1. 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

2. 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 )

3. vdW density functional Dion, Rydberg, Schröder, Langreth, and Lundqvist, PRL 92, 246401 (2004)

4. Non-local correlation kernel D=(q 1 +q 2 )r 12 /2  =(q 1 -q 2 )r 12 /2 General-purpose, ‘seamless’ functional

5. 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)

6. Results for adsorption S.D. Chakarova-Käck et al, PRL 96, 146107 (2006) Benzene/GrapheneNaftalene/Graphene Experiments

7. Results for solids PolyethyleneSilicon Reasonable results for molecular systems Keeps GGA accuracy for covalent systems  General purpose functional

8. 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

9. Factoring  (q 1,q 2,r 12 )

10. 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

11. Factoring by interpolation

12. Functional derivative

13. 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

14. Algorithm accuracy Ar 2 GGA vdW Lines: Dion et al Circles: our results

15. 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%