Permutational symmetry group theoretical properties

Oppenheimer approximation, 517-542 Coulomb interaction, 527-542 first-order derivatives, 529-535 second-order derivatives, 535-542 normalization factor, 517 nuclei interaction terms, 519-527 overlap integrals, 518-519 permutational symmetry, group theoretical properties, 670—674... 

Dirac bra-ket notation, permutational symmetry, group theoretical properties, 672-674... 

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

Systems containing more than one identical particles are invariant under the interchange of these particles. The permutations form a symmetry group. If these particles have several degrees of freedom, the group theoretical analysis is essential to extract symmetry properties of the permissible physical states. Examples include Bose-Einstein, Fermi-Dirac, Maxwell-Boltzmann statistics, Pauli exclusion principle, etc. 

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 ... 

The wave functions are often labelled with N, the number of quanta, with /, the total angular momentum and parity, and with the dimension of the SU(6) representation. The Hamiltonian (3.43) has, indeed, a symmetry of structure U(6) = SU(6) x U(l), where U(l) is associated with the number of quanta N [6]. In fact, SU(6) also denotes another symmetry combining spin and flavour. It becomes exact when one neglects hyperfine effects and takes the limit of equal quark masses m = m = m. We refer to the specialized literature [6] for these group-theoretical aspects of the harmonic oscillator. For our purpose, it is sufficient to know the following correspondence between SU(6) representations and permutation properties ... 

Since the related Hamiltonian needs to remain invariant under all the symmetry operations of the molecular symmetry (point) group, the potential energy expansion, see equation (5), may contain only those terms which are totally symmetric under all symmetry operations. Consequently, a simple group theoretical approach, based principally on properties of the permutation groups can be devised, " which yields the number and symmetry classification of anharmonic force constants. The burgeoning number of force constants at higher orders can be appreciated from the entries given in Table 4. 

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