Hamiltonian, invariant

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

Assume that there exists a unitary operator U(it) which maps the Heisenberg operator Q(t) at time t into the operator (—<). Assume further that this mapping has the property of leaving the hamiltonian invariant, i.e., that U(it)SU(it)" 1 = H. Consider then the equation satisfied by the transformed operator... 

For Hamiltonians invariant under rotational and time-reversal transformations the corresponding ensemble of matrices is called the Gaussian orthogonal ensemble (GOE). It was established that GOE describes the statistical fluctuation properties of a quantum system whose classical analog is completely chaotic. 

If e. = 0, then there will be no displacements in these directions whether is zero or not. Since the energy is invariant to translations and rotations, the center of coordinates and moments about these coordinates will be conserved. In the former case it can be shown that X = 0 in the case of rotation the corresponding X may not equal zero except at the extreme point, but the corresponding e. = 0. Since the energy is also invariant to symmetry operations that leave the Hamiltonian invariant, the gradient vector will have nonvanishing components only along totally symmetric modes, "A" (see [43] for a more detailed proof). 

It is easily seen that the complete set of operators Mi, which leave the nuclear Hamiltonian invariant (7), forms a group [21]. Indeed, if [H,Mi] = 0 and [H, M ] = 0, the product M Mj will also commute ... 

In this Appendix, Htickel theory calculations are used to demonstrate that the polynomials of Appendix 1 can be applied as basis functions for the irreducible subspaces of a Hamiltonian invariant under icosahedral point symmetry, while extended Htickel theory calculations on cubium cages of cubic point symmetry are used to demonstrate the same result for the kubic harmonics, since single bond-length regular orbits are not possible in all cases. 

Abstract The time has come to see how the concept of irreducible representations ties in with quantum chemistry. After a brief introduction to the prequantum principles of symmetry, we will show that eigenfunctions of the Hamiltonian are also eigenfunctions of the symmetry operators that commute with the Hamiltonian. We further analyze the concept of a degeneracy and show how the degenerate components can be characterized by canonical symmetry relationships. The final section will then provide a detailed account of the symmetry operations that leave the Hamiltonian invariant. 

So far, the symmetry of the Hamiltonian was defined as the set of all operations that leave the Hamiltonian invariant. This invariance group was assumed to coincide with the point group of the nuclear frame of the molecule, but it is now time to provide a clear explanation of this connection. This section relies on the definition of the... 

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. 

To derive the selection rules for vibrational transitions, it is necessary to take the symmetry of the normal modes into account. We assume that the polyatomic molecule belongs to some point group G of symmetry operations By definition, all in G leave the vibrational Hamiltonian invariant, so that [J , = 0. If each normal mode transforms as an irreducible representation (IR) of G, one can set up matrices R with elements Rij and H with elements Hij in a basis of vibrational states which are eigenvectors of R (i.e., vibrational states that transform as IRs of G). Since [R, i vib] = it follows that [4]... 

Our experience with inversion is related to the nonrelativistic case and operations in R3. We can always apply the operator in the nonrelativistic form, but our aim is to find operations that leave the Dirac Hamiltonian invariant, and there is no guarantee that the nonrelativistic inversion operator will do this. However, the operator we are looking for should have the same effect as the familiar inversion when applied only to R3. We may therefore assume that the inversion operator we are looking for may be written as the product of one part that acts only in R3 in the manner described in (6.99) and one part that does not act in R3, but which may affect other coordinates. We write the relativistic inversion operator as... 

A symmetry operation is thus characterized mathematically as one that leaves the Hamiltonian invariant. This invariance property is not necessarily shared by the eigenfunctions, however, and symmetry plays a large part in determining their forms and transformation properties. 

Applications of quantum mechanics to chemistry invariably deal with systems (atoms and molecules) that contain more than one particle. Apart from the hydrogen atom, the stationary-state energies caimot be calculated exactly, and compromises must be made in order to estimate them. Perhaps the most useful and widely used approximation in chemistry is the independent-particle approximation, which can take several fomis. Conuiion to all of these is the assumption that the Hamiltonian operator for a system consisting of n particles is approximated by tlie sum... 

For the Hamiltonian //we identify a synnnetry group, and this is a group of synnnetry operations of /7a synnnetry operation being defmed as an operation that leaves //invariant (i.e., that coimmites with //). In our example, the synnnetry group is K (spatial). 

We hope that by now the reader has it finnly in mind that the way molecular symmetry is defined and used is based on energy invariance and not on considerations of the geometry of molecular equilibrium structures. Synnnetry defined in this way leads to the idea of consenntion. For example, the total angular momentum of an isolated molecule m field-free space is a conserved quantity (like the total energy) since there are no tenns in the Hamiltonian that can mix states having different values of F. This point is discussed fiirther in section Al.4.3.1 and section Al.4.3.2. 

The Hamiltonian considered above, which connmites with E, involves the electromagnetic forces between the nuclei and electrons. However, there is another force between particles, the weak interaction force, that is not invariant to inversion. The weak charged current mteraction force is responsible for the beta decay of nuclei, and the related weak neutral current interaction force has an effect in atomic and molecular systems. If we include this force between the nuclei and electrons in the molecular Hamiltonian (as we should because of electroweak unification) then the Hamiltonian will not conuuiite with , and states of opposite parity will be mixed. However, the effect of the weak neutral current interaction force is mcredibly small (and it is a very short range force), although its effect has been detected in extremely precise experiments on atoms (see, for... 

One can regard the Hamiltonian (B3.6.26) above as a phenomenological expansion in temis of the two invariants Aiand//of the surface. To establish the coimection to the effective interface Hamiltonian (b3.6.16) it is instnictive to consider the limit of an almost flat interface. Then, the local interface position u can be expressed as a single-valued fiinction of the two lateral parameters n(r ). In this Monge representation the interface Hamiltonian can be written as... 

If 4>(t) is a wave function amplitude arising from a Hamiltonian that is time-inversion invariant, then we can choose = 4> (0 for real f, where the star... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

Let us introduce a suitably simple example in order to illustrate the notion of almost invariant sets and the performance of our algorithm for Hamiltonian systems. For p = pi,P2),q = (91,92) consider the potential... 

Moreover, our Hamiltonian system possesses an additional symmetry — it is equivariant under the transformation (52,P2) —(92, 2). In other words each of these sets is a candidate for a set B mentioned in the assumptions of Corollary 4. Thus, by this result, both of these sets are almost invariant with... 

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