Potentials surface

B. Volta Potentials, Surface Potential Differences, and the Thermionic Work Function... 

Hutson J M and Howard B J 1980 Spectroscopic properties and potential surfaces for atom-diatom van der Waals molecules Mol. Phys. 41 1123... 

Figure Al.6.21. Bra and ket wavepacket dynamics which detennine the coherence overlap, (( ) ( ) ). Vertical arrows mark the transitions between electronic states and horizontal arrows indicate free propagation on the potential surface. Full curves are used for the ket wavepacket, while dashed curves indicate the bra wavepacket. (a) Stimulated emission, (b) Excited state (transient) absorption (from [41]).

Hollenstein H, Marquardt R, Quack M and Suhm M A 1994 Dipole moment function and equilibrium structure of methane In an analytical, anharmonic nine-dimenslonal potential surface related to experimental rotational constants and transition moments by quantum Monte Carlo calculations J. Chem. Phys. 101 3588-602... 

With the frequency removed from the sum, (B1.1.9) has just a sum over vibrational integrals. Because all the vibrational wavefiinctions for a given potential surface will fomi a complete set, it is possible to apply a sum rule to simplify the resulting expression ... 

Cohen R C and Saykally R J 1991 Multidimensional intermolecular potential surfaces from VRT spectra of van der Waals complexes Ann. Rev. Rhys. Ohem. 42 369-92... 

Pople J A, Krishnan R, Schlegel H B and Binkley J S 1978 Electron correlation theories and their application to the study of simple reaction potential surfaces int. J. Quantum Chem. 14 545-60... 

For both first-order and continuous phase transitions, finite size shifts the transition and rounds it in some way. The shift for first-order transitions arises, crudely, because the chemical potential, like most other properties, has a finite-size correction p(A)-p(oo) C (l/A). An approximate expression for this was derived by Siepmann et al [134]. Therefore, the line of intersection of two chemical potential surfaces Pj(T,P) and pjj T,P) will shift, in general, by an amount 0 IN). The rounding is expected because the partition fiinction only has singularities (and hence produces discontinuous or divergent properties) in tlie limit i—>oo otherwise, it is analytic, so for finite Vthe discontinuities must be smoothed out in some way. The shift for continuous transitions arises because the transition happens when L for the finite system, but when i oo m the infinite system. The rounding happens for the same reason as it does for first-order phase transitions whatever the nature of the divergence in thennodynamic properties (described, typically, by critical exponents) it will be limited by the finite size of the system. 

Figure B3.4.10. Schematic figure of a ID double-well potential surface. The reaction probabilities exliibit peaks whenever the collision energy matches the energy of the resonances, which are here the quasi-bound states in the well (with their energy indicated). Note that the peaks become wider for the higher energy resonances—the high-energy resonance here is less bound and Teaks more toward the asymptote than do the low-energy ones.

The classical counterpart of resonances is periodic orbits [91, 95, 96, 97 and 98]. For example, a purely classical study of the H+H2 collinear potential surface reveals that near the transition state for the H+H2 H2+H reaction there are several trajectories (in R and r) that are periodic. These trajectories are not stable but they nevertheless affect strongly tire quantum dynamics. A study of tlie resonances in H+H2 scattering as well as many other triatomic systems (see, e.g., [99]) reveals that the scattering peaks are closely related to tlie frequencies of the periodic orbits and the resonance wavefiinctions are large in the regions of space where the periodic orbits reside. 

Figure B3.4.16. A generic example of crossing 2D potential surfaces. Note that, upon rotating around the conic intersection point, the phase of the wavefunction need not return to its original value.

Walker R B, Stechel E B and Light J C 1978 Accurate Hg dynamics on an accurate Hg potential surface J. Chem. Phys. 69 2922... 

Mandelshtam V A and Moiseyev N 1996 Complex scaling of ab initio molecular potential surfaces J. Chem. Phys. 104 6192... 

Most gradient optimization methods rely on a quadratic model of the potential surface. The minimum condition for the... 

It is usually not efficient to use the methods described above to refine the transition state to full accuracy. Starting from a qualitatively correct region on the potential surface, in particular one where the Hessian has the right signature, efficient gradient optimization teclmiques, with minor modifications, are usually able to zero in on the transition state quickly. 

Koga N and Morokuma K 1985 Determination of the lowest energy point on the crossing seam between two potential surfaces using the energy gradient Chem. Phys. Lett. 119 371... 

Northby J A 1987 Structure and binding of Lennard-Jones clusters 13< W < 147 J. Chem. Phys. 87 6166 Berry R S 1993 Potential surfaces and dynamics what clusters tell us Chem. Rev. 93 2379... 

The occurrence of predissociation opens up a new family of observable quantities. It is possible to measure not only linewidths or lifetimes, but also the internal state distributions of the fragments. All these quantities are sensitive to the intennolecular potential and can be used to test or refine proposed potential surfaces. 

In light of tire tlieory presented above one can understand tliat tire rate of energy delivery to an acceptor site will be modified tlirough tire influence of nuclear motions on tire mutual orientations and distances between donors and acceptors. One aspect is tire fact tliat ultrafast excitation of tire donor pool can lead to collective motion in tire excited donor wavepacket on tire potential surface of tire excited electronic state. Anotlier type of collective nuclear motion, which can also contribute to such observations, relates to tire low-frequency vibrations of tire matrix stmcture in which tire chromophores are embedded, as for example a protein backbone. In tire latter case tire matrix vibration effectively causes a collective motion of tire chromophores togetlier, witliout direct involvement on tire wavepacket motions of individual cliromophores. For all such reasons, nuclear motions cannot in general be neglected. In tliis connection it is notable tliat observations in protein complexes of low-frequency modes in tlie... 

Figure 1. Adiabatic potential surfaces (a) for the linear E x e case and (b) for a state with linear Jahn-Teller coupling and spin-orbit coupling to a state,...

---