Potential Energy Surfaces

Deborah Crittenden, Keiran Thompson
(with Prof Mick Collins, ANU)

Molecular potential energy surfaces (PES) describe how the energy of a molecule changes as the positions of its constituent atoms change. Accurate PES are crucial to our understanding of molecular structure and reactivity. We have developed new interpolation techniques for constructing PES that allow the PES to be iteratively improved until convergence in any desired molecular property occurs.

Our interpolation technique is a modified Shepard interpolation. It uses local Taylor expansions at a set of PES data points, where energies and derivatives have been calculated using ab initio theory. A weighting function is used to interpolate between the Taylor expansions and provide an analytic expression for the global, multidimensional, PES. Dynamical calculations are performed on the interpolated PES and these are used to ‘sample’ the important regions of configuration space for any given property of the system. These ‘sampled’ configurations are analysed and new data points are chosen. Ab initio calculations at the new data points add to the PES data set and produce a new interpolated PES. This process is repeated and further points are added to the PES data set until convergence is achieved in the desired property. As the number of PES data points increases the interpolated PES converges to the exact PES.

The PES interpolation is illustrated below:

interp1

The PES is expanded in terms of 2nd order Taylor series about each data point in a data set. Here there are 2 data points.

interp2

The simplest weighting formula is the distance to the points in the data set. That is, the energy exactly midway between the 2 data points is given as 50% of the Taylor series energy about data point 1 plus 50% of the Taylor series energy about data point 2.

interp3

The energy of a point one quarter of the way between data point 1 and data point 2 is dominated (90%) by the Taylor series energy about data point 1, wherewas the Taylor series energy about data point 2 contributes only 10% of the interpolated energy.

interp4

In this way the energy at any point can be determined, with the interpolated energy being equal to the exact energy at each data point.

This scheme makes no assumption about the nature of the PES, it does not involve fitting functional forms and the coordinates chosen for the Taylor expansion ensure correct asymptotic behaviour of the PES.

We have used classical trajectories and quantum wavefunctions to sample configuration space and examine both classical and quantum properties. These include reaction cross section, the various product distributions for a reaction, the zero point energy and bond length and bond angle distribution functions.

We have released a computer package “Grow” that makes our techniques freely available. The “Grow” package has recently been combined with quantum diffusion Monte Carlo methods for determining the ground state nuclear wavefunction of a molecule. The current implementation of “Grow” is “Grow2.2” and it is available by clicking the “Grow2.2” tab in the page header.

Selected publications:

D. L. Crittenden and M. J. T. Jordan “Interpolated Potential Energy Surfaces: How Accurate do the Second Derivatives Have to Be?” J. Chem. Phys., 122, 044102 (2005). pdf (86 kB)

D. L. Crittenden, K. C. Thompson, M. Chebib and M. J. T. Jordan “Efficiency Considerations in the Construction of Interpolated Potential Energy Surfaces for the Calculation of Quantum Observables by Diffusion Monte Carlo” J. Chem. Phys. 121, 9844-9854 (2004). pdf (320 kB)

R. P. A. Bettens, D. H. Zhang, M. J. T. Jordan, and M. A. Collins “Ab initio potential energy surface for the reactions between H₂O and H”, J. Chem. Phys., 112, 10162-72 (2000).

K. C. Thompson, M. J. T. Jordan and M. A. Collins, “Polyatomic molecular potential energy surfaces by interpolation in local internal coordinates”, J. Chem. Phys., 108, 8302-8316 (1998).

M. J. T. Jordan and M. A. Collins, “An Interpolated UHF Potential Energy Surface for the OH+H₂ ® H₂O+H Reaction”, J. Chem. Phys.,104, 4600-4610 (1996).

M. J. T. Jordan, K. C. Thompson and M. A. Collins, “The Utility of Higher Order Derivatives in Constructing Molecular Potential Energy Surfaces by Interpolation”, J. Chem. Phys., 103, 9669, (1995).

M. J. T. Jordan, K. Thompson and M. A. Collins, “Convergence of Molecular Potential Energy Surfaces by Interpolation: Application to the OH+H₂ ® H₂O+H Reaction”, J. Chem. Phys., 102, 5647, (1995).