If all atoms in a molecule are allowed to vibrate, molecular crystals could be treated in the same w ay as above. However, the equations must be adapted slightly when the rigid molecule approxim ation is used, so that only intermolecular lattice vibrations are treated.
The derivation for molecular crystals is similar to that leading to Equations (4.10) and (4.11), but the atom ic motions are restricted such that the molecular geometry is held rigid. Unlike the point m asses in the preceding theory, rigid molecular units have rotational as well as translational degrees o f freedom. To describe the molecular degrees o f freedom, we introduce internal coordinates for molecules; a molecule-fixed axis system is assigned to each molecule, w ith the origin at the centre o f mass and the axes pointing along the principal inertial axes. The 3« degrees o f freedom o f a molecule can then be separated into three translational degree o f freedom o f the molecular centre o f mass, three rotations about the inertial axes and (3« - 6) internal modes describing intram olecular motion. The choice o f coordinates to describe the translational and rotational m otion o f rigid molecules is discussed in detail by Walmsleyl'^-^l and Califano, Schettino and Neto.t^-^1 For rigid molecules, in
L attice D y n am ical S tudies o f M olecular C rystals w ith A p p licatio n to P olym orpltism and S tructure Prediction
G raem e M . D ay 2003
Chapter 4. Lattice Dynamics o f M olecular Crystals____________________________________________^ contrast to individual atoms, the dynamical matrix (4.10) refers to molecular motions, i.e. translations and librations and becomes
£>^.(k) = (M ,,,M ,.,.)'2 2 ]0 " :(y )e x p (ik -r'')
(4.13)
r
(4.14) /o
d u l { i ) d u 'X j )
where the molecular displacements, « ^ ( 0 correspond to translations for r = 1 , 2 , 3 and librations for r = 4, 5, 6. The matrix is weighted by the masses and mom ents o f inertia o f the rigid molecules according to
and the eigenvalue problem becomes
= 0 . (4.16)
JJy
Equations (4.11) to (4.16) allow the evaluation o f the modes o f vibration o f the crystal. A unit cell containing m rigid molecules has 6m vibrational modes at each value o f k. The internal vibrations o f the m olecules are separated from the translations and librations. We make this rigid body approxim ation now and assess its applicability in the following chapters.
Because o f the long-range nature o f Coulombic interactions in crystals, evaluation o f (k • r ) in equation (4,13) can be the most problematic part o f calculating phonon frequencies. Because o f their conditional convergence, the electrostatic sums up to dipole-dipole are treated by the Ewald sum m ation techniquet'^-^1 in our modelling software, DMAREL.t'^-^] However, the Ewald sum m ust be adapted to account for the e x p (/k -r)in (4.10). The procedure for this summation has been
described by Bom and H u a n g , U but the implementation and testing o f these calculations for m ultipolar electrostatics has, unfortunately, been beyond the scope o f this thesis. However, the exponential term disappears at k = 0, so the zone-centre frequencies are, computationally, the easiest to evaluate. At k = 0, the acoustic modes vanish and there are (6m - 3) modes o f vibration. These calculations have been implemented in DM AREL and are evaluated and discussed for a range o f m olecular crystals in Chapter 6.
Lattice D y n am ical Studies o f M olecular C rystals G raem e M . D ay w ith A p p licatio n to Polym orphism and Structure P rediction 2003
Chapter 4. Lattice Dynamics o f M olecular Crystals____________________________________________^
4.2.3
Anharmonicity and the quasi-harmonic approximation
M any important effects cannot be explained in the harmonic approximation because the lattice structure and phonon frequencies are invariant with temperature. Anharmonic corrections can be m ade by including higher order terms in the Taylor series expansion o f the lattice energy (4.1) but their com putation quickly becomes very complex. Many approximate methods for introducing the anharm onic corrections have been suggested and these have been summarised in great detail.[^-^>^î The tem perature dependence o f phonon frequencies in a perfect crystal originates from two effects - m ulti-phonon processes, which are described by the higher terms in the Taylor series, and the indirect effect o f therm al expansion o f the crystal lattice. A crystal normally expands with increasing tem perature, so the intermolecular contacts are expanded and the effective force constants weaken.
We have not been able to study the effects o f anharmonicity in detail and have only used one o f the sim plest approaches to include some o f its effects, the quasi-harmonic (QH) approxim ation (see, e.g. refs [4.8,9] for a more detailed discussion). The approximation simply ignores interactions between phonons, but allows the phonon frequencies to change with temperature. Therm al expansion can be modelled in the quasi-harmonic approximation by varying the lattice param eters to minimise the quasi-harm onic free energy, <E> + Fvib (the expressions for Fyjb are derived in the follow ing section). The lattice expands with temperature because o f the balance between 0 and Fvib- Expansion o f the lattice moves the structure away from the minimum in lattice energy, but the normal m ode vibrations now have different equilibrium distances and, so different force constants. Hence, expansion lowers the frequencies, which then contribute more to the entropy. In Chapter 6, we use the quasi-harm onic approxim ation to compare calculated phonon frequencies to measured spectra at different tem peratures, but, instead o f calculating the equilibrium structure by minim ising the free energy, we take the tem perature dependence o f the lattice constants from experiment.