V12V21

(3.29)

The corresponding potential energy curves are shown as Ui in Fig. 3.5. The en-ergy levels are driven apart, and their crossing is prevented, This is called anavoided crossing. A second general feature can also be seen in Fig. 3.5: the effectof the coupling is greater the smaller the energy separation of the uncoupled levels.For instance, when the two energies of the diabatic potential curves have the sameenergy (Y1 = Y2) at RC , then

U1 − U2 = 2VC (3.30)

whereVC ≡

√V12(RC)V21(RC). (3.31)

Eq. (3.29) also shows that the stronger the coupling Vij , the stronger the effectiverepulsion of the levels. When the is coupling absent, Vij = 0 and Ui = Yi, thepotential energy curves correspond to the non-coupled states.

The fact that the diabatic two-level system results in potential curves Ui whenthe static coupling Vij is implemented, might be confusing. Ui are denoted adi-abatic potential curves, which I come back to in the next Section. Adiabaticpotential energy curves can indeed be obtained, even in the diabatic framework,if the diabatic system is coupled. If influence of nuclear motion is neglected, i. e.Ti ≡ 0, the same result will be obtained in the diabatic framework as in the adi-abatic case: The system is well-defined on one adiabatic level solely, associatedwith a single Ui. The ansatz Ti ≡ 0 is hereby equivalent to the Born-Oppenheimerapproximation in the adiabatic framework.

31

3.3.3 The adiabatic framework

Now a second approach will be used, complementary to the diabatic one describedabove. It is based upon the fact that electrons move much faster than the nuclei dueto the large difference in mass (mel/Mnuc < 10−3). Here, the electronic degreesof freedom can be considered to respond instantaneously to any changes in thenuclear configuration, i. e. their wave function always corresponds to a stationarystate. In other words, the interaction between nuclei and electrons, Vel−nuc, ismodified due to the motion of the nuclei only adiabatically and does not causetransitions between different stationary electronic states. Thus, it is reasonableto define an electronic Hamiltonian which has a parametric dependence on thenuclear coordinates:

Hel(R) = Tel + Vel−nuc + Vel−el. (3.32)

As a consequence the solutions of the time-independent electronic Schrodingerequation describing the motion of the electrons in the electrostatic field of thestationary nuclei will parametrically depend on the set of nuclear coordinates:

Hel(R)φi(r; R) = Ei(R)φi(r; R) i = 1, 2, (3.33)

(note the that the same notation as above is used: φi(r; R)). Since the discussionhas been restricted to a two-level system, the index i, which labels the adiabaticelectronic states, is limited to 1 and 2 only. The adiabatic electronic wave func-tions φi(r; R) = 〈r; R|φi〉 define a two-level basis. Hence, given the solutions toEq. (3.33) the molecular wave function can be expanded in this basis set as follows

ψ(r; R) = χ1(R)φ1(r; R) + χ2(R)φ2(r; R). (3.34)

The expansion coefficients χi(R) depend on the configuration of the nuclei. It ispossible to derive an equation for their determination after inserting Eq. (3.34)into Eq. (3.3). One obtains

Hmolψ(r; R) = (Hel(R) + T + V )(χ1(R)φ1(r; R) + χ2(R)φ2(r; R))= [E(R) + V ](χ1(R)φ1(r; R) + χ2(R)φ2(r; R))

+T (χ1(R)φ1(r; R) + χ2(R)φ2(r; R))= E(χ1(R)φ1(r; R) + χ2(R)φ2(r; R)). (3.35)

Multiplication of Eq. (3.35) by φ∗i (r; R) from the left and integration over all

electronic coordinates yield the following equation for the expansion coefficientχi(R) (using the orthogonality of the adiabatic basis and ket notation)

〈φ∗i (r; R)|Hmol|χ1(R)φ1(r; R) + χ2(R)φ2(r; R)〉 =

= [Ei(R) + Vi]χi(R)

+∑

j=1,2

〈φi(r; R)|T |χj(R)φj(r; R)〉

= Eχi(R). (3.36)

32

Since the electronic wave functions depend on the nuclear coordinates, Eq. (3.5)is used, where Pa = −ih∇a, and using the product rule for differentiation (theunderlined expression is the same as in Eq. (3.36))

Tφj(r; R)χj(R) =

=∑

n=1,2

12Mn

︷ ︸︸ ︷[P 2

nφj(r; R)] χj(R)

+ 2 [Pnφj(r; R)]Pn︸ ︷︷ ︸χj(R)

+ φj(r; R)P 2nχj(R). (3.37)

The last term is simply the kinetic energy operator acting on χj. The other termscan be comprised into the so-called nonadiabaticity operator (the expressions hereand in Eq. (3.37) in the braces are related),

Θij = 〈φi(r; R)|︷ ︸︸ ︷T |φj(r; R)〉+

∑n=1,2

1Mn

〈φi(r; R)|Pn|φj(r; R)〉Pn︸ ︷︷ ︸ . (3.38)

Thus, from Eq. (3.36) one obtains an equation for the coefficient χi(R) which reads

(Ti + Ei(R) + Vi + Θii + Θij)χi(R) = Eχi(R). (3.39)

This result can be interpreted as the stationary Schrodinger equation for the mo-tion of the nuclei with the χi(R) being the respective wave functions. The solutionof Eq. (3.39) is still exact, and requires a knowledge of the electronic spectrum forall configurations of the nuclei which are covered during their motion.

It is convenient to introduce the following effective potential for nuclear motionif the electronic system is in its adiabatic state |φi〉:

Ui = Ei(R) + Vnuc i(R) + Θii. (3.40)

This function defines the adiabatic potential curves Ui=1,2 in Fig. (3.5). Eq. (3.39)can now for the two-level system be described as:[(

T1 00 T2

)+

(U1(R) 0

0 U2(R)

)+

(0 Θ12

Θ21 0

)](χ1(R)χ2(R)

)= E

(χ1(R)χ2(R)

),

(3.41)

Given the solutions to Eq. (3.41), χi,v′(R), labelled by the index v′ the molecularwave function is

ψv′(r, R) = χ1,v′(R)φ1(r; R) + χ2,v′(R)φ2(r; R). (3.42)

In analogy with the discussion about the operator V in the diabatic framework,the result of the operator Θ (see Eq. (3.38)) can now be identified:

• Θii gives a contribution to the adiabatic potential energy curve Ui, seeEq. (3.40).

• Θij (i = j), couples the adiabatic electronic states. It is the origin of thedynamic coupling.

33

Until now, no approximation has been implemented. Eq. (3.41) represents the com-plete solution for the system. Solving the coupled Eqs. (3.41) is often a formidabletask. To simplify the computations, approximations have to be made. Oftenthe nonadiabatic couplings Θij are rather small. The neglect of these couplings,i. e. Θij ≡ 0, is called the Born − Oppenheimer approximation. For this caseEq. (3.41) is completely diagonalized. If the system starts on one adiabatic energycurve, i. e. |χi〉 = 1 and |χj =i〉 = 0, then the system remains well-defined onthat energy curve. The solutions of Eq. (3.42) are again labelled by v′. The totaladiabatic wave function the becomes

ψi,v′(r, R) = χi,v′(R)φi(r; R). (3.43)

3.3.4 The Landau-Zener formula

The Landau-Zener formula, which has been used to qualitatively explain the resultsin Papers I, II and III, stems from the early 30’s, and was developed independentlyby Lev Landau and Clarence Zener. Landau considered a scattering of two atomswhereas Zener focused on the electronic levels of a diatomic molecule.

In the below discussion, the diabatic base with states ϕi is used (see Fig. 3.6).In Fig. 3.6, the Fi are the gradients of Yi at RC :

Fi = − δYi(R)δR

∣∣∣∣R=RC

. (3.44)

The potential curves can be expanded around RC :

Yi = Yi(RC) − Fi(RC)∆R, (3.45)

where ∆R = R − RC . The distance ∆R depends on the unknown velocity vC atthe crossing point and time:

∆R = vCt. (3.46)

Now the description of the system has an explicit time dependency. With theseapproximations, the Hamilton operator for the motion of the nuclei in the elec-tronic state |ϕi〉 (see Eq. (3.26)) can then be written:

Hi = Ti + Yi(RC) + FivCt + Vij , (3.47)

The two-level system in Eq. (3.26) then becomes:[(T1 00 T2

)+

(Y1(RC) 0

0 Y2(RC)

)+(

F1vCt 00 F2vCt

)+

(0 V12

V21 0

)](a1(R)a2(R)

)= E

(a1(R)a2(R)

)(3.48)

where ai(R) are the coefficients for the diabatic electronic states ϕi(r; R(0)):

ψ = a1(R)ϕ1(r; R(0)) + a2(R)ϕ2(r; R(0)) (3.49)

If the molecular system starts well-defined on the first diabatic state |ψ〉 = |ϕ1〉,i. e. |a1〉 = 1 and |a2〉 = 0, the interstate coupling V12 allows transitions to state

34

Ene

rgy

E

RRc

2VC

lLZ

F2(RC)

F1(RC)Y1

Y2

Figure 3.6: Two diabatic potential curves Yi (dashed lines) corresponding to diabaticstates |ϕi〉 as a function of internuclear distance R. The diabatic states are degenerate atan internuclear distance Rc. VC is the diabatic static coupling at RC . The Landau-Zenerlength lLZ is shown. Fi are the gradients of Yi at RC .

|ϕ2〉. The asymptotic value of the survival probability of the system for remainingin the state |ϕ1〉, P|ϕ1〉 ≡ P|ϕ1〉(t = ∞), is the square of the transition amplitude

Pϕ1 = |〈a1|〈ϕ1 |U(−∞,∞)| a1〉|ϕ1〉|2 , (3.50)

where the time-evolution operator U(t′, t) is given by F1vCt+V12. Interestingly, thepresent model allows calculation of this transition amplitude exactly, see Ref. [12]for details. Here, only the ϕ1 survival probability is quoted:

Pϕ1 = e−ξ, (3.51)

which is denoted the Landau-Zener formula. It depends on the so-called Masseyparameter ξ which is defined as

ξ =2π

vC

|V12|2|F1 − F2| . (3.52)

Introducing the Landau-Zener length

lLZ =2π |V12|2|F1 − F2| , (3.53)

the Massey parameter becomes

ξ = lLZ|V12|vc

. (3.54)

35

Now let us see what happens with a small velocity vC around the crossing: theMassey parameter ξ becomes large, and the survival probability Pϕ1 in Eq. (3.51)gets a value close to zero. The system leaks out to the dissociative diabatic elec-tronic state |ϕ2〉. With a large velocity vC , the system would get more blend fromthe diabatic bound state |ϕ1〉.

The Landau-Zener length lLZ can be understood as the distance from Rc,where the difference Y1−Y2 is equal to the electronic coupling V12: Y1 −Y2 = V12,see Fig. 3.6.

3.3.5 Adiabatic/diabatic framework: short repetition

The molecular dynamics section has so far been loaded with expressions. Here Iwill try to make a short summary in simple words without using any mathematicalexpressions.

The complete Schrodinger equation for a molecule contains all information(position and momentum) about all particles (electrons and nuclei) in the system.How does one describe this system in a transparent way?

The first, diabatic, approach assumes two diabatic non-coupled electronic statesassociated with two electronic configurations (4σ)−1 and (5σ)−2(6σ∗)1. The as-sociated diabatic potential curves Yi are degenerate at some internuclear distanceRC , where the potential curves cross. If the overall geometry of the molecular sys-tem is the same, the degeneracy at RC is forbidden. The potential energy curvesrepel, and the result is an avoided crossing. In the diabatic framework, this is dueto a static coupling term Vij . When the static coupling term Vij ≡ 0, which is thecase for diabatic electronic states with different symmetries, the diabatic potentialcurves Yi are sufficient for the description of the system.

The second, adiabatic, approach is based upon the assumption that the elec-trons move infinitely fast as compared to the nuclei due to their lower mass. Forthe adiabatic electronic states, the electrons adapt instantaneously to the nuclearconfiguration changes. The adiabatic potential curves associated with the elec-tronic states repel each other if the symmetry of the states is the same. Therepelling results in an avoided crossing. If the nuclei move fast enough, one hasdynamic coupling between adiabatic states. Note that in the diabatic case, thestatic coupling results in repelling of the electronic states. On the contrary, forthe adiabatic case, the dynamic coupling results in mixing of the electronic states.In Table 3.1, an overview of the two complementary approaches is shown.

Both frameworks give the same result in complementary ways. If the nuclearmotion can be neglected, both approaches give the same avoided crossing aroundRC . In the diabatic case, if a static coupling between originally non-coupleddiabatic curves is implemented, the same adiabatic curves are obtained as in theadiabatic case, where the dynamic coupling between adiabatic curves is neglected.

Why is there a need for two complementary frameworks? The system in con-sideration determines the framework which gives the ”cheapest” representation.If the static coupling Vij is strong, and the dynamic coupling Θij is small, thenthe adiabatic representation is favourable. The system can be described with twonon-coupled Schrodinger equations, and the two-level Eq. (3.41) can be simpli-fied in two separate expressions. On the other hand, if the static coupling Vij is

36

Table 3.1: An overview of the diabatic and adiabatic framework. Index i = 1, 2for a two-level system. Some general definitions are shown as well.Diabatic AdiabaticWavefunction for el. state:ϕ ≡ ϕ(r; R) φ ≡ φ(r; R)Wavefunction for nuclear state (acts as coefficient to corresponding el. state):a ≡ a(R) χ ≡ χ(R)Electronic energy:ε ≡ ε(R(0)) E ≡ E(R)Effective potential energy for the nuclei (Potential energy curves):Yi ≡ Yi(R) Ui ≡ Yi(R)Coupling between electronic states:Vij ≡ Vij(R, R(0)); i = j Θij ≡ Θij(R); i = jCoupling at internuclear distance RC :VC ≡

√V12(RC , R(0))V21(RC , R(0))

Contribution to potential curves:Vii ≡ Vii(R, R(0)) Θii ≡ Θii(R)Approximation implemented for obtaining potential curves:Vij ≡ 0; i = j Θij ≡ 0; i = jBorn-Oppenheimer approximation:Ti ≡ 0 Θij ≡ 0; i = j

General definitionsψi(r, R) ≡ 〈r, R|ψi〉 Mol. w. f. obtained by projecting

the state vector onto (r,R) representationHmol Hamiltonian for moleculeHel ≡ Hel(R) Hamiltonian for electronsV (R, R(0)) = Hel(R) − H(0)(R(0)) ”Perturbation”H(0) ≡ Hel(R(0)) Hamiltonian for el. at fixed R(0)

E Energy for total systemVel−nuc Pot. energy between el.-nucleiVel−el Pot. energy between el.Vi ≡ Vnuc−nuc,i Pot. energy between nuclei for el. state iTel Kin. energy op. for el.Ti ≡ Tnuc i Kin. energy op. for nuclei for el. state i

F = − δYi(R)δR

∣∣∣R=RC

Gradients of Y at R = RC

ra, pa Position and momentum of el. aRa, Pa Position and momentum of nucleus ar ≡ (r1, r2, ..., rNel

) Multi-indices of pa

R ≡ R1 − R2 Internuclear distanceRC Internuclear distance where Y1 = Y2

37

weak, and the dynamic coupling Θij is strong, the diabatic framework will simplifyEq. (3.22) in two non-coupled expressions.

The term adiabatic stems from thermodynamics. In a science dictionary, theword is explained as without loss or gain of heat. In a small quantum mechanicalsystem, heat is not a relevant quantity, but can be understood in terms of thevelocity, for which the system develops. A thermodynamic system does not loseor gain heat to the surrounding during a sufficiently fast, adiabatic process. For aquantum mechanical system an adiabatic process, the electrons move sufficientlyfast compared to the nuclear motion for the wave function to remain on the adi-abatic potential curves, the thick lines in Fig. 3.5, corresponding to the adiabaticelectronic states |φi〉. The term diabatic can be understood as the negation ofadiabatic, where the electron/nuclear motion velocity ratio not is sufficiently largefor the system to remain on an adiabatic potential curve. Instead it keeps theelectronic configuration related to diabatic electronic state |ϕi〉, see Fig. 3.5.

A major goal in this Thesis has been to study the validity of the adiabaticand diabatic approximations in a non-trivial case. Papers I, II and III deal with aphenomenon observed in the inner-valence photoionization bands of HCl and DCl,where the understanding is based upon the insight that neither the diabatic northe adiabatic approximations are valid here. This is an intermediate case.

3.4 Franck-Condon projections

The potential energy of a given electronic wavefunction φ can be calculated fordifferent interatomic distances R for a diatomic molecule, see Section 3.3. The cal-culated potential energy can be plotted vs the distance R. Characteristic potentialenergy curve plots are shown in Fig. 3.7.

For some φi, the potential curves have local minima. Re is the equilibrium dis-tance corresponding to the minimum of the potential energy curve, around whichthe molecule vibrates. Other φ have no minima, and the potential energy decreasesmonotonously with R. These φ are associated with continuum states, where themolecule dissociates until the fragments no longer interact. When solving theSchrodinger equation for bound states, the allowed energies are distributed non-continuously. For continuum states, all energies are allowed. The nuclear wavefunction χ characteristics for the associated states are shown on the potentialcurves in Fig. 3.7. The indexes v′ = m are the same as in Eq. (3.43). For theground electronic state φg, for small molecular systems as HCl, the only energylevel populated at room temperature is the lowest lying nuclear state χ v = 0.Since the molecule vibrates around Re, the nuclear wave function χ for the groundstate will be distributed around Re. Assuming a harmonic oscillator potential,which is a good approximation for the lower part of a bound potential curve, thewave function will look like a Gaussian for the lowest lying state, as shown inFig. 3.7. At the two classical limits of the wave function, where the kinetic energypart is zero, one can assume that the wave function is so small as to be negligibleoutside the classical limits. (Even though an exponential tail is allowed outsideof the classical limits, due to quantum mechanical arguments.) One way to de-fine the Franck-Condon region, shown in Fig. 3.7, which is used for showing the

38

Re R

Groundneutral state

Finalcation states

Franck-Condonregion

0B

indi

ngen

ergy

Pot

entia

lene

rgy

Fran

ck-C

ondo

nfa

ctor

spec

trum

φg

φ1

φ2

χ v’=0χ v’=1χ v’=2

χ v=0

Figure 3.7: Potential energy curves corresponding to the electronic wavefunctions φi forthe ground state of the neutral molecule and two states of the ionized molecule. For thebound final state, three nuclear wavefunctions χ v′ = m are pointed out with indeces.By convention, the nuclear wavefunctions have indeces v for the electronic ground state,and v′ for electronic ionized states.

ground state extension, is to use the classical limits. When exciting the molecularsystem, using light, electrons or some other external energy source, the changeof electron configuration is assumed to occur instantaneously, which is referred toas the ”sudden approximation”. R can be assumed to be fixed during excitation.This is called the Franck-Condon principle.

In a molecule, the electric dipole moment operator µ depends on the config-urations of the electrons and nuclei, represented by the separated wavefunctionsφ and χ. The separation of the wavefunctions is permitted within the Born-Oppenheimer approximation. The transition moment between initial and finalstates can be written as

〈φiχv′=m |µ|φgχv=0〉 = µφiφg〈χv′=m|χv=0〉 = µφiφgS(χv′=m, χv=0) (3.55)

where µφiφg is a constant which stem from the operator µ working on the electronicstates φ, and

S(χv′=m, χv=0) =∫

χ∗v′=m(R)χv=0(R)dR (3.56)

is the overlap integral between the two nuclear states in their respective electronicstates. The transition dipole moment is therefore largest between nuclear statesthat have the greatest overlap. The relative intensities of the lines are proportional

39

to the square of the transition dipole moments and hence to the Franck-Condonfactors, |S(χv′=m, χv=0)|2. When plotting the Franck-Condon factor intensities vsthe energy difference between the ground state and the final state, one gets theFranck-Condon factor spectrum, which is shown in the left part of Fig. 3.7. Bycomparing this spectrum with the one obtained with the correlation diagram inFig. 3.3 (see Section 3.2), one gets more information, since one now knows whetherfinal cation state φ is associated with bound or continuum states χ.

3.5 Fano resonances

If the Eqs. (3.22) and (3.41) are regarded, what happens to the energy E observablewhen two states are involved? The amplitudes of both states add and give rise tointerferences that produce the characteristic Fano shape seen in the cross-sectionand the related observables [22]. For the case in Papers I, II and III, isolated

0

-8 -6 -4 -2 0 2 4 6 8Ered (dim. less)

q=3

q=0

q=2

q=1

1+E

red2

(Ere

d+q)

2

24

68

10

(dim

.les

s)

q=-3

Figure 3.8: Line profiles of a Fano resonance for different values of the q-parameter. Inthe special case of q=0, there is no maximum, but a window resonance.

discrete states interacting with one continnum of a different electronic state, theabsorption cross-section σ can be expressed as [22, 23]:

σ ∝ (q + Ered)2

1 + E2red

(3.57)

where q is called the Fano parameter and

Ered =E − Er

Γ(3.58)

where Er is the resonance position, Γ is the resonance width related to the lifetimeτ of the bound state in the Heisenberg uncertainty formula

Γ =h

τ. (3.59)

Some typical Fano profiles are shown in Fig. 3.8.

40

3.6 Hydrogen chloride

Hydrogen chloride, HCl, the molecule which is studied in this Thesis is a colourlessgas. It is used in the industry for chlorination of unsaturated organic compounds,e. g. chloroethene in polymerization, for isomerization and for alkylation, as acatalyst. By dissolving HCl in water, hydrochloric acid is produced.

An acid in its most general form is a proton donor. In the reaction formula

HCl + H2O Cl− + H3O+, (3.60)

HCl is the acid which donates the proton to the base (which is a proton acceptor)H2O. On the right side, H3O+ is the acid to the base Cl−. Due to the highly reac-tive positively charged protons in the solution, it dissolves many metals, formingchlorides and releasing hydrogen. We all carry around strong hydrochloric acid(Ph-value ≈ 2) in our bellies, which dissolves the long sugar- and fatchains, andkills bacteria, in our food.

Inte

nsity

(arb

.uni

ts)

35 30 25 20 15Binding energy (eV)

x0.2

X-state(2π)-1

A-state(5σ)-1

B-state"(4σ)-1"

HCl valence-band spectrum

Innervalence

Outervalence

Figure 3.9: Photoelectron valence band spectrum of HCl.

Gas-phase HCl bottles for experiments can be bought commercially. Since thisThesis deals with dynamical effects in the dissociation process of the molecules, itis of high interest to compare the case when hydrogen is replaced by deuterium,the heavy isotope with twice the mass of hydrogen (proton+neutron). However,deuterium chloride, DCl, can not easily be bought. The group has chosen toproduce it in the chemistry laboratory. The reaction used is:

2NaCl + D2SO4 → 2DClgas + Na2SO4, (3.61)

41

where liquid deuterated sulphuric acid, D2SO4, is dropped on chemically cleanNaCl, sodium cloride, table salt. Here, NaCl is the base (proton acceptor) andD2SO4 the acid (proton donor). The arrow to the left in formula (3.61) is miss-ing, since DCl is a gas, and therefore not available for any backward reaction.Reaction (3.61) is of course equivalent when changing deuterium to hydrogen. Aphotoelectron valence band spectrum of HCl, obtained at BL I411 at MAX-labis shown in Fig. 3.9. The outer valence bands X- and A-states are associatedwith two bound states (see the discussion about Franck-Condon projections inSection 3.4), whereas the inner valence bands is a much more complicated matter,with various bound and continuum states. The HCl molecule has been extensivelystudied in the electron spectroscopy community. The setup used in this Thesisis only one of a large variety. Below, a short overview of the work being doneover the last decades is presented, with connections to the work presented in thisThesis.

The Aksela group in Oulo, Finland has contributed a great deal to the liter-ature. Their main concern has been excitation, and ionization of the Cl 2p coreelectrons. Molecular fragmentation patterns after 2p → 6σ∗/Rydberg orbitals ex-citation have been observed in Refs. [24], using a PEPIPICO spectrometer (thesame type of spectrometer setup as in Paper VII). Normal and resonant Augerdecay has been studied using photoelectron spectroscopy in Refs. [25, 26, 27, 28,29, 30, 31, 32], where information about Auger decay channels and angular distri-butions for photo- and Auger electrons has been obtained. The theoretical studiesin Refs. [33, 34] concern the orientation of the 2p electron core holes, and their in-fluence on Auger decay channels. This work is of major importance for Paper VI,discussed in Section 3.7.4.

Uwe Hergenhahn and Uwe Becker at BESSY I in Berlin, Germany have per-formed resonant Auger angular-distribution measurements around the HCl 2p →6σ∗ resonance, using synchrotron radiation together with a time-of-flight electronspectrometer [35]. Andrew J. Yencha and co-workers from New York have spe-cialized in threshold photoelectron spectroscopy, TPES. For the experiment, thesynchrotron photon energy is swept, while only the zero-kinetic energy electrons(in practice <∼ 40meV) are detected. Whereas conventional photoelectron spect-roscopy generally involves only direct transitions from the ground state which arewell-described by Franck-Condon factors, TPES provides information about bothdirect and indirect (non-Franck-Condon controlled) ionization processes. The lat-ter processes involve autoionization of initially formed superexcited Rydberg statesproduced in a photoabsorption process.

When performing TPES on the HCl and DCl valence bands, many vibrationalprogressions that are not seen in conventional PES appear due to the auto-ionizingRydberg states. The vibrational progression in Paper VI (Section 3.7.4), wherethe X-state vibrational levels are populated to very high levels after exciting the2p → 6σ∗ resonance, can be seen in TPES as well [36, 37]. Here, auto-ionizationis responsible for the population of the higher vibrational states in the X-state.For the ”(4σ)−1” band in the inner-valence region, where this Thesis presents theobservation of an interference pattern between a broad continuum band and avibrational progression (see Fig. 3.11), TPES shows both the the continuum bandand the vibrational progression [36, 37]. The inner-valence spectra in Ref. [37] are

42

reused in Paper IV in this Thesis, where the differences between TPES and PESspectra for the same molecular systems (HCl and DCl) are discussed.

When discussing TPES in the literature, the work of H. Frohlich et al should bementioned too. They performed the first TPES made on HCl, observing rotationalstructure of bands 2Π3/2 and 2Π3/2. Autoionization from Rydberg states with a2Π3/2 core into the 2Π3/2 continuum was observed.

A. J. Yencha has also participated in flourescence measurements on HCl, us-ing VUV lamps and Penning ionization (using meta-stable atoms as an ionizationsource) [38]. Ion-pair formation of HCl and DCl, (HCl+, H+ + Cl−) as a functionof photon energy using synchrotron light has been presented in Ref. [39]. A tun-able VUV light source, based upon lasers, was used with an electron time-of-flightspectrometer, to perform rotationally selective state-to-state photoelectron spec-troscopy in the autoionization region between the 2Π-ionization thresholds in theX-state of HCl [40]. A. J. Yencha has as well been dealing with spectroscopy abovethe double ionization threshold. In Ref. [41], two TPES spectrometers were usedin coincidence, using synchrotron radiation. Further reading on investigations ondouble ionization of HCl can be found in references in that Paper.

Helene Lefebvre-Brion et al. have performed both theoretical and experimentalresearch on auto-ionization on HCl and DCl. In Ref. [42], competition betweenautoionization and predissociation of HCl and DCl in the A-state is being dis-cussed. Auto-ionization between rotational levels in 2Π1/2 to continuum 2Π3/2

states is presented in Refs. [43]. Rotationally resolved spin-orbit autoionizationspectra are presented in Refs. [44, 45].

Rohana Liyanage, Robert J. Gordon, Robert W. Field et al. have studied pre-dissociation of electronically excited bound states, below the ionization threshold,of HCl and DCl [46, 47, 48]. Theoretical and experimental work has been per-formed. A pump- and probe laser was used for the experimental work. Theadiabatic/diabatic picture plays an important role for the understanding of thefragment branching ratios, just as in Papers I and II (Section 3.7.1).

The resonace-enhanced multiphoton ionization, REMPI, scheme, has been usedby E. de Beer, C. A. de Lange et al. Observations of the competition betweenelectronic autoionization and dissociation via superexcited states below the ion-ization threshold is presented in Ref. [49]. REMPI, and Zero-Electron KineticEnergy-Pulsed Field Ionization, ZEKE-PFI, setups were used in Ref. [50], whereRotationally resolved photoelectron spectra for HCl were obtained for the firsttime. D. L. Hansen, D. W. Lindle et al. have used a PEPIPICO spectrometer (thesame spectrometer setup as in Paper VII) for determination of decay channels forthe fragments when exciting HCl from the Cl K-shell [51, 52, 53]. Neutral H-atomemission was found to be a significant decay channel for excited states with veryshort lifetimes (1 fs).

P. Natalis, P. Pennetreau, L. Longton and J. E. Collin has performed photo-electron spectra on the outer valence bands X and A of HCl using Ne(I) and He(I)light [54, 55].

In Uppsala, electron spectroscopy has been performed on HCl and DCl, usingmonochromatized x-ray, ultraviolet, electron beam excitation and synchrotron ra-diation. The inner- and outer valence bands in HCl were presented and discussedin Refs. [56, 57, 58, 59]. The possibility to record the inner-valence spectra for

43

HCl and DCl with higher resolution has made Paper I (Section 3.7.1) possible.Resonant Auger decay concerning atomic vs molecular decay channels, dependingon the detuning around the 2p3/2 → σ∗ resonance, using synchrotron radiation, ispresented in Ref. [60]. Spectra around the same resonance was re-recorded, withsignificantly better statistics, since the photon flux is orders of magnitude higherat MAX II as compared to MAX I. Due to the better statistics, the continuum-continuum interference presented in Paper V could be recorded for HCl and DCl(see Section 3.7.1).

Theoretical work for calculations of potential curves and transition dipole mo-ments of HCl has been carried out by Atul D. Pradhan, Kate P. Kirby and A. Dal-garno [61]. These potential curves could have been used for the discussion inPaper I, see Fig. 3.11. Instead, the potential curves of Miyabi Hiyama and Sue-hiro Iwata [13] were chosen, because theoretical Franck-Condon projections (justas in Fig. 3.11) are provided, and comparison is made to old experimental datafrom the Uppsala group [57]. The continuum and discrete states, which interfere inPaper I, are shown in Franck-Condon projections. The potential curves providedby M. Hiyama were used as a starting point for Paper II, but coupling parameterswere tuned so that agreement between experiment and theory was improved fromreasonable to excellent, see Section 3.7.1.

Werner von Niessen, Lorenz S. Cederbaum et al. have computed pole strengths,probabilities for transitions to 1h, 2h1p, etc states for different energies in theinner-valence region of HCl [62]. Their work made it possible to identify thatthe 2h1p state associated to the vibrational progression B in Fig. 1 in Paper I(Section 3.7.1) is of (2π)−2(1δ)1 character. Meanwhile, there is disagreement con-cerning the character of the state corresponding to vibrational progression A be-tween W. von Niessen, who suggests (2π)−2(6σ∗)1 character, and M. Hiyama, whosuggests (5σ)−2(6σ∗)1 character.

Theoretical work done by Pawel Salek, Faris Gel’mukhanov and Hans Agren [63]and Zbigniew W. Gortel, Robert Teshima and Dietrich Menzel [64] predicted theexistence of interference between molecular and atomic Auger decay for a dissoci-ating molecule. Both Papers took HCl as a showcase. Elke Pahl, L. S. Cederbaumet al. contributed to the discussion, using HF as a showcase [65]. In Paper V, ex-perimental evidence for the existence for such an interference pattern was shown,see Section 3.7.3.

3.7 Summary of Papers

In Fig. 3.10, the potential energy curves for HCl, HCl+ and HCl*, related to thisThesis, are shown. When comparing with the correlation diagram in Fig. 3.3, theelectronic configurations (2π)−1, ”(4σ)−1”, and (2p)−1(6σ∗)1 are included. Theblack photoionization arrow to final state ”(4σ)−1” shows the direct photoelectronchannel, which is related to Papers I, II and III. Section 3.7.1 discusses a dynamiccoupling between two electronic states in that region. The quotation marks aroundthe electronic configuration ”(4σ)−1”, which is frequently used in the literature, isto show that there is more than one electronic configuration involved.

The direct channel, related to the black arrow in Fig. 3.10 contributes to the

44

H-Cl distance

Ene

rgy

(eV

)

ground state

(2π)-1

"(4σ)-1"

(2p)-1(6σ∗)1

Papers I, II, III, IV and V

Papers V, VI

Paper VI

Paper V

130

2620

2

1.27A

Figure 3.10: Potential energy curves for HCl related to the results presented in this The-sis. The excitation and deexcitation channels, related to Papers I, II, III, VIII, V and VIare pointed out.

effect shown in Paper V. Furthermore in Paper V, a second resonant channelwas opened, by tuning the synchrotron light to the resonance of (2p)−1(6σ∗)1.The dark grey arrow in Fig. 3.10 corresponds to the excitation. The excitedstate is dissociative, and if the electronic de-excitation occurs after dissociationof the molecule (light grey arrow to the right in Fig. 3.10), one has an atomicAuger decay. In Section 3.7.3, an interference phenomenon between atomic andmolecular channels is discussed.

In Paper VI, the channel of interest has been excitation to the same dissociativestate (dark grey arrow), followed by de-excitation (light grey arrow to the left inFig. 3.10) to the bound final state (2π)−1. The experimental results are beingdiscussed in Section 3.7.4.

45

3.7.1 Papers I, II and III

Sections 3.3 and 3.4 have been included in the Thesis to provide a background forthe understanding of the experimental result in Papers I and III. The potential

1 2 3 4 5Internuclear distance H/D-Cl (A)

0

HCl+DCl+

24

26

2826

28

Pot

entia

lene

rgy

(eV

)

Bin

ding

ener

gy(e

V)

Franck-Condonregion

1h (Diabatic)

2h1p (Diabatic)

φ2 (Adiabatic)

avoidedcrossing

ground state

φ1 (Adiabatic)

Figure 3.11: Photoelectron spectra of the inner valence band in HCl and DCl, withcorresponding potential curves. Labeling of the adiabatic potential curves are done inanalogy with Fig. 3.5. The spectra and the potential curves are joined in a Franck-CondonProjection scheme.

curves shown in Fig. 3.11 are taken from Fig. 2 in Ref. [13], where the avoidedcrossing region is enlarged. The adiabatic curves 32Σ+ and 42Σ+ are denotedφ1 and φ2, respectively, analogous to Fig. 3.5. and so are the diabatic curvescorresponding to the electronic configurations 1h (one hole: (4σ)−1) and 2h1p(two holes one particle: (5σ)−2(6σ∗)1), ϕ1 and ϕ2 respectively. See Section 3.4for an explanation of the Franck-Condon region. The broad band between 25 and28 eV in the photoelectron spectrum of HCl, corresponding to a continuum state,can be understood as the Franck-Condon projection on the adiabatic 32Σ+(φ1)state in Fig. 3.7. The vibrational progression on the upper half of the broad bandis partially composed by dips, instead of peaks. The dips suggest a destructiveinterference between wavefunctions of the 32Σ+(φ1) state and a bound state. Theinterpretation is that the bound state is the adiabatic φ2 state. The dynamiccoupling between states is strong enough to violate the purely adiabatic framework.Mn in Eq. (3.38) are small enough for HCl+ to give the coupling terms non-negligible size.

46

If one looks on the Landau-Zener Eqs. (3.51) and (3.52), which are based onthe diabatic framework, one see that the velocity vC around the avoided crossingcomes into play. vC is large enough for the system to make the survival probabilityPϕ1 of the bound diabatic state ϕ1 non-negliglible. For DCl the hydrogen atomis replaced by the twice as heavy isotope deuterium, while the electronic statesare identical. The only change in the photoelectron spectra is therefore due tothe dynamics of the system. M for deuterium in Eq (3.38) is twice as large ascompared to hydrogen, and the dynamic coupling between states is weakened. Andwhereas the kinetic energy for HCl+ and DCl+ is equal when developing on thedissociative potential curve, the velocity is less for DCl+, so the probability P inEq. (3.51) for non-avoided crossing is less for DCl. Consequently, the vibrationalprogression almost vanishes for DCl+.

Inte

nsity

(arb

.uni

ts)

29 28 27 26Binding energy (eV)

"e+e - +"

+

+

+

"1/2

Cl

DClTCl

HClHCl+"

Adi

abat

icap

prox

imat

ion

fully

valid

Dia

batic

appr

oxim

atio

nfu

llyva

lidIn

betw

een

Figure 3.12: Theoretical spectra with infinite resolution for HCl+, DCl+, TCl+ and theimaginary isotope systems ”1/2HCl+”, the ”1/2” isotope of hydrogen, and ”PositroniumCloride+”, where the Hydrogen proton is replaced by a positron, the positively chargedtwin of the electron. The spectra span from one limiting case to another: for the TCl+

cation, the adiabatic approximation is fully valid, whereas for ”e+e−HCl+”, it is thediabatic approximation that is valid. Note that for improved resolution, the DCl+ systemwould show a vibrational structure.

The observant reader of Papers I and II should have one objection to theFranck-Condon projection discussion: The potential curves are too high in energyfor the projection to fit the bands in the spectra. The explanation lies in theab initio calculations in Ref. [13], which typically have a constant error of 1-2 eVas compared to experiments. However, the assignment of the bands in the spectrawith relation to the potential energy curves presented in Ref. [13] is in agreementwith the results presented here.

As Paper I was written, feedback from theorists was needed for further un-derstanding of the dynamical coupling between electronic states. The Quantum

47

Binding energy (eV)27.5 27.0 26.5 26.0 25.5

Inte

nstit

y(a

rb.u

nits

)0

0

A

B

AB

HCl+

DCl+

Figure 3.13: Experimental (dots) and simulated (lines) photoelectron spectra for theinner-valence band of HCl and DCl. The experimental DCl spectrum was recorded with10 meV at BESSY in Berlin, Germany, and the HCl spectrum with 23 meV resolution atALS in California, USA. The agreement with the simulated spectra for DCl+ is reasonableenough to make the statement that the experiment confirms the theoretical predictionthat the vibrational progression A would appear for 10 meV experimental resolution.

Chemistry department in Uppsala was interested in a collaboration, and resultedin Paper II, where spectra produced by numerical simulations confirm the qualita-tive discussion in Paper I. In Fig. 3.12, theoretical spectra with infinite resolutionare shown for HCl, DCl, TCl (T for tritium, the radioactive isotope 3H), ”1/2HCl”,the ”1/2” isotope of hydrogen, ”light” hydrogen, and ”e+e−Cl+”, ”positroniumchloride”, where the hydrogen proton is replaced by a positron, where the two lastcases of course only exist in theory. Experimental measurements for TCl+ are notallowed due to its radioactivity. However, the theoretical spectrum shows that forTCl+, the vibrational progression vanishes completely. The adiabatic approxima-tion, without dynamic couplings (see Eqs. (3.41) and (3.38)), is sufficient for theunderstanding of the dissociation dynamics of TCl+ in this case. Furthermore, dueto the reduced Franck-Condon region, the continuum feature shrinks considerablyon the binding energy scale.

The imaginary isotope system ”1/2HCl” is shown for illustration. The vibrat-ional progression is further spread, due to lower reduced mass, and the interferenceeffects are even further pronounced. Even more pronounced differences are seen for

48

”positronium chloride+”, where the hydrogen proton is replaced by the 1836 timeslighter positron, the positively charged twin of the electron. Here, the vibrationalprogression is even further spread and the continuum band can no longer be seen.The dynamic coupling between the adiabatic electronic states φi in Fig. 3.11 isso strong that there is reason to talk about a ”diabatic approximation”, since thepositron has the same mass as the electrons. It can be said to be the diabatic lim-iting case. Another striking feature for ”e+e−Cl” is the broadening of the peaks,which is due to the short lifetime of the system. One can see with the Heisen-berg uncertainty formula in Eq. (3.59) that if τ increases, Γ will decrease. I. e.for ”e+e−Cl”, the bound state corresponding to the vibrational progression has ashorter lifetime compared to HCl+, due to more dynamic coupling, and therefore,Γ increases. The same argument holds for DCl+, but with the opposite effect. Thepeaks get narrower due to longer lifetime of the predissociative state φ2. Theorypredicted, that the vibrational progression for DCl+ should appear in the exper-imental photoelectron spectra if the experimental resolution could be improvedto ≈ 10 meV, compared to the resolution achieved (≈ 30 meV). After Papers Iand II were published, the experimental goal was to confirm that prediction. InSection 2.4, the experimental work for the achievement of the 10 meV spectrumrecorded at BESSY in Berlin, Germany has been discussed. The result is shownin the upper spectrum in Fig. 3.13. One can claim that the confirmation of theappearence of the vibrational progression A has been achieved for DCl+.

3.7.2 Paper IV

In Paper IV, photoelectron (PE) spectra of HCl and DCl are presented for theinner-valence region, which were recorded for various photon energies (40-64 eV),and for different angles (54.7 and 90). Furthermore, previously published thresh-old photoelectron (TPE) spectra on the inner-valence region are included in thePaper for comparison. The main conclusions in the Paper are:

• In the 90 PE spectrum, features appear which are hidden below the dom-inant main inner-valence band for the 54.7 spectra. The most interestingfeature, which is revealed in the 90 PE spectrum, can be seen in Fig. refH-ClAngComp. A second vibrational progression A’ (denoted G1 in Paper IV)appears in the same energy region as the main vibrational progression A (de-noted G2 in Paper IV). Furthermore, the bands in vibrational progressionA’ have Fano-type behaviour, similar to vibrational progression A.

• The angular-resolved spectra reveal information about the asymmetry pa-rameter β for the different features in the spectra. The main inner-valenceband ”(4σ)−1” exhibits a large β value, close to the maximum value 2. Theother vibrational bands exhibit a positive β value between the main inner-valence band and 0.

• The vibrational band structures have different intensities, but roughly thesame energy positions, when comparing the PE and TPE spectra. The in-terference phenomenon discussed in Papers I, II and III can not be seen in

49

the TPE spectra. The differences are attributed to the sudden approxima-tion: for PES, the photoelectron has a large enough kinetic energy (tens ofeV’s) to be considered as decoupled from the cation, whereas for TPES, thethreshold photoelectron with a few meV kinetic energy, the electron ejec-tion occurs the same time scale as the nuclear motion, and the cation andphotoelectron can not be regarded as isolated.

27 26 25

A

BPES hν=50eV

54.790

HCl+

Binding energy / eV

Inte

nsity

(arb

.uni

ts)

A’

Figure 3.14: Comparison between photoelectron spectra of HCl measured at 54.7 and90 with respect to the plane of polarization of the undulator radiation. For the 90

spectrum, a second vibrational progression A’ appears, with similar Fano-type behaviouras the main vibrational progression A.

3.7.3 Paper V

In Paper V, a puzzling interference phenomenon is shown. The HCl molecule isexcited to a dissociative (2p)−1(6σ∗)1 state, which corresponds to the dark grey ex-citation arrow in Fig. 3.10. The excited molecule can either de-excite as a molecule,and a molecular Auger decay results. On the same femtosecond (10−15 s) time-scale as the de-excitation process, the excited molecule can dissociate, resulting inan excited Cl∗ atom and an H atom. When the Cl* atom de-excites, an atomicAuger electron results. The term ultra-fast dissociation is used for dissociation onthe same timescale as the Auger electron decay. The atomic Auger de-excitationchannel is shown as a light grey arrow on the right side of Fig. 3.10. A third chan-nel to the final ”(4σ)−1” state is the direct channel, corresponding to the blackarrow in Fig. 3.10, studied extensivelly in Papers I, II and III. Since the potentialenergy curves corresponding to electronic states (2p)−1(6σ∗)1 and ”(4σ)−1” havea well-defined potential energy difference for long interatomic H-Cl distances, theoutgoing atomic Auger electron will exhibit a kinetic energy independent of thephoton energy used for the excitation process. The atomic Auger electrons arenon− dispersive with respect to the photon energy. In Fig. 3.15, peaks A, B and

50

Inte

nsity

(arb

.uni

ts)

176

on res.

0.4

1.0

1.1

1.2

1.4

1.6

1.9

+Ω(eV)HCl+

on res.

0.4

1.01.1

1.4

1.7

1.2

1.6

1.8

2.2

1.92.0

+Ω(eV)DCl+

190180170160150Kinetic energy (eV)

Ω =-13eVon res.

AB

C

A A

HCl2p-1σ*Auger

176 178178

Figure 3.15: Resonant Auger electron spectra for HCl and DCl, as the photon energy istuned over the (2p)−1(6σ∗)1 resonance. Ω is defined as the photon energy relative to themaximum of the (2p3/2)

−1(6σ∗)1 resonance. Peaks A, B and C in the resonant spectrumin the lower panel correspond to atomic Auger lines. Also in the lower panel, a spectrumrecorded 13 eV below resonance, which looks like Fig. 3.9. When the photon energy Ωis tuned over the resonance for HCl, upper left panel, the direct channel ”(4σ)−1” isdispersive in kinetic energy with Ω, whereas the atomic Auger peak A is non-dispersive.When the kinetic energy for the two channels coincides, and an interference patternresults. For DCl, the dissociation on the excited state is slower, resulting in a differentinterference pattern.

C in the lower panel correspond to atomic Auger decay. Photoelectrons from thedirect channel are dispersive, according to Eq (1.1). In the upper left panel inFig. 3.15, the ”(4σ)−1” band gains kinetic energy as the photon energy (Ω, detun-ing energy relative to the (2p3/2)−1(6σ∗)1 resonance) is increased. Atomic Augerpeak A has fixed kinetic energy, and when the channels have the same kineticenergy, an interference pattern is observed. Furthermore, interference between thedirect and resonant molecular channels can be observed in the spectra on reso-nance and Ω =0.4 eV, where the profile of the broad molecular band is heavily

51

distorted, and the shape is very dependent on the photon energy. The vibrationalprogression A in Fig. 3.13 can not be seen here, since the experimental resolutionwas too limited.

In Section 3.5, Fano resonances were discussed on the basis of one discrete andone continuum state, which interfere. In this work, the first example was presentedwhere two continuum states exhibit a Fano-type interference pattern.

The HCl system has to be considered both fragmented, and molecular. For thedeuterated system DCl, shown in the right panel in Fig. 3.15, two changes in thespectra relative to HCl are obvious:

• The relative intensity for the atomic peak compared to the molecular bandis smaller for DCl. This is easily seen for detuning energy Ω=1.0 eV. This isdue to the larger mass, which makes the molecule dissociates slower in theexcited (2p)−1(6σ∗)1 state. The result is less atomic Auger decay, relativeto the molecular Auger decay.

• The complete destructive interference (sometimes referred to as an ”atomichole”) for HCl for detuning energy Ω=1.9 eV is changed to another inter-ference pattern for DCl. This is due to change of the relative intensity andthe phase relation between the channels, related to the slower dissociationfor DCl.

52

3.7.4 Paper VI

In Section 3.2, the basics of electron quantum numbers in molecules is discussed,which is important for the understanding of the results in Paper VI. In the lowerpanel of Fig. 3.16, the direct-channel (see Fig. 1.1) photoelectron spectrum for theX-state, which is related to the HOMO (see Fig. 3.3), of HCl is shown. Three spin-orbit split (Λ −Σ-coupling, see Section 3.2) vibrational states can be seen. In the

Table 3.2: Calculated properties of the 2p−16σ∗ excited states with resultant angu-lar momentum Ω: Excitation energies (E) from the HCl ground state, their excitationprobabilities f , the contribution of the jj coupled (2p−1

j,mj6σ∗) configurations to the inter-

mediate state wavefunction, and Auger transition rates Γf (2ΠΩ) in µau for the transitionsto the 2ΠΩ final ionic states. The figures related to the states and transitions in Fig. 3.16are underlined with solid and dashed lines, in analogy with Fig. 3.16. N. b.: the Ω quan-tum number in the second column for the core-excited states is given by the sum of theprojections of the spin angular momenta of the 2p core-hole and the 6σ∗ valence electronspins, whereas for the two last columns, only the 2π valence hole contributes to Ω.

contribution to the wavefunction

No. Ω E(eV) f · 103 2p−11/2,1/2

6σ∗ 2p−13/2,1/2

6σ∗ 2p−13/2,3/2

6σ∗ Γf (2Π1/2) Γf (2Π3/2)

1 0+ 201.4 0.640 0.021 0.979 2552 25522 0+ 202.9 0.597 0.979 0.021 2813 28131 1 201.1 0.044 0.000 0.686 0.314 404 4972 1 201.2 8.096 0.002 0.314 0.684 68 7753 1 202.8 5.034 0.998 0.000 0.002 836 36

upper panel of Fig. 3.16, the spectra obtained as the HCl system is excited to the(2p)−1(6σ∗)1 state, are shown. A behaviour was observed, where the populatedfinal state could be chosen depending on which core-excited state ((2p3/2)−1(6σ∗)1

or (2p1/2)−1(6σ∗)1) molecule was excited to. The explanation for the behaviourin the Auger transitions resulting in the spectra in Fig. 3.16 is shown in Table 3.2.When exposing the HCl molecule to E=201.2 eV photon energy, mainly the in-termediate state (2pj=3/2,mj=3/2)−1(6σ∗)1 is populated. The magnetic quantumnumber mj = 3/2 of the core-hole is then transferred to the final-state valence holemj=3/2. For photon excitation energy 202.8 eV, the magnetic quantum numbermj = 1/2 of the core hole in electronic state (2pj=1/2,mj=1/2)(6σ∗)1 is transferredto the final valence-hole state mj=1/2. The dashed (solid) underlined quantumnumbers in Fig. 3.16 and figures in Table 3.2 are related.

53

x0.1x0.01

Inte

nsity

(arb

.uni

ts)

012345678910111213v’

2p j=1/2,mj=1/2-1 6σ*

2p j=3/2,mj=3/2-1 6σ*

HCl-1σ* Auger

X(2Π1/2)

X(2Π3/2)

+

2p2π mj=1/2

-1

16 15 14 13Binding energy (eV)

HCloff res.(h =190 eV)ν

X(2Π1/2)

X(2Π3/2)

E(e

V)

No.

201

202

121

23

Ω

110+

10+

Gro

und

stat

eC

ore-

exci

ted

stat

es

Fina

lsta

tes

013

-162Π1/2

2Π3/2

mj=1/2

mj=3/2

Main contributionto thewave function

203

mj=3/22π-1

Figure 3.16: In the lower panel, an off-resonance direct channel photoelectron spectrumfor the X state is shown. In the upper panel, two Auger electron spectra of the X-statein HCl+, after (2p)−1(6σ∗)1 photoexcitation, is shown. When exciting the moleculeto electronic state (2p1/2)

−1(6σ∗)1 [(2p3/2)−1(6σ∗)1], the final state becomes primarily

2Π1/2 [2Π3/2]. The quantum number mj , which is transferred from the intermediatestate to the final state, is underlined. On the right-hand side, the electronic transitionsare shown. The sizes of the excitation arrows are roughly proportional to the excitationprobabilities (column f ·1000 in Table 3.2). The No. and Ω columns here are equivalentto the corresponding columns in Table 3.2. The energy scale on the right-hand sidecorresponds to the energies in the E(eV) column in Table 3.2. The quantum numbermj , which is transferred from the intermediate state to the final state, is underlined, asin the graph.

3.7.5 Papers VII and VIII

In Paper VII, a user manual and an overview of a new data acquisition systemfor Photo-Electron PhotoIon PhotoIon COincidence (PEPIPICO) Time-Of-Flight(TOF) measurements is presented. At the LNLS Synchrotron in Campinas, Brazil,and at MAX-lab in Lund, Sweden, two groups have TOF spectrometers, and re-cently, there was need for a new data acquisition system. Both groups boughta Time-to-Digital-Converter, TDC card, which transforms flight times to digitalinformation, from the FASTComTec company [66]. Traditionally, the TOF com-munity has used the data software program Igor Pro [67] as a user interface. The

54

aim of this project has been to use the TDC card provided by FASTComTec,with Igor as a user interface. At this stage, at LNLS, the acquisition systemcan be used for running the monochromator for near-edge x-ray absorption finestructure (NEXAFS) measurements. A full coincidence record can be obtainedfor each energy step. At MAX-lab, this feature can be implemented with networkprogramming.

The PEPIPICO spectrometer at LNLS has been characterized in Paper VIII,to provide a solid experimental basis for the interpretation of the experimentalresults.

55

Comments on my own participation

One of the things that makes me really enjoy the work of experimental physicsis the variety of tasks that have to be done between first thought and finishedPaper. I have been working as a chemist in the chemistry lab for production ofgases, primarily DCl. I have been struggling with voltage divider boxes for thepower supply of the PEPIPICO spectrometer in Lund. I have been doing SimIonsimulations, for improvement of the efficiency of the PEPIPICO spectrometer inCampinas. I have vented a beamline (and I did the dirty work needed afterwards).I have done programming for a data acquisition setup. I have participated in thephysics discussions concerning the material presented in this Thesis, and in thewriting of the Papers. This has always been in collaboration with people, whohave taught me things I did not know, but sometimes also vice versa. The Paperswhere I am the first author, I have had the main responsibility for the projectsand the writing of the manuscripts. One exception is Paper IV, where the groupin Uppsala was invited by Andrew J. Yencha to participate in the writing of theoutcome of the experiments he had performed. The collaboration with the Quan-tum Chemistry department, which resulted in Papers II and III was initiated byme. Concerning Papers V and VI, I was deeply involved in the measurements andthe discussion about the material presented in the Papers, but I was not the mainperson responsible for the writing of the manuscripts.

Why fundamental science?

Why should people pay taxes for the financing of fundamental science, like theresearch field of which this Thesis is a humble piece? The question is justified, andresearchers have to be able to give an answer.

History of science is full of unexpected results, where the scientist has dis-covered something completely different, and useful, compared to the expectedoutcome. Two examples, both of which every one of us have benefited from inour lives: X-rays (Wilhelm Conrad Rontgen in 1895) and penicillin (Sir AlexanderFleming in 1928) were pure chance discoveries. Such spin-off effects are impossi-ble to predict. So instead of directing all money to particular ”hotspots”, todaybiotech, yesterday IT, science should be a boiling kettle.

Below, I present two examples closely related to the work done in my researchgroup, which have been proven to give valuable information for non-fundamentalresearch interests, even though the group deals purely with purely fundamentalresearch. In the fifties, Electron Spectroscopy for Chemical Analysis, ESCA, wasstarted in Uppsala by Kai Siegbahn [68]. X-rays were used to ”pull electronsoff” a material, and the electron kinetic energy gave valuable insight into thecomposition of Elements of materials, and the chemical environment of atoms.Now the technique is used in the industry for probing the quality of materials.Our electron spectroscopy group is a child of the ESCA family.

A few years ago, a concept named the ”core-hole clock” was developed inour group. The life-time of an electron core hole in an atom of a molecule isknown. This information can be used for determining the dissociation time formolecules [60]. The core-hole clock technique has now been used to determine the

56

charge transfer time in the Gratsel cell [69, 70], a new type of solar cell, which isextensively studied due to the possibilities of different types of applications.

I find it hard to belive that the results presented in the Papers presented inthis Thesis will give such spin-off effects though. But even without technological,or medical future applications, physical phenomena can have their beauty per se,just as a fine piece of music, or a painting. It is just that the audience is far morelimited for a Paper concerning Adiabatic Crossing then a Britney Spears CD. Butthen again, my salary is far more limited than Britney Spears’ too. A referee ofPaper V called the result ”a textbook example”, and a referee of Papers I andII said that, besides the very nice experimental and theoretical results presented,the Papers give a didactic introduction to the field. I do not know if the spectrapresented in this Thesis will show up in a textbook, but I would be as glad thatday, as Britney Spears for a number 1 Billboard hit.

Acknowledgments

I had finished my Master Thesis at UNICAMP in Campinas, Brazil late 1998.I returned to Uppsala without knowing what to do next. I just knew that I wantedto go back to Brazil, but without losing contact with family and friends in Sweden.Chances for finding something which suited these demands were about as close tozero as they can get. By chance I got to know Olle Bjorneholm at Fysikum, andfound out that he had a collaboration going on with people in Campinas. Aftersome small talk with Olle, he realized that I was interested in going to Brazil, andI realized that he needed a student going to Brazil. Finding this collaborationOlle Bjorneholm- Arnaldo Naves de Brito at LNLS in Campinas is one of thosepure luck events of my life. I have been waiting for the backlash since day one:There has got to be something wrong. This is too good. But this backlash dayhas never come.

A esposa do Arnaldo, Maria de Lourdes ajudou um monte para fazer a vidanao-profissional tao boa como a vida do trabalho, voce merece um abracao. Opessoal no LNLS sempre estava atento de ajudar com mil coisas, e eu adorei ofutebol e os churrascos nas sexta-feiras: Christiano, Elaine, James, Jose, Mario,Pacheco, Paulo, Ruben, Yvone e Yuri. E sempre e um grande prazer bater umpapo com o Paulinho sobre a musica maravilhosa brasileira.

The person who is to thank in the first place for the Uppsala-Campinas col-laboration is Svante Svensson, who invited Arnaldo to do his PhD in Uppsalain the late 80’s. I do not know if it is common in succeeding to invite researchseniors to dinner in student rooms (and drinking wine out of coffee cups), but Iam glad to say that Svante did not hesitate (which says much about the relaxedatmosphere in the group). Other people in Uppsala: Gunnar Ohrwall (thanksfor the 10 meV!), Dimitri Arvanitis, Paul Bruhwiler, Karl-Einar Ericsson, Jan-Olof Forsell, Birgit Gravahs, Gunnel Ingelog, Charlie Karis, Leif Karlsson, AnnikaKollstedt, Anne Kronquist, Maria-Novella Piancastelli, Carla Puglia and Jan-ErikWahlund all deserve a thanksalot for everything.

We have had nice collaborations with three theory groups, and I am gratefulfor the input, knowledge and patient teaching they have provided: At the quan-tum chemistry group two floors upstairs: Mauritz Andersson, Hans Karlsson and

57

Osvaldo Goscinski. In Stockholm: Pawel Salek, Faris Gel’mukhanov and HansAgren. And in Lund Reinhold Fink. I hope for continuing collaboration with allof you!

In Lund, Stacey Sorensen, Mathieu Gisselbrecht (thanks for the working acqui-sition setup!), Joachim Schulz, Jonathan Hunter-Dunn, Magnus Lundin, HelenaUllman, Maxim, Lidia, Reinhold Fink and Nils Martensson have been helping meout at work and contributed to some fine days and nights of friendship.

Also in Lund, Margit Bassler has worked harder than hard for making mea-surements possible at BL I411. Glad you’re back Margit! The MAX-lab staff,in my case in particular Anders Mansson and Stefan Wiklund, have helped andtaught me a lot about practical issues that are essential for experimental work,but easily forgotten when the Paper is written.

At BESSY in Berlin, I have found many friends during the last year: KaiGodehusen, Tobias Richter, Michael Martins, Peter Zimmermann, Ralph Puttner,Uwes Becker and Hergenhahn, Hans Kleinpoppen, Matthias Neeb. I have had theopportunity to discuss physics with people outside of the team at conferences andsimilar occasions. Andrew J. Yencha from Albany, New York, USA, NobuhiroKosugi from Okazaki, Japan, Gerson from Rio de Janeiro, and Valentin Ostrovskyfrom St Petersburg, Russia have all taken their time for discussions, which havebeen valuable for me.

With all respect to seniors, the most important contribution for my joy of goingto work, even on Monday mornings, are my fellow students: Achim, Alexandra,Ana, Andre, Anna, Annemieke, Anders, Andreas, Annika, Arantxa, Ceara, Chris-tine, Cecilia, Cissi, Dennis, Emerson, Erik, Fabio, Gribel, Henrik, Ingela, Jan,Joe, Joachim, Johan, John, Katharina, Karin, Karine, Karo, Klas, Larsa, Lucia,Marcus, Maria, Mario, Micke, Ola, Patrik, Per-Erik, Rafael, Raimund, Ricardo,Timo, Torbjorn, Tobias and Ylvi.

All these people have proven that professional collaboration and friendship canin deed be combined, if anyone has stated the contrary.

Meine Familie Dietrich, Gabriele, Jennifer, Oma Eva, Tante Ellen, Hans undBrigitte, Julia und Barbara, und meine dritte Oma Senhora Emma Zulauf sindimmer bei mir, obwohl das umgekehrte leider zu selten wahr ist.

Tack for allt Anton, Ove, Rasmus och Ulla.Skal Emma!

Florian Burmeister, Uppsala, Winter 2003

58

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