Overview of molecular forces: preliminary considerations

Oliver Smart
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# Preliminary considerations

### Interactions, forces and energies.

It is conventional to talk of the different classes of interactions which act between atoms as "forces" (e.g., the van der Waals force). This can be somewhat confusing as it is normal to go on and give an equation for the potential energy of the particular interaction. In this section an attempt will be made to stick to the terms interaction and energy, reserving "force" for the derivative of the potential energy (unfortunately I did not choose the section's name!).

## Electromagnetic interactions dominate on the molecular scale

The interactions of atoms are primarily governed by electromagnetic interactions, the nuclear and gravitational interactions being completely negligible on the scale of chemistry. The behaviour of a molecule can be completely described by the Schrödinger equation (given here in time independent form):

where x are the positions of the nuclei and electrons of the molecule, 'Psi' is the wavefunction which contains all information about the dynamical properties of the system and E is the energy for the state. The Hamiltonian operator is given by:

Where m_i is the mass of particle i, Z_i the charge and the other symbols have their conventional meaning. The former term equates to the classical kinetic energy and the latter to the electrostatic (Coulomb) energy. Do not worry too much if you do not understand the equation! - it is just important to know that it exists. The properties of all molecules, including proteins, are governed by this equation (excluding the small effects due to relativistic considerations). Unfortunately the solution of this equation is not possible even for simplest molecule H_2^+ (i.e., two protons and an electron), without making an approximation

## The Born-Oppenheimer approximation

Born and Oppenheimer (Ann. Phys. 84:457-484 1927) showed that to a good approximation the electronic and nuclear distributions of a molecule can be treated separately. The electronic distribution can be worked for some set of fixed nuclear positions. The approximation relies on the great difference in mass between nuclei and electrons. It allows the description of a molecular structural configuration in terms of the nuclear position of each atom: the description of a molecule in terms of bond lengths or angles relies on the validity of the Born-Oppenheimer approximation.

Subject to the approximation, the energy for a molecule of N nuclei and n electrons is given by:

where R are the position vectors of the nuclei and the Z_i the charges. The first term represents the contribution made to the potential energy by interactions involving the electrons, the second is simply a classical Coloumb term giving the replusive interaction between the charged nuclei of the molecule. The electronic wavefunction and potential energy are given by:

The Hamiltonian operator for the electrons' contribution to the energy is given by:

where m is the mass of the electron and r_i are the position vectors for the electrons. The first term of the sum corresponds to the kinetic energy of the electrons, the second the interaction between electrons and nuclei and the third the (tricky) electron-electron interactions.

## Quantum chemistry can (theoretically) deal with all molecular interactions

The solution to the above equations gives rise to a large field of study: quantum chemistry. As the procedures are complicated and not terribly relevant to the topic of interest a full desription will not be given here. If you are interested in the subject take the link to advanced topic: A brief introduction to quantum chemistry.

The basic idea of quantum chemical methods is to find the electron distribition for a fixed set of nuclear positions describing the molecule (see the Born-Oppenheimer approximation ). By the application of an energy minimization procedure the geometry of the molecule can be optimized. Ab initio quantum chemistry starts from first principles using no outside or experimental information. This allows the consideration of "exotic" species as well as more conventional molecules. Chemical reactions can be followed and the methods are very useful in organic chemistry.

Suppose one could solve the Schrödinger equation for all the nuclei and electrons in the system of interest (e.g., an enzyme, its substrate and a surrounding set of water molecules) for any given set of nuclear positions. If this could be done in a reasonably short time then allied with molecular dynamics (and other techniques) it would be possible to fairly comprehensively treat the structural properties, dynamics and chemical reactions of a protein from first principles.

Unfortunately, it is impossible apply the methods to molecules the size of proteins - computational limits mean that practically only a few tens of atoms can be considered. Even semiemprical quantum mechanics, in which approximations and experimental data are introduced, are limited to the consideration of systems the size of a few aminoacids. However, quantum chemistry can be extremely useful in providing parameters for molecular mechanics methods and following chemical reactions in proteins. If you are interested have a look at advanced topic: The application of quantum chemistry to protein simulation..