Two-particle matrix

The unitary decomposition may be applied to any Hermitian, antisymmetric two-particle matrix including the 2-RDM, the two-hole RDM, and the two-particle reduced Hamiltonian. The decomposition is also readily generalized to treat p-particle matrices [80-82]. The trial 2-RDM to be purified may be written... 

These operators can be averaged in the same manner as in Chapter 14 where we have introduced the average operator of electrostatic interaction of electrons in a shell. The main departure of the case at hand is that the Pauli exclusion principle, owing to the fact that electrons from different shells are not equivalent, imposes constraints neither on the pertinent two-particle matrix elements nor on the number of possible pairing states, which equals (4/i + 2)(4/2 + 2). The averaged submatrix element of direct interaction between the shells will then be... 

This paper explores the use of the two-particle matrix in terms of two typical applications, both of which bear on Henry Eyring s work. They are (a) bond strength using a molecule-in-molecules approach, and (b) static indices for Diels-Alder reactions. We consider these separately. But in view of the different definitions of the density matrices, a preliminary section is desirable. 

We now imagine the two-particle matrix to be expanded in terms of 9 -, (p j and other atomic or ethylenic orbitals and we measure the coefficient with which the term 9 ,-,(l) (2)9P (l ), (2 ) occurs. We choose this term because, for a pure ethylenic double bond, the coefficient of this term is unity. By seeking the coefficient of this term in the full molecular two-particle matrix, we are measuring some kind of double-bond character for the bond. For a closed-shell molecule it is not difficult to show that this coefficient is... 

If we are considering the two-particle matrix the obvious thing to do is to expand in a basis consisting of at least the two atomic orbitals 9 and on the two sites of attack, and look for the coefficient (C b) in the spinless density matrix of the form 9 a(1) 9b(2) 95a(10 b(2 ). The appropriate term in the one-particle matrix would then. be [Pg.312]

The angular decomposition of the two-particle matrix element Pijki is easily carried out and leads to... 

Here a etc. are excitation operators with respect to an orthonormal spin-orbital basis V p, while AJ and 9II are one- and two-particle matrix elements... 

In course of the evaluation of the vacuum amplitudes, only those terms survive eventually in this step which are completely contracted, i.e. those with no creation or annihilation operators at all in the final expression. In other words, each (non-zero) vacuum amplitude is written as a superposition of just one- and two-particle matrix elements of various kinds (due to the one- and two-particle character of all atomic and molecular interactions), and including summations over the core, core-valence, valence and/or virtual orbitals. In certain cases, it has been found useful to combine the steps (1) and (3) and to evaluate the vacuum amplitudes directly from the rhs of the full matrix elements in second quantization. 

In addition, the two-particle matrix elements could be obtained from... 

The eigenfiinctions of a system of two particles are detemiined by their positions x and j, and the density matrix is generalized to... 

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

To study the structure of the exchange-correlation energy functional, it is useful to relate this quantity to the pair-correlation function. The pair-correlation function of a system of interacting particles is defined in terms of the diagonal two-particle density matrix (for an extensive discussion of the properties of two-particle density matrices see [30]) as... 

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. 

Then, in the Old Ages (1940 or 1951-1967) some ingenious people became aware that, in the case of two-body interactions, it is the two-particle reduced density matrix (2-RDM) that carries in a compact way all the relevant information about the system (energy, correlations, etc.). Early insight by Husimi (1940) and challenges by Charles Coulson were completed by a clear realization and formulation of the A-representability problem by John Coleman in 1951 (for the history, see his book [1] and Chapters 1 and 17 of the present book). Then a series of theorems on A-representability followed, by John Coleman and many... 

While all three matrices are interconvertible, the nonnegativity of the eigenvalues of one matrix does not imply the nonnegativity of the eigenvalues of the other matrices, and hence the restrictions Q>0 and > 0 provide two important 7/-representability conditions in addition to > 0. These conditions physically restrict the probability distributions for two particles, two holes, and one particle and one hole to be nonnegative with respect to all unitary transformations of the two-particle basis set. Collectively, the three restrictions are known as the 2-positivity conditions [17]. 

The three complementary representations of the reduced Hamiltonian offer a framework for understanding the D-, the Q-, and the G-positivity conditions for the 2-RDM. Each positivity condition, like the conditions in the one-particle case, correspond to including a different class of two-particle reduced Hamiltonians in the A-representability constraints of Eq. (50). The positivity of arises from employing all positive semidefinite in Eq. (50) while the Q- and the G-conditions arise from positive semidefinite and B, respectively. To understand these positivity conditions in the particle (or D-matrix) representation, we define the D-form of the reduced Hamiltonian in terms of the Q- and the G-representations ... 

If the G-matrix is positive semidefinite, then the above expectation value of the G-matrix with respect to the vector of expansion coefficients must be nonnegative. Similar analysis applies to G, operators expressible with the D- or Q-matrix or any combination of D, Q, and G. Therefore variationally minimizing the ground-state energy of n (H Egl) operator, consistent with Eq. (70), as a function of the 2-positive 2-RDM cannot produce an energy less than zero. For this class of Hamiltonians, we conclude, the 2-positivity conditions on the 2-RDM are sufficient to compute the exact ground-state A-particle energy on the two-particle space. 

D. A. Mazziotti, Variational minimization of atomic and molecular ground-state energies via the two-particle reduced density matrix. Phys. Rev. A 65, 062511 (2002). 

T. Juhasz and D. A. Mazziotti, Perturbation theory corrections to the two-particle reduced density matrix variational method. J. Chem. Phys. 121, 1201 (2004). 

Nakatsuji [37] in 1976 first proved that with the assumption of N-representability [3] a 2-RDM and a 4-RDM will satisfy the CSE if and only if they correspond to an A-particle wavefunction that satishes the corresponding Schrodinger equation. Just as the Schrodinger equation describes the relationship between the iV-particle Hamiltonian and its wavefunction (or density matrix D), the CSE connects the two-particle reduced Hamiltonian and the 2-RDM. However, because the CSE depends on not only the 2-RDM but also the 3- and 4-RDMs, it cannot be solved for the 2-RDM without additional constraints. Two additional types of constraints are required (i) formulas for building the 3- and 4-RDMs from the 2-RDM by a process known as reconstruction, and (ii) constraints on the A-representability of the 2-RDM, which are applied in a process known as purification. 

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... 

Any two-particle Hermitian matrix may be decomposed into three components that exist in different subspaces of the unitary group. These components reveal the structure of the matrix with respect to the contraction operation [4, 76-80],... 

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