Permutation functionals

Each pair of unique, external hole or particle lines introduces a permutation function, P(pq) (as defined earlier in Eq. [154]), to ensure antisymmetry of the final expression. 

One might suppose that any linear combination of such permuted functions is suitable. Actually, because of the Pauli exclusion principle only combinations which are antisymmetric with respect to electron interchange are allowed. Slater has shown how to formalize these with the determinant below as the basis ... 

Tables 4.4-4.7 show the comparison for an assembly language program which performs the permutation function, and the entire program when replaced by a...

Fig. 1.5 Graphical example of mappings produced by the permutation functions n and their...

The subscript notation 7f i) in the mathematical expression given in Eq. (1.40) is based on the mathematical theory of permutations [101], where the permutation function value n i) gives the rank of the (th molecule and the unique inverse r j) designates they th molecule in the overall ranking. A graphic example of how these functions operate is provided in Fig. 1.5. It is important to note that while the permutations determine the rank order of the compormds, it is the similarity values themselves that are combined using the MEAN fusion rule in similarity fusion. 

Note the order of the subscripts on D[R] which follows from the fact that we use the N-convention of (equation A1.4.56) to define the effect of a permutation on a function. 

Stabilizing resonances also occur in other systems. Some well-known ones are the allyl radical and square cyclobutadiene. It has been shown that in these cases, the ground-state wave function is constructed from the out-of-phase combination of the two components [24,30]. In Section HI, it is shown that this is also a necessary result of Pauli s principle and the permutational symmetry of the polyelectronic wave function When the number of electron pairs exchanged in a two-state system is even, the ground state is the out-of-phase combination [28]. Three electrons may be considered as two electron pairs, one of which is half-populated. When both electron pahs are fully populated, an antiaromatic system arises ("Section HI). 

A symmetry that holds for any system is the permutational symmetry of the polyelectronic wave function. Electrons are fermions and indistinguishable, and therefore the exchange of any two pairs must invert the phase of the wave function. This symmetry holds, of course, not only to pericyclic reactions. 

With 4) containing a normalization factor and all permutations over the atomic orbital wave functions i (1 = 1,2,... 2n). Likewise, if all electron pairs were exchanged in a cyclic manner, the product wave function, 4>b, has the fonn ... 

IT. Total Molecular Wave Functdon TIT. Group Theoretical Considerations TV. Permutational Symmetry of Total Wave Function V. Permutational Symmetry of Nuclear Spin Function VT. Permutational Symmetry of Electronic Wave Function VIT. Permutational Symmetry of Rovibronic and Vibronic Wave Functions VIIT. Permutational Symmetry of Rotational Wave Function IX. Permutational Symmetry of Vibrational Wave Function X. Case Studies Lis and Other Systems... 

Let us examine a special but more practical case where the total molecular Hamiltonian, H, can be separated to an electronic part, W,.(r,s Ro), as is the case in the usual BO approximation. Consequendy, the total molecular wave function fl(R, i,r,s) is given by the product of a nuclear wave function X uc(R, i) and an electronic wave function v / (r, s Ro). We may then talk separately about the permutational properties of the subsystem consisting of electrons, and the subsystemfs) formed of identical nuclei. Thus, the following commutative laws Pe,Hg =0 and =0 must be satisfied X =... 

As pointed out in the previous paragraph, the total wave function of a molecule consists of an electronic and a nuclear parts. The electrons have a different intrinsic nature from nuclei, and hence can be treated separately when one considers the issue of permutational symmetry. First, let us consider the case of electrons. These are fermions with spin and hence the subsystem of electrons obeys the Fermi-Dirac statistics the total electronic wave function... 

Let us discuss further the pemrutational symmetry properties of the nuclei subsystem. Since the elechonic spatial wave function t / (r,s Ro) depends parameti ically on the nuclear coordinates, and the electronic spacial and spin coordinates are defined in the BF, it follows that one must take into account the effects of the nuclei under the permutations of the identical nuclei. Of course. 

VII. PERMUTATIONAL SYMMETRY OF ROVIBRONIC AND VIBRONIC WAVE FUNCTIONS... 

Since the total wave function must have the correct symmetry under the permutation of identical nuclei, we can determine the symmetiy of the rovi-bronic wave function from consideration of the corresponding symmetry of the nuclear spin function. We begin by looking at the case of a fermionic system for which the total wave function must be antisynmiebic under permutation of any two identical particles. If the nuclear spin function is symmetric then the rovibronic wave function must be antisymmetric conversely, if the nuclear spin function is antisymmebic, the rovibronic wave function must be symmetric under permutation of any two fermions. Similar considerations apply to bosonic systems The rovibronic wave function must be symmetric when the nuclear spin function is symmetric, and the rovibronic wave function must be antisymmetiic when the nuclear spin function is antisymmetric. This warrants... 

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

The permutational symmetry of the rotational wave function is determined by the rotational angular momentum J, which is the resultant of the electronic spin S, elecbonic orbital L, and nuclear orbital N angular momenta. We will now examine the permutational symmetry of the rotational wave functions. Two important remarks should first be made. The first refers to the 7 = 0 rotational... 

Next, we address some simple eases, begining with honronuclear diatomic molecules in E electronic states. The rotational wave functions are in this case the well-known spherical haimonics for even J values, Xr( ) symmetric under permutation of the identical nuclei for odd J values, Xr(R) is antisymmetric under the same pemrutation. A similar statement applies for any type molecule. 

However, drastic consequences may arise if the nuclear spin is 0 or In these cases, some rovibronic states cannot be observed since they are symmetry forbidden. For example, in the case of C 02, the nuclei are spinless and the nuclear spin function is symmetric under permutation of the oxygen nuclei. Since the ground electronic state is only even values of J exist for the ground vibrational level (vj, V3) = (OO O), where (vi,V2,V3) are the... 

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