is used, which represents exactly any reasonable function in the limit of using acomplete set of basis functions. In quantum chemistry, Slater–type basis functions(STOs)f S m(r) = N S m r mxxrymyrz mzexp [−ζ m |r|] (98)with an exponentially decaying radial part <strong>and</strong> Gaussian–type basis functions(GTOs)f G m(r) = N G m r mxxrymyrz mzexp [ −α m r 2] (99)have received widespread use, see e.g. Ref. 292 for a concise overview–type presentation.Here, N m , ζ m <strong>and</strong> α m are constants that are typically kept fixed duringa <strong>molecular</strong> electronic structure calculation so that only the orbital expansion coefficientsc iν need to be optimized. In addition, fixed linear combinations of theabove–given “primitive” basis functions can be used for a given angular momentumchannel m, which defines the “contracted” basis sets.The Slater or Gaussian basis functions are in general centered at the positions ofthe nuclei, i.e. r → r − R I in Eq. (98)–(99), which leads to the linear combinationof atomic orbitals (LCAO) ansatz to solve differential equations algebraically. Furthermore,their derivatives as well as the resulting matrix elements are efficientlyobtained by differentiation <strong>and</strong> integration in real–space. However, Pulay forces(see Sect. 2.5) will result for such basis functions that are fixed at atoms (or bonds)if the atoms are allowed to move, either in geometry optimization or <strong>molecular</strong><strong>dynamics</strong> schemes. This disadvantage can be circumvented by using freely floatingGaussians that are distributed in space 582 , which form an originless basis set sinceit is localized but not atom–fixed.2.8.2 Plane WavesA vastly different approach has its roots in solid–state theory. Here, the ubiquitousperiodicity of the underlying lattice produces a periodic potential <strong>and</strong> thus imposesthe same periodicity on the density (implying Bloch’s Theorem, Born–von Karmanperiodic boundary conditions etc., see e.g. Chapt. 8 in Ref. 27 ). This heavilysuggests to use plane waves as the generic basis set in order to exp<strong>and</strong> the periodicpart of the orbitals, see Sect. 3.1.2. Plane waves are defined asf PWG (r) = N exp [iGr] , (100)where the normalization is simply given by N = 1/ √ Ω; Ω is the volume of theperiodic (super–) cell. Since plane waves form a complete <strong>and</strong> orthonormal set offunctions they can be used to exp<strong>and</strong> orbitals according to Eq. (97), where thelabeling ν is simply given by the vector G in reciprocal space / G–space (includingonly those G–vectors that satisfy the particular periodic boundary conditions). Thetotal electronic energy is found to have a particularly simple form when expressedin plane waves 312 .It is important to observe that plane waves are originless functions, i.e. theydo not depend on the positions of the nuclei {R I }. This implies that the Pulayforces Eq. (67) vanish exactly even within a finite basis (<strong>and</strong> using a fixed number39
of plane waves, see the discussion related to “Pulay stress” in Sect. 2.5), whichtremendously facilitates force calculations. This also implies that plane waves area very unbiased basis set in that they are “delocalized” in space <strong>and</strong> do not “favor”certain atoms or regions over others, i.e. they can be considered as an ultimately“balanced basis set” in the language of quantum chemistry. Thus, the only wayto improve the quality of the basis is to increase the “energy cutoff” E cut , i.e. toincrease the largest |G|–vector that is included in the finite expansion Eq. (97).This blind approach is vastly different from the traditional procedures in quantumchemistry that are needed in order to produce reliable basis sets 292 . Anotherappealing feature is that derivatives in real–space are simply multiplications in G–space, <strong>and</strong> both spaces can be efficiently connected via Fast Fourier Transforms(FFTs). Thus, one can easily evaluate operators in that space in which they arediagonal, see for instance the flow charts in Fig. 6 or Fig. 7.According to the well–known “No Free Lunch Theorem” there cannot be onlyadvantages connected to using plane waves. The first point is that the pseudopotentialapproximation is intimately connected to using plane waves, why so? A planewave basis is basically a lattice–symmetry–adapted three–dimensional Fourier decompositionof the orbitals. This means that increasingly large Fourier componentsare needed in order to resolve structures in real space on decreasingly small distancescales. But already orbitals of first row atoms feature quite strong <strong>and</strong> rapid oscillationsclose to the nuclei due to the Pauli principle, which enforces a nodal structureonto the wavefunction by imposing orthogonality of the orbitals. However, mostof chemistry is ruled by the valence electrons, whereas the core electrons are essentiallyinert. In practice, this means that the innermost electrons can be takenout of explicit calculations. Instead they are represented by a smooth <strong>and</strong> nodelesseffective potential, the so–called pseudopotential 296,297,484,485,139 , see for instanceRefs. 487,578,221 for reviews in the context of “solid state theory” <strong>and</strong> Refs. 145,166 forpseudopotentials as used in “quantum chemistry”. The resulting pseudo wavefunctionis made as smooth as possible close to the nuclear core region. This also meansthat properties that depend crucially on the wavefunction close to the core cannotbe obtained straightforwardly from such calculations. In the field of plane wavecalculations the introduction of “soft” norm–conserving ab <strong>initio</strong> pseudopotentialswas a breakthrough both conceptually 274 <strong>and</strong> in practice 28 . Another importantcontribution, especially for transition metals, was the introduction of the so–calledultrasoft pseudopotentials by V<strong>and</strong>erbilt 661 . This approaches lead to the powerfultechnique of plane wave–pseudopotential electronic structure calculations in theframework of density functional theory 312,487 . Within this particular frameworkthe issue of pseudopotentials is elaborated in more detail in Sect. 3.1.5.Another severe shortcoming of plane waves is the backside of the medal of beingan unbiased basis set: there is no way to shuffle more basis functions into regions inspace where they are more needed than in other regions. This is particularly bad forsystems with strong inhomogeneities. Such examples are all–electron calculationsor the inclusion of semi–core states, a few heavy atoms in a sea of light atoms, <strong>and</strong>(semi–) finite systems such as surfaces or molecules with a large vacuum region inorder to allow the long–range Coulomb interactions to decay. This is often referredto as the multiple length scale deficiency of plane wave calculations.40
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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.