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

176	Y. G. SMEYERS ET AL

2.Theory

2.1.GENERAL APPROACH

The HPHF wavefunction for an 2n electron system, in a ground state of S quantum number, even or odd, is written as a linear combination of only two DODS Slater determinants, built up with spinorbitals which minimize the total energy [1-2]:

where and are two spinorbitals of opposite spin belonging to a same electron pair, so that

This linear combination is obtained by projection of one of the determinants on the spin eigenstates with S even or odd:

where	is a permutation operator which interchanges all the and spinfunctions
in the	initial determinant.

Since the HPHF wavefunction for singlet states does not contain any triplet contamination, this model was seen to produce relatively good results for singlet ground states, very close to those of the fully projected one [1-10].

The Brillouin’s theorem has been shown to hold in the case of the HPHF function [2]. As a result, any variations of the orbitals which minimizes the HPHF total energy, can be expressed as:

where	is the HPHF function in which an occupied orbital has been replaced
for an	virtual one.

Introducing the HPHF wave-function expression (1) in (3), and taking into account the idempotency of operator the following equation may be obtained:

where is a Slater determinant in which one i occupied orbital has been substituted by a t virtual one.

In order to solve equation (4), the following matrix elements between Slater Determinants have to be considered:

Since

and

are constructed with the same set of orthonormal spinorbitals, the

two first matrix elements can easily rewritten, according to the Slater’s rules [13], as:

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177

where	is a Slater determinant in which an		orbital is replaced by an	one.
In this expression, the		operator is the usual	Fock operator of the Unrestricted

Hartree-Fock method [14]:

In this equation, the Mulliken notation for the repulsion integrals is used, that is:

and stands for the well known monoelectronic operator:

A similar operator as (8) can be written when a orbital is substituted by a virtual one:

The calculation of the cross matrix elements (6) is somewhat more difficult, be- cause the Slater Determinants involved in them are constructed with two sets of non-orthonormal spinorbitals. This calculation, however, may be greatly simplified, if the two sets are assumed to be corresponding, that is, if they fulfill the following condition [14]:

As well known, this condition is not a restriction whenever the wavefunction is invariant under an unitary transformation [2].

Taking into account (12), the matrix elements (6), with their phase factor in (4), may be written in the following way:

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and

In these equations, S is the overlap between the two determinants built up with corresponding orbitals (12)(multiplied by a phase factor according the multiplicity required):

and

is defined as a cross operator analogous to the Fock operator (8):

A similar operator can be written for the case in which a bi orbital is replaced by a virtual one,

Introducing now expressions (13) and (14) in (4) , the following equation is obtained:

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179

which is the expression of the Generalized Brillouin Theorem for the HPHF function written as a function of the orbitals. A similar equation can be deduced when a orbital is replaced by a one. The next step will now be to write equations (19) as pseudo-eigenvalue equations to be solved in an iterative way just as in the

Unrestricted Hartree-Fock method.

For this purpose, let us define the following density projection operators:

and let us introduce them in (19). After some straightforward operations, we obtain:

From this equation the following HPHF Fock operator for determining the	orbitals
can be extracted:

A similar	operator for determining the	orbitals can be obtained in the same
way.

Let us remark that operator (21) is not symmetric. But, it can be symmetrized easily just by adding the adjoint of the asymmetric part:

In addition, since the action of	operator on a virtual orbital is zero, it is seen
that this adjoint will not affect	the results. So that the complete	operator may
be written as:

2.2. APPLICATION TO EXCITED STATES

The HPHF wavefunction for an excited state	is constructed by substi-
tuting in the HPHF ground state wavefunction (1) an	occupied spinorbital by an

virtual one. In order to avoid the possible collapsing of to so-constructed excited wavefunction onto the ground state one during the variational process, it is conve- nient that the excited function should be orthogonal to the former. In some cases,

180	Y. G. SMEYERS ET AL.

this orthogonality requirement is automatically achieved, when both wavefunctions exhibit different multiplicities or different spatial symmetries. In the second case, the promoved and excited spinorbitals, and possess also different symmetries. When both wavefunctions exhibit the same multiplicity and the same spatial symmetry, it is convenient that the excited function should be orthogonal to the fundamental one [15]. One way to achieve partially this requirement is orthogonalized the excited

orbital	to its companion	at each step of the iterative procedure. Remember
that	and possess the same symmetry.

In any cases, the orthogonality requirement applied to the orbitals:

implies some modifications in the formulae of the previous paragraph in order to avoid some singularities [7]. In particular, new cross Fock operators have to be redefined:

and

in which the sumations are restricted to the nonorthogonal orbitals.

In addition, partial cross Fock operators are also to be defined for evaluating the matrix elements in which the orthogonal orbitals are involved:

and

For the same reason, new density projection operators are redefined:

as well as limited projection operators to the k or u orbital space:

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181

Finally, a restricted overlap between the two determinants limited to the nonorthog- onal orbitals is defined:

In order to deduce the new pseudo-eigenvalue equations (22), we have to distinguish the two possibilities:

When the Brillouin Theorem equation (20) is reduced to:

where the cross energy term,		between the two Slater determinants, takes the form
of a simple repulsion integral:
In contrast, when	the following expression is found:

From equations (33) and (35), a general HPHF Fock operator for determining the orbitals of excited states can be extracted after some straightforward transformations:

Since equation (36) is not symmetric, it is symmetrized by addition of the ad joint of the asymmetric part. We obtain the new expression:

A similar equation can be deduced for the bi orbitals.

3. Calculation

In order to determine the HPHF wave-functions, the HPHF Fock operators (24) and

(37) for the ground and excited states, respectively, have to be expressed in matrix

form, in which the orbitals are developed on a basis function set. So, we have for the ground state:

182			Y. G. SMEYERS ET AL.
where		,.. are the matrix representations of the corresponding oper-
ators. In particular,		is defined as	, where A is now the inverse diagonal
matrix	of the overlap	between the occupied	corresponding orbitals,	and	and
_ and	the coefficient matrices of two set of occupied orbitals. Finally, s is the

overlap matrix between the basis functions.

For the excited states, with an orthogonal orbital pair, we have:

where	is defined as before except for the diagonal element,				of the , corre-
sponding to the orthogonal orbitals which is missing, and where
are the column vectors of the coefficients of the			and	orbitals, respectively.
3.1.	THE METHYLENE MOLECULE
As well known, the methylene lowest state is a			triplet, with electronic configu-
ration		which lies somewhat below the fundamental
singlet state,		In addition, the companion singlet state,			is also known. To

in degrees, r in and energy in Hartrees.
study the potential energy functions of both	states, as a function of the bending

HARTREE-FOCK MODEL TO THE LOWEST SINGLET AND TRIPLET EXCITED STATES

183

angle, was for many years an attractive problem for many scientists, in order to analyze the spectrum structure of the transition which is allowed [16,17].

Next, and as an example of HPHF calculations, the potential energy curves for the bending in the and states of methylene were determined into the HPHF approximation, using the Huzinaga-Dunning valence basis set with polarization orbitals

(d on the C atom, and p on the H atoms with 0.8 and 1.1 as exponent respectively),

and full geometry optimization. For this purpose,		an	orbital was substituted by
an	one, into the fundamental configuration.	The energy variations are given in

Table 1 for both states. The geometrical parameters encountered in this way are in very good agreement with the experimental data. These are gathered in Table 2.

3.2.THE METHANAL MOLECULE

Methanal (formol) presents an additional degree of freedom: the out-of-plane wagging

of the oxygen atom. In its singlet ground,		this molecule is planar. But, in its
triplet and singlet lowest exited	and	this molecule is pyramidal
with the possibility of inversion,	because its	electronic system is destabilized by

antibonding orbital. In order to test the performance of the HPHF model, to determine the potential energy curves for the inversion seems to be illustrative. HPHF calculations with full geometry optimization were performed for these lowest excited states using the Huzinaga-Dunning valence basis set with polarization orbitals. The potential energy curves are given in Figure 1. As expected, a double potential energy well is obtained for both excited states of methanal. In Table 3, the geometrical parameters, as well as the inversion barrier, obtained with different

basis sets and different approaches are given for the first singlet state,

It

is seen that the values for the barrier height are very basis dependent. Anyway, the HPHF approach gives the best values into the same basis set, except for the SCF-CI calculation which involves 17,000 configurations [18].

In Table 4, similar results are gathered for the first triplet excited state,

Here, the UHF model is seen to furnish slightly better results. The HPHF model, however, yields a better value for

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185

3.3.TRANS-BIACETYL MOLECULE

In order to test the HPHF model with a larger system, we have considered the transbiacetyl molecule, which contains six atoms of the second row and six hydrogen atoms. The energy values of the two extremal configurations for the double rotation of the methyl groups were determined on the potential energy surface, with full geometry optimization except for the molecular frame which is constrained to be planar. In the same calculation, the barrier height was determined.