First singlet (S1) excitations are of primary importance in the photoluminescence spectra of organic chromophores. However, due to the multi-determinantal nature of the singlet excited states, standard Kohn-Sham density-functional theory (DFT) is not applicable. While linear-response time-dependent DFT is the method of choice for the computation of excitation energies, it fails severely for excitations with charge-transfer character. Becke’s recent virial exciton model [A. D. Becke, J. Chem. Phys. 148, 044112 (2018)] offers a promising solution to employ standard DFT for calculation of the S1 excitation energy in molecular systems. Here, it is shown that the virial exciton model is free of charge-transfer error. It is equally reliable for S1 excitations with significant charge-transfer character as for other classes of transitions.

The photoluminescence of organic chromophores plays a fundamental role in nature, with prominent examples being photosynthesis,1 vision,2 and bioluminescence.3 Recent applications of photoluminescent materials include development of organic light-emitting diodes,4,5 fluorescent sensors,6,7 lasers,8 waveguides,9 and biomedical imaging.7,10 The first singlet (S1) electronic excitation is of primary importance in photoluminescence spectra. Computational modeling of these excitations is complicated as standard Kohn-Sham (KS) density-functional theory (DFT)11 is not applicable to the S1 excited state due to its multi-determinant nature. Linear-response time-dependent density-functional theory (TD-DFT)12–14 is the predominant method employed for the calculation of S1 and higher excitation energies. However, TD-DFT typically exhibits a severe underestimation of the excitation energy (frequently in excess of 1 eV) when the excitation is of charge-transfer (CT) character.15–19 This problem can be ameliorated using long-range-corrected hybrid functionals, but the optimum range-separation parameter in these functionals is extremely system dependent.20–22 Also, time-independent methods as exemplified by the very recent work of the Van Voorhis group (see Ref. 23 and references therein) are known which can handle CT excitations well. We recommend Ref. 24 for extensive reviews of both time-dependent and time-independent approaches to excited states in DFT.

Becke recently derived a simple model25 for the energy splitting between the first singlet and triplet (S1—T1) excited states, and hence the S1 excitation energy itself, based on the virial theorem.26 This “virial exciton model” requires only conventional DFT calculations for the S0 ground state and the (single-determinant) T1 excited state. It therefore represents a simple alternative to TD-DFT for the calculation of the S1 excitation energy in molecular systems. For Thiel’s benchmark set27 of 28 small-molecule excitation energies, the virial exciton model achieves a mean absolute error (MAE) for S1 on par with TD-B3LYP (0.26 and 0.24 eV, respectively), relative to high-level correlated wavefunction reference data. Remarkably, it significantly out-performs TD-B3LYP for S1 excitation energies of polycyclic aromatic hydrocarbons,28 achieving a MAE of 0.13 eV, versus the TD-B3LYP value of 0.31 eV.25 

In this work, the performance of the virial exciton model for systems that feature S1 excitations of significant CT character will be assessed for the first time. It is a two-step method, beginning with a conventional T1 excitation-energy computation, followed by a simple two-electron integral correction. The first step ensures, in large part, that the method does not suffer the CT failures of TD-DFT. Our benchmark set (see Fig. 1) consists of three subsets: (i) the ethylene-tetrafluoroethylene intermolecular CT dimer that has been used as a classic demonstration of TD-DFT charge-transfer error;16 (ii) four intermolecular CT dimers consisting of tetrafluoroethylene and aromatic hydrocarbons, for which experimental S1 excitation energies are available;29,30 and (iii) three donor-acceptor molecules featuring S1 excitations with intramolecular CT, for which high-level correlated wavefunction benchmark data are available.31,32 The results show that the virial exciton model is free of CT error.

FIG. 1.

The chemical systems investigated in this work. Shown are the ethylene-tetrafluoroethylene (C2H4—C2F4) complex; the donor-acceptor molecules 4-dimethylamino-benzonitrile (DMABN), para-nitroaniline (p-NA), and N,N-dimethyl-4-nitroaniline (DAN); and the intermolecular CT dimers between tetracyanoethylene (TCNE) and each of benzene, toluene, o-xylene, and naphthalene.

FIG. 1.

The chemical systems investigated in this work. Shown are the ethylene-tetrafluoroethylene (C2H4—C2F4) complex; the donor-acceptor molecules 4-dimethylamino-benzonitrile (DMABN), para-nitroaniline (p-NA), and N,N-dimethyl-4-nitroaniline (DAN); and the intermolecular CT dimers between tetracyanoethylene (TCNE) and each of benzene, toluene, o-xylene, and naphthalene.

Close modal

In the virial exciton model, the difference between the S1 and T1 excitation energies is given by the following two-electron integral:

ΔEST=Kif=  d3r1d3r2ϕi(r1)ϕf(r1)ϕi(r2)ϕf(r2)r12,
(1)

where ϕi and ϕf are the initial and final Kohn-Sham (KS) orbitals involved in the single-electron excitation, respectively. This expression is the result of adding a correlation correction to the uncorrelated S1-T1 splitting. In the following, we briefly summarise how this result is derived.

For a non-interacting system, the S1-T1 excitation-energy difference is

ΔEST0=12  d3r1d3r2ΔΠST0(r1,r2)r12,
(2)

where ΔΠST0(r1,r2) is the non-interacting pair-density difference between the S1 and T1 states

ΔΠST0(r1,r2)=4ϕi(r1)ϕf(r1)ϕi(r2)ϕf(r2).
(3)

Substituting Eq. (3) into Eq. (2), one obtains

ΔEST0=2  d3r1d3r2ϕi(r1)ϕf(r1)ϕi(r2)ϕf(r2)r12=2Kif.
(4)

A correlation correction, ΔESTcorr, must be added to ΔEST0 to recover the correlated S1-T1 splitting, ΔEST,

ΔEST=ΔEST0+ΔESTcorr.
(5)

ΔESTcorr consists of kinetic and potential energy contributions,

ΔESTcorr=ΔTSTcorr+ΔVSTcorr.
(6)

The quantum virial theorem states that, for a system at equilibrium, its kinetic (T) and potential (V) energies have the simple relation 2T = −V. This theorem is valid for both the ground and excited states. It also equally applies to both the correlated and uncorrelated systems. Therefore, this theorem can be used to simplify Eq. (6) and write

ΔESTcorr=12ΔVSTcorr.
(7)

Becke argued25 that electron correlation would have the effect of “smoothing out” the S1-T1 non-interacting pair-density difference [Eq. (3)], reducing it to zero everywhere. Correlation would then lower the potential energy of the S1 state, relative to the T1 state, by ΔVSTcorr=2Kif. Thus,

ΔESTcorr=Kif
(8)

and substitution into Eq. (5) gives the correlated S1-T1 splitting,

ΔEST=2KifKif=Kif,
(9)

which is the result in Eq. (1).

The S1 energy, ES1, and the corresponding excitation energy, ΔE0S=ES1ES0, can therefore be obtained from the energies of the S0 and T1 states and the Kif integral by

ES1=ET1+Kif,
(10a)
ΔE0S=ET1+KifES0=ΔE0T+Kif,
(10b)

where ΔE0T=ET1ES0 is the triplet excitation energy. To evaluate ΔE0S, the virial exciton model requires the energy of the S0 state as well as a restricted-open-shell (RO) calculation for the T1 state. The calculation must be RO in order to uniquely define ψi and ψf.

The geometries of the four tetracyanoethylene (TCNE)-aromatic dimers (B3LYP/cc-pVDZ)29 and DMABN (B3LYP/6-31G*)32 were taken from the literature. The geometries of p-NA and DAN were optimized using B3LYP/6-311G(d,p), consistent with Ref. 31. The C2H4—C2F4 dimer geometry (C2v symmetry) was optimized using B3LYP/6-31+G* at a fixed intermolecular separation of 4 Å. This intermolecular separation, R, was defined by the distance between the midpoints of the two C=C bonds, as shown in Fig. 1, and was varied from 4.0 to 10.0 Å in 0.5 Å increments. Ground-state, unrestricted and RO triplet-state, and TD-DFT single-point calculations were performed on the optimized geometries of all species using B3LYP33,34/cc-pVTZ. Configuration interaction singles (CIS)35 calculations were also performed using the cc-pVTZ basis set for the C2H4—C2F4 dimer. The Gaussian 09 package36 was employed throughout. An in-house “postG” program was used to compute the Kif integrals employing the numerical method of Becke and Dickson.37 

We first apply the virial exciton model to the C2H4—C2F4 intermolecular dimer,16 which is an established test of CT-excitation errors. The S1 excitation energy was calculated for a range of intermolecular separations with the virial exciton model, TD-B3LYP, and CIS. The results are shown in Fig. 2.

FIG. 2.

Calculated S1 excitation energy (ΔE0S) as a function of the intermolecular separation, R, for the C2H4—C2F4 dimer. The B3LYP functional was used for both the TD-DFT and virial exciton model calculations.

FIG. 2.

Calculated S1 excitation energy (ΔE0S) as a function of the intermolecular separation, R, for the C2H4—C2F4 dimer. The B3LYP functional was used for both the TD-DFT and virial exciton model calculations.

Close modal

CIS theory, which will serve as our benchmark, predicts a localized ππ transition on the ethylene molecule as the lowest-energy singlet excitation. By contrast, various TD-DFT calculations erroneously predict the intermolecular CT state to lie lower in energy.16,38 This causes TD-B3LYP to drastically underestimate the S1 excitation energy over the entire range of intermolecular separations. Moreover, because the TD-B3LYP S1 excitation has CT character, the excitation energy shows a strong dependence on the intermolecular distance, as seen in Fig. 2. Conversely, the lowest-energy triplet excitation is localized on the ethylene molecule and is of ππ character. As a result, the virial exciton model is in good agreement with CIS over the entire range of intermolecular separations and does not share the same CT breakdown displayed by TD-B3LYP. Calculations were also attempted on the bacteriochlorin-zinc bacteriochlorin intermolecular dimer, which is a second complex popularized as a demonstration of CT error.38 However, due to the near degeneracy of the first three excited states,38 we were not able to converge the RO triplet calculations required for the virial exciton model.

We now turn to a set of systems for which the S1 excitation does correspond to a CT state. The S1 excitation energies were computed for four TCNE-aromatic CT dimers and three donor-acceptor molecules featuring intramolecular CT excitations. The resulting excitation energies, and related quantities required for the virial exciton model, are tabulated in Table I. The S1 excitation energies are compared to experimental or high-level theoretical reference values.29,31,32

TABLE I.

Calculated excitation energies, and related quantities, in eV. Absolute errors from the literature reference values (ΔE0SRef.) are given in parentheses. Tabulated values are the unrestricted and restricted T1 excitation energies (ΔE0TU and ΔE0TRO), the Kif integral, the unrestricted and restricted S1 excitation energies (ΔE0SU and ΔE0SRO), and the TD-B3LYP S1 excitation energies (ΔE0STD).

SystemΔE0TUΔE0TROKifΔE0SUΔE0SROΔE0STDΔE0SRef.
TCNE-benzene 2.27 2.41 1.55 3.82 (0.23) 3.96 (0.37) 1.91 (−1.68) 3.5929  
TCNE-toluene 2.21 2.33 1.30 3.51 (0.15) 3.63 (0.27) 1.74 (−1.62) 3.3629  
TCNE-o-xylene 2.14 2.25 1.12 3.26 (0.11) 3.37 (0.22) 1.48 (−1.67) 3.1529  
TCNE-naphthalene 1.61 1.72 1.10 2.71 (0.11) 2.82 (0.22) 0.81 (−1.79) 2.6029  
DMABN 3.33 3.41 1.66 4.99 (0.27) 5.07 (0.35) 4.31 (−0.41) 4.7232  
p-NA 3.12 3.17 1.46 4.58 (0.19) 4.63 (0.24) 3.50 (−0.89) 4.3931  
DAN 2.91 2.97 1.37 4.28 (0.34) 4.34 (0.40) 3.19 (−0.75) 3.9431  
MAE    0.20 0.29 1.26 … 
SystemΔE0TUΔE0TROKifΔE0SUΔE0SROΔE0STDΔE0SRef.
TCNE-benzene 2.27 2.41 1.55 3.82 (0.23) 3.96 (0.37) 1.91 (−1.68) 3.5929  
TCNE-toluene 2.21 2.33 1.30 3.51 (0.15) 3.63 (0.27) 1.74 (−1.62) 3.3629  
TCNE-o-xylene 2.14 2.25 1.12 3.26 (0.11) 3.37 (0.22) 1.48 (−1.67) 3.1529  
TCNE-naphthalene 1.61 1.72 1.10 2.71 (0.11) 2.82 (0.22) 0.81 (−1.79) 2.6029  
DMABN 3.33 3.41 1.66 4.99 (0.27) 5.07 (0.35) 4.31 (−0.41) 4.7232  
p-NA 3.12 3.17 1.46 4.58 (0.19) 4.63 (0.24) 3.50 (−0.89) 4.3931  
DAN 2.91 2.97 1.37 4.28 (0.34) 4.34 (0.40) 3.19 (−0.75) 3.9431  
MAE    0.20 0.29 1.26 … 

To verify that the T1 excited states in question indeed possess CT character, we computed density differences relative to the S0 ground state. The results are presented in Fig. 3. For each of the four TCNE-aromatics dimers, notable intermolecular CT is observed, with the electron density shifting from the aromatic moiety to the TCNE molecule. DMABN, p-NA, and DAN all show typical intramolecular push-pull CT from the electron-donating to the electron-withdrawing substituent.

FIG. 3.

Computed T1—S0 density differences for the TCNE-aromatic dimers and donor-acceptor molecules. Violet (green) isosurfaces represent an increase (decrease) in electron density in the T1 state relative to the S0 state. The isovalues are ±0.001 a.u.

FIG. 3.

Computed T1—S0 density differences for the TCNE-aromatic dimers and donor-acceptor molecules. Violet (green) isosurfaces represent an increase (decrease) in electron density in the T1 state relative to the S0 state. The isovalues are ±0.001 a.u.

Close modal

Returning to Table I, the S1-T1 energy splitting, given by the Kif integral, ranges from 1.1 to 1.7 eV for these systems. One might expect a vanishing Kif integral for CT excitations, as the ground-state frontier orbitals will be localised on either the donor or acceptor moieties and will consequently have negligible overlap. However, this is not the case, as the two singly-occupied molecular orbitals in the RO triplet calculations are delocalised over both moieties and have substantial overlap.

Table I shows that TD-B3LYP drastically underestimates the CT excitation energies, as expected, with a MAE of 1.26 eV. For all seven systems, the virial exciton model provides significantly more accurate CT excitation energies than TD-B3LYP, with a MAE of 0.29 eV. An even lower MAE of 0.20 eV can be achieved by adding Kif (which must be computed from the RO triplet orbitals) to the unrestricted T1 excitation energy. Contrary to the typical underestimation by TD-DFT methods, the virial exciton model systematically overestimates the CT excitation energies in Table I. This is possibly a result of using the cc-pVTZ basis set, which lacks diffuse functions. The CT nature of the present excitations results in anionic moieties in the excited states, which will be preferentially stabilised by the addition of diffuse functions. Unfortunately, the RO triplet calculations are somewhat difficult to converge with the present basis set and addition of diffuse functions greatly exacerbates the problem. This emphasizes the importance of improving self-consistent-field algorithms for RO calculations, as one must be able to efficiently converge to the correct triplet state before applying the virial correction. Regardless, the performance of the virial exciton model is impressive and confirms that it does not suffer from the same intrinsic CT errors as TD-DFT.

In this work, the accuracy of Becke’s virial exciton model was assessed for CT excitation energies. The results demonstrate that the model is free of the systematic CT errors that plague conventional TD-DFT methods. For a benchmark set consisting of four intermolecular TCNE-aromatic dimers and three donor-acceptor molecules, the virial exciton model achieves an overall MAE of 0.29 eV (or 0.20 eV using unrestricted T1 energies) compared to the literature reference data, significantly improving upon the accuracy of the widely used TD-B3LYP method. This error is roughly on par with the MAE of 0.26 eV previously obtained25 for Thiel’s small-molecule excitation data set.27 We therefore conclude that the virial exciton model can be reliably used to predict S1 excitation energies in molecular systems, even for excitations with CT-character. See, also, the very recent application of the model to computation of the optical gap in polyacetylene.39 

E.R.J. and A.D.B. thank the Natural Sciences and Engineering Research Council of Canada (NSERC), and X.F. acknowledges the Government of Nova Scotia for financial support. Computational resources were provided by ACEnet, the Atlantic Computational Excellence Network, funded by Compute Canada.

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