EOM-CCSDT

Watts JD, Bartlett RJ (1996) Iterative and non-iterative triple excitation corrections in coupled-cluster methods for excited electronic states The EOM-CCSDT-3 and EOM-CCSD(r) methods. Chem Phys Lett 258 581-588. 

J. D. Watts and R. J. Bartlett, Chem. Phys. Lett., 258, 581 (1996). Iterative and Noniterative Triple Excitation Corrections in Coupled-Cluster Methods for Excited Electronic States— The EOM-CCSDT-3 and EOM-CCSD(T) Methods. 

Although length of the many-body expansion of Hn,open is enormous, very few terms enter a particular approximate scheme. For example, the EOM-CCSD method needs one- and two-body components of H and certain types of the three-body H3 terms [33,34] (for further comments, see Ref. 60 cf., also, Sections 4 and 5). The situation gets complicated if we want to go beyond the EOMCCSD approximation and represent T as a sum of, for example, I), T2, and T3 components. For this reason, the only higher-than-two-body terms included in the current implementation of the EOM-CCSDT scheme are the three-body terms of the EOMCCSD method [46]. This approximation seems to work very well, even though the EOMCCSDT procedure defined in this way remains an n8 method. 

A few years ago we extended EOM-CCSD to the full EOM-CCSDT method and made some fairly large basis set ( 90 function) calculations, based upon the full triple excitation. CCSDT ground state, and the inclusion of all triple excitations in h [149]. At the same time others [150] reported a study of Hg. These authors also looked at simple potential energy curves with EOM-CCSDT and its active orbital modification, EOM-CCSDT [151,152]. Now, by virtue of their automated procedures, [153] EOM-CC can be taken to any level. Hirata has similarly done EOM-CCSDTQ [154], and Kallay EOM-CCSDTQP. 

As one would expect, if we describe ionization potentials with IP-EOM-CCSD, we will have similar behavior. We should do quite well for most principal ionizations where the eigenstate is dominated by single excitations, meaning linear combinations of i lO) determinants, with i a.y Q) playing most of the correlation and relaxation role but when the latter shake-up is dominant, then we would logically need i ayb k )Q in our space to do as well for the shake-up eigenvalues. The latter requires IP-EOM-CCSDT [156], and the complementary EA-EOM-CCSDT method [157] and higher [166]... 

Fig. 1. The diagrammatic representation of the elements of 7/jv used in the derivation of the IP-, EA-, and EE-EOM-CCSDT equations. ...

In case of EE we have one more variant, i.e., EOM-CCSDT-3 which scales as EOM-CCSDT-3. 

The first of the considered approximate models, termed EOM-CCSDT-3, is defined by the equations (9) and (10) ... 

This suggests a possibility to introduce for the EE problem the same treatment of the Tj operator as in the remaining two cases. In other words we eliminate the eomponent from the / element occurring in the R2 equations. The variant is denoted in the respective tables as EOM-CCSDT-3. In this variant the Ri and Rj equations are the same as in the EOM-CCSDT-3 method but in the R2 equation the operator is added only in the eonstruetion of the /j elements of H, see in Fig. 1, and components, respeetively. 

In the next approximate model, EOM-CCSDT-3a, we construct elements of without T2 operator. Therefore, the R and R equations are the same as in the EOM-CCSDT-3 variant but we need to modify the R2 equation which now takes the form ... 

In the last two variants, denoted as EOM-CCSDT-3b and EOM-CCSDT-3c, we allow for the operator in the R equation. 

Finally, we have considered one more group of approximate variants (denoted as EOM-CCSDT-3c) which is very close to the previous one. In these methods we add within the 7 3 equation term or terms (it depends on the objective, i.e., EE, IP, EA) which contain the elements of H, see in Fig. 1 CfkJlum components. 

As a last approach we have investigated the performance of the hybrid method - referred to as CCSDT - which uses for the ground state the standard CCSDT-3 method and - at the EOM level - it includes the 7 3 operator in a rigorous manner. This approach is justified for the IP and EA calculations, since in the full EOM-CCSDT approach the computational bottleneck occurs at the ground state n ) step. So approximating the ground state with the CCSDT-3 method we make the scaling for the two steps comparable. 

Using the latter method (or the T-3) for the previous two states would increase the errors by 0.053, and 0.065 eV for the Bi and A2 states, respectively. For 62, replaeing the methods would result in a much larger error (by ca. 0.2 eV). So, on the whole, the test calculations would indicate that the EOM-CCSDT-3 approaeh is the most reliable approximate variant. 

In Table 6 we list the results for the C2 molecule obtained with the EOM-CCSDT-3 and full T approaehes for several basis sets. We observe that the differences between the approximate and the rigorous scheme are stable indicating that the mutual interrelations between the method do not depend on the basis set quality and size. Due to the cancellation of the errors the approximate scheme gives results eloser to the experimental values, but - of course - this is not meaningful for the general case. 

The results collected in Tables 2-5 indicate that the inclusion of more terms in the final EOM equations do not necessarily improve the results. Eor the EE scheme the variants which most closely reproduce the rigorous method are those which either the Tj, operator in the H elements (EOM-CCSDT-3a scheme) or allow for it only in the R2 equation (EOM-CCSDT-3 scheme), see equation (19). Similar conclusions could be presented with respect to the IP and EA problems, although we should be aware that using the approximate versions (i.e., those which require no higher than scaling) is not so crucial here, since the full approaches need n, which is the same as the CCSDT-3 method employed for the ground state. Hence we recommend using the hybrid methods which - in view of the collected results - seems to be preferable over the rest of the considered schemes. 

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