Self-consistent averaging

We can now give the explicit forms of the self-consistent averaged and first-order oscillating parts of the ground-state energy of the open-shell system ... 

Benjamin and Wilson used classical molecular dynamics to study the photodissociation of model ICN in the gas phase and Xe solution. One of the novel features of this study involved the treatment of molecular dynamics on coupled surfaces in solution using the Meyer-Miller method.i For sufficiently high excitation (Benjamin and Wilson use 266 nm), ICN may be excited to a linear state that correlates with an I product. However, this excited state may couple to another state that has a bent configuration and correlates to ground state I. Benjamin and Wilson developed an extension of the Meyer-Miller method, which takes into account the interaction between the ICN and the Xe solvent. The result is a Hamiltonian that describes motion on a self-consistent average of the two excited state potentials. 

Finally, the total RHF band energy per cell, band> calculated. Eband defined by the standard atomic expressions but is computed with the renormalized wave functions, including the self-consistent average 5d and 6s conduction band wave functions and occupation numbers. Eband differs from E ai of eq. (3) only by the exclusion of the Hund-rule correction for the 4f electrons discussed in section 2.2. 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

The result of this approximation is that each mode is subject to an effective average potential created by all the expectation values of the other modes. Usually the modes are propagated self-consistently. The effective potentials governing die evolution of the mean-field modes will change in time as the system evolves. The advantage of this method is that a multi-dimensional problem is reduced to several one-dimensional problems. 

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

For the case of nonzero temperatures the vacuum averages in Eq.(7) should be replaced by thermal averages over phonon populations. Using (7) and (5) we obtain that the scattering of an exciton in the effective medium by the perturbation fi — v z)) is described by the following self-consistent condition... 

Cl calculations can be used to improve the quality of the wave-function and state energies. Self-consistent field (SCF) level calculations are based on the one-electron model, wherein each electron moves in the average field created by the other n-1 electrons in the molecule. Actually, electrons interact instantaneously and therefore have a natural tendency to avoid each other beyond the requirements of the Exclusion Principle. This correlation results in a lower average interelectronic repulsion and thus a lower state energy. The difference between electronic energies calculated at the SCF level versus the exact nonrelativistic energies is the correlation energy. 

We will actually use the idea that the interaction between electrons can somehow be averaged in a later chapter you will see how this idea forms the basis for the self-consistent field (SCF) model. 

At the energy minimum, each electron moves in an average field due to the Other electrons and the nuclei. Small variations in the form of the orbitals at this point do not change the energy or the electric field, and so we speak of a self-consistent field (SCF). Many authors use the acronyms HF and SCF interchangeably, and I will do so from time to time. These HF orbitals are found as solutions of the HF eigenvalue problem... 

After we obtained the self-consistent electronic structure of the magnetic multilayers we calculated the non-local conductivity by evaluating the quantum mechanical linear response of the current to the electric field using an approach developed by Kubo and Greenwood. In this approach the conductivity is obtained from a configurational average of two one-electron Green functions ... 

The last term in Eq. 11.47 gives apparently the "average one-electron potential we were asking for in Eq. 11.40. The Hartree-Fock equations (Eq. 11.46) are mathematically complicated nonlinear integro-differential equations which are solved by Hartree s iterative self-consistent field (SCF) procedure. 

Data are given in Table 10-7 to illustrate certain facets of the Marcus cross relation. They refer to six reactions in which the cage complex Mn(sar)3+ is reduced or Mn(sar)2+ oxidized.34 These data were used to calculate the EE rate constant for this pair. The results of the calculation, also tabulated, show that there is a reasonably self-consistent value of fcEE for Mn(sar)3+/Mn(sar)2+ lying in the range 3-51 L mol-1 s-1. When values34 for an additional 13 reactions were included the authors found an average value of kEE = 17 L mol 1 s l. 

Indeed, the self-consistent model averages the stresses and strains in either phase of a two-phase material, and it determines them, by solving separate problems, whose superposition yields the final configuration of the model 7). 

In the self-consistent model the matrix material outside the inclusion is assumed as possessing the effective macroscopic properties of the composite. Moreover, two consecutive problems were solved by assuming either phase of the composite as occupying its position and surrounded by this average material. In both cases the average values of the composite are determined from the values of the characteristic... 

For planar unsaturated and aromatic molecules, many MO calculations have been made by treating the a and n electrons separately. It is assumed that the o orbitals can be treated as localized bonds and the calculations involve only the tt electrons. The first such calculations were made by Hiickel such calculations are often called Hiickel molecular orbital (HMO) calculations Because electron-electron repulsions are either neglected or averaged out in the HMO method, another approach, the self-consistent field (SCF), or Hartree-Fock (HF), method, was devised. Although these methods give many useful results for planar unsaturated and aromatic molecules, they are often unsuccessful for other molecules it would obviously be better if all electrons, both a and it, could be included in the calculations. The development of modem computers has now made this possible. Many such calculations have been made" using a number of methods, among them an extension of the Hiickel method (EHMO) and the application of the SCF method to all valence electrons. ... 

The electron energy density We = tiff (the product of the electron density and average electron energy e) is calculated self-consistently from the second moment of the Boltzmann equation ... 

The physical reasoning for why these densities were frequently employed in the earlier days of density functional theory was that in this way the degeneracy of the partially filled d-orbitals could be retained. A technical reason why these densities still have to be employed in some recent investigations is that calculations with integral orbital occupations simply do not converge in the self consistent field procedure (see, e. g., Blanchet, Duarte, and Salahub, 1997). Such densities correspond to a representation of a particular state 2S+1L with Mg = S and a spherical averaging over ML. 

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