Cumulant 2-RDMs

Rosina s theorem states that for an unspecified Hamiltonian with no more than two-particle interactions the ground-state 2-RDM alone has sufficient information to build the higher ROMs and the exact wavefunction [20, 51]. Cumulants allow us to divide the reconstruction functional into two parts (i) an unconnected part that may be written as antisymmetrized products of the lower RDMs, and (ii) a connected part that cannot be expressed as products of the lower RDMs. As shown in the previous section, cumulant theory alone generates all of the unconnected terms in RDM reconstruction, but cumulants do not directly indicate how to compute the connected portions of the 3- and 4-RDMs from the 2-RDM. In this section we discuss a systematic approximation of the connected (or cumulant) 3-RDM [24, 26]. 

Additional insight may be obtained by writing the system of equations in the natural-orbital basis set, that is, the basis set that diagonalizes the 1-RDM. In this basis set the two terms with the connected 3-RDM may be collected to obtain the formula for the elements of the connected (or cumulant) 3-RDM [26],... 

The two types of cumulant 3-RDM elements that cannot easily be constructed from elements of the 2-RDM are [24, 26]... 

Upon simplihcation it is not difficult to show that Eq. (57) is equivalent to Eqs. (48) and (55). Therefore the position of the number operator does not affect the relation that we have derived for the cumulant 3-RDM. More general relations for the cumulant p-RDM may similarly be derived by contracting the (p + 1)-RDM to the p-RDM. 

Importantly, the anti-Hermitian CSE may be evaluated through second order of a renormalized perturbation theory even when the cumulant 3-RDM is neglected in the reconstruction. The anti-Hermitian part of the CSE [27, 31, 63] is the stationary condition for two-body unitary transformations of the A-particle wave-function [31, 32], and hence the two-body unitary transformations may easily be evaluated with the anti-Hermitian CSE and RDM reconstruction without the many-electron Schrodinger equation. The contracted Schrodinger equation in conjunction with the concepts of reconstruction and purification provides a new, important approach to computing the 2-RDM directly without the many-electron wavefunction. 

The theory of cumulants allows us to partition an RDM into contributions that scale differently with the number N of particles. Because aU of the particles are connected by interactions, the cumulant RDMs scale linearly with the number N of particles. The unconnected terms in the p-RDM reconstruction formulas scale between N and W according to the number of connected RDMs in the wedge product. For example, the term scales as NP since all p particles are statistically independent of each other. By examining the scaling of terms with N in the contraction of higher reconstruction functionals, we may derive an important set of relations for the connected RDMs. 

As in the previous section, by connected we mean all terms that scale linearly with N. Wedge products of cumulant RDMs can scale linearly if and only if they are connected by the indices of a matrix that scales linearly with N transvec-tion). In the previous section we only considered the indices of the one-particle identity matrix in the contraction (or number) operator. In the CSE we have the two-particle reduced Hamiltonian matrix, which is defined in Eqs. (2) and (3). Even though the one-electron part of scales as N, the division by A — 1 in Eq. (3) causes it to scale linearly with N. Hence, from our definition of connected, which only requires the matrix to scale linearly with N, the transvection... 

New optimism was brought into the field of RDMs by Hiroshi Nakatsuji, Carmela Valdemoro, and David Mazziotti with their cumulant expansion, the hierarchy of equations connecting the 2-RDM with 4-RDMs, and the contracted Schrd-dinger equation. John Coleman continues to be the motor for further progress. 

There are several other studies of cumulant expansions of the ROMs. Thus Kutzelnigg and Mukheijee also published in 1999 [64] an RDM expansion that is similar to Mazziotti s. An extended study of this cumulant approach was given by Ziesche [65]. Also, a particularly interesting analysis of the cumulant expansions was given by Harris [66], who proposed a systematic way for obtaining the different terms of the expansion. 

The CSE allows us to recast A-representability as a reconstruction problem. If we knew how to build from the 2-RDM to the 4-RDM, the CSE in Eq. (12) furnishes us with enough equations to solve iteratively for the 2-RDM. Two approaches for reconstruction have been explored in previous work on the CSE (i) the explicit representation of the 3- and 4-RDMs as functionals of the 2-RDM [17, 18, 20, 21, 29], and (ii) the construction of a family of higher 4-RDMs from the 2-RDM by imposing ensemble representability conditions [20]. After justifying reconstmction from the 2-RDM by Rosina s theorem, we develop in Sections III.B and III.C the functional approach to the CSE from two different perspectives—the particle-hole duality and the theory of cumulants. 

The coefficients of the multivariable Taylor series expansion of G J) about the point where the Schwinger probes vanish are elements of the ROMs. Thus G J) is known as the generating functional for ROMs. Mathematically, the RDMs of the functional G J) are known as the moments. The moment-generating functional G(y) may be used to define another functional W J), known as the cumulant-generating functional, by the relation... 

Because the reconstruction of the 3-RDM with A = 0 has a second-order error, the evaluation of the CSE with the unconnected 3- and 4-RDM cumulant... 

Cumulant theory offers a systematic approach to reconstructing the 3- and 4-RDMs within the CSE from the 2-RDM, but it also provides insight into the stmcrnre of the CSE. Let us define C as the connected part of the left-hand side... 

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. 

The 2-RDM is automatically antisymmetric, but it may require an adjustment of the trace to correct the normalization. The functionals in Table I from cumulant theory allow us to approximate the 3- and the 4-RDMs from the 2-RDM and, hence, to iterate with the contracted power method. Because of the approximate reconstruction the contracted power method does not yield energies that are strictly above the exact energy. As in the full power method the updated 2-RDM in Eq. (116) moves toward the eigenstate whose eigenvalue has the largest magnitude. 

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]. 

In this work, we derive—via explicit cancellation of unconnected terms in the CSE—a pair of simultaneous, connected equations that together determine the 1- and 2-RDMCs, which in mrn determine the 2-RDM in a simple way. Because the cancellation of unconnected terms is exact, we have in a sense done nothing the connected equations are equivalent to the CSE and, given Al-representabUify boundary conditions, they are also equivalent to the electronic Schrodinger equation. The important difference is that the connected equations for the cumulants automatically yield a size-consistent 2-RDM, even when solved approximately, because every term in these equations is manifestly extensive. 

In the present context, the way to ensure extensivity is to reformulate the CSE so that the RDMCs and not the RDMs are the basic variables. One can always recover the RDMs from the cumulants, but only the cumulants satisfy connected equations that do not admit the possibility of mixing noninteracting subsystems. Connected equations are derived in Section V. Before introducing that material, we first provide a general formulation of the p-RDMC for arbitrary p. 

Using either Eq. (23) or Mazziotti s adaptation of it, one may derive exact expression for the RDMs in terms of their cumulants. The first few such expressions are... 

The combinatorial point of view is reminiscent of the classical cumulant formalism developed by Kubo [39], and indeed the structure of Eqs. (25) and (28) is essentially the same as the equations that define the classical cumulants, up to the use of an antisymmetrized product in the present context. In further analogy to the classical cumulants, the p-RDMC is identically zero if simultaneous p-electron correlations are negligible. In that case, the p-RDM is precisely an antisymmetrized product of lower-order RDMs. 

A alone is insufficient to specify any of the other cumulants, or any RDMs at all (save for the trivial n = case). 

The basic diagram elements representing D = Ai, g, h, and Ap (for p >2) are illustrated in Fig. 2. Recall that CSE(p) is given by the equation Op = 0, where ftp is the p-electron kernel defined in Eq. (10). The terms in this kernel consist of h and g acting on ROMs, followed in some cases by a trace over one or two coordinate indices. Upon replacing the RDMs with their cumulant... 

It does not appear that the ICSEs can be solved by self-consistent iteration, however. In Eq. (68), CSE(2) is expressed in a form that affords the 2-RDM as an explicit functional of the 2-, 3-, and 4-RDMs, but no analogous formulation of ICSE(l) or ICSE(2) is possible, since the 1- and 2-RDMCs appearing in these equations are always acted upon by or g (cf. Eqs. (66) and (67)). Thus the ICSEs are implicit equations for the cumulants. 

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). 

Natural orbitals are used. The 2-RDM is then decomposed into the I-RDM through the cumulant expansion. 

Piris [73] has recently proposed an explicit form for the cumulant [74, 75] of the 2-RDM in terms of two symmetric matrices, A and A. The suggested form of... 

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