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Theory and implementation

The molecular reorientational correlation function is defined as the conditional probability to find a molecule with orientation at time , given it had the orientation at time . In the following this probability will be denoted by . Here denotes a set of angular coordinates, as Euler angles or quaternion parameters. The joint probability which gives the probablity to find a molecule with orientation at time and with orientation at time , can be expressed as . Here is the probability to find a molecule with orientation at time . If we consider an isotropic system in thermal equilibrium the reorientational correlation function depends only on the time difference, , and the change in orientation, , i.e. . In addition we have , where is the volume of the angular space.

The reorientational correlation function may now be expanded in Wigner rotation matrices [62] which form a complete set of basis functions in [63,64]:

| (4.110)

In the following the coefficients are called p-coefficients. Using the orthogonality of the Wigner functions,

| (4.111)

Eq. (4.110) can be inverted to give:

| (4.112)

Writing the reorientational correlation function as

| (4.113)

where is the orientation of molecule with respect to its initial orientation and is a thermal average, relation (4.112) can be written as

| (4.114)

The p-coefficients can also be expressed as time correlation functions of irreducible tensor components. This is convenient for numerical purposes since time correlation functions of discrete and finite time series can be very efficiently computed by Fast Fourier Transform techniques (see Section A). Consider the general form of the time correlation function

| (4.115)

where are the components of an irreducible tensor [63,64]. From the transformation properties of irreducible tensors it follows that

| (4.116)

For an isotropic system in thermal equilibrium we may now write

| (4.117)

Inserting this in (4.115), performing a change in the integration variables from to , and using the orthogonality of the Wigner functions one can show that

| (4.118)

where the components are referred to a convenient reference frame. In practice only tensors with integer are relevant. In this case, the well known spherical harmonics [63,64] may be used to define irreducible tensors. They are related to the Wigner functions by

| (4.119)

where are Euler angles. Following ROSE [65] the Wigner functions can be expressed as complex polynomials in the quaternion parameters:

| (4.120)

Here the quaternion parameters describe the rotation of the space-fixed coordinate system into the body-fixed coordinate system. The corresponding rotation matrix is given in Eq. (4.52). According to Eq. (4.119) the spherical harmonics are just special cases of the Wigner functions,

| (4.121)

Using the normalization of the spherical harmonics and Eq. (4.118) one arrives at the following expression for the p-coefficients

| (4.122)

The following relations for the p-coefficients hold:

|  |  | (4.123)
|  |  | (4.124)

The coefficients are the components of the unit matrix. The initial value of the the p-coefficients is an immediate consequence of definition (4.112) and . Eq. (4.124) follows from the symmetry of the Wigner functions and the symmetry of classical time correlation functions.

Since measurable quantities must be real it follows from (4.124) that only p-coefficients with can be directly measured. is measured by infrared spectroscopy (dipole-dipole correlation function) and by relaxation NMR experiments. Here one measures in most cases the integral over .

Next: Parameters Up: Reorientational Correlation Function Previous: Reorientational Correlation Function Contents