Interaction energies

Interactions between molecules occur at various levels, from straight Coulombic energies that vary as 1/r  , to van der Waals energies that vary as 1/r⁶.  More complex combinations of these forces are of interest as well.  For example, how does a charged particle interact with a dipole or how do two dipoles interact?

Point Charge Interaction with a Dipole

When a charge interacts with a dipole, we can in fact, calculate all the interactions as if they were merely a series of point charges.  We start with the Coulombic potential energy of two charges

We need to add up all the attractions and repulsions in this three charge system to arrive at a potential of a system pictured above of a charge q₂ and a dipole m₁ with charges ±q₁.

And finally by using the expansion^* of the terms in x and keeping only the first term, we get

now, using the equation for the dipole, we get

So, we see that the energy of interaction of a dipole with a point charge decreases as 1/r²

Fig. 18.3  (from Atkins):  There are two contributions to the diminishing field of an electric dipole with distance (here seen from the side). The potentials of the charges decrease (shown here by a fading intensity) and the two charges appear to merge, so their combined effect approaches zero more rapidly than by the distance effect alone.

Dipole - Dipole interaction

If we have two dipoles interacting with each other, we need to take into account all the different Coulombic interactions (two attractive and two repulsive).  If the two are arranged in a line, as shown to the right (10) then we find the final equation for potential energy of the system is

What would the equation look like if the two dipoles are arranged as shown in 11?

Generally, the more complex the multipole, the faster it's interaction diminishes with distance. 

n-pole.

Monopole: n= 1, total charge ≠ 0, no net dipole moment


Dipole: n= 2 total charge = 0, net dipole moment ≠ 0.



Quadrupole, n=3 total charge = 0, net dipole moment = 0.










Octupole: n=4, total charge = zero, net dipole moment = 0.

The energy of interaction of an n-pole with an m-pole varies as 1/(r^n+m−1) 

Dipole − Induced-Dipole interactions

The potential energy of a system where a dipole (polar molecule) interacts with a polarizable molecule (but with no permanent dipole) varies as 1/r⁶ and can be written as:

where a' is the polarizability of the non-polar molecule.

Induced-Dipole − Induced-Dipole interactions

The potential energy of a system where a dipole (polar molecule) interacts with a polarizable molecule (but with no permanent dipole) also varies as 1/r⁶ but the relation is a bit more complex.

where I_j is the polarizability of the non-polar molecule j.

Hydrogen Bonding

A hydrogen bond, unlike all the previous interactions only occur in certain molecules.  Molecules that have a hydrogen covalently bonded to a highly electronegative element (N,O, or F), where the electronegative element has lone pairs.  This type of interaction cannot be approximated using point charges as is done for the other van der Waals' and dipolar interactions.  A hydrogen bond is actually caused by the overlap of an orbital from the hydrogen and the atom to which it is bonded (A) and also from the atom to which the hydrogen bond is formed (B) such that

ψ = c₁ψ₁ +  c₂ψ₂ +  c₃ψ₃

The final molecular orbital set will consist of three MOs, with different values of the coefficients for each.  These three MOs must be independent linear combinations of the AOs (LCAO).  Each MO will have a different distribution of the negative charge around the nuclei and as a result, will have different energies using equation 9.15 and integrating over the orbital space.  The net result is that the four electrons (originally, the pair in the A-H bond and the lone pair on B can now redistribute themselves in to the new MO space and lower their energy.  This is still all electrostatic energy but it is much more complicated to calculate than simple point-charge estimates.

The hydrophobic interaction

When non-polar molecules pass from a non-polar medium into a polar medium, there tends to be several significant changes in energies.  Generally, such a transition is endergonic (positive Gibbs energy change) but many times, it is also exothermic (negative enthalpy change).  This must mean that the entropy of the system is lowered by quite a bit to account for this.

Consider the following transition at room temp.

CH₄(in CCl₄) -->  CH₄(aq)

D_transG = + 12 kJ/mol

D_transH = – 10 kJ/mol

D_transS = – 75 J/mol·K

The thermodynamic changes can be accounted for at the molecular level.  When non-polar molecules dissolve in water, a solvation shell (clathrate cage) forms around the non-polar molecule because of the large difference in intermolecular forces.  Since the solvation shell involves the creation of more hydrogen bonds, the enthalpy change is negative.  The large drop in entropy is due to loss of randomness as the solvent molecules organize themselves into these shells.

Molecules such as methane for which the transition to water from a polar solvent is endergonic are called hydrophobic.  Molecules for which this transition is exergonic are called hydrophilic.

Parts of large molecules can also be hydrophobic or hydrophilic.  For example, we are all familiar with the idea that while methane is hydrophobic and we would also readily agree that methanol is hydrophilic (it dissolves readily in water).  The difference in these two is that one of the H-atoms from methane was replaced by the group -OH.

We would term the hydroxyl group hydrophilic, since it increases the solubility of the methyl main group in water, compared to just hydrogen.

We can recognize some hydrophilic groups easily.  Those with electronegative heteroatoms generally are hydrophilic, such as -OH, =O, -NH₂, -F, … .

The thermodynamic parameter p called the Hydrophobicity constant gives us a measure of the hydrophobic or hydrophilic nature of substituents R on a larger molecule A

Here, S is the ratio of the molar solubility of the solute R-A in octanol to that in water,

and S₀ is the ratio of the molar solubility of the solute H-A in octanol to that in water.

If p is positive then the substituent R is hydrophobic while negative values of p indicate that R is hydrophilic.

if we look at a series of alkane branches, we see that the p value becomes more positive, the larger the group.

* Expansions used do derive equation 9.19: