Molecular Dynamics Simulations

• Type of physical science: Chemistry
• Field of study: Chemistry of molecules: nature of chemical bonds

Molecular dynamics is the study of the effects of forces and motions on the behavior of atoms and molecules. In a molecular dynamics simulation, these forces and motions are described mathematically and then used to provide a testable model of chemical behavior.

Overview

Chemical change occurs because of the motions of atoms and molecules. Molecules are constantly in motion. Humans are constantly bombarded by molecules (oxygen and nitrogen) in the atmosphere that are traveling in excess of 400 meters per second. Even in solid materials, the molecules that compose the solid are in constant motion. In this microscopic world of atoms and molecules, in which motion is unceasing, change is the natural consequence.

Change occurs in a chemical reaction in which reactants are converted to products. The motion of the reactants leads to collisions that stretch and distort the molecules. The bonds holding atoms together are broken, and new bonds linking different atoms are formed. This is the essence of a chemical reaction. In the freezing of a liquid, the molecules slow down as the temperature is lowered. The slow-moving molecules become subject to electrical forces of attraction and start to stick together. As more and more of these molecules become stuck together, the liquid is converted to a solid.

These detailed pictures of chemical change are the result of simulations of the dynamics of molecular motion. Molecular dynamics simulations allow the motions of individual molecules to be described and then use this information to understand and predict how chemical change occurs. The fundamental principle of molecular dynamics simulations is that the forces acting on a particle determine its motion and hence, its behavior.

For example, consider the motion of a ball attached to the end of a spring. As the spring is stretched, a force acts on the ball to return it to its resting point. The speed that the ball attains is determined by the mass of the ball and the strength (the force constant) of the spring. The motion of the ball carries it past the resting point of the spring and causes the spring to compress.

The compressed spring produces another force (now in the opposite direction) that sends the ball back. By knowing the forces involved and the mass of the ball, one can mathematically describe its motion and understand how it behaves in any given situation.

Understanding the relation between force and motion provides insight into the behavior of atoms and molecules. A spring provides a good model for understanding the motions of atoms within a molecule. A water molecule (H₂O) consists of two hydrogen atoms connected to an oxygen atom. The bonds (forces) holding these atoms together are electronic; however, the net result of these forces is similar to what one would expect if they were connected by springs. To simulate the complex motion of the atoms in water, one can represent these molecular forces mathematically. By representing these molecular forces mathematically and using the masses of the individual atoms, one can simulate the actual motion of an individual water molecule. The same approach is used to simulate the motion of a large number of water molecules in liquid water or even a chemical reaction; however, the mathematical representation of the forces becomes much more complicated.

The representation of the forces is often given in terms of a potential energy surface.

Potential energy is that form of energy that depends upon position. Force is related to the change of potential energy resulting from a change in position. As a spring is stretched, its potential energy is being increased. The further it is stretched, the higher the potential energy. The greater the rate of change of potential energy, the greater the force. The potential energy surface is a graph of how potential energy changes as the positions of all the atoms in a system change. This potential energy surface is very helpful in interpreting the dynamics. For example, the potential energy graph of the spring has the shape of a parabola (the geometric shape similar to the letter U) with its minimum point set to zero. Stretching the spring corresponds to going up the right-hand side of the parabola. As an example, consider what would happen if a small ball were placed on the parabola at this point. The ball would fall, then go back up the left-hand side of the parabola, and then fall again. The movement of the ball simulates the vibration of the spring. The potential energy surfaces used in molecular dynamics simulations are more complex than the parabola of a spring, but the dynamics can be visualized as the movement of a ball on a surface.

Determining these potential energy surfaces is often the most difficult part of a molecular dynamics simulation. Since the simulation involves very small particles--atoms and molecules--the most reliable method for obtaining potential energies is through a quantum mechanical calculation. Unfortunately, quantum mechanical calculations are not straightforward and require very sophisticated mathematical techniques to achieve reliable results. The difficulty in doing accurate quantum mechanical calculations increases greatly as the number of atoms increases. There are a number of approximate methods that are used to obtain potential energy surfaces. One approach is to assume that the potential energy has a particular form—a parabola, for example—and then to modify the potential form with adjustable parameters. These are known as empirical potential energy surfaces because they are based on experimental (empirical) data and not determined by theoretical calculation. For a parabola, the force constant, k, could be assumed to be an adjustable parameter. The parabola could be changed, by changing k, from having a very flat form to a very steep one. The values of the adjustable parameters are obtained by adjusting the potential surface until it agrees with certain known experimental quantities, like bond lengths of chemical bonds. Another approach to obtaining potential energy surfaces is to make approximations to the quantum mechanical equations themselves. This approach is a hybrid between empirical and rigorous theoretical calculations, and these surfaces are known as semi-empirical methods. Whichever approach is used, it is imperative that the potential energy surface be reasonable in order for the molecular dynamics simulation to model reality effectively.

Once the potential energy surface is obtained, each particle in the simulation must be given an initial position and an initial velocity. The initial velocity specifies the direction and speed at which the particle is moving. It is important that these initial values of position and velocity be well-chosen. For example, to do a molecular dynamics simulation of a gas at a particular temperature, one would need to choose velocities that were consistent with the known distribution of velocities of gas particles. Subject to the appropriate overall statistical distributions, the initial values of individual particles are assigned randomly. The next step is to follow the motion of the particles in time. This involves solving a set of mathematical equations known as the equations of motion. These equations come from classical mechanics in physics.

For all but the very simplest potential energy surfaces, these equations of motion cannot be solved exactly; however, there are a number of well-known numerical methods that allow these equations to be solved by a computer to specified accuracy at discrete points in time. Using these numerical methods often requires large amounts of computer time.

The final step of the simulation is the analysis of the results. In order for the results to be meaningful, the motions of the particles for a given set of initial conditions must be followed for a long enough period of time to sample the behavior adequately. The calculations must be repeated with a large number of different sets of initial conditions to obtain statistically meaningful results. The analysis involves calculating averages of a large number of events. For example, in a study of the reaction of hydrogen (H) with H₂, the analysis would require doing thousands of collisions of hydrogen atoms with hydrogen molecules in order for the simulation to account properly for how often the actual reaction would take place in nature.

Applications

One of the major contributions of molecular dynamics simulations is the ability to describe the detailed behavior of simple and complex systems at the microscopic level. These simulations provide a means for understanding how atoms and molecules interact. As a part of the molecular dynamics simulations, computer graphics can be used to provide visualizations of molecular motion. These visualizations can give chemists the same type of insight that biologists gain in looking through a microscope.

From the use of molecular dynamics simulations, chemists have learned much about the nature of chemical reactions. As their knowledge increases, chemists grow closer to being able to control and facilitate the outcome of chemical reactions. Molecular dynamics simulations have shown that reactivity for chemical reactions in which the products have higher potential energy than the reactants can be enhanced by vibrationally exciting the reactants. For these reactions, increasing the energy of collisions does not greatly increase reactivity. On the other hand, reactions in which products have lower potential energy react best with high-energy collisions. The reactivity of these reactions is largely independent of the amount of vibrational energy. With this kind of detailed understanding of how to make the best use of energy, chemists can develop methods to supply energy in the appropriate form to promote reaction. The microwave radiation used in a microwave oven, for example, is the most efficient form of energy for heating food that contains water molecules.

Another area that has benefited from molecular dynamics simulations is chemical lasers. A laser is a device that produces a very intense (high-energy) beam of light over a very narrow range of wavelengths. Lasers have a very wide range of uses, from medical surgery to fundamental chemical research. The first step in making a chemical laser is to find a chemical reaction that produces products with an unusual energy distribution. Molecular dynamics simulations have guided the discovery of these reactions, and they also indicate how chemical lasers can be used. For example, lasers can be used to excite reactants vibrationally in reactions for which vibrational energy is the best form of energy for the reaction.

Molecular dynamics simulations have provided biochemists with a new appreciation of the importance of motion to the function of biologically important molecules. One example is the protein myoglobin, which is used to store oxygen in muscle tissue. If myoglobin were a rigid molecule, the absorption and release of an oxygen molecule would be far too slow for proper biological function. This would prevent myoglobin from functioning properly in muscle tissue. Molecular dynamics calculations have shown that the movement of various parts of the myoglobin molecule opens a pathway that allows oxygen to enter and leave the myoglobin molecule easily. Understanding how myoglobin and other biological molecules function helps scientists discover the deficiencies in molecules that lead to certain diseases. Sickle-cell anemia is a disease caused by an abnormality in the hemoglobin molecule in red blood cells. It is hoped that this kind of knowledge will lead to cures for these diseases. Another area of biochemistry in which molecular dynamics simulations offer new insight is the role of enzyme molecules in catalyzing (speeding up) chemical reactions. Calculations have shown the importance of the interplay of the motions of the enzyme, the reactants, and even the surrounding water molecules in the reaction.

One of the possible outcomes of this detailed understanding of enzymes would be that chemists will be able to design specialized molecules with the ability to catalyze a wide variety of chemical reactions.

Sometimes, knowledge in one area of science leads to progress in a completely different area. One example of this is a newly developed mathematical technique for solving a special class of optimization problems that includes the famous traveling salesperson problem of finding the shortest cyclic path for a salesperson to visit each of his clients. This numerical method is known as the method of simulated annealing because it mimics the way nature finds the optimal arrangement of atoms in a crystal as a liquid or solid is slowly cooled (a process known as annealing). It can be surmised that the physical concept of annealing inspired mathematicians to develop a new approach to solving difficult optimization problems.

Context

Molecular dynamics simulations take a deterministic approach to describing molecular behavior in the sense that motion is assumed to be completely determined by the forces acting on the particles. This is one of the basic principles of classical physics, which traces its roots to the work of Sir Isaac Newton in the seventeenth century.

Being deterministic is both a strength and a weakness of molecular dynamics simulations. It is a strength in that these calculations can provide a detailed picture of the behavior of atoms. This picture of atoms and molecules as particles that move through space, colliding and interacting, fits into the way one normally comprehends the world. While this may be an appealing picture, it is not completely correct. Therein lies an important weakness of most molecular simulations. Quantum mechanics states that objects of small mass, such as electrons and atoms, exhibit both particle-like and wave-like behavior. The concept of atoms as particles moving through space should be joined with the concept of waves, expanding, contracting, and interfering with one another. This points out that molecular dynamics simulations based on classical mechanics can, at best, be only approximately correct and must always be interpreted carefully. There have been some molecular dynamics simulations of very simple systems (for example, H + H₂) that have used the correct quantum mechanical equations to describe the dynamics. Unfortunately, these calculations are mathematically complex, and the results are difficult to visualize and incorporate. For these reasons, molecular dynamics simulations based on classical mechanics will continue to be a valuable means for understanding chemical behavior at the molecular level. Advances in machine learning have enabled molecular dynamics simulations to achieve near quantum-level accuracy while retaining the speed of classical methods.

Often, they will be the only way to obtain detailed information. Scientific advances have allowed molecular dynamics simulations to be combined with coarse-grained models and machine learning methods to study larger systems, including entire biological cells.

Molecular dynamics simulations are not the only way to simulate chemical behavior. Developments have introduced differentiable simulation methods that integrate molecular dynamics, Monte Carlo techniques, and machine learning, allowing simulations to be optimized and trained alongside prediction.

Another approach, which is based on probability theory and statistical mechanics, is the stochastic simulation method, sometimes known as Monte Carlo simulation. Instead of solving Newtonian equations of motion, these methods predict change from the laws of probability. By not solving equations of motion, these methods avoid some complexities associated with deterministic approaches, but they do not eliminate the need to account for quantum mechanical effects when relevant. Stochastic simulations focus on the distribution of possible arrangements of atoms and molecules, deemphasizing how the system evolves from one arrangement to another. They are ideally suited for processes that are largely probabilistic in nature, such as diffusion, and for describing equilibrium behavior. Because they do not deal with how change occurs, however, Monte Carlo methods cannot be used to determine the step-by-step details of a chemical reaction.

In order to understand the world, one must be able to develop concepts that point to the underlying nature of things. Molecular dynamics simulations play an important role in doing this by helping scientists uncover the dynamic nature of atoms and molecules.

Principal terms

CLASSICAL MECHANICS: the branch of physics that deals with the action of force on objects of mass; classical mechanics accounts for this action in terms of Sir Isaac Newton’s three laws of motion

CHEMICAL BOND: the attractive forces resulting from positive and negative charges that hold atoms together in a molecule

EQUATIONS OF MOTION: the set of mathematical equations (from Newton’s laws of motion) that can be solved to determine the position and velocity of a particle at any point in time

MOLECULE: a discrete collection of atoms held together by chemical bonds

POTENTIAL ENERGY SURFACE: a graphical representation of how potential energy (the energy caused by position) depends on the distances between particles; the shape of the potential energy surface determines the forces acting on the particles

QUANTUM MECHANICS: the branch of physics developed by Erwin Schrödinger and Werner Heisenberg in 1926 to describe atomic and subatomic particles; classical mechanics is not valid for these small particles

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