Commutator expansion

Finally, to assess the convergence of the commutator expansion in the effective Hamiltonian as the bond is stretched, we computed a second-order energy using the L-CTSD amplitudes, denoted CASSCF/L-CTSD(2). Here the energy expression is evaluated as As seen from Tables VI... 

In addition to the encouraging numerical results, the canonical transformation theory has a number of appealing formal features. It is based on a unitary exponential and is therefore a Hermitian theory it is size-consistent and it has a cost comparable to that of single-reference coupled-cluster theory. Cumulants are used in two places in the theory to close the commutator expansion of the unitary exponential, and to decouple the complexity of the multireference wave-function from the treatment of dynamic correlation. 

As we have taken the groupings A,B etc., to refer to true linked clusters TA, Tg etc., the operators T"A, Tg must appear as physically connected entities in the occupation number representations. Eft, Eg etc., will also then appear as connected entities - as a consequence of the multi-commutator expansion generated by eqs. (3.8). Since the groupings are... 

Finally we present results of SE and VP calculations for ns valence electrons in heavy and superheavy atoms with n up to 8 and Z up to 119 [13], [14]. For the calculation of the SE contribution the PWR approach described in Sec.4.2 based on the multiple commutator expansion [71] was used. The corrections are given in Table 1. Since the B-spline approach requires the employment of the local potential, the local approximation to the DHF potential obtained by the direct parametrization [77] was used. The VP contribution was treated in the Uehling approximation. One can expect that the Uehling term will suffice not only for highly charged ions but in screened systems as well. The Uehling potential was corrected for the extended nucleus [78] - [80]. The Uehling potential for the point-like nucleus (233) was replaced by the expression ... 

If the cluster operator is connected, one can easily show that the dressed Hamiltonian and the matrix elements are also connected via multi-commutator expansion. Hence, the proof of the connectedness of the first term of Eq. (7) is quite... 

Since ij/Q is a CAS-type function, the first term and the dressed Hamiltonian of the second term of the left-hand side of Eq. (30) are manifestly connected involving H, T and via commutator expansion, and it is hence enough to show that the third term of the equation is connected. According to Baker-Campbell formula, the second term of the above equation can be written as follows ... 

This form is convenient because the transformed operators can then be obtained using the commutator expansion [32,101]... 

Now we will look at the second mathematical fact, which is a commutator expansion ... 

However, it would not be possible to apply the commutator expansion and instead of the four terms in Eq. (10.53) we would have an infinite number. Thus, the non-variational CC method benefits from the very economical condition of the intermediate normalization. For tins reason, we prefer the non-variational approach. 

How does the CC machinery work Let us show it for a relatively simple case, T =. Equation (10.54), written without the commutator expansion, takes the form... 

One might recognize the modification as being proportional to one part of the commutator expansion in the Verlet method, in fact... 

Here we have to use the operators E q and not the X q. The reason is that to ensure that the commutator expansions in coupled-cluster theory truncate, the wave operator must be expressed in terms of excitation operators between two disjoint one-particle spaces. This expression and the corresponding one for T2 form the basis for the Kramers-restricted CCSD (KRCCSD) method (Visscher et al. 1995). 

The only way to get a wave operator that is symmetric under time-reversal symmetry is to impose the restriction from the beginning. While this fixes the relations between the amplitudes, it also forces the occupied and the unoccupied Kramers components of the open shell to be treated equivalently. This equivalence is what introduces the ambiguity in the treatment of the open shell the open-shell Kramers pair must behave as both a particle and a hole, and the result is that the truncated commutator expansions in the coupled-cluster equations are much longer than in closed-shell theory. The alternative is to use an unrestricted wave operator with the Kramers-restricted spinors. The use of the latter provides some reduction in the work due to the relations between the integrals, but not a full reduction (Visscher et al. 1996). 

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