Empirical Fit of the Surface

Contents

Introduction

Solving Eq. 2-3 is important if you are interested in the structure or time evolution of a molecule. As written, Eq. 2-3 is the Schrödinger equation for the motion of the nuclei on the potential energy surface. In principle, Eq. 2-2 could be solved for the potential energy E, and then Eq. 2-3 could be solved. However, the effort required to solve Eq. 2-2 is extremely large, so usually an empirical fit to the potential energy surface is used. In any case, the solution of the quantum mechanical form of Eq. 2-3 is called quantum dynamics, but since the nuclei are relatively heavy objects, the quantum mechanical effects are often insignificant, in which case Eq. 2-3 can be replaced by Newton's equation of motion:

Eq. 2-4:
The solution of Eq. 2-4 using an empirical fit to the potential energy surface E(R) is called molecular dynamics. Molecular mechanics ignores the time evolution of the system and instead focuses on finding particular geometries and their associated energies or other static properties. This includes finding equilibrium structures, transition states, relative energies, and harmonic vibrational frequencies.

The Forcefield

The empirical fit to the potential energy surface is the forcefield. The forcefield defines the coordinates used, the mathematical form of the equations involving the coordinates, and the parameters adjusted in the empirical fit of the potential energy surface. The forcefields commonly used for describing molecules employ a combination of internal coordinates (bond distances, bond angles, and torsions) to describe the bond part of the potential energy surface, and interatomic distances to describe the van der Waals and electrostatic interactions between atoms. The functional forms range from simple quadratic forms to Morse functions, Fourier expansions, Lennard-Jones potentials, etc. The goal of a forcefield is to describe entire classes of molecules with reasonable accuracy. In a sense, the forcefield interpolates and extrapolates from the empirical data of the small set of molecules used to parameterize the forcefield to a larger set of related molecules and structures.

The physical significance of most of the types of interactions in a classical forcefield is easily understood. Describing a molecule's internal degrees of freedom in terms of bonds, angles, and torsions is natural. The analogy of vibrating balls connected by springs to describe molecular motion is equally familiar. However, it must be remembered that such classical models have limitations. Consider for example the difference between a classical and a quantum mechanical ``bond''.

Covalent bonds can, to a first approximation, be described by a harmonic oscillator, both in quantum and classical theory. Consider the classic harmonic oscillator in the left panel of Figure 2-1. A ball poised at the intersection of the pale grey horizontal line with the parabolic energy surface (thick magenta line) would begin to roll away, converting its potential energy to kinetic energy and achieving a maximum velocity as it passes the minimum. Its velocity (kinetic energy) is then converted back into potential energy until, at the exact same height as it had started, it would pause momentarily before rolling back. The interconversion of kinetic and potential energy in such a classical system is familiar and intuitive. The probability that the ball is at any point along its trajectory is inversely proportional to its velocity at that point. This probability is plotted above the parabolic curve (thin blue line). The probability is greatest near the high-energy limits of its trajectory (where it is moving slowly) and lowest at the energy minimum (where it is moving quickly). Because the total energy cannot exceed the initial potential energy defined by the starting point, the probability drops to zero outside the limit defined by the intersection of the total energy (pale grey horizontal line) with the parabola.

Describing a quantum mechanical ``trajectory'' is impossible, because the uncertainty principle prevents an exact, simultaneous specification of both position and momentum. However, the probability that the quantum mechanical ball will be at a given point on the parabola can be quantified. The quantum mechanical probability function plotted in the right panel of Figure 2-1 is very different from the classical system. First, the highest probability is at the energy minimum, which is the opposite of the classical system. Second, the quantum mechanical ball can actually be found beyond the classical limits imposed by the total energy of the system (tunneling). Both these properties can be attributed to the uncertainty principle.

With such a different qualitative picture of fundamental physical principles, is it reasonable to use a classical approach for obviously quantum mechanical entities like bonds? In practice, many experimental properties such as vibrational frequencies, sublimation energies, and crystal structures can be reproduced with a classical forcefield, not because the systems behave classically, but because the forcefield is fit to reproduce relevant observables and therefore includes most of the quantum effects empirically. Nevertheless, it is important to appreciate the fundamental limitations of a classical approach.

Applications beyond the capability of most classical methods include:

The Energy Expression

The actual coordinates of a molecule combined with the forcefield data create the energy expression or target function for the molecule. This energy expression is the equation that describes the potential energy surface of a particular molecule as a function of its atomic coordinates. As a simple example, consider the following equation, which might be used to describe the potential energy surface of a water molecule:

Eq. 2-5:
where Koh, boh0, Khoh, and hoh0 are parameters of the forcefield, b is the current bond length of one OH bond, b' is the length of the other OH bond, and is the HOH angle.

In this example, the forcefield defines:

Eq. 2-5 is an example for a simple case. Eq. 2-6 is a more realistic energy expression:

Eq. 2-6:
The first four terms in this equation are sums that reflect the energy needed to stretch bonds (b), bend angles () away from their reference values, rotate torsion angles () by twisting atoms about the bond axis that determines the torsion angle, and distort planar atoms out of the plane formed by the atoms they are bonded to (). The next five terms are cross terms that account for interactions between the four types of internal coordinates. The final term represents the nonbond interactions as a sum of repulsive and attractive Lennard-Jones terms as well as Coulombic terms, all of which are a function of the distance between atom pairs rij. The forcefield defines the functional form of each term in this equation as well as the parameters such as Db, , and b0. The forcefield also defines the internal coordinates such as b, , , and as a function of the Cartesian atomic coordinates, although this is not explicitly obvious in Eq. 2-6. Finally, the energy expression in Eq. 2-6 is cast in a general form. The true energy expression for a specific molecule includes information about the coordinates that are included in each sum. For example, it is common to exclude interactions between bonded and 1-3 atoms in the summation representing the nonbond interactions. Thus, a true energy expression might actually use a list of allowed interactions rather than the full summation implied in Eq. 2-6.

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