Permutational symmetry invariant operators

Invariant operators, permutational symmetry, conical intersection, adiabatic state, 735-737... 

Since the nuclear coordinates appearing in Eq. 3.2 are fixed parameters, as indicated by using the upper-case symbol R for the intemuclear distances, the spatial symmetry of the Hamiltonian is reduced to those operations that leave the nuclear framework invariant. (Permutational symmetry among the electrons is retained and will be considered in Chapter 6.)... 

A common idea underlying particular forms of symmetry is the invariance of a system under a certain set (group) of transformations. The normally considered forms of symmetry are rotational symmetry, which is based on the equivalence of all directions in space, and permutation symmetry, which is caused by identical particles. The operations of the geometrical symmetry group are responsible for appropriate conservation laws. So, the rotational symmetry of a closed system gives rise to the law of conservation of angular momentum. 

The regular orbit of a point symmetry group is the set of positions for which the only operation that leaves each vertex label invariant is the identity all other operations permute... 

Of particular importance in the physical sciences is the fact that the symmetry operations of any symmetrical system constitute a group under the operators that effect symmetry transformations, such as rotations or reflections. A symmetry transformation is an operation that leaves a physical system invariant. Thus any rotation of a circle about the perpendicular axis through its centre is a symmetry transformation for the circle. The permutation of any two identical atoms in a molecule is a symmetry transformation... 

Altmann considered two types of operations that belong to the Schrodinger subgroup the Euclidean and the discrete symmetry operations. Euclidean operations are those that change the laboratory axes, leaving the Hamiltonian operator invariant. They are translations and rotations of the whole molecule, in free space, in which the x,y,z molecular axes are kept constant. A discrete symmetry operation is a change of the molecular axes in such away as to induce permutations of the coordinates of identical particles [10]. 

The nuclear-attraction integrals contain the nuclear attraction terms from the electrostatic Hamiltonian, which, of course( ), has the symmetry of the nuclear framework and so is left invariant by exactly the same permutation operations which we are considering. Again, we may consider only the symmetry properties of the permutations of the basis functions. 

The quantum number v embodies the set of nuclear dynamic states with their labels (see below) and /c stands for the electronic quantum state. Thus, the nuclear wave function is always determined relatively to particular electronic states which, in turn, must be correlated to the (point) symmetries of the system. This stationary wave function may define, for particular cases, a class of geometric elements having an invariant center of mass. Actually, the (equivalence) class of configurations are those for which symmetry operations leave invariant this center of mass. This framework shares the discrete symmetries, such as permutation and space reflection invariances that are properties of the molecular eigenstates. There exists, then, a specific geometric framework pok- At this point, the expectation value ofH ,. taken with respect to the universal wave function is stationary to any geometric variation. 

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 ... 

In continuation, as outlined in Appendix 8, label 14 has only one site where it must go, which is also the case with label 13. The next smallest labels have two possibilities, but all the remaining labels have unique locations and could not be assigned to alternative neighboring locations. Thus, one finds that the total number of symmetry operations for diamantane, which is equal to the number of alternative labeling of vertices that will produce the same adjacency matrix, is 6 x 2 = 12. In Appendix 9, we show the 12 possible canonical labels for diamantane, and in Table 2.3, we have listed all 12 permutations of labels that leave the adjacency matrix invariant. As one... 

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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