Decomposition cumulant

The appropriate modification of the 2-RDM may be accomphshed by combining A-representability constraints, known as positivity conditions, with both the unitary and the cumulant decompositions of the 2-RDM. 

Much of the recent literature on RDM reconstruction functionals is couched in terms of cumulant decompositions [13, 27-38]. Insofar as the p-RDM represents a quantum mechanical probability distribution for p-electron subsystems of an M-electron supersystem, the RDM cumulant formalism bears much similarity to the cumulant formalism of classical statistical mechanics, as formalized long ago by by Kubo [39]. (Quantum mechanics introduces important differences, however, as we shall discuss.) Within the cumulant formalism, the p-RDM is decomposed into connected and unconnected contributions, with the latter obtained in a known way from the lower-order -RDMs, q < p. The connected part defines the pth-order RDM cumulant (p-RDMC). In contrast to the p-RDM, the p-RDMC is an extensive quantity, meaning that it is additively separable in the case of a composite system composed of noninteracting subsystems. (The p-RDM is multiphcatively separable in such cases [28, 32]). The implication is that the RDMCs, and the connected equations that they satisfy, behave correctly in the limit of noninteracting subsystems by construction, whereas a 2-RDM obtained by approximate solution of the CSE may fail to preserve extensivity, or in other words may not be size-consistent [40, 42]. 

The cumulant formulas in Eqs. (28a)-(28c) can be generated easily using a convenient mnemonic introduced by Harris [62]. To obtain the cumulant decomposition of Dp+ from that of Dp, one sums—for each term in the Dp—all possible ways in which the particle number can be increased by one. 

Thus far we have discussed connectivity and extensivity in terms of the ROMs and RDMCs, but our ultimate goal is to apply these concepts to CSE(2). Replacing the ROMs in fl2 with their cumulant decompositions elucidates the unconnected terms in CSE(2). Consider, as an example, the following term in... 

Using cumulant reconstruction functionals A3[Ai, A2] and A4[Ai, A2], one can certainly derive closed, nonlinear equations for the elements of Ai and A2, which could be solved using an iterative procedure that does not exploit the reconstruction functionals at each iteration. Of the RDM reconstruction functionals derived to date, several [7, 8, 11] utilize the cumulant decompositions in Eqs. (25c) and (25d) to obtain the unconnected portions of D3 and D4 exactly (in terms of the lower-order RDMs), then use many-body perturbation theory to estimate the connected parts A3 and A4 in terms of Aj and A2, the latter essentially serving as a renormalized pair interaction. Reconstruction functionals of this type are equally useful in solving ICSE(l) and ICSE(2), but the reconstruction functionals introduced by Valdemoro and co-workers [25, 26] cannot be used to solve the ICSEs because they contain no connected terms in D3 or D4 (and thus no contributions to A3 or A4). 

This is not quite what we need, as in the current context, we require a cumulant decomposition of a three-particle operator. We can construct an... 

With the above decomposition, the commutator H, A is reduced to an expression containing only terms of the form we wish to keep (i.e., one- and two-particle operators). Let us denote this approximate form of the commutator as 2)> to indicate that the cumulant decomposition retains only one- and... 

We now discuss (ii), the evaluation of operator expectation values with the reference ho- We are interested in multireference problems, where may be extremely complicated (i.e., a very long Slater determinant expansion) or a compact but complex wavefunction, such as the DMRG wavefunction. By using the cumulant decomposition, we limit the terms that appear in the effective Hamiltonian to only low-order (e.g., one- and two-particle operators), and thus we only need the one- and two-particle density matrices of the reference wavefunction to evaluate the expectation value of the energy in the energy expression (7). To solve the amplitude equations, we further require the commutator of which, for a two-particle effective Hamiltonian and two-particle operator y, again involves the expectation value of three-particle operators. We therefore invoke the cumulant decomposition once more, and solve instead the modihed amplitude equation... 

To summarize the theory dynamic correlations are described by the unitary operator exp A acting on a suitable reference funchon, where A consists of excitation operators of the form (4). We employ a cumulant decomposition to evaluate all expressions in the energy and amphtude equations. Since we are applying the cumulant decomposition after the first commutator (the term linear in the amplimdes), we call this theory linearized canonical transformation theory, by analogy with the coupled-cluster usage of the term. The key features of the hnearized CT theory are summarized and compared with other theories in Table II. 

First consider a Hartree-Fock reference function and transform to the Fermi vacuum (aU occupied orbitals are in the vacuum). Then all particle density matrices are zero and the cumulant decomposition, Eq. (23), based on this reference corresponds to simply neglecting aU three and higher particle-rank operators generated by commutators. This type of operator truncation is used in the canonical diagonalization theory of White [22]. 

Now consider the effect of the cumulant decomposition on the different orders of energy contribution. First, no decomposition is involved in computing EP. For E, the cumulant decomposition corresponds to... 

In the expression for E, we apply the cumulant decomposition twice for the double commutator [[W, A2](i 2), 2](i 2)- Once again, only the fully contracted term contributes to the energy. The only way fully contracted terms arise is from double contractions in [W,A2] to produce a two-particle operator, which then doubly contracts with the final A2 commutator, to contribute to the energy. Since double contractions are involved in each step, no cumulant decomposition is involved for this term. There is no contribution from the three-particle... 

Figure 3. A diagrammatic representation of the cumulant decomposition ([IV, A2]p 2)) for the three-particle operator drawn in Fig. 2. Four kinds of one- and two-particle operators are obtained. The double line is the contraction for the particle-rank reduction (closure), where the correlation is averaged with the effective field (i.e., density matrices).

In the expression for E we hnd our first error from using the cumulant decomposition. Here, the three-particle operator arising from the first commutator [W, A2],... 

Figure 5. A diagram in ([[[iy,A2],A2],A2]) that yields nonzero energy in 4 (Eq. (32)) and that is missed in the cumulant decomposition in L-CTSD theory. In this diagram, the three-particle operator arising from [W,A2] contracts successively with two other A2 terms.

A further point is of interest in the formal discussion of the canonical transformation theory. So far we have assumed that the reference function is fixed and have considered only solving for the amplitudes in the excitation operator. We may also consider optimization of the reference function itself in the presence of the excitation operator A. This consideration is useful in understanding the nature of the cumulant decomposition in the canonical transformation theory. 

Using the energy functional (7) and the cumulant decomposition, and making the energy stationary with respect to variations in ihQ, we hnd that the optimal reference iho satisfies... 

To understand this more clearly, consider a simpler model where A consists of single excitations, only single-particle operators are retained in the effective Hamiltonian, and we choose the reference function iho to be a single determinant. Then, from a cumulant decomposition of the two-particle terms, the effective Hamiltonian becomes... 

Aspects of size extensivity in unitary group adapted multi-reference coupled cluster theories the role of cumulant decomposition of spin-free reduced density matrices... 

In the spin-free formulations of the UGA-MRCC theories, the use of CSFs entails that both the MRCC equations are in matrix form and the associated effective Hamiltonians will involve various w-body spin-free reduced density matrices (n-RDMs). n-RDMs are product separable and hence not size-extensive. From now on, we will refer to the spin-free RDMs as simply the ROMs. When spinorbital-based RDMs are discussed, we will explicitly indicate this. So, no confusions should arise. It is non-trivial to establish the extensivity of both the cluster operators and the effective Hamiltonian in spite of the occurrence of these n-RDMs. This paper will briefly review the formulation of the UGA-MRCC theories mentioned above and will present a comprehensive account of the aspects of connectivity which leads to extensivity. Although in some of our earlier papers [47] we sketched how size extensivity emerges after the cumulant decomposition of the n-RDMs, we will present here a detailed and thorough analysis of the underlying issues. 

In the next step, we analyze the structure of the various terms generated after the application of the WT to the matrix element in our working equations and establish that we can systematically eliminate the disconnected portion of M, if we keep track of which components of the composites containing F and G are connected. This particular analysis requires the concept of cumulant decomposition [75, 80, 88, 89] of the density matrix elements of Fjt for various ranks k. Since the final working equations are connected after the elimination of the disconnected terms, the cluster amplitudes of F are connected and are compatible with the connectivity of G. ... 

The general structure of the cumulant decomposition of a spin-free RDM of given rank is discussed in some detail in the Appendix. In our discussions here, we will make use of the expressions explained therein and for the sake of continuity of reading refer to the appendix for further details. The cumulant decomposition of a 2-body RDM, is given by Eq. (35)... 

Fig. 2 Pictorial reiwesentatioii of the diagrams generated from Fig. Ic by the cumulant decomposition of two-body RDM. b, c Belong to the connectivity class C, as introduced in the text, while a does not...

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