Number of configurations

CASSCF is a version of MCSCF theory in which all possible configurations involving the active orbitals are included. This leads to a number of simplifications, and good convergence properties in the optimization steps. It does, however, lead to an explosion in the number of configurations being included, and calculations are usually limited to 14 elections in 14 active orbitals. 

Unfortunately, this is not a feasible approach for calculating thermodynamic properties di to the large number of configurations that have extremely small (effectively zero) Boltzmar... 

Filter collectors available in a large number of configurations, resulting in a range of dimensions and inlet and outlet flange locations to suit installment requirements... 

We therefore have to generate a significant number of configurations, and we might (for example) envisage moving each of the particles in succession according to the prescription... 

The number of excited determinants thus grows factorially with the size of the basis set. Many of these excited determinants will of course have different spin multiplicity (triplet, quintet etc. states for a singlet HF determinant), and can therefore be left out in the calculation. Generating only the singlet CSFs, the number of configurations at each excitation level is shown in Table 4.1. 

The Multi-configuration Self-consistent Field (MCSCF) method can be considered as a Cl where not only the coefficients in front of the determinants are optimized by the variational principle, but also the MOs used for constructing the determinants are made optimum. The MCSCF optimization is iterative just like the SCF procedure (if the multi-configuration is only one, it is simply HF). Since the number of MCSCF iterations required for achieving convergence tends to increase with the number of configurations included, the size of MCSCF wave function that can be treated is somewhat smaller than for Cl methods. 

RHF to UHF, or to a TCSCF, is almost pure static correlation. Increasing the number of configurations in an MCSCF will recover more and more of the dynamical correlation, until at the full Cl limit, the correlation treatment is exact. As mentioned above, MCSCF methods are mainly used for generating a qualitatively correct wave function, i.e. recovering the static part of the correlation. 

Table 4.3 Number of configurations generated in a [ , ]-CASSCF wave function...

The internal energy will fluctuate around a mean value which may be calculated by averaging over the number of configurations, U)... 

Notice that the averaging is done over the number of particles N and to, but not the number of configurations M. Since an MD simulation produces a set of time-connected configurations, the number of a given configuration is directly related to the simulation time. 

Many workers have employed pre-cracked specimens in a number of configurations that permit interchangeable data to be obtained from an analysis derived with fracture mechanics. The pre-crack provides an ideal crevice for... 

Once the classification is fixed, the first principal branch can be chosen in /4j, the second in and the third in different ways. Thus the number of configurations of principal branches is... 

Calculating the number of configurations which are nonequivalent with respect to the three groups we find ... 

Burnside s Lemma in its simplest form states that, under conditions similar to those of the general Polya problem given above, the number of inequivalent configurations can be obtained as follows. For each element of G count the number of configurations that are invariant under G, and take the average of these numbers. In symbols, the number of inequivalent configurations is... 

If we disallow reflections, thus heeding the fact that the molecule exists in three-dimensional space, we must use the alternating group instead of S. In this case, as the reader can verify, the total number of configurations becomes 36. This shows that in the latter case there are two distinct configurations which become equivalent if reflection is allowed these are precisely the two configurations in which the substituents are all different. 

The different applications led to an immense number of configurations and sizes, for example small round cells for hearing aids or large prismatic cells for the lead accumulators used in trucks. Here the great variety of demands has the consequence... 

Thus we assume that an ice crystal can exist in any one of a large number of configurations,6 each corresponding to certain orientations of the water molecules. The crystal can change from one configuration to another by rotation of some of the molecules or by the motion of some of the hydrogen nuclei, each moving a distance of about 0.86 A. from a potential minimum 0.95 A. from... 

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