Single-reference cluster amplitudes

The CR-EOMCCSD(T) approach is a purely single-reference, blackbox method based on the MMCC(2,3) approximation, in which the wave function jH/ ) entering Eq. (67) is designed by using the singly and doubly excited cluster amplitudes tl and defining Ti and T2, respectively, obtained in the CCSD calculations, and the zero-, one- and two-body amplitudes ro p), rl pL) and r pi), defining R, 1, and respectively. 

Results. Calculations were carried out at two internuclear separations, the equilibrium Re = 2.0844 A as in Ref. [89], and 2.1 A, for comparison with Ref. [127]. The relativistic coupled cluster (RCC) method [130, 131] with only single (RCC-S) or with single and double (RCC-SD) cluster amplitudes is used (for review of different coupled cluster approaches see also [132, 133] and references). The RCC-S calculations with the spin-dependent GRECP operator take into account effects of the spin-orbit interaction at the level of the one-configurational SCF-type method. The RCC-SD calculations include, in addition, the most important electron correlation effects. 

Similarly the CC amplitudes are determined by projecting the Schrodinger equation from the left against all excitations included in the cluster operator, for example, all single and double excitations from the HF reference function. If we denote this set of excited states by >, the cluster amplitude equations have the form... 

It is important to note, however, that there are fundamental differences between FSCC and SRCC with respect to the nature of their excitation operators. For a given truncation of the cluster operators beyond simple double excitations, the determinantal expansion space available in an FSCC calculation is smaller than those of SRCC calculations for the various model space determinants. A class of excitations called spectator triple excitations must be added to the FSCCSD method to achieve an expansion space that is in some sense equivalent to that of the SRCC. But even then, the FSCC amplitudes are restricted by the necessity to represent several ionized states simultaneously. Thus, we should not expect the FSCCSD to produce results identical to a single reference CCSD, nor should we expect triple excitation corrections to behave in the same way. The differences between FSCC and SRCC shown in Table I and others, below, should be interpreted as a manifestation of these differences. 

MRCEPA(O) by us, is the simplest among the CEPA-hke approximants to the SS-MRCC theory which is extensive and also avoids intruders. However, the appearance of the CAS energy Eq, rather than the ground state energy E itself renders it rather approximate. As we already mentioned, the complete linearity of the SS-MRCEPA(O) equations in the cluster amplitudes lends the same invariance property to it as in the MR-CISD. This parallels the situation in the single reference CEPA(O), which also possesses the invariance. In our earlier papers [58,59], we suggested other schemes where E appears which, however, did not have the orbital invariance property as that of the SS-MRCEPA(O). 

However, other attempts have been made to improve on the treatment of electron correlation in SOPPA. Three SOPPA-like methods have thus been presented. All are based on the fact that a coupled cluster wavefunction gives a better description than the Mpller-Plesset first- and second-order wavefunctions, Eqs. (9.66) and (9.70). In the second-order polarization propagator with coupled cluster singles and doubles amplitudes-SOPPA(CCSD)-method (Sauer, 1997), the reference state in Eqs. (3.160) to (3.163) is approximated by a linearized CCSD wavefunction... 

In SOPPA the equations for the polarization propagator are solved by retaining all terms second order in the fluctuation potential. The reference state is a correlated Moller Plesset wavefunction with the corresponding correlation coefficients. The zeroth-order wavefunction is a single reference SCF ground state. Correlation is introduced via the fluctuation potential. SOPPA includes dynamical correlation, but not nondynamical effects. The same technique can be applied to other methods, e.g., coupled cluster, giving CCSDPPA where the cluster amplitudes replace the correlation coefficients used in SOPPA. 

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