Density matrix treatment theory

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. 

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. 

S. A. Rice I agree with Prof. Kohler that the use of a density matrix formalism by Wilson and co-workers generalizes the optimal control treatment based on wave functions so that it can be applied to, for example, a thermal ensemble of initial states. All of the applications of that formalism I have seen are based on perturbation theory, which is less general than the optimal control scheme that has been developed by Kosloff, Rice, et al. and by Rabitz et al. Incidentally, the use of perturbation theory is not to be despised. Brumer and Shapiro have shown that the perturbation theory results can be used up to 20% product yield. Moreover, from the point of view of generating an optimal control held, the perturbation theory result can be used as a first guess, for which purpose it is very good. 

A quantum mechanical treatment combined with the density matrix formalism extends the description to include the dynamic spectra of spin-coupled systems. (34-38) Further developments in the theory and presentation thereof, in a form suitable for computer calculations, are due to Binsch et al., and to Kaplan et al. (14, 15, 39) However, even the recent theories are not rigorous in certain aspects and contain some errors. This is particularly true in the case of the intermolecular exchange of spins. 

The fast desorption of CO in CO/Cu(OOf) has been measured [33] and also calculated. [30,31] The collision induced vibrational excitation and following relaxation of CO on Cu(001) has also been experimentally explored using time-of-flight techniques, and has been analyzed in experiments [34] and theory. [23,32] Our previous treatment of instantaneous electronic de-excitation of CO/Cu(001) after photoexcitation is extended here to include delayed vibrational relaxation of CO/Cu(001) in its ground electronic state. We show results for the density matrix, from calculations with the described numerical procedure for the integrodifferential equations. 

A general treatment of spin coupled with quadrupolar spins was given using density matrix theory.27 This formulation enables one to calculate, on the same theoretical basis, the lineshapes of the systems with different ratios of the quadrupolar interaction to the Zeeman interaction. Additionally, it includes the spinning sidebands very naturally. 

In this section we provide a summary of the commonly used 2D methods, principally in liquids, but with some reference also to solids. Our objectives are (1) to describe the purpose of the particular type of study (known in 2D NMR jargon as an experiment), (2) to indicate qualitatively the sort of data that can be obtained, and (3) to provide an explanation of the spin physics within the framework of the theory that we have at hand so far. In some instances we shall find that we need to apply density matrix or product operator treatments and that we must revisit these experiments in Chapter 12. 

In our introduction to the physics of NMR in Chapter 2, we noted that there are several levels of theory that can be used to explain the phenomena. Thus far we have relied on (1) a quantum mechanical treatment that is restricted to transitions between stationary states, hence cannot deal with the coherent time evolution of a spin system, and (2) a picture of moving magnetization vectors that is rooted in quantum mechanics but cannot deal with many of the subder aspects of quantum behavior. Now we take up the more powerful formalisms of the density matrix and product operators (as described very briefly in Section 2.2), which can readily account for coherent time-dependent aspects of NMR without sacrificing the quantum features. 

The treatment of INEPT by the density matrix is more satisfactory than the development in Section 9.6 in that it follows in a logical way from the initial theory without the need to pull together in an ad hoc way certain features from classical and quantum mechanics. However, we have not really gained any new insights into the physics of the processes. Let s now look at some aspects in more detail. 

In this review, we begin with a treatment of the functional theory employing as basis the maximum entropy principle for the determination of the density matrix of equilibrium ensembles of any system. This naturally leads to the time-dependent functional theory which will be based on the TD-density matrix which obeys the von Neumann equation of motion. In this way, we present a unified formulation of the functional theory of a condensed matter system for both equilibrium and non-equilibrium situations, which we hope will give the reader a complete picture of the functional approach to many-body interacting systems of interest to condensed matter physics and chemistry. 

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,... 

Yet, some theoretical problems are left to be discussed to seek for the ultimate and idealistic features as a nonadiabatic-transition theory Although a trajectory thus hopping plural times converges to run on an adiabatic potential surface asymptotically, the off-diagonal density matrix element Pij t) does not vanish practically, as in the original SET. This is ascribed to an incomplete treatment of the nuclear-electronic entanglement. This issue, often referred to as the problem of decoherence, is originated from the nuclear wavepacket bifurcation due to different slopes of potential surfaces, which will be discussed more precisely below. 

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

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