Hamiltonian similarity transformed

For the second-order term in the Hamiltonian, the rotation of the rotor implies similar transformations, which in the fast-spinning case yields an average operator... 

As seen from equation (50), the ESC Hamiltonian is energy dependent and Hermitian. For a fixed value of E, the ESC Hamiltonian can be diagonalized and the resulting solutions, in principle, form a complete orthonormal set. The eigenfunctions of are identical to the large component of the Dirac spinor. When Z — 0, equations (38) and (44) give us the similarity transformed Hamiltonian... 

Now, substituting for X in equation (40) we get the FW or equivalent Hermitian similarity transformed Hamiltonian which is an expansion in terms of... 

The aim of our similarity-transformed Hamiltonian is to improve the computation of the correlation energy of conventional configuration-interaction (Cl) calculations. In this framework, the conventional wave function is multiplied by the correlation function 14,15,16)... 

The implementation of the above described correlation function is described in detail in Ref. (77). Figure 4 illustrates the efficiency of the method for the He ground-state energy as a function of the basis sets. Already for basis sets with s-and p-fimctions, the similarity-transformed Hamiltonian shows a better accuracy than the conventional Cl method. 

The similarity-transformed Hamiltonian method has so far been applied only to two-electron systems. Using closure (i.e., RI) approximations, this technique will be generalized to many-electron systems (IS). 

Of the methods listed above, only the noniterative CC approaches based on the partitioning of the similarity-transformed Hamiltonian (24-28) and the (C)R-CC approaches of refs 9,13-18,20,21, which employ the MMCC formalism (P. 13, 14, 18, 19, 21, 45, 106, 107), retain the simplicity and the black-box character of the standard CCSD(T) or CCSD(TQf) methods. One of the two goals of the present work is the development of a new class of the MMCC-based black-box methods for multiple bond breaking. 

Note that in contrast to a general similarity transformation (e.g., as found in the usual coupled-cluster theory) the canonical transformation produces a Hermitian effective Hamiltonian, which is computationally very convenient. When U is expressed in exponential form, the effective Hamiltonian can be constructed termwise via the formally infinite Baker-Campbell-Hausdorff (BCH) expansion,... 

We are now equipped with all of the basic concepts of the CC/EOMCC theory which are necessary to explain the noniterative MMCC approaches to ground and excited electronic states. In this section, we focus on the exact MMCC theory. The approximate MMCC schemes for excited electronic states, including the externally corrected MMCC approaches and the CR-EOMCCSD(T) theory, and their most recent analog based on the left eigenstates of the similarity-transformed Hamiltonian, are discussed in Section 3. 

CCSD/EOMCCSD equations defined by Eqs. (58) and (59). These moments are easy to calculate. As implied by Eqs. (58) and (59), their determination requires the explicit consideration of the triples-reference, triples-singles, and triples-doubles blocks of the matrix representing the CCSD/EOMCCSD similarity-transformed Hamiltonian Eq. (13). 

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