Rotational averaging describes the contribution to the potential energy from the rotational orientation of a charge-dipole interaction. Expectation values are utilized to give a single optimal value for the system's potential energy due to rotation.
For example, take a charged particle and a molecule with a permanent dipole. When they interact, the potential energy of this interaction can easily be calculated. For a dipole of length , with a radius of between the dipole centre and the charged particle, the energy of interaction can be described by:
Potential Energy of a Charge Dipole Interaction
where is the charge of the particle, is the dipole moment, is the angle between and the dipole vector, is the vacuum permittivity constant, and is the radius between the particle and dipole.
Geometrically, this interaction is dependant on the radius and the length of dipole, as well as the orientation angle. If the radius between the ion and dipole is taken to be a fixed value, the angle still has the ability to change. This varied orientation of results in rotation of the dipole about its center, relative to the interacting charged particle. The weights of various orientations are described by a Boltzmann distribution expectation value, described generally by:
Expectation value (discrete states)
Expectation value (continuous states)
where is the expectation value, is the energy value for a particular configuration, is Boltzmann's constant, and T is temperature. This Boltzmann-described weighting is the sum over the quantum mechanical energy levels of the system. Therefore, the probability is directly proportional to , indicating that at a specific temperature lower energy configurations are more probable. An equation can then be derived from this general expression, in order to relate it to the geometry and energy of a charge-dipole interaction.
Derivation of Rotationally-Averaged Charge-Dipole Interaction Potentialedit
The orientationally averaged potential energy is the expectation value of the charge-dipole potential energy averaged over
Starting with the potential energy of a charge-dipole interaction
The average over the dipole orientation using the expectation value in classical statistical mechanics is:
Note: When integrating over an angle, the variable of integration becomes
To solve this integral we must first use first order Taylor's series approximation because integrals of exponential of do not have analytical solutions.
The first order Taylor's series approximations is as follows:
Using the Taylor series with gives:
The integral now becomes:
Multiplying into the brackets will give:
All terms that do not depend on are constants and can be factored out of the integral. The terms can be expressed as 4 integrals:
We must use trigonometric integrals to solve each of these for integrals:
Plugging each trigonometric solved integral back into the equation gives:
Finally replace with to give:
The Orientational Average of the Charge-Dipole Interaction Energy