# Prediction of Excitation Energies for Conjugated Oligomers and Polymers from Time-Dependent Density Functional Theory

^{1}

^{2}

^{*}

## Abstract

**:**

**PACS**78.67.-n, 71.15.Mb, 78.40-q, 71.45.Gm

## 1. Introduction

## 2. Time-dependent Density Functional Theory

#### 2.1. TDDFT linear response theory

#### 2.2. Adiabatic approximation for the time-dependent XC potential

## 3. Excitation Energies of Atoms and Small Molecules

_{2}, H

_{2}O, CH

_{2}O (formaldehyde), (CH

_{3})

_{2}CO (acetone), C

_{2}H

_{4}(ethylene), C

_{6}H

_{6}(benzene), and C

_{5}H

_{5}N (pyridine). Since the TPSS meta-GGA is constructed from the PBE GGA [79], and PBE GGA is constructed from the LSDA, the LSDA and PBE GGA were also included in this test. The results are compared to both experiment and those obtained with two popular hybrid functionals PBE0 and B3LYP.

#### 3.1. Atoms

#### 3.2. Small molecules

_{2}, H

_{2}O) and five organic (CH

_{2}O, (CH

_{3})

_{2}CO, C

_{2}H

_{4}, benzene, pyridine) molecules. The results are reported in Tables II–IX, respectively.

_{2}, and H

_{2}O. For the CO molecule, as shown in Table II, the adiabatic TPSS functional produces the vertical (low-lying) excitation energies in better agreement with the experimental values [98] than the adiabatic PBE GGA, while it is slightly less accurate than the adiabatic LSDA. As expected, the adiabatic TPSSh yields further improvement over the TPSS meta-GGA. Mixing small amount of the exact exchange into a semilocal functional improves the asymptotic behavior of the XC potential and the description of nodal regions, both of which a pure density functional has difficulty to treat. Similar results are observed for the N

_{2}molecule, an iso-electron series of the CO molecule. As observed in Table IV, both TPSS and TPSSh functionals describe the vertical excitations of water molecule well and produce the low-lying excitation energies more accurately than the adiabatic LSDA and PBE GGA. As expected, the best results are obtained with the adiabatic hybrid functionals PBE0, B3LYP, and TPSSh. We can see from the mean errors in Tables II–IV that all the density functionals tend to underestimate the molecular excitation energies.

## 4. Absorption Spectra of Blue-Light Emitting Oligoquinolines

_{3}) solution, respectively, with the adiabatic TDDFT methods. The excitation energies of oligoquinolines in solution were calculated with PCM (polarizable continuum model) [112]. Our calculations show that the first peak of optical absorption corresponds to the lowest singlet excited state, whereas several excited states that are degenerate or nearly-degenerate, contribute to the experimentally observed higher-frequency peak. We find that the lowest excitation energies of oligoquinolines in chloroform (CHCl

_{3}) solution calculated with the adiabatic hybrid functional PBE0 are in good agreement with experiment. We also calculated the oscillator strengths and dipole moments of the oligoquinoline molecules both in gas phase as well as in chloroform solution. We see that both oscillator strength and dipole moment are larger in solution than in gas phase, as expected. These two quantities are directly related to the peak magnitude or absorption intensity in the UV/visible absorption spectra. By comparing the simulated absorption spectra in gas phase with those in chloroform solution, we find that, relative to the excitation energy in gas phase, there is a consistent redshift in excitation energy in solution, due to the solute-solvent interaction.

#### 4.1. B1PPQ and BtBPQ

#### 4.2. B2PPQ and BDBPQ

#### 4.3. BPYPQ and B3PPQ

## 5. Excitation Energies of Conjugated Polymers

_{0}− T

_{1}) absorption coefficient [129] and low phosphorescence quantum yield [121] (< 10

^{−6}). The major approaches to probe triplet states in conjugated polymers are the charge recombination energy transfer, and singlet-triplet (T

_{1}− S

_{0}or S

_{1}− T

_{1}) intersystem crossing [130,131,132].

_{T}is the triplet excitation energy and E

_{S}is the singlet-singlet excitation energy.

_{2})

_{n}CH

_{3}has little effect on the optical properties of the polymers [46,47]. These side chains only affect some physical and chemical properties, such as phase transition temperature, solubilities, etc. and thus can be removed from the backbone of polymers in calculations.

_{S}− E

_{T}, is ∼ 0 − 0.1 eV for LSDA, ∼ 0 − 0.2 eV for TPSS, ∼ 0.5 eV for TPSSh, ∼ 0.6 eV for B3LYP, and ∼ 0.8 eV for PBE0. The difference increases as the amount of exact exchange increases. However, some studies suggest [110,138] that for semilocal density functionals (LSDA, GGA, and meta-GGA), this difference may vanish in the limit of infinite chain length, a result similar to the performance of semilocal functionals for solids. Mixing exact exchange into a semilocal functional will (i) partly correct the errors from self interaction, (ii) improve the asymptotic behavior of the XC potential, (iii) improve the description of nodel regions of the Kohn-Sham orbitals, and (iv) build in other many-body properties such as excitonic effects [110,138] which have not been taken into account properly in pure density functional approximations and thus will lead to a finite difference in this limit.

_{0}− S

_{1}transition dipole moment and is consistent with what we have observed for oligomers [46,47]. However, this trend does not apply to the triplet excitation which has no dipolar strength. Triplet excitation energies are nearly the same whether the polymer is in gas phase or in solution.

## 6. Conclusion

## Acknowledgements

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**Figure 2.**Normalized absorption I of Equation (13) (in arbitrary units) (right side) and oscillator strength f (left side) of B1PPQ. The solid and dashed curves represent the normalized absorption in gas phase and solution, while the solid and dashed “sticks” represent the oscillator strength in gas phase and solution, respectively. The absorption wavelength ${\lambda}^{\text{abs}}$ (in units of nm) may be obtained from the relation ${\lambda}^{\text{abs}}=(1239.84\mathrm{eV}/{\omega}^{\text{abs}})\left(\mathrm{nm}\right)$, where ${\omega}^{\text{abs}}$ is the absorption frequency (in units of eV).

**Figure 3.**The same as Figure 2, but for BtBPQ.

**Figure 4.**The same as Figure 2, but for B2PPQ.

**Figure 5.**The same as Figure 2, but for BDBPQ.

**Table 1.**Two lowest-lying singlet excitation energies (in eV) of atoms calculated using six functionals with the basis set 6-311++G(3df,3pd). The mean error (m.e.) (with the sign convention that error = theory - experiment) and the mean absolute error (m.a.e.) are also shown. The mean experimental value of these atoms is 8.06 eV. (1 hartree = 27.21 eV).

Atom | Transition | LSDA | PBE | TPSS | TPSSh | PBE0 | B3LYP | Expt^{a} |

He | 1s → 2s | 19.59 | 19.73 | 20.27 | 20.58 | 20.62 | 20.50 | 20.62 |

1s → 2s | 22.99 | 23.41 | 24.04 | 24.23 | 24.05 | 23.95 | 21.22 | |

Li | 2s → 2p | 1.98 | 1.98 | 1.99 | 1.97 | 1.95 | 1.98 | 1.85 |

2s → 3s | 3.12 | 3.09 | 3.09 | 3.13 | 3.23 | 3.16 | 3.37 | |

Be | 2s → 2p | 4.84 | 4.91 | 5.06 | 5.05 | 4.94 | 4.88 | 5.28 |

2s → 3s | 6.11 | 6.12 | 6.29 | 6.35 | 6.32 | 6.21 | 6.78 | |

Ne | 2p → 3s | 17.45 | 17.21 | 17.55 | 17.94 | 18.27 | 17.88 | 16.62 |

2p → 3p | 19.82 | 19.46 | 19.74 | 20.16 | 20.59 | 20.11 | 18.38 | |

Na | 3s → 3p | 2.25 | 2.12 | 2.02 | 2.02 | 2.08 | 2.23 | 2.10 |

3s → 4s | 3.05 | 2.91 | 2.87 | 2.90 | 3.02 | 3.02 | 3.19 | |

Mg | 3s → 3p | 4.24 | 4.18 | 4.18 | 4.19 | 4.20 | 4.23 | 4.35 |

3s → 4s | 5.02 | 4.93 | 5.01 | 5.06 | 5.08 | 5.00 | 5.39 | |

Ar | 3p → 4s | 11.32 | 11.27 | 11.59 | 11.81 | 11.90 | 11.56 | 11.55 |

3p → 4p | 12.68 | 12.50 | 12.74 | 13.00 | 13.22 | 12.89 | 12.91 | |

K | 4s → 4p | 1.70 | 1.50 | 1.36 | 1.36 | 1.45 | 1.64 | 1.61 |

4s → 5s | 2.52 | 2.35 | 2.28 | 2.30 | 2.42 | 2.43 | 2.61 | |

Ca | 4s → 3d | 1.88 | 1.88 | 1.87 | 2.02 | 2.24 | 2.16 | 2.71 |

4s → 4p | 3.09 | 2.98 | 2.90 | 2.90 | 2.96 | 3.03 | 2.93 | |

Zn | 4s → 4p | 5.80 | 5.67 | 5.59 | 5.52 | 5.51 | 5.65 | 5.80 |

2s → 5s | 6.38 | 6.12 | 6.10 | 6.12 | 6.20 | 6.22 | 6.92 | |

Kr | 4p → 5s | 9.52 | 9.43 | 9.72 | 9.92 | 10.01 | 9.69 | 9.92 |

4p → 5s | 10.84 | 10.64 | 10.85 | 11.10 | 11.30 | 10.98 | 11.30 | |

m.e. | -0.06 | -0.14 | -0.02 | 0.12 | 0.19 | 0.09 | … | |

m.a.e. | 0.47 | 0.51 | 0.49 | 0.50 | 0.50 | 0.47 | … |

^{a}From Ref. [97].

**Table 2.**Low-lying excitation energies (in eV) of CO calculated using six functionals with the basis set 6-311++G(3df,3pd). Calculations are performed using the geometry optimized on respective functionals with the same basis. The mean error (m.e) and the mean absolute error (m.a.e.) are also shown. The mean experimental value is 9.58 eV.

Symmetry | LSDA | PBE | TPSS | TPSSh | PBE0 | B3LYP | Expt^{a} |

^{3}Π | 5.98 | 5.68 | 5.75 | 5.78 | 5.77 | 5.89 | 6.32 |

^{3}Σ^{+} | 8.45 | 7.97 | 7.88 | 7.88 | 7.96 | 8.03 | 8.51 |

^{1}Π | 8.19 | 8.19 | 8.40 | 8.50 | 8.49 | 8.47 | 8.51 |

^{3}Δ | 9.21 | 8.59 | 8.53 | 8.59 | 8.70 | 8.71 | 9.36 |

^{3}Σ^{−} | 9.90 | 9.31 | 9.64 | 9.92 | 9.89 | 9.80 | 9.88 |

^{1}Σ^{−} | 9.94 | 9.79 | 10.05 | 10.15 | 9.89 | 9.86 | 9.88 |

^{1}Δ | 9.90 | 9.72 | 9.96 | 10.01 | 10.29 | 10.26 | 10.23 |

^{3}Σ^{+} | 9.55 | 9.72 | 9.96 | 10.01 | 10.05 | 9.92 | 10.40 |

^{3}Σ^{+} | 10.48 | 10.21 | 10.60 | 10.86 | 10.94 | 10.85 | 11.30 |

^{1}Σ^{+} | 10.73 | 10.62 | 10.89 | 11.15 | 11.31 | 11.32 | 11.40 |

m.e. | -0.35 | -0.60 | -0.41 | -0.30 | -0.25 | -0.28 | … |

m.a.e. | 0.36 | 0.60 | 0.45 | 0.36 | 0.27 | 0.28 | … |

^{a}From Ref. [98].

Symmetry | LSDA | PBE | TPSS | TPSSh | PBE0 | B3LYP | Expt^{a} |

${}^{3}\Sigma _{\mathrm{u}}^{+}$ | 7.96 | 7.42 | 7.22 | 7.12 | 7.14 | 7.25 | 7.75 |

^{3}Π_{g} | 7.62 | 7.34 | 7.43 | 7.54 | 7.64 | 7.68 | 8.04 |

^{3}Δ_{u} | 8.90 | 8.19 | 8.05 | 8.01 | 8.06 | 8.12 | 8.88 |

^{1}Π_{g} | 9.11 | 9.04 | 9.23 | 9.37 | 9.43 | 9.37 | 9.31 |

${}^{3}\Sigma _{\mathrm{u}}^{-}$ | 9.73 | 9.58 | 9.82 | 9.79 | 9.53 | 9.47 | 9.67 |

${}^{1}\Sigma _{\mathrm{u}}^{-}$ | 8.73 | 9.58 | 9.82 | 9.79 | 9.53 | 9.47 | 9.92 |

^{1}Δ_{u} | 10.28 | 9.98 | 9.95 | 9.98 | 10.05 | 10.86 | 10.27 |

^{3}Π_{u} | 10.39 | 10.37 | 10.65 | 10.79 | 10.79 | 10.68 | 11.19 |

m.e. | -0.29 | -0.44 | -0.36 | -0.33 | -0.36 | -0.27 | … |

m.a.e. | 0.36 | 0.44 | 0.40 | 0.38 | 0.39 | 0.43 | … |

^{a}From Ref. [99].

Symmetry | LSDA | PBE | TPSS | TPSSh | PBE0 | B3LYP | Expt^{a} |

^{3}B_{1} | 6.30 | 6.06 | 6.30 | 6.59 | 6.80 | 6.56 | 7.14 |

^{1}B_{1} | 6.60 | 6.44 | 6.65 | 6.96 | 7.24 | 6.96 | 7.49 |

^{3}A_{2} | 7.99 | 7.72 | 7.90 | 8.24 | 8.57 | 8.31 | 9.1 |

^{1}A_{2} | 8.08 | 7.88 | 8.05 | 8.39 | 8.77 | 8.47 | 9.2 |

^{3}A_{1} | 8.26 | 8.10 | 8.36 | 8.64 | 8.84 | 8.58 | 9.35 |

^{1}A_{1} | 8.67 | 8.62 | 8.86 | 9.15 | 9.43 | 9.10 | 9.73 |

^{3}B_{2} | 9.94 | 9.75 | 9.95 | 10.26 | 10.55 | 10.28 | 9.93 |

^{1}B_{2} | 10.14 | 10.04 | 10.23 | 10.57 | 10.93 | 10.59 | 10.0 |

m.e. | -0.75 | -0.92 | -0.71 | -0.39 | -0.10 | -0.39 | … |

m.a.e. | 0.78 | 0.93 | 0.77 | 0.62 | 0.49 | 0.62 | … |

**Table 5.**The same as Table 2, but for formaldehyde (H

_{2}CO). The mean experimental value is 6.90 eV.

Symmetry | LSDA | PBE | TPSS | TPSSh | PBE0 | B3LYP | Expt^{a} |

^{3}A_{2} | 3.15 | 3.09 | 3.26 | 3.30 | 3.22 | 3.26 | 3.5 |

^{1}A_{2} | 3.75 | 3.82 | 4.06 | 4.12 | 4.02 | 3.99 | 4.1 |

^{3}A_{1} | 6.37 | 5.75 | 5.57 | 5.46 | 5.43 | 5.58 | 6.0 |

^{3}B_{2} | 5.89 | 5.68 | 5.95 | 6.27 | 6.53 | 6.38 | 7.09 |

^{1}B_{2} | 5.99 | 5.89 | 6.11 | 6.45 | 6.77 | 6.53 | 7.13 |

^{3}B_{2} | 7.10 | 6.91 | 7.17 | 7.44 | 7.62 | 7.46 | 7.92 |

^{1}B_{2} | 7.18 | 7.07 | 7.29 | 7.58 | 7.82 | 7.61 | 7.98 |

^{3}A_{1} | 6.86 | 6.63 | 6.87 | 7.21 | 7.50 | 7.35 | 8.11 |

^{1}A_{1} | 6.95 | 6.82 | 7.01 | 7.36 | 7.72 | 7.47 | 8.14 |

^{1}B_{1} | 8.86 | 8.82 | 9.01 | 9.15 | 9.22 | 9.09 | 9.0 |

m.e. | -0.69 | -0.87 | -0.69 | -0.49 | -0.31 | -0.43 | … |

m.a.e. | 0.77 | 0.87 | 0.69 | 0.52 | 0.36 | 0.44 | … |

**Table 6.**The same as Table 2, but for acetone ((CH

_{3})

_{2}CO). The mean experimental value is 6.17 eV.

Symmetry | LSDA | PBE | TPSS | TPSSh | PBE0 | B3LYP | Expt^{a} |

^{3}A_{2} | 3.70 | 3.59 | 3.69 | 3.73 | 3.81 | 3.81 | 4.18 |

^{1}A_{2} | 4.22 | 4.21 | 4.37 | 4.41 | 4.49 | 4.44 | 4.43 |

^{3}A_{1} | 6.13 | 5.70 | 5.97 | 5.96 | 5.60 | 5.70 | 5.88 |

^{3}A_{2} | 6.28 | 6.11 | 6.27 | 6.26 | 6.01 | 5.75 | 6.26 |

^{1}B_{2} | 5.09 | 5.00 | 5.22 | 5.22 | 6.08 | 5.80 | 6.36 |

^{1}A_{2} | 6.30 | 6.14 | 6.30 | 6.30 | 7.18 | 6.92 | 7.36 |

^{1}A_{1} | 6.08 | 5.92 | 6.08 | 6.08 | 7.02 | 6.72 | 7.41 |

^{1}B_{1} | 6.51 | 6.36 | 6.53 | 6.52 | 7.37 | 7.12 | 7.49 |

m.e. | -0.63 | -0.79 | -0.62 | -0.61 | -0.23 | -0.39 | … |

m.a.e. | 0.70 | 0.79 | 0.64 | 0.53 | 0.24 | 0.39 | … |

^{a}From Ref. [71].

**Table 7.**The same as Table 2, but for ethylene (C

_{2}H

_{4}). The mean experimental value is 7.40 eV.

Symmetry | LSDA | PBE | TPSS | TPSSh | PBE0 | B3LYP | Expt^{a} |

^{3}B_{1u} | 4.81 | 4.26 | 4.12 | 4.02 | 3.97 | 4.17 | 4.36 |

^{3}B_{3u} | 6.75 | 6.45 | 6.58 | 6.74 | 6.86 | 6.65 | 6.98 |

^{1}B_{3u} | 6.82 | 6.58 | 6.67 | 6.84 | 7.01 | 6.75 | 7.15 |

^{1}B_{1u} | 7.58 | 7.44 | 7.53 | 7.59 | 7.61 | 7.48 | 7.66 |

^{3}B_{1g} | 6.95 | 6.99 | 7.17 | 7.34 | 7.39 | 7.27 | 7.79 |

^{3}B_{2g} | 7.34 | 7.02 | 7.12 | 7.31 | 7.52 | 7.26 | 7.79 |

^{1}B_{1g} | 7.36 | 7.16 | 7.25 | 7.43 | 7.60 | 7.34 | 7.83 |

^{1}B_{2g} | 7.41 | 7.13 | 7.21 | 7.40 | 7.64 | 7.34 | 8.0 |

^{3}A_{g} | 8.39 | 8.03 | 8.20 | 8.33 | 8.37 | 8.25 | 8.15 |

^{1}A_{g} | 8.71 | 8.48 | 8.56 | 8.70 | 8.85 | 8.63 | 8.29 |

m.e. | -0.22 | -0.47 | -0.37 | -0.25 | -0.12 | -0.29 | … |

m.a.e. | 0.41 | 0.50 | 0.42 | 0.35 | 0.27 | 0.37 | … |

^{a}From Ref. [102].

Symmetry | LSDA | PBE | TPSS | TPSSh | PBE0 | B3LYP | Expt^{a} |

^{3}B_{1u} | 4.47 | 3.98 | 3.84 | 3.73 | 3.68 | 3.84 | 3.94 |

^{3}E_{1u} | 4.82 | 4.61 | 4.67 | 4.70 | 4.75 | 4.72 | 4.76 |

^{1}B_{2u} | 5.33 | 5.22 | 5.32 | 5.42 | 5.52 | 5.41 | 4.90 |

^{3}B_{2u} | 5.05 | 4.89 | 4.98 | 5.06 | 5.12 | 5.07 | 5.60 |

^{1}B_{1u} | 6.07 | 5.94 | 6.00 | 6.09 | 6.18 | 6.05 | 6.20 |

^{1}E_{1g} | 6.12 | 5.89 | 5.99 | 6.18 | 6.38 | 6.11 | 6.33 |

^{3}E_{1g} | 6.09 | 5.84 | 5.95 | 6.14 | 6.32 | 6.07 | 6.34 |

^{1}A_{2u} | 6.70 | 6.43 | 6.50 | 6.69 | 6.90 | 6.62 | 6.93 |

^{1}E_{2u} | 6.71 | 6.44 | 6.50 | 6.70 | 6.95 | 6.65 | 6.95 |

^{3}E_{1u} | 6.66 | 6.37 | 6.45 | 6.63 | 6.82 | 6.57 | 6.98 |

m.e. | -0.09 | -0.33 | -0.27 | -0.16 | -0.03 | -0.18 | … |

m.a.e. | 0.30 | 0.40 | 0.36 | 0.26 | 0.17 | 0.28 | … |

^{a}From Ref. [104].

**Table 9.**The same as Table 2, but for pyridine (C

_{5}H

_{5}N). The mean experimental value is 5.07 eV.

Symmetry | LSDA | PBE | TPSS | TPSSh | PBE0 | B3LYP | Expt^{a} |

^{3}B_{1} | 3.69 | 3.68 | 3.84 | 3.99 | 3.81 | 3.97 | 4.1 |

^{3}A_{1} | 4.59 | 4.11 | 3.97 | 3.86 | 4.08 | 4.05 | 4.1 |

^{1}B_{1} | 4.22 | 4.33 | 4.55 | 4.74 | 4.86 | 4.76 | 4.59 |

^{3}B_{2} | 4.62 | 4.41 | 4.44 | 4.49 | 4.54 | 4.52 | 4.84 |

^{3}A_{1} | 5.04 | 4.78 | 4.81 | 4.86 | 4.92 | 4.88 | 4.84 |

^{1}B_{2} | 5.46 | 5.33 | 5.41 | 5.53 | 5.63 | 5.52 | 4.99 |

^{3}A_{2} | 4.19 | 4.30 | 4.57 | 4.83 | 5.03 | 4.93 | 5.40 |

^{1}A_{2} | 4.29 | 4.43 | 4.71 | 4.99 | 5.20 | 5.07 | 5.43 |

^{3}B_{2} | 5.45 | 5.40 | 5.65 | 6.06 | 5.72 | 5.64 | 6.02* |

^{1}A_{1} | 6.03 | 5.97 | 6.18 | 6.31 | 6.41 | 6.23 | 6.38 |

m.e. | -0.31 | -0.40 | -0.26 | -0.08 | -0.05 | -0.11 | … |

m.a.e. | 0.54 | 0.47 | 0.34 | 0.25 | 0.25 | 0.26 | … |

LSDA | PBE | TPSS | TPSSh | PBE0 | B3LYP | |

m.a.r.e. (%) | 7.3 | 8.3 | 7.1 | 5.7 | 4.4 | 5.3 |

**Table 11.**B1PPQ: Singlet and triplet vertical excitation energies (${\omega}_{\mathrm{S}}^{n}$, ${\omega}_{\mathrm{T}}^{n}$, n = the n-th excited state) in eV, the transition oscillator strength (${f}^{\text{abs},\mathrm{n}}$), and the dipole moment of the ground state in Debye of B1PPQ molecule in gas phase (${\mu}_{\mathrm{g}}$) and chloroform solution (${\mu}_{\text{sol}}$), calculated using the five adiabatic density functionals with the basis set 6-31G(2df,p) and the geometry optimized on the respective density functionals with the same basis. (1 eV = 8065.5 cm

^{−1}= 0.03675 hartree; The energy (in units of eV) of wave length λ (in units of nm) is hc/λ = (nm/λ)1239.84 eV, where h is Planck’s constant and c is the speed of light). Experimental values measured in chloroform are obtained from Ref. [1].

gas | gas | gas | gas | gas | gas | gas | gas | sol | sol | sol | sol | sol | sol | sol | sol | |

${\omega}_{\mathrm{S}}^{\text{abs},1}$ | ${f}^{\text{abs},1}$ | ${\omega}_{\mathrm{S}}^{\text{abs},10}$ | ${f}^{\text{abs},10}$ | ${\omega}_{\mathrm{S}}^{\text{abs},11}$ | ${f}^{\text{abs},11}$ | ${\omega}_{\mathrm{T}}^{\text{abs}}$ | ${\mu}_{\mathrm{g}}$ | ${\omega}_{\mathrm{S}}^{\text{abs},1}$ | ${f}^{\text{abs},1}$ | ${\omega}_{\mathrm{S}}^{\text{abs},9}$ | ${f}^{\text{abs},9}$ | ${\omega}_{\mathrm{S}}^{\text{abs},12}$ | ${f}^{\text{abs},12}$ | ${\omega}_{\mathrm{T}}^{\text{abs}}$ | ${\mu}_{\text{sol}}$ | |

LSDA | 2.70 | 0.745 | 3.37 | 0.197 | 3.49 | 0.121 | 2.26 | 0.708 | 2.66 | 0.948 | 3.38 | 0.122 | 3.50 | 0.194 | 2.67 | 1.052 |

${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},10}$ | ${f}^{\text{abs},10}$ | ${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},10}$ | ${f}^{\text{abs},10}$ | |||||||||

TPSS | 2.81 | 0.747 | 3.50 | 0.187 | 3.60 | 0.124 | 2.18 | 0.838 | 2.78 | 0.938 | 3.50 | 0.155 | 3.61 | 0.231 | 2.19 | 1.231 |

${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | ${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},11}$ | ${f}^{\text{abs},11}$ | |||||||||

TPSSh | 3.12 | 0.916 | 3.90 | 0.205 | 4.15 | 0.919 | 2.23 | 0.865 | 3.08 | 1.137 | 3.89 | 0.174 | 4.09 | 1.144 | 2.24 | 1.261 |

${\omega}_{\mathrm{S}}^{\text{abs},10}$ | ${f}^{\text{abs},10}$ | ${\omega}_{\mathrm{S}}^{\text{abs},11}$ | ${f}^{\text{abs},11}$ | ${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},9}$ | ${f}^{\text{abs},9}$ | |||||||||

B3LYP | 3.33 | 1.040 | 4.37 | 0.623 | 4.40 | 0.422 | 2.35 | 0.901 | 3.28 | 1.288 | 4.20 | 0.650 | 4.31 | 0.856 | 2.37 | 1.300 |

${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},9}$ | ${f}^{\text{abs},9}$ | ${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},9}$ | ${f}^{\text{abs},9}$ | |||||||||

PBE0 | 3.45 | 1.112 | 4.41 | 0.786 | 4.54 | 0.744 | 2.27 | 0.902 | 3.40 | 1.359 | 4.38 | 1.279 | 4.50 | 0.464 | 2.29 | 1.305 |

${\omega}_{1\mathrm{st}}^{\text{abs}}$ | ${\omega}_{2\mathrm{nd}}^{\text{abs}}$ | |||||||||||||||

Expt | 3.48 | 4.43 |

gas | gas | gas | gas | gas | gas | gas | gas | sol | sol | sol | sol | sol | sol | sol | sol | |

${\omega}_{\mathrm{S}}^{\text{abs},1}$ | ${f}^{\text{abs},1}$ | ${\omega}_{\mathrm{S}}^{\text{abs},10}$ | ${f}^{\text{abs},10}$ | ${\omega}_{\mathrm{S}}^{\text{abs},11}$ | ${f}^{\text{abs},11}$ | ${\omega}_{\mathrm{T}}^{\text{abs}}$ | ${\mu}_{\mathrm{g}}$ | ${\omega}_{\mathrm{S}}^{\text{abs},1}$ | ${f}^{\text{abs},1}$ | ${\omega}_{\mathrm{S}}^{\text{abs},9}$ | ${f}^{\text{abs},9}$ | ${\omega}_{\mathrm{S}}^{\text{abs},12}$ | ${f}^{\text{abs},12}$ | ${\omega}_{\mathrm{T}}^{\text{abs}}$ | ${\mu}_{\text{sol}}$ | |

LSDA | 2.64 | 0.896 | 3.33 | 0.316 | 3.61 | 0.3685 | 2.24 | 0.699 | 2.61 | 1.095 | 3.34 | 0.535 | 3.59 | 0.485 | 2.25 | 1.010 |

${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},15}$ | ${f}^{\text{abs},15}$ | ${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | |||||||||

TPSS | 2.77 | 0.868 | 3.47 | 0.458 | 3.73 | 0.339 | 2.18 | 0.844 | 2.74 | 1.058 | 3.47 | 0.536 | 3.69 | 0.207 | 2.19 | 1.220 |

${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | ${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},11}$ | ${f}^{\text{abs},11}$ | |||||||||

TPSSh | 3.08 | 1.093 | 3.87 | 0.427 | 4.10 | 0.627 | 2.23 | 0.872 | 3.04 | 1.300 | 3.88 | 0.337 | 4.06 | 0.815 | 2.24 | 1.248 |

${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},11}$ | ${f}^{\text{abs},11}$ | ${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},11}$ | ${f}^{\text{abs},11}$ | |||||||||

B3LYP | 3.29 | 1.255 | 4.17 | 0.764 | 4.36 | 0.753 | 2.35 | 0.916 | 3.25 | 1.473 | 4.16 | 0.931 | 4.32 | 0.704 | 2.36 | 1.300 |

${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},11}$ | ${f}^{\text{abs},11}$ | ${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},11}$ | ${f}^{\text{abs},11}$ | |||||||||

PBE0 | 3.41 | 1.346 | 4.36 | 1.160 | 4.56 | 0.415 | 2.27 | 0.909 | 3.37 | 1.575 | 4.33 | 1.431 | 4.53 | 0.299 | 2.28 | 1.286 |

${\omega}_{1\mathrm{st}}^{\text{abs}}$ | ${\omega}_{2\mathrm{nd}}^{\text{abs}}$ | |||||||||||||||

Expt | 3.44 | 4.35 |

gas | gas | gas | gas | gas | gas | gas | gas | sol | sol | sol | sol | sol | sol | sol | sol | |

${\omega}_{\mathrm{S}}^{\text{abs},1}$ | ${f}^{\text{abs},1}$ | ${\omega}_{\mathrm{S}}^{\text{abs},10}$ | ${f}^{\text{abs},10}$ | ${\omega}_{\mathrm{S}}^{\text{abs},11}$ | ${f}^{\text{abs},11}$ | ${\omega}_{\mathrm{T}}^{\text{abs}}$ | ${\mu}_{\mathrm{g}}$ | ${\omega}_{\mathrm{S}}^{\text{abs},1}$ | ${f}^{\text{abs},1}$ | ${\omega}_{\mathrm{S}}^{\text{abs},9}$ | ${f}^{\text{abs},9}$ | ${\omega}_{\mathrm{S}}^{\text{abs},12}$ | ${f}^{\text{abs},12}$ | ${\omega}_{\mathrm{T}}^{\text{abs}}$ | ${\mu}_{\text{sol}}$ | |

LSDA | 2.48 | 1.168 | 3.04 | 0.235 | 3.19 | 0.306 | 2.14 | 0.708 | 2.44 | 1.403 | 3.03 | 0.404 | 3.20 | 0.277 | 2.14 | 1.025 |

${\omega}_{\mathrm{S}}^{\text{abs},5}$ | ${f}^{\text{abs},5}$ | ${\omega}_{\mathrm{S}}^{\text{abs},9}$ | ${f}^{\text{abs},9}$ | ${\omega}_{\mathrm{S}}^{\text{abs},5}$ | ${f}^{\text{abs},5}$ | ${\omega}_{\mathrm{S}}^{\text{abs},15}$ | ${f}^{\text{abs},15}$ | |||||||||

TPSS | 2.62 | 1.135 | 3.19 | 0.366 | 3.35 | 0.261 | 2.11 | 0.828 | 2.59 | 1.332 | 3.17 | 0.590 | 3.62 | 0.193 | 2.12 | 1.227 |

${\omega}_{\mathrm{S}}^{\text{abs},7}$ | ${f}^{\text{abs},7}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | ${\omega}_{\mathrm{S}}^{\text{abs},7}$ | ${f}^{\text{abs},7}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | |||||||||

TPSSh | 2.95 | 1.446 | 3.73 | 0.336 | 4.05 | 0.607 | 2.18 | 0.847 | 2.91 | 1.695 | 3.75 | 0.442 | 4.01 | 0.863 | 2.19 | 1.244 |

${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | ${\omega}_{\mathrm{S}}^{\text{abs},6}$ | ${f}^{\text{abs},6}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | |||||||||

B3LYP | 3.18 | 1.690 | 4.01 | 0.692 | 4.32 | 0.876 | 2.31 | 0.908 | 3.14 | 1.952 | 3.98 | 0.851 | 4.29 | 0.885 | 2.32 | 1.308 |

${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | ${\omega}_{\mathrm{S}}^{\text{abs},8}$ | ${f}^{\text{abs},8}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | |||||||||

PBE0 | 3.31 | 1.812 | 4.18 | 1.155 | 4.52 | 0.808 | 2.23 | 0.889 | 3.26 | 2.082 | 4.18 | 0.753 | 4.49 | 0.688 | 2.25 | 1.287 |

${\omega}_{1\mathrm{st}}^{\text{abs}}$ | ${\omega}_{2\mathrm{nd}}^{\text{abs}}$ | |||||||||||||||

Expt | 3.39 | 4.22 |

gas | gas | gas | gas | gas | gas | gas | gas | sol | sol | sol | sol | sol | sol | sol | sol | |

${\omega}_{\mathrm{S}}^{\text{abs},1}$ | ${f}^{\text{abs},1}$ | ${\omega}_{\mathrm{S}}^{\text{abs},10}$ | ${f}^{\text{abs},10}$ | ${\omega}_{\mathrm{S}}^{\text{abs},11}$ | ${f}^{\text{abs},11}$ | ${\omega}_{\mathrm{T}}^{\text{abs}}$ | ${\mu}_{\mathrm{g}}$ | ${\omega}_{\mathrm{S}}^{\text{abs},1}$ | ${f}^{\text{abs},1}$ | ${\omega}_{\mathrm{S}}^{\text{abs},9}$ | ${f}^{\text{abs},9}$ | ${\omega}_{\mathrm{S}}^{\text{abs},12}$ | ${f}^{\text{abs},12}$ | ${\omega}_{\mathrm{T}}^{\text{abs}}$ | ${\mu}_{\text{sol}}$ | |

LSDA | 2.63 | 0.683 | 2.70 | 0.173 | 3.26 | 0.149 | 2.25 | 0.772 | 2.61 | 0.896 | 2.70 | 0.140 | 3.31 | 0.131 | 2.26 | 1.092 |

${\omega}_{\mathrm{S}}^{\text{abs},12}$ | ${f}^{\text{abs},12}$ | ${\omega}_{\mathrm{S}}^{\text{abs},14}$ | ${f}^{\text{abs},14}$ | ${\omega}_{\mathrm{S}}^{\text{abs},10}$ | ${f}^{\text{abs},10}$ | ${\omega}_{\mathrm{S}}^{\text{abs},14}$ | ${f}^{\text{abs},14}$ | |||||||||

TPSS | 2.76 | 0.797 | 3.36 | 0.120 | 3.46 | 0.246 | 2.18 | 0.891 | 2.74 | 0.957 | 3.37 | 0.143 | 3.44 | 0.266 | 2.19 | 1.231 |

${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | ${\omega}_{\mathrm{S}}^{\text{abs},15}$ | ${f}^{\text{abs},15}$ | ${\omega}_{\mathrm{S}}^{\text{abs},6}$ | ${f}^{\text{abs},6}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | |||||||||

TPSSh | 3.08 | 1.069 | 3.85 | 0.217 | 3.91 | 0.138 | 2.22 | 0.922 | 3.05 | 1.236 | 3.57 | 0.131 | 3.86 | 0.284 | 2.23 | 1.262 |

${\omega}_{\mathrm{S}}^{\text{abs},12}$ | ${f}^{\text{abs},12}$ | ${\omega}_{\mathrm{S}}^{\text{abs},14}$ | ${f}^{\text{abs},14}$ | ${\omega}_{\mathrm{S}}^{\text{abs},5}$ | ${f}^{\text{abs},5}$ | ${\omega}_{\mathrm{S}}^{\text{abs},12}$ | ${f}^{\text{abs},12}$ | |||||||||

B3LYP | 3.31 | 1.227 | 4.14 | 0.446 | 4.27 | 0.153 | 2.36 | 1.014 | 3.28 | 1.400 | 3.83 | 0.171 | 4.13 | 0.436 | 2.37 | 1.361 |

${\omega}_{\mathrm{S}}^{\text{abs},10}$ | ${f}^{\text{abs},10}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | ${\omega}_{\mathrm{S}}^{\text{abs},10}$ | ${f}^{\text{abs},10}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | |||||||||

PBE0 | 3.42 | 1.321 | 4.31 | 0.644 | 4.45 | 0.270 | 2.27 | 0.982 | 3.39 | 1.501 | 4.29 | 0.842 | 4.44 | 0.407 | 2.28 | 1.335 |

${\omega}_{1\mathrm{st}}^{\text{abs}}$ | ${\omega}_{2\mathrm{nd}}^{\text{abs}}$ | |||||||||||||||

Expt | 3.45 | 4.40 |

gas | gas | gas | gas | gas | gas | gas | gas | sol | sol | sol | sol | sol | sol | sol | sol | |

${\omega}_{\mathrm{S}}^{\text{abs},1}$ | ${f}^{\text{abs},1}$ | ${\omega}_{\mathrm{S}}^{\text{abs},4}$ | ${f}^{\text{abs},4}$ | ${\omega}_{\mathrm{S}}^{\text{abs},14}$ | ${f}^{\text{abs},14}$ | ${\omega}_{\mathrm{T}}^{\text{abs}}$ | ${\mu}_{\mathrm{g}}$ | ${\omega}_{\mathrm{S}}^{\text{abs},1}$ | ${f}^{\text{abs},1}$ | ${\omega}_{\mathrm{S}}^{\text{abs},4}$ | ${f}^{\text{abs},4}$ | ${\omega}_{\mathrm{S}}^{\text{abs},12}$ | ${f}^{\text{abs},12}$ | ${\omega}_{\mathrm{T}}^{\text{abs}}$ | ${\mu}_{\text{sol}}$ | |

LSDA | 2.10 | 0.558 | 2.48 | 0.358 | 3.06 | 0.286 | 1.90 | 1.031 | 2.08 | 0.714 | 2.47 | 0.499 | 3.03 | 0.384 | 1.90 | 1.645 |

${\omega}_{\mathrm{S}}^{\text{abs},3}$ | ${f}^{\text{abs},3}$ | ${\omega}_{\mathrm{S}}^{\text{abs},12}$ | ${f}^{\text{abs},12}$ | ${\omega}_{\mathrm{S}}^{\text{abs},5}$ | ${f}^{\text{abs},5}$ | ${\omega}_{\mathrm{S}}^{\text{abs},12}$ | ${f}^{\text{abs},12}$ | |||||||||

TPSS | 2.21 | 0.473 | 2.58 | 0.392 | 3.17 | 0.334 | 1.85 | 1.112 | 2.19 | 0.610 | 2.56 | 0.536 | 3.15 | 0.400 | 1.86 | 1.761 |

${\omega}_{\mathrm{S}}^{\text{abs},5}$ | ${f}^{\text{abs},5}$ | ${\omega}_{\mathrm{S}}^{\text{abs},11}$ | ${f}^{\text{abs},11}$ | ${\omega}_{\mathrm{S}}^{\text{abs},5}$ | ${f}^{\text{abs},5}$ | ${\omega}_{\mathrm{S}}^{\text{abs},10}$ | ${f}^{\text{abs},10}$ | |||||||||

TPSSh | 2.61 | 0.852 | 3.16 | 0.506 | 3.48 | 0.271 | 1.88 | 1.138 | 2.58 | 1.081 | 3.14 | 0.503 | 3.45 | 0.314 | 1.89 | 1.784 |

${\omega}_{\mathrm{S}}^{\text{abs},5}$ | ${f}^{\text{abs},5}$ | ${\omega}_{\mathrm{S}}^{\text{abs},9}$ | ${f}^{\text{abs},9}$ | ${\omega}_{\mathrm{S}}^{\text{abs},5}$ | ${f}^{\text{abs},5}$ | ${\omega}_{\mathrm{S}}^{\text{abs},7}$ | ${f}^{\text{abs},7}$ | |||||||||

B3LYP | 2.89 | 1.275 | 3.44 | 0.501 | 3.64 | 0.092 | 1.98 | 1.122 | 2.85 | 1.998 | 3.42 | 0.513 | 3.61 | 0.122 | 1.98 | 1.744 |

${\omega}_{\mathrm{S}}^{\text{abs},5}$ | ${f}^{\text{abs},5}$ | ${\omega}_{\mathrm{S}}^{\text{abs},9}$ | ${f}^{\text{abs},9}$ | ${\omega}_{\mathrm{S}}^{\text{abs},5}$ | ${f}^{\text{abs},5}$ | ${\omega}_{\mathrm{S}}^{\text{abs},14}$ | ${f}^{\text{abs},14}$ | |||||||||

PBE0 | 3.04 | 1.516 | 3.62 | 0.448 | 3.79 | 0.071 | 1.88 | 1.190 | 3.00 | 1.817 | 3.60 | 0.479 | 4.09 | 0.096 | 1.89 | 1.839 |

${\omega}_{1\mathrm{st}}^{\text{abs}}$ | ${\omega}_{2\mathrm{nd}}^{\text{abs}}$ | ${\omega}_{3\mathrm{rd}}^{\text{abs}}$ | ||||||||||||||

Expt | 3.26 | 3.60 | 4.34 |

gas | gas | gas | gas | gas | gas | gas | gas | sol | sol | sol | sol | sol | sol | sol | sol | |

${\omega}_{\mathrm{S}}^{\text{abs},1}$ | ${f}^{\text{abs},1}$ | ${\omega}_{\mathrm{S}}^{\text{abs},4}$ | ${f}^{\text{abs},5}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | ${\omega}_{\mathrm{T}}^{\text{abs}}$ | ${\mu}_{\mathrm{g}}$ | ${\omega}_{\mathrm{S}}^{\text{abs},1}$ | ${f}^{\text{abs},1}$ | ${\omega}_{\mathrm{S}}^{\text{abs},5}$ | ${f}^{\text{abs},5}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | ${\omega}_{\mathrm{T}}^{\text{abs}}$ | ${\mu}_{\text{sol}}$ | |

LSDA | 2.34 | 1.235 | 2.78 | 0.598 | 3.27 | 0.575 | 2.09 | 1.137 | 2.31 | 1.421 | 2.75 | 0.721 | 3.26 | 0.593 | 2.09 | 1.457 |

${\omega}_{\mathrm{S}}^{\text{abs},4}$ | ${f}^{\text{abs},4}$ | ${\omega}_{\mathrm{S}}^{\text{abs},13}$ | ${f}^{\text{abs},13}$ | ${\omega}_{\mathrm{S}}^{\text{abs},4}$ | ${f}^{\text{abs},4}$ | ${\omega}_{\mathrm{S}}^{\text{abs},14}$ | ${f}^{\text{abs},14}$ | |||||||||

TPSS | 2.49 | 1.209 | 2.93 | 0.425 | 3.42 | 0.503 | 2.079 | 1.234 | 2.46 | 1.378 | 2.89 | 0.430 | 3.42 | 0.391 | 2.09 | 1.602 |

${\omega}_{\mathrm{S}}^{\text{abs},5}$ | ${f}^{\text{abs},5}$ | ${\omega}_{\mathrm{S}}^{\text{abs},16}$ | ${f}^{\text{abs},16}$ | ${\omega}_{\mathrm{S}}^{\text{abs},10}$ | ${f}^{\text{abs},10}$ | ${\omega}_{\mathrm{S}}^{\text{abs},15}$ | ${f}^{\text{abs},15}$ | |||||||||

TPSSh | 2.86 | 1.779 | 3.42 | 0.502 | 3.95 | 1.047 | 2.16 | 1.267 | 2.83 | 1.998 | 3.74 | 0.772 | 3.92 | 0.856 | 2.18 | 1.636 |

${\omega}_{\mathrm{S}}^{\text{abs},7}$ | ${f}^{\text{abs},7}$ | ${\omega}_{\mathrm{S}}^{\text{abs},12}$ | ${f}^{\text{abs},12}$ | ${\omega}_{\mathrm{S}}^{\text{abs},6}$ | ${f}^{\text{abs},6}$ | ${\omega}_{\mathrm{S}}^{\text{abs},11}$ | ${f}^{\text{abs},11}$ | |||||||||

B3LYP | 3.12 | 2.197 | 3.86 | 0.542 | 4.09 | 0.923 | 2.29 | 1.265 | 3.08 | 2.429 | 3.81 | 0.498 | 4.06 | 0.642 | 2.30 | 1.632 |

${\omega}_{\mathrm{S}}^{\text{abs},6}$ | ${f}^{\text{abs},6}$ | ${\omega}_{\mathrm{S}}^{\text{abs},12}$ | ${f}^{\text{abs},12}$ | ${\omega}_{\mathrm{S}}^{\text{abs},6}$ | ${f}^{\text{abs},6}$ | ${\omega}_{\mathrm{S}}^{\text{abs},12}$ | ${f}^{\text{abs},12}$ | |||||||||

PBE0 | 3.27 | 2.373 | 4.02 | 0.957 | 4.30 | 0.962 | 2.23 | 1.306 | 3.23 | 2.609 | 4.02 | 0.842 | 4.28 | 1.099 | 2.24 | 1.676 |

${\omega}_{1\mathrm{st}}^{\text{abs}}$ | ${\omega}_{2\mathrm{nd}}^{\text{abs}}$ | |||||||||||||||

Expt | 3.32 | 4.04 |

**Table 17.**TDDFT natural transition orbital analysis for the three excited states with the largest oscillator strengths in B1PPQ in gas phase. ΔE is the excitation energy, f is the corresponding oscillator strength, and W is the weight of the plotted orbital in the respective transition density matrix.

Excited state | Electron | Hole |

|1〉 ΔE = 3.45 eV f = 1.112 W = 97.4% | ||

|8〉 ΔE = 4.41 eV f = 0.786 W = 53.5% | ||

|9〉 ΔE = 4.54 eV f = 0.744 W = 81.8% |

**Table 18.**The same as Table 17, but for BtBPQ.

Excited state | Electron | Hole |

|1〉 ΔE = 3.41 eV f = 1.346 W = 97.2% | ||

|8〉 ΔE = 4.36 eV f = 1.160 W = 58.7% | ||

|11〉 ΔE = 4.56 eV f = 0.415 W = 82.5% |

**Table 19.**Excitation energies of singlet-singlet (S

_{0}− S

_{1}) and singlet-triplet (S

_{0}− T

_{1}) gaps (in units of eV) of polymers of length of ∼ 10 nm in gas phase calculated using the adiabatic TDDFT methods with the ground-state geometries optimized on the respective density functionals. Basis set 6-31G is used in all calculations. The number in parentheses is the number of rings included in our calculations. 1 hartree = 27.21 eV.

S_{0} − S_{1} | ${S}_{0}-{T}_{1}^{a}$ | ||||||||||||

Polymer | Expt^{a} | LSDA | TPSS | TPSSh | B3LYP | PBE0 | Expt^{a} | LSDA | TPSS | TPSSh | B3LYP | PBE0 | |

P3OT(28) | 2.8-3.8 | 0.99 | 0.99 | 1.35 | 1.59 | 1.76 | 1.7-2.2 | 0.90 | 0.80 | 0.88 | 0.96 | 0.95 | |

PBOPT(32) | 2.52 | 1.49 | 1.55 | 1.96 | 2.26 | 2.39 | 1.60 | 1.37 | 1.31 | 1.42 | 1.57 | 1.54 | |

MEHPPV(16) | 2.48 | 1.14 | 1.27 | 1.66 | 1.94 | 2.07 | 1.30 | 1.04 | 1.08 | 1.18 | 1.31 | 1.24 | |

PFO(36) | 3.22 | 2.30 | 2.45 | 2.89 | 3.13 | 3.30 | 2.30 | 2.22 | 2.23 | 2.34 | 2.45 | 2.43 | |

DHOPPV(16) | 2.58 | 1.14 | 1.27 | 1.67 | 1.95 | 2.07 | 1.50 | 1.04 | 1.08 | 1.18 | 1.32 | 1.24 | |

PPY(24) | 3.4-3.9 | 1.82 | 2.10 | 2.61 | 2.87 | 3.03 | 2.4-2.5 | 1.82 | 1.99 | 2.11 | 2.23 | 2.20 | |

CN-MEHPPV(16) | 2.72 | 1.10 | 1.34 | 1.84 | 2.16 | 2.27 | N/A | 1.06 | 1.22 | 1.34 | 1.48 | 1.43 | |

PANi(20) | 2.00 | 2.34 | 2.53 | 3.05 | 3.30 | 3.44 | < 0.9 | 2.31 | 2.43 | 2.63 | 2.75 | 2.73 |

Polymer | Expt | PBE0 | Energy |

P3OT^{a} | ∼ 24 ° | ∼ 0 ° | redshift |

PBOPT | ∼ 35 ° | ∼ 40 ° | On experiment |

MEHPPV^{b} | ∼ 30 ° | ∼ 1 ° | redshift |

PFO^{c} | ∼ 40 ° | ∼ 38 ° | On experiment |

DHOPPV | ∼ 30 ° | ∼ 0 ° | redshift |

PPY^{d} | ≳ 0 ° | ∼ 0 − 1 ° | slightly redshift |

CN-MEHPPV | ∼ 30 ° | ∼ 0 ° | redshift |

PANi | ∼ 0 ° | ∼ 18 − 26 ° | too blueshift |

**Table 21.**The same as Table 19, but in benzene solution. The solvent effects are taken into account through PCM (polarizable continuum model) method.

S_{0} − S_{1} | ${S}_{0}-{T}_{1}^{b}$ | ||||||||||||

Polymer | Expt^{a} | LSDA | TPSS | TPSSh | B3LYP | PBE0 | Expt^{a} | LSDA | TPSS | TPSSh | B3LYP | PBE0 | |

P3OT(28) | 2.8-3.8 | 0.97 | 0.97 | 1.32 | 1.56 | 1.73 | 1.7-2.2 | 0.89 | 0.80 | 0.87 | 0.95 | 0.94 | |

PBOPT(32) | 2.52 | 1.60 | |||||||||||

MEHPPV(16) | 2.48 | 1.12 | 1.25 | 1.64 | 1.91 | 2.04 | 1.30 | 1.03 | 1.07 | 1.18 | 1.32 | 1.25 | |

PFO(36) | 3.22 | 2.30 | 2.45 | 2.88 | 3.12 | 3.29 | 2.30 | 2.22 | 2.24 | 2.35 | 2.46 | 2.43 | |

DHOPPV(16) | 2.58 | 1.12 | 1.25 | 1.64 | 1.92 | 2.04 | 1.50 | 1.03 | 1.07 | 1.18 | 1.32 | 1.25 | |

PPY(24) | 3.4-3.9 | 2.08 | 2.16 | 2.61 | 2.85 | 3.01 | 2.4-2.5 | 2.02 | 1.99 | 2.11 | 2.23 | 2.20 | |

CN-MEHPPV(16) | 2.72 | 1.10 | 1.32 | 1.80 | 2.10 | 2.21 | N/A | 1.05 | 1.21 | 1.34 | 1.48 | 1.43 | |

PANi(20) | 2.00 | 2.33 | 2.53 | 3.03 | 3.27 | 3.41 | < 0.9 | 2.30 | 2.42 | 2.62 | 2.75 | 2.73 |

© 2010 by the authors; licensee MDPI, Basel, Switzerland. This article is an Open Access article distributed under the terms and conditions of the Creative Commons Attribution license http://creativecommons.org/licenses/by/3.0/.

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**MDPI and ACS Style**

Tao, J.; Tretiak, S.; Zhu, J.-X.
Prediction of Excitation Energies for Conjugated Oligomers and Polymers from Time-Dependent Density Functional Theory. *Materials* **2010**, *3*, 3430-3467.
https://doi.org/10.3390/ma3053430

**AMA Style**

Tao J, Tretiak S, Zhu J-X.
Prediction of Excitation Energies for Conjugated Oligomers and Polymers from Time-Dependent Density Functional Theory. *Materials*. 2010; 3(5):3430-3467.
https://doi.org/10.3390/ma3053430

**Chicago/Turabian Style**

Tao, Jianmin, Sergei Tretiak, and Jian-Xin Zhu.
2010. "Prediction of Excitation Energies for Conjugated Oligomers and Polymers from Time-Dependent Density Functional Theory" *Materials* 3, no. 5: 3430-3467.
https://doi.org/10.3390/ma3053430