The density matrix approach

In general the exciton dynamics exhibits both coherent and incoherent behaviour, where the incoherence arises from the couphng of the system to a dissipative environment. This is conveniently modelled by an equation of motion for the reduced density operator, p, defined by 

In the localized exciton basis m), defined by eqn (9.17), the matrix elements of the reduced density operator are 

The diagonal elements, p , represent the classical populations, P , while the off-diagonal elements, pmn, describe the coherences between the quantum states 

For a system in contact with a thermal heat bath equilibrium is achieved provided that detailed balance is satisfied, namely 

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The treatment developed here is based on the density matrix of quantum mechanics and extends previous work using wavefunctions.(42 5) The density matrix approach treats all energetically accessible electronic states in the same fashion, and naturally leads to average effective potentials which have been shown to give accurate results for electronically diabatic collisions. 19) The approach is taken here for systems where the dynamics can be described by a Hamiltonian operator, as it is possible for isolated molecules or in models where environmental effects can be represented by terms in an effective Hamiltonian. 

The crisis of the GME method is closely related to the crisis in the density matrix approach to wave-function collapse. We shall see that in the Poisson case the processes making the statistical density matrix become diagonal in the basis set of the measured variable and can be safely interpreted as generators of wave function collapse, thereby justifying the widely accepted conviction that quantum mechanics does not need either correction or generalization. In the non-Poisson case, this equivalence is lost, and, while the CTRW perspective yields correct results, no theoretical tool, based on density, exists yet to make the time evolution of a contracted Liouville equation, classical or quantum, reproduce them. 

As in previous chapters, we do not attempt to provide a rigorous treatment here, but rather develop the concept of a density matrix, which is often unfamiliar to chemists, and show how it may easily be used to understand the behavior of one-spin and two-spin systems. As in the treatment of complex spectra in Chapter 6, we shall see that the density matrix approach can be readily extended to larger spin systems, but with a great deal of algebra and often with little physical insight. However, in the course of treating the simple spin systems, we will notice that some of the results can be obtained more succinctly by certain manipulations of the spin operators Ix, Iy, and Iz, with which we are familiar. 

We now have a formula for constructing the density matrix for any system in terms of a set of basis functions, and from Eq. 11.6 we can determine the expectation value of any dynamical variable. However, the real value of the density matrix approach lies in its ability to describe coherent time-dependent processes, something that we could not do with steady-state quantum mechanics. We thus need an expression for the time evolution of the density matrix in terms of the Hamiltonian applicable to the spin system. 

Most earlier theoretical work on multiple resonance used either the Bloch equations or the spin-Hamiltonian. That is, relaxation is considered separately from other effects. The tendency in recent years has been to adopt the density-matrix approach but on some occasions simpler methods have still been appropriate. In the interpretation of multiple resonance experiments it is often necessary to consider the labelling of the energy level diagram in detail. 

In the case of strong coupling, 72 < Wc/ i, MEG by a short light pulse is accompanied by quantum beats with the frequency Wc/ i. These oscillations are damped in the opposite case of weak coupling, 72 > Wc/ i however, the MEG process can still be very efficient. The density matrix approach is the only self-consistent method that takes into account the diverse processes responsible for MEG in NCs. 

The detection of NMR signals is based on the perturbation of spin systems that obey the laws of quantum mechanics. The effect of a single hard pulse or a selective pulse on an individual spin or the basic understanding of relaxation can be illustrated using a classical approach based on the Bloch equations. However as soon as scalar coupling and coherence transfer processes become part of the pulse sequence this simple approach is invalid and fails. Consequently most pulse experiments and techniques cannot be described satisfactorily using a classical or even semi-classical description and it is necessary to use the density matrix approach to describe the quantum physics of nuclear spins. The density matrix is the basis of the more practicable product operator formalism. 

In NMR-SIM the simulation of an NMR experiment is based on the density matrix approach with relaxation phenomena implemented using a simple model based on the Bloch equations. Spectrometer related difficulties such as magnetic field inhomogenity, acoustic ringing, radiation damping or statistical noise cannot be calculated using the present approach. Similarly neither can some spin system effects such as cross-relaxation and spin diffusion can be simulated. 

Back in the 1960s, hopes for future progress in electronic structure theory were associated with correlated wave function techniques and the tantalizing possibility of variational calculations based on the two-electron reduced density matrix [294]. DFT was not on the quantum chemistry agenda at that time. The progress of wave function techniques has been remarkable, as documented elsewhere in this volume. In contrast, the density matrix approach has not yet materialized into a competitive computational method, despite many persistent efforts [295]. Meanwhile, approximate DFT has become the most widely used method of quantum chemistry, offering an unprecedented accuracy-to-cost ratio. 

The BPP expression was derived under the assumptions that the dipolar interactions formed a perturbation on the Zeeman levels and that the time dependent part of the dipolar interaction could be treated by time dependent perturbation theory or equivalently the density matrix approach to determine the relaxation expression. This does not rule out a BPP type expression for relaxation in the rotating frame. Specifically, a BPP type approach can be used to derive the following expression for T due to rotational motion under basically... 

As mentioned, the convergence of the 2x2 scheme is slow and not so reliable, especially for delocalized covalently bonded systems. It was strongly desired to develop a more reliable and more efficient localization scheme, both reliable and efficient. Gu et al worked out a new localization scheme based on the density matrix approach on the regional localized molecular orbitals (RLMOs). This new localization scheme works very well even for delocalized systems, and tests on some model systems have been reported. 

As described in Section 2.6, the dynamics of an ensemble of non-interacting nuclei with spin 1 /2 is described with use of the density matrix approach in a two-fold matrix space, which allows the understanding of the behavior of such collection of nuclei when submitted to static magnetic field, RF pulses, free evolution times, and so on. This description is complete only if one has a material composed of identical units (molecules, ionic groups, etc.) with just one chemically distinct NMR-active nucleus in each unit and with all inter-... 

Spin Evolution and Relaxation The Density Matrix Approach... 

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