Density matrix form

We now connect the analysis given above with the equation of motion displayed in Eq. (5.5). That equation of motion follows from subdivision of a system into an open subsystem S and a complementary reservoir R. When the coupling between S and R is weak, the evolution of the open system 5, due to the internal dynamics of 5 and the interaction with the reservoir R, can be described in density matrix form by Eq. (5.5). Now writing... 

Given (7.12) it is straightforward to obtain the corresponding density matrix form of the equation of motion. The time dependence of the density matrix element for the mth-state population in the P subspace is found to be... 

Fig. IX. 1. Product operators for a two-spin system. The density matrix form is shown along with the vector representation (adapted from [1]).

The initial density matrix formed by the preparatory pulse [Pg.174]

We shall use the density matrix form of the equations so that we do not have to think about the arbitrariness of the choice of orbital basis in the occupied and virtual spaces. Recall the equations satisfied by the self-consistent JJ-matrix ... 

This equation may be brought into a density matrix form by using the usual... 

A number of procedures have been proposed to map a wave function onto a function that has the form of a phase-space distribution. Of these, the oldest and best known is the Wigner function [137,138]. (See [139] for an exposition using Louiville space.) For a review of this, and other distributions, see [140]. The quantum mechanical density matrix is a matrix representation of the density operator... 

So only the two-electron integrals wilh p. > v. and I>aand [p.v > 7.a need to he computed and stored. Dp.v.la on ly appears m Gpv, and Gvp, w hereas ih e original two-electron integrals con tribute to other matrix elemen is as well. So it is m iich easier to form ih e Fock matrix by using the siipermairix D and modified density matrix P th an the regular format of the tw O-electron in tegrals and stan dard den sity m atrix. 

Thus, HyperChem occasionally uses a three-point interpolation of the density matrix to accelerate the convergence of quantum mechanics calculations when the number of iterations is exactly divisible by three and certain criteria are met by the density matrices. The interpolated density matrix is then used to form the Fock matrix used by the next iteration. This method usually accelerates convergent calculations. However, interpolation with the MINDO/3, MNDO, AMI, and PM3 methods can fail on systems that have a significant charge buildup. 

Naturally, fibers and whiskers are of little use unless they are bonded together to take the form of a structural element that can carry loads. The binder material is usually called a matrix (not to be confused with the mathematical concept of a matrix). The purpose of the matrix is manifold support of the fibers or whiskers, protection of the fibers or whiskers, stress transfer between broken fibers or whiskers, etc. Typically, the matrix is of considerably lower density, stiffness, and strength than the fibers or whiskers. However, the combination of fibers or whiskers and a matrix can have very high strength and stiffness, yet still have low density. Matrix materials can be polymers, metals, ceramics, or carbon. The cost of each matrix escalates in that order as does the temperature resistance. 

Form an initial guess for the molecular orbital coefficients, and construct the density matrix. 

Form the Fock matrix as the core (one-electron) integrals + the density matrix times the two-electron integrals. 

Form the new density matrix. If it is sufficiently close to the previous density matrix, we are done, otherwise go to step (4). 

A certain answer may be found (Lowdin 1955) by considering the first-order density matrix y(x x1) defined by Eq. II.9. By means of the basis y>k and formula III. 14, this matrix may be expressed in the form... 

This theorem follows from the antisymmetry requirement (Eq. II.2) and is thus an expression for Pauli s exclusion principle. In the naive formulation of this principle, each spin orbital could be either empty or fully occupied by one electron which then would exclude any other electron from entering the same orbital. This simple model has been mathematically formulated in the Hartree-Fock scheme based on Eq. 11.38, where the form of the first-order density matrix p(x v xx) indicates that each one of the Hartree-Fock functions rplt y)2,. . ., pN is fully occupied by one electron. 

Next, consider an ensemble defined in configuration space, so that the density matrix has the form of Eq. (8-190). We assume that the eigenvectors X> are not eigenvectors of the hamiltonian. We have... 

Density matrix and ensembles, 465 Bloch s equation, 475 Derogatory form, 73... 

The HRO s are holes density operators and operate by first filling orbitals with electrons (i.e. they annihilate holes) and then removing electrons from orbitals (i.e. they create holes). These operators generate the Holes Reduced Density Matrix (HRDM) which in our notation takes the form ... 

We notice that neither the momentum distribution nor the reciprocal form factor seems to carry any information about the translational part of the space group. The non diagonal elements of the number density matrix in momentum space, on the other hand, transform under the elements of the space group in a way which brings in the translational parts explicitly. 

In the first method, known as the diagonal density matrix method, we followed Mo and Suzuki s analysis25 by assuming the ion angular distribution is of the form... 

The spin Hamiltonian also forms the theoretical basis for describing the temporal response of the spin system to a pulse sequence and/or mechanical manipulations of the sample via calculations of the evolution of the density matrix. Computer... 

Suppose that the density matrix has been brought to diagonal form, whereby it is immediately apparent that all its eigenvalues should be either zero or one. More precisely, since trg=l, only one of the eigenvalues can be equal to one and all the others are zero. 

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