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16

J. TOMASI

2.Papers from the Conference on Molecular Quantum Mechanics held at the

University of Colorado, Boulder, June 21-27 (1959), Rev. Mod. Phys. 32,169 (1960).

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5.C.A. Coulson, Rev. Mod. Phys. 32,171 (1960).

6.J. Tomasi, Int. J. Mol Struct. (Theochem) 179, 273 (1988).

7.C. Trindle, Croat. Chim. Acta 57, 1231 (1984).

8.Z. Maksic, "Theoretical Models of Chemical Bonding" vol 1, Springer, Berlin (1991)

9.G. del Re, Adv. Quant. Chem. 8, 95 (1974).

10.P. Durand and J.P. Malrieu, Adv, Chem. Phys. 67,321 (1987).

11.A. Amman and W. Cans, Angew. Chem. Int Engl. Ed. 28, 268 (1988).

12.W.G. Runcimann, "A Treatise on Social Theory", Cambridge Univ. Press, London (1983).

13.H. Primas, "Chemistry, Quantum Mechanics and Reductionism", Springer, Berlin (1983).

14.R. G. Wolley, Adv. Phys. 25, 27 (1976).

R.G. Wolley, Structure and Bonding 52,1 (1982).

15.P. Claverie and S. Diner, Israel J. Chem. 19, 54 (1980)

P.Claverie, in "Symmetries and Properties of Non-rigid Molecules". J. Maruani and J. Serre Eds. Elsevier, Amsterdam (1983).

16.B. Sutcliffe and Z. Maksic "Theoretical Models of Chemical Bonding". vol 1, Springer, Berlin (1991).

17.M. Eyring, J. Walter and G.E. Kimball "Quantum Chemistry". Wiley, N. York (1944).

18.G. Fogarasi and P. Pulay , Ann. Rev. Phys. Chem. 35, 191 (1984).

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19.P.A.M. Dirac, Proc. Roy. Soc. A123, 714 (1929).

20.A.E. Reed, L.A. Curtiss and F. Weinhold, Chem. Rev. 88, 899 (1988).

21.E. Hollauer and M.A.C. Nascimento, Chem. Phys. Letters, 184, 470 (1991).

22.D.L. Cooper, J. Gerratt and M. Raimondi, Adv. Chem. Phys. 65, 319 (1987).

23.R. Mc Weeny, Int.J. Quant. Chem. 34. 25 (1988).

24.B. Pullman , Int J. Quant. Chem. Q.Biol S. 17, 81 (1990).

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27.J.T. Fraser, F.C. Haber and G.M. Miiller "The Study of Time". Springer, Berlin (1972; J.T. Fraser and N. Lawrence "The Study of Time II". Springer, Berlin (1972)

28.H. Primas, ref [13], chapter 6.

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Toute tentative de faire rentrer les questions chimiques dans le domaine des doctrines mathématiques doit être réputée jusqu'ici, et sans doute à jamais, profondement irrationnelle, comme étant antipathique à la nature des phénomènes: elle ne pourrait découler que d'hypothèses vagues et radicalement arbitraires sur la constitution intime des corps, ainsi que j'ai eu occasion de l'indiquer dans les prolégomènes de cet ouvrage.

A. Comte

Cours de Philosophic Positive

Tome Troisième, Trente-cinquième Leçon1838

Strategies and Formalisms

Some 150 years after ....

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Theory of Orbital Optimisation in SCF and MCSCF Calculations

C. CHAVY, J. RIDARD and B.LEVY

Groupe de Chimie Quantique, Laboratoire de Physico-Chimie des Rayonnements,

(UA CNRS 75), bât. 337, Université Paris Sud, 91405, Orsay Cedex, France

The aim of the present article is to present a qualitative description of the ’optimised’ orbitals of molecular systems i.e. of the orbitals resulting from SCF calculations or from MCSCF calculations involving a valence CI : we do not present here a new formal development (although some formalism is necessary), nor a new computational method, nor an actual calculation of an observable quantity ... but merely the description of the orbitals.

In fact, it turns out that the orbitals resulting from SCF or valence MCSCF calculations in molecules can be described in extremely simple terms by comparing them with the RHF orbitals of the separated atoms.

In the case of a valence MCSCF calculation the difference between the optimised orbitals and these atomic RHF orbitals simply represents the way in which the atoms are distorted by the molecular environment. Thus, this difference is closely related to the idea of ’atoms in molecules’ ( l ) . However, here, the atoms are represented only at the RHF level, and the difference concerns only the orbitals, not the intraatomic correlation.

The starting step of the present work is a specific analysis of the solution of the Schrödinger equation for atoms (section 1). The successive steps for the application of this analysis to molecules are presented in the section 2 (description of the optimised orbitals near of the nuclei), 3 (description of the orbitals outside the molecule), and

4 (numerical test in the case of

). The study of other molecules will be presented

elsewhere.

1. The atomic case

We briefly recall here a few basic features of the radial equation for hydrogen-like atoms. Then we discuss the energy dependence of the regular solution of the radial equation near the origin in the case of hydrogen-like as well as polyelectronic atoms. This dependence w i l l t u r n out to be the most significant aspect of the radial equation for the description of the optimum orbitals in molecules.

19

Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 19–37.

© 1996 Kluwer Academic Publishers. Printed in the Netherlands.

20	C. CHAVY ET AL.

1.1. HYDROGEN-LIKE ATOMS

In the case of hydrogen-like atoms the Schrödinger equation can be written as (in atomic units) :

where T represents the kinetic energy				operator, Z the nuclear charge, -Z/r				the
Coulomb electron-nuclear attraction,			e the energy and		the orbital.
The solution	of this	equation can	be	factorised into the product of a radial				part
and an angular part (spherical harmonic				where the radial		part	depends of
the quantum number l but not of m (2).
Inserting this form of		into the eq.(l) gives the equation			to be	satisfied	by	the

so called radial equation :

It can be demonstrated (2) that two linearly independent solutions of this equation can be chosen in general (i.e. except for some values of e) in such a way that one of them (the so called ’regular’ solution) is continuous at the origin and diverges at infinity, and the other one (the so called ’irregular’ solution) diverges at the origin and tends to zero at i n f i n i t y .

Neither of these two solutions is square summable in general. However for some values of e (the ’eigen values’) these two solutions coincide and can be accepted physically for atoms since they both are continuous at the origin and they both tend to zero at infinity.

It should be emphasized that we are not interested here specifically by these particular values of e. On the contrary , what is useful here i.e. for the description of optimum orbitals in molecules is to study the variation of the regular solution when e varies continuously.

To solve that problem, we depart here from the development used for instance in (2) and we write in the form :

(4)

Substituting this form of into the eq.(3) leads to :

(5)

But we are interested here only by the ’regular’ solution, and we can write

in the

THEORY OF ORBITAL OPTIMIZATION IN SCF AND MCSF CALCULATIONS

21

form of a power expansion

where the	are numerical coefficients depending of l.
Substituting this form of			into the eq.(5) gives a recursion relation which allows to
determine all the		for any arbitrary choice of one of them. Choosing		, one
gets

These expressions of the	will allow us now to	discuss the energy dependence of
and then to derive some consequences from this		dependence.

1.2.THE VALLEY THEOREM

We first note that the choice particular norm of (and thus of

made in deriving the eq.(7) simply consists in a ). In fact the standard norm cannot

be used here since for most		values of e the orbital is not square summable. The
choice	is a convenient	alternative for

Next we consider the value of . It implies the relation :

which is the well known ’Cusp’ theorem (see e.g. the ref.3).

An other aspect of the eq.(7) concerns the energy dependence of							.	In fact one
deduces from this equation that :
The meaning of the eq.			(9) can be stated as : the energy dependence of					vanishes
like	near the	origin	(or even faster than	since	there is a	partial		cancellation
between the		and	terms). Therefore the energy		dependence	of	vanishes like
or	faster.

This statement will be referred to here as the ’Valley’ theorem. It constitutes the formal basis of our description of the optimum orbitals in molecular systems.

In fact, the Valley theorem is a simple extension of the Cusp theorem. However, the Cusp theorem provides only a local information (for r=0), while the Valley theorem

22	C. CHAVY ET AL.

is the extension providing a qualitative information (weak e dependence) valid inside a finite volume . This last aspect (finite volume) is the one that allows the description of the optimum orbitals in molecular systems.

The Cusp and the Valley theorems express the same aspect of the Schrödinger equa-

tion, eq.(l) : since		has no pole for r=0, the pole of		can be compensated
only by	; but a pole of		with a residue equal to -Z implies the Cusp theorem

(at the origin) and the Valley theorem (inside a finite volume around the origin).

It should be noted that the weak energy dependence of the orbitals inside a finite volume around the nucleus has already been noted and used in different contexts : the numerical determination of atomic orbitals (4) as well as the scattering of electrons by atoms (5).

1.3. ORBITAL OPTIMISATION IN POLYELECTRONIC SYSTEMS

The equation determining the optimum orbitals of polyelectronic systems in the case of the SCF and MCSCF theories can be written in the form :

where
-	are the creation operators corresponding to the orbitals		and	and j, l the
	anihilation operators for the orbitals	and

-h is the one electron part of the total Hamiltonian.

-is a local operator :

- the factors are the Lagrange multipliers that take care of the orthonormality constraints.

1.4. POLYELECTRONIC ATOMS

We consider here only the SCF case where the off diagonal factors vanish. In addition, we assume that the orbitals satisfy the usual symmetry constraint i.e. that they are pure s, p, d ... functions (RHF approach). On the other hand, no spin

constraint is assumed. Then the eq.(10) is most conveniently written as :

THEORY OF ORBITAL OPTIMIZATION IN SCF AND MCSF CALCULATIONS

23

with

The eq.(12) is similar to the eq.(l) in the sense that it requires a compensation between T and -Z/r. The main difference comes from the presence of and that might reduce the range of that compensation. In order to solve the eq.(12) one writes

and in the form :

We now study the		dependence of the solution of the eq.(14)		using the following
scheme :
-	we first determine normalised		by using some standard program of Quantum
	Chemistry ;
-	using these normalised		we determine the functions	and	;
-	then we set up the equation :

where f is an unknown function, e is a variable parameter,		and	are the func-
tions evaluated at the preceding step using the normalised		and	f (0),
are the values of f and	at the origin (note that f (0) is unknown) .

24	C. CHAVY ET AL.

-finally we solve the eq.(15) with various values of e but always with the same functions

The factor	ensures that the solution of the eq.(15) is independent of a
multiplicative factor (if	f is a solution, then		is also a solution for any		number )
and that f is proportional to		when	It turns out that no useful comparison
with the molecular case can be made in the absence of this factor.
The eq.(15) can be solved by mean of a power expansion of f,				and of	in the
same way as the eq.(5)	:

Substituting the eq.(16) in the eq.(15) gives a recursion relation which allows to determine the Owing to the factor it is possible here to choose as done in the eq.(7), so that one gets :

etc ...

The main aspect of the eq.(17) is that the orbital energy e occurs only in the coefficients w i t h Therefore we obtain here the same results as the one obtained in the case of hydrogen-like atoms (§1.1 and §1.2) :

-	the energy dependence of the RHF orbitals of polyelectronic atoms decrease faster
	than	in the region close to the nucleus (Valley theorem);
-	and the corollary that these orbitals depend very weakly of the orbital energy in a
	finite volume around the nucleus. The range of that volume, which depends of the
	magnitude of		and	, will be now determined numerically.

1.5 NUMERICAL ILLUSTRATIONS

We present here numerical results illustrating that the solutions of the radial equations (eq.(5) for the hydrogen-like case and eq.(14) for polyelectronic atoms) are ’weakly’ dependent of e in a finite volume.

In the case of polyelectronic atoms we have calculated the	and	parameters
as described in the preceding section (see above, the §1.4) i.e.	using the normalised

orbitals resulting from a RHF calculation of the atom in a gaussian basis (11).

The radial equations was then solved using the Runge-Kutta method (7).

We present in Fig. (1-6) the function defined in the eq.(2) (or defined in the eq.(13)), in the case of the orbital 1s of Hydrogen (Fig.l), 2s and 2p of Carbon

THEORY OF ORBITAL OPTIMIZATION IN SCF AND MCSF CALCULATIONS

25

(Fig. 2 and 4) , 3s and 3p of Silicon (Fig. 3 and 5), and 3d of Scandium (Fig.6). In each case three values of e have been chosen : the RHF value, one value higher by 0.2 H and one value lower by 0.2 H. Thus we can study the deviation of the orbitals when e varies by around the RHF value.

It is seen on the fig.(l) - 1s orbital of the Hydrogen atom - that this deviation is smaller than 5% of the orbital for (close of the covalent radius of the H atom). In the case of 2s(C) and 3s(Si), similar deviations (less than 5%) are observed for r smaller than the position of the last extremum of the function (the one obtained with the largest r) i.e. for in the case of Carbon, in the case of Si. These distances are smaller than the covalent radii of these atoms (ca. 1.5 B for C and 2 B for Si). But close to the covalent radius, (at 1.4 B for C and 1.8 B