Fourth-order energy

By a similar although more elaborate process, the third and fourth order energy corrections can be derived. For further details, consult the references. 

AT is a constant, fi and v denote compound indices representing a maximum of four actual indices. The intermediates and gp differ only in that in the latter there is an additional denominator factor. A number of calculations of the quadruple-excitation component of the fourth-order energy have been reported.131-140... 

The triple-excitation fourth-order energy, in contrast to the quadruple-excitation component, arises from connected wave function diagrams. The algorithm required to evaluate this energy component is considerably less tractable than that for the quadruple-excitation energy, depending on 7, where n is the number of basis functions. The triple-excitation diagrams can be written in terms of the intermediates. 

D. J. Baker, D. Moncrieff, V. R. Saunders, and S. Wilson, Comput. Phys. Commun., 62, 25 (1991). Diagrammatic Many-Body Perturbation Expansion for Atoms and Molecules. VII. Experiments in Vector and Parallel Processing for Fourth-Order Energy Terms Involving Triply Excited Intermediate States. 

The formula for the second-order correction to the wave function (eq, (4.39)) contains products of the type ( y H >()(, H o)- The o is the HF determinant and the bracket can only be non-zero if, is a doubly excited determinant. This means that the first bracket only can be non-zero if is either a singly, doubly, triply or quadruply excited determinant (H is a two-electron operator). The second-order wave function allows calculation of the fourth- and fifth-order energies, these terms therefore have contributions from determinants which are singly, doubly, triply or excited. The computational cost of the fourth-order energy without th contribution from the triply excited determinants, MP4(SDQ), increases as M , while the triples contribution increases as M . MP4 is still a computationally feasible model for many molecular. systems, requiring a time similar to CTSD. Tn... 

Perturbative approaches to the electron correlation problem have proved to be successful even when calculating second-order corrections only but finer results require fourth order energy corrections, as we will see later in this text. The reliability of a perturbation expansion greatly depends on the partitioning of the exact Hamiltonian H = H0 + W to an unperturbed part H0 and a perturbation W. Good quality approximations are to be expected if the N-particle operator H0 is chosen as the sum of equivalent one-particle Fock operators... 

The appropriate SCEP formulas for the fourth-order energy, excluding the triples contribution, have also been given by Saebo and Pulay [31]. Although there are some new aspects as far as the singles term is concerned, much of the treatment is very similar to what we have already presented. For example, the equation for the second-order pair coefficient matrix, C , is the same as Eq. (26) except that one must substitute... 

The extension of the SCEP formalism to account for the computationally demanding triples contribution to the fourth-order energy has not previously been addressed either on its own or in connection with the local correlation treatment. However, it is not too difficult to do so since an expression (albeit messy) for the key tensor quantity, (w,jk)pqr, has already been given by Krishnan et al. [37], In analogy with Eq. (21) we write the triples coefficients in the form... 

Finally, the local fourth-order energy due to triple excitations is... 

Some of the examples of the fourth-order energy diagrams in the perturbation theory of nuclear and electronic motion are displayed in Figure 12. The three diagrams in the top row are associated with excited states which are only doubly excited with respect... 

Figure 12. Some examples of fourth-order energy diagrams in the perturbation theory of nuclei and electrons.

E. Computational Scheme for the Fourth-Order Energy Terms. . 310... 

The question arises whether we need T4 amplitudes in order to generate all fifth-order diagrams. The answer depends on the approach. If we solve the CCSDT equations and obtain converged results, then we have the full fourth-order energy, but not the fifth-order energy, since we miss 168 diagrams, denoted here as Ef1 and EfQ. We do include some other diagrams that are formally of quadruple excitation type that are termed as... 

The analysis of the several different CC approaches in terms of the fifth-order energy contributions points out that within an n6 dependent scheme, i.e., LCCD to CCSD in Table I, the CCSD is much preferred since it accounts for nearly one-third of all terms and avoids potential singularities in LCCD.42 It may also be observed that it pays off to include, even partially, the triple contribution, as was done in the CCSDT-1 method.10 In this model the number of terms is nearly doubled as compared to CCSD, and, of course, this method is correct through the fourth-order energy and the second-order wave function. Also, the connected T contributions are numerically important.11-34... 

This kind of factorization can be performed for all diagrams from the fourth-order energy and up. It is obvious that these stepwise-type calculations, presented in Fig. 25 are, in fact, exercised in the coupled-cluster... 

Now we see the consequences of (size)-extensivity for the first time. The last term, which depends on <2), arises from the second term on the right in Eq. [29c] after multiplying on the left by and using the fourth-order energy formula. This term plays a critical role in distinguishing MBPT from Cl. If we continued approximating the CID coefficients by higher order perturbation theory, we would have... 

The fourth order energy component takes the form... 

When the single determinant many-electron functions are constructed from canonical Hartree-Fock orbitals, the excited functions, and , are doubly excited with respect to the reference function . The second term in the third order energy expression cancels diagonal components for which p = v in the first term. The principal term in the fourth order energy expression has the form... 

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... 

Table 6 Fourth order energy components involving singly excited intermediate states...

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