9.1 Introduction 265becomesνvibr =13.64× 1013 Hz, to be compared with the experimental value
νvibr=12.48× 1013 Hz[1].^1
The harmonic approximation works well for low energies. It is used for stretch,
bend and torsional energies; see also Section 8.6.1. For higher energies, anharmonic
terms can be included in the potentials – see the Morse potential in the hydrogen
example. For energies much higher than the spacing between the energy levels,
quantum effects do not affect physical properties and a fully classical description
is appropriate.
For interacting molecules the force field procedure becomes unfeasible because
we would have to calculate energies and variations in energies for all possible rel-
ative positions and orientations for pairs or sets of two and more molecules, which
becomes exceedingly tedious and (computer) time-consuming. Therefore, in these
systems, the intramolecular interactions are modelled by force fields and the inter-
molecular interactions by atomic pair-interactions as we have seen throughout the
previous chapter. Although this approach yields rather accurate results, in particu-
lar when the density is not too high, the use of these pair-potentials is not justified
for dense systems. Moreover, the energy differences between different atomic con-
formations are often very small, so that high accuracy is needed for predicting
equilibrium structures.
To achieve accuracy in these cases, it is necessary to calculate forces and energies
from quantum electronic structure calculations; if this is unfeasible for all pos-
sible configurations, we take the more economical approach of calculating forces
and energies only for those configurations which actually occur in the simulation.
We must therefore perform an electronic structure calculation at every molecular
dynamics time step, and derive theforceon the nuclei from that calculation. The
word force is emphasised because the methods described in the first few chapters of
this book aimed at calculating the energies and not the forces. Of course, it would
be possible to derive the forces from the energies by studying the variations in the
latter with nuclear positions but that would require an exceedingly large number of
energy calculations. It is better therefore to try to find methods for calculating the
forces directly.
The energy of a system of electrons in its ground stateψGfor a fixed configur-
ation of nucleiS=(R 1 ,...,RM), whereRnis the position of thenth nucleus, is
given by^2
E=
〈ψG|H(S)|ψG〉
〈ψG|ψG〉. (9.1)
(^1) In atomic units, the unit of frequency isαc/a 0 =4.13414× 1014 Hz;αis the fine structure constant.
(^2) We use the letterSin order to avoid confusion withR=(r 1 ,...,rN).