Novel Recipe for Double-Hybrid Density Functional Computations of

Article pubs.acs.org/JPCA

Novel Recipe for Double-Hybrid Density Functional Computations of Linear and Nonlinear Polarizabilities of Molecules and Nanoclusters Mojtaba Alipour* Department of Chemistry, College of Sciences, Shiraz University, Shiraz, Iran ABSTRACT: Double-hybrid (DH) density functionals are now among the most applied methods for quantum chemical calculations within density functional theory (DFT). In this work, a new DH density functional is developed for linear and nonlinear optical properties of molecules and hydrogen-bonded nanoclusters. The proposed functional, denominated as PBEDH-P (-P stands for polarizability), is based on Perdew−Burke−Ernzerhof (PBE) exchange and correlation functionals and includes 68% Hartree−Fock exchange and 31% correlation from second-order Møller−Plesset perturbation theory (MP2). From the obtained results, PBEDH-P is shown to be accurate for the calculations of hyperpolarizability, isotropic and anisotropic polarizabilities, and dipole moment of molecules and hydrogen-bonded nanoclusters of H2O (neutral, protonated, and deprotonated), NH3, HF, and binary mixtures of HF−H2O. This novel DH functional not only reveals a considerable improvement in comparison to the recently proposed parameter-free and parametrized DHs but also seems to be superior to the MP2 method in some cases. Moreover, we find that using only contributions of electron pairs with opposite spin for the perturbative part within scaled opposite-spin scheme does not represent a great improvement over PBEDHP. On the whole, our study nominates PBEDH-P as a promising model for the calculations of electric response properties, where the DH density functionals again come into play and further evidence of the quality of these approximations are highlighted.

1. INTRODUCTION The advent of density functional theory (DFT)1−3 has enabled electronic structure calculations to be performed on medium to large systems. The most remarkable characteristic of DFT is the exchange-correlation (XC) energy part that requires being approximated and therefore the results of DFT depend on the form of this term. Although no systematic route for improving on a given DFT approximation exists, a classification for XC functionals has been suggested, the “Jacob’s ladder” approach,4 connecting the earth (Hartree theory) to the heaven (chemical accuracy). This is based on the types of information entailed in the construction of density functional approximations (DFAs): first rung, local spin density approximations (LSDAs); second rung, generalized gradient approximations (GGAs); third rung, meta-GGAs; fourth rung, hybrid-GGAs and hybrid metaGGAs; and fifth rung, the approximations in which virtual (unoccupied) orbitals make their entry. A special case of fifthrung functionals, which is also our concern in the present investigation, corresponds to double-hybrids (DHs) that combine correlation energy arising from second-order Møller−Plesset perturbation theory (MP2) together with a DFT correlation energy, in addition to the hybrid combination of orbital-dependent Hartree-Fock (HF) exchange energy and an exchange energy functional from GGA (or meta-GGA). We mention in passing that another scheme to combine DFT correlation with wave function based correlation is the DFT− random-phase approximation (DFT-RPA).5 © 2014 American Chemical Society

Given the rich collection of references on the proposals and tests of DHs to predict various energetic properties in recent years (see the following section), few are found for applications of these approximations to electric response properties. To the best knowledge of the present author, no DH density functional for linear and nonlinear optical properties of hydrogen-bonded networks has been proposed as of yet. The study of hydrogen-bonded nanoclusters is a topic of academic and technological interest because of their importance in both applied and fundamental sciences. We are lately witnessing the blooming of a large number of studies on various properties of these types of systems. One of the fundamental response properties of nanoclusters containing hydrogen bonds is their polarizabilities. Dipole polarizability as a fundamental characteristic of each system determines the dynamical response of a bound system to external fields and can provide valuable information on the microscopic features of nanoclusters and the behavior of low-dimensional systems. In light of the important applications of this central property, the theory of electric dipole polarizabilities is an essential issue to the rational approach and interpretation of large classes of phenomena. In this respect, development of computational procedures to predict accurate polarizabilities is of great attention. Hereby, following our more recent study6 on the Received: April 23, 2014 Revised: June 11, 2014 Published: July 8, 2014 5333

dx.doi.org/10.1021/jp503959w | J. Phys. Chem. A 2014, 118, 5333−5342

The Journal of Physical Chemistry A

Article

Figure 1. Geometrical structures of hydrogen-bonded nanoclusters under study.

2. THEORETICAL FRAMEWORK As mentioned previously, in double-hybrid density functional theory (DHDFT), not only the exchange is a hybrid of GGA (or meta-GGA) and HF exchange but also the correlation has a hybrid nature, DFT and MP2 correlation, in the basis of the Kohn−Sham orbitals. Using a combination of all these terms, , is expressed as follows the total XC energy, EDHDFT xc

superior performance of DHs compared to the conventional functionals from previously available rungs for the electric response properties, we propose a new DH functional for the linear and nonlinear optical properties of molecules and hydrogen-bonded nanoclusters and demonstrate its superiority to the earlier DHs. 5334



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ExcDHDFT = wxHFExHF + (1 − wxHF)ExDFT + wcDFTEcDFT + wcMP2EcMP2

(1)

where EHF is the HF exchange energy, EDFT and EDFT are, x x c respectively, semilocal exchange and correlation energies, and EMP2 is the MP2 correlation energy; c EcMP2

1 =− 4

∑∑ ij

ab

|⟨φφ || φφ ⟩|2 i j a b

αiso =

1 (αxx + αyy + αzz) 3

αaniso =

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

εa + εb − εi − εj

(2)

βtot = (β x 2 + βy 2 + βz 2)1/2

Here, i and j label occupied and a and b virtual Kohn−Sham orbitals with orbital energies ε and ⟨φiφj∥φaφb⟩ are the DFT MP2 antisymmetrized two-electron integrals. wHF x , wc , and wc are mixing coefficients to be determined. In order to reduce the number of linearly independent parameters, wDFT is usually c defined as 1 − wMP2 c . DH density functionals differ by construction and underlying physics. Historical reviews on the design of functionals within the DH category and related developments can be found in refs 7−42. For details on the theoretical basis of the DH density functionals we refer the interested reader to refs 23−25 and 38. Considering a linear scaling for correlation energy functional, i.e., Ec[ρ1/λ] ≈ λEc[ρ], where Ec[ρ1/λ] is the correlation energy functional evaluated at the scaled density ρ1/λ(r) = (1/λ)3ρ (r/ λ), Brémond and Adamo22 have proposed a new formulation for one-parameter double-hybrid (1DH) approximations as follows:

(5)

(6) (7)

In eq 7, βx, βy, and βz are explicitly written, in terms of components of the hyperpolarizability tensor, as βx = βxxx + βxyy + βxzz, βy = βyxx + βyyy + βyzz, and βz = βzxx + βzyy + βzzz where the values of the components can be extracted from the results output in Multiwfn program.48 As a test case of hydrogen-bonded systems, we consider the neutral water clusters (H2O)n (n = 2−6) with the structures optimized at the CCSD/aug-cc-pVDZ level. To examine the effect of the shape of clusters on dipole polarizabilities and applicability of the proposed functional to obtain the corresponding polarizabilities, we have also included ionized water clusters and four structures for water hexamer.6 Furthermore, other hydrogen-bonded nanoclusters have been benchmarked: dimer and trimer of NH3, tetramer up to hexamer of HF, and six binary mixtures of HF−H2O.49,50 In addition, the linear and nonlinear polarizabilities of protonated and deprotonated water nanoclusters, H3O+ (H2O)n and OH−(H2O)n,51 have also been considered as another test set. The corresponding geometrical structures for the considered nanoclusters are shown in Figures 1 and 2. Finally, dipole− dipole polarizabilities for a series of small molecules have been considered as the last test set. Calculations of (hyper)polarizabilities and dipole moments of nanoclusters and molecules have been performed using the aug-cc-pVDZ and 6-311++G(3df,3pd) basis sets, respectively. Moreover, in cases

1DH, λ Exc = λExHF + (1 − λ)ExDFT + (1 − λ 3)EcDFT + λ 3EcMP2

(3)

in which, as compared to eq 1, the number of parameters has been reduced. These parameters can be determined using a fitting process (leading to parametrized DHs) or on the basis of some proper physical arguments (leading to parameter-f ree DHs). In the present work, in light of our earlier findings on the superior performance of Perdew−Burke−Ernzerhof (PBE)43−45 based functionals for dipole polarizability calculations, we follow the protocol of DH approximation (eq 3) to propose a novel DH functional based on combining PBE exchange and correlation functionals. The technical details to construct the form of the proposed functional will be explained in section 4.

3. COMPUTATIONAL DETAILS The perturbed energy of an atom or molecule in the presence of a weak uniform external static electric field can be expanded as follows,46,47 z

E(F) = E(0) −

∑ μi Fi − i=x

1 − 24

z

z

z

1 2

z

z

∑ ∑ αijFF i j − i=x j=x

1 6

z

z

z

∑ ∑ ∑ βijkFFF i j k i=x j=x k=x

z

∑ ∑ ∑ ∑ γijklFFF i j kFl − ... i=x j=x k=x l=x

(4)

Here, E(F), E(0), μi, αij, βijk, and γijkl are, respectively, the energy of the atomic or molecular system in the presence of the static electric field F, energy in the absence of the field, dipole moment of the system, static polarizability, first hyperpolarizabilities, and second hyperpolarizabilities. The response properties that we report in this work are isotropic polarizability (αiso), anisotropic polarizability (αaniso), and total first order hyperpolarizability (βtot). These quantities are related to the experiment and in terms of the Cartesian components are defined as

Figure 2. Geometrical structures of protonated and deprotonated water nanoclusters under study. 5335



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combining PBE exchange and correlation terms. Recently, Cortona55 has suggested that four values 1/2, 1/3, 1/4, and 1/5 for parameter λ in eq 3 can be chosen and that all of these values can be justified using the same theoretical arguments. Since we are interested in proposing the best combination of PBE exchange and correlation terms based on eq 3 for response properties, all theoretical mixing coefficients have been tested. Shown in Figure 3 is the relationship between the values of λ and MADs of the computed anisotropic polarizabilities of water nanoclusters using the resulting functionals. This figure represents how MADs decrease with HF exchange energy increasing. Moreover, from Figure 3, it is seen that there exists a linear correlation (R2 = 0.99) between MADs and λ values. We find that the parameter λ can be approximately extrapolated to be 68% to approach the MAD to zero. Employing this value of HF exchange and the resulting contribution for MP2 correlation (≈ 31%) with eq 3, the energy expression for our proposed functional, denoted as PBEDH-P (-P stands for polarizability), contains mixing parameters as follows: 68% for PBE PBE MP2 EHF x , 32% for Ex , 69% for Ec , and 31% for Ec ,

where the experimental data are not available the results of CCSD/aug-cc-pVDZ model chemistry are employed as a reference. In order to compute the corresponding components of each tensor in eq 4 and consequently the quantities defined 52,53 in eqs 5−7, the field strength of 0.001 Eh e−1 a−1 0 was used. Details on the employed computational philosophy for nanoclusters have been presented in sufficient clarity in our previous work6 and will not be repeated here. We have used the Gaussian09 suite of codes for all of the calculations.54 In order to analyze the results, we have employed the usual statistical measures as the mean signed deviation (MSD), mean absolute deviation (MAD), maximum absolute deviation (MaxAD), and the related metrics. With Δi as the deviation from the corresponding reference values and n given values, these measures are defined respectively as MSD = (1/n)∑ni Δi, MAD = (1/n)∑ni |Δi|, and MaxAD = maxi|Δi|.

4. RESULTS AND DISCUSSION As noticed in the previous sections, the proposed functional in this work has been based on the DH approximation with

ExcPBEDH ‐ P = 0.68ExHF + 0.32ExPBE + 0.69EcPBE + 0.31EcMP2 (8)

Zooming in the values of coefficients for ab initio and DFT parts, we find that this new functional shares almost similar mixing contributions for HF exchange and PBE correlation, ≈70% (ExHF + EcPBE), and for PBE exchange and MP2 correlation, ≈30% (EPBE + EMP2 x c ). Plus, considering the higher percent of the HF exchange term in PBEDH-P functional with respect to the low-HF exchange DH models, the problem of the self-interaction error is expected to be reduced. In Figure 4, we compare the performance of the PBEDH-P with other DH functionals for the prediction of isotropic polarizabilities of water nanoclusters. Moreover, since DHs include a proportion of nonlocal dynamical electron correlation, the results of MP2 have also been included for the sake of completeness. From Figure 4 it is observed that the most accurate values of isotropic components are obtained from the PBEDH-P model with statistical measures as MAD = 0.34 au and MaxAD = 0.44 au. Next, the predicted isotropic

Figure 3. Mean absolute deviation (MAD) of the computed anisotropic polarizabilities of water nanoclusters as a function of the parameter λ in eq 3. λ = 1/2, 1/3, 1/4, and 1/5 correspond to PBE0DH-1/2 (denoted as PBE0-DH in ref 22), PBE0DH-1/3, PBE0DH-1/4, and PBE0DH-1/5 functionals, respectively.

Figure 4. Graphical representation of the MAD and MaxAD for the computed values of isotropic polarizabilities of water nanoclusters at DHDFT levels with respect to references values. 5336



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Figure 5. Trend of MADs variations in the calculation of isotropic polarizabilities of water nanoclusters using PBE-based functionals upon climbing up the Jacob’s ladder.

Table 1. Performance of B2GP-PLYP, B2-PLYP, PBE0DH-1/2, and PBEDH-P for the Prediction of Anisotropic Polarizabilities of Hydrogen-Bonded Nanoclusters under Studya deviation nanocluster

reference

(H2O)2+ (H2O)3+ (H2O)4+ (HF)4 (HF)5 (HF)6 (NH3)2 (NH3)3 HF-H2O HF−(H2O)2 HF−(H2O)3 (HF)2−H2O (HF)3−H2O (HF)2−(H2O)2

12.41 10.83 13.77 6.97 9.55 11.00 4.85 8.02 4.52 5.34 8.28 4.97 7.69 8.66

MAD MaxAD

b

B2GP-PLYP

B2-PLYP

PBE0DH-1/2

PBEDH-P

1.60 1.24 1.73 0.17 0.20 0.24 0.31 0.50 0.16 0.19 0.24 0.13 0.15 0.15

3.22 2.64 3.21 0.32 0.40 0.49 0.59 0.98 0.29 0.32 0.42 0.24 0.31 0.33

1.09 1.22 1.63 0.13 0.14 0.16 0.12 0.15 0.10 0.18 0.22 0.11 0.11 0.09

0.14 0.35 0.67 0.02 0.01 0.00 0.02 0.08 0.01 0.08 0.10 0.02 0.00 0.02

0.50 1.73

0.98 3.22

0.39 1.63

0.11 0.67

a

Statistical descriptors (shown in boldface) are mean absolute deviation (MAD) and maximum absolute deviation (MaxAD). All values are in atomic units. bReference data are from CCSD/aug-cc-pVDZ model chemistry.

significantly reduced when going from pure to hybrid and from hybrid to double-hybrid functional, highlighting the importance of dynamical electron correlation for polarizability calculation of hydrogen-bonded systems. Concerning now the effect of the clusters shape and hydrogen bonding networks on response properties, we examine the validity of PBEDH-P functional for isotropic and anisotropic polarizabilities and dipole moments of other hydrogen-bonded systems such as nanoclusters of (H2O)+, NH3, HF, and HF−H2O (see Figure 1) that were not included in the process of parameters optimization. In Tables 1 and 2, the performance of PBEDH-P has been tested in comparison to the parametrized DH functionals B2-PLYP and B2GP-PLYP and the parameter-free DH functional PBE0-DH (called PBE0DH-1/2 in our discussion) for the anisotropic and isotropic polarizabilities, respectively. As shown by data in Table 1 and as expected from the process of obtaining the mixing coefficients using anisotropic polarizabilities of water

polarizabilities from B2GP-PLYP (MAD = 0.96 au and MaxAD = 1.23 au) and MP2 (MAD = 1.03 au and MaxAD = 1.32 au) are the closest ones to the reference values. According to the MADs, the following classification from lower to higher deviations is established for the other tested methods: SDPBEP86 > mPW2-PLYP > B2-PLYP. Finally, a closer look to Figure 4 reveals that the polarizabilities are well predicted by PBE-based DHs as compared to other forms of functionals. On the other hand, the comparison between DHs and conventional functionals should also be taken into account. In fact, considering the functionals from previous rungs of Jacob’s ladder constitutes the way for further discussing the relative performance across the hierarchy of functionals. To verify this point, Figure 5 exhibits the performance of three categories of = 0 and wMP2 = 0), PBE-based functionals, i.e., pure (wHF x c HF MP2 hybrid (wx ≠ 0 and wc = 0), and double-hybrid (wHF x ≠ 0 ≠ 0), in the calculation of isotropic polarizabilities of and wMP2 c water nanoclusters. It can be seen that the MADs are 5337



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Table 2. Performance of B2GP-PLYP, B2-PLYP, PBE0DH-1/2, and PBEDH-P for the Prediction of Isotropic Polarizabilities of Hydrogen-Bonded Nanoclusters under Studya deviation nanocluster

referenceb

B2GP-PLYP

B2-PLYP

PBE0DH-1/2

PBEDH-P

(H2O)2+ (H2O)3+ (H2O)4+ (HF)4 (HF)5 (HF)6 (NH3)2 (NH3)3 HF−H2O HF−(H2O)2 HF−(H2O)3 (HF)2−H2O (HF)3−H2O (HF)2−(H2O)2

24.60 34.95 46.23 21.07 26.59 32.13 27.60 42.02 14.03 23.57 33.26 19.44 24.95 28.95

1.25 1.54 2.20 0.48 0.56 0.67 0.54 0.89 0.28 0.51 0.68 0.43 0.51 0.58

2.33 2.90 3.95 0.88 1.06 1.28 0.96 1.58 0.52 0.93 1.27 0.78 0.96 1.09

0.18 0.22 0.60 0.09 0.09 0.10 0.04 0.09 0.02 0.00 0.04 0.03 0.02 0.02

0.22 0.26 0.05 0.10 0.15 0.18 0.20 0.20 0.11 0.16 0.25 0.12 0.17 0.22

0.79 2.20

1.46 3.95

0.11 0.60

0.17 0.26

MAD MaxAD a

Statistical descriptors (shown in boldface) are mean absolute deviation (MAD) and maximum absolute deviation (MaxAD). All values are in atomic units. bReference data are from CCSD/aug-cc-pVDZ model chemistry.

Figure 6. Comparison of the general performance of PBEDH-P with parameter-free and parametrized DH functionals based on SMAD and TSMAD.

the SMAD values have also been calculated. Since other functionals are compared with PBEDH-P, its SMAD is set to be 1. If an SMAD is lower than 1, the corresponding functional has a better performance than PBEDH-P, while the functional for which SMAD > 1 has a worse performance with respect to PBEDH-P. Figure 6 displays the general performance of functionals based on SMAD and TSMAD values. Not only for each of the studied properties but also from a more general point of view we can see evidence of the superiority of PBEDHP in comparison to other parametrized and parameter-free DHs. With the aim of reducing the computational cost for predicting the dipole polarizability of larger clusters, we have also considered another tack on our functional in which the perturbative treatment is carried out within a scaled oppositespin (SOS) scheme;56,57 see refs 13, 19, 20, 26, and 35 for some applications in this context. Practically, in this approach it is

nanoclusters, the PBEDH-P functional (MAD = 0.11 au and MaxAD = 0.67 au) outperforms other tested methods for the anisotropic component. Moreover, comparing the behavior of the two PBE-based DH functionals in more detail we find that by increasing the HF and MP2 contributions from 50% and 12.5% in PBE0DH-1/2 to 68% and 31% in PBEDH-P, the numerical performance is largely improved. In the case of the isotropic component, Table 2, MAD for PBEDH-P is larger than that of PBE0DH-1/2 but is smaller than that provided by B2GP-PLYP and B2-PLYP. However, in total, it appears that the PBE-based functionals PBE0DH-1/2 and PBEDH-P dominate over B2GP-PLYP and B2-PLYP. In order to have a more general comparison between PBEDH-P and other functionals over all of the computed properties (isotropic and anisotropic polarizabilities and also dipole moments), for each functional i the scaled MAD (SMAD) as SMADi = MADi/ MADPBEDH−P and the total SMAD (TSMAD) as the average of 5338



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Table 3. Performance of the PBEDH-P Version with Only Contributions of Electron Pairs with Opposite Spin for the Perturbative Part on the Prediction of Anisotropic and Isotropic Polarizabilities and Dipole Moments of Hydrogen-Bonded Nanoclusters under Studya deviation nanocluster

reference (αaniso/αiso/μ)

anisotropic polarizability

isotropic polarizability

dipole moment

(H2O)2+ (H2O)3+ (H2O)4+ (HF)4 (HF)5 (HF)6 (NH3)2 (NH3)3 HF−H2O HF−(H2O)2 HF−(H2O)3 (HF)2−H2O (HF)3−H2O (HF)2−(H2O)2

12.41/24.60/2.86 10.83/34.95/3.92 13.77/46.23/3.32 6.97/21.07/0.00 9.55/26.59/0.00 11.00/32.13/0.00 4.85/27.60/0.74 8.02/42.02/0.00 4.52/14.03/1.68 5.34/23.57/0.90 8.28/33.26/0.79 4.97/19.44/1.15 7.69/24.95/1.09 8.66/28.95/0.81

0.55 0.17 0.03 0.11 0.17 0.23 0.27 0.56 0.10 0.02 0.06 0.07 0.14 0.20

0.99 1.30 1.46 0.50 0.65 0.79 0.60 0.86 0.35 0.60 0.88 0.48 0.64 0.76

0.01 0.01 0.04 0.00 0.00 0.00 0.02 0.00 0.03 0.02 0.01 0.02 0.02 0.01

0.19 0.56

0.78 1.46

0.01 0.04

MAD MaxAD a

Statistical descriptors (shown in boldface) are mean absolute deviation (MAD) and maximum absolute deviation (MaxAD). All values are in atomic units.

Table 4. Relative Errorsa in the Computed Values of Hyperpolarizabilities, Anisotropic Polarizabilities, and Isotropic Polarizabilities for Protonated and Deprotonated Water Nanoclusters Using Parameter-Free and Parametrized DH Functionalsb

possible to completely neglect the same-spin component contribution, and due to a Laplace transformation algorithm,58 the formal computational cost is reduced from N 5 to N 4 (N is the size of the system), which is the same as the scaling of common hybrid density functionals. Accordingly, we have checked another version of the PBEDH-P functional in which only contributions of electron pairs with opposite spin are included for the perturbative part. Corresponding results are reported in Table 3, where the statistical measures on various test sets are as follows: anisotropic polarizability, MAD = 0.19 au and MaxAD = 0.56 au; isotropic polarizability, MAD = 0.78 au and MaxAD = 1.46 au; and dipole moment, MAD = 0.01 au and MaxAD = 0.04 au. The issues of reduced CPU time aside, the obtained results for anisotropic components are comparable to that of PBEDH-P, and in the case of isotropic polarizabilities we can conclude that the use of the SOS approach deteriorates the accuracy of the results. On the dipole moments, neglecting the same-spin contributions we obtain the results very close to that of PBEDH-P (MAD = 0.01 au and MaxAD = 0.05 au). Altogether, what we would like to highlight the most here is that the same-spin and opposite-spin MP2 are related, respectively, to long-range correlations and short-range interactions and consequently a functional lacking each of them will not be able to predict accurate results for response properties. In another test set we consider protonated and deprotonated water nanoclusters H3O+(H2O)n and OH−(H2O)n, respectively. In addition to anisotropic and isotropic polarizabilities, the total first order hyperpolarizabilities have also been computed for these systems. In Table 4, the performance of PBEDH-P is compared with the standard DH functional B2-PLYP and the PBE0DH-1/2 model as a parameter-free functional. On the hyperpolarizability and anisotropic polarizability, PBEDH-P provides the lowest relative errors as 0.05 and 0.01, respectively, in comparison to other functionals. In the case of isotropic polarizability, however, the error of PBEDH-P is larger than PBE0DH-1/2 but its results are still better than B2-PLYP.

relative error nanocluster OH−(H2O) OH−(H2O)2 OH−(H2O)3 H3O+ (H2O) H3O+(H2O)2 H3O+ (H2O)3

OH−(H2O) OH−(H2O)2 OH−(H2O)3 H3O+(H2O) H3O+(H2O)2 H3O+(H2O)3

OH−(H2O) OH−(H2O)2 OH−(H2O)3 H3O+(H2O) H3O+(H2O)2 H3O+(H2O)3

reference

c

B2-PLYP

PBE0DH-1/2

Hyperpolarizability 58.34 1.914 76.13 0.840 40.79 0.854 9.03 0.292 21.22 0.049 6.90 0.060 0.668 Anisotropic Polarizability 11.96 0.164 12.49 0.150 6.28 0.112 6.35 0.008 7.57 0.044 6.20 0.074 0.092 Isotropic Polarizability 29.57 0.085 38.04 0.069 46.43 0.059 15.79 0.016 25.11 0.023 34.43 0.027 0.046

PBEDH-P

0.872 0.273 0.329 0.002 0.019 0.112 0.268

0.168 0.010 0.018 0.004 0.040 0.092 0.055

0.011 0.042 0.030 0.021 0.009 0.016 0.022

0.024 0.003 0.003 0.025 0.004 0.003 0.010

0.004 0.002 0.003 0.012 0.008 0.006 0.006

0.015 0.012 0.011 0.014 0.011 0.010 0.012

Relative error = (calc − ref)/ ref. bMean absolute relative errors are shown in boldface. cReference data are from CCSD/aug-cc-pVDZ model chemistry. a

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Lastly, although in this work the dipole polarizabilities of hydrogen-bonded nanoclusters have been studied, we further extend the validity of the proposed functional to a representative series of molecules containing main group elements as a last test case. The dipole−dipole polarizabilities have been computed at the experimental molecular geometries collected from the U.S. National Institute of Stantards and Technology (NIST) database (see http://cccbdb.nist.gov/). The performance of PBEDH-P to predict the experimental gasphase static mean dipole−dipole polarizabilities59 for molecules of this set has been tested in Table 5. As can be seen from Table 5, the values of signed deviations are negative, leading to a tendency of underestimating the dipole−dipole polarizabilities with MSD = −1.08 au on this set of molecules for the PBEDH-P functional. Moreover, we find the MAD and MaxAD values of 1.08 au and 1.93 au, respectively, on this set. To confirm the strong correlation between computed and experimental polarizabilities, shown in Figure 7 is the linear relationship between these two quantities with the regression coefficient of 0.997. Therefore, in addition to response properties of nanoclusters, we expect that the PBEDH-P functional can also perform well for the prediction of dipole polarizabilities of molecular systems. Further testing on other properties is mandatory for the reliability of a method, but our main focus in this work has been based on response properties and we hope that, from the viewpoint of performance to computational cost ratio, the research field of linear and nonlinear optical properties could benefit from the findings of the present study.

Table 5. Signed Deviations in the Computed Values of Mean Static Dipole−Dipole Polarizabilities of Some Molecules Using PBEDH-P Functionala molecule

referenceb

signed deviation

BF3 C2H2 C2H4 CF4 CH2F2 CH3Cl CH3F CH3OH CH4 CHF3 Cl2 CO CO2 F2 H2 H2O H2S HBr HCl HCN HF N2 N2O NF3 NH3 NO O2 O3 PH3 SiH4

16.27 22.96 27.72 19.10 17.89 29.98 17.46 21.94 17.24 18.63 30.48 13.04 17.50 8.38 5.43 9.64 24.68 23.74 17.40 16.75 5.60 11.74 19.70 18.95 14.56 11.53 10.59 18.68 30.40 31.97

−1.36 −0.83 −0.52 −1.14 −1.31 −1.19 −1.25 −1.57 −1.09 −1.09 −1.58 −0.34 −1.12 −0.77 −0.43 −0.84 −1.93 −1.53 −1.49 −0.38 −0.81 −0.37 −1.02 −1.34 −1.30 −0.51 −0.84 −1.59 −1.22 −1.76

MSD MAD MaxAD

5. FINAL COMMENTS In summary, we have proposed a density functional from DH category based on the PBE exchange and correlation terms for the linear and nonlinear optical properties of molecules and hydrogen-bonded systems. This new functional, named PBEDH-P, contains contributions as 68% HF exchange and 31% MP2 correlation. The performance of PBEDH-P has been compared to those of several parametrized and parameter-free DH models for the computations of hyperpolarizability, anisotropic and isotropic polarizabilities, and dipole moment of H2O, NH3, HF, and HF−H2O hydrogen-bonded nanoclusters. From the obtained results, PBEDH-P seems to represent a considerable improvement with respect to the recently proposed DHs and MP2 method. It was observed that considering contributions of electron pairs with opposite spin for the perturbative term within the SOS scheme does not represent a great improvement. The results of PBEDH-P functional reveal further evidence of the quality of functionals blending DFT and perturbation theory to study the electric response properties. Nonetheless, the quest for a universal functional continues, and this issue remains to be addressed in the years ahead.

−1.08 1.08 1.93

a

Statistical descriptors (shown in boldface) are mean signed deviation (MSD), mean absolute deviation (MAD), and maximum absolute deviation (MaxAD). All values are in atomic units. bReference data are the experimental electronic dipole−dipole polarizabilities from a most recent compilation in ref 59.

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AUTHOR INFORMATION

Corresponding Author

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

Figure 7. Linear relationship between the computed values of dipole polarizabilities using PBEDH-P functional and experimental data for the considered set of molecules. The fitted line is αexperiment = 1.033αPBEDH−P + 0.517 with R2 = 0.997.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.A. acknowledges Shiraz University for computing facilities. 5340



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GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent Interactions. J. Chem. Theory Comput. 2011, 7, 291−309. (21) Sharkas, K.; Toulouse, J.; Savin, A. Double-Hybrid DensityFunctional Theory Made Rigorous. J. Chem. Phys. 2011, 134, 0641131−064113-9. (22) Brémond, E.; Adamo, C. Seeking for Parameter-Free DoubleHybrid Functionals: The PBE0-DH Model. J. Chem. Phys. 2011, 135, 024106-1−024106-6. (23) Toulouse, J.; Sharkas, K.; Brémond, E.; Adamo, C. Rationale for a New Class of Double-Hybrid Approximations in Density-functional Theory. J. Chem. Phys. 2011, 135, 101102-1−101102-3. (24) Fromager, E. Rigorous Formulation of Two-Parameter DoubleHybrid Density Functionals. J. Chem. Phys. 2011, 135, 244106-1− 244106-8. (25) Zhang, I. Y.; Xu, X. Doubly Hybrid Density Functional for Accurate Description of Thermochemistry, Thermochemical Kinetics and Nonbonded Interactions. Int. Rev. Phys. Chem. 2011, 30, 115−160. (26) Kozuch, S.; Martin, J. M. L. DSD-PBEP86: In Search of the Best Double-Hybrid DFT with Spin-Component Scaled MP2 and Dispersion Corrections. Phys. Chem. Chem. Phys. 2011, 13, 20104− 20107. (27) Zhang, I. Y.; Su, N. Q.; Brémond, E.; Adamo, C.; Xu, X. Doubly Hybrid Density Functional xDH-PBE0 from a Parameter-Free Global Hybrid Model PBE0. J. Chem. Phys. 2012, 136, 174103-1−174103-8. (28) Chai, J.-D.; Mao, S.-P. Seeking for Reliable Double-Hybrid Density Functionals without Fitting Parameters: The PBE0−2 Functional. Chem. Phys. Lett. 2012, 538, 121−125. (29) Mohajeri, A.; Alipour, M. B2-PPW91: A Promising DoubleHybrid Density Functional for the Electric Response Properties. J. Chem. Phys. 2012, 136, 124111-1−124111-4. (30) Yu, F. Intermolecular Interactions of Formic Acid with Benzene: Energy Decomposition Analyses with ab Initio MP2 and DoubleHybrid Density Functional Computations. Int. J. Quantum Chem. 2013, 113, 2355−2360. (31) Peverati, R.; Head-Gordon, M. Orbital Optimized DoubleHybrid Density Functionals. J. Chem. Phys. 2013, 139, 024110-1− 024110-6. (32) Zhang, I. Y.; Xu, X. Reaching a Uniform Accuracy for Complex Molecular Systems: Long-Range-Corrected XYG3 Doubly Hybrid Density Functional. J. Phys. Chem. Lett. 2013, 4, 1669−1675. (33) Sancho-García, J. C.; Adamo, C. Double-Hybrid Density Functionals: Merging Wavefunction and Density Approaches to Get the Best of Both Worlds. Phys. Chem. Chem. Phys. 2013, 15, 14581− 14594. (34) Bousquet, D.; Brémond, E.; Sancho-García, J. C.; Ciofini, I.; Adamo, C. Is There Still Room for Parameter Free Double Hybrids? Performances of PBE0-DH and B2PLYP over Extended Benchmark Sets. J. Chem. Theory Comput. 2013, 9, 3444−3452. (35) Kozuch, S.; Martin, J. M. L. Spin-Component-Scaled Double Hybrids: An Extensive Search for the Best Fifth-Rung Functionals Blending DFT and Perturbation Theory. J. Comput. Chem. 2013, 34, 2327−2344. (36) Meo, F. D.; Trouillas, P.; Adamo, C.; Sancho-García, J. C. Application of Recent Double-Hybrid Density Functionals to LowLying Singlet-Singlet Excitation Energies of Large Organic Compounds. J. Chem. Phys. 2013, 139, 164104-1−164104-6. (37) Su, N. Q.; Adamo, C.; Xu, X. A Comparison of Geometric Parameters from PBE-Based Doubly Hybrid Density Functionals PBE0-DH, PBE0−2, and xDH-PBE0. J. Chem. Phys. 2013, 139, 174106-1−174106-11. (38) Cornaton, Y.; Franck, O.; Teale, A. M.; Fromager, E. Analysis of Double Hybrid Density-Functionals along the Adiabatic Connection. Mol. Phys. 2013, 111, 1275−1294. (39) Alipour, M. Validation of Double-Hybrid Density Functionals for Electric Response Properties of Transition-Metal Systems: A New Paradigm Based on Physical Considerations. J. Phys. Chem. A 2013, 117, 2884−2890.

REFERENCES

(1) Hohenberg, P.; Kohn, W. Inhomogeneus Electron Gas. Phys. Rev. 1964, 136, B864−B871. (2) Kohn, W.; Sham, L. J. Self-consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140, A1133− A1138. (3) Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford: New York, 1989. (4) Perdew, J. P.; Schmidt, K. Density Functional Theory and its Application to Materials. AIP Conf. Proc. 2001, 577, 1−20. (5) Eshuis, H.; Bates, J. E.; Furche, F. Electron Correlation Methods Based on the Random Phase Approximation. Theor. Chem. Acc. 2012, 131, 1084−1101. (6) Alipour, M. 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. J. Phys. Chem. A 2013, 117, 4506−4513. (7) Zhao, Y.; Lynch, B. J.; Truhlar, D. G. Doubly Hybrid Meta DFT: New Multi-Coefficient Correlation and Density Functional Methods for Thermochemistry and Thermochemical Kinetics. J. Phys. Chem. A 2004, 108, 4786−4791. (8) Zhao, Y.; Lynch, B. J.; Truhlar, D. G. Multi-Coefficient Extrapolated Density Functional Theory for Thermochemistry and Thermochemical Kinetics. Phys. Chem. Chem. Phys. 2005, 7, 43−52. (9) Grimme, S. Semiempirical Hybrid Density Functional with Perturbative Second-Order Correlation. J. Chem. Phys. 2006, 124, 034108-1−034108-16. (10) Tarnopolsky, A.; Karton, A.; Sertchook, R.; Vuzman, D.; Martin, J. M. L. Double-Hybrid Functionals for Thermochemical Kinetics. J. Phys. Chem. A 2008, 112, 3−8. (11) Karton, A.; Tarnopolsky, A.; Lamère, J.-F.; Schatz, G. C.; Martin, J. M. L. Highly Accurate First-Principles Benchmark Data Sets for the Parametrization and Validation of Density Functional and Other Approximate Methods. Derivation of a Robust, Generally Applicable, Double-Hybrid Functional for Thermochemistry and Thermochemical Kinetics. J. Phys. Chem. A 2008, 112, 12868−12886. (12) Sancho-García, J. C.; Pérez-Jiménez, A. J. Assessment of Doublehybrid Energy Functionals for π-Conjugated Systems. J. Chem. Phys. 2009, 131, 084108-1−084108-11. (13) Benighaus, T.; DiStasio, R. A.; Lochan, R. C.; Chai, J.-D.; HeadGordon, M. Semiempirical Double-Hybrid Density Functional with Improved Description of Long-Range Correlation. J. Phys. Chem. A 2008, 112, 2702−2712. (14) Á ngyán, J. G.; Gerber, I. C.; Savin, A.; Toulouse, J. van der Waals Forces in Density Functional Theory: Perturbational LongRange Electron Interaction Corrections. Phys. Rev. A 2005, 72, 012510-1−012510-9. (15) Schwabe, T.; Grimme, S. Towards Chemical Accuracy for the Thermodynamics of Large Molecules: New Hybrid Density Functionals Including Non-Local Correlation Effects. Phys. Chem. Chem. Phys. 2006, 8, 4398−4401. (16) Graham, D. C.; Menon, A. S.; Goerigk, L.; Grimme, S.; Radom, L. Optimization and Basis-Set Dependence of a Restricted-Open-Shell Form of B2-PLYP Double-Hybrid Density Functional Theory. J. Phys. Chem. A 2009, 113, 9861−9873. (17) Zhang, Y.; Xu, X.; Goddard, W. A., III. Doubly Hybrid Density Functional for Accurate Descriptions of Nonbond Interactions, Thermochemistry, and Thermochemical Kinetics. Proc. Natl. Acad. Sci. U. S. A. 2009, 106, 4963−4968. (18) Kozuch, S.; Gruzman, D.; Martin, J. M. L. DSD-BLYP: A General Purpose Double Hybrid Density Functional Including Spin Component Scaling and Dispersion Correction. J. Phys. Chem. C 2010, 114, 20801−20808. (19) Zhang, Y.; Xu, X.; Jung, Y.; Goddard, W. A., III. A Fast Doubly Hybrid Density Functional Method Close to Chemical Accuracy Using a Local Opposite Spin Ansatz. Proc. Natl. Acad. Sci. U. S. A. 2011, 108, 19896−19900. (20) Goerigk, L.; Grimme, S. Efficient and Accurate Double-HybridMeta-GGA Density Functionals-Evaluation with the Extended 5341



Article

(40) Chan, B.; Radom, L. Accurate Quadruple-ζ Basis-Set Approximation for Double-Hybrid Density Functional Theory with an Order of Magnitude Reduction in Computational Cost. Theor. Chem. Acc. 2014, 133, 1426−1410. (41) Souvi, S. M. O.; Sharkas, K.; Toulouse, J. Double-Hybrid Density-Functional Theory with Meta-Generalized-Gradient Approximations. J. Chem. Phys. 2014, 140, 084107-1−084107-6. (42) Su, N. Q.; Xu, X. Construction of a Parameter-Free Doubly Hybrid Density Functional from Adiabatic Connection. J. Chem. Phys. 2014, 140, 18A512-1−18A512-15. (43) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (44) Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0 Model. J. Chem. Phys. 1999, 110, 6158−6170. (45) Ernzerhof, M.; Scuseria, G. E. Assessment of the Perdew− Burke−Ernzerhof Exchange-Correlation Functional. J. Chem. Phys. 1999, 110, 5029−5036. (46) Buckingham, A. D. Permanent and Induced Molecular Moments and Long-Range Intermolecular Forces. Adv. Chem. Phys. 1967, 12, 107−142. (47) McLean, A. D.; Yoshimine, M. Theory of Molecular Polarizabilities. J. Chem. Phys. 1967, 47, 1927−1935. (48) Lu, T.; Chen, F. Multiwfn: A Multifunctional Wavefunction Analyzer. J. Comput. Chem. 2012, 33, 580−592. (49) Janeiro-Barral, P. E.; Mella, M. Study of the Structure, Energetics, and Vibrational Properties of Small Ammonia Clusters (NH3)n (n = 2−5) Using Correlated ab Initio Methods. J. Phys. Chem. A 2006, 110, 11244−11251. (50) Baburao, B.; Visco, D. P., Jr.; Albu, T. V. Association Patterns in (HF)m(H2O)n (m + n = 2−8) Clusters. J. Phys. Chem. A 2007, 111, 7940−7956. (51) Bryantsev, V. S.; Diallo, M. S.; Van Duin, A. C. T.; Goddard, W. A., III. Evaluation of B3LYP, X3LYP, and M06-Class Density Functionals for Predicting the Binding Energies of Neutral, Protonated, and Deprotonated Water Clusters. J. Chem. Theory Comput. 2009, 5, 1016−1026. (52) Cohen, H. D.; Roothaan, C. C. J. Electric Dipole Polarizability of Atoms by Hartree-Fock Method. I. Theory for Closed-shell Systems. J. Chem. Phys. 1965, 43, S34−S39. (53) Mohammed, A. A. K.; Limacher, P. A.; Champagne, B. Finding Optimal Finite Field Strengths Allowing for a Maximum of Precision in the Calculation of Polarizabilities and Hyperpolarizabilities. J. Comput. Chem. 2013, 34, 1497−1507. (54) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; et al. Gaussian 09, Revision A.02; Gaussian: Wallingford, CT, USA, 2009. (55) Cortona, P. Note: Theoretical Mixing Coefficients for Hybrid Functionals. J. Chem. Phys. 2012, 136, 086101-1−086101-2. (56) Jung, Y.; Lochan, R. C.; Dutoi, A. D.; Head-Gordon, M. Scaled Opposite-Spin Second Order Møller−Plesset Correlation Energy: An Economical Electronic Structure Method. J. Chem. Phys. 2004, 121, 9793−9802. (57) Grimme, S. Improved Second-Order Møller−Plesset Perturbation Theory by Separate Scaling of Parallel- and Antiparallel-Spin Pair Correlation Energies. J. Chem. Phys. 2003, 118, 9095−9102. (58) Almlöf, J. Elimination of Energy Denominators in Møller Plesset Perturbation Theory by a Laplace Transform Approach. Chem. Phys. Lett. 1991, 181, 319−320. (59) Hohm, U. Experimental Static Dipole-Dipole Polarizabilities of Molecules. J. Mol. Struct. 2013, 1054, 282−292.

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