Until recently first principles electronic calculations where based on techniques which required the computationally expensive matrix diagonalisation methods. Car and Parrinello formulated a new more efficient method which can be expressed in the language of molecular dynamics. The essential step was to treat the expansion coefficients of the wavefunction as dynamical variables.
In a conventional molecular dynamics simulation a Lagrangian can be written in terms of the dynamical variables which normally are atomic positions and the unit cell dimensions . The Car-Parrinello Lagrangian can similarly be written, but also includes a term for the electronic wavefunction. Ignoring any constraints for the moment, it is
where is a fictitious mass which is associated this the expansion coefficients of the Kohn-Sham electronic wavefunctions . E is the Kohn-Sham energy functional. This is analogous to the usual form of the Lagrangian where the kinetic energy term is replaced with the fictitious dynamics of the wavefunctions and the Kohn-Sham energy functional replaces the potential energy.
The Kohn-Sham electronic orbitals are subject to the orthonormal constraints
These constraints can be simply incorporated into the Car-Parrinello Lagrangian as follows:
where are the Lagrange multipliers ensuring the wavefunctions remain orthonormal. In terms of molecular dynamics, these can be thought of as additional forces on the wavefunctions which maintain orthonormality throughout the calculation.
From , it follows that the Lagrange equations of motion
where H is the Kohn-Sham Hamiltonian and the force is the gradient of the Kohn-Sham energy functional at the point in Hilbert space that corresponds to the wavefunction .