THE JOURNAL OF CHEMICAL PHYSICS 134, 114110 (2011)
A simple but fully nonlocal correction to the random phase approximation Adrienn Ruzsinszky,1,a) John P. Perdew,1 and Gábor I. Csonka1,2 1 2
Department of Physics and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118, USA Department of Chemistry, Budapest University of Technology and Economics, Budapest H-1111, Hungary
(Received 28 October 2010; accepted 2 March 2011; published online 21 March 2011) The random phase approximation (RPA) stands on the top rung of the ladder of ground-state density functional approximations. The simple or direct RPA has been found to predict accurately many isoelectronic energy differences. A nonempirical local or semilocal correction to this direct RPA leaves isoelectronic energy differences almost unchanged, while improving total energies, ionization energies, etc., but fails to correct the RPA underestimation of molecular atomization energies. Direct RPA and its semilocal correction may miss part of the middle-range multicenter nonlocality of the correlation energy in a molecule. Here we propose a fully nonlocal, hybrid-functional-like addition to the semilocal correction. The added full nonlocality is important in molecules, but not in atoms. Under uniform-density scaling, this fully nonlocal correction scales like the second-order-exchange contribution to the correlation energy, an important part of the correction to direct RPA, and like the semilocal correction itself. For the atomization energies of ten molecules, and with the help of one fit parameter, it performs much better than the elaborate second-order screened exchange correction. © 2011 American Institute of Physics. [doi:10.1063/1.3569483] I. INTRODUCTION A. Density functional theory and the random phase approximation
The random phase approximation (RPA) (Refs. 1–7) has recently emerged as an attractive method to describe many situations which are still challenges for semilocal or hybrid density functionals. Direct RPA, sometimes with a semilocal density-functional correction, appears to work well for many energy differences,8–13 but does not give a satisfactory account of molecular atomization energies.9, 14 Here we will propose a fully nonlocal addition to the semilocal correction. It is designed to improve mainly the energies of molecules. Kohn–Sham density functional theory15, 16 is a widely used method of electronic-structure calculation for atoms, molecules, and solids. It achieves computational efficiency by replacing the problem of interacting electrons by one of the fully occupied or fully unoccupied self-consistent oneelectron orbitals, in a way that is formally exact for the ground-state energy and electron density. In practice, the exact density functional for the exchange-correlation energy must be approximated, and much effort has gone into refining the approximations. Most such refinements fall on a ladder17, 18 in which higher rungs construct the exchangecorrelation energy density by including successively more complex ingredients: (1) the local spin densities, in the local spin density approximation (LSDA),15, 19–21 (2) the gradients of the local spin densities in the generalized gradient approximation (GGA),22 (3) the orbital kinetic energy density, in the meta-GGA,23 (4) exact-exchange ingredients, in hybrid functionals or hyper-GGAs,24 and (5) the unoccupied orbitals and the orbital energies, in the direct RPA and its generalizations. a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]. 0021-9606/2011/134(11)/114110/6/$30.00
The local spin densities and their gradients are common to rungs (2)–(4). The ascent of the ladder leads to increased accuracy (in an overall but not uniform sense), but at increasing computational cost. This increase is modest over rungs (1)–(3). The semilocal functionals on these three rungs can be constructed without empiricism, by the satisfaction of exact constraints,18 and may be sufficient for the description of many atoms, molecules, and solids near equilibrium. Computational cost increases steeply for the fully nonlocal functionals on rungs (4) and (5), but full nonlocality is needed to describe strongly correlated systems and stretched bonds, long-range van der Waals effects on rung (5), etc. So far, some empiricism is needed on rung (4), but the need for empirical parameters is reduced or eliminated on rung (5). The direct RPA in a density functional context (using the Kohn–Sham orbitals and orbital energies) was proposed by Langreth and Perdew,3–5 who followed the earlier work of Pines and co-workers1, 2 for the uniform electron gas. Within the adiabatic-connection fluctuation dissipation theorem,3, 4, 20 it amounts to replacing the exact density response function by its time-dependent Hartree approximation. In the electron gas21 and at the jellium surface,3, 4, 8 the correction to direct RPA is a short-ranged correlation energy amounting to about +0.02 hartree per electron, which arises because the RPA correlation hole is too deep close to its electron. Yan et al.8 proposed the RPA+ approximation, in which the correction is treated semilocally, in LSDA or GGA. They expected that this correction would be generally accurate, and that it would tend to cancel out of isoelectronic energy differences (with no change of electron number). No such cancellation was found in the early molecular RPA calculations of Furche,9 where RPA atomization energies of small molecules were found to be too small by about 10 kcal/mol (0.4 eV or 0.02 hartree) on average. But the cancellation is evident in applications of direct RPA to many properties and systems, including the
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jellium surface.8, 10–13 We14 have interpreted this to mean that the correction to RPA in a molecule has multicenter full nonlocality, with both short- and middle-range contributions to the correlation hole, as discussed in Sec. II. Compared to finite-order perturbation theory, the RPA sums up diagrams to infinite order, thus eliminating the divergence of finite-order perturbation series for metallic systems. The RPA often has an accurate long-range correlation, which gives it the capacity to account for long-range van der Waals interactions. The RPA can also accurately predict lattice constants for metallic, ionic, or covalent bonds.10, 11 RPA correctly describes the static correlation for the H2 molecule in the dissociation limit, without spin-symmetry breaking.25 For all these reasons, in recent years there has been a strong interest in RPA. Direct RPA is exact for the exchange energy, but not for the correlation energy. It can be regarded as the summation to infinite order of the ring diagrams, which provide the dominant part of the correlation energy for the highdensity electron gas. It can be derived from the “ring-coupledcluster” theory by removal of the exchange terms from the two-particle Hamiltonian.26 The absence of these terms is responsible for a self-correlation error and for the dramatic failure of the RPA in the case of stretched radicals, such as + + 27, 28 The spurious one-electron RPA corH+ 2 , He2 , and Ne2 . relation energy is −0.021 hartree for H (Ref. 29) and −0.18 27 hartree for infinitely stretched H+ 2. There are other ways to correct RPA, beyond the semilocal corrections already mentioned. For example, one can include an approximate exchange-correlation kernel30–33 into the integral of the adiabatic-connection fluctuation dissipation theorem, or even include the exact exchange kernel.34, 35 The inhomogeneous Singwi–Tosi– Land–Sjoelander approximation36 also provides a kind of self-correlation-free generalization of RPA, as does the additive second-order screened exchange (SOSEX) approximation.28, 37 So far, none of these corrections has been demonstrated to work very well for the atomization energies of molecules. Range-separated hybrids with long-range RPA and shortrange LSDA or GGA have also been constructed.28, 38 They provide a correction to RPA which is itself a fifth-rung functional, with one or more empirical parameters, like the correction we proposed earlier.14 The new correction we propose here is a fourth-rung functional with one empirical parameter, and with better scaling properties than that of Ref. 14. We are also exploring the possibility of a nonempirical fourth-rung correction based upon a model for the correlation hole.39 II. MULTICENTER NONLOCALITY AND THE CORRECTION TO RPA
In the valence region of an atom, the exchange energy and the correlation energy are separately semilocal because the exponential decay of the electron density around one nuclear center leads to exchange and correlation holes that are largely short ranged. However, in the valence region of a molecule, there are strong multicenter nonlocalities of exchange and correlation, which tend to cancel one another
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largely but not completely. Since direct RPA employs exact nonlocal exchange and approximate nonlocal correlation, it can fail to achieve the proper degree of cancellation between these nonlocalities in a molecule. The exact exchange-correlation energy can be written3, 4, 20 as the interaction between the electron denr , r ) at r of the sity n( r ) at position r and the density n xc ( exchange-correlation hole around an electron at r: r ) d 3r n xc ( r , r ) 1 d 3r n( . (1) E xc = 2 | r − r| Here n xc ( r , r ) = n x ( r , r ) + n c ( r , r ) is the sum of exchange and correlation hole densities, which satisfy the exact constraints3, 4, 20 3 n x ( r , r ) < 0, d r n x ( r , r ) = −1, d 3r n c ( r , r ) = 0. (2) The exact exchange hole density is 2 occup − σ ψασ ( r )ψασ ( r ) α n x ( r , r ) = , n( r)
(3)
where ψασ is an occupied Kohn–Sham orbital with spin σ =↑ or ↓. The exact correlation hole density vanishes in oneelectron regions. But, in many-electron regions, it tends to be negative close to its electron and positive far away, with a tendency to cancel the exact exchange hole density in the long-range (large | r − r|) limit. We have argued14 that the RPA correlation hole, even after semilocal or short-range correction, is not correct at mid-range in a molecule. Semilocal density-functional approximations for E x and r , r ) and n c ( r , r ) (or E c have their own models40–43 for n x ( more precisely for the system averages thereof), which satisfy various constraints including those of Eq. (2), with hole densities that (for strongly nonuniform electron densities) decay rapidly at large | r − r|. But the inputs to these models involve only the local spin densities and possibly their gradients or the orbital kinetic energy density at r. They can describe the short-range behavior of the hole density, but not what we shall call “multicenter nonlocality,” which arises when r is located on one attractive center (atom), r is on a neighboring center, r , r ) or n c ( r , r ) at r is considerably bigger in magand n x ( nitude than the semilocal approximations predict. The most clear-cut example of multicenter nonlocality arises in the infinitely stretched spin-unpolarized H2 molecule, where half of the exact exchange hole around an electron on one H atom is located on the other H atom, infinitely far away. In this case, the positive lobe of the exact correlation hole on the other H atom exactly cancels the negative lobe of the exact exchange hole there. As a result, the semilocal exchange energies are far too negative, and the semilocal correlation energies are not nearly negative enough; the semilocal exchange-correlation energy is however less unrealistic than either its exchange or its correlation contribution. Some residue of multicenter nonlocality persists in molecules at equilibrium. As a result, the semilocal (sl) functionals for E xc tend to overestimate the atomization
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energies of molecules (with the overestimation decreasing from LSDA to GGA to meta-GGA). The global hybrid (gh) functionals24, 44 improve the atomization energies via gh = E xsl + a(E xexact − E xsl ) + E csl , E xc
(4)
where a is a positive empirical parameter in the range 0.1 < a < 0.6. The fully nonlocal correction in Eq. (4) can also be written as exact sl r )εxsl ( r ) f ( r ), (5) a(E x − E x ) = −a d 3r n( where 1 2
εx ( r) =
r , r ) d 3 r n x ( | r − r|
(6)
is the exchange energy per particle at r and f ( r) =
r ) − εxexact ( r) εxsl ( sl εx ( r)
(7)
is a dimensionless ratio that we expect to be small in an atom, but more significantly positive in the valence region of r ) < εxexact ( r ) < 0. (The first inequality a molecule where εxsl ( is a consequence of the multicenter spreading of the exact exchange hole in a molecule.) Thus the nonlocal correction of Eq. (5) raises the energy of a molecule relative to the energy of its constituent atoms and so corrects the semilocal overestimation of the molecular atomization energy. We propose the following correction to the RPA correlation energy: r )[εcGGA ( r ) − εcRPA−GGA ( r )][1 − α F( f ( r ))], (8) d 3r n( where α is an empirical parameter and F(0) = 0. The part of Eq. (8) that remains when α = 0 is our best or GGA semilocal RPA+ correction8 to RPA. It is not very different from the LSDA correction; both increase the correlation energy by about 0.02 hartree/electron for typical electron densities. We have no meta-GGA for this correction, since our meta-GGA for beyond-RPA correlation is just a self-interaction correction to GGA correlation, and RPA is not self-correlation-free. The contribution from F in Eq. (8) is our correction for multicenter full nonlocality. By analogy with Eq. (5), our first try is F( f ) = f,
(9)
where α > 0. This will lower the energy of the molecule relative to its separated atoms, and so correct the underestimation of molecular atomization energies that is found from direct RPA with or without its semilocal correction. Thus our correction to RPA makes the correlation energy more negative in the molecule, presumably by making the correlation hole density deeper near the electron and higher further away (on the neighboring centers), all in the middle range, and thus making the exchange-correlation hole more localized around its electron.14 In principle, the semilocal εxsl ( r ) in Eq. (7) should be r ) in an atom. Most GGA’s and a good model for εxexact ( meta-GGA’s are designed to model the integrated E xexact , and r ). An exception is the Becke–Roussel not necessarily εxexact (
FIG. 1. Plot of the odd function F(f) of Eq. (10) vs f. In the region of small f relevant to atoms and to molecules at equilibrium, F is nearly equal to f. But F is designed to vanish for stretched spin-unpolarized H2 , where f = 0.373, and to remain well bounded for all f.
meta-GGA,45 which we have used here forεxsl , and for which α ≈ 9 provides a good fit to the atomization energies of Furche.9 But we have also found that, in practice, one could r ) inuse the Perdew–Burke–Ernzerhof (PBE) GGA for εxsl ( stead. The simple expression (9) is good enough to fit the atomization energies of Ref. 9. It might even remove a spurious positive bump25 in the binding energy curve for spinunpolarized H2 , but it would produce a spurious negative dissociation limit for this curve. This dissociation is a longrange limit where RPA with or without a semilocal correction needs no further correction.25 In this limit, when r is on either atom, f ≈ 0.373, much larger than in molecules at equilibrium. To zero out our correction in this limit, without much changing the atomization energies from those of Eq. (9), and to keep the magnitude of F well bounded for all f , we have used, in preference over Eq. (9), F( f ) = f [1 − 7.2 f 2 ][1 + 14.4 f 2 ] exp(−7.2 f 2 ),
(10)
whose small- f expansion is f + O( f ). The function of Eq. (10) is plotted in Fig. 1. We stress that the only fit parameter to the atomization energies in Eq. (8) is α ≈ 9. Finally, we consider the electron-number-conserving uniform density scaling:46 5
n( r ) → n λ ( r ) = λ3 n(λ r ),
(11)
where λ is a positive scale factor. Under this scaling, the exact and semilocal exchange energies E x scale like λ, while the exact, semilocal, and RPA correlation energies E c (in the absence of degeneracy between occupied and unoccupied orbital energies) scale like λ0 in the high-density (λ → ∞) limit. Our present correction to RPA has the correct correlation-energy behavior in the high-density limit, while our nonlocal empirical correction of Ref. 14 does not.
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Perhaps more relevantly, our fully nonlocal correction to RPA, starting from a physical density, changes very weakly with λ for λ ≥ 1, like the semilocal correction. The second-order exchange contribution to the Goerling–Levy47 (second-order or high-density-limit) correlation energy, which is believed to be a dominant piece of the correction, does not change at all with λ. The fully nonlocal part of our correction to the RPA correlation energy of a small N-electron molecule is only about −2% to −10% of the total correction (≈N × 0.02 hartree) to its RPA correlation energy. Similarly, the fully nonlocal part of our correction to the RPA high-density limit is only about −2% to −10% of the total correction (≈N × 0.02 hartree) to that limit. However, the true RPA high-density limit for the correlation energy of a molecule is not known numerically. It is worth mentioning that standard global hybrids of the form of Eq. (5) do not work as a correction to RPA. This fact shows the relevance of proper scaling in the highdensity limit. The error that RPA makes in a molecule is not a long-range or static-correlation error, which can scale like exchange over a range of λ≈1, but a short- and middle-range error.
grid, which is conceptually simple but computationally inefficient. The nearest-neighbor grid-point distances varied between 0.16 and 0.20 a.u. The extent of the box containing the grid points and the species similarly varied between 16 and 20 a.u. giving about 1 × 106 integration points. We eliminated those grid points from the integration where the electron densities were below 6 × 10−5 in order to speed up calculations. Even after this simplification the computation time is at least 3 orders of magnitude larger than is usual in density functional work. We plan to make the code more efficient for future studies. Even this grid would not be fine enough if we did not cut off the high electron densities in the ion cores of atoms other than H. For details of how we did that, and for a demonstration that the 0.16 a.u. grid converges the GGA correction to the atomization energy, we refer to our earlier work.14 Table I suggests that our fully nonlocal correction of Eq. (8) is also reasonably converged on the 0.16 a.u. grid. To achieve maximum error cancellation in each atomization energy, we positioned each of a molecule’s free atoms with respect to the grid in the same way as in the molecule.14 For Si2 , which has the smallest ionization energy and the most diffuse electron density of all the molecules in Table I, the finest grid we could use had a mesh with cube side 0.18 a.u. The current state-of-the-art for RPA is a non-selfconsistent calculation using PBE GGA orbitals and orbital energies.9–13 Our correction requires only the occupied orbitals, which we took from the GAUSSIAN03 code,48 outputting the basis functions and PBE expansion coefficients in a wavefunction file. The basis sets were ccpVTZ, and the molecular geometries were the optimized PBE geometries for this basis set. For Si2 , we used Hartree– Fock orbitals to minimize the diffuseness of the electron density.
III. TECHNICAL DETAILS OF THE CALCULATION
The exact exchange energy per electron at position r, 3 exact r , r ) 1 d r n x ( , (12) εxexact ( r) = 2 | r − r| is itself an integral over r . In some codes, this integration can be done analytically for each combination of basis functions in Eq. (3). Because we did not have access to this option, we had to evaluate our correction of Eq. (8) as a numerical double integral over a real-space grid. We chose a simple cubic
TABLE I. Static-nucleus atomization energies (kcal/mol) of the ten molecules of Ref. 9, in RPA and with various corrections to RPA. RPA+ is the nonempirical GGA correction of Ref. 8, as implemented in Ref. 14. SOSEX is a nonempirical orbital-based correction implemented in Ref. 28. Our simple but fully nonlocal correction of Eq. (8), with one empirical parameter α, has been implemented here in the versions of Eqs. (9) and (10), using cubic meshes of sides 0.16 and 0.20 bohr. The mean error (ME) and mean absolute error (MAE) are shown for each approximation. Eq. (9), α = 8.6
Eq. (10), α = 9.3
RPAa
RPA+b
[PBE+RPA+]/2c
SOSEXd
Mesh = 0.2
0.16
0.2
0.16
Expt.a
H2 N2 O2 F2 Si2 CO CO2 H2 O C2 H2 HF
109 223 113 30 70 244 364 223 381 133
110 223 111 29 70 242 360 222 378 132
108 234 128 41 76 266 388 228 397 137
... 215 103 25 72 252 375 229 398 139
111 243 126 39 73 258 386 227 393 133
111 239 125 39 78e 260 384 227 393 133
111 244 127 40 78 259 387 227 396 133
111 242 126 39 77e 259 385 227 400 133
109 228 121 38 75 259 389 232 405 141
ME MAE
−10.7 10.7
−12.0 12.2
0.6 4.2
−8.9 8.9
−0.8 5.4
−0.8 5.2
0.5 5.3
0.2 4.6
... ...
Molecule
a
Reference 9. Reference 8. c Reference 14. d Reference 28. e mesh = 0.18 bohr b
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IV. RESULTS AND CONCLUSIONS
Our results for the static-nucleus atomization energies of the ten molecules of Ref. 9 are displayed in Table I. The first column is the RPA atomization energy as computed by Furche.9 The next seven columns add various approximate corrections to the RPA atomization energies, and the last column is the experimental atomization energy after removal of its zero-point vibrational contribution.9 As found in Refs. 8 and 9, the nonempirical GGA or RPA+ correction of Ref. 8 greatly improves the total energies over RPA but hardly changes the atomization energies and so cannot correct the non-negligible underbinding of RPA in comparison with the experiment. The nonempirical orbital-based SOSEX correction28 is not much better for these ten molecules. It reduces the mean absolute error of the atomization energies from 10.7 kcal/mol (RPA underbinding) to 8.9 kcal/mol (SOSEX, mixed underand overbinding). However, these 10 molecules may be particularly difficult for SOSEX. For the 55 G2–1 molecules of Ref. 28, SOSEX reduces the mean absolute error of the atomization energies from 9.9 (RPA) kcal/mol to 5.7 (SOSEX) kcal/mol. Our simple, fully nonlocal correction of Eq. (8) reduces the mean absolute error of RPA for the atomization energies of our ten molecules from 11 to 5 kcal/mol in the versions of Eqs. (9) and (10), giving almost the best correction in the table. However, it should be remembered that Eq. (8) has one empirical parameter fitted to the experimental atomization energies in Table I. The good results we have found could only be a consequence of fitting to a small data set. That the results are not better may only reflect the simplified and approximate nature of our correction. In Table I, we also show the results from our oneparameter hybrid of Ref. 14, which approximates the exchange-correlation energy as the average of the PBE GGA and RPA+ estimates. It is slightly more accurate in this test than our present Eq. (8). The reason for this is unclear, since the [PBE+RPA+]/2 correction to the RPA exchange-correlation energy does not properly scale to a constant in the high-density limit. However, it may be related to the fact that the unconventional hybrid [PBE+RPA+]/2 gives a fifth-rung correction to RPA, while our Eq. (8) is only a fourth-rung correction. In summary, the RPA correlation energy in a molecule can make not only a short-range error correctable by GGA, but also a middle-range multicenter nonlocality error that seems to be corrected by our Eq. (8), a hybridlike expression which scales like the GGA correction to RPA. Before a firmer conclusion can be drawn, our correction to RPA needs to be implemented more efficiently and tested more broadly. For example, it could be applied to the 55 G2–1 molecular atomization energies of Ref. 28, to the bindingenergy curve of spin-unpolarized H2 ,25 and to the six BH6 barrier heights to chemical reactions.28 For the BH6 barriers, SOSEX was found28 to worsen the mean absolute error from 1.8 kcal/mol (RPA) to 4.2 kcal/mol (SOSEX, all barriers too high).
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It would also be interesting to test our corrected RPA for solids, including strongly correlated ones like NiO. NOTE ADDED IN PROOF
We have just received a preprint49 on the correction to RPA. ACKNOWLEDGMENTS
This work was supported in part by the U.S. National Science Foundation (NSF) under Grant No. DMR-0854769. Computational resources were provided by the Center for Computational Science at Tulane University and by the Louisiana Optical Network Institute. This work is connected to the scientific program of the “Development of qualityoriented and harmonized R+D+I strategy and functional model at BME” project, supported by the New Hungary Development Plan (Project ID: TÁMOP-4.2.1/B-09/1/KMR2010-0002). This work was supported in part by the Hungarian Academy of Science (NSF/104).
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