Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200

26	C.CHAVY ET AL.
for Si) the deviation	is still smaller than 10% of the value of the orbital at the last
extremum.

These results illustrate the fact that the orbital is weakly dependent of the energy e at the peak for r=0 (Cusp theorem) but also down in the valley and even on the next hill if any (Valley theorem).

In the case of the p and d orbitals (fig. 4-6) the deviations are larger than the deviations obtained with s orbitals. This is simply because the magnitudes of the p and d orbitals are larger than those of the s orbitals for r ca. 1.0 B due to the

THEORY OF ORBITAL OPTIMIZATION IN SCF AND MCSF CALCULATIONS	27
particular norm chosen here. In fact, it can be checked that deviations	smaller than

10% of the value of the last extremum, are obtained for r values up to a limit close

to the covalent radius of the atom in all three cases (		for the 2p(C) orbital,
for the 3p(Si) orbital and	for the 3d(Sc) orbital).

We conclude from these numerical examples that it is possible to give a quantitative and probably rather general expression of the Valley theorem (weak e dependence of the orbital in a finite volume around the nucleus) : a variation of the energy of

ca. 0.2 H results in a variation of the function		smaller than 10% of the last
extremum of	until a distance of the nucleus equal to ca. 90% of the covalent
radius of the corresponding atom.

2. Molecular systems

We arrive now at the main purpose of the present work : to find a qualitative description of the optimum orbitals (obtained by SCF or MCSCF calculations) of molecular systems.

To that end, we will start with the same equation as the one used above in the case of polyelectronic atoms,viz. the eq.(10), and we will try to use the equivalent of the compensation between the kinetic energy and the nuclear attraction (T and -Z/r) found in the atomic case.

In fact, it turns out that the compensation between the kinetic energy and the nuclear attraction does lead to a qualitative description of the optimum orbitals in molecular systems, but only in the frame of the following restrictive conditions.

i) Global versus local description. In the case of molecular systems, the one electron part of the electronic hamiltonian includes a sum over the electron-nuclear attraction of all the nuclei:

Therefore it appears that the above mentionned compensation takes place separatly in the vicinity of each atom. We can arrive to a description of the optimum orbitals ; however this description is not global, but local in the sense that it concerns separatly the regions around each atom. Thus, we will hereafter consider only the region of a single atom, say A, and study the effect of the compensation between T and

ii)	Natural versus non	natural orbitals. The	factor is always combined in the
eq.	(10) with the	factor according to

If appears in several terms corresponding to different orbitals, and it is difficult to demonstrate directly that the compensation occurs separatly for each orbital. Therefore, we will consider here only the cases

28	C. CHAVY ET AL.
where	i.e. we will consider only natural orbitals.

iii) Strongly versus weakly occupied orbitals. It is then seen on the expression 19 that

appears in the eq.(10) multiplied by			when natural orbitals are
used. Thus, if	is small	then	dominate

the remaining terms of the eq.(10) only in a very small volume around the nucleus of A. In the remaining part of the volume occupied by the molecular system the description of this orbital cannot be deduced from the Valley theorem. Therefore, we will consider here only strongly occupied orbitals with

In fact, a simple description of the weakly occupied orbitals resulting from valence MCSCF calculations has already been presented (12) .

iv) Canonical versus non canonical orbitals. Let us now consider the right hand side of the eq.(10) which depends of the off diagonal Lagrange multipliers through terms like Such terms may present very steep variations with so that the Valley theorem may lead to no special conclusion. Therefore, we consider here only

the cases where one can have	. A similar restriction has not been made
in the atomic case (section 1.4	above) because it turns out that	is very
small in all useful cases.

v) Partial waves versus orbital . Finally it is worth noting already that the present approach will tell us nothing concerning the orbitals themselves! It will tell us something only on each of the partial wave around A separatly : the relative weights of the different partial waves in the total orbital do not result from the local compensation between T and . It appears rather as a global property of the molecular system .

Let us note that the two conditions can be satisfied only w i t h canonical SCF orbitals. Thus, in fact, the present theory can be applied only i n such cases. However it has been demonstrated (12) that in most systems, the strongly occupied MCSCF orbitals and the SCF orbitals are extremely close one to the others. Therefore, in practice, the present theory also applies to the strongly occupied MCSCF orbitals.

On the all, the limitations coming from the above hypotheses i-v are :

-one can find a description of the partial waves of the optimum orbitals near each atom separatly , not of the orbitals themselves;

-these descriptions concern only the strongly occupied canonical orbitals, not any type of orbitals.

We now return to the eq.(10). In the frame of the hypotheses i-v it writes :

THEORY OF ORBITAL OPTIMIZATION IN SCF AND MCSF CALCULATIONS

29

with

Let us now introduce the partial waves expansion of		with the origin on the atom
A :
and the expansions given by the eq.(13) for	and
Thus, the eq.(20) becomes

where the summations over l" and m" in the expression of			arc restricted	by the
conditions :	and/or
The presence of	the	term is the only formal difference	between the	eq.(14),

obtained in the atomic case, and the eq.(22). This term comes from the fact that

the partial wave expansion of		includes several terms here instead of a single term
in the atomic case. In fact		and	are two components of the Coulomb type
potential	is diagonal	in the	partial wave	while	gives rise	to a
coupling between different partial waves					of the	same
orbital

We now transform the eq.(22) in the same way as done for the eq.(14) : we assume

that the (normalised) optimum orbitals		have been determined by some existing
Quantum Chemistry program along with the partial waves			and with the poten-
tial terms	. Using these quantities we then set up the equation

30	C. CHAVY ET AL.

This equation is similar to the eq.(15) obtained in the atomic case. Thus one can switch at will between the atomic and the molecular cases : if we give to the param-

eters	the values determined for the atom as described in the
above section 1.4 (this implies		), then f is proportional to the RHF orbital of

the atom A with the quantum numbers l and m and the energy e ; if alternatively we give to the parameters the values obtained for a molecular system as just explained, then f is proportional to the l , m partial wave of the orbital of the molecular system with the energy e.

We now use the Valley theorem : the atomic function f depends weakly of the e parameter in a large region near the nucleus. It can be seen by inspection of the eqs.(15),(16),(17) and (23) that the critical parameter in the molecular case is

an effective energy	where	are the
differences between the values of		at the origin in the molecular
case and in the atomic case. Therefore, if	is not too different from the atomic

orbital energy,then the two f functions obtained with the atomic and molecular values of the parameters arc extremely close to be proportional one of the other in a finite region near the nucleus. Stated differently : in a finite region near the nucleus of an atom A, the partial waves of the optimum orbitals centered on A are proportional to the corresponding RHF orbitals of the atom A , unless the atomic and molecular parameters are very different from each other (i.e. unless the difference is much larger than the variations mentionned in the section 1.5).

3. Asymptotic conditions

The Valley theorem leads to simple conditions for the optimised orbitals near the nuclei. However these conditions are not sufficient to characterize these orbitals : one needs in addition to take the asymptotic form of the equations into account.

In the asymptotic region, an electron approximately experiences a		potential,
where	is the charge of the molecule-minus-one-electron (	in the case of a

neutral molecule) and r the distance between the electron and the center of the charge repartition of the molecule-minus -one-electron. Thus the orbital describing the state of that electron must be close to the asymptotic form of the irregular solution of the Schrödinger equation for the hydrogen-like atom with atomic number

(see for instance the Eq.13.5.2 of Ref.9) where e is the orbital energy. Since e is differ- ent in the molecule and in the separated atoms, this asymptotic behaviour cannot be represented properly if the molecular orbital is approximated by a linear combination of the RHF orbitals of the separated atoms.

4. The case of

The interest of

in the present context is that it provides a good test for the present

orbital optimisation theory because one knows the exact solution.

THEORY OF ORBITAL OPTIMIZATION IN SCF AND MCSF CALCULATIONS	31
Thus we will use the result of calculations of the wave function of	expanded in a

gaussian basis to provide numerical tests of the qualitative discussion on the orbital optimisation theory presented in the above sections 2 and 3.

We have calculated several approximations of the energy of (ground electronic state) using various GTO bases (Table 1). In all cases the intcrnuclear distance used was equal to 2 B, close to the experimental equilibrium distance

The accuracy of the results obtained here using gaussian bases - and the usefulness of the numerical tests based on these results - can be seen from the values given in the Table 1. It is seen that the dissociation energy De obtained in the largest basis used here is excellent (error equal to 0.01 eV). On the other hand, the error on the value obtained using the minimum basis is as high as 1.35 eV (or 48% in relative value). This proves, if need be, the importance of the orbital optimisation studied in the present article.

It is also useful to note that the major part (77%) of the effect of the orbital optimisation is obtained in the intermediate basis where no polarisation orbital is used.

4.1.OPTIMISATION IN THE VICINITY OF A NUCLEUS

We first consider what happens when comparing directly the optimum orbital of

the un-optimised orbital of

(i.e. the sum of the two 1s orbitals of the H atoms)

and the orbitals of the H atom itself. The comparison between the values of these orbitals along the bond axis is presented on the fig.(7).

It is seen that in the inner region (positive values of the abscissae), the atomic orbital is close neither to the optimal orbital nore to the un-optimised orbital. On the contrary, the atomic orbital is very close of the un-optimised orbital but not of the

optimised one in the outer region (negative values of the abscissae). The inverse con-

32	C. CHAVY ET AL.

clusion is obtained in the perpendicular direction presented in the fig.(8) : the atomic orbital is very close to the optimal molecular orbital but not of the un-optimised one. Thus, no clear conclusion can be reached in this way.

Let us now consider what happens when comparing the orbital of the H atom, no longer with the orbitals of the system, but with the partial waves of these latter orbitals.

In the case of the s wave (l = 0) of the optimised orbital the effective energy defined

in the section 2 is given here by

(R is the internuclear

distance). According to the analysis of that section it is seen on the fig.(9) that the s wave of the optimum orbital obtained in the gaussian basis is actually very close to the numerical regular atomic s orbital with while the s wave of the un-optimised orbital is significantly different from these two functions.

Here the effective energy is very close to the energy of the genuine atomic orbital

(-0.602 H to be compared to -0.5 H). Correspondingly, it can be seen on the fig.(9) that the s wave of the optimum orbital is also very close to the genuine 1s orbital of the H atom. In fact, the difference between these two functions is smaller than 2.4% in all the considered range of r.

A similar conclusion cannot be reached concerning the p waves (l = 1). In fact the coupling term between the s and p waves (the term of the eq.(22)) is not small here and correspondingly the p wave of the molecular system cannot be expected to

THEORY OF ORBITAL OPTIMIZATION IN SCF AND MCSF CALCULATIONS

33

be close of any atomic-like orbital.

4.2.OPTIMISATION IN THE ASYMPTOTIC REGION

When expanding the orbital in partial waves with origin at the midpoint of the molecule (center of charge of the molecule-minus-one-electron) the p wave vanishes, and only the s wave has to be considered. According to Sec.3, this partial wave must be proportionnal to the irregular solution of the hydrogen-like system with atomic number Z’= 2 and with e equal to the exact orbital energy (-1.102 H).

We present in the Table 2 the ratio of the irregular solution of the hydrogen-like sys- tem with the s wave of the optimised orbital, and with the s wave of the unoptimised orbital. It is seen that the irregular numerical solution is actually much closer to be proportional to the s wave of the optimised orbital than to that of the unoptimised orbital.

In fact, the ratio between the numerical and the optimised orbital is nearly constant (relative variation smaller than 11%) for 2< r <6 B, while the ratio with the s wave of the un-optimised orbital is multiplied by ca. 5 when r increases from 2 B to 6 B

( r=distance to the midpoint of the two nuclei). The decrease of the ratio at larger

34	C. CHAVY ET AL.

distances in the case of the optimised orbital just comes from the fall off of gaussian functions at large distances.

4.3.CONSTRUCTION OF THE ORBITAL OF

In the two preceding sections (4.1 and 4.2) we have presented numerical test of the following description (resulting from the analysis of the sections 2 and 3) of the

optimised orbital of			:
-	near of a nucleus, the s wave (with origin on that nucleus ) of the optimised orbital
	of	is proportional to the s regular solution of the radial equation (eq.(2)) with
	Z=l and a shifted energy			given by	(e=orbital energy, R=inter-
	nuclear distance);
-	outside the molecule, the s wave ( with origin at the midpoint of the two nuclei ) of
	the optimised orbital of			is proportional to the s irregular solution of the radial
	equation with Z=2 and the actual energy of the orbital.

We examine now a numerical test of the reciprocal of this description : if a function satisfies this description, then it is the optimised orbital of . If both the description and the reciprocal are true we can conclude that the description is complete.

To that end we first introduce the following notations :

- the ’ internal zone ’ corresponding to the nucleus A is defined by the condition

(in	there are two ’internal zones’, but , due to the symmetry, only one

of them will be considered here) ;

- the ’external zone’ is the region outside the molecule ;

is the orbital to be determined in the form of an expansion in a gaussian basis :

where	is a gaussian function and			the numerical coefficients to be determined;
is the s partial wave of			with origin on the nucleus A;
is the σ partial wave of			with origin at the middle of the bond;
and		are the regular and irregular solutions of the two radial equations
corresponding to the internal and external zones (assuming					and e to be known).
Then	and	are obtained by mean of the partial wave expansion of the gaussian
functions	:

THEORY OF ORBITAL OPTIMIZATION IN SCF AND MCSF CALCULATIONS					35
where	is a spherical harmonic centered on the nucleus A		and	is	a gaussian
centered on either of the two nuclei. A similar relation		holds for		with	in the
place of	, where M is the midpoint of the two nuclei.
We introduce now two unknown numerical constants		and	and we try to check
that the two conditions

directly by diagonalising the hamiltonian matrix.

To do that, we first guess starting values of and ; secondly we determine the coefficients by minimising the quantity Q given by

where		and	are two set of points i n the internal and external regions respectively;
thirdly we evaluate the energy				of		using the	coefficients just determined. The
steps two and			three are repeated	w i t h d i f f e r e n t values of and				until the energy is
minimised.
It is seen that this process is essentially a least square fit of								and	by
and	,	subject to a m i n i m u m energy condition					which allows to	determine	and
. Note		that	are related by the norm of				so that there is in fact a single

parameter in this minimisation.

This calculation has been made here using the 4s basis set (which includes no polarisation p gaussian orbitals). The energy obtained in this way is very good : it reproduces the energy obtained by diagonalisation (viz. -0.59088 H ; cf the Table 1) w i t h an error equal to 0.02 eV.

Concerning the expansion coefficients, the most significant comparison concerns the values of the two orbitals : the one obtained by the fitting process just described and the one obtained by diagonalising the matrix of the hamiltonian in the gaussian basis.

In fact we have found that the difference between these two orbitals never exceeds 3% in the internal region as well as in the external region.

We conclude that the description of the orbital by the proportionality between

and

on the one hand and between

and

on the other hand is supported

by the present calculation and that it is indeed complete.

5. Conclusion

We have demonstrated formally that the optimum orbitals of any given molecular system (canonical SCF orbitals or strongly occupied MCSCF orbitals that are closed

to the SCF ones) can be described very simply in the regions surrounding each nucleus