Symmetry and Spin in Molecules

Symmetry and Spin in Molecules These results are collected below... 

We expect that S s SB(molecule) with minor corrections for differences in symmetry and spin thus, AS — S fatom). Since the absolute entropies of atoms are in the range 25-35 cal/mol-K (Chase et al., 1985), — AS would be expected to be in this range without any corrections. 

For diatomic systems with states of different symmetry and spin, the probability of crossing was derived independently by Landau and Zener. Even for diatomics the Landau-Zener formula has been shown to be inapplicable in certain instances and has been revised. For polyatomic molecules the Landau-Zener formula is not applicable as such, and modifications have been made to permit its use for specified cases. 

The search for conical intersections, avoided crossings and seams of intersection between two PESs are also tasks involving optimization [115-123]. If the two surfaces represent different spin states or have different spatial symmetry, they can cross. If they are the same symmetry and spin, they can interact and the crossing is avoided. Where the matrix element coupling the two surfaces is zero, they touch and give rise to a conical intersection, as illustrated in Eig. 10.3. To study the mechanisms of photochemical reactions, we often wish to hnd the lowest energy point on a seam or conical intersection. For a seam of intersection between two adiabatic surfaces and E2, we require ] = Ei-Since a (non-linear) molecule has 6 internal degrees of freedom, a seam of... 

PERMUTATIONAL SYMMETRY AND THE ROLE OF NUCLEAR SPIN IN THE VIBRATIONAL SPECTRA OF MOLECULES IN DOUBLY DEGENERATE ELECTRONIC STATES ... 

Atoms, linear molecules, and non-linear molecules have orbitals which can be labeled either according to the symmetry appropriate for that isolated species or for the species in an environment which produces lower symmetry. These orbitals should be viewed as regions of space in which electrons can move, with, of course, at most two electrons (of opposite spin) in each orbital. Specification of a particular occupancy of the set of orbitals available to the system gives an electronic configuration. For example,... 

Recall that the symmetry labels e and o refer to the symmetries of the orbitals under reflection through the one Cy plane that is preserved throughout the proposed disrotatory closing. Low-energy configurations (assuming one is interested in the thermal or low-lying photochemically excited-state reactivity of this system) for the reactant molecule and their overall space and spin symmetry are as follows ... 

It is possible to construct a HF method for open-shell molecules that does maintain the proper spin symmetry. It is known as the restricted open-shell HF (ROHF) method. Rather than dividing the electrons into spin-up and spin-down classes, the ROHF method partitions the electrons into closed- and open-shell. In the easiest case of the high-spin wavefunction ( op = — electrons in op... 

Excited states formed by light absorption are governed by (dipole) selection rules. Two selection rules derive from parity and spin considerations. Atoms and molecules with a center of symmetry must have wavefunctions that are either symmetric (g) or antisymmetric (u). Since the dipole moment operator is of odd parity, allowed transitions must relate states of different parity thus, u—g is allowed, but not u—u or g—g. Similarly, allowed transitions must connect states of the same multiplicity—that is, singlet—singlet, triplet-triplet, and so on. The parity selection rule is strictly obeyed for atoms and molecules of high symmetry. In molecules of low symmetry, it tends to break down gradually however,... 

In the theory of deuteron spin-lattice relaxation we apply a simple model to describe the relaxation of the magnetizations T and (A+E), for symmetry species of four coupled deuterons in CD4 free rotators. Expressions are derived for their direct relaxation rate via the intra and external quadrupole couplings. The jump motion between the equilibrium positions averages the relaxation rate within the same symmetry species. Spin conversion transitions couple the relaxation of T and (A+E). This mixing is included in the calculations by reapplying the simple model under somewhat different conditions. The results compare favorably with the experimental data for the zeolites HY, NaA and NaMordenite [6] and NaY presented here. Incoherent tunnelling is believed to dominate the relaxation process at lowest temperatures as soon as CD4 molecules become localized. 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

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