Equations for the Cluster Amplitudes

The cluster amplitudes Q are determined by insisting that exp(T)l0 satisfy the usual Schrodinger equation (or at least certain projections of this equation) 

Hexp(r) 0 = EcxplDlO ) which upon premultiplying by exp( — 7 ) gives 

The above exponential series gives, when expanded and collected together as commutators. 

The series truncates (exactly) after four commutators regardless of the level at which (T ) T is truncated (if at all). This exact truncation is a result of the fact that // contains at most two-electron operators, which involve four general (particle or hole) operators i j lk. Therefore [//, T] contains at most three general operators, [[f/, T], T] contains two, and [[[[ J 

It is the presence of exp(—7), rather than exp(T ), in Eq. (4.12) that gives rise to the exactly closed quartic equation for T, Moreover, the presence of the commutators in the expression for E and the fact that T contains only particle creation and hole annihilation operators makes the CC-calculated energy contain only linked terms (in the sense discussed in Chapter 3). This then makes contain only size-consistent terms. 

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The second-order response equations for the cluster amplitudes and the Lagrangian multipliers are ... 

Eqs. (57), (59) and (60) are the working equations for the cluster amplitudes. We should note that they are coupled also to the combining coefhcients c s, which are obtained as the elements of the eigenvector from the diagonalization of W dehned in the IMS, in Eq. (50). This is similar to what we had in the SS-MRCC theory for the CMS [38, 39]. Usually, one may get the coefficients from Eq. (50) in a macro-iteration, and get the cluster amplitudes in an inner, micro-iteration. 

The above first-order equation for 7 [Eq. (5.29)] expresses a set of linear gebraic equations for the cluster amplitudes in which can be written... 

The explicit equations for the cluster amplitudes and defining the CCSD approach are determined by projecting the electronic Schrodinger equation on the singly and doubly exc respectively, to obtain... 

The CC wavefunction had been considered by Coester and Kummel [10] as the exponential S ansatz at about the same time in the nuclear physics literature however, none of these authors took the next step to develop explicit equations for the cluster amplitudes which appear in the cluster operators. 

We have shown in our parent UGA-SSMRCC paper [45] that there are two natural ways of choosing the sufficiency conditions, and each leads to a set of UGA-SSMRCC equations for the cluster amplitudes which is inequivalent with the other. However, a study of the numerical performance of both the variants leads us to the conclusion that they produce very close results and no criteria for choosing one over the other can be established [44]. Since the version A involves fewer terms and is simpler in structure, our subsequent applications focused mainly on this alternative. 

Although the similarity-transformed Hamiltonian is quartic in the cluster amplitudes, the equations for the cluster amplitudes (13.2.32) need not contain all the amplitudes to this order. In Section 13.2.8, we use the cluster-commutation condition (13.2.36) to show that, for a general operator O of particle rank mo, the state... 

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