The Coulomb cusp

In the present section, we consider the behaviour of the exact wave function for coinciding particles. At such points, the electronic Hamiltonian becomes singular and gives rise to a characteristic cusp in the wave function. To simplify the discussion, we shall again examine the ground state of the helium atom, for which highly accurate approximations to the true wave function are easily generated. 

In Cartesian coordinates with the origin at the nucleus, the nonrelativistic electronic Hamiltonian of the helium-like atoms may be written in the usual manner as 

To see the implications of the Hamiltonian singularities for the wave function, it is convenient to express the Hamiltonian in a different set of coordinates. Since the ground-state wave function of the helium atom is totally synunetric, it can be expressed in terms of the three ratlial coordinates 

Note that the two one-electron parts of this Hamiltonian each have the same form as the radial Hamiltonian for an electron of zero angular momentum in a spherical potential (6.3.10). The singularities at the nucleus are now seen to be balanced by the kinetic-energy terms proportional to 1/r,  

Likewise, the tmns that multiply l/ri2 at ri2 = 0 must vanish in H if, imposing the additional condition 

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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. 

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 ... 

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. 

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... 

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... 

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... 

The Coulombic potential becomes infinitely negative when an electron and a nucleus coalesce and, because of this, the state function for an atom or molecule must exhibit a cusp at a nuclear position. That is, as shown by Kato (1957), the first derivative of the function is discontinuous at the position of a nucleus. Thus, while the charge density is a maximum at the position of a nucleus, this point is not a true critical point because Vp, like is discontinuous there. However, as discussed in Election E2.1, this is not a problem of practical import and the nuclear positions behave topologically as do (3, — 3) critical points in the charge distribution and hereafter they will be referred to as such. 

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. 

Bingel i ) has investigated the consequence of Kato s theorems for the pair density. It seems that generally the pair density has a cusp like that shown in Fig. 3. The Coulomb hole is not as deep as the Fermi hole and it has a cusp. 

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