f(G) are related by three–dimensional Fourier transformsThe Fourier transforms are defined by[inv FT [f(G)]] uvw=[fw FT [f(R)]] jkl=f(R) = inv FT [f(G)] (127)f(G) = fw FT [f(R)] . (128)N∑x−1j=0N∑ y−1k=0N∑z−1l=0f G jkl[exp i 2π ] [j u exp i 2π ] [k v exp i 2π ]l wN x N y N zN∑x−1u=0N∑ y−1v=0N∑z−1w=0f R uvw[exp −i 2π ] [j u exp −i 2π ] [k v exp −i 2π ]l wN x N y N zwhere the appropriate mappings of q <strong>and</strong> g to the indices(129), (130)[u, v, w] = q (131){j, k, l} = g s if g s ≥ 0 (132){j, k, l} = N s + g s if g s < 0 (133)have to be used. From Eqs. (129) <strong>and</strong> (130) it can be seen, that the calculationof the three–dimensional Fourier transforms can be performed by a series of onedimensional Fourier transforms. The number of transforms in each direction isN x N y , N x N z , <strong>and</strong> N y N z respectively. Assuming that the one-dimensional transformsare performed within the fast Fourier transform framework, the number ofoperations per transform of length n is approximately 5n logn. This leads to anestimate for the number of operations for the full three-dimensional transform of5N logN, where N = N x N y N z .3.1.5 PseudopotentialsIn order to minimize the size of the plane wave basis necessary for the calculation,core electrons are replaced by pseudopotentials. The pseudopotential approximationin the realm of solid–state theory goes back to the work on orthogonalizedplane waves 298 <strong>and</strong> core state projector methods 485 . Empirical pseudopotentialswere used in plane wave calculations 294,703 but new developments have considerablyincreased efficiency <strong>and</strong> reliability of the method. Pseudopotential are requiredto correctly represent the long range interactions of the core <strong>and</strong> to producepseudo–wavefunction solutions that approach the full wavefunction outside a coreradius r c . Inside this radius the pseudopotential <strong>and</strong> the wavefunction should be assmooth as possible, in order to allow for a small plane wave cutoff. For the pseudo–wavefunction this requires that the nodal structure of the valence wavefunctionsis replaced by a smooth function. In addition it is desired that a pseudopotentialis transferable 238,197 , this means that one <strong>and</strong> the same pseudopotential can be47
used in calculations of different chemical environment resulting in calculations withcomparable accuracy.A first major step to achieve all this conflicting goals was the introduction of”norm–conservation” 622,593 . Norm–conserving pseudopotentials have to be angularmomentum dependent. In their most general form they are semi–localV PP (r,r ′ ) = ∑ lmY lm (r)V l (r)δ r,r ′Y lm (r ′ ) , (134)where Y lm are spherical harmonics. A minimal set of requirements <strong>and</strong> a constructionscheme for soft, semi–local pseudopotentials were developed 274,28 . Since thenmany variations of the original method have been proposed, concentrating eitheron an improvement in softness or in transferability. Analytic representations of thecore part of the potential 326,626,627,509 were used. Extended norm-conservation 564was introduced to enhance transferability <strong>and</strong> new concepts to increase the softnesswere presented 659,509,369 . More information on pseudopotentials <strong>and</strong> theirconstruction can be found in recent review articles 487,578,221 .Originally generated in a semi-local form, most applications use the fully separableform. Pseudopotentials can be transformed to the separable form using atomicwavefunctions 335,73,659 . Recently 239,288 a new type of separable, norm-conservingpseudopotentials was introduced. Local <strong>and</strong> non-local parts of these pseudopotentialshave a simple analytic form <strong>and</strong> only a few parameters are needed to characterizethe potential. These parameters are globally optimized in order to reproducemany properties of atoms <strong>and</strong> ensure a good transferability.A separable non-local pseudopotential can be put into general form (this includesall the above mentioned types)V PP (r,r ′ ) = (V core (r) + ∆V local (r)) δ r,r ′ + ∑ k,lP ⋆ k(r)h kl P l (r ′ ) . (135)The local part has been split into a core ( ∼ 1/r for r → ∞ ) <strong>and</strong> a short-rangedlocal part in order to facilitate the derivation of the final energy formula. Theactual form of the core potential will be defined later. The local potential ∆V local<strong>and</strong> the projectors P k are atom-centered functions of the formthat can be exp<strong>and</strong>ed in plane wavesϕ(r) = ϕ(|r − R I |) Y lm (θ, φ) , (136)ϕ(r) = ∑ Gϕ(G) exp[iG · r] S I (G) Y lm (˜θ, ˜φ) , (137)R I denote atomic positions <strong>and</strong> the so–called structure factors S I are defined asS I (G) = exp[−iG · R I ] . (138)The functions ϕ(G) are calculated from ϕ(r) by a Bessel transform∫ ∞ϕ(G) = 4π (−i) l dr r 2 ϕ(r) j l (Gr) , (139)048
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- Page 44 and 45: disposable parameters that can be o
- Page 46 and 47: The index i runs over all states an
- Page 50 and 51: where j l are spherical Bessel func
- Page 52 and 53: andE self = ∑ I1√2πZ 2 IR c I.
- Page 54 and 55: ¢££¤¤¢¢¢n tot (G)inv FTn to
- Page 56 and 57: correlation energyΩ ∑E xc = ε x
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- Page 62 and 63: The local part of the pseudopotenti
- Page 64 and 65: ¢¢¢¢¢i = 1 . . .N b¢c i (G)¢
- Page 66 and 67: ¢¢¢¢¢c i (G)123g, E self∆V I
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- Page 88 and 89: ab = 2 * sdot(2 * N p D ,A(1),1,B(1
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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.