Symmetry Group of the Hamiltonian

The foregoing discussion allows to state the theorem The full isometric group JP( ) is a proper or improper subgroup ofthe symmetry group ft of the rotation internal motion hamiltonian H = T + V 

Since the dynamical problem (3.10) refers to the LS, the primitive period isometric transformations are to be included in ( ). A proof of this important theorem has been given earlier14 9. 5 H represents symmetry of H w.r.t. to operations of the 

8 For the sake of brevity the set apy will often be abbreviated by e and the velocity vector by e. 

The set of properly orthogonal transformations R1 forms the group SO(3), the reflexion Z1 at the origin of the LS likewise leaves A symmetric, since the eulerian angles remain unaffected by Z1. Therefore, H is symmetric w.r.t. the full rotation group 0(3/. However, in agreement with the usual conventions we will omit the elements Z R1 0(3). As a consequence we will consider hence -forward the group 

For a number of SRMs C H, i.e. ( ) isa proper subgroup of This fact has been established by determining the group H directly. In the case where a principle axis coincides with an internal rotation axis for all values of the internal rotation angle, but not being a covering symmetry axis, we have 

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Each operation in a symmetry group of the Hamiltonian will generate such an / x / matrix, and it can be shown (see, for example, appendix 6-1 of [1]) that if three operations of the group T 2 and / j2 related by... 

In the early sixties, it was shown by Roothaan [ 1 ] and Lowdin [2] that the symmetry adapted solution of the Hartree-Fock equations (i.e. belonging to an irreducible representation of the symmetry group of the Hamiltonian) corresponds to a specific extreme value of the total energy. A basic fact is to know whether this value is associated with the global minimum or a local minimum, maximum or even a saddle point of the energy. Thus, in principle, there may be some symmetry breaking solutions whose energy is lower than that of a symmetry adapted solution. 

Such a set of eigenfunctions must form the basis for a representation of the symmetry group of the Hamiltonian, because for every symmetry operation S, Tipi = pi implies that H Spi) = Sp>i) and hence that the transformed wave function Spi must be a linear combination of the basic set of eigenfunctions (/ ,... Pn-... 

Selection rules for radiationless transitions may also be derived if it is known how T transforms under the symmetry groups of the Hamiltonian. We make some brief remarks on three broad types of radiationless transitions ... 

Ga is a rotation by the angle a in the spin space Va.) There is a two-to-one correspondence from SF El J a d to or SF. This inner subdirect product group is115>117 i7a a Symmetry group of the Hamiltonian H(Qeq),... 

The algebraic approach begins with the notion of a zeroth-order description based on a dynamical symmetry, a concept which is a generalization of the usual definition of the symmetry group of the Hamiltonian. What a dynamical symmetry means in practice is that one constructs a zeroth-order Hamiltonian for which there is a full set of quantum numbers for labeling the eigenstates and that the energy is an analytical function of these quantum numbers. In the infrared or Raman spectroscopy of polyatomic molecules (1) one knows that to zeroth order it is practical to represent the spectrum by a Dunham-type formula... 

To summarize, we have seen that it is possible to explain the degeneracy pattern characteristic of a three-dimensional harmonic oscillator by introducing, as a proper symmetry group of the Hamiltonian operator (here referred to as the degeneracy group), a group of unitary transformations in a three-dimensional complex space. 

A symmetry orbital is a linear eombination of atomie orbitals that transforms aeeording to an irredueible representation Y of the symmetry group of the Hamiltonian (see Appendix C available at booksite.elsevier.eom/978-0-444-59436-5). In order to obtain sueh a funetion, we may use the eorresponding projeetion operator [see Eq. (C.13)]. 

Two molecules, when isolated (say at infinite distance), are independent and the wave function of the total system might be taken as a product of the wave functions for the individual molecules. When the same two molecules are at a finite distance, then any product-like function represents only an approximation (sometimes a very poor one ), because according to a postulate of quantum mechanics, the wave function has to be antisymmetric with respect to the exchange of electronic labels, while the product does not fulfill that. More exactly, the approximate wave function has to belong to the irreducible representation of the symmetry group of the Hamiltonian (see Appendix C available at booksite.elsevier.com/978-0-444-59436-5, p. el7), to which the ground-state wave function belongs. This means, first of all, that the Pauli exclusion principle is to be satisfied. 

Sometimes it is said that the barrier results from an avoided crossing (cf. Chapter 6) of two diabatic hypersurfaces that belong to the same irreducible representation of the symmetry group of the Hamiltonian (in short of the same q mmetry ). This, however, caimot be taken literally, because, as we know from Cliapter 6, the non-crossing rule is valid for diatomics only. The solution to this dilemma is the conical intersection described in Chapter 6 (cf. Fig. 6.15). Instead of diabatic we have two adiabatic hypersurfaces ("upper and lower ), each consisting of the diabatic part I and the diabatic part II. A thermic reaction takes place as a rule on the lower hypersurface and corresponds to crossing the border between I and n. 

Fig. C.4. Each energy level corresponds to an irredueible representation of the symmetry group of the Hamiltonian. Its linearly independent eigenfunctions corresponding to a given level form the basis of the irreducible representation, or in other words, transform according to this representation. The number of the basis functions is equal to the degeneracy of the level.

A symmetry atomic orbital (SAO) represents such hnear combination of equivalent-by-symmetry AOs that transforms according to one of the irreducible representations of the symmetry group of the Hamiltonian. Then, when molecular orbitals (MOs) are formed in the LCAO MO procedure, any given MO is a linear combination of the SAOs belonging to a particular irreducible representation. For example, the water molecule exhibits a symmetry plane ([Pg.530]

Fig. C.4. Each energy level corresponds to an irreducible representation of the symmetry group of the Hamiltonian. Its linearly...

The transformation g is said to be a symmetry operation of the Hamiltonian H if H(p r) = H(pr), i.e. if H is invariant under g. Symmetry operations g that leave H invariant form a group. This group is called the symmetry group of the Hamiltonian or of the Schrodinger equation (3.1). 

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