Two-electron matrix

In an ab initio method, all the integrals over atom ic orbital basis function s are com puted and the bock in atrix of th e SCK com puta-tiori is formed (equation (6 1) on page 225) from the in tegrals. Th e Kock matrix divides inui two parts the one-electron Hamiltonian matrix, H, and the two-electron matrix, G, with the matrix ele-m en ts... 

There is a clear one-to-one correspondence between the theoretical expressions and the computational implementation in terms of one- and two-electron matrix elements. Implementations of the expressions are therefore facilitated. 

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 matrix K is the reduced Hamiltonian [22, 25] and has the same symmetry properties as the two-electron matrix that is. 

Table 15.4. Comparison of some one- and two-electron matrix elements for pure and SCVB 2pz orbitals. All energies are in hartrees.

If we assume that the difference between the two electronic matrix elements varies only slowly with Q, then... 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

If the specific tensorial structure (14.58) of a two-electron operator is known, then we can obtain its representation in terms of the product of operators (14.30) acting in the space of states of one shell. In fact, if we substitute into the two-electron matrix element which enters into (13.23), the operator (14.58) in the form... 

Among all the possible two-particle operators for physical quantities for lN configuration we have only considered in detail the electrostatic interaction operator for electrons here too we shall confine ourselves to the examination of this operator. The explicit form of the two-electron matrix elements of the electrostatic interaction operator for electrons (the... 

Formula (17.16) is the most general form of the two-electron matrix element in which all four one-electron wave functions have different quantum numbers. We shall put it into general formula (13.23), whereupon the creation and annihilation operators will be rearranged to place side by side those second-quantization operators whose rank projections enter into the same Clebsch-Gordan coefficient. Summing over the projections then gives... 

Matrix elements of the operators of the interaction energy between two shells of equivalent electrons may be expressed, with the aid of the CFP, in terms of the corresponding two-electron quantities. Substituting in such formula the explicit expression for the two-electron matrix element, after a number of mathematical manipulations and using the definition of submatrix elements of operators composed of unit tensors, we get convenient expressions for the matrix elements in the case of two shells of equivalent electrons. The corresponding details may be found in [14], here we present only final results. 

In a similar manner the particular case of two-electron matrix elements may be deduced from the two-shell expressions presented above. 

As was mentioned at the beginning of this chapter, the matrix element of the interaction between subshells may be expressed in terms of the CFP with one detached electron and two-electron matrix elements of the operator considered. The corresponding formula for jj coupling is as follows ... 

However, formulas of the kind (20.26) are rather inconvenient for calculations. Therefore, one has usually to insert explicit expressions for two-electron matrix elements, to perform, where it turns out to be possible, the summations necessary and to find finally the representation of the energy matrix element of the interaction between two subshells in the form of direct and exchange parts. Thus, for the electrostatic interaction we find... 

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

A two-electron matrix element of the operator of the electrostatic interaction energy is equal to... 

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... 

Here (aa/ W/j is the symmetrized two-electron matrix element of the electron-electron Coulomb repulsion. 

The theoretical tools of quantum chemistry briefly described in the previous chapter are numerously implemented, sometimes explicitly and sometimes implicitly, in ab initio, density functional (DFT), and semi-empirical theories of quantum chemistry and in the computer program suits based upon them. It is usually believed that the difference between the methods stems from different approximations used for the one- and two-electron matrix elements of the molecular Hamiltonian eq. (1.177) employed throughout the calculation. However, this type of classification is not particularly suitable in the context of hybrid methods where attention must be drawn to the way of separating the entire molecular system (eventually - the universe itself) into parts, of which some are treated explicitly on a quantum mechanical/chemical level, while others are considered classically and the rest is not addressed at all. That general formulation allows us to cover both the traditional quantum chemistry methods based on the wave functions and the DFT-based methods, which generally claim... 

The basic elements of the diagrams are shown in Figure 1. Figure 1 (a) shows the diagrammatic representation of a one-electron operator matrix element. Figure 1 (b) shows the representation of a two-electron matrix which in the Brandow scheme includes permutation of the two electrons involved. Upward (downward) directed lines represent particles (holes) created above (below) the Fermi level when an electron is excited. 

Here, h is the one electron matrix in the non-orthogonal Gaussian basis and G (P ) is the two electron matrix for Hartree-Fock calculations, but for DFT it represents the Coulomb potential. The term Exc is the DFT exchange-correlation functional (for Hartree-Fock Exc = 0), while Vmn represents the nuclear repulsion energy. In the orthonormal basis, these matrices are h = etc., where the overlap... 

The convention for writing two-electron matrix elements is that ju and a belong to one electron, v and p to the other. 

In obtaining (5.16) from (5.15) we have used the fact that we can exchange the orbitals in both the bra and ket of a two-electron matrix element, which represents an integration over dummy coordinate—spin variables. Hhf is a one-electron operator since we sum over orbitals C). Equation (5.16) shows that we can diagonalise Hhf in the space of occupied orbitals only, since it has no matrix elements connecting occupied and unoccupied orbitals. Performing the diagonalisation we have... 

This equation is true because P may operate either on the bra or ket vectors of a two-electron matrix element with the same result since only a redefinition of dummy integration coordinates is involved. From (5.19,5.21) we form the matrix equations... 

It is significant that the relevant two-electron matrix elements are those for double excitations, which are of the same order in the number of excitations as those usually considered in the configuration basis of a configuration-interaction calculation. 

H o.V G v and P v are the one-electron matrix, two-electron matrix and the density mtrix, respectively. Note that in this derivation the number of electrons (njO in a MO may take on a non-integer value (0physical system non-integer electrons have been used previously in, for example, the so-called half-electron method (Dewar, Hashmall et al., 1968). 

Since the FCI corrections are based on different truncated virtual orbital spaces, it is very important to have an efficient algorithm for the transformation of two-electron integrals. A Cholesky decomposition of the two-electron matrix is then very convenient [31, 32], A two-electron integral [juv Ao] is related to the integral tables (obtained by the Cholesky decomposition) L t = 1,rs by the relation n... 

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