Vibrationally bound states

Figure 4- A-trajectory calculations of the complex eigenvalues of Re H) — tA/m(H) where H is a matrix representation of the nuclear Hamiltonian with complex electron potential surface. For A = 0 the autoionization is artificially suppressed and the real eigenvalues obtained are divided into vibrationally bound states (denoted by B and into dissociative continuum solutions denoted by C (note that the discretization is due to the finite box approximation). The inset shows that very close to A = 1 (the physical solution) one of the continuum solutions and the 8-th B solution coalesce.

Above the possible applicability of a number of techniques to the challenging problem of the ro-vibrational bound states of polyatomic Van der Waals clusters have been considered. Of course to do such calculations it is necessary to have an appropriate potential function. 

H1] Qiu Y and Bai Z 1998 Vibration-rotation-tunneling dynamics of (HF)2 and (HCI)2 from fulldimensional quantum bound state calculations Advances in Moiecuiar Vibrations and Coiiision Dynamics, Voi. i-ii Moiecuiar dusters ed J Bowman and Z Bai (JAI Press) pp 183-204... 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

Figure 6. The vibrational levels of the lowest 40 bound states of A[ symmetry for Li3 calculated without consideration and with consideration of GP effect.

A computational method which is suitable for studies of this nature should fulfill certain basic requirements (a) it should be sufficiently economical to allow computation of full potential-energy curves for comparatively large number of states, (b) the calculated potential curves for bound states should give rise to vibrational and rotational constants which are in reasonable agreement with experiment when a comparison is possible, (c) the calculated total energies of all the states should be of comparable accuracy, and (d) the ordering of the states should be correct. 

Next, we discuss the J = 0 calculations of bound and pseudobound vibrational states reported elsewhere [12] for Li3 in its first-excited electronic doublet state. A total of 1944 (1675), 1787 (1732), and 2349 (2387) vibrational states of A, Ai, and E symmetries have been computed without (with) consideration of the GP effect up to the Li2(63 X)u) +Li dissociation threshold of 0.0422 eV. Figure 9 shows the energy levels that have been calculated without consideration of the GP effect up to the dissociation threshold of the lower surface, 1.0560eV, in a total of 41, 16, and 51 levels of A], A2, and E symmetries. Note that they are genuine bound states. On the other hand, the cone states above the dissociation energy of the lower surface are embedded in a continuum, and hence appear as resonances in scattering experiments or long-lived complexes in unimolecular decay experiments. They are therefore pseudobound states or resonance states if the full two-state nonadiabatic problem is considered. The lowest levels of A, A2, and E symmetries lie at —1.4282,... 

The spectrum of Eq. (2.42) is shown in Figure 2.1. This is the spectrum of the one-dimensional truncated harmonic oscillator with a maximum vibrational quantum number equal to N. Thus N + 1 represents the number of bound states. When N - oo one recovers the full oscillator spectrum. 

Note how the finite number of bound states arises very naturally in the algebraic approach. This example also illustrates the role of the extra quantum number, N. All possible truncated oscillators are described by the same algebra, for different values of the quantum number N. In any given problem, the value of N is fixed, and nz plays the role of the vibrational quantum number. 

The numbers N, N2 are the vibron numbers of each bond. As discussed in Chapter 2 they are related to the number of bound states for bonds 1 and 2, respectively. For Morse rovibrators they are given by Eq. (2.111) that is, they are related to the depth of the potentials. They are fixed numbers for a given molecule. The numbers (0], co, X], X2 are related to the vibrational quantum numbers, as discussed explicitly in the following sections. We have written the O] (4) representations as (C0i, 0) and not simply as aq, since for coupled systems one can have representations of 0(4) in which the second quantum number is not zero. The values of (iq and (02 are given by the rule (2.102),... 

The semiclassical mapping approach outlined above, as well as the equivalent formulation that is obtained by requantizing the classical electron-analog model of Meyer and Miller [112], has been successfully applied to various examples of nonadiabatic dynamics including bound-state dynamics of several spin-boson-type electron-transfer models with up to three vibrational modes [99, 100], a series of scattering-type test problems [112, 118, 120], a model for laser-driven... 

Photodissociation has been referred to as a half-collision. The molecule starts in a well-defined initial state and ends up in a final scattering state. The intial bound-state vibrational-rotational wavefunction provides a natural initial wavepacket in this case. It is in connection with this type of spectroscopic process that Heller [1-3] introduced and popularized the use of wavepackets. 

Bound state calculations with J = 0 are carried out for the above CCSD(T) surface. The linear He-Br-Br structure is found to be the most stable isomer with a binding energy of Do=16.02 cm , while the T-shaped isomer is predicted to lie only 1.1 cm above, indicates the coexistence of them even at low temperatures. The vibrationally averaged structures for these isomers are determined to be (R=4.88 A, 0= 0°) and (R=4.14 A, 0=90°), respectively. The above values are in good accordance with earlier experimental observations and in excellent agreement with recent LIF experimental data available. ... 

The purpose of this work is to study the electronic predissociation from the bound states of the excited A and B adiabatic electronic states, using a time dependent Golden rule (TDGR) method, as previously used to study vibrational pre-dissociation[32, 33] as well as electronic predissociation[34, 35], The only difference with previous treatments[34, 35] is the use of an adiabatic representation, what requires the calculation of non-adiabatic couplings. The method used is described in section II, while the corresponding results are discussed in section III. Finally, some conclusions are extracted in section IV. 

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