Nuclear cusp

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... 

Approximate linear dependence of AO-based sets is always a numerical problem, especially in 3D extended systems. Slater functions are no exceptions. We studied and recommended the use of mixed Slater/plane-wave (AO-PW) basis sets [15]. It offers a good compromise of local accuracy (nuclear cusps can be correctly described), global flexibility (nodes in /ik) outside primitive unit cell can be correct) and reduced PW expansion lengths. It seems also beneficial for GW calculations that need low-lying excited bands (not available with AO bases), yet limited numbers of PWs. Computationally the AOs and PWs mix perfectly mixed AO-PW matrix elements are even easier to calculate than those involving AO-AO combinations. 

Fig. 7. Local on-top exchange-correlation hole at full coupling strength divided by density as a function of r in the He atom. The nuclear cusp produces greater LSD error for r-0...

As mentioned in the text, the local maxima in p at the positions of nuclei are not true (3, — 3) critical points because the gradient vector of the charge density is discontinuous at the nuclear cusp that is present in both the state function and the density (Kato 1957). However, there always exists a function homeomorphic to p(r X) which coincides with p almost everywhere and for which the nuclear positions are (3, — 3) critical points. In this sense, the nuclear positions behave topologically as do (3, — 3) critical points in the charge distribution. 

The fact that po identifies the molecular Hamiltonian should not come as a surprise. Indeed, the nuclear cusps [116] of the electron density in an atom, molecule or solid, in the neighborhood of the atomic nuclei, necessary to avoid divergences in // P of the Schrodinger equation for r— Ra, i.e., r,s = Ir — Rj = lral — 0,... 

In principle, a finite GTO basis can describe neither the firee-space tail of an electronic orbital nor the nuclear cusp correctly. Because there are relatively few physically or chemically interesting properties that depend on detailed behavior of orbitals in an arbitrarily small neighborhood of a nucleus or at arbitrarily large distances from all the nuclei, the intrinsic deficiencies of a GTO basis have not proved to be a major drawback in practice. For most observables, a rich GTO basis obviates the formal limitations. More discxxssion of the long-ranged behavior issue is given below in connection with calculated work functions. 

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. 

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. 

If accurate solutions for an atom are desired, they can be obtained to any desired accuracy in practice by expanding the core basis functions in a sufficiently large number of Gaussians to ensure their correct behavior. Furthermore, properties related to the behavior of the wavefunction near nuclei can often be predicted correctly, even without an accurately cusped wavefunction [461]. In most molecular applications the asymptotic behavior of the density far from the nuclei is considered much more important than the nuclear cusp [458). The molecular wavefunction for a bound state must fall off exponentially with distance, whenever the Hamiltonian contains Goulomb electrostatic interaction between particles. However, even though an STFs basis would, in principle, be capable of providing such a correct exponential decay, this occurs in practice only when the smallest exponent in the basis set is Cmin = where Imi is the first ionization potential. Such a restriction on... 

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. 

One strategy decomposes the orbitals into sums of one-center contributions in local spherical coordinate systems, including one around each nucleus. Each local contribution is not just one atomic-like function (see Section 3.2) but a partial-wave sum analogous to equation (8). Equation (3) becomes a system of coupled ODE for the local radial functions, where the coupling arises from the off-center (nonspher-ical) contributions. Accurate numerical solution is possible because each nuclear cusp can be represented in the local coordinate system, but representing the off-center contributions numerically is not easy. 

Certain physical properties, such as NMR shielding tensor calculations directly involve the nuclear cusp and correct treatment of radial nodes, which indicates that basis sets such as Coulomb Sturmians are better suited to their evaluation than gaussians [4,16,33]. 

Some tests show that Slater type orbitals (STO) or B-functions (BTO) are less adequate basis functions that Coulomb Sturmians, because only the Sturmians possess the correct nuclear cusp and radial behavior. 

For NMR work, it is essential to use a basis set which comprises orbitals with the correct nuclear cusp behavior. This implies a non-zero value of the function at the... 

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 ... 

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