Symmetric properties permutational symmetry

For a system of either bosons or fermions, the wavefunction must have the correct properties of symmetry and antisymmetry. Particles with half-integral spin, such as electrons, are fermions and require antisymmetric wavefunctions. Particles with integral spin, such as photons, are bosons and require symmetric wavefunctions. The complete space-spin wavefunction of a system of two or more electrons must be antisymmetric to the permutation of any two electrons. Except in the simplest cases, the wavefunction for a system of n fermions is positive and negative in different regions of the 3 -dimensional space of the fermions. The regions are separated by one or more (3 - 1 )-dimensional hypersurfaces that cannot be specified except by solution of the Schrodinger equation. 

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. 

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. 

Central to freeon dynamics is the indistinguishability of electrons this property is a symmetry which is expressed in terms of the symmetric group, Sn> the group of permutations on the indices of the N identical electrons. The irreducible-representation-spaces (IRS) of Sn are uniquely labeled by Young diagrams denoted YD[X] where [X] is a partition of N and where YD[X] is an array of N boxes in columns of nondecreasing lengths. The Hamiltonian for a system of N identical particles commutes with the elements of Sn- By the... 

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 ... 

One may note that quasienergy derivatives dt listed above are symmetric with respect to the capital indices Ai... A . It must be emphasized that the indices must be thought of as a double index specifying a specific perturbation strength a, as well as a frequency cOa, and, as discussed previously, they must be permuted at the same time in order to retain symmetry. Further simplifications of the expressions for molecular properties are possible using the response equations, e.g. (48), (49) and (50), but generally at the expense of index symmetry and numerical precision [28,9]. As an example, using the response equation the expression for second-order molecular properties (52) can be rewritten as... 

In the case of the electron-repukion integrals, we noted that the electron-repulsion operator (l/ri2) was sphe ic dly symmetric" and so it is only the permutation (transformation) properties of the basis functions which mattered in using molecular symmetry. The situation is similar in the case of the one-electron integrals ... 

Notice that each operation appears exactly once, and only once, in each row and in each column. The group describing symmetry operations on an equilateral triangle has precisely the same structure as the 3 = 6 possible permutations of three objects. The latter is known as the symmetric group of order 3, designated S3. These symmetry and permutation groups are said to be isomorphous— their abstract properties are identical although they apply to completely different sorts of objects. 

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