<strong>and</strong> a matrix form of the Gram–Schmidt procedureX ji = (G T ) −1ji , (226)where S is the overlap matrix <strong>and</strong> G is its Cholesky decompositionS = GG T . (227)Recently new methods that avoid the orthogonalization step have been introduced.One of them 483 relies on modified functionals that can be optimized withoutthe orthogonality constraint. These functionals, originally introduced in the contextof linear scaling methods 417,452 , have the property that their minima coincidewith the original Kohn–Sham energy functional. The methods described above canbe used to optimize the new functional.Another approach 309 is to use a variable transformation from the expansioncoefficients of the orbitals in plane waves to a set of non–redundant orbital rotationangles. This method was introduced in quantum chemistry 618,149,167 <strong>and</strong> is usedsuccessfully in many optimization problems that involve a set of orthogonal orbitals.A generalization of the orbital rotation scheme allowed the application also for caseswhere the number of basis functions is orders of magnitudes bigger than the numberof occupied orbitals. However, no advantage is gained over the st<strong>and</strong>ard methods,as the calculation of the gradient in the transformed variables scales the same as theorthogonalization step. In addition, there is no simple <strong>and</strong> efficient preconditioneravailable for the orbital rotation coordinates.3.6.4 Fix-Point MethodsOriginally all methods to find solutions to the Kohn–Sham equations were usingmatrix diagonalization methods. It became quickly clear that direct schemes canonly be used for very small systems. The storage requirements of the Kohn–Shammatrix in the plane wave basis <strong>and</strong> the scaling proportional to the cube of the basisset size lead to unsurmountable problems. Iterative diagonalization schemes can beadapted to the special needs of a plane wave basis <strong>and</strong> when combined with a properpreconditioner lead to algorithms that are comparable to the direct methods, bothin memory requirements <strong>and</strong> over all scaling properties. Iterative diagonalizationschemes are abundant. Methods based on the Lanczos algorithm 357,151,489 canbe used as well as conjugate gradient techniques 616,97 . Very good results havebeen achieved by the combination of the DIIS method with the minimization ofthe norm of the residual vector 698,344 . The diagonalization methods have to becombined with an optimization method for the charge density. Methods based onmixing 153,4 , quasi-Newton algorithms 92,77,319 , <strong>and</strong> DIIS 495,344,345 are successfullyused. Also these methods use a preconditioning scheme. It was shown that the optimalpreconditioning for charge density mixing is connected to the charge dielectricmatrix 153,4,299,658,48 . For a plane wave basis, the charge dielectric matrix can beapproximated by expressions very close to the ones used for the preconditioning inthe direct optimization methods.Fix-point methods have a slightly larger prefactor than most of the direct methods.Their advantage lies in the robustness <strong>and</strong> capability of treating systems withno or small b<strong>and</strong> gaps.69
3.7 Molecular DynamicsNumerical methods to integrate the equations of motion are an important partof every <strong>molecular</strong> <strong>dynamics</strong> program. Therefore an extended literature exists onintegration techniques (see Ref. 217 <strong>and</strong> references in there). All considerationsvalid for the integration of equations of motion with classical potentials also applyfor ab <strong>initio</strong> <strong>molecular</strong> <strong>dynamics</strong> if the Born–Oppenheimer <strong>dynamics</strong> approach isused. These basic techniques will not be discussed here.A good initial guess for the Kohn–Sham optimization procedure is a crucialingredient for good performance of the Born–Oppenheimer <strong>dynamics</strong> approach. Anextrapolation scheme was devised 24 that makes use of the optimized wavefunctionsfrom previous time steps. This procedure has a strong connection to the basic ideaof the Car–Parrinello method, but is not essential to the method.The remainder of this section discusses the integration of the Car–Parrinelloequations in their simplest form <strong>and</strong> explains the solution to the constraints equationfor general geometric constraints. Finally, a special form of the equations ofmotion will be used for optimization purposes.3.7.1 Car–Parrinello EquationsThe Car–Parrinello Lagrangian <strong>and</strong> its derived equations of motions were introducedin Sect. 2.4. Here Eqs. (41), (44), <strong>and</strong> (45) are specialized to the case ofa plane wave basis within Kohn–Sham density functional theory. Specifically thefunctions φ i are replaced by the expansion coefficients c i (G) <strong>and</strong> the orthonormalityconstraint only depends on the wavefunctions, not the nuclear positions. Theequations of motion for the Car–Parrinello method are derived from this specificextended LagrangianL = µ ∑ i∑|ċ i (G)| 2 + 1 ∑M I Ṙ 2 I2− E KS [{G}, {R I }]GI+ ∑ ( )∑Λ ij c ⋆ i (G)c j(G) − δ ij , (228)ij Gwhere µ is the electron mass, <strong>and</strong> M I are the masses of the nuclei. Because ofthe expansion of the Kohn–Sham orbitals in plane waves, the orthonormality constraintdoes not depend on the nuclear positions. For basis sets that depend onthe atomic positions (e.g. atomic orbital basis sets) or methods that introduce anatomic position dependent metric (ultra–soft pseudopotentials 661,351 , PAW 143,347 ,the integration methods have to be adapted (see also Sect. 2.5). Solutions that includethese cases can be found in the literature 280,351,143,310 . The Euler–Lagrangeequations derived from Eq.( 228) areµ¨c i (G) = −∂E∂c ⋆ i (G) + ∑ jΛ ij c j (G) (229)M I ¨R I = − ∂E∂R I. (230)70
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John von Neumann Institute for Comp
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2500Number200015001000CP PRL 1985AI
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The goal of this section is to deri
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¡the Newtonian equation of motion
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Ehrenfest molecular dynamics is cer
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the Car-Parrinello approach 108 , s
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According to the Car-Parrinello equ
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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.