Breit potential functions

To evaluate the two-electron integrals within the Cl and SCF equations the functions W)a (r) and Vkh,(r) can be calculated in close analogy to the Coulomb case [201]. With the definition of a general density distribution 

there is an efficient way for computing these integrals [201] which can be easily shown using the additional definition 

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This equation does not have to be evaluated for all possible e,j. The condition ij = ji proved by Hinze [292] allows one to control the numerical solution of the SCF equations and is actually an identical solution condition to obtain the MCSCF spinors. It can be fulfilled (in terms of machine precision and numerical accuracy) only if the SCF iteration is converged. On the other hand, a large discrepancy could result in convergence problems. Of course, Eq. (9.111) has to be adjusted if the Breit interaction enters the two-electron interaction terms and hence modifies the potential functions. These changes, however, are straightforward [201]. 

Assuming, for example, a point-like atomic nucleus and taking into account the pure Coulomb potential, where the wave functions are analytically known, yields for the Is state (Breit 1928)... 

All structures were optimized at the CASSCF(8,8) level with the cc-pVDZ basis set. For multireference calcidations involving bromine and iodine, the Cowan-Griffin ab initio model potential with a relativistic effective core potential was used.222 CASPT2 calculations were performed on all optimized CASSCF(8,8)/cc-pVDZ geometries, using the CASSCF wave functions as the reference wave functions. SOCs were computed by using the Pauli-Breit Hamiltonian. 

The terms etc. in (10) represent the one-body mean-field potential, which approximates the two-electron interaction in the Hamiltonian, as is the practice in SCF schemes. In the DFB equations this interaction includes the Breit term (4) in addition to the electron repulsion l/rjj. The radial functions Pn ( ) and Qn/c( ) may be obtained by mmierical integration [20,21] or by expansion in a basis (for more details see recent reviews [22,23]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [24,25], where admixture of negative-energy solutions may yield energies much below experimental. To avoid this failure, the basis sets used for expanding the large and small components must maintain kinetic balance [26,27]. In the nonrelativistic limit (c oo), the small component is related to the large component by [24]... 

The complex energy, 2 of Eq. (5) is normally understood in fhe context of resonance scattering theory as the complex pole in the Breit-Wigner amplitude, or in the S-matrix, or in the optical potential of Feshbach s fheory," or in the Green s function, e.g.. Refs. [2,6-8]. 

One of the most recent theoretical advances has been to correlate almost all n-p and p-p data below 300 Mev in terms of boundary conditions on S, P and D states with only one energy dependent parameter (see Breit and Feshbach ). However, there is still no answer to whether it is possible to write an analytical expression as a function of energy for the law of force between two nucleons. It may be that the need of a Lorentz invariant expression for the interaction is the fundamental reason why potentials (even of the repulsive core type (Jastrow )] specifically have not been able to fit all of the data. 

Hay and Wadt (1985a, b) have published ECPs which are in form identical to the averaged RECPs of Christiansen, Ermler and co-workers. However, there are differences. First, the Hay-Wadt potentials are derived from the Cowan-GriflSn adaptation of the Breit-Pauli Hamiltonian into a variational computation of the atomic wave-function. From these solutions the ECPs are generated. It should be noted that the spin-orbit coupling is not included in the Hay-Wadt ECPs. Consequently, molecular calculations done using these ECPs would not include spin-orbit coupling. 

As a final remark we may comment on the fact that we need to study the two-electron problem in the attractive external potential of an atomic nucleus, hence, as a bound-state problem. It is immediately seen that this affects, for instance, the expansion in terms of zeroth-order state functions in Eq. (8.22), where bound states of the one-electron problem rather than free-particle states become the basis for the construction of the wave function (and wave-function operators in second quantization). The situation is, however, more delicate than one might think and reference is usually made to the discussion of this issue provided by Furry [216] (Furry picture). Of course, it is of fundamental importance to the QED basis of quantum chemistry. However, as a truly second-quantized QED approach, we abandon it in our semi-classical picture and refer to Schweber for more details [165, p. 566]. Instead, we may adopt from this section only the possibility to include either the Gaunt or the Breit operators in a first-quantized many-particle Hamiltonian. 

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