Ab initio molecular dynamics: Theory and Implementation

in order to compute first the diagonal elements ρ ii (∆τ) of the “high–temperature”Boltzmann operator ρ(∆τ); here ∆τ = β/P and P is the Trotter “time slice” asintroduced in paragraph Ab Initio Path Integrals: Statics. To this end, the kineticand potential energies can be conveniently evaluated in reciprocal and real space, respectively,by using the split–operator / FFT technique 183 . The Kohn–Sham eigenvaluesɛ i are finally obtained from the density matrix via ɛ i = −(1/∆τ) lnρ ii (∆τ).They are used in order to compute the occupation numbers {f i }, the density n(r),the band–structure energy Ω KS , and thus the free energy Eq. (295).In practice a diagonalization / density–mixing scheme is employed in order tocompute the self–consistent density n(r). Grossly speaking a suitably constructedtrial input density n in (see e.g. the Appendix of Ref. 571 for such a method) is usedin order to compute the potential V KS [n in ]. Then the lowest–order approximantto the Boltzmann operator Eq. (300) is diagonalized using an iterative Lanczos–type method. This yields an output density n out and the corresponding free energyF KS [n out ]. Finally, the densities are mixed and the former steps are iterated until astationary solution n scf of F KS [n scf ] is achieved, see Sect. 3.6.4 for some details onsuch methods. Of course the most time–consuming part of the calculation is in theiterative diagonalization. In principle this is not required, and it should be possibleto compute the output density directly from the Fermi–Dirac density matrix evenin a linear scaling scheme 243 , thus circumventing the explicit calculation of theKohn–Sham eigenstates. However, to date efforts in this direction have failed, orgiven methods which are too slow to be useful 9 .As a method, molecular dynamics with the free energy functional is most appropriateto use when the excitation gap is either small, or in cases where thegap might close during a chemical transformation. In the latter case no instabilitiesare encountered with this approach, which is not true for ground–state abinitio molecular dynamics methods. The price to pay is the quite demanding iterativecomputation of well–converged forces. Besides allowing such applicationswith physically relevant excitations this method can also be straightforwardly combinedwith k–point sampling and applied to metals at “zero” temperature. Inthis case, the electronic “temperature” is only used as a smearing parameter ofthe Fermi edge by introducing fractional occupation numbers, which is known toimprove greatly the convergence of these ground–state electronic structure calculations220,232,185,676,680,343,260,344,414,243 .Finite–temperature expressions for the exchange–correlation functional Ω xc areavailable in the literature. However, for most temperatures of interest the correctionsto the ground–state expression are small and it seems justified to use one ofthe various well–established parameterizations of the exchange–correlation energyE xc at zero temperature, see Sect. 2.7.4.3.3 A Single Excited State: S 1 –DynamicsFor large–gap systems with well separated electronic states it might be desirableto single out a particular state in order to allow the nuclei to move on the associatedexcited state potential energy surface. Approaches that rely on fractionaloccupation numbers such as ensemble density functional theories – including the104