Self-consistent solutions generalized

General expression for the fluctuation contribution to the specific heat is given by the first line (15) and can be resolved with the help of the self-consistent solution of (11), (19) (or (20), (21), (22) in the limiting cases). Assuming for rough estimates that fluctuations can be described pcrturbativcly and putting m2 — r/ vip u, from the second line of (15) we find for T <. 

Recently, Ya.B. has been working on a complete cosmological theory which would incorporate the creation of the Universe (1982) [45 ]. Let us mention finally that it was Ya.B. who recently gave a profound formulation of the question of the cosmological constant, i.e., the energy density in Minkovsky space (see Section 9). More precisely, the question is formulated thus is the Minkovsky space a self-consistent solution of the equations for all possible fields and the equations of general relativity ... 

Here Ho is the kinetic energy operator of valence electrons Vps is the pseudopotential [40,41] which defines the atomic core. V = eUn(r) is the Hartree energy which satisfies the Poisson equation ArUn(r) = —4nep(r) with proper boundary conditions as discussed in the previous subsection. The last term is the exchange-correlation potential Vxc [p which is a functional of the density. Many forms of 14c exist and we use the simplest one which is the local density approximation [42] (LDA). One may also consider the generalized gradient approximation (GGA) [43,44] which can be implemented for transport calculations without too much difficulty [45]. Importantly a self-consistent solution of Eq. (2) is necessary because Hks is a functional of the charge density p. One constructs p from the KS states Ts, p(r) = (r p r) = ns Fs(r) 2, where p is the density matrix,... 

Electron transport simulation is performed using earlier developed ensemble Monte Carlo algorithms and procedures, which include self-consistent solution of Poisson and Boltzmann equations [3,4]. In general, in both types of MOSFETs the normal component of electric field at Si/Si02 interface in a certain jc-point of the channel may be calculated using the following expression... 

In general terms, we may say that the atomic or molecular calculations in quantum chemistry have the aim of finding self-consistent solutions of the Schrodinger equation and Poisson equation for the distributions of nuclei and electrons making up the system of our interest in the chosen state. 

The Gauss law of electrostatics is used to calculate the electric field produced by the total charge distribution obtained in the previous step. The new function V(r) generally differs from the estimation made in the first step. The procedure is repeated until two consecutive values of V(r) become essentially the same. Then the obtained self-consistent solution describes the electrons in the ground state of the multi-electron atom. 

The most widely used method in the study of interfaces and OPVs in general is ab initio quantum mechanics (QM). While there are a variety of QM flavors, they all involve solving for approximate solutions to the Schrodinger equation and require a self-consistent solution. The biggest differentiator between the various flavors is the way in which they incorporate approximations to electron-electron interactions. 

Theessence of the procedure is that Halpin and Tsai [3-17] showed that Hermans solution [3-14] generalizing Hill s self-consistent model [3-13] can be reduced to the approximate form... 

Finally, we note that we have mostly limited attention so far to the self-consistent reaction field limit of dynamical solvent polarization, which is the only one that has been generally implemented (see next Section). Nevertheless, there are problems where the solute-solvent dynamical correlation must be considered, and we will address that topic in Section 5. 

In this contribution, we describe and illustrate the latest generalizations and developments[1]-[3] of a theory of recent formulation[4]-[6] for the study of chemical reactions in solution. This theory combines the powerful interpretive framework of Valence Bond (VB) theory [7] — so well known to chemists — with a dielectric continuum description of the solvent. The latter includes the quantization of the solvent electronic polarization[5, 6] and also accounts for nonequilibrium solvation effects. Compared to earlier, related efforts[4]-[6], [8]-[10], the theory [l]-[3] includes the boundary conditions on the solute cavity in a fashion related to that of Tomasi[ll] for equilibrium problems, and can be applied to reaction systems which require more than two VB states for their description, namely bimolecular Sjy2 reactions ],[8](b),[12],[13] X + RY XR + Y, acid ionizations[8](a),[14] HA +B —> A + HB+, and Menschutkin reactions[7](b), among other reactions. Compared to the various reaction field theories in use[ll],[15]-[21] (some of which are discussed in the present volume), the theory is distinguished by its quantization of the solvent electronic polarization (which in general leads to deviations from a Self-consistent limiting behavior), the inclusion of nonequilibrium solvation — so important for chemical reactions, and the VB perspective. Further historical perspective and discussion of connections to other work may be found in Ref.[l],... 

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