Density matrix method

In the first method, known as the diagonal density matrix method, we followed Mo and Suzuki s analysis25 by assuming the ion angular distribution is of the form... 

Z. Zhao, B. J. Braams, H. Fukuda, M. L. Overton, and J. K. Percus, The reduced density matrix method for electronic structure calculations and the role of three-index representabihty conditions. J. Chem. Phys. 120, 2095 (2004). 

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D. A. Mazziotti, Geminal functional theory a synthesis of density and density matrix methods. J. Chem. Phys. 112, 10125 (2000). 

R. Benesch and V. H. Smith, Jr., Density matrix methods in X-ray scattering and momentum space calculations, in Wave Mechanics—the First Fifty Years, W. C. Price, S. S. Chissick, and T. Ravensdale, eds. (Butterworths, London, 1973), pp. 357-377. 

M.M. Mestechkin. Density Matrix Method in Theory of Molecules. Naukova Dumka, Kiev, 1977. [in Russian]. 

IR Density matrix methods Excited states (adiabatic)... 

The next two chapters are devoted to ultrafast radiationless transitions. In Chapter 5, the generalized linear response theory is used to treat the non-equilibrium dynamics of molecular systems. This method, based on the density matrix method, can also be used to calculate the transient spectroscopic signals that are often monitored experimentally. As an application of the method, the authors present the study of the interfadal photo-induced electron transfer in dye-sensitized solar cell as observed by transient absorption spectroscopy. Chapter 6 uses the density matrix method to discuss important processes that occur in the bacterial photosynthetic reaction center, which has congested electronic structure within 200-1500cm 1 and weak interactions between these electronic states. Therefore, this biological system is an ideal system to examine theoretical models (memory effect, coherence effect, vibrational relaxation, etc.) and techniques (generalized linear response theory, Forster-Dexter theory, Marcus theory, internal conversion theory, etc.) for treating ultrafast radiationless transition phenomena. 

The theory of the multi-vibrational electron transitions based on the adiabatic representation for the wave functions of the initial and final states is the subject of this chapter. Then, the matrix element for radiationless multi-vibrational electron transition is the product of the electron matrix element and the nuclear element. The presented theory is devoted to the calculation of the nuclear component of the transition probability. The different calculation methods developed in the pioneer works of S.I. Pekar, Huang Kun and A. Rhys, M. Lax, R. Kubo and Y. Toyozawa will be described including the operator method, the method of the moments, and density matrix method. In the description of the high-temperature limit of the general formula for the rate constant, specifically Marcus s formula, the concept of reorganization energy is introduced. The application of the theory to electron transfer reactions in polar media is described. Finally, the adiabatic transitions are discussed. 

In this chapter, the ultrafast radiationless transition processes are treated theoretically. The method employed is based on the density matrix method, and specifically, a generalized linear response theory is developed by applying the projection operator technique on the Liouville equation so that non-equilibrium cases can be handled properly. The ultrafast molecular... 

The density matrix method is useful in treating relaxation processes, linear and non-linear laser spectroscopies and non-equilibrium statistical mechanics. In this chapter, the definition of density matrix and the equation of motion (EOM) it follows are introduced. The projection operator technique, which makes the density matrix method a very powerful tool in non-equilibrium statistical mechanics, is presented. 

In the above, the principles of how to study the dynamics of an isolated system by using the density matrix method have been shown. However, most experiments are performed for the system (or subsystem) embedded in a heat bath in this case the isolated system consists of the system plus heat bath. In the following the MEs shall be derived for the system embedded in heat bath. In this case, instead of pm, pnsnbfls b will be employed. Here s and b describe the system and heat bath, respectively. For the case in which the bath is much larger than the system, it may be assumed that the bath maintains thermal equilibrium and... 

In Section 2, the density matrix method is applied to an isolated system. In this section, the system embedded in a heat bath shall be treated that is, the isolated system shall be divided into two parts, one is referred to as the system which is under observation and the other part the heat bath. 

In this section, the density matrix method shall be applied to study the ultrafast dynamics of the system embedded in a heat bath. Due to the use of the ultrashort pulse in the pumping lasers the dynamic behaviors of both population and coherence have to be considered [1,19-21],... 

Spectroscopic methods are very useful for determining molecular properties. Time-resolved spectroscopic methods are useful for monitoring the evolution of the molecular properties in real time. Moreover, time-resolved spectroscopic techniques have the best time resolution available among all kinds of time-resolved experimental techniques. Thus, very often time-resolved spectroscopic methods reveal the dynamics of a molecular system in the non-equilibrium regime. In this section, the density matrix method is applied to calculate the spectroscopic properties of molecular systems. These include the linear and non-linear optical processes, in equilibrium or non-equilibrium cases. The approach is based on the susceptibility theory. 

Developments in the density matrix method still remain to be done, in both the fundamental aspect and the technical aspect. For example, the properties of the damping operator T are yet poorly known. Usually, only very simple parameters are input by hands. Such approach can help us simplify the process of model setup. However, there might be artifacts in the computation, as will be explained below. 

Technically, the density matrix for a molecular system can be very large, especially when many nuclear degrees of freedom are taken into account, or when more than two electronic levels are involved. In the future, to make the density matrix method a realistic tool for large scale problem, computational techniques have to be further developed. 

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