Page 254 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

APPLICATIONS OF NESTED SUMMATION SYMBOLS TO QUANTUM CHEMISTRY 237

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.
Suppose a r-electron operator to be written as with the r-dimensional
vector r representing the coordinates of the canonically ordered electron set:
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,
which will be the ones connected with the r-electron operator:

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