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Relativistic calculation of the SeH2 and TeH2 photoelectron spectra
Chemical Physics 329 (2006) 256–265 www.elsevier.com/locate/chemphys
Relativistic calculation of the SeH2 and TeH2 photoelectron spectra Markus Pernpointner Theoretische Chemie, Universita¨t Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany Received 28 April 2006; accepted 13 July 2006 Available online 20 July 2006
Abstract Photoelectron (PE) spectra provide detailed insight into the electronic structure of atoms, molecules and solids. Hereby electron correlation and relativistic effects influence the structure of the PE spectrum in a complicated way necessitating a consistent theoretical treatment. By embedding the one-particle propagator technique in a four-component framework the interplay between relativistic and correlation effects can be described correctly. In this article the Dirac–Hartree–Fock algebraic diagrammatic construction scheme (DHF-ADC) together with recent applications is reviewed and fully relativistic PE spectra of SeH2 and TeH2 in combination with basis set studies are presented. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Photoelectron spectra; Relativity; Spin–orbit effects; Hydrogen selenide; Hydrogen telluride; One-particle-propagator
1. Introduction In the PE spectra of light elements a pronounced spin– orbit splitting is already observable (e.g., in the neon atom DSO(2p1/2/2p3/2) amounts to 0.12 eV at the DHF level and to 0.1 eV at the experimental level) and additional scalar relativistic effects can further shift the peak centers (0.15 eV for Ne 2s). Except for the scalar relativistic effects that can be taken into account by relativistic effective core potentials (RECP) an outer or inner valence spin–orbit structure together with electron correlation effects could not be properly described by the methods available so far. In general, ionization energies and electron affinities can be obtained by the one-particle propagator (Green’s function) technique [1–4]. Amongst various approximation techniques the algebraic diagrammatic construction (ADC) scheme [5] is an elegant approach to the one-particle propagator and was embedded in the four-component formalism [6,7]. The PE spectra of various systems have already been calculated at the DHF-ADC(3) level where the number in parentheses indicates the order of perturbation theory to which the ADC scheme is complete. As a first application fully relativistic outer valence PE spectra of the noble gas E-mail address: [email protected]. 0301-0104/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2006.07.019
atoms, carbon monoxide and cyanogen iodide were calculated [6]. It was observed that the outer valence energies of the noble gas atoms deviate in a systematic manner from the experimental results. Whereas in the Ne atom the experimental value is overestimated by +0.1 eV an increasing underestimation for the remaining elements Ar, Kr and Xe of 0.05 eV, 0.10 eV, 0.27 eV has occurred. One might be tempted to explain this trend with a size-consistency problem but the ADC method was shown to be inherently size-consistent [5,8], a property that can be tracked down to the cancellation of unlinked diagrams in the diagrammatic expansion of the Green’s function [9,1]. More recent investigations have shown that for mediumsize systems substantial extensions to the basis sets are necessary in order to achieve the same quality as for smaller systems. Especially the presence of numerous polarization and diffuse functions is hereby required (see below). However, the experimentally observed spin–orbit splittings in the noble gases are well reproduced in all cases (see Table 1). It should also be noted that ADC(3) bears a systematic error of approximately 0.2 eV due to its restriction to third order and quickly becomes inaccurate for inner valence and core orbitals due to their large relaxation effect. The smallness of the relativistic effects in CO allowed for a direct comparison with nonrelativistic (NR) ADC(3)
M. Pernpointner / Chemical Physics 329 (2006) 256–265 Table 1 Calculated and experimental values (in eV) for the spin–orbit splittings in the noble gas atoms Level
DHF-ADC(3)
Experiment
Ne 2p1/2,3/2 Ar 3p1/2,3/2 Kr 3d3/2,5/2 Kr 4p1/2,3/2 Xe 4d3/2,5/2 Xe 5p1/2,3/2
0.11 0.19 1.19 0.67 1.99 1.3
0.1 0.2 1.2 0.68 2.0 1.3
results. Only an error of 0.08 eV for the X2R+ and A2P cationic final states was obtained with respect to experiment. For these states the molecular orbital picture of ionization is an accurate approximation [10] and the final X2R+ and A2P states mainly correspond to an ionization from the 5r and 1p orbital. The validity of the MO description does not imply the absence of electron correlation in the outer valence region, it just reflects the fact that the 1h configuration has a dominant coefficient in a possible CI expansion of the true state and that the other coefficients of coupling 2h1p configurations are small. For inner valence states a much stronger coupling between the 1h and 2h1p configurations takes place which is accompanied by the occurrence of numerous satellite lines of low intensity (breakdown of the MO picture). A detailed analysis of the satellite states can also be found in [6]. Cyanogen iodide is a molecule with one ‘relativistic’ element and was extensively studied in the one-component framework [11] enabling a direct comparison to the fully relativistic treatment. A pronounced spin–orbit splitting of the outer valence X2P1/2,3/2 states of 0.54 eV is experimentally observed and well reproduced by DHF-ADC(3) as 0.57 eV [7] despite a considerable absolute error of 0.36 eV. The single nonrelativistic ADC(3) value of 10.89 eV for the first ionization energy perfectly matches the first spectral peak position at 10.90 eV but does not correspond to the weighted average of both spin–orbit split states that should lie at 11.08 eV. Insofar the one-component result bears an error of 0.19 eV being slightly superior in accuracy. The reason for the deviation in the relativistic case has its origin in a possible lack of more diffuse basis functions but computational restrictions did not allow for a further basis set extension at that time. For all systems discussed so far the spin–orbit splittings were all well reproduced even in case of more substantial deviations from the experimental peak position. One could therefore assume similar correlation contributions to the individual spin–orbit states corresponding to additivity of both effects. For not too heavy systems as the ones described above large deviations from this approximation could not be found. However, the situation changes significantly in heavy (Z > 70) systems such as in PtCl2 6 whose PE spectrum was calculated in [12]. It was observed that electron correlation now contributes to a different extent to individual spin–orbit components resulting in a substantial alteration of the level order with respect to one-compo-
257
nent calculations. The overall spectral features were then better reproduced than in the one-component scalar relativistic calculations which further corroborates the importance of taking spin–orbit coupling into account. Noble gas compounds exhibit unusual bonding situations (see for example [13–15] and references therein) and are therefore an ideal subject for photoelectron spectroscopy in order to gain more insight into the electronic structure. Extensive DHF-ADC(3) calculations of ionization spectra were performed on the xenon fluorides XeFn (n = 2,4,6) [16] and the influence of electron correlation studied. Corresponding one-component results for the xenon fluorides were already available [17] but without a consistent inclusion of relativistic effects. For these PE spectra the role of spin–orbit splitting could be clarified and a better resemblance of the experimental features was achieved. When one asks about relativistic effects there is also the correction to the electron–electron interaction. In the conventional four-component approach one applies a Hamiltonian consisting of a sum of one-electron Diracoperators in combination with the conventional Coulomb repulsion operator representing the instantaneous interaction between electrons. The Coulomb interaction term is not invariant under a Lorentz transformation and in order to account for this deficiency corrections derived from quantum electrodynamics (QED) are introduced [18]. The correction term describes the exchange of virtual photons between the electrons and is frequency-dependent. If one expands these terms in the limit of low frequencies one obtains an expression analogous to the one introduced by Breit in order to account for the retardation effects and for the fine-structure separation in He [19–23]. The general correction term reads as ! ai aj 1 ðai rij Þðaj rij Þ Bij ¼ þ ai aj : ð1Þ 2rij rij r2ij The ai are the Dirac matrices in the standard representation. It should be noted that Eq. (1) is already an approximation to the frequency-dependent expression for small electron velocities resp. light atoms and corrects up to an order of (a Z)2 [24]. The magnetic (Gaunt) term a i aj Gij ¼ ð2Þ rij and the retardation term can clearly be identified where the latter is gauge dependent. In order to investigate the relevance of these corrections for the PE spectra only the magnetic term was taken into account as a first approximation. In general this term has significant influence on the ground state energy (see for example [14]) and it was found that for light systems the contribution of the Gaunt term can be safely neglected and that in heavy systems basis set deficiencies count stronger to the peak positions than the Gaunt corrections [25]. If ionization from the core is considered, however, these corrections may become more significant. Another QED correction, namely the Lamb shift,
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was not taken into account but may be of similar importance for an ionization process from the inner shells. In this paper we investigate the fully relativistic ionization spectra of SeH2 and TeH2 and compare the results to experimental findings. A nonrelativistic theoretical spectrum of SeH2 is available [26] allowing for a direct comparison with the DHF-ADC(3) spectrum. To the best of our knowledge a nonrelativistic theoretical spectrum at the ADC(3) level is not yet available for TeH2. We therefore performed a NR-ADC(3) calculation of the TeH2 spectrum employing the ND-ADC3 program package [27]. In contrast to the monovalent iodine compounds a spin–orbit splitting of the outer valence peak was not observed experimentally motivating further theoretical investigations. In the Section 2 we will summarize the essentials of the relativistic ADC method and afterwards discuss the results for the SeH2 and TeH2 ionization spectra. 2. Methodology In the following we will briefly outline the method of the fully relativistic one-particle propagator method DHFADC(3) for the calculation of PE spectra and refer to review articles and more detailed presentations where necessary. The calculation of nonrelativistic and relativistic ionization energies including electron correlation is based on the formalism of the one-particle Green’s function Gpq(t,t 0 ) (electron propagator) which is defined [9] by the equation þ ihWN0 jcyq ðt0 Þcp ðtÞjWN0 iHðt0 tÞ:
ð3Þ
jWN0 i is the exact (non-degenerate) ground state of the considered N-particle system, cyr ðtÞ [cr(t)] denote creation (destruction) operators for one-particle states ju(r)i and H(s) is the Heaviside step function. In the energy representation Gpq takes on the form Gpq ðxÞ ¼
þ
G pq ðxÞ;
ð4Þ
where Gþ pq ðxÞ G pq ðxÞ
¼
hWN0 jcp
¼
hWN0 jcyq
^ þ EN þ ig xH 0 ^ xþH
EN0
ig
1 1
ð5Þ
cp jWN0 i:
ð6Þ
^ is the Hamiltonian of the system and Here H is the ground state energy, g is a positive infinitesimal required to define the Fourier transform between the time and energy representation of Eqs. (3) and (4). The parts G+(x) and G(x) contain physical information on the (N + 1) and (N 1) particle systems, respectively. This becomes explicit in the spectral representation [9,1], which for the (N 1) particle part can be written as X xnq xnp : ð7Þ G ðxÞ ¼ pq x xn ig n EN0
The pole positions
xnp ¼ hWnN 1 jcp jWN0 i;
ð9Þ
are related to spectral intensities. In Eq. (7) summation is carried out over all (N 1)-particle states jWnN 1 i. In the ADC method [5] the pole positions xn and residue amplitudes xnp are obtained as eigenvalues and eigenvectors of an Hermitian matrix whose entries are determined by comparison of the diagrammatic expansions of G with the terms of an algebraic series expansion of the nondiagonal resolvent matrix that is related to the spectral representation of G. It should be noted that in the ADC(3) scheme the ionization energies are treated consistently through third order within a 1h/2h1p configuration space. Those classes of Feynman diagrams that derive from the first n orders are implicitly summed up to infinity by application of the Dyson equation [1] for the Green’s function: GðxÞ ¼ Gð0Þ ðxÞ þ Gð0Þ ðxÞRðxÞGð0Þ ðxÞ þ
ð10Þ
The Dyson equation is a concise expression for the infinite diagrammatic expansion of the full Green’s function that contains the dynamic self energy part R(x) consisting of all irreducible diagrams. R(x) itself is the sum of a purely dynamic and an x-independent part [4] according to ð11Þ
Beyond the second-order treatment x-independent diagrams R(1) occur and their determination is based on a self-consistent procedure [4] starting from the equation I X 1 Rpq ð1Þ ¼ Gsr ðxÞdx nr drs ; V pr;qs ð12Þ 2pi rs with the occupation numbers nr taking on the value +1 for hole states and 0 for particle states. The totally antisymmetric two-electron integral Vpq,rs is hereby defined as V pq;rs ¼ hpqjrsi hpqjsri:
cyq jWN0 i;
ð8Þ
are the (negative) ionization energies of the system, while the residue amplitudes
RðxÞ ¼ MðxÞ þ Rð1Þ:
Gpq ðt; t0 Þ ¼ ihWN0 jcp ðtÞcyq ðt0 ÞjWN0 iHðt t0 Þ
Gþ pq ðxÞ
xn ¼ ðEnN 1 EN0 Þ;
ð13Þ
In contrast to the full Dyson formulation where the (N 1) and (N + 1) parts of G(x) are treated simultaneously [5] one can also resort to the (N 1) particle part exclusively and apply the ADC scheme to G(x) directly (non-Dyson formulation) [28]. For the purpose of calculating ionization energies the restriction to G(x) is therefore sufficient. A few aspects of the Green’s function method deserve attention for the embedding into a many-particle relativistic theory. The Green’s function formalism makes use of the general apparatus of quantum electrodynamics (QED). Whereas the latter deals with interactions between electrons, positrons and photons including pair creation and annihilation processes, the many-body Green’s function method for electron ionization and attachment treats
M. Pernpointner / Chemical Physics 329 (2006) 256–265
the electron–electron interaction only. In the field-theoretical framework the powerful perturbative diagrammatic techniques become applicable to the electron correlation problem. Since the particle-hole formalism in the electronic Green’s function method does not refer to electron/positron pair creation processes as in QED, but represents an elegant way for the formulation of secondquantized perturbation theory, a note on the definition of the vacuum state and operators used in the following is required. In the electronic Green’s function treatment the hole lines refer to the Dirac bound states in the energy range 2mc2 < E 6 0 but not to the negative continuum below and particle lines comprise the energy range 0 < E < E, where E is the highest available virtual orbital. The Fermi level in this context has the same definition as in the nonrelativistic counterpart. A natural consequence is therefore the application of the Dirac operator in its ‘no pair’ form which can be consistently derived from QED (see [29] and references therein). The latter neglects processes related to the presence of the negative eigenstates in the Dirac spectrum. The negative energy branch of the Dirac spectrum then does not enter the perturbation formalism associated with the Dirac–Fock framework but can be included by extending the diagrams over the negative eigenspinors as well. Relativistic many-body energy calculations including the effect of virtual electron-positron pair creation processes have already been reported by several authors [30–33]. In the ADC(3) method only the transformed MO integrals Vpq,rs and the corresponding orbital (spinor) energies ep enter the working equations. This is a key point connecting the Green’s function method in the ADC realization to the four-component Dirac–Hartree–Fock theory because all relevant relativistic information is transported via these entities (see [6] for the details). Once the ADC matrix is diagonalized one obtains the energetic positions (ionization energies) of the spectral lines as the eigenvalues where each line corresponds to an individual eigenstate of the cation. If we worked in the independent-particle picture exclusively the resulting cationic spectrum would give one line with a spectroscopic factor of 1 for each occupied molecular orbital (MO) juii. The energies of these lines would then correspond to the orbital energies ei according to Koopmans’ theorem. Actually, the electrons exhibit correlation not described by the independent particle model of ionization and the resulting cationic state is to be seen as a linear combination of 1h, 2h1p, 3h2p, . . . excited configurations. The ADC matrix up to a certain order is constructed within a subset of these configurations and the resulting eigenvectors then provide information about the contributing configurations, i.e., the strength of the coupling of a 1 h configuration to other excited configurations. In [10] the interpretation of Green’s function calculations and a description of the various satellite types can be found and need not be repeated here.
259
3. Computations and basis set studies 3.1. Computational details At first we optimized a (20s14p11d) dual family basis for the Se atom with respect to the average Dirac–Hartree– Fock ground state energy using the program system GRASP [34,35]. The resulting exponent set is not necessarily well suited for the description of an outer valence ionization process. It is therefore mandatory to extend the basis by additional polarization and diffuse functions in combination with a properly chosen correlation space (in the following also termed as active spinors which should not be confused with the definition of active spaces in CASSCF calculations). The duality condition hereby helps to reduce the number of integrals because the d exponents are a subset of the s exponents leading to a common set of small component p functions when the kinetic balance condition is applied (for a detailed discussion of kinetic balance that is required for variational stability in the four-component case see [36–41]). The same relation holds for the f primitives being a subset of the p primitives. For a comparison also a nondual Se (20s13p10d) primitive set was generated possessing the same quality with respect to the ground state energy. It turned out that the accuracy of the ADC results is not affected by the duality restriction (see below). In the next subsection we outline the systematic procedure we followed for the generation of a good basis for the description of ionization processes. In the case of Te a (23s19p13d) dual family set was generated (see Table 2) and subsequently enhanced by polarization and diffuse functions and for hydrogen a Dunning cc-pVTZ basis [42] was used and extended by one soft p function. For the generation of the relativistic MO integrals the Dirac–Hartree–Fock program DIRAC version 4.0 [43] was employed and the basis sets were all used in their uncontracted form. In DIRAC a standardized interface for the transformed one- and two-electron integrals is available and all subsequent modules for correlated methods do not depend on any DIRAC-specific structures any longer. In the SCF step the integrals of the (SSjSS) type that represent Coulombic integrals over the small component were included but not taken into account in the AO/MO transformation step. The calculation of the ionization spectra of SeH2 and TeH2 were done at their equilibrium geometries [44], ˚ , \(H–Se–H) = 91°, d(Te–H) = namely d(Se–H) = 1.460 A ˚ 1.690 A, \(H–Te–H) = 90°. 3.2. Basis set studies for SeH2 and TeH2 In order not to start with a too inappropriate basis two f-type polarization functions (f1 = 1.00088, f2 = 0.51209) as part of the p exponent set were added to the energy-optimized basis obtained in the first step. The strategy was as follows: at first a correlation space that comprises the
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M. Pernpointner / Chemical Physics 329 (2006) 256–265
Table 2 Energy-optimized dual exponent sets for Se and Te obtained by the GRASP program suite [34,35] Se dual basis s,d 2.412627869E + 07 3.527776528E + 06 7.422103093E + 05 1.901676051E + 05 5.680649549E + 04 1.910913060E + 04 7.009468885E + 03 2.751853067E + 03 1.143648278E + 03 4.963698930E + 02 2.198378559E + 02 9.816622052E + 01 4.459587426E + 01 2.011418048E + 01 9.060304640E + 00 3.956022277E + 00 1.680472861E + 00 6.444324161E 01 2.962976612E 01 1.221728266E 01
Se dual basis p s s s s s s s sd sd sd sd sd sd sd sd sd sd sd s s
5.420203591E + 04 8.496324889E + 03 2.144041996E + 03 6.992176332E + 02 2.672220844E + 02 1.141029240E + 02 5.335687975E + 01 2.666379242E + 01 1.280700006E + 01 6.307631145E + 00 2.668175870E + 00 1.000888362E + 00 5.120943310E 01 1.546909615E 01
Te dual basis s,d p p p p p p p p p p p p p p
Te dual basis p
6.010151307E + 07 1.298550130E + 07 3.505397192E + 06 1.055748622E + 06 3.479219430E + 05 1.231601721E + 05 4.632571625E + 04 1.832279372E + 04 7.548288115E + 03 3.214543350E + 03 1.411079801E + 03 6.382691634E + 02 2.963948139E + 02 1.406463743E + 02 6.817005831E + 01 3.366726516E + 01 1.688529261E + 01 8.607947908E + 00 4.265325810E + 00 2.052560453E + 00 9.477608871E 01 3.823433551E 01 1.449916525E 01
s s s s s s s s s sd sd sd sd sd sd sd sd sd sd sd sd sd s
2.708778860E + 06 4.041179360E + 05 8.397248222E + 04 2.176939044E + 04 6.812080471E + 03 2.474936186E + 03 1.003182477E + 03 4.396465520E + 02 2.037094186E + 02 9.832202189E + 01 4.859779632E + 01 2.359257386E + 01 1.176171870E + 01 5.799142226E + 00 2.870109707E + 00 1.372047125E + 00 5.446504854E 01 2.288029135E 01 8.837091988E 02
p p p p p p p p p p p p p p p p p p p
For the additional diffuse and polarization functions see section ‘Basis set studies’. The nondual exponent sets can be obtained on request from the author.
valence d spinors and all virtuals up to the corresponding positive d spinor energy was chosen. After that the active virtual space was successively enlarged until the first IP did not change within a certain threshold anymore. Of course, due to computational limits the correlation space could not be made arbitrarily large in order to possibly cover all relevant virtuals but it turned out that appropriately chosen basis functions have considerably more impact on the IPs than an increase of the virtual space to high-lying unoccupied spinors. In this respect a successive inclusion or omission of basis functions in combination with properly chosen active spaces led to a final basis that was used for the spectrum. During the optimization process the importance of certain additional basis functions became evident and might provide some orientation for further studies. Because in the SeH2 spectrum the outer three peaks have main state character the change in the energies of all these three spinors was monitored and included in Table 3. The nomenclature of the basis and active space is BxSy where x denotes the degree of basis set extension and y the successive inclusion of additional virtual orbitals. B1 is then our initial (20s14p11d2f) primitive set and S1 an active space comprising all virtual spinors up to an energy of approximately +3.0 a.u. The Se 1s2s2p3s3p spinors were kept frozen. An inclusion of the complete n = 3 shell in the correlation space did not lead to significant changes in the IPs and we will therefore always keep this spinor set frozen. The experimental values for the first three outer valence peaks are 9.897 eV, 12.825 eV and 14.626 eV [45]. As can be seen immediately the initial basis B1 is not appropriate for a reasonable description of the ionization
Table 3 The first three lines in the calculated SeH2 photoelectron spectrum (in eV) for various basis sets and correlation spaces B1 S1
9.295 12.518 14.249
B3 S1
9.478 12.692 14.423
B7 S1
9.656 12.787 14.490
B1 S2
9.366 12.601 14.355
B4 S1
9.310 12.532 14.263
B8 S1
9.703 12.824 14.526
B1 S3
9.369 12.610 14.356
B5 S1
9.528 12.739 14.451
B8 S2
9.729 12.838 14.550
B2 S1
9.478 12.692 14.423
B5 S2
9.588 12.809 14.545
B2 S2
9.538 12.769 14.522
B6 S1
9.628 12.773 14.466
process. By extending the active space to S2 (virtuals up to +5.0 a.u.) and S3 (virtuals up to +9.5 a.u.) the contributions from additional virtual spinors become evident and the spinors above +5.0 a.u. obviously improve the first IP by only 0.003 and can therefore be excluded from the correlation space. Finding an upper bound for the virtual spinors is essential because in uncontracted calculations a large number of virtuals is generated up to very high energies. By adding more basis functions the density of virtual states within a certain energy range increases significantly and necessitates a stable cutoff. The addition of one more soft p function (f = 0.05, B1 ! B2) improves the first IP by 0.18 eV provided the
M. Pernpointner / Chemical Physics 329 (2006) 256–265
same energy cutoff in the virtual space was maintained (S1) and the transition from S1 to S2 yields another 0.06 eV being only one third of the energy obtained by adding one more soft p function. One might now naturally ask about a further improvement by additional diffuse p functions (B2 ! B3). At this stage one more p function with f = 0.01 had no effect (B2 S1 ! B3 S1) and, similarly, the change by one more diffuse s function (f = 0.05) added to B1 can also be neglected (see B4 S1 entry with respect to the B1 S1 entry). So far we found the most important effect by one soft p function on Se (B2) and restart our optimization from this basis. Before we further enlarge the function space one soft p function (f = 0.1) was added to the cc-pVTZ basis (B5 S1) on hydrogen yielding an improvement of 0.05 eV. This change is significant enough to keep the extended hydrogen basis for the future calculations. Additionally, the larger active space (S2) also improved the first IP by 0.06 eV, again pointing to S2 as being superior to the space S1. It is also observed that the second and third peak are stronger influenced by the extension of the correlation space. Certainly, all these changes are not drastic but they will add up at the end and contribute to the final result. By separately adding and omitting functions in combination with the variation of active spaces a full account of the intricate interplay between functions and correlation spaces is not possible but by this procedure the basis could be kept at a tractable size and linear dependencies avoided. Linear dependencies often occur when large even-tempered sequences of very soft functions are added to the original basis. Starting from the basis B5 we next investigated the influence of one additional soft d function on Se (f = 0.296298 from the s space, see B6 results) and obtained a substantial improvement of 0.1 eV in the first IP. In order to provide suitable polarization functions for the soft d functions we omitted the hardest f function and added the next soft f primitive from the p space (ff now: 0.51209 and 0.15469, see B7). Finally, the effect of g functions is not to be neglected as can be seen from the results where one g function of fg = 0.296298 has been added (B8 S1) and the active space enlarged to a positive spinor energy of +4.0 a.u. (B8 S2). The (B8 S2) basis/space combination was then chosen for the calculation of the valence and subvalence ionization spectrum and the quality of the results is now comparable to the nonrelativistic ADC(3) study by Powis et al. [26]. A similar study was undertaken with the selenium nondual basis set in its uncontracted form but no improvement could be achieved. We therefore conclude that the duality condition does not introduce further errors but allows for a reduction of the computational effort. Additionally, the inclusion of very soft s, p and d exponents as proposed by Kaufmann et al. [46] did not lead to a better reproduction of the experimental value but destabilized the fourcomponent calculations. In the optimization procedure for a suitable Te basis we started from a (23s19p13d2f) (B1) primitive set with
261
ff = 0.54465,0.22880. As it was described for SeH2 in detail the same strategy with respect to the first IP was followed for the creation of a suitable Te basis where the experimental value for the first TeH2 ionization energy was determined as 9.14268 eV from a Rydberg series [47] and as 9.14 eV from a photoelectron spectrum [48]. For the final calculation a (24s19p15d2f1g) basis was applied where the softest s and p exponents were 0.05 and 0.08371 and fg amounted to 0.382343. Using this basis a first IP of 8.91 eV was obtained bearing an error of 0.23 eV lying within the methodological error bars. It was observed that the g function in the basis set improved the result from 8.84 eV to 8.91 eV reflecting the relevance of higher angular momentum functions. A further extension of the basis certainly would lead even closer to the experimental value but due to computational limitations a compromise had to be found. 4. Results and discussion In [26] the outer valence PE spectrum of SeH2 was analyzed at the one-component ADC(3)/OVGF level and compared to results [49] obtained by other methods such as SAC-CI [50–52]. We therefore restrict ourselves to the comparison of the nonrelativistic and Dirac–Hartree–Fock ADC(3) results. The nonrelativistic valence electron configuration of SeH2 can be written as ð8a1 Þ2 ð4b2 Þ2 ð9a1 Þ2 ð4b1 Þ2 : In the Se atom a Dirac–Hartree–Fock average of configurations calculation yielded spin–orbit splittings of 0.969 eV for the 3d and 0.4 eV for the 4p spinors. A possible spin– orbit structure in the outer valence region with participating Se 4p functions was not obtained in the calculations and the bonding situation can be well described as in the nonrelativistic case in terms of Se px,py, pz and H 1s orbitals as a corresponding population analysis of the molecular spinors at the DHF level showed (see Table 4). We therefore use the ordinary C2v symbols also for the relativistic valence peak assignment. As in the nonrelativistic treatment only a very weak correlation contribution for the outermost peak of 2B1 symmetry can be observed whereas for the 2A1 and 2B2 states electron correlation causes a shift of approximately 0.24 eV resp. 0.5 eV (see Table 5). Table 4 Mulliken population analysis of the four outer valence spinors at the DHF-level MO-spinor
Population Se px
Se py
Se pz
Se s
Hs
4b1 9a1 4b2 8a1
97.4 – – –
– – 44.0 –
– 58.3 – 1.4
– 12.5 – 79.6
– 27.2 53.0 17.5
Contributions smaller than 1% are not included.
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M. Pernpointner / Chemical Physics 329 (2006) 256–265
Table 5 Nonrelativistic and relativistic SeH2 ionization potentials (in eV) at the independent particle and correlated level for the three outermost states MO-spinor
HF[26]
DHF
ADC(3) [26]
DHF-ADC(3)
Experiment [45]
4b1 9a1 4b2
9.78 13.01 15.09
9.772 13.083 15.048
9.74 12.84 14.79
9.729 (0.95) 12.838 (0.94) 14.550 (0.92)
9.897 12.825 14.626
The numbers in parentheses for the DHF-ADC(3) results are the pole strengths.
The first calculated satellites occur at 16.94 eV (0.004) and 17.2 eV (0.007) both borrowing intensity from the 2 B2 hole state corresponding to a (4b2)1 configuration. The next prominent peak corresponds to an ionization from the 8a1 orbital at 21.93 eV with a substantially reduced pole strength of 0.42. The intensity is distributed over a multitude of satellite states clearly indicating a breakdown of the one-particle model of ionization [10]. It was observed that the 8a1 satellite states are distributed over a wide energetic range where the last one lies at 57 eV. Most of the satellite states between 18.5 and 57.3 eV are related to the ionization from the 8a1 MO spinor and only a few of them could be attributed to the other three valence states: 9a1 (23.5 eV, 25.6 eV, 27.8 eV), 4b2 (28.5 eV, 32.2 eV) and 4b1 (31.8 eV, 43,7 eV). The DHFADC(3) description of the inner valence region between 20 and 25 eV yields a slightly different picture than in the one-component case which is possibly due to basis set deficiencies in the s function space and the sensitivity of satellite states to the basis set. The doublet structure of the d ionization is well reproduced and dense line bundles are observed mimicking Lorentzian shape (see Fig. 1). This specific shape is observable as soon as the initial cationic hole state is unstable against electronic decay [53] where the discrete basis functions are used for a description of the continuum states. In this work we focus on the ionization spectra and did not further analyze subsequent decay processes. The lowering of symmetry from Oh in the Se atom to C2v in SeH2 lifts the degeneracy of the d3/2 and d5/2 spinors resulting in
two groups of close-lying DHF-MO spinors with a 100% Se d population. In Fig. 2 only the spectrum of this inner valence region together with the Koopmans lines is shown. A considerable shift due to electron correlation is visible and the formerly isolated lines are now resolved in a dense Lorentzian structure providing an indication of electronic decay. These features could not be obtained in the nonrelativistic description of [26]. The bonding situation in TeH2 is identical to that of SeH2 except that all the participating Te orbitals now have a principal quantum number increased by unity. The outer valence structure according to a nonrelativistic calculation is then 2
2
2
ð12a1 Þ ð6b2 Þ ð13a1 Þ ð6b1 Þ
2
and this similarity is reflected in the comparable outer valence orbital energies and in the corresponding ionization spectrum (see Fig. 3). A more significant energetic difference exists, however, in the d features. In the TeH2 spectrum the 4d features lie approximately 15 eV lower in energy than the corresponding 3d peaks of SeH2 and a very pronounced Lorentzian envelope can also be observed in the TeH2 (4d3/2,5/2)1 final state. The 4d features are broader and the original intensity is distributed over more satellite states than in SeH2. Despite the sensitivity of the satellites to basis set quality a more efficient decay process can be assumed in the TeH2 case and corresponding life time calculations would be of high value in order to clarify this issue. For the nonrelativistic and relativistic numerical values for the outer valence IPs see Table 6.
1
4b 1
Hydrogen selenide DHF – ADC(3)
Pole strength
0.8
9a 1
4b 2
0.6
3d 5/2 3d 3/2
0.4
Sat.
8a 1
0.2
0
0
10
20
30
40
50
60
70
Ionization energy (eV) Fig. 1. The fully relativistic (DHF) ionization spectrum of hydrogen selenide at the ADC(3) level.
M. Pernpointner / Chemical Physics 329 (2006) 256–265
263
1
Hydrogen selenide DHF–ADC(3)
Pole strength
0.8
3d 5/2 Koopmans lines 0.6
3d
3/2
ADC(3) lines 0.4
0.2
0 62
63
64
65
66
67
68
69
70
71
Ionization energy (eV) Fig. 2. Enlarged ionization spectrum for the 3d range. A large correlation contribution compared to the Koopmans’ lines is visible together with the typical Lorentzian structure of the two spin–orbit split line bundles describing a decaying state. 1
6b 1
Pole strength
0.8
6b 2
13a 1
Hydrogen telluride DHF–ADC(3)
0.6
0.4
4d 3/2
12a 1 0.2
4d 5/2
0 1
6b 1 6b 2
0.8
Pole strength
13a 1
Hydrogen telluride NR–ADC(3) 0.6
4d 12a 1
0.4
0.2
0
0
10
20
30
40
50
Ionization energy (eV) Fig. 3. The fully relativistic (DHF) ionization spectrum of hydrogen telluride at the ADC(3) level.
In Fig. 3 both the nonrelativistic (NR) and fully relativistic TeH2 ionization spectra are plotted. For the outermost 6b1 state no relativistic shift is observable and the
next two subsequent states show only slight alterations from the nonrelativistic result. This is insofar remarkable that in neighboring iodine compounds a pronounced
M. Pernpointner / Chemical Physics 329 (2006) 256–265
Table 6 Nonrelativistic and relativistic TeH2 ionization potentials (in eV) at the independent particle and correlated level for the three outermost states MO-spinor 6b1 13a1 6b2
HF 8.88 11.87 13.84
DHF 8.85 12.00 13.71
ADC(3) 8.94 (0.95) 11.66 (0.94) 13.20 (0.87)
DHF-ADC(3) 8.91 (0.95) 11.79 (0.94) 13.12 (0.88)
Experiment [47,48] 9.14 12.00 13.25
The numbers in parentheses for the ADC(3) results are the pole strengths.
9 8.8 8.6 8.4 8.2 8
60 80 100 120 140 160 180 1.5
nd Bo
e gl an
spin–orbit splitting occurs in the outermost state [54] whereas in TeH2 no such features are observed even with a comparable size of the atomic 5p spin–orbit splitting in tellurium and iodine (0.88 eV resp. 1.13 eV). This can be explained by the availability of two p-type lone pairs (px and py) in monovalent iodine compounds that can be mixed differently under the influence of the spin–orbit operator in a molecular environment whereas in TeH2 only one px lone pair is available being represented by a single j,mj component of a quasi-atomic spinor. The outermost peak in the experimental spectrum does not exhibit vibrational structure [48] in contrast to the relatively broad a1 and b2 bands. In [48] both the vertical and adiabatic ionization potentials are given and we take the vertical IP values as the experimental reference. The DHF-ADC(3) values show a maximum deviation of 0.2 eV lying within the methodological error range. For the a1 and b2 peaks a vibrational fine structure was not calculated. The TeH2 satellite structure around 18–30 eV bears some similarity to the nonrelativistic case but due to different coupling schemes resulting from the application of double group symmetry the intensities will be distributed differently among the individual peaks (see also [7] for a coupling analysis in the carbon monoxide case). A striking difference in the spectra happens in the 4d region around 50 eV. Here only one 4d line bundle is obtained in the NR-ADC(3) treatment whereas two bundles shifted to lower ionisation energy result in the DHFADC(3) calculation. For TeH2 we also investigated the dependence of the first IP on the molecular geometry. Hereby the H–Te–H angle was varied from 65o to 180o and the Te–H bond dis˚ to 1.8 A ˚ in steps of 0.1 A ˚ always keeping tance from 1.5 A C2v symmetry. For all bond lengths the first IP is lowest for the linear molecular conformation and has its minimum at ˚ . (see Fig. 4). Enlarging the bond angle leads to a 1.8 A higher repulsion between the bonding MO’s and the px lone pair increasing its energy. It can also be seen that there is only a weak dependence of the IP on the bond length around the equilibrium bond angle and for larger bond angles the sensitivity to bond stretching becomes more pronounced. Due to computational limitations these calculations were performed without g functions on Te but we believe that the observed trend is correctly reproduced with the smaller basis as well.
First IP of hydrogen telluride at DHF-ADC(3) level Ionization potential (eV)
264
1.55
1.6
1.65
1.7
1.75
1.8
tr.)
nce (Angs
Bond dista
Fig. 4. The first ionization potential of TeH2 with respect to variations of the Te–H bond distance and H–Te–H bond angle calculated at the DHFADC(3) level of theory.
5. Outlook For an accurate theoretical description of PE spectra a consistent treatment of electron correlation and relativity is necessary. Especially in the PtCl2 6 complex nonadditivity of spin–orbit coupling and electron correlation became evident. In this respect an investigation of the PtF2 6 ionization spectrum would also be of great interest. To the best of our knowledge no experimental ionization spectrum of PtF2 6 is available so far and this calculation would have some predictive character. Furthermore, lifetime calculations of the (3d)1 resp. (4d)1 final states could provide valuable information for a subsequent analysis of the decay processes in SeH2 resp. TeH2. References [1] A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, 1971. [2] D.J. Thouless, The Quantum Mechanics of Many-Body Systems, Academic, New York, 1972. ¨ hrn, Propagators in Quantum Chemistry, [3] J. Linderberg, Y. O Academic, New York, 1973. [4] L.S. Cederbaum, W. Domcke, Adv. Chem. Phys. 36 (1977) 205. [5] J. Schirmer, L.S. Cederbaum, O. Walter, Phys. Rev. A 28 (1983) 1237. [6] M. Pernpointner, A.B. Trofimov, J. Chem. Phys. 120 (2004) 4098. [7] M. Pernpointner, J. Chem. Phys. 121 (2004) 8782. [8] J. Schirmer, Phys. Rev. A 43 (1991) 4647. [9] A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, Englewood Cliffs, 1963. [10] L.S. Cederbaum, W. Domcke, J. Schirmer, W. von Niessen, Adv. Chem. Phys. 65 (1986) 115. [11] D.M.P. Holland, I. Powis, L. Karlsson, A.B. Trofimov, J. Schirmer, W. von Niessen, Chem. Phys. 297 (2004) 55. [12] M. Pernpointner, J. Breidbach, L.S. Cederbaum, J. Chem. Phys. 122 (2005) 064311. [13] M. Kaupp, C. van Wu¨llen, R. Franke, F. Schmitz, W. Kutzelnigg, J. Am. Chem. Soc. 118 (1996) 11939.
M. Pernpointner / Chemical Physics 329 (2006) 256–265 [14] J. Styszyn´ski, X. Cao, G.L. Malli, L. Visscher, J. Comp. Chem. 18 (1997) 601. [15] M.-S. Liao, Q.-E. Zhang, J. Phys. Chem. A 102 (1998) 10647. [16] M. Pernpointner, L.S. Cederbaum, J. Chem. Phys. 122 (2005) 214302. [17] C. Buth, R. Santra, L.S. Cederbaum, J. Chem. Phys. 119 (2003) 7763. [18] H. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Twoelectron Systems, Springer, Berlin, 1957. [19] G. Breit, Phys. Rev. 34 (1929) 553. [20] G. Breit, Phys. Rev. 36 (1930) 383. [21] G. Breit, Phys. Rev. 38 (1931) 463. [22] G. Breit, Proc. Nat. Acad. Sci. USA 17 (1931) 70. [23] G. Breit, Phys. Rev. 39 (1932) 616. [24] W. Kutzelnigg, Phys. Scr. 36 (1987) 416. [25] M. Pernpointner, J. Phys. B: At. Mol. Phys. 38 (2005) 1955. [26] I. Powis, J.D. Thrower, A.B. Trofimov, T.E. Moskovskaya, J. Schirmer, A.W. Potts, D.M.P. Holland, F. Bruhn, L. Karlsson, Chem. Phys. 315 (2005) 121. [27] ND-ADC3 Version 1.1.0 (2004): program for the calculation of the non-Dyson ADC(3)-Matrix and the ISR representation of a oneparticle operator, written by J. Breidbach. [28] J. Schirmer, A.B. Trofimov, G. Stelter, J. Chem. Phys. 109 (1998) 4734. [29] I.P. Grant, H.M. Quiney, Adv. Atom. Mol. Phys. 23 (1988) 37. [30] H.M. Quiney, Relativistic effects in atoms and molecules, in: S. Wilson (Ed.), Methods in Computational Chemistry, vol. 2, Plenum, New York, 1988, p. 223. [31] H.M. Quiney, I.P. Grant, S. Wilson, Phys. Scr. 36 (1987) 460. [32] H.M. Quiney, I.P. Grant, S. Wilson, Many-body methods, in: U. Kaldor (Ed.), Quantum Chemistry, Springer, Berlin, Heidelberg, New York, 1989. [33] S. Wilson, The Effects of Relativity on Atoms, Molecules and the Solid State, Plenum Press, New York, 1991.
265
[34] K.D. Dyall, I.P. Grant, F.A. Parpia, E.P. Plummer, Comput. Phys. Commun. 55 (1989) 425. [35] K.G. Dyall, K. Fægri, Theor. Chim. Acta 94 (1996) 39. [36] F. Mark, W.H.E. Schwarz, Phys. Rev. Lett. 48 (1982) 673. [37] W.H.E. Schwarz, H. Wallmeier, Mol. Phys. 46 (1982) 1045. [38] R.E. Stanton, S.J. Havriliak, Chem. Phys. 81 (1984) 1910. [39] K. Dyall, I.P. Grant, S. Wilson, J. Phys. B: At. Mol. Phys. 17 (1984) 1910. [40] P.J.C. Aerts, W.C. Nieuwpoort, Chem. Phys. Lett. 125 (1986) 83. [41] W. Kutzelnigg, Int. J. Quantum Chem. 25 (1984) 107. [42] J.T.H. Dunning, J. Chem. Phys. 90 (1989) 1007. [43] Dirac, a relativistic ab initio electronic structure program, Release DIRAC04.0 (2004), written by H.J. Aa. Jensen, T. Saue, L. Visscher with contributions from V. Bakken, E. Eliav, T. Enevoldsen, T. Fleig, O. Fossgaard, T. Helgaker, J. Laerdahl, C.V. Larsen, P. Norman, J. Olsen, M. Pernpointner, J.K. Pedersen, K. Ruud, P. Salek, J.N.P. van Stralen, J. Thyssen, O. Visser, T. Winther. (http://dirac.chem.sdu.dk). [44] G. Herzberg, Electronic Spectra and Electronic Structure of Polyatomic Molecules, Van Nostrand, New York, 1966. [45] K. Bo¨rlin, M. Jungen, L. Karlsson, R. Maripuu, Chem. Phys. 113 (1987) 309. [46] K. Kaufmann, W. Baumeister, M. Jungen, J. Phys. B: At. Mol. Phys. 22 (1989) 2223. [47] C.A. Mayhew, J.P. Connerade, M.A. Baig, J. Phys. B: At. Mol. Phys. 19 (1986) 4149. [48] A.W. Potts, W.C. Price, Proc. R. Soc. A 326 (1972) 181. [49] M. Ehara, M. Ishida, H. Nakatsuji, J. Chem. Phys. 114 (2001) 8990. [50] H. Nakatsuji, K. Hirao, Chem. Phys. Lett. 47 (1977) 569. [51] H. Nakatsuji, Chem. Phys. Lett. 67 (1979) 329. [52] H. Nakatsuji, Chem. Phys. Lett. 67 (1979) 334. [53] L.S. Cederbaum, J. Zobeley, F. Tarantelli, Phys. Rev. Lett. 79 (1997) 4778. [54] M. Pernpointner, S. Knecht, Chem. Phys. Lett. 410 (2005) 423.
Relativistic calculation of the SeH2 and TeH2 photoelectron spectra Markus Pernpointner Theoretische Chemie, Universita¨t Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany Received 28 April 2006; accepted 13 July 2006 Available online 20 July 2006
Abstract Photoelectron (PE) spectra provide detailed insight into the electronic structure of atoms, molecules and solids. Hereby electron correlation and relativistic effects influence the structure of the PE spectrum in a complicated way necessitating a consistent theoretical treatment. By embedding the one-particle propagator technique in a four-component framework the interplay between relativistic and correlation effects can be described correctly. In this article the Dirac–Hartree–Fock algebraic diagrammatic construction scheme (DHF-ADC) together with recent applications is reviewed and fully relativistic PE spectra of SeH2 and TeH2 in combination with basis set studies are presented. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Photoelectron spectra; Relativity; Spin–orbit effects; Hydrogen selenide; Hydrogen telluride; One-particle-propagator
1. Introduction In the PE spectra of light elements a pronounced spin– orbit splitting is already observable (e.g., in the neon atom DSO(2p1/2/2p3/2) amounts to 0.12 eV at the DHF level and to 0.1 eV at the experimental level) and additional scalar relativistic effects can further shift the peak centers (0.15 eV for Ne 2s). Except for the scalar relativistic effects that can be taken into account by relativistic effective core potentials (RECP) an outer or inner valence spin–orbit structure together with electron correlation effects could not be properly described by the methods available so far. In general, ionization energies and electron affinities can be obtained by the one-particle propagator (Green’s function) technique [1–4]. Amongst various approximation techniques the algebraic diagrammatic construction (ADC) scheme [5] is an elegant approach to the one-particle propagator and was embedded in the four-component formalism [6,7]. The PE spectra of various systems have already been calculated at the DHF-ADC(3) level where the number in parentheses indicates the order of perturbation theory to which the ADC scheme is complete. As a first application fully relativistic outer valence PE spectra of the noble gas E-mail address: [email protected]. 0301-0104/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2006.07.019
atoms, carbon monoxide and cyanogen iodide were calculated [6]. It was observed that the outer valence energies of the noble gas atoms deviate in a systematic manner from the experimental results. Whereas in the Ne atom the experimental value is overestimated by +0.1 eV an increasing underestimation for the remaining elements Ar, Kr and Xe of 0.05 eV, 0.10 eV, 0.27 eV has occurred. One might be tempted to explain this trend with a size-consistency problem but the ADC method was shown to be inherently size-consistent [5,8], a property that can be tracked down to the cancellation of unlinked diagrams in the diagrammatic expansion of the Green’s function [9,1]. More recent investigations have shown that for mediumsize systems substantial extensions to the basis sets are necessary in order to achieve the same quality as for smaller systems. Especially the presence of numerous polarization and diffuse functions is hereby required (see below). However, the experimentally observed spin–orbit splittings in the noble gases are well reproduced in all cases (see Table 1). It should also be noted that ADC(3) bears a systematic error of approximately 0.2 eV due to its restriction to third order and quickly becomes inaccurate for inner valence and core orbitals due to their large relaxation effect. The smallness of the relativistic effects in CO allowed for a direct comparison with nonrelativistic (NR) ADC(3)
M. Pernpointner / Chemical Physics 329 (2006) 256–265 Table 1 Calculated and experimental values (in eV) for the spin–orbit splittings in the noble gas atoms Level
DHF-ADC(3)
Experiment
Ne 2p1/2,3/2 Ar 3p1/2,3/2 Kr 3d3/2,5/2 Kr 4p1/2,3/2 Xe 4d3/2,5/2 Xe 5p1/2,3/2
0.11 0.19 1.19 0.67 1.99 1.3
0.1 0.2 1.2 0.68 2.0 1.3
results. Only an error of 0.08 eV for the X2R+ and A2P cationic final states was obtained with respect to experiment. For these states the molecular orbital picture of ionization is an accurate approximation [10] and the final X2R+ and A2P states mainly correspond to an ionization from the 5r and 1p orbital. The validity of the MO description does not imply the absence of electron correlation in the outer valence region, it just reflects the fact that the 1h configuration has a dominant coefficient in a possible CI expansion of the true state and that the other coefficients of coupling 2h1p configurations are small. For inner valence states a much stronger coupling between the 1h and 2h1p configurations takes place which is accompanied by the occurrence of numerous satellite lines of low intensity (breakdown of the MO picture). A detailed analysis of the satellite states can also be found in [6]. Cyanogen iodide is a molecule with one ‘relativistic’ element and was extensively studied in the one-component framework [11] enabling a direct comparison to the fully relativistic treatment. A pronounced spin–orbit splitting of the outer valence X2P1/2,3/2 states of 0.54 eV is experimentally observed and well reproduced by DHF-ADC(3) as 0.57 eV [7] despite a considerable absolute error of 0.36 eV. The single nonrelativistic ADC(3) value of 10.89 eV for the first ionization energy perfectly matches the first spectral peak position at 10.90 eV but does not correspond to the weighted average of both spin–orbit split states that should lie at 11.08 eV. Insofar the one-component result bears an error of 0.19 eV being slightly superior in accuracy. The reason for the deviation in the relativistic case has its origin in a possible lack of more diffuse basis functions but computational restrictions did not allow for a further basis set extension at that time. For all systems discussed so far the spin–orbit splittings were all well reproduced even in case of more substantial deviations from the experimental peak position. One could therefore assume similar correlation contributions to the individual spin–orbit states corresponding to additivity of both effects. For not too heavy systems as the ones described above large deviations from this approximation could not be found. However, the situation changes significantly in heavy (Z > 70) systems such as in PtCl2 6 whose PE spectrum was calculated in [12]. It was observed that electron correlation now contributes to a different extent to individual spin–orbit components resulting in a substantial alteration of the level order with respect to one-compo-
257
nent calculations. The overall spectral features were then better reproduced than in the one-component scalar relativistic calculations which further corroborates the importance of taking spin–orbit coupling into account. Noble gas compounds exhibit unusual bonding situations (see for example [13–15] and references therein) and are therefore an ideal subject for photoelectron spectroscopy in order to gain more insight into the electronic structure. Extensive DHF-ADC(3) calculations of ionization spectra were performed on the xenon fluorides XeFn (n = 2,4,6) [16] and the influence of electron correlation studied. Corresponding one-component results for the xenon fluorides were already available [17] but without a consistent inclusion of relativistic effects. For these PE spectra the role of spin–orbit splitting could be clarified and a better resemblance of the experimental features was achieved. When one asks about relativistic effects there is also the correction to the electron–electron interaction. In the conventional four-component approach one applies a Hamiltonian consisting of a sum of one-electron Diracoperators in combination with the conventional Coulomb repulsion operator representing the instantaneous interaction between electrons. The Coulomb interaction term is not invariant under a Lorentz transformation and in order to account for this deficiency corrections derived from quantum electrodynamics (QED) are introduced [18]. The correction term describes the exchange of virtual photons between the electrons and is frequency-dependent. If one expands these terms in the limit of low frequencies one obtains an expression analogous to the one introduced by Breit in order to account for the retardation effects and for the fine-structure separation in He [19–23]. The general correction term reads as ! ai aj 1 ðai rij Þðaj rij Þ Bij ¼ þ ai aj : ð1Þ 2rij rij r2ij The ai are the Dirac matrices in the standard representation. It should be noted that Eq. (1) is already an approximation to the frequency-dependent expression for small electron velocities resp. light atoms and corrects up to an order of (a Z)2 [24]. The magnetic (Gaunt) term a i aj Gij ¼ ð2Þ rij and the retardation term can clearly be identified where the latter is gauge dependent. In order to investigate the relevance of these corrections for the PE spectra only the magnetic term was taken into account as a first approximation. In general this term has significant influence on the ground state energy (see for example [14]) and it was found that for light systems the contribution of the Gaunt term can be safely neglected and that in heavy systems basis set deficiencies count stronger to the peak positions than the Gaunt corrections [25]. If ionization from the core is considered, however, these corrections may become more significant. Another QED correction, namely the Lamb shift,
258
M. Pernpointner / Chemical Physics 329 (2006) 256–265
was not taken into account but may be of similar importance for an ionization process from the inner shells. In this paper we investigate the fully relativistic ionization spectra of SeH2 and TeH2 and compare the results to experimental findings. A nonrelativistic theoretical spectrum of SeH2 is available [26] allowing for a direct comparison with the DHF-ADC(3) spectrum. To the best of our knowledge a nonrelativistic theoretical spectrum at the ADC(3) level is not yet available for TeH2. We therefore performed a NR-ADC(3) calculation of the TeH2 spectrum employing the ND-ADC3 program package [27]. In contrast to the monovalent iodine compounds a spin–orbit splitting of the outer valence peak was not observed experimentally motivating further theoretical investigations. In the Section 2 we will summarize the essentials of the relativistic ADC method and afterwards discuss the results for the SeH2 and TeH2 ionization spectra. 2. Methodology In the following we will briefly outline the method of the fully relativistic one-particle propagator method DHFADC(3) for the calculation of PE spectra and refer to review articles and more detailed presentations where necessary. The calculation of nonrelativistic and relativistic ionization energies including electron correlation is based on the formalism of the one-particle Green’s function Gpq(t,t 0 ) (electron propagator) which is defined [9] by the equation þ ihWN0 jcyq ðt0 Þcp ðtÞjWN0 iHðt0 tÞ:
ð3Þ
jWN0 i is the exact (non-degenerate) ground state of the considered N-particle system, cyr ðtÞ [cr(t)] denote creation (destruction) operators for one-particle states ju(r)i and H(s) is the Heaviside step function. In the energy representation Gpq takes on the form Gpq ðxÞ ¼
þ
G pq ðxÞ;
ð4Þ
where Gþ pq ðxÞ G pq ðxÞ
¼
hWN0 jcp
¼
hWN0 jcyq
^ þ EN þ ig xH 0 ^ xþH
EN0
ig
1 1
ð5Þ
cp jWN0 i:
ð6Þ
^ is the Hamiltonian of the system and Here H is the ground state energy, g is a positive infinitesimal required to define the Fourier transform between the time and energy representation of Eqs. (3) and (4). The parts G+(x) and G(x) contain physical information on the (N + 1) and (N 1) particle systems, respectively. This becomes explicit in the spectral representation [9,1], which for the (N 1) particle part can be written as X xnq xnp : ð7Þ G ðxÞ ¼ pq x xn ig n EN0
The pole positions
xnp ¼ hWnN 1 jcp jWN0 i;
ð9Þ
are related to spectral intensities. In Eq. (7) summation is carried out over all (N 1)-particle states jWnN 1 i. In the ADC method [5] the pole positions xn and residue amplitudes xnp are obtained as eigenvalues and eigenvectors of an Hermitian matrix whose entries are determined by comparison of the diagrammatic expansions of G with the terms of an algebraic series expansion of the nondiagonal resolvent matrix that is related to the spectral representation of G. It should be noted that in the ADC(3) scheme the ionization energies are treated consistently through third order within a 1h/2h1p configuration space. Those classes of Feynman diagrams that derive from the first n orders are implicitly summed up to infinity by application of the Dyson equation [1] for the Green’s function: GðxÞ ¼ Gð0Þ ðxÞ þ Gð0Þ ðxÞRðxÞGð0Þ ðxÞ þ
ð10Þ
The Dyson equation is a concise expression for the infinite diagrammatic expansion of the full Green’s function that contains the dynamic self energy part R(x) consisting of all irreducible diagrams. R(x) itself is the sum of a purely dynamic and an x-independent part [4] according to ð11Þ
Beyond the second-order treatment x-independent diagrams R(1) occur and their determination is based on a self-consistent procedure [4] starting from the equation I X 1 Rpq ð1Þ ¼ Gsr ðxÞdx nr drs ; V pr;qs ð12Þ 2pi rs with the occupation numbers nr taking on the value +1 for hole states and 0 for particle states. The totally antisymmetric two-electron integral Vpq,rs is hereby defined as V pq;rs ¼ hpqjrsi hpqjsri:
cyq jWN0 i;
ð8Þ
are the (negative) ionization energies of the system, while the residue amplitudes
RðxÞ ¼ MðxÞ þ Rð1Þ:
Gpq ðt; t0 Þ ¼ ihWN0 jcp ðtÞcyq ðt0 ÞjWN0 iHðt t0 Þ
Gþ pq ðxÞ
xn ¼ ðEnN 1 EN0 Þ;
ð13Þ
In contrast to the full Dyson formulation where the (N 1) and (N + 1) parts of G(x) are treated simultaneously [5] one can also resort to the (N 1) particle part exclusively and apply the ADC scheme to G(x) directly (non-Dyson formulation) [28]. For the purpose of calculating ionization energies the restriction to G(x) is therefore sufficient. A few aspects of the Green’s function method deserve attention for the embedding into a many-particle relativistic theory. The Green’s function formalism makes use of the general apparatus of quantum electrodynamics (QED). Whereas the latter deals with interactions between electrons, positrons and photons including pair creation and annihilation processes, the many-body Green’s function method for electron ionization and attachment treats
M. Pernpointner / Chemical Physics 329 (2006) 256–265
the electron–electron interaction only. In the field-theoretical framework the powerful perturbative diagrammatic techniques become applicable to the electron correlation problem. Since the particle-hole formalism in the electronic Green’s function method does not refer to electron/positron pair creation processes as in QED, but represents an elegant way for the formulation of secondquantized perturbation theory, a note on the definition of the vacuum state and operators used in the following is required. In the electronic Green’s function treatment the hole lines refer to the Dirac bound states in the energy range 2mc2 < E 6 0 but not to the negative continuum below and particle lines comprise the energy range 0 < E < E, where E is the highest available virtual orbital. The Fermi level in this context has the same definition as in the nonrelativistic counterpart. A natural consequence is therefore the application of the Dirac operator in its ‘no pair’ form which can be consistently derived from QED (see [29] and references therein). The latter neglects processes related to the presence of the negative eigenstates in the Dirac spectrum. The negative energy branch of the Dirac spectrum then does not enter the perturbation formalism associated with the Dirac–Fock framework but can be included by extending the diagrams over the negative eigenspinors as well. Relativistic many-body energy calculations including the effect of virtual electron-positron pair creation processes have already been reported by several authors [30–33]. In the ADC(3) method only the transformed MO integrals Vpq,rs and the corresponding orbital (spinor) energies ep enter the working equations. This is a key point connecting the Green’s function method in the ADC realization to the four-component Dirac–Hartree–Fock theory because all relevant relativistic information is transported via these entities (see [6] for the details). Once the ADC matrix is diagonalized one obtains the energetic positions (ionization energies) of the spectral lines as the eigenvalues where each line corresponds to an individual eigenstate of the cation. If we worked in the independent-particle picture exclusively the resulting cationic spectrum would give one line with a spectroscopic factor of 1 for each occupied molecular orbital (MO) juii. The energies of these lines would then correspond to the orbital energies ei according to Koopmans’ theorem. Actually, the electrons exhibit correlation not described by the independent particle model of ionization and the resulting cationic state is to be seen as a linear combination of 1h, 2h1p, 3h2p, . . . excited configurations. The ADC matrix up to a certain order is constructed within a subset of these configurations and the resulting eigenvectors then provide information about the contributing configurations, i.e., the strength of the coupling of a 1 h configuration to other excited configurations. In [10] the interpretation of Green’s function calculations and a description of the various satellite types can be found and need not be repeated here.
259
3. Computations and basis set studies 3.1. Computational details At first we optimized a (20s14p11d) dual family basis for the Se atom with respect to the average Dirac–Hartree– Fock ground state energy using the program system GRASP [34,35]. The resulting exponent set is not necessarily well suited for the description of an outer valence ionization process. It is therefore mandatory to extend the basis by additional polarization and diffuse functions in combination with a properly chosen correlation space (in the following also termed as active spinors which should not be confused with the definition of active spaces in CASSCF calculations). The duality condition hereby helps to reduce the number of integrals because the d exponents are a subset of the s exponents leading to a common set of small component p functions when the kinetic balance condition is applied (for a detailed discussion of kinetic balance that is required for variational stability in the four-component case see [36–41]). The same relation holds for the f primitives being a subset of the p primitives. For a comparison also a nondual Se (20s13p10d) primitive set was generated possessing the same quality with respect to the ground state energy. It turned out that the accuracy of the ADC results is not affected by the duality restriction (see below). In the next subsection we outline the systematic procedure we followed for the generation of a good basis for the description of ionization processes. In the case of Te a (23s19p13d) dual family set was generated (see Table 2) and subsequently enhanced by polarization and diffuse functions and for hydrogen a Dunning cc-pVTZ basis [42] was used and extended by one soft p function. For the generation of the relativistic MO integrals the Dirac–Hartree–Fock program DIRAC version 4.0 [43] was employed and the basis sets were all used in their uncontracted form. In DIRAC a standardized interface for the transformed one- and two-electron integrals is available and all subsequent modules for correlated methods do not depend on any DIRAC-specific structures any longer. In the SCF step the integrals of the (SSjSS) type that represent Coulombic integrals over the small component were included but not taken into account in the AO/MO transformation step. The calculation of the ionization spectra of SeH2 and TeH2 were done at their equilibrium geometries [44], ˚ , \(H–Se–H) = 91°, d(Te–H) = namely d(Se–H) = 1.460 A ˚ 1.690 A, \(H–Te–H) = 90°. 3.2. Basis set studies for SeH2 and TeH2 In order not to start with a too inappropriate basis two f-type polarization functions (f1 = 1.00088, f2 = 0.51209) as part of the p exponent set were added to the energy-optimized basis obtained in the first step. The strategy was as follows: at first a correlation space that comprises the
260
M. Pernpointner / Chemical Physics 329 (2006) 256–265
Table 2 Energy-optimized dual exponent sets for Se and Te obtained by the GRASP program suite [34,35] Se dual basis s,d 2.412627869E + 07 3.527776528E + 06 7.422103093E + 05 1.901676051E + 05 5.680649549E + 04 1.910913060E + 04 7.009468885E + 03 2.751853067E + 03 1.143648278E + 03 4.963698930E + 02 2.198378559E + 02 9.816622052E + 01 4.459587426E + 01 2.011418048E + 01 9.060304640E + 00 3.956022277E + 00 1.680472861E + 00 6.444324161E 01 2.962976612E 01 1.221728266E 01
Se dual basis p s s s s s s s sd sd sd sd sd sd sd sd sd sd sd s s
5.420203591E + 04 8.496324889E + 03 2.144041996E + 03 6.992176332E + 02 2.672220844E + 02 1.141029240E + 02 5.335687975E + 01 2.666379242E + 01 1.280700006E + 01 6.307631145E + 00 2.668175870E + 00 1.000888362E + 00 5.120943310E 01 1.546909615E 01
Te dual basis s,d p p p p p p p p p p p p p p
Te dual basis p
6.010151307E + 07 1.298550130E + 07 3.505397192E + 06 1.055748622E + 06 3.479219430E + 05 1.231601721E + 05 4.632571625E + 04 1.832279372E + 04 7.548288115E + 03 3.214543350E + 03 1.411079801E + 03 6.382691634E + 02 2.963948139E + 02 1.406463743E + 02 6.817005831E + 01 3.366726516E + 01 1.688529261E + 01 8.607947908E + 00 4.265325810E + 00 2.052560453E + 00 9.477608871E 01 3.823433551E 01 1.449916525E 01
s s s s s s s s s sd sd sd sd sd sd sd sd sd sd sd sd sd s
2.708778860E + 06 4.041179360E + 05 8.397248222E + 04 2.176939044E + 04 6.812080471E + 03 2.474936186E + 03 1.003182477E + 03 4.396465520E + 02 2.037094186E + 02 9.832202189E + 01 4.859779632E + 01 2.359257386E + 01 1.176171870E + 01 5.799142226E + 00 2.870109707E + 00 1.372047125E + 00 5.446504854E 01 2.288029135E 01 8.837091988E 02
p p p p p p p p p p p p p p p p p p p
For the additional diffuse and polarization functions see section ‘Basis set studies’. The nondual exponent sets can be obtained on request from the author.
valence d spinors and all virtuals up to the corresponding positive d spinor energy was chosen. After that the active virtual space was successively enlarged until the first IP did not change within a certain threshold anymore. Of course, due to computational limits the correlation space could not be made arbitrarily large in order to possibly cover all relevant virtuals but it turned out that appropriately chosen basis functions have considerably more impact on the IPs than an increase of the virtual space to high-lying unoccupied spinors. In this respect a successive inclusion or omission of basis functions in combination with properly chosen active spaces led to a final basis that was used for the spectrum. During the optimization process the importance of certain additional basis functions became evident and might provide some orientation for further studies. Because in the SeH2 spectrum the outer three peaks have main state character the change in the energies of all these three spinors was monitored and included in Table 3. The nomenclature of the basis and active space is BxSy where x denotes the degree of basis set extension and y the successive inclusion of additional virtual orbitals. B1 is then our initial (20s14p11d2f) primitive set and S1 an active space comprising all virtual spinors up to an energy of approximately +3.0 a.u. The Se 1s2s2p3s3p spinors were kept frozen. An inclusion of the complete n = 3 shell in the correlation space did not lead to significant changes in the IPs and we will therefore always keep this spinor set frozen. The experimental values for the first three outer valence peaks are 9.897 eV, 12.825 eV and 14.626 eV [45]. As can be seen immediately the initial basis B1 is not appropriate for a reasonable description of the ionization
Table 3 The first three lines in the calculated SeH2 photoelectron spectrum (in eV) for various basis sets and correlation spaces B1 S1
9.295 12.518 14.249
B3 S1
9.478 12.692 14.423
B7 S1
9.656 12.787 14.490
B1 S2
9.366 12.601 14.355
B4 S1
9.310 12.532 14.263
B8 S1
9.703 12.824 14.526
B1 S3
9.369 12.610 14.356
B5 S1
9.528 12.739 14.451
B8 S2
9.729 12.838 14.550
B2 S1
9.478 12.692 14.423
B5 S2
9.588 12.809 14.545
B2 S2
9.538 12.769 14.522
B6 S1
9.628 12.773 14.466
process. By extending the active space to S2 (virtuals up to +5.0 a.u.) and S3 (virtuals up to +9.5 a.u.) the contributions from additional virtual spinors become evident and the spinors above +5.0 a.u. obviously improve the first IP by only 0.003 and can therefore be excluded from the correlation space. Finding an upper bound for the virtual spinors is essential because in uncontracted calculations a large number of virtuals is generated up to very high energies. By adding more basis functions the density of virtual states within a certain energy range increases significantly and necessitates a stable cutoff. The addition of one more soft p function (f = 0.05, B1 ! B2) improves the first IP by 0.18 eV provided the
M. Pernpointner / Chemical Physics 329 (2006) 256–265
same energy cutoff in the virtual space was maintained (S1) and the transition from S1 to S2 yields another 0.06 eV being only one third of the energy obtained by adding one more soft p function. One might now naturally ask about a further improvement by additional diffuse p functions (B2 ! B3). At this stage one more p function with f = 0.01 had no effect (B2 S1 ! B3 S1) and, similarly, the change by one more diffuse s function (f = 0.05) added to B1 can also be neglected (see B4 S1 entry with respect to the B1 S1 entry). So far we found the most important effect by one soft p function on Se (B2) and restart our optimization from this basis. Before we further enlarge the function space one soft p function (f = 0.1) was added to the cc-pVTZ basis (B5 S1) on hydrogen yielding an improvement of 0.05 eV. This change is significant enough to keep the extended hydrogen basis for the future calculations. Additionally, the larger active space (S2) also improved the first IP by 0.06 eV, again pointing to S2 as being superior to the space S1. It is also observed that the second and third peak are stronger influenced by the extension of the correlation space. Certainly, all these changes are not drastic but they will add up at the end and contribute to the final result. By separately adding and omitting functions in combination with the variation of active spaces a full account of the intricate interplay between functions and correlation spaces is not possible but by this procedure the basis could be kept at a tractable size and linear dependencies avoided. Linear dependencies often occur when large even-tempered sequences of very soft functions are added to the original basis. Starting from the basis B5 we next investigated the influence of one additional soft d function on Se (f = 0.296298 from the s space, see B6 results) and obtained a substantial improvement of 0.1 eV in the first IP. In order to provide suitable polarization functions for the soft d functions we omitted the hardest f function and added the next soft f primitive from the p space (ff now: 0.51209 and 0.15469, see B7). Finally, the effect of g functions is not to be neglected as can be seen from the results where one g function of fg = 0.296298 has been added (B8 S1) and the active space enlarged to a positive spinor energy of +4.0 a.u. (B8 S2). The (B8 S2) basis/space combination was then chosen for the calculation of the valence and subvalence ionization spectrum and the quality of the results is now comparable to the nonrelativistic ADC(3) study by Powis et al. [26]. A similar study was undertaken with the selenium nondual basis set in its uncontracted form but no improvement could be achieved. We therefore conclude that the duality condition does not introduce further errors but allows for a reduction of the computational effort. Additionally, the inclusion of very soft s, p and d exponents as proposed by Kaufmann et al. [46] did not lead to a better reproduction of the experimental value but destabilized the fourcomponent calculations. In the optimization procedure for a suitable Te basis we started from a (23s19p13d2f) (B1) primitive set with
261
ff = 0.54465,0.22880. As it was described for SeH2 in detail the same strategy with respect to the first IP was followed for the creation of a suitable Te basis where the experimental value for the first TeH2 ionization energy was determined as 9.14268 eV from a Rydberg series [47] and as 9.14 eV from a photoelectron spectrum [48]. For the final calculation a (24s19p15d2f1g) basis was applied where the softest s and p exponents were 0.05 and 0.08371 and fg amounted to 0.382343. Using this basis a first IP of 8.91 eV was obtained bearing an error of 0.23 eV lying within the methodological error bars. It was observed that the g function in the basis set improved the result from 8.84 eV to 8.91 eV reflecting the relevance of higher angular momentum functions. A further extension of the basis certainly would lead even closer to the experimental value but due to computational limitations a compromise had to be found. 4. Results and discussion In [26] the outer valence PE spectrum of SeH2 was analyzed at the one-component ADC(3)/OVGF level and compared to results [49] obtained by other methods such as SAC-CI [50–52]. We therefore restrict ourselves to the comparison of the nonrelativistic and Dirac–Hartree–Fock ADC(3) results. The nonrelativistic valence electron configuration of SeH2 can be written as ð8a1 Þ2 ð4b2 Þ2 ð9a1 Þ2 ð4b1 Þ2 : In the Se atom a Dirac–Hartree–Fock average of configurations calculation yielded spin–orbit splittings of 0.969 eV for the 3d and 0.4 eV for the 4p spinors. A possible spin– orbit structure in the outer valence region with participating Se 4p functions was not obtained in the calculations and the bonding situation can be well described as in the nonrelativistic case in terms of Se px,py, pz and H 1s orbitals as a corresponding population analysis of the molecular spinors at the DHF level showed (see Table 4). We therefore use the ordinary C2v symbols also for the relativistic valence peak assignment. As in the nonrelativistic treatment only a very weak correlation contribution for the outermost peak of 2B1 symmetry can be observed whereas for the 2A1 and 2B2 states electron correlation causes a shift of approximately 0.24 eV resp. 0.5 eV (see Table 5). Table 4 Mulliken population analysis of the four outer valence spinors at the DHF-level MO-spinor
Population Se px
Se py
Se pz
Se s
Hs
4b1 9a1 4b2 8a1
97.4 – – –
– – 44.0 –
– 58.3 – 1.4
– 12.5 – 79.6
– 27.2 53.0 17.5
Contributions smaller than 1% are not included.
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M. Pernpointner / Chemical Physics 329 (2006) 256–265
Table 5 Nonrelativistic and relativistic SeH2 ionization potentials (in eV) at the independent particle and correlated level for the three outermost states MO-spinor
HF[26]
DHF
ADC(3) [26]
DHF-ADC(3)
Experiment [45]
4b1 9a1 4b2
9.78 13.01 15.09
9.772 13.083 15.048
9.74 12.84 14.79
9.729 (0.95) 12.838 (0.94) 14.550 (0.92)
9.897 12.825 14.626
The numbers in parentheses for the DHF-ADC(3) results are the pole strengths.
The first calculated satellites occur at 16.94 eV (0.004) and 17.2 eV (0.007) both borrowing intensity from the 2 B2 hole state corresponding to a (4b2)1 configuration. The next prominent peak corresponds to an ionization from the 8a1 orbital at 21.93 eV with a substantially reduced pole strength of 0.42. The intensity is distributed over a multitude of satellite states clearly indicating a breakdown of the one-particle model of ionization [10]. It was observed that the 8a1 satellite states are distributed over a wide energetic range where the last one lies at 57 eV. Most of the satellite states between 18.5 and 57.3 eV are related to the ionization from the 8a1 MO spinor and only a few of them could be attributed to the other three valence states: 9a1 (23.5 eV, 25.6 eV, 27.8 eV), 4b2 (28.5 eV, 32.2 eV) and 4b1 (31.8 eV, 43,7 eV). The DHFADC(3) description of the inner valence region between 20 and 25 eV yields a slightly different picture than in the one-component case which is possibly due to basis set deficiencies in the s function space and the sensitivity of satellite states to the basis set. The doublet structure of the d ionization is well reproduced and dense line bundles are observed mimicking Lorentzian shape (see Fig. 1). This specific shape is observable as soon as the initial cationic hole state is unstable against electronic decay [53] where the discrete basis functions are used for a description of the continuum states. In this work we focus on the ionization spectra and did not further analyze subsequent decay processes. The lowering of symmetry from Oh in the Se atom to C2v in SeH2 lifts the degeneracy of the d3/2 and d5/2 spinors resulting in
two groups of close-lying DHF-MO spinors with a 100% Se d population. In Fig. 2 only the spectrum of this inner valence region together with the Koopmans lines is shown. A considerable shift due to electron correlation is visible and the formerly isolated lines are now resolved in a dense Lorentzian structure providing an indication of electronic decay. These features could not be obtained in the nonrelativistic description of [26]. The bonding situation in TeH2 is identical to that of SeH2 except that all the participating Te orbitals now have a principal quantum number increased by unity. The outer valence structure according to a nonrelativistic calculation is then 2
2
2
ð12a1 Þ ð6b2 Þ ð13a1 Þ ð6b1 Þ
2
and this similarity is reflected in the comparable outer valence orbital energies and in the corresponding ionization spectrum (see Fig. 3). A more significant energetic difference exists, however, in the d features. In the TeH2 spectrum the 4d features lie approximately 15 eV lower in energy than the corresponding 3d peaks of SeH2 and a very pronounced Lorentzian envelope can also be observed in the TeH2 (4d3/2,5/2)1 final state. The 4d features are broader and the original intensity is distributed over more satellite states than in SeH2. Despite the sensitivity of the satellites to basis set quality a more efficient decay process can be assumed in the TeH2 case and corresponding life time calculations would be of high value in order to clarify this issue. For the nonrelativistic and relativistic numerical values for the outer valence IPs see Table 6.
1
4b 1
Hydrogen selenide DHF – ADC(3)
Pole strength
0.8
9a 1
4b 2
0.6
3d 5/2 3d 3/2
0.4
Sat.
8a 1
0.2
0
0
10
20
30
40
50
60
70
Ionization energy (eV) Fig. 1. The fully relativistic (DHF) ionization spectrum of hydrogen selenide at the ADC(3) level.
M. Pernpointner / Chemical Physics 329 (2006) 256–265
263
1
Hydrogen selenide DHF–ADC(3)
Pole strength
0.8
3d 5/2 Koopmans lines 0.6
3d
3/2
ADC(3) lines 0.4
0.2
0 62
63
64
65
66
67
68
69
70
71
Ionization energy (eV) Fig. 2. Enlarged ionization spectrum for the 3d range. A large correlation contribution compared to the Koopmans’ lines is visible together with the typical Lorentzian structure of the two spin–orbit split line bundles describing a decaying state. 1
6b 1
Pole strength
0.8
6b 2
13a 1
Hydrogen telluride DHF–ADC(3)
0.6
0.4
4d 3/2
12a 1 0.2
4d 5/2
0 1
6b 1 6b 2
0.8
Pole strength
13a 1
Hydrogen telluride NR–ADC(3) 0.6
4d 12a 1
0.4
0.2
0
0
10
20
30
40
50
Ionization energy (eV) Fig. 3. The fully relativistic (DHF) ionization spectrum of hydrogen telluride at the ADC(3) level.
In Fig. 3 both the nonrelativistic (NR) and fully relativistic TeH2 ionization spectra are plotted. For the outermost 6b1 state no relativistic shift is observable and the
next two subsequent states show only slight alterations from the nonrelativistic result. This is insofar remarkable that in neighboring iodine compounds a pronounced
M. Pernpointner / Chemical Physics 329 (2006) 256–265
Table 6 Nonrelativistic and relativistic TeH2 ionization potentials (in eV) at the independent particle and correlated level for the three outermost states MO-spinor 6b1 13a1 6b2
HF 8.88 11.87 13.84
DHF 8.85 12.00 13.71
ADC(3) 8.94 (0.95) 11.66 (0.94) 13.20 (0.87)
DHF-ADC(3) 8.91 (0.95) 11.79 (0.94) 13.12 (0.88)
Experiment [47,48] 9.14 12.00 13.25
The numbers in parentheses for the ADC(3) results are the pole strengths.
9 8.8 8.6 8.4 8.2 8
60 80 100 120 140 160 180 1.5
nd Bo
e gl an
spin–orbit splitting occurs in the outermost state [54] whereas in TeH2 no such features are observed even with a comparable size of the atomic 5p spin–orbit splitting in tellurium and iodine (0.88 eV resp. 1.13 eV). This can be explained by the availability of two p-type lone pairs (px and py) in monovalent iodine compounds that can be mixed differently under the influence of the spin–orbit operator in a molecular environment whereas in TeH2 only one px lone pair is available being represented by a single j,mj component of a quasi-atomic spinor. The outermost peak in the experimental spectrum does not exhibit vibrational structure [48] in contrast to the relatively broad a1 and b2 bands. In [48] both the vertical and adiabatic ionization potentials are given and we take the vertical IP values as the experimental reference. The DHF-ADC(3) values show a maximum deviation of 0.2 eV lying within the methodological error range. For the a1 and b2 peaks a vibrational fine structure was not calculated. The TeH2 satellite structure around 18–30 eV bears some similarity to the nonrelativistic case but due to different coupling schemes resulting from the application of double group symmetry the intensities will be distributed differently among the individual peaks (see also [7] for a coupling analysis in the carbon monoxide case). A striking difference in the spectra happens in the 4d region around 50 eV. Here only one 4d line bundle is obtained in the NR-ADC(3) treatment whereas two bundles shifted to lower ionisation energy result in the DHFADC(3) calculation. For TeH2 we also investigated the dependence of the first IP on the molecular geometry. Hereby the H–Te–H angle was varied from 65o to 180o and the Te–H bond dis˚ to 1.8 A ˚ in steps of 0.1 A ˚ always keeping tance from 1.5 A C2v symmetry. For all bond lengths the first IP is lowest for the linear molecular conformation and has its minimum at ˚ . (see Fig. 4). Enlarging the bond angle leads to a 1.8 A higher repulsion between the bonding MO’s and the px lone pair increasing its energy. It can also be seen that there is only a weak dependence of the IP on the bond length around the equilibrium bond angle and for larger bond angles the sensitivity to bond stretching becomes more pronounced. Due to computational limitations these calculations were performed without g functions on Te but we believe that the observed trend is correctly reproduced with the smaller basis as well.
First IP of hydrogen telluride at DHF-ADC(3) level Ionization potential (eV)
264
1.55
1.6
1.65
1.7
1.75
1.8
tr.)
nce (Angs
Bond dista
Fig. 4. The first ionization potential of TeH2 with respect to variations of the Te–H bond distance and H–Te–H bond angle calculated at the DHFADC(3) level of theory.
5. Outlook For an accurate theoretical description of PE spectra a consistent treatment of electron correlation and relativity is necessary. Especially in the PtCl2 6 complex nonadditivity of spin–orbit coupling and electron correlation became evident. In this respect an investigation of the PtF2 6 ionization spectrum would also be of great interest. To the best of our knowledge no experimental ionization spectrum of PtF2 6 is available so far and this calculation would have some predictive character. Furthermore, lifetime calculations of the (3d)1 resp. (4d)1 final states could provide valuable information for a subsequent analysis of the decay processes in SeH2 resp. TeH2. References [1] A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, 1971. [2] D.J. Thouless, The Quantum Mechanics of Many-Body Systems, Academic, New York, 1972. ¨ hrn, Propagators in Quantum Chemistry, [3] J. Linderberg, Y. O Academic, New York, 1973. [4] L.S. Cederbaum, W. Domcke, Adv. Chem. Phys. 36 (1977) 205. [5] J. Schirmer, L.S. Cederbaum, O. Walter, Phys. Rev. A 28 (1983) 1237. [6] M. Pernpointner, A.B. Trofimov, J. Chem. Phys. 120 (2004) 4098. [7] M. Pernpointner, J. Chem. Phys. 121 (2004) 8782. [8] J. Schirmer, Phys. Rev. A 43 (1991) 4647. [9] A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, Englewood Cliffs, 1963. [10] L.S. Cederbaum, W. Domcke, J. Schirmer, W. von Niessen, Adv. Chem. Phys. 65 (1986) 115. [11] D.M.P. Holland, I. Powis, L. Karlsson, A.B. Trofimov, J. Schirmer, W. von Niessen, Chem. Phys. 297 (2004) 55. [12] M. Pernpointner, J. Breidbach, L.S. Cederbaum, J. Chem. Phys. 122 (2005) 064311. [13] M. Kaupp, C. van Wu¨llen, R. Franke, F. Schmitz, W. Kutzelnigg, J. Am. Chem. Soc. 118 (1996) 11939.
M. Pernpointner / Chemical Physics 329 (2006) 256–265 [14] J. Styszyn´ski, X. Cao, G.L. Malli, L. Visscher, J. Comp. Chem. 18 (1997) 601. [15] M.-S. Liao, Q.-E. Zhang, J. Phys. Chem. A 102 (1998) 10647. [16] M. Pernpointner, L.S. Cederbaum, J. Chem. Phys. 122 (2005) 214302. [17] C. Buth, R. Santra, L.S. Cederbaum, J. Chem. Phys. 119 (2003) 7763. [18] H. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Twoelectron Systems, Springer, Berlin, 1957. [19] G. Breit, Phys. Rev. 34 (1929) 553. [20] G. Breit, Phys. Rev. 36 (1930) 383. [21] G. Breit, Phys. Rev. 38 (1931) 463. [22] G. Breit, Proc. Nat. Acad. Sci. USA 17 (1931) 70. [23] G. Breit, Phys. Rev. 39 (1932) 616. [24] W. Kutzelnigg, Phys. Scr. 36 (1987) 416. [25] M. Pernpointner, J. Phys. B: At. Mol. Phys. 38 (2005) 1955. [26] I. Powis, J.D. Thrower, A.B. Trofimov, T.E. Moskovskaya, J. Schirmer, A.W. Potts, D.M.P. Holland, F. Bruhn, L. Karlsson, Chem. Phys. 315 (2005) 121. [27] ND-ADC3 Version 1.1.0 (2004): program for the calculation of the non-Dyson ADC(3)-Matrix and the ISR representation of a oneparticle operator, written by J. Breidbach. [28] J. Schirmer, A.B. Trofimov, G. Stelter, J. Chem. Phys. 109 (1998) 4734. [29] I.P. Grant, H.M. Quiney, Adv. Atom. Mol. Phys. 23 (1988) 37. [30] H.M. Quiney, Relativistic effects in atoms and molecules, in: S. Wilson (Ed.), Methods in Computational Chemistry, vol. 2, Plenum, New York, 1988, p. 223. [31] H.M. Quiney, I.P. Grant, S. Wilson, Phys. Scr. 36 (1987) 460. [32] H.M. Quiney, I.P. Grant, S. Wilson, Many-body methods, in: U. Kaldor (Ed.), Quantum Chemistry, Springer, Berlin, Heidelberg, New York, 1989. [33] S. Wilson, The Effects of Relativity on Atoms, Molecules and the Solid State, Plenum Press, New York, 1991.
265
[34] K.D. Dyall, I.P. Grant, F.A. Parpia, E.P. Plummer, Comput. Phys. Commun. 55 (1989) 425. [35] K.G. Dyall, K. Fægri, Theor. Chim. Acta 94 (1996) 39. [36] F. Mark, W.H.E. Schwarz, Phys. Rev. Lett. 48 (1982) 673. [37] W.H.E. Schwarz, H. Wallmeier, Mol. Phys. 46 (1982) 1045. [38] R.E. Stanton, S.J. Havriliak, Chem. Phys. 81 (1984) 1910. [39] K. Dyall, I.P. Grant, S. Wilson, J. Phys. B: At. Mol. Phys. 17 (1984) 1910. [40] P.J.C. Aerts, W.C. Nieuwpoort, Chem. Phys. Lett. 125 (1986) 83. [41] W. Kutzelnigg, Int. J. Quantum Chem. 25 (1984) 107. [42] J.T.H. Dunning, J. Chem. Phys. 90 (1989) 1007. [43] Dirac, a relativistic ab initio electronic structure program, Release DIRAC04.0 (2004), written by H.J. Aa. Jensen, T. Saue, L. Visscher with contributions from V. Bakken, E. Eliav, T. Enevoldsen, T. Fleig, O. Fossgaard, T. Helgaker, J. Laerdahl, C.V. Larsen, P. Norman, J. Olsen, M. Pernpointner, J.K. Pedersen, K. Ruud, P. Salek, J.N.P. van Stralen, J. Thyssen, O. Visser, T. Winther. (http://dirac.chem.sdu.dk). [44] G. Herzberg, Electronic Spectra and Electronic Structure of Polyatomic Molecules, Van Nostrand, New York, 1966. [45] K. Bo¨rlin, M. Jungen, L. Karlsson, R. Maripuu, Chem. Phys. 113 (1987) 309. [46] K. Kaufmann, W. Baumeister, M. Jungen, J. Phys. B: At. Mol. Phys. 22 (1989) 2223. [47] C.A. Mayhew, J.P. Connerade, M.A. Baig, J. Phys. B: At. Mol. Phys. 19 (1986) 4149. [48] A.W. Potts, W.C. Price, Proc. R. Soc. A 326 (1972) 181. [49] M. Ehara, M. Ishida, H. Nakatsuji, J. Chem. Phys. 114 (2001) 8990. [50] H. Nakatsuji, K. Hirao, Chem. Phys. Lett. 47 (1977) 569. [51] H. Nakatsuji, Chem. Phys. Lett. 67 (1979) 329. [52] H. Nakatsuji, Chem. Phys. Lett. 67 (1979) 334. [53] L.S. Cederbaum, J. Zobeley, F. Tarantelli, Phys. Rev. Lett. 79 (1997) 4778. [54] M. Pernpointner, S. Knecht, Chem. Phys. Lett. 410 (2005) 423.