Ab initio molecular dynamics: Theory and Implementation

The corresponding equations of motion follow from the Euler–Lagrange equationsand yield a microcanonical ensemble in the extended phase space where theHamiltonian of the extended system is strictly conserved. In other words, theHamiltonian of the physical (sub–) system is no more (strictly) conserved, and theproduced ensemble is no more the microcanonical one. Any extended system dynamicsis constructed such that time–averages taken in that part of phase space thatis associated to the physical degrees of freedom (obtained from a partial trace overthe fictitious degrees of freedom) are physically meaningful. Of course, dynamicsand thermodynamics of the system are affected by adding fictitious degrees of freedom,the classic examples being temperature and pressure control by thermostatsand barostats, see Sect. 4.2.In the case of Car–Parrinello molecular dynamics, the basic Lagrangian forNewtonian dynamics of the nuclei is actually extended by classical fields {ψ i (r)},i.e. functions instead of coordinates, which represent the quantum wavefunction.Thus, vector products or absolute values have to be generalized to scalar productsand norms of the fields. In addition, the “positions” of these fields {ψ i } actuallyhave a physical meaning, contrary to their momenta { ˙ψ i }.2.5 What about Hellmann–Feynman Forces ?An important ingredient in all dynamics methods is the efficient calculation of theforces acting on the nuclei, see Eqs. (30), (32), and (44). The straightforwardnumerical evaluation of the derivativeF I = −∇ I 〈Ψ 0 |H e |Ψ 0 〉 (62)in terms of a finite–difference approximation of the total electronic energy is bothtoo costly and too inaccurate for dynamical simulations. What happens if the gradientsare evaluated analytically? In addition to the derivative of the Hamiltonianitself∇ I 〈Ψ 0 |H e |Ψ 0 〉 = 〈Ψ 0 |∇ I H e |Ψ 0 〉+ 〈∇ I Ψ 0 |H e |Ψ 0 〉 + 〈Ψ 0 |H e |∇ I Ψ 0 〉 (63)there are in general also contributions from variations of the wavefunction ∼ ∇ I Ψ 0 .In general means here that these contributions vanish exactlyF HFTI = − 〈Ψ 0 |∇ I H e |Ψ 0 〉 (64)if the wavefunction is an exact eigenfunction (or stationary state wavefunction) ofthe particular Hamiltonian under consideration. This is the content of the often–cited Hellmann–Feynman Theorem 295,186,368 , which is also valid for many variationalwavefunctions (e.g. the Hartree–Fock wavefunction) provided that completebasis sets are used. If this is not the case, which has to be assumed for numericalcalculations, the additional terms have to be evaluated explicitly.In order to proceed a Slater determinant Ψ 0 = det{ψ i } of one–particle orbitalsψ i , which themselves are expandedψ i = ∑ νc iν f ν (r; {R I }) (65)24