Long-range van der Waals attraction and alkali-metal lattice constants

Jianmin Tao,1 John P. Perdew,2 and Adrienn Ruzsinszky2

1Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA2Department of Physics and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118, USA

�Received 5 May 2010; published 11 June 2010�

While the short-range part of the van der Waals �vdW� interaction can be described by semilocal densityfunctionals, the long range cannot. By respecting two uniform electron gas and other exact limits, we constructa nonlocal density functional for the long-range coefficient C6. C8 and C10 may be determined empirically fromC6. Then we estimate the effect of the core-core vdW attraction upon the lattice constants of the alkali metals,including dynamic valence-electron screening. This attraction is important for the softest metals, shrinking thelattice constant of Cs by 0.1 Å.

DOI: 10.1103/PhysRevB.81.233102 PACS number�s�: 71.15.Mb, 31.15.E�, 71.45.Gm

A central task of electronic structure theory is to predictproperties of materials and ultimately to control and optimizedevice functionalities. For example, successful prediction oflattice constants is crucial in the design and fabrication ofelectronic devices. The most efficient way to calculate theseproperties is density-functional theory �DFT�,1,2 in which allcomponents are treated exactly except for the exchange-correlation energy which must be approximated as a func-tional of the electron density. With the rapid development ofreliable semilocal density-functional approximations, DFThas become a standard electronic-structure method. How-ever, despite its practical success for normally bonded sys-tems, its accuracy is limited3 for weakly bound molecules�e.g., rare-gas dimers� and solids �e.g., graphite�, due to theomission of the long-range van der Waals �vdW� interaction,which cannot be included in semilocal functionals. Since thebulk moduli of K, Rb, and Cs are as small as those of therare-gas solids Ar and Kr, Ref. 4 recently proposed that thesesimple metals are also “soft matter.”

The vdW interaction is an important correlation affectingthe properties of molecules, liquids, and solids.5 Numericalstudy shows6 that, while semilocal density functionals candescribe the short-range part of the vdW interaction well,they fail to describe the long-range part in inhomogeneousdensities. This inadequacy has a direct impact on the perfor-mance of these functionals in ground-state electronic-structure calculations. To date, many density functionals7–14

have been proposed to simulate this interaction. However, nomethod has been proposed to estimate the effect of this long-range vdW interaction on the lattice constants of the alkalimetals; most methods, including Ref. 15, are not intended formetals.

In the simulation of the long-range vdW interaction, non-local density functionals are usually developed to achievegood accuracy for molecules10,11,13 or for systems of slowlyvarying densities.7,8 The former approach makes use of mo-lecular information while the latter starts with the second-order perturbative treatment of the interaction between twofragments of slowly varying density. There are two uniform-gas long-range vdW interactions: one is for the interactionbetween nonoverlapped spherical pieces of a uniform elec-tron gas and the other is for the interaction between twononoverlapped spheres of uniform density in empty space.The former is the standard uniform gas while the latter is

relevant to a Wigner crystal16 and seems more like and thusmore relevant to the atom-atom interaction in empty space.In this work, we present a nonlocal density functional tosimulate this long-range vdW interaction. Our functional sat-isfies the exact zero- and high-frequency limits of the dy-namic polarizability imposed by Lima and Caldas12 but inaddition it is constrained to be exact for C6 in the latteruniform-gas limit. �Note that the uniform-gas limit we im-pose here differs from that which has been imposed on thesemilocal meta-generalized gradient approximation �meta-GGA� of Tao, Perdew, Staroverov, and Scuseria �TPSS��Ref. 17� and its revised version �revTPSS�.4� Then we de-rive a simple formula for the effect of the long-range vdWinteraction on the DFT lattice constants of metals. In evalu-ating the vdW coefficient for the ion pair M+ , . . . ,M+ in met-als, the screening effect from the valence electrons is calcu-lated via second-order time-dependent perturbation theory.18

Finally we illustrate our approach with the revTPSS func-tional, without losing generality. Our calculations show that,with the vdW correction, the maximum error �Cs� of therevTPSS lattice constants of the alkali metals drops signifi-cantly from 0.151 to 0.056 Å.

The vdW interaction arises from density fluctuations. Inthe large separation �R= �r2−r1�→�� limit, the leading termof the long-range part of the interaction between two frag-ments of nonoverlapped densities n�r1� and n�r2� can bewritten as EvdW

LR =−C6 /R6, where

C6 = �V1

dr1�V2

dr2��r1,r2�n�r1�n�r2� �1�

with n=n↑+n↓ and � being the pair interaction. Based on thesecond-order perturbation theory, Rapcewicz and Ashcroft7

proposed the uniform-gas equivalent effective interaction �= �3 /4���e2 /m�2 / ��p�r1 ,r2��3, where �p=�4�neff�e2 /m� isthe average plasmon frequency, and neff=�n�r1�n�r2� is theeffective density. Because the fluctuating interaction is car-ried out by plasmons, which become Landau damped whenq��p /vF, where q is the local wave vector characterizingthe spatial variation in the density �q���n� /6n�,19,20 and vFis the local Fermi velocity, a cutoff ���p /vF−q� must beimposed on each integral of Eq. �1�. To take into account thelimit of separated atoms, Andersson, Langreth, and

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Lundqvist8 found a better choice for neff�r1 ,r2�.In order to make our nonlocal functional for the long-

range vdW interaction accurate for both molecules and sol-ids, we respect the paradigm densities in condensed matterphysics �slowly varying densities� and quantum chemistry�one- or two-electron densities�. For this purpose, we startwith the exact expression5

C6 =3�

��

0

�

du1�iu�2�iu� , �2�

where j��� is the atomic dynamical polarizability. Whilethe exact frequency dependence of the dynamical polariz-ability remains unknown, several exact properties have beendiscovered. For example, in the high-frequency �u→��limit, j��� has the asymptotic form5 j���→drn�r� /u2. Inthe low-frequency limit, j��� reduces to the static polariz-ability j�0�. For the interaction between two nonoverlappedspheres of uniform density, the pair interaction is given by16

�̃unif = �3b2/4��4�n/b/�4�n�2. �3�

�Atomic units e2=�=m=1 are used from now on unless oth-erwise explicitly stated.� b=1 if the sphere belongs to part ofthe uniform electron gas and 3 if it is an isolated sphere. Thepair potential �̃unif for b=3 is larger than that for b=1 by afactor of 5.2. �The size of this factor strongly suggests that avalence-valence C6 in a metal would be seriously overesti-mated by the free-atom value.�

To satisfy the above conditions, we assume

j�iu� =3

4�aj� dr

f j2��̃p�

u2 + f j2��̃p�

��rcj − r� , �4�

where �̃p=�4�n /3 is the vibrational frequency of an iso-lated uniform sphere,16 � is the step function, aj is a constantto be fixed below, and rcj is the cut-off radius determined byrequiring that the static polarizability be reproduced

j�0� =3

4�aj� dr��rcj − r� . �5�

In order to recover the uniform-gas and separate-atom limits,we choose f j =aj

2�̃p. Substituting Eq. �4� into Eq. �2� andperforming the integration over the frequency u yields thedesired result

C6 =27a1a2

32�2 � dr1�1� dr2�2�̃p�r1��̃p�r2�

a12�̃p�r1� + a2

2�̃p�r2�, �6�

where � j =��rcj −r j�. For spherical systems, the radial cutoffis rcj = �aj j�0��1/3. The parameter aj may be found by requir-ing that the exact high-frequency limit of the dynamical po-larizability be reproduced. Thus, from the spherical densityof system j

aj = �0

�

dr4�r2n�r���0

rcj

dr4�r2n�r��1/3

� 1. �7�

To better understand Eqs. �4�–�7� with Eqs. �1� and �3�, con-sider two uniform-gas limits: �a� for a sphere of uniformdensity with radius R in empty space, �0�=R3.16 Then a

=1, f =�4�n̄ /3, and �̃unif is given by Eq. �3� with b=3. �b�For a sphere of uniform density with radius R, which is partof a uniform electron gas, �0�=R3 /3.16 Then a=31/4, f=�4�n̄, and �̃unif is given by Eq. �3� with b=1. Thus thecharacteristic frequency f and the pair interaction �̃unif areright in both uniform-gas limits. For the other paradigm den-sity, Fig. 1 shows that the proposed j�iu� of Eq. �4� almostreproduces the exact one21 for the H atom.

As a simple test, we apply our nonlocal functional of Eq.�6� to evaluate C6 for diverse atom pairs. The results areshown in Table I. In our calculations, spin-restricted Hartree-Fock densities22 are used. The static polarizabilities �0� aretaken, respectively, from Ref. 23 for H, Mg, and Ca, Ref. 24for He, Ne, Ar, and Kr, Ref. 25 for Xe, Ref. 26 for Li, Na,and K, and Ref. 27 for Be. From Table I we can see that thepresent nonlocal functional is remarkably accurate with amean absolute relative error of only 6.2%. Note that, whilethe Lima-Caldas ansatz for the form of j�iu� �their Eqs.�1�–�6�� differs from ours in its form and in its avoidance ofa sharp cutoff, we find that it is also exact in both uniform-gas limits. Thus we have found an important reason whyboth functionals are accurate. Generalization to nonsphericaldensities, as in Ref. 30, is possible.

Near equilibrium, the higher-order contributions −C8 /R8

and −C10 /R10 can also be important. They can be estimatedfrom the empirical formulas

C8 = 10C61.25, C10 = 121C6

1.50, �8�

which fit the reference many-body-perturbation-theory val-ues of Ref. 13 within about 15%.

To estimate the effect of the long-range vdW attraction onthe revTPSS lattice constants of the alkali metals, we startwith the revTPSS equation of state for the energy per atomas a function of the volume per atom, E�v�. Then around theequilibrium volume v0, we have

E�v� = E�v0� + E��v0�v +1

2E��v0��v�2, �9�

where v v−v0 is the volume shift per atom. At equilib-rium, the revTPSS energy minimizes so that E��v0�=0. Thesecond derivative of the energy at equilibrium is related to

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 1 2 3 4 5

α(iu

)(a

.u.)

u

FIG. 1. �Color online� Dynamical polarizability �iu� �in atomicunits� of the H atom: Exact �solid red line�, and Eq. �4� with a=1.143 and rc=1.726 �dashed green line�.

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the bulk modulus by B0=v0E��v0�. Since the lattice of thealkali metals is bcc, we have v=a3 /2, where a is the latticeconstant. Thus Eq. �9� can be rewritten as

E�v� − E�v0� = �9/4�B0a0�a�2, �10�

where a a−a0 is the lattice constant shift relative to theoriginal revTPSS lattice constant a0. The vdW interaction ofone ion core with all others is

EvdW = − b6C6+/a6 − b8C8

+/a8 − b10C10+ /a10, �11�

where bn= �1 /2��i=1�a /Ri0�n �b6=14.52, b8=16.36, and b10=20.15� and Cn

+ is the Cn coefficient for the interaction be-tween two singly positive ion cores in the metal, as screenedby the valence electrons. Ri0 is the distance from atom 0 toany other atom i. Expansion of EvdW around a0 to secondorder in a, and minimization of the sum of Eqs. �10� and�11�, leads straightforwardly to a linear equation for thecorrection31 a to the revTPSS equilibrium lattice constant.The quantity that determines the importance of Cn

+ is xn=bnCn

+ / �B0a0n+3�.

To evaluate C6+, we let the valence electrons screen the

interaction between two cores, according to the simple for-mula of Rehr, Zaremba, and Kohn,18,31 who showed that onecould use the long-wavelength limit of the dynamic dielec-tric function of the uniform valence-electron density �withplasma frequency �p= �4�nvalence�1/2�. For like-pair ionic in-teractions, we find

C6+ =

27

32�2� dr1�1� dr2�2�̃p�r1��̃p�r2�

�̃p�r1� + �̃p�r2�S�r1,r2� ,

S = a6�̃p�r1��̃p�r2���p��̃p�r1� + �̃p�r2��/2 + a2�̃p�r1�

� �̃p�r2�����p + a2�̃p�r1��2��p + a2�̃p�r2��2�−1, �12�

where a is given by Eq. �7� and S accounts for the screeningof valence electrons. The accuracy of our C6

+ is tested for freeion pairs, for which S=1, using the static polarizabilities32,33

of the free ions. The results are 10% higher than thoseestimated33 with the time-dependent local density approxi-mation. See Ref. 31 for a1�a2.

The alkali metals are well-suited to this proposed correc-tion because their valence densities are nearly uniform out-side nearly nonoverlapping cores.34 The full valence-valencevdW interaction is already included in the revTPSS energy,which is exact for any uniform density. We calculate thevdW-corrected lattice constants of the alkali metals using theexpressions in Ref. 31. The C6

+ are calculated from Eq. �12�using the Hartree-Fock densities22 of the ion cores �in thecorresponding free atoms� and the polarizabilities of thesecores in a uniform electron gas.32 Only the low frequency�u �2�p� contribution to C6

+ �about 30% of the un-screened value� gets screened out by Eq. �12�. C8

+ and C10+ are

estimated from Eq. �8�, and increase the long-range vdWeffect by about 50%. The results are summarized in Table II.The revTPSS lattice constants and bulk moduli are calcu-lated with the all-electron BAND code,35 as in Ref. 4. Weobserve that the long-range vdW attraction systematicallyshrinks the revTPSS lattice constants. The vdW effect growswhen we pass from Li �harder� to Cs �softer� due largely tothe increase of x6=b6C6

+ / �B0a09�. As a result, the vdW-

corrected revTPSS lattice constants are considerably closerto experiment than the original revTPSS lattice constants.

TABLE I. The vdW coefficients C6 �in atomic units� calculatedwith Eq. �6� using spin-restricted Hartree-Fock densities �Ref. 22�.The mean absolute relative error is 6.2%.

Reference Present Reference Present

He-He 1.46c 1.44 Ar-H 20.5d 19.8

Ne-Ne 6.38c 7.35 Ar-Li 171d 180

Ar-Ar 64.3c 67.8 Ar-Na 189d 198

Kr-Kr 130c 132 Ar-K 269d 317

Xe-Xe 286c 295 Kr-H 28.0d 28.0

He-Ne 3.03c 3.22 Kr-Li 255d 265

He-Ar 9.54c 9.81 Kr-Na 282d 289

He-Kr 13.4c 13.6 Kr-K 403d 462

He-Xe 19.5c 20.0 Xe-H 40.7d 42.4

Ne-Ar 19.5c 21.5 Xe-Li 404d 422

Ne-Kr 27.3c 29.6 Xe-Na 448d 456

Ne-Xe 39.7c 43.3 Xe-K 642d 729

Ar-Kr 91.1c 94.4 He-Be 13.0a 13.6

Ar-Xe 135c 140 He-Mg 21.1a 21.2

Kr-Xe 192c 197 He-Ca 32.5a 36.7

H-H 6.50a 6.28 Ne-Be 27.5c 27.7

Li-Li 1389a 1334 Ne-Mg 42.9c 43.3

Na-Na 1540a 1363 Ne-Ca 94.1c 74.8

K-K 3945a 3701 Ar-Be 102c 101

H-Li 65.9a 67.2 Ar-Mg 162c 158

H-Na 72.1a 71.7 Ar-Ca 346c 276

H-K 105a 114 Kr-Be 149c 146

Li-Na 1460a 1346 Kr-Mg 238c 228

Li-K 2334a 2214 Kr-Ca 503c 400

Na-K 2443a 2238 Xe-Be 228c 226

Be-Be 213b 213 Xe-Mg 367c 355

Mg-Mg 618a 569 Xe-Ca 775c 624

Ca-Ca 2005a 1971 H-Be 34.4a 34.7

Be-Mg 362a 346 H-Mg 57.8a 54.8

Be-Ca 619a 630 H-Ca 93.0a 96.9

Mg-Ca 1112a 1051 Li-Be 467a 473

He-H 2.81a 2.75 Li-Mg 854a 811

He-Li 22.0a 23.3 Li-Ca 1615a 1570

He-Na 24.2a 26.0 Na-Be 505a 491

He-K 34.4a 41.7 Na-Mg 920a 837

Ne-H 5.60d 5.79 Na-Ca 1723a 1608

Ne-Li 42.6d 46.4 K-Be 755a 786

Ne-Na 47.0d 52.4 K-Mg 1390a 1347

Ne-K 66.3d 84.2 K-Ca 2663a 2608

aFrom Ref. 23.bFrom Ref. 27.cFrom Ref. 28.dFrom Ref. 29.

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In summary, we use constraint satisfaction to construct awell-motivated nonlocal density functional for the long-range vdW interaction, showing the power of two uniform-gas limits. Then we derive a simple formula to evaluate thecore-core vdW correction to the lattice constants of the alkalimetals calculated with a semilocal density functional. We

find that the vdW-corrected revTPSS meta-GGA lattice con-stants of the alkali metals agree better with experiment,showing errors like those of revTPSS for other solids.4

This work was supported by DOE under Grant No. DE-AC52-06NA25396 and NSF under Grants No. DMR-0501588 and No. DMR-0854769.

1 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 �1965�.2 G. I. Csonka et al., Phys. Rev. B 79, 155107 �2009�.3 P. Haas et al., Phys. Rev. B 79, 085104 �2009�.4 J. P. Perdew et al., Phys. Rev. Lett. 103, 026403 �2009�.5 H. Margenau and N. R. Kestner, Theory of Interatomic Forces,

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20 M. van Faassen et al., Phys. Rev. Lett. 88, 186401 �2002�.21 D. M. Bishop and J. Pipin, J. Chem. Phys. 97, 3375 �1992�.22 E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14, 177

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10.1103/PhysRevB.81.233102 for equations and explanations.32 R. M. Nieminen and M. J. Puska, Phys. Scr. 25, 952 �1982�.33 G. D. Mahan, J. Chem. Phys. 76, 493 �1982�.34 The core-core vdW effect can also be important in the alkaline

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TABLE II. Alkali-metal lattice constants calculated with the revTPSS meta-GGA and their core-corevdW-corrected values. Experimental lattice constants from Ref. 2. The last columns show the long-rangevdW corrections to the cohesive energy per atom and to the bulk modulus. 1 bohr=0.5292 Å,1 hartree /bohr3=2.9424�104 GPa, 1 hartree=27.2 eV.

SolidC6

+

�a.u.�

B0

�revTPSS��GPa�

a0

�revTPSS���

a0

�correc���

a0

�expt.���

�cohesive

�eV�B /B0

�revTPSS�

Li 0.072 12.63 3.425 3.424 3.449 0.000 −0.002

Na 1.69 7.43 4.213 4.207 4.210 0.003 −0.015

K 19.5 3.11 5.312 5.285 5.212 0.010 −0.039

Rb 43.5 2.64 5.689 5.646 5.576 0.015 −0.056

Cs 104. 1.59 6.190 6.095 6.039 0.022 −0.108

Meanerror 0.069 0.034

Meanabsolute error 0.078 0.045

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