Configuration generating function

ThcorcnL The configuration generating function is obtained by substituting the figure generating function in the cycle index, by which is meant replacing every occurrence of s in the cycle index by f(x ). Thus... 

The figure generating function for the two sets are // + F + / and X + y (in an obvious notation), and a complete description of all distinct possibilities is given by the configuration generating function... 

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

But since x — 1) contains a constant term, if A x 1) is a conventional polynomial, then A x-, )/ x — 1) must also be a conventional polynomial. Thus, all reachable configurations represented by the generating function 1) have the form... 

That is, F(x,y,z) is the generating function of the number of nonequivalent configurations. The solution of our problem consists in expressing the generating function F(x,y,z) in terms of the generating function /(x,y,z) of the collection of figures and the cycle index of the permutation group H. 

It is easy to see that a combination with no repetitions gives rise to exactly two transitivity systems with respect to Ag. Summarizing the results, we have the rule the number of different transitivity systems of configurations with respect to Ag is the sum of the respective numbers of combinations with and without repetitions. Therefore, the generating function of the permutations which are nonequivalent with respect to Ag is... 

Theorem. The generating function for the configurations [] which are nonequivalent with respect to H is obtained by substituting the generating function of [4>] in the cycle index of U. 

Let jp(5 ) be the number of those configurations with content k,tl,m) which remain invariant under the permutation S in (1.6). Thus, the generating function of interest is... 

Making use of the relationship discussed above, "the number of noncongruent planted trees equals the number of nonequivalent configurations of three planted trees", of the generating function and the main theorem of Chapter 1 (Sec. 16) and taking the special case n 0 into account, we establish for each of the three situations an equation ... 

From the quantum mechanical standpoint the appearance of the factor 1/2 = 1/s for the diatomic case means the configurations generated by a rotation of 180° are identical, so the number of distinguishable states is only one-half the classical total. Thus the classical value of the partition function must be divided by the symmetry number which is 1 for a heteronuclear diatomic and 2 for a homonuclear diatomic molecule. 

The theory of statistical mechanics provides the formalism to obtain observables as ensemble averages from the microscopic configurations generated by such a simulation. From both the MC and MD trajectories, ensemble averages can be formed as simple averages of the properties over the set of configurations. From the time-ordered properties of the MD trajectory, additional dynamic information can be calculated via the time correlation function formalism. An autocorrelation function Caa( = (a(r) a(t + r)) is the ensemble average of the product of some function a at time r and at a later time t + r. 

Let us assume that the orbit-generating wavefunction for the nth state is given in the form of a configuration interaction wavefunction (i.e., a linear combination of configuration state functions) ... 

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