Self Consistent Method

Three such methods have been proposed by Morokuma and coworkers. The integrated MO + MM (IMOMM) method combines an orbital-based technique with an MM technique. The integrated MO + MO method (IMOMO) integrates two different orbital-based techniques. The our own n-layered integrated MO and MM method (ONIOM) allows for three or more different techniques to be used in successive layers. The acronym ONIOM is often used to refer to all three of these methods since it is a generalization of the technique. 

This technique can be used to model a complete system as a small model system and the complete system. The complete system would be computed using only the lower level of theory. The model system would be computed with both levels of theory. The energy for the complete system, combining both levels of theory, would then be 

Likewise, a three-layer system could be broken down into small, medium, and large regions, to be computed with low, medium, and high levels of theory (L, M, and // respectively). The energy expression would then be 

One disadvantage is that the lower levels of theory must be able to describe all atoms in the inner regions of the molecule. Thus, this method cannot be used to incorporate a metal atom into a force field that is not parameterized for it. The effect of one region of the molecule causing polarization of the electron density in the other region of the molecule is incorporated only to the extent that the lower levels of theory describe polarization. This method requires more CPU time than most of the others mentioned. However, the extra time should be minimal since it is due to lower-level calculations on smaller sections of the system. 

Bersuker and coworkers have proposed a technique whereby the atoms on the boundary between regions are included in both calculations. In this procedure. 

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Once the format of the Fock matrix is known, the semiempirical molecular problem (and it is a considerable one) is finding a way to make valid approximations to the elements in the Fock matrix so as to avoid the many integrations necessary in ab initio evaluation of equations like Fij = J 4>,F4> dx. After this has been done, the matrix equation (9-62) is solved by self-consistent methods not unlike the PPP-SCF methods we have already used. Results from a semiempirical... 

Shimada J, Ishchenko AV, Shakhnovich El. Analysis of knowledge-based protein-ligand potentials using a self-consistent method. Protein Sci 2000 9 765-75. 

The electric field that each dipole feels depends on all other induced dipoles requiring the use of a self-consistent method as mentioned above. To achieve this, Eq. (9-13) can be rearranged and written in matrix form considering that A = W1 (I - aT),... 

The problem of time evolution for a Hamiltonian bilinear in the generators (Levine, 1982) has been extensively discussed. The proposed solutions include the use of variational principles (Tishby and Levine, 1984), mean-field self-consistent methods (Meyer, Kucar, and Cederbaum, 1988), time-dependent constants of the motion (Levine, 1982), and numerous others, which we hope to discuss in detail in a sequel to this volume. 

Laws and McLaughlin30 discuss viscoelastic creep compliances of composite materials using another approach to the problem of the elastic properties of heterogeneous materials - the self-consistent method. 

For aligned transversely isotropic elements the self consistent method gives (Walpole58 ) the relations... 

Laws and McLaughlin30 solve the problem of the viscoelastic ellipsoidal inclusion in anisotropic materials and then use the self consistent method to calculate the overall viscoelastic compliances for a composite. 

The earliest works of trying to model different length scales of damage in composites were probably those of Halpin [235, 236] and Hahn and Tsai [237]. In these models, they tried to deal with polymer cracking, fiber breakage, and interface debonding between the fiber and polymer matrix, and delamination between ply layers. Each of these different failure modes was represented by a length scale failure criterion formulated within a continuum. As such, this was an early form of a hierarchical multiscale method. Later, Halpin and Kardos [238] described the relations of the Halpin-Tsai equations with that of self-consistent methods and the micromechanics of Hill [29],... 

Einding the inducible dipoles requires a self-consistent method, because the field that each dipole feels depends on all of the other induced dipoles. There exist three methods for determining the dipoles matrix inversion, iterative methods, and predictive methods. We describe each of these in turn. 

When the charges are treated adiabatically, a self-consistent method must be used to solve for the shell displacements, d, (just as with the dipoles fi, in the previous section). Combining Eqs. [26], [28], and [29], we can write the total energy of the shell model system as... 

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. 

We analyze in this chapter recent developments in the area of ion-solid interactions, describing in particular the non-linear method to study the energy loss of ions in solids. We consider the non-perturbative scheme provided by quantum scattering theory, using self-consistent methods to... 

In Table 7.4, the differences between the experimental values for A/ and Ga for the first lines compared to B and In can be attributed to the above-mentioned phonon resonances. There are also non-negligible differences between the measured and calculated spacings as well as between the calculated spacings for the deepest levels. As already mentioned, it is possible to use a self-consistent method to obtain experimental acceptor energy levels in... 

In this determinant y>j (ft) indicates that electron /t has been assigned to orbital with spin and (fi) represents the assignment of electron ft to orbital with spin —In accordance with the Pauli exclusion principle, this wave function is antisymmetric under a permutation of any two electrons. The principle of the self-consistent method is then to minimize the integral... 

There are cases for which more than one solution is found, and it is possible that both may possess physical reality under certain conditions [12] (this will arise again in chapter 11). Furthermore, the Hartree-Fock method can be made multiconfigurational, i.e. several configurations can be mixed or superposed. An electron is then shared between different states, which goes beyond the independent particle approximation. The self-consistent method allows the mixing coefficients to be determined, but the configurations to be included must be specified at the outset, and there is no simple prescription as to which ones should be chosen or left out. 

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