Electric properties of molecules

For a discrete charge distribution the force on charge b due to charge a is given by Coulomb's law:

r_ab is a unit vector point in the direction r_b - r_a. The vector notation defines an attractive force for unlike charges and a repulsive force for like charges. The above equation is written in MKS units. The permittivity of vacuum e₀ = 8.854 x 10^-12 C²N^-1m^-2 in MKS units. The unit of charge is the Coulomb and the force is given in Newtons. If there is more than one charge surrounding b then the force is the sum over all of the charges (principle of superposition).

The electric field is the force per unit charge. More precisely, it is the force on a test charge in the limit that the test charge is infinitesimally small.

The direction of the field is that of the force on a positive charge. The field lies point in the direction that a positive charge wants to go.

For a static charge distribution, the electric field is the negative gradient of the scalar potential f.

The potential at a distance r from a charge q is

The physical significance of the potential is easier to see when we consider work. The work to move a charge is W = q(f₂ - f₁). In MKS units the potential is expressed in volts (V) where 1V = 1 J/C.

Dipoles

A dipole is defined when two opposite charges are separated by a distance, m = qd, where d is the distance. In MKS units dipoles are given in Cm (Coulomb-meters). Chemists most often use the unit of the Debye (D). 1 Debye is equivalent to a full charge separation over 0.208 Å. Put another way, 1 electron displaced through 1 Å results in a dipole moment of 4.8 D. The conversion to MKS units is given by

1 Debye = 3.336 x 10^-30 Cm

We can compare expressions for the charge and dipole moment of a given set of charges

There are higher moments, e.g. quadrupole, Q etc. The properties of dipole moments are particularly important for spectroscopy since it is the interaction of the transition dipole with radiation that leads to the most strongly allowed transitions among states. In general we can consider a multipole expansion of the energy of a collection of charges as

Again it is the dipole term that will be most important in understanding spectroscopic transitions. In spectroscopy one can think of the electric field as the electric vector of the light. We will differentiate between a static dipole of a molecule and a transition dipole.

The static dipole can be calculated from the nuclear charge distribution. In a classical picture we imagine the electrons a particles around the nuclei. The dipole moment is

The charge on each nucleus is Z_a, where Z_a is the atomic number of nucleus a. However, the electrons are not point charges. In quantum mechanics they are represented as probability distributions. The average dipole moment (due to the electrons) is

for state n. The dipole operator is still given above and thus the practical formula for the calculation of the dipole moment is

The permanent dipole moment (ground state dipole moment) of a molecule is important for vibrational and rotational transitions. However, transitions between electronic states involve the transition dipole. A transition dipole is defined by

Notice that in the transition dipole the two quantum states are not the same (indices m and n). A transition dipole represents the charge displacement between two states.

Returning now to the static dipole moment, if we consider a test charge in the vicinity of the dipole

the potential due to the dipole is given by

As above, for a static dipole

So that the electric field is

The energy of a dipole in a field is W = - m·E. Thus the interaction energy of two dipoles is

Polarizability

In the presence of an external field there is also a dipole moment induced in the system. This can occur for a molecule in the field of its neighbors. For example, the dipole moment of water is 1.8 D in the gas phase. However, in the electric field of the neighboring solvent dipoles the dipole moment becomes 2.3 D for water in the liquid state. An induced dipole moment arises from an interaction of the molecular polarizability with an applied electric field. Expressing the dipole moment as an expansion in powers of the field

The term m₀ is the permanent dipole moment. The term a is the polarizability, b is the hyperpolarizability etc. The polarizability of a molecule represents the tendency for an electric moment to be induced in an external field. The polarizability is a second-rank tensor:

The induced dipole moment is

The mean polarizability is

The symbol Tr signifies the trace which is the sum of the diagonal elements, a_xx, a_yy, and a_zz. The ground state polarizability increases with the number of electrons in the molecule. We usually refer to polarizability in volume units a/4pe₀ which has units of m³. For example, the polarizability of CH₄ is 2.6 x 10^-30 m³ or 2.6 Å³ while CCl₄ is 11.2 x 10^-30 m³ or 11.2 Å³.