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Fractional Charge Behavior and Band Gap Predictions with the XYG3 Type of Doubly Hybrid Density Functionals Neil Qiang Su,† Weitao Yang,‡ Paula Mori-Sánchez,§ 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 ‡ Department of Chemistry, Duke University, Durham, North Carolina 27708, United States § Departamento de Quı ́mica, Universidad Autónoma de Madrid, Madrid 28049, Spain ABSTRACT: In this work, we examine the fractional charge behaviors of doubly hybrid (DH) functionals. By plotting the ground-state energies E and energy derivatives for atoms and molecules with fractional electron numbers N, we directly quantify the delocalization errors of some representative DH functionals such as B2PLYP, XYG3, and XYGJ-OS. Numerical assessments on ionization potentials (IPs), electron affinities (EAs), and fundamental gaps, from either integer number calculations or energy derivative calculations, are provided. It is shown that the XYG3 type of DH functionals gives good agreement between their energy derivatives and the experimental IPs, EAs, and gaps, as expected from their nearly straight line fractional charge behaviors.

1. INTRODUCTION Many theoretical studies in density functional theory (DFT) have been devoted to the development of the exchange− correlation (xc) functionals.1−4 As a result, many xc functionals have been proposed,5−44 such as the local density approximation (LDA, e.g., SVWN55,6), the generalized gradient approximation (GGA, e.g., BLYP7,8 and PBE13), meta-GGA (e.g., TPSS16 and VSXC29), and the hybrid functionals (e.g., B3LYP12,30 and PBE014,15). The hybrid functionals represent the current mainstream of the xc functionals, which include a fraction of Hartree−Fock (HF) exchange.11 More recently, a new type of functional, the so-called doubly hybrid (DH) functionals,17,22,24 appeared, which, in addition to the hybridization of the HF exchange, also hybridize the perturbative energy to the second order as a certain portion of the correlation energy in functionals.17,22,24−28,33−44 The DH functionals enable accurate calculation of a wide range of properties, including standard enthalpies of formation, bond dissociation energies, transition barrier heights, nonbounded interactions, geometries and so forth,32−62 at reasonable computational cost. Electron removal and addition are two fundamental electronic processes in chemistry and material science, leading to essential molecular properties,3,63−65 that is, the ionization potential (IP) and electron affinity (EA), respectively, which, in turn, define many useful concepts such as electronegativity,66 chemical potential,67 hardness and softness,68 as well as electrophilicity and nucleophilicity,69 and so forth. For solids, the difference between the IP and EA corresponds to the fundamental band gap, which plays a critical role in determining the properties of electron transport, structure and energetics of defects and interfaces, and many electromagnetic responses.70 © 2014 American Chemical Society

However, there still remain major challenges in the accurate prediction of band gaps and related properties,71−94 which affect the application of DFT to many important systems. The main problem of DFT for energy gaps can be traced to the delocalization error89−94 because common approximate functionals tend to overdelocalize the added electron or hole, exhibiting a convex behavior for the energy of systems with fractional charges, which violates the exact linearity condition.72 Such deviation from linearity is also known as the manyelectron self-interaction error.95−99 In order to overcome this difficulty, the linearity condition has been imposed in the construction of some xc functionals. A typical example is the MCY3 functional developed in Yang’s group.96 More recently, a nonempirical scaling approach was proposed by Zheng et al., which largely restores the linearity for conventional DFT functionals.93 On the basis of the observation that the HF theory typically leads to an opposite concave behavior for the energy of systems with fractional charges, a pragmatic approach is to simply tune the proportion of the mixed HF exchange, which tends to cancel the incorrect convex behavior of pure DFT functionals. This approach has been successfully applied to either global100−102 or rangeseparated hybrid functionals,86,103−107 although some recent works have recently discussed a number of consequences and caveats for such an approach.108,109 Special Issue: International Conference on Theoretical and High Performance Computational Chemistry Symposium Received: March 26, 2014 Revised: May 16, 2014 Published: May 20, 2014 9201

dx.doi.org/10.1021/jp5029992 | J. Phys. Chem. A 2014, 118, 9201−9211

The Journal of Physical Chemistry A

Article

DH functionals have not been tested for their noninteger charge behaviors to the best of our knowledge. However, the second-order Møller−Plesset perturbation theory (MP2), with ingredients also found in DH functionals, has been shown to have minimal delocalization errors.92 The aim of the present work is to examine DH functionals. By plotting the groundstate energies E and energy derivatives for fractional electron numbers N, we directly quantify the delocalization errors of some representative DH functionals B2PLYP,22 XYG3,24 and XYGJ-OS. 25 Numerical assessments on IPs, EAs, and fundamental gaps of these methods, from either integer numbers or energy derivatives, are provided.

∂E ∂N

(1)

EA = E(N0) − E(N0 + 1)

(2)

is just obtained from the Kohn−Sham (KS) eigenvalues of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO). Equations 9−11 are also applicable to generalized Kohn−Sham (GKS) calculations, for example, the B3LYP hybrid functional89,94 GKS deriv GKS GKS Egap = εgap = εLUMO − εHOMO

integer deriv Δstraight = Egap − Egap

For a system with a fractional number of electrons N = N0 ± δ with 0 ≤ δ ≤ 1, it is known that in the grand canonical ensemble, the energy is a straight line connecting the total energies at the adjacent integers72

EcMP2

∂E ∂N

= N0 + δ

E(N0 + δ) − E(N0) = −EA δ

− N0 + δ

∂E ∂N

⎫ ⎪ ⎬ ⎪ N0 − δ ⎭

(ia|jb) =

=

deriv Egap

(6)

(7)

∬ ϕi( r1⃗)ϕa( r1⃗)r12−1ϕj( r2⃗ )ϕb( r2⃗ ) d r1⃗ d r2⃗

(15)

(16)

The KS orbitals and orbital eigenvalues thus obtained are subsequently used in the same way as those in MP2 (eq 14) for a perturbative correlation energy evaluation. The energy from the SCF part is finally augmented by the perturbative term with a scaling factor (1 − a2), leading to the full expression for B2PLYP

(8)

SCF ExcB2PLYP = Exc + (1 − a 2)EcMP2

KS = εHOMO N0 − δ

ijab

|(ij || ab)|2 (1 − na)(1 − nb) εi + εj − εa − εb

SCF Exc = a1ExHF + (1 − a1)(Exs + ΔExB88) + a 2EcLYP

The analytic expressions for the energy derivatives in eqs 5−7 have been derived, which can be different for different types of approximate xc functionals.89,94 In the case where Exc is an explicit functional of density ρ (e.g., LDAs or GGAs), one finds89,94 ∂E ∂N

∑ ninj

A simplified, but useful, analytic expression for the MP2 singleparticle energy has been derived along different routes 80,92,113,114 and has been used to calculate the quasiparticle band gap of solids.115,116 The full derivative of the MP2 energy can be obtained via the finite difference calculations with the minimizing HF potential at a given electron number N. This provides the tool with which the derivatives of the total energies from the DH functionals can be calculated. 2.2. DH Functionals. Grimme proposed B2PLYP,22 which is now a widely recognized DH functional. B2PLYP employed a hybrid GGA functional, defined in eq 16, which performs a standard self-consistent field (SCF) calculation in the first place

(5)

For the exact energy from the exact xc functional where the linearity condition (i.e., eq 4) holds, one has integer Egap

all

where (ij∥ab) = (ia|jb) − (ib|ja) is an antisymmetrized combination of the regular two-electron repulsion integrals112

Hence, the fundamental gap can also be given by the derivative difference ⎧ ⎪ ∂E deriv = IP − EA = ⎨ Egap ⎪ ⎩ ∂N

1 = 4

(14)

(4)

The pure state proof has also been given in consideration of finite systems in the dissociation limit.110 This linear relation means that

N0 − δ

(13)

Here, Δstraight measures the nonlinearity through the difference between the gap predicted as energy differences from integer points (eqs 1−3) and as the differences of derivatives at N0 (eqs 5−7 or 9−12). With these perspectives, the fractional charge behaviors of MP2, a very widely used wave function method, have been recently examined.92 The canonical MP2 correlation energy from a HF reference energy has been generalized to include occupation numbers, nf, for the ϕf molecular orbital80,92,111

(3)

E(N0) − E(N0 − δ) = = −IP δ

(12)

However, an approximate functional may not have the correct straight line behavior between the integers. This erroneous nonlinearity behavior has important implications for the KS orbital energies and their relationship to IPs and EAs89,94

integer Egap = IP − EA = E(N0 − 1) + E(N0 + 1) − 2E(N0)

∂E ∂N

(11)

Ederiv gap

where E(N0) is the ground-state energy of the N0-electron system, with N0 being an integer. The fundamental band gap for such a system is then given by

E(N0 ± δ) = (1 − δ) ·E(N0) + δ·E(N0 ± 1)

(10)

N0 + δ

deriv KS KS KS Egap = εgap = εLUMO − εHOMO

2. THEORETICAL BACKGROUND 2.1. IP, EA, and Fundamental Gap. IP and EA are defined as IP = E(N0 − 1) − E(N0)

KS = εLUMO

(17)

The parameters {a1, a2} = (0.53, 0.73) in eqs 16 and 17 were determined by parametrization against heats of formation

(9) 9202



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Table 1. IP,a EA,b\ and Fundamental Band Gapc (Einteger gap ) of an N-Electron System (in eV), Calculated as Energy Differences from Integer Points Li

Be

B

C

N

O

F

F2

OH

NH2

CH3

CN

MADf MADf MADf

Einteger gap IP EA Einteger gap IP EA Einteger gap IP EA Einteger gap IP EA Einteger gap IP EA Einteger gap IP EA Einteger gap IP EA Einteger gap IP EA Einteger gap IP EA Einteger gap IP EA Einteger gap IP EA Einteger gap IP EA Einteger gap IP EA

refd

HF

MP2

SVWN5

BLYP

B3LYP

MCY3e

B2PLYP

XYG3

XYGJ-OS

4.77 5.39 0.62 9.68 9.32 −0.36 8.02 8.30 0.28 10.00 11.26 1.26 14.76 14.54 −0.22 12.15 13.61 1.46 14.02 17.42 3.40 15.28 15.70 0.42 11.18 13.01 1.83 11.23 12.00 0.77 9.76 9.84 0.08 9.74 13.60 3.86

5.51 5.34 −0.17 8.96 8.04 −0.92 8.47 8.04 −0.43 10.47 10.80 0.33 16.15 13.89 −2.26 12.89 12.02 −0.88 14.75 15.65 0.90 16.10 16.13 0.03 11.95 11.36 −0.59 11.84 10.45 −1.39 10.94 8.98 −1.96 13.34 16.17 2.83 1.02 1.09 1.49

5.05 5.38 0.32 9.63 8.87 −0.75 8.26 8.31 0.04 10.21 11.30 1.09 15.49 14.63 −0.87 12.47 13.41 0.95 14.23 17.37 3.14 15.40 15.44 0.04 11.55 13.06 1.51 11.59 11.99 0.40 10.22 9.76 −0.47 8.14 12.93 4.80 0.41 0.16 0.42

4.92 5.47 0.55 9.42 9.03 −0.39 8.19 8.64 0.45 10.19 11.69 1.50 15.30 15.00 −0.30 12.52 13.98 1.45 14.42 17.93 3.51 15.40 15.63 0.23 11.74 13.49 1.75 11.65 12.30 0.65 10.20 10.09 −0.11 10.90 14.78 3.88 0.40 0.40 0.11

5.13 5.53 0.40 9.55 8.98 −0.57 8.51 8.62 0.11 10.39 11.40 1.00 14.77 14.49 −0.28 12.97 14.13 1.16 14.68 17.68 3.00 15.35 15.31 −0.04 12.00 13.28 1.28 11.95 12.18 0.23 10.32 9.81 −0.52 11.02 14.63 3.61 0.53 0.31 0.33

5.11 5.63 0.52 9.57 9.11 −0.46 8.52 8.74 0.21 10.43 11.54 1.11 14.96 14.65 −0.31 12.98 14.12 1.15 14.73 17.71 2.98 15.59 15.90 0.32 12.02 13.31 1.28 11.98 12.24 0.26 10.38 9.91 −0.46 11.22 15.16 3.94 0.59 0.37 0.25

5.29 5.83 0.54 9.63 9.26 −0.37 8.51 8.85 0.34 10.46 11.71 1.25 15.30 14.85 −0.45 12.86 14.02 1.16 14.68 17.77 3.09 15.61 16.24 0.63 11.98 13.35 1.37 11.96 12.29 0.33 10.48 10.08 −0.40 11.33 15.67 4.34 0.63 0.50 0.26

5.04 5.48 0.44 9.66 9.02 −0.64 8.46 8.51 0.05 10.37 11.35 0.97 15.09 14.52 −0.57 12.82 13.79 0.97 14.57 17.47 2.90 15.44 15.59 0.16 11.87 13.12 1.25 11.86 12.08 0.22 10.32 9.78 −0.54 10.56 14.39 3.83 0.46 0.17 0.36

4.92 5.47 0.55 9.84 9.16 −0.68 8.43 8.46 0.02 10.37 11.32 0.95 15.16 14.51 −0.65 12.76 13.68 0.92 14.52 17.39 2.87 15.28 15.49 0.21 11.80 13.08 1.28 11.82 12.08 0.26 10.30 9.78 −0.51 10.04 13.76 3.72 0.39 0.10 0.37

4.84 5.52 0.68 9.91 9.29 −0.62 8.39 8.43 0.04 10.40 11.30 0.90 15.07 14.45 −0.62 12.69 13.65 0.96 14.52 17.40 2.88 15.78 15.74 −0.04 11.80 13.09 1.29 11.81 12.10 0.29 10.20 9.74 −0.46 9.75 13.51 3.76 0.38 0.07 0.37

a Ionization energy calculated as IP = E(N0 − 1) − E(N0). bEA calculated as EA = E(N0) − E(N0 + 1). cFundamental band gap calculated as Einteger = gap IP − EA = E(N0 − 1) + E(N0 + 1) − 2E(N0). dReferences are experimental data taken from refs 117, 118, 124, and 129, except EAs for Be, N, and F2, which are CCSD(T) data from ref 86. eTaken from ref 89. fMAD: mean absolute deviation.

(HOFs) of the G2/97 set.117,118 There are many DH functionals of the B2PLYP type having been proposed in the literature.22,26,28,33−43 On the basis of the adiabatic connection formalism119−121 and Görling−Levy (GL)122,123 coupling-constant perturbation expansion to the second order, Xu’s group proposed another type of the DH functional, coined as XYG324

functional relies.32 Parameters of (b1 = 0.8033, b2 = 0.2107, b3 = 0.3211) in eq 18 were determined empirically by fitting only to the data of HOFs in the G3/99 set115,116,124 but were tested independently with different systems including reaction barrier heights and nonbonded interactions.32,45−62 More recently, a new version of the XYG3 type of functional, XYGJ-OS, was developed, where the MP2 contributions are simplified with the opposite-spin (OS) ansatz25

ExcXYG3 = b1ExHF + (1 − b1)Exs + b2ΔExB88 + (1 − b3)EcLYP + b3EcMP2

ExcXYGJ‐OS = c1ExHF + (1 − c1)Exs + (c 2EcVWN + c3EcLYP)

(18)

MP2 + c4Ec,OS

A unique feature is that all energy terms associated with XYG3 are evaluated by using the SCF 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

(19)

The final optimized XYGJ-OS parameters against HOF of the G3/99 set115,116,124 are {c1, c2, c3, c4} = {0.7731, 0.2309, 0.2754, 0.4364}. As compared to the conventional fifth-order DH functionals, XYGJ-OS has a favorable fourth-order scaling, 9203



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Table 2. IP,a EA,b and Fundamental Band Gapc (Ederiv gap ) of an N-Electron System (in eV), Calculated as the Derivative Differences Li

Be

B

C

N

O

F

F2

OH

NH2

CH3

CN

MADf MADf MADf

Ederiv gap IP EA Ederiv gap IP EA Ederiv gap IP EA Ederiv gap IP EA Ederiv gap IP EA Ederiv gap IP EA Ederiv gap IP EA Ederiv gap IP EA Ederiv gap IP EA Ederiv gap IP EA Ederiv gap IP EA Ederiv gap IP EA Ederiv gap IP EA

exptd

HF

MP2

SVWN5

BLYP

B3LYP

MCY3e

B2PLYP

XYG3

XYGJ-OS

4.77 5.39 0.62 9.68 9.32 −0.36 8.02 8.30 0.28 10.00 11.26 1.26 14.76 14.54 −0.22 12.15 13.61 1.46 14.02 17.42 3.40 15.28 15.70 0.42 11.18 13.01 1.83 10.37 11.14 0.77 9.76 9.84 0.08 9.74 13.60 3.86

5.64 5.34 −0.29 9.60 8.41 −1.19 9.76 8.67 −1.08 12.72 11.94 −0.78 18.89 15.52 −3.37 16.83 14.18 −2.64 20.01 18.47 −1.54 20.49 18.12 −2.37 16.56 13.95 −2.61 15.48 12.59 −2.89 13.30 10.47 −2.83 13.29 14.14 0.85 3.51 0.81 2.84

5.15 5.37 0.22 9.43 8.69 −0.74 8.28 8.17 −0.11 10.22 11.10 0.88 15.60 14.27 −1.33 12.38 12.93 0.55 13.37 16.36 2.99 12.39 13.51 1.13 10.62 12.00 1.39 10.93 11.17 0.24 9.96 9.21 −0.75 8.58 13.67 5.09 0.66 0.64 0.64

1.04 3.17 2.12 3.51 5.60 2.09 0.20 4.10 3.90 0.09 6.10 6.01 4.07 8.36 4.29 0.31 7.35 7.04 0.14 10.25 10.11 3.47 9.56 6.09 0.18 7.28 7.10 2.00 7.11 5.11 1.84 5.37 3.53 1.67 9.75 8.08 9.34 5.00 4.34

1.32 3.03 1.71 3.55 5.47 1.92 0.62 4.05 3.43 0.62 5.91 5.29 3.70 8.03 4.33 0.89 7.51 6.63 0.94 10.19 9.25 3.73 9.36 5.63 0.85 7.22 6.37 2.59 7.06 4.48 2.21 5.18 2.98 1.75 9.48 7.73 8.99 5.12 3.86

2.07 3.65 1.58 4.91 6.32 1.41 2.49 5.16 2.67 3.06 7.31 4.25 6.69 9.74 3.05 4.10 9.19 5.09 4.77 12.20 7.43 7.05 11.45 4.40 4.03 8.90 4.87 5.19 8.48 3.30 4.36 6.46 2.10 3.80 10.76 6.96 6.51 3.70 2.81

5.72 6.10 0.37 10.10 9.31 −0.79 8.65 8.57 −0.08 10.03 11.12 1.09 14.31 13.84 −0.48 11.57 13.01 1.44 12.30 16.18 3.88 13.75 15.17 1.42 11.23 12.70 1.47 12.10 12.15 0.05 10.90 10.08 −0.82 10.19 14.34 4.14 0.74 0.47 0.44

3.27 4.30 1.04 6.91 7.30 0.40 5.05 6.42 1.37 6.23 8.90 2.67 10.40 11.65 1.25 7.74 10.88 3.14 8.72 14.12 5.41 9.96 12.64 2.68 7.02 10.36 3.35 7.73 9.72 1.99 6.76 7.68 0.92 6.51 12.36 5.86 3.69 2.30 1.39

4.17 4.98 0.80 8.84 8.37 −0.48 7.47 7.70 0.23 9.26 10.51 1.25 13.82 13.58 −0.24 11.34 12.76 1.42 12.87 16.32 3.45 13.64 14.49 0.85 10.22 12.13 1.91 10.37 11.22 0.85 9.10 9.07 −0.03 9.68 13.95 4.27 0.82 0.80 0.13

4.07 5.01 0.94 8.79 8.37 −0.42 7.45 7.61 0.15 9.62 10.57 0.95 13.78 13.82 0.04 10.65 12.22 1.57 12.54 16.00 3.45 14.10 14.63 0.53 10.00 11.87 1.87 10.16 11.01 0.85 9.00 9.04 0.05 9.60 13.92 4.31 0.90 0.88 0.16

IP calculated as IP = (−∂E/∂N)|N0−δ. bEA calculated as EA = (−∂E/∂N)|N0+δ. cFundamental band gap calculated as Ederiv gap = IP − EA = {(∂E/∂N)|N0+δ − (∂E/∂N)|N0−δ}. dReferences are experimental data taken from refs 117, 118, 124, and 129, except EAs for Be, N, and F2, which are CCSD(T) data from ref 86. eTaken from ref 89. fMAD: mean absolute deviation. a

100 radial shells and 1202 angular points (for the first row elements) was used for numerical integration of the DFT xc potential. The basis sets used were cc-pVQZ127,128 with the Cartesian angular functions, which facilitates the direct comparison between the results of the present work and those in ref 89. Basis set dependence has been tested. A larger basis set (e.g., aug-cc-pVQZ or cc-pV5Z) was found to be important to reach converged results, in particular, for EAs, but was not able to change the statistics qualitatively. In this paper, we considered the direct extension of the KS formalism to systems with noninteger total electron number. The orbitals were occupied according to the aufbau principle. Only the highest-occupied (HO) spin orbital was allowed to be fractionally occupied, as prescribed by the Janak’s theorem.71 The occupation number in HO was increased and decreased from the neutral to the anion and the cation in the increment of

which can be further reduced to the third order by exploiting the spatial locality of electron correlation.25,125 This extends the applicability of the method to substantially larger molecular systems. Instead of B3LYP, other DFT methods can also be used to generate the orbitals and densities, albeit the mixing coefficients in the DH functionals for final energy expression shall be reparameterized.

3. COMPUTATIONAL DETAILS A development version of the NorthWest computational Chemistry (NWChem) software package (version 6.1.1)126 was used for all of the calculations in this study. The calculations were performed self-consistently in the spinunrestricted formalisms to generate orbitals and densities with conventional DFT methods. The NWChem “Xfine” grid with 9204



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δ = ±0.1. No spherical symmetry requirements were imposed on electron densities in atoms. Hence, a fractionally occupied HO spin orbital ended up being nondegenerate in all cases that we studied. In the numerical calculations of the derivatives (∂E/ ∂N)|N0±δ, δ was chosen as 0.001 according to ref 92. Previously, the numerical differentiation was examined for δ = 0.1, 0.01, 0.001, 0.0001, and 0.00001 with the orbital-specific global functional of BLYP, and it was found that stable numerical differentiation results can be achieved for δ ≤ 0.01.107 The data set used in the present work follows that of Yang’s group.89 It includes atoms of Li, Be, B, C, N, O, and F, as well as molecular species of F2, OH, NH2, CH3, and CN. Reference data for IPs and EAs were generally the experimental data taken from ref 129, whereas there are no experimental data available for EAs of Be, N, and F2; hence, the CCSD(T) data from ref 86 were used instead. The experimental geometries of the neutral molecules,86,130,131 that is, F2 (1.4119 Å), OH (0.9697 Å), NH2 (1.02402 Å, 103.40°), CH3 (1.079 Å, 120.0°), and CN (1.1718 Å), were adopted, which were fixed in the calculations for the cations and anions. The tested DH functionals include B2PLYP,22 XYG3,24 and XYGJ-OS.25 For comparison, SVWN5, BLYP, and B3LYP along with HF and MP2 calculations were also performed.

accumulations. This is confirmed by the present work for BLYP and B3LYP, where MADs are increased by ∼0.2 eV to 0.53 and 0.59 eV, respectively, for the gap calculations. For the methods studied with the present set, MADs for the gap calculations are between 0.38 and 0.63 eV, with the exception of HF, whose MAD is 1.02 eV. 4.2. IPs, EAs, and Fundamental Band Gaps from Derivative Differences. The main focus of the present work is to examine the fractional charge dependence of some representative DH functionals and to see its consequence on the IP, EA, and band gap calculations. The calculation results for IPs, EAs, and the fundamental band gaps from the derivative differences are summarized in Table 2. It is now well-established that the energy derivatives are exactly the eigenvalues in a KS and GKS calculation.89,92 Hence, data in Table 2 demonstrate how well the orbital energies from an approximate functional can be used to approximate IPs and EAs and how well the (G)KS gaps correspond to the fundamental gaps. Conventional functionals display significant errors in IP and EA predictions, and these errors add up in the gap predictions. SVWN5, BLYP, and B3LYP yield MADs of 5.00, 5.12, and 3.70 eV, respectively, for IPs and those of 4.34, 3.86, and 2.81 eV for EAs. MADs associated with SVWN5, BLYP, and B3LYP for the gaps are 9.34, 8.99, and 6.51 eV, respectively. These represent dramatic degradation as compared to their own performance from integer calculations. HF gives quite reasonable derivative IPs (MAD = 0.81 eV), as approximated by orbitals energies; the MAD for the derivative EAs is as high as 2.84 eV, leading to a MAD of 3.51 eV for the gaps as approximated by orbitals energies. MP2 gives the best prediction of the gaps (MAD = 0.66 eV), with similar accuracy for both derivative IPs and EAs (MAD = 0.64 eV), once again showing the importance for the inclusion of the correlation effects. As shown in Table 2, there is a clear improvement in gap prediction from SVWN5 to BLYP by 0.35 eV and to B3LYP by another 2.48 eV. The DH functionals provide further improvement in this perspective. MADs for the derivative IPs and EAs are both reduced by ∼1.4 eV from B3LYP to 2.30 and 1.39 eV, respectively, for B2PLYP, which adds up to an improvement of 2.82 eV in the derivative gap calculations, although MAD associated with B2PLYP is still as high as 3.69 eV. XYG3 and XYGJ-OS display a clear improvement over the conventional DFT methods. MADs are 0.82 and 0.92 eV, respectively, for XYG3 and XYGJ-OS in the derivative gap calculations. The errors for a functional in the derivative gaps Δderiv gap as compared to the integer gaps Δinteger can be traced to the gap deviations from the nonlinearity,89 where eq 13 provides a quantitative measurement. From data in Tables 1 and 2, one finds that the averaged Δstraight follows the trend such that 9.69(SVWN5) > 9.49(BLYP) > 7.08(B3LYP) > 4.15(B2PLYP) > 1.28(XYGJ-OS) > 1.21 eV(XYG3). In fact, for all of these > Δderiv methods, one has Δinteger gap gap for every system, leading to an overall positive Δstraight. Tables 1 and 2 also list the data for MCY3 taken from ref 89. MCY3 is a functional that has taken into account the linearity condition by construction. MCY3 leads to the smallest averaged Δstraight = 0.60 eV among the functionals studied in the present work, with MADs for Δinteger and Δderiv gap gap being 0.63 and 0.74 eV, respectively.

4. RESULTS AND DISCUSSION 4.1. IPs, EAs, and Fundamental Band Gaps from Integer Points. For atomic and molecular species, the validations for the functional performance in predicting IPs and EAs from integer points have been performed quite often in the literature.18,19,25,29,32,35,58,86,132,133 In a recent publication,58 36 functionals have been examined, where B2PLYP and XYG3 have been included. Using the well-established G2-1 set asa reference,117,118,124 which contains 14 atoms and 24 molecules for IPs, along with 7 atoms and 18 molecules for EAs, it was found that XYG3 leads to mean absolute deviations (MADs) of 0.057 and 0.080 eV for IPs and EAs, respectively. In comparison, MADs for IPs/EAs are 0.109/0.090 (B2PLYP) and 0.163/0.166 eV (MP2). These are the results for the adiabatic IPs and EAs.58 Table 1 summarizes the results from the present work for IPs and EAs, where the data set may be considered as a subset of the G2-1 set. While XYG3 and XYGJ-OS give MADs of 0.10 and 0.07 eV, respectively, for IPs and 0.37 eV for EAs, B2PLYP gives a MAD that is about 0.1 eV higher than that of XYGJ-OS for IPs and behaves similarly on average as XYG3 for EAs. Conventional DFT methods such as SVWN5, BLYP, and B3LYP lead to larger MADs for IPs (0.40, 0.31, and 0.37 eV, respectively), whereas the corresponding MADs for EAs turn out to be smaller (0.11, 0.33, and 0.25 eV, respectively) for this data set. Prediction of the IP for the CN radical is a challenging case, where conventional DFT methods all give MADs higher than 1 eV. Error associated with B2PLYP is still as high as 0.79 eV, which has been satisfactorily reduced to 0.16 and 0.09 eV for XYG3 and XYGJ-OS, respectively. HF is the least useful, which leads to MADs of 1.09 and 1.49 eV, respectively, for IPs and EAs. MP2 successfully reduces the corresponding MADs to 0.15 and 0.42 eV, showing the importance of the inclusion of the correlation effects. Table 1 also shows the results for the corresponding fundamental gaps. It is known that for some conventional DFT methods, errors are larger for the gap predictions than those for the IP and EA calculations due to error 9205



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Figure 1. Fractional charge behaviors for the carbon atom with methods of (a) HF, MP2, (b) SVWN5, BLYP, B3LYP, and (c) B2PLYP, XYG3, XYGJ-OS.

4.3. Fractional Charge Behavior for Energy E versus Electron Number N. We are now in a position to explicitly check the fractional charge behaviors against the linearity condition.89,92,95,96,98,106,107 Figure 1 shows, as an example, the ground-state energy E of the C atom as a function of the electron number N. The exact straight lines are obtained using the experimental IP and EA of the C atom. From Figure 1a, one finds that the HF result is a nearly straight line connecting N = 5 and 6. This is in accord with the fact that HF has a quite reasonable description of IPs. On the other hand, HF has a quite poor description of EAs. The HF curve is clearly in a concave manner. This gives too small values for EAs and even negative EAs for many systems, as shown in Table 2, which results in too large derivative gaps. From Figure 1b, one sees that even though conventional DFT methods yield very reasonable values for IPs and EAs from integer numbers, they exhibit strongly convex behaviors in between integers. Thus, these methods tend to give too small values for IPs and too large values for EAs, leading to gaps that are too small. Such a fractional charge behavior is in line with the findings summarized in Tables 1 and 2 for conventional DFT methods such as SVWN5, BLYP, and B3LYP. As HF and conventional DFT methods show opposite errors, there is a possibility that mixing these two in a proper portion will result in error cancellation. This has highlighted the importance of the HF exchange, leading to several useful methods in the literature.100−109 On the other hand, MP2 clearly exhibits a very good linearity for the C atom, as shown in Figure 1a, consistent with earlier findings.92 This emphasizes the importance of the correlation effects. Figure 1c displays the fractional charge behaviors for some representative DH functionals. Even though the convexity is still clearly seen, B2PLYP shows a clear improvement over the conventional DFT methods shown in Figure 1b. Due to this convexity, B2PLYP still tend to give IP values that are too small and EA values that are too large, leading to an underestimation of the gap values from energy derivatives. For XYG3 and XYGJ-OS, as their lines are nearly straight, they give a good description of the fractional charge behaviors. The use of their energy derivatives to predict IPs, EAs, and gaps is most satisfactory among the functionals studied in this work. This explains the data in Table 2 and provides theoretical support to the generally good performance of the XYG3 type of

functionals in many properties,45−62 in particular, transition barrier heights. Figure 2 shows the deviations of the methods from the linearity in another way, which plots the difference of the

Figure 2. Deviations from the corresponding linear interpolations for the carbon atom with methods of (a) HF, MP2, (b) SVWN5, BLYP, B3LYP, and (c) B2PLYP, XYG3, XYGJ-OS. Note that the scale changes for the vertical coordinates from (a) to (b) and to (c).

energy predicted by the method and its own straight line interpolation between adjacent integer values for the C atom (i.e., C+ ← C → C−). Note that the scale changes for the vertical coordinates from Figure 2a to b and to c. The positive deviations mean a concave error, while the negative deviations suggest a convex error. MP2 exhibits the smallest error for the C atom. Interestingly, the MP2 error changes sign at N = 5.5 and 6.5 with minimum errors, whereas curves from other methods are almost quadratic, showing the largest deviation at 9206



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these middle points. Analogous figures for OH and F2 are shown in Figure 3a and b, respectively, where the MP2

Figure 4. Eigenvalues (in eV) for the carbon atom, estimated from finite differences at each N with δ = 0.001. {E(N) − E(N − δ)}/δ for 5 < N ≤ 6 may correspond to the HOMO of the carbon atom, while {E(N + δ) − E(N)}/δ for 6 ≤ N < 7 may correspond to the LUMO of the carbon atom.

Figure 3. Deviations from the corresponding linear interpolations with various methods for (a) OH and (b) F2 species.

−5.29 and −5.91 eV, respectively. With B3LYP, the discontinuity at N0 = 6 is clearly seen (εB3LYP LUMO = −4.25 versus εB3LYP HOMO= −7.31 eV), but the gap is less than 1/3 of the exact one. Instead of staying in constancy, the frontier orbitals from these conventional DFT methods are erroneously destabilized with the increase of the fractional number of electrons (see Figure 4b). Figure 4c clearly demonstrates that the DH functionals represent an undisputed improvement over the conventional DFT methods. In B2PLYP, the discontinuity at N0 = 6 is clearly widened, although the energy derivatives still undergo steady changes with the fractional number of electrons. The linearity condition has been served as a rigorous litmus test for correct xc functionals.72,89−108 It is encouraging to see that XYG3 and XYGJ-OS, which have not been tuned according to the linearity condition, tend to obey such a condition.

performance degrades as compared to its own performance on the C atom. Higher-order correlation than that of MP2 is expected to restore the linearity. 4.4. Fractional Charge Behavior for Eigenvalues versus Electron Number N. For conventional DFT methods, as well as HF, it is known that the energy derivatives (∂E/ ∂N)|N0±δ correspond to eigenvalues of the LUMO and HOMO of the N0 system, as shown in eqs 9 and 10.89,94 The linearity condition demands that the exact energy derivative is a constant that is equal to −IP for N0 − 1 < N ≤ N0 and to −EA for N0 ≤ N < N0 + 1, which displays the derivative discontinuity at N0. Figure 4 checks this fractional charge behavior for eigenvalues of the frontier orbitals versus electron number N using the carbon atom as a representative. Figure 4a checks the behaviors of the HF and MP2 methods. As HF shows a concave E versus N behavior, the frontier orbitals from HF are stabilized with the increase of the fractional number of electrons. By inclusion of the correlation effects, MP2 successfully corrects this error. Both its HOMO and LUMO energies stay constant with the increase of the fractional number of electrons with a clear and nearly exact gap. As SVWN5 is strongly convex, its derivative discontinuity at N0 = 6 is almost missing (see Figure 4b). The LUMO energy estimated by {E(6.001) − E(6)} is −6.10 eV, which is almost identical to the HOMO energy, −6.01 eV, estimated by {E(6.001) − E(6)}. BLYP does not improve too much over SVWN5. The LUMO and HOMO energies are estimated to be

5. CONCLUSIONS There remain major challenges in using DFT to accurately predict some important chemical properties such as IPs, EAs, and their difference (i.e., fundamental band gaps). The main problem of relating KS frontier orbital energies to these properties has been traced to the delocalization errors by Yang and co-workers89−94 because common approximate functionals exhibit a convex behavior in violation of the exact linearity condition for fractional charges. It is not clear, however, how a new class of approximate xc functionals, namely, DH 9207



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functionals, behaves in this situation. We present here such an analysis and show that the XYG3 type of DH functionals, for example, XYG324 and XYGJ-OS,25give good agreement between their energy derivatives and the experimental IPs, EAs, and gaps, as expected from their nearly straight line fractional charge behaviors. It was shown that there is a large fractional spin error in MP2,92 which DH functionals shall inherit. Removal of such an error is a future direction to go for the construction of a better DH functional.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Telephone number: +86-2165643529. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS X.X. has been supported by National Natural Science Foundation of China (91027044, 21133004) and the Ministry of Science and Technology (2013CB834606, 2011CB808505). W.Y. has been supported by the National Science Foundation (No. CHE-09-11119). P.M.S. has been supported by Ramón y Cajal and the Spanish Ministry of Science (FIS2009-12721).

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NOTE ADDED AFTER ASAP PUBLICATION This paper was published ASAP on June 3, 2014. The data in the spreadsheet were incorrectly (one column shifted) copied into Table 2. X.X. thanks Yihan Shao for pointing out this error. The corrected version was reposted on September 15, 2014.

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