Electronic overlap matrix element

The third type of matrix element may be called "electronic overlap matrix element" E... 

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. 

Table 2.1. Numerical values for overlap, kinetic energy, nuclear attraction, and electron repulsion matrix elements in the two-state calculation.

S v are elements of the overlap matrix. Similar types of expressions may be constructed for density functional and correlated models, as well as for semi-empirical models. The important point is that it is possible to equate the total number of electrons in a molecule to a sum of products of density matrix and overlap matrix elements. ... 

For first- and second-row atoms, the Is or (2s, 2p) or (3s,3p, 3d) valence-state ionization energies (au s), the number of valence electrons ( Elec.) as well as the orbital exponents (es, ep and ec ) of Slater-type orbitals used to calculate the overlap matrix elements SP V corresponding are given below. 

Here, S denotes the inverse overlap matrix element between basis functions and v. With Eqs. (102) and (103) the one-electron terms in Eq. (100) reduce to... 

The matrix element is understood to be on-the-energy-shelF, i.e., the energy e of the photoelectron has to be calculated according to equ. (1.29a). Due to the different binding energies of electrons ejected from different shells of the atom, it is therefore possible to restrict the calculation of the matrix element to the selected process in the present example to photoionization in the Is shell only. As a consequence, the matrix element factorizes into two contributions, a matrix element for the two electrons in the Is shell where one electron takes part in the photon interaction, and an overlap matrix element for the other electrons which do not take part in the photon interaction (passive electrons). The overlap matrix element is given by... 

The single-electron matrix elements for the passive and the active electrons contain different projections of the electron spin. Since neither the photon operator nor the unity operator (in the overlap matrix element) acts on the spin, the quantum numbers M, and ms are fixed by the corresponding spin of the formerly bound ls-electron. This yields... 

This result shows that the original matrix element containing the orbitals of all electrons factorizes into a two-electron Coulomb matrix element for the active electrons and an overlap matrix element for the passive electrons. Within the frozen atomic structure approximation, the overlap factors yield unity because the same orbitals are used for the passive electrons in the initial and final states. Considering now the Coulomb matrix element, one uses the fact that the Coulomb operator does not act on the spin. Therefore, the ms value in the wavefunction of the Auger electron is fixed, and one treats the matrix element Mn as... 

Using, for example, Slater wavefunctions for the initial and final states, this matrix element can be evaluated to yield a one-particle matrix element for the active electron and an overlap matrix element for the passive electrons (see equ. (2.4)). Of interest in the present discussion is the one-particle matrix element of the active electron ... 

From Eq. [239], it is apparent that the size of a particular is not only determined by the magnitude of the electronic coupling matrix element but also by the overlap of the vibrational wave functions v,- and i/. Squared overlap integrals of the type (Xi/, (Q) IXt/ (Q))q 2 are frequently called Franck-Con-don (FC) factors. In contrast to radiative processes, FC factors for nonradiative transitions become particularly unfavorable if two states differing considerably in their electronic energies exhibit similar shapes and equilibrium coordinates of their potential curves. Due to the near-degeneracy requirement, an upper state vibrational wave function, with just a few nodes... 

Slater s bond eigenfunctions constitute one choice (out of an infinite number) of a particular sort of basis function to use in the evaluation of the Hamiltonian and overlap matrix elements. They have come to be called the Heitler-London-Slater-Pauling (HLSP) functions. Physically, they treat each chemical bond as a singlet-coupled pair of electrons. This is the natural extension of the original Heitler-London approach. In addition to Slater, Pauling[12] and Eyring and Kimbal[13] have contributed to the method. Our following description does not follow exactly the discussions of the early workers, but the final results are the same. 

In this section, a new function, called paired-permanent-determinant (PPD), is introduced, which is an algebrant. An overlap matrix element in the spin-free VB method may be obtained by evaluating a corresponding PPD, while the Hamiltonian matrix element is expressed in terms of the products of electronic integrals and sub-PPDs. 

As shown in the last section, Hamiltonian and overlap matrix elements are expressed in terms of PPDs. A practical VB package highly depends on an efficient routine for the evaluation of a PPD. Although a PPD may be expressed in terms of sub-PPDs of any given order and their complementary minors, in the present version of Xiamen-99, an algorithm of 2x(V-2) expansion is used. This is because the 1-e and 2-e electron integrals may be built as effective 2x2 PPDs. 

From Eq. (50), an overlap matrix element is exactly a PPD and can easily be evaluated from the routine for PPDs, while Hamiltonian matrix elements may be obtained by a similar routine to that for PPDs, where 2x2 sub-PPDs are replaced with effective sub-PPDs of one-electron and two-electron integrals. 

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