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Validation of Double-Hybrid Density Functionals for Electric Response Properties of Transition-Metal Systems: A New Paradigm Based on Physical Considerations Mojtaba Alipour* Department of Chemistry, College of Sciences, Shiraz University, Shiraz 71454, Iran S Supporting Information *

ABSTRACT: Double-hybrid density functional approximations are increasingly popular for electronic structure calculations within density functional theory. However, despite much progress in numerous interesting efforts in this respect, further extension of this approach to the chemistry and physics of transition-metal compounds poses major challenges that remain to be addressed. In the present article, without the use of any empirical fitting to experimental or high-level ab initio data, we propose a new parameter-free double-hybrid density functional, called mPWPW91DH, for the electric response properties of transition-metalcontaining molecules. It is based on a mixing of modified Perdew−Wang (mPW) and Perdew−Wang91 (PW91) generalized gradient approximations for exchange and correlation, respectively, along with Hartree−Fock (HF) exchange and a perturbative correlation term obtained from the Kohn−Sham orbitals and eigenvalues. The performance of this functional was tested on a number of representative test sets of static dipole polarizabilities and dipole moments of molecules containing transition metals and main-group elements. From our analysis, mPWPW91DH seems to represent a significant improvement in comparison to functionals on the different rungs of Jacob’s ladder. Moreover, scrutinizing the role of exchange and correlation and their contributions in the functionals shows evidence of the superiority of this new functional with respect to other parameter-free and parametrized double-hybrid functionals. The results of the present study are encouraging in terms of further improvements in double-hybrid approximations for investigating the response properties of more complex transition-metal systems.

1. INTRODUCTION Over the past two decades, from a broad array of electronic structure methods, Kohn−Sham density functional theory (KSDFT) has become one of the mainstream formalisms in modern quantum chemistry and solid-state physics.1−8 This is due to a favorable balance between the accuracy of the results and the computational economy of KS-DFT as compared to wave-function-theory- (WFT-) based methods. In KS-DFT, the exchange−correlation (XC) energy is the only term in the expression of the total energy that requires some approximations, and consequently, the major problem lies in finding a reliable expression for this contribution. Accordingly, it is not surprising that both the development of new XC functionals and their benchmarking are currently flourishing, with the aim of an increasingly accurate description of various properties of systems. The hierarchy of XC approximations has been formulated as a ranking, the so-called Jacob’s ladder, in which each rung corresponds to a different density functional approximation (DFA):9 local spin density approximation (LSDA), generalized gradient approximation (GGA), metaGGA, hybrid and hybrid meta-GGA, and nonlocal correlation approximations using virtual KS orbitals. Functionals on higher rungs include more complicated ingredients and are generally expected (but not guaranteed) to give higher accuracy. © XXXX American Chemical Society

A significant step toward the top of Jacob’s ladder has been made by incorporating not only a portion of the Hartree−Fock (HF) exchange energy into the exchange part but also a portion of the correlation energy calculated using perturbation theory truncated at second order (PT2), giving the so-called doublehybrid (DH) density functionals. This rung is our main focus in this study and is outlined in the next section in more detail. To date, all DH functionals have been specifically developed with the aim of providing improved performance for structural and energetic properties such as atomization energies, ionization potentials and electron affinities, bond dissociation enthalpies, reaction barrier heights, and interaction energies. Therefore, it is worthwhile to investigate the ability of DH approximations to predict some properties other than those reported earlier. Herein, we further investigate how well DH density functionals can describe the electric response properties of molecules. More specifically, the main goal of the present work is to develop a new DH functional, without the use of any empirical fitting to experimental or high-level ab initio data, for the dipole polarizabilities and dipole moments of transition-metalReceived: February 16, 2013 Revised: March 11, 2013

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dx.doi.org/10.1021/jp4016596 | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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connection formalism and Görling−Levy coupling constant perturbation expansion to the second order.22,23 The theoretical foundations of DH functionals were recently analyzed by several authors.24−27 Neglecting the density scaling in the correlation functional, that is, Ec[ρ1/λ] ≈ Ec[ρ], where Ec[ρ1/λ] is the correlation energy functional evaluated at the scaled density ρ1/λ(r) = (1/λ)3ρ(r/λ), Sharkas et al.24 provided a rigorous reformulation of the DH XC functionals, leading to the one-parameter double-hybrid (1DH) approximations

containing molecules. In recent years, transition-metal compounds have attracted great attention because they provide knowledge of chemical bonding involving electrons in d orbitals. Recent interest in these species is motivated by both basic research and practical applications. However, although numerous studies have been carried out on these systems, works on their electric response properties such as dipole polarizabilities are still few. The rest of this article is organized as follows: In section 2, the theoretical framework and technical details are explained. The results and a discussion of the general trends of the benchmark calculations are covered in section 3. Finally, we conclude the article by highlighting the main inferences in section 4.

1DH, λ Exc = λExHF + (1 − λ)ExDFT + (1 − λ 2)EcDFT

+ λ 2EcMP2

Later, by considering a linear scaling for the correlation energy functional, namely, Ec[ρ1/λ] ≈ λEc[ρ], and consequently modifying the λ dependence of the MP2 term, Brémond and Adamo25 proposed a new class of DH approximations as

2. THEORETICAL CONSIDERATIONS AND COMPUTATIONAL DETAILS 2.1. Double-Hybrid Density Functional Theory. From a practical point of view, to obtain the double-hybrid density functional theory (DH-DFT) energy expression, in the first step, a standard self-consistent KS-DFT calculation is performed using a hybrid density functional containing a semilocal generalized gradient approximation for exchange and correlation. Subsequently, the second-order Møller−Plesset (MP2) energy is calculated in the space of the converged KS orbitals. The total XC energy for the DH-DFT procedure, EDH‑DFT , is then obtained as xc

1DH, λ Exc = λExHF + (1 − λ)ExDFT + (1 − λ 3)EcDFT

+ λ 3EcMP2

ExDFT

(1)

EcDFT

where and are the semilocal exchange and correlation energies, respectively; EHF is the HF exchange x energy; and ax and ac are empirical parameters to be determined. The MP2 correlation energy expression, EMP2 c , is given by EcMP2

1 =− 4

∑∑ ij

ab

|⟨ϕϕ || ϕϕ ⟩|2 i j a b εa + εb − εi − εj

(4)

Equations 3 and 4 show that only one independent parameter λ is needed instead of the two parameters ax and ac in eq 1. However, these parameters can be determined by fitting using some reference data sets, as in the functionals proposed in refs 13−21, or it can be chosen on the basis of some physical convincing arguments, as in the PBE0-DH25 and PBE0-228 functionals. 2.2. mPWPW91DH Model. Earlier, we published a benchmarking study to assess the behaviors of a large number of density functionals in polarizability calculations of 4d transition-metal monohalides.29 Following that work, we also performed a complementary investigation to explore the effects of long-range corrections for computing the electric dipole polarizabilities of similar compounds.30 In our previous works, in addition to assessing DFT methods, a comparison between different exchange and correlation terms was carried out with the aim of determining the exchange and correlation functionals that can yield quantitatively accurate dipole polarizabilities. We found the modified Perdew−Wang (mPW) exchange functional31 to be the best in comparison to other exchange terms. Another important conclusion of our recent investigations is that the Perdew−Wang91 (PW91) correlation32 is the overall best correlation functional. The superiority of mPW and PW91 with respect to other exchange and correlation functionals for polarizability and hyperpolarizability calculations of transition-metal complexes has also been advocated in other studies.33−35 In this work, we made use of the superior performance of the DH-DFT procedure, which stems from a combination between the DFT and MP2 parts of the functional. DFT handles the short-range correlation and provides a more stable reference for the calculation of the MP2 energy, whereas the MP2 contribution provides the benefit of including the nonlocal dynamical electron correlation responsible for long-range interactions. Therefore, we follow the protocol of DH approximations (see eqs 3 and 4) to develop a new DH density functional based on combining mPW and PW91 terms for the electric dipole polarizability calculations. On the other hand, in a recent publication on the theoretical mixing coefficients for hybrid functionals, Cortona36 showed that the four values 1/2, 1/3, 1/4, and 1/5 for parameter λ in eqs 3 and 4 can be chosen that actually have the same theoretical basis.

ExcDH‐DFT = a x ExHF + (1 − a x )ExDFT + (1 − ac)EcDFT + acEcMP2

(3)

(2)

where i and j label occupied KS orbitals and a and b label virtual KS orbitals with corresponding orbital energies ε and ⟨ϕiϕj|| ϕaϕb⟩ represents the antisymmetrized two-electron integrals. The DH density functionals currently available can be classified into three groups according to which orbitals are used to evaluate the correlation perturbative terms. Zhao et al.10,11 originally coined the term double-hybrid and proposed the MC3BB and MC3MPW methods, in which the conventional MP2 energy is mixed with the DFT energy. Their purpose was to generalize the multicoefficient methods to allow mixing of wave-function-based methods with hybrid density functionals. Grimme12 developed the second class of DH functionals by proposing the B2PLYP method whereby the KS orbitals and eigenvalues from the B2LYP functional are used in the same way as in MP2 for a perturbative correlation energy evaluation. This approach has been demonstrated to be very successful for many chemical applications, and it has generated DH functionals such as mPW2PLYP, B2TPLYP, B2GPPLYP, B2πPLYP, ROB2PLYP, B2P3LYP, and B2PPW91.13−19 For some advances in the construction of DH density functionals as well as their performances, we refer the reader to a recent review by Zhang and Xu.20 The XYG3 functional21 represents another type of DH functional based on the adiabatic B



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were selected for study here because of their previously demonstrated relatively high accuracy for dipole polarizability calculations.29,30 The results of the coupled cluster methodology with single, double, and perturbative triple excitations, CCSD(T), were used as reference values for assessment of the performance of the functionals. In the context of basis set assessment and the construction of large, flexible, and adequately polarized basis in electronic property and dipole polarizability calculations, many efforts have been reported in the literature.48−52 In all calculations, for main-group elements and 3d transition metals, we used Dunning’s correlation consistent valence triple-ζ basis set augmented by diffuse polarization functions for all angular momenta in the basis set added in an even-tempered fashion.53−55 Moreover, for 4d and 5d transition metals, another triple-ζ basis set, namely, the Stuttgart−Dresden relativistic effective core potential (SDD basis set), was used.56−59 2.3.2. Polarizability Calculations. The dipole polarizabilities are response properties that characterize the ability of an electric field to distort the electronic distribution of a molecule. In their static form, they can be defined by the Taylor series expansion of the perturbed energy of an atom or molecule in the presence of a weak uniform external static electric field60,61

However, in light of the effects of nonlocal dynamical electron correlation on the polarizabilities, we are interested in proposing an MP2 model using DFT orbitals, namely, a functional with a significant contribution of nonlocal electron correlation, rather than a full DFT approach. Therefore, from the choices of eqs 3 and 4 with four possible values for parameter λ, only one option, namely, eq 3 with λ = 1/2, provides our desired form of DH approximations. It should be pointed out, however, that, for the selection of λ = 1/2, which can also be considered as a mixing parameter of the Becke halfand-half single-hybrid functional,37 a strong theoretical support using the original model of the adiabatic connection formula can be presented. For more details, please also see eqs 12−14 in ref 25, where the same value for parameter λ with eq 4 was used to present the PBE0-DH functional. Accordingly, the final XC energy expression for our proposed parameter-free DH density functional, EmPWPW91DH , takes the xc form ExcmPWPW91DH = 0.5ExHF + 0.5ExmPW + 0.75EcPW91 + 0.25EcMP2

(5)

where EmPW and EPW91 are the mPW exchange and PW91 x c correlation energy density functionls, respectively. This new functional is denoted in the following discussion as mPWPW91DH. 2.3. Details of Calculations. 2.3.1. Benchmark Sets, Functionals Studied, and Basis Sets. The performances of mPWPW91DH in comparison to other functionals and DHs were assessed on a set of benchmarks including static dipole polarizabilities and dipole moments of molecules containing 3d, 4d, and 5d transition metals and main-group elements. The functionals for which we present tests in this article, along with their types and percentages of HF exchange (HFX), are listed in Table 1. Apart from mPWPW91DH, the other functionals

z

1 E(F) = E(0) − ∑ μi Fi − 2 i=x −

typea

HFXb

ref(s)

mPWPW91DH B2PLYP B2PPW91 B3LYP BPW91 CAMB3LYP LC-ωPBE M05-2X mPW2PLYP PBE SVWN5 TPSS ωB97X-D

DHGGA DHGGA DHGGA HGGA GGA RSHGGA RSHGGA HMGGA DHGGA GGA LSDA MGGA RSHGGA-D

50 53 20 20 0 19−65c 0−100c 56 55 0 0 0 22.2−100c

present work 12 19 38, 39 32, 38 40 41 42 18 43 44, 45 46 47

z

z

∑ ∑ αijFF i j i=x j=x

z

∑ ∑ ∑ βijkFF i jFk

1 − 24

i=x j=x k=x z z z

z

∑ ∑ ∑ ∑ γijklFF i jFkFl − ··· i=x j=x k=x l=x

(6)

E(F) is the energy of the atomic or molecular system in the presence of the static electric field F; E(0) is its energy in the absence of the field; μi corresponds to the dipole moment of the system; αij to the static dipole polarizability tensor; and βijk and γijkl correspond to the first and second dipole hyperpolarizabilities, respectively. The properties of interest in this article are the dipole moments and static dipole polarizabilities. The measured data in experiments are usually mean dipole polarizabilities, which can be obtained by the trace of the polarizability tensor αxx + αyy + αzz α̅ = (7) 3

Table 1. List of Density Functionals Tested in This Investigation functional

1 6

z

z

To compute the components of polarizability tensor, we relied on the finite-field (FF) approach using a weak electric field strength of 0.001 Ehe−1a0−1.62 All calculations were carried out using the Gaussian 03 and Gaussian 09 codes.63,64

3. RESULTS AND DISCUSSION As a first test, we compare the performance of the newly mPWPW91DH functional with those of several functionals from different categories (see Table 1) on a set of static dipole polarizabilities of 4d transition-metal monohalides MX (M = Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, and Cd; X = F and Cl), SDP4dTMX set. Results for the statistical analysis of these functionals are collected in Table 2. Moreover, to make the key trends more visible, in Figure 1, we provide a graphical representation of the patterns for the mean signed errors (MSEs) and mean unsigned errors (MUEs) for the computed values of dipole polarizabilities using various functionals with

a

DHGGA, double-hybrid generalized gradient approximation; GGA, generalized gradient approximation; HGGA, hybrid generalized gradient approximation; HMGGA, hybrid meta generalized gradient approximation; LSDA, local spin density approximation; MGGA: meta generalized gradient approximation; RSHGGA, range-separated hybrid generalized gradient approximation; RSHGGA-D, range-separated hybrid generalized gradient approximation including empirical damped atom-pairwise dispersion terms. bPercentage of HF exchange in the functional. cPercentage of HF exchange in these functionals is distance-dependent. C



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Table 2. Statistical Analysis of the Relative Errorsa for the Computed Static Dipole Polarizabilities of the SDP4dTMX Set functional

MSEb

MUEc

R2 d

mPWPW91DH B2PLYP B2PPW91 B3LYP BPW91 CAMB3LYP LC-ωPBE M05-2X mPW2PLYP PBE SVWN5 TPSS ωB97X-D

−0.0022 −0.0446 −0.0196 −0.0200 0.0276 −0.0289 −0.0392 −0.0521 −0.0496 0.0590 0.0387 0.0292 −0.0097

0.0575 0.0735 0.0617 0.0738 0.1018 0.1168 0.0734 0.0655 0.0745 0.1168 0.1205 0.0905 0.1011

0.990 0.979 0.969 0.975 0.970 0.874 0.959 0.990 0.983 0.960 0.962 0.968 0.861

along with mPW exchange, yields substantially better results than the LYP correlation for polarizability calculations. In the next test, we evaluated the performance of mPWPW91DH in comparison to the recently proposed parameter-free DH functionals PBE0-2 and PBE0-DH and the parametrized DH functional B2PPW91. These functionals were assessed for predicting response properties including the static dipole polarizabilities of molecules containing 3d, 4d, and 5d transition metals as well as main-group elements, SDPTMM set, and dipole moments of some 4d transition-metal monohalides for which experimental data were available, DM4dTMX set.65−68 Table 3 summarizes the statistical Table 3. Comparison of the Performance of Parameter-Free and Parameterized DH Functionals for the Studied Properties of SDPTMM and DM4dTMX Test Sets SDPTMM

Relative errors were computed as {α̅ (DFT) − α̅ [CCSD(T)]}/ α̅ [CCSD(T)]. bMean signed error. cMean unsigned error. dRegression coefficient obtained using a standard least-squares linear fitting procedure. a

a

DM4dTMX

functional

MUEa

MaxUEb

MUEa

MaxUEb

mPWPW91DH B2PPW91 PBE0-2 PBE0-DH

0.046 0.067 0.073 0.065

0.095 0.113 0.245 0.099

0.051 0.053 0.078 0.064

0.094 0.083 0.120 0.120

Mean unsigned error. bMaximum unsigned error.

measures for the performance of mPWPW91DH, B2PPW91, PBE0-2, and PBE0-DH for these test sets. Moreover, the corresponding results are depicted in Figure 2. A more detailed presentation of each set comprising molecules and the computed values of their properties is available in the Supporting Information (Table S1).

Figure 1. Graphical representation of the patterns of the mean signed errors (light) and mean unsigned errors (dark) for the computed dipole polarizabilities of the SDP4dTMX set with respect to reference values.

respect to reference values. First, a glance at these results is sufficient to conclude that inclusion of a perturbative correction leads to an improvement in the computed static dipole polarizabilities, which, in turn, confirms the greater potentiality of DHs. From these results, it is evident that the mPWPW91DH functional (MSE = −0.0022 and MUE = 0.0575) yields an overall better description for the static dipole polarizabilities of 4d transition-metal monohalides not only with respect to DHs, but also in comparison to other functionals. We also compared the results of mPWPW91DH with those obtained by the DH functionals B2PLYP and mPW2PLYP in more detail. Considering the MP2 correlation and HF exchange contributions, it is observed that the physically determined parameter λ and, consequently, λ2 in mPWPW91DH (0.50, 0.25) are close to the empirical parameters ax and ac in B2PLYP (0.53, 0.27) and mPW2PLYP (0.55, 0.25). This, in turn, indicates that the parametrized functionals are also theoretically supported. However, despite this similarity, the results of these functionals are different, and according to the MUEs, the trend observed for their performance was mPWPW91DH > B2PLYP > mPW2PLYP, highlighting the importance of the type of exchange and correlation terms. In agreement with earlier studies,29,30,33−35 we found that the use of PW91 correlation,

Figure 2. Pictorial representation of the MUEs in the computed static dipole polarizabilities and dipole moments obtained using parameterfree and parametrized DH density functionals for the considered benchmark sets.

Focusing on the different test sets and using the results in Table 3, we found that, for the SDPTMM set, the error for mPWPW91DH was about 40% lower than that for PBE0-2 and 30% better than those for both B2PPW91 and PBE0-DH. For the DM4dTMX set, the largest error appeared for PBE0-2 (MUE = 0.078), and a large improvement was observed in going from PBE0-2 to the new functional mPWPW91DH, with the errors being reduced by up to 35%. In short, we arrived at the following trends for the functionals performance: mPWPW91DH > PBE0-DH > B2PPW91 > PBE0-2 for the SDPTMM set and mPWPW91DH > B2PPW91 > PBE0-DH > PBE0-2 for the DM4dTMX set. Overall, it is encouraging to again see the superior performance of mPWPW91DH D



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compared to that of any parameter-free and parametrized DHs to which it was compared in these tests. However, although the newly developed mPWPW91DH functional seems to be a promising starting point for investigating the response properties of transition-metal compounds within the framework of DH-DFT, this work raises some issues that ought to be addressed in the near future: • The application of DHs should be extended not only to large molecules and clusters but also to other problems such as the determination of interaction-induced electric properties for which more attention is required (see, for example, ref 69). For more details, we refer the interested reader to some recent reviews in this context.70,71 • Higher-order terms in eq 6 and, consequently, nonlinear optical properties should be considered, as they can provide a very solid basis for the clear distinction of the predictive capability of approximate computational methods on electric response properties. • The effect of dipole polarizability components should be taken into account. However, it is important to note that, in some cases in which large polarizability anisotropies are encountered,72−74 relying on the mean polarizability values does not necessarily lead to a reliable benchmarking. • Further studies should be performed in the direction of obtaining accurate results at a reasonable computational cost by using a more general DH density functional.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +98 711 6137160. Fax: +98 711 6460788. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Computer facilities from Shiraz University are gratefully acknowledged. REFERENCES

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4. CONCLUDING REMARKS In this work, on the basis of physical considerations, a new parameter-free DH density functional has been presented for the electric response properties of transition-metal-containing molecules. This functional contains contributions as 25% MP2, 50% HF exchange, 75% PW91 correlation, and 50% mPW exchange. We denote this functional mPWPW91DH. The performance of this functional was tested on a number of representative test sets as follows: static dipole polarizabilities for 4d transition-metal monofluorides and monochlorides, 5d transition-metal monofluorides, and some other transitionmetal-containing molecules and dipole moments for some 4d transition-metal monohalides. According to the obtained results, mPWPW91DH not only displays a significant improvement with respect to other functionals from different rungs of Jacob’s ladder, but also shows superior performance in comparison to the recently proposed parameter-free and parametrized DH functionals. However, further insight into the nature of DH density functionals targeted toward a fair judgment between the two philosophies of parameter-free and parametrized (empirical and nonempirical) approaches and the development of a more general functional to study the linear and nonlinear optical properties of transition-metal compounds still need to be investigated.

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ASSOCIATED CONTENT

S Supporting Information *

Table S1 and references on the structural analysis of all considered molecules in the benchmark sets. This material is available free of charge via the Internet at http://pubs.acs.org. E



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