Weyl formula

The conceptual simplicity of the CASSCF model lies in the fact that once the inactive and active orbitals are chosen, the wave function is completely specified. In addition such a model leads to certain simplifications in the computational procedures used to obtain optimized orbitals and Cl coefficients, as was illustrated in the preceding chapters. The major technical difficulty inherent to the CASSCF method is the size of the complete Cl expansion, NCAS. It is given by the so-called Weyl formula, which gives the dimension of the irreducible representation of the unitary group U(n) associated with n active orbitals, N active electrons, and a total spin S ... 

The Weyl formula gives the dimensions Ps (iV, Ne) of the basis for N orbitals with Ne electrons and total spin 5,... 

The number of configuration functions that can be generated from m orbitals for a state with n electrons and total spin S is given by the Weyls formula... 

How many triplet structures can you find for the system in Problem 7.6 And how many Weyl tableaux Show that in each case the number agrees with that predicted by the Weyl formula (7.6.18). Show also that of four apparently distinct short-bonded non-polar structures only three are linearly independent. Construct the branching-diagram functions and show that the second and third, in inverse last-letter sequence, are linear combinations of the spin functions in the remaining short-bonded structures. Proceed as in Problem 7.7 to obtain combinations belonging to the irreps of C, . 

Here the energy E and the wave number k are related as E = h2k2/ 2rn. When inserted into Krein s formula, the product over the single-scatterer determinants generates just the bulk (or Weyl term) contribution to the density of states... 

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... 

The number of configurations that can be formed from allocating N electrons to n orbitals with spin S is given by a formula due to Weyl and Robinson, but before giving the formula it is useful to establish some similarities and differences to the symmetric group approach. Again, the valid partitions of N (for fermions) must be of the form [2° l6], as we saw above, so we immediately have... 

To determine the interplay between the spectral properties, both boundary conditions, we return to Weyl s theory [32]. The key quantity in Weyl s extension of the Sturm-Liouville problem to the singular case is the m-function or ra-matrix [32-36]. To define this quantity, we need the so-called Green s formula that essentially relates the volume integral over the product of two general solutions of Eq. (1), u and v with eigenvalue X and the Wronskian between the two solutions for more details, see Appendix C. The formulas are derived so that it immediately conforms to appropriate coordinate separation into the... 

Realizing that Eq. (13) gives an explicit solution of (1) with an appropriate V, in terms of logarithmic derivatives, it is possible to identify u with the well-known Jost solution denoted as/(r, 2), see more below and Ref. [44], which here must be proportional to the Weyl s solution x(f, )- With this identification, we obtain the generalized Titchmarsh formula (generalized since it applies to all asymptotically convergent exponential-type solutions commensurate with Weyl s limit point classification)... 

In summary, we have derived formulas for the Weyl-Titchmarsh m-function, where the imaginary part serves as a spectral function of the differential equation in question. Before we look at the full m-function, we will see how it works in connection with the spectral resolution of the associated Green s function... 

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]... 

The decomposition (2.42) of one exponential into a product of exponentials is a generalization of the formula due to Weyl and Glauber.187 One finds the expressions for /, g, h, ip by differentiating the two members of (2.42) with respect to r and using (A.26) to eliminate the exponentials. One finds a system of differential equations in the variable r, whose solution satisfying the required limit conditions is (2.43). 

The CAS Cl procedure is thus a method which can be used to partition the full Cl space into one part that comprises the most important CFs and another much larger part, which it is believed that one can treat using other quantum chemical approaches, like for example perturbation theory. One problem with the approach is the size of the Cl expansion. The number of CFs of a given spin symmetry S for N electrons and m orbitals is given by Weyl s formula ... 

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... 

In this work 2 was a sphere of radius R and the nucleus was placed at the center of the sphere. This reduced the problem to that of the radial function only. In 1911, H. Weyl solved some vibrational problems [3], which now may be interpreted as describing the structure of the highly excited part of the spectrum of a free particle in a bounded region 2 with Dirichlet boundary conditions. Weyl s famous asymptotic formulae for the density of states in a region of large volume, that depends on the volume but not on the form of the region 2 (see e.g. Sect. VI.4. in [4], or Sect. XIII.15 in [5]), are usually used in physical chemistry when the partition function is calculated for translational motion of an ideal gas. Nowadays the next term in this asymptotic expression is usually studied in the theory of chaos (see e.g. Sect. 7.2 of [6]). 

We shall return to this formula later the method of proving it is general, and not confined merely to elastic proper vibrations. Moreover, as has been shown by Weyl, the formula is true whatever may be the form of the volume F. 

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. 

Finally, it is possible to derive a simple formula (Weyl, 1956 Mulder, 1966) for the total number of linearly independent CFs of given S, M, with N electrons and m available orbitals it is... 

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