Density matrix description

The density matrix description is useful when discussing the electron correlations. The statement that the motion of electrons is correlated can be given an exact sense only if the two-electron density matrices eqs. (1.199) and (1.200) are used. In terms of the wave function, the statement of the correlated character of electron motions sounds like a negative statement the non-correlated (Hartree-Fock) wave function is one which is represented by a single Slater determinant, and the correlated one... 

The /nfpr-molecular NOE is mentioned above briefly. It is of course this which makes it desirable to use samples in a solvent without nuclei of high for these experiments. There have, however, been few detailed studies of this effect. (234) A density matrix description (235) has been successfully applied to the effect upon the solute (1,1,2-trichloroethane) protons of irradiating the solvent (Me4Si) resonance (235) and differential effects upon the resonances of the [AB]2 spin system provided by the protons of o-dichlorobenzene have been used to aid the assignment. (236)... 

A density matrix description of the polarization inversion process has been presented enabling the visualization of the role of the process in the suppression of zero-frequency peaks in SLF-2D NMR experiments based on the dipolar oscillations during CP. It has been shown that, during this process, a doubling of the amplitude of the Oscillatory component occurs accompanied by a reduction of the initial intensity of the non-oscillatory component to zero. 

B. A Density Matrix Description of Hole-Burning in a Three-Level System. 435... 

A density matrix approach makes it possible to examine more general models for systems like this [38,39]. For instance, unlike superexchange theory, this approach admits models that include strong couplings between the closely associated species in the intercalated DNA/donor-acceptor complex. More important, the density matrix description allows a proper treatment of quantum coherences, which is critical in bridging the tunneling and free-conduction regimes of transport. On the one hand,... 

In the case of a reduced density-matrix description of the system dynamics, the reduced probability density of coordinate Ri is given by... 

Fane U 1957 Description of states in quantum mechanics by density matrix and operator techniques Rev. Mod. Phys. 29 74-93... 

Dead time A very short delay introduced before the start of acquisition that allows the transmitter gate to close and the receiver gate to open. Density matrix A description of the state of nuclei in quantum mechanical terms. 

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. 

Abstract. The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples. 

Beginning way back in the 20s, Thomas and Fermi had put forward a theory using just the diagonal element of the first-order density matrix, the electron density itself. This so-called statistical theory totally failed for chemistry because it could not account for the existence of molecules. Nevertheless, in 1968, after years of doing wonders with various free-electron-like descriptions of molecular electron distributions, the physicist John Platt wrote [2] We must find an equation for, or a way of computing directly, total electron density. [This was very soon after Hohenberg and Kohn, but Platt certainly was not aware of HK by that time he had left physics.]... 

In Section 8.4.2, we considered the problem of the reduced dynamics from a standard DFT approach, i.e., in terms of single-particle wave functions from which the (single-particle) probability density is obtained. However, one could also use an alternative description which arises from the field of decoherence. Here, in order to extract useful information about the system of interest, one usually computes its associated reduced density matrix by tracing the total density matrix p, (the subscript t here indicates time-dependence), over the environment degrees of freedom. In the configuration representation and for an environment constituted by N particles, the system reduced density matrix is obtained after integrating pt = T) (( over the 3N environment degrees of freedom, rk Nk, ... 

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. 

We conclude that the QCL description represents a promising approach to the treatment of multidimensional curve-crossing problems. The density-matrix... 

A description of the different terms contributing to the correlation effects in the third order reduced density matrix faking as reference the Hartree Fock results is given here. An analysis of the approximations of these terms as functions of the lower order reduced density matrices is carried out for the linear BeFl2 molecule. This study shows the importance of the role played by the homo s and lumo s of the symmetry-shells in the correlation effect. As a result, a new way for improving the third order reduced density matrix, correcting the error ofthe basic approximation, is also proposed here. 

In modest sized systems, we can treat the nondynamic correlation in an active space. For systems with up to 14 orbitals, the complete-active-space self-consistent field (CASSCF) theory provides a very satisfactory description [2, 3]. More recently, the ab initio density matrix renormalization group (DMRG) theory has allowed us to obtain a balanced description of nondynamic correlation for up to 40 active orbitals and more [4-13]. CASSCF and DMRG potential energy... 

A complicating factor is tlrat each spitr density matrix element is multiplied by the corresponding basis function overlap at tire nuclear positions. The orbitals having maximal amplitude at the nuclear positions are tire core s orbitals, which are usually described with less flexibility than valence orbitals itr typical electronic structure calculations. Moreover, actual atomic s orbitals are characterized by a cusp at tire nucleus, a feature accurately modeled by STOs, but only approximated by the more commonly used GTOs. As a result, tlrere are basis sets in the literature tlrat systematically improve tire description of the core orbitals in order to improve prediction of h.f.s., e.g. IGLO-III (Eriksson et al. 1994) and EPR-III (Barone 1995). 

Many problems appear to be ripe for a more quantitative discussion. What is the error involved in the introduction of unstable states as asymptotic states in the frame of the 5-matrix theory 16 What is the role of dissipation in mass symmetry breaking What is the consequence of the new definition of physical states for conservation theorems and invariance properties We hope to report soon about these problems. We would like, however, to conclude this report with some general remarks about the relation between field description and particles. The full dynamical description, as given by the density matrix, involves both p0 and the correlations pv. However, the particle description is expressed in terms of p (see Eq. (50)). Now p has only as many elements as p0. Therefore the... 

The theory described so far is based on the Master Equation, which is a sort of intermediate level between the macroscopic, phenomenological equations and the microscopic equations of motion of all particles in the system. In particular, the transition from reversible equations to an irreversible description has been taken for granted. Attempts have been made to derive the properties of fluctuations in nonlinear systems directly from the microscopic equations, either from the classical Liouville equation 18 or the quantum-mechanical equation for the density matrix.19 We shall discuss the quantum-mechanical treatment, because the formalism used in that case is more familiar. 

The density matrix provides a complete description in the sense that all averages can be expressed in it but it does not uniquely identify the j/iv) and wv used for the construction of the ensemble. 

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