l?
vibrations remain invariant. Therefore, the heat of sublimation is
an important source of information on intermolecular energy poten-
tials. Note, however, that here again, when energy calculations of
crystal packing are compared with experimental heat of sublimation,
enthalpy corrections, for translations,rotations and the pV term
in the gas and for lattice vibrations in the solid, are required.
Such corrections may be estimated satisfactorily by classical
thermodynamicslb.
Eguilibrium Structure of Molecules and Crystals
_ The equilibrium structure of molecules, as well as that of
molecular crystals is calculated from the second term of the Taylor
expansion of Vtotalﬂr), Equation (18). Since the equilibrium
structure is the Structure for which the energy function Vtotal
obtains a minimum value, all derivatives of Vtotal with respect to
the components of E, namely the gradient of Vtotal (denoted grad
vtotal) must vaniSH at ro- The set of equations
~
avtotal(3)/ari = O 1=l,...,3n (19)
may therefore be used to solve for go. Computer programs are
available to obtain numerical solutions of Equation (19), and
various algorithms for such solutions have been studied extensively.
The general idea of all such algorithms is the same: One starts
at any non—equilibrium value 3 and calculates grad vtotal. If the
gradient does not vanish, r is changed to r+§g, where or is
calculated in such a way as to move 3 in the direction of 30. The
process is iterated until finally the minimum is reached to the
desired precision.
The choice of the appropriate minimization method is very
important in energy calculations. The simplest method of solving
Equation (19) is the "steepest descent" method. At each iteration
the vector grad Vtotal is calculated, and a “line search” is per-
formed along the direction of the gradient. That is to say that
the value of Vtotal is calculated at several points along the
direction of the gradient, and a minimum point is obtained by
interpolation between these points. The method is "stable", in
the sense that each iteration leads to a new point of lower value
of Vtotal. However it usually gets stuck by progressively slow
convergence.
Fast convergence is obtained if grad Vtotal is expanded in a
Taylor series around the (yet unknown) point of minimum r0. In
matrix notation such expansion is written as ”
grad Vt (r0) = grad Ute
otal..r talgf) + FEF)§E + "' (20)