Transpositional symmetries

B2) 888 (Aj) methyl groups in transposition assignment acc. to local symmetry C21,. Compounds with higher alkyl substance are known, too (43)... 

A ) Every achiral molecule of the class must have at least one symmetry operation which in permutational form corresponds to a transposition. 

In the case of permutation isomers with some indistinguishable ligands and/or with skeletal symmetry there are different but equivalent ligand permutations leading to the reference isomer of the configuration. Among the numerically ordered equivalent ligand index permutations that one is chosen for the descriptor which contains the lowest number of transpositions ). 

The symmetry properties of the quantities used in the theory of complex atomic spectra made it possible to establish new important relationships and, in a number of cases, to simplify markedly the mathematical procedures and expressions, or, at least, to check the numerical results obtained. For one shell of equivalent electrons the best known property of this kind is the symmetry between the states belonging to partially and almost filled shells (complementary shells). Using the second-quantization and quasispin methods we can generalize these relationships and represent them as recurrence relations between respective quantities (CFP, matrix elements of irreducible tensors or operators of physical quantities) describing the configurations with different numbers of electrons but with the same sets of other quantum numbers. Another property of this kind is the symmetry of the quantities under transpositions of the quantum numbers of spin and quasispin. 

This expression for /-electrons can be derived using the phase relations established for isoscalar parts of factorized CFP with different parities of the seniority number [24]. It turned out [91] that phase (16.55) provides sign relations between the CFP in the tables for d- and /-electrons, but it is unsuitable for the p-electrons. In this connection, in what follows all the relationships derived using the symmetry properties under transposition of the quantum numbers of spin and quasispin are provided up to the sign. 

CFP (9.11) also have a simple algebraic form. In the previous paragraph we discussed the behaviour of coefficients of fractional parentage in quasispin space and their symmetry under transposition of spin and quasispin quantum numbers. The use of these properties allows one, from a single CFP, to find pertinent quantities in the interval of occupation numbers for a given shell for which a given state exists [92]. 

Using (16.16) and the above relations, we can work out algebraic expressions for SCFP, and hence for CFP, in the entire interval of the number of electrons in the shell existing for given v. Taking account of the symmetry of CFP under transposition of spin and quasispin quantum numbers further expands the number of such expressions. Formulas of this kind can be established also for larger v, but with v = 5 and above they become so unwieldy and difficult to handle that this limits their practical uses. They may be found in [108]. 

Thus we see that even at v = 3 the expressions for CFP become rather involved. In a similar way, we can construct CFP for higher v, but the form of the resultant formulas will be more complex. For example, at v = 4 we obtain eight expressions for the CFP. They contain 3n -coefficients of high order. Expressions for the CFP with S = 2 and any Q(v) can, however, be derived using the symmetry properties under transposition of Q and S. 

The most direct way to represent the electronic structure is to refer to the electronic wave function dependent on the coordinates and spin projections of N electrons. To apply the linear variational method in this context one has to introduce the complete set of basis functions k for this problem. The complication is to guarantee the necessary symmetry properties (antisymmetry under transpositions of the sets of coordinates referring to any two electrons). This is done as follows. 

In this set the functions can be classified into two types in the right column the spatial multiplier is symmetric with respect to transpositions of the spatial coordinates and the spin multiplier is antisymmetric with respect to transpositions of the spin coordinates in the left column the spatial multiplier is antisymmetric with respect to transpositions of the spatial coordinates and the spin multipliers are symmetric with respect to transpositions of the spin coordinates. Because in the second case the spatial (antisymmetric) multiplier is the same for all three spin-functions, the energy of these three states will be the same i.e. triply degenerate - a triplet. The state with the antisymmetric spin multiplier is compatible with several different spatial wave functions, which probably produces a different value of energy when averaging the Hamiltonian, thus producing several spin-singlet states. From this example one may derive two conclusions (i) the spin of the many electronic wave function is important not by itself (the Hamiltonian is spin-independent), but as an indicator of the symmetry properties of the wave function (ii) the symmetry properties of the spatial and spin multipliers are complementary - if the spatial part is symmetric with respect to permutations the spin multiplier is antisymmetric and vice versa. 

Multilin utilizes transpositional symmetry for any pairs of indices. Thus, if A is declared symmetric, antisymmetric, Hermi-tean, or anti-Hermitean in any index pair, an appropriate (multiply) triangular storage and addressing scheme is invoked. 

Transpositional symmetries are declared by appending the attribute ((dummy indices=opi(transposed dummy indices)=...= opn (transposed dummy indices)), where op. ..opn may be either "CONJG", or "-CONJG". For example, the two-particle reduced density matrix of quantum chemistry would be declared... 

The Pauli principle must also apply to the nuclei so that the transposition of the nuclei must lead to an antisymmetric function. Let us now consider the symmetry character of these functions with respect to the nuclei. Transposition of the nuclei in equation 8.10 gives ... 

We conclude this subsection by asking what happens when the permutational symmetry is lower but still non-trivial. One case is when 5i = S2, but S3 is not necessarily the same, and is invariant merely to the transposition (12). The corresponding group P2 has two classes Ai and. 4 2 say. The character of (12) was given before and we also notice the "branching rules" for representations of P3, which are... 

Note the symmetry of the last term with respect to the transposition of the a and p indexes ... 

The proper combinations of X functions for naphthalene are obtained for the separate F s as the dot product of each horizontal row of the character table with the table of transpositions under the symmetry operations. Thus, for Fj, we have... 

Here, or e S is an element of the permutation group of the n electron labels, and sgn(a) is its parity. Equation (6.43) indicates that this permutation can equally well be applied to the component labels, since the determinant is invariant under matrix transposition. We can now calculate the matrix element in the symmetry operator ... 

The transposition (23) means a reflection, and its representing matrix (with respect to the chosen particular basis) can be evaluated simileirly. This is a bit cumbersome, but we do not need to evaluate all 24 of these matrices. All we need is to check that there is a symmetry operation that is an improper rotation, for example, a reflection or an inversion. This is true in our case, as we just saw. 

The transposition to the tetrahedral coordination symmetry (T ) can be also directly obtained from the 4th column (in Table 2.16) for the metallic... 

The transposition from the LFSE Tj)I values for the ions in the valence state (II) to those corresponding to the ions in the valence state (III), for the Tj symmetry, will be done, again, through the one-to-one multiplication of the 6th column with the factor 3/2 thus obtaining the 7th column of Table 2.16. 

In evaluating the elements of H and M, the special form of the spatial function may now be recognized. By assuming that electrons 1 and 2 occupy the first orbital, 3 and 4 the second, and so on, we impose a symmetry on the spatial function 0. If 0 is symmetric under transposition (12), it will be necessary to ensure that the spin factor is anrisymmetric under (12) this must be so for each doubly occupied orbital, and the first g columns of any Young tableau describing an associated spin eigenfunction will thus read... 

Reconsider the derivation of 82 in Problem 8.8, noting that when all quantities are real only two types of 2-electron terms remain—both of which may be expressed in terms of the matrices J and K defined in (6.2.30). Hence obtain (8.4.11). [Hint For real matrices the dagger is equivalent to transposition, and a quantity may thus be regarded as the rt-element of the matrix e,el, where c, and c are the s and u columns of T. Remember also that for real orbitals (rs g / ) = (rt, su) has an 8-fold symmetry, as follows from the definition.]... 

In cyclic symmetry, there is a -fold rotational axis, n being the number of subunits. Figure 2.38 shows the arrangements of protomers for dimer, trimer, tetramer, pentamer, and hexamer. The dimer has the Cj symmetry (each protomer coincides to the other by a rotation of 360°/2), the trimer has the C3 symmetry (rotation of 360°/3) and is required for the protomer transposition. The cyclic symmetry is referred to as C in the Shoenflies notation, n being the number of protomers. In the International Hermann-Mauguin notation it is referred to as n. 

---