A-particle density matrices

The two updates differ only by a factor of one-half before the first-order change from A and the second-order change. Unlike the wavefunction power method, the A -particle density matrices from each iteration in Eq. (Ill) are not exactly positive semidehnite until convergence. 

This is the desired result which may be substituted into the scattering amplitude formula (6). The resulting scattering formula is the same as found by other authors [5], except that in this work SI units are used. The contributions to the Fourier component of magnetic field density are seen to be the physically distinct (i) linear current JL and (ii) the magnetisation density Ms associated with the spin density. A concrete picture of the physical system has been established, in contrast to other derivations which are heavily biased toward operator representations [5]. We note in passing that the treatment here could be easily extended to inelastic scattering if transition one particle density matrices (x x ) were used in Equations (12)—(14). 

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... 

Direct minimization of the energy as a functional of the p-RDM may be achieved if the p-particle density matrix is restricted to the set of Al-represen-table p-matrices, that is, p-matrices that derive from the contraction of at least one A-particle density matrix. The collection of ensemble Al-representable p-RDMs forms a convex set, which we denote as P. To define P, we first consider the convex set of p-particle reduced Hamiltonians, which are... 

D. A. Mazziotti, Contracted Schrodinger equation determining quantum energies and two-particle density matrices without wave functions. Phys. Rev. A 57, 4219 (1998). 

Consequently, we can carry out the BCH expansion to arbitrarily high order without any increase in the complexity of the terms in the effective Hamiltonian. In practice, the expansion is carried out until convergence in a suitable norm of the operator coefficients is achieved, as illustrated in Table I. Rapid convergence is usually observed. Note that through the decomposition (23), the effective Hamiltonian depends on the one- and two-particle density matrices and therefore becomes state specific, much like the Fock operator in Hartree-Fock theory. 

We now discuss (ii), the evaluation of operator expectation values with the reference ho- We are interested in multireference problems, where may be extremely complicated (i.e., a very long Slater determinant expansion) or a compact but complex wavefunction, such as the DMRG wavefunction. By using the cumulant decomposition, we limit the terms that appear in the effective Hamiltonian to only low-order (e.g., one- and two-particle operators), and thus we only need the one- and two-particle density matrices of the reference wavefunction to evaluate the expectation value of the energy in the energy expression (7). To solve the amplitude equations, we further require the commutator of which, for a two-particle effective Hamiltonian and two-particle operator y, again involves the expectation value of three-particle operators. We therefore invoke the cumulant decomposition once more, and solve instead the modihed amplitude equation... 

Consequently, with the simplihcations above, all the working equations of the canonical transformation theory can be evaluated entirely in terms of a limited number of reduced density matrices (e.g., one- and two-particle density matrices) and no explicit manipulation of the complicated reference function is required. 

First consider a Hartree-Fock reference function and transform to the Fermi vacuum (aU occupied orbitals are in the vacuum). Then all particle density matrices are zero and the cumulant decomposition, Eq. (23), based on this reference corresponds to simply neglecting aU three and higher particle-rank operators generated by commutators. This type of operator truncation is used in the canonical diagonalization theory of White [22]. 

Here, we will focus our attention on the 2-RDM and the 3-RDM which are the matrices appearing in the l-CSE aiming at describing them in terms of a single particle density matrices. 

In this paper we have reported a series of theoretical results which contribute to clarify how the many-body effects can be expressed in terms of the single particle density matrices 1-RDM and 1-TRDM. Whether it is possible to approximate a solution to this system of non-linear equations remains an open question after our initial study of this problem. 

Finally, px and rY, with X = A or B, are the conventional one- and two-particle density matrices for monomer X, normalized to Nx and NX(NX — 1), respectively. In Eqs. (1-75) and (1-76) qj = (r Sj) denotes the space and spin coordinates of the /th electron. Since theoretical methods for the evaluation of the density matrices px and Yx for many-electron molecules are well developed, Eqs. (1-75) and (1-76) enable practical calculations of the first-order exchange energy using accurate electronic wave functions of the monomers A and B116. 

The other terms of Eq. (1-225) follow by permutation of A, B and C. Similarly as in the two-body case the first-order exchange nonadditivity can be expressed through one- and two-particle density matrices of the isolated monomers314,... 

All matrix elements in the Newton-Raphson methods may be constructed from the one- and two-particle density matrices and transition density matrices. The linear equation solutions may be found using either direct methods or iterative methods. For large CSF expansions, such micro-iterative procedures may be used to advantage. If a micro-iterative procedure is chosen that requires only matrix-vector products to be formed, expansion-vector-dependent effective Hamiltonian operators and transition density matrices may be constructed for the efficient computation of these products. Sufficient information is included in the Newton-Raphson optimization procedures, through the gradient and Hessian elements, to ensure second-order convergence in some neighborhood of the final solution. 

In the proof essential use is made of the fact that the density and the potential are conjugate variables. For the same reason we can, for instance, prove that the two-particle interaction is a unique functional of the diagonal two-particle density matrix. The general mapping between /-particle density matrices and /V-body potentials is discussed by De Dominicis and Martin [6]. 

The higher order particle density matrices are adl factorizable into antisymmetric products of one body a s. Mixed densities with particle as well as valence have a similar factorizability involving one-body cr s and R s with valence labels. 

For SCF methods, p represents elements of the one- and two-particle density matrices. In correlated methods, the one-particle part of p is the sum of the aaual reduced density and a contribution that is proportional to the derivative of the energy with respect to orbital rotations. For the MCSCF method, the orbitals are variationally optimum, and this latter term vanishes. Similarly, for FCI there is no orbital contribution. However, it is required for other Cl, CC, and MBPT correlated methods, because the energy in these approaches is not stationary with respect to first-order changes in the molecular orbitals. This... 

The n-particle density matrices are widely used in molecular quantum mechanics (e.g.[10-12], where further references may be found. A well known property of the... 

In the normal (probability theory) use of the term, two probability distributions are not correlated if their joint (combined) probability distribution is just the simple product of the individual probability distributions. In the case of the Hartree-Fock model of electron distributions the probability distribution for pairs of electrons is a product corrected by an exchange term. The two-particle density function cannot be obtained from the one-particle density function the one-particle density matrix is needed which depends on two sets of spatial variables. In a word, the two-particle density matrix is a (2 x 2) determinant of one-particle density matrices for each electron ... 

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