Supplemental Material to: Asymptotic Correction Schemes for Semilocal Exchange-Correlation Functionals

Semilocal Exchange-Correlation Functionals

Chi-Ruei Pan,¹ Po-Tung Fang,¹ and Jeng-Da Chai^{1, 2,⇤}

1Department of Physics, National Taiwan University, Taipei 10617, Taiwan

2Center for Theoretical Sciences and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan

Reference (page S2).

Proof of the size-consistency of the LFA scheme (page S3).

FIGURE S1. Exchange potentials for the Ne atom (page S5).

FIGURE S2. Exchange potentials for the Ar atom (page S6).

TABLE S1. Total energies of several atoms and molecules (page S7).

TABLE S2. Vertical ionization potentials of IP131 database (page S7).

TABLE S3. Reaction energies of 30 chemical reactions (page S13).

TABLE S4. Valence and Rydberg excitation energies (page S14).

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

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

[2] J.-D. Chai and M. Head-Gordon, J. Chem. Phys. 128, 084106 (2008).

[3] S. Hirata and M. Head-Gordon, Chem. Phys. Lett. 314, 291 (1999).

I. PROOF OF THE SIZE-CONSISTENCY OF THE LFA SCHEME

The LFA scheme, which contains the sum of E_x^LFA and EDC, can be shown to be size- consistent for systems composed of atoms (e.g, atoms, molecules, and solids). Consider a system composed of two well-separated subsystems, S₁ and S₂, and each subsystem contains a number of atoms. Assume that ⇢^S1(r) and ⇢^S2(r) are the -spin densities of S1 and S2, respectively. Since ⇢^S1(r) and ⇢^S2(r) are not overlapped, their product should vanish.

Similarly, {wA(r)⇢^S1(r)} should vanish if the atom A belongs to S2, and {wA(r)⇢^S2(r)} should vanish if the atom A belongs to S1.

The LFA exchange energy of the system can be shown to be equal to the sum of the LFA exchange energies of the two separate subsystems as follows:

X

=↵,

E_x,^LFA[⇢^S1 + ⇢^S2] = X

=↵,

X

A

1 2NA,

ZZ

⇢A, (r)⇢A, (r⁰)erf(!|r r⁰|)

|r r⁰| drdr⁰

= X

=↵,

X

A

1 2NA,

ZZ

{wA(r)⇢ (r)}{wA(r⁰)⇢ (r⁰)}erf(!|r r⁰|)

|r r⁰| drdr⁰

= X

=↵,

X

A

1 2N_A,

ZZ

{w^A(r)[⇢^S1(r) + ⇢^S2(r)]}{w^A(r⁰)[⇢^S1(r⁰) + ⇢^S2(r⁰)]}

⇥ erf(!|r r⁰|)

|r r⁰| drdr⁰

= X

=↵,

X

A

1 2NA,

ZZ

{w^A(r)⇢^S1(r)}{w^A(r⁰)⇢^S1(r⁰)}erf(!|r r⁰|)

|r r⁰| drdr⁰ X

=↵,

X

A

1 2N_A,

ZZ

{w^A(r)⇢^S2(r)}{w^A(r⁰)⇢^S2(r⁰)}erf(!|r r⁰|)

|r r⁰| drdr⁰

= X

=↵,

X

A2S1

1 2NA,

ZZ

{wA(r)⇢^S1(r)}{wA(r⁰)⇢^S1(r⁰)}erf(!|r r⁰|)

|r r⁰| drdr⁰ X

=↵,

X

A2S2

1 2NA,

ZZ

{wA(r)⇢^S2(r)}{wA(r⁰)⇢^S2(r⁰)}erf(!|r r⁰|)

|r r⁰| drdr⁰

= X

=↵,

E_x,^LFA[⇢^S1] + X

=↵,

E_x,^LFA[⇢^S2],

(1)

whereP

A2S1 sums over all the atoms that belong to S1, andP

A2S2 sums over all the atoms that belong to S2.

The other term, EDC, which is linear in the number of electrons N , can be easily shown

to be size-consistent:

EDC= ! p⇡N

= !

p⇡(N1+ N2)

= !

p⇡N1

p!

⇡N2

= E_DC^S1 + E_DC^S2 ,

(2)

where N1 is the number of electrons in S1, and N2 is the number of electrons in S2. E_DC^S1 and E_DC^S2 are the DC energies for the isolated S1 and S2 subsystems, respectively.

FIG. S1. Exchange potentials for the Ne atom (in atomic units). Inset shows the di↵erences between the exchange potentials calculated by RILFA-PBE and LFA-PBE. ! = 0.15 Bohr ¹ is adopted for all the LFA-corrected PBE functionals.

FIG. S2. Same as Fig. S1, but for the Ar atom.

TABLE S1. Absolute percentage errors of the LFA-PBE, RILFA-PBE, and LFAs-PBE energies with respect to the PBE energy (in hartree). ! = 0.15 Bohr ¹ is adopted for all the LFA-corrected PBE functionals.

PBE LFA-PBE RILFA-PBE LFAs-PBE

System Energy error % error % error %

C -37.793956 1.30 1.30 1.32

N -54.528970 1.06 1.06 1.07

O -75.004532 0.88 0.88 0.89

F -99.661035 0.75 0.75 0.76

Ne -128.845640 0.65 0.65 0.65

HCl -460.629551 0.32 0.32 0.33

NaCl -622.267085 0.37 0.37 0.38

CO₂ -188.469386 0.97 0.97 0.98

HOOH -151.456785 0.98 0.98 0.99

CH₃OH -115.627227 1.28 1.28 1.30

C6H6 -232.010609 1.49 1.49 1.51

TABLE S2: Vertical ionization potentials (in eV) for the IP131 database [1]. ! = 0.15 Bohr ¹ is adopted for all the LFA-corrected LDA and PBE functionals. The notation used for characterizing statistical errors is as follows: mean signed errors (MSEs), mean absolute errors (MAEs), and root-mean- square (RMS) errors.

Molecule Reference PBE LFA-PBE RILFA-PBE LFAs-PBE LDA LFA-LDA RILFA-LDA LFAs-LDA

H 13.60 7.59 11.94 11.94 12.03 7.32 11.64 11.64 11.75

He 24.59 15.63 20.24 20.13 20.28 15.52 19.88 19.88 20.04

Li 5.39 3.22 7.05 7.04 7.22 3.16 6.99 6.99 7.15

Be 9.32 5.61 9.71 9.71 9.82 5.60 9.71 9.71 9.81

B 8.30 4.17 8.32 8.32 8.43 4.11 8.27 8.27 8.36

C 11.26 6.10 10.36 10.36 10.45 6.13 10.39 10.39 10.48

N 14.53 8.31 12.65 12.65 12.72 8.42 12.76 12.76 12.83

O 13.62 7.60 12.00 12.00 12.03 7.47 11.86 11.86 11.89

F 17.42 10.32 14.74 14.74 14.78 10.40 14.82 14.82 14.86

Ne 21.57 13.38 17.82 17.82 17.86 13.60 18.06 18.06 18.09

Na 5.14 3.03 6.91 6.91 6.98 3.08 6.96 6.96 7.04

Mg 7.65 4.70 8.73 8.73 8.79 4.77 8.81 8.81 8.87

Al 5.99 3.09 7.05 7.04 7.12 3.00 6.94 6.96 7.02

Si 8.15 4.61 8.68 8.68 8.76 4.56 8.62 8.65 8.71

P 10.49 6.30 10.47 10.47 10.55 6.30 10.47 10.47 10.53

S 10.36 6.15 10.39 10.39 10.43 6.16 10.39 10.39 10.43

Cl 12.97 8.14 12.43 12.43 12.48 8.22 12.51 12.51 12.56

Ar 15.76 10.30 14.63 14.63 14.67 10.41 14.74 14.74 14.78

CH₃ 9.84 5.42 9.74 9.74 9.80 5.39 9.68 9.68 9.76

CH₄ 13.60 9.45 13.82 13.82 13.89 9.48 13.84 13.84 13.91

NH 13.49 7.92 12.27 12.27 12.33 7.98 12.32 12.32 12.39

NH₂ 12.00 7.22 11.56 11.56 11.63 7.21 11.56 11.56 11.62

NH₃ 10.82 6.18 10.53 10.53 10.58 6.28 10.61 10.61 10.68

OH 13.02 7.38 11.75 11.75 11.81 7.43 11.80 11.80 11.85

H₂O 12.62 7.24 11.61 11.61 11.67 7.40 11.75 11.75 11.82

HF 16.12 9.65 14.04 14.04 14.10 9.83 14.23 14.23 14.28

SiH₃ 8.74 5.37 9.57 9.57 9.65 5.31 9.52 9.52 9.57

SiH₄ 12.30 8.52 12.81 12.81 12.89 8.53 12.81 12.81 12.89

PH₃ 10.59 6.72 10.99 10.99 11.05 6.77 11.02 11.02 11.09

SH₂ 10.50 6.31 10.55 10.55 10.62 6.40 10.64 10.66 10.71

HCl 12.77 8.05 12.35 12.35 12.40 8.16 12.46 12.46 12.50

HCCH 11.49 7.20 11.51 11.51 11.56 7.38 11.67 11.67 11.74

CH₂CH₂ 10.68 6.74 11.04 11.04 11.12 6.93 11.23 11.23 11.31

CH₃CH₃ 11.99 8.17 12.54 12.54 12.62 8.16 12.54 12.54 12.60

HCN 13.61 9.02 13.36 13.36 13.41 9.20 13.55 13.55 13.60

CO 14.01 9.04 13.38 13.38 13.44 9.13 13.46 13.46 13.52

HCO 9.31 5.16 9.49 9.49 9.55 5.12 9.47 9.47 9.52

CH₂O 10.89 6.26 10.61 10.61 10.68 6.35 10.72 10.72 10.77

CH₃OH 10.96 6.26 10.61 10.61 10.68 6.36 10.72 10.72 10.79

N₂ 15.58 10.28 14.63 14.63 14.69 10.43 14.80 14.80 14.85

NH₂NH₂ 8.98 5.30 9.66 9.66 9.71 5.36 9.71 9.71 9.77

NO 9.26 4.52 8.89 8.89 8.94 4.57 8.92 8.92 8.99

O₂ 12.30 6.84 11.23 11.23 11.29 6.97 11.37 11.37 11.42

HOOH 11.70 6.46 10.83 10.83 10.89 6.61 10.99 10.99 11.04

F₂ 15.70 9.48 13.93 13.93 13.96 9.69 14.12 14.12 14.16

CO₂ 13.78 9.09 13.46 13.46 13.51 9.33 13.71 13.71 13.75

P₂ 10.62 7.15 11.34 11.34 11.39 7.26 11.45 11.45 11.50

S₂ 9.55 5.83 10.06 10.06 10.13 5.83 10.06 10.06 10.13

Cl₂ 11.49 7.33 11.61 11.61 11.67 7.44 11.72 11.75 11.79

NaCl 9.80 5.30 9.47 9.47 9.51 5.44 9.60 9.60 9.65

SiO 11.61 7.48 11.72 11.72 11.77 7.61 11.83 11.83 11.89

CS 11.34 7.40 11.67 11.67 11.73 7.45 11.72 11.72 11.78

ClO 11.01 6.30 10.61 10.64 10.67 6.37 10.69 10.69 10.74

ClF 12.77 7.86 12.19 12.19 12.24 8.00 12.32 12.32 12.38

SiH₃SiH₃ 10.53 7.19 11.45 11.45 11.53 7.26 11.51 11.53 11.59

CH₃Cl 11.29 7.12 11.42 11.42 11.48 7.20 11.51 11.51 11.57

CH₃SH 9.44 5.57 9.85 9.85 9.90 5.65 9.93 9.93 9.98

SO₂ 12.50 8.08 12.40 12.40 12.44 8.28 12.59 12.59 12.64

BF₃ 15.96 10.07 14.44 14.44 14.50 10.32 14.69 14.69 14.75

BCl₃ 11.64 7.72 12.02 12.02 12.06 7.85 12.13 12.13 12.19

AlCl₃ 12.01 8.02 12.27 12.27 12.33 8.16 12.40 12.43 12.47

CF₄ 16.20 10.42 14.82 14.82 14.87 10.68 15.07 15.10 15.13

CCl₄ 11.69 7.69 12.00 12.00 12.04 7.82 12.10 12.10 12.16

OCS 11.19 7.50 11.78 11.78 11.84 7.66 11.94 11.94 12.00

CS₂ 10.09 6.82 11.07 11.07 11.13 6.94 11.18 11.18 11.24

CF₂O 13.60 8.52 12.89 12.89 12.95 8.77 13.14 13.14 13.20

SiF₄ 16.40 10.69 15.04 15.04 15.10 10.96 15.31 15.31 15.36

N₂O 12.89 8.40 12.78 12.78 12.83 8.65 13.03 13.03 13.07

NF₃ 13.60 8.45 12.87 12.87 12.90 8.63 13.03 13.03 13.08

PF₃ 12.20 7.36 11.64 11.64 11.70 7.51 11.80 11.80 11.85

O₃ 12.73 8.02 12.40 12.40 12.46 8.25 12.62 12.62 12.68

F₂O 13.26 7.72 12.13 12.13 12.18 7.91 12.32 12.32 12.38

ClF₃ 13.05 8.00 12.38 12.38 12.41 8.18 12.57 12.57 12.60

CF₂CF₂ 10.69 6.31 10.69 10.69 10.74 6.56 10.93 10.93 10.99

CF₃CN 14.30 9.57 13.93 13.93 13.98 9.77 14.12 14.12 14.18

CH₃CCH 10.37 6.49 10.80 10.80 10.87 6.64 10.96 10.96 11.02

CH₂CCH₂ 10.20 6.56 10.88 10.88 10.95 6.72 11.02 11.02 11.10

cylC₃H₄ 9.86 6.11 10.45 10.44 10.51 6.23 10.55 10.55 10.63

cylC₃H₆ 10.54 7.07 11.42 11.42 11.48 7.23 11.56 11.56 11.64

CH₃CH₂CH₃ 11.51 7.75 12.10 12.13 12.19 7.73 12.10 12.10 12.17

CH₃CCCH₃ 9.79 5.93 10.25 10.25 10.32 6.06 10.39 10.39 10.45

cylC₄H₆ 9.43 6.04 10.36 10.36 10.43 6.19 10.50 10.50 10.58

isobutane 11.13 7.58 11.94 11.94 12.01 7.59 11.94 11.94 12.02

benzene 9.25 6.33 10.64 10.64 10.71 6.54 10.85 10.85 10.92

CH₂F₂ 13.27 8.15 12.54 12.54 12.60 8.25 12.65 12.65 12.70

CF₃H 15.50 9.35 13.74 13.74 13.80 9.51 13.90 13.90 13.96

CH₂Cl₂ 11.40 7.38 11.70 11.70 11.74 7.48 11.78 11.78 11.83

CCl₃H 11.50 7.42 11.72 11.72 11.77 7.55 11.83 11.83 11.89

CH₃NO₂ 11.29 6.92 11.29 11.29 11.34 7.11 11.48 11.48 11.54

CH₃SiH₃ 11.60 7.92 12.21 12.24 12.30 8.01 12.32 12.32 12.38

HCOOH 11.50 6.73 11.07 11.07 11.13 6.92 11.26 11.26 11.32

CH₃CONH₂ 10.00 5.79 10.12 10.15 10.19 5.95 10.28 10.28 10.35

cylNHC₂H₄ 9.85 5.78 10.12 10.12 10.18 5.88 10.23 10.23 10.28

NCCN 13.51 9.39 13.74 13.74 13.79 9.64 13.98 13.98 14.03

CH₃NHCH₃ 8.95 5.06 9.41 9.41 9.47 5.13 9.47 9.47 9.54

CH₂CO 9.64 5.91 10.23 10.23 10.29 6.09 10.42 10.42 10.48

cylOC₂H₄ 10.57 6.27 10.64 10.64 10.69 6.39 10.74 10.74 10.81

OCHCHO 10.60 6.39 10.74 10.74 10.80 6.53 10.88 10.88 10.94

CH₃CH₂OH 10.64 6.15 10.50 10.50 10.58 6.27 10.64 10.64 10.69

CH₃OCH₃ 10.10 5.83 10.17 10.17 10.25 5.93 10.28 10.28 10.34

cylSC₂H₄ 9.05 5.37 9.63 9.63 9.70 5.46 9.74 9.74 9.79

CH₃SOCH₃ 9.10 5.38 9.68 9.68 9.74 5.52 9.82 9.82 9.88

CH₂CHF 10.63 6.52 10.85 10.85 10.92 6.72 11.04 11.04 11.11

CH₃CH₂Cl 11.06 6.97 11.29 11.29 11.34 7.07 11.37 11.37 11.43

CH₂CHCl 10.20 6.42 10.72 10.72 10.79 6.59 10.88 10.88 10.95

CH₃CClO 11.03 7.13 11.45 11.45 11.52 7.29 11.61 11.61 11.68

prplCl 10.88 6.94 11.26 11.26 11.31 7.04 11.34 11.34 11.40

NC₃H₉ 8.54 4.85 9.17 9.17 9.25 4.92 9.25 9.25 9.32

cylOC₄H₄ 8.90 5.67 9.98 9.98 10.05 5.88 10.20 10.20 10.27

cylNHC₄H₄ 8.23 5.13 9.44 9.44 9.51 5.33 9.63 9.63 9.71

NO₂ 11.23 6.50 10.88 10.88 10.92 6.60 10.96 10.96 11.02

SF₆ 15.70 10.14 14.53 14.52 14.57 10.40 14.80 14.80 14.84

CFCl₃ 11.76 7.75 12.05 12.05 12.10 7.88 12.19 12.19 12.23

CClF₃ 13.08 8.56 12.89 12.89 12.93 8.73 13.06 13.06 13.10

CBrF₃ 12.08 7.80 12.10 12.10 12.15 8.00 12.29 12.29 12.33

HCCF 11.50 7.01 11.34 11.34 11.40 7.22 11.53 11.53 11.60

HCCCN 11.75 7.87 12.19 12.19 12.24 8.08 12.40 12.40 12.46

NCCCCN 11.84 8.44 12.76 12.76 12.82 8.68 13.00 13.00 13.07

C₂N₂ 13.51 9.39 13.74 13.74 13.79 9.64 13.98 13.98 14.03

C₃O₂ 10.80 7.27 11.61 11.61 11.66 7.50 11.83 11.83 11.90

FCN 13.65 8.73 13.08 13.08 13.13 8.97 13.30 13.30 13.37

HCCCCH 10.30 6.64 10.93 10.93 11.01 6.83 11.12 11.12 11.20

H₂CS 9.38 5.53 9.79 9.79 9.86 5.61 9.87 9.87 9.93

HCONH₂ 10.40 6.02 10.36 10.36 10.42 6.17 10.50 10.50 10.56

CH₂CHCHO 10.10 6.00 10.34 10.34 10.41 6.15 10.50 10.50 10.55

CH₂CCl₂ 10.00 6.44 10.74 10.74 10.80 6.61 10.91 10.91 10.97

CHFCF₂ 10.62 6.22 10.58 10.58 10.64 6.45 10.80 10.80 10.87

CH₂CF₂ 10.70 6.57 10.91 10.91 10.97 6.78 11.12 11.12 11.18

CH₃F 13.04 8.09 12.49 12.48 12.53 8.19 12.57 12.57 12.63

CF₂Cl₂ 12.24 8.08 12.40 12.40 12.44 8.23 12.54 12.54 12.59

SiF₂ 11.08 7.14 11.40 11.40 11.44 7.27 11.51 11.51 11.57

MSE 4.42 0.10 0.10 0.03 4.30 0.01 0.01 0.09

MAE 4.42 0.66 0.66 0.67 4.30 0.68 0.68 0.69

RMS 4.52 0.93 0.93 0.93 4.41 0.93 0.93 0.94

TABLE S3. Comparison of errors (in kcal/mol) of the reaction energies of 30 chemical reactions [2], calculated by the PBE and LFA-corrected PBE functionals (! = 0.15 Bohr ¹). (1 kcal/mol = 0.0434 eV = 0.00159 hartree.)

Reactions Eref PBE LFA-PBE RILFA-PBE LFAs-PBE

H + N2O! OH + N2 65.08 22.59 22.53 22.52 22.57

H + FCH3 ! HF + CH3 26.64 4.16 4.22 4.23 4.24

H + F2 ! HF + F 103.91 13.19 12.95 12.95 13.09

CH3+ FCl! CH3F + Cl 52.74 4.64 4.24 4.24 4.42

F + CH3Cl! FCH3 + Cl 32.65 0.83 3.67 3.69 2.41

F · · · CH3Cl! FCH3· · · Cl 26.73 4.61 6.14 6.15 5.45

OH + CH3F! HOCH3+ F 20.11 0.38 0.70 0.70 0.27

OH · · · CH3F! HOCH3· · · F 36.24 3.09 4.11 4.11 3.68

H + N2 ! HN2 3.97 7.86 8.26 8.25 8.06

H + CO! HCO 19.51 6.84 7.48 7.47 7.15

H + C2H4! CH3CH2 40.03 0.31 1.12 1.13 0.73

CH3+ C2H4 ! CH3CH2CH2 26.12 2.14 2.74 2.74 2.48

HCN! HNC 15.05 0.06 0.47 0.47 0.29

H + HCl! H2+ Cl 3.0 5.51 4.78 4.76 5.11

OH + H2 ! H + H2O 16.1 3.95 3.47 3.45 3.68

CH3+ H2! H + CH4 3.2 2.30 2.10 2.09 2.20

OH + CH4 ! CH3+ H2O 12.9 1.66 1.38 1.37 1.47

OH + NH3! H2O + NH2 9.5 1.73 1.78 1.78 1.75

HCl + CH3 ! Cl+ CH4 6.2 3.21 2.69 2.67 2.90

OH + C2H6! H2O + C2H5 16.5 3.28 3.01 3.00 3.11

F + H2 ! HF + H 31.6 6.49 6.03 6.01 6.20

O + CH4! OH + CH3 5.6 5.83 5.55 5.54 5.63

H + PH3! PH2+ H2 20.1 0.40 0.00 0.01 0.19

H + HO! H2+ O 2.4 8.13 7.65 7.62 7.84

H + H2S! H2 + HS 13.8 3.48 2.79 2.80 3.11

O + HCl! OH + Cl 0.6 2.61 2.87 2.87 2.73

NH2+ CH3 ! CH4 + NH 14.4 4.55 4.24 4.23 4.36

NH2+ C2H5! C2H6+ NH 10.8 6.17 5.88 5.86 5.99

C2H6+ NH2! NH3+ C2H5 7.0 1.54 1.23 1.22 1.36

NH2+ CH4 ! CH3 + NH3 3.3 0.02 0.30 0.31 0.18

MSE 1.08 1.02 1.03 1.05

MAE 4.38 4.48 4.47 4.42

RMS 6.24 6.24 6.24 6.23

TABLE S4. The 19 valence and 23 Rydberg excitation energies (in eV) of N₂, CO, water, ethylene, and formaldehyde, calculated by the PBE and LFA-corrected PBE functionals (! = 0.15 Bohr ¹).

The molecular geometries and experimental reference values are taken from Ref. [3].

Expt. PBE LFA-PBE RILFA-PBE LFAs-PBE

N2 V¹⇧g 9.31 9.12 9.15 9.15 9.15

V¹⌃u 9.97 9.70 9.72 9.72 9.72

V¹ u 10.27 10.12 10.14 10.14 10.14

V³⌃⁺u 7.75 7.56 7.56 7.56 7.56

V³⇧g 8.04 7.42 7.43 7.43 7.43

V³ u 8.88 8.37 8.38 8.38 8.38

V³⌃u 9.67 9.70 9.72 9.72 9.72

V³⇧u 11.19 10.40 10.42 10.42 10.42

CO V¹⇧ 8.51 8.25 8.33 8.33 8.33

V¹⌃ 9.88 9.86 9.91 9.91 9.91

V³⇧ 6.32 5.74 5.75 5.75 5.75

V³⌃⁺ 8.51 8.11 8.14 8.14 8.14

V³ 9.36 8.77 8.80 8.80 8.80

V³⌃ 9.88 9.86 9.91 9.91 9.91

H2O R¹B1 7.4 6.33 6.77 6.77 6.72

R¹A2 9.1 7.44 8.42 8.42 8.27

R¹A1 9.7 8.18 8.91 8.91 9.21

R¹B1 10.0 7.84 9.52 9.52 9.25

R¹A1 10.17 8.48 9.52 9.52 8.85

R³B1 7.2 6.00 6.35 6.35 6.31

C2H4 R¹B3u 7.11 6.40 7.07 7.07 7.00

V¹B1u 7.60 7.29 7.60 7.60 7.60

R¹B1g 7.80 6.87 7.68 7.68 7.59

R¹B2g 8.01 6.83 7.69 7.69 7.58

R¹Ag 8.29 7.18 8.71 8.71 8.40

R¹B3u 8.62 7.42 8.91 8.91 8.76

V³B1u 4.36 4.26 4.26 4.26 4.26

R³B3u 6.98 6.31 6.91 6.91 6.86

R³B1g 7.79 6.83 7.17 7.17 7.53

R³B2g 7.79 6.78 7.54 7.54 7.45

R³Ag 8.15 7.06 8.37 8.37 8.51

CH2O V¹A2 4.07 3.74 3.77 3.77 3.77

R¹B2 7.11 5.77 6.40 6.40 6.37

R¹B2 7.97 6.54 7.62 7.62 7.52

R¹A1 8.14 7.11 9.16 9.16 8.94

R¹A2 8.37 6.69 8.24 8.24 7.96

R¹B2 8.88 6.81 8.99 9.00 8.63

V³A2 3.50 3.00 3.03 3.03 3.03

V³A1 5.86 5.57 5.58 5.58 5.58

R³B2 6.83 5.63 6.16 6.16 6.13

R³B2 7.79 6.47 7.44 7.44 7.36

R³A1 7.96 6.35 7.22 7.23 7.17

MAE Valence (19) 0.32 0.29 0.29 0.29

Rydberg (23) 1.30 0.46 0.46 0.49