How Well Can Parametrized and Parameter-Free Double-Hybrid

Article pubs.acs.org/JPCA

How Well Can Parametrized and Parameter-Free Double-Hybrid Approximations Predict Response Properties of Hydrogen-Bonded Systems? Dipole Polarizabilities of Water Nanoclusters as a Working Model Mojtaba Alipour* Department of Chemistry, College of Sciences, Shiraz University, Shiraz 71454, Iran ABSTRACT: The development of double-hybrid (DH) density functional approximations has been a crucial ingredient in the success of density functional theory (DFT). To further extend the range of applicability of these approximations, their suitability for investigating the response of hydrogen-bonded systems to external static electric fields is explored here. As a case study, we focus on the performance of parametrized and parameter-free DH density functionals for calculations of isotropic and anisotropic dipole polarizabilities of water nanoclusters. The functionals considered in our study are B2GP-PLYP, B2-PLYP, mPW2-PLYP, and SD-PBEP86 as parametrized functionals, and PBE0-DH and PBE0-2 as parameter-free models. The second-order Møller−Plesset perturbation theory (MP2) as a wave function theory (WFT) based method is also included for the sake of comparison. As a reference for methods benchmarking, the results of coupledcluster method with single and double excitations (CCSD) are used. Scrutinizing the role of exchange, correlation, and their contributions in the functionals reveals that PBE0-DH is the best method that predicts the most accurate isotropic polarizabilities by a large margin with respect to other tested functionals. Moreover, we found that the PBE0-DH has a smaller mean absolute deviation than MP2. On the other hand, our results show that PBE0-2 outperforms other methods for anisotropic polarizabilities of clusters studied. On the whole, among all the tested DH density functionals for dipole polarizabilities of water clusters, the parameter-free PBE-based models are found to offer the best overall performance.

I. INTRODUCTION Kohn−Sham density functional theory (KS-DFT) has become a much used tool for electronic structure calculations of atoms, molecules and solids in areas of chemistry and physics.1−7 In KSDFT, 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. Therefore, it is necessary to test the accuracy of density functionals to determine the range of their applicability and identify areas for improvement. The purpose of the current study is the benchmarking of a new class of approximations termed double-hybrid (DH) density functionals (section II) for electric response properties of water nanoclusters. Water clusters exist naturally in the atmosphere8 and can also be produced experimentally.9 Numerous experimental and theoretical investigations have been carried out on water clusters to gain insight into fascinating physical and chemical properties of water. Hydrogen bonds between water molecules provide the cohesive force that makes water a liquid at room temperature and favors extreme ordering of water molecules in ice. Moreover, these hydrogen bonds are responsible for the observed unusual © 2013 American Chemical Society

properties of water. Hence, water has received more attention than any other substance by researchers.10 One of the fundamental response properties of water clusters is their dipole polarizabilities, which is also our primary focus in this work. In general, a thorough understanding of the polarizabilities obtained from theoretical calculations is important in molecular science. The interplay between theory and experiment is a powerful tool that serves to identify which compound is observed in the experiment by a comparison of the calculated polarizabilities with the experimental values. Moreover, polarizability is sensitive to the structural geometry and delocalization of the valence electrons of molecules and can provide information on the electronic properties and geometrical features of the system. However, considering the difficulty and complexity of the experimental work in this respect, development and benchmarking of relatively low-cost computational methods such as DFT to predict accurate response properties is clearly of great interest. In this respect, there have been several Received: March 17, 2013 Revised: May 1, 2013 Published: May 15, 2013 4506

dx.doi.org/10.1021/jp402659w | J. Phys. Chem. A 2013, 117, 4506−4513

The Journal of Physical Chemistry A

Article

Table 1. Summary of the DH Density Functionals Benchmarked in This Work DH functionals

exchange term

parameterized B2GP-PLYPa B2-PLYPb mPW2-PLYPc SD-PBEP86d

Becke88 Becke88 modified Perdew−Wang Perdew−Burke−Ernzerhof

parameter-free PBE0-DHe PBE0-2f

Perdew−Burke−Ernzerhof Perdew−Burke−Ernzerhof

correlation term

contribution of HF exchange

contribution of MP2 correlation

Lee−Yang−Parr Lee−Yang−Parr Lee−Yang−Parr Perdew86

ax in eq 1 0.65 0.53 0.55 0.70

Perdew−Burke−Ernzerhof Perdew−Burke−Ernzerhof

λ in eq 4 0.50 0.79

ac in eq 1 0.36 0.27 0.25 0.25 same spin 0.53 opposite spin λ3 in eq 4 0.125 0.50

a

Reference 22. bReference 20. cReference 26. dReference 27 (without dispersion correction); in this spin component scaled double-hybrid (SD) functional additional flexibility has been provided by setting different weights to the same spin and opposite spin MP2 correlation (25% and 53%, respectively) in the spin component scaled MP2 (SCS-MP2) method. eReference 34. fReference 37.

B2-PLYP method where the KS orbitals and eigenvalues from B2LYP functional are used in the same way as in MP2 for a perturbative correlation energy evaluation. This approach has been proven to be very successful for many chemical applications and it has generated various types of DH functionals in recent years.21−28 For some advances in construction of DH density functionals as well as their performances, we refer the reader to a recent review by Zhang and Xu.29 XYG330 represents another type of DH functionals, which is based on the adiabatic connection formalism and Görling−Levy coupling constant perturbation expansion to the second order.31,32 The theoretical foundations of the DH functionals have recently been analyzed by several authors.33−36 Neglecting the density scaling in the correlation functional, i.e., 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.33 have provided a rigorous reformulation of the DH functionals, leading to the one-parameter double-hybrid (1DH) approximations,

studies on the performance of conventional density functionals for electric properties of water clusters.11−17 However, to the best knowledge of the present author, benchmarking of DH approximations for prediction of isotropic and anisotropic dipole polarizabilities of water clusters has not previously been reported in the literature.

II. THEORETICAL FRAMEWORK AND CONSIDERED FUNCTIONALS Double-hybrid density functional theory (DH-DFT) is an increasingly popular approach for electronic structure calculations within DFT. This theory not only incorporates Hartree− Fock (HF) exchange admixture in the same way that hybrid functionals do but also combines contributions from the semilocal correlation density functional with a proportion of second-order Møller−Plesset (MP2) correlation derived using the KS reference orbitals. Operationally, to obtain DH-DFT energy expression, in the first step a standard self-consistent KSDFT calculation is performed using a hybrid density functional containing a semilocal generalized gradient approximation for exchange and correlation. Subsequently, the MP2 energy is calculated in the space of the converged KS orbitals. The total XC energy for the DH-DFT method, EDH‑DFT , is then obtained by xc

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

+ λ 2EcMP2

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

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

(1)

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

where EDFT and EDFT are, respectively, semilocal exchange and x c correlation energies, EHF x is the HF exchange 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

+ λ 3EcMP2

(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 process using some reference data sets,21−29 or it can be chosen on the basis of some physical convincing arguments.34,37 The former are known as parametrized DH functionals and the latter are called parameter-f ree models. The functionals from each lineage for which we present tests in this article and the type of exchange and correlation terms used in each method as well as percentage of HF exchange and MP2 correlation are listed in Table 1. Because DH functionals include a proportion of nonlocal dynamical electron correlation, we have also included a WFT-based method, MP2, for the sake of completeness.

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

(3) 34

(2)

where i and j label occupied and a and b virtual KS orbitals with orbital energies ε and ⟨φiφj || φaφb⟩are the antisymmetrized twoelectron 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.18,19 coined, for the first time, the word double-hybrid and proposed MC3BB and MC3MPW methods, where conventional MP2 energy was mixed with DFT energy. Their purpose was to generalize the multicoefficient methods to allow mixing of the wave function based methods with hybrid density functionals. Grimme20 made the second class of DH functionals by proposing

III. COMPUTATIONAL DETAILS The dipole polarizabilities are response properties, which characterize the ability of an electric field to distort the electronic 4507



Article

Figure 1. Geometrical structures of water clusters (H2O)n with n = 2−6 optimized at the CCSD/aug-cc-pVDZ level of theory. See text for the labeling conventions below the structures.

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 to the first and second dipole hyperpolarizabilities, respectively. The response properties that we present and discuss in this study are isotropic polarizability, αiso, and anisotropic polarizability, αaniso. These quantities are related to the experiment and in terms of the Cartesian components are as follows

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 field,38,39 z

E(F) = E(0) − − −

1 6

z

∑ μi Fi − i=x z

1 2

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

αiso =

(5) 4508

1 (αxx + αyy + αzz) 3

(6)



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Table 2. Values of Isotropic and Anisotropic Dipole Polarizabilities (au) of the Water Clusters Computed Using Parameterized and Parameter-Free Double-Hybrid Density Functionals Compared with CCSD Results W2

W3

W4

W5

W6-B

W6-C

W6-P

W6-R

Isotropic Polarizabilities parameterized DHs B2GP-PLYP B2-PLYP mPW2-PLYP SD-PBEP86 parameter-free DHs PBE0-DH PBE0-2 CCSD parameterized DHs B2GP-PLYP B2-PLYP mPW2-PLYP SD-PBEP86 parameter-free DHs PBE0-DH PBE0-2 CCSD

αaniso =

18.99 19.32 19.15 19.06

28.76 29.27 28.99 28.88

38.64 39.42 39.06 38.73

48.69 49.55 49.10 48.91

58.24 59.28 58.73 58.50

57.60 58.63 58.08 57.88

57.26 58.28 57.74 57.53

58.60 59.64 59.10 58.86

18.53 18.55

28.12 28.11

37.88 38.96

47.61 47.59

56.97 56.94

56.41 56.37

56.07 56.02

57.30 57.27

18.59

28.12

37.92 47.66 Anisotropic Polarizabilities

57.00

56.39

56.04

57.36

3.63 3.74 3.70 3.60

5.46 5.60 5.56 5.39

7.10 7.72 7.62 7.23

9.60 9.94 9.82 9.58

9.23 9.55 9.43 9.20

8.00 8.22 8.15 7.92

5.32 5.37 5.35 5.22

11.11 11.56 11.39 11.13

3.60 3.51

5.47 5.31

7.41 7.28

9.53 9.29

9.16 8.93

7.94 7.76

5.31 5.22

11.00 10.73

3.47

5.22

7.10

9.15

8.80

7.70

5.21

10.55

⎛ 1 ⎞1/2 ⎜ ⎟ [(αxx − αyy)2 + (αxx − αzz)2 ⎝2⎠ + (αzz − αyy)2 + 6(αxy 2 + αxz 2 + αzy 2)]1/2

deviation (MaxAD) with respect to reference values (CCSD results) are used as statistical measures of density functionals performance. Table 2 contains the values of isotropic and anisotropic polarizabilities of the water clusters computed at the CCSD and DHs levels of theory. The accuracy plots in Figure 2 show the deviation of computed αiso and αaniso using parametrized and parameter-free DH density functionals (points) from CCSD values (line y = x). Moreover, we have provided a graphical representation of the signed deviations in the computed isotropic and anisotropic dipole polarizabilities in Figure 3a,b, respectively. As can be seen from Figure 3a, all the parametrized DH functionals display relatively similar trends for SDs in isotropic polarizabilities of water clusters. It is clear that the values of SDs increase monotonically with cluster size up to water hexamer and show a linear increasing pattern. However, for different hexamer conformations, the change in the computed SDs using various parametrized functionals is very smooth. On the other hand, the parameter-free models predict SDs which have a smooth variation with clusters size and consequently are almost stable for all considered clusters. Variation of SDs is more pronounced in the case of anisotropic polarizabilities, Figure 3b, where the values of αaniso and their SDs computed by parametrized and parameter-free DHs are highly dependent on the patterns of hydrogen bonding in various clusters. Moreover, our results reveal that for both αiso and αaniso, the values of SDs do not change systematically for all clusters. Although most of the DHs overestimate polarizabilities, we see that the polarizabilities are underestimated by some methods for a few of clusters. SDs (and mean SDs which are average over positive and negative values) by themselves cannot be used as estimators for assessment of density functionals. Therefore, for the purpose of the evaluation of methods performance, we used MAD and MaxAD. The relevant statistical measures of our investigation are reported in Table 3. The corresponding results for MP2 have also been reported. Moreover, absolute deviations of the computed

(7)

A detailed investigation on the isotropic and anisotropic dipole polarizabilities of water clusters at the coupled-cluster (CC) levels of theory, assessment of several density functionals, and the effect of basis set convergence through comparing the results of basis sets from the Dunning, Pople and Sadlej families has been performed by Hammond et al.14 They concluded that the use of the aug-cc-pVDZ basis set for the benchmarking study is a reasonable choice with considerable efficiency. Moreover, from various high-level correlation methods and basis sets tested, the use of CCSD/aug-cc-pVDZ results as reference data has been advocated. This methodology combines the advantages of both yielding highly accurate polarizabilities and being able to be applied to large clusters.40−42 Accordingly, the same methodology has been employed in the present work. For calculation of the corresponding components of each polarizability tensor, the field strength of 0.001 au was used.43−45 All DH-DFT and CCSD calculations were carried out using the Gaussian09 suite of codes.46

IV. RESULTS AND DISCUSSION Figure 1 depicts the geometries of the water clusters (H2O)n with n = 2−6 optimized at the CCSD/aug-cc-pVDZ level of theory.47 The labeling conventions that we will use in the figures and tables are as follows: (H 2 O) 2 :W2, (H 2 O) 3 :W3, (H 2 O) 4 :W4, (H 2 O) 5 :W5, (H 2 O) 6 -Book:W6-B, (H 2 O) 6 -Cage:W6-C, (H2O)6-Prism:W6-P, and (H2O)6-Ring:W6-R. To examine the effect of the clusters shape and structure on their polarizabilities and ability of DH density functionals for reproducing the corresponding dipole polarizabilities, we have included more than one structure for water hexamer. Signed deviation (SD), mean absolute deviation (MAD), and maximum absolute 4509



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Figure 2. Accuracy plots for isotropic (panel A) and anisotropic (panel B) polarizabilities computed using various DH density functionals. Position of points indicates positive or negative deviation with respect to CCSD data (line y = x).

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Figure 4. Graphical representation of the mean absolute deviation (dark bar charts) and maximum absolute deviation (light bar charts) for the computed isotropic (a) and anisotropic (b) polarizabilities using DH density functionals and MP2 method with respect to CCSD results. Figure 3. Trend of signed deviations in the computed isotropic (a) and anisotropic (b) polarizabilities using parametrized and parameter-free DH density functionals with respect to CCSD results.

values (0.102 and 0.174, respectively) in comparison to MP2 and other DHs. Because DH density functional calculations are less expensive than MP2 for large clusters, it is very encouraging that some of the DHs outperform MP2 for isotropic and anisotropic polarizabilities calculations. The next DH density functional that perform well for anisotropic polarizabilities is SD-PBEP86 with MAD = 0.259 au and MaxAD = 0.574 au. Although SD-PBEP86 method is constructed in the same spirit as other parametrized functionals by testing different types of exchange and correlation terms with the aim of good performance for various properties,27 additional flexibility has also been provided by setting different weights to the same spin and opposite spin MP2 (25% and 53%, respectively) in the spin component scaled MP2 (SCS-MP2) method.48 Note that the same spin MP2 reflects mostly longrange correlations and the opposite spin is related to short-range interactions. The ranking of the SD-PBEP86 highlights the utility of application of SCS-MP2 method in DH approaches for dipole polarizability calculations. Finally, the functionals PBE0-DH and B2GP-PLYP provide deviations comparable to each other and outperform mPW2-PLYP and B2-PLYP for polarizability anisotropy of water clusters. Let us analyze more in detail the results of DHs benchmarking. Comparing the behavior of exchange and correlation terms, one sees that the isotropic polarizabilities are well described by DHs including the PBE functional as exchange and/or correlation part. Moreover, we could not reach to a general conclusion about the relationship between performance of different DH functionals for isotropic polarizabilities and their percentage of HF exchange and MP2 correlation. However, if we restrict ourselves to parametrized functionals for anisotropic polarizabilities, it seems that the higher the percentage of HF exchange the better the performance of functional. It should be noted that although the high percentage of HF exchange leads to better performance of functionals for dipole polarizabilities, this is not a general conclusion. Observe, from Table 3, that the PBE0-DH functional

Table 3. MAD and MaxAD of the Computed Isotropic and Anisotropic Polarizabilities of Water Clusters under Investigation (All Values in au) αiso B2GP-PLYP B2-PLYP mPW2-PLYP SD-PBEP86 PBE0-DH PBE0-2 MP2

αaniso

MAD

MaxAD

MAD

MaxAD

0.961 1.787 1.358 1.156 0.039 0.169 1.033

1.234 2.280 1.736 1.499 0.065 1.036 1.318

0.284 0.562 0.476 0.259 0.279 0.102 0.122

0.559 1.002 0.834 0.574 0.449 0.174 0.320

values of isotropic and anisotropic dipole polarizabilities using all considered methods with respect to CCSD results are shown in Figure 4a,b, respectively. We first analyze the results for isotropic polarizabilities. It is evident that the most accurate values for αiso of water clusters are obtained by the recently introduced parameter-free DH models PBE0-DH (MAD = 0.039 au and MaxAD = 0.065 au) and PBE02 (MAD = 0.169 au and MaxAD = 1.036 au) with statistical measures that are even smaller than that of MP2. Note that in these methods the ratio of HF, KS, and MP2 terms has been determined on the basis of physical considerations. The next functional is B2GP-PLYP, as a general purpose (GP) method,22 with MAD = 0.961 au and MaxAD = 1.234 au. For the rest of functionals we find a trend as SD-PBEP86 > mPW2-PLYP > B2PLYP. Concerning the corresponding results for anisotropic polarizabilities of water clusters, we see that the PBE0-2 as a parameter-free functional gives the lowest MAD and MaxAD 4511



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Notes

with 50% HF exchange does better than other high-exchange methods for isotropic polarizabilities. Also, for instance, in PBE0DH as a top ranked method for isotropic polarizability calculations the resulting coefficient for the exchange is close to that obtained upon optimization in other DH approaches such as B2-PLYP and mPW2-PLYP whereas the one for the MP2 correlation is significantly lower than those of other functionals (Table 1). Indeed, the optimum value of HF, KS, and MP2 terms is both method- and system-dependent. Ultimately, we see that the results of DH functionals for electric response properties, as for various physical and chemical properties reported in the literature, approach the accuracy of ab initio composite methods at a fraction of their computational cost. Thanks to the availability of DH approximations up to now and, however, if the computational cost is not a serious obstacle for DHs (because of the added perturbational correlation term like MP2 to the DFT correlation), the double-hybrid DFT-based methodologies and particularly parameter-free ones should receive more considerations to investigate the electric response properties of hydrogen-bonded systems.

The authors declare no competing financial interest.

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V. CONCLUSIONS AND OUTLOOK In this paper, the performance of double-hybrid approximations including four parametrized density functionals (B2GP-PLYP, B2-PLYP, mPW2-PLYP, and SD-PBEP86) and two parameterfree models (PBE0-DH and PBE0-2), and MP2 as a representative example of WFT based methods has been assessed with respect to CCSD for the static dipole polarizabilities of water nanoclusters (H2O)n with n = 2−6. The main conclusions emerged from our study, including the recommended functionals for similar applications, are listed below: 1. From the values of signed deviations we found that although most of the DHs overestimate the isotropic and anisotropic polarizabilities, the dipole polarizabilities are underestimated by some methods for a few of clusters. 2. Considering the role of exchange, correlation, and their contributions in the functionals, it turned out that PBE0DH is the best method that predicts the most accurate isotropic polarizabilities by a large margin with respect to others. Moreover, the PBE0-DH has a smaller mean absolute deviation than MP2. 3. Our results show that the PBE0-2 outperforms other methods for polarizability anisotropy calculations of water clusters. 4. As in conventional density functionals,49,50 we could not reach to a general conclusion about the relationship between performance of different DHs and their percentage of HF exchange and MP2 correlation. 5. Looking at the overall performance, the most recommended methods to predict dipole polarizabilities of water clusters are parameter-free functionals PBE0-DH and PBE0-2. Hopefully, such benchmarks can pave the way for developers to make further improvements on existing models in the direction of proposing a generally applicable DH functional for electric response properties of systems in which hydrogen bond plays an important role.

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