Property gradients#
The first-order nuclear derivatives of properties other than the energy can be related to intensities in vibrational spectroscopies. More specifically, the derivative of the dipole moment and polarizability with respect to nuclear coordinates are needed to determine infrared (IR) intensities and Raman activities, respectively. Their calculation is described in more detail in the following.
IR intensities#
In order to calculate intensities in the IR spectrum,
one needs to know how the electric dipole moment
with the charge
The nuclear gradient of the dipole moment can again be calculated either numerically or analytically.
Since the dipole moment itself as a first-order property can be considered a first derivative of the energy, its gradient corresponds to a mixed second derivative of the energy (once with respect to an electric field and once with respect to a nuclear coordinate, and thus a second-order property),
and its calculation thus resembles the molecular Hessian calculation requiring the solution of a set of coupled-perturbed SCF (CPSCF) equations. While the analytic derivative of the nuclear contribution
where the perturbed density and the derivatives of the dipole integrals are needed.
The IR transition dipole moment is then calculated by taking the dot product of the dipole moment gradient with the Cartesian normal modes
Raman activities#
Intensities in the vibrational Raman spectrum are calculated in an analogous manner, except that the nuclear derivative of the electric-dipole polarizability
The main differences include the following:
all the density matrices (
and ) depend also on two Cartesian components, , corresponding to the element of the polarizability tensor , and need to be symmetrized with respect to and .the right-hand side of the linear response equation needs to be included in the Lagrangian, corresponding in this case to the contraction of response vector component
with component of the dipole integrals (also to be symmetrized with respect to and ).in the dynamic case (
), also the term needs to be taken into account (for all combinations of vector components and ).
For randomly oriented molecules and linearly polarized incident light, the Raman differential cross-section is calculated in practice as [Gut19]
where
where
The rotational invariants can further be used to calculate the parallel (or “polarized”) and perpendicular (or “depolarized”)
intensities as