Of nondegenerate

Six-Coordinate Systems with Ground States of Nondegenerate Octahedral... 

In the four-orbital model (1 ), low-lying ir-ir states of free-base porphyrins (symmetry D2h) are considered as resulting from single electron excitation from a pair of nondegenerate highest occupied molecular orbitals (bi, bo) to a pair of nondegenerate lowest unoccupied molecular orbitals (ci, cg). In the case of symmetry D2h mutually perpendicular electric transition dipoles X and Y are not equivalent and, therefore, in the visible absorption spectra of free-base porphyrins two different electronic bands Qx(0>0) and Qy(0,0) are observed (Table 1 and Fig. 10). 

The interaction of nondegenerate molecular or charge-transfer states is insufficient to describe the stability of photoassociation products of molecules with different electronic energy levels, ionization potentials, and electron affinities. On the other hand, treatments26-26 of the exciplex as a pure charge-transfer state afford a quantitative description of the shift in fluorescence peak with solvent polarity and with electron affinity of the (fluorescent) donor in the same quencher-solvent system (Eq. 13) moreover, estimated values for the dipole moment of the emitting species (Table VI) confirm its pronounced charge-transfer character. 

We can now show that the eigenfunctions for a molecule are bases for irreducible representations of the symmetry group to which the molecule belongs. Let us take first the simple case of nondegenerate eigenvalues. If we take the wave equation for the molecule and carry out a symmetry operation, / , upon each side, then, from 5.1-1 and 5.1-2 we have... 

Payne, S. E. Symmetric representations of nondegenerate generalized n-gons, Proc. Amer. Math. Soc. 19, 1321-1326 (1968)... 

ANALYTIC SOLUTION OF NONDEGENERATE QUANTUM CONTROL PROBLEM 215... 

Thus, there is a very simple and analytic pulse shaping recipe for achieving ) complete population transfer between two arbitrary superposition states [VF) am P ), composed of nondegenerate energy eigenstates. 

Thus the excitation pulse can create a superposition of i), 2) consisting of two states of different reflection symmetry. The resultant superposition possesses no symmetry properties with respect to reflection [78]. We now show that the broken symmetry created by this excitation of nondegenerate bound states translates into a nonsymmetry in the probability of populating the degenerate , n, D ), , n, L ) continuum states upon subsequent excitation. To do so we examine the properties of the bound-free transition matrix elements ( , n, q de,g Ek) that enter into the probability of dissociation. Note first that although the continuum states , n, q ) are nonsymmetric with respect to reflection, we can define symmetric and antisymmetric continuum eigenfunctions , n, s ) and , n, a ) via the relations... 

In the perturbation theory for degenerate states the resonant hyperpolarizability is determined by the tensor part of polarizability [9] and may be extracted out of the fourth-order terms self-consistently in the case of nondegenerate perturbation theory the resonant part appears for separate sublevels of an atomic multiplet. The numerical results are listed in Table 2. 

It follows from the preceding results that the electro-optical properties of molecules in degenerate electronic states should have unusual temperature dependence, which is absent in the case of nondegenerate states. Even for nondipolar degenerate electronic states (e.g., for states in which the reduced matrix elements of the dipole moment are zero) for certain relationships between the vibronic constant and the temperature, there may be a quadratic dependence of the Kerr effect on p, similar to that observed in the case of molecules that are simultaneously anisotropic polarizable and possess a proper dipole moment. The nonlinear dependence on p under consideration is due exclusively to the vibronic interaction that redetermines the vibronic spectrum and leads to different polarizability in different vibronic states. This dependence on p has to be distinguished from that which arises due to the nonzero value of the dipole moment in the initial ground electronic state (e.g., as in the case of the E term in molecules with D3h symmetry). The two sources of the... 

Hence, mappings satisfying conservation B preserve norms of arbitrarily chosen degenerate eigenvectors [154], but not necessarily those of nondegenerate eigenfunctions. Thus, conservation B does not imply conservation C. The converse is also true since the condition ( <, ) = implied by conservation C for a set of model eigenfunctions that includes the <, )o, differs from Eq. (B.12). 

The dressed states (129) group into manifolds of nondegenerate triplets unless A = and then the states +,1V), 3,1V) in each manifold are degenerate. 

A model Hamiltonian that describes the excitation spectrum of the crystal in the energy region E = 2MI can be readily constructed on the basis of the qualitative considerations presented above. As a matter of fact, the Hamiltonian of the crystal, describing the effect of intermolecular interaction on the spectrum, for example, of nondegenerate molecular vibrations can be written in the harmonic approximation as follows ... 

Let us now pick an arbitrary density out of the set A of densities of nondegenerate ground states. The Hohenberg-Kohn theorem then tells us that there is a unique external potential v (to within a constant) and a unique ground state wavefunction I W[ri]) (to within a phase factor) corresponding to this density. This also means that the ground state expectation value of any observable, represented by an operator O. can be regarded as a density functional... 

The properties of a nondegenerate stationary point, and its stability, derive from the properties of nondegenerate critical points of the potential. Note that the classification of stationary points described in Section 5.1 applies directly to a nondegenerate stationary point of a gradient system. Hence, in the case of gradient systems, the requirement of lack of degeneracy of a critical point constitutes the criterion of applicability of this classification. 

The basic effects of nondegenerate vibrational motions on the rotational constants, molecular g-values, and susceptibility anisotropies may be demonstrated by the model of a diatomic molecule with its center of mass fixed in space and with the nuclear motion restricted to a plane perpendicular to the exterior magnetic field. 

Selection rules for overtones of nondegenerate vibrations can be obtained using the following relation ... 

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