Excited-state descriptors based on the one-particle transition density matrix referring to the exciton picture have been implemented for time-dependent density functional theory. State characters such as local, extended ππ, Rydberg, or charge transfer can be intuitively classified by simple comparison of these descriptors. Strong effects of the choice of the exchange-correlation kernel on the physical nature of excited states can be found and decomposed in detail leading to a new perspective on functional performance and the design of new functionals.

Linear-response time-dependent density functional theory (TDDFT) is certainly one of the most widely used quantum-chemical methods for excited-state calculations of medium-size and large molecules.1–3 At the same time, the failure of TDDFT to correctly describe charge-transfer (CT) states is well-known.4–6 Also Rydberg states, excited states of extended π-conjugated systems,7,8 and multiply excited states9,10 cause difficulties. To solve conceptual short-comings, great effort has been made to search for improved functionals that reliably describe CT states, and in particular, long-range corrected (LRC) functionals featuring varying amounts of non-local exchange were developed for this purpose. The first LRC functional proposed by Yanai et al.11 was a Coulomb-attenuated variant of the prominent Becke-3-Lee-Yang-Parr functional12,13 (CAM-B3LYP), which contains non-local orbital exchange from 19% at short range up to 65% at long range.

The search for tools to diagnose the failures of TDDFT in combination with different functionals is an ongoing challenge and several descriptors measuring orbital overlaps14–16 and centroid distances17–19 have been proposed. However, some cases like extended symmetric π-conjugated systems, e.g., acenes, cause significant but hard-to-detect errors.7,8,20 For these systems, the overlap of the canonical orbitals is not a good predictor of the performance of TDDFT20–23 and a more complex dynamical view has to be adopted.

In recent work,24–27 we established a series of excited-state descriptors based on the physical picture of an exciton wavefunction, which is constructed from the one-particle transition density matrix (1TDM) (see also Refs. 28–30). Information about the spatial distributions of electron and hole over the molecular system as well as statistical descriptors provide a solid basis for analyzing excited states. This opens a route to assign excited-state characters without the need for orbital inspection, which is beneficial for identifying problematic cases. Furthermore, the effect of a chosen exchange-correlation (xc) functional on an excited state can be investigated in detail. High-level benchmarks are accessible as the same descriptors can be computed for any method providing 1TDMs. This allows detailed benchmarking of TDDFT results going beyond a simple comparison of energies.

The central idea of our approach is the definition of an exciton wavefunction, which is identified with the 1TDM between the ground and I-th excited states.26,31 In TDDFT, the 1TDM is usually constructed from the excitation and de-excitation amplitudes X i a I and Y i a I using the form

(1)

where ϕi and ϕa are the occupied and virtual molecular orbitals and rh and re denote the coordinates of the electron and hole quasi-particles. The form of Eq. (1) is used in the current implementation, but we note for completeness that this is not a rigorous assignment as wavefunctions do not exist in TDDFT, and a different formalism suggests the use of additional scaling factors.2,32,33 In the case of the Tamm-Dancoff approximation (TDA),34 the same formalism is applied with the exception that the Y i a I terms vanish.

Properties of the exciton wavefunction can be directly calculated as operator expectation values

(2)

for any operator O ˆ , whose matrix elements in the underlying orbital basis are available. The properties and relation of these expectation values to natural transition orbital analysis35 have been discussed in detail recently.27 

The main focus of this study is the description of spatial distributions and correlation effects of exciton wavefunctions, a task which can be achieved by computing multipole moments of the electron and hole position operators.27 Two measures of charge transfer are investigated: the mean

(3)

and root-mean-square (rms)

(4)

electron-hole separations. While the first is a simple distance of the charge centroids, the second also includes dynamic charge separation contributions and we refer to it as exciton size. The two quantities are related by27 

(5)

where σh and σe are the rms sizes of the hole and electron distributions and

(6)

marks the covariance between the quasi-particles. It also follows that

(7)

While these descriptors bear some resemblance to previous charge-transfer measures,17,18 they stand out due to their well-defined physical meaning in the exciton picture. Furthermore, the lack of any explicit dependence on the orbitals avoids the problem of choosing the optimal representation of the orbitals in invariant subspaces.23,36

A recently introduced idea27 is to quantify correlation effects between electron and hole in analogy to the Pearson correlation coefficient

(8)

obtained as the normed covariance of the electron-hole distribution ranging from −1 to 1. A negative correlation corresponds to the picture of two charges avoiding each other dynamically. A value of zero indicates no linear correlation, i.e., electron and hole “move” independently, which is typically taken for granted when a state is analyzed with respect to molecular orbitals. Positive correlation means a joint electron-hole motion which can be thought of as a bound exciton. All presented excited-state descriptors are implemented in the Q-Chem 4.3 release version.

We will first examine the relation between rms exciton sizes dexc and charge centroid distances dhe for Tozer’s benchmark set14 as presented in Fig. 1. In these and all following computations, three functionals of different classes are applied: PBE37 is employed as a representative of the generalized gradient approximation (GGA), B3LYP12,13 as hybrid, and CAM-B3LYP as long-range corrected functional. The exciton descriptors are plotted against the error in excitation energy compared to experimental data, where the same geometries, basis sets, and reference data are used as in Ref. 14. Starting with exciton sizes computed by TDA/PBE, four different groups can be identified. First, locally excited states with well-described excitation energies and small exciton sizes appear in the upper left corner of Fig. 1(a) marked in black. Second on the main diagonal, charge-transfer states with strongly underestimated excitation energies are found, where the error in excitation energies seems to scale linearly with the exciton size (also marked in black). To highlight some notoriously difficult cases, the exciton sizes for acenes and polyacetylenes are marked in red. As their energies are rather well-described using the Tamm-Dancoff approximation,38 they appear at small deviations in ΔEexp but possess large exciton sizes with a linear dependency between molecular size and exciton size. In addition, the Rydberg states are marked in blue. They have rather large exciton sizes due to their diffuse electron sizes σe. At the same time, their description is somewhat poorer compared to the previous group and they show a systematic underestimation of excitation energies on the order of 2 eV.

FIG. 1.

Exciton sizes (dexc, Å, first row) and center-of-mass electron-hole distances (dhe, Å, second row) for Tozer’s benchmark set14 plotted against the error of excitation energy compared to experimental data (ΔEexp, eV). All presented states are singlet excited states and the Tamm-Dancoff approximation was applied. Excited states with local or directed charge-transfer character are marked in black (×), states for extended π systems (polyacenes and polyenes) in red (△), and Rydberg states in blue (♢).

FIG. 1.

Exciton sizes (dexc, Å, first row) and center-of-mass electron-hole distances (dhe, Å, second row) for Tozer’s benchmark set14 plotted against the error of excitation energy compared to experimental data (ΔEexp, eV). All presented states are singlet excited states and the Tamm-Dancoff approximation was applied. Excited states with local or directed charge-transfer character are marked in black (×), states for extended π systems (polyacenes and polyenes) in red (△), and Rydberg states in blue (♢).

Close modal

Going to the charge centroid separation dhe plotted in Fig. 1(d), the overall distribution of the black data group is preserved. The interpretation of this finding is simple: Locally excited states show only small electron-hole separations, however, some values above zero may occur as in some cases charges are promoted between adjacent functional groups. For the charge-transfer states, the values of dhe and dexc are quite similar as expected from Eq. (5) in the case for insignificant electron and hole sizes, as well as correlation effects. For the two other groups, the large conjugated systems and the Rydberg states, the situation differs dramatically. With a few exceptions for some Rydberg states, the static electron-hole separation dhe is zero, which means that no static charge transfer occurs. In particular in extended π systems, an almost equal and symmetric distribution of electron and hole densities over the excited molecules and a superposition of their centroids is present. The spatial extent of the charge clouds, as measured by σh and σe, and correlation effects are responsible for the large exciton sizes. In contrast for the Rydberg states, the dominant factor for large exciton sizes is clearly the electron size σe.

Proceeding to the B3LYP functional, locally excited states show similar exciton sizes compared to PBE. In contrast, the other states show in average a shift to smaller exciton sizes, especially in the cases of extended π systems and Rydberg states. While the huge errors of the directed charge-transfer states are attenuated at this level, there is only a slight improvement for the extended π system states. For the Rydberg states, the excitation energies are substantially corrected by about +1 eV compared to PBE.

For the CAM-B3LYP functional, the best performance in excitation energies is found, see Fig. 1(c). Examining the exciton sizes of the locally excited states, we find their sizes to be preserved independent of the functional choice. This comes along with almost constant excitation energy shifts with respect to experiment. In contrast, as observed before in the case of B3LYP, the states for extended π systems and Rydberg states have a tendency to significantly smaller exciton sizes compared to PBE. To better understand this localization effect, it is worth revisiting Eq. (5). Especially for the cases where dhe is zero, the exciton size depends on the sizes of electronσe and holeσh and their covariance. Going from PBE to B3LYP to CAM-B3LYP for these states, a general trend is that σe,PBE > σe,B3LY P > σe,CAM−B3LY P and the same is observed for σh which is logical considering that a diminished self-interaction error results in more compact electron-hole distributions. However, the variations in σh and σe only explain a small part of the variations in dexc and the covariance becomes the decisive factor for explaining the differences between the functionals. In the case of a positive electron-hole correlation, the resulting attraction leads to an additional compression of dexc. In contrast for a negative correlation, the exciton size is enlarged because of the effective repulsion and charge avoidance. To gain deeper insight into this effect, the correlation coefficient Reh will be discussed for acenes.

We will now examine the linear electron-hole correlation effects with respect to the choice of functionals and compare standard linear-response TDDFT with the Tamm-Dancoff approximation. The 1La and 1Lb states of acenes from naphthalene to hexacene are calculated at the TDDFT/cc-pVTZ and TDA/cc-pVTZ levels of theory employing the PBE, B3LYP, and CAM-B3LYP functionals and the correlation coefficient Reh is plotted against the system size (Fig. 2). First, the 1La states are discussed (Fig. 2(a)). For the TDDFT results indicated by solid lines, the correlation coefficient Reh for the PBE functional (blue) is the smallest for all molecules ranging between −0.20 and −0.15 with a trend to lower values for larger systems. This behaviour can be interpreted with the picture of two charges avoiding each other dynamically. Going to B3LYP (green), the values of Reh become closer to zero staying slightly below −0.05 and are almost constant for the systems larger than naphthalene. Finally, in the case of CAM-B3LYP (red) the correlation coefficient starts from approximately zero and constantly increases for larger systems to a value of 0.08 for hexacene. This positive correlation can be understood as the onset of exciton binding. This result sheds new light on the observations by Tretiak et al.39 that bound excitons in large conjugated systems can only be described by LRC functionals.

FIG. 2.

Correlation coefficients Reh for (a) 1La and (b) 1Lb states for the acene series from naphthalene (n = 2) to hexacene (n = 6) calculated at the TDDFT/cc-pVTZ (solid lines) and TDA/cc-pVTZ (dotted lines) levels of theory employing the PBE, B3LYP, and CAM-B3LYP functionals and ADC(3)/SV(P) reference values.

FIG. 2.

Correlation coefficients Reh for (a) 1La and (b) 1Lb states for the acene series from naphthalene (n = 2) to hexacene (n = 6) calculated at the TDDFT/cc-pVTZ (solid lines) and TDA/cc-pVTZ (dotted lines) levels of theory employing the PBE, B3LYP, and CAM-B3LYP functionals and ADC(3)/SV(P) reference values.

Close modal

Going from TDDFT to TDA (dotted lines), the electron-hole correlation behaviour is significantly altered in favour of a stronger electron-hole attraction or at least a decrease of their repulsion. In the case of PBE, the average shift is about +0.06 still containing significant amount of dynamic electron-hole repulsion. For the B3LYP functional, an increase of Reh from clearly negative values of −0.05 for naphthalene to zero correlation for tetracene and larger molecules is found. This zero correlation is considered as linearly independent dynamical motion of electron and hole which is the standard assumption when inspecting molecular orbitals. Going to the long-range corrected CAM-B3LYP functional, the positive correlation increases almost constantly for all systems and ranges between 0.02 and 0.15 which is significantly higher than for TDDFT.

The values for the 1Lb states are shown in Fig. 2(b) and again we will start the discussion with the TDDFT results. For PBE, the Reh values are approximately zero for all systems independent of their sizes. In contrast, for B3LYP and CAM-B3LYP, positive values are found with a constant increase for larger system sizes. At the same time for CAM-B3LYP, the slope of the curve is larger and a stronger trend towards bound excitons is encountered. Going to the TDA results, essentially the same trends can be observed for all functionals. However, an almost constant shift towards higher correlation values is observed with a shift of +0.02 for B3LYP and an even larger shift of +0.05 for CAM-B3LYP. The Reh values were recomputed at the ab initio level using the algebraic-diagrammatic construction method for the polarization propagator to third order (ADC(3)/SV(P)),41 showing excellent agreement with the TDA/CAM-B3LYP level.

With the knowledge of the electron-hole correlation within the 1La and 1Lb states, the problems encountered to calculate excitation energies for the two states at the TDDFT level can be easily explained. For the 1La state, a long-range corrected hybrid functional is absolutely necessary to correctly describe a bound exciton. In contrast, the physically correct picture for the 1Lb state emerges already for B3LYP and in the case of PBE, at least no dynamical electron-hole repulsion is found. This explains the increased sensitivity of the 1La state with respect to the choice of functional.20,38 The level of theory and the choice of the exchange-correlation functional are very critical to obtain a physically reasonable description. In this respect, there is a decisive difference between TDDFT and TDA that Peach et al. related to the singlet-triplet instability issue.40,42

In summary, a series of excited-state descriptors based on the exciton picture has been presented for TDDFT. A wealth of information about excited-state properties can be obtained such as static electron-hole separations, exciton sizes, and correlation coefficients. Tozer’s benchmark set was studied to illustrate how states can be characterized and different types can be distinguished based on exciton descriptors. Furthermore, the dependence of exciton properties with respect to the choice of xc-functionals can be analyzed in detail. The correlation coefficient has proven to be a powerful descriptor for the emerging electron-hole picture of an excited state as shown for acenes. The erroneous description of excitation energies for PBE is associated with repulsive electron-hole interactions. In contrast for the long-range corrected CAM-B3LYP functional, not only much better excitation energies are obtained but also particularly bound excitonic states with attractive electron-hole interactions are present. This effect is even enlarged when the TDA is applied and the poorer results from full TDDFT can be traced back to more weakly bound electron-hole pairs.

We are convinced that this detailed decomposition of exciton-related properties will further our understanding of the performance of xc-functionals in TDDFT. The methodology presented should not only be suitable for diagnosing methodological shortcomings but should, ultimately, also provide a new viewpoint for generating improved functionals. Due to the general formulation used, the quantities are also accessible in an ab-initio context26,27 offering new ways for benchmarking. Finally, the computed excited-state properties open a new route for property-based diabatization schemes.43–45 

S.A.M. acknowledges funding of the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences and Landesgraduiertenförderung Baden-Württemberg. F.P. acknowledges funding of the Alexander-von-Humboldt-Stiftung and the Vienna Scientific Cluster research school.

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