N electronic states

II. n-ELECTRONIC STATE ADIABATIC REPRESENTATION A. Born-Huang Expansion... 

Let us define x (R>.) as an n-dimensional nuclear motion column vector, whose components are Xi (R i) through X (R )- The n-electronic-state nuclear motion Schrodinger equation satisfied by (Rl) can be obtained by inserting Eqs. (12)... 

In the n-electronic-state case, n n — l)/2 such matrices (j > i with... 

This can be used to rewrite the diabatic nuclear motion Schrodinger equation for an incomplete set of n electronic states as... 

In this section, it was shown how an optimal ADT matrix for an n-electronic-state problem can be obtained. In Section in.D, an application of the method outlined above to a two-state problem for the H3 system is described. 

Appendix A Perturbative Handling of the Renner—Teller Effect and Spin-Orbit Coupling in n Electronic States of Tetraatomic Molecules... 

APPENDIX A PERTURBATIVE HANDLING OF THE RENNER-TELLER EFFECT AND SPIN-ORBIT COUPLING IN n ELECTRONIC STATES OF TETRAATOMIC MOLECULES... 

Magnetic Hamiltonians are defined for a desired group of N electronic states obtained in the ab initio calculation, to which a pseudospin S (it reduces to a true spin S in the absence of spin-orbit coupling) is subscribed according to the relation N = 2S + 1. For instance, the two wave functions of a I[Pg.161]

DCCS radical, Renner-Teller effect, tetraatomic molecules, n electronic states, 633—640 Degenerate states ... 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)—(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the n-electronic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth referred to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

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 p2A, being thus mostly pronounced in n electronic state. Renner developed the system of two coupled Schrodinger equations and solved it for n 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 n electronic states of symmetric tetraatomic molecules. The situation in n states of asymmetric tetraatomics is briefly discussed in Section IV.B, where we present the handling of a concrete case, the X2 u state of the HCCS radical. For A states the reader is referred to original references [18,149,150,153],... 

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

In the lowest order (quadratic) approximation for n electronic states of asymmetrical (ABCD) tetraatomics, the electronic matrix elements (60) have the forms [18,152,153] ... 

The stationarity condition (170) does not give any information on the diagonal elements yP, not even on the yP with Cp = Zq. We must get this information from another source. Fortunately, for an n-electron state, vanishing of >.2 is only compatible with Lt = 0 for > 2 and with idempotency of y. 

Let us consider the Density Matrix of an N-electron state . Its A, Q, element in the iV-electron space representation is ... 

Interchanging the nuclear coordinates does not affect R, but it does affect the electronic spatial coordinates since they are defined with respect to the molecule-fixed xyz axes, which are rigidly attached to the nuclei. To find the effect on el of interchanging the nuclear coordinates, we will first invert the space-fixed coordinates of the nuclei and the electrons, and then carry out a second inversion of the space-fixed electronic coordinates only the net effect will be the interchange of the space-fixed coordinates of the two nuclei. We found in the last section that inversion of the space-fixed coordinates of all particles left //e, unchanged for 2+,n+,... electronic states, but multiplied it by —1 for 2, II ,... states. Consider now the effect of the second step, reinversion of the electronic space-fixed coordinates. Since the nuclei are unaffected by this step, the molecule-fixed axes remain fixed for this inversion, so that inversion of the space-fixed coordinates of the electrons also inverts their molecule-fixed coordinates. But we noted in Section 1.19 that the electronic wave functions of homonuclear diatomics could be classified as g or m, according to whether inversion of molecule-fixed electronic coordinates multiplies ptl by + 1 or -1. We conclude that for 2+,2,7,11, IV,... electronic states, i//el is symmetric with respect to interchange of nuclear coordinates, whereas for... 

The generalization to the control of the dynamics of a molecule with n electronic states is straightforward. For the purpose of deducing the control conditions we will examine the extreme case in which every possible pair of these electronic states is connected via the radiation field and a nonzero transition dipole moment. If the molecule is coupled to a radiation field that is a superposition of individual fields, each of which is resonant with a dipole allowed transition between two surfaces, the density operator of the system can be represented in the form... 

In this manner, the electronic transitions of some molecules are sensitive probes of the solute environment. Since the probe molecules selected by Kamlet and Taft have n electronic states which are more polar than the ground state, a change in the polar-ity/polarizability of the solvent medium changes the electronic energy gap, and thus the position of the absorption band. Kamlet and Taft have developed an empirical relationship between measured solute absorption maxima in a solvent and the polarity/polar-izability of that solvent ... 

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