Ab initio molecular dynamics: Theory and Implementation

3.7 Molecular DynamicsNumerical methods to integrate the equations of motion are an important partof every molecular dynamics program. Therefore an extended literature exists onintegration techniques (see Ref. 217 and references in there). All considerationsvalid for the integration of equations of motion with classical potentials also applyfor ab initio molecular dynamics if the Born–Oppenheimer dynamics approach isused. These basic techniques will not be discussed here.A good initial guess for the Kohn–Sham optimization procedure is a crucialingredient for good performance of the Born–Oppenheimer dynamics approach. Anextrapolation scheme was devised 24 that makes use of the optimized wavefunctionsfrom previous time steps. This procedure has a strong connection to the basic ideaof the Car–Parrinello method, but is not essential to the method.The remainder of this section discusses the integration of the Car–Parrinelloequations in their simplest form and explains the solution to the constraints equationfor general geometric constraints. Finally, a special form of the equations ofmotion will be used for optimization purposes.3.7.1 Car–Parrinello EquationsThe Car–Parrinello Lagrangian and its derived equations of motions were introducedin Sect. 2.4. Here Eqs. (41), (44), and (45) are specialized to the case ofa plane wave basis within Kohn–Sham density functional theory. Specifically thefunctions φ i are replaced by the expansion coefficients c i (G) and the orthonormalityconstraint only depends on the wavefunctions, not the nuclear positions. Theequations of motion for the Car–Parrinello method are derived from this specificextended LagrangianL = µ ∑ i∑|ċ i (G)| 2 + 1 ∑M I Ṙ 2 I2− E KS [{G}, {R I }]GI+ ∑ ( )∑Λ ij c ⋆ i (G)c j(G) − δ ij , (228)ij Gwhere µ is the electron mass, and M I are the masses of the nuclei. Because ofthe expansion of the Kohn–Sham orbitals in plane waves, the orthonormality constraintdoes not depend on the nuclear positions. For basis sets that depend onthe atomic positions (e.g. atomic orbital basis sets) or methods that introduce anatomic position dependent metric (ultra–soft pseudopotentials 661,351 , PAW 143,347 ,the integration methods have to be adapted (see also Sect. 2.5). Solutions that includethese cases can be found in the literature 280,351,143,310 . The Euler–Lagrangeequations derived from Eq.( 228) areµ¨c i (G) = −∂E∂c ⋆ i (G) + ∑ jΛ ij c j (G) (229)M I ¨R I = − ∂E∂R I. (230)70