H electronic states

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p, being thus mostly pronounced in H electronic state. Renner developed the system of two coupled Schrbdinger equations and solved it for H states in the harmonic approximation by means of the perturbation theory. 

In Section IV.A.4, we show what this general model looks like in the case of H electronic states of symmehic tetraatomic molecules. The situation in H states of asymmetric tetraatomics is briefly discussed in Section IV.B, where we present the handling of a concrete case, the X Uu state of the HCCS radical. For A states the reader is referred to original references [18,149,150,153]. 

In this section, we consider H electronic state (A = 1) of ABBA type molecules. The additional Hamiltonian H is of the form... 

As in the case of H electronic states of tetraatomic molecules, because of generally high degeneracy of zeroth-order vibronic leves only several particular (but important) coupling cases can be handled efficiently in the framework of the pertnrbation theory. We consider the following paiticnlai" cases ... 

The rotational/fine-structure levels of the lower, H electronic state in figure B2.3.12 are drawn for a molecule near the case (a) limit since NO falls into this coupling scheme. Also indicated in the figure are the electric-dipole allowed rotational lines, indicated with conventional spectroscopic notation [34] In the... 

For reasons which will become clear later, y -doubling effects are largest for molecules in H electronic states. The electronic orbital part of the wave function for such a state can be represented by the pair of functions vl = +1) and = — 1 >, which... 

If we confine attention to molecules in a H electronic state, there are four magnetic hyperfine parameters, a, b, c and d. The first of these describes the strength of the nuclear-spin/electron-orbital interaction and gives information on the spatial distribution of the unpaired electrons. The other three parameters give information on the electron spin distribution within the molecule. Though often similar, these two distribution functions are not identical. 

The same experimental approach has also been applied to accurately measure the lifetime of the metastable a H electronic state of CO molecules. CO molecules in this state can only decay to the electronic ground state (giving rise to the Cameron bands), via a spin-forbidden transition, weakly allowed because of spin-orbit mixing of the a H state with H states [78]. The spin-orbit mixing... 

Longuet-Higgins H C 1956 The electronic states of composite systems Proc. R. Soc. A 235 537... 

Aspects of the Jahn-Teller symmetry argument will be relevant in later sections. Suppose that the electronic states aie n-fold degenerate, with symmetry at some symmetiical nuclear configuration Qq. The fundamental question concerns the symmetry of the nuclear coordinates that can split the degeneracy linearly in Q — Qo, in other words those that appeal linearly in Taylor series for the matrix elements A H B). Since the bras (/1 and kets B) both transform as and H are totally symmetric, it would appear at first sight that the Jahn-Teller active modes must have symmetry Fg = F x F. There... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

H at m energy of 1.2 eV in the center-of-mass frame. By using an atomic orbital basis and a representation of the electronic state of the system in terms of a Thouless determinant and the protons as classical particles, the leading term of the electronic state of the reactants is... 

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

The expression for the force on the nuclei, Eq. (89), has the same form as the BO force Eq. (16), but the wave function here is the time-dependent one. As can be shown by perturbation theory, in the limit that the nuclei move very slowly compared to the electrons, and if only one electronic state is involved, the two expressions for the wave function become equivalent. This can be shown by comparing the time-independent equation for the eigenfunction of H i at time t... 

The energies of this Cl and of the other ones calculated in this work are listed in Table III. The calculated CASSCF values of the energies of the two lowest electronically states are 9.0 eV (5i, vertical) and 10.3 eV ( 2, vertical) [99]. They are considerably higher than the expenmental ones, as noted for this method by other workers [65]. In all cases, the computed conical intersections lie at much lower energies than the excited state, and are easily accessible upon excitation to Si. In the case of the H/allyl Cl, the validity confirmation process recovered the CHDN and 1,3-CHDN anchors. An attempt to approach the third anchor [BCE(I)] resulted instead in a biradical, shown in Figure 43. The bhadical may be regarded as a resonance hybrid of two allyl-type biradicals. 

Vo + V2 and = Vo — 2 (actually, effective operators acting onto functions of p and < )), conesponding to the zeroth-order vibronic functions of the form cos(0 —4>) and sin(0 —(()), respectively. PL-H computed the vibronic spectrum of NH2 by carrying out some additional transformations (they found it to be convenient to take the unperturbed situation to be one in which the bending potential coincided with that of the upper electi onic state, which was supposed to be linear) and simplifications (the potential curve for the lower adiabatic electi onic state was assumed to be of quartic order in p, the vibronic wave functions for the upper electronic state were assumed to be represented by sums and differences of pairs of the basis functions with the same quantum number u and / = A) to keep the problem tiactable by means of simple perturbation... 

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