Pair correlation effects

In the case of molten salts, no obvious model based on statistical mechanics is available because the absence of solvent results in very strong pair correlation effects. It will be shown that the fundamental properties of these liquids can be described by quasi-chemical models or, alternatively, by computer simulation of molecular dynamics (MD). 

J. Y. Hsu, C. H. Lin, and C. Y. Cheng, Pair-correlation effect and virial theorem in the self-consistent density-functional theory. Phys. Rev. A 71, 052502 (2005). 

It is clear that an enlargement of the active subspace in NiH, which tries to account for the 3d pair correlation effects, will run into balance problems similar to those experienced in FH. The conclusion seems rather clear structure-dependent dynamical correlation effects in systems with high electron density cannot be accounted for in an MCSCF treatment in a balanced way. Large Cl or MBPT treatments then become necessary and the calculations have to include a large fraction of the total correlation energy, in order to give reliable results for relative energies. 

The physical interpretation of terms arising in C Mq Ci in terms of orbital relaxation and electron pair correlation effects has been carried out by several workers. To give some feeling for the physical content of the terms in (P2)ij we... 

Whereas the paramagnetic shift of the nuclear magnetic resonance frequency for a given applied field is related to the strength of the local hyperfine field at the nuclear site, induced by the electronic moments, the nuclear spin-lattice relaxation rate yields information about the low-frequency spectrum of thermally induced spin fluctuations. The influence of pair-correlation effects on the NMR relaxation in paramagnets was analysed experimentally and theoretically by... 

There are three important types of pair correlation effects that should always be considered, namely left-right correlation, angular (up-down) correlation, and in-out correlation. 

By including the most important left-right, angular, and in-out pair correlation effects the Li2 bond dissociation energy (De = 1.05 eV) is reproduced with a small error of 6% while at the HF level (De = 0.17 eV) the error is larger than 80%. Mostly it is sufficient to consider just the correlation effects of valence electron pairs. Inner-shell correlation effects can be ignored (frozen core description). However, for larger atoms the core is polarized and inner-shell correlation becomes important. 

However, when this is done the HF orbitals are no longer the best orbitals for the new physical situation, with (partially) correlated electron pairs. Accordingly, the occupied orbitals should be reoptimized, which corresponds to a new mixing with the virtual (unoccupied) orbitals. This is done in a MCSCF method, but not in methods such as MP that describe Just dynamic electron correlation. Instead, single (S) excitations f are mixed (via D excitations) into the wavefunction these describe orbital relaxation effects, i.e., the orbitals are partially readjusted to accommodate correlated electron pairs. S excitations cannot compensate for an orbital reoptimization within a MCSCF calculation, however they represent an useful orbital relaxation correction, which compared to pair correlation effects is considerably smaller. 

In the case of the third-order correction, again only D excitations are included, which means that the third-order energy covers just pair correlation effects. However due to the fact that matrix elements Vst are included in the correction formula (equation 26), where , and , represent excitations D and D, the third-order correction includes a coupling between different D excitations, which leads to a correction of pair correlation effects overestimated at the second-order level. 

Mpller and Plesset were the first to derive the formula for the second-order correlation energy. Calculation of the MP2 correlation energy requires just O(M ) operations and, therefore, MP2 calculations represent one of the cheapest ways of getting correlation corrections. As shown before, D excitations de.scribe pair correlation effects and, therefore, the MP2 correction covers the largest part of the correlation energy defined... 

Analysing S, D, T, Q, P, and H contributions to the low-order MP correlation energies, Cremer and He came to the conclusion that differences in their contributions reflect the fact that the electronic systems of class A and B basically differ with regard to their electron distribution. Class A covers those molecules for which bond electron and lone pairs are well separated and distributed over the whole space of the molecule. For example, in BH, E" ", core electron pair, bonding electron pair and lone pair are localized in different parts of the molecule. The same is true in the case of alkanes, boranes, Li or Be compounds, etc. Because the electron pairs of class A systems are well separated, the importance of three-electron correlations and couplings between the correlation modes of the various election pairs is moderate and the molecular correlation energy is dominated by pair correlation effects. 

T and Q correlation effects can be exaggerated at MP4 for the same reason pair correlation effects are exaggerated at MP2. MP5 introduces the coupling between S, D, T, and... 

Changes comprise a n -electron transfer from C to S, transfer of (T-electrons from outer valence to inner valence space at C and vice versa at S, a transfer of cr-electronic charge from S to C and depopulation (population) of the lone-pair region at S (C). These changes lead to a decrease of the CS bond polarity and decreased atomic charges relative to MP2 (Table 9). Pair correlation effects included at MP2 are reduced by the coupling of pair correlations covered at MP3,... 

It is clear from this equation, that the explicitly-correlated terms do only describe pair-correlation effects. This means that only for pair correlation contributions, we can expect an improved basis set convergence. As pair correlation covers the greatest part of the correlation energy, this seems sufficient for the time being, see also Section 3.6. In the following, we will focus on the coupled-cluster singles and doubles (CCSD) method and its F12 variant CCSD-F12, i.e. f = fi + fz -f fy-The usual projection technique is employed to obtain the energy expression (projection on o))... 

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