Coulomb basis functions

If Coulomb basis functions are used such that hola) = eo o), as is the case here for two-electron systems, Kij is reduced to... 

Next, we shall consider four kinds of integrals. The first is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at that nucleus. The second is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at a different point (usually another nucleus). Then, we will consider the matrix element of a Coulomb term between two primitive basis functions at different centers. The third case is when one basis function is centered at the nucleus considered. The fourth case is when both basis functions are not centered at that nucleus. By that we mean, for two Gaussian basis functions defined in Eqs. (73) and (74), we are calculating... 

When the Coulomb and exchange operators are expressed in terms of the basis functions and the orbital expansion is substituted for xu then their contributions to the Fock matrix element take the following form ... 

The original FMM has been refined by adjusting the accuracy of the multipole expansion as a function of the distance between boxes, producing the very Fast Multipole Moment (vFMM) method. Both of these have been generalized tc continuous charge distributions, as is required for calculating the Coulomb interactioi between electrons in a quantum description. The use of FMM methods in electronic structure calculations enables the Coulomb part of the electron-electron interaction h be calculated with a computational effort which depends linearly on the number of basi functions, once the system becomes sufficiently large. 

Thus, we can alternatively express the Coulomb contribution in equation (7-12) solely in terms of the basis functions as the following four-center-two-electron integrals (since the four basis functions qfl, r v, q/ r a can be attached to a maximum of four different atoms)... 

Note again the formal simplicity of equation (7-17) as compared to equation (7-18) in spite of the fact that the former is exact provided the correct Vxc is inserted, while the latter is inherently an approximation. The calculation of the formally L2/2 one-electron integrals contained in hllv, equation (7-13) is a fairly simple task compared to the determination of the classical Coulomb and the exchange-correlation contributions. However, before we turn to the question, how to deal with the Coulomb and Vxc integrals, we want to discuss what kind of basis functions are nowadays used in equation (7-4) to express the Kohn-Sham orbitals. 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

The definition of the matrix in equation (60) requires some explanation The minus sign is motivated by the fact that H(x) is assumed to be an attractive potential. Division by Po is motivated by the fact that for Coulomb systems, when is so defined, it turns out to be independent of po, as we shall see below. The Sturmian secular equation (61) has several remarkable features In the first place, the kinetic energy has vanished Secondly, the roots are not energy values but values of the parameter po, which is related to the electronic energy of the system by equation (52). Finally, as we shall see below, the basis functions depend on pq, and therefore they are not known until solution... 

The method ofmany-electron Sturmian basis functions is applied to molecules. The basis potential is chosen to be the attractive Coulomb potential of the nuclei in the molecule. When such basis functions are used, the kinetic energy term vanishes from the many-electron secular equation, the matrix representation of the nuclear attraction potential is diagonal, the Slater exponents are automatically optimized, convergence is rapid, and a solution to the many-electron Schrodinger eqeuation, including correlation, is obtained directly, without the use ofthe self-consistent field approximation. 

The problem of evaluating matrix elements of the interelectron repulsion part of the potential between many-electron molecular Sturmian basis functions has the degree of difficulty which is familiar in quantum chemistry. It is not more difficult than usual, but neither is it less difficult. Both in the present method and in the usual SCF-CI approach, the calculations refer to exponential-type orbitals, but for the purpose of calculating many-center Coulomb and exchange integrals, it is convenient to expand the ETO s in terms of a Cartesian Gaussian basis set. Work to implement this procedure is in progress in our laboratory. 

The first summation is over nuclei A. Z are atomic numbers and Rap are distances between the nuclei and the point charge. The second pair of summations is over basis functions, ( ). P is the density matrix (equation 16 in Chapter 2), and the integrals reflect Coulombic interactions between the electrons and the point charge, where rp is the distance separating the electron and the point charge. 

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