Density matrix generalized

Remarkably, when our general ME is applied to either AN or PN in Section 4.4, the resulting dynamically controlled relaxation or decoherence rates obey analogous formulae provided the corresponding density matrix (generalized Bloch) equations are written in the appropriate basis. This underscores the universality of our treatment. It allows us to present a PN treatment that does not describe noise phenomenologically, but rather dynamically, starting from the ubiquitous spin-boson Hamiltonian. 

Starting with a density matrix generalized to different values J and J, the... 

A more intuitive, and more general, approach to the study of two-level systems is provided by the Feynman-Vemon-Flellwarth geometrical picture. To understand this approach we need to first introduce the density matrix. 

In the above discussion of relaxation to equilibrium, the density matrix was implicitly cast in the energy representation. However, the density operator can be cast in a variety of representations other than the energy representation. Two of the most connnonly used are the coordinate representation and the Wigner phase space representation. In addition, there is the diagonal representation of the density operator in this representation, the most general fomi of p takes the fomi... 

Figure Al.6.32. (a) Initial and (b) final population distributions corresponding to cooling, (c) Geometrical interpretation of cooling. The density matrix is represented as a point on generalized Bloch sphere of radius R...

The eigenfiinctions of a system of two particles are detemiined by their positions x and j, and the density matrix is generalized to... 

The Bloch equation approach (equation (B2.4.6)) calculates the spectrum directly, as the portion of the spectrum that is linear in a observing field. Binsch generalized this for a frilly coupled system, using an exact density-matrix approach in Liouville space. His expression for the spectrum is given by equation (B2.4.42). Note that this is fomially the Fourier transfomi of equation (B2.4.32). so the time domain and frequency domain are coimected as usual. 

Lengsfield B H III 1980 General second-order MC-SCF theory a density matrix directed algorithm J. Chem. Phys. 73 382... 

The density matrix p describes the pure state, as seen from the equality p = p, while p does not. The transition from (2.35a) to (2.35b) describes a strong collision , which fully localizes the particle, but in general the off-diagonal elements may not completely vanish. This however does not affect the qualitative picture. 

In order to evaluate the expectation value of the energy for an electronic system it is hence sufficient to know the generalized second-order density matrix r(x x 2 x1x2), from which the first-order density matrix may be obtained by using the formula... 

McWeeny, R., Proc. Roy. Soc. [London) A241, 239, (ii) The density matrix in self-consistent field theory. III. Generalizations of the theory/ The method of Parts I and II is generalized so as to apply to a system in which some orbitals are doubly and others singly occupied. 

The treatment presented so far is quite general and formally exact. It combines the eikonal representation for nuclear motions and the time-dependent density matrix in an approach which could be named as the Eik/TDDM approach. The following section reviews how the formalism can be implemented in the eikonal approximation of short wavelengths for the nuclear motions, and for specific choices of electronic states leading to the TDHF equations for the one-electron density matrix, and to extensions of TDHF. 

The calculation of the density operators over time requires integration of the sets of coupled differential equations for the nuclear trajectories and for the density matrix in a chosen expansion basis set. The density matrix could arise from an expansion in many-electron states, or from the one-electron density operator in a basis set of orbitals for a given initial many-electron state a general case is considered here. The coupled equations are... 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

Approximations have been reviewed in the case of short deBroglie wavelengths for the nuclei to derive coupled quantal-semiclassical computational procedures, by choosing different types of many-electron wavefunctions. Time-dependent Hartree-Fock and time-dependent multiconfiguration Hartree-Fock formulations are possible, and lead to the Eik/TDHF and Eik/TDMCHF approximations, respectively. More generally, these can be considered special cases of an Eik/TDDM approach, in terms of a general density matrix for many-electron systems. 

The Eik/TDDM approximation can be computationally implemented with a procedure based on a local interaction picture for the density matrix, and on its propagation in a relax-and-drive perturbation treatment with a relaxing density matrix as the zeroth-order contribution and a correction due to the driving effect of nuclear motions. This allows for an efficient computational procedure for differential equations coupling functions with short and long time scales, and is of general applicability. 

Combining the inverses of (III. 14) and (III. 16) we get the natural expansion for a general element of the number density matrix in momentum space ... 

However, billiard balls are a pretty bad model for electrons. First of all, as discussed above, electrons are fermions and therefore have an antisymmetric wave function. Second, they are charged particles and interact through the Coulomb repulsion they try to stay away from each other as much as possible. Both of these properties heavily influence the pair density and we will now enter an in-depth discussion of these effects. Let us begin with an exposition of the consequences of the antisymmetry of the wave function. This is most easily done if we introduce the concept of the reduced density matrix for two electrons, which we call y2. This is a simple generalization of p2(x1 x2) given above according to... 

In general the spectral one-particle density matrix p(r, rE) describes the mutual coherence of the wave field of high-energy electrons at the points r and r. For the simplest case of time-independent interaction potential the diagonal elements of... 

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... 

A rather general method of the calculation of the tunneling taking account of the dissipation was given in Ref. 82. The cases of rather strong dissipation were considered in Refs. 81 and 82, where it was assumed that a thermodynamical equilibrium in the initial potential well exists. The case of extremely weak friction has been considered using the equations for the density matrix in Ref. 83. A quantum analogue of the Focker-Planck equation for the adiabatic and nonadiabatic processes in condensed media was obtained in Refs. 105 and 106. 

The usual reactivity indices, such as elements of the first-order density matrix, are also incapable of distinguishing properly between singlet and triplet behavior. Recently, French authors 139,140) have discussed the problem and shown how electron repulsion terms can be introduced to obtain meaningful results. The particular case of interest to them was excited state basicity, but their arguments have general applicability. In particular, the PMO approach, which loses much of its potential appeal because of its inability to distinguish between singlet and triplet behavior 25,121) coui(j profit considerably from an extension in this direction. 119,122)... 

Chaos provides an excellent illustration of this dichotomy of worldviews (A. Peres, 1993). Without question, chaos exists, can be experimentally probed, and is well-described by classical mechanics. But the classical picture does not simply translate to the quantum view attempts to find chaos in the Schrodinger equation for the wave function, or, more generally, the quantum Liouville equation for the density matrix, have all failed. This failure is due not only to the linearity of the equations, but also the Hilbert space structure of quantum mechanics which, via the uncertainty principle, forbids the formation of fine-scale structure in phase space, and thus precludes chaos in the sense of classical trajectories. Consequently, some people have even wondered if quantum mechanics fundamentally cannot describe the (macroscopic) real world. 

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). 

Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg,... 

Of course, the Coulomb interaction appears in the Hamiltonian operator, H, and is often invoked for interpreting the chemical bond. However, the wave function, l7, must be antisymmetric, i.e., must satisfy the Pauli exclusion principle, and it is the only fact which explains the Lewis model of an electron pair. It is known that all the information is contained in the square of the wave function, 1I7 2, but it is in general much complicated to be analyzed as such because it depends on too many variables. However, there have been some attempts [3]. Lennard-Jones [4] proposed to look at a quantity which should keep the chemical significance and nevertheless reduce the dimensionality. This simpler quantity is the reduced second-order density matrix... 

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