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Car-Parrinello Lagrangian

Until recently first principles electronic calculations where based on techniques which required the computationally expensive matrix diagonalisation methods. Car and Parrinello[46] 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 tex2html_wrap_inline5750 and the unit cell dimensions tex2html_wrap_inline5752 . 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 tex2html_wrap_inline5754 is a fictitious mass which is associated this the expansion coefficients of the Kohn-Sham electronic wavefunctions tex2html_wrap_inline5478 . 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 tex2html_wrap_inline5478 are subject to the orthonormal constraints


These constraints can be simply incorporated into the Car-Parrinello Lagrangian as follows:


where tex2html_wrap_inline5762 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 tex2html_wrap_inline5764 , it follows that the Lagrange equations of motion




where H is the Kohn-Sham Hamiltonian and the force tex2html_wrap_inline5768 is the gradient of the Kohn-Sham energy functional at the point in Hilbert space that corresponds to the wavefunction tex2html_wrap_inline5478 .

Stewart Clark
Thu Oct 31 19:32:00 GMT 1996