Higher-order RDMs

All these steps are bnilt into an iterative procednre whose success pivots on the approximation of the higher order RDM s in terms of the 2-RDM. This important part of the method will be addressed in the next section. 

The convention followed here is that the row/column indices of the RDM s coincide respectively with the creator/annihilator labels. The form of all kinds of higher order RDM s can be inferred by extension ofthe definitions given above. 

Higher-Order RDMs Construction Algorithms Other Approaches A Unifying Algorithm 1. A Criterion for Selecting the Parameter Value for the 4-RDM... 

The arguments just described for the construction of the 2-RDM were extended without difficulty to the higher-order RDMs. The algorithms for these high-order RDMs were originally reported by Colmenero et al. [20] in a spin-free basis and Valdemoro et al. [46] obtained later on the algorithms in a spin-orbital basis. For the 3-RDM, the algorithm in a spin-orbital basis is... 

In 2001 Valdemoro et al. [54] proposed a generalization of the VCP basic approach. It exploits the fact that the 2-RDM has more information than the 1-RDM and instead of replacing one Kronecker delta in terms of 1-RDM and 1-HRDM, one can replace functions of Kronecker deltas in terms of higher-order RDMs and HRDMs. In this way, one partly avoids the cancelation of the correction terms pointed out by Mazziotti [45]. 

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. 

Nakatsuji and Yasuda [56, 57] derived the 3- and 4-RDM expansions, in analogy with the Green function perturbation expansion. In their treatment the error played the role of the perturbation term. The algorithm that they obtained for the 3-RDM was analogous to the VCP one, but the matrix was decomposed into two terms one where two A elements are coupled and a higher-order one. Neither of these two terms can be evaluated exactly thus, in a sense, the difference with the VCP is just formal. However, the structure of the linked term suggested a procedure to approximate the A error, as will be seen later on. 

Although a formal solution of the A-representability problem for the 2-RDM and 2-HRDM (and higher-order matrices) was reported [1], this solution is not feasible, at least in a practical sense [90], Hence, in the case of the 2-RDM and 2-HRDM, only a set of necessary A-representability conditions is known. Thus these latter matrices must be Hermitian, Positive semidefinite (D- and Q-conditions [16, 17, 91]), and antisymmetric under permutation of indices within a given row/column. These second-order matrices must contract into the first-order ones according to the following relations ... 

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]. 

Each of the reconstructions contains many contributions from higher orders of perturbation theory via the 1- and 2-RDMs and thus may be described as highly renormalized. The CSE requires a second-order correction of the 3-RDM functional to generate second-order 2-RDMs and energies, but the ACSE can produce second-order 2-RDMs and third-order energies from only a first-order reconstruction of the 3-RDM. 

In the 1Y case there is, I think, no problem since, provided the higher order conditions are fulfilled, each iteration should improve the 1 -RDM, and consequently the estimation of the 1Y would also improve. 

The unknowns in these three equations axe not only the Energy and the 2-RDM but also the higher order matrices - the 3-RDM and the 4-RDM. This, as has been mentioned, is what causes the indeterminacy of these equations. In spite of this drawback an approximate solution may be obtained (12, 13) by estimating the 3- and 4-RDM from an initial 1-and 2-RDM, replacing them into the equation and continue iterating until consistency. As can be observed, these three equations axe coupled since the same high order RDM appears in more than one equation which, as will be discussed later on, renders necessary their simultaneous solution. 

Next, we present some observations concerning the connection between the reconstruction process and the iterative solution of either CSE(p) or ICSE(p). The perturbative reconstruction functionals mentioned earlier each constitute a finite-order ladder-type approximation to the 3- and 4-RDMCs [46, 69] examples of the lowest-order corrections of this type are shown in Fig. 3. The hatched squares in these diagrams can be thought of as arising from the 2-RDM, which serves as an effective pair interaction for a form of many-body perturbation theory. Ordinarily, ladder-type perturbation expansions neglect three-electron (and higher) correlations, even when extended to infinite order in the effective pair interaction [46, 69], but iterative solution of the CSEs (or ICSEs) helps to... 

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