Weyl’s number

The dependence of Weyl s number on the number of basis functions, M. A five-electron system with a total spin of i is considered as an... 

In general, for basis sets which are capable of supporting the level of accuracy required in chemical studies, the full configuration interaction problem is computationally intractable. Even when configuration state functions are employed in place of determinants, the total number of terms in the full configuration interaction expansion is prohibitive. The number of configuration state functions is given by Weyl s number... 

In closing the section on non-Hermitean approaches to continuum processes in atomic and molecular physics, we will also mention accurate examinations on resonance parameters in molecular predissociation displaying unexpected resonance overlapping [46,89]. The phenomenon of predissociation by rotation in HgH was analyzed via an isotopically combined potential due to Stwalley [120]. The potential, i.e., a relatively shallow energy curve with a nonzero /-value giving rise to a rotational barrier, supported novel metastable states above the dissociation limit. The Weyl s method was able to resolve the closely lying vibrational states v = 3 and v = 4 for the rotational quantum number K = 9. 

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,... 

A more brief coverage of much of the same material is found in Houben-Weyl s Methoden der organische Chemie, particularly in the first three volumes, as mentioned above. A number of books dealing with specific techniques are also available, and many of the more recent of these sources have been listed under the reading references at the end of each chapter. 

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... 

There are a number of large works in organic chemistry which deal with laboratory synthesis and technique. A very comprehensive German work is Houben-Weyl s Methoden der organischen Chemie (Fig. 34) edited by E. Muller (G. Thieme Verlag, 4th edn., 1952-) of which 16 volumes have been planned, some in several parts. Volume 1 deals with general methods of laboratory practice in organic chemistry, vol. 2 is a summary of analytical... 

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... 

Now that we know the number of linearly independent -particle fimctions for a particular y, we can ask for the total number of linearly independent -particle functions that can be generated from m orbitals. Weyl[38] gave ageneral expression for all partitions and we will only quote his result for our two-column tableaux. The total number of functions, D n, m, S), i.e., the size of a full Cl calculation is... 

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... 

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... 

Table 2.2 Synthesis of macromolecules by chain growth polymerization. Note The citation of literature in the right-hand column of the following tables should be understood as follows the number(s) in the first line(s) refer(s) to this book. For example, 3.2 means experiment number 2 in Chap. 3. The following line refers to Houben Weyl, Methoden der Organischtai Chemie, Vol. E20, Makromolekulare Stoffe Teil 1-3, Thieme Verlag, Stuttgart, New York, 1987 (abbreviated HW). For example, the citation HW 135 refers to an experimental procedure described on page 135 thereof. The bottom line refers to H. R. Kricheldorf, Handbook of Polymer Synthesis, Vols. 1 and 2, Dekker, New York, 1992 (abbreviated K). For example, K 34 refers to further references given on page 34 of that book...

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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