Situation
Sajesh Thomas posed these questions to me, based on his research directed by Mark Spackman.

How can we estimate the vibrational frequencies of a molecule when it is embedded in a crystal?

How much does the molecular geometry change from the gas phase?

What other properties can we calculated for the embeded molecule?
Context
The electron density distribution (EDD) had been derived for several hostguest complexes via Xray diffraction. From this EDD, Sajesh could calculate the electrostatic potential and the electic fields at any point in the crystal — including at sites within the embedded molecule.
Initial approach
Sajesh’s approach was to optimize the geometry of the molecule in the presence of these experimentally derived fields using a quantum chemistry program such as Gaussian. But the problem was that, one the fields were applied, the molecule would rotate out of its experimentally determined position.
Solution
During beers last Friday night we came up with a solution based on a Taylor expansion of the energy. Using the quantum chemistry programs to obtain various energy derivatives, the outcome is that we can estimate
 The gasphase molecular geometry
 The change in the force constants due to the external electric fields
 The molecular dipole moment derivatives (with additional assumptions)
Notation
In order to understand the solution some notation has to be introduced.
The most important quantity is the generalized molecular potential energy surface \( E(\B{R},\B{F}) \):
 \( \B{R} = (R\sub{I\alpha}) \) are the nuclear coordinates
 \( \B{F} = (F\sub{I\alpha}) \) are the external electric fields
 Capital roman letters \( I, I=1\ldots N\sub{n} \) refer to atom indices
 Greek letters \(\alpha \) are cartesian coordinates
The coordinates \(\color{red} \B{R}^c \) and fields \( \color{red}\B{F}^c \) refer refer to experimental incrystal cooordinates and fields, respectively. Note that below we will always use:

\( \color{red}\textrm{red for experimental data} \),

\( \color{green} \textrm{green for quantum chemical data} \).
The coordinates \( \B{R}^e \) refers to gas phase equilibrium coordinates and are to be determined.
To make the Taylor expansions less cluttered understand a repeated index summation convention is used.
For the same reason, partial derivative symbols are eliminated, as follows:
 Positional derivatives of the molecular energy \( E \) are written as
 Derivatives involving one electric field are dipole moment derivatives:
 Derivatives involving two electric fields are polarizability derivatives:
Importantly, all of these quantities may be evaluated using quantum chemistry program packages, if not directly, then by using finite differences.
Some of these quantities may also be estimated from experimental information using the input of quantum chemical information — the essence of the original approach to the problems.
Gas phase structure from crystal
We expand \( E\sub{I\alpha}({\color{red}\B{R}^c},{\color{red} \B{F}^c}) \) around the gas phase geometry with zero field,
The first term on the LHS is zero because it is the gradient at the equilibrium geometry. The last term on the LHS is the cartesian force constant matrix at the equilibrium geometry. This term, and the second term, the dipole moment derivatives, may be obtained from ab initio caculations, so are colored green. Therefore these are \( 3N\sub{n}\times 3N\sub{n} \) linear equations for the gas phase equilibrium geometry,
Note that we could have chosen to expand around the experimental geometry. We chose to use the gas pjase geometry because quantum chemistry methods are much better validated under these conditions.
Change in the harmonic frequency
To evaluate the change in the harmonic frequency we need to estimate the force constant matrix at the crystal geometry when external fields are applied, i.e. we need \( E\sub{I\alpha J\beta}({\color{red}\B{R}^c},{\color{red}\B{F}^c})\) which may be evaluated by the following expansion
Note that the actual force constants involve coupling terms between the atom coordinates of the embedded molecule and those in the environment. Therefore the above formula is only a best firstorder approximation for treating the molecule as a separate entity in the crystal. Intuitively one might expect that the lower frequency modes to be most affected by these separability considerations.
Dipole derivatives from local fields
The molecular dipole moment derivatives are fundamental molecular properties related to the intensities for fundamental vibrational transitions. The following argument shows that they are related directly to the local crystal fields.
By definition a molecule in the crystal is at a stationary point i.e. there are no forces on its nuclei,
Expanding the left hand side in a Taylor series around the crystal geometry at zero field strength gives,
These are \( 3 N\sub{n} \) equations in the \( 9 N\sub{n} \) unknown molecular dipole moment derivatives \( \mu\sub{\delta, I\alpha}(\B{R}^c,\B{0}) \). (Note that the dipole derivatives here pertain to the experimental crystal geometry not to the gas phase). Therefore it is not possible to determine these quantities from a single experiment unless some of the unknown coefficients are known to be zero by symmetry — for example the molecule is linear or uncharged.
… Reduction in the number of unknowns
Actually, there are only \( 9 N\sub{n}  6 \) unknown molecular dipole moment derivatives (or \( 9 N\sub{n}  5 \) for linear molecules) as the following argument shows.
For an uncharged molecule the dipole moment does not depend on the coordinate system i.e. \( \B{\mu}(\B{R},\B{F}) = \B{\mu}(\B{R}+\B{\Delta},\B{F}) \) where \( \B{\Delta} \) is a constant shift applied to all nuclear coordinates. From this we can deduce the relation
which provide three extra conditions on the dipole moment derivatives. A further three conditions could also be obtained (two for linear molecules) if the dipole moment of the molecule is known.
These conditions should not be used if the molecule has acquired a charge.
In order to determine the dipole moment derivatives some additional models are required to reduce the number of unknowns.
… Local charge model for dipole derivatives
Suppose that the dipole moment in the molecule at different geometries is well modelled by fixed atomic charges \( q_I \) at each atomic site \( I \),
This is not a good model because it does not account for the change in the charges with change in geometry. Such charges are also called Born effective charges. Nevertheless, it is straightforward to show that the atomic charges in this may be approximated at the trace of of the dipole moment derivatives, which in this case are better known as the atomic polar tensors,
The important thing to note is that this model reduces the number of unknown dipole derivatives to just \( N_n \) effective charges \( q\sub{I} \). With the translational and rotational conditions on the dipole derivatives, we now have an overdetermined systems of equations which can be solved by least squares. Without the translation and rotational constraints these equations read:
… Are the local fields reasonable?
This can easily be tested by evaluating how different the predicted and actual forces are at the nuclear centers,
Acknowledgement
These notes grew directly out of a conversation with Sajesh Thomas and Marcus Kettner.
Some of the ideas go back to a memorable conversation with Kersti Hermansson at the third European Charge Density Meeting (ECDM) conference in Sandbjerg Estate, where apart from this we discussed the islands around Stockholm in the summer.
Of course, all of this is based on the ideas of A. David Buckingham, the master of molecular properties.