Vydrov and Van Voorhis reply
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Vydrov, Oleg A., and Troy Van Voorhis. “Vydrov and Van
Voorhis Reply:.” Physical Review Letters 104.9 (2010): 099304.
© 2010 The American Physical Society.
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http://dx.doi.org/10.1103/PhysRevLett.104.099304
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Final published version
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Detailed Terms
PRL 104, 099304 (2010)
PHYSICAL REVIEW LETTERS
Vydrov and Van Voorhis Reply: The central result of our
Letter [1] is Enl
c of Eq. (13). In our schematic derivation of
Eq. (13), we used an admittedly limited model polarization
operator Sq;q0 of Eq. (10). We do not recommend using this
S for any purpose outside the context of Ref. [1]. Langreth
and Lundqvist [2] point out that if our Eq. (11) is used in
the standard expression for the exchange-correlation energy of a uniform system, one obtains an infinite result.
This observation is of little consequence for our Eq. (13):
not only is it nondivergent, but it vanishes for a uniform
system. For nonuniform systems, an effective wave vector
dependence of excitation energies is included in our model
via !g of Eq. (9).
Equation (3) in Ref. [1], used as the definition of the
nonlocal correlation energy Enl
c in our work and in Ref. [3],
has a clear physical meaning only in the limit of wellseparated fragments. Enl
c was derived [3] using secondorder perturbation theory and a number of other approximations. Higher-order and many-body (Axilrod-Tellertype) terms, neglected in Enl
c , can be significant at intermediate distances. Moreover, Enl
c is never used by itself: It
, which contributes sizably and unpreis paired with ELSDA
c
dictably to binding energies whenever fragment densities
overlap even slightly [4]. Hence, in this formalism, only
the asymptotic dispersion interactions are treated rigorously, and there are uncontrolled sources of errors at
shorter distances. In this context, the significance of enforcing the correct short- and intermediate-range behavior
for Sq;q0 is questionable. This is why in Ref. [1], we focused
on the small-q limit of S. Our Eq. (10) obeys the f-sum rule
for q2 þ q02 k2s , i.e., in the relevant long-range regime.
Enl
c , defined by Eq. (3), is not expected to capture the
physics of short-range electron correlations. Indeed, both
vdW-DF-04 [3] and VV09 [1] give poor correlation enerþ Enl
gies (ELSDA
c
c ) for atoms, which indicates that these
functionals are inaccurate at short range. We see no physical reason for the nonlocal correlation kernel to diverge as
lnR for R ! 0, as it does in vdW-DF-04. This R ! 0
divergence causes difficulties in practical implementations.
Omitting the singular r ¼ r0 terms in the double sum over a
real-space grid leads to numerical instabilities, such as
errors in the gradients with respect to nuclear displacements. It also causes substantial grid superposition errors
in binding energies, if atom-centered quadrature grids are
used. Implementational tricks have been devised to deal
with the singularity [5], but we would rather eliminate the
source of the problem. The singularity in the kernel results
from the q4 behavior of Sq;q0 in the q ! 1 limit, enforced
in Ref. [3]. By lifting this inessential [in the context of
Eq. (3)] constraint, we obtained [1] an expression that is
finite at R ¼ 0.
Neither VV09 nor its precursor [3] are free of arbitrariness in their construction [6], but that may be unavoidable.
Our nonlocal correlation functional of Eq. (13) was designed [1] to have the following properties: (1) It has an
0031-9007=10=104(9)=099304(1)
week ending
5 MARCH 2010
analytic form convenient for implementation; (2) it gives
accurate asymptotic C6 coefficients for molecules [7]; (3) it
vanishes for a uniform system, i.e., the negative long-range
contribution is exactly cancelled by the positive shortrange part—see Fig. 1 in Ref. [1]; (4) it is strictly nonnegative in nonuniform systems, i.e., the short-range portion prevails; (5) it has a constant second-order gradient
coefficient in the slowly varying limit. Points 3 and 4, and
to a lesser degree point 2, are shared by vdW-DF-04. Points
1 and 5 are novel in VV09. Recovery of the second-order
density gradient expansion for correlation may not be
important, yet it can result in successful functionals [8,9].
The value of the gradient coefficient varies from one
correlation functional to another, but it is typically a constant [8–10]. Cancellation of the correlation gradient coefficient by its exchange counterpart is not necessary for
good performance [9]. The positive second-order gradient
term of Enl
c is largely produced by short-range interactions
(see point 4 above). The spin-dependence in VV09 also
enters at shorter range: asymptotic C6 coefficients do not
depend on . The length scale in VV09 is given by K of
Eq. (14). In a uniform system, K / ks R, which is the
scaled distance relevant for correlation energy [11].
Overall, we believe that the advantages of our theory,
summarized above, outweigh its imperfection—the unaesthetic short-range damping mechanism—pointed out by
Langreth and Lundqvist [2].
Oleg A. Vydrov and Troy Van Voorhis
Department of Chemistry,
Massachusetts Institute of Technology,
Cambridge, Massachusetts, 02139, USA
Received 2 December 2009; published 4 March 2010
DOI: 10.1103/PhysRevLett.104.099304
PACS numbers: 31.15.E, 34.20.Gj, 71.15.Mb
[1] O. A. Vydrov and T. Van Voorhis, Phys. Rev. Lett. 103,
063004 (2009).
[2] D. C. Langreth and B. I. Lundqvist, preceding Comment,
Phys. Rev. Lett. 104, 099303 (2010).
[3] M. Dion et al., Phys. Rev. Lett. 92, 246401 (2004).
[4] Approximate exchange functionals can have an even
greater impact on binding energy curves.
[5] G. Román-Pérez and J. M. Soler, Phys. Rev. Lett. 103,
096102 (2009).
[6] See relevant discussions in O. A. Vydrov and T. Van
Voorhis, J. Chem. Phys. 130, 104105 (2009).
[7] The ad hoc coupling constant integration performed in
Eq. (4) of Ref. [1] enabled us to reproduce the correct C6
coefficients for jellium spheres. By contrast, vdW-DF-04
fails even qualitatively for these model systems.
[8] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.
77, 3865 (1996).
[9] L. S. Pedroza, A. J. R. da Silva, and K. Capelle, Phys.
Rev. B 79, 201106(R) (2009).
[10] J. P. Perdew et al., Phys. Rev. Lett. 100, 136406 (2008).
[11] J. P. Perdew and Y. Wang, Phys. Rev. B 46, 12947 (1992).
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Ó 2010 The American Physical Society