Unified molecular dynamics and density functional calculations for group VIA clusters

Hohl, D.; Jones, Robert O.

Book/Report

FZJ-2018-03212

Unified molecular dynamics and density functional calculations for group VIA clusters

Hohl, D.FZJ* ; Jones, R. O.FZJ*

1989
Kernforschungsanlage Jülich, Verlag Jülich

Jülich : Kernforschungsanlage Jülich, Verlag, Berichte der Kernforschungsanlage Jülich 2336, 74 p. (1989)

Please use a persistent id in citations: http://hdl.handle.net/2128/18707

Report No.: Juel-2336

Abstract: The determination of ground state structures and other properties of clusters and molecules is an important goal in molecular physics and chemistry, and the associated problems fall into two main categories : (i) finding a reliable method for calculating the total energy E for fixed atomic positions IRI}, (ii) minimizing E with respect to variations in {R$_{I}$}. While the difficulties in calculating E are well known, systematic procedures to locate global minima in configuration space - ${min}_{\lbrace R_{I}\rbrace }$E[{R$_{I}$}] - have received less attention. Assuming N atoms in a cluster, the total number of internal coordinates is 3N - 6, since the energy of the (nonlinear) cluster is invariant to rotations and translations. In a 10-atom cluster, for example, there are 24 independent interatomic degrees of freedom and the magnitude of the problem is apparent: (1) Finding the energy minimum by calculating the energy for all possible geometries is obviously impracticable, since 10 calculations per coordinate would lead to 10$^{24}$ calculations in a cluster where one seems already to be very difficult. (2) If one takes any particular geometry and locates the next local minimum, there is still little prospect of success. As we now show, the number of local minima in the energy surface can be very large. Tab. 1: Number M of local minima in clusters of N particles interacting with a Lennard-Jones potential (Hoare and McInnes). [...] Hoare and McInnes$^{1}$ have studied clusters where the atoms interact with each other with a simple, pairwise Lennard-Jones potential: V$_{Lennard-Jones}$ = 4$_{\epsilon_0}$[($\frac{\sigma}{r})^{12}$ - ($\frac{\sigma}{r})^{6}$] For small clusters it is possible to locate all the minima, and the results are shownfor clusters of different sizes in Tab. 1. The number of minima increases rapidly with increasing N, showing signs that the increase is exponential. For clusters of identical atoms interacting with a pairwise potential, Wille and Vennik$^{2}$ have shown, in fact, that there is no known algorithm that grows with time as power of N and with which one can determine the ground state energy and structure. Such problems are classified as "NP - hard"$^{3}$ (non-polynomial) and are considered "intractable". For sufficiently small clusters, the global minimum can still be determined by locating and enumerating all the local minima. However, the search is confined in practice to small parts of configuration space based on intuition, experimental information and/or symmetry restrictions. [...]

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