Coulomb cusp

To illustrate the convergence of the FCI principal expansion with respect to short-range electron correlation, we have in Fig. 1.1 plotted the ground-state He wavefunction with both electrons fixed at a distance of 0.5 ao from the nucleus, as a function of the angle 0i2 between the position vectors ri and r2 of the two electrons. The thick grey lines correspond to the exact nonrelativistic wavefunction, whereas the FCI wavefunctions are plotted using black lines. Clearly, the description of the Coulomb cusp and more generally the Coulomb hole is poor in the orbital approximation. In particular, no matter how many terms we include in the FCI wavefunction, we will not be able to describe the nondifferentiability of the wavefunction at the point of coalescence. 

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. 

The interelectronic Coulomb cusp can be analyzed by transforming a two-electron Hamiltonian to relative coordinates. The one-electron potential function is regular at the singularity ri2 -> 0 and does not affect the cusp behavior. Given coordinates ri and r2, mean and relative coordinates are defined, respectively, by... 

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... 

Fig. 25.5. The Coulomb cusp, showing the functional form of the exact wave function, 12), in an area where the distance between two electrons is small.

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. 

Clearly, with this ansatz, very high accuracy is attained with only a few terms in the expansion. Indeed, the plots of the Coulomb hole, the kinetic energy and the Coulomb cusp in this chapter... 

In this exercise, we consider the use of correlating functions - that is, functions that depend only on the interelectronic distances - to describe the Coulomb cusp of many-electron systems. Consider a two-electron system with the Hamiltonian... 

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