Total dissociation wavefunction

For fixed total energy E, Equation (2.59) defines one possible set of Nopen degenerate solutions I/(.R, r E, n),n = 0,1,2,..., nmax of the full Schrodinger equation. As proven in formal scattering theory they are orthogonal and complete, i.e., they fulfil relations similar to (2.54) and (2.55). Therefore, the (R,r E,n) form an orthogonal basis in the continuum part of the Hilbert space of the nuclear Hamiltonian H(R, r) and any continuum wavefunction can be expanded in terms of them. Since each wavefunction (R, r E, n) describes dissociation into a specific product channel, we call them partial dissociation wavefunctions. 

After having defined the partial dissociation wavefunctions l>(R,r E,n) as basis in the continuum, the derivation of the absorption rates and absorption cross sections proceeds in the same way as outlined for bound-bound transitions in Sections 2.1 and 2.2. In analogy to (2.9), the total time-dependent molecular wavefunction T(t) including electronic (q) and nuclear [Q = (R, r)] degrees of freedom is expanded within the Born-Oppenheimer approximation as... 

For illustration purposes we find it useful to define the total dissociation wavefunction... 

However, the total dissociation wavefunction is useful in order to visualize the overall dissociation path in the upper electronic state as illustrated in Figure 2.3(a) for the two-dimensional model system. The variation of the center of the wavefunction with r intriguingly illustrates the substantial vibrational excitation of the product in this case. As we will demonstrate in Chapter 5, I tot closely resembles a swarm of classical trajectories launched in the vicinity of the ground-state equilibrium. Furthermore, we will prove in Chapter 4 that the total dissociation function is the Fourier transform of the evolving wavepacket in the time-dependent formulation of photodissociation. The evolving wavepacket, the swarm of classical trajectories, and the total dissociation wavefunction all lead to the same general picture of the dissociation process. 

Fig. 3.2. Two-dimensional potential energy surface V(R, 7) (dashed contours) for the photodissociation of C1CN, calculated by Waite and Dunlap (1986) the energies are given in eV. The closed contours represent the total dissociation wavefunction tot R,l E) defined in analogy to (2.70) in Section 2.5 for the vibrational problem. The energy in the excited state is Ef = 2.133 eV. The heavy arrow illustrates a classical trajectory starting at the maximum of the wavefunction and having the same total energy as in the quantum mechanical calculation. The remarkable coincidence of the trajectory with the center of the wavefunction elucidates Ehrenfest s theorem (Cohen-Tannoudji, Diu, and Laloe 1977 ch.III). Reprinted from Schinke (1990).

Fig. 4.2. Evolution of a two-dimensional wavepacket (R, r t). 4>(i) is shown at several instants. The model system is the same as the one in Figure 2.3. According to (4.11), the wavepacket is related to the time-independent total dissociation wavefunction depicted in Figure 2.3(a), by a Fourier trans-...

The wavepacket in Figure 4.2 follows essentially the same route as the time-independent total dissociation wavefunction tot Ef), defined in Equation (2.70), which is shown for a particular energy in Figure 2.3(a). This coincidence does not come as a surprise, however. If we multiply f(t) with elEft/h and integrate over t we obtain... 

The time-independent total dissociation wavefunction tot Ef) is the Fourier transform of the time-dependent wavepacket y (t). 

The individual partial cross sections are even more structured than the total cross sections and a simple explanation of the energy dependences is probably impossible, except at the low-energy tail of the spectrum. For total energies below the barrier of the A-state PES (E = —2.644 eV in this normalization), the dissociative wavefunction is mainly confined to the two H + OH channels with little amplitude in the intermediate region. It therefore overlaps only that part of the ground-state wavefunction which extends well into the two exit channels. There, however, the 40-) and the 31-) states behave completely differently (see Figure 13.4),... 

If the total energy increases and eventually exceeds the barrier, the two dissociation wavefunctions begin to spread over both channels. This causes the overlap of with d+OH to increase while at the same... 

The spin of the radical is characterized by two spin quantum numbers, the total spin S and the component of the total spin along the z-axis M. The simplest type of radical has one unpaired electron, and hence S = and M = , where the sign of M indicates the orientation of the electron spin in the z-direction. The dissociated singlet molecule, described by the N + l)-electron wavefunction, consists of the radical and a hydrogen atom in orbital (j) at infinity, ... 

Figure 3.2 shows the same trajectory as in Figure 5.1 in a different representation together with the time-independent total dissociation wave-function StotiR,7)- The energy for the trajectory is the same as for the wavefunction. It comes as no surprise that, on the average, the wave-function closely follows the classical trajectory. Another example of this remarkable coincidence between quantum mechanical wavefunctions on one hand and classical trajectories on the other hand is shown in Figure 4.2. There, the time-dependent wavepacket (f) follows essentially the trajectory which starts at the equilibrium geometry of the ground electronic state. This is not unexpected in view of Equations (4.29) which state that the parameters of the center of the wavepacket obey the same equations of motion as the classical trajectory provided anharmonic effects are small. Figures 3.2 and 4.2 elucidate the correspondence between classical and quantum mechanics, at least for short times and fast evolving systems ...

Fig. 8.9. Contour plot of the potential energy surface of H2O in the BlA state as a function of the H-OH dissociation bond Rh-oh and the HOH bending angle a the other O-H bond is frozen at the equilibrium value in the ground electronic state. The energy normalization is such that E = 0 corresponds to H(2S ) + OH(2E, re). This potential is based on the ab initio calculations of Theodorakopulos, Petsalakis, and Buenker (1985). The structures at short H-OH distances are artifacts of the fitting procedure. The cross marks the equilibrium in the ground state and the ellipse indicates the breadth of the ground-state wavefunction. The heavy arrow illustrates the main dissociation path and the dashed line represents an unstable periodic orbit with a total energy of 0.5 eV above the dissociation threshold.

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

Figure 9 compares transitions probabilities for zero total angular momentum J resulting from (2S-DIABATIC) and (IS-GP) calculations with initial and final states chosen to be representative of two cases (A) initial and/or final states have low rovibrational excitation, (B) initial and final states have both high rovibrational excitation. In case (A), both (2S-DIABATIC) and (IS-GP) results almost coincide, even for energies close to the three body dissociation limit. Case (A) corresponds to the kind of transitions already considered in Sect. 4.1 where it was shown that it is valid to use the single rovibrational wavefunction associated to the ground electronic state... 

VB functions have been championed by Warshel and his co-workers for use in studying reactions in enzymes and in solution. The method, which they term the empirical VB (EVB) method, supposes that the wave-function for a particular problem, i/>,can be written as a linear combination of the wavefunctions of resonant forms, v i, which are postulated to be important in the process. For example, for a bond breaking reaction, AB — A+ + B, which produces ionic products, the contributing resonant forms could be the covalent, AB, and ionic, A B+,states that dissociate to atoms and ions respectively. The total wavefunction is ... 

In this case the doorway component of the total wavefunction determines the dissociation rates. Based on our previous discussion, we anticipate that the decay rates will fluctuate considerably for the regular states, since the relative magnitude of the doorway channel component is expected to vary drastically from one state to another. In contrast, chaotic states should have a more homogeneous representation of the doorway component. As a result, chaotic states yield a diminished sensitivity of the decay rate to the exact eigenstate studied. Furthermore, individual T, are expected to be insensitive to the identity of the doorway channel. 

The Faddeev-Watson equations are suitable to the study of permutational symmetry for identical nuclei. This has been done (Micha, 1974) for the three cases in which (1) C = B (2) C = A" and (3) B = A and C = A", to obtain transition amplitudes for direct, atom-exchange and dissociative processes. Nuclear spin variables were included, and amplitudes were found by successively reexpressing symmetrized amplitudes in terms of unsymmetrized ones, reducing nuclear-spin dependences and uncoupling the equations required for calculations. For example in case (2), the terms in the total wavefunction... 

The initial state-selected total dissociation probability of the diatom is obtained by projecting out the energy-dependent reactive flux. If i /,) denotes the time-independent (TI) full scattering wavefunction, where the labels i and E denote the initial state and energy, the total dissociation probability from an initial state i can be obtained by the flux formula (95). We choose the diatomic distance r to be the 5 coordinate in Eq. (96). The full TI scattering wave function is normalized as (if/,) NV/. ) = 2/nhh(E - E ). The total dissociation probability, according to Eq. (98), is given by... 

The system considered in this paper is, again in the words of Bohm [3], a molecule containing two atoms in a state in which the total spin is zero and the spin of each atom is but the restriction to a singlet state and to spin- atoms has now been removed. The molecule is dissociated by separating the nuclei to a large distance and the course of the reaction AB -t A -I- B is followed using standard quantum mechanics. In the general case, the wavefunction 1 for AB is taken to be a spin-coupled antisymmetrized product of two factors which axe exact ... 

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