Second-order correlation energy for

Distributed Gaussian basis sets in correlation energy studies the second order correlation energy for the ground state of the hydrogen molecule... 

The results of the distributed basis set study of the second-order correlation energy for the nitrogen molecule(83) is compared with the second-order correlation energies obtained by employing other recently reported basis sets for this system in Figure 1. 

Every term in this expansion (as indicated by the integer at the end of the line) represents a Feyman-Goldstone diagram and could be drawn also graphically. These are the (three) well-known diagrams of the second-order correlation energy for closed-shell systems as found at different places in the literature. If we assume a Hartree-Fock (one-particle) spectrum, instead,... 

Now if we look at the perturbation theory description of this system using wavefunctions which are localized on the individual atoms we find that the second-order correlation energy for two helium atoms is just twice that of a single helitun atom and in general... 

Table 5.8. Second-Order Correlation Energy for Rings of H22 Atoms (11 H2 Molecules) for Various Degrees of Disorder"... 

For all intents and purposes then, we are concerned here with the CCSD (coupled cluster with all single and double substitutions [30]) correlation energy. Its convergence is excruciatingly slow Schwartz [31] showed as early as 1963 that the increments of successive angular momenta l to the second-order correlation energy of helium-like atoms converge as... 

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... 

As a first try, we have elected to follow our treatment of the SCF and second-order correlation energies described above, and employ Eq. (6.2) to provide a linear extrapolation of the cc-pVDZ and cc-pVTZ total CBS-CCSD(T) energies obtained with Eq. (2.2), including the interference correction. These total energies reproduce the CCSD(T) limits estimated by Martin [55] via an (lmax + 5)-3 extrapolation of the CCSD(T)/cc-pVDZ, TZ, QZ, 5Z, and 6Z basis sets to within 0.96 kcal/mol RMS error. The agreement with Martin s energies for a small set of chemical reactions is even better (Table 4.8). The use of the cc-pVnZ basis sets for PNO-(Zmax + 5)-3 double extrapolations is indeed promising. 

The effects of adding secondary off axis basis functions are smnmarized in Table 4. The results obtained with the largest basis set considered in Table 3, 30s ac 30s be 28s oa(ac) p. = 4] 26s oa bc) [n j = 4], are used as a reference with respect to which the results presented in Table 4 are analyzed. The addition of secondary off axis basis functions reduces the error in the matrix Hartree-Fock energy to 4 Hartree. For the second order correlation energy the addition of secondary off axis basis subsets reduces the error to 723 / Hartree, thereby achieving the target accuracy of the present study. 

The second order correlation energy component, E Ne [IVe]) calculated for the ground state of the neon atom using systematic sequences of even-tempered basis sets of Gaussian functions designed for the Ne atom and designated [2nsnp] with n = 3,4,..., 13 are also collected in Table 1. 

In Table 3 the total energy through second-order and the second-order correlation energy component for the ground state of the fluoride anion determined with a sequence of basis sets designed for the neutral F atom are presented. Values of A [p/o](F [F])[2 p and AE2 F [N]) 2 results presented in Table 3 should be compared with... 

Figure 1 Comparison of the convergence behaviour of the second order correlation energy component for the F anion with sequences of even-tempered Gaussian basis sets containing s- and p-type functions designed for the neutral F and Ne atoms with the behaviour of this component for the Ne atom. The cmrves are labelled as follows - (a) AE2 Ne [A e]) (b) AE2 F -, [iVe]) (c) AEtiF--, [F]).

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. 

Ai [p/o](A e [iVe])(spd] and i Epresent work to determine accurate correlation energies. However, it is known [45] [46] [47] that the exact second order energy for the neon atom ground state is —0.3879 Hartree. The largest basis set of s- and p- type functions considered in Table 1, therefore, recovers 37.7% of the exact second order correlation energy component whilst the largest basis set of functions with s-, p- and d-symmetries considered in Table 8 corresponds to 66.8% of the exact value. 

Thus, the numerical procedure involving the treatment of both the first- and the second-order correlation energies in a complete finite basis demonstrates its high accuracy and flexibility in MBPT calculations. The MBPT version, described in Chapters 3 and 29, provides very accurate results, at least in the range of Z = 10—26. Accounting for the correlation energy due to the residual electrostatic interaction in the second-order of... 

The results of some preliminary calculations of the second order correlation energy component for the water molecule, a prototype non-linear polyatomic system, are displayed in Table 5. AE2 is again the difference between the correlation energy estimate obtained with a given basis set and that obtained with the 0 sp H s set. It can be seen that, whereas the 0 sp H sp OH sp HH sp set gave a lower energy than the 0 spd H.sp set in matrix Hartree-Fock calculations, the situation is reversed for the correlation energy studies. 

Some second order correlation energy components for the water... 

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