Correlation energy variational-perturbation calculations

In ab initio methods (which, by definiton, should not contain empirical parameters), the dynamic correlation energy must be recovered by a true extension of the (single configuration or small Cl) model. This can be done by using a very large basis of configurations, but there are more economical methods based on many-body perturbation theory which allow one to circumvent the expensive (and often impracticable) large variational Cl calculation. Due to their importance in calculations of polyene radical ion excited states, these will be briefly described in Section 4. 

In this section we shall discuss an approach which is neither variational nor perturba-tional. This approach also has its origin in nuclear physics and was introduced to quantum chemistry by Sinanoglu47, It is based on a cluster expansion of the wave function. A systematic method for the calculation of cluster expansion components of the exact wave function was developed by C ek48 The characteristic feature of this approach is the expansion of the wave function as a linear combination of Slater determinants. Formally, this expansion is similar to the ordinary Cl expansion. The cluster expansion, however, gives us not only the physical insight of the correlation energy but it also shows the connections between the variational approaches (Cl) and the perturbational approaches (e.g. MB-RSPT). 

Only the computationally cheapest post-HF method can be currently applied on zeolites. Computationally the fastest and most popular post-HF method is perturbation theory considering up to the second order terms (MP2 method, using a Moller-Plesset formulation for the correlation energy).[8] This method is not variational and typically it overestimates the effect of the electron correlation. When the resolution of identity (RI) approximation ] is used the RI-MP2 method can be used for calculations on systems consisting of more than hundred atoms. 

The development of correlation methods which are both accurate and economical has been, and continues to be, one of the thorniest problems in modern quantum chemistry. There are a number of traditional approaches to the correlation problem. The correlation energy represents a small fraction of the total energy (usually less than one percent) and so Moller and Plesset proposed [6] that it be calculated via a perturbation technique. Another prominent class of correlation methods is configuration interaction (Cl) methods, which involve the variational addition to the wavefunction of substituted configurations, obtained by replacing occupied orbitals in the HF determinant with unoccupied (virtual) orbitals, i.e. 

The second-order perturbation calculations following the CAS-SCF treatments previously described are intended to supply the deficiency of correlation, if any, included in the latter. To compute the energy corrections to be added to the variational zero-order values obtained by diagonalizing the active-space Hamiltonian matrices (see tables 3a and 3b of section 2.2), we have performed two different kinds of multireference perturbation Cl treatments, both of them being based on improvements of the general CIPSI algorithm (26] recently developed by the Ferrara-Pisa group[27,28,29,30]. 

The last important contribution to intermolecular energies that will be mentioned here, the dispersion energy (dEnis). is not accessible in H. F. calculations. In our simplified picture of second-order effects in the perturbation theory (Fig. 2), d mS was obtained by correlated double excitations in both subsystems A and B, for which a variational wave function consisting of a single Slater determinant cannot account. An empirical estimate of the dispersion energy in Li+...OH2 based upon the well-known London formula (see e.g. 107)) gave a... 

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