The orthonormal constraints in the equations of motion are vital to give the correct electronic states in the molecular dynamics method. If the equations are solved in the absence of the orthonormal constraints the time evolution of the one electron wavefunctions are found to be either oscillatory or convergent to a single degenerate ground state, depending on initial conditions. The initial wavefunctions will only converge to different eigenstates if orthogonality is imposed.
Equation (3.46) ensures that the remain orthogonal at all instants in time. However, to ensure this, the Lagrange multipliers must vary continuously with time and so the implementation of these equations require the evaluation of at infinitely small time separations. To make the calculation possible, in practice, are held constant throughout each timestep of integration. This leads to non-orthogonality of the wavefunctions at the end of each timestep which then requires a separate orthogonalisation step. This is done by Gram-Schmit orthogonalisation
where the orthonormalised set is generated from the linearly independent set obtained from integration of the equations of motion.
As a result of this, the constraints of orthogonality are imposed each time the electronic equations of motion are integrated. can the be approximated by the expectation values of the eigenstates, giving the equation of motion of the form
The wavefunctions are then an exact eigenstate when their accelerations are zero which are found self-consistently.