Electron repulsion matrix

The accepted wisdom is that the Hiickel Hamiltonian matrix should be identified with the matrix (h where G is the electron repulsion matrix of Chapter 6. The basis for this belief is that that the matrix (h has eigenvalues that do sum correctly to the electronic energy. 

In this latter formula, the two electron repulsion integral is written following Mulliken convention and the one electron integrals are grouped in the matrix e. In this way, the one-electron terms of the Hamiltonian are grouped together with the two electron ones into a two electron matrix. Here, the matrix is used only in order to render a more compact formalism. 

The usual reactivity indices, such as elements of the first-order density matrix, are also incapable of distinguishing properly between singlet and triplet behavior. Recently, French authors 139,140) have discussed the problem and shown how electron repulsion terms can be introduced to obtain meaningful results. The particular case of interest to them was excited state basicity, but their arguments have general applicability. In particular, the PMO approach, which loses much of its potential appeal because of its inability to distinguish between singlet and triplet behavior 25,121) coui(j profit considerably from an extension in this direction. 119,122)... 

Diagonal matrix elements of the P3 self-energy approximation may be expressed in terms of canonical Hartree-Fock orbital energies and electron repulsion integrals in this basis. For ionization energies, where the index p pertains to an occupied spinorbital in the Hartree-Fock determinant,... 

Here the symbol e represents the one-electron integral matrix and iaja lKia ) is the usual two-electron repulsion integral in the Condon and Shortley notation. 

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.

To construct the Fock matrix, one must already know the molecular orbitals ( ) since the electron repulsion integrals require them. For this reason, the Fock equation (A.47) must be solved iteratively. One makes an initial guess at the molecular orbitals and uses this guess to construct an approximate Fock matrix. Solution of the Fock equations will produce a set of MOs from which a better Fock matrix can be constructed. After repeating this operation a number of times, if everything goes well, a point will be reached where the MOs obtained from solution of the Fock equations are the same as were obtained from the previous cycle and used to make up the Fock matrix. When this point is reached, one is said to have reached self-consistency or to have reached a self-consistent field (SCF). In practice, solution of the Fock equations proceeds as follows. First transform the basis set / into an orthonormal set 2 by means of a unitary transformation (a rotation in n dimensions),... 

The largest arrays which occur in calculations are of two types. One arises from the electron repulsion integrals and grows in size like the fourth power of the number of basis functions. The other is the configuration interaction hamiltonian matrix which grows like the square of the number of configurations. Many other smaller arrays, whose size is proportional to the square of the number of basis functions, occur throughout the calculation. 

In terms of the ITO approach, the reduced matrix elements of the electron repulsion operator (which gives rise to terms) are expressed as follows ... 

The matrix elements of the electron repulsion operator match the corresponding energies of terms in the case of noninteracting terms (Table 50) and can also be found elsewhere. For interacting terms, however, other sources are silent in their tabulations these are compiled in Table 51. 

Let us construct the Hamiltonian matrix by adding the CF matrix to the electron repulsion matrix ... 

Table 51 Matrix element of the electron repulsion for interacting termsa...

First, let the unitary transformation diagonalizes the interaction matrix that involves the operators of the electron repulsion, the crystal field, and the... 

The expansion coefficients, YJr are obtained by the diagonalization of the Hamiltonian matrix constructed in the basis of the intermediate states. It is a crucial feature of the ADC approach that the Hamiltonian matrix elements of the type ( Tj H T / can be expressed analytically via the orbital energies and the electron repulsion integrals if one performs the orthonormalization... 

The initial Hiickel calculations can be employed to obtain preliminary values for the electron densities and bond orders, from which the self-consistent field matrix elements can be evaluated by introduction of the chosen core potentials and electron repulsion integrals.11 Table I lists the ionization potentials, electron affinities and nuclear charges employed in the present calculations. 

Each element of the electron repulsion matrix G has eight 2-electron repulsion integrals, and of these 32 there appear to be 14 different ones ... 

The two-electron matrix G, the electron repulsion matrix (Eq. 5.104), is calculated from the two-electron integrals (Eqs. 5.110) and the density matrix elements (Eq. 5.81). This is intuitively plausible since each two-electron integral describes one interelectronic repulsion in terms of basis functions (Fig. 5.10) while each density matrix element represents the electron density on (the diagonal elements of P in Eq. 5.80) or between (the off-diagonal elements of P) basis functions. To calculate the matrix elements Grs (Eqs. 5.106-5.108) we need the appropriate integrals (Eqs. 5.110) and density matrix elements. These latter are calculated from... 

From the G values based on the initial guess c s the initial-guess electron repulsion matrix is... 

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