Hamiltonian symmetry group

This definition is because there are cases in which the Hamiltonian symmetry group has more elements (double) than the point symmetry group of the active center. Those cases deal with rare earth ions with half-integer J values for instance, the Nd + ion. They will be treated in Section 7.7. 

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... 

Since space is isotropic, K (spatial) is a symmetry group of the molecular Hamiltonian v7in that all its elements conmuite with // ... 

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. 

Taking into account that Bq parameters represent the coefficient of an operator related to the spherical harmonic ykq then the ranges of k and q are limited to a maximum of 27 parameters (26 independent) Bq with k = 2,4,6 and q = 0,1,. .., k. The B°k values are real and the rest are complex. Due to the invariance of the CF Hamiltonian under the operations of the symmetry groups, the number of parameters is also limited by the point symmetry of the lanthanide site. Notice that for some groups, the number of parameters will depend on the choice of axes. In Table 2.1, the effect of site symmetry is illustrated for some common ion site symmetries. 

The hfs (or quadrupole) tensors of geometrically (chemically) equivalent nuclei can be transformed into each other by symmetry operations of the point group of the paramagnetic metal complex. For an arbitrary orientation of B0 these nuclei may be considered as nonequivalent and the ENDOR spectra are described by the simple expressions in (B 4). If B0 is oriented in such a way that the corresponding symmetry group of the spin Hamiltonian is not the trivial one (Q symmetry), symmetry adapted base functions have to be used in the second order treatment for an accurate description of ENDOR spectra. We discuss the C2v and D4h covering symmetry in more detail. 

Gv( f) covering symmetry67. For orientations of B0 in the mirror plane S, the symmetry group of the spin Hamiltonian is < 9f = C2h(e2f). The direct product base of the nuclear spin functions of two geometrically equivalent nuclei reduces to two classes, containing six A-type and three B-type functions, respectively. Second order perturbation theory applied to H = UtHU, where U symmetrizes the base functions of the Hamil-... 

If there is a molecular symmetry group whose elements leave the hamiltonian 36 invariant, then the closed-shell wavefunction belongs to the totally symmetric representation of both the spin and symmetry groups.8 It is further true that under these symmetry operations the molecular orbitals transform among each other by means of an orthogonal transformation, such as mentioned in Eq. (5) 9) and, therefore, span a representation of the molecular symmetry group. In general, this representation is reducible. 

The eigenfunctions of the Hamiltonian H related to an energy level E belong to one of the irreducible representations F of the symmetry group G, and the eigenfunctions belonging to another irreducible representations r ofG are related to another energy level Em-... 

The importance of symmetry in the study of the electronic structure of atoms and molecules depends on the fact that wave functions must transform according to one of the symmetry species of the symmetry group of the molecule. Stated precisely, the eigenfunctions of a Hamiltonian form bases for irreducible representations of the symmetry group of the Hamiltoirian. This principle allows wave functions to be classified according to symmetry species it assists... 

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-... 

The theoretical basis for a conserved quantity is the commutation of an effective Hamiltonian with the elements of some symmetry group. If this condition exists, then the irreducible representations of the group are good quantum numbers, i.e., are conserved. Conversely, the existence of good quantum numbers implies a Hamiltonian which commutes with an appropriate group. The most general molecular A-electron Hamiltonians... 

Quantum numbers are in general associated with symmetry groups of effective Hamiltonians. Often the group theoretical nature of certain quantum numbers is not emphasized, or perhaps even realized. A case in point is provided by systems well described by a spin-free Hamiltonian in which case the symmetric group S%F yields the analogs of the usual spin quantum numbers. Often this problem is treated in a spin-oriented manner despite the fact that a spin-free Hamiltonian is used. A consequence of the use of a spin-oriented language is that many chemists implicitly assume... 

The transition operator J t) is determined by the perturbed Hamiltonian H and, in particular, transforms in the same manner as T does under the symmetry group of H0. The manner in which J(t) and Y transform determines selection rules, through the use of the Wigner-Eckart theorem. 

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 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... 

From the fact that r Ncf 3% is the isometric group of the NC (4.1) it follows by the same reasoning as for SRMs that e is the symmetry group of the rotation-vibration hamiltonian. Though the representation (4.11) is commonly used in vibrational spectroscopy55-S7 it only seldom has been characterized as a group of isometric transformations57. ... 

Here ( )is the single particle energy and v(k,k ) the interaction. The symmetry of the superconducting state can be derived from that of the Hamiltonian. In general, the symmetry group Q is the direct product... 

---