Hamiltonian operator symmetry

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

The Spin adapted Reduced Hamiltonian SRH) is the contraetion to a p-electron space of the matrix representation of the Hamiltonian Operator, 2 , in the N-electron space for a given Spin Symmetry [17,18,25,28], The basis for the matrix representation are the eigenfunctions of the operator. The block matrix which is contracted is that which corresponds to the spin symmetry selected In this way, the spin adaptation of the contracted matrix is insnred. 

The Hamiltonian operators can be written in terms of the symmetry coordinates... 

Consider first chain (I). A dynamical symmetry corresponding to this route implies that the Hamiltonian operator contains only invariant operators of the chain,... 

The construction of the symmetry-adapted operators and of the Hamiltonian operator of polyatomic molecules will be illustrated using the example of the benzene molecule. In order to do the construction, draw a figure corresponding to the geometric structure of the molecule (Figure 6.1). Number the degrees of freedom we wish to describe. 

The Hamiltonian operator that preserves the symmetry of the molecule can now be constructed. Since all the bonds in Figure 6.1 are equivalent, the most general lowest order Hamiltonian for C-H stretching vibrations of C6Hg is... 

The Hamiltonian operator that preserves the symmetry of octahedral molecules can now be constructed. For XY6 molecule it is... 

After the energy is expressed as a functional of the 2-RDM, a systematic hierarchy of V-representabihty constraints, known as p-positivity conditions, is derived [17]. We develop the details of the 2-positivity, 3-positivity, and partial 3-positivity conditions [21, 27, 34, 33]. In Section II.E the formal solution of V-representability for the 2-RDM is presented through a convex set of two-particle reduced Hamiltonian matrices [7, 21]. It is shown that the positivity conditions correspond to certain classes of reduced Hamiltonian matrices, and consequently, they are exact for certain classes of Hamiltonian operators at any interaction strength. In Section II.F the size of the 2-RDM is reduced through the use of spin and spatial symmetries [32, 34], and in Section II.G the variational 2-RDM method is extended to open-shell molecules [35]. 

In this chapter we introduce the SchrSdinger equation this equation is fundamental to all applications of quantum mechanics to chemical problems. For molecules of chemical interest it is an equation which is exceedingly difficult to solve and any possible simplifications due to the symmetry of the system concerned are very welcome. We are able to introduce symmetry, and thereby the results of the previous chapters, by proving one single but immensely valuable fact the transformation operators Om commute with the Hamiltonian operator, Jf. It is by this subtle thread that we can then deduce some of the properties of the solutions of the Schrodinger equation without even solving it. 

From our point of view the most significant thing about the Hamiltonian operators H9l and f/nno is that they both commute with the operators Og, we say that i/el and f/nuc are invariant under all symmetry transformation operators of the point group of the molecular framework... 

An important case is where F in (9.183) is replaced by the Hamiltonian operator H. The Hamiltonian is invariant under all symmetry operations, so that the function belongs to the irreducible representation Tp. For... 

Here x) stands for the positional coordinates of all the particles in the system, E is the energy of the system, and 77 is the Hamiltonian operator. Since a symmetry operator merely rearranges indistinguishable particles so as to leave the system in an indistinguishable configuration, the Hamiltonian is invariant under any spatial symmetry operator R. Let tpi denote a set of eigenfunctions of H so that... 

The operators so and ss are compound tensor operators of rank zero (scalars) composed of vector (first-rank tensor) operators and matrix (second-rank tensor) operators. We will make use of this tensorial structure when it comes to selection rules for the magnetic interaction Hamiltonians and symmetry relations between their matrix elements. Similar considerations apply to the molecular rotation and hyperfine splitting interaction... 

In this paper, both theories will be briefly reviewed presenting their differences. From these comparisons, the Non-rigid Molecule Group (NRG) will be stricktly defined as the complete set of the molecular conversion operations which commute with a given Hamiltonian operator [21]. The operations of such a set may be written either in terms of permutations and permutations-inversions, just as in the Longuet-Higgins formalism, or either in terms of physical operations just as in the formalism of Altmann. But, the order, the structure, the symmetry properties of the group will depend exclusively on the Hamiltonian operator considered. 

The coordinates are expressed in the molecular axes x,y,z, whidi are rigidly attached to the molecule. These coordinates and masses are labelled in some laboratory axes, X,Y,Z, fixed in space. Under this definition, a symmetry operation is a change of axes that leaves the Hamiltonian operator (14) invariant, and the group of all such operations is the Schrodinger subgroup. 

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]. 

Such a group is strictly equivalent to the Longuet-Higgins Molecular Symmetry Group, provided that the motions considered in the Hamiltonian operator are those described by the Longuet-Higgins Group. 

The energy potential energy function of the Hamiltonian operator for the double Csv internal rotation and wagging may be written in terms of Ai symmetry eigenvectors [37] ... 

Let us return to the example of the distorted methylhalide (Fig.8) in which a Cs symmetry plane is retained. The full Hamiltonian operator of such a system will be written as (100). As found in Section 5, the full NRG of such a molecular system is written as (55) ... 

Throughout this paper, we give potential energy functions, symmetry eigenvectors, as examples, for systems of one, two and three internal degrees of freedom, in the formalism of the restricted Hamiltonian operator, as well as in the local one. A generalization of these ideas can be found in the scientific literature [21,22] and [30-37]. 

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