Relaxed Core Hartree Fock Approximation Research Articles

Molecular-frame photoelectron angular distributions during double core-hole generation in and molecules

Angular distributions of photoelectrons emitted upon double core-hole (DCH) generation in nitrogen and oxygen molecules are studied theoretically in the frame of a molecular reference. The respective electronic structure calculations are performed by the single center method for photoelectron kinetic energies up to 40 eV in the relaxed-core Hartree–Fock approximation. The molecular frame photoelectron angular distributions are computed for single-site and two-site DCH creation processes and further analyzed for different orientations of the molecular axis with respect to the electric field vector of linearly polarized incident light and for localized or delocalized emitting atomic site scenarios. The present theoretical results provide reliable predictions for future experiments with high-repetition free-electron lasers.

Photoelectron circular dichroism of O 1s-photoelectrons of uniaxially oriented trifluoromethyloxirane: energy dependence and sensitivity to molecular configuration.

The photoelectron circular dichroism (PECD) of the O 1s-photoelectrons of trifluoromethyloxirane (TFMOx) is studied experimentally and theoretically for different photoelectron kinetic energies. The experiments were performed employing circularly polarized synchrotron radiation and coincident electron and fragment ion detection using cold target recoil ion momentum spectroscopy. The corresponding calculations were performed by means of the single center method within the relaxed-core Hartree-Fock approximation. We concentrate on the energy dependence of the differential PECD of uniaxially oriented TFMOx molecules, which is accessible through the employed coincident detection. We also compare the results for the differential PECD of TFMOx to those obtained for the equivalent fragmentation channel and similar photoelectron kinetic energy of methyloxirane (MOx), studied in our previous work. Thereby, we investigate the influence of the substitution of the methyl group by the trifluoromethyl group at the chiral center on the molecular chiral response. Finally, the presently obtained angular distribution parameters are compared to those available in the literature.

New developments in the theory of molecular K‐shell photoionization

AbstractA new version of the relaxed core Hartree–Fock approximation with a fractional charge of the molecular ion state is applied for calculations of molecular photoionization processes. The set of continuous spectrum wave functions obtained by that method is used when taking many‐electron correlations into account in the framework of the random‐phase approximation. The vibrational motion is also included into consideration. In that way, the most reliable results for the vibrationally resolved partial photoionization cross sections and the angular asymmetry parameter β for the K‐shells of CO and N2 molecules have been obtained. A good agreement with the state‐of‐the‐art experimental data is demonstrated. From our calculations it follows that the nondipole effects for the K‐shells of N2 and CO molecules are small near the ionization thresholds, in agreement with the latest experimental data. Two predictions have been made based on these calculations. One of them is the appearance of the correlational maximum in the photoionization cross section of the 1σu shell of N2 molecule near the σ* shape resonance, which is now fully confirmed by the experiment. The second prediction concerns the behavior of the nondipole parameters for the 1σg and 1σu shells of N2. According to our calculation, they oscillate in antiphase with rather large amplitudes around the unresolved values, so that after summation of the contributions of the two K‐shells, the oscillations practically disappear. This prediction does not yet have an experimental confirmation. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Nondipole effects in the angular distribution of photoelectrons from the CKshell of the CO molecule

Measurements and calculations of a contribution of the nondipole terms in the angular distribution of photoelectrons from the C $K$ shell of randomly oriented CO molecules are reported. In two sets of measurements, the angular distribution in the plane containing the photon polarization and the photon momentum vectors of linearly polarized radiation and the full three-dimensional photoelectron momentum distribution after absorption of circularly polarized light have been measured. Calculations have been performed in the relaxed core Hartree-Fock approximation with a fractional charge. Both theory and experiment show that the nondipole terms are very small in the photon energy region from the ionization threshold of the $K$ shell up to about $70\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ above it.

Complete experiments in photoionization of atoms and molecules

The general discussion of possible complete experiments in photoionization of atoms and molecules is presented. In atoms the complete experiment must include measurements of the angular distribution and spin polarization of photoelectrons and the alignment or/and orientation of the residual ion. Two relations connecting the spin polarization parameters and the parameters characterizing the ion polarization state are presented. The complete experiments performed for Xe atom demonstrated a good agreement between theory and experiment. The complete experiments in photoionization of diatomic molecules can be performed by measuring photoelectron angular distributions for fixed-in-space molecules with circularly or elliptically polarized light. The use of additional experimental data like the angular asymmetry parameter β and the ratio of π to σ symmetry resolved cross sections made it possible to extract as many as 16 dipole matrix elements and phase shift differences for the C K-shell of CO molecule. Comparison with the recent calculations in the relaxed core Hartree–Fock approximation revealed discrepancies between theory and experiment.

Vibrationally resolved photoionization of the O K-shell of CO molecule

Simultaneous theoretical and experimental study of the angular asymmetry parameter β for the O K-shell of CO molecule has been performed. Transitions from the ground vibrational level of the neutral molecule to the vibrational states of the ion with v ′ = 0 , 1 , 2 , were studied in the region of the σ ∗ shape resonance. Calculations in the relaxed core Hartree–Fock approximation with a fractional charge are in a qualitative agreement with the experiment. The theory demonstrates that the dipole amplitudes with different orbital angular momentum l have different behavior with increasing v ′ which explains the observed variation of β .

O K-shell photoemission of the CO molecule: comparison between theory and experiment

Photoionization cross sections, dipole matrix elements and phase shifts are calculated for the O K-shell of the CO molecule using two theoretical models. One of them is a new version of the relaxed core Hartree–Fock approximation with a fractional charge. The value of this charge is considered as a fitting parameter. In this paper, the fractional charge 0.7 was used which gives a correct position of the maximum of the σ* shape resonance in the photoionization cross section. The other model is a modification of the multiple scattering method with overlapping atomic spheres. The dipole matrix elements and phase shift differences calculated by these two methods are compared with the corresponding values determined from the complete experiment by Cherepkov et al (2000 J. Phys. B: At. Mol. Opt. Phys. 33 4213).

Interplay of different partial waves on vibrationally resolved photoionization of the OKshell of the CO molecule

The asymmetry parameter $\ensuremath{\beta}$ for the O $K$ shell photoemission of the CO molecule has been measured for different vibrational states of the residual ion. The corresponding calculations performed by us in the relaxed core Hartree-Fock approximation are in qualitative agreement with the experiment. From the analysis of theoretical data, we made the conclusion that the variation of $\ensuremath{\beta}$ with the vibrational quantum number is connected with different dependences of the dipole matrix elements in the $s\ensuremath{\sigma}$, $d\ensuremath{\sigma}$, and $f\ensuremath{\sigma}$ channels on the internuclear distance $R$ in the $\ensuremath{\sigma}*$ shape resonance.

Theoretical study of vibrationally resolved photoionization for the C K-shell of the CO molecule

Partial photoionization cross sections and the angular asymmetry parameter β for the C K-shell of the CO molecule have been calculated in the relaxed core Hartree–Fock approximation. In contrast with all previous calculations, we introduced the relaxed core potential with a fractional charge of the ion which was fitted from the condition to get an agreement with the experimental position of the σ* shape resonance for the K-shell cross section. We found the optimal value of the fractional charge to be 0.5. After performing calculations within the Born–Oppenheimer approximation for several fixed internuclear distances R, we averaged the R-dependent dipole amplitudes over R with the vibrational wavefunctions of the initial and final states. Transitions from the ground vibrational level to several vibrationally excited states of the residual molecular ion have been studied within this approximation. A good agreement was found with the recently published vibrationally resolved experimental data.

Shape-resonance-enhanced vibrational effects in the angular distributions of C 1s photoelectrons from fixed-in-space CO molecules.

Angular distributions of C 1s photoelectrons from fixed-in-space CO molecules have been measured with vibrational resolution. A strong dependence of the angular distributions on the vibrational states of the residual molecular ion has been found for the first time in the region of the shape resonance. Calculations in the relaxed core Hartree-Fock approximation have reproduced the angular distributions fairly well in the general shapes of the angular distributions due to the correct description of nuclear motion as an average of the internuclear-distance-dependent dipole amplitudes.

Generalization of atomic random-phase-approximation method for diatomic molecules. II.N2K-shell photoionization

Partial and total photoionization cross sections of the K shell of the ${\mathrm{N}}_{2}$ molecule are calculated using the generalization of the random-phase approximation that has been applied earlier to the valence shells of ${\mathrm{N}}_{2}$ [Phys. Rev. A 61, 032704 (2000)]. At zero order the relaxed core Hartree-Fock approximation is used. It is demonstrated that due to strong intershell correlations the $\ensuremath{\sigma}*$ shape resonance reveals itself not only in the $1{\ensuremath{\sigma}}_{g}\ensuremath{\rightarrow}\ensuremath{\epsilon}{\ensuremath{\sigma}}_{u}$ channel as it takes place in all single-particle approximations, but also in the $1{\ensuremath{\sigma}}_{u}\ensuremath{\rightarrow}\ensuremath{\epsilon}{\ensuremath{\sigma}}_{g}$ channel. The influence of vibrational motion on the cross sections is taken into account. Good agreement with the most recent experimental data for different partial cross sections is achieved.

K-shell photoionization of CO: I. Angular distributionsof photoelectrons from fixed-in-space molecules

Angular distributions of photoelectrons from both C and O K-shells ofthe fixed-in-space COmolecule have been measured using the angle-resolvedphotoelectron-photoioncoincidence technique. The measurements have been performed at severalphotonenergies from the ionization thresholds up to about 30 eV above them,where theσ* shape resonances occur. Experimental results are comparedwith the multiple-scattering calculations of Dill et al (1976 J. Chem. Phys. 653158) and with ournew calculations in the relaxed-core Hartree-Fock approximation. Ourcalculations are ina better agreement with the experimental data though numericaldiscrepancies remain.The experimental angular distributions are fitted by the expansion inLegendrepolynomials containing up to ten terms and the extracted parameters arecompared withthe corresponding theoretical values.

Photoelectron Angular Distributions from OKShell of Oriented CO Molecules: A Critical Comparison between Theory and Experiment

The dynamical information (ten dipole matrix elements and eight phase differences) has been deduced from the measured angular distributions of photoelectrons from O K shell of oriented CO molecules near the ionization threshold in the region of a sigma(*) shape resonance. Light polarization parallel and perpendicular to the molecular axis has been used. An important contribution of six lsigma partial waves with 0</=l</=5 to the sigma(*) shape resonance is demonstrated. A comparison with our calculations in the relaxed core Hartree-Fock approximation reveals only a qualitative agreement, therefore a more advanced theory is needed.

Calculation of the dipole-allowed excitations of the3p44s4pand3p43d4pconfigurations in argon

The energy spectrum of the argon ${3p}^{4}4s4p$ and ${3p}^{4}3d4p$ configurations with a total $J=1$ was calculated within the single-configuration relaxed-core Hartree-Fock approximation. The spin-orbit interaction was included in the calculation. The results indicate strong mixing of the ${}^{1}P$ terms which are the ones allowed to couple to the ground state via electric-dipole excitation. Most experimental lines could be identified using this calculation. The calculated spin-orbit splittings within groups of levels are in very good agreement with experiment. However, differences of the order of 1 eV between experiment and theory were found for the absolute positions of groups of levels.

Theoretical study of inner shell photoabsorption spectra of simple molecules

Carbon and oxygen photoabsorption K- spectra of the CO and O 2 molecules are calculated within a one-centre relaxed core Hartree-Fock approximation with inclusion of both single and multiple photoabsorption. Photoabsorption cross sections are calculated by the methods of non-orthogonal orbitals theory. Results of the calculations are compared with the experiment.

Molecular K-shell excitation spectra in the relaxed-core Hartree-Fock approximation

The relaxed-core Hartree-Fock (RCHF) approximation is applied to discrete K-shell (or deep inner shell) excitations in molecules. As an example, the O 1s excitations in H 2O are considered. A recently proposed method is used to evaluate the N-electron transition moments and to deal with the concomitant non-orthogonality problem arising from the “frozen” and “relaxed” orbital representations of the intiial and final states, respectively. The comparison of the RCHF results for H 2O with those of a largescale polarization propagator treatment shows a very satisfactory agreement both for term values and oscillator strengths. By contrast, only poor results are obtained in the frozen-core Hartree-Fock (FCHF) scheme neglecting the relaxation of the valence electrons in the final state. Due to its simplicity the RCHF approximation allows one to consider large molecules and to employ rather extended SCF basis sets. Thus, this method may be particularly useful to determine quantum defect parameters for core-excited Rydberg states in polyatomic molecules.

Molecular K-shell photoionization cross sections in the relaxed-core Hartree-Fock approximation.

The relaxed-core Hartree-Fock (RCHF) approach to the calculation of K-shell photoionization cross sections is analyzed and applied to K-shell single-hole ionization in CO. A direct method based on the Schwinger variational principle and single-center-expansion techniques is used to generate the continuum orbitals associated with the motion of the photoelectron in the direct and exchange potential of the relaxed ion. A method is presented for evaluating the N-electron transition moment, a step that has posed a considerable computational obstacle due to the lack of orthogonality between the frozen and relaxed orbitals in the initial and final N-electron states, respectively. Besides being very practical and efficient, this formulation establishes the distinction between the ``direct'' and ``conjugate'' part of the transition moment, introducing bound-free dipole and overlap integrals, respectively. Whereas for large photoelectron energies the conjugate terms can be neglected, they become important near threshold, contributing, for example, up to 30% to the 1s cross sections in CO. An analysis by means of low-order perturbation theory shows that the RCHF model correctly describes the effect of ionic relaxation, that is, essentially the screening of the 1s hole by the valence electrons. As a consequence the ${\ensuremath{\sigma}}^{\mathrm{*}}$ shape resonance is substantially shifted to higher energy and broadened compared with the frozen-core Hartree-Fock picture where the more attractive unscreened 1s-hole potentials are used. The remaining discrepancies with the experimental results are attributed to the neglect of target polarization in the RCHF model.