Nuclear permutation operator

We use the nuclear permutation operations (123) and (12) to show what happens when we apply two operations in succession. We write the successive effect of these two permutations as (remember that we are using the N-convention see (equation Al.4,14))... 

The nuclear permutation operator for a homonuclear diatomic molecule... 

Fortunately we are able to simplify this expression somewhat by using symmetry arguments. The functions [Pg.221]

Note that in two different molecules a given common point symmetry operator R can be associated with two different permutation operators P and P depending on the nuclear arrangements. Consequently, the framework groups do contain more information on molecular shapes than point symmetry groups. 

The Hamiltonian /lclcc(f f) has the same invariance under the rotation-reflection group 0(3) as does the full translationally invariant Hamiltonian (6), and it has a somewhat extended invariance under nuclear permutations, since it contains the nuclear masses only in symmetrical sums. Since it contains the translationally invariant nuclear coordinates as multiplicative operators, its domain is of... 

Employing these base operators the permutation operator of a homo nuclear spin 1 pair is given by... 

Nuclear permutations in the N-convention (which convention we always use for nuclear permutations) and rotation operations relative to a nuclear-fixed or molecule-fixed reference frame, are defined to transform wavefunctions according to (equation A 1.4.56). These S5mimetry operations involve a moving reference frame. Nuclear permutations in the S-convention, point group operations in the space-fixed axis convention (which is the convention that is always used for point group operations see section A 1.4.2.2 and rotation operations relative to a space-fixed frame are defined to transform wavefunctions according to (equation A 1.4.57). These operations involve a fixed reference frame. 

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. 

As an example, in Fig. 5.1 we return to our favored ammonia molecule and list all nuclear permutations, with and without the all-particle inversion operator, that leave the full Hamiltonian invariant. Nuclear permutations are defined here in the same way as in Sect. 3.3. A permutation such as (ABC) means that the letters A, B, and C are replaced by B, C, and A, respectively. The inversion operator, E, inverts the positions of all particles through a common inversion center, which can be conveniently chosen in the mass origin. In total, 12 combinations of such operations are found, which together form a group that is isomorphic to Ds. How is this related to our previous point group At this point it is very important to recall that the state of a molecule is not only determined by its Hamiltonian but also, and to an equal extent, by the boundary conditions. The eigenvalue equation is a differential equation that has a very extensive set of mathematical solutions, but not all these solutions are also acceptable states of the physical system. The role of the boundary conditions is to define constraints that Alter out physically unacceptable states of the system. In most cases these constraints also lead to the quantization of the energies. 

Table 5.1 Embedding of the Csu point group in the Longuet-Higgins group. The symmetry elements of the point group act on the electrons. They are identified as the product of nuclear permutations, inversion of all particles (star operation), and bodily rotations of aU particles (Q operators) along particular directions...

The subgroup of the complete nuclear permutation-inversion group that contains all feasible permutation-inversion operations. 

For the homonuclear (HON) species, the permutation-symmetry operator had the following form Y = 83) <8) Ye S2), where 83) is a Young operator for the third-order symmetric group which permutes the nuclear coordinates and 82) is a Young operator for the second-order symmetric group which permutes the electronic coordinates. For the fermionic nuclei (H and T, spin = 1/2) the Young operators corresponded to doublet-type representations, while for the bosonic D nuclei we use operators that correspond to the totally symmetric representation. In all cases the electronic operator corresponded to a singlet representation. 

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.)... 

The choice of translationally invariant electronic coordinates makes He(te) trivially invariant under permutations of the original electronic coordinates and independent of any particular choice of translationally invariant nuclear coordinates. Similarly Hn(tn) is independent of any particular choice of translationally invariant electronic coordinates and can be shown, after some algebra, to be invariant under any permutation of the original coordinates of identical nuclei. The interaction operator Hen(tn,te) is obviously invariant under a permutation of the original electronic coordinates and, again after a little algebra, can be shown to be invariant under any permutation of the original coordinates of identical nuclei. 

Ptj, the symmetry operation involving interchange of identical particles (nuclei or electrons). All particles axe either Bosons or Fermions, and the total wavefunction must, respectively, be even or odd upon interchange of any pair of identical particles. The total wavefunction of a homonuclear molecule, exclusive of the nuclear spin part, is classified s or a according to whether it is even or odd with respect to nuclear exchange. Since electrons are Fermions, the total molecular wavefunction must be odd with respect to permutation of any two electrons. This requirement is satisfied by the determinantal form of the electronic wavefunction (see Section 3.2.4). 

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