Cusp correction

Noga, J., Kutzelnigg, W., Klopper, W. CC-R12, a correlation cusp corrected coupled-cluster method with a pilot application to the Be2 potential curve. Chem. Phys. Lett. 1992, 199, 497-504. 

The generalization of MP2-R12 from two-electron to many-electron states is straightforward, at least as long as we use a single coefficient c for the entire cusp correction . Instead of (47) we now have... 

If one does not care to describe the correlation cusp correctly, but uses a Cl-like expansion of the wave function, the just-mentioned singularity would not show up, so it is not surprising that so far it has not plagued any numerical calculation. In particular, everything remains regular in the framework of Hartree-Fock or MC-SCF theory. 

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 hydrogen, trial wavefunctions identical to those utilized by Goodisman were chosen. These, most conveniently expressed in terms of mixed inter-particle-confocal elliptical coordinates, are cusp corrected. They are composed of terms of the type ... 

The factor g is the cusp-correcting correlation factor which imparts the proper behavior to the wavefunction near the singularity. The ranges for the original coordinates used are, respectively,... 

Keywords Cusp correction Electron correlation Quantum Monte Carlo ... 

The behavior of the local energy around the nucleus is more severe for the ZORA-QMC results, because relativistic effects cause the contraction of s and p orbitals and increase the electron density near the nucleus. Thus, an adequate relativistic cusp correction scheme is necessary to treat heavier elements where the relativistic effects are important. Recently, we proposed a cusp correction scheme for the ZORA-VMC formalism [15]. Strictly speaking, the term cusp correction may not be adequate for our orbital correction scheme because relativistic wave function does not have a cusp but divergences at electron-nucleus collision. However, we call our relativistic orbital correction scheme as a relativistic cusp correction scheme because the term cusp correction makes clearer that our scheme is a relativistic extension of the NR cusp correction scheme. 

Let us consider the cusp correction scheme for the Jastrow-Slater wave function. This type of wave function allows us to use a MO correction scheme. The s type component of the Is MO inside a given radius is replaced with the correction function,... 

In Fig. 10.1, the Is orbitals of the Ne atom are plotted with and without the cusp correction. GTO and STO orbitals are used in this calculation. The condition (0.08 30), where Tc = 0.08a.u. and rmn is 30% of r, is used. Note that the corrected orbitals are not normalized. The original and corrected orbitals have similar shapes except in the region of r 0. This observation in the relativistic case is the same as that seen for the case of NR cusp correction scheme of Ma et al. in the GTO case. 

Fig. 10.1 Relativistic Is orbital of Ne atom with and without the cusp correction. Both GTO and STO orbitals are obtained from the ZORA-HF calculation...

We have developed a new relativistic treatment in the QMC technique using the ZORA Hamiltonian. We derived a novel relativistic local energy using the ZORA Hamiltonian and tested its availability in the VMC calculation. In addition, we proposed a relativistic electron-nucleus cusp correction scheme for the relativistic ZORA-QMC method. The correction scheme was a relativistic extension of the MO correction method where the 1 s MO was replaced by a correction function satisfying the cusp condition. The cusp condition for the ZORA wave function in electron-nucleus collisions was derived by the expansion of the ZORA local energy and the condition required the weak divergence of the orbital itself. The proposed relativistic correction function is the same as the NR correction function of Ma et al. in the NR... 

Actually, all of the above results are in contradiction to the currently conventional view [32-35] that solvent dynamical effects for electronically adiabatic ET reactions are determined by solvent dynamics in the R and P wells, and not the barrier top region. This misses the correct picture, even for fairly cusped barrier. Instead, it is the solvent dynamics occurring near the barrier top, and the associated time dependent friction, that are the crucial aspects. It could however be thought possible that, for cusped barrier adiabatic ET reactions in much more slowly relaxing solvents, the well dynamics could begin to play a significant role. However, MD simulations have now been carried out for the same ET solute in a solvent where the... 

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. 

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. 

Successful density functional approximations such as the PW91 GGA or the self-interaction correction (SIC) [57] to LSD recover [19] LSD values for the on-top hole density and cusp. The weighted density approximation (WDA) [41,42], which recovers the LSD exchange hole density but not the LSD correlation hole density [19] in the limit u -> 0, needs improvement in this respect. 

STOs have a number of features that make them attractive. The orbital has the correct exponential decay with increasing r, the angular component is hydrogenic, and the Is orbital has, as it should, a cusp at the nucleus (i.e., it is not smooth). More importantly, from a practical point of view, overlap integrals between two STOs as a function of interatomic distance are readily computed (Mulliken Rieke and Orloff 1949 Bishop 1966). Thus, in contrast to simple Huckel theory, overlap matrix elements in EHT are not assumed to be equal to the Kronecker delta, but are directly computed in every instance. 

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