Ab initio molecular dynamics: Theory and Implementation

utilization of such functionals in the framework of ab initio molecular dynamicsis for instance given in Ref. 588 . Those exchange–correlation functionals that willbe considered in the implementation part, Sect. 3.3, belong to the class of the“Generalized Gradient Approximation”∫E GGAxc [n] =dr n(r) ε GGAxc (n(r); ∇n(r)) , (88)where the unknown functional is approximated by an integral over a function thatdepends only on the density and its gradient at a given point in space, see Ref. 477and references therein. The combined exchange–correlation function is typicallysplit up into two additive terms ε x and ε c for exchange and correlation, respectively.In the simplest case it is the exchange and correlation energy density ε LDAxc (n) of aninteracting but homogeneous electron gas at the density given by the “local” densityn(r) at space–point r in the inhomogeneous system. This simple but astonishinglypowerful approximation 320 is the famous local density approximation LDA 338(or local spin density LSD in the spin–polarized case 40 ), and a host of differentparameterizations exist in the literature 458,168 . The self–interaction correction 475SIC as applied to LDA was critically assessed for molecules in Ref. 240 with adisappointing outcome.A significant improvement of the accuracy was achieved by introducing the gradientof the density as indicated in Eq. (88) beyond the well–known straightforwardgradient expansions. These so–called GGAs (also denoted as “gradient corrected”or “semilocal” functionals) extended the applicability of density functional calculationto the realm of chemistry, see e.g. Refs. 476,42,362,477,478,479 for a few “popularfunctionals” and Refs. 318,176,577,322 for extensive tests on molecules, complexes,and solids, respectively.Another considerable advance was the successful introduction of “hybrid functionals”43,44 that include to some extent “exact exchange” 249 in addition to astandard GGA. Although such functionals can certainly be implemented within aplane wave approach 262,128 , they are prohibitively time–consuming as outlined atthe end of Sect. 3.3. A more promising route in this respect are those functionalsthat include higher–order powers of the gradient (or the local kinetic energydensity) in the sense of a generalized gradient expansion beyond the first term.Promising results could be achieved by including Laplacian or local kinetic energyterms 493,192,194,662 , but at this stage a sound judgment concerning their “prize /performance ratio” has to await further scrutinizing tests. The “optimized potentialmethod” (OPM) or “optimized effective potentials” (OEP) are another routeto include “exact exchange” within density functional theory, see e.g. Sect. 13.6in Ref. 588 or Ref. 251 for overviews. Here, the exchange–correlation functionalExcOPM = E xc [{φ i }] depends on the individual orbitals instead of only on the densityor its derivatives.2.7.3 Hartree–Fock TheoryHartree–Fock theory is derived by invoking the variational principle in a restrictedspace of wavefunctions. The antisymmetric ground–state electronic wavefunctionis approximated by a single Slater determinant Ψ 0 = det{ψ i } which is constructed36