4596
J. Phys. Chem. A 1998, 102, 4596-4605
Representation of Intermolecular Potential Functions by Neural Networks Helmut Gassner,† Michael Probst,*,† Albert Lauenstein,‡ and Kersti Hermansson‡ Institute of General and Inorganic Chemistry, Innsbruck UniVersity, Innrain 52a, A-6020 Innsbruck, Austria, and Inorganic Chemistry, The Ångstro¨ m Laboratory, Uppsala UniVersity, Box 538, S-75121 Uppsala, Sweden ReceiVed: July 8, 1997; In Final Form: March 17, 1998
We have investigated how a neural network representation of intermolecular potential functions can be used to elevate some of the problems commonly encountered during fitting and application of analytical potential functions in computer simulations. For this purpose we applied feed-forward networks of various sizes to reproduce the three-body interaction energies in the system H2O-Al3+-H2O. In this highly polarizable system the three-body interaction terms are necessary for an accurate description of the system, and it proved difficult to fit an analytical function to them. Subsequently we performed Monte Carlo simulations on an Al3+ ion dissolved in water and compared the results obtained using the neural network type potential function with those using a conventional analytical potential. The performance and results of our calculations lead to the conclusion that, for suitable systems, the advantages of a neural network type representation of potential functions as a model-independent and “semiautomatic” potential function outweigh the disadvantages in computing speed and lack of interpretability.
1. Introduction It is now rather commonly appreciated that so-called “artificial neural networks” (which will further be abbreviated as NNs) can be useful in various contexts. Among the most important are storage and interpolation of data as well as pattern recognition in the sense of extracting important features from sets of data. The interest in NNs has increased much during the past decade, although the concept has been known for 50 years.1,2 Examples of applications in the field of chemistry include medical chemistry,3 electrostatic potential comparison,4,5 and, generally, structure-activity relationship studies of various kinds.4 Whereas in most such applications semiquantitative results are sufficient and a very accurate reproduction or prediction of data is not necessary, the simple mathematical structure of feed-forward NNs also makes them a suitable alternative for classical function approximation. Furthermore, due to their relatively simple structure and their universal applicability, NNs can be implemented in computer hardware, and specialized NN coprocessors are available (this is a situation similar to that in the case of digital signal processing). A second technical aspect is that they are very well suited for parallel processing. The conceptual and practical importance of the potential energy surface (PES) within the framework of “Born-Oppenheimer chemistry” is well established. In the following, we will use the terms “potential energy surface” and “potential energy function” interchangeably. PESs are needed, for example, as input for molecular mechanics, molecular dynamics, and Monte-Carlo simulations. The feasibility of performing such simulations critically depends on an accurate and fast representation of the system energy as a function of the atomic or molecular coordinates. In the many cases where it is not necessary or feasible to calculate energies (or forces) by quantum * Corresponding author. † Innsbruck University. ‡ Uppsala University.
chemical means, the potential energy surface is commonly approximated by analytical potential energy formulas. These have usually been obtained by parametrization, either toward quantum chemically calculated energies or toward different experimental data, and should allow fast and accurate retrieval and interpolation of the energies and forces. In this sense, analytical potential functions can be viewed simply as a means of storing and retrieving data and to map discrete data points into continuous functions, as can also be accomplished, for example, via interpolation from look-up tables or via splines or Bezier curves. It often turns out that the fitting procedure to construct such analytical functions is a labor-intensive and cumbersome task which requires a lot of experience since, in a real chemical system, a multitude of bonding effects interact to form the potential surface. From a more formal point of view, the total energy of a molecular system can be expressed as a many-body expansion:
Etot )
(2) (3) (m) Ei,j + ∑ Ei,j,k + ... + ∑ Ei,j,k...,n ∑i E(1)i + ∑ i
