Triple-excitation energy

Table V CPU times on the CRAY-1 and the IBM 360/195 computer required to execute the inner loop in evaluation of triple excitation energy component.

The self-consistent-field energies are given in hartree the triple-excitation energy, Ei t, the quadruple-excitation energy,... 

The innermost loop in the program written to evaluate the triple-excitation energy component described in the previous subsection has the form... 

It can be seen that the linked triple excitation energy is in fact quite large and certainly chemically significant. There is clearly a danger in any technique which attempts to make a partial evaluation of higher order terms in the perturbation series.181 154... 

Rendell et al. compared three previously reported algorithms to the fourth-order triple excitation energy component in MBPT." The authors investigated the implementation of these algorithms on current Intel distributed-memory parallel computers. The algorithms had been developed for shared-... 

The weight is the sum of coefficients at the given excitation level, eq. (4.2). The Cl method determines the coefficients from the variational principle, thus Table 4.2 shows that the doubly excited determinants are by far the most important in terms of energy. The singly excited determinants are the second most important, then follow the quadruples and triples. Excitations higher than 4 make only very small contributions, although there are actually many more of these highly excited determinants than the triples and quadruples, as illustrated in Table 4,1. 

We present in Table 3 the excitation energies needed to produce a valence state with all orbitals singly occupied. The largest excitation energy is for Ac. The price to pay for forming a triple bond between two Ac atoms is 2.28 eV for Th, only 1.28 eV is needed, which can then, in principle, form a quadruple bond. Note that in these two cases only 7s and 6d orbitals are involved. For Pa, 1.67eV is needed, which results in the possibility of a quintuple bond. The uranium case was already described above where we saw that, despite six unpaired atomic orbitals, only a quintuple bond is formed with an effective bond order that is closer to four than five. 

To improve on the CCSD description, we go to the next level of coupled-cluster theory, including corrections from triple excitations -see the third row of Table 1.4, where we have listed the triples corrections to the energies as obtained at the CCSD(T) level. The triples corrections to the molecular and atomic energies are almost two orders of magnitude smaller than the singles and doubles corrections. However, for the triples, there is less cancellation between the corrections to the molecule and its atoms than for the doubles. The total triples correction to the AE is therefore only one order of magnitude smaller than the singles and doubles corrections. 

A third class of compound methods are the extrapolation-based procedures due to Martin [5], which attempt to approximate infinite-basis-set URCCSD(T) calculations. In the Wl method [16] calculations are performed at the URCCSD and URCCSD(T) levels of theory with basis sets of systematically increasing size. Separate extrapolations are then performed to determine the SCF, URCCSD valence-correlation, and triple-excitation components of the total atomization energy at... 

The Wlc total atomization energy at 0 K of aniline, 1468.7 kcal/mol, is in satisfying agreement with the value obtained from heats of formation in the NIST WebBook 39), 1467.7 0.7 kcal/mol. (Most of the uncertainty derives from the heat of vaporization of graphite.) The various contributions to this result are (in kcal/mol) SCF limit 1144.4, valence CCSD correlation energy limit 359.0, connected triple excitations 31.7, inner shell correlation 7.6, scalar relativistic effects -1.2, atomic spin-orbit coupling -0.5 kcal/mol. Extrapolations account for 0.6, 12.1, and 2.5 kcal/mol, respectively, out of the three first contributions. 

The relative energies of the three protonated species are well reproduced by all methods from the Gn family. This can largely be explained by (a) the fact that all these methods involved CCSD(T) or QCISD(T) steps (and apparently triple excitations are quite important here) (b) the relatively rapid basis set convergence noted above, which means that it is not really an issue that the CCSD(T) and QCISD(T) steps are carried out in relatively small basis sets. CBS-QB3 likewise reproduces the relative energetics quite well. 

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