Ab initio molecular dynamics: Theory and Implementation

changes in the “effective basis set” as a result of variations of the supercell), whichis related to the Pulay stress 219,660 .Finally, the strength of the smoothing A > 0 should be kept as modest as possiblesince the modification Eq. (284) of the kinetic energy leads to an increase of thehighest frequency in the electronic power spectrum ∝ A. This implies a decreaseof the permissible molecular dynamics time step ∆t max according to Eq. (55). Itis found that a suitably tuned set of the additional parameters (A, σ, Ecut) eff leadsto an efficiently converging constant–pressure scheme in conjunction with a fairlysmall number of plane waves 202,55 . Note that the cutoff was kept strictly constantin applications of the Born–Oppenheimer implementation 679 of variable–cellmolecular dynamics 681,682 , but the smoothing scheme presented here could beimplemented in this case as well. An efficient method to correct for the discontinuitiesof static total energy calculations performed at constant cutoff was proposedin Ref. 211 . Evidently, the best way to deal with the incomplete–basis–set problemis to increase the cutoff such that the resulting artifacts become negligible on thephysically relevant energy scale.4.3 Beyond Ground States4.3.1 IntroductionExtending ab initio molecular dynamics to a single non–interacting excited state isstraightforward in the framework of wavefunction–based methods such as Hartree–Fock 365,254,191,379,281,284,316,293 , generalized valence bond (GVB) 282,283,228,229,230 ,complete active space SCF (CASSCF) 566,567 , or full configuration interaction(FCI) 372 approaches, see Sect. 2.7. However, these methods are computationallyquite demanding – given present–day algorithms and hardware. Promising stepsin the direction of including several excited states and non–adiabatic couplings arealso made 385,386,387,71 .Density functional theory offers an alternative route to approximately solvingelectronic structure problems and recent approaches to excited–state propertieswithin this framework look promising. In the following, two limiting and thuscertainly idealistic situations will be considered, which are characterized by either• many closely–spaced excited electronic states with a broad thermal distributionof fractional occupation numbers, or by• a single electronic state that is completely decoupled from all other states.The first situation is encountered for metallic systems with collective excitations orfor materials at high temperatures compared to the Fermi temperature. It is notedin passing that associating fractional occupation numbers to one–particle orbitals isalso one route to go beyond a single–determinant ansatz for constructing the chargedensity 458,168 . The second case applies for instance to large–gap molecular systemswhich complete a chemical reaction in a single excited state as a result of e.g. avertical homo / lumo or instantaneous one–particle / one–hole photoexcitation.100