Van der Waals density functional from multipole dispersion interactions

Van der Waals density functional from multipole dispersion interactions

Neemias Alves de Lima

Citation: The Journal of Chemical Physics 132, 014110 (2010); doi: 10.1063/1.3282265 View online: http://dx.doi.org/10.1063/1.3282265

View Table of Contents: http://aip.scitation.org/toc/jcp/132/1 Published by the American Institute of Physics

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Van der Waals density functional from multipole dispersion interactions

Neemias Alves de Limaa兲

Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte, Caixa Postal 1524, CEP 59072–970 Natal, RN, Brazil

共Received 17 August 2009; accepted 9 December 2009; published online 6 January 2010; publisher error corrected 7 January 2010兲

We present a van der Waals density functional from high order multipole dispersion interactions between pairs of atoms. Calculated C_2mⱕ16 dispersion coefficients for dimers involving alkali, alkaline-earth, and noble gas atoms show mean absolute deviations in the range of 2%–6% from highly accurate calculations. This successful test indicates that this approach can yield efficient algorithms for calculation of van der Waals forces. © 2010 American Institute of Physics. 关doi:10.1063/1.3282265兴

Although, in principle, density functional theory共DFT兲1 within the Kohn–Sham scheme2is able to provide the exact ground state energy of an electronic system, present local and semilocal density functional approximations关local spin-density approximation, generalized gradient approximation 共GGA兲, and meta-GGA兴3

are inappropriate to describe long-range electron correlation and consequently fail for disper-sion or van der Waals 共vdW兲 interactions.4This problem of DFT was emphasized by Kohn et al.,5that proposed an ap-proach in which traditional DFT methods are modified by treating long-range interactions separately in terms of the susceptibility. In the limit of large distances, they get the usual −C6R−6 form for the dipole dispersion attraction be-tween pairs of atoms.

In an earlier work6 we have proposed a DF for the asymptotic dispersion energy by replacing the electric dipole susceptibility model based on the uniform electron gas7–9by one that satisfies a sum rule and fitting exact dipole dynamic polarizability for the hydrogen atom. Calculated C₆ coeffi-cients for several dimers involving alkali, alkaline-earth, and noble gas atoms showed an unbelievably mean absolute de-viation of only 2% from results of highly accurate calcula-tions. The current paper extends the approach for all multi-pole dispersion interactions. In addition, we propound a simple vdW DF for the whole range of intermolecular sepa-rations.

When the charge distributions of the interacting atoms do not overlap appreciably, the second-order dispersion energy for two atoms A and B in their ground state can be expanded in terms of a series of inverse powers of the inter-nuclear separation R共Refs.10and11兲

E共2兲= −

兺

C_2mAB

R2m, 共1兲

where the multipole dispersion coefficients are given by12 共we use atomic units throughout兲

C_2mAB=

兺

冕

A/B_{共i␻兲 being the dynamic 2}l_{-pole electric} polariz-ability of atom A or B at the imaginary frequency i␻.

Here we generalize for all dynamic multipole polariz-ability the local approximation8

␣l A_共i ␻兲 =

冕

where␹_lA共i␻, n兲 is the dynamic electric 2l_{-pole susceptibility} at angle-averaged density n共rA兲, and rA is the distance from the nucleus of atom A. The integration is over the atom.

The dynamic multipole polarizabilities, in the limit of infinite frequency, must satisfy13

␣l A_共i

␻兲 →l具rA2l−2典

␻2 . 共4兲

The brackets represent the average over the ground-state atomic wave function. A simple input susceptibility for Eq.

共3兲 that satisfies this constraint is the Padé approximation

␹l A_{共i␻,n兲 =} ␹l A_共0,n兲 1 +

冋

册

Substituting Eqs. 共3兲and共5兲 in Eq.共2兲we obtain after inte-gration over frequencies

C_2mAB关n兴 =

兺

冕

冕

cAB_p,q关n兴 = ␹p A_共0,n兲 ␹q B_共0,n兲

冑

冑

Thus, the DF关Eq.共6兲兴 yields any C_2mABdispersion coeffi-cients provide that the static electric 2l_{-pole susceptibility}

␹l A_共0,n

A兲 is known. Inspired in our earlier work

6 we write ␹l A_共0,n A兲 = ␣l A_共0兲

冋

冉

冊

册

冕

冋

冉

冊

册

-pole polarizability for atom A, and fA S_{共n兲 = a +}共1 − a兲关1 − e−␭A n共rA兲_兴 1 + e−␭A关nH c_{−n共rA兲兴} 共9兲

sets to a = 0.054 inside a core radius where the density is very

large; nH c

= 1.000; and ␭A=共ncusp

A _{− 0.682兲}5_{/3 is a parameter} characterizing the smoothness of the step function and n_cuspA ⬅n共rA= 0兲 is the cusp density for atom A.

In performing DF calculations for C6 coefficients6 with

␯1= 1 we found that the results are sensible mainly to param-eter n_cutoffA which plays the rule of low density cutoff instead of one criterion used by Rapcewicz–Aschroft,7 Andersson

et al.8 and Dobson–Dinte.9 The simple choice of n_cutoffA as being the density at “polarizability radius”14–16

冑

冑

3

␣1

A_{共0兲兲 for all multipole susceptibilities and we set}

␯2= 0.850, ␯3= 0.827, ␯4= 0.833, ␯5= 0.870, and ␯6= 1.000 such that Eq. 共6兲 fitting the exact C2mⱕ16 dispersion coeffi-cients for hydrogen dimer.17 Finally, we get a DF for all dispersion coefficients from a parameterization based only on the exact results for hydrogen dimer. We will test this DF applying it to calculation of vdW coefficients for dimers in-volving alkali, alkaline-earth, and noble gas atoms. We ex-tract the atomic charge densities from available Roothaan– Hartree–Fock wave functions18–20 and the static multipole polarizabilities from references.21–29

TablesIandIIshow that both mean absolute deviations for C8and C10coefficients between ours and highly accurate calculations17,23–25,27–29 are only 3%. Also, we calculate the second-order C₁₂, C₁₄, and C₁₆ coefficients for hydrogen-alkali and hydrogen-alkali-hydrogen-alkali dimers and, similarly, the results 共Table III兲 are very close to those currently considered as

reference values,17,29 with mean absolute deviations of 4%, 5% and 6%, respectively.

Our accurate DF 关Eqs. 共6兲–共9兲兴 can be useful of two ways:共i兲 It can substitute the semiempirical fits of dispersion coefficients present in the dispersion corrected DFT approach30–42 and, more important, 共ii兲 it can be used to mapping the nonlocal part of the correlation energy43–47by

TABLE I. C8and C10dispersion coefficients共in atomic units兲 calculated in

this work and from accurate many-body calculations.共Refs.17,23–25, and

27–29兲, X共n兲=X⫻10n_.

Atom pairs C8 Ref. C10 Ref.

H–H 124.4 124.4 3285.7 3285.8 –Li 3150 3280 217418 223017 –Na 3887 4059 2.87共5兲 2916 –K 8073 7970 7.54共5兲 7.344共5兲 –Rb 10031 9540 9.90共5兲 9.301共5兲 –Cs 14499 1.57共6兲 Li–Li 8.14共4兲 8.34共4兲 7.16共6兲 7.35共6兲 –Na 9.52共4兲 9.88共4兲 8.84共6兲 9.16共6兲 –K 1.93共5兲 1.95共5兲 2.08共7兲 2.10共7兲 –Rb 2.35共5兲 2.34共5兲 2.64共7兲 2.61共7兲 –Cs 3.36共5兲 3.21共5兲 4.03共7兲 3.84共7兲 Na–Na 1.10共5兲 1.16共5兲 1.08共7兲 1.13共7兲 –K 2.17共5兲 2.24共5兲 2.48共7兲 2.53共7兲 –Rb 2.63共5兲 2.66共5兲 3.13共7兲 3.13共7兲 –Cs 3.72共5兲 3.62共5兲 4.72共7兲 4.55共7兲 K–K 4.15共5兲 4.20共5兲 5.37共7兲 5.37共7兲 –Rb 4.95共5兲 4.93共5兲 6.68共7兲 6.60共7兲 –Cs 6.86共5兲 6.62共5兲 9.86共7兲 9.40共7兲 Rb–Rb 5.87共5兲 5.77共5兲 8.26共7兲 7.96共7兲 –Cs 8.06共5兲 7.69共5兲 12.1共7兲 11.3共7兲 Cs–Cs 10.9共5兲 10.2共5兲 17.6共7兲 15.9共7兲 Be–Be 1.04共4兲 1.022共4兲 5.23共5兲 5.165共5兲 –Mg 2.06共4兲 2.082共4兲 1.22共6兲 1.232共6兲 –Ca 5.20共4兲 5.007共4兲 3.83共6兲 3.713共6兲 –Sr 7.03共4兲 6.645共4兲 5.62共6兲 5.354共6兲 Mg–Mg 3.99共4兲 4.164共4兲 2.73共6兲 2.817共6兲 –Ca 9.83共4兲 9.807共4兲 8.12共6兲 8.088共6兲 –Sr 1.31共5兲 1.294共5兲 1.17共7兲 1.149共7兲 Ca–Ca 2.32共5兲 2.260共5兲 2.28共7兲 2.200共7兲 –Sr 3.05共5兲 2.956共5兲 3.22共7兲 3.068共7兲 Sr–Sr 4.00共6兲 3.854共6兲 4.50共7兲 4.250共7兲

TABLE II. C8and C10dispersion coefficients共in atomic units兲 calculated in

this work and from accurate many-body calculations.共Refs.17,23–25, and

27–29兲, X共n兲=X⫻10n_.

Atom pairs C8 Ref. C10 Ref.

He–He 14.47 14.11 189.7 183.6 –Ne 36.82 36.18 557.2 545.1 –Ar 168.8 167.5 3797 3701 –Kr 274.7 280.0 7349 7257 –Xe 508.8 525.0 16823 16674 Ne–Ne 90.98 90.34 1556 1536 –Ar 392.8 390.1 9529 9335 –Kr 624.3 638.1 17772 17658 –Xe 1126 1162 39138 38978 Ar–Ar 1642 1623 50096 49063 –Kr 2570 2617 88334 88260 –Xe 4540 4669 183535 184250 Kr–Kr 3985 4187 151869 155450 –Xe 6957 7389 306308 316030 Xe–Xe 11928 12807 594677 619840

Ec nl_{关n兴 = −}

冕

冕

冋

兺

兺

册

where rAB=兩rជA− rជB兩 and ⍀A_/B is the atomic basin48,49 of the atom A/B.

If we set D2m共r兲=1, this long-range functional is just an

ad hoc generalization of the dipole-dipole dispersion

interac-tion DF, derived first by Rapcewicz–Aschroft,7 for all 共second-order兲 multipole dispersion interactions. The need for damping functions,50,51 D2m共r兲, arises from the fact that the expansion 关Eq. 共1兲兴 becomes physically unrealistic at small interatomic distances where charge overlap effects are relevant. Using the triplet H₂ as reference system, Douketis

et al.52 propose for these damping functions the expression

D_2m共r兲 =

冋

冉

冑

冊

册

Recent studies41,53,54suggest truncating the multipole

expan-sion in m = 5, the dipole-octopole, and

quadrupole-quadrupole interactions.

The great appeal of the functional关Eq.共10兲兴 is that 共i兲 it reproduces the first-principles dipole-dipole dispersion DF 共Refs.6–9兲; 共ii兲 it recovers the standard multipole expansion

关Eq. 共1兲兴, with damping correction, in the asymptotic limit

rAB⬇R ; and 共iii兲 it allows to calculate all second-order mul-tipole dispersion coefficients for atom-atom interaction with a very good accuracy.

Recently Becke and Johnson55–57 showed how to use Hirshfeld partition to get atomic polarizability to be used for dispersion calculations, these and other studies49,58–61 sug-gest that the extension of Eq.共10兲for vdW interactions be-tween molecules is straightforward if we generalize the Hir-shfeld scheme for all multipole polarizabilities. Moreover, Kong et al.62have shown that a postself-consistent-field ap-proach for vdW interactions constitutes a good approxima-tion. This finding greatly simplifies the implementation of the functional.

In summary, we have presented a DF approach to obtain all high-order multipole dispersion coefficients for two atoms separated by a large distance. From the encouraging results obtained for the dispersion coefficients, we proposed a DF for the long-range part of the correlation energy that can be extended for interactions between molecules.

The author is very grateful to Marilia Junqueira Caldas for helpful discussions. This work was supported by the Bra-zilian agencies CNPq, INEO/INCT-MCT, and FAPESP.

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