288 Quantum molecular dynamics
The resulting states|ψnJ〉are not yet orthogonal; an orthogonalisation of the states
must be performed after each of the steps described above.
9.4.4 Large systemsPeriodic or other type of boundary conditions are not imposed a priori in QMD
simulations. Boundary conditions may, however, be introduced by the basis set,
which might consist of periodic functions, or functions vanishing at some bound-
ary. There is an increasing interest in nonperiodic systems containing large numbers
of atoms; these systems are said to bemesoscopic. Examples of mesoscopic sys-
tems and phenomena are the scanning tunnelling microscope tip and surface, grain
boundaries, quantum dots and wires, biological macromolecules, etc. Often these
systems are periodically continued; in the case of grain boundaries, for example,
we might consider a system cell with a linear size corresponding to 10 atoms, and
containing a grain boundary. Imposing periodic boundary conditions means that
we are considering a system containing a periodic array of such grain boundaries.
If the system cell contains large numbers of atoms, it becomes very important to
use a method for solving electronic structure and dynamics for which the computer
time scales favourably with time. In this subsection we analyse the scaling behaviour
for a few methods. Two parameters are important: first, the numberNatof atoms
in the system, and second, the sizeNBof the basis set. Of course, these numbers
are not independent: they are usually proportional to each other. However, as the
number of basis states exceeds the number of atoms sometimes by a factor of 100
or more, it is important to distinguish between the two in the time scaling.
It depends strongly on the type of basis functions used how the time required by
a quantum molecular dynamics simulation scales with the number of atoms. In the
case of plane wave basis sets, FFT techniques (see Appendix A9) can be used to
increase the efficiency of the calculations. The kinetic energy is diagonal in a plane
wave basis set; for evaluating the potential energy the FFT transforms the states
into real space where the potential energy is diagonal. The plane wave basis leads
to a time scaling (complexity) for one integration step in the quantum molecular
dynamics method ofNatNBlnNB.
Other methods use localised orbitals as basis functions so that the Hamiltonian
couples only orbitals on neighbouring atoms – hence the Hamiltonian becomes
sparse, and sparse matrix methods can be very efficient, giving essentially a scaling
behaviour ofNatNB. Obviously, this can be used in the calculation of the elec-
tronic structure using the conventional self-consistent approach where the Hamilton
matrix has to be diagonalised several times. Using localised basis functions, the
recursion methodof Haydock [ 25 – 27 ] is particularly useful as it allows for cal-
culating the density in a number of steps independent ofNat, and as long as the