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Efficient factorization of van der Waals DFT functional Guillermo Roman and Jose M. Soler Departamento de Física de la Materia Condensada Universidad Autónoma de Madrid
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Van der Waals and DFT Essential for molecular solids and liquids, biological systems, physisorption, etc (Semi)local LDA and GGA functionals cannot describe the nonlocal dispersion correlation. Usual semiempirical add-on: E xc = E LDA/GGA + E vdW E vdW = - ij C ij f(r ij ) / r ij 6 True van der Waals density functional: E xc [ (r)] = E x GGA + E c LDA + E c nl E c nl = (1/2) dr 1 dr 2 (r 1 ) (r 2 ) (q 1,q 2,r 12 )
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vdW density functional Dion, Rydberg, Schröder, Langreth, and Lundqvist, PRL 92, 246401 (2004)
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Non-local correlation kernel D=(q 1 +q 2 )r 12 /2 =(q 1 -q 2 )r 12 /2 General-purpose, ‘seamless’ functional
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Results for simple dimers Ar 2 and Kr 2 (C 6 H 6 ) 2 Binding distances 5-10% too long Binding energies 50-100% too large M. Dion et al, PRL 92, 246401 (2004)
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Results for adsorption S.D. Chakarova-Käck et al, PRL 96, 146107 (2006) Benzene/GrapheneNaftalene/Graphene Experiments
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Results for solids PolyethyleneSilicon Reasonable results for molecular systems Keeps GGA accuracy for covalent systems General purpose functional
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The double integral problem (q 1,q 2,r 12 ) decays as r 12 -6 E c nl = (1/2) d 3 r 1 d 3 r 2 (r 1 ) (r 2 ) (q 1,q 2,r 12 ) can be truncated for r 12 > r c ~ 15Å In principle O(N) calculation for systems larger than 2r c ~ 30Å But... with x ~ 0.15Å (E c =120Ry) there are ~(2 10 6 ) 2 = 4 10 12 integration points Consequently, direct evaluation of vdW functional is much more expensive than LDA/GGA
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Factoring (q 1,q 2,r 12 )
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Interpolation as an expansion f1p1f1p1 f2p2f2p2 f3p3f3p3 f4p4f4p4 x1x1 x2x2 x3x3 x4x4 = General recipe: f j = ij f(x)=p i (x) x f f1f1 f2f2 f3f3 f4f4
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Factoring by interpolation
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Functional derivative
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O( N log(N) ) algorithm do, for each grid point i find i and i find q i =q( i, i ) find i = i p (q i ) end do Fourier-transform i k do, for each reciprocal vector k find u k = (k) k end do Inverse-Fourier-transform u k u i do, for each grid point i find i, i, and q i find i, i / i, and i / i find v i end do No SIESTA-specific: Input: i on a regular grid Output: E xc, v i xc on the grid No need of supercells in solids No cutoff radius of interaction
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Algorithm accuracy Ar 2 GGA vdW Lines: Dion et al Circles: our results
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Algorithm efficiency Conclusion If you can simulate a system with LDA/GGA, you can also simulate it with vdW-DFT SystemAtoms CPU time in GGA-XC CPU time in vdW-XC vdW/GGA overhead Ar 2 20.75 s (44%)7.5 s (89%)400% MMX polymer1241.9 s (2%)10.6 s (16%)17% DWCN16811.9 s (0.6%)109 s (5.2%)4%
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