Cite this: Phys. Chem. Chem. Phys., 2014, 16, 21564
Assessment of asymptotically corrected model potentials for charge-transfer-like excitations in oligoacenes
Wei-Tao Pengaand Jeng-Da Chai*abc
We examine the performance of the asymptotically corrected model potential scheme on the two lowest singlet excitation energies of acenes with different numbers of linearly fused benzene rings (up to 5), employing both the real-time time-dependent density functional theory and the frequency-domain formulation of linear-response time-dependent density functional theory. The results are compared with the experimental data and those calculated using long-range corrected hybrid functionals and others. The long-range corrected hybrid scheme is shown to outperform the asymptotically corrected model potential scheme for charge-transfer-like excitations.
I. Introduction
Over the past two decades, time-dependent density functional theory (TDDFT)1 has been a popular method for the study of excited-state and time-dependent properties of large systems, due to its favorable balance between accuracy and efficiency.2,3 However, the exact exchange–correlation (XC) potential vxc(r,t) in TDDFT remains unknown, and needs to be approximated for practical applications.
For a system subject to a slowly varying external potential, the most popular approximation for vxc(r,t) is the adiabatic approximation,
vxcðr; tÞ #dExc½r%
drðrÞ
!!
!!
rðrÞ¼rðr;tÞ
; (1)
where vxc(r,t) is approximated by the functional derivative of the XC energy functional Exc[r] evaluated at the instantaneous density r(r,t). In the adiabatic approximation, memory effects, whereby vxc(r,t) may depend on the density at all previous times (t0o t), are completely neglected. Surprisingly, results obtained from the adiabatic approximation can be accurate in many cases, even if the system considered is not in this slowly varying regime.
However, as the exact Exc[r], which appears in both Kohn–Sham density functional theory (KS-DFT)4(for ground-state properties) and adiabatic TDDFT (for excited-state and time-dependent
properties), has not been known, the development of a generally accurate density functional approximation for Exc[r] remains an important and challenging task.5,6
Functionals based on the localized model XC holes, such as the local density approximation (LDA) and generalized gradient approximations (GGAs), are reliably accurate for applications governed by short-range XC effects, such as low-lying valence excitation energies. However, they can produce erroneous results in situations where the accurate treatment of nonlocal XC effects is important. In particular, some of these situations occur in the asymptotic regions of atoms and molecules, where the LDA or GGA XC potential exhibits an exponential decay, instead of the correct '1/r decay. Accordingly, LDA and GGAs (i.e., semilocal density functionals) severely underestimate high-lying Rydberg excitation energies,7–10and completely fail for charge-transfer (CT) excitation energies10–15and excitations in completely symmetrical systems where no net CT occurs.16
With the aim of resolving the asymptote problem, long-range corrected (LC) hybrid functionals17–23and asymptotically corrected (AC) model potentials,24–28which are two distinct density functional methods with the correct asymptotic behavior, have been actively developed over the past few years. The LC hybrid scheme, which adopts 100% Hartree–Fock (HF) exchange for long-range electron–
electron interactions, thereby provides an AC XC potential (i.e., a local multiplicative XC potential), when the optimized effective potential (OEP) method is employed.5,29–31Similar to other orbital- dependent XC energy functionals, a generalized Kohn–Sham (GKS) method (i.e., using orbital-specific XC potentials) has been frequently employed in the LC hybrid scheme to circumvent the computational complexity of the OEP method, as the density, energy, and highest- occupied orbital energy obtained from the GKS method are generally similar to those obtained from the OEP method.5,31In our recent
aDepartment of Physics, National Taiwan University, Taipei 10617, Taiwan.
E-mail: [email protected]
bCenter for Theoretical Sciences and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan
cPhysics Division, National Center for Theoretical Sciences (North), National Taiwan University, Taipei 10617, Taiwan
Received 5th July 2014, Accepted 14th August 2014 DOI: 10.1039/c4cp02946a
www.rsc.org/pccp
PCCP
PAPER
work, the performance of the LC hybrid scheme (i.e., using the GKS method) and the AC model potential scheme has been examined on a very wide range of applications.32In particular, we have shown that LC hybrid functionals could be reliably accurate for various types of excitation energies, including valence, Rydberg, and CT excitation energies, in the frequency-domain formulation of linear-response TDDFT (LR-TDDFT).33Nevertheless, due to the inclusion of long- range HF exchange, the LC hybrid scheme can be computationally expensive for large systems.
On the other hand, in the AC model potential scheme, an AC XC potential is directly modeled, maintaining computational complexity similar to the efficient semilocal density functional methods. However, as most popular AC model potentials are not found to be functional derivatives,34,35 the associated XC energies and XC kernels (i.e., the second functional derivative of Exc[r]) are not well-defined. Accordingly, an adiabatic LDA or GGA XC kernel (i.e., not a self-consistent adiabatic XC kernel) has been frequently adopted for the LR-TDDFT calculations using the AC model potential scheme. Such combined approaches have been shown to perform well for both valence and Rydberg excitations, but very poorly for CT excitations,16,32,36–38 due to the lack of a space- and frequency-dependent discontinuity in the adiabatic LDA or GGA XC kernel adopted in LR-TDDFT.39Nevertheless, it remains unclear whether the AC model potential scheme can accurately describe CT or CT-like excitations, when a self-consistent adiabatic XC kernel (if available) is adopted in LR-TDDFT.
To circumvent this problem, in this work, we examine the performance of the AC model potential scheme on various types of excitation energies in the real-time formulation of TDDFT (RT-TDDFT), since LR-TDDFT is typically a good approximation to RT-TDDFT. As an absorption spectrum (and hence, excitation energies) can be obtained by explicitly propagating the time- dependent Kohn–Sham (TDKS) equations, the knowledge of the XC kernel is not needed within the framework of RT-TDDFT.
Particularly, we like to address if the AC model potential scheme is able to accurately describe CT-like excitations in RT-TDDFT, which to the best of our knowledge has never been addressed in the literature. The rest of this paper is organized as follows. In Section II, we describe our test sets and computational details.
The excitation energies calculated using the AC model potential scheme, the LC hybrid scheme, and others in both RT-TDDFT and LR-TDDFT are compared with the experimental data and the results obtained from a highly accurate ab initio method in Section III. Our conclusions are given in Section IV.
II. Test sets and computational details
Linear n-acenes (C4n+2H2n+4), consisting of n linearly fused benzene rings (see Fig. 1), are important molecules for a variety of devices, such as organic light-emitting diodes,40solar cells,41 and field-effect transistors.42 Recently, the two lowest singlet p - p* transitions of n-acenes, commonly labelled as the1La
(the lowest excited state of B2usymmetry) and1Lb(the lowest excited state of B3usymmetry) states in Platt’s nomenclature,43 have received considerable attention.44–52
The1Lastate is dominated by a transition between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), i.e., the HOMO - LUMO transition, with polarization along the molecular short axis, while the1Lbstate is dominated by a combination of the two nearly degenerate configurations HOMO ' 1 - LUMO and HOMO - LUMO + 1, with polarization along the molecular long axis. From a valence- bond point of view, the 1La state is mainly ionic in character, whereas the1Lbstate is mainly covalent in character.44,45
Kuritz et al. described the1Lastate as a CT-like excitation,49 and showed that through a unitary transformation, the coupling between the HOMO and LUMO is weak, supporting the surmise of Richard and Herbert that the1Lastate has CT character in disguise.48 Note that as the1Lastate is not a pure CT excitation (i.e., for well- separated donor–acceptor systems),12,13the terminology ‘‘CT-like’’
may be controversial.46,50,52However, in this work, the1Lastate is regarded as a CT-like excitation, as suggested in ref. 48 and 49.
By contrast, the1Lbstate is a valence excitation with substantial double-excitation character.48In LR-TDDFT, although LDA and GGAs can accurately predict the 1Lb excitation energy, they substantially underestimate the1Laexcitation energy.44–49,51On the other hand, LC hybrid functionals are reliably accurate for the1Lastate, but less accurate for the1Lbstate.46–49,51Note that the efficient AC model potential scheme has never been examined on the1Laand1Lbstates of n-acenes in the literature.
To examine the performance of several density functional methods on various types of excitation energies in both RT-TDDFT and LR-TDDFT, the1Laand1Lbstates of n-acenes (up to 5-acene) are adopted as our test sets. For the LR-TDDFT calculations, we adopt LDA,53,54PBE55(a popular GGA functional), and LB9424(a popular AC model potential) with the 6-31+G(d,p) basis set, to calculate the
1Laand1Lbexcitation energies on the ground-state geometries of n-acenes obtained at the oB97X/6-31G(d) level19with a develop- ment version of Q-Chem 4.0.56 For the LDA and PBE calcula- tions, the adiabatic LDA and PBE XC kernels (i.e., the second functional derivatives of the LDA and PBE XC energy functionals, respectively) are adopted, respectively. For the LB94 calculations, the adiabatic LDA XC kernel is adopted, due to the lack of a self- consistent adiabatic XC kernel for the LB94 model potential.
The RT-TDDFT calculations, employing the adiabatic LDA, PBE, and LB94 XC potentials, are performed using the program Fig. 1 Structures of the n-acenes investigated.
package Octopus 4.0.1.57The system, which is described on a real-space grid with a 0.2 Å spacing, is constructed by adding spheres created around each atom, of radius 6 Å. Troullier–Martins pseudopotentials are adopted to describe the complicated effects of the motion of core electrons.58 To obtain spectroscopic information, n-acene, which starts from the ground state, is excited via a linearly polarized delta kick in then-direction:
vext(r,t) = !hkd(t)rn, (2) where rnis one of the Cartesian coordinates (x, y, and z), and the perturbation strength k = 0.01 bohr'1is adopted to obtain the linear spectra.59 To propagate the TDKS equations, we adopt a time step of Dt = 0.001 !h per eV (0.658 as) and run up to 100 !h per eV (65.8 fs), which corresponds to 105time steps.
The approximated enforced time-reversal symmetry (AETRS) algorithm is employed to numerically represent the time evolution operator.60
III. Results and discussion
The absorption spectra of n-acenes, calculated using LDA, PBE, and LB94 in RT-TDDFT, are plotted in Fig. 2–5, where the spectra close to the position of the 1La and 1Lb peaks are highlighted in the subfigures, and the corresponding LR-TDDFT results are marked with the red lines. Note that the 1Lb state exhibits weak intensity compared with the1Lastate. For 2-acene, as the oscillator strengths of the1Lbstate calculated using LDA, PBE, and LB94 in LR-TDDFT are found to be vanishingly small, the total propagation time adopted in our RT-TDDFT calculations may not be long enough to detect the1Lbstate.
For a comprehensive comparison, the1Laand1Lbexcitation energies, calculated using LDA, PBE, and LB94 in both RT-TDDFT and LR-TDDFT, are summarized in Tables 1 and 2, respectively, where the results calculated using BNL18(a popular LC hybrid functional) are taken from ref. 51, and those calculated using
time-dependent coupled-cluster theory with single and double excitations (albeit with an approximate treatment of the doubles, CC261) and the experimental data are taken from ref. 44.
Owing to the CT-like character, LDA and PBE significantly underestimate the excitation energies of the ionic1Lastates in both RT-TDDFT and LR-TDDFT. Compared to the highly accurate CC2 results and experimental data, the errors of LDA and PBE increase with the acene length. LB94 performs similarly to LDA and PBE, indicating that the LB94 model potential is also inappropriate for CT-like excitations. As the LB94 results obtained from both the RT-TDDFT and LR-TDDFT calculations are very similar, the adiabatic LDA XC kernel adopted in the LR-TDDFT calculations appears to be appropriate. In RT-TDDFT, the failure of LB94 may be attributed to the lack of the step and peak structure in the Fig. 2 Absorption spectra of 2-acene calculated using various functionals
in RT-TDDFT. Subfigures (left top: LDA; left bottom: PBE; right: LB94) show the spectra close to the position of the1Lapeaks, where the corresponding LR-TDDFT results are marked with the red lines.
Fig. 3 Absorption spectra of 3-acene calculated using various functionals in RT-TDDFT. Subfigures (left top: LDA; left bottom: PBE; right: LB94) show the spectra close to the position of the1La and1Lbpeaks, where the corresponding LR-TDDFT results are marked with the red lines.
Fig. 4 Absorption spectra of 4-acene calculated using various functionals in RT-TDDFT. Subfigures (left top: LDA; left bottom: PBE; right: LB94) show the spectra close to the position of the1La and1Lbpeaks, where the corresponding LR-TDDFT results are marked with the red lines.
adiabatic LB94 XC potential, which has recently been shown to be essentially important for CT excitations.62In LR-TDDFT, the failure of LB94 may be attributed to the lack of a space- and frequency-dependent discontinuity in the adiabatic LDA XC kernel adopted.11,16,37,39Based on the above reasons, we expect that the CT failures may not be remedied by other AC model
potentials exhibiting the same features as LB94 or a pure density functional whose functional derivative has the correct '1/r asymptote.63By contrast, BNL performs very well on the excitation energies of the1Lastates in both RT-TDDFT and LR-TDDFT. Fully nonlocal (i.e., orbital-dependent) functionals, in particular, LC hybrid functionals, can be essential for the accurate description of CT-like excitations.
On the other hand, for the covalent 1Lbstates, LDA, PBE, and LB94 accurately predict the1Lbexcitation energies in both RT-TDDFT and LR-TDDFT, yielding quantitative agreement with the experimental data. Due to the inclusion of a large fraction of HF exchange, BNL yields noticeable errors on the excitation energies of the 1Lb states in both RT-TDDFT and LR-TDDFT, which may be attributed to the pronounced double- excitation character of the1Lbstates.48It remains very difficult to accurately describe both the1Laand1Lbstates of n-acenes with existing density functionals.
IV. Conclusions
In this work, we have examined the performance of a variety of density functionals on the two lowest singlet p - p* transition energies (i.e., the1Laand1Lbstates) of n-acenes (up to 5-acene) in both RT-TDDFT and LR-TDDFT. Our results have shown that the LB94 model potential performs similarly to LDA and PBE on both the1La(CT-like) and1Lb(valence) states. The excitation energies of the1Lastates are severely underestimated by LDA, PBE, and the LB94 model potential, and the errors have been shown to increase with the acene length. Despite its computational efficiency, our results suggest that the LB94 model potential may not accurately describe CT-like excitations in both RT-TDDFT (due to the lack of the step and peak structure in the adiabatic LB94 XC potential) and LR-TDDFT (due to the lack of a space- and frequency-dependent discontinuity in the adiabatic LDA XC kernel adopted). Although only the LB94 model potential has been examined in this work, we expect that other AC model potentials exhibiting the same features as LB94 or a pure density functional whose functional derivative has the correct asymptote may not resolve the CT problems.
On the other hand, the LC hybrid scheme, which can be computationally expensive for large systems, is reliably accurate for the excitation energies of the1Lastates due to the inclusion of long-range HF exchange, but less accurate for the excitation energies of the1Lbstates due to the significant double-excitation character of the1Lbstates. It remains very challenging to develop a generally accurate density functional for the ground-state, excited-state, and time-dependent properties of large systems.
Acknowledgements
This work was supported by the National Science Council of Taiwan (Grant No. NSC101-2112-M-002-017-MY3), National Taiwan University (Grant No. NTU-CDP-103R7855), the Center for Quantum Science and Engineering at NTU (Subproject No.:
NTU-ERP-103R891401 and NTU-ERP-103R891403), and the National Center for Theoretical Sciences of Taiwan.
Fig. 5 Absorption spectra of 5-acene calculated using various functionals in RT-TDDFT. Subfigures (left top: LDA; left bottom: PBE; right: LB94) show the spectra close to the position of the1La and1Lbpeaks, where the corresponding LR-TDDFT results are marked with the red lines.
Table 1 Excitation energies (in eV) for the1Lastates of n-acenes, calculated using various functionals in both RT-TDDFT and LR-TDDFT. The BNL results are taken from ref. 51, and the CC2 and experimental results are taken from ref. 44. The mean absolute errors (MAEs) of these methods are provided for comparisons (error = theoretical value ' experimental value)
n-Acene Experiment CC2
RT-TDDFT LR-TDDFT
LDA PBE LB94 BNL LDA PBE LB94 BNL 2 4.66 4.88 4.11 4.11 4.12 4.79 4.16 4.16 4.09 4.86 3 3.60 3.69 2.96 2.97 2.97 3.68 3.00 3.01 2.96 3.72 4 2.88 2.90 2.20 2.22 2.21 2.91 2.25 2.26 2.21 2.94 5 2.37 2.35 1.67 1.70 1.69 2.41 1.71 1.73 1.69 2.39 MAE 0.09 0.64 0.63 0.63 0.07 0.60 0.59 0.64 0.10
Table 2 Excitation energies (in eV) for the 1Lb states of n-acenes, calculated using various functionals in both RT-TDDFT and LR-TDDFT.
The BNL results are taken from ref. 51, and the CC2 and experimental results are taken from ref. 44. The mean absolute errors (MAEs) of these methods are provided for comparisons (error = theoretical value ' experimental value)
n-Acene Experiment CC2
RT-TDDFT LR-TDDFT
LDA PBE LB94 BNL LDA PBE LB94 BNL
2 4.13 4.46 4.61 4.29 4.29 4.21 4.64
3 3.64 3.89 3.65 3.64 3.68 4.03 3.69 3.69 3.61 4.07 4 3.39 3.52 3.26 3.26 3.28 3.68 3.29 3.30 3.23 3.70 5 3.12 3.27 3.00 2.99 3.01 3.42 3.03 3.03 2.97 3.44 MAE 0.22 0.09 0.09 0.09 0.37 0.10 0.10 0.11 0.39
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