see e.g. the discussion following Eq. (35) for a similar variational procedure. Thisleads to the Kohn–Sham equations{− 1 2 ∇2 + V ext (r) + V H (r) + δE }xc[n]φ i (r) = ∑ Λ ij φ j (r) (81)δn(r)j{− 1 }2 ∇2 + V KS (r) φ i (r) = ∑ Λ ij φ j (r) (82)jHeKS φ i (r) = ∑ jΛ ij φ j (r) , (83)which are one–electron equations involving an effective one–particle HamiltonianHeKS with the local potential V KS . Note that HeKS nevertheless embodies the electronicmany–body effects by virtue of the exchange–correlation potentialδE xc [n]δn(r)= V xc (r) . (84)A unitary transformation within the space of the occupied orbitals leads to thecanonical formH KSe φ i = ɛ i φ i (85)of the Kohn–Sham equations, where {ɛ i } are the eigenvalues. In conventional staticdensity functional or “b<strong>and</strong> structure” calculations this set of equations has to besolved self–consistently in order to yield the density, the orbitals <strong>and</strong> the Kohn–Sham potential for the electronic ground state 487 . The corresponding total energyEq. (75) can be written asE KS = ∑ iɛ i − 1 2∫∫dr V H (r) n(r) + E xc [n] −dr δE xc[n]δn(r)n(r) , (86)where the sum over Kohn–Sham eigenvalues is the so–called “b<strong>and</strong>–structure energy”.Thus, Eqs. (81)–(83) together with Eqs. (39)–(40) define Born–Oppenheimer<strong>molecular</strong> <strong>dynamics</strong> within Kohn–Sham density functional theory, see e.g.Refs. 232,616,594,35,679,472,36,343,344 for such implementations. The functional derivativeof the Kohn–Sham functional with respect to the orbitals, the Kohn–Shamforce acting on the orbitals, can be expressed asδE KSδφ ⋆ i= f i H KSe φ i , (87)which makes clear the connection to Car–Parrinello <strong>molecular</strong> <strong>dynamics</strong>, seeEq. (45). Thus, Eqs. (59)–(60) have to be solved with the effective one–particleHamiltonian in the Kohn–Sham formulation Eqs. (81)–(83). In the case of Ehrenfest<strong>dynamics</strong> presented in Sect. 2.2, which will not be discussed in further detailat this stage, the Runge–Gross time–dependent generalization of density functionaltheory 258 has to be invoked instead, see e.g. Refs. 203,617,532 .Crucial to any application of density functional theory is the approximation ofthe unknown exchange <strong>and</strong> correlation functional. A discussion focussed on the35
utilization of such functionals in the framework of ab <strong>initio</strong> <strong>molecular</strong> <strong>dynamics</strong>is 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 <strong>and</strong> its gradient at a given point in space, see Ref. 477<strong>and</strong> references therein. The combined exchange–correlation function is typicallysplit up into two additive terms ε x <strong>and</strong> ε c for exchange <strong>and</strong> correlation, respectively.In the simplest case it is the exchange <strong>and</strong> 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 ), <strong>and</strong> 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” <strong>and</strong> Refs. 318,176,577,322 for extensive tests on molecules, complexes,<strong>and</strong> 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 ast<strong>and</strong>ard 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 <strong>Theory</strong>Hartree–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
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over processors. All processors sho
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ab = 2 * sdot(2 * N p D ,A(1),1,B(1
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ENDCALL ParallelFFT3D("INV",scr1,sc
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are the improved load-balancing for
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situations where• it is necessary
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espectively), but completely analog
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4.2.3 Imposing Pressure: BarostatsK
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in the previous section. The isobar
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4.3.2 Many Excited States: Free Ene
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down the generalization of the Helm
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free energy functional discussed in
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Figure 16. Four patterns of spin de
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and electrons r = {r i } can be wri
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The effective classical partition f
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up e.g. in Refs. 132,37,596,597,428
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The eigenvalues of A when multiplie
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frequency of the electronic degrees
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5.2 Solids, Polymers, and Materials
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the penetration of the oxidation la
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in terms of their electronic struct
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culations on very accurate global p
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AcknowledgmentsOur knowledge on ab
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57. M. Bernasconi, G. L. Chiarotti,
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Superiore di Studi Avanzati (SISSA)
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175. E. Ermakova, J. Solca, H. Hube
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244. H. Goldstein, Klassische Mecha
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313. T. Ikeda, M. Sprik, K. Terakur
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384. N. A. Marks, D. R. McKenzie, B
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442. S. Nosé and M. L. Klein, Mol.
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502. L. M. Ramaniah, M. Bernasconi,
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562. F. Shimojo, K. Hoshino, and Y.
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620. A. Tongraar, K. R. Liedl, and
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682. R. M. Wentzcovitch, Phys. Rev.