Consistency problems

Thereby the solution of the electronic-structure problem for an N-atomic system is decomposed into N locally self-consistent problems including only the M atoms in the LIZ associated with each atom in the system, and the computational effort now scales linearly with N, i.e. exhibits 0 N) scaling. 

With those people who develop drug problems, they do not necessarily experience consistent problems over time. Some clients do have continuous problems, a situation often called chronic drug dependence by therapists. However, some... 

Using this expression and a model for in (6.145) and (6.146) yields a complete description of reactive scalar mixing without the consistency problems associated with CMC. Moreover, since it is not necessary to supply boundary conditions for the conditional moments,116 the FP model can be applied to partially mixed regions of the flow117 where the CMC boundary conditions cannot be predicted a priori. 

To summarize, it has been found that the SH method is able to at least qualitatively describe the complex photoinduced electronic and vibrational relaxation dynamics exhibited by the model problems under consideration. The overall quality of SH calculations is typically somewhat better than the quality of the mean-field trajectory results. In particular, this holds in the case of several curve crossings (see Fig. 2) as well as when the dynamics and the observables of interest are essentially of adiabatic nature— for example, for the calculation of the adiabatic population dynamics associated with a conical intersection (see Figs. 3 and 12). Furthermore, we have briefly discussed various consistency problems of a simple quasi-classical SH description. It has been shown that binned electronic population probabilities and no momentum adjustment for classically forbidden transitions help us to improve this matter. There have been numerous suggestions to further improve the hopping algorithm [70-74] however, the performance of all these variants seems to depend largely on the problem under consideration. 

Surface area A deviations from 30% to 70% were calculated for identical runs of methane and ethane, suggesting a self-consistency problem. 

Because of the incongruity in the desired extension of "bond order" (using a discrete variable as the measure for a concept that requires a continuous variable), any nomenclature will have to lump together structures having "similar" bond orders. Here the heuristics of "similar" is assumed — with all of the consistency problems that could arise by such an assumption. In the proposed system,... 

Most discrete approximations that have been mentioned in this book are consistent, with the exception of one. This is the DuFort-Frankel method [216], described on page 153 in Chap. 9. It is stable for all A, yet it has a consistency problem. Giving (9.19) the same treatment as above, one ends with... 

This property is no longer true for thin films, where ( ) and hence the shape of the profile also depend on the boundary condition. Obviously, Eqs. (14), (19) then define a non-trivial self-consistency problem, which requires an iterative numerical treatment [58]. We note, however, that Eqs. (14), (18) yield a simple relation between ( > and ( )0=< )(z=+D/2),... 

Equation 4.7 has the form of a self-consistency problem. The solution to the equation is rffj(fi), but the exact form of the equation is determined by (/ y(/r) itself. It must be solved by an iterative procedure (called the self-consistent held, or SCF approach), in which convergence is taken to occur at the step where both sj and (/ y(/r) do not differ appreciably from the prior step. 

One notable problem with termination of the full Cl expansion has been referred to as the size-consistency problem. What this means is that the same treatment of a complex is fundamentally different than that of the subunits of which it is composed. Consider for example the CISD level. Such a treatment of the first subunit would permit double excitations within it similarly for the other subunit. A consistent theory should hence permit quadruple excitations within the complex, as this would account for simultaneous double excitations in each of the subunits. But CISD terminates the excitation list at doubles in the complex. Truncated Cl treatments are therefore fundamentally poorly disposed to handle molecular interactions such as hydrogen bonds. 

There have been a number of correction algorithms formulated over the years to help improve this size consistency problem but they do not entirely resolve it °. Another means of introducing size consistency is by a quadratic approximation, QCISD. The approach achieves this size consistency by sacrificing its variational character. It can be considered as a simplified approximate form of CCSD (see below) the method may cease to remain size consistent on going to higher levels of substitution. ... 

In ref. 36 Song et al. introduced an analogue of Davidson s correction for VBCISD wavefunctions, which provides an estimate of the contribution of quadruple excitations that are products of double excitations to the correlation energy. This has been shown to alleviate, just as in the MO case, the size-consistency problem in calculations on small molecules. 

If Japanese beetles are a consistent problem in your area, avoid planting raspberry cultivars particularly attractive to these pests. Cultivars to avoid include Fall Gold , Festival , Heritage, Latham , Newburgh , Reveille , Ruby , and Skeena . In the future, gardeners may be able to buy parasitic nematodes that can be applied to the soil to control the grub stage of this pest. 

A great improvement can be achieved by using Newton-Raphson methods (Bendt and Zunger, 1982 Srivastava, 1984) for solving our self-consistency problem, which consists of finding the value of input potential u(r) for which the output potential u(o)C)> a functional of u(r), satisfies ... 

Strong coupling is implied when the following occurs. If, in the process of solving the self-consistent problem, one successively shifts states of the ion and positions of the wall sources to the point that... 

Here is the difference between the MRCI energy and the energy obtained with the reference CSFs, and Cj, is the coefficient of reference CSF R in the MRCI wave function. We shall use the notation i -I- Q) to denote the use of the corrections given by Eq. (7) or (8). A problem with such approximations is that as there are no useful bounds on the adjusted energy expression, some numerical noise can be introduced across a large set of calculations (say, for a potential energy surface), while the MRCI values themselves would be much smoother. " (This noise can also be a problem with those MRCI methods that apply selection to all CSFs used, rather than just reference CSFs.) A more sophisticated approach to the size-consistency problem is to... 

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