Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200
.pdf156 |
M. DEFRANCESCHI ET AL. |
References
1 . P.A.M. Dirac, The principles of Quantum Mechanics. 4th ed, The Clarendon Press,
Oxford, 1958, pp. 89-97
2.P. Roman, Advanced Quantum Theory, Addison Wesley, Reading, 1965, p. 11
3.J. H. Moore, J. A. Tossel, M. A. Coplan, Acc. Chem. Res., 15, 192, (1982)
4.C. E. Brion, Int. J. Quantum Chem., 29, 1397, (1986)
5.E. Weigold, Momentum Space Wave Functions. American Institute of Physics, vol. 86, Adelaide, 1982
6.B. K. Novosadov, J Mol. Struct., 54, 269, (1979)
7.J. Navaza, G. Tsoucaris, Phys. Rev., A24, 683, (1981)
8.E.A. Hylleraas, Z. Physik, 74, 216, (1932)
9.H. A. Bethe, E.E. Salpeter. Quantum Mechanics of One and Two-electrons Atoms .
Springer-Verlag, Berlin, 1957, pp. 36-47
10.R. McWeeny, C.A. Coulson, Proc. Phys. Soc., 162, 509, (1949)
1 1. R. McWeeny, Proc. Phys. Soc., 162, 519, (1949)
12.N.V. Svartholm, The Binding Energies of the Lightest Atomic Nuclei. Thesis, Lund, 1945
13.E.N. Lassettre, J. Chem. Phys., 58, 1991, (1973)
14.C. Carter, Proc. Roy. Soc., A235, 321, (1956)
15.E.A. McCullough Jr., J. Chem. Phys., 62, 3991, (1975) and L. Laaksonen, P. Pyykkö, D. Sundholm, Chem. Phys. Lett., 96, 1, (1983)
16.M. Defranceschi, M. Suard, G. Berthier, C. R. Acad. Sci. Paris, 299, 9, (1984)
17.M. Defranceschi, M. Suard, G. Berthier, C. R. Acad. Sci. Paris, 296, 1301,
(1983) and Int. J. Quantum Chem, 25, 863, (1984)
18.M. Defranceschi, J. Delhalle, C.R. Acad. Sc. Paris, 301, 1405, (1985)
19.M. Defranceschi, J. Delhalle, Phys. Rev. B34, 5862, (1986)
20.J. Delhalle, J.G. Fripiat and M. Defranceschi, Annales Soc. Scient. Bruxelles ,
101, 9, (1987)
2 1 . J.G. Fripiat, M. Defranceschi, J. Delhalle in Numerical Determination of the
Electronic Structure of Atoms. Diatomic and Polyatomic Molecules. M.Defranceschi,
J. Delhalle (eds), NATO-ASI Series C vol. 271, Kluwer Academic Publishers,
Dordrecht, 1989, pp. 245-250
QUANTUM CHEMISTRY COMPUTATIONS IN MOMENTUM SPACE |
157 |
22.M. Defranceschi, L. De Windt, J.G. Fripiat, J. Delhalle, J. Mol. Struct.(Theochem),
258, 179, (1992)
23.J. Delhalle, J.G. Fripiat, M. Defranceschi, Bull. Soc. Chim. Belg., 99, 135, (1990)
24.M. Defranceschi, J. Delhalle, Eur. J. Phys., 11, 172, (1990)
25.J. Delhalle and M. Defranceschi, Int. J. Quantum Chem., S 21, 425, (1987)
26.M. Defranceschi, A Lahmam-Bennani, J. Elect. Spectr. Rel. Phenom., 48, 1,(1989)
27.L. De Windt, M. Defranceschi, J. Delhalle, Theoret. Chim. Acta , 86, 487, (1993)
28.L. De Windt, P. Fischer, M. Defranceschi, J. Delhalle, J.G. Fripiat, J. Comp. Phys., 1 1 1 , 266, (1994)
29.L. De Windt, J.G. Fripiat, J. Delhalle, M. Defranceschi, J. Mol. Struct.(Theochem),
254, 145, (1992)
30.L. De Windt, M. Defranceschi, J. Delhalle, Int. J. Quantum Chem., 45, 609, (1993)
31. L. De Windt, M. Defranceschi, J.G. Fripiat, J. Delhalle, Annales Soc. Scient.
Bruxelles , 105, 89, (1991)
32.G. Berthier, M. Defranceschi, J. Navaza, M. Suard, G. Tsoucaris, J. Mol. Struct.
(Theochem) , 120, 243, (1985)
33.G. Berthier, M. Defranceschi, J. Delhalle, in Numerical Determination of the Electronic Structure of Atoms. Diatomic and Polyatomic Molecules. M. Defranceschi and J.Delhalle (eds) NATO-ASI Series C vol 271, Kluwer Academic Publishers,
Dordrecht, 1989, pp. 209-238
34.G. Berthier, M. Defranceschi, J. Delhalle in Self-Consistent Field : Theory and Applications. R. Carbo, M. Klobukowski Eds, Elsevier, Amsterdam 1990, "Studies in Physical and Theoretical Chemistry ", n 70, p. 387
35.P. Fischer, M. Defranceschi, J. Delhalle, Numerische Mathematik , 63, 67, (1992)
36.M. Defranceschi, M. Suard, G. Berthier, Folia Chimica Theoretica Latina, XVII,
65, (1990)
37.P. Fischer, L. De Windt, M. Defranceschi, J. Delhalle, J. Chem. Phys., 99, 7888, (1994)
38.L. De Windt, M. Defranceschi, J. Delhalle, to be published
39.C.C.J. Roothaan, Rev. Mod. Phys., 23, 69, (1951)
40.G.G. Hall, Proc. Roy. Soc., A205, 541, (1951)
4 1 . G. Berthier, C. R. Acad. Sci. Paris, 238, 91, (1954) ; J. Chim. Phys., 51, 363,
(1954) and 52, 141, (1955)
158 |
M. DEFRANCESCHI ET Al |
42.J.A. Pople, R.K. Nesbet, J. Chem. Phys., 22, 571, (1954)
43.C.C.J. Roothaan, Rev. Mod. Phys., 32, 179, (1960)
44.J. Hinze, J. Chem. Phys., 59, 6424, (1973)
45.A. Veillard, Theoret. Chim. Acta, 4, 22, (1967)
46.D.R. Hartree, W. Hartree, B. Swirles, Phil. Trans. Roy. Soc., A238, 229, (1939)
47.G. Das, A.C. Wahl, J. Chem. Phys., 44, 87, (1970)
48.A. Veillard, E. Clementi, Theoret. Chim. Acta, 7, 133, (1967)
49.G. Del Re, J. Ladik, G. Biczo, Phys. Rev., 155, 977, (1967)
50.J.M. André, L. Gouverneur, G. Leroy, Int. J. Quantum Chem., 1, 427, (1967)
51. G. Fonte, Theoret. Chim. Acta , 59, 533, (1981)
52.E.R. Davidson, D. Feller, Chem. Rev., 86, 681, (1986)
53.B. Klahn, J.D. Morgan, J. Chem. Phys., 81, 410, (1984)
54.C.L. Davis, H.J. Monkhorst, Chem. Phys. Lett, 111, 526, (1984)
55.L. Laaksonen, P. Pyykkö, D. Sundholm, Comput. Phys. Rep., 4, 313, (1986)
56.E.A. McCullough, Comput. Phys. Rep., 4, 265, (1986)
57.P.O. Löwdin, J. Chem. Phys., 18, 365, (1950)
58.S.A. Alexander, H.J. Monkhorst, Int. J. Quantum Chem., 32, 361, (1987)
59.S.A. Alexander, R.L. Coldwell, H.J. Monkhorst, J. Comput. Phys., 76, 263,
(1988).
60.W. Rodriguez, Y. Ishikawa, Chem. Phys. Lett., 146, 515, (1988)
61. Y. Ishikawa, I.L. Aveponte-Allevanet, S.A. Alexander, Int. J. Quantum Chem,
S23, 209, (1989).
62.N.N. Lebedev, Special Functions and their Applications. Dover, New York, 1972
63.J. Spanier, K.B. Oldham, An Atlas of Functions. Hemisphere, Washington, 1987
64.M. Defranceschi, M. Sarrazin, Comput. Phys. Com, 52, 409, (1989)
65.Gausfit, M. Herman, R.E. Stanton, Quantum Chemistry Program Exchange, 11,
237, (1973)
66.R.F. Stewart, J. Chem. Phys., 50, 2485, (1969)
Core-Valence Separation in the Study of Atomic Clusters
O. SALVETTI
Dipartimento di Chimica e Chimica Industriale. Università di Pisa
Via Risorgimento 35, 56126 Pisa, Italy
The study of clusters containing an increasing number of atoms provides an interesting theoretical way of understanding the properties of solid matter.
In particular it allows us to consider in a simple way possible irregularities of structure, the existence of non stoichiometric compounds, and the possibility of replacing one atom by another.
A study of the variation of properties with cluster size is also of great importance, especially in view of experimentally observed variations, which may amount to almost a change of phase, in clusters ranging from 10 to 50 atoms [1].
The main difficulty in the theoretical study of clusters of heavy atoms is that the number of electrons is large and grows rapidly with cluster size. Consequently, ab initio "brute force" calculations soon meet insuperable computational problems. To simplify the approach, conserving atomic concept as far as possible, it is useful to exploit the classical separation of the electrons into "core" and "valence" electrons and to treat explicitly only the wavefunction of the latter. A convenient way of doing so, without introducing empirical parameters, is provided by the use of generalyzed product function, in which the total electronic wave function is built up as antisymmetrized product of many group functions [2-6].
This scheme is very appealing, since it allows us to reduce drastically the numbers of electrons to be considered, thus making possible essentially "ab initio" calculation, even for large systems,.
If a cluster is built from various separated atoms A, B, ... with |
... "core" electrons, |
|
descibed by the functions |
..., the generalized product for the |
|
total number of electrons will be given by the following expression:
where |
is the total number of "core" electrons, |
and |
are the total number and the |
|
wave |
function of the "valence" electrons, |
is the operator that antisymmetrizes the |
||
product, and M is a normalization factor.
The strong orthogonality requirement among the wave functions of different groups, is satisfied for the "core" groups, because they are localized in different spatial sites, but it
must be imposed between and each "core" function. It is well known that this last condition is equivalent to assuming that the function is built up using spin-orbitals drawn from a set orthogonal to all orbitals of the "core" functions.
159
Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 159–164.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.
160 |
O. SALVETTI |
Let us first consider a single atom, A. We can study this atom with high accuracy and prepare some atomic quantities useful in subsequent calculations. Also for this atom we suppose that the total wave function is given by a product
where the meaning of the symbols is obvious.
We suppose that the core functions |
is built up from orbitals |
, ... which satisfy |
|
the following relation |
|
|
|
where |
means that the integration is extended to a sphere SA around A. This sphere |
||
SA is supposed much smaller than the Van der Waals sphere of the atom A.
The |
function is then built up from the orbitals |
|
... , where the |
functions extend well beyond SA. |
|
|
|
Since we are dealing with a monocentric problem, the function |
can be easily studied |
||
as accurately as required. In order to determine the valence function |
|
we generally use |
|
an open shell H.F. method and so obtain a set of orthonormal orbitals |
which satisfy |
||
the conditions of strong orthogonality to "core" functions. Each of these orbitals will be of the following form
and if |
is one of the orbitals appearing in |
the following relation holds: |
|
The "core" |
orbital |
is built up from function |
|
The following atomic quantities can now be calculated
CORE-VALENCE SEPARATION IN THE STUDY OF ATOMIC CLUSTERS |
161 |
To build up |
in the cluster function (1) we use the functions |
all of which satisfy the strong orthogonality condition in the sense of to (2), but do not satisfy the strong orthogonality needed for (1) We therefore consider the linear combination
and require that this function be a linear combination of the functions in each sphere SP. This condition can be only approximatively satisfied and it is useful to have a measure of the goodness of the approximation. To obtain such a criterion we consider the quantity
This quantity measures the error in the orthogonality of fi Since the urij are arbitrary coefficients, we can put
to all the group functions. and so obtain
As noted in previous papers [7–11], by considering the matrix |
|
we can find the minimum of (15) by diagonalizing the |
matrix. The eigenvectors, |
ordered according to the corresponding increasing eigenvalues, give functions less and less orthogonal to the "core" orbitals. The associated eigenvalues give us a measure of the goodness of the functions obtained. One must keep only functions corresponding to eigenvalues smaller than some chosen threshold.
In this way we obtain n functions
with the property
162 |
O. SALVETTI |
From (1) and (6), (8) we then obtain
The |
in (21), are the coefficients of the two center expansion of the potential energy |
|
of interaction of two non overlapping charge distributions. [12]. |
||
To obtain the |
contribution only from calculations over type functions and from |
|
atomic data one needs a more detailed analysis of the equations. Let us consider the following form of Fock operator
To obtain matrix elements of we have
But from (18-19) and (7-13) one obtains
In order to derive Gv matrix elements, we consider the bielectronic integral
which can be written
CORE-VALENCE SEPARATION IN THE STUDY OF ATOMIC CLUSTERS |
163 |
since
one has
From (24) and (25) it follows that all the matrix elements can be obtained by
calculation over the set of "valence" functions, with the addition of terms relating to single atoms.
Numerical applications to particular clusters are wery encouraging [7-11].
164 |
O. SALVETTI |
References
1.U.Even, N. Ben-Horin and J.Jortner, Phys. Rev.Lett. 62, 140, (1989).
2.R. McWeeny, Proc.R.Soc. London Ser. A 253, 242, (1959).
3.R. McWeeny and B.T. Sutcliffe, Proc. R.Soc.London Ser. A 273, 103, (1963).
4.Y. Ohrn and R.McWeeny, Arch. Phys. 31, 461, (1966).
5.P.D. Drace and R.McWeeny, Proc.R.Soc.London Ser. A 317, 435, (1970).
6.M. Klessinger and R.McWeeny, J.Chem.Phys. 42, 3343, (1965).
7.R. Colle, A.Fortunelli and O. Salvetti, J. Chem.Phys. 80, 2654, (1984).
8.R. Colle, A.Fortunelli and O. Salvetti, Mol.Phys. 57, 1305, (1986).
9.A. Fortunelli and O. Salvetti, Mol.Phys. 75, 1191, (1992).
10.A. Fortunelli, O. Salvetti and G. Villani, Surface Sci. 244, 355, (1991).
11.A. Fortunelli, A. Desalvo, O.Salvetti and E. Albertazzi, Cluster Models for Surface and Bulk Phenomena, Plenum Press,(1992).
12.J.O. Hirschfelder, C.F. Curtiss and R.B. Bird, Molecular Theory of Gases and Liquids, Wiley (1965).
Core-Hole States and the Koopmans Theorem
C. AMOVILLI and R. McWEENY
Dipartimento di Chimica e Chimica Industriale, Via Risorgimento 35, 56100 Pisa, Italy
1. Introduction
The theorems of Brillouin [1,2] and Koopmans [3], in both their original and generalized forms, have provided a recurring theme in the work of Gaston Berthier who always showed a profound appreciation of their significance and importance (see, for example, [4,5]). Both theorems have been of immense value in the calculation and interpretation of a wide range of molecular properties. But both are ‘first-order’ theorems, based originally on the Hartree-Fock model, and refer to the first-order effect of perturbations that are considered ‘small’. When the perturbations become large the theorems lose their value, except as a basis for rough approximations, but the violations themselves are also of considerable practical importance. In particular, as every quantum chemist knows, the ionization energy for removal of an electron from orbital is related to the Hartree-Fock orbital energy according to the Koopmans theorem, by
where is calculated using the ‘zero-order’ for the unperturbed (neutral) system. The perturbation of the orbitals in passing from the neutral to the ionized system is irrelevant to the first-order result. To calculate secondand higher-order corrections to equation (1), however, it is necessary to allow the orbitals of the ionized system to ‘relax’ in order to describe the perturbation of the Hartree-Fock field caused by the change in occupation number of orbital Such relaxation effects are often rather small and the Koopmans result (1) can give a fairly satisfactory interpretation of the ionization processes observed in valence-electron photoelectron spectroscopy (PES); but for ‘deep’ ionizations, as observed in ESCA experiments (see, for example, Siegbahn et al [6]) where electrons are knocked out of atomic inner shells, the relaxation effects can be very large. The electron distribution tends to ‘collapse’ towards the
‘core hole’ – roughly equivalent to an increased nuclear charge – and the use of (1) commonly yields ionization energies in error by 20–30 eV.
This note is concerned with the alternative procedure in which (1) is replaced by
where E (the electronic energy of a neutral molecule) and (that for the molecule in a ‘core-hole’ state) are both calculated independently. It must be remarked at
165
Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 165–173.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.