Coulomb singularities

To improve on the wave function one has to accept that the standard multideterminantal expansion [Eq. (13.3)] is unsuitable for near-exact but practical approximations to the electronic wavefunction. The problem is dear from a simple analysis of the electronic Hamiltonian in Eq. (13.2) singularities in the Coulomb potential at the electron coalescence points necessarily lead to irregularities in first and higher derivatives of the exact wave function with respect to the interpartide coordinate, rj 2. The mathematical consequences of Coulomb singularities are known as electron-electron (correlation) and electron-nuclear cusp conditions and were derived by... 

As shown by Chang, Pelissier and Durand (CPD) [41] a regular expansion, however, can be deduced by isolating the Coulomb singularity by infinite summations. Let us rewrite the equation (38), when z — 0... 

From this equation one finds that has a Coulombic singularity near the atomic nucleus. Furthermore it follows that if the GGA reproduces the correct asymptotic behavior of the Slater potential... 

However, in practice, the reliance on the complete (or an extremely large) basis set is avoided by exploiting the analytical cancellation of the Coulomb singularities by the correlation factor in evaluating the necessary integrals of the Hamiltonian ... 

Although a Slater-determinant reference state 4> cannot describe such electronic correlation effects as the wave-function modification required by the interelec -tronic Coulomb singularity, a variationally based choice of an optimal reference state can greatly simplify the -electron formalism. 4> defines an orthonormal set of N occupied orbital functions occupation numbers = 1. While () = 1 by construction, for any full A-electron wave function T that is to be modelled by it is convenient to adjust (T T) > 1 to the unsymmetrical... 

It is relevant to the discussion of Sect. 4 to note that if V were simply the electric field term —eFz in Eq. (22) and this was switched on the free particle form at Eq. (21), then the additional factor multiplying would be exp(PFz). This is precisely the factor present in the diagonal form of Eq. (24). However, potentials K(r) in atomic ions evidently have Coulomb singularities at nuclei, so that... 

Since the factors ( , + mc2) l grow asymptotically (for p, -> oo, i.e. r, — 0) like I/ p, , all contributions of momentum operators in the numerator (leading to the 1/r3 divergence in the case of the Breit-Pauli operator) are cancelled asymptotically, and only a Coulomb singularity remains. Recently, Brummelhuis et al (2002) have formally proved that the operator is variationally stable. 

The behavior of relativistic wave functions at the Coulomb singularities of the Hamiltonian have been studied [84]. The nuclear attraction potentials don t cause any problem. There are weak singularities of the type r with p slightly smaller than 0, as they are familiar for the H-like ions. The limits r —> 0 and oo commute, and the Kato cusp conditions [85] arise in the nrl. For the coalescence of two electrons the two limits do not commute. An expansion in powers of c is possible to the lowest orders and leads to results consistent with those reported above. 

One must probably conclude that we do not know the correct electron interaction for small distance between the electrons, and that this interaction ought to be somehow regularized. The Coulomb singularity of the potential, which can be tamed in the nrl [34], is apparently too strong in the relativistic context. 

It should be noted that in this model, where the electron appears as a quasi-Bohr subsystem with radius rc, there is no Coulomb singularity, according to Gauss theorem, and no cusp condition is required if the wave equation is reformulated to account for the electron size. 

Much as for the ID solvent-solvent correlations, the renormalization of the 3D-RISM/HNC equations is not necessary in respect to convergence. Nor is it required for the 3D fast Fourier transform (3D-FFT) employed to evaluate the convolution in Eq. (4.A.41). For a periodic solute neutralized by a compensating background charge, the Coulomb potential of the solute charge is screened at a supercell length. Therefore the 3D site direct, and hence total correlation functions, are free from the Coulomb singularity at fc = 0 and can be transformed directly by the 3D-FFT. 

According to Eq. (12), the coulombic singularities have the form of a second-order pole and a confluent first-order pole. The residues of the poles can in principle be calculated exactly, from the D 1 limit of the Schrodinger equation, as we discussed in Chapter 4.1. This suggests that we use an approximant of the form... 

Some flexibility remains in the selection of terms to be included in Eq. 9.36. An assortment of ECPs are possible within this framework because the form of the pseudo-orbital within the cutoff radius is not completely defined. So-called soft ECPs have been designed so that wi cancels the Coulomb singularity at the nucleus [133, 135]. This is valuable for QMC calculations because their efficiency is sensitive to rapid changes of the potential. Several sets of soft ECPs have been designed specifically for QMC so that Gaussian basis function can be used in QMC calculations without special consideration of the electron-nucleus cusp conditions [136, 137]. 

Hylleraas was thus not only the first person to experience the slow convergence of the Cl expansion. He also solved the problem - at least for the two-electron system. However, Hylleraas had not arrived at his wave function from a consideration of the singularities of the Hamiltonian. In fact, in 1928, Slater had analyzed the properties of the helium wave function and had found that the Coulomb singularity in the Hamiltonian imposes a certain behaviour on the wave function when the electrons coincide and had suggested that the wave function be multiplied by a factor of exp(ri2/2) in order to model this behaviour [8] however, less interested in numerical solutions and applied mathematics than Hylleraas, he did not attempt to include this factor in the wave function to resolve the discrepancy with experiment for... 

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