Lower-order RDMs

By identifying the holes part ofthe l.h.s. of this relation with the holes part ofthe r.h.s. and similarly for the electrons, Valdemoro, Colmenero and Perez del Valle [5-7] built up a p-RDM in terms ofthe lower order RDM s. This construction procedure will be referred to in what follows as VCP and denoted in the formulae as D. Since the value of the norm of the VCP matrix was very close - but not exactly equal - to ( ), different procedures of renormalization were proposed [8,9,12]. This step should not be under-... 

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 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. 

Before deriving equations that determine the RDMCs, we ought to clarify precisely which are the RDMCs of interest. It is clear, from Eqs. (25a) and (25b), that Ai and A2 contain the same information as D2 and can therefore be used to calculate expectation values (IT), where W is any symmetric two-electron operator of the form given in Eq. (1). Whereas the 2-RDM contains all of the information available from the 1-RDM, and affords the value of (IT) with no additional information, the 2-RDMC in general does not determine the 1-RDM [43, 65], so both Ai and A2 must be determined independently in order to calculate (IT). More generally, Ai,...,A are all independent quantities, whereas the RDMs Dj,..., D are related by the partial trace operation. The u-RDM determines all of the lower-order RDMs and lower-order RDMCs, but... 

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). 

Equivalent expressions for approximating the 4-RDM in terms of the lower order RDM s can be obtained in a similar way. 

The results for approximating the 4-RDM in terms of the lower order RDM s are slightly inferior but still very good. In consequence, I expect that the iterative procedure proposed in the previous section may prove to be a realistic one. 

The method for approximating an RDM in terms of the lower order ones is based on equation (49). The working hypothesis which has been put forward [46] is ... 

The different approaches proposed in the literature for constructing the high order RDM s as function of the lower order ones are summarized here, although some of their features will be later considered in more detail. 

An analysis of the stmcture of the electron correlation terms in which the reference was the antisymmetrized products of FCI -RDM elements was reported in [12], The advantage of using correlated lower order matrices for building a high order reference matrix is that in an iterative process the reference is renewed in a natural way at each iteration. However, if the purpose is to analyse the structure of the electron correlation terms in an absolute manner that is, with respect to a fixed reference with no correlation, then the Hartree Fock p-RDM"s are the apropriate references. An important argument supporting this choice is that these p-RDM s are well behaved A-representable matrices and, moreover, (as has been discussed in [15]) the set of 1-, 2-, and 3-Hartree Fock-RDM constitute a solution of the 1 -CSE. 

The dependence of the 2-CSE on the 3- and 4-RDMs renders indeterminate this equation [18, 52]. This is the reason why the interest in this equation was initially lost. However, after the encouraging results obtained in the construction of the 2-RDM in terms of the 1-RDM [19], the possibility arose of constmcting good approximations of the high-order RDMs in terms of the lower-order ones. This permitted one to remove the indeterminacy and opened the way to build an iterative method for solving the 2-CSE. 

Therefore the 4-MCSE is not only determinate but, when solved, its solution is exact. As already mentioned, the price one has to pay is the fact of working in a four-electron space and the difficulty, as in the 2-CSE case, is that the matrices involved must be A-representable. Indeed, in order to ensure the convergence of the iterative process, the 4-RDM should be purified at each iteration, since the need for its A-representability is crucial. In practice, the optimizing procedure used is to antisymmetrize the at each iteration. This operation would not be needed if aU the matrices were A-representable but, if they are not, this condition is not satisfied. In order to impose that the 4-RDM, from which all the lower-order matrices are obtained, be positive semidefinite, the procedure followed by Alcoba has been to diagonalize this matrix and to apply to the eigenvalues the same purification as that applied to the diagonal elements in the 2-CSE case, by forcing the trace to also have a correct value. 

The indeterminacy of the 2-CSE had caused this equation to be overlooked for many years. In 1992 Valdemoro proposed a method to approximate the 2-RDM in terms of the 1-RDM [108], which was extended in order to approximate the 3- and 4-RDMs in terms of the lower-order matrices [46, 47] and in 1994 Colmenero and Valdemoro [48] applied these approximate constructing algorithms to avoid the indeterminacy problem and solve iteratively the 2-CSE. Since then, the smdy of improved constructing algorithms as well as alternative strategies in the iterative procedure have been proposed [1,6, 15, 18, 36, 49-57, 59, 60, 62-65, 68, 70, 71, 79-85, 109-111]. Also, good results of several calculations have been reported [6, 49-52, 55, 56, 68, 70, 88, 109, 111]. 

As can be seen, the 2-CSE depends not only on the 2-RDM but also on the 3- and 4-RDMs. This fact lies at the root of the indeterminacy of this equation [63, 107]. As already mentioned, in the method proposed by Colmenero and Valdemoro [46 8] and in those further proposed by Nakatsuji and Yasuda [49, 51] and by Mazziotti [52, 111], a set of algorithms for approximating the higher-order ROMs in terms of the lower-order ones [46, 47, 108] allows this equation to be solved iteratively until converging to a self-consistent solution. In the approach considered in this work, the spin-adapted 2-CSE has been used. This equation is obtained by coupling the 2-CSE with the second-order contracted spin equation [50]. 

We have proposed a method for approximating a high order RDM in terms of the lower order ones [15] and we can use this approximation to replace the former ones, appearing on the r.h.s. of eq. (3) by the lower order ones forming a matrix M [9,10]. In this case, the trace of matrix M must be equal to S times No, which provides a new value for . Then, by dividing M by this new value of 5, a new l-RDM is found. The iterative procedure may then start again. 

The Contracted Schrodinger Equation is studied here in a spin-orbital representation coupled with the S2 eigenvalue equation as an auxiliary condition. A set of new algorithms for approximating RDM s in terms of the lower order ones are reported here. These new features improve significantly the method. 

The possibility to decouple the CSE arose when a systematic way to approximate high order RDM s in terms of the lower order ones was found (10, 11, 12). Thus the iterative solution of the CSE was proposed by Colmenero and Valdemoro (11), who later on obtained the first results with this method (13). 

Recently, Nakatsuji and Yasuda. have proposed an alternative way to approximate the higher order RDM s in terms of the lower order ones (14) proceeding then, like us, to solve the DE iteratively. 

By replacing the Kronecker deltas by their value in terms of the expectation values of the lower order similar expressions, a new expression is obtained from which all the mixed products of elements of holes RDM s and of RDM s may be eliminated without difficulty. As a result, one obtains a relation which can be summarized as ... 

The 4-RDMs for the ground state and first excited singlet state were calculated with the UA (Eq. (71)) for different values of the parameter. The first three columns correspond, respectively, to the GP-H, the VCP, and to the NYM algorithms. The different RDM elements were analyzed in order to find out if they satisfied the bounds inequalities. If this were not the case, these elements values would be given the values corresponding to the upper or lower bound which had been unsatisfied. In order to examine the performance of the different algorithms and of the bounds without the influence of the quality of the data used, the 1-, 2- and 3-RDMs employed to calculate the 4-RDM were obtained in an FCI treatment. The different results were then compared with the FCI... 

The reconstruction functionals may be understood as substantially renormalized many-body perturbation expansions. When exact lower RDMs are employed in the functionals, contributions from all orders of perturbation theory are contained in the reconstructed RDMs. As mentioned previously, the reconstruction exactly accounts for configurations in which at least one particle is statistically isolated from the others. Since we know the unconnected p-RDM exactly, all of the error arises from our imprecise knowledge of the connected p-RDM. The connected nature of the connected p-RDM will allow us to estimate the size of its error. For a Hamiltonian with no more than two-particle interactions, the connected p-RDM will have its first nonvanishing term in the (p — 1) order of many-body perturbation theory (MBPT) with a Hartree-Fock reference. This assertion may be understood by noticing that the minimum number of pairwise potentials V required to connectp particles completely is (p — 1). It follows from this that as the number of particles p in the reconstmcted RDM increases, the accuracy of the functional approximation improves. The reconstmction formula in Table I for the 2-RDM is equivalent to the Hartree-Fock approximation since it assumes that the two particles are statistically independent. Correlation corrections first appear in the 3-RDM functional, which with A = 0 is correct through first order of MBPT, and the 4-RDM functional with A = 0 is correct through second order of MBPT. 

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