Photoelectron Spectroscopy on HCl and DCl: Synchrotron Radiation Based Studies of Dissociation Dynamics

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(10) Dissertation for the Degree of Doctor of Philosophy in Physics presented at Uppsala University in 2003 Abstract Burmeister, F. 2003. Photoelectron spectroscopy on HCl and DCl: Synchrotron radiation based studies of dissociation dynamics. Acta Universitatis Upsaliensis. Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 805. 61 pp. Uppsala. ISBN 91-554-5531-X The dissociation dynamics of the ionized molecules hydrogen chloride (HCl) and deuterium chloride (DCl) have been studied in gas-phase using synchrotron based photoelectron spectroscopy (PES). The main inner-valence photoionization band for DCl and HCl was recorded using maximum resolution in order to probe an interference pattern between a dissociative and a bound electronic state. For HCl+ , distorted Fano-type peaks were observed even for modest resolution, whereas for DCl+ , the pattern was hardly discernible. The observation in HCl+ has been explained by a coupling between two adiabatic electronic states, where the bound state was populated through non-adiabatic curve-crossing. The nuclear motion of HCl+ is too fast for the Born-Oppenheimer approximation to be fully valid in this case, whereas for DCl+ , with larger reduced mass and therefore slower nuclear motion, the non-adiabatic coupling 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 differences in intensities and shapes of the vibrational bands. These differences were attributed to the fact that the photoelectron can be regarded as isolated from the cation 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 Augerand photoelectron channels. The atomic features are associated with ultra-fast dissociation of the molecules, on the same time scale asf the Auger decay. The observation shows that the excited molecular system has to be regarded as a superposition of fragmented and molecular states. ˜ A study of the X-state of HCl+ , populated via a core-excited state, shows a selective population of the final state. The explanation is that that the magnetic orientation 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 COincidence (PEPIPICO) measurements using a Time-Of-Flight (TOF) spectrometer has been developed. A Time-to-Digital Converter (TDC) card has been linked together with the data treatment program Igor Pro as a user interface. Furthermore, the PEPIPICO spectrometer has been characterized to provide a solid basis for the analysis of future experimental data. Florian Burmeister, Department of Physics, Uppsala University, Box 530 SE-751 21 Uppsala, Sweden c Florian Burmeister 2003 ISSN 1104-232X ISBN 91-554-5531-X Printed in Sweden by Kopieringshuset AB, Uppsala 2003.

(11) Till Emma♥. S˚ a mjuk, s˚ a varm, s˚ a fin! Jorge Ben (¨ overs¨ attning Mats Hallgren).

(12) 4. List of papers This Thesis is based on the following Papers, which will be referred to in the text by their Roman numerals. I. Nonadiabatic effects in photoelectron spectra of HCl and DCl. I. Experiment F. Burmeister, S. L. Sorensen, O. Björneholm, A. Naves de Brito, R. F. Fink, R. Feifel, I. Hjelte, K. Wiesner, A. Giertz, M. B¨ assler, C. Miron, H. Wang, M. N. Piancastelli, L. Karlsson, and S. Svensson Phys. Rev. A 65, 012704 (2001). II. Nonadiabatic effects in the photoelectron spectra of HCl and DCl. II. Theory L. M. Andersson, F. Burmeister, H. O. Karlsson, and O. Goscinski Phys. Rev. A 65, 012705 (2001). III. Confirmation of non-adiabatic vibrational progression in the main inner-valence photoionization band of DCl and HCl ¨ F. Burmeister, L. M. Andersson, G. Ohrwall, A. J. Yencha, T. Richter, P. Zimmermann, K. Godehusen, M. Martins, H. O. Karlsson, S. L. Sorensen, O. Bj¨ orneholm, R. Feifel, K. Wiesner, O. Goscinski, L. Karlsson, and S. Svensson Submitted to Phys. Rev. A IV. PES/TPES Comparative Study of the Inner Valence Ionization Region in HCl and DCl ¨ F. Burmeister, G. Ohrwall, L. Karlsson, O. Bj¨ orneholm, S. Svensson, and A. J. Yencha In manuscript. V. Observation of a Continuum-Continuum Interference Hole in Ultrafast Dissociating Core-Excited Molecules R. Feifel, F. Burmeister, P. Salek, M. N. Piancastelli, M. B¨ assler, S. L. Sorensen, C. Miron, H. Wang, I. Hjelte, O. Bj¨ orneholm, A. Naves de Brito, F. Kh. Gel’mukhanov, H. ˚ Agren, and S. Svensson Phys. Rev. Lett 85, 3133 (2000). ˜ 2 Π) state using VI. Spin-orbit selectivity observed for the HCl + (X resonant photoemission R. F. Fink, F. Burmeister, R. Feifel, M. B¨ assler, O. Bj¨ orneholm, L. Karlsson, C. Miron, M.-N. Piancastelli, S. L. Sorensen, H. Wang, K. Wiesner, and S. Svensson Phys. Rev. A, 65, 034705 (2002).

(13) 5. VII. Using Igor Pro as the user interface together with a FAST TDC card for PEPIPICO data acquistion: manual and overview F. Burmeister, M. Gisselbrecht, J. R. Piton, E. S. Cardoso, S. L. Sorensen, and A. Naves de Brito LNLS-CT, 06, 01 (2002) VIII. Description and performance of an electron-ion coincidence TOF spectrometer used at the Brazilian synchrotron facility LNLS F. Burmeister, L. H. Coutinho, R. R. T. Marinho, K. Wiesner, M. A. A. de Morais, A. Mocellin, O. Björneholm, S. L. Sorensen, P. de Tarso Fonseca, J. G. Pacheco, and A. Naves de Brito In manuscript. The following is a list of papers to which I have contributed but that are not included in this Thesis. Development of a four-element conical electron lens dedicated to high resolution Auger electron-ion(s) coincidence experiments K. Le Guen, D. Ceolin, R. Guillemin, C. Miron, N. Leclercq, M. Bougeard, M. Simon, P. Morin, A. Mocellin, F. Burmeister, A. Naves de Brito, and S. L. Sorensen Rev. Sci. Instrum. 73, 3885 (2002) Filtering core excitation spectra: vibrationally resolved constant ionic state studies of N 1s → π core-excited NO H. Wang, R. F. Fink, M.-N. Piancastelli, I. Hjelte, K. Wiesner, M. B¨ assler, R. Feifel, O. Bjorneholm, C. Miron, A. Giertz, F. Burmeister, S. L. Sorensen, and S. Svensson J. Phys. B 34, 4417 (2001) The dynamic Auger-Doppler effect in HF and DF: control of fragment velocities in femtosecond dissociation through photon energy detuning K. Wiesner, A. Naves de Brito, S. L. Sorensen, F. Burmeister, M. Gisselbrecht, S. Svensson, and O. Bj¨ orneholm Chem. Phys. Lett. 354, 382 (2002) Experimental study of photoionization of ozone in the 12 to 21 eV region A. Mocellin, K. Wiesner, F. Burmeister, O. Bjorneholm, and A. Naves de Brito J. Chem. Phys. 115, 5041 (2001) A vibrationally resolved experimental study of the sulfur L-shell photoelectron spectrum of the CS2 molecule H. Wang, M. B¨ assler, I. Hjelte, F. Burmeister, and L. Karlsson J. Phys. B 34, 1745 (2001).

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(15) 7. Contents 1 Introduction 2 Experimental 2.1 Synchrotron radiation 2.2 Beamline I411 . . . . . 2.3 Experiment endstation 2.4 Why BESSY? . . . . .. 9. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 11 11 13 15 16. 3 Molecular physics 3.1 Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Atoms, molecules and their electronic structure . . . . 3.2.1 The electronic structure of atoms . . . . . . . . 3.2.2 Electronic structure of HCl . . . . . . . . . . . 3.3 Molecular dynamics . . . . . . . . . . . . . . . . . . . 3.3.1 The Schr¨ odinger equation . . . . . . . . . . . . 3.3.2 The diabatic framework . . . . . . . . . . . . . 3.3.3 The adiabatic framework . . . . . . . . . . . . 3.3.4 The Landau-Zener formula . . . . . . . . . . . 3.3.5 Adiabatic/diabatic framework: short repetition 3.4 Franck-Condon projections . . . . . . . . . . . . . . . 3.5 Fano resonances . . . . . . . . . . . . . . . . . . . . . . 3.6 Hydrogen chloride . . . . . . . . . . . . . . . . . . . . 3.7 Summary of Papers . . . . . . . . . . . . . . . . . . . . 3.7.1 Papers I, II and III . . . . . . . . . . . . . . . . 3.7.2 Paper IV . . . . . . . . . . . . . . . . . . . . . 3.7.3 Paper V . . . . . . . . . . . . . . . . . . . . . . 3.7.4 Paper VI . . . . . . . . . . . . . . . . . . . . . 3.7.5 Papers VII and VIII . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. 19 19 21 21 22 25 25 27 31 33 35 37 39 40 43 45 48 49 52 53. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. Comments on my own participation. 55. Why fundamental science?. 55. Acknowledgments. 56. Bibliography. 61.

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(17) 9. Chapter 1. Introduction The name photoelectron spectroscopy reveals the basic fundament of the research method, of which the work presented here is based upon. A photon interacts with a 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. The kinetic energy of the electron(s) is analyzed with a spectrometer. The kinetic energy is related to the binding energy of the electrons in the system according to hν = Ekin + Ebin ,. (1.1). Core Valence electrons electrons. where hν is the photon energy, Ekin is the kinetic energy of the electron and Ebin is the binding energy of the electron. Eq. (1.1) is a modified version of the 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. Augerelectron. Photoelectron. Direct channel. Resonant 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 the emission of one electron from the molecule. The extracted electron is called.

(18) 10. a photoelectron. The direct channel is non-resonant, i. e. the process occurs whenever the photon energy exceeds the Ebin on the photon energy, see Eq. (1.1). The direct photoelectron channel is shown in the left-hand side of Fig. 1.1. 2. The resonant channel: Here, the photon excites the molecule. A core-electron is 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. The probability for an electron to carry the excess energy is orders of magnitude larger than for a photon, except for deep core levels in heavy atoms (much heavier than in the here discussed light diatomic cases HCl, CO, etc), where the photon channel becomes significant. The discoverer of the decay process in 1925, Pierre Auger, has given the name to the Auger process and the Auger electron. The resonant channel is highly sensitive to the photon energy since it 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 VI deals 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 photoelectron spectroscopy using synchrotron radiation. Chapter 3 gives a background to and an extension of the physics that lie behind the results presented in the Papers in this Thesis. 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 dissociation dynamics can be probed and analyzed in photoelectron spectra, revealing intricate details..

(19) 11. Chapter 2. Experimental The understanding of physics in nature is based upon experimental studies. No theory, no matter how beautiful it might appear, can stand the falsification of an experiment. And theory can be strengthened (but strictly never proven) by experiments. Over the centuries, experimental setups have grown in size and complexity. The laboratory where experiments similar to those discussed here is one of the largest in Sweden, the National Synchrotron Radiation Facility MAXlab in Lund. This Chapter gives a general overview of the laboratory employed for the present investigations.. 2.1. Synchrotron radiation. Synchrotron radiation has its origin from accelerated charged particles. The particles accelerated can be electrons, positrons (the positively charged twins of electrons), protons, etc. The acceleration is achieved by electric and magnetic fields. There are natural sources for synchrotron light in universe, for example neutron stars quickly spinning around their axis. Here the accelerated charges are electrons accelerated by the magnetic field around the neutron stars. Charged particles from the sun which are deflected by the magnetic field of the earth is another synchrotron 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 Electric laboratories in New York, where emitted light was observed from an electron beam [1]. The principle of the synchrotron is to synchronously change the generator frequency of the accelerating stages together with the magnetic field in such a way that the electrons, whose orbital frequencies and momenta are increasing as a result of the acceleration, always feel an accelerating force and are simultaneously kept on their assigned orbits inside the vacuum tube. In 1949, J. Schwinger published a paper characterizing synchrotron light theoretically [2]. There it was shown that the energy lost in one turn is proportional to the fourth power of the electron energy, i. e. doubling the particle beam energy means sixteen times more photons..

(20) 12. Undulator. BL I411. MAX II. MAX I. 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 magnets used are primarily dipoles and quadrupoles. Quadrupoles are used for focussing the electron beam, in analogy with optical systems, whereas dipoles (called bending magnets) are used for making the electrons turn around in the ring. Since the magnetic fields are not changed with time, see below, the name synchrotron light source has lost its original meaning. A more appropriate name used for the accelerator device today is electron storage ring. As the electrons are accelerated by the centripetal magnetic force, they emit photons in the forward direction, synchrotron light. Between the magnets there are magnet-free sections. In one of these sections, the electron bunches are pushed to keep the kinetic energy each turn by a oscillating electric field. The device is called a radio frequency cavity, due to the high frequency of the field (50-1500 MHz). The 1st generation of synchrotron light research was ”parasitic” activity on behalf of the nuclear physicists accelerators. The 2nd generation of light sources, starting up in the 70’s, were electron storage rings dedicated to synchrotron light research. One of these 2 nd generation 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-purpose Accelerator for X-rays. A microtron is used for pre-acceleration of the electrons before they are ejected into the storage ring. The microtron at MAX-lab is placed under MAX I. MAX II is a 3rd generation synchrotron light source, which has been operational since 1998. Whereas for 2nd generation light sources one mainly uses the synchrotron light from the dipole bending magnets, more sophisticated magnetic devices have been developed for 3rd generation light sources. Arrays of dipole magnets force the electrons to proceed in a sinusoidal trajectory, emitting more photons than the one-turn bending magnets. These arrays are called wigglers or undulators. Wigglers are used when high photon energies (keV range) are required. The higher the magnetic field, the higher the photon energy. Sometimes super-conducting electromagnets are used to produce as high magnetic field as possible. The wiggler photon intensity versus photon energy characteristics is.

(21) 13. Table 2.1: MAX II, and beamline I411 parameters Parameter MAX II: Electron energy Electron speed Storage ring circumference Radio Frequency of Cavity Electron bunch length Electron bunch periodicity Beam lifetime Max. Current Undulator at BL I411: Period between magnets Number of magnet poles Total undulator length Magnet gap Peak field. Value 1.5 GeV 0.999999% 90 m 500 MHz 20 ps 2 ns > 10 h >200 mA 60 mm 87 2.65m 22 - 300 mm 0.65 T. continuous over a long energy range. Undulators typically have a larger number of shorter dipole magnets in one array. Unlike wigglers, the undulator photon intensity versus photon energy characteristics is non-continuous: for certain energies, the photon intensity increases by orders of magnitude. This is due to constructive interference of the superimposed photons emitted on the undulator. The wavelength for which the constructive interference occurs depends on the deviation of the electrons from a straight path, which in its turn depends on the strength of the magnetic field. By changing the size of the undulator magnet gap, and thereby changing the strength of the magnetic field, the photon energy corresponding to positive interference can be changed. With the development of wigglers and undulators, the demands on the electron beam quality has increased. Source size, emittance and beam stability are parameters that are important this aspect. Some basic parameters of MAX II and undulator ”Finnen” of beamline insertion device 411 (BL I411) are shown in Table 2.1 [3, 4].. 2.2. Beamline I411. In the previous Section, the creation of synchrotron light has been discussed. The light 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 I411 can 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.

(22) End station with Scienta analyzer. One-meter section. Refocusing mirror. Exit slit. SX700 Monochromator. BL I411 Water cooled baffles. Cylindrical mirror. Undulator source. 14. 2m. Figure 2.2: An overview of beamline I411 at MAX-lab. from the electron storage ring. All optics elements (mirrors, gratings) work at grazing incidence, due to the fact that no material has appreciable reflectivity at normal 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 in horizontal direction. Baffles are used for reducing the size of the photon beam to the 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 photon energy out of the undulator spectrum of light (mono=one; chromos=colour). The monochromator of SX700-type at BL I411 consists of three optical components: a large plane mirror, a plane diffraction grating, and an ellipsoidal mirror. The diffraction grating disperses the photons at different angles, depending on the photon energy, in analogy with a prism. By changing the angle of the grating, the outgoing photon energy passing through the fixed exit angle can be varied. The exit slit, mounted behind the monochromator, has a variable aperture. Depending on the aperture size (1-800µm), the photon energy band width can be chosen to be narrow for high-resolution experiments, or broad, for higher flux and better statistics. After one last refocusing mirror, the photon beam enters the part of the beamline dedicated to the experiments. The baffles, the first cylindrical mirror, and the diffraction grating in the monochromator are water cooled, because they are heated up by the light. The experiments can be performed at either the endstation with the stationary Scienta electron analyzer, see next Section, or at the one-meter section. At the one-meter section, the scientific groups can use their portable equipment. For instance, the collaboration with the Stacey Sorensen group in Lund has used the one-meter section for measurements with a time-of-flight spectrometer for coincidence measurements. This spectrometer has the same overall characteristics as the spectrometer described in Paper VIII. In the BL I411 case, the beamline meets a second requirement, besides the function of the monochromator: the ultra-high vacuum (≈ 10−10 mbar) in the electron storage ring has to be protected from the relatively high pressure at the experiment (≈ 10−3 mbar). Different kinds of vacuum pumps are installed along the beamline. They make it possible to maintain the pressure difference between the experiment and the monochromator and the electron storage ring. Capillary tubes 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..

(23) 15. 2.3. Experiment endstation. The experimental research presented in Papers I, V and VI in this Thesis, was performed 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. Magnetic shields. Analyser spheres. Photons. MCP Detector Phosphor screen CCD camera. 5- element lens system Gas cell. Figure 2.3: The principal parts of a Scienta electron spectrometer. the HCl/DCl gas-phase molecules in a gas-cell with differential pumping. The pressure inside the gas cell during the measurements has been estimated to be in the 10−3 mbar range, whereas outside the gas cell, the pressure is in the 10−5 mbar range. The emitted photo- and Auger electrons are detected by the electron spectrometer. The setup works in a crossed-beam configuration where the gas source, the axis of the Scienta electron spectrometer lens and the direction of propagation of the synchrotron light form an orthogonal set. The electron spectrometer can be rotated from 0◦ to 90◦ with respect to the linear polarization of the synchrotron light. The measurements at BL I411 were done at the magic angle, 54.7 ◦, at which the electron intensity is proportional to the case where electrons are collected over all angles. The electron spectrometer consists of two main parts, the electron lens and the analyzer. The electron lens is a complex five-element system, made to collect electrons, transporting and focussing them on the entrance slit of the analyzer, and also to accelerate/retard the electrons to a pre-set kinetic energy before entering the analyzer, the pass energy. The analyzer is composed of one inner and one outer metal hemisphere, with 200 mm mean-radius, hereby the name SES200, where SES stands for Scienta Electron Spectrometer. Electrons with kinetic energy equal to the pass energy at the entrance fulfill the half-turn between the hemispheres, and hit the centre of a multi-channel plate (MCP) detector on the other side of the analyzer. Electrons with less or more kinetic energy are detected on a smaller or larger radius relative to the centre of the MCP. The electrons enter the hemisphere at different angles. Electrons with identical kinetic energy but different entrance angles are focused onto the detection plane (the MCP). The.

(24) 16. MCP detector amplifies the impact of the photoelectron to a measurable pulse. A phosphor screen on the back-side of the MCP detector transforms the MCP pulse to a visible dot. The position of the phosphor screen dot is monitored by a chargecoupled-device (CCD) camera, and the corresponding kinetic energy is calculated by a computer. The computer also controls the potentials of the electron lens and the 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 the Scienta 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 FASTCOMTec company Time-to-Digital Converter (TDC) card with Igor Pro as user interface. More detailed information about the data acquisition setup is presented in Paper VII. Furthermore, the characterization of the PEPIPICO spectrometer has been made and is available in Paper VIII. This Paper provides a solid basis for the understanding of the experimental results.. 2.4. Why BESSY?. There is a state-of-the-art setup for high-resolution photoelectron spectroscopy using synchrotron light at MAX-lab, as shown by Papers I, V and VI. Why does one then choose to go to the Berlin synchrotron BESSY II for the DCl measurement published in Paper III? After Paper I had been written, contact was established with the theory group of Quantum Chemistry. One of the results of the collaboration with theory was the statement ”From the simulations we observe that two additional peaks in the experimental DCl should appear if the resolution were to be enhanced to around 10 meV” in the last sentence in the abstract in Paper II. For the experimental and theoretical discussion about the Physics of related Papers, see further in Section 3.7.1. Is a photoelectron spectrum of the inner-valence band of HCl/DCl at BL I411 at MAX-lab of 10 meV within reach, or are the characteristics of beamline BUS-SGM U125 at BESSY II significantly favourable? 1. The so called Doppler broadening has to be considered, which is an effect which is particularly important for light molecules such as HCL/DCl. The initial kinetic energy of the photoelectrons varies, since the atoms or molecules from which they are ejected have a thermal motion. If the motion is totally random, this gives rise to an energy spread that has a Gaussian distribution with a full width at half maximum (FWHM): Ekin T , (2.1) ∆E = 0.722 M 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 in atomic units, a. u.[9]. Hereby, the Doppler broadening for the experiments in Paper I with Ekin = hν − Ebin =(64-26) eV=38 eV, T =293 K, and.

(25) 17. M =37 a. u. will be 13 meV. This is enough to obliterate the resolution required for the expected outcome of the experiment. With hν=40 eV, the Doppler broadening is reduced to 7.6 meV. The main advantage with beamline 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 energy down 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 included, the Drift Region option. The DCl experiment in Paper III will be taken as an example. The region of interest is the inner-valence band of DCl, which has a low cross section, and the band can not be seen on the CCD screen. Only after hours of recording and adding swept spectra, the structure of the band is visible to the eye. By experience, the kinetic energy of an electron for a fixed binding energy is known to drift with time. This is due to drifting potentials in the interaction region, the surrounding surfaces might not be perfectly grounded, etc. This problem can be solved by sweeping a second region (called Drift Region in the software), of an intense ˜ peak, in the present case the peak corresponding to X-state (2 Π3/2 ) v = 0, in the outer valence band for HCl/DCl. The cross section for ionization of this state is orders of magnitude larger than for the inner-valence band states and can be seen immediately on the CCD screen. The kinetic energy of the intense peak is chosen, here (40-12.74) eV=27.26 eV. The two regions are swept consecutively, and the software corrects for possible drifting of the kinetic energy of both regions automatically. In the present experiment a drift of 20 meV over a few hours was noticed..

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(27) Chapter 3. Molecular physics In Chapter 2 the importance of experiments for a validation of theory was mentioned. However, it works in the other direction as well. An experimental observation is sterile without a proper theoretical model, which puts the observation in a more general context. The scientific projects presented in this Thesis were developed 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 particular is presentedd in Section 3.2.2, • the core of this Thesis, which is an overview of the diabatic/adiatic frameworks, which lead to the Born-Oppenheimer approximation is given in Section 3.3, • a fundamental tool for the understanding of electron spectra of molecules is the Franck-Condon projection concept, presented in Section 3.4, • When electronic states interfere, they give rise to Fano-profiles, which will be discussed briefly in Section 3.5, • an overall view of the literature concerning the HCl molecule is presented in Section 3.6.. 3.1. Light. Light, when described in the classical (i. e. pre-quantum mechanical) sense of J. Clerk Maxwell, can be considered to be an electromagnetic (EM) wave. Energy is transferred continuously. In contrast to this classical electrodynamical view, Quantum Electrodynamics describes electromagnetic interactions in terms of massless elementary ”particles” known as photons. Both views have raison d etre, 19.

(28) 20. depending on which experimental observation one discusses. For geometrical optics, such as the understanding of phenomena interference and diffraction of light through macroscopic slits, the classical view is sufficient. This is due to the fact that 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 well as other numerous experimental observations, can not be explained by the classical wave view. Instead the interpretation of the observations require a quantization of the EM field for the interaction between light and matter, and the quanta are called photons. Photons are stable, chargeless, massless elementary particles that exist only at the speed of light, c. Moreover, photons are bosons, they are spin-1 particles. An arbitrary large number of bosons can be in the same state, contrary to f ermions, see Section 3.2. Each photon has an energy given by the product of Planck’s Constant and the frequency of the radiation field: E = hν. (3.1) When many photon progress with the same state, with the same momentum and. -state L. k, p. -state L. k, p. Figure 3.1: Angular momentum, L, linear momentum, p, and wave vector, k, of a photon.. direction, they form a monochromatic plane wave. According to the quantummechanical description, a wave transfers energy in quantized packets, i. e. photons such that E = hν, and spin angular momentum L of a photon which is either −1 or +1. The signs indicate right- (L-state), and left-handedness (R-state), respectively. In Fig. 3.1, the two alternatives are shown. The angular momentum of a photon is completely independent of its energy and linear momentum. Whenever a charged particle emits or absorbs electromagnetic radiation, along with changes in its energy and linear momentum, it will undergo a change of ±1 in its angular momentum. If all photons of an EM wave are right- or left-handed, then the light is purely right- or left-circularly polarized. A purely left-circularly polarized plane wave.

(29) 21. will impart angular momentum to the target as if all the constituent photons in the beam had their spins aligned in the direction of propagation. Now, how can linearly polarized light be explained, which is the case for BL I411 [5], see Section 2.2? Here each individual photon exists in either spin state with equal likelihood [11].. 3.2. Atoms, molecules and their electronic structure. This Section aims at giving an understanding of the electronic structure of HCl needed for the discussion about the results presented in the Papers and in the latter part of this Thesis. How can a complicated system composed of two nuclei and 18 electrons be treated in a comprehensive way? First, the electronic structure of a simpler system, namely an atom, is discussed. The results will then be implemented 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 one proton. The characteristics of the electron can be described by a set of quantum numbers. n=1, 2,... defines the shell. Within the shell, there are n subshells with quantum numbers l=0, 1, 2, (n-1). l is the orbital angular momentum quantum number 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 angular momentum. ml is the quantization of the orbital angular momentum projected on 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 the boundary 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 a given n-value is triply degenerate, with three values for ml (0,±1), where ml =0 is associated with the pz -orbital,while a linear combination of ml =±1 are associated with the px and py orbitals. Besides orbital angular momentum, the electron exhibits a second property of fundamental importance for understanding the energy levels of electrons, namely the spin angular momentum. The spin momentum quantum number for an electron is 12 and the projection of the electron spin along the magnetic field direction is ms = ± 21 . The sum of the orbit and spin momenta is called j, l + s = j, and the 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 the shells in a multielectron atom, certain restrictions have to be considered. The Pauli principle restricts the system to allow only one electron for a given set of quantum numbers. Hund’s rules describe the building-up order for a given subshell for minimization of the energy of the total system. For a given many-electron atom, these rules will determine which atomic orbitals are populated. When considering multielectron atoms, l, s and j are replaced by the vector sum of all the individual electron moments and designated L, S and J. The.

(30) 22. y x z s. y. y. x. x z. z px. y. py. x z. pz. 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 of a many-electron atom is expressed by a term symbol by using these quantities: 2S+1 LJ . 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 complex units. This Thesis deals with the understanding of the diatomic molecule hydrogen chloride (HCl) and different inherent degrees of freedom which are related to the electronic structure in this system. In Fig. 3.3, the atomic orbitals (AO) are presented for the chlorine (Cl) and hydrogen (H) atoms. The orbital scheme in Fig. 3.3 is simplified, since only the quantum numbers n (1, 2, and 3 for Cl; 1 for H) and l (0:s, 1:p) are considered for the atoms. The spin will be considered later in this Section. When considering a diatomic 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 orbitals. This approach is called the linear combination of atomic orbitals-molecular orbitals (LCAO-MO). It has been shown to be powerful as a tool for calculations of molecular orbitals. A schematic view of the energy levels, and how they change between 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.

(31) 23. electrostatic force field is cylindrically symmetric with respect to the internuclear axis. For one electron in a diatomic molecule, a precession of l takes place about h), where ml can take only the values the axis with constant component ml (¯ ml = l, l − 1, l − 2, ..., −l.. (3.2). l is projected onto the internuclear axis, the projection ml is denoted λ. The corresponding molecular states to λ=0, 1, 2, 3, ..., are designated σ, π, δ, φ, ... states, analogous to the mode of designation for atoms. Electron orbitals with rotational symmetry around the internuclear axis (z-axis by convention) are denoted 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 case is removed for the molecular case. The px,y orbitals are perpendicular to each. HCl. H. 13 17 26 277 208. Cl3pz+H1s(LUMO). 6σ* 3p. Ebin(eV) 2840. Core electrons. Valence electrons. 0. Cl. 2π 5σ. 3s. 1s. Cl3px,y(HOMO) Cl3pz+H1s Cl3s+H1s. 4σ Cl2p 2p. 3σ,1π. 2s. 2σ. 1s. 1σ. AO. MO. Cl2s. Cl1s AO composition. Figure 3.3: Schematic presentation of atomic orbitals, AO, and their linear combinations forming molecular orbitals, MOs in the HCl molecule. The composition of the MOs of AOs is also indicated. The solid (dashed) lines represent occupied (unoccupied) electronic orbitals. The coupling between the orbital and spin momenta of the electrons is not included in this description. other and the z-axis, forming propeller-like probability densities. These electron orbitals are denoted π-orbitals. An example is the highest occupied molecular orbital (HOMO) in HCl, 2π. For HCl, the 2π electron orbitals are non-bonding since they are localized around the Cl atom, and no orbitals of the same symmetry.

(32) 24. exist around the H atom for any molecular electronic orbital formation. Electron orbitals around an atom without any bonding/antibonding character to neighbours are denoted lone-pair orbitals. The HCl molecule has a (1σ)2 (2σ)2 (3σ)2 (1π)4 (4σ)2 (5σ)2 (2π)4 electronic configuration in the neutral ground state. An electronic configuration is the setup of occupied orbitals of the molecule. The superscript numbers give the number of electrons in each molecular orbital. When adding the numbers together one gets 18 electrons for the HCl molecule, as expected.. (a). 2Π 1/2. Ω=3/2 Λ=1 Ω=1/2. Σ=+1/2 Σ=-1/2. 2Π. 3/2 2Π 1/2. Ebin. 2Π 3/2. (b). Λ=1 Figure 3.4: (a) Vector diagrams and (b) Energy Level diagram for a 2 Π State (Λ = 1,. S = 12 ). 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 Λ and Σ. In (b), to the left the term is drawn without taking the interaction of Λ and Σ ˜ into account; to the right, taking account of it. The X-state in HCl+ is composed by a doublet which can be described in these terms.. When the electronic configuration of an atom/molecule is different from the ground state, then one often uses the convention to write only the orbitals which are 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 is added to the LUMO, then one writes (5σ)−2 (6σ ∗ )1 , where the ∗ marks the antibonding character of the orbital. The electrons in molecules with larger binding energies have orbitals which are localized around one atom, and the energy levels resemble the case of the electronic orbitals in atoms. These electrons are called core-electrons. In Fig. 3.3, the lowest lying MO orbitals 1σ, 2σ, 3σ and 1π are core-electrons, and the binding energy levels remain approximately the same as for the Cl atom. So far only the orbital angular momenta of the electrons have been considered for the HCl molecule. Now the intrinsic spin of the electron will be regarded. For one electron, the projection of the spin resultant s onto the internuclear axis, with h) = ± 21 , is denoted σ. (this quantum number must a constant component ms (¯ not be confused with the symbol σ for terms with λ = 0). The total electronic angular momentum about the internuclear axis, denoted ω, (or mj , which will be used in this Thesis), is obtained by adding λ and σ, just as the total electronic.

(33) 25. angular momentum j for atoms is obtained by adding l and s. Whereas, however, a vector addition has to be carried out for atoms, for molecules an algebraic addition is 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 the lower-case letters above are substituted by capital letters. An example is given in ˜ Fig. 3.4, where the (2π)−1 X-state is described. Just as for atoms, the multiplicity 2S+1 is added to the term symbol as a left superscript. Furthermore the value of Λ+Σ is added as a subscript (similar to J for atoms). The example in Fig. 3.4 deals with a 2 Π term whose components are designated 2 Π3/2 and 2 Π1/2 .. 3.3. Molecular dynamics. This Section aims at giving a better understanding of molecular dynamics. I have chosen a two-level system in the inner-valence region of HCl+ as a showcase. Two electronic configurations (4σ)−1 and (5σ)−2 (6σ ∗ )1 are associated with diabatic potential energy curves, see below. The molecular symmetry is the same for the electronic states, and the degeneracy of states of the same symmetry is prohibited for any internuclear distance. The electronic states are coupled, i. e. they are dependent of each other. Mixing of electronic configurations is possible, which results in configuration interaction (CI). For the coupled system, the resulting potential energy curves repel each other, which results in an avoided crossing. The consequence is adiabatic potential curves. To obtain overview, I have neglected other electronic configurations in the discussion, and the matter is certainly more complicated than in this showcase indicates. In Table 3.1 (see Section 3.3.5) the definitions for the adiabatic and diabatic frameworks are provided for guidance. The main reference for this Section is Ref. [12]. The two-level treatment was implemented on electronic configurations and adiabatic energy curves which stem from Ref. [13], since they play a central role for the understanding of the results presented 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 Schr¨ odinger equation. The Schr¨ odinger equation Hψi = Ei ψi ,. (3.3). describes a quantum system, in this case an HCl+ cation. The total energy Ei of the system is calculated by applying the Hamilton operator H to the wave function ψi . The HCl+ molecular ion is composed of two atoms having atomic numbers Z1 =1 and Z2 =17. The Cartesian coordinates and conjugate momenta for the N el electrons are denoted ra and pa , respectively. For the nuclei, R1,2 and P1,2 are used. When disregarding the electron spin, the Hamilton operator has the general form Hmol = Tnuc + Vnuc−nuc + Tel + Vel−nuc + Vel−el .. (3.4).

(34) 26. here the kinetic energy of the nuclei is T ≡ Tnuc =. P12 P2 + 2 2M1 2M2. (3.5). with M1,2 being the masses of the first and second nuclei. The kinetic energy of the electrons is given by N el p2a Tel = , (3.6) 2mel a=1 where mel is the electron mass and p = i¯ h∇. Since both kinds of particles are charged they interact via Coulomb forces. The repulsive Coulomb pair interaction between the electrons is e2 1 Vel−el = (3.7) 2 |ra − rb | a=b. (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 =. Z1 Z2 e2 . |R1 − R2 |. The attractive interaction between electrons and nuclei is given by Z2 e2 Z1 e2 Vel−nuc = − + . |ra − R1 | |ra − R2 | a a. (3.8). (3.9). All quantum mechanical information about the stationary properties of the molecular system defined so far is contained in the solutions of the time-independent nonrelativistic Schr¨ odinger equation Hmol ψi (r, R) = Ei ψi (r, R).. (3.10). Here and in the following, the set of electronic Cartesian coordinates are combined in the multi-indices r = (r1 , r2 , ..., rNel ). A similar notation is introduced for the nuclear Cartesian coordinates R = (R1 , R2 ). As it stands, Eq. (3.10) does not tell much about what one is aiming at, namely electronic excitation spectra, equilibrium geometries etc. However, some general points can be made immediately: 1. The solution of Eq. (3.10) will provide an energy spectrum Ei and corresponding eigenfunctions ψi (r, R). The energetically lowest state E0 is denoted the ground state. In the following, the more formal notation where the 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 distribution of electrons as well as on the arrangement of the nuclei..

(35) 27. 3. In the Hamilton operator in Eq. (3.4), the intrinsic spin of the electrons and nuclei have been neglected, since they are not needed for the discussion concerning the molecular dynamics in the following sections and related Papers I, II and III. However, in many cases, a Hamilton operator with the electron and even nuclear spin included is necessary for a description of the molecular system. Paper VI is an obvious case, where the consideration of electron 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 one treat that system in the context of the Schrödinger equation? I will start with a diabatic, static approach, based on a single configuration model. Only the electronic configuration (4σ)−1 is considered initially, and the nuclei are fixed at some point R(0) . In Fig. 3.5, R(0) has been chosen at the minimum of the diabatic curve Y1 . (In principle, any other R(0) can be chosen.) This simplifies the Schr¨ odinger equation considerably, since the nuclear dynamics in the Hamiltonian in Eq. (3.4) can be neglected. One has to consider only the electronic Schr¨ odinger equation, from which the eigenstate ϕ1 (r; R(0) ) and eigenvalue ε1 can be calculated: U (R(0) )ϕ1 (r; R(0) ) = ε1 ϕ1 (r; R(0) ), (3.11) Hel U where Hel (R(0) ) is the electronic Hamiltonian for the uncoupled diabatic electronic U state at R = R(0) . The superscript U in Hel (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) , due U (R(0) ). This is the meaning the notation throughout the text. to the operator Hel The same treatment can be done for the electronic configuration (5σ)−2 (6σ ∗ )1 , for which the energy ε2 and the eigenfunction ϕ2 (r; R(0) ) are obtained. The electronic U at R(0) . A second fundamental states ϕi (r; R(0) ) are only eigenfunctions of Hel restriction is that the electronic states have to be independent of each other: The symmetries of the states have to be different. If one wants to define ψ(r; R) at other internuclear distances than R(0) , and the symmetry of the states is the same, the following ansatz can be made for the Hamiltonian: U Hel (R) = Hel (R(0) ) + V (R, R(0) ). (3.12). where V (which will result in the coupling of the diabatic states) is: U V (R, R(0) ) = Hel (R) − Hel (R(0) ).. (3.13). U Two eigenstates ϕ1 (r; R(0) ) and ϕ2 (r; R(0) ) are known for Hel . These states form (0) a two-dimensional orthonormal basis for R = R for independent, uncoupled states. They correspond to energies ε1 and ε2 at R(0) . The states of the coupled system differ only slightly from those of the uncoupled system represented by the uncoupled diabatic electronic states ϕi , and one can hope to solve the equation for the complete molecular system:. Hmol ψ(r, R) = Eψ(r, R). (3.14).

(36) 28. Energy E. avoided crossing U2. Θ12 R(0) RC. Y1 Y2 U1. 2VC. V12 R. Figure 3.5: Schematic figure of an avoided crossing for a diatomic molecule with a twolevel electronic state system. The diabatic potential energy curves Yi corresponding to a fixed electronic configuration described by wave function ϕi (r; R(0) ) have degenerate values for a given internuclear distance RC . R(0) is an internuclear distance where the uncoupled diabatic electronic states have well-defined eigenenergies εi . If the symmetry of the states is the same, a static coupling Vij between the states will make coefficients ai (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 eigenstates φi (r; R) for all R if R can be assumed to change so slow that the electrons always can adapt adiabatically to the nuclear configuration changes. However, if this adiabatic approximation 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 nuclear wave functions. This is reasonable, since the complete system ψ(r, R) is composed of the electronic and nuclear wavefunctions ϕi (r; R(0) ) and ai (R). Eq. (3.15) is the 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) U (R(0) ) + V (R, R(0) ), = T + Vnuc + Hel. (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.

(37) 29 resulting Schr¨ odinger equation by ϕi | from the left and integration over the two electronic coordinates yields the following equation for the expansion coefficients ai (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) (3.17) = Eai (R). where Ti = ϕi |T |ϕi . (3.18). Vi = ϕi |Vnuc |ϕi . (3.19). and N. b.: The index i for the kinetic and potential energy matrix elements for the nuclei Ti and Vi (see Eq. 3.5) is related to the index of the electronic state ϕi , not the nuclei with coordinates Ra . Note that the off-diagonal terms U U (R(0) )|ϕj = Hel (R(0) )ϕi |ϕj ϕi |Hel ϕi |T |ϕj = T ϕi |ϕj . ≡ 0 ≡ 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 coefficients ai=1,2 (R) is obtained: ε1 0 V1 0 T1 0 + + + 0 T2 0 ε2 0 V2 a1 (R) a1 (R) V11 V12 = E , (3.22) + a2 (R) a2 (R) V21 V22 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 one introduces the Hamiltonian for the motion of the nuclei in the electronic state |ϕi as (3.24) Hϕi = Ti + Yi (R) one gets. Yi = 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 diabatic U electronic energies related to Hel (R(0) ) in Eq. (3.11). Eq. (3.22) then simplifies to: a1 (R) a1 (R) Y1 0 0 V12 T1 0 =E . + + V21 0 0 Y2 a2 (R) a2 (R) 0 T2 (3.26).

(38) 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 static coupling, since it stems from the potential energy operator in Eq. (3.13). What is the relation between the diabatic electronic energies Yi and the eigenvalues E in Eq. (3.26)? The condition for the existence of non-trivial solutions of this pair of equations is that the determinant of the coefficients of the constants a1 (R) and a2 (R) should disappear:. T1 + Y1 − E. V12. =0 (3.27). V21 T2 + Y2 − E This condition is satisfied for the following values of E: E± =. 1 1. [(T1 + Y1 ) + (T2 + Y2 )] ± [(T1 + Y1 ) − (T2 + Y2 )]2 + V12 V21 2 2. (3.28). If one assumes fixed nuclei: T1 = T2 = 0, the only energy E, which remains in Hmol is the potential energy, which for the coupled system is defined as U , i. e. U = E. Then Eq. (3.28) simplifies to. U1 = 21 (Y1 + Y2 ) + 12 (Y1 − Y2 )2 + 4V12 V21 (3.29) U2 = 12 (Y1 + Y2 ) − 12 (Y1 − Y2 )2 + 4V12 V21 The corresponding potential energy curves are shown as Ui in Fig. 3.5. The energy levels are driven apart, and their crossing is prevented, This is called an avoided crossing. A second general feature can also be seen in Fig. 3.5: the effect of 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 same energy (Y1 = Y2 ) at RC , then U1 − U2 = 2VC where VC ≡. V12 (RC )V21 (RC ).. (3.30) (3.31). Eq. (3.29) also shows that the stronger the coupling Vij , the stronger the effective repulsion of the levels. When the is coupling absent, Vij = 0 and Ui = Yi , the potential energy curves correspond to the non-coupled states. The fact that the diabatic two-level system results in potential curves U i when the static coupling Vij is implemented, might be confusing. Ui are denoted adiabatic potential curves, which I come back to in the next Section. Adiabatic potential 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 adiabatic case: The system is well-defined on one adiabatic level solely, associated with a single Ui . The ansatz Ti ≡ 0 is hereby equivalent to the Born-Oppenheimer approximation in the adiabatic framework..

(39) 31. 3.3.3. The adiabatic framework. Now a second approach will be used, complementary to the diabatic one described above. It is based upon the fact that electrons move much faster than the nuclei due to the large difference in mass (mel /Mnuc < 10−3 ). Here, the electronic degrees of freedom can be considered to respond instantaneously to any changes in the nuclear configuration, i. e. their wave function always corresponds to a stationary state. In other words, the interaction between nuclei and electrons, Vel−nuc , is modified due to the motion of the nuclei only adiabatically and does not cause transitions between different stationary electronic states. Thus, it is reasonable to define an electronic Hamiltonian which has a parametric dependence on the nuclear coordinates: Hel (R) = Tel + Vel−nuc + Vel−el .. (3.32). As a consequence the solutions of the time-independent electronic Schrödinger equation describing the motion of the electrons in the electrostatic field of the stationary 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 discussion has been restricted to a two-level system, the index i, which labels the adiabatic electronic states, is limited to 1 and 2 only. The adiabatic electronic wave functions φi (r; R) = r; R|φi define a two-level basis. Hence, given the solutions to Eq. (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 is possible to derive an equation for their determination after inserting Eq. (3.34) into Eq. (3.3). One obtains Hmol ψ(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 coefficient χi (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) + φi (r; R)|T |χj (R)φj (r; R) j=1,2. = Eχi (R).. (3.36).

(40) 32. Since the electronic wave functions depend on the nuclear coordinates, Eq. (3.5) h∇a , and using the product rule for differentiation (the is used, where Pa = −i¯ underlined expression is the same as in Eq. (3.36)) T φj (r; R)χj (R) = =. 1 2

(41) {[Pn φj (r; R)] χj (R) 2Mn n=1,2 . + 2 [Pn φj (r; R)]Pn χj (R).

(42) + φj (r; R)Pn2 χj (R)}.. (3.37). The last term is simply the kinetic energy operator acting on χj . The other terms can be comprised into the so-called nonadiabaticity operator (the expressions here and in Eq. (3.37) in the braces are related),

(43) . Θij = φi (r; R)| T |φj (r; R) +. 1 φi (r; R)| Pn |φj (r; R)Pn . Mn.

(44) n=1,2. (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 Schr¨ odinger equation for the motion of the nuclei with the χi (R) being the respective wave functions. The solution of Eq. (3.39) is still exact, and requires a knowledge of the electronic spectrum for all configurations of the nuclei which are covered during their motion. It is convenient to introduce the following effective potential for nuclear motion if 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 0. 0 T2. +. U1 (R) 0. 0 U2 (R). +. 0 Θ21. Θ12 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 molecular wave 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 , see Eq. (3.40). • Θij (i = j), couples the adiabatic electronic states. It is the origin of the dynamic coupling..

(45) 33. Until now, no approximation has been implemented. Eq. (3.41) represents the complete solution for the system. Solving the coupled Eqs. (3.41) is often a formidable task. To simplify the computations, approximations have to be made. Often the nonadiabatic couplings Θij are rather small. The neglect of these couplings, i. e. Θij ≡ 0, is called the Born − Oppenheimer approximation. For this case Eq. (3.41) is completely diagonalized. If the system starts on one adiabatic energy curve, i. e. |χi = 1 and |χj=i = 0, then the system remains well-defined on that energy curve. The solutions of Eq. (3.42) are again labelled by v . The total adiabatic wave function the becomes ψi,v (r, R) = χi,v (R)φi (r; R).. 3.3.4. (3.43). The Landau-Zener formula. The Landau-Zener formula, which has been used to qualitatively explain the results in Papers I, II and III, stems from the early 30’s, and was developed independently by Lev Landau and Clarence Zener. Landau considered a scattering of two atoms whereas 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 :. δYi (R) . (3.44) Fi = − δR R=RC 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 at the crossing point and time: ∆R = vC t. (3.46) Now the description of the system has an explicit time dependency. With these approximations, the Hamilton operator for the motion of the nuclei in the electronic state |ϕi (see Eq. (3.26)) can then be written: Hi = Ti + Yi (RC ) + Fi vC t + Vij ,. (3.47). The two-level system in Eq. (3.26) then becomes: . T1 0. 0 T2. F1 vC t 0. + 0 F2 vC t. Y1 (RC ) 0. . . +. 0 V21. 0 Y2 (RC ) V12 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.

(46) 34. Energy E. lLZ. Y1. F1(RC). 2VC. Y2. F2(RC). Rc. R. Figure 3.6: Two diabatic potential curves Yi (dashed lines) corresponding to diabatic. states |ϕi as a function of internuclear distance R. The diabatic states are degenerate at an internuclear distance Rc . VC is the diabatic static coupling at RC . The Landau-Zener length lLZ is shown. Fi are the gradients of Yi at RC .. |ϕ2 . The asymptotic value of the survival probability of the system for remaining in the state |ϕ1 , P|ϕ1 ≡ P|ϕ1 (t = ∞), is the square of the transition amplitude 2. Pϕ1 = |a1 |ϕ1 |U(−∞, ∞)| a1 |ϕ1 | ,. (3.50). . where the time-evolution operator U(t , t) is given by F1 vC t+V12 . Interestingly, the present 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 Massey parameter ξ which is defined as 2. ξ=. 2π |V12 | . vC |F1 − F2 |. (3.52). Introducing the Landau-Zener length 2. lLZ =. 2π |V12 | , |F1 − F2 |. (3.53). the Massey parameter becomes ξ = lLZ. |V12 | . vc. (3.54).

(47) 35. Now let us see what happens with a small velocity vC around the crossing: the Massey 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 electronic state |ϕ2 . With a large velocity vC , the system would get more blend from the 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 I will try to make a short summary in simple words without using any mathematical expressions. The complete Schr¨ odinger 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 states associated with two electronic configurations (4σ)−1 and (5σ)−2 (6σ ∗ )1 . The associated diabatic potential curves Yi are degenerate at some internuclear distance RC , where the potential curves cross. If the overall geometry of the molecular system is the same, the degeneracy at RC is forbidden. The potential energy curves repel, and the result is an avoided crossing. In the diabatic framework, this is due to a static coupling term Vij . When the static coupling term Vij ≡ 0, which is the case for diabatic electronic states with different symmetries, the diabatic potential curves Yi are sufficient for the description of the system. The second, adiabatic, approach is based upon the assumption that the electrons move infinitely fast as compared to the nuclei due to their lower mass. For the adiabatic electronic states, the electrons adapt instantaneously to the nuclear configuration changes. The adiabatic potential curves associated with the electronic states repel each other if the symmetry of the states is the same. The repelling results in an avoided crossing. If the nuclei move fast enough, one has dynamic coupling between adiabatic states. Note that in the diabatic case, the static coupling results in repelling of the electronic states. On the contrary, for the 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 nuclear motion can be neglected, both approaches give the same avoided crossing around RC . In the diabatic case, if a static coupling between originally non-coupled diabatic curves is implemented, the same adiabatic curves are obtained as in the adiabatic case, where the dynamic coupling between adiabatic curves is neglected. Why is there a need for two complementary frameworks? The system in consideration determines the framework which gives the ”cheapest” representation. If the static coupling Vij is strong, and the dynamic coupling Θij is small, then the adiabatic representation is favourable. The system can be described with two non-coupled Schr¨ odinger equations, and the two-level Eq. (3.41) can be simplified in two separate expressions. On the other hand, if the static coupling Vij is.

(48) 36. Table 3.1: An overview of the diabatic and adiabatic framework. Index i = 1, 2 for a two-level system. Some general definitions are shown as well. Diabatic Adiabatic Wavefunction 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: E ≡ E(R) ε ≡ ε(R(0) ) Effective potential energy for the nuclei (Potential energy curves): Ui ≡ Yi (R) Yi ≡ Yi (R) Coupling between electronic states: Θij ≡ Θij (R); i = j Vij ≡ Vij (R, R(0) ); i = j Coupling at internuclear distance R : C. VC ≡ V12 (RC , R(0) )V21 (RC , R(0) ) Contribution to potential curves: Θii ≡ Θii (R) Vii ≡ Vii (R, R(0) ) Approximation implemented for obtaining potential curves: Θij ≡ 0; i = j Vij ≡ 0; i = j Born-Oppenheimer approximation: Ti ≡ 0 Θij ≡ 0; i = j General definitions ψi (r, R) ≡ r, R|ψi Hmol Hel ≡ Hel (R) V (R, R(0) ) = Hel (R) − H (0) (R(0) ) H (0) ≡ Hel (R(0) ) E Vel−nuc Vel−el Vi ≡ Vnuc−nuc,i Tel Ti ≡ Tnuc i i (R) F = − δYδR. R=RC. ra , p a Ra , Pa r ≡ (r1 , r2 , ..., rNel ) R ≡ R1 − R2 RC. Mol. w. f. obtained by projecting the state vector onto (r,R) representation Hamiltonian for molecule Hamiltonian for electrons ”Perturbation” Hamiltonian for el. at f ixed R(0) Energy for total system Pot. energy between el.-nuclei Pot. energy between el. Pot. energy between nuclei for el. state i Kin. energy op. for el. Kin. energy op. for nuclei for el. state i Gradients of Y at R = RC Position and momentum of el. a Position and momentum of nucleus a Multi-indices of pa Internuclear distance Internuclear distance where Y1 = Y2.

(49) 37. weak, and the dynamic coupling Θij is strong, the diabatic framework will simplify Eq. (3.22) in two non-coupled expressions. The term adiabatic stems from thermodynamics. In a science dictionary, the word is explained as without loss or gain of heat. In a small quantum mechanical system, heat is not a relevant quantity, but can be understood in terms of the velocity, for which the system develops. A thermodynamic system does not lose or gain heat to the surrounding during a sufficiently fast, adiabatic process. For a quantum mechanical system an adiabatic process, the electrons move sufficiently fast compared to the nuclear motion for the wave function to remain on the adiabatic potential curves, the thick lines in Fig. 3.5, corresponding to the adiabatic electronic states |φi . The term diabatic can be understood as the negation of adiabatic, where the electron/nuclear motion velocity ratio not is sufficiently large for the system to remain on an adiabatic potential curve. Instead it keeps the electronic 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 adiabatic and diabatic approximations in a non-trivial case. Papers I, II and III deal with a phenomenon observed in the inner-valence photoionization bands of HCl and DCl, where the understanding is based upon the insight that neither the diabatic nor the 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 for different interatomic distances R for a diatomic molecule, see Section 3.3. The calculated potential energy can be plotted vs the distance R. Characteristic potential energy curve plots are shown in Fig. 3.7. For some φi , the potential curves have local minima. Re is the equilibrium distance corresponding to the minimum of the potential energy curve, around which the molecule vibrates. Other φ have no minima, and the potential energy decreases monotonously with R. These φ are associated with continuum states, where the molecule dissociates until the fragments no longer interact. When solving the Schr¨ odinger equation for bound states, the allowed energies are distributed noncontinuously. For continuum states, all energies are allowed. The nuclear wave function χ characteristics for the associated states are shown on the potential curves in Fig. 3.7. The indexes v = m are the same as in Eq. (3.43). For the ground electronic state φg , for small molecular systems as HCl, the only energy level 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 ground state will be distributed around Re . Assuming a harmonic oscillator potential, which is a good approximation for the lower part of a bound potential curve, the wave function will look like a Gaussian for the lowest lying state, as shown in Fig. 3.7. At the two classical limits of the wave function, where the kinetic energy part is zero, one can assume that the wave function is so small as to be negligible outside the classical limits. (Even though an exponential tail is allowed outside of the classical limits, due to quantum mechanical arguments.) One way to define the Franck-Condon region, shown in Fig. 3.7, which is used for showing the.

(50) Potential energy. Franck-Condon region 0 Binding energy. Franck-Condon factor spectrum. 38. φ2. Final cation states. φ1 χ v’=2 χ v’=1 χ v’=0 Ground φg neutral state χ v=0 Re. R. Figure 3.7: Potential energy curves corresponding to the electronic wavefunctions φi for the ground state of the neutral molecule and two states of the ionized molecule. For the bound 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 molecular system, using light, electrons or some other external energy source, the change of electron configuration is assumed to occur instantaneously, which is referred to as 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 configurations of the electrons and nuclei, represented by the separated wavefunctions φ and χ. The separation of the wavefunctions is permitted within the BornOppenheimer approximation. The transition moment between initial and final states can be written as φi χv =m |µ| φg χv=0 = µφi φg χv =m |χv=0 = µφi φg S(χv =m , χv=0 ). (3.55). where µφi φg is a constant which stem from the operator µ working on the electronic states 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 electronic states. The transition dipole moment is therefore largest between nuclear states that have the greatest overlap. The relative intensities of the lines are proportional.

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