Permutation parity

Of course, PP = P" has the parity (—l)f+P. because this is how such a permutation parity is to be calculated first, we make p transpositions in order to get P, and next, making p transpositions, we obtain the permutation PP. Note, that when keeping P fixed and taking P from all possible permutations, we ate rutming with PP over all possible permutations as well. This is because the complete set of penmitations is obtmned independently of what the starting pemuitation looks like (i.e.. independent of P ). 

Figure 2-77. Determination of parity value, a) First the structure is canonicalized. Only the Morgan numbers at the stereocenter are displayed here, b) The listing starts with the Morgan numbers of the atoms next to the stereocenter (1), according to certain rules. Then the parity value is determined by counting the number of permutations (odd = 1, even = 2).

Pc represents the parity of the permutations belonging to class C The symbol Fis given by ... 

According to the argument presented above, any molecule must be described by wavefunctions that are antisymmetric with respect to the exchange of any two identical particles. For a homonuclear diatomic molecule, for example, thepossibility of permutation of the two identical nuclei must be considered. Although both the translational and vibrational wavefunctions are symmetric under such a permutation, die parity of the rotational wavefunction depends on the value of 7, the rotational quantum number. It can be shown that the wave-function is symmetric if J is even and antisymmetric if J is odd The overall... 

Where the summation is over all 2n permutations P each with parity ep. We use a short-hand notation ... 

Configurations of Ruch s class A occur pairwise, e.g., asymmetric C-atoms. The parities of their permutation descriptors can be used as their -parity descriptors. A permutation containing an even number of transpositionsP has a parity of +1, conversely, odd permutations have parities of — 1. Since a-atom RASI (see Section 5.4.1) correspond in most casses to the CIP sequences, an assignment of +1 to an asymmetric C-atom is equivalent to an R-configuration, and — 1 to the S, if the skeleton is indexed as shown in 25. 

In each product function, the same set of one-electron quantum numbers is arranged in the same order (usually in the standard order 1,2,..., N) but the electron coordinates ri,r2,r3,... have been rearranged into some new order r i,ry2,ry3,. The summation in (10.8) is over all N possible permutations P = jij2h jN of the normal coordinate ordering 12 3. .. N, and p is the parity of the permutation P (p = 0 if P is obtained from the normal ordering by an even number of interchanges, and p = 1 if an odd number of interchanges is involved). 

SU(3) symmetry in hypernuclear physics Radicati, Wigner s supermul-tiplet theory100 Fraunfelder, Parity and Time Reversal in Nuclear Physics Wilkinson, the isobaric analogue symmetry Aage Bohr, the permutation group in light nuclei and J. P. Elliot, the shell model symmetry. 

The antisymmetrized state function for N electrons is the sum of products such as j/ i(1) 02(2). .. i7n N) and similar terms with the variables permuted between the same set of one-electron eigenfunctions i/ i Vb /VI- Thus each term contains the same product of spherical harmonics and the state therefore has parity... 

The overall symmetry of a given level must be antisymmetric with respect to the permutation P 2 of the two H nuclei to satisfy the Pauli principle. The + and -parity combinations defined by equation (8.202) are antisymmetric and symmetric respectively with respect to P 2 (because the electronic wavefunction has u character, see equation (8.251)). Since the ortho and para nuclear spin states are symmetric and antisymmetric respectively, we see that the + parity states combine with the ortho... 

In relation to permutation inversion symmetry species the superscript + or — may be used to designate parity. 

Examples Ax+ totally symmetric with respect to permutation, positive parity A - totally symmetric with respect to permutation, negative parity... 

The component Mz belongs to the species 4" in the Dah group because fiz is not changed by pure permutations and it changes sign by permutation—inversion operations (Section 4.1). The overall symmetry selection rule therefore allows transitions only between vibration—inversion-rotation states with opposite parity with respect to the operation of inversion (cf. Fig. 6). 

For group-theoretical selection of nonzero matrix elements of the hyperpolarizability components, one has to know the symmetry properties of the operator / < ( w). Owing to the definition (153), /3y (w) is symmetric with respect to the permutation of the indices./ and k, but it has no definite symmetry with respect to the permutation of all the indices and has no definite parity with respect to the operation of time reversal ... 

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