Hamiltonian selection rules

The vanishing integral rule is not only usefi.il in detemiining the nonvanishing elements of the Hamiltonian matrix H. Another important application is the derivation o selection rules for transitions between molecular states. For example, the hrtensity of an electric dipole transition from a state with wavefimction "f o a... 

Two states /a and /b that are eigenfunctions of a Hamiltonian Hq in the absence of some external perturbation (e.g., electromagnetic field or static electric field or potential due to surrounding ligands) can be "coupled" by the perturbation V only if the symmetries of V and of the two wavefunctions obey a so-called selection rule. In particular, only if the coupling integral (see Appendix D which deals with time independent perturbation theory)... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

The zeroth-order Hamiltonian and the spin-orbit part of the perturbation are diagonal with respect to the quantum numbers K, S, P, Ur, It, t>c, and lc-The terms of H involving the parameters aj, ac, and bo are diagonal with respect to both the lT and lc quantum numbers, while the hi term connects with one another the basis functions with l T = lT 2, l c = Zc T 2. The c terms couple with each other the electronic species —A and A. The selection rules for the vibrational quantum numbers are v Tjc = vT/c, t)j/c 2, vT/c 4. 

Upon entering the interaction frame of the rf irradiation for the CNvn or RN n sequences [cf. (14)] and taking the first-order effective Hamiltonians [cf. (17a) and (18a)], it is possible to establish the following selection rules for the averaging (and conversely recoupling) of the various interactions described in (45) as... 

Because many physical systems possess certain types of symmetry, its adaptation has become an important issue in theoretical studies of molecules. For example, symmetry facilitates the assignment of energy levels and determines selection rules in optical transitions. In direct diagonalization, symmetry adaptation, often performed on a symmetrized basis, significantly reduces the numerical costs in diagonalizing the Hamiltonian matrix because the resulting block-diagonal structure of the Hamiltonian matrix allows for the separate... 

The model fundamental to all analyses of vibrational motion requires that the atoms in the system oscillate with small amplitude about some defined set of equilibrium positions. The Hamiltonian describing this motion is customarily taken to be quadratic in the atomic displacements, hence in principle a set of normal modes can be found in terms of these normal modes both the kinetic energy and the potential energy of the system are diagonal. The interaction of the system with electromagnetic radiation, i.e. excitation of specific normal modes of vibration, is then governed by selection rules which depend on features of the microscopic symmetry. It is well known that this model can be worked out in detail for small molecules and for crystalline solids. In some very favorable simple cases the effects of anharmonicity can be accounted for, provided they are not too large. 

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 ... 

There are point-group selection rules in the presence of spin interactions.73,115117 172 We recall that a spin-free Hamiltonian //SF(Qeq) for a rigid nuclear framework Qeq has a point group SF which acts on electronic spatial coordinates, and that... 

Furthermore, there is a potential surface for each set of excited states for the N nuclei, i.e., for each set of singlets, triplets, etc. (assuming a spin-free Hamiltonian). Transitions from one surface to the next will, instantaneously, still be governed by the Franck-Condon principle and selection rules. Thus, the important question concerning purity of states in an electronic transition can be dismissed. 

Since the many-electron Hamiltonian, the effective one-electron Hamiltonian obtained in Hartree-Fock theory (the Fock operator ), and the one- and two-electron operators that comprise the Hamiltonian are all totally symmetric, this selection rule is extremely powerful and useful. It can be generalized by noting that any operator can be written in terms of symmetry-adapted operators ... 

Low-symmetry LF operators are time-even one-electron operators that are non-totally symmetric in orbit space. They thus have quasi-spin K = 1, implying that the only allowed matrix elements are between 2P and 2D (Cf. Eq. 28). Interestingly in complexes with a trigonal or tetragonal symmetry axis a further selection rule based on the angular momentum theory of the shell is retained. Indeed in such complexes two -orbitals will remain degenerate. This indicates that the intra-t2g part of the LF hamiltonian has pseudo-cylindrical D h symmetry. As a result the 2S+1L terms are resolved into pseudo-cylindrical 2S+1 A levels (/l = 0,1,..., L ). It is convenient to orient the z axis of quantization along the principal axis of revolution. In this way each A level comprises the ML = A components of the L manifold. In a pseudo-cylindrical field only levels with equal A are allowed to interact, in accordance with the pseudo-cylindrical selection rule ... 

Hamiltonian for the X2Eig — E2B2u electronic manifold of Bz+, involving extended group theoretical considerations, is given in Ref. [23]. The relevant submatrices for the cases treated below will be quoted in subsequent sections. Here we only note a general symmetry selection rule for the mode / in order to linearly couple the states i and j ... 

Using the basis functions which follow from the approximate Hamiltonian H° of equ. (1.3), it is the residual interaction H — H° which causes the Auger transitions. This operator, however, reduces to the Coulomb interaction if more than one electron changes its orbital.) Within the LS-coupling scheme this transition operator requires the following selection rules... 

The electric-dipole transition is determined by the symmetry properties of the initial-state and the final-state wave functions, i.e., their irreducible representations. In the case of electric-dipole transitions, the selection rules shown in table 7 hold true (n and a represent the polarizations where the electric field vector of the incident light is parallel and perpendicular to the crystal c axis, respectively. Forbidden transitions are denoted by the x sign). In the relativistic DVME method, the Slater determinants are symmetrized according to the Clebsch-Gordan coefficients and the symmetry-adapted Slater determinants are used as the basis functions. Therefore, the diagonalization of the many-electron Dirac Hamiltonian is performed separately for each irreducible representation. 

A. Mn(II) EPR. The five unpaired 3d electrons and the relatively long electron spin relaxation time of the divalent manganese ion result in readily observable EPR spectra for Mn2+ solutions at room temperature. The Mn2+ (S = 5/2) ion exhibits six possible spin-energy levels when placed in an external magnetic field. These six levels correspond to the six values of the electron spin quantum number, Ms, which has the values 5/2, 3/2, 1/2, -1/2, -3/2 and -5/2. The manganese nucleus has a nuclear spin quantum number of 5/2, which splits each electronic fine structure transition into six components. Under these conditions the selection rules for allowed EPR transitions are AMS = + 1, Amj = 0 (where Ms and mj are the electron and nuclear spin quantum numbers) resulting in 30 allowed transitions. The spin Hamiltonian describing such a system is... 

The fact that the magnetic interaction Hamiltonians are compound tensor operators can be exploited to derive more specific selection rules than the one given above. Furthermore, as we shall see later, the number of matrix elements between multiplet components that actually have to be computed can be considerably reduced by use of the Wigner-Eckart theorem. 

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... 

Spin-Orbit Coupling For the derivation of selection rules, it is sufficient to employ a simplified Hamiltonian. To this end, we rewrite each term in the microscopic spin-orbit Hamiltonians in form of a scalar product between an appropriately chosen spatial angular momentum 2 and a spin angular momentum S... 

A similar interaction would be observed between all Fermi polyads containing sets of vibrational levels related by the selection rule A tv = 2, A tv = +1, and the hamiltonian matrix should be diagonalized for each Fermi polyad without the use of perturbation theory. If, on the other hand, the interaction (63) were smaller, or the separation between the unperturbed levels were larger, the interaction could be treated by perturbation theory it can be shown that, in second-order perturbation theory, equation (63) would contribute a term to the vibrational anharmonic constants... 

The infrared transitions obey the following selection rules Ap — 1, and A./ = 1 or 0. The wave functions for the initial and final states obtained by solving the Schrodinger equation with the Hamiltonian of Eq. (1-261) or (1-265) can be used to compute the infrared absorption intensities for the complex. The infrared absorption coefficient J(J" ->./ ) for the transition J" J is proportional to,... 

In the present treatment, we retain essentially all the diagonal matrix elements of X these are the first-order contributions to the effective electronic Hamiltonian. There are many possible off-diagonal matrix elements but we shall consider only those due to the terms in Xrot and X o here since these are the largest and provide readily observable effects. The appropriate part of the rotational Hamiltonian is —2hcB(R)(NxLx + NyLy). The matrix elements of this operator are comparatively sparse because they are subject to the selection rules AA = 1, A,Y=0 and AF=0. The spin-orbit coupling term, on the other hand, has a much more extensive set of matrix elements allowed... 

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