Nuclear cusp condition

Secondly, it can be shown (Davidson, 1976) that the so-called nuclear cusp condition for nucleus A with position vector Ra gives... 

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... 

For the electron density of closed-shell atoms, the Thomas-Fermi kinetic energy is in error by about 5%. Variational minimization of the density increases this error to about 10% and leads to electron densities that diverge at the atomic nucleus. Adding constraints to force the correct electron-nuclear cusp conditions on the electron density, like other attempts to improve the simple Thomas-Fermi model, gives disappointing results. 

The large maxima of the electron density are expected and are found at the nuclear positions Ra. These points are m-limits for the trajectories of Vp(r), in this sense they are attractors of the gradient field although they are not critical points for the exact density because the nuclear cusp condition makes Vp(Ra) not defined. The stable manifold of the nuclear attractors are the atomic basins. The non-nuclear attractors occur in metal clusters [59-62], bulk metals [63] and between homonu-clear groups at intemuclear distances far away from the equilibrium geometry [64]. In the Quantum Theory of Atoms in Molecules (QTAIM) an atom is defined as the union of a nucleus and of the electron density of its atomic basin. It is an open quantum system for which a Lagrangian formulation of quantum mechanics [65-70] enables the derivation of many theorems such as the virial and hypervirial theorems [71]. As the QTAIM atoms are not overlapping, they cannot share electron pairs and therefore the Lewis s model is not consistent with the description of the matter provided by QTAIM. 

Let us examine the implications of the nuclear cusp condition for the first electron, assuming that the wave function does not vanish at n = 0 (which holds for the helium ground state). The cusp condition is satisfied if the wave function exhibits an exponential dependence on r close to the nucleus ... 

As discussed in Chapter 6, molecular electronic wave fiuictions are usually expanded in simple analytical functions centred on the atomic nuclei (AOs). Assuming that, close to a given nucleus, the behaviour of the electrons is determined completely by the analytical form of the one-electron basis functions at that nucleus, the nuclear cusp condition imposes constraints on the form of these basis functions close to the nucleus. By employing atomic basis functions that satisfy these constraints, it is possible to construct approximate wave functions that satisfy the nuclear cusp condition exactly. In particular, the STOs of Section 6.5 are compatible with the nuclear cusp condition but not so the GTOs of Section 6.6, which depend on the square of the electronic distance to the atomic centre. In passing, we note that the nuclear cusp condition applies only to point-charge nuclei such as that pre.sent in (7.2.1). 

However, it is indeed fortunate that the IV-representability problem for the electron density p(r) greatly simplifies itself. In fact, the necessary and sufficient conditions that a given p(r) be /V-representable are actually given by Equation 4.5 above. Nevertheless, question remains Can the single-particle density contain all information about a many-electron system, at least in its ground state An affirmative answer to this question can be given from Kato s cusp condition for a nuclear site in the ground state of any atom, molecule, or solid, viz.,... 

So, STOs give "better" overall energies and properties that depend on the shape of the wavefunction near the nuclei (e.g., Fermi contact ESR hyperfine constants) but they are more difficult to use (two-electron integrals are more difficult to evaluate especially the 4-center variety which have to be integrated numerically). GTOs on the other hand are easier to use (more easily integrable) but improperly describe the wavefunction near the nuclear centers because of the so-called cusp condition (they have zero slope at R = 0, whereas Is STOs have non-zero slopes there). 

This important property has been proven by Ayers in two steps. The fact that nuclear positions, Ra, and the atomic numbers of the nuclei, Za, can be determined from the cusp conditions... 

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. 

For He 12-figure accuracy was reported [36, 16]. This is surprising since this ansatz neither fulfills the nuclear cusp nor the correlation cusp conditions. Although it is not yet fully understood why this works, some preliminary comments can be made. 

The functional forms of the one- and two-electrons terms x and u are chosen such that they model the nuclear-electron and electron-electron cusp conditions, respectively, and the parameters inherent in these functions are variationally optimized by the QMC procedure. 

These are maxima, minima, and saddle points. If we start from an arbitrary point and follow the direction of Vp, we end up at a maximum of p. Its position may correspond to any of the nuclei or to a non-nuclear concentration distribution (Fig. 11.2). Formally, positions of the nuclei are not the stationary points because Vp has a discontinuity here connected to the cusp condition (see Chapter 10, p. 585), but the largest maxima correspond to the positions of the nuclei. Maxima may appear not only at the positions of the nuclei, but also elsewhere (nonnuclear attractors, (Fig. 11.2a). The compact set of starting points which converge in this way... 

Cusp condition. In a molecule, the density has cusps at each nuclear position, These cusps satisfy the relation... 

Strictly speaking the nuclear attractors do not represent critical points, because of the cusp condition (Kato 1957). 

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. 

---