Performances of the XYG3 Type of Doubly Hybrid ... - ACS Publications

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Error Accumulations in Adhesive Energies of Dihydrogen Molecular Chains: Performances of the XYG3 Type of Doubly Hybrid Density Functionals Neil Qiang Su and Xin Xu* Collaborative Innovation Center of Chemistry for Energy Materials, Shanghai Key Laboratory of Molecular Catalysis and Innovative Materials, MOE Laboratory for Computational Physical Science, Department of Chemistry, Fudan University, Shanghai 200433, China S Supporting Information *

ABSTRACT: We have systematically analyzed the error accumulations in the adhesive energies for a series of hydrogen molecular chains calculated by various kinds of density functional theory (DFT) methods. In particular, we have focused on some representative doubly hybrid (DH) functionals of either the B2PLYP type (B2PLYP, B2PLYP-D, and B2GP-PLYP) or the XYG3 type (XYG3, XYGJ-OS, and xDH-PBE0). The hydrogen molecular chain models have recently been proposed by Zheng et al. (J. Chem. Phys. 2012, 137, 214106) to identify the delocalization errors (DEs) in thermodynamic properties. From the perspective of DEs, it is shown here that the XYG3 type of DH functionals yield good performance on the calculated adhesive energies due to the minimizing effects of DEs, highlighting the underlying physics for the successes or failures of the approximate functionals. Examination was also extended to HF-DFT, where DFT energies are evaluated with the Hartree−Fock (HF) densities.

I. INTRODUCTION Density functional theory (DFT) has gained a great success in addressing a wide range of problems in physics, chemistry, and materials science, etc. Substantial errors may still arise due to the fact that certain density functional approximation (DFA) for the exchange-correlation functional has to be adopted in the application of DFT in the framework of the Kohn−Sham (KS) scheme.1 Some infamous errors associated with conventional DFAs involve predictions of (1) geometries and nonlinear optical properties of conjugated organic molecules, oligomers, and polymers,2−6 (2) thermochemistry of reactions involving ring and cagelike molecules,7−12 molecules of extended conjugation,13 and highly branched alkanes,14−16 (3) reaction barrier heights,17−24 (4) band gaps of materials,25−36 (5) longrange charge-transfer excited states,37 and (6) dissociation energy curves for covalent bonds and noncovalent bonds in closed-shell or radical molecules,38−43 etc. Self-interaction error (SIE)44 is often considered as the major source of error for many failures of DFAs. This concept, which is widely used in the literature, is well-defined only for oneelectron systems. SIE has been extended to many-electron systems. The so-called many-electron SIE45,46 refers to as the deviation of the energy for fractionally charged systems from the exact linearity conditions.25,47 However, it is important to differentiate the deviations as convex or concave behaviors of fractional charges, which lead to distinctly different impact on the calculated properties.23,31−36,42,45,46 The concept of the delocalization error (DE)32 was thus defined and much studied by Mori-Sánchez, Cohen, and Yang as the convex deviation of the energy for fractionally charged systems associated with © XXXX American Chemical Society

conventional DFAs. In contrast, the Hartree−Fock (HF) method (a DFT method in view of the generalized KS scheme48) suffers from the localization error (LE)32 and results in a concave deviation from the linearity conditions. The concept of DE unravels the physical origin for a large class of failures in DFT calculations with conventional DFAs, including the underestimation of reaction barriers, band gaps of materials, and charge-transfer excitation energies, and the overestimation of binding energies of charge-transfer complexes, conductance of molecular junctions, and polarizabilities of molecules and solids.2−43 A number of efforts have been made to correct the problem by imposing the linearity condition in the construction of the exchange-correlation functionals.49−53 The emphasis has often been laid on the exchange functional, leading to some global49,50 or range-separated hybrid functionals,51−53 such that the hybridization of the HF exchange with a concave fractional charge behavior tends to cancel the incorrect convex behavior of pure DFT functionals. However, the second-order Møller−Plesset perturbation theory (MP2) has been shown to have largely restored the linearity as compared to the HF theory,54 which, in fact, emphasizes the role played by the correlation. More recently, a new type of functionals, the soSpecial Issue: 25th Austin Symposium on Molecular Structure and Dynamics Received: July 31, 2014 Revised: October 4, 2014

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dx.doi.org/10.1021/jp507711t | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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called doubly hybrid (DH) functionals,55−57 appeared, which, in addition to the hybridization of the HF exchange, also hybridize the MP2 term as a certain portion of the correlation energy in functionals. We have tested,58 for the first time, the fractional charge behaviors of some representative DH functionals such as B2PLYP,56 XYG3,57 and XYGJ-OS59 by plotting the ground state energies E and energy derivatives for several atoms and molecules with fractional electron numbers N. It was shown that the XYG3 type of DH functionals give good agreement between their energy derivatives and the experimental ionization potentials (IPs), electron affinities (EAs), and fundamental gaps, as expected from their nearly straight line fractional charge behaviors.58 As pointed out by Zheng et al.,60 it remains scarce and nontrivial to resolve or quantify the amount of DEs in more complex chemical species and to analyze the influence of DEs on the calculated thermodynamic properties. In order to examine systematically the DEs associated with various DFAs, Zheng et al.60 have designed a series of chemical systems consisting of hydrogen molecular chains, where a number of hydrogen molecules are aligned head-to-head in an equally displaced manner. The DEs associated with approximate functionals were extracted by analyzing the deviation of calculated adhesive energies from the results of the coupledcluster method with single, double, and noniterative triple excitations [i.e., CCSD(T)], which were considered as highly accurate thermodynamic data for the hydrogen systems.60 The underlying relationship between the error magnitude and the degree of electron delocalization in hydrogen molecular chains was explored, which highlights the underlying physics for the successes or failures of the approximate functionals. Their attention has been focused on the examination of the performance of some conventional DFAs, such as SVWN5,61,62 BLYP,63,64 PBE,65 B3LYP,66,67 and, in particular, rCAM-B3LYP.49 In this paper, we extend their work to the DH functionals.

while others are omitted. For d = 1.50 and 2.25 bohr, a set of hydrogen molecular chains with n = 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 are studied.60 For n = 10, a set of hydrogen molecular chains with d = 1.50, 1.75, 2.00, 2.25, 2.50, 2.75, and 3.00 bohr are studied.60 These provide two ways that the length and strength for the electron delocalization can be varied. Figure 1b plots a bending hydrogen dimer, where α is the bending angle formed by an H−H bond and the intermolecular connecting line. Previously, Zheng et al.60 found that the bending hydrogen dimer has a local minimal at r = 1.404 bohr, d = 5.941 bohr, and α = 176.6° at the level of CCSD(T) with the basis set of cc-pVTZ.75 This corresponds to a van der Waals well with binding energy of only ∼0.02 kcal/mol. To examine the effects associated with DEs, a set of dimers with varying d are studied, where r and α are fixed at 1.404 bohr and 176.6°, respectively.60 Zheng et al. tuned d in the range of 1.20 and 2.60 bohr with uneven increments.60 Here in the present work, d is adjusted in the same range between 1.20 and 2.60 bohr but with an even interval of 0.20 bohr. B. Calculation Methods. Several representative DH functionals have been examined in the present work, which include B2PLYP,56 B2PLYP-D,76 B2GP-PLYP,77 XYG3,57 XYGJ-OS,59 and xDH-PBE0.78 The former three are the B2PLYP type of functionals (bDH for short), while the latter three are the XYG3 type of functionals (xDH for short).79 The three bDH functionals share the same functional form: ExcbDH = b1ExHF + (1 − b1)ExB88 + (1 − b2)EcLYP + b2EcMP 2 (1)

where (b1, b2) = (0.53, 0.27) for B2PLYP and B2PLYP-D, while (b1, b2) = (0.65, 0.36) for B2GP-PLYP. The scaled MP2 energy is added in a manner of post self-consistent-field (SCF) based on orbitals that are obtained from the solution of the KS equations associated with a functional of the first three terms in eq 1. It is often useful to add a force-field-like dispersion correction in applications of bDH to the complex systems, such that a D term is added to B2PLYP to arrive at B2PLYP-D.76 Here the dispersion effects are minimized in the hydrogen systems.60 Hence B2PLYP-D and B2PLYP should lead to the same results. The XYG3 functional has the functional form as

II. COMPUTATIONAL DETAILS A. Chemical Systems. Most mainstream DFAs are also haunted by other types of errors (e.g., errors in treating strongly correlated systems and chemical bond breaking), as well as noncovalent interactions.41,47,68−74 The hydrogen molecular systems were designed by Zheng et al., trying to distinguish DEs from other sources of errors.60 Figure 1a displays a schematic diagram for the models of the linear hydrogen molecular chains (H2)n, where the intramolecular H−H bond length is fixed at r = 1.50 bohr (around the equilibrium bond length), while the interunit distance d is allowed to vary. Figure 1a only shows two H2 units (n = 2),

ExcXYG3 = a1ExHF + (1 − a1)ExS + a 2ΔExB88 + (1 − a3)EcLYP + a3EcMP2

(2)

with (a1, a2, and a3) = (0.8033, 0.2107, and 0.3211). A unique feature that differs from the bDH is that all energy terms associated with xDH are evaluated by using the SCF orbitals from a standard functional. In particular, XYG3 used orbitals from B3LYP. It was believed that the well-established B3LYP functional provides a good approximation to the real (yet unknown) KS functional to construct the zeroth order Hamiltonian upon which the XYG3 functional relies.57,71,72 More recently, two new versions of xDHs, XYGJ-OS,59 and xDH-PBE078 were developed, where the MP2 contributions are simplified with the opposite-spin (os) ansatz:80 ExcXYGJ − OS = c1ExHF + (1 − c1)ExS + (c 2EcVWN + c3EcLYP) + c4EcMP2 , os

Figure 1. (a) Linear hydrogen molecular chains (H2)n with n being the number of H2 units.60 Here two H2 units are explicitly shown, while others are not displayed. (b) A bending H2 dimer model.60

(3)

ExcxDH − PBE0 = d1ExHF + (1 − d1)ExPBE + d 2EcPBE + d3EcMP2 ,os (4) B



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with (c1, c2, c3, and c4) = (0.7731, 0.2309, 0.2754, and 0.4364), and (d1, d2, and d3) = (0.8335, 0.5292, and 0.5428). While XYGJ-OS uses B3LYP66,67 orbitals, xDH-PBE0 uses PBE081,82 orbitals. As compared to the standard fifth order DH functionals, these two functionals have a favorable fourth order scaling, which can be further reduced to the third order by exploiting the spatial locality of electron correlation.59,80 This extends the applicability of the methods to substantially larger molecular systems. For comparison purpose, we include several conventional functionals, BLYP, PBE, B3LYP, and PBE0, where the former three have already been examined by Zheng et al.60 In particular, we also examine the performances of HF-BLYP and HF-PBE,83,84 which, instead of solving the KS equations selfconsistently, we perform the SCF calculations using the HF method in the first place and then evaluate the final DFT energies on the HF densities. This method was recently named as density-corrected density functional theory (DC-DFT),85 as the HF densities, although lacking correlation effects, are SIEfree and deemed as more accurate than the self-consistent densities from BLYP and PBE which suffer from SIE84,85 or DE. It was shown that, while BLYP and PBE yield wrongly shaped surfaces and/or incorrect minima for HO·Cl− and HO·H2O complexes when calculated self-consistently, HF-BLYP and HFPBE yield almost identical results as compared to the CCSD(T) reference data.85 Previously, it was also found that HF-DFT leads to substantially improved results for the predictions of reaction barrier heights.84,86,87 Can HF-BLYP and HF-PBE reduce the DEs suffered by BLYP and PBE, giving good performances in the hydrogen molecular chains? For the basis sets, Zheng et al.60 adopted a modified aug-ccpVDZ basis set75 for the linear chain models, where the original outermost S-shell is removed to avoid near linear dependence problems. For the hydrogen dimer models, they used Pople’s 6311++G(3df,3pd)88,89 for DFAs and cc-pVTZ75 for CCSD(T). They found that the basis set superposition error (BSSE) is negligible.60 We follow their strategy, using the same modified aug-cc-pVDZ for the linear chain models and 6-311+ +G(3df,3pd) for the bending dimer models. A development version of the NorthWest computational Chemistry (NWChem) software package (version 6.1.1)90 is used for all the DFT calculations in this study. The CCSD(T) calculations are done using the GAUSSIAN 09.91

(5)

Eat, n = 2E(H ) − E((H2)n )/n

(6)

(7)

Ead, n = E(H2) − E((H2)n )/n

(8)

ΔEDFA ad,n

It was found that can be used as a measure of DEs, which are more prominent for more diffuse electron distributions with larger n.60 DFA We would like to point out that ΔEDFA at,n and ΔEad,n are correlated: DFA DFA DFA ΔEat, n = ΔEad, n /2 + Eat,1

(9)

Hence error accumulation over n found in atomization energy is actually a result of ΔEDFA ad,n dependence on n. Although Zheng DFA et al.60 studied both ΔEDFA at,n and ΔEad,n , we will focus on the analysis of ΔEDFA . ad,n B. Linear (H2)n Chains with r = d = 1.50 Bohr. Figure 2 plots the dependence of ΔEDFA ad,n on n in the linear chain models

Figure 2. Dependence of adhesive energy error per H2 unit on n in the linear chain models (H2)n where r = d = 1.50 bohr. Note that the scale changes for the vertical coordinates from (a) to (b). Scattered data are the calculation results, while lines are drawn to guide the eye.

III. RESULTS AND DISCUSSION A. Atomization Energy versus Adhesive Energy. Atomization energy (at) is one of the most important thermodynamic properties. It was shown that errors in conventional DFAs increase as the system size increases.92−94 For a (H2)n system, the error, with respect to the CCSD(T) reference, of atomization energy averaged over the number of H atoms can be defined as DFA DFA CCSD(T ) ΔEat, )/2 n = (Eat, n − Eat, n

DFA DFA CCSD(T) ΔEad, n = Ead, n − Ead, n

(H2)n where r = d = 1.50 bohr. Previously, it was found that the local density approximation (LDA) predicts too low energies for systems with diffuse electron distributions, while HF shows the opposite trend, suffering from DEs and LEs, respectively.32 The results for ΔEad,n associated with LDA (e.g., SVWN5) and HF are not shown in Figure 2 but can be found in Table S3 of the Supporting Information. SVWN5 leads to ΔEad,n which increase monotonically from 6.05 kcal/mol at n = 2 to 11.75 kcal/mol at n = 10, whereas HF leads to ΔEad,n which decrease monotonically from −3.27 kcal/mol at n = 2 to −7.26 kcal/mol at n = 10. The curves shown in Figure 2a from the generalized gradient approximations (GGAs, for example, PBE and BLYP) and hybrid GGAs (e.g., PBE0 and B3LYP) also exhibit substantial errors. Nevertheless, the magnitudes are reduced to ΔEad,10 of 8.53 and 6.58 kcal/mol for PBE and BLYP, respectively, which are further reduced to 5.86 and 4.96 kcal/ mol for PBE0 and B3LYP, respectively.

For n = 1, eq 6 defines the bond dissociation energy (EDFA at,1 ) of an H2 molecule calculated by each method. Hence eq 5 at n = 1 measures essentially the correlation error in description of an H−H bond by each method. The error in adhesive energy (ad) per H2 units can be defined as C



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It is intriguing to ask to what extent HF-DFT reduce ΔEad,n in the hydrogen chain models. Figure 2a shows that HF-DFT follow the same trend of the original DFT methods, yielding a larger ΔEad,n for a larger n. Still, the use of HF densities reduces ΔEad,2 from 4.30/3.35 for PBE/BLYP to 3.42/2.43 kcal/mol for HF-PBE/HF-BLYP and ΔEad,10 from 8.53/6.58 for the former two to 6.89/4.81 kcal/mol for the latter two. Importantly, the amounts of improvements from PBE/BLYP to HF-PBE/HFBLYP are monotonically increasing from ΔEad,2 by 0.88/0.93 kcal/mol to ΔEad,10 by 1.65/1.77 kcal/mol. This supports that ΔEad,n is indeed a good indicator of DEs, which become larger for a more diffuse electron distribution at a larger n, while the use of SIE-free HF densities removes a larger amount of errors in ΔEad,n for a larger n. As shown in Figure 2a, HF-BLYP behaves very similarly as B3LYP, although HF-PBE is inferior to PBE0. Figure 2b shows the performance of the DH functionals. As the errors are positive for all chains associated with bDHs, the adhesive energies are overall overestimated, which increase to a greater extent with larger n. Note, however, the scale for the vertical axis in Figure 2b is reduced as compared to that in Figure 2a. Hence, ΔEad,10 for B2PLYP-D, B2PLYP, and B2GPPLYP are 3.03, 2.79, and 1.96 kcal/mol, respectively, being a significant improvement over conventional DFAs. The xDH functionals perform even better. Although the tendency that ΔEad,n increase as the system size increases is still clearly seen in XYG3, the error magnitude is considerably reduced. ΔEad,10 for XYG3 is 0.85 kcal/mol. The xDH-PBE0 functional shows a negative error, indicating that the adhesive energies are overall underestimated, which actually saturate to around −0.25 kcal/ mol. XYGJ-OS also performs satisfactorily. It predicts very accurate adhesive energies for all chains with an average error of 0.23 kcal/mol. C. Linear (H2)n Chains with r = 1.50 and d = 2.25 Bohr. Figure 3 plots the dependence of ΔEDFA ad,n on n in the linear chain models (H2)n where r = 1.50 and d = 2.25 bohr. This corresponds to a situation where the adjacent H2 units interact weakly as compared to that in Figure 2. As shown in Figure 3a, the trend for the dependence of ΔEDFA ad,n on n is nearly the same HF−DFT as that in Figure 2a, namely, ΔEGGA > ΔEhybrid . On ad,n ad,n ∼ ΔEad,n the other hand, PBE-based functionals lead to more errors, such PBE BLYP PBE0 B3LYP HF−PBE > ΔEad,n , ΔEad,n > ΔEad,n , ΔEad,n > that ΔEad,n . Again, we see that the amounts of improvements ΔEHF−BLYP ad,n from PBE/BLYP to HF-DFT are monotonically increasing from ΔEad,2 by ∼0.65 kcal/mol to ΔEad,10 by ∼1.35 kcal/mol (c.f., Table S6 of the Supporting Information). This shows that DEs degrade the quality of the self-consistent densities, whereas SIE-free HF densities improve the situation even though the same energy functional (PBE or BLYP) is utilized.85 The trend shown in Figure 3b for bDHs is obvious: > ΔEB2PLYP > ΔEB2GP−PLYP . Adding the dispersion ΔEB2PLYP−D ad,n ad,n ad,n correction as in B2PLYP-D further overestimates the adhesive energies that have already been overestimated by B2PLYP. The same as in Figure 2b, the xDH-PBE0 functional shows a negative error, underestimating the adhesive energies. ΔExDH−PBE0 gradually decrease to around −0.35 kcal/mol for ad,n ΔEad,10. Both XYG3 and XYGJ-OS are satisfactory. These two xDHs yield very accurate adhesive energies for all chains with an average error of 0.24 and 0.21 kcal/mol, respectively (see Table S6 of the Supporting Information). D. Linear (H2)10 Chains at r = 1.50 Bohr with Varying d. Besides varying the number of H2 units, one may also change the interunit distance d which tunes the length and strength for

Figure 3. Dependence of adhesive energy error per H2 unit on n in the linear chain models (H2)n where r = 1.50 and d = 2.25 bohr. Note that the scale changes for the vertical coordinates from (a) to (b). Scattered data are the calculation results, while lines are drawn to guide the eye.

the electron density distribution. Note that there are two opposing effects: Increasing d, although increasing the length for the electron density distribution, tends to reduce the interactions between the adjacent H2 units (Table S2 vs Table S5 of the Supporting Information), which, in turn, reduces the magnitude of ΔEDFA ad,n as already seen by comparison of the results displayed in Figure 2 and Table S3 of the Supporting Information for d = 1.50 and Figure 3 and Table S6 of the Supporting Information for d = 2.25 bohr with varying n. Figure 4 (panels a and b) depicts the evolution of ΔEDFA ad,10 with respect to d, where n is chosen as 10 (see Table S9 of the Supporting Information for numeric details). Indeed, the magnitude of ΔEDFA ad,10 decreases monotonically with increasing d for all DFAs with the exception of XYGJ-OS and xDH-PBE0, where ΔEDFA ad,10 are nearly constant over different d. At each individual value of d, the errors still follow the same pattern as PBE0 HF−PBE displayed in Figures 2 and 3 (e.g., ΔEPBE ad,10 > ΔEad,10 ∼ ΔEad,10 B2PLYP−D B2PLYP B2GP−PLYP xDH and ΔEad,10 > ΔEad,10 > ΔEad,10 > ΔEad,10). As pointed out by Zheng et al.,60 one needs to consider the relative error in order to more suitably assess the significance of the role played by DEs at different interunit separations. Following their work,60 a ratio of the adhesive energy error is CCSD(T) defined as ΔEDFA |, where the numeric details are ad,n /|Ead,n collected in Table S10 of the Supporting Information. Figure 5 CCSD(T) plots the dependence of ΔEDFA | for a series of d in ad,n /|Ead,n the range of 1.50 and 3.00 bohr with an increment of 0.25 bohr. As the interactions between H2 units are actually repulsive rather than adhesive, ECCSD(T) yields a negative value in all cases ad,10 and its magnitude also decreases with increasing d down to ECCSD(T) = −2.57 kcal/mol at d = 3.00 bohr (see Table S8 of ad,10 the Supporting Information). As shown by Zheng et al.,60 the error ratio of SVWN5 increases drastically as d is enlarged due to its severe DE, which eventually leads to a stabilized linear (H2)10 chain at d = 3.00 bohr with a positive adhesive energy of ESVWN5 = 0.42 kcal/mol (see Table S8 of the Supporting ad,10 D



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ΔEad,10 for a larger interunit d. BLYP, on the other hand, performs better than PBE. This is in accordance with the observations shown in Figures 2 and 3. The situation is improved for PBE0 and B3LYP, as compared to the corresponding GGAs, due to the inclusion of the HF exchange. Zheng et al.60 found that the error ratio for the HF exchange is all the way negative and varies downward with increasing d, showing the effects of LE (see also data in Table S10 of the Supporting Information). As shown in Figure 5a, HF-DFT improves all the way the corresponding GGA, and the improvement is enlarged at larger d, highlighting the importance to use the SIE-free densities. The curve of HF-BLYP is interesting. The error ratio for HF-BLYP is positive, a sign for DE,60 which is almost constant for d = 1.50−2.25 bohr. The error ratio decreases in the range of d = 2.25−3.00 bohr. Afterward, ΔEHF−BLYP /|ECCSD(T) | shall become ad,10 ad,10 negative, predicting that the linear (H2)10 chain at d > 3.00 bohr is less stable than that by CCSD(T) as does by HF, which is a sign of LE.60 Figure 5b shows the performance of the representative DHs. The error ratios are also more prominent for a larger interunit d for bDHs (see also Table S10 of the Supporting Information). The magnitude of the error ratios is reduced for B2PLYP and B2GP-PLYP as compared to those of the conventional functionals shown in Figure 5a. However, B2PLYP-D degrades as compared to B2PLYP, in particular, for larger d between 2.50−3.00 bohr. The performance of either XYG3 or XYGJ-OS is quite satisfactory, where the error ratios are small and nearly constant for various d, indicating a negligible DE effect in this case. The xDH-PBE0 functional actually displays an effect from LE instead of from DE. Hence, our results show that additional information on DE/LE is revealed as compared with the results by tuning either n or d. E. Hydrogen Molecular Dimers. We now focus on the hydrogen molecular dimer (H2)2, for which CCSD(T) predicts a stabilized structure in the Cs symmetry. From the perspective of DEs, we are interested in how symmetry change, which leads to a different average electron density in the interstitial region between H2 units, affects the DEs, which, in turn, affect the calculated thermodynamic properties, as compared to those of the linear (H2)n chains. Figure 6 (panels a and b) depict the evolution of ΔEDFA ad,2 with respect to d for the bending H2 dimer, which, indeed, exhibits the same trend as displayed in Figure 4 for the linear (H2)10 chains. The magnitude of ΔEDFA ad,2 decreases monotonically with increasing d for all DFAs with the exception of xDH, where ΔExDH ad,2 are nearly constant over different d. At each value of d, the errors still follow the same pattern as displayed in Figures PBE0 HF−PBE 2−4 (e.g., ΔEPBE and ΔEB2PLYP−D > ad,2 > ΔEad,2 ∼ ΔEad,2 ad,2 B2PLYP B2GP−PLYP xDH > ΔEad,2 ). We refer to Table S13 of ΔEad,2 > ΔEad,2 the Supporting Information for numeric details. Figure 7 (panels a and b) depicts the error ratios, ΔEDFA ad,2 /| CCSD(T) Ead,2 |, for the bending H2 dimers, which again follow the trend as displayed in Figure 5 for the linear (H2)10 chains. Note, CCSD(T) | for XYG3 and XYGJ-OS are both however, ΔEDFA ad,2 /|Ead,2 more accentuated at larger d. This shows the pervasive and resilient effects of DEs in the approximate functionals. The xDH-PBE0 performs most satisfactorily in this situation. Interestingly, unlike that in Figure 5b, which indicates LE, xDH-PBE0 now displays a small but clearly visible DE in Figure 7b. F. Extrapolation of the Adhesive Energy Error to n → ∞. Up to now, we have been studying how DEs affect the

Figure 4. Dependence of adhesive energy error per H2 unit on d in the linear chain models of (H2)10, where r = 1.50 and d varies from 1.50 to 3.00 bohr. Note that the scale changes for the vertical coordinates from (a) to (b). Scattered data are the calculation results, while lines are drawn to guide the eye.

Figure 5. Dependence of the adhesive energy error ratio, with respect to the CCSD(T) reference data, per H2 unit on d in the linear chain models of (H2)10, where r = 1.50 and d varies from 1.50 to 3.00 bohr. Note that the scale changes for the vertical coordinates from (a) to (b). Scattered data are the calculation results, while lines are drawn to guide the eye.

Information).60 Figure 5a shows that PBE displays the same trend as that for SVWN5. The linear (H2)10 chain predicted by PBE at each d is always more stable than that by CCSD(T); the error ratio is all the way positive and varies upward with increasing d, indicating a more prominent role played by E



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against 1/n (see also Figure 8). By fitting the data to straight lines and then extrapolating to 1/n = 0, the bulk values of errors

Figure 6. Dependence of adhesive energy error per H2 unit on d in the H2 dimer models, where r = 1.404 bohr, α = 176.6°, and d varies from 1.20 to 2.60 bohr. Note that the scale changes for the vertical coordinates from (a) to (b). Scattered data are the calculation results, while lines are drawn to guide the eye.

Figure 8. Extrapolation from calculated adhesive energy error for (H2)n (r = 1.5 and d = 2.25 bohr) with n ≤ 10 to n → ∞. Note that the scale changes for the vertical coordinates from (a) to (b).

may be estimated. For instance, ΔEDFA ad for (H2)∞ as extracted from linear extrapolation are 7.10, 4.10, and 2.81 kcal/mol for SVWN5, PBE, and BLYP, respectively (see Table S6 of the Supporting Information). For conventional DFAs, one can actually perform a calculation with periodic boundary condition (PBC). For this purpose, we use Gaussian 0991 with the same basis set using ΔECCSD(T) = −11.95 kcal/mol as the reference (c.f., Table S5 of ad,∞ the Supporting Information). We arrive at ΔEDFA ad,∞ = 7.11, 4.11, and 2.81 kcal/mol for SVWN5, PBE, and BLYP, respectively. Thus, there is a quantitative agreement for ΔEDFA ad,∞ between the data estimated by linear extrapolations and PBC calculations, which demonstrate that errors converge and are most prominent at n → ∞. For PBC calculations with HF, PBE0, and B3LYP, we meet technical problems such that the diffuse p function in the modified aug-cc-pVDZ basis set75 has to be removed, which actually gives back to the standard cc-pVDZ basis set. The PBC calculations with cc-pVDZ yield ΔEDFA ad,∞ = −3.79, 2.36, and 1.59 kcal/mol for HF, PBE0, and B3LYP, respectively. The linear extrapolations give ΔEDFA ad,∞ = −3.59, 2.74, and 2.18 kcal/mol for HF, PBE0, and B3LYP, respectively, with the modified aug-ccpVDZ basis set. Figure 8a also shows how ΔEDFA ad,n associated with HF-DFT converge to the bulk values as n →∞. ΔEDFA ad,∞ for HF-PBE and HF-BLYP are reduced, being 2.58 and 1.30 kcal/mol, respectively, as compared to the corresponding values of PBE and BLYP due to the utilization of the SIE-free densities from HF. Figure 8b shows how the DH functionals perform for (H2)∞. Errors are all reduced with ΔEDH ad,∞ being 1.53 and 1.18 kcal/mol for B2PLYP-D and B2PLYP, respectively. In particular, XYG3

Figure 7. Dependence of the adhesive energy error ratio, with respect to the CCSD(T) reference data, per H2 unit on d in the H2 dimer models, where r = 1.404 bohr, α = 176.6°, and d varies from 1.20 to 2.60 bohr. Note that the scale changes for the vertical coordinates from (a) to (b). Scattered data are the calculation results, while lines are drawn to guide the eye.

calculated thermodynamic data for molecular systems. We will now examine how DEs influence the bulk values for crystals. The linear (H2)n models at r = 1.50 and d = 2.25 bohr have been taken as a prototype for a one-dimensional (1D) crystal, where the bulk is achieved when n tends to infinity.60 It was shown that60 the data for ΔEDFA ad,n exhibit a remarkable linearity F



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and XYGJ-OS are very satisfactory. ΔExDH ad,∞ are 0.27 and 0.20 kcal/mol for XYG3 and XYGJ-OS, respectively, being nearly independent of the chain lengths. ΔExDH−PBE0 is −0.40 kcal/ ad,∞ mol, which is also quite satisfactory. G. Fractional Charge Behavior for Energy E versus Electron Number N. We will now explicitly check the fractional charge behaviors against the linearity condition.31−36,45,46,50−53,95,96 Figure 9 (panels a and b) plots the

Figure 10. Deviations from the corresponding linear interpolations for the hydrogen atom with methods of (a) BLYP, PBE, B3LYP, PBE0, HF-BLYP, HF-PBE, and (b) B2PLYP, B2GP-PLYP, XYG3, XYGJ-OS, xDH-PBE0. Note that the scale changes for the vertical coordinates from (a) to (b).

performs satisfactorily for the anion system, which is a case that is density-driven.85 Hence, the improvements shown in HFDFT for the hydrogen molecule chains come from the reduction of errors in densities, as DEs degrade the selfconsistent densities from the original GGAs. Figures 9b and 10b display the fractional charge behaviors for some representative DH functionals. Even though the convexity is still clearly visible, B2PLYP already shows a significant improvement over the conventional DFT methods, and there is a clear and steady improvement from B2PLYP to B2GP-PLYP and to xDHs (XYGJ-OS, XYG3, and xDH-PBE0). The xDH functionals are all quite satisfactory, displaying a near linearity. Interestingly, while deviation curves (Figure 10) from most other methods are almost quadratic, showing the largest deviation at the middle points, xDH-PBE0 error changes sign at N = 1.5 with the minimum error. Note that the positive deviations mean a concave error, while the negative deviations suggest a convex error. This provides an explanation that xDHPBE0 may exhibit the effect of DE on the thermodynamic properties as in the bending H2 dimers with varying d, or that of LE as in the linear (H2)n chains with varying d.

Figure 9. Fractional charge behaviors for the hydrogen atom with methods of (a) BLYP, PBE, B3LYP, PBE0, HF-BLYP, HF-PBE, and (b) B2PLYP, B2GP-PLYP, XYG3, XYGJ-OS.

ground state energy, E, of the H atom as a function of the electron number, N. The exact straight lines are obtained using the experimental IP and EA of the H atom. Figure 10 (panels a and b) shows the deviations of the methods from the linearity in another way, which plots the difference of the energy predicted by the method and its own straight line interpolation between adjacent integer values for the H atom (i.e., H+ ← H → H−). The basis set used is aug-cc-pVQZ.75 We do not depict the HF curve for fractional charges, as it is now well-known that the HF curve is in a concave manner.32 The BLYP, PBE, PBE0, and B3LYP curves are displayed in Figure 9a for the H atom, which confirm the previous observation, showing that conventional DFAs exhibit strongly convex behaviors in between integers.32 We are interested in examining the fractional charge behaviors of HF-BLYP and HFPBE, which, to the best of our knowledge, have not been investigated before. As shown in Figure 9a, HF-DFT also exhibits a strong convex fractional charge behavior to the left of N = 1 for H → H+; its convexity is much more mild to the right of N = 1 for H → H−. This is clearly seen in Figure 10. HFBLYP and HF-PBE are nearly identical to BLYP and PBE in the magnitude of the deviation from the linearity for the range of N = 0−1, whereas HF-BLYP and HF-PBE significantly improve over BLYP and PBE for the range of N = 1−2, where the deviation from the linearity is much reduced for the HF-DFT. This is in agreement with the observation that HF-DFT

III. CONCLUDING REMARKS We have systematically analyzed the error accumulations of the adhesive energies, calculated by various kinds of DFT methods, for a series of hydrogen molecular chains and dimers with respect to number of H2 units n or interunit distance d. In particular, we have focused on some selected DH functionals and the analyses are extended to HF-DFT. These two types of functionals share a common feature that one functional is used to generate orbitals and densities with which another functional is used to evaluate the final energy, while conventional G



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functionals (e.g., BLYP, PBE, PBE0, and B3LYP) solve the (generalized) KS equations self-consistently. There is a clear and steady improvement in the calculated adhesive energies from conventional functionals to B2PLYP to B2GP-PLYP and to xDHs (XYGJ-OS, XYG3, and xDH-PBE0) with respect to the CCSD(T) reference values. The errors associated with a DFA are more prominent in systems with more delocalized electron distributions, which converge to the bulk values calculated for the 1D hydrogen molecular crystal as n → ∞. The good performance of the xDH functionals shown here highlights their potentials as a useful tool for studying molecular crystals. Plots of the error evolution with respect to number of H2 units n or interunit distance d provide an intuitive visualization of electron-density delocalization and how its extent is affected by choice of functionals. From the perspective of DEs/LEs, hybrid functionals (PBE0 and B3LYP) improve over pure DFAs (PBE and BLYP) and HF due to the mixture of the opposing effects of the convex and concave behaviors of the latter two methods at a delocalized electron distribution. HFDFT display an improvement in cases which are densitydriven,85 where DEs degrade the electron density distribution to affect the energy. The use of the SIE-free HF densities in HF-DFT corrects such density errors to reduce DEs and to improve the energy. In general, the functional performances on adhesive energies in the hydrogen molecular chains are in line with the trends predicted by their deviations from the exact energy-linearity condition as represented by the H atom. As has been emphasized by Zheng et al.,60 it is nontrivial to resolve or quantify the amount of DEs in a complex chemical system near its equilibrium geometry and to analyze the influence of DEs on the calculated thermodynamic properties. Different sources of errors are often intercalated, while the hydrogen chain systems minimize the dispersion effects. DE (or LE) is a distinctly different source of error from the deficiency of DFAs in describing dispersive interactions, and adding dispersion correction (e.g., B2PLYP-D versus B2PLYP) will not help in minimizing DEs. DEs, which are pervasive and resilient, degrade the performance of approximate DFAs for various properties. Identifying and minimizing DEs remain vitally important for the development of better and better functionals.

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ASSOCIATED CONTENT

S Supporting Information *

Absolute energies from all methods examined in the present work. This material is available free of charge via the Internet at http://pubs.acs.org.

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +86-21-65643529. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Prof. Xiao Zheng for insightful discussion. Financial support from the Ministry of Science and Technology (Grants 2013CB834606 and 2011CB808505) and the National Natural Science Foundation of China (Grants 91027044 and 21133004) are gratefully appreciated. H



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Performances of the XYG3 Type of Doubly Hybrid ... - ACS Publications

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