Two-particle density operator

B. Binary Collision Approximation for the Two-Particle Density Operator— Kinetic Equations for Free Particles and Atoms... 

Because of the interaction this equation is not closed it is coupled with the two-particle density operator. Therefore, it is necessary to find an... 

It is essential to note that the contribution to the potential energy comes from the retardation correction of the two-particle density operator, which describes three-particle interactions of the self-energy type. 

We will carry out our program in two steps. In this section we will derive the two-particle density operator Fn in a three-particle collision approximation for the application in the collision integral of Fl. As compared with Section II.2, the main difference will be the occurrence of bound states and, especially, the generalization of the asymptotic condition, which now has to account for bound states too. For the purpose of the application in the kinetic equation of the atoms (bound states) we need an approximation of the next-higher-order density matrix, that is, F 23 This quantity will be determined under inclusion of certain four-particle interaction. 

The spin-free two-particle excitation operators and density matrices are symmetric with respect to simultaneous exchange of the upper and lower indices, but neither symmetric nor antisymmetric with respect to exchange of either upper or lower indices separately ... 

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... 

Obviously we may expect that the simple two- and three-particle collision approximation discussed in the previous sections is not appropriate, because a large number of particles always interact simultaneously. Formally this approximation leads to divergencies. In the previous sections we used in a systematic way cluster expansions for the two- and three-particle density operator in order to include two-particle bound states and their relevant interaction in three- and four-particle clusters. In the framework of that consideration we started with the elementary particles (e, p) and their interactions. The bound states turned out to be special states, and, especially, scattering states were dealt with in a consistent manner. 

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. 

An important consequence of the inclusion of an intraatomic repulsion term is the possibility of modeling magnetic features in partially filled, narrow bands. Some elements of the two-particle density matrix, for instance , derive from the propagator matrix and give an indication of the local spin features. The use of the additional operators requires some further algebra. The metric measures... 

To extract the linearly independent excitations, we shall have to use the so-called singular value decomposition of the valence density matrices generated by the creation/annihilation operators with valence-labels which axe present in the particular excitation operator in T. To illustrate this aspect, let us take an example. For any excitation operator containing the destruction of a pair of active orbitals from V o the overlap matrix of all such excited functions factorize, due to our new Wick s theorem, into antisymmetric products of one-body densities with non-valence labels and a two-particle density matrix ... 

Just as for Green s functions, expressions like two-particle operator and two-particle density matrix refer to the number of particles involved in the definition of the operator (two in the case of an interaction, one for a potential energy, etc.), not to the total number of particles present in the system. 

Note that all multiplicative two-particle interaction operators cancel in the commutator. Hence, the definitions of density and current density do not depend on the choice of the electron-electron interaction operators as long as they do not depend on momentum operators. The right-hand side of the... 

We expanded the X operators into the primitive products of creation and annihilation operators to achieve this factorization. Instead of first expanding the x in terms of X, we can expand them directly in the primitive products of creation and annihilation operators, to obtain the sums of products of parentage coefficients. As an example, consider the second off-diagonal block, for which the two-particle density matrix factorizes naturally ... 

Between the limits of small and large r, the pair distribution function g(r) of a monatomic fluid is detemrined by the direct interaction between the two particles, and by the indirect interaction between the same two particles tlirough other particles. At low densities, it is only the direct interaction that operates through the Boltzmaim distribution and... 

Conditional probability, 267 density function, 152 Condon, E. U., 404 Configuration space amplitude, 501 Heisenberg operator, 507 operators, 507, 514, 543 Conservation laws for light particles (leptons), 539 for heavy particles (baryons), 539 Continuous memoryless channels, 239 Contraction symbol for two time-labelled operators, 608 Control of flow, 265 Converse to coding theorem, 215 Convex downward function, 210 Convex upward function, 209 Cook, L. F 724... 

In general, the equations for the density operator should be solved to describe the kinetics of the process. However, if the nondiagonal matrix elements of the density operator (with respect to electron states) do not play an essential role (or if they may be expressed through the diagonal matrix elements), the problem is reduced to the solution of the master equations for the diagonal matrix elements. Equations of two types may be considered. One of them is the equation for the reduced density matrix which is obtained after the calculation of the trace over the states of the nuclear subsystem. We will consider the other type of equation, which describes the change with time of the densities of the probability to find the system in a given electron state as a function of the coordinates of heavy particles Pt(R, q, Q, s,...) and Pf(R, q, ( , s,... ).74,77 80... 

M. Rosina, (a) Direct variational calculation of the two-body density matrix (b) On the unique representation of the two-body density matrices corresponding to the AGP wave function (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions (d) Hermitian operator method for calculations within the particle-hole space in Reduced Density Operators with Applications to Physical and Chemical Systems—II (R. M. Erdahl, ed.), Queen s Papers in Pure and Applied Mathematics No. 40, Queen s University, Kingston, Ontario, 1974, (a) p. 40, (b) p. 50, (c) p. 57, (d) p. 126. 

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