Electronic overlap matrix

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. 

Here, Flffl are matrix elements of a zeroth-order Hamiltonian, which is chosen as a one-electron operator in the spirit of MP2. is an overlap matrix The excited CFs are not in general orthogonal to each other. Finally, Vf)(i represents the interaction between the excited function and the CAS reference function. The difference between Eq. [2] and ordinary MP2 is the more complicated structure of the matrix elements of the zeroth-order Hamiltonian in MP2 it is a simple sum of orbital energies. Here H is a complex expression involving matrix elements of a generalized Fock operator F combined with up to fourth-order density matrices of the CAS wave function. Additional details are given in the original papers by Andersson and coworkers.17 18 We here mention only the basic principles. The zeroth-order Hamiltonian is written as a sum of projections of F onto the reference function 0)... 

The CBS methods of Petersson and coworkers (22,25,25-30) extrapolate the basis set and add empirical terms to extrapolate to experiment. Two especially powerful versions of the CBS approach are CBS-4 (30) and CBS-Q (30). For CBS-Q, the empirical terms are based on the overlap matrix and the spin contamination (which arises in some partial treatments of the electron correlation). CBS-4 also includes these terms plus a term based on the number of electrons in the system. 

In the previous section we examined the variational result of the two-term wave function consisting of the covalent and ionic functions. This produces a 2 x 2 Hamiltonian, which may be decomposed into kinetic energy, nuclear attraction, and electron repulsion terms. Each of these operators produces a 2 x 2 matrix. Along with the overlap matrix these are... 

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 ... 

The matrix element Prs is summed over all filled MOs (from fi to ij/n for the ground electronic state of a 2 -election closed-shell molecule) an example of the calculation of P was given in Section 5.2.3.6.5. The elements of the overlap matrix S are simply the overlap integrals ... 

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