Hamiltonian disconnected

The valence correlation component of TAE is the only one that can rival the SCF component in importance. As is well known by now (and is a logical consequence of the structure of the exact nonrelativistic Bom-Oppenheimer Hamiltonian on one hand, and the use of a Hartree-Fock reference wavefunction on the other hand), molecular correlation energies tend to be dominated by double excitations and disconnected products thereof. Single excitation energies become important only in systems with appreciable nondynamical correlation. Nonetheless, since the number of single-excitation amplitudes is so small compared to the double-excitation amplitudes, there is no point in treating them separately. 

Obviously, we gain precisely the same expressions as in the multi-root theory since we postulated the same form of the effective Hamiltonian. We recall that all matrix elements of the effective Hamiltonian are expressed by means of connected diagrams only in the case of diagonal elements just connected vacuum diagrams may come into consideration and in the case of off-diagonal elements at least one part of a disconnected diagram would correspond to an internal excitation. 

Indeed, when Mr < Mt, the disconnected component of the left-hand side of Eq. (97), i.e. the expression P Ck,open Hn,oPen )i vanishes, since cluster amplitudes defining T, Eq. (41), satisfy equations (78) with n = 1,..., Mr-Equation (99) represents a generalization of the exact Eq. (88) to truncated EOMXCC schemes. Again, the only significant difference between the EOMXCC equations (98) and (99) and their EOMCC analogs (48) and (47), respectively, is the similarity transformed Hamiltonian used by both theories. As in the EOMCC theory, Eqs. (98) and (99) have the same general form (in particular, they rely on the same similarity transformed Hamiltonian) for all the sectors of Fock space. 

Some properties of the Fock space transformations W and effective Hamiltonians h and, thus, of the resulting h, appear to differ from those obtciined by Hilbert space transformations. For example, their canonical unitary W is not separable and yields an h and, thus, an h with disconnected diagrams on each degenerate subspace. However, the analogous U of Eq. (5.13) may be shown to be separable 71), and the resulting He on each complete subspace flo is fully linked , as proven by Brandow [8]. These differences are not explained. 

Exploring the cluster analysis of a finite set of FCI wave functions based on the SU CC Ansatz [224], we realized that by introducing the so-called C-conditions ( C implying either constraint or connectivity , as will be seen shortly), we can achieve a unique representation of a chosen finite subset of the exact FCI wave functions, while preserving the intermediate normalization. (In fact, any set of MR Cl wave functions can be so represented and thus reproduced via an MR CC formalism.) These C-conditions simply require that the internal amplitudes (i.e. those associated with the excitations within the chosen GMS) be set equal to the product of aU lower-order cluster amplitudes, as implied by the relationship between the Cl and CC amplitudes [223], rather than by setting them equal to zero, as was done in earlier IMS-based approaches [205,206] (see also Ref. [225]). Remarkably, these conditions also warrant that all disconnected contributions, in both the elfective Hamiltonian and the coupling coefficients, cancel out, leaving only connected terms [202,223]. 

A further point which should be mentioned concerns the treatment of incomplete model spaces. Meissner and his collaborators [61], have shown that the cancellation of terms corresponding to disconnected diagrams in the equations for amplitudes is, in general, not a sufficient condition for extensivity. In any Hilbert-space MRCC method, extensivity may be destroyed by diagonalization of the effective Hamiltonian matrix. For the complete model spaces employed in the studies reviewed here, the diagonalization has a full Cl-like character and, therefore, extensivity is ensured. Although Meissner and collaborators [61,62] devised an approach which should also be extensive for the case of an incomplete model space, practical implementation appears somewhat complicated. 

The quantum mechanical information that follows from a normal mode analysis must reveal the same mechanical equivalence to a set of disconnected oscillators as the classical analysis. Each such oscillator (normal mode of vibration) can exist in any of the states possible for a one-dimensional harmonic oscillator. Each has its own contribution to the energy of the system, and thus, the Hamiltonian in Equation 7.35 corresponds to a quantum mechanical energy level expression... 

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