Electron-nuclear cusps

But the proof is deceptively simple [31]. Because the shape function is proportional to the electron density, it inherits the characteristic electron-nuclear coalescence cusps at the positions of the atomic nuclei [32,33]. The location of those cusps determines the positions of the nuclei, R the steepness of the cusps determines the atomic charges, Za. So the shape function determines the external potential for any molecular system [31]. 

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... 

Fig. 1. The Hirshfeld electron densities (Hh) of bonded hydrogen atoms obtained from the molecular density (H2). The free hydrogen densities (H°) and the resulting electron density of the promolecule (H2) are also shown for comparison. The density values and inter-nuclear distances are in a.u. The zero cusp at nuclear positions is the artifact of the Gaussian basis set used in DFT calculations.

To hold the electron-nuclear Kato cusp, the nuclear position is invariant off. If/is a uniform scaling/i, the latter equation takes the form... 

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. 

A completely different type of property is for example spin-spin coupling constants, which contain interactions of electronic and nuclear spins. One of the operators is a delta function (Fermi-Contact, eq. (10.78)), which measures the quality of the wave function at a single point, the nuclear position. Since Gaussian functions have an incorrect behaviour at the nucleus (zero derivative compared with the cusp displayed by an exponential function), this requires addition of a number of very tight functions (large exponents) in order to predict coupling constants accurately. ... 

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). 

Moreover, because the nuclei are effectively point charges, it should be obvious that their positions correspond to local maxima in tlie election density (and these maxima are also cusps), so the only issue left to completely specify the Hamiltonian is die assignment of nuclear atomic numbers. It can be shown diat diis information too is available from the density, since for each nucleus A located at an electron density maximum Fa... 

A complicating factor is tlrat each spitr density matrix element is multiplied by the corresponding basis function overlap at tire nuclear positions. The orbitals having maximal amplitude at the nuclear positions are tire core s orbitals, which are usually described with less flexibility than valence orbitals itr typical electronic structure calculations. Moreover, actual atomic s orbitals are characterized by a cusp at tire nucleus, a feature accurately modeled by STOs, but only approximated by the more commonly used GTOs. As a result, tlrere are basis sets in the literature tlrat systematically improve tire description of the core orbitals in order to improve prediction of h.f.s., e.g. IGLO-III (Eriksson et al. 1994) and EPR-III (Barone 1995). 

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. 

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,... 

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. 

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. 

Finally, we examine the cusp values for the helium atom. Table 3 shows the nuclear-electron and electron-electron cusp values at the distance r = 1.0 a.u., which was explained below Eq. (24). Both nuclear-electron and electron-electron cusp values approach the exact values of —2.0 and 0.5, respectively, as the order n of the FC calculation increases. At = 27, the cusp values are correct to 22 digits, which is about a half of the correct digits of the variational energy, 41 digits, given in Table 1. This result is natural from a theoretical point of view. 

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