Dimension formula

This is frequently called the Weyl dimension formula. For small S and large m and n, D can grow prodigiously beyond the capability of any current computer. 

The Weyl dimension formula (Eq. (5.115)) tells us that six electrons in six orbitals in a singlet state yield 175 basis functions. These may be combined into 22 Aig symmetry functions. Table 15.1 shows the important HLSP functions for a rr-only calculation of benzene for the SCF optimum geometry in the same basis. The a orbitals are all treated in the core , as described in Chapter 9, and the tt electrons are subjected to its SEP. We discuss the nature of this potential farther in the next section. The functions numbered in the first row of Table 15.1 have the following characteristics. 

Our treatments of ethylene are all carried out with two methylene fragments that have the a la b parts of both of their configurations doubly occupied in all VB stmctures used. The 12 electrons involved can be placed in the core as described in Chapter 9, which means that there are only four electrons, those for the C—C a and 7t bonds, that are in the MCVB treatment. For simplicity we shall rename the other two methylene orbitals a, and tti, where / = 1,2 for the two ends of the molecule. The Weyl dimension formula tells us that there are 20 linearly independent tableaux from four electrons distributed in four orbitals. When we use D2h s mimetry, however, only 12 of them are involved in eight Ai functions. 

The wave function is an extension of the one we used for the dissociation of ethylene. We now have 18 electrons in nine core orbitals, and six electrons in the three a and three tt orbitals that will make up the C—C bonds. As before, the valence orbitals are allowed to breathe (see Eqs. (16.1) and (16.2) for the linear combinations) as the system changes. According to the Weyl dimension formula... 

Therefore the dimension of the model space is equal to the dimension of the pertinent representation and is given by the Weyl-Paldus dimension formula [20, 21]... 

Here, the brackets are used to indicate an operation using the basic dimension of the variables. It is not difficult to obtain the dimension formulae for the variables presented in the previously discussed examples these are ... 

Let us apply this formula to the Cantor set. The Cantor set consists of two pieces, each of which can be magnified by a factor m = 3 and yields a complete copy of the original set. Therefore, we have k = 2 and m = 3 and d — ln(2)/ln(3) according to (2.3.8). This result agrees with what we computed using the dimension formula (2.3.3). 

In contrast to MO approaches, having more than one basis function on an atomic centre is a major problem for classical VB theory. For example, if in the above-mentioned 7t-only VB description of benzene we decide to switch f rom a single- to a double-C basis, the number of covalent structures increases from 5 to 2 X 5 = 320 and, according to Weyl s dimension formula which gives the number of linearly independent configurations for N electrons distributed between M orbitals,... 

The size of the full Cl space in CSFs can be calculated (including spin symmetry but ignoring spatial symmetry) by Weyl s dimension formula.82 If N is the number of electrons, n is the number of orbitals, and S is the total spin, then the dimension of the Cl space in CSFs is given by... 

The discussion of other MCSCF wavefunctions will be preceded by a brief discussion of the full Cl expansion. The full Cl expansion for a molecular system consists of all possible orbital occupations and all possible spin couplings consistent with the overall molecular spatial and spin symmetry. If the orbital basis set is complete (i.e. an arbitrary function of the three spatial coordinates may be represented exactly with the basis), the CSF expansion space is also complete and is called complete Cl. Only finite, and therefore incomplete, basis sets are considered in this discussion. If the reductions due to spatial symmetry are ignored, the number of expansion terms is given by the WeyF dimension formula... 

Presently, the widely used post-Hartree-Fock approaches to the correlation problem in molecular electronic structure calculations are basically of two kinds, namely, those of variational and those of perturbative nature. The former are typified by various configuration interaction (Cl) or shell-model methods, and employ the linear Ansatz for the wave function in the spirit of Ritz variation principle (c/, e.g. Ref. [21]). However, since the dimension of the Cl problem rapidly increases with increasing size of the system and size of the atomic orbital (AO) basis set employed (see, e.g. the so-called Paldus-Weyl dimension formula [22,23]), one has to rely in actual applications on truncated Cl expansions (referred to as a limited Cl), despite the fact that these expansions are slowly convergent, even when based on the optimal natural orbitals (NOs). Unfortunately, such limited Cl expansions (usually truncated at the doubly excited level relative to the IPM reference, resulting in the CISD method) are unable to properly describe the so-called dynamic correlation, which requires that higher than doubly excited configurations be taken into account. Moreover, the energies obtained with the limited Cl method are not size-extensive. 

This conclusion becomes even clearer if we replace c by another quantity, the volume fraction of a polymer in the solution. Let v be the volume of a single segment, then, with c such segments per unit volume, the fraction of the whole volume occupied by the segments is = cv. The advantage of (compared to c) is that it has no dimensions. Formulas become even... 

The intensity theory will be explained with emphasis on the relationships between theoretical quantities and experimental results. It may look confusing that the molar absorptivities obtained from optical absorption spectra are expressed in terms of mor lcm", whereas the phenomenological intensity parameters are expressed in cm. Therefore, we will emphasize the dimensions of the diflerent quantities and, if appropriate, units will be mentioned. For the dimensions of the quantities, only length (L), mass (M) and time (T) are used with dimensionless quantities we use a slash (/). In order to simplify certain formulae, the dimension of a charge (M L VT) is not always written explicitly, but e will be given instead. The same is true for the dimension of an energy (M LVt ), which will be reported in the dimension formulae as energy. 

It has been recognized for some years that the unit cell dimensions of micas are dependent on their chemistry i.e., when isomorphous substitutions occur in micas, the cell dimensions change in ways that depend upon ionic radii and should, therefore, be predictable. A number of attempts have been made to develop cell-dimension formulas or algebraic relations between cell parameters and the ionic proportions as expressed in the structural chemical formulas. 

The reasons for seeking to establish such cell-dimension formulas are several (1) They should provide quantitative relations between two properties determined independently, and hence give a check on each determination (2) in some cases, the cell parameters will broadly indicate the chemistry, at least by distinguishing biotites from phlogopites from muscovites (3) the development of adequate cell formulas for all micas (including some of very unusual composition) has refined our understanding of their crystal structures. 

Cell-dimension formulas developed so far have, therefore, been almost entirely empirical, though it has usually been shown that the formulas make physical sense as well. The best-known formulas are probably those of Brown [1951], Brindley and MacEwan [1953], Radoslovich [1962], and Donnay et al. [1964]. These formulas are largely Z>-axis formulas because a = b/, /3 very closely indeed and does not require a separate expression. For the micas, it has proved quite difficult to develop cell-dimension formulas relating the layer thickness [i.e., isomorphous substitutions indeed, it has only been attempted by Donnay et al. [1964], and then only for the one-layer trioctahedral micas. 

The b dimension formula above does not include a term for AP substituting for tetrahedral Si , because the coefficient was found to be not significantly different from zero. This simply reflects the ease with which a tetrahedral layer can contract by articulating at the shared or bridging oxygens. 

Donnay et al. [1964] have carefully reviewed the basic postulates for cell-dimension formulas in the case of trioctahedral one-layer micas. For these micas, Donnay et al. offer formulas to enable the detailed prediction of atomic coordinates on the basis of known composition, the experimental values of b and of d (001) = c sin jS, and average values of tetrahedral and octahedral metal-oxygen distances taken from the literature. Their predictions certainly give quite close agreement with atomic positions and bond lengths where these are known experimentally. Their formulas are limited in application to the one-layer trioctahedral micas as Brown [1965] has commented, the Donnay formulas do not allow for ordering of octahedral cations or for incomplete occupancy of the octahedral sites, partly because both factors would be very difficult to deal with theoretically. 

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