Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200
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C. AMOVILLI AND R. McWEENY |
the outset that since the energy of a ‘core-hole’ state normally lies high in the con- tinuum, relative to the lowest energy state in which the hole has been filled by an
Auger transition from a valence orbital, there are severe problems in calculating the energy by conventional bound-state methods: indeed the corresponding ‘state’ is not a true bound state at all, being at best metastable and subject to spontaneous decay, with filling of the hole and ejection of a second (Auger) electron. A completely satisfactory calculation would thus require the inclusion in the basis of continuum functions, to admit the possibile presence of a scattered electron, and would employ propagator methods which are well adapted to the description of such processes (see, for example, Agren [7]). Nevertheless, bound-state methods have been widely and successfully used in the interpretation of PES, ESCA and Auger spectra. In particular, the formulation of SCF methods for systems containing incompletely occupied shells (McWeeny [8]) was applied by Firsht and McWeeny [9] to free atoms and ions, with inner-shell holes, yielding results of much higher accuracy than those based on the Koopmans theorem. The present paper reports applications of similar methods to some small molecules.
2. Formulation
For inner-shell ionizations, where the energy change may be several hundred eV, it is sufficient to use ensemble averaging (Slater [10]; McWeeny [8]) over the various states of a configuration - which differ relatively little in energy. The corresponding formulation of many-shell SCF theory is fully described elsewhere (McWeeny [11])
and will be summarized only briefly. We use |
to |
denote |
the |
orbitals of Shell |
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K, containing |
electrons, and express the orbitals of all shells in terms of a common |
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set of TO basis functions { |
}: thus, collecting the functions in row matrices, |
||||||
where |
is an |
matrix, n being the total |
number of orbitals employed. It is |
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also convenient to partition the row matrix into subsets |
and |
the rectangular |
|||||
matrix |
into corresponding |
blocks |
. |
The set |
of occupation numbers |
||
|
|
then defines the electron configuration, while the average energy |
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(for all states with the same partitioning of electrons among shells) is given by
Here |
|
is the fractional occupation number of the |
spin-orbitals |
of Shell K and |
is a suitably averaged electron interaction matrix (cf. the usual |
||
Roothaan |
matrix') and depends on the density matrices |
of all |
|
shells: in |
fact |
|
|
where the modified occupation number (removing the self-interaction when |
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is |
|
The matrix G(2R) coincides with the usual G matrix |
|
for a closed-shell system, while h in (4) is the usual 1-electron Hamiltonian matrix.
All matrices are defined with respect to the basis functions in
The energy expression (4) applies when the orbitals are orthonormal and in seeking a stationary value it is thus necessary to introduce constraints to maintain orthonor-
mality during a variation. When this is done, the orbitals that give a stationary
CORE-HOLE STATES AND THE KOOPMANS THEOREM |
167 |
point turn out to be eigenfunctions of a certain ‘effective’ Hamiltonian (which embodies the constraints); and this leads to an iterative procedure parallel to that of the usual closed-shell SCF theory. For molecules, these ‘canonical’ orbitals are normally the delocalized MOs which extend over the whole molecular framework; but
is invariant against unitary mixing of the orbitals within each shell (which leaves
the matrices |
unchanged) and |
this freedom may be exploited in the usual way |
to obtain alternative orbitals with |
a high degree of localization in different regions |
|
of the molecule |
(e.g. inner shells, |
bonds, lone pairs). Clearly, in discussing phe- |
nomena related to physically well-defined regions, we shall be more concerned with the localized orbitals than the canonical MOs. The question that then arises is that of what localization criterion to adopt: the one to be used in working with atomic inner shells is simply that the inner-shell orbital be constructed from basis functions located on the atom in question. All other orbitals are easily orthogonalized against inner shells (e.g. by the Schmidt method) and among themselves (e.g. by the Löwdin
transformation).
Instead of using repeated solution of a suitable eigenvalue equation to optimize the orbitals, as in conventional forms of SCF theory, we have found it more convenient to optimize by a gradient method based on direct evaluation of the energy functional (4), orthonormalization being restored after every parameter variation1. Although many iterations are required, the energy evaluation is extremely rapid, the process is very stable, and any constraints on the parameters (e.g. due to spatial symmetry or choice of some type of localization) are very easily imposed. lt is also a simple matter to optimize with respect to non-linear parameters such as orbital exponents.
3. Some results
We have considered K-shell ionizations from the atoms of carbon, oxygen, and nitrogen in a series of small molecules, typically using basis sets of ‘double-zeta’ quality (as tabulated by, for example, Dunning [12]), with the addition of polarization functions for the smallest systems. The total energies for the neutral molecules and some of their core-hole positive ions are collected in Table 1. Energies for the molecular ground states, as calculated by standard (RHF) SCF methods, are also shown for comparison. It is evident that the localization constraint for the inner shells has a
negligible effect on the energies. |
|
The energies in the last column of |
1 show the effect of modifying the basis |
set, after the ionization, to allow for the increased central field to which the valence electrons are then exposed. Some of the early work on the interpretation of ESCA and Auger spectroscopy employed an ‘equivalent-core’ approximation (Shirley [13]) in which, with a minimal basis set, valence orbitals were given exponents appropriate to an effective atomic number Z + 1 instead of Z: the inner shell, with only one Is electron, was thus ‘modelled’ by an 'equivalent core' with two 1s electrons but one extra unit of positive charge on the nucleus. This simple model has been found equally effective in the case of a DZ basis: in describing the valence electrons of an atom with a core hole it is sufficient to use contracted gaussians with tabulated exponents and contraction coefficients for the atom of atomic number Z + 1 instead of Z. For the Is orbital, on the other hand, appeal to ‘screening constant’ rules
(Slater [14]; Clementi and Raimondi [15]) suggests that the Gaussian exponents for
1The valence set is orthogonalized against the core set, so as not to ‘contaminate’ the core
orbitals, while symmetric orthonormalization is employed w i t h i n each set.
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C. AMOVILLI AND R. McWEEN |
the neutral atom should be multiplied by a scale factor
following removal of an electron, |
being the recommended effective nuclear charge |
for the atom in question. This value (close to the actual nuclear charge) proves to be perfectly satisfactory. In fact, the procedure just described leads to energy values which are not appreciably affected by further parameter optimization. The results in the Table confirm the need to re-optimize the basis following ionization, the resultant drop in energy being quite significant. The agreement with experiment at this level is now probably as good as can be expected, bearing in mind the extreme simplicity of the theoretical model on which the calculations are based.
The results for homonuclear molecules are of particular interest in so far as they exhibit “symmetry breaking”. For N2 , for example, removal of an electron from the
MO of a ground-state SCF calculation, with re-optimization of all orbitals subject to symmetry constraints, leads to an energy value of –93.49612 hartree for the positive ion in which the core hole is symmetrically ‘shared’ between the two atoms. But when the symmetry constraint is relaxed the energy falls to –93.88673 hartree, corresponding to localization of the hole on one centre alone: this is the result expected on physical grounds, given a sufficiently short time scale for the process of electron removal – the valence-electron distribution responding immediately to the enhanced attraction towards the core with the hole. Of course, as already remarked, the resultant metastable state will decay rapidly: the symmetry constrained wavefunction describes the stationary state, in a long-time limit, of a fictitious model system in which the hole appears on either centre with equal probability. The spectroscopic
CORE-HOLE STATES AND THE KOOPMANS THEOREM |
169 |
observations are in fact consistent with the short-time situation, before relaxation of the electron distribution has taken place.
The inner-shell ionization energies are collected in Table 2 and compared with the
Koopmans estimates (which are seen to be seriously in error) and the best available experimental values. Whilst the Koopmans approximation is clearly incapable of giving good ionization energies and must therefore be used with caution in predicting the ‘chemical shifts’ in going from one molecule to another, the ionization energies based on (2) are rather satisfactory.
In view of possible applications of the method to much larger molecules, where the use of extended basis sets may be impracticable, it is worth asking whether good results might also be obtained with only a minimal basis: in this case, with the reduced flexibility of the basis, it would clearly be desirable to optimize the orbital exponents. To investigate this possibility, a study has been made of the inner-shell ionization energies of the carbon atom in the series of fluoro-substituted methanes
using the MIDI-4 basis of Tatewaki and Huzinaga [16]. The approximation (6) was used for the 1s orbitals, but the scale factors for the valence
orbitals had to be optimized (for both the neutral and ionized systems) since the tabulated MIDI-4 values refer to the isolated atoms. Table 3 shows the resultant scale factors for the carbon 2s and 2p orbitals, which are most strongly affected by
170 C. AMOVILLI AND R. McWEENY
the core ionization. The absolute energy values are of course somewhat inferior to those obtained with a basis (cf. the results for in Table 1) but the ionization energies, shown in Table 4, are still in very satisfactory agreement with experiment. Even the chemical shifts for fluoro-substitution are very close to those observed: what is more surprising is that the shifts predicted by the Koopmans theorem are also quite satisfactory, even though the ionization energies are in error by 10-15 eV. Whether the Koopmans theorem will retain its apparent predictive value in situations where the chemical shift is much smaller remains an open question.
4. Conclusions and further applications
It is evident that the method of calculation used in this work provides an extremely simple approach to the interpretation of ESCA results for ionization from an atomic
K shell, in spite of the fact that the state of the ion plus the ejected electron lies high in the energy continuum of the neutral molecule. More sophisticated methods of dealing with such states are of course available (see Agren et al [7]) but, whilst capable of giving excellent results for valence electron ionizations (including also intensities and vibrational fine structure), encounter considerable difficulty in treating core-hole states, where relaxation effects are very severe. The simple model used here, on the other hand, is particularly well adapted to the study of these ‘deep’ ionizations and gives an immediate and transparent interpretation of the relaxation effects in
terms of scaling (contraction) of the valence orbitals. It is also possible to extend the approach in various ways; for example, for open-shell molecules, the states of the
CORE-HOLE STATES AND THE KOOPMANS THEOREM |
171 |
configuration with a core hole (which arise from differences of spin coupling between the core and valence electrons) can readily be studied by using the optimized orbitals for the configurational average energy to set up the secular equations that will lead to the individual states.
Another application is to the study of the ‘Auger states’ in which a further electron ionization of attachment may occur, leaving the system with holes in more than one shell. Such states were considered in some detail by Firsht and McWeeny [9] for free atoms: here we have made a preliminary application to the nitrogen molecule. The initial aim is simply to identify and assign the principal peaks and satellites in the Auger spectrum of gaseous
The calculations were performed using a double-zeta basis set with addition of a polarization function and lead to the results reported in Table 5. The notation used for each state is of typical hole-particle form, an asterisc being added to an orbital (or shell) containing a hole, a number (1) to one into which an electron is promoted. In the same Table we show also the frequently used 'letter' symbolism in which ‘K’ indicates an inner-shell hole, ‘L’ a hole in the valence shell, and ‘e’ represents an excited electron. The more commonly observed ionization processes in the Auger spectra of
are of the type K—LL (a ‘normal’ process, ‘core-hole state’ ‘double-hole state’);
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C. AMOVILLI AND R. McWEENY |
KL-LLL (‘monopole ionizations’); and KLe–LL or Ke–L (‘high-energy satellites’). From Table 5 it is possible to estimate the energies of transition for various pairs of states, even though the ‘state’ energies are stricly speaking configurational averages. Figure 1 shows (vertical lines) the estimated values of these transition energies, superimposed on the experimental spectrum. It is noteworthy that, even if the vertical lines are not actually coincident with the peak positions, the assignment of the peaks for various processes is substantially in accord with that made by Moddeman et al [21] on the basis of the experimental data.
References
1.L. Brillouin, Actualités Sci. et Ind., No. 71, (1933).
2.L. Brillouin, Actualités Sci. et Ind., No. 159, (1934).
3.T. A. Koopmans, Physica 1, 104, (1933).
4.G. Berthier, in Current Aspects of Quantum Chemistry, R. Carbó (Ed.), Elsevier, Amsterdam, 1982, pp. 145-156.
CORE-HOLE STATES AND THE KOOPMANS THEOREM |
173 |
5.G. Berthier, in Quantum Chemistry - Basic Aspects, Actual Trends. R. Carbó (Ed.), Elsevier, Amsterdam, 1989, pp. 91–102.
6.K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P. F. Heden, K.
Hamrin, U. Gelius, T. Bergmark, L. O. Werme, R. Manne and Y. Baer,
ESCA Applied to Free Molecules, North-Holland, Amsterdam, 1969.
7.H. Agren, A. Cesar and C. M. Liegener, Adv. Quant. Chem. 23, 1, (1992).
8.R. McWeeny, Mol. Phys. 28, 1273, (1974).
9.D. Firsht and R. McWeeny, Mol. Phys. 32, 1637, (1976).
10.J. C. Slater, Quantum Theory of Atomic Structure, Vol. I, McGraw-Hill, New
York, 1960.
11.R. McWeeny, Methods of Molecular Quantum Mechanics, 2nd ed., Academic,
London, 1989.
12.T. II. Dunning, J. Chem. Phys. 53, 2823, (1970).
13.D. A. Shirley, Chem. Phys. Lett. 16, 220, (1972).
14.J. C. Slater, Phys. Rev. 36, 57, (1930).
15.E. dementi and D. L. Raimondi, J. Chem. Phys. 38, 2686, (1963).
16.H. Tatewaki and S. Huzinaga, J. Comput. Chem. 1, 205, (1980).
17.S. Huzinaga, J. Chem. Phys. 42, 1293, (1965).
18.T. X. Carroll and T. D. Thomas, J. Electron Spectrose. Relat. Phenom. 10, 215, (1977).
19.D. B. Adams and D. T. Clark, Theor. Chim. Acta 31, 171, (1973).
20.M. Nakamura, M. Sasanuma, S. Sato, M. Watanabe, H. Yamashita, Y. Iguchi,
A.Ejiri, S. Nakai, S. Yamaguchi, T. Sagawa, Y. Nakai and T. Oshio, Phys. Rev. 178, 80, (1969).
21.W. E. Moddeman, T. A. Carlson, M. O. Krause, B. P. Pullen, W. E. Bull and
G.K. Schweitzer, J. Chem. Phys. 55, 2317, (1971).
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An Application of the Half-Projected Hartree-Fock Model to the Direct Determination of the Lowest Singlet and Triplet Excited States of Molecular Systems
Y. G. SMEYERS, P. FERNANDEZ-SERRA1 and M. B. RUIZ
Instituto de Estructura de la Materia, C.S.I.C., c/Serrano, 123, E-28006-Madrid, Spain
1. Introduction
Among the many ways to go beyond the usual Restricted Hartree-Fock model in order to introduce some electronic correlation effects into the ground state of an electronic system, the Half-Projected Hartree-Fock scheme, (HPHF) proposed by
Smeyers [1,2], has the merit of preserving a conceptual simplicity together with a relatively straigthforward determination. The wave-function is written as a DODS Slater determinant projected on the spin space with S quantum number even or odd.
As a result, it takes the form of two DODS Slater determinants, in which all the spin functions are interchanged. The spinorbitals have complete flexibility, and should be determined from applying the variational principle to the projected determinant.
The difficulty of determining the Half-Projected Hartree-Fock function has somewhat hampered its utilization [3-10]. Some calculations, however, exist in literature. At present time, because of the increasing computing facilities, as well as the introduction of more powerful convergence techniques, the HPHF model is expected to play a more important role, especially in the field of medium size molecules, in which the use of more sophisticated procedure are not yet possible [9-10].
In addition, since the HPHF wavefunction exhibits a two-determinantal form, this model can be used to describe singlet excited states or triplet excited states in which
the projection of the spin momentum |
The HPHF approximation appears thus |
|
as a simple method for the direct determination of excited states (with |
such |
|
as the usual Unrestricted Hartree Fock model does for determining triplet excited states with
In the present paper, we propose the use of the HPHF approximation for the direct
calculation of excited states, in which |
just as Berthier [11], and Pople and |
|
Nesbet [12] did for the determination of states in which |
We give some ex- |
|
amples of such calculations, either when the excited state wavefunction is orthogonal or not by symmetry to that of the ground state.
1Permanent Address: Departamento de Ingeniería de Circuitos y Sistemas, E.U.I.T. de Teleco- municacion, Universidad Politécnica de Madrid, E-28031-Madrid, Spain
175
Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 175-188.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.