Population Analysis Based on Basis Functions

The electron density (probability of finding an electron) at a certain position r from a single molecular orbital containing one electron is given as the square of the MO. 

Assuming that the MO is expanded in a set of normalized, but non-orthogonal, basis functions, this can be written as (eq. (3.48)). 

Integrating and summing over all occupied MOs gives the total number of electrons, N. 

We may generalize this by introducing an occupation number (number of electrons), n, for each MO. For a single determinant wave function this will either be 0, 1 or 2, while it may be a fractional number for a correlated wave function (Section 9.5). 

The sum of the product of MO coefficients and the occupation numbers is the density matrix defined in Section 3.5 (eq. (3.51)). The sum over the product of the density and overlap matrix elements is the number of electrons. 

Introduction to Computational Chemistry, Second Edition. Frank Jensen. 2007 John Wiley Sons, Ltd 

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Population analyses based on basis functions (such as Mulliken or Lowdin) require insignificant computational time. The NAO analysis involves only matrix diagonaliza-tion of small subsets of the density matrix, and also requires a negligible amount of... 

In linear variational problems, one way of satisfying Hurley s conditions is to make the basis set closed with respect to the differential operators d/dp. Such a basis set is in principle infinite. Practically, however, the Hellmann-Feynman theorem will be approximately satisfied if, for each significantly populated basis function x, its derivatives with respect to the orbital centers, X, x are included in the basis set (Pulay, 1969). The use of augmented basis sets in conjunction with the Hellmann-Feynman theorem was considered by Pulay (1969, 1977) but dismissed as expensive. Recently, Nakatsuji et al. (1982) have recommended such a procedure. However, an analysis of their procedure (Pulay, 1983c Nakatsuji et al., 1983) reveals that it is not competitive with the traditional gradient technique. Much of the error in the Hellmann-Feynman forces is due to core orbitals. Therefore, methods based on the Hellmann-Feynman theorem presumably work better for effective core... 

The examples in Section 9.1 illustrate that it would be desirable to base a population analysis on properties of the wave function or electron density itsell and not on the basis set chosen for representing the wave function. The electron density is the square of the wave function integrated over Aeiec - 1 coordinates (it does not matter which coordinates since the electrons are indistinguishable). 

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