SCS-CCSD

Just as one can use spin-component scaling to improve the performance of MP2, one can multiply the contribution of the same-spin terms to the CCSD energy by an empirical parameter and multiply the contribution of the opposite-spin terms by another parameter. This gives the SCS-CCSD method [T. Takatani et al., J. Chem. Phys., 128,124111 (2008)]. The same-spin and opposite-spin parameter values 1.13 and 1.27, respectively, were found by fitting a set of known reaction energies. SCS-CCSD performs quite well for intermo-lecular interactions. 

Table 6 SCS-CCSD CCSD(T) Results" and Other Correlated Methods in Comparison to Estimated...

We may now stress some formal similarities and differences between both approaches when the model space used in (SC) CAS-SDCI and the source of corrections in ec-CCSD is the same (a more mathematically oriented comparison will be showed in the next section). 

SC) CAS-SDCI is a size-extensive method, while ec-CCSD is almost size-extensive (CC procedure is itself size-extensive, but the corrections are not, provided they come from a truncated variational Cl wavefunction). 

SC) CAS-SDCI may be viewed as a set of d equations to be solved simultaneously (actually, the implementation of the method is a recursive diagonalization process). In contrast, ec-CCSD implies a unique diagonalization and a further resolution of a rather small set of non-linear equations. 

Comparing (SC) CAS-SDCI and Externally Corrected CCSD Methods... 

SC) CAS-SDCI is, in general, computationally heavier than ec-CCSD. A full dressing method (equivalent to SS-MRCC) would be much heavier. 

For a detailed comparison of (SC)2CAS-SDCI and ec-CCSD methods, equations corresponding to single and double excitations should be written explicitly for both approaches. For the former one, we can substitute eq. (10) in (4) and, using operator notation, we have... 

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 ... 

For sake of comparison, in all studied cases, we run calculations for those geometries and basis sets with a FCI (or near FCI) available. The methods we deal with are CCSD, CAS-SDCI, (SC)2CAS-SDCI and ec-CCSD corrected from both CAS-SDCI and (SC) CAS-SDCI. The performance of the methods is examined from two aspects the total energy and the quality of the potential energy surface (PES), being this quality measured by the so-called non-parallelity error (NPE). For a given set of calculations in a dissociative curve, the NPE is defined as the difference between the maximal and minimal deviation from the exact FCI PES. 

We compare the performance of a variational CAS-SDCI with its size-extensive version (SC) CAS-SDCI and the standard size-extensive CCSD. Additionally, we correct CCSD using the non size-extensive T and T amplitudes yielded by a cluster analysis of the CAS-SDCI wavefunction and also using the size-extensive and T amplitudes coming from (SC) CAS-SDCI. 

The results in Table 1 for [2e/2o] space clearly indicate that at the equilibrium geometry both, CAS-SDCI and (SC) CAS-SDCI, do not reach the good performance of standard CCSD (errors 14.6 mHa, 10.9 mHa and 3.1 mHa, respectively). Additionally, we see that by dressing CAS-SDCI we quite improve the energy. On the other hand, ec-CCSD takes the best from both, CCSD and the external source, yielding the most accurate result (only 2.4mHa away from nFCI, irrespective of the external source used). 

We can also see from Table 1 that very accurated values of NPE are yielded by CAS-SDCI (1.0 mHa), ec-CCSD(CAS-SDCI) (1.3 mHa), (SC)2CAS-SDCI (1.6 mHa) and ec-CCSD((SC) CAS-SDCI) (1.6 mHa as well). It is not the case of standard CCSD which failure at stretched geometries yields 18.6 mHa NPE error. 

The same trends are observed with the larger space [8e/7o]. The differences between CAS-SDCI and nFCI range from 7.9 mHa at 3Re up to 8.9 mHa at Re. In the case of (SC) CAS-SDCI it goes from 6.6 mHa to 7.3 mHa, while CCSD, as stated above, goes from 21.6 mHa to 3.1 mHa. Finally, ec-CCSD error lies between 0.5 mHa and 0.7 mHa (roughly the same with the two external sources employed). 

We have studied the performance of the so-called (SC)2CAS-SDCI and ec-CCSD methods versus standard CAS-SDCI method on one hand, and CCSD on the other hand, in quasidegenerate situations that arise when exploring dissociation channels of simple molecules. In particular, we dissociate a single bond, two single bonds simultaneously and a triple bond. 

All of the calculations have been performed at the experimental equilibrium distance R = 1.128 A, in order to enable a proper comparison with the EOM-CCSD reference. In so far as there are neither largely interacting excited states nor special reasons for expecting a breakdown of the Born Oppenheimer approximation, great changes in the MAE are not expected if one takes the (SC) SDCI ground state equilibrium value for Re which is Re = 1.140 A (very close to the CCSD value, as expected Cfr. table 1). We have performed a separate calculation of the whole set of VEE with the aug-cc-pVDZ basis set at the Rg distance, in any case. The results have not been included in table II for the sake of clarity, but the total MAE values where 2.34 eV for the MR-SDCI and 0.17 eV for (SC)2mR-SDCI. 

SDCI (SC)2- SDCI MR- SDCI (SC)2 MR-SDCE SDCI (SC) - SDCl MR- SDCI (scy MR-SDCl EOM- CCSD ... 

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