Extended Lagrange techniques Car-Parrinello methods

We now add contributions corresponding to treating the orbital expansion coefficients as variables with fictive masses Pi. 

The two potential energies can be combined to a single term depending on both the nuclear positions and the orbital coefficients. 

The orbital orthogonality constraints can be included by addition of terms involving Lagrange multipliers. 

The constraint forces are handled iteratively, analogously to the constraint of fixed bond lengths in the SHAKE algorithm. 

The CP method may be considered as a semi-classical dynamics approach where the electrons are treated quantum mechanically while the nuclear motion is treated classically. The latter implies that for example zero point vibrational effects are not included, nor can nuclear tunnelling effects be described this requires fully quantum methods, as described in the next section. 

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