Triatomic molecules Hamiltonian equations

Podolsky method, Renner-Teller effect, triatomic molecules, Hamiltonian equations, 612—615 Poincare sphere, phase properties, 206 Point group symmetry ... 

Floquet theory principles, 35—36 single-surface nuclear dynamics, vibronic multiplet ordering, 24—25 Barrow, Dixon, and Duxbury (BDD) method, Renner-Teller effect tetraatomic molecules, Hamiltonian equations, 626-628 triatomic molecules, 618-621 Basis functions ... 

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Hamiltonian equations, 627-628 II electronic states, 632-633 triatomic molecules, 587-598 minimal models, 615-618 Hartree-Fock calculations ... 

II electronic states, 638-640 vibronic coupling, 628-631 triatomic molecules, 594-598 Hamiltonian equations, 612-615 pragmatic models, 620-621 Kramers doublets, geometric phase theory linear Jahn-Teller effect, 20-22 spin-orbit coupling, 20-22 Kramers-Kronig reciprocity, wave function analycity, 201 -205 Kramers theorem ... 

ABBA models, 631-633 Hamiltonian equations, 626-628 HCCS radical, 633-640 perturbative handling, 641-646 theoretical background, 625-626 triatomic molecules ... 

Let us again consider the photodissociation of the linear triatomic molecule with coordinates R and r (Figure 2.1). We want to solve the time-dependent Schrodinger equation (4.1) with the Hamiltonian given in (2.39) and the initial condition (4.4). 

The extension to more than one dimension is rather straightforward within the time-dependent approach (Heller 1978a, 1981a,b). For simplicity we restrict the discussion to two degrees of freedom and consider the dissociation of the linear triatomic molecule ABC into A and BC(n) as outlined in Section 2.5 where n is the vibrational quantum number of the free oscillator. The Jacobi coordinates R and r are defined in Figure 2.1, Equation (2.39) gives the Hamiltonian, and the transition dipole function is assumed to be constant. The parent molecule in the ground electronic state is represented by two uncoupled harmonic oscillators with frequencies ur and ur, respectively. 

In order to illustrate electronic transitions we discuss the simple two-dimensional model of a linear triatomic molecule ABC as depicted in Figure 2.1. R and r are the appropriate Jacobi coordinates to describe the nuclear motion and the vector q comprises all electronic coordinates. The total molecular Hamiltonian Hmoi, including all nuclear and electronic degrees of freedom, is given by Equation (2.28) with Hei and Tnu being the electronic Hamiltonian and the kinetic energy of the nuclei, respectively. 

Previous fully quantum mechanical studies of predissociation phenomena in triatomic molecules do not, to our knowledge, use a Hamiltonian that has a non-zero total angular momentum. Tennyson et al[43, 44, 45, 46, 47, 48, 49, 50, 51] solve the same equations as we do but have not yet, to our knowledge, treated any predissociation problems. The adiabatic rotation approximation method of Carter and Bowman[52] plus a complex C2 modification have, on the other hand, been used to compute rovibrational energies and widths in the HCO[53, 54] and HOCl[55, 56, 57] molecules. This method is based upon the the Wilson and Howard[58], Darling and Dennison[59] and Watson[60] formalism. It is less transparent but the exact formalism in refs.[58, 59, 60] is equivalent to the one presented here and in ref [43]. While both we and Tennyson et al[43] include the exact Hamiltonian in our formalism the latter authors 152] use an approximate method which they have analysed and motivated. 

Each vibration a molecule exhibits corresponds to two quadratic terms in the Hamiltonian as in Equation 7.35. There is a quadratic momentum term and a quadratic term in the associated displacement coordinate. Thus, each vibrational mode contributes 2 X NkT/2 to U. We can collect this information concisely by saying that for a molecular gas, U is 3NkT/2 (translation), plus either NkT/2 for linear molecules or 3NkT/2 for nonlinear molecules (rotation), plus NkT times the number of modes of vibration. The gas of an ideal diatomic molecule has U = 7NkT/2 as in Equation 11.59. A linear triatomic molecule has U = 13NkT/2 because there are three vibrational modes, one of which is twofold degenerate. 

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