Cusp condition

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

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.,... 

Secondly, information is obtained on the nature of the nuclei in the molecule from the cusp condition [11]. Thirdly, the Hohenberg-Kohn theorem points out that, besides determining the number of electrons, the density also determines the external potential that is present in the molecular Hamiltonian [15]. Once the number of electrons is known from Equation 16.1 and the external potential is determined by the electron density, the Hamiltonian is completely determined. Once the electronic Hamiltonian is determined, one can solve Schrodinger s equation for the wave function, subsequently determining all observable properties of the system. In fact, one can replace the whole set of molecular descriptors by the electron density, because, according to quantum mechanics, all information offered by these descriptors is also available from the electron density. 

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... 

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]. 

While any approximate wave function O that consists of Slater determinants violates this cusp condition, the product (1 + (1 /2)ri2)4> may satisfy it exactly. Inspired by this knowledge and by the fact that a reference wave function (O0) still dominates in the exact correlated wave function, Kutzelnigg has suggested... 

Another critical advance responsible for the success of MP2-R12 is the introduction of nonlinear correlation functions, in particular, the Slater-type correlation function, 1— exp(—yr 2), of Ten-no [29]. It is asymptotically linear in ru near the coalescence point and, hence, satisfies the cusp condition in the leading order. Unlike the linear correlation function, the Slater-type correlation function... 

An even more radical yet effective approximation to the R12 method was proposed by Ten-no [28,43], in which the coefficients multiplying the correlation function were held fixed at the values implied by the first-order cusp condition and hence were not to be determined iteratively or noniteratively. Several variants of the CCSD(T)-R12 methods were developed on the basis of this promising approximation by Adler et al. [68], Tew et al. [69], Bokhan et al. [70], and Torheyden et al. [66]. 

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

Bokhan, D., Ten-no, S., Noga, J. Implementation of the CCSD(T)-F12 method using cusp conditions. Phys. Chem. Chem. Phys. 2008, 10, 3320-6. 

If the electron-nucleus cusp condition for the electron density [171,172]... 

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). 

The highest-order singularity in this system cannot satisfy the winged cusp condition F = Fx = Fxx = Fz = Fxz = 0. For one thing the system of equations only has four quantities x, ires, gad, and tn, whereas the winged cusp... 

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. 

---