Coulomb operator functionals

N.2 Computational speedup for the direct and reciprocal sums Computational speedups can be obtained for both the direct and reciprocal contributions. In the direct space sum, the issue is the efficient evaluation of the erfc function. One method proposed by Sagui et al. [64] relies on the McMurchie-Davidson [57] recursion to calculate the required erfc and higher derivatives for the multipoles. This same approach has been used by the authors for GEM [15]. This approach has been shown to be applicable not only for the Coulomb operator but to other types of operators such as overlap [15, 62],... 

The Bethe logarithm is formally defined as a certain normalized infinite sum of matrix elements of the coordinate operator over the Schrodinger-Coulomb wave functions. It is a pure number which can in principle be calculated with arbitrary accuracy, and high accuracy results for the Bethe logarithm can be found in the literature (see, e.g. [13, 14] and references therein). For convenience we have collected some values for the Bethe logarithms [14] in Table 3.1. 

As in the case of corrections of order a (Za) m, not only the diagrams in Fig. 7.3 with insertions of polarization operators in one and the same external Coulomb line but also the reducible diagrams Fig. 7.5 with polarization insertions in different external Coulomb lines generate corrections of order Respective contributions were calculated in [18] with the help of the subtracted Coulomb Green function from [20]... 

In order to calculate the matrix elements with the Coulomb operator Vc, one again uses Slater determinantal wavefunctions, for the intermediate state xp(Mp, t) as well as for the complete final state which contains the doubly charged ion, f, and the two ejected electrons, x<, (Ka, Kb). Assuming that there is no correlation between the two escaping electrons and that their common boundary condition applies separately to each single-particle function, the directional emission property is included in the factors f( ka) and f( kb), and one gets for this Coulomb matrix element C... 

The Coulomb operator is a two-particle operator, i.e., it describes an interaction between at most two different orbitals on each side of its matrix element. Therefore, these matrix elements vanish unless the energies and spatial parts of the wave-function in the orbitals a> or b> coincide with t>. This gives... 

For d = 3, the solution of Eq. (7) is the familiar Coulomb wave function in 3 dimensions. For d 3 this is not true. This represents an additional problem since the non-relativistic calculations have to be done without an explicit knowledge of the wave function. Fortunately, cancelation of all divergences can be ensured on the operator level using the Schrodinger equation in d-dimensions. Once divergences are canceled, the limit d —> 3 can be taken and a non-trivial (but now finite) matrix elements can be easily computed. 

Here the physical and time-reversed Coulomb scattering functions are denoted respectively by The time-reversal operator is given by... 

As molecular applications of the extended DK approach, we have calculated the spectroscopic constants for At2 equilibrium bond lengths (RJ, harmonic frequencies (rotational constants (B ), and dissociation energies (Dg). A strong spin-orbit effect is expected for these properties because the outer p orbital participates in their molecular bonds. Electron correlation effects were treated by the hybrid DFT approach with the B3LYP functional. Since several approximations to both the one-electron and two-electron parts of the DK Hamiltonian are available, we dehne that the DKnl -f DKn2 Hamiltonian ( 1, 2= 1-3) denotes the DK Hamiltonian with DKnl and DKn2 transformations for the one-electron and two-electron parts, respectively. The DKwl -I- DKl Hamiltonian is equivalent to the no-pair DKwl Hamiltonian. For the two-electron part the electron-electron Coulomb operator in the non-relativistic form can also be adopted. The DKwl Hamiltonian with the non-relativistic Coulomb operator is denoted by the DKwl - - NR Hamiltonian. 

The expression for the lowest order contribution to the parity violating potential within the Dirac Hartree-Fock framework is identical to that within the relativistically parameterised extended Hiickel approach in eq. (146). The difference is, however, that in DHF typically atomic basis sets with fixed radial functions are employed (see [161]) and that the molecular orbital coefficients are obtained in a self-consistent Dirac Hartree-Fock procedure. Computations of parity violating potentials along these lines have occasionally been called fully relativistic, although this term is rather unfortunate. In the four-component Dirac Hartree-Fock calculations by Quiney, Skaane and Grant [155] as well as in those by Schwerdtfeger, Laerdahl and coworkers [65,156,162,163] the Dirac-Coulomb operator has been employed, which is for systems with n electrons given by... 

The combination of the Dirac-Kohn-Sham scheme with non-relativis-tic exchange-correlation functionals is sometimes termed the Dirac-Slater approach, since the first implementations for atoms [13] and molecules [14] used the Xa exchange functional. Because of the four-component (Dirac) structure, such methods are sometimes called fully relativistic although the electron interaction is treated without any relativistic corrections, and almost no results of relativistic density functional theory in its narrower sense [7] are included. For valence properties at least, the four-component structure of the effective one-particle equations is much more important than relativistic corrections to the functional itself. This is not really a surprise given the success of the Dirac-Coulomb operator in wave function based relativistic ab initio theory. Therefore a major part of the applications of relativistic density functional theory is done performed non-rela-tivistic functionals. 

Equation (6), can be considered as an overlap-like QSM [33] between a molecular eDF p r) and a point charge located at position R, represented by the Dirac delta function 5(r - R). At the same time, equation (6) bears the same structure as equations (I) and (5) have. Thus, it can be also said that the eDF itself, as eEMP with respect to Coulomb operator, could formally be considered as the expectation value of a Dirac delta function acting as an operator. 

Nuclear magnetic moment or spin Ionization potential Spin-spin coupling matrix Coulomb operator Coulomb integral Coulomb functional Lagrange multiplier Transmission coefficient Compressibility constant Boltzmann s constant Rate constant... 

It would be good now to get rid of the non-diagonal Lagrange multipliers in order to obtain a beautiful one-electron equation analogous to the Fock equation. To this end, we need the operator in the curly brackets in Eq. (11.33) to be invariant with respect to an arbitrary unitary transformation of the spinorbitals. The sum of the Coulomb operators (Ucoui) is invariant, as has been demonstrated on p. 406. As to the unknown functional derivative SE c/Sp (i.e., potential Uxc), its invariance follows from the fact that it is a functional of p [and p of Eq. (11.6) is invariant]. Finally, after applying such a unitary transformation that diagonalizes the matrix of Sij,v/e obtain the Kohn-Sham equation (su = Si) ... 

Function n appears in a natural way, when we compute the mean value of the total electronic repulsion ( 17 F) with the Coulomb operator U = and a normalized N-electron... 

The difference between this Fock operator and the Hartree-Fock counterpart in Eq. (2.51) is only the exchange-correlation potential functional, Exc, which substitutes for the exchange operator in the Hartree-Eock operator. That is, in the electron-electron interaction potential, only the exchange operator is replaced with the approximate potential density functionals of the exchange interactions and electron correlations, while the remaining Coulomb operator, Jj, which is represented as the interaction of electron densities, is used as is. The point is that the electron correlations, which are incorporated as the interactions between electron configurations in wavefunction theories (see Sect. 3.3), are simply included... 

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