Interaction Energy Between Two Molecules

The interaction energy between two molecules or molecular fragments is obtained as a sum over all pairwise atom-atom interactions. The atom-atom potential expressions implicitly assume that the interactions are two-body interactions, undisturbed by other bodies in the vicinity, and that they are isotropic about the atomic centers. 

Like the interaction energy between two molecules, the total lattice energy of a molecular crystal contains several contributions from the different types of interactions. We may write... 

However, parameter a, which stands for the interaction energy between two molecules, is proportional to the product of their polarizabilities and to the product of their dipole moments, and would not be linearly additive. 

Equation 6.64 refers to the electrostatic interaction of one pair of atoms, A and B, and the overall Coulomb interaction energy between two molecules, X and Y, would include all possible pairs of atoms from the different molecules ... 

A very useful tool to rationalize the nature of these intermolecular bonds will be the analysis of the dominant components of the interaction energy. Using classical mechanics, when the two molecules are far enough the interaction energy between two molecules A and B fixed in the space can be approximated as the electrostatic interaction energy (E ) between two sets of multipoles [1,24, 25]. The leading terms of the electrostatic interaction energy between two undistorted A and B molecules can be written as a power series ... 

The coefficient Ce in the long-range interaction energy between two molecules A and B is given in terms of the oscillator strengthsand by... 

Figure 3.3-5 The interaction energy between two molecules as a function of their separation distance. Since the molecules cannot overlap, there is a strong repulsion (positive interaction energy) at small separation distances. At larger separation distances the interactions between the electrons result in an attraction between the molecules (negative interaction energy), which vanishes at very large separations.

The non-covalent forces have a quantum mechanical basis, but much of their nature can be understood using a classical or semi-classical picture. Studies of intermolecu-lar interactions in the gas phase [1] suggest that the intermolecular (non-covalent) interaction energy between two molecules can be broken down into 5 main components (1) electrostatic, (2) exchange repulsion, (3) dispersion, (4) polarization and (5) charge transfer. 

The interaction energy between molecules depends not only upon their separation but also on their relative orientations and, where appropriate, their conformations. It is usual to calculate the van der Waals interaction energy between two molecules using a site model in which the interaction is determined as the sum of the interactions between all pairs of sites on the two molecules. The sites are often identified with the nuclear positions, but this need not necessarily be the case. 

The most common approach used to model intermolecular interactions in crystals is the atom-atom method, where the interaction between molecules is approximated as a sum of atom-atom contributions. The approach was pioneered by Kitaigorodskii [14, 15] and has been an invaluable tool for the modelling of molecular crystals. In its most common form, the interaction energy between two molecules is expressed as ... 

The interaction energy between two molecules i and j lying at a distance of H may be calculated from the equation [41]... 

This induced moment is added to the permanent electric moment of the second molecule and Debye showed that the interaction energy between two molecules, due to this polarizability effect, is independent of temperature and takes the form ... 

It is possible to find the interaction energy between two molecules by integrating their polarizabilities multiplied by the dipole interaction tensor along the imaginary frequency axis. Since in Eq. (2.24) we use the electrostatic interaction tensor T,- the frequency integration actually affects only the polarizabilities. These quantities are strictly real on the imaginary frequency axis. 

In a perturbation approach the interaction energy between two molecules is treated as a small perturbation to the ground state wavefunction of the isolated molecules. For weak perturbations, such as intermolecular interactions, one can separate the interaction terms into components that depend only on the properties of isolated molecules, a primary approximation in virtually all fields of simulations of molecules. The general functions of these interaction terms will also give the appropriate functional forms of the potential surface used in most molecular simulations. A division of the total molecular interaction can schematically be written... 

The expression for the interaction energy between two molecules can be thus supplemented by the electronic interactions between induced dipoles and between induced dipole and charges of both molecules A and B ... 

Thus the total interaction energy between two molecules is given by... 

Besides the most basic and predominant nonpolar interactions (dispersion forces), there are polarization or polar interactions between molecules of counter bodies, such as dipole-dipole interactions (Keesom 1922) and dipole-induced dipole interactions (Debye 1921). The essential difference between dispersion and polarization forces is that, while the former involve simultaneous excitation of both molecules, those for the latter involve only a passive partner. The Keesom orientation interaction energy between two molecules with permanent dipoles is temperature dependent and proportional to the dipole moments as follows ... 

Debye argued that if the attraction energy was simply due to a Keesom effect, then the interaction energy should be drastically reduced at high temperatures. Since experimental results were contrary to the prediction, he concluded that an additional attractive effect should be involved. He showed that an additional polar interaction should he induced between a permanent dipole and an induced dipole. The Dehye induction interaction energy between two molecules with permanent dipoles is proportional to the square of the dipole moments and to the polarizabilities as follows ... 

---