Coulomb cusp condition

If the Hellmann-Feynman theorem is to be valid for forces on nuclei, the Coulomb cusp condition must be satisfied. However, if the nuclei are displaced, the orbital Hilbert space is modified. Hurley [179] noted this condition for finite basis sets, and introduced the idea of floating basis functions, with cusps that can shift away from the nuclei, in order to validate the theorem for such forces. 

The electronic Coulomb interaction u(r 12) = greatly complicates the task of formulating and carrying out accurate computations of iV-electron wave functions and their physical properties. Variational methods using fixed basis functions can only with great difficulty include functions expressed in relative coordinates. Unless such functions are present in a variational basis, there is an irreconcilable conflict with Coulomb cusp conditions at the singular points ri2 - 0 [23, 196], No finite sum of product functions or Slater determinants can satisfy these conditions. Thus no practical restricted Hilbert space of variational trial functions has the correct structure of the true V-electron Hilbert space. The consequence is that the full effect of electronic interaction cannot be represented in simplified calculations. 

To avoid a singularity that cannot be cancelled by any one-electron potential, the coefficient ofV/-1 must vanish when t = 0. This implies the Coulomb cusp condition fo(q) = food + q H------). A similar expansion is valid for any i > 0. Because the... 

The Fade function has a cusp at r = 0 that can be adjusted to match the Coulomb cusp conditions by adjusting the a parameter. The Sun form also has a cusp, but approaches its asymptotic value far more quickly than the Fade function, which is useful for the linear scaling methods. An exponential form proposed by Manten and Luchow is similar to the Sun form, but shifted by a constant. By itself, the shift affects only the normalization of the Slater-Jastrow function, but has other consequences when the function is used to construct more elaborate correlation functions. The polynomial Fade function does not have a cusp, but its value goes to zero at a finite distance. 

Whilst CASSCF and related methods give a qualitatively accurate description of static correlation, the effects of dynamic correlation are largely neglected. The inclusion of dynamical correlation is critical for the quantitatively correct simulation of f-element complexes. This can be recovered through the application of full Cl but, as already discussed, this method is intractable for all but very small systems. In fact. Cl expansions converge on the full Cl limit very slowly. The Coulomb cusp condition specifies a relationship between the two-electron wavefunction and its first derivative when the interelectronic separation is equal to zero ... 

Owing to the presence of the Coulomb potential, the molecular electronic Hamiltonian becomes singular when two electrons coincide in space. To balance this singularity, the exact wave function exhibits a characteristic nondifferentiable behaviour for coinciding electrons, giving rise to the electronic Coulomb cusp condition [3]... 

The Coulomb cusp condition at ri2 = 0 has an even more severe implication for the approximate wave function. Consider the ground-state helium wave function for a coUinear arrangement of the nucleus and the two electrons. Expanding the wave function around rz = n and ri2 = 0, we obtain... 

The Coulomb cusp condition therefore leads to a wave function that is continuous but, because of the last term in (7.2.10), not smooth at ri2 = 0. Consequently, the wave function has discontinuous first derivatives for coinciding electrons. 

To illustrate the nuclear and electronic Coulomb cusp conditions, we have in figure 7.5 plotted the ground-state helium wave function with one electron fixed at a point 0.5ao from the nucleus. On the left, the wave function is plotted with the free electron restricted to a circle of radius 0.5ao centred at the nucleus (with the fixed electron at the origin of the plot) on the ri t, the wave function is plotted on the straight line through the nucleus and the fixed elearon. The wave function is differentiable everywhere except at the points where the particles coincide. 

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... 

Another more subtle condition is the electron-electron cusp condition. As two electrons approach each other, their Coulomb interaction dominates, and this leads to a cusp in the exchange-correlation hole at zero separation[15]. It is most simply expressed in terms of the pair distribution function. We define its spherically-averaged derivative at zero separation as... 

Physically, C, which controls the large wavevector decay of ( / i, depends on the behavior of the system at small interelectronic separations. In fact, C is proportional to the system-average of the cusp in the exchange-correlation hole at zero separation. If smooth function of r - r, then g x(r, r) = 0, and C would vanish, as it does at the exchange-only level (i.e., to first order in e2). However, as we saw in section 2.2, the singular nature of the Coulomb interaction between the electrons leads to the electron-coalescence cusp condition, Eq. (31). For the present purposes, we wish to keep track explicitly of powers of the coupling constant, so we rewrite Eq. (31) as... 

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. 

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. 

The Coulomb hole that wavefunctions are predicted to have for close anti-parallel-spin electrons is also called a correlation hole. As a condition for wavefunctions containing correlation holes, Kato proposed a correlation cusp condition (Kato 1957),... 

The Hartree-Fock wavefunction violates this condition, because it gives zero for the left-hand side of this equation. As shown in Fig. 3.1, a wavefunction satisfying this condition contains a correlation hole, which contains a sharp dip, called a cusp, near ri2 = 0. This correlation hole causes anti-parallel-spin electrons to be further apart, and therefore reduces Coulomb interactions, thus lowering the total electronic energies. Sinanoglu named this electron correlation in the correlation cusp condition as dynamical correlation (Sinanoglu 1964). 

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]. 

The problem with conventional configuration interaction or coupled-cluster wavefunctions is that the cusp conditions cannot be fulfilled, or more precisely that the overall shape of the wavefunction in the vicinity of electron coalescence, the Coulomb hole, converges extremely slowly with the size of the underlying basis set expansion. Figure 1 demonstrates that the expansion in orbital products does not provide terms which are linear in While at large separation, the wavefunction shape is well described even by the shortest Cl expansion, even the largest Cl expansion is far off at rj2 0. 

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