van der Waals forces in density functional theory - Laboratoire de ...
van der Waals forces in density functional theory - Laboratoire de ...
van der Waals forces in density functional theory - Laboratoire de ...
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ÁNGYÁN et al.attenuated short-range electron <strong>in</strong>teraction, the local <strong><strong>de</strong>nsity</strong>approximation to the exchange <strong>functional</strong> becomes exact.Heyd, Scuseria, and Ernzerhof applied an <strong>in</strong>verse rangeseparation<strong>in</strong> or<strong><strong>de</strong>r</strong> to get rid of the convergence problems ofthe exact exchange <strong>in</strong> solid-state calculations 30,31. TheirHSE03 <strong>functional</strong> is a generalization of the PBE0 hybrid<strong>functional</strong> 32 where the long-range portion of the exactexchange is replaced by the long-range component of thePer<strong>de</strong>w, Burke, and Ernzerhof PBE exchange <strong>functional</strong>33.In the context of the calculation of <strong>van</strong> <strong><strong>de</strong>r</strong> <strong>Waals</strong> energies,the i<strong>de</strong>a of separat<strong>in</strong>g the electron <strong>in</strong>teraction operatorto short- and long-range components has already been exploredby the work of Kohn, Meier, and Makarov, who appliedthe adiabatic connection–fluctuation-dissipation approachfor long-range electron <strong>in</strong>teractions 34, lead<strong>in</strong>g toan asymptotically correct expression of the dispersion <strong>forces</strong>.It has also been shown 35 that the artificial m<strong>in</strong>imum of therare gas dimer potential curves can be removed by an exacttreatment of the long-range exchange.The second-or<strong><strong>de</strong>r</strong> perturbational treatment of the full Coulomb<strong>in</strong>teraction has already been used by several authors forthe <strong>van</strong> <strong><strong>de</strong>r</strong> <strong>Waals</strong> problem 5,36,37, and it was shown thatthe result<strong>in</strong>g asymptotic potential has the qualitatively correct1/R 6 form. As shown very recently, quantitatively reliableasymptotic form of the potential energy curve can beexpected from adiabatic connection–fluctuation-dissipation<strong>theory</strong> calculations 38.The general theoretical framework is outl<strong>in</strong>ed <strong>in</strong> Sec. II,<strong>de</strong>scrib<strong>in</strong>g the RSH scheme and the second-or<strong><strong>de</strong>r</strong> perturbationaltreatment of long-range correlation effects. As <strong>de</strong>scribed<strong>in</strong> Sec. III, our approach has been tested on rare gasdimers. These systems are typical <strong>van</strong> <strong><strong>de</strong>r</strong> <strong>Waals</strong> complexes,where the attractive <strong>in</strong>teractions are exclusively due to Londondispersion <strong>forces</strong>. They constitute a str<strong>in</strong>gent test of themethod, s<strong>in</strong>ce the potential curves have very shallow m<strong>in</strong>imaof the or<strong><strong>de</strong>r</strong> of about 100 H.Unless otherwise stated, atomic units are assumedthroughout this work.II. THEORYA. Multi<strong>de</strong>term<strong>in</strong>antal extension of the Kohn-Sham schemeWe first recall the pr<strong>in</strong>ciple of the multi<strong>de</strong>term<strong>in</strong>antal extensionof the KS scheme based on a long-range/short-range<strong>de</strong>composition see, e.g., Ref. 24, and references there<strong>in</strong>.The start<strong>in</strong>g po<strong>in</strong>t is the <strong>de</strong>composition the Coulombelectron-electron <strong>in</strong>teraction w ee r=1/r asw ee r = w lr, ee r + w sr, ee r,1where w lr, ee r=erfr/r is a long-range <strong>in</strong>teraction andw sr, ee r is the complement short-range <strong>in</strong>teraction. This <strong>de</strong>compositionis controlled by a s<strong>in</strong>gle parameter . For =0, the long-range <strong>in</strong>teraction <strong>van</strong>ishes, w lr,=0 ee r=0, and theshort-range <strong>in</strong>teraction reduces to the Coulomb <strong>in</strong>teraction,w sr,=0 ee r=w ee r. Symmetrically, for →, the short-range<strong>in</strong>teraction <strong>van</strong>ishes, w sr,→ ee r=0, and the long-range <strong>in</strong>teractionreduces to the Coulomb <strong>in</strong>teraction, w lr,→ ee r=w ee r. Physically, 1/ represents the distance at which theseparation is ma<strong>de</strong>.The Coulombic universal <strong><strong>de</strong>nsity</strong> <strong>functional</strong> Fn=m<strong>in</strong> →n Tˆ +Ŵ ee 39, where Tˆ is the k<strong>in</strong>etic energyoperator, Ŵ ee =1/2dr 1 dr 2 w ee r 12 nˆ 2r 1 ,r 2 is the Coulombelectron-electron <strong>in</strong>teraction operator expressed withthe pair-<strong><strong>de</strong>nsity</strong> operator nˆ 2r 1 ,r 2 , is then <strong>de</strong>composed asFn = F lr, n + E sr, Hxc n,where F lr, n=m<strong>in</strong> →n Tˆ +Ŵ lr, ee is a long-range universal<strong><strong>de</strong>nsity</strong> <strong>functional</strong> associated to the <strong>in</strong>teraction operatorŴ lr, ee =1/2dr 1 dr 2 w lr, ee r 12 nˆ 2r 1 ,r 2 , and E sr, Hxc n=E sr, H n+E sr, xc n is by <strong>de</strong>f<strong>in</strong>ition the correspond<strong>in</strong>gcomplement short-range energy <strong>functional</strong>, composed by atrivial short-range Hartree contribution E sr, H n=1/2dr 1 dr 2 w sr, ee r 12 nr 1 nr 2 and an unknown shortrangeexchange-correlation contribution E sr, xc n. At=0,the long-range <strong>functional</strong> reduces to the usual KS k<strong>in</strong>eticenergy <strong>functional</strong>, F lr,=0 n=T s n, and the short-range<strong>functional</strong> to the usual Hartree-exchange-correlation <strong>functional</strong>,E sr,=0 Hxc n=E Hxc n. In the limit →, the long-range<strong>functional</strong> reduces to the Coulombic universal <strong>functional</strong>,F lr,→ n=Fn, and the short-range <strong>functional</strong> <strong>van</strong>ishes,E sr,→ Hxc n=0.The exact ground-state energy of a N-electron system <strong>in</strong>an external nuclei-electron potential v ne r, E=m<strong>in</strong> n→N Fn+drv ne rnr where the search is over allN-representable <strong>de</strong>nsities, can be re-expressed us<strong>in</strong>g thelong-range/short-range <strong>de</strong>composition of Fn:sr, n + drv ne rnrE = m<strong>in</strong>n→NF lr, n + E HxcPHYSICAL REVIEW A 72, 012510 2005= m<strong>in</strong> + Ŵ ee→NTˆ lr, + drv ne rn r + E sr, Hxc n ,where the last search is carried out over all N-electron normalizedmulti<strong>de</strong>term<strong>in</strong>antal wave functions . InEq.3,n r is the <strong><strong>de</strong>nsity</strong> com<strong>in</strong>g from the wave function , i.e.,n r=nˆr, where nˆr is the <strong><strong>de</strong>nsity</strong> operator.The m<strong>in</strong>imiz<strong>in</strong>g wave function <strong>in</strong> Eq. 3 is given bythe correspond<strong>in</strong>g Euler-Lagrange equationTˆ + Ŵ lr, ee + Vˆ ne + Vˆ sr, Hxcn = E ,where Vˆ ne=drv ne rnˆr, Vˆ sr, Hxcn=drv sr, Hxc rnˆr with theshort-range Hartree-exchange-correlation potential v sr, Hxc r=E sr, Hxc n/nr, and E is the Lagrange multiplier associatedto the constra<strong>in</strong>t of the normalization of the wave function.Equation 4 <strong>de</strong>f<strong>in</strong>es a long-range <strong>in</strong>teract<strong>in</strong>g effectiveHamiltonian Ĥ =Tˆ +Ŵ lr, ee +Vˆ ne+Vˆ sr, Hxcn that must besolved iteratively for its multi<strong>de</strong>term<strong>in</strong>antal ground-statewave function , which gives, <strong>in</strong> pr<strong>in</strong>ciple, the exact physicalground-state <strong><strong>de</strong>nsity</strong> nr=n r= nˆr , <strong>in</strong><strong>de</strong>-234012510-2
ÁNGYÁN et al.PHYSICAL REVIEW A 72, 012510 2005˜ 1 =−Rˆ 0Ŵ˜ 0 − Rˆ 0Ĝ 0 ˜ 1 , A22whereĜ 0 =2drdrnˆr˜ 0 2 Fn 0 ˜ 0 nˆr.nrnrA23The f<strong>in</strong>al expression of the second-or<strong><strong>de</strong>r</strong> energy correctioncan be written as the seriesE 2 =−˜ 0 Ŵ1+Rˆ 0Ĝ 0 −1 Rˆ 0Ŵ˜ 0 =−˜ 0 ŴRˆ 0Ŵ˜ 0 + ˜ 0 ŴRˆ 0Ĝ 0 Rˆ 0Ŵ˜ 0 − ¯ . A24Further <strong>de</strong>tails and higher-or<strong><strong>de</strong>r</strong> expressions will be given <strong>in</strong>a forthcom<strong>in</strong>g publication.1 J. F. Dobson, K. McLennan, A. Rubio, J. Wang, T. Gould, H.M. Le, and B. P. D<strong>in</strong>te, Aust. J. Chem. 54, 513 2002.2 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 1964.3 W. Kohn and L. J. Sham, Phys. Rev. 140, 1133 1965.4 S. Kristyán and P. Pulay, Chem. Phys. Lett. 229, 175 1994.5 E. Engel, A. Hock, and R. M. Dreizler, Phys. Rev. A 61,032502 2000.6 G. Jansen and A. Heßelmann, Phys. Chem. Chem. Phys. 5,5010 2003.7 A. J. Misquitta, B. Jeziorski, and K. Szalewicz, Phys. Rev.Lett. 91, 033201 2003.8 J. F. Dobson and J. Wang, Phys. Rev. Lett. 82, 2123 1999.9 M. Dion, H. Rydberg, E. Schrö<strong><strong>de</strong>r</strong>, D. C. Langreth, and B. I.Lundqvist, Phys. Rev. Lett. 92, 246401 2004.10 D. C. Patton and M. R. Pe<strong><strong>de</strong>r</strong>son, Phys. Rev. A 56, R24951997.11 J. Harris, Phys. Rev. B 31, 1770 1985.12 P. Nozières and D. P<strong>in</strong>es, Phys. Rev. 111, 442 1958.13 W. Kohn and W. Hanke unpublished.14 H. Stoll and A. Sav<strong>in</strong>, <strong>in</strong> Density Functional Methods <strong>in</strong> Physics,edited by R. M. Dreizler and J. d. Provi<strong>de</strong>ncia Plenum,New York, 1985, p. 177.15 I. Panas, Chem. Phys. Lett. 245, 171 1995.16 R. D. Adamson, J. P. Dombroski, and P. M. W. Gill, Chem.Phys. Lett. 254, 329 1996.17 A. Sav<strong>in</strong> and H.-J. Flad, Int. J. Quantum Chem. 56, 3271995.18 A. Sav<strong>in</strong>, <strong>in</strong> Recent Ad<strong>van</strong>ces <strong>in</strong> Density Functional Theory,edited by D. P. Chong World Scientific, S<strong>in</strong>gapore, 1996.19 A. Sav<strong>in</strong>, <strong>in</strong> Recent Developments of Mo<strong><strong>de</strong>r</strong>n Density FunctionalTheory, edited by J. M. Sem<strong>in</strong>ario Elsevier, Amsterdam,1996, pp. 327–357.20 T. Le<strong>in</strong><strong>in</strong>ger, H. Stoll, H.-J. Werner, and A. Sav<strong>in</strong>, Chem.Phys. Lett. 275, 151 1997.21 R. Pollet, A. Sav<strong>in</strong>, T. Le<strong>in</strong><strong>in</strong>ger, and H. Stoll, J. Chem. Phys.116, 1250 2002.22 A. Sav<strong>in</strong>, F. Colonna, and R. Pollet, Int. J. Quantum Chem.93, 166 2003.23 J. K. Pe<strong><strong>de</strong>r</strong>sen and H. J. A. Jensen <strong>in</strong> press.24 J. Toulouse, F. Colonna, and A. Sav<strong>in</strong>, Phys. Rev. A 70,062505 2004.25 R. Baer and D. Neuhauser, Phys. Rev. Lett. 94, 0430022005.26 H. Iikura, T. Tsuneda, T. Yanai, and K. Hirao, J. Chem. Phys.115, 3540 2001.27 Y. Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai, and K. Hirao,J. Chem. Phys. 120, 8425 2004.28 T. Yanai, D. P. Tew, and N. C. Handy, Chem. Phys. Lett. 393,51 2004.29 P. M. W. Gill, R. Adamson, and J. A. Pople, Mol. Phys. 88,1005 1996.30 J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys.118, 8207 2003.31 J. Heyd and G. E. Scuseria, J. Chem. Phys. 121, 1187 2004.32 C. Adamo and V. Barone, J. Chem. Phys. 110, 6158 1999.33 J. P. Per<strong>de</strong>w, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,3865 1996.34 W. Kohn, Y. Meir, and D. E. Makarov, Phys. Rev. Lett. 80,4153 1998.35 M. Kamiya, T. Tsuneda, and K. Hirao, J. Chem. Phys. 117,6010 2002.36 M. Le<strong>in</strong>, J. F. Dobson, and E. K. U. Gross, J. Comput. Chem.20, 121999.37 V. F. Lotrich, R. J. Bartlett, and I. Grabowski, Chem. Phys.Lett. 405, 492005.38 F. Furche and T. <strong>van</strong> Voorhis, J. Chem. Phys. 122, 1641062005.39 M. Levy, Proc. Natl. Acad. Sci. U.S.A. 76, 6062 1979.40 J. Toulouse, A. Sav<strong>in</strong>, and H.-J. Flad, Int. J. Quantum Chem.100, 1047 2004.41 J. Toulouse, F. Colonna, and A. Sav<strong>in</strong>, J. Chem. Phys. 122,14110 2005.42 J. G. Ángyán and P. R. Surján, Phys. Rev. A 44, 2188 1991.43 J. G. Ángyán, Int. J. Quantum Chem. 47, 469 1993.44 X. Gonze, Phys. Rev. A 52, 1096 1995.45 M. Schütz, G. Hetzer, and H.-J. Werner, J. Chem. Phys. 111,5691 1999.46 G. Hetzer, M. Schütz, H. Stoll, and H.-J. Werner, J. Chem.Phys. 113, 9443 2000.47 M. Sierka, A. Hogekamp, and R. Ahlrichs, J. Chem. Phys.118, 9136 2003.48 C. Pisani, M. Busso, G. Capecchi, S. Casassa, R. Dovesi, L.Maschio, C. Zicovich-Wilson, and M. Schütz, J. Chem. Phys.122, 094113 2005.49 L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople,J. Chem. Phys. 109, 1063 1997.50 L. A. Curtiss, K. Raghavachari, P. C. Redfern, V. Rassolov,and J. A. Pople, J. Chem. Phys. 109, 7764 1998.51 I. C. Gerber and J. G. Ángyán unpublished.52 H.-J. Werner and P. J. Knowles, MOLPRO, a package of ab <strong>in</strong>itio012510-8
VAN DER WAALS FORCES IN DENSITY FUNCTIONAL…programs, Version 1.6, 2002.53 T.-H. Tang and J. P. Toennies, J. Chem. Phys. 118, 49762003.54 A. Kumar and W. J. Meath, Mol. Phys. 54, 823 1985.55 T. H. Dunn<strong>in</strong>g Jr., J. Chem. Phys. 90, 1007 1989.56 D. E. Woon and T. H. Dunn<strong>in</strong>g Jr., J. Chem. Phys. 99, 1358PHYSICAL REVIEW A 72, 012510 20051993.57 D. E. Woon and T. H. Dunn<strong>in</strong>g Jr., J. Chem. Phys. 100, 29751994.58 A. K. Wilson, D. E. Woon, K. A. Peterson, and T. H. Dunn<strong>in</strong>gJr., J. Chem. Phys. 110, 7667 1999.012510-9