Microscopic expression

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

The present work aims to derive fully microscopic expressions for the nucleation rate J and to apply the results to realistic estimates of nucleation rates in alloys. We suppose that the state with a critical embryo obeys the local stationarity conditions (9) dFjdci — p, but is unstable, i.e. corresponds to the saddle point cf of the function ft c, = F c, — lN in the ci-space. At small 8a = c — cf we have... 

Explicit forms for the stress tensors d1 are deduced from the microscopic expressions for the component stress tensors and from the scheme of the total stress devision between the components [164]. Within this model almost all essential features of the viscoelastic phase separation observable experimentally can be reproduced [165] (see Fig. 20) existence of a frozen period after the quench nucleation of the less viscous phase in a droplet pattern the volume shrinking of the more viscous phase transient formation of the bicontinuous network structure phase inversion in the final stage. 

We have shown that the microscopic expression for the polymer diffusion coefficient. Equation 2, is the starting point for a discussion of diffusion in a wide range of polymer systems. For the example worked out, polymer diffusion at theta conditions, the resulting expresssion describes the experimental data without adjustable parameters. It should be possible to derive expressions for diffusion... 

Am = (Gm — 2Gm) is the vibrational anharmonicity. This Hamiltonian describes excitons as oscillator (quasiparticle) degrees of freedom. Hb represents a bath Hamiltonian. We shall not specify it and merely require that it conserves the number of excitons. The bath induces relaxation kernels. The structure of the final expression is independent of the specific properties of the bath the latter only affects the microscopic expression for the relaxation kernels (51). 

One considers a particle interacting linearly with an environment constituted by an infinite number of independent harmonic oscillators in thermal equilibrium. The particle equation of motion, which can be derived exactly, takes the form of a generalized Langevin equation, in which the memory kernel and the correlation function of the random force are assigned well-defined microscopic expressions in terms of the bath operators. 

Use the expression derived above for to obtain the following microscopic expression for the Henry s law coefficient ... 

To derive microscopic expressions for the reaction rate constants we first use the solution of the rate equations to determine the fraction of molecules of type B at time t under the condition that initially all of the N molecules where of type A, i.e. Na 0) = N and Nb(0) = 0,... 

In this Appendix we justify the approach followed in Sect. 5 to derive microscopic expressions for the reaction rate constants. The standard way to do this for the reaction... 

We already know that the average kinetic energy is 3/2)NkBT, so once the average potential energy has been calculated we will have a full microscopic expression for the macroscopic energy of the system. This average potential energy is... 

Thus, Eqs (18.98)—(18.109) provide a microscopic expression for the dielectric response function in a system of noninteracting particles... 

IV. Density Matrix Approach, the Diagrammatic Technique, and the Microscopic Expressions for Wp (r) and... 

IV. DENSITY MATRIX APPROACH, THE DIAGRAMMATIC TECHNIQUE, AND THE MICROSCOPIC EXPRESSIONS FOR... 

The diagrammatically obtained electrical susceptibility expressions can then be used to find the microscopic expression for as given... 

Having derived the microscopic expression for polarization, the focus is now on the macroscopic formulation of the dielectric constant for a cubic crystal. The relative dielectric constant r (the ratio of absolute dielectric constant eabs with 0) can be introduced through the average electric field E acting on a crystal unit cell as ... 

The general microscopic expression for the nth-order susceptibility contains n + 1 dipole moment matrix elements, involving n intermediate states. For the linear susceptibility there is only one intermediate state, and if the latter is a hybrid one, the corresponding dipole matrix elements are determined mainly by the Frenkel component of the hybrid state. Thus, the linear susceptibility of the hybrid structure contains the factor (dp/ap)2, as is seen from eqn (13.77). For the second-order nonlinear susceptibility x one must have two intermediate states or three virtual transitions. One of them may be a hybrid one, and as long... 

Since a portion of this chapter is devoted to the derivation of rate laws and various microscopic expressions for the rate coefficients of condensed-phase chemical reactions, it is useful to first write down the phenomenological rate law we expect to obtain, to define the various rate coefficients and relaxation times, and to present the different points of view that we shall adopt in describing the system. 

The previous sections have attempted to provide some insight into the form of the microscopic expressions for the rate kernels and rate coefficients that characterize condensed-phase reactions. Although the equilibrium one-way flux rate coefficient ky is relatively easy to calculate and under certain circumstances may yield an adequate description of the rate, a variety of important dynamic effects are contained in the relaxing part of the rate kernel, In this section, we describe a kinetic theory that... 

STEP 5. Determine the relationships between the off-diagonal elements of the r matrix. Detailed microscopic expressions for the nonzero elements can be obtained from Eq. (11.4.4). 

The isothermal compressibility of liquids gives a measure of the change in volume of the system due the change in pressure applied at a constant temperature. A microscopic expression of isothermal compressibility is given in Appendix 2.A. The expression shows that compressibility is related to the natural fluctuations in the total volume of the system. 

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