Wavefunctions nuclear

To remedy this diflSculty, several approaches have been developed. In some metliods, the phase of the wavefunction is specified after hopping [178]. In other approaches, one expands the nuclear wavefunction in temis of a limited number of basis-set fiinctions and works out the quantum dynamical probability for jumping. For example, the quantum dynamical basis fiinctions could be a set of Gaussian wavepackets which move forward in time [147]. This approach is very powerfLil for short and intemiediate time processes, where the number of required Gaussians is not too large. 

The fact that the set of hi is, in principle, complete in r-space allows the full (electronic and nuclear) wavefunction h to have its r-dependence expanded in terms of the hp... 

The concept of a potential energy surface has appeared in several chapters. Just to remind you, we make use of the Born-Oppenheimer approximation to separate the total (electron plus nuclear) wavefunction into a nuclear wavefunction and an electronic wavefunction. To calculate the electronic wavefunction, we regard the nuclei as being clamped in position. To calculate the nuclear wavefunction, we have to solve the relevant nuclear Schrddinger equation. The nuclei vibrate in the potential generated by the electrons. Don t confuse the nuclear Schrddinger equation (a quantum-mechanical treatment) with molecular mechanics (a classical treatment). 

This threshold prevents basis functions with small population (which are only negligibly contributing to the nuclear wavefunction in any case) from giving rise to new basis functions. The ideal of Fmin = 0 is usually computationally wasteful, leading to many unpopulated basis functions. However, it is also important to note that the uncertainty in branching ratios incurred by finite Fnun is dependent on the average population of a basis function in the wavepacket. Second, it... 

Because the AIMS method associates a unique nuclear wavefunction with each electronic state, one has direct access to dynamical quantities on individual states. This is unlike mean-field based approaches that use only one nuclear wavefunction for all electronic states [59]. One can therefore calculate branching ratios... 

In this approximation the nuclear wavefunctions are a product of N harmonic oscillator functions, one for each normal mode ... 

This equation is in accord with the Frank-Condon principle The nuclei stand still during an electronic transition, so that a good overlap between the nuclear wavefunctions is required for the transition. 

In the harmonic approximation the functions Xi and Xf are products of harmonic oscillator functions. We therefore specify the initial state by a set of quantum numbers n — (ni, ri2,..., n/v), and those for the final state by m = (mi,m2,..., tun)- So the nuclear wavefunctions are henceforth denoted by Xi,n and Xf,m- Equation (19.21) tells us how to calculate the rate of transition from one particular initial quantum mode n to a final quantum state m. This is more than we want to know. All we are interested in is the total rate from any initial state to any final state. The ensemble of reactants is in thermal equilibrium therefore... 

The function G in eq 1 is the Franck-Condon factor which accounts for the contribution of nuclear degrees of freedom and represents the thermal average of the overlap integrals between nuclear wavefunctions with respect to conservation of energy, and is given by (2, 3, 8, 9)... 

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

In this section we derive certain rules which will determine whether or not an integral over given electronic or nuclear wavefunctions vanishes from such rules we can deduce spectroscopic selection rules. Consider the integral... 

Here, the pt are the permanent dipoles of molecules i = 1 and 2, and the ptj( r, i 2, Rij) are the dipoles induced by molecule i in molecule j the are the vectors pointing from the center of molecule i to the center of molecule j and the r, are the (intramolecular) vibrational coordinates. In general, these dipoles are given in the adiabatic approximation where electronic and nuclear wavefunctions appear as factors of the total wavefunction, 0(rf r) ( ). Dipole operators pop are defined as usual so that their expectation values shown above can be computed from the wavefunctions. For the induced dipole component, the dipole operator is defined with respect to the center of mass of the pair so that the induced dipole moments py do not depend on the center of mass coordinates. For bigger systems the total dipole moment may be expressed in the form of a simple generalization of Eq. 4.4. In general, the molecules will be assumed to be in a electronic ground state which is chemically inert. 

In molecules which contain only C-D bonds, the situation is different because the heavier D atom leads to much smaller spacings of vibrational levels, something like 2000 cm-1. The crossing from Ti (v = 0) to S0(v = n) must reach a vibrational level of much higher quantum number so that the overlap of the nuclear wavefunctions is much smaller. The phosphorescence quantum yields and lifetimes are therefore greater in the deuterated compounds. To take one example, the observed phosphorescence lifetime of naphthalene-A8 is 2.3 s, but that of naphthalene-rf8 is 18.4 s, both measured in a rigid glass at 77 K. 

Here ipn, ipn are the electronic wavefunctions corresponding to the initial and final electronic states, nv ancI f n v are the corresponding nuclear wavefunctions, and v and v are the sets of quantum numbers, which describe the nuclear motion. Substituting eqs. 23 and 24 into eq. 22 leads to... 

As a next step, one should take into account the features of the electronic and nuclear wavefunctions. The nuclear wave-... 

We now turn to the analysis of indirect photodissociation. We shall see that this case can also be reduced to the analysis of nuclear dynamics. The problem of the evaluation of the nuclear wavefunctions and the FC factors will be discussed below in Sections III.C and III.D. 

Bound and Quasi-Bound States, it was shown in the last section how the problem of the evaluation of photofragment energy distributions can be reduced to the analysis of nuclear dynamics. Indeed, it is necessary to evaluate integrals containing the nuclear wavefunctions (see eqs. 28, 34, and 57). 

The nuclear wavefunctions and (jp are bound or quasibound states. In the harmonic approximation each can be written as a product... 

One can develop a rigorous adiabatic approach in the framework of the nuclear problem which facilitates the evaluation of the nuclear wavefunction for the D state (29,42). Note that the problem of the description of the D state was raised by Landau about 50 years ago in a classic paper (43). According to his development, the correct description of the D state should contain adiabatic coupling of the bound and continuous parts of the wavefunction. 

Fragments A and B, and p = RB - RA is the distance between their centers of mass. In order to obtain the expression for the nuclear wavefunction D(p,q, qB), one solves the Schroedinger equation associated with the Hamiltonian (66). 

---