Overcoming Low Orbital Overlap and Triplet Instability Problems in

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Overcoming Low Orbital Overlap and Triplet Instability Problems in TDDF Michael J G Peach, and David James Tozer J. Phys. Chem. A, Just Accepted Manuscript • Publication Date (Web): 12 Sep 2012 Downloaded from http://pubs.acs.org on September 19, 2012

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The Journal of Physical Chemistry

Overcoming Low Orbital Overlap and Triplet Instability Problems in TDDFT Michael J G Peach⇤ and David J Tozer⇤ Department of Chemistry, Durham University, South Road, Durham, DH1 3LE, UK E-mail: [email protected]; [email protected]

⇤ To

whom correspondence should be addressed

1

ACS Paragon Plus Environment


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Abstract Low orbital overlap and triplet instability problems in TDDFT are investigated for a new benchmark set, encompassing challenging singlet and triplet excitation energies of local, chargetransfer and Rydberg character. The low orbital overlap problem is largely overcome for both singlet and triplet states by the use of a Coulomb-attenuated functional. For all the categories of functional considered, however, errors associated with triplet instability problems plague high overlap excitations, as exemplified by the excited states of acenes and polyacetylene oligomers. Application of the Tamm–Dancoff approximation reduces these errors for both singlet and triplet states, whilst leaving low-overlap excitations unaffected. The study illustrates the synergy between overlap and stability and highlights the success of a combined, Coulomb-attenuated Tamm–Dancoff approach.

Introduction Time-dependent 1–3 density functional theory 4–6 (TDDFT) in the adiabatic approximation is the most widely used first-principles technique for studying the electronic excited states of molecules. For real orbitals, vertical excitation energies wTDDFT are computed via the generalised eigenvalue equation 2,7 0

10 1 0 BA BC BXC B1 A @ A = wTDDFT @ @ B A Y 0

10 1 0 C BX C A@ A , 1 Y

(1)

where A and B are Hessian matrices and X and Y define the character of the excited state in terms of occupied–virtual (X) and virtual–occupied (Y) orbital rotations. In previous work, we introduced 8 a quantity L that measures the degree of spatial overlap between the canonical Kohn–Sham occupied and virtual orbitals involved in an excitation. Specif-

2


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ically, L=

2O Âi,a kia ia , 2 Âi,a kia

(2)

where kia = Xia + Yia is the contribution of a particular occupied–virtual (i–a) orbital pair to the excitation and Oia =

Z

|ji (r)||ja (r)| dr

(3)

measures the spatial overlap of those orbitals. The normalisation of Eq. (2) is such that 0 < L 6 1; values close to 0 indicate a ‘low-overlap’ excitation while values close to 1 indicate a ‘high-overlap’ excitation. We demonstrated 8 that as L becomes small, so singlet excitation energies from GGA/ hybrid functionals 9,10 become significantly underestimated. This led us to suggest that L could be used in a diagnostic manner for these functionals, to identify problematic low-overlap excitations, such as those of long-range charge-transfer (CT) (or Rydberg) character; it has been used successfully in a variety of studies of the well-known ‘CT problem’ of approximate TDDFT, e.g. see Refs. 11–15. We then demonstrated 8 that the low-L breakdown is largely overcome by using a Coulomb-attenuated functional, 16–24 where the 1/r12 operator in the exchange term is partitioned into short- and long-range components and the long-range exchange is evaluated using an exact orbital expression. Other studies have also highlighted the success of Coulomb-attenuated functionals in this context. 11,20,25–28 More recently, we have considered the role of L in the calculation of Rydberg states, 29 oscillator strengths, 30,31 and potential energy surfaces. 32,33 We subsequently turned our attention 34 to the calculation of triplet excitation energies in TDDFT, 35 highlighting the influence of the triplet instability problem. 7,36–46 The stabilities wSTAB of the Kohn–Sham determinant are determined from the Hermitian analogue of Eq. (1), 47,48 0

10 1 0 1 BA B C BX C BX C @ A @ A = wSTAB @ A . B A Y Y 3


(4)


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Due to the inherent similarities between the TDDFT and stability equations, 38,40 they often share eigenvectors of similar occupied–virtual character and so it is possible to associate a given triplet wTDDFT with that of a triplet wSTAB . Throughout this work, the notation wSTAB will always refer to the triplet stability measures, which are the eigenvalues of A

B. 7,46 We highlighted the fact

that as wSTAB reduces towards zero, so the corresponding triplet wTDDFT also approaches zero, often resulting in significantly underestimated triplet excitation energies; when wSTAB becomes negative, i.e., when there is an actual triplet instability, the triplet wTDDFT becomes imaginary. This problem is well-known from Hartree–Fock theory, but is not widely appreciated in the TDDFT user community, despite being highlighted in several studies. 7,39–42,45,46,49 This can be traced to the fact that the extent of the problem is dependent on the amount of exact orbital exchange in the functional, becoming more severe as the amount increases. 34 Until relatively recently, most TDDFT calculations were performed using functionals with relatively modest amounts of exact orbital exchange and so such errors in triplet wTDDFT were not significant. However, given the recent surge of interest in functionals containing larger amounts of exact exchange—e.g. Coulomb-attenuated approximations or others that explicitly include 100% exact exchange, which are necessary for an accurate description of the low-L situations mentioned above—these triplet errors are becoming much more pronounced. Consistent with earlier studies, 41,42 we demonstrated that application of the Tamm–Dancoff approximation 41,42,50,51 (TDA), which corresponds to setting B = 0 in Eq. (1) and solving AX = wTDA X ,

(5)

largely eliminates the errors in triplet excitation energies associated with the triplet instability problem. Interestingly, our results 34 suggested that the singlet states of the same occupied–virtual spatial character could also be improved by the TDA. The TDA has the advantage that it is computationally and conceptually simpler 34,41,42 than conventional TDDFT and unlike Eq. (1), it cannot yield imaginary excitation energies due to the Hermitian nature of Eq. (5). We do note that con-

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cern has been expressed 7,52 regarding the validity of transition intensities (oscillator strengths) computed from the TDA because they do not satisfy the Thomas–Reiche–Kuhn 53–55 sum rule. The present study focuses only on excitation energies; oscillator strengths will be considered in future work. For the excitations we consider here, the TDDFT and TDA oscillator strengths exhibit qualitatively similar trends. To date, our investigations have considered the low orbital overlap and triplet instability problems independently. The aim of the present study is to consider these two problems in a single investigation for a diverse range of molecules, excitations, and functionals, considering the relationship between overlap and stability and quantifying the extent to which the two problems can be simultaneously overcome.

Benchmark Set and Computational Details In order to allow the accuracy of equivalent singlet and triplet excitations (those of a given spatial character) to be assessed together, we have compiled a new benchmark set of molecules, drawn from those investigated in Ref. 8. The set comprises a model dipeptide, two acenes (naphthalene and anthracene), N-phenylpyrrole (PP), dimethylaminobenzonitrile (DMABN), four polyacetylene oligomers (PAO) with 4–10 carbon atoms, N2 , CO, and H2 CO, at the geometries of that earlier study. To further assess geometrical dependence, we also include structures obtained by twisting the dimethylamino group in DMABN by 90 degrees relative to the plane of the ring, and by twisting the relative planes of the two rings in PP also by 90 degrees, and performing symmetryconstrained MP2/6-31G* optimisations. Following Ref. 8, all calculated excitations are vertical and are categorised as either local, CT, or Rydberg, based on earlier classifications. For the small molecules, N2 , CO, and H2 CO, the TDDFT calculations were performed using the diffuse d-aug-cc-pVTZ basis, in order to minimise the artificial basis set confinement of Rydberg states. To quantify the accuracy of these TDDFT excitation energies, reference values were determined using all-electron third-order approximate coupled cluster 56 (CC3) calculations,

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with the same extensive basis set. For the remaining, larger molecules, the TDDFT calculations were performed using aug-cc-pVTZ, which is sufficiently diffuse because no Rydberg states are considered. For these molecules, reference values were determined using all-electron equationsof-motion coupled-cluster with single and double excitations 57,58 (EOM-CCSD). Due to the computational cost of applying this method to these-sized molecules, we were unable to use the augcc-pVTZ basis set. We therefore determined excitation energies using cc-pVTZ and added an approximate, double-z correction to account for the effect of diffuse functions, defined as the difference between excitation energies determined using aug-cc-pVDZ and cc-pVDZ. Specifically, we define our reference values to be cc-pVTZ

wREF = wCCSD

aug-cc-pVDZ

+ wCCSD

cc-pVDZ

wCCSD

.

(6)

TDDFT, CC3 and EOM-CCSD excitation energies were determined using the Dalton, 59 NWChem 60 and CFour 61 programs. Values of the associated stabilities wSTAB were determined using Gaussian 09. 62 Given the importance of long-range exact orbital exchange in both the low orbital overlap and triplet instability problems, calculations were performed using three representative exchange– correlation functionals with differing amounts of exact exchange: the PBE 9 GGA (no exact exchange), the B3LYP 63–67 hybrid functional (20% fixed exact exchange at all r12 ) and the CAMB3LYP 20 Coulomb-attenuated functional (increasing from 19% to 65% exact exchange at large r12 ). Details of the conventional TDDFT, TDA TDDFT and reference excitation energies, errors, values of L and wSTAB , and selected density difference plots, are presented in the supporting information.

Results and Discussion Figure 1 presents errors in DFT singlet (left panel) and triplet (right panel) excitation energies, relative to the reference values (error defined as calculated minus reference value), as a function 6


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of the corresponding singlet and triplet L values, respectively; the L values differ little between singlet and triplet excitations since they have equivalent spatial character. The three categories of excitation (local, CT, Rydberg) are distinguished by the use of different symbols and colours. First consider the conventional TDDFT results, which are indicated using light symbols. The behaviour for singlet and triplet states is very similar, indicating that the findings of Ref. 8, which only considered singlets, are readily generalised to triplets; of course, in the L ! 0 limit, the two become degenerate. The utility of L as a diagnostic quantity for GGA/ hybrid functionals is clearly evident—low L excitations are significantly underestimated for PBE and B3LYP. As the amount of long-range exact orbital exchange increases from PBE to B3LYP to CAM-B3LYP, the low-L breakdown (CT and Rydberg) is largely overcome, reflecting the importance of long-range exact exchange for such excitations. In order to ascertain whether the TDDFT errors in Figure 1 can also be related to triplet instability problems, Figure 2 plots wSTAB for each excitation, as a function of the associated (triplet) L values. The elements of the matrix B depend on the degree of occupied–virtual orbital overlap and the quantity L provides a direct measure of the overlap relevant to a given excitation. It follows that for excitations where L is small, the contribution from B to Eqs. (1) and (4) is small and wSTAB is close to wTDDFT . The functional dependence of wSTAB at low-L in Figure 2 therefore simply reflects the different excitation energies from the functionals. As L increases, the contribution from B also increases and wSTAB differs from wTDDFT . For all three functionals, wSTAB broadly reduces with increasing L, with the minimum value decreasing as the amount of long-range exact orbital exchange increases. This suggests the excitations are increasingly likely to be affected by triplet instability problems, most notably for the highest L excitations (L ⇠ 0.8) where wSTAB is smallest, and this is consistent with the fact that these excitations are uniformly and significantly underestimated. It is therefore pertinent to recompute the excitation energies using the Tamm– Dancoff approximation; these results are indicated using dark symbols in Figure 1, plotted against the corresponding TDDFT L values. The TDA results support the triplet instability hypothesis. For the excitations with low L val7



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ues, the TDA singlet and triplet excitation energies are indistinguishable from the conventional TDDFT values; the points in the figure are coincident. However, as L increases, the differences become much more pronounced. For the local singlet and triplet states with the highest L values, application of the TDA largely eliminates the underestimation observed with conventional TDDFT. Once again, the diminishing effect of the TDA with decreasing L reflects the decreasing contribution from B in Eq. (1), meaning wTDDFT is close to wTDA from Eq. (5). To further quantify the effect of the TDA and the dependence on functional and spin state, Figure 3 plots the difference between TDA and TDDFT excitation energies as a function of wSTAB . For triplet states, the difference becomes increasingly pronounced as the amount of exact orbital exchange increases from PBE to B3LYP to CAM-B3LYP, whilst for singlet states, the trend is reversed, becoming more significant as the amount of exact exchange decreases. The above analysis highlights a synergy between wSTAB and L. An excitation characterised by a small wSTAB and small L (as observed for a CT state in Figure 2(a)) is unaffected by the TDA, whereas one characterised by a small wSTAB and large L is significantly affected due to a triplet instability problem. Of the three categories of excitation, it is the local excitations that are most affected by the TDA and its effect is to improve the average accuracy in all cases. For the local singlet states, application of the TDA reduces the mean absolute errors by 0.14, 0.11, and 0.08 eV for PBE, B3LYP and CAM-B3LYP, respectively. For triplet states, the reductions are 0.07, 0.16, and 0.25 eV. These are significant error reductions, on the order of 20–60% of the TDDFT mean absolute errors for the local excitations. It is noteworthy that the highest L excitations—those most affected by the triplet instability problem and TDA—occur in twisted DMABN and PP and in the acenes and polyacetylene oligomers, i.e., systems characterised by delocalised electronic structures. 48 Such systems are representative of many technologically relevant molecules involved in luminescence and studies have previously highlighted the challenge of accurately describing such excitations using TDDFT, 8,68–73 particularly in the acenes. It has been shown 8,69,70,72 that Coulomb-attenuated functionals are able to correctly reproduce the acene chain-length evolution (singlet Lb state) and the singlet La –Lb state ordering, a challenge first highlighted by Grimme and Parac, 68 and that 8


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the TDA is able to additionally correct the state ordering deficiency with GGA/ hybrid functionals, 34,44,71 although the chain-length evolution remains problematic. Our present results are consistent with these earlier studies but interpret the observations in terms of the triplet instability problem. Finally, we observe that the TDA leads to significant improvements in singlet excitation energies from PBE and B3LYP (Figure 1), which is somewhat unexpected, given that these functionals do not contain significant amounts of exact orbital exchange. The importance of simultaneously describing low-overlap and low-stability excitations within a single molecule—i.e., requiring both Coulomb-attenuation and Tamm–Dancoff—is exemplified by the triplet calculations on twisted DMABN (see supporting information). For the low-overlap 3A 1

excitation, B3LYP underestimates the excitation energy by more than one electronvolt; this

error is reduced to less than 0.4 eV for CAM-B3LYP. However, for the 3A2 low-stability excitation, the accuracy degrades from B3LYP to CAM-B3LYP, with the latter giving an error of 0.47 eV. Upon application of the TDA to CAM-B3LYP, the 3A1 excitation energy is essentially unaffected whereas the 3A2 error reduces to just 0.05 eV. Similar observations are made in the twisted PP molecule and in the corresponding singlet states of the two molecules. Figure 4 plots the mean absolute errors over all the excitations, summarising the improvement in conventional TDDFT from PBE to B3LYP to CAM-B3LYP (due primarily to low-overlap CT and Rydberg excitations) and the subsequent improvement with TDA (due primarily to lowstability local excitations), for singlet and triplet excitations. Whilst this study has focused on singlets and triplets individually, we note that TDA CAM-B3LYP also yields the most accurate singlet–triplet energy differences, with a mean absolute error of 0.17 eV across all the systems. Once again, the latter assessment predominantly samples the large L excitations, since those with small L necessarily exhibit small singlet–triplet differences.

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Conclusions We have investigated low orbital overlap and triplet instability problems in TDDFT for a new benchmark set, encompassing challenging singlet and triplet excitation energies of local, CT and Rydberg character. The low orbital overlap problem is largely overcome for both singlet and triplet states by the use of a Coulomb-attenuated functional. For all the categories of functional considered, however, errors associated with triplet instability problems plague high overlap excitations, as exemplified by the excited states of acenes and polyacetylene oligomers. Application of the Tamm–Dancoff approximation reduces these errors for both singlet and triplet states, whilst leaving low-overlap excitations unaffected. The study illustrates the synergy between overlap and stability and highlights the success of a combined, Coulomb-attenuated Tamm–Dancoff approach.

Acknowledgement The authors thank the EPSRC for financial support and Trygve Helgaker for helpful discussions regarding basis set extrapolation.

Supporting Information Available Details of the molecules and excitations we consider, including tables of singlet and triplet excitation energies computed both conventionally and in the TDA for all functionals, together with L and wSTAB values; mean errors, mean absolute errors and standard deviations relative to reference values; details of the EOM-CCSD results used in the derivation of the wREF values; CAM-B3LYP density difference plots for selected molecules highlighting the character of the excitations. This material is available free of charge via the Internet at http://pubs.acs.org/.

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!

!ref / eV

singlets

triplets

(a) PBE

1

1

0

0

1

1

2

2

3

3

4

4

!

!ref / eV

(b) B3LYP 1

1

0

0

1

1

2

2

3

3

4

4

!ref / eV

(c) CAM-B3LYP

!

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

1

0

0

1

1

2

2

3

3

4

4 0.0

0.2

0.4

⇤

0.6

0.8

1.0

0.0

0.2

!tddft : Local Rydberg CT !tda : Local Rydberg CT

0.4

⇤

0.6

0.8

Figure 1: Errors in conventional TDDFT (light symbols) and TDA TDDFT (dark symbols) excitation energies, relative to reference values, as a function of the TDDFT L values. 11


1.0


(a) PBE 12

!stab / eV

10 8 6 4 2 0

(b) B3LYP 12

!stab / eV

10 8 6 4 2 0

(c) CAM-B3LYP 12 10 !stab / eV

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8 6 4 2 0 0.0

0.2

0.4

⇤

0.6

0.8

1.0

Local Rydberg CT

Figure 2: Triplet stabilities wSTAB plotted as a function of triplet L values.

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triplets

(a) PBE

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

!tda

!tddft / eV

singlets

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

!tda

!tddft / eV

(b) B3LYP 1.0

!tddft / eV

(c) CAM-B3LYP 1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

!tda

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

2

4

6 8 !stab / eV

10

12

0

2

Local Rydberg CT

4

6 8 !stab / eV

10

12

Figure 3: Differences between TDA and conventional TDDFT excitation energies, as a function of triplet stabilities wSTAB . 13



singlets

triplets

1.2

1.2

0.9

0.9

MAE / eV

MAE / eV

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.6 0.3 0

PBE

0.6 0.3 0

B3LYP CAM

PBE

B3LYP CAM

Figure 4: Mean absolute errors in conventional TDDFT (pink) and TDA TDDFT (blue) excitation energies, for the full assessment set. CAM denotes CAM-B3LYP.

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TDA CAM-B3LYP

OVERLAP

CAM-B3LYP STABILITY B3LYP

Keywords: density functional theory, singlet excited states, triplet excited states, benchmark excitation energies, TDDFT diagnostics, Tamm–Dancoff approximation

19


Overcoming Low Orbital Overlap and Triplet Instability Problems in

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