Density matrix determination

These relations show that the Fock-Dirac density matrix is identical with the first-order density matrix, and that consequently the first-order density matrix determines all higher-order density matrices and then also the entire physical situation. This theorem is characteristic for the Hartree-Fock approximation. 

Van der Spoel,D., Berendsen, H.J.C. Determination of proton transfer rate constants using ab initio, molecular dynamics and density matrix evolution calculations. Pacific Symposium on Biocomputing, World Scientific, Singapore (1996) 1-14. 

The electronic contribution to the dipole moment is thus determined from the density matrix and a series of one-electron integrals J dr< (-r)0. The dipole moment operator, r, h.)-components in the x, y and z directions, and so these one-electron integrals are divided into their appropriate components for example, the x component of the electronic contribution to the dipole moment would be determined using ... 

Mulliken population analysis is a trivial calculation to perform once a self-consistent field has been established and the elements of the density matrix have been determined. 

Ceramic matrix composites are candidate materials for high temperature stmctural appHcations. Ceramic matrices with properties of high strength, hardness, and thermal and chemical stabiUty coupled with low density are reinforced with ceramic second phases that impart the high toughness and damage tolerance which is required of such stmctural materials. The varieties of reinforcements include particles, platelets, whiskers and continuous fibers. Placement of reinforcements within the matrix determines the isotropy of the composite properties. 

A more practical problem21 is to determine the density matrix... 

As a consequence, field methods, which consist of computing the energy or dipole moment of the system for external electric field of different amplitudes and then evaluating their first, second derivatives with respect to the field amplitude numerically, cannot be applied. Similarly, procedures such as the coupled-perturbed Hartree-Fock (CPHF) or time-dependent Hartree-Fock (TDHF) approaches which determine the first-order response of the density matrix with respect to the perturbation cannot be applied due to the breakdown of periodicity. 

The main problem is to calculate (/ (q, H-r)/(q, t- -r)) of Eq. (2). To achieve this goal, one first considers E(r,f) as a well-defined, deterministic quantity. Its effect on the system may then be determined by treating the von Neumann equation for the density matrix p(f) by perturbation theory the laser perturbation is supposed to be sufficiently small to permit a perturbation expansion. Once p(i) has been calculated, the quantity... 

Density functions can be obtained up to any order from the manipulation of the Slater determinant functions alone as defined in section 5.1 or from any of the linear combinations defined in section 5.2. Density functions of any order can be constructed by means of Lowdin or McWeeny descriptions [17], being the diagonal elements of the so called m-th order density matrix, as was named by Lowdin the whole set of possible density functions. For a system of n electrons the n-th order density function is constructed from the square modulus of any n-electron wavefunction attached to the n-electron system somehow. 

The dynamical behaviors of p(At) v and p(At)av av, have to be determined by solving the stochastic Liouville equation for the reduced density matrix the initial conditions are determined by the pumping process. For the purpose of qualitative discussion, we assume that the 80-fs pulse can only pump two vibrational states, say v = 0 and v = 1 states. In this case we obtain... 

In the case of an equilibrium system the Hamiltonian is the same as that of an ensemble of conservative systems in statistical equilibrium. If the energy of the system is measured to lie between Ek and EK + AE, then the representative ensemble is also restricted to the energy shell [AE K. From the hypotheses of equal a priori probabilities and random a priori phases it then follows that the diagonal elements of the density matrix lying inside [AE]k are all equal and that all others vanish. The density matrix of the quantum statistical microcanonical ensemble is thereby determined as... 

Here N is the normalization constant and the coefficients Ai,A2 and B will be determined by Schrodinger equation. We are interested in the reduced density matrix for the long wavelength mode , which is obtained by integrating the density matrix To(i, 2, t) over the short wavelength mode 2 = reduced density matrix for the long wavelength mode can be written in the form... 

Time-dependent response theory concerns the response of a system initially in a stationary state, generally taken to be the ground state, to a perturbation turned on slowly, beginning some time in the distant past. The assumption that the perturbation is turned on slowly, i.e. the adiabatic approximation, enables us to consider the perturbation to be of first order. In TD-DFT the density response dp, i.e. the density change which results from the perturbation dveff, enables direct determination of the excitation energies as the poles of the response function dP (the linear response of the KS density matrix in the basis of the unperturbed molecular orbitals) without formally having to calculate a(co). 

In Bohmian mechanics, the way the full problem is tackled in order to obtain operational formulas can determine dramatically the final solution due to the context-dependence of this theory. More specifically, developing a Bohmian description within the many-body framework and then focusing on a particle is not equivalent to directly starting from the reduced density matrix or from the one-particle TD-DFT equation. Being well aware of the severe computational problems coming from the first and second approaches, we are still tempted to claim that those are the most natural ways to deal with a many-body problem in a Bohmian context. 

The calculation of the indices requires the overlap matrix S of atomic orbitals and the first-order density (or population) matrix P (in open-shell systems in addition the spin density matrix Ps). The summations refer to all atomic orbitals /jl centered on atom A, etc. These matrices are all computed during the Hartree-Fock iteration that determines the molecular orbitals. As a result, the three indices can be obtained... 

Nuclear spin relaxation is considered here using a semi-classical approach, i.e., the relaxing spin system is treated quantum mechanically, while the thermal bath or lattice is treated classically. Relaxation is a process by which a spin system is restored to its equilibrium state, and the return to equilibrium can be monitored by its relaxation rates, which determine how the NMR signals detected from the spin system evolve as a function of time. The Redfield relaxation theory36 based on a density matrix formalism can provide... 

The eigenvalue equation corresponding to the Hamiltonian of Eq. [57] can be solved self-consistently by an iterative procedure for each orientation of the spin magnetization (identified as the z direction). The self-consistent density matrix is then employed to calculate the local spin and orbital magnetic moments. For instance, the local orbital moments at different atoms i are determined from... 

Terms containing the W intermediates no longer contain a factor of The energy-independent, third-order term, Epp (oo), is a Coulomb-exchange matrix element determined by second-order corrections to the density matrix, where... 

The origins of density functional theory (DFT) are to be found in the statistical theory of atoms proposed independently by Thomas in 1926 [1] and Fermi in 1928 [2]. The inclusion of exchange in this theory was proposed by Dirac in 1930 [3]. In his paper, Dirac introduced the idempotent first-order density matrix which now carries his name and is the result of a total wave function which is approximated by a single Slater determinant. The total energy underlying the Thomas-Fermi-Dirac (TFD) theory can be written (see, e.g. March [4], [5]) as... 

For a system of N identical fermions in a state ij/ there is associated a reduced density matrix (RDM) of order p for each integer p, 1 Hermitian operator DP, which we call a reduced density operator (RDO) acting on a space of antisymmetric functions of p particles. The case p = 2 is of particular interest for chemists and physicists who seldom consider... 

M. Rosina and M. V. Mihailovic, The determination of the particle—hole excited states by using the variational approach to the ground state two-body density matrix, in International Conference on Properties of Nuclear States, Montreal 1969, Les Presses de I Universite de Montreal, 1969. 

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