Long range contributions

For molecules being separated by a relatively large distance, the overlap between their wavefunctions can be approximately ignored (the overlap is never exactly zero, but decreases exponentially with increasing distance between the molecules). Therefore all electron density can be rigorously assigned to one or the other molecule. This simplifies the mathematical approach in that sense that all terms which represent electron exchange between the molecules can be neglected and therefore the calculation of the overall wavefunction for the complex can be done without antisymmetrisation. The long range perturbation theory has first been discussed by London ^ and has been re-formulated by others since then.^"^^

In the Rayleigh-Schrodinger Perturbaton Theory, the infinitely separated molecules A and B with associated groundstate wavefunctions ( ,m = 0 and ^ g , n = 0), have Hamiltonians Ha and Mb which add up to the unperturbed Hamiltonian of the system: H^ = Ha + Hg. The energy of zeroth order, Uab^ o f such a system is therefore equal to the sum o f the monomer energies É^a and

% / = < K > + < > = E f ’A + 2 .3

All the important long range contributions to the intermolecular forces between molecules arise ultimately from the electrostatic (Coulombic) interaction of the charge distributions between the particles (electrons and nuclei) which make up these

C hapter!. The theory o f intermolecular forces

molecules. The difference V (the perturbation operator) between the unperturbed Hamiltonian and the one for the perturbed system H can therefore be defined as

B

2. 4

A /vrr* ^

T nj

In this equation is the inter-particle separation between particles i and j in the different molecules, and the charge e/^ refers to the particle (electron or nucleus) associated with molecule A and e f to j on B. £o is the permittivity o f vacuum. The interaction energy can be expanded in the series

Uab — Uab ^ + Uab ^ + Uab ^ • + Uab ” 2 . 5

The definitions of the different contributions to the intermolecular forces, such as the electrostatic terms, the induction and the dispersion, correspond to different terms in the perturbation series expansion. The first-order perturbation represents the electrostatic energy, i. e. the change in energy due to the electrostatic interaction between the permanent multipole moments of the molecules. It is calculated as the expectation value of the electrostatic interaction V for the ground state (m = « = 0) with m, n being the order of the wavefunction

u,b' = < « | y | « >=

2

6

This classical interaction of the undistorted, non-spherical charge distributions and p® of the ground state wavefunctions in isolation of molecules A and B respectively can be calculated by an integration over the ab initio charge densities of the interacting molecules at each relative orientation.

Chapter 2. The theory o f intermolecular forces

U e , e a r o s , a , c =

J

2

•'

k; -r.

The elements of the charge distributions of A and B are separated by jri-rzl and as they are not distorted in the first-order energy correction, this contribution to the intermolecular forces is strictly pairwise additive.

This electrostatic term is zero for spherical neutral molecules (atoms), but significant for the majority of organic molecules. It is the only major contribution to the intermolecular potential that is either attractive or repulsive and is usually the interaction that persists over the longest range. Furthermore, it is the most sensitive potential contribution towards the relative orientation of the molecules and is therefore likely to dominate the orientation dependence of the intermolecular potential for polar molecules.

The second-order perturbation energy

= -

E

<

Induction In duction D ispersion

represents two physical effects, the induction and the dispersion interaction, and it does not contain terms with both molecules in the ground state.

The induction energy can be described as the distortion o f the charge distribution on one molecule (molecular polarisability) due to the electric field o f all surrounding (undistorted) molecules. In the induction energy of molecule A, where the field from molecule B polarises A, the distortion of A is described by the excited state wave functions o f energy and molecule B in the ground (undistorted) state

^ I n d u c t i o n 2 .9

Chapter 2. The theory o f intermolecular forces

The sum in this equation is over all the excited states o f molecule A.

The induction (polarisation) energy of B where the field from molecule A polarises B and which has B in an excited state and A in the ground state is equal to

^ I n d u c t i o n

2

2.10

The induction energy between ground state molecules is always attractive, as the distortions only occur to lower the energy of the pair. This highly non-pairwise additive energy term is important for the intermolecular interactions o f ions, but it is often one of the smallest long range terms for neutral molecules, and zero for neutral spherical molecules (atoms).

The final term of the second order perturbation energy represents the dispersion energy, the energy lowering associated with the polarisation by instantaneous fluctuations in the charge distributions of the monomers A and B. The excited states of molecules A and B are used to describe this non-classical phenomenon.

u i - u U u : - u l )

For spherical neutral molecules (for example argon), which consequently have no permanent dipole or higher multipole moments, the electrostatic and induction (polarisation) contributions to the intermolecular interaction energy are absent. The only attractive interaction between such molecules at long range is the dispersion interaction, first identified by London ^ in the 1930s. This long range term has recently been described as the 'universal attractive glue that leads to the formation of condensed phases'.

Chapter 2. The theory o f intermolecular forces