<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math>-representability problem within the framework of the contracted Schrödinger equation

Valdemoro, C.; Tel, Luis M.; Pérez‐Romero, E.

doi:10.1103/physreva.61.032507

Cited by 66 publications

(55 citation statements)

References 37 publications

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“…Although a definition of the q-CRDM is possible at all orders, we are concerned here by the cases q ¼ 1 and q ¼ 2 only, for which a simpler expression is available (see Eqs. (4) and (5)). …”

Section: Theoretical Backgroundmentioning

confidence: 96%

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Locality of the reduced-density-matrices: a numerical study

Bories

Evangelisti

Leininger

et al. 2004

Chemical Physics Letters

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Section: Theoretical Backgroundmentioning

confidence: 96%

“…It would be attractive to solve variationally the nonrelativistic Schrodinger equation by minimizing the electronic energy with the help of the 2-RDM. But this problem has not been entirely solved yet [3] although large progress has been made for years in the framework of the contracted Schrodinger equation (CSE) [4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Locality of the reduced-density-matrices: a numerical study

Bories

Evangelisti

Leininger

et al. 2004

Chemical Physics Letters

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“…Recently, density matrix reconstruction methods have required estimation of the three-and four-body density matrix [5][6][7][8][9][10]. Some N-representability conditions have been suggested in the recent literature for these higher-order density matrices [11][12][13]. This has suggested that it would be useful to extend our previous results to higher-order density matrices.…”

mentioning

confidence: 87%

“…For a wave function with definite N, the terms in † that do not conserve particle number give zero and the rest can be evaluated with the q-matrix formed from . This general form includes all of the inequalities report by Mazziotti and Erdahl [11] and Valdemoro et al [12,13].…”

mentioning

confidence: 96%

Linear inequalities for density matrices: III

Davidson

2002

Int J of Quantum Chemistry

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Diagonal linear inequalities are derived for three-and four-body reduced density matrices. These give a different insight into the previous derivations of linear inequalities for the two-body reduced density matrix. Key words: N-representability; density matrices; linear inequalities I n the two previous articles in this series [1, 2], we derived some necessary linear inequalities for the N-representability of the two-body reduced density matrix. These results extended the inequalities known from the previous work of Coleman [3] and Weinhold and Wilson [4]. Recently, density matrix reconstruction methods have required estimation of the three-and four-body density matrix [5][6][7][8][9][10]. Some N-representability conditions have been suggested in the recent literature for these higher-order density matrices [11][12][13]. This has suggested that it would be useful to extend our previous results to higher-order density matrices. As will be shown here, doing this actually provides a simpler derivation of our previous results.The most interesting inequalities among the previous results were derived by considering the socalled Slater hull ensemble. If F is a polynomial of maximum degree q in products of second-quantized number operators, n i , involving up to r orbitals, then the eigenfunctions of F are Slater determinants with N electrons. The average of F over a variable-N ensemble of arbitrary wave functions reduces to a weighted average of the eigenvalues of F and consequently cannot be lower than the lowest eigenvalue for any N Յ r. This average can be expressed in terms of the diagonal elements of the q-body reduced density matrix (q-matrix for short). Consequently, each F provides a necessary N-representability condition on the q-matrix. Considering all F generates an infinite set of inequalities continuous in the coefficients with which these operators enter into F. It can be shown that these provide redundant information, and it is possible to find a finite discrete set of essential inequalities that guarantee that the q-matrix will satisfy the inequalities generated by considering every F.The construction of these essential inequalities begins by considering a space where the variables are the average values of each operator 1, n i , n i n j , . . . that could occur in an F of maximum degree q involving r orbitals. The average of these operators with each Slater determinant with N Յ r can be plotted as a point in this space. The convex

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“…This important property is counterbalanced by the fact that it depends on the 4-CM. On the other hand, it is far more convenient to solve this 3-order GHV equation than the CCSE/2-CSE since, as will be seen below, the iterative procedure for solving this hypervirial equation preserves the N-representability of the resulting solutions, which is not the case in the CCSE/2-CSE methods where an N-representability purification procedure must be combined with the iterative process [21,27,31,33,43,48,[63][64][65]. As mentioned earlier, although a similar theorem has not yet been demonstrated for the (2-order) GHV equation, the results it yielded have proved to be highly accurate, which is why in what follows we will focus on this second-order equation.…”

Section: The Anti-hermitian Part Of the 3-order Ccsementioning

confidence: 98%

Theoretical and applicative properties of the correlation and G‐particle‐hole matrices

Valdemoro

Alcoba

Tel

et al. 2009

Int J of Quantum Chemistry

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We have recently (Valdemoro et al., Sixth International Congress of theInternational Society for Theoretical Chemical Physics, 2008; Alcoba et al., Int J Quantum Chem, in press) reported the form of the G-particle-hole hypervirial equation, which can be identified with the anti-Hermitian part of the correlation contracted Schrödinger equation (Alcoba, Phys Rev A, 2002, 65, 032519), as a tool to obtain the second-order reduced density matrix of an N-electron system without previous knowledge of the wave-function. The results which have been obtained when solving the G-particle-hole hypervirial equation with an iterative method also described in (Valdemoro et al., Sixth International Congress of the International Society for Theoretical Chemical Physics, 2008; Alcoba et al., Int J Quantum Chem, in press) have been highly accurate. The convergence of these test calculations has been very smooth, though rather slow. One of the factors which determines the performance of the method is the accuracy with which the 3-order correlation matrices (3-CM) involved in the calculations are approximated. It is, therefore, necessary to optimize to the utmost the construction algorithms of these 3-order matrices in terms of the 2-CM. In this article, the main theoretical features of the p-CM are described. Also, some aspects of the correlation contracted Schrödinger equation and of the G-particle-hole hypervirial equation are revisited. A new theorem, concerning the sufficiency of the hypervirial of the 3-order correlation operator to guarantee a correspondence between its solution and that of the Schrödinger equation, and some preliminary results concerning the constructing algorithms of the 3-CM in terms of the 2-CM, are reported in the second part of this article.

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N-representability problem within the framework of the contracted Schrödinger equation

Cited by 66 publications

References 37 publications

Locality of the reduced-density-matrices: a numerical study

Locality of the reduced-density-matrices: a numerical study

Linear inequalities for density matrices: III

Theoretical and applicative properties of the correlation and G‐particle‐hole matrices

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