Single density

We ve specified five excited states with NSlotes=5 (the reasons will be clear in a moment). The DensitysAll keyword tells Gaussian to perform the population analysis using all available densities the SCF (ground state) density, the Cl one-partide density, and the Cl (Cl-Singles) density. The population analyses using excited state densities will be performed for the first excited state (the default if the Root option is not included), which is the one in which we are interested. 

This appendix contains some of the data generated for the SIMCA and PLS analyses. The complete data set is available from the authors. Upon request the data will be provided on 8 single density single sided floppy disks in IBM 3740 format for CP/M based systems or on 5 1/4 double sided double density floppy disk for the IBM/PC or other MS/DOS based computers. The requestor, however, must supply a properly formated floppy disk. 

In closing, it is emphasized that we are basing this work upon the simplified single-density formula for information, as is permitted by Liouville s measure-preserving theorem, which gives unity for the second (basic) density.11... 

Analogously the Lotka-Volterra model, let us write down the fundamental equation of the Markov process in a form of the infinite hierarchy of equations for the many-point densities. Thus equations for the single densities (m + m ) — 1 read ... 

The critical point condition is obtained similarly. For a single density variable, Eq. (56) reduces to... 

At some low enough density threshold a3, the methane molecule has a single density domain that contains all five nuclei C, H, H, H, and H. At this density threshold a3 the set of density domains has again only a single element,... 

Localized density range, [amax, ag) amax >s the maximum electron density in the molecule, whereas ag is the threshold below which only a single density domain is found. 

Skinny molecular range, [af, a< ) af is defined above, whereas is the maximum threshold at and below which all locally nonconvex domains on the surface of density domains are simply connected. In simpler terms, in the skinny molecular range all nuclei are found within a single density domain, but there are formal "neck regions on the surface of density domains. In the terminology of shape group analysis [2], rings of D) type can be found on the surface of density domains. 

A finite number of the coefficients could be found from a hierarchy of differential equations for single densities, joint densities etc. These series were found well-behaved with the coefficients all positive and monotonously increasing. Even moderate length series of 5-6 terms gave exponents quite close to those analytically estimated for above-mentioned basic bimolecular reactions (see Section 5.3). 

Figure 2.2 Selected families (DDj(aj)) of density domains of the water molecule, as calculated with the GAUSSIAN 90 ab initio program [253] and the GSHAPE 90 molecular shape analysis program [254], using a 6-3IG basis set. There are only two topologically different types of families of density domains either a single density domain, or a family of three density domains. The sequence of topologically distinct cases provides a topological description of chemical bonding.

There are only two topologically different types of families of density domains of water either a single density domain,... 

The first family displayed, DDj(ai))= (DDi(a )), contains only a single density domain DD (ai) of density threshold value a =0.45 at the given 6-31G level of ab initio calculations. The corresponding body DD(a ) has only one component, it is a topological ball, and the only nucleus it encloses is that of the oxygen atom. At a lower density threshold value a2, separate density domains about the hydrogen nuclei appear, hence a topologically different family... 

Although these shapes are all different, they all represent the same topological molecular body of a single density domain. 

The shape changes of this single density domain can be described in detail using the Shape Group Method (SGM) [155-158], discussed in later chapters of this book. 

In any of these alternatives, it is expected that those fragments serve best within the MEDLA approach which possess a well recognizable identity, as manifested by the existence of a single density domain for the nuclei of the fragment within the parent molecule. 

Consider the conformations Ki and K2 of interacting molecules M, and M2, respectively, and their local shape complementarity, with reference to the MIDCOs G(Ki,aQ) and G(K2,a0) for the given interaction and a contact density value aQ. Naturally, shape complementarity is limited to MIDCOs of a single density threshold value aQ, and, generally, one should consider the local shape complementarities of all MIDCO pairs G(Ki,a0—a ) and G(K2,a0 + a ) in some narrow density interval surrounding the reference density threshold ... 

Proponents of this method claim that by using a fixed temperature the variation of conductivity within the sample as well as the occupation statistics are also fixed so that these spectra are more easily interpreted. Proponents of the standard DLTS method point out the enhanced ability to vary the thermal emission energy by sweeping temperature (a DLTS scan with 10 sec" rate window taken over the temperature range 100-450°Kin a-Si H is equivalent to a time scale range at 300°K of from 3 nsec to 65 hr. To the extent that the DLTS spectra for a variety of rate windows can be fit consistently to a single density of states, there should be no difference between these... 

Numerical implementation of the boundary conditions is conducted by the use of the IN-FORM tool in PHOENICS 2008, which increases the simulation time, as higher computational effort is demanded. On the other hand, it represents a simpler and more intuitive manner of dealing with complex initial or boundary conditions. A point-history object is used to track the plume concentration reached on a location at a distance of about 750 meters far from the source point. A lighter-than-air gas is considered and a single density calculation is conducted for the whole domain, as a fimction of the gas fraction (mixture air and gas). A monitoring time of 30 minutes is simulated by the use of adequate time and space grids 9000 time steps (0.2 s for each) and about 18,000 hexahedral cells. 

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