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

where the usual abbreviated form for a Slater determinant has been used as in equation (11). Both determinants D(j) and D(k) can be considered built up in the same manner. The number of different spinorbitals appearing in both determinants, can produce a zero result for the matrix element, as it is well known for one and two electron operators, see reference [9]. Generalization to integrals over any number of electrons can be performed as follows.

The matrix element between two Slater determinants can be written as:

where the symbol j[p] means that a permutation p has been performed over the parameter vector j subindices. Here must be noted that the expression (14) above can be written with a unique summation symbol, using the property outlined in equation (2). Then, the integral over the spinorbital products, appearing as the rightmost term of equation (14) can be now simplified. Because in the spinorbital products appearing in equation (11), the canonical ordering of the electrons is preserved by convention in equation (12), as discussed before, one can write the integral using only the first r spinorbitals of the successive products,

The logical Kronecker delta, which appears when integration is performed over the coordinates of the remaining n-r electrons, can be easily substituted by the equivalent logical expression:

where the Minkowski norm of the difference, between the permuted vectors j[p] and k[q], must be equal to the sum of the absolute values of the differences between the first r-th components of both vectors.

The right hand part of the last equality (16), may be substituted in equation

(15) and the resulting formula transferred into the expression (14). The final result indicates fairly well one can have at least r differences between the spinorbitals involved in constructing both determinants in order that the integral becomes not automatically

zero. This result encompass the well described zero-, oneand two-electron operator cases [9], generalizing in this way the rules governing the calculation of operator matrix elements between two Slater determinants. One can say that the general rule in order to prevent automatic integral nullity is: "r-electron operators allow a maximal amount of r spinorbital differences". This rule is connected to the Brillouin theorem [15].

The same expression can be used with the appropriate restrictions to obtain matrix elements over Slater determinants made from non-orthogonal one-electron functions. The logical Kronecker delta expression, appearing in equation (15) as defined in (16)] must be substituted by a product of overlap integrals between the involved spinorbitals.

5.2 CI WAVEFUNCTIONS

Using the approach already described for combination generation, one can formulate in a short but completely general form the CI wavefunctions [16].

This kind of wavefunctions, in the complete CI framework, as Knowles and

Handy [16e] have proved feasible, for a system of m spin-orbitals and n electrons can be written within the NSS formalism:

where the logical vector L is defined according to the combinations generation and the terms D(j) are Slater determinants constructed as the one defined in equation (13). The

C(j) factors are the variational coefficients attached to each Slater determinant.

Also, an alternative formulation of equation (17) can be conceived if one wants to distinguish between ground state, monoexcitations, biexcitations, ... and so on.

Such a possibility is symbolized in the following CI wavefunction expression for n

Then, the CI wavefunction is written in this case as the linear combination:

where the index e, appearing in the first classical sum, signals the excitation order. That is, for e=0 one has the ground state, for e=l the monoexcitations are obtained and so on.

parameters attached to the involved nested sums and to the occupied-unoccupied orbitals respectively. That is:

This two general CI function expressions, along with the results obtained in the section 5.1 above, permit to compute the expected value form of any quantum mechanical operator in a most complete general way.

5.3 DENSITY FUNCTIONS

Density functions can be obtained up to any order from the manipulation of the Slater determinant functions alone as defined in section 5.1 or from any of the linear combinations defined in section 5.2. Density functions of any order can be constructed by means of Löwdin or McWeeny descriptions [17], being the diagonal elements of the so called m-th order density matrix, as was named by Löwdin the whole set of possible density functions. For a system of n electrons the n-th order density function is constructed from the square modulus of any n-electron wavefunction attached to the n- electron system somehow.

5.3.1.Density functions over Slater determinants

Using a unnormalized n-electron Slater determinant D(j) as system wavefunction, constructed as discussed in section 5.1, then one can write the n-th order density function

A recurrent procedure can be defined in order to obtain the remaining lesser order density functions. The (n-l)-th order density function is obtained from the n-th order one, integrating over the coordinates of the n-th electron (or the first) the right hand side of equation (20). The result is:

where the primed index vectors mean that the n-th element has been erased from the initial unprimed vector.

Thus, there is the possible relationship between both n-th and (n-l)-th order density functions:

It is straightforward to deduce, in general, how to obtain the (n-m)-th term of the sequence:

The zero-th order term being, finally, the norm of the Slater determinant, which by means of equation (23) becomes n!, a well known result.

Generalization of this one determinant function to linear combinations of Slater determinants, defined for example as these discussed in the previous section 5.2, is also straightforward. The interesting final result concerning m-th order density functions, constructed using Slater determinants as basis sets, appears when obtaining the general structure, which can be attached to these functions, once spinorbitals are described by means of the LCAO approach.

5.3.2.LCAO expression of density functions

Taking into account equation (23), and supposing the Slater determinants normalized, one can write, calling the initial constant factor v(n,m)=1/(n-m)!:

and using the LCAO approach for the spinorbitals, written as:

where each spinorbital has been expressed as a linear combination of atomic spinorbitals from a given M-dimensional basis set Then, a product of spinorbitals like (12) can be structured by means of the linear combination (25) as:

where C(a,j[p]) and X(a) are products of the coefficients and the basis functions respectively, appearing in the linear combinations (25) for every spinorbital. Now using (26) in the spinorbital product appearing in the rightmost side of (24), one obtains using a simplified NSS notation:

Finally the density function of (n-m)-th order can be expressed in terms of the atomic spinorbitals as:

using the hypermatrix elements:

The equation (28) has the same structure as the well known LCAO form of the first order density function [9]. Thus, it can be concluded that density functions of any order exhibit the same formal structure. In this manner, it can be seen that NSS’s lead to an interesting mnemotechnical rule.

5.4 PERTURBATION THEORY

In order to define the notation which we will use from now on, let us consider the application of the perturbation theory to a system which has a perturbed hamiltonian

From here, the goal consists to find the eigenvalues and the eigenvectors of the perturbed system, which we denote as the sets respectively. That is, the target is focused into solving the eigenvalue problem:

The eigenvalues and eigenvectors of the unperturbed hamiltonian are assumed to be known:

and the ket stands for the unperturbed eigenfunction of the i-th state and is the corresponding energy. Also it is assumed that this system has an energy spectrum with a simple structure.

The perturbed energies for the i-th state can be expressed as:

and the corresponding wavefunction is:

where the index n signals the correction order in expressions (34) and (35).

On the other hand, the n-th order energy correction can be written using the

form:

provided that the orthogonality condition holds between the unperturbed state wavefunction and the corrections of any order:

5.4.1.Brillouin-Wigner perturbation theory

In the Brillouin-Wigner perturbation formalism, the following identity is used

[18]:

Combining equations (36) and (38) it can be easily found that the n-th order wavefunction correction is given by [18]:

Thus, the n-th order energy correction for the i-th system’s state can be

written as:

provided that the orthogonality condition:

holds between the unperturbed state wavefunction and their perturbation corrections up to any order.

The wavefunction corrections can be obtained similarly through a resolvent operator technique which will be discussed below. The n-th wavefunction correction for the i-th state of the perturbed system can be written in the same manner as it is customary when developing some scalar perturbation theory scheme: by means of a linear combination of the unperturbed state wavefunctions, excluding the i-th unperturbed state. That is:

Using expression (52) into equation (49), after some straightforward manipulation, one can obtain the equivalent rule in order to construct the n-th order wavefunction correction:

with the weighted projector sum Zi (0) defined in turn as:

completely general perturbation theory for nondegenerate systems, although a recent development permits to extend the formalism to degenerate states [1e].