Quadruple excitations

Bartlett R J and Purvis G D 1978 Many-body perturbation theory coupled-pair many-electron theory and the importance of quadruple excitations for the correlation problem int. J. Quantum Chem. 14 561-81... 

It is particularly desirable to use MCSCF or MRCI if the HF wave function yield a poor qualitative description of the system. This can be determined by examining the weight of the HF reference determinant in a single-reference Cl calculation. If the HF determinant weight is less than about 0.9, then it is a poor description of the system, indicating the need for either a multiple-reference calculation or triple and quadruple excitations in a single-reference calculation. 

Note 0 The CISD energies, which include Davidson correction for unlinked quadruple excitations, where obtained ot the MCSCF/6-31 G(d)optimized geometries and used a TC-SCF reference for closed-shell singlet states and a single configuration reference for triplet and open-shell singlet states. The SCF/6-31G(d) geometry was employed for the A state. See Ref. 55. 

Further improvement of accuracy, as well as applicability to systems exhibiting nondynamical correlation, will almost certainly require some level of treatment of connected quadruple excitations. 

Usually, SD-CI wavefiinctions recover about 90-95% of the correlation energy, inclusion of triply and quadruply substituted configurations is adequate for many purposes, and inclusion of quintuply and sextuply excited configurations is sufficient for describing most chemical reactions. Since the explosion of the number of configurations typically starts with the quadruple excitations, our first priority is the predictive deletion of deadwood from this group. 

Furthermore, ec-CCSD only considers triple and quadruple excitations appearing in the Cl wavefunction (i.e., those 3- and 4-body excitations included in Sm) and they are treated non-iteratively. (SC) CAS-SDCl uses the same subset of triples and quadruples in an iterative process, and, additionally, an approximation to the disconnected components of the external triple and quadruple excitations (i.e., those belonging to S ). These terms are labelled as ext in eqs. (20) and (21). This is, indeed, an approximation, because the exact disconnected C]C2 and terms should be written as ... 

Eq. (95). Again, as in the MMCC(2,3)/CI case, one can easily extend the above MMCC(2,3)/PT approximation to higher-order MRMBPT-corrected MMCC(mA,ms) schemes, such as the MMCC(2,4)/PT approach which describes the combined effect of selected triple and quadruple excitations introduced by the MRMBPT wave functions [78]. 

If the CISD wave functions for two identical molecules are multiplied, to give the wave function for the pair of molecules, there are terms in the resulting wave function in which both molecules are doubly excited. Since these terms represent quadruple excitations from the HF configuration, they are not included in the CISD wave function for two identical molecules at infinity. Consequently, the CISD energy for a pair of identical molecules is higher than twice the CISD energy for an individual molecule. 

Clearly, the way to remedy the lack of size consistency in a CISD calculation is to include the missing terms that contain quadruple excitations. Because these terms are products of simultaneous double excitations, their sizes can be determined from information that is available from CISD. Two methods that include quadruple excitations in this manner and, hence, are size consistent are discussed in Section 3.2.3.4. 

The CCSD calculations are similar, both in methodology and in the accuracy of the results obtained, to calculations performed with Pople s quadratic Cl method. Like CCSD calculations, QCISD calculations also explicitly include single and double excitations and the effects of quadruple excitation in QCISD are obtained from quadrature of the effects of double excitations. However, CCSD does contain terms for the effects of excitations beyond quadruples, which are absent from QCISD. 

Various approaches to overcoming the size extensivity problem have been proposed. Owing to its simplicity, one of the more popular methods is that of Langhoff and Davidson (1974), which estimates the energy associated with the missing quadruple excitations as... 

CCSDTQ Coupled cluster including single through quadruple excitations... 

In fact, for tightly localized electron pairs, the dominant excitation level is the value of k nearest "vO.OlN (i.e., for about 200 electrons the double excitations in aggregate are more important than the SCF configuration and for 400 electrons quadruple excitations should dominate). Even for molecules with only 40 electrons quadruple and higher excitations must be considered in order to reproduce excitation energies (30) or potential surfaces to an accuracy of 0.1 eV. Thus, configuration interaction calculations for very large molecules are hopeless unless perturbation theory can be used to correct for unlinked cluster effects. 

An interesting point of the coupled cluster method concerns the treatment of quadruple excitations. If the CCD method is considered, in which only the T2 operator is retained in the exponent, the amplitudes for these excitations are given as products of amplitudes for double excitations according to the term Ij. In fact is a sum of the 18 products of type (with phase factors) which can be formed... 

The above description is one way to realize that the SD-CI method is not size-consistent. Another way is to look in detail at what happens when this method is used on the composite and on the separated systems. It is clear that if the energy of A and B should be additive the corresponding wavefunction for (A+B) should be equal to the wavefunction of A times the wavefunction of B. This means that since in the calculations on the separated systems there are local double excitations on both A and B, the product wavefunction will contain certain quadruple excitations. In the SD-CI calculation on the composite system these quadruple excitations are clearly missing and this is the reason for the size-inconsistency. It is also clear that for the SD-CI method E(A)+E(B) must be lower in energy than E(A+B). 

The convergence behaviour of the MP expansion is obviously crucial to the success of the method, and numerous investigations have been carried out [9, 10]. Neither MP2 nor MP3 are entirely satisfactory. These implicitly require the first-order wave function, which involves only double excitations. MP4, if carried out completely, involves single, double, triple and quadruple excitations. It is more expensive than CISD, say, but often produces better results. MP5 and higher orders are likely to be impractically expensive. 

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