## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 3. Basic Energetics: Intermolecular Interactions

All bonding and molecular interaction characteristics are determined by quantum mechanics. In liquids, we deal mainly with intermolecular forces and these may be described to quite some extent via classical considerations. In this chapter, we describe the long-range electrostatic intermolecular interactions using the multipole expansion. To that we add the induction and the dispersion interactions. In this approach, short-range repulsion, due to orbital overlap, is usually treated in an empirical way. We end with a brief discussion of some often-used model potentials, together with a few remarks on hydrogen bonding and three-body interactions and the virial theorem. For further details, reference should be made to the books in the section “Further Reading” of this chapter, which have been used freely.

### 3.1. Preliminaries

In principle, molecules and their interactions should be described by the Schrödinger equation. As this formidable problem cannot be solved exactly we use approximate methods, and in this chapter we describe some of them for intermolecular interactions. From the *Born–Oppenheimer approximation* (see Section 2.4) we know that an (*effective*)* potential energy* *E*(**r**) for the motion of the nuclei with coordinates **r** exists. In fact, this is the potential energy employed in all of the discussions on intermolecular interactions. To emphasize the fact that *E*(**r**) acts as a potential energy, from now on, we label it as *Φ*(**r**).

Experiments together with quantum mechanics have made clear that all interaction potentials between molecules have a shape, as presented in __Figure 3.1__, which shows for short distances a strong repulsive part and for longer distances an attractive part of which the magnitude depends heavily on the type of bonding. We need thus to explain the origin of the attractive part and of the repulsive part.

** Figure 3.1** The shape of the Mie pair potential for a diatomic molecule in units of

*ε*and

*σ*for

*n*= 12 and

*m*= 6. The equilibrium distance is

*r*

_{0}, practically equal to for the vibrational ground state at

*T*= 0 K. This distance increases with

*T*due to the asymmetry of the potential and the increase in population of excited vibrational states, as indicated by and . The

*equilibrium energy*is

*D*

_{e}, while the

*dissociation energy*is

*D*

_{0}=

*D*

_{e}−½

*hν*(vibrational state spacing exaggerated).

The balance between the long-range, attractive interactions, collectively denoted as the *van der Waals interactions* *Φ*_{vdW}, and the short-ranged repulsion forces labeled as *Φ*_{rep}, determine whether a complex of molecules is stable, or not. Obviously, the total potential energy *Φ* is given by *Φ* = *Φ*_{vdW} + *Φ*_{rep}. The equilibrium distances^{1)}**r**_{0} are given by the solution of ∂*Φ*/∂**r** = 0, while the force constant matrix reads **K** = ∂^{2}*Φ*/∂**r**∂**r ^{′}**|

_{r}_{=r0}(or

*K*= ∂

_{ij}^{2}

*Φ*/∂

*r*∂

_{i}*r*|

_{j}

_{r}_{=r0}).

Before we discuss these interactions, let us spend a few words on the moments of a charge distribution. In molecular interactions the (continuum) charge distribution *ρ*(** r**) of the molecule is the relevant distribution, where

**is the vector of coordinates. Both, nuclei and electrons contribute to**

*r**ρ*(

**) and we assume that**

*r**ρ*(

**) is given. Alternatively, the charge distribution is described by a (discrete) set of charges**

*r**q*at positions

_{i}

*r**. The zeroth moment of this distribution is a scalar and given by*

_{i}(3.1)

It is essentially the *charge*^{2)}*q* of the molecule. A molecule with *q* = 0 is called *neutral*. The first moment obviously becomes

(3.2)

It is a vector called the *dipole moment*__ ^{3)}__ with components

*μ*,

_{x}*μ*and

_{y}*μ*. Essentially, the magnitude of the dipole moment is given by

_{z}*μ*= |

**| =**

*μ**qδ*, where

*δ*is the distance between the centers of gravity of the positive and negative charges +

*q*and −

*q*in the molecule. The different location of these centers in the molecule renders the dipole moment a vector, leading to strong orientational forces. The

*ideal dipole moment*is obtained by decreasing the distance

*δ*to zero, while the charge

*q*increases in such a way that the product

*qδ*remains

*μ*. A molecule with

*μ*≠ 0 (

*μ*= 0) is called

*polar*(

*apolar*). One also defines the

*quadrupole moment*by

(3.3)

It is a symmetrical tensor with components *Q _{xx}*,

*Q*

_{yy},

*Q*

_{zz},

*Q*,

_{xy}*Q*

_{yz}, and

*Q*. Higher-order moments can be defined similarly, but we will need only the charge and dipole moment. The importance of the moment expansion lies in the fact that, if a full series of moments is given, it is equivalent to the original distribution. Only the first non-zero moment is independent of the location of the origin, which is usually taken as the center of gravity of the molecule.

_{zx}