Vibrational wavefunction phase

The unit cell group description of the normal modes of vibration within a unit cell, many of which are degenerate, given above is adequate for the interpretation of IR or Raman spectra. The complete interpretation of vibronic spectra or neutron inelastic scattering data requires a more generalized type of analysis that can handle 30N (N=number of unit cells) normal modes of the crystal. The vibrations, resulting from interactions between different unit cells, correspond to running lattice waves, in which the motions of the elementary unit cells may not be in phase, if ky O. Vibrational wavefunctions of the crystal at vector position (r+t ) are described by Bloch wavefunctions of the form [102]... 

However, if by chance there is a near degeneracy between the O spin-component of the nth level and the fi = fi 1 components of the (v + l)th level of the same electronic state, then the S-uncoupling operator can cause a perturbation between these levels. In the harmonic approximation and using the phase choice that all vibrational wavefunctions are positive at the inner turning point,... 

Each vibrational wavefunction appears twice thus the phase of the vibrational wave function is irrelevant to whether the upper or the lower eigenstate has the... 

In general the bound (predissociated) potential curve is much better characterized experimentally than the repulsive (predissociating) curve. Predissociation linewidths and shifts are usually the best available experimental informal tion about the repulsive state. Indeed, as for bound bound interactions, the vibrational variation of the overlap factor is related to the relative locations of the nodes of the bound and continuum vibrational wavefunctions near Rq (the point of stationary phase of the product x jXe, j, which is where x jXe>,j oscillates most slowly). The Xv,J and Xe,j functions are solutions of the following nuclear Schrodinger equations expressed in atomic units ... 

Figure 7.19 Pictorial descriptions of the phase difference between bound and continuum vibrational wavefunctions. The top part of the figure shows the crossing bound and repulsive potential curves and the two paths between which the phase shift is to be determined. The lower part of the figure represents the classical phase-space trajectories for motion on Vj (ellipse) and V2 (parabola). The shaded area is the phase difference between the two paths, (o) Outer wall crossing. Path I (single arrows) is the most direct dissociation path ai to Rc on Vj, Rc to oo on V2. Path II (double arrows) is the shortest indirect path 01 to i to Rc on Vi, Rc to a2 to oo on V2. (6) Inner wall crossing. The phase difference is between the shortest ( i to Rc on V, Rc to a2 to oo on V2) and next longer ( i to 01 to Rc on Vi,Rc to oo on V2) path.

Note that q is dimensionless because (xe,j xq) and Xv,j xe,j) have units of (energy)-1/2 and (< i H 02) has units of (energy)"1-1. Again, as for bound bound perturbations, the sign of the interference index, q, does not depend on arbitrary phase choices for vibrational wavefunctions exactly twice. 

Figure 1 gives an example of derivative vibrational wavefunctions obtained from the DNC procedure. Using a quite realistic ab initio vibrational potential, Uo R), for the HF molecule, the basic NC approach jdelded the vibrational wavefunaions [Po(R)] for the ground and first two excited states, and these are plotted in the top parts of Figure 1. The first derivatives of the wavefunctions with respect to the strength of an axial electric field [Po (R)] are plotted for each state immediately below, and then the second derivatives of the wavefunaions [Po (R)] are below those. Notice that the first derivative of the ground state wavefunaion is similar (ignore phase) to the unperturbed wavefunaion for the first excited state. This is because the mixing of zeroeth-order vibrational states on application of an external field is primarily with adjacent level states, i.e., one level above or one level below.

B state, and 1.8-2.5 A for the C state. In these ranges, electronic transition takes place as though it makes a copy of vibrational wavefunction in the common regions between the relevant electronic states. This is a reflection of the Condon principle. We note that there is counter-intuitive decrease of population in the potential well of the X state when omitting ionization, especially in the panels for Pulse 2 and Pulse 6, indicating there is some deexcitation from the excited electronic states via Rabi-oscillation like coupling with the ion continuum. Thus, complicated transfer, overlapping, and dispersion of the vibrational wavepackets proceed in each electronic state in a stepwise manner. Very fine information in the attosecond time scale is thus folded in the complicated structures and phases of the set of vibrational wavepackets [305]. 

We haven t yet considered what these vibrations might actually look like. In any system of vibrating objects, such as a molecule, there is a set of equations of motion (in classical physics) or vibrational wavefunctions (in quantum mechanics) called normal modes that describe the lowest-energy motions of the system. In the normal modes, each atom in the molecule oscillates (if it moves at all) back and forth across its equilibrium position at the same frequency and phase as every other atom in the molecule. At higher vibrational energy, the motions can be more complicated, but we can write those motions as a combination of different normal modes. Any vibration of the system can be expressed as a sum of the normal modes they are one possible basis set of vibrational coordinates. 

From a theoretical perspective, the object that is initially created in the excited state is a coherent superposition of all the wavefunctions encompassed by the broad frequency spread of the laser. Because the laser pulse is so short in comparison with the characteristic nuclear dynamical time scales of the motion, each excited wavefunction is prepared with a definite phase relation with respect to all the others in the superposition. It is this initial coherence and its rate of dissipation which determine all spectroscopic and collisional properties of the molecule as it evolves over a femtosecond time scale. For IBr, the nascent superposition state, or wavepacket, spreads and executes either periodic vibrational motion as it oscillates between the inner and outer turning points of the bound potential, or dissociates to form separated atoms, as indicated by the trajectories shown in Figure 1.3. 

So far, this discussion of selection rules has considered only the electronic component of the transition. For molecular species, vibrational and rotational structure is possible in the spectrum, although for complex molecules, especially in condensed phases where collisional line broadening is important, the rotational lines, and sometimes the vibrational bands, may be too close to be resolved. Where the structure exists, however, certain transitions may be allowed or forbidden by vibrational or rotational selection rules. Such rules once again use the Born-Oppenheimer approximation, and assume that the wavefunctions for the individual modes may be separated. Quite apart from the symmetry-related selection rules, there is one further very important factor that determines the intensity of individual vibrational bands in electronic transitions, and that is the geometries of the two electronic states concerned. Relative intensities of different vibrational components of an electronic transition are of importance in connection with both absorption and emission processes. The populations of the vibrational levels obviously affect the relative intensities. In addition, electronic transitions between given vibrational levels in upper and lower states have a specific probability, determined in part... 

For nonlinear (magneto-) optical properties, calculations of an accuracy close to that of modern gas phase experiments require - similar to what has also been found for other properties like structures [79, 109], reaction enthalpies [79, 110, 111], vibrational frequencies [112, 113], NMR chemical shifts [114], etc. - at least an approximate inclusion of connected triple excitations in the wavefunction. This has been known for years now from calculations of static hyperpolarizabilities with the CCSD(T) approximation [9-13]. CCSD(T) accounts rather efficiently for connected triples through a perturbative correction on top of CCSD. For the reasons pointed out in Section 2.1 CCSD(T) is, as a two-step approach, not suitable for the calculation of frequency-dependent properties. Therefore, the CC3 model has been proposed [56, 58] as an alternative to CCSD(T) especially designed for use in connection with response theory. CC3 is an approximation to CCSDT - alike CCSDT-la and related methods - where the triples equations are truncated such that the scaling of the computational efforts with system size is reduced to as for CCSD(T),... 

To obtain hyperpolarizabilities of calibrational quality, a number of standards must be met. The wavefunctions used must be of the highest quality and include electronic correlation. The frequency dependence of the property must be taken into account from the start and not be simply treated as an ad hoc add-on quantity. Zero-point vibrational averaging coupled with consideration of the Maxwell-Boltzmann distribution of populations amongst the rotational states must also be included. The effects of the electric fields (static and dynamic) on nuclear motion must likewise be brought into play (the results given in this section include these effects, but exactly how will be left until Section 3.2.). All this is obviously a tall order and can (and has) only been achieved for the simplest of species He, H2, and D2. Comparison with dilute gas-phase dc-SHG experiments on H2 and D2 (with the helium theoretical values as the standard) shows the challenge to have been met. 

Remember that for every observable quantity, the sign of each wavefunction (electronic and vibrational) always appears twice. This is the reason why the phase convention adopted is irrelevant provided that the same phase convention is used both times each wavefunction appears. It is dangerous to derive an observable quantity, using one matrix element taken from the literature without knowing the phase convention that has been used (see one example in Section 6.2.1). 

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