Wavefunction cusps

The expression for the anisotropic part of hyperfine coupling involves an integral over the spatial distribution of the unpaired electron, which is relatively easy to compute accurately even at a relatively low level of theory. The contact term, however, includes a delta-function that chips out the wave function amplitude at the nucleus point. The latter is quite difficult to compute both because standard Gaussian basis sets do not reproduce the wavefunction cusp at the nucleus point and because additional flexibility has to be introduced into the core part of the basis to account for the now essential core valence interaction. " ... 

The STO-3G wavefunction does not have a cusp at the nucleus. Very few molecular properties depend on the exact shape of the wavefunction at the nucleus ... 

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. 

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 differences between the single-configuration wavefunctions are more clearly illustrated by comparing their plots of the intracule function h(ri2), also shown in Fig. 1. This plot reveals the absence of an electron-electron cusp for both the closed and split-shell functions, but shows that the inclusion of exp( —yri2) causes the distribution to have a minimum at ri2=0, forming a cusp (of the correct sign) at that point. This feature will be important for the description of phenomena that depend upon the coincidence probability. 

The results of the previous section suggest that a good hybrid method might treat antiparallel spin using a GGA, while using a wavefunction treatment for parallel-spin [54]. Since the parallel-spin hole has no cusp and is of greater spatial extent, this contribution should be more accessible to a wavefunction method beginning from one-particle wave functions, such as Cl. 

From the last column of the table, we see that the ratio of the parallel-spin to the total correlation energy is remarkably independent of the size of the basis set. Contrary to expectation, the parallel-spin correlation contribution appears to be about as difficult to account for within a finite basis-set approach as the antiparallel-spin correlation. Our investigation does not provide a careful study of the basis-set saturation behavior in MP2 calculations, such as is given in Refs. [74,72,75,33]. However, our results show that, with small- and moderate-sized basis sets which are sufficiently flexible for most purposes and computationally tractable in calculations on larger systems, there is no evidence that the parallel-spin correlation contribution converges more rapidly than the antiparallel-spin contribution. A plausible explanation for this effect is that, for small interelectronic separations, the wavefunction becomes a function of the separation, which is difficult to represent in a finite basis-set approach for either spin channel. The cusp condition of Eq. (19) is a noticeable manifestation of this dependence, but does not imply that the antiparallel-spin channel is more difficult to describe with a moderate-sized basis set than the parallel channel. In fact, in the parallel correlation hole, there is a higher-order cusp condition, relating the second and third derivatives with respect to u [76]. 

Pack, R.T., Beyers Brown, W. Cusp conditions for molecular wavefunctions. J. Chem. Phys. 1966, 45, 556-9. 

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). 

Although at the nuclei there is a singularity in the wavefunction not reflected by either STO s or GTO s, but this is a different problem. This cusp problem is the reason for the sometimes poor values of calculated properties near the nucleus. 

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