Page 256 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

APPLICATIONS OF NESTED SUMMATION SYMBOLS TO QUANTUM CHEMISTRY 239

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: