Supplemental Material to: Restoration of the Derivative Discontinuity in Kohn-Sham Density Functional Theory: An E

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Supplemental Material to: Restoration of the Derivative Discontinuity in Kohn-Sham Density Functional Theory: An

Efficient Scheme for Energy Gap Correction

Jeng-Da Chai

^{1, 2,}^⇤

and Po-Ta Chen

¹

1

Department of Physics, National Taiwan University, Taipei 10617, Taiwan

2

Center for Theoretical Sciences and Center for Quantum Science and Engineering,

National Taiwan University, Taipei 10617, Taiwan

(2)

⇤

Author to whom correspondence should be addressed. Electronic mail: [email protected].

[1] D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., (Prentice Hall, Upper Saddle River, NJ, 2004).

[2] Y.-S. Lin, C.-W. Tsai, G.-D. Li, and J.-D. Chai, J. Chem. Phys. 136, 154109 (2012).

[3] P. A. M. Dirac, Proc. Cambridge Philos. Soc. 26, 376 (1930); J. P. Perdew and Y. Wang, Phys.

Rev. B 45, 13244 (1992).

[4] R. van Leeuwen and E. J. Baerends, Phys. Rev. A 49, 2421 (1994).

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DERIVATIVE DISCONTINUITY FROM THE PERSPECTIVE OF PERTURBATION THEORY

As discussed in our paper, we aim to express ✏

_{N +1}

(N + 1) in terms of {✏

i

(N ),

_i

(r) }, based on perturbation theory [1]. Here, we provide the full derivation (the notations used here are the same as those used in the paper). As Eq. (12) of our paper gives

(H

KS

+ H

⁰⁽⁰⁾

+

²

H

⁰⁽¹⁾

+ · · · )(

i⁽⁰⁾

+

_i⁽¹⁾

+

² _i⁽²⁾

+ · · · )

= (✏

⁽⁰⁾_i

+ ✏

⁽¹⁾_i

+

²

✏

⁽²⁾_i

+ · · · )(

i⁽⁰⁾

+

_i⁽¹⁾

+

² _i⁽²⁾

+ · · · ),

(1)

expanding Eq. (1) and comparing coefficients of each power of yields an infinite series of simultaneous equations.

Zeroth-Order Theory

To zeroth order (

⁰

) in Eq. (1), the equation is

H

_KS _i⁽⁰⁾

(r) = ✏

⁽⁰⁾_i ⁽⁰⁾_i

(r), (2) which is simply the KS equation for the N -electron system (i.e., the unperturbed system).

We then have

_i⁽⁰⁾

(r) =

i

(r) and ✏

⁽⁰⁾_i

= ✏

i

(N ). Therefore, ✏

N +1

(N + 1) = ✏

_{N +1}⁼¹

(N + 1) ⇡

✏

⁽⁰⁾_{N +1}

= ✏

N +1

(N ). Correspondingly,

xc

= ✏

N +1

(N +1) ✏

N +1

(N ) ⇡ ✏

N +1

(N ) ✏

N +1

(N ) = 0, and E

g

=

KS

+

xc

⇡

^KS

. Therefore, to obtain a nonvanishing

xc

, it is necessary to go beyond the zeroth-order theory.

First-Order Theory

To first order (

¹

) in Eq. (1), the equation is H

KS (1)

i

(r) + H

⁰⁽⁰⁾ _i⁽⁰⁾

(r) = ✏

⁽⁰⁾_i _i⁽¹⁾

(r) + ✏

⁽¹⁾_i ⁽⁰⁾_i

(r). (3) Taking the inner product of Eq. (3) with

_i⁽⁰⁾

,

h

⁽⁰⁾i

| H

KS

|

i⁽¹⁾

i + h

⁽⁰⁾i

| H

⁰⁽⁰⁾

|

i⁽⁰⁾

i

= ✏

⁽⁰⁾_i

h

i⁽⁰⁾

|

i⁽¹⁾

i + ✏

⁽¹⁾i

h

i⁽⁰⁾

|

i⁽⁰⁾

i.

(4)

As H

_KS

is Hermitian, the first term on the left-hand side is the same as that on the right- hand side. Moreover, we have h

⁽⁰⁾i

|

i⁽⁰⁾

i = h

i

|

i

i = 1, so the first-order correction to the orbital energy is

✏

⁽¹⁾_i

= h

i⁽⁰⁾

| H

⁰⁽⁰⁾

|

i⁽⁰⁾

i = h

ⁱ

| H

⁰=0

|

ⁱ

i. (5) Rewriting Eq. (3), we have

(H

KS

✏

⁽⁰⁾_i

)

⁽¹⁾_i

(r) = (H

⁰⁽⁰⁾

✏

⁽¹⁾_i

)

⁽⁰⁾_i

(r). (6) Since {

⁽⁰⁾j

(r) } constitute a complete set,

i⁽¹⁾

(r) can be expressed as a linear combination of them:

(1)

i

(r) = X

j6=i

a

⁽ⁱ⁾_j _j⁽⁰⁾

(r). (7)

Note that it is unnecessary to include j = i in the sum, for if

_i⁽¹⁾

(r) satisfies Eq. (6), so too does (

⁽¹⁾_i

(r) + ↵

⁽⁰⁾_i

(r)), for any constant ↵, and we can use this freedom to subtract o↵

the

_i⁽⁰⁾

(r) term so that ˜

_i

(r) is normalized at first order in . Inserting Eq. (7) into Eq.

(6), we have

X

j6=i

(✏

⁽⁰⁾_j

✏

⁽⁰⁾_i

)a

⁽ⁱ⁾_j _j⁽⁰⁾

(r) = (H

⁰⁽⁰⁾

✏

⁽¹⁾_i

)

_i⁽⁰⁾

(r). (8)

Taking the inner product of Eq. (8) with

_k⁽⁰⁾

, we get X

j6=i

(✏

⁽⁰⁾_j

✏

⁽⁰⁾_i

)a

⁽ⁱ⁾_j

h

k⁽⁰⁾

|

⁽⁰⁾j

i

= h

⁽⁰⁾k

| H

⁰⁽⁰⁾

|

⁽⁰⁾i

i + ✏

⁽¹⁾i

h

k⁽⁰⁾

|

i⁽⁰⁾

i.

(9)

If k = i, the left-hand side is zero, we recover Eq. (5); If k 6= i, we get (✏

⁽⁰⁾k

✏

⁽⁰⁾_i

)a

⁽ⁱ⁾_k

= h

k⁽⁰⁾

| H

⁰⁽⁰⁾

|

i⁽⁰⁾

i, or a

⁽ⁱ⁾j

=

^h

(0)

j |H⁰⁽⁰⁾| ⁽⁰⁾_i i

✏⁽⁰⁾_i ✏⁽⁰⁾_j

, so the first-order correction to the orbital is

(1)

i

(r) = X

j6=i

h

j⁽⁰⁾

| H

⁰⁽⁰⁾

|

i⁽⁰⁾

i

✏

⁽⁰⁾_i

✏

⁽⁰⁾_j

(0)

j

(r). (10)

Note that ˜ ⇢

₌₀

(r) = P

N +1

i=1

| ˜

i⁼⁰

(r) |

²

= P

N +1

i=1

|

i⁽⁰⁾

(r) |

²

= P

N +1

i=1

|

i

(r) |

²

= ⇢(r) + ⇢

_L

(r), where ⇢

L

(r) ⌘ |

N +1

(r) |

²

is the KS LUMO density of the N -electron system. Consequently, we have

H

⁰₌₀

= e

²

Z ⇢

L

(r

⁰

)

| r r

⁰

| dr

⁰

+ v

xc

([⇢ + ⇢

L

]; r) v

xc

([⇢]; r). (11)

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As ✏

N +1

(N + 1) = ✏

_{N +1}⁼¹

(N + 1) ⇡ ✏

⁽⁰⁾N +1

+ ✏

⁽¹⁾_{N +1}

, we have

xc

= ✏

N +1

(N + 1) ✏

N +1

(N )

⇡ ✏

⁽¹⁾N +1

= h

N +1

| H

⁰=0

|

N +1

i

= e

²

ZZ ⇢

L

(r)⇢

L

(r

⁰

)

|r r

⁰

| drdr

⁰

+

Z

⇢

L

(r)

⇢

v

xc

([⇢ + ⇢

L

]; r) v

xc

([⇢]; r) dr,

(12)

and E

g

=

KS

+

xc

⇡

^KS

+ ✏

⁽¹⁾_{N +1}

. Eq. (12) is a key result, showing that the DD can be approximately expressed as an explicit universal functional of ⇢(r) and ⇢

L

(r), and can be calculated in the standard KS method without extra computational cost.

Second-Order Theory

To second order (

²

) in Eq. (1), the equation is

H

KS (2)

i

(r) + H

⁰⁽⁰⁾ ⁽¹⁾_i

(r) + H

⁰⁽¹⁾ _i⁽⁰⁾

(r)

= ✏

⁽⁰⁾_i ⁽²⁾_i

(r) + ✏

⁽¹⁾_i _i⁽¹⁾

(r) + ✏

⁽²⁾_i _i⁽⁰⁾

(r).

(13)

Taking the inner product of Eq. (13) with

_i⁽⁰⁾

, we get

h

⁽⁰⁾i

| H

KS

|

⁽²⁾i

i + h

i⁽⁰⁾

| H

⁰⁽⁰⁾

|

i⁽¹⁾

i + h

⁽⁰⁾i

| H

⁰⁽¹⁾

|

⁽⁰⁾i

i

= ✏

⁽⁰⁾_i

h

i⁽⁰⁾

|

i⁽²⁾

i + ✏

⁽¹⁾i

h

i⁽⁰⁾

|

i⁽¹⁾

i + ✏

⁽²⁾i

h

⁽⁰⁾i

|

⁽⁰⁾i

i.

(14)

As H

_KS

is Hermitian, the first term on the left-hand side is the same as that on the right- hand side. Moreover, we have h

i⁽⁰⁾

|

⁽⁰⁾i

i = 1 and h

⁽⁰⁾i

|

i⁽¹⁾

i = 0 (see Eq. (10)), so the second-order correction to the orbital energy is

✏

⁽²⁾_i

= h

i⁽⁰⁾

| H

⁰⁽⁰⁾

|

i⁽¹⁾

i + h

i⁽⁰⁾

| H

⁰⁽¹⁾

|

⁽⁰⁾i

i

= X

j6=i

|h

j

| H

⁰=0

|

i

i|

²

✏

i

(N ) ✏

j

(N ) + h

i

| H

⁰⁽¹⁾

|

i

i, (15)

(6)

where

H

⁰⁽¹⁾

= @H

⁰

@

₌₀

=

Z ⇢ e

²

| r r

⁰

| + v

xc

([˜ ⇢ ]; r)

˜

⇢ (r

⁰

)

@ ˜ ⇢ (r

⁰

)

@

₌₀

dr

⁰

=

Z ⇢ e

²

| r r

⁰

| + f

_xc

([⇢ + ⇢

_L

]; r, r

⁰

)

⇥

⇢ 2

N +1

X

i=1

<(

i^⇤

(r

⁰

)

_i⁽¹⁾

(r

⁰

)) dr

⁰

.

(16)

Here f

xc

([⇢]; r, r

⁰

) ⌘ v

^xc

([⇢]; r)/ ⇢(r

⁰

) is the XC kernel, the asterisk (*) denotes complex conjugate, and <(· · · ) denotes the real part of (· · · ). From Eq. (15), we have

✏

⁽²⁾_{N +1}

= X

j6=N+1

|h

^j

| H

⁰=0

|

^{N +1}

i|

²

✏

N +1

(N ) ✏

j

(N ) + h

N +1

| H

⁰⁽¹⁾

|

N +1

i. (17) Correspondingly, ✏

N +1

(N + 1) = ✏

_{N +1}⁼¹

(N + 1) ⇡ ✏

⁽⁰⁾N +1

+ ✏

⁽¹⁾_{N +1}

+ ✏

⁽²⁾_{N +1}

. This gives

xc

=

✏

N +1

(N + 1) ✏

N +1

(N ) ⇡ ✏

⁽¹⁾N +1

+ ✏

⁽²⁾_{N +1}

and E

g

=

KS

+

xc

⇡

KS

+ (✏

⁽¹⁾_{N +1}

+ ✏

⁽²⁾_{N +1}

).

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TABLES

TABLE I. Fundamental gaps (in eV) of the FG115 database [2], calculated by three schemes (see text for details) using the LDA [3] and LB94 [4] functionals.

LDA LB94

Molecule Reference KS KS+ xc Eg KS KS+ xc Eg

H (Hydrogen atom) 12.86 4.71 11.05 9.19 9.48 14.26 11.16

He (Helium atom) 27.23 18.01 22.01 21.81 20.71 27.60 27.40

Li (Lithium atom) 4.22 1.13 5.49 3.98 -2.55 6.13 4.72

Be (Beryllium atom) 9.66 3.50 10.44 7.51 3.34 12.05 8.93

B (Boron atom) 7.99 0.19 10.30 6.19 -0.16 11.09 7.74

C (Carbon atom) 9.97 0.08 13.38 8.08 -0.52 14.28 9.59

N (Nitrogen atom) 14.74 4.02 14.02 11.94 3.28 17.01 13.15

O (Oxygen atom) 12.14 0.29 14.44 10.27 -0.28 17.79 12.20

F (Fluorine atom) 13.98 0.12 18.95 12.13 -0.81 22.36 13.99

Ne (Neon atom) 26.91 17.25 23.74 23.40 18.77 27.98 27.69

Na (Sodium atom) 4.14 0.91 5.44 3.88 -1.92 6.86 4.93

Mg (Magnesium atom) 7.76 3.39 8.15 6.20 3.68 9.71 8.04

Al (Aluminum atom) 5.53 0.09 6.74 4.54 -0.41 7.26 5.68

Si (Silicon atom) 6.73 -0.02 8.37 5.76 -0.52 8.91 6.52

P (Phosphorus atom) 9.78 2.16 9.56 8.13 0.65 10.87 8.49

S (Sulfur atom) 8.23 0.04 9.85 7.25 -0.64 11.79 8.34

Cl (Chlorine atom) 9.30 -0.07 12.47 8.38 -0.76 13.29 9.35

Ar (Argon atom) 18.65 11.23 15.78 15.52 11.85 18.38 18.23

CH₃ (Methyl radical) 9.86 1.79 8.25 8.04 0.67 11.21 9.35

CH₄ (Methane) 15.06 9.21 12.19 11.47 10.08 15.55 14.78

NH (Imidogen) (³⌃ ) 13.17 3.10 11.72 11.04 1.99 15.71 12.11

NH₂ (Amino radical) 11.34 1.97 10.56 9.87 1.17 14.46 11.27

NH₃ (Ammonia) 11.54 5.64 9.42 8.47 6.59 12.96 11.91

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OH (Hydroxyl radical) 11.27 0.16 11.52 9.79 -0.63 16.50 11.38

H₂O (Water) 13.35 6.57 11.03 9.85 7.65 15.08 13.66

HF (Hydrogen fluoride) 16.91 9.00 14.23 12.50 10.40 19.39 17.10

SiH₃(Silyl) 7.95 1.32 8.48 6.80 0.15 9.91 7.49

SiH₄(Silane) 14.03 8.14 11.36 10.55 8.43 13.69 13.11

PH₃ (Phosphine) 11.82 6.13 9.63 8.72 6.32 14.08 11.34

H₂S (Hydrogen sulfide) 11.00 5.57 9.69 8.44 5.56 14.12 11.06

HCl (Hydrogen sulfide) 13.36 7.02 13.00 10.49 7.06 15.89 13.30

C₂H₂(Acetylene) 13.43 6.78 14.94 9.23 6.70 16.23 13.61

C₂H₄(Ethylene) 12.57 5.60 14.41 9.01 5.46 15.42 12.30

C₂H₆(Ethane) 13.41 7.80 10.79 10.06 8.61 13.93 13.24

HCN (Hydrogen cyanide) 14.31 7.87 17.57 11.53 7.71 18.65 15.17

CO (Carbon monoxide) 15.57 6.87 18.28 12.93 6.54 19.07 15.35

HCO (Formyl radical) 9.56 1.31 12.24 7.99 0.97 12.83 9.56

H₂CO (Formaldehyde) 11.56 3.41 14.51 8.94 3.27 15.37 11.86

CH₃OH (Methyl alcohol) 11.67 5.75 9.76 8.28 6.70 13.87 12.07

N₂ (Nitrogen diatomic) 17.88 8.21 20.51 15.84 7.85 20.94 17.30

N₂H₄ (Hydrazine) 10.29 4.43 8.49 7.44 5.33 11.80 10.74

NO (Nitric oxide) 10.11 -0.04 13.79 8.76 -0.24 14.38 10.15

O₂(Oxygen diatomic) (³⌃g) 12.52 2.06 13.55 11.42 1.81 16.91 12.44 H₂O₂(Hydrogen peroxide) 12.65 4.66 17.70 8.99 4.57 19.47 13.08

F₂(Fluorine diatomic) 15.53 3.42 20.95 14.02 3.24 21.51 15.40

CO₂ (Carbon dioxide) 14.58 8.41 13.44 11.90 8.51 20.97 15.64

P₂ (Phosphorus diatomic) 10.19 3.71 11.25 9.37 3.46 11.58 9.96

S₂ (Sulfur diatomic) (³⌃g) 7.96 1.05 9.64 7.40 0.55 9.65 7.75

Cl₂ (Chlorine diatomic) 10.93 2.96 12.67 9.55 2.85 13.00 10.55

NaCl (Sodium Chloride) 8.64 3.30 7.55 6.24 3.35 11.30 8.95

SiO (Silicon monoxide) 11.60 4.61 12.44 9.96 4.74 13.48 11.67

CS (Carbon monosulfide) 11.58 3.85 13.59 10.42 3.62 13.77 11.26

ClO (Monochlorine monoxide) 8.85 -0.03 12.41 8.16 -0.26 12.58 8.86

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ClF (Chlorine monofluoride) 12.43 3.28 14.96 10.86 3.08 15.57 12.07

Si₂H₆ (Disilane) 11.33 6.56 10.09 9.12 6.37 13.13 11.05

CH₃Cl (Methyl chloride) 12.01 6.25 12.16 9.16 6.28 15.05 12.09

CH₃SH (Methanethiol) 10.01 4.76 9.10 7.53 4.89 11.62 10.14

SO₂ (Sulfur dioxide) 11.74 3.64 13.66 10.99 3.51 13.95 11.78

BF₃ (Borane, trifluoro-) 17.22 9.55 13.80 12.70 10.04 21.46 18.10 BCl₃(Borane, trichloro-) 12.07 4.85 13.24 10.80 4.68 14.44 11.31 AlCl₃ (Aluminum trichloride) 12.13 5.66 11.57 9.67 5.73 12.73 11.27 CF₄(Carbon tetrafluoride) 17.85 10.68 14.18 13.60 12.60 18.35 18.12 CCl₄(Carbon tetrachloride) 11.97 4.90 11.90 9.86 4.86 12.23 10.54

OCS (Carbonyl sulfide) 12.13 5.51 15.04 10.09 5.20 15.17 12.69

CS₂ (Carbon disulfide) 10.19 3.94 12.06 9.50 3.74 12.83 10.21

CF₂O (Carbonic difluoride) 16.08 6.63 18.68 11.46 6.39 19.13 15.78 SiF₄ (Silicon tetrafluoride) 16.95 9.66 14.84 13.30 10.87 18.38 17.03

N₂O (Nitrous oxide) 15.01 6.81 18.48 11.89 6.56 18.74 15.49

NF₃ (Nitrogen trifluoride) 15.76 7.28 21.00 13.00 7.17 21.65 16.54 PF₃ (Phosphorus trifluoride) 13.00 6.43 14.13 10.16 6.72 16.22 13.48

O₃(Ozone) 11.06 1.72 14.19 10.40 1.46 14.27 10.95

F₂O (Difluorine monoxide) 13.82 3.22 17.96 12.34 3.13 18.16 13.52 ClF₃(Chlorine trifluoride) 11.79 3.02 14.48 10.69 3.02 14.98 11.72 C₂F₄ (Tetrafluoroethylene) 12.45 5.88 16.43 9.13 5.72 16.80 13.97

CH₃CCH (Propyne) 11.69 6.21 9.29 8.32 6.49 15.16 11.54

CH₂CCH₂ (Allene) 10.83 5.69 13.81 8.55 5.55 14.75 11.89

C₃H₄(Cyclopropene) 11.87 4.90 13.67 8.13 4.79 14.78 11.94

C₃H₆(Cyclopropane) 11.64 6.97 9.91 9.21 7.64 12.86 12.25

CH₂F₂ (Methane, difluoro-) 14.15 7.94 11.27 10.37 9.39 18.17 14.57 CHF₃(Methane, trifluoro-) 15.44 9.23 12.97 11.94 10.78 17.62 16.11 CH₂Cl₂(Methylene chloride) 12.18 5.86 13.28 9.52 5.80 14.27 11.91

CHCl₃ (Chloroform) 12.38 5.31 12.67 9.75 5.26 12.92 11.20

CH₃NO₂(Methane, nitro-) 11.94 3.67 15.16 9.24 3.44 15.25 11.52

(10)

CH₃SiH₃ (Methyl silane) 12.35 7.30 11.00 9.87 7.60 13.18 12.42

HCOOH (Formic acid) 11.98 5.13 16.09 9.03 4.95 16.84 13.35

CH₃CONH₂(Acetamide) 10.05 4.98 11.35 7.59 4.89 15.37 10.91

C₂N₂(Cyanogen) 13.90 5.83 14.73 12.67 5.74 14.98 13.28

CH₂CO (Ketene) 10.32 3.72 14.32 8.11 3.42 14.70 11.32

C₂H₄O (Ethylene oxide) 11.68 6.10 9.16 8.37 7.34 13.25 12.10

C₂H₂O₂ (Ethanedial) 10.04 1.80 11.15 8.63 1.57 11.37 9.16

CH₃CH₂OH (Ethanol) 11.38 5.64 9.37 8.07 6.63 13.30 11.60

CH₃OCH₃ (Dimethyl ether) 10.79 5.47 8.59 7.71 6.64 12.29 11.31

C₂H₄S (Thiirane) 9.93 4.34 11.64 7.41 4.13 12.53 10.12

CH₂CHF (Ethene, fluoro-) 11.55 5.46 14.50 8.78 5.41 15.41 12.73 CH₃CH₂Cl (Ethyl chloride) 11.74 6.31 11.37 8.90 6.36 15.13 11.85 CH₂CHCl (Ethene, chloro-) 11.35 4.92 13.61 8.50 4.76 14.23 11.59 CH₃COCl (Acetyl Chloride) 11.97 4.72 15.12 9.29 4.48 15.16 11.79

NO₂ (Nitrogen dioxide) 9.79 1.13 12.94 8.85 0.93 13.37 10.08

CFCl₃ (Trichloromonofluoromethane) 12.61 5.21 12.99 10.26 5.11 13.32 11.26 CF₃Cl (Methane, chlorotrifluoro-) 14.27 7.18 16.96 11.44 6.89 17.67 14.30

HCCF (Fluoroacetylene) 12.04 6.50 11.49 9.12 6.98 14.93 12.64

HCCCN (Cyanoacetylene) 12.20 5.16 13.29 11.03 5.05 13.63 12.00

C₄N₂(2-Butynedinitrile) 11.52 4.44 11.76 10.28 4.37 11.94 10.70

C₂N₂(Cyanogen) 13.90 5.83 14.73 12.67 5.74 14.98 13.28

C₃O₂ (Carbon suboxide) 11.64 4.67 13.37 11.42 4.47 13.53 11.82

FCN (Cyanogen fluoride) 14.33 7.31 14.25 11.41 7.76 17.03 14.59

C₄H₂(Diacetylene) 11.00 4.77 12.17 9.94 4.69 12.67 11.09

H₂CS (Thioformaldehyde) 9.18 1.80 11.02 8.20 1.48 11.27 8.87

CHONH₂ (Formamide) 10.81 5.15 15.38 8.13 5.00 16.16 11.67

CH₂CCl₂ (Ethene, 1,1-dichloro-) 11.17 4.69 13.08 8.66 4.58 14.11 11.01 C₂HF₃ (Trifluoroethylene) 11.11 5.39 15.16 8.62 5.31 15.96 13.10 CH₂CF₂ (Ethene, 1,1-difluoro-) 11.81 5.77 15.06 8.87 5.66 15.99 13.12

CH₃F (Methyl fluoride) 14.09 7.88 11.02 10.21 9.02 17.02 14.36

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CF₂Cl₂ (Difluorodichloromethane) 13.33 6.06 14.58 10.86 5.87 15.08 12.56 SiF₂ (Silicon difluoride) 11.04 4.09 12.17 9.62 4.26 13.40 11.50

MSE -7.22 0.78 -2.33 -7.20 2.73 0.03

MAE 7.22 2.11 2.33 7.20 2.74 0.47

rms 7.44 2.45 2.57 7.48 3.13 0.63

TABLE II. Comparison of errors (in eV) of the 115 derivative discontinuities (in eV) of the FG115 database [2], calculated by the DD(0th) and DD(1st), using the LDA [3] and LB94 [4] functionals.

Here DD(0th) = 0 is obtained from the zeroth-order perturbation theory, and DD(1st), which is calculated by the di↵erence between the

_KS

+

xc

and

_KS

gaps, is obtained from the first- order perturbation theory (i.e., “frozen orbital approximation”). The reference (DD) values are calculated by the di↵erence between the E

g

and

_KS

gaps, using the respective functionals. Note that since LB94 yields very accurate E

_g

, and possibly accurate

_KS

(due to the correct asymptote of the LB94 potential), the reference DD values calculated by LB94 should be more accurate than those calculated by LDA.

LDA LB94

Molecule DD DD(0th) DD(1st) DD DD(0th) DD(1st)

H (Hydrogen atom) 4.48 -4.48 1.86 1.68 -1.68 3.10

He (Helium atom) 3.80 -3.80 0.20 6.70 -6.70 0.20

Li (Lithium atom) 2.85 -2.85 1.51 7.27 -7.27 1.40

Be (Beryllium atom) 4.01 -4.01 2.93 5.58 -5.58 3.12

B (Boron atom) 6.00 -6.00 4.12 7.90 -7.90 3.36

C (Carbon atom) 8.01 -8.01 5.29 10.11 -10.11 4.69

N (Nitrogen atom) 7.92 -7.92 2.08 9.87 -9.87 3.86

O (Oxygen atom) 9.98 -9.98 4.17 12.48 -12.48 5.59

F (Fluorine atom) 12.02 -12.02 6.82 14.80 -14.80 8.37

Ne (Neon atom) 6.15 -6.15 0.33 8.92 -8.92 0.30

Na (Sodium atom) 2.97 -2.97 1.56 6.85 -6.85 1.93

Mg (Magnesium atom) 2.81 -2.81 1.95 4.36 -4.36 1.67

Al (Aluminum atom) 4.44 -4.44 2.20 6.10 -6.10 1.58

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Si (Silicon atom) 5.78 -5.78 2.61 7.03 -7.03 2.40

P (Phosphorus atom) 5.96 -5.96 1.44 7.84 -7.84 2.38

S (Sulfur atom) 7.20 -7.20 2.60 8.98 -8.98 3.45

Cl (Chlorine atom) 8.46 -8.46 4.09 10.11 -10.11 3.93

Ar (Argon atom) 4.29 -4.29 0.26 6.38 -6.38 0.15

CH₃ (Methyl radical) 6.24 -6.24 0.21 8.67 -8.67 1.86

CH₄ (Methane) 2.26 -2.26 0.72 4.71 -4.71 0.77

NH (Imidogen) (³⌃ ) 7.94 -7.94 0.68 10.12 -10.12 3.60

NH₂ (Amino radical) 7.90 -7.90 0.70 10.10 -10.10 3.19

NH₃ (Ammonia) 2.83 -2.83 0.95 5.32 -5.32 1.05

OH (Hydroxyl radical) 9.63 -9.63 1.74 12.01 -12.01 5.12

H₂O (Water) 3.28 -3.28 1.19 6.01 -6.01 1.41

HF (Hydrogen fluoride) 3.50 -3.50 1.73 6.70 -6.70 2.29

SiH₃(Silyl) 5.48 -5.48 1.68 7.34 -7.34 2.42

SiH₄(Silane) 2.41 -2.41 0.81 4.67 -4.67 0.58

PH₃ (Phosphine) 2.59 -2.59 0.92 5.01 -5.01 2.74

H₂S (Hydrogen sulfide) 2.88 -2.88 1.25 5.50 -5.50 3.06

HCl (Hydrogen sulfide) 3.47 -3.47 2.51 6.25 -6.25 2.59

C₂H₂(Acetylene) 2.45 -2.45 5.71 6.91 -6.91 2.62

C₂H₄(Ethylene) 3.41 -3.41 5.41 6.84 -6.84 3.12

C₂H₆(Ethane) 2.26 -2.26 0.73 4.62 -4.62 0.69

HCN (Hydrogen cyanide) 3.66 -3.66 6.04 7.46 -7.46 3.48

CO (Carbon monoxide) 6.06 -6.06 5.35 8.81 -8.81 3.72

HCO (Formyl radical) 6.68 -6.68 4.25 8.59 -8.59 3.27

H₂CO (Formaldehyde) 5.52 -5.52 5.58 8.59 -8.59 3.51

CH₃OH (Methyl alcohol) 2.53 -2.53 1.48 5.37 -5.37 1.79

N₂ (Nitrogen diatomic) 7.63 -7.63 4.67 9.45 -9.45 3.64

N₂H₄ (Hydrazine) 3.00 -3.00 1.05 5.41 -5.41 1.06

NO (Nitric oxide) 8.80 -8.80 5.03 10.39 -10.39 4.23

O₂(Oxygen diatomic) (³⌃g) 9.36 -9.36 2.13 10.63 -10.63 4.47

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H₂O₂(Hydrogen peroxide) 4.33 -4.33 8.71 8.51 -8.51 6.39

F₂(Fluorine diatomic) 10.60 -10.60 6.94 12.16 -12.16 6.11

CO₂ (Carbon dioxide) 3.49 -3.49 1.53 7.14 -7.14 5.33

P₂ (Phosphorus diatomic) 5.66 -5.66 1.88 6.50 -6.50 1.61

S₂ (Sulfur diatomic) (³⌃g) 6.35 -6.35 2.24 7.20 -7.20 1.90

Cl₂ (Chlorine diatomic) 6.59 -6.59 3.12 7.70 -7.70 2.45

NaCl (Sodium Chloride) 2.94 -2.94 1.31 5.60 -5.60 2.35

SiO (Silicon monoxide) 5.35 -5.35 2.48 6.93 -6.93 1.81

CS (Carbon monosulfide) 6.56 -6.56 3.17 7.64 -7.64 2.51

ClO (Monochlorine monoxide) 8.19 -8.19 4.24 9.11 -9.11 3.73

ClF (Chlorine monofluoride) 7.58 -7.58 4.10 8.99 -8.99 3.50

Si₂H₆ (Disilane) 2.56 -2.56 0.98 4.69 -4.69 2.08

CH₃Cl (Methyl chloride) 2.91 -2.91 3.01 5.81 -5.81 2.96

CH₃SH (Methanethiol) 2.76 -2.76 1.58 5.25 -5.25 1.48

SO₂ (Sulfur dioxide) 7.36 -7.36 2.66 8.27 -8.27 2.17

BF₃ (Borane, trifluoro-) 3.15 -3.15 1.10 8.06 -8.06 3.35

BCl₃(Borane, trichloro-) 5.95 -5.95 2.44 6.63 -6.63 3.12

AlCl₃ (Aluminum trichloride) 4.01 -4.01 1.89 5.55 -5.55 1.46

CF₄(Carbon tetrafluoride) 2.91 -2.91 0.59 5.52 -5.52 0.23

CCl₄(Carbon tetrachloride) 4.96 -4.96 2.04 5.68 -5.68 1.70

OCS (Carbonyl sulfide) 4.59 -4.59 4.95 7.49 -7.49 2.48

CS₂ (Carbon disulfide) 5.56 -5.56 2.56 6.47 -6.47 2.62

CF₂O (Carbonic difluoride) 4.83 -4.83 7.22 9.40 -9.40 3.35

SiF₄ (Silicon tetrafluoride) 3.64 -3.64 1.53 6.16 -6.16 1.35

N₂O (Nitrous oxide) 5.08 -5.08 6.59 8.93 -8.93 3.25

NF₃ (Nitrogen trifluoride) 5.72 -5.72 8.00 9.38 -9.38 5.11

PF₃ (Phosphorus trifluoride) 3.73 -3.73 3.96 6.75 -6.75 2.74

O₃(Ozone) 8.68 -8.68 3.79 9.49 -9.49 3.33

F₂O (Difluorine monoxide) 9.12 -9.12 5.63 10.40 -10.40 4.64

ClF₃(Chlorine trifluoride) 7.67 -7.67 3.79 8.70 -8.70 3.27

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C₂F₄ (Tetrafluoroethylene) 3.25 -3.25 7.30 8.25 -8.25 2.83

CH₃CCH (Propyne) 2.11 -2.11 0.97 5.06 -5.06 3.61

CH₂CCH₂ (Allene) 2.86 -2.86 5.26 6.34 -6.34 2.86

C₃H₄(Cyclopropene) 3.23 -3.23 5.55 7.16 -7.16 2.83

C₃H₆(Cyclopropane) 2.24 -2.24 0.70 4.62 -4.62 0.61

CH₂F₂ (Methane, difluoro-) 2.43 -2.43 0.89 5.18 -5.18 3.60

CHF₃(Methane, trifluoro-) 2.71 -2.71 1.03 5.34 -5.34 1.51

CH₂Cl₂(Methylene chloride) 3.67 -3.67 3.75 6.11 -6.11 2.36

CHCl₃ (Chloroform) 4.44 -4.44 2.92 5.94 -5.94 1.73

CH₃NO₂(Methane, nitro-) 5.57 -5.57 5.92 8.08 -8.08 3.73

CH₃SiH₃ (Methyl silane) 2.57 -2.57 1.14 4.83 -4.83 0.76

HCOOH (Formic acid) 3.90 -3.90 7.07 8.40 -8.40 3.50

CH₃CONH₂(Acetamide) 2.61 -2.61 3.76 6.01 -6.01 4.46

C₂N₂(Cyanogen) 6.84 -6.84 2.06 7.54 -7.54 1.70

CH₂CO (Ketene) 4.39 -4.39 6.21 7.90 -7.90 3.38

C₂H₄O (Ethylene oxide) 2.27 -2.27 0.79 4.76 -4.76 1.16

C₂H₂O₂ (Ethanedial) 6.83 -6.83 2.52 7.59 -7.59 2.21

CH₃CH₂OH (Ethanol) 2.44 -2.44 1.29 4.97 -4.97 1.70

CH₃OCH₃ (Dimethyl ether) 2.24 -2.24 0.88 4.67 -4.67 0.99

C₂H₄S (Thiirane) 3.07 -3.07 4.23 5.99 -5.99 2.41

CH₂CHF (Ethene, fluoro-) 3.31 -3.31 5.73 7.32 -7.32 2.67

CH₃CH₂Cl (Ethyl chloride) 2.59 -2.59 2.47 5.49 -5.49 3.28

CH₂CHCl (Ethene, chloro-) 3.58 -3.58 5.10 6.83 -6.83 2.64

CH₃COCl (Acetyl Chloride) 4.57 -4.57 5.83 7.31 -7.31 3.36

NO₂ (Nitrogen dioxide) 7.72 -7.72 4.08 9.15 -9.15 3.29

CFCl₃ (Trichloromonofluoromethane) 5.05 -5.05 2.73 6.15 -6.15 2.06 CF₃Cl (Methane, chlorotrifluoro-) 4.27 -4.27 5.52 7.41 -7.41 3.37

HCCF (Fluoroacetylene) 2.62 -2.62 2.37 5.67 -5.67 2.28

HCCCN (Cyanoacetylene) 5.86 -5.86 2.26 6.94 -6.94 1.63

C₄N₂(2-Butynedinitrile) 5.84 -5.84 1.47 6.34 -6.34 1.24

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C₂N₂(Cyanogen) 6.84 -6.84 2.06 7.54 -7.54 1.70

C₃O₂ (Carbon suboxide) 6.75 -6.75 1.95 7.35 -7.35 1.71

FCN (Cyanogen fluoride) 4.10 -4.10 2.84 6.83 -6.83 2.43

C₄H₂(Diacetylene) 5.17 -5.17 2.23 6.40 -6.40 1.58

H₂CS (Thioformaldehyde) 6.40 -6.40 2.82 7.39 -7.39 2.40

CHONH₂ (Formamide) 2.98 -2.98 7.25 6.66 -6.66 4.49

CH₂CCl₂ (Ethene, 1,1-dichloro-) 3.97 -3.97 4.42 6.43 -6.43 3.10

C₂HF₃ (Trifluoroethylene) 3.23 -3.23 6.54 7.79 -7.79 2.86

CH₂CF₂ (Ethene, 1,1-difluoro-) 3.10 -3.10 6.19 7.46 -7.46 2.87

CH₃F (Methyl fluoride) 2.33 -2.33 0.81 5.34 -5.34 2.66

CF₂Cl₂ (Difluorodichloromethane) 4.80 -4.80 3.72 6.69 -6.69 2.52

SiF₂ (Silicon difluoride) 5.53 -5.53 2.55 7.24 -7.24 1.91

MSE -4.89 3.11 -7.23 2.70

MAE 4.89 3.11 7.23 2.70

rms 5.36 3.73 7.48 3.01