The Higher-Order Correlation Energy

The higher-order contributions to the correlation energy [such as CCSD(T)-MP2] are more than an order of magnitude smaller than their second-order counterparts. However, the basis set convergence to the CCSD(T)-R12 limit does not follow the simple linear behavior found for the second-order correlation energy. This is a consequence of the interference effect described in Eq. (2.2). The full Cl or CCSD(T) basis set truncation error is attenuated by the interference factor (Fig. 4.9). The CBS correction to the higher-order components of the correlation energy is thus the difference between the left-hand sides of Eqs. (2.2) and 

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The Dunning cc-pVnZ basis sets can be used with our PNO extrapolations to form a potent new combination. We shall consider the SCF energy first, then the MP2 correlation energy, and finally higher-order correlation energy through CCSD(T). 

The accurate description of correlation effects requires the inclusion of functions of higher symmetry than those required for the matrix Hartree-Fock model. The most important of these functions for the F anion are functions of d-type. In this section, the convergence of the total energy through second order and the second order correlation energy component for a systematic sequence of even-tempered basis sets of Gaussian functions of s-, p-and d-type is investigated. 

Complete Basis Set Methods Petersson et al.61-63 developed a series of methods, referred to as complete basis set (CBS) methods, for the evaluation of accurate energies of molecular systems. The central idea in the CBS methods is an extrapolation procedure to determine the projected second-order (MP2) energy in the limit of a complete basis set. This extrapolation is performed pair by pair for all the valence electrons and is based on the asymptotic convergence properties of pair correlation energies for two-electron systems in a natural orbital expansion. As in G2 theory, the higher order correlation contributions are evaluated by a sequence of calculations with a variety of basis sets. 

QCISD(T) and MP2 energies is taken as an estimate of the higher-order correlation correlation ... 

Higher Order Correlation Energy Components. - 2.5.1 Fourth order energy components. - The general fourth order term for the correlation energy expansion of the closed-shell system described in zero order by a single determinant is... 

For all higher-order correlation energies (of closed-shell systems), moreover, we can follow similar lines and evaluate the vacuum expectation values EI"I = (0c HX2 " 0c). This results in the standard Moeller-Plesset expressions for the energy corrections. 

Scaling of the second-order correlation energy has been used to estimate the effects of basis set extension. Consider a calculation performed using a basis set designated Sa- The relation between the modified second-order correlation component and the higher-order approximation to the total correlation energy then takes the form... 

Although the EOMCCSD approach is known to provide an accurate description of excited states dominated by one-electron transitions, such as the r —k transition in cis-7HQ and its complexes, there have been cases of similar states reported in the literamre, where the EOMCCSD level has not been sufficient to obtain high-quality results [84,85]. Moreover, the small energy differences defining the environment-induced shifts Aco j may be sensitive to the higher-order correlation effects neglected in the EOMCCSD calculations. For these reasons, we also examined the effect of triple excitations on the and AcOn n values... 

The thennodynamic properties of a fluid can be calculated from the two-, tln-ee- and higher-order correlation fiinctions. Fortunately, only the two-body correlation fiinctions are required for systems with pairwise additive potentials, which means that for such systems we need only a theory at the level of the two-particle correlations. The average value of the total energy... 

MP2 correlation energies (Table 4.6), and the higher-order contributions to the correlation energy (Table 4.7), we can now combine these components to obtain total electronic energies. There are many plausible combinations of basis sets and extrapolation procedures that must ultimately be explored. Efficient methods should use smaller basis sets for the CCSD(T) component than for the SCF and MP2 ones. The use of intermediate basis sets for the MP4(SDQ) component should also be explored, since we found this effective for the CBS-QB3 model (Table 4.2). 

These higher-order correlation functions play a large role in determining many physical properties of polyatomic systems. For example, the vibrational relaxation can, in some cases, be expressed in terms of the rotational kinetic energy autocorrelation function.27... 

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