Potential multidimensional

Flammer B, Scheffler M, Jacobsen K W and Norskov J K 1994 Multidimensional potential energy surface for H2 dissociation over Cu(111) Phys.Rev. Lett. 73 1400... 

The fitting parameters in the transfomi method are properties related to the two potential energy surfaces that define die electronic resonance. These curves are obtained when the two hypersurfaces are cut along theyth nomial mode coordinate. In order of increasing theoretical sophistication these properties are (i) the relative position of their minima (often called the displacement parameters), (ii) the force constant of the vibration (its frequency), (iii) nuclear coordinate dependence of the electronic transition moment and (iv) the issue of mode mixing upon excitation—known as the Duschinsky effect—requiring a multidimensional approach. 

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... 

An alternate and fonnally very powerfiil approach to resonance extraction is complex scaling [7, 101. 102. 103. 104. 105. 106 and 107] whereby a new Hamiltonian is solved. In this Hamiltonian, tlie grid s multidimensional coordinate (e.g., x) is multiplied by a complex constant a. The kinetic energy gains a constant complex factor > (1/a )(d /dx )), while the potential needs to be evaluated at points with a complex... 

In this chapter, we look at the techniques known as direct, or on-the-fly, molecular dynamics and their application to non-adiabatic processes in photochemistry. In contrast to standard techniques that require a predefined potential energy surface (PES) over which the nuclei move, the PES is provided here by explicit evaluation of the electronic wave function for the states of interest. This makes the method very general and powerful, particularly for the study of polyatomic systems where the calculation of a multidimensional potential function is an impossible task. For a recent review of standard non-adiabatic dynamics methods using analytical PES functions see [1]. 

On the basis of tbe above analyses, it follows that them is no need to compute multidimensional potential surfaces if one wishes to handle the R-T effect in the framework of the model proposed. In spite of that, such computations were carried out in [152] in order to demonstrate the reliability of the model for handling the R-T effect and to estimate the range in which it can safely be applied in its lowest order (quadratic) approximation. The 3D potential surfaces involving the variation of the bending coordinates pi, p2 and the relative azimuth angle 7 = 4 2 1 were computed for both component of the state. 

However, in many applications the essential space cannot be reduced to only one degree of freedom, and the statistics of the force fluctuation or of the spatial distribution may appear to be too poor to allow for an accurate determination of a multidimensional potential of mean force. An example is the potential of mean force between two ions in aqueous solution the momentaneous forces are two orders of magnitude larger than their average which means that an error of 1% in the average requires a simulation length of 10 times the correlation time of the fluctuating force. This is in practice prohibitive. The errors do not result from incorrect force fields, but they are of a statistical nature even an exact force field would not suffice. 

R. M. Levy, O. de la Luz Rojas, and R. A. Friesner. Quasi-harmonic method for calculating vibrational spectra from classical simulations on multidimensional anharmonic potential surfaces. J. Phys. Chem., 88 4233-4238, 1984. 

The additional integration in the final equation is similar to integration of potential of mean force. Equation (29) is more difficult to compute since Xq is a multidimensional vector. 

For a multidimensional function, the variable x is replaced by the vector x and matrices are used for the various derivatives. Thus if the potential energy is a function of 3N... 

KD Ball, RS Beii y, RE Kunz, E-Y Li, A Proykova, DJ Wales. Erom topographies to dynamics of multidimensional potential energy surfaces of atomic clusters. Science 271 963-966, 1996. RS Berry, N Elmaci, JP Rose, B Vekhter. Linking topography of its potential surface with the dynamics of folding of a protein model. Proc Natl Acad Sci USA 94 9520-9524, 1997. Z Guo, D Thii-umalai. J Mol Biol 263 323-343, 1996. 

OM Becker, M Karplus. The topology of multidimensional potential energy surfaces Theory and application to peptide stiaicture and kinetics. I Chem Phys 106 1495-1517, 1997. 

RE Kunz, RS Berry. Statistical interpretation of topographies and dynamics of multidimensional potentials. J Chem Phys 103 1904-1912, 1995. 

If all the PES coordinates are split off in this way, the original multidimensional problem reduces to that of one-dimensional tunneling in the effective barrier (1.10) of a particle which is coupled to the heat bath. This problem is known as the dissipative tunneling problem, which has been intensively studied for the past 15 years, primarily in connection with tunneling phenomena in solid state physics [Caldeira and Leggett 1983]. Interaction with the heat bath leads to the friction force that acts on the particle moving in the one-dimensional potential (1.10), and, as a consequence, a> is replaced by the Kramers frequency [Kramers 1940] defined by... 

The most general problem should be that of a particle in a nonseparable potential, linearly coupled to an oscillator heat bath, when the dynamics of the particle in the classically accessible region is subject to friction forces due to the bath. However, this multidimensional quantum Kramers problem has not been explored as yet. 

The situation simplifies when V Q) is a parabola, since the mean position of the particle now behaves as a classical coordinate. For the parabolic barrier (1.5) the total system consisting of particle and bath is represented by a multidimensional harmonic potential, and all one should do is diagonalize it. On doing so, one finds a single unstable mode with imaginary frequency iA and a spectrum of normal modes orthogonal to this coordinate. The quantity A is the renormalized parabolic barrier frequency which replaces in a. multidimensional theory. In order to calculate... 

A multidimensional PES for the reaction (6.45a) has been calculated by Wight et al. [1993] with the aid of the atom-atom potential method combined with the semiempirical London-Eyring-Polanyi-Sato method (see, e.g., Eyring et al. [1983]). Because of high exoergicity, the PES... 

E. Regnier and G. Huang, Euture potential of targeted component analysis by multidimensional liquid chromatogr aphy-mass spectrometry , ]. Chromatogr. 750 3-10 (1996). 

---