Does a Molecule-Specific Density Functional Give an Accurate

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Letter pubs.acs.org/JPCL

Does a Molecule-Specific Density Functional Give an Accurate Electron Density? The Challenging Case of the CuCl Electric Field Gradient Monika Srebro†,‡ and Jochen Autschbach*,† †

Department of Chemistry, University at Buffalo, State University of New York, Buffalo, New York 14260-3000, United States Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian University, 30-060 Krakow, Poland

‡

S Supporting Information *

ABSTRACT: In the framework of determining system-specific long-range corrected density functionals, the question is addressed whether such functionals, tuned to satisfy the condition −εHOMO = IP or other energetic criteria, provide accurate electron densities. A nonempirical physically motivated two-dimensional tuning of range-separated hybrid functionals is proposed and applied to the particularly challenging case of a molecular property that depends directly on the ground-state density: the copper electric field gradient (EFG) in CuCl. From a continuous range of functional parametrizations that closely satisfy −εHOMO = IP and the correct asymptotic behavior of the potential, the one that best fulfills the straight-line behavior of E(N), the energy as a function of a fractional electron number N, was found to provide the most accurate electron density as evidenced by calculated EFGs. The functional also performs well for related Cu systems. SECTION: Molecular Structure, Quantum Chemistry, General Theory

K

local exchange-correlation potential. Those have been called generalized Kohn−Sham (GKS) equations.4 The parameters α and β are typically determined empirically, quantifying the exact exchange contribution in the short-range/long-range region with its fraction changing from α to α + β as r12 changes from 0 to ∞. In eq 1, γ is the range-separation parameter. Since it is strongly system-dependent, it may not be particularly compatible with a universal functional parametrization. The system-dependence of γ calls for determining this parameter system-specifically (functional “tuning”).4 In the approach developed by Baer et al.,4 γ is determined such that the system fulfills a fundamental DFT requirement, namely, that in exact Kohn−Sham theory, the negative of the highest occupied molecular orbital (HOMO) energy is equal to the ionization potential (IP), i.e. −εHOMO(N) = IP(N).5 The IP for the N-electron system can be calculated with the same functional as a ground-state energy difference relative to the N − 1 electron system, IP(N) = Egs(N − 1) − Egs(N). Reference 4 gives an argument that the same criterion applies in GKS theories with hybrid and range-separated hybrid functionals. For an approximate functional and a given system, γ may hence be determined by minimizing the target functional6 J = |εHOMO(N) + IP(N)| to be internally consistent with the

ohn−Sham density functional theory (DFT) is widely used to determine the electronic structure of molecules and solids in an efficient manner. Approximate functionals may, however, lead to spectacular failures for predicted properties of “difficult” cases or for certain classes of properties. Many of the problems with pure (nonhybrid) functionals (local density approximation (LDA), gradient density approximation, (GGA)), and with standard hybrids, range-separated hybrids, and meta-GGAs, can be traced back to the static correlation problem, and to the delocalization error, which is intimately related to the spurious electron self-repulsion present in commonly used functional approximations.1 One of the better-known consequences of self-interaction, in a broader sense, is the breakdown of time-dependent DFT for chargetransfer (CT) excitations when using conventional pure and hybrid functionals (see citations in ref 2). This drawback can be alleviated or even avoided by using range-separated hybrids where the exchange is split into long-range and short-range parts, for instance, via separating the interelectronic distance r12 as in3 1 r12

=

α + β erf(γr12) 1 − [α + β erf(γr12)] + r12 r12

(1)

The Hartree−Fock (HF) exchange orbital functional is then used for the long-range part of the exchange interaction (with DFT, not HF orbitals). Variational minimization yields equations for the orbitals with a nonlocal exchange term of the same structure as is found in HF but accompanied by a © 2012 American Chemical Society

Received: December 22, 2011 Accepted: February 6, 2012 Published: February 6, 2012 576

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linearly as N varies between integers, with the slope of E(N) changing discontinuously at integer N. Too much or too little delocalization due to approximations in the functional gives positive or negative curvature, respectively, in plots of E(N).1 When selecting the parameters in eq 1 to simultaneously minimize J′2 and provide as close to straight-line E(N) behavior as possible, we obtain the most accurate electron density as far as the calculated Cu EFGs are concerned. All calculations were performed using a combination of the Tsuchiya−Abe−Nakajima−Hirao relativistic basis sets16 along with polarization functions from the TZVPP basis.17 Scalar relativistic computations with point-charge nuclei were performed with a locally modified developer’s version of the NWChem code. For Cu and Cl, finite nucleus corrections are minor compared to the functional variations of the EFGs. A detailed description of the computational details together with the corresponding literature references were previously provided in ref 18, which reports the EFG implementation in NWChem within the four-component density corrected approach based on the two-component relativistic zerothorder regular approximation (ZORA-4) framework utilized in the present work. Full computational details and references are also provided in the SI. Table 1 collects field gradients represented by V33, the largest magnitude principal component of the negative EFG. The

chosen functional and therefore without empirical fitting. A more refined version7,8 J ′2 =

1

∑ [εHOMO(N + i) + IP(N + i)]2 i=0

(2)

considers the IP of the N + 1 system as well, which may benefit properties that depend on the lowest unoccupied molecular orbital (LUMO) energy of the N-electron system; CT excitations come to mind. The tuned range-separated DFT approach has been applied successfully to problems that are considered challenging for DFT, such as Rydberg and CT excitations as well as response properties that relay on their proper description.2,4,8,9 In such problems, the orbital energies are of crucial importance. The question arises whether a system-specific approximate functional, tuned to satisfy certain energy-related DFT requirements, in fact gives a better electron density than a global parametrization. Herein, we examine the physically motivated tuning approach and apply it to a very challenging case of a molecular property that depends directly on the ground-state density: the copper electric field gradient (EFG) in the CuCl molecule.10−12 We also examine the dipole moment and the chlorine EFG. For this seemingly simple diatomic, most popular functionals fail to reproduce the sign and/or magnitude of the Cu EFG as determined by experiment and accurate wave function-based ab initio theory (coupledcluster). A series of careful studies by Schwerdtfeger and coworkers showed that the errors in the electronic density are mainly caused by drawbacks in the exchange part of the functional.10−12 This makes CuCl an appropriate test case in the context of tuning the parameters α, β, γ in range-separated functionals. In conjunction with nonempirical exchange-correlation (XC) functionals such as PBE, which we adopted here to represent the class of GGA XC functionals, the range-separation of eq 1 leads to a rather flexible functional with only few parameters. We aim to explore how these parameters can be determined nonempirically, in the aforementioned “tuning” sense, and the extent to which one obtains a fairly accurate electron density for the challenging case of CuCl in return. In recent work9 we have obtained some numerical evidence that a range-separated functional should be fully long-range corrected in order to be tunable for γ in the sense of eq 2, i.e., α + β = 1. The issue may be related to the finding that continuum spin-dependent XC potentials would afford VXC(∞) ≠ 0 = εHOMO + IP.13 The Supporting Information (SI) provides data for the CAMB3LYP functional (α + β = 0.65) showing that for CuCl J′2 does not reach a minimum as γ is varied within a reasonable range. We therefore adopt the constraint α + β = 1, which ensures that the potential has the desired asymptotic behavior going as −1/r. Even with this constraint, the numerical results reported below exhibit a continuous range of minima for J′2 in the α, γ plane, and therefore an additional criterion is needed to fix α and/or β. A two-dimensional tuning is performed herein, whereby the optimal α is selected such that the functional satisfies another DFT requirement. The DFT delocalization error can be considered as particularly important for deficiencies in calculated ground-state electron densities of closed-shell systems. Therefore, the aim is to eliminate delocalization error as best as possible.14,15 A criterion is furnished by the requirement that the energy E(N) should vary

Table 1. Cu and Cl EFGs (V33) and Dipole Moments (μ) Calculated for CuCl with HF and Different Density Functionals V33/a.u.

parametrization method

#

HF BLYP CAMB3LYP* LC-PBE0a

LC-PBE0-#b

DKCCSD(T)c Exp.d

1 2 3 4 5 6 7

α

Cl

μ/ Debye

−0.872 0.730 0.99 −0.301

1.235 2.570 1.771

6.504 4.349 5.782

−0.120 −0.649 −0.565 −0.462 −0.344 −0.222 −0.103 0.004 −0.341

1.852 1.510 1.576 1.650 1.727 1.799 1.863 1.915 1.715

5.625 6.170 6.067 5.950 5.824 5.707 5.607 5.530 5.319

−0.313

1.675

β

γ

1.00 0.00 0.40

0.18

0.25 0.712 0.649 0.570 0.473 0.362 0.239 0.107

0.75 0.288 0.351 0.430 0.527 0.638 0.761 0.893

0.30 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Cu

a

Default parametrization. bParametrizations determined via twodimensional tuning procedure in this work. Compare panel a of Figure 1. cDouglas−Kroll relativistic coupled-cluster data from ref 10. d Calculated from experimental NQCCs (ref 19) and currently recommended NQMs (ref 20).

functionals are HF, the nonhybrid functional BLYP, and various parametrizations of a fully long-range corrected (LC, α + β = 1) hybrid variant of PBE.21 Further, wave function-based ab initio and experimental data are provided. The CAM-B3LYP* entry is based on a parametrization proposed by Thierfelder et al.12 who fitted the functional parameters to reproduce the experimental Cu EFG in CuCl. The problem with exact exchange in the functional is evident in the huge variations between HF, BLYP which has the wrong sign, and among the LC-PBE0 variants. The comparison of HF and coupled-cluster 577

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Figure 1. (a) System-specific tuning of LC-PBE0 for CuCl. Contour plot of J′2 of eq 2 as function of α and γ, with α + β = 1. The thick (orange) line corresponds to J′2 ≈ 0. (b) Energy of CuCl as a function of fractional electron number ΔN relative to neutral system (ΔN = 0). Numerical values provide measures of curvature obtained from fitting quadratic functions to the data sets. Left: Comparison of HF, BLYP, CAM-B3LYP*, and LCPBE0 (default parametrization). Right: Comparison of the seven LC-PBE0-# parametrizations determined in this work.

Table 2. LMO Analysis of Cu V33 (a.u.) for CuCl at Various Levels of Theorya LC-PBE0-#

a

LMO

HF

BLYP

CAM*

1

2

3

4

5

6

7

σ(Cu−Cl) Cu Σ s Cu Σ 2p Cu Σ 3p Cu dδ (ea.) Cu dπ (ea.) Cu dσ Cu Σ 3d Cl Σ all total calcd σ (Cu−Cl) % Cu %p % dσ % 4s in dσ

−0.456 −0.002 −0.036 −0.763 8.609 −4.256 −8.086 0.620 −0.238 −0.872

−0.466 −0.051 0.086 0.041 8.774 −4.317 −7.569 1.345 −0.221 0.730

−0.467 −0.020 0.017 −0.458 8.730 −4.304 −7.998 0.854 −0.226 −0.301

−0.470 −0.009 −0.015 −0.624 8.659 −4.272 −8.076 0.698 −0.232 −0.649

−0.471 −0.010 −0.007 −0.583 8.667 −4.274 −8.048 0.738 −0.229 −0.565

−0.474 −0.015 0.001 −0.532 8.676 −4.277 −8.012 0.786 −0.228 −0.462

−0.482 −0.019 0.010 −0.472 8.689 −4.282 −7.971 0.843 −0.225 −0.344

−0.494 −0.025 0.022 −0.409 8.705 −4.287 −7.929 0.907 −0.222 −0.222

−0.512 −0.030 0.032 −0.343 8.725 −4.295 −7.892 0.968 −0.218 −0.103

−0.533 −0.036 0.042 −0.280 8.749 −4.304 −7.866 1.024 −0.213 0.004

7.74 4.34 2.22 1.81

20.08 0.59 7.76 8.05

11.77 2.54 3.51 3.38

9.51 3.75 2.55 2.31

10.10 3.50 2.76 2.56

10.77 3.25 3.07 2.90

11.49 3.04 3.46 3.33

12.18 2.91 3.91 3.83

12.76 2.84 4.39 4.36

13.22 2.83 4.85 4.87

Natural localized molecular orbitals; see ref 18 for details. CAM* = CAM-B3LYP*. For LC-PBE0-# parametrization, see Table 1.

is readily apparent from the large positive curvature. HF is too localized, giving negative curvatures in the E(N) plots. For the seven selected parametrizations that minimize J′2, the curvatures are significantly smaller than those seen for HF and BLYP, but there are still variations to which the density reacts sensitively, as shown below in the discussion of Table 2. Parametrization #4 overall gives the least curvature in E(N) when considering both the electrondeficient (ΔN < 0) and the electron-rich (ΔN > 0) portions of the plots. Parametrization #3 is better in the electrondeficient regime but worse in the electron-rich regime. With all criteria taken together, parametrization #4 emerges as the best one. It remains to be seen if tuning by satisfying multiple criteria can be achieved more generally with other functionals and for other molecules. The curvature parameters in Figure 1 indicate that the CAMB3LYP* functional affords a significantly stronger delocalization error than any of the LC-PBE0 variants. The CAMB3LYP* parametrization is difficult to justify on ab initio grounds. Compared to the coupled-cluster data in Table 1, the

data also highlights the large effect of electron correlation on the Cu EFG, which is in excess of 100% relative to the correlated ab initio value. The criterion J′2 of eq 2 is plotted in Figure 1a for varying α and γ with the constraint α + β = 1, using a 10 × 10 grid followed by interpolation. The thick orange line roughly in the center of the plot indicates the minimum region where J′2 is very close to zero (see SI). Within the numerical noise of the interpolation, there is a continuous range of α parameters for which γ can be tuned to minimize J′2. Seven points along this minimum line with γ ranging from 0.05 and 0.35 were chosen for EFG calculations. The corresponding data are given in Table 1 indicated by #1−7 and show that the Cu EFG is very sensitive to the parameter set α, β, γ. Figure 1b shows E(N) calculated for fractional electron numbers for these seven parametrizations as well as for the other functionals listed in Table 1. The data were fitted to quadratic functions. The coefficients of (ΔN)2 from the fits are provided as relative measures of the curvature in each set of E(N). The delocalization error of BLYP for the CuCl molecule 578

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Figure 2. Contours maps of electronic density difference for CuCl. (a−c) ΔρM−ΔρLC‑PBE0‑4, where M = HF, BLYP, CAM-B3LYP*. The numerical Δ values correspond to the difference in EFGs. (d) ρCuCl − ρCu+ − ρCl− for LC-PBE0-4. The contours values are ± p·2n, n = 0, 1, 2, ... with p = 0.0005 for panels a−c and p = 0.0014 for panel d.

LC-PBE0-4 functional furnishes very accurate EFGs, both for Cu and for Cl (the latter being less sensitive on a relative scale). The dipole moment is overestimated by 10% relative to DKCCSD(T) which, however, is likely related to differences in the basis sets in the outer valence regions to which the EFGs are less sensitive. The results of an analogous but nonrelativistic functional tuning are collected in the SI, showing similarly good agreement with nonrelativistic CC reference data. Although a protocol of using system-specific functionals to determine energy differences between different systems has not been established yet6 the calculated LC-PBE0-4 atomization energy appears to be in reasonable agreement with the DK-CCSD(T) reference data. For percent-deviations of EFGs from reference data and atomization energies for the set of LC-PBE0-# see the SI. We emphasize that the functional has not been fitted to experimental or calculated reference data in a semiempirical fashion. Figure 2 displays contour maps of the electron densities calculated with HF and different functionals relative to the LCPBE0-4 density, which is deemed to be the most accurate. The figure also shows the LC-PBE0-4 deformation density of CuCl relative to a Cu+Cl− promolecule with spherical atoms. Compared to the magnitude of the deformation density, the difference between the densities for CAM-B3LYP* and LCPBE0-4 is indeed comparatively small, particularly around Cu, showing that CAM-B3LYP* gives the correct Cu EFG because the density around Cu is of good quality in terms of contributions from p and d angular momenta. Most of the residual difference between the two functionals exhibits approximately spherical symmetry around the two atoms in CuCl and therefore it is not contributing much to the EFGs. The deviations of the calculated EFGs for HF and BLYP relative to LC-PBE0-4, which are of opposite sign, manifest in the contour maps as patterns with roughly opposite signs. A detailed analysis of the calculated Cu EFG in CuCl in terms of contributions from localized molecular orbitals (LMOs) is provided in Table 2. The trends in the calculated EFGs across the set of functionals is evidently linked to trends in the contributions from the Cu 3d- and 3p-shells; all other significant contributions remain fairly constant across the table. BLYP and the hybrid functionals with residual delocalization error afford comparatively large contributions from Cu 3d in the Cu−Cl bond. As a consequence, there is not enough charge

Table 3. Cu EFGs (V33) Calculated for Selected Diatomics with HF and Different Functionals V33/a.u. method HF BLYP CAMB3LYP* LC-PBE0a LC-PBE0-4b exp.c

CuH

CuF

CuCl

CuBr

CuI

OCCuCl

−0.625 0.801 −0.045

−1.272 0.917 −0.458

−0.872 0.730 −0.301

−0.730 0.700 −0.229

−0.566 0.643 −0.144

−1.951 −0.718 −1.453

0.107 −0.081 −0.016

−0.115 −0.473 −0.425

−0.120 −0.344 −0.313

−0.089 −0.281 −0.249

−0.052 −0.203 −0.153

−1.314 −1.511 −1.370

a

Default parametrization. bOptimal parametrization as determined in this work. cCalculated from experimental NQCCs (refs 19, 22−25) and currently recommended NQMs (ref 20).

density in the 3dσ orbital close to Cu, and the negative EFG contribution from 3dσ is not large enough in magnitude. For HF, the 3dσ involvement in the bond is smaller than with any of the density functionals. This trend is likely linked to the too weak delocalization (see curvatures in Figure 1). As a consequence, the EFG caused by the formally nonbonding 3dσ orbital is too negative. The right density is created by a delicate balance of the small covalent contribution of 3dσ in the predominantly ionic Cu−Cl bond. The picture is not complete without consideration of the contributions from the 3p semicore shell. With BLYP, the 3p contribution to the Cu EFG is almost zero and indicates an atom-like spherical p-shell. For HF there is a sizable negative EFG from the Cu 3p shell. Across the set of functionals, the larger the Cu p fraction in the Cu−Cl bond, the more strongly the Cu 3p shell contributes to the EFG. Further analysis of the LMOs and their “parent” natural bond orbitals shows that the Cu 3p are pure one-center orbitals without indications of delocalization. Therefore, the Cu 3p EFG is easiest explained not by accumulation or loss of charge density in 3pσ versus 3pπ but via different spatial extensions of 3pσ versus 3pπ in response to the formation of the Cu−Cl bond. In this sense, the EFG resulting from the Cu 3p shell is attributable to valence− semicore orthogonalization and semicore polarization driven by the mixing of Cu 4p in the valence orbitals. Regarding the good performance of the CAM-B3LYP* parametrization, the analysis gives orbital contributions that are very similar to those for 579

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LC-PBE0-4. The analysis therefore corroborates the visual assessment based on the agreement of the densities (Figure 2). A final test of the functional tuning is given for a set of six Cu containing compounds (including CuCl), which shows similarly pronounced functional dependencies of the copper field gradient. The EFGs listed in Table 3 show that the LCPBE-4 parametrization offers improvements over other common functionals for the whole set, similar to what has been found previously for CAM-B3LYP*.12 Additional data are provided in the SI. Thus, the conclusion drawn in ref 12, that a functional that gives an accurate density around Cu for CuCl also performs well for related Cu systems, transfers to the nonempirically determined (tuned) functional. This indicates that for a set of related molecules it may not be necessary to perform the time-consuming task of functional tuning for each species individually but only for selected members of the set. In summary, we have shown that for the difficult case of CuCl a functional that is “optimal” by virtue of certain energetic criteria rooted in DFT produces a fairly accurate electron density. The Cu EFG is caused by a delicate balance of d and p contributions in the partially covalent Cu−Cl bond that is sensitive to the exchange component of the functional. The delocalization error appears to play an important role in the failure of many functionals to predict even the correct sign of the Cu EFG. A two-dimensional tuning of the functional allows to determine the parameters α, β, and γ in the range-separated exchange kernel subject to one additional constraint, in our case α + β = 1 to ensure correct asymptotic behavior of the potential.

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REFERENCES

(1) Cohen, A. J.; Mori-Sanchés, P.; Yang, W. Insights into Current Limitations of Density Functional Theory. Science 2008, 321, 792− 794. (2) Autschbach, J. Charge-Transfer Excitations and Time-Dependent Density Functional Theory: Problems and Some Proposed Solutions. ChemPhysChem 2009, 10, 1−5. (3) Yanai, T.; Tew, D. P.; Handy, N. C. A New Hybrid ExchangeCorrelation Functional Using the Coulomb-Attenuating Method (CAM-B3LYP). Chem. Phys. Lett. 2004, 393, 51−57. (4) Baer, R.; Livshits, E.; Salzner, U. Tuned Range-Separated Hybrids in Density Functional Theory. Annu. Rev. Phys. Chem. 2010, 61, 85− 109. (5) Levy, M.; Perdew, J. P.; Sahni, V. Exact Differential Equation for the Density and Ionization Energy of a Many-Particle System. Phys. Rev. A 1984, 30, 2745−2748. (6) Livshits, E.; Baer, R. A Well-Tempered Density Functional Theory of Electrons in Molecules. Phys. Chem. Chem. Phys. 2007, 9, 2932−2941. (7) Stein, T.; Kronik, L.; Baer, R. Prediction of Charge-Transfer Excitations in Coumarin-Based Dyes Using a Range-Separated Functional Tuned from First Principles. J. Chem. Phys. 2009, 131, 244119(1)−244119(5). (8) Kuritz, N.; Stein, T.; Baer, R.; Kronik, L. Charge-Transfer-like π→π* Excitations in Time-Dependent Density Functional Theory: A Conundrum and Its Solution. J. Chem. Theory Comput. 2011, 7, 2408− 2415. (9) Srebro, M.; Autschbach, J. Tuned Range-Separated TimeDependent Density Functional Theory Applied to Optical Rotation. J. Chem. Theory Comput. 2012, 8, 245−256. (10) Schwerdtfeger, P.; Pernpointner, M.; Laerdahl, J. K. The Accuracy of Current Density Functionals for the Calculation of Electric Field Gradients: A Comparison with Ab Initio Methods for HCl and CuCl. J. Chem. Phys. 1999, 111, 3357−3364. (11) Bast, R.; Schwerdtfeger, P. The Accuracy of Density Functionals for Electric Field Gradients. Test Calculations for ScX, CuX, and GaX (X = F, Cl, Br, I, H and Li). J. Chem. Phys. 2003, 119, 5988−5994. (12) Thierfelder, C.; Schwerdtfeger, P.; Saue, T. 63Cu and 197Au Nuclear Quadrupole Moments from Four-Component Relativistic Density-Functional Calculations Using Correct Long-Range Exchange. Phys. Rev. A 2007, 76, 034502(1)−034502(4). (13) Tozer, D. J.; Handy, N. C. Improving Virtual Kohn−Sham Orbitals and Eigenvalues: Application to Excitation Energies and Static Polarizabilities. J. Chem. Phys. 1998, 109, 10180−10189. (14) Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Development of Exchange-Correlation Functionals with Minimal Many-Electron SelfInteraction Error. J. Chem. Phys. 2007, 126, 191109(1)−191109(5). (15) Sai, N.; Barbara, P. F.; Leung, K. Hole Localization in Molecular Crystals from Hybrid Density Functional Theory. Phys. Rev. Lett. 2011, 106, 226403(1)−226403(4). (16) Tsuchiya, T.; Abe, M.; Nakajima, T.; Hirao, K. Accurate Relativistic Gaussian Basis Sets for H through Lr Determined by Atomic Self-Consistent Field Calculations with the Third-Order Douglas−Kroll Approximation. J. Chem. Phys. 2001, 115, 4463−4472. (17) Weigend, F.; Ahlrichs, R. Balanced Basis Sets of Split Valence, Triple Zeta Valence and Quadruple Zeta Valence Quality for H to Rn: Design and Assessment of Accuracy. Phys. Chem. Chem. Phys. 2005, 7, 3297−3305. (18) Aquino, F.; Govind, N.; Autschbach, J. Electric Field Gradients Calculated from Two-Component Relativistic Density Functional Theory Including Spin−Orbit Coupling. J. Chem. Theor. Comput. 2010, 6, 2669−2686. (19) Low, R.; Varberg, T.; Connelly, J.; Auty, A.; Howard, B.; Brown, J. The Hyperfine Structures of CuCl and CuBr in Their Ground States Studied by Microwave Fourier Transform Spectroscopy. J. Mol. Spectrosc. 1993, 161, 499−510. (20) Pyykkö, P. Year-2008 Nuclear Quadrupole Moments. Mol. Phys. 2008, 106, 1965−1974.

ASSOCIATED CONTENT

S Supporting Information *

Cu and Cl EFGs and dipole moments calculated for CuCl. Example tuning of γ. LC-PBE0 parametrizations from twodimensional tuning. LC-PBE0 parametrizations from nonrelativistic calculations. Average relative deviations from reference data. Tests of energy for fractional charges. LCPBE0-# atomization energies. NMLO analysis of Cu EFGs for CuCl. Metal and halide EFGs and dipole moments calculated for selected Cu diatomics. Radial orbital expectation values ⟨r−3⟩ for Cu atom calculated using different nuclear models. Computational details. This material is available free of charge via the Internet at http://pubs.acs.org.

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Letter

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been supported by grant No. CHE 0952253 from the National Science Foundation. A portion of this work has also been supported by a U.S. Department of Energy grant (Basic Energy Sciences DE-SC0001136). M.S. is grateful for financial support from the Foundation for Polish Science (“START” scholarship) and from the Polish Ministry of Science and Higher Education (“Mobility Plus” program). The authors would like to acknowledge the Center for Computational Research (CCR) at the University at Buffalo for providing computational resources. 580

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(21) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (22) Okabayashi, T.; Tanimoto, M. Laboratory Measurement of the J = 1−10 Transition of Copper Hydride. Astrophys. J. 1997, 487, 463−465. (23) Evans, C. J.; Gerry, M. C. L. Noble Gas−Metal Chemical Bonding? The Microwave Spectra, Structures, and Hyperfine Constants of Ar−CuX (X = F, Cl, Br). J. Chem. Phys. 2000, 112, 9363−9374. (24) Batten, S. G.; Ward, A. G.; Legon, A. Observation of the Rotational Spectra of CuI and AgI Molecules Generated by Laser Ablation in a Pulsed-Jet, Fourier-Transform Microwave Spectrometer. J. Mol. Struct. 2006, 780−781, 300−305. (25) Walker, N. R.; Gerry, M. C. L. Microwave Spectra, Geometries, and Hyperfine Constants of OCCuX (X = F, Cl, Br). Inorg. Chem. 2001, 40, 6158−6166.

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NOTE ADDED AFTER ASAP PUBLICATION This paper was published ASAP on February 14, 2012. The Supporting Information was updated. The revised paper was reposted on February 23, 2012.

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