A-representability conditions

The RDM s are therefore much simpler objects than the A-electron Wave Function (WF) which depends on the variables of N electrons. Unfortunately, the search for the 7V-representability conditions has not been completed and this has hindered the direct use of the RDM s in Quantum Chemistry. In 1963 A. J. Coleman [4] defined the A -representability conditions as the limitations of an RDM due to the fact that it is derived by contraction from a matrix represented in the N-electron space. In other words, an antisymmetric A-electron WF must exist from which this RDM could have been derived by integrating with respect to a set of electron variables. 

The basic relations for stndying the properties of the RDMs are the anticommu-tation/commntation relations of gronps of fermion operators since their expectation values give a set of A-representability conditions of the RDMs. Thus,... 

Since both the RDMs and the HRDMs are positive matrices, this relation says that the eigen-value pi of the 1 -RDM, must be 0 < p, < 2 which is the well known ensemble A-representability condition for the -RDM [10] represented in an orbital basis (in a spin-orbital representation the upper bound would be 1 instead of 2). 

As we show later, the energy of the state of any system of N indistinguishable fermions or bosons can be expressed in terms of the Hamiltonian and D (12,1 2 ) if its Hamiltonian involves at most two-particle interactions. Thus it should be possible to find the ground-state energy by variation of the 2-matrix, which depends on four particles. Contrast this with current methods involving direct use of the wavefunction that involves N particles. A principal obstruction for this procedure is the A-representability conditions, which ensure that the proposed RDM could be obtained from a system of N identical fermions or bosons. 

Others (e.g., Fukashi Sasaki s upper bound on eigenvalues of 2-RDM [2]). Claude Garrod and Jerome Percus [3] formally wrote the necessary and sufficient A -representability conditions. Hans Kummer [4] provided a generalization to infinite spaces and a nice review. Independently, there were some clever practical attempts to reduce the three-body and four-body problems to a reduced two-body problem without realizing that they were actually touching the variational 2-RDM method Fritz Bopp [5] was very successful for three-electron atoms and Richard Hall and H. Post [6] for three-nucleon nuclei (if assuming a fully attractive nucleon-nucleon potential). 

The partial sucess/failure slowed down the applications, especially as the computers at that time were too slow to manage larger model spaces and additional, more complicated, A -representability conditions. Some hope was offered by applying symmetries—orbital rotation, spin, isobaric spin—and it was stimulating to explore them with Bob Erdahl and my younger collaborator Bojan Golli [32], However, new ideas were needed. 

Both the energy as well as the one- and two-electron properties of an atom or molecule can be computed from a knowledge of the 2-RDM. To perform a variational optimization of the ground-state energy, we must constrain the 2-RDM to derive from integrating an A -electron density matrix. These necessary yet sufficient constraints are known as A -representability conditions. 

General p-particle A -representability conditions on the 2-RDM are derivable from metric (or overlap) matrices. From the ground-state wavefunction lih) and a set of p-particle operators. of basis functions can be defined. 

In contrast to the and metric matrices in 3-positivity, the strength of the T2 matrix as a 2-RDM A -representability condition is not completely invariant upon altering the order of the second-quantized operators in C, j - For example, a slightly different metric matrix T2 can be defined by exchanging the operators a, and a in Eq. (40) to obtain... 

When N = p, the set B simply contains the p-particle reduced Hamiltonians, which are positive semidefinite, but when N = p + 1, because the lifting process raises the lowest eigenvalue of the reduced Hamiltonian, the set also contains p-particle reduced Hamiltonians that are lifted to positive semidefinite matrices. Consequently, the number of Wrepresentability constraints must increase with N, that is, B C B. To constrain the p-RDMs, we do not actually need to consider all pB in B, but only the members of the convex set B, which are extreme A member of a convex set is extreme if and only if it cannot be expressed as a positively weighted ensemble of other members of the set (i.e., the extreme points of a square are the four corners while every point on the boundary of a circle is extreme). These extreme constraints form a necessary and sufficient set of A-representability conditions for the p-RDM [18, 41, 42], which we can formally express as... 

The A -representability conditions on the 2-RDM can be systematically strengthened by adding some of the 3-positivity constraints to the 2-positivity conditions. For three molecules in valence double-zeta basis sets Table II shows... 

Following Ref. [5] the T1 condition is obtained by considering an operator A = Y ij gij,kaiajak, where the gij k are arbitrary real or complex coefhcients totally antisymmetric in the three indices. (We view g as a vector of dimension (0, where r is the size of the one-electron basis.) The contractions (t / A+A t /) and (t / AA+ t /) both involve the 3-RDM, but with opposite sign, and so the nonnegativity of (tk 4 4 -f AA I ) for all three-index functions g provides a representability condition involving only the 1-RDM and 2-RDM. In exphcit form the condition is of semidefinite form, 0 T, where the Hermitian matrix T is... 

The inclusion of other known A-representability conditions like G, Tl, and T2 [14] in the variational calculation can be embedded into the primal SDP problem in a similar way. 

A -representability conditions [28]. Let us start this description by focusing on the RDM s properties, which may be deduced from their definition as expectation values of density fermion operators. Thus the ROMs are Hermitian, are positive semidefinite, and contract to finite values that depend on the number of electrons, N, and in the case of the HRDMs on the size of the one-electron basis of representation, 2/C. Thus... 

The structure of the parametric UA for the 4-RDM satisfies the fourth-order fermion relation (the expectation value of the commutator of four annihilator and four creator operators [26]) for any value of the parameter which is a basic and necessary A-representability condition. Also, the 4-RDM constructed in this way is symmetric for any value of On the other hand, the other A-representability conditions will be affected by this value. Hence it seems reasonable to optimize this parameter in such a way that at least one of these conditions is satisfied. Alcoba s working hypothesis [48] was the determination of the parameter value by imposing the trace condition to the 4-RDM. In order to test this working hypothesis, he constructed the 4-RDM for two states of the BeHa molecule in its linear form Dqo/,. The calculations were carried out with a minimal basis set formed by 14 Hartree-Fock spin orbitals belonging to three different symmetries. Thus orbitals 1, 2, and 3 are cr orbitals 4 and 5 are cr and orbitals 6 and 7 are degenerate % orbitals. The two states considered are the ground state, where... 

As shown in the second line, like the expression for the energy as a function of the 2-RDM, the energy E may also be expressed as a linear functional of the two-hole reduced density matrix (2-HRDM) and the two-hole reduced Hamiltonian K. Direct minimization of the energy to determine the 2-HRDM would require (r — A)-representability conditions. The definition for the p-hole reduced density matrices in second quantization is given by... 

Some of the most important A-representability conditions on the 2-RDM arise from its relationship with the 1-RDM. A 2-RDM must contract to a 1-RDM that is A-representable,... 

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 ... 

The remarkable fact, first demonstrated by Nakatsuji [18], is that for each p >2, CSE(p) is equivalent (in a necessary and sufficient sense) to the original Hilbert-space eigenvalue equation, Eq. (2), provided that CSE(p) is solved subject to boundary conditions (A -representability conditions) appropriate for the (p + 2)-RDM. CSE(p), in other words, is a closed equation for the (p+ 2)-RDM (which determines the (p + 1)- and p-RDMs by partial trace) and has a unique A -representable solution Dp+2 for each electronic state, including excited states. Without A -representability constraints, however, this equation has many spurious solutions [48, 49]. CSE(2) is the most tractable reduced equation that is still equivalent to the original Hilbert-space equation, and ultimately it is CSE(2) that we wish to solve. Importantly, we do not wish to solve CSE(2) for... 

Earlier iterative solutions of the CSE for the 2-RDM often required that the 2-RDM be adjusted to satisfy important A-representabUity conditions in a process called purification [18, 24]. The solution of the ACSE automatically maintains the A-representability of the 2-RDM within the accuracy of the 3-RDM reconstruction. Necessary A-representability conditions require keeping the eigenvalues of three different forms of the 2-RDM, known as the D, Q, and... 

A direction for improving DPT lies in the development of a functional theory based on the one-particle reduced density matrix (1-RDM) D rather than on the one-electron density p. Like 2-RDM, the 1-RDM is a much simpler object than the A-particle wavefunction, but the ensemble A-representability conditions that have to be imposed on variations of are well known [1]. The existence [10] and properties [11] of the total energy functional of the 1-RDM are well established. Its development may be greatly aided by imposition of multiple constraints that are more strict and abundant than their DPT counterparts [12, 13]. 

To satisfy the known necessary A -representability conditions for the 2-RDM, the matrix elements of A have to conform to the following analytic constraints ... 

A -representability conditions on the g-density were originally intended for use in variational optimization with respect to the g-electron reduced density matrix. 

The similarities and differences between methods based on the electron density, electron-pair density, and reduced density matrices have recently been reviewed [5]. This chapter is not intended as a comprehensive review, but as a focused consideration of A-representability constraints that are applicable to diagonal elements of reduced density matrices. Such constraints are useful both to researchers working with the Q-density and to researchers working with g-electron reduced density matrices, and so we shall attempt to review these constraints in a way that is accessible to both audiences. Our focus is on inequalities that arise from the Slater hull because the Slater hull provides an exhaustive list of A-representability conditions for the diagonal elements of the density matrix, Although the Slater hull constraints... 

The argument in Eq. (77) can be generalized to higher-order electron distribution functions [28]. Unfortunately, the other A -representability conditions in Section III.G do not seem amenable to this approach. 

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