Article pubs.acs.org/JPCA
Electronic Circular Dichroism of Highly Conjugated π‑Systems: Breakdown of the Tamm−Dancoff/Configuration Interaction Singles Approximation Christoph Bannwarth and Stefan Grimme* Mulliken Center for Theoretical Chemistry, Institut für Physikalische und Theoretische Chemie der Universität Bonn, Beringstr. 4, D-53115 Bonn, Germany. S Supporting Information *
ABSTRACT: We show that the electronic circular dichroism (ECD) of delocalized π-systems represents a worst-case scenario for Tamm−Dancoff approximated (TDA) linear response methods. We mainly consider density functional theory (TDA-DFT) variants together with range-separated hybrids, but the conclusions also apply for other functionals as well as the configuration interaction singles (CIS) approaches. We study the effect of the TDA for the computation of ECD spectra in some prototypical extended π-systems. The C76 fullerene, a chiral carbon nanotube fragment, and [11]helicene serve as model systems for inherently chiral, π-chromophores. Solving the full linear response problem is inevitable in order to obtain accurate ECD spectra for these systems. For the C76 fullerene and the nanotube fragment, TDA and CIS approximated methods yield spectra in the origin-independent velocity gauge formalism of incorrect sign which would lead to the assignment of the opposite (wrong) absolute configuration. As a counterexample, we study the ECD of an α-helix polypeptide chain. Here, the lowest-energy transitions are dominated by localized excitations within the individual peptide units, and TDA methods perform satisfactorily. The results may have far-reaching implications for simple semiempirical methods which often employ TDA and CIS for huge molecules. Our recently presented simplified time-dependent DFT approach proves to be an excellent low-cost linear response method which together with range-separated density functionals like ωB97X-D3 produces ECD spectra in very good agreement with experiment. problems that can occur in TD-DFT,13,14 but it is well-known that the sum rules for the oscillator and rotatory strengths are no longer fulfilled by TDA-DFT.15,16 Oscillator strengths are unsigned quantities; therefore, only their magnitude changes when going from TD-DFT to TDA-DFT. The resulting discrepancy between TD-DFT and TDA-DFT is often beyond the accuracy necessary for comparison of experimental and theoretical UV spectra. For rotatory strengths, which are the signed quantities required for electronic circular dichroism (ECD) spectra, the consequence of the Tamm−Dancoff approximation (TDA) may be more severe. Thus, TDA-DFT is generally not used to compute ECD spectra, as long as a TDDFT calculation is still feasible. For an overview on current state-of-the-art computation of ECD spectra, see refs 15 and 17−19. To our knowledge, it has not been studied where TDA-DFT might still be applicable or how bad its performance actually is for so-called “worst cases”. Similar considerations also hold for the configuration interaction singles (CIS) approach which is
1. INTRODUCTION Kohn−Sham density functional theory (KS-DFT) is now the most widely used method for electronic ground-state calculations of molecules and solids. Because of the moderate computational cost and reasonable accuracy, time-dependent density functional theory (TD-DFT)1−4 based on a KS-DFT reference has become the leading method used to calculate excited-state properties and electronic spectra (see refs 5−9 for reviews; for TD-DFT calculations of ground-state properties like dispersion coefficients, see refs 10 and 11). It is routinely applicable to large systems (about a few hundred atoms) in which correlated wave-function-based methods of similar accuracy are not feasible. In very large systems with about 2000 atoms, the calculation of a few excited states by TD-DFT applying special software on graphics processing units has also been reported.12 However, the theoretical treatment of an entire electronic spectrum in a typical excitation energy range (e.g., up to 7 eV) for systems with several hundred atoms is still too demanding, in particular if hybrid TD-DFT is used. Applying the Tamm− Dancoff approximation to TD-DFT (TDA-DFT) generally reduces the computational cost but sacrifices the gaugeinvariance of TD-DFT. TDA-DFT is more robust to instability © 2015 American Chemical Society
Received: February 18, 2015 Revised: March 23, 2015 Published: March 23, 2015 3653
DOI: 10.1021/acs.jpca.5b01680 J. Phys. Chem. A 2015, 119, 3653−3662
Article
The Journal of Physical Chemistry A
spectra from TD-DFT while the earlier reported simplified TDA-DFT variant (sTDA)44 behaves like its nonsimplified counterpart TDA-DFT.43 Because of its good and representative performance and the possibility of combining it with common hybrid density functionals, this method will be used in the present work. For another approach to facilitate the computation of entire absorption spectra, see ref 45. 1.2. Systems under Consideration. In the present work, we study the effect of the Tamm−Dancoff approximation on the computed ECD spectra of three inherently chiral and structurally rigid chromophores. The first is the D2 symmetric C76 fullerene; the second is a cutout from a the chiral (11,7) carbon nanotube. The smallest chromophore studied is the [11]helicene. All systems are highly conjugated π-systems with delocalized orbitals. We first noticed a discrepancy between ECD spectra from sTDA and sTD-DFT for a substituted, chiral C60 derivative in our original paper on the sTD-DFT method.43 Therein, the spectra (obtained from the velocity form, vide infra) computed by both approaches differed in sign over a wide energy range. Therefore, a further study of this phenomenon seems appropriate. In the present study, C76 was chosen because, due to its rigidity, we can ignore conformational effects on the ECD spectrum. In that respect, substituted chiral C 60 derivatives (see e.g., ref 46.) are less suited. Furthermore, we may exploit its symmetry allowing us to additionally perform standard TD-DFT/TDA-DFT and TD-HF/CIS calculations on this system. According to the isolated pentagon rule,47 there is only one chiral isomer of the C76 fullerene allowing direct comparison to the experimental ECD spectrum based on a single structure. This is a big advantage compared to the previously studied C84 fullerene for which the occurrence of other isomers might have caused differences in the calculated and experimental ECD intensities.48 The second model system is a cutout from a chiral (11,7) carbon nanotube which has a diameter of about 13 Å.49 It contains 180 carbon atoms, and the edges are capped by hydrogens (18 on each edge); we will denote it as (11,7)hCNTh. This system is different compared to fullerenes because the π-system is noncontinuous in one direction. The second difference is the connectivity within the π-system. Fullerenes contain hexagonic as well as pentagonic structural motifs while carbon nanotubes contain only hexagons. [11]Helicene is the third inherently chiral chromophore to be studied here. An experimental ECD spectrum was presented a few years ago.50 In our previous work, we have shown that most of the ECD bands are reproduced correctly by either sTDA with a range-separated hybrid (RSH)51 or sTD-DFT with a global hybrid (BHLYP).43 Even though the negative bands in the range of 280-340 nm were qualitatively reproduced, the ECD at 250−280 nm was improperly described. In the present study, we investigate this system by combining a gauge-invariant method (i.e., sTD-DFT) with an RSH functional. Another conceptually different case to be studied is a polypeptide. Our model system is Ac(Ala)19Me, i.e., an Nterminus acylated, C-terminus methyl-aminated poly-Ala chain (in a right-handed α-helix conformation) containing in total 20 peptide chromophores. Here, the chromophores are not conjugated and instead separated by the α-carbons. The lowlying excitations may therefore be constructed from coupled, localized states.52 This system with rather high-lying excitations
equivalent to Tamm−Dancoff approximated time-dependent Hartree−Fock (TD-HF).7 It is often used in combination with semiempirical variants of Hartree−Fock (HF). Typically applied semiempirical approaches employ the zero differential overlap (ZDO) approximation20,21 and have been particularly parametrized for the calculation of electronic spectra such as CNDO/S,22−24 INDO/S,25,26 or MSINDO-sCIS.27−29 For a very recent variant (dubbed INDO/X) particularly designed for electronic excited states, see ref 30. The CIS formalism has been used in previous studies to calculate the ECD spectra of various systems either in combination with an independent chromophore approach (then often termed “matrix method”)31−35 or in combination with the above-mentioned semiempirical methods.36−38 However, the limited applicability of these approaches to particular systems due to the use of the TDA/CIS approximation is rarely discussed. It is shown here that TDA/CIS methods can lead to computed ECD spectra of reasonable shape but with an almost mirror-image relation to the correct spectra, which would lead to incorrect absolute configuration assignments. 1.1. Suitable Methods for Computing Full Range ECD Spectra of Large Systems. There are only a few gaugeinvariant approaches that are capable of computing a whole spectrum up to e.g., 7 eV, in particular for large molecules with several hundred or even more than a thousand atoms. The fastest variant is the time-dependent density functional tight binding (TD-DFTB) approach of Elstner and co-workers.39 In terms of the computational cost, it is comparable to the semiempirical ZDO approaches because of the use of a minimal basis set as well as integral approximations. In contrast to the HF-based semiempirical CIS approaches, the linear-response time-dependent (LR-TD) eigenvalue problem is solved instead of the Tamm−Dancoff approximated one, retaining the gaugeinvariance. The benefit from using a DFT-derived (i.e., from density functional tight-binding) reference is the diagonal nature of the (A − B) matrix (see below), considerably reducing the computational cost. Some significant drawbacks are, however, the small basis set (as in most semiempirical methods) and particularly the lack of exact Fock-exchange. The semilocal exchange results in a wrong description of chargetransfer (CT) and Rydberg states. For a discussion of the same problem in semilocal TD-DFT, see refs 40 and 41. Attempts have been made to remove these spurious “ghost states” by removing configuration state functions (CSFs) with small electric transition dipole moments.42 Such an approach may lead to removal of electrically forbidden, yet physically important CSFs as well (e.g., nπ* transitions). It should also be noted that such a criterion is not applicable for ECD as the rotatory strength depends on both the electric and magnetic transition dipole moment. As for most semiempirical methods, the parametrization of (TD-)DFTB is nontrivial and is thus restricted to a certain small set of elements. Another method suitable for computing entire ECD spectra of large systems is the simplified TD-DFT (sTD-DFT) approach that has recently been proposed by our group.43 Similar to regular TD-DFT, it uses a (hybrid) KS-DFT reference but is orders of magnitude faster than TD-DFT in the post-SCF procedure because of a drastic simplification of the two-electron integrals as well as a massive reduction of the single excitation configuration space (vide infra). The computational bottleneck is the preceding KS-DFT procedure to obtain the ground-state orbitals. It could be shown that this method produces electronic excitation spectra of similar quality as 3654
DOI: 10.1021/acs.jpca.5b01680 J. Phys. Chem. A 2015, 119, 3653−3662
Article
The Journal of Physical Chemistry A (A − B)1/2 (A + B)(A − B)1/2 Z = ω 2 Z
(200 nm, 200 nm,
Electronic Circular Dichroism of Highly Conjugated π‑Systems: Breakdown of the Tamm−Dancoff/Configuration Interaction Singles Approximation Christoph Bannwarth and Stefan Grimme* Mulliken Center for Theoretical Chemistry, Institut für Physikalische und Theoretische Chemie der Universität Bonn, Beringstr. 4, D-53115 Bonn, Germany. S Supporting Information *
ABSTRACT: We show that the electronic circular dichroism (ECD) of delocalized π-systems represents a worst-case scenario for Tamm−Dancoff approximated (TDA) linear response methods. We mainly consider density functional theory (TDA-DFT) variants together with range-separated hybrids, but the conclusions also apply for other functionals as well as the configuration interaction singles (CIS) approaches. We study the effect of the TDA for the computation of ECD spectra in some prototypical extended π-systems. The C76 fullerene, a chiral carbon nanotube fragment, and [11]helicene serve as model systems for inherently chiral, π-chromophores. Solving the full linear response problem is inevitable in order to obtain accurate ECD spectra for these systems. For the C76 fullerene and the nanotube fragment, TDA and CIS approximated methods yield spectra in the origin-independent velocity gauge formalism of incorrect sign which would lead to the assignment of the opposite (wrong) absolute configuration. As a counterexample, we study the ECD of an α-helix polypeptide chain. Here, the lowest-energy transitions are dominated by localized excitations within the individual peptide units, and TDA methods perform satisfactorily. The results may have far-reaching implications for simple semiempirical methods which often employ TDA and CIS for huge molecules. Our recently presented simplified time-dependent DFT approach proves to be an excellent low-cost linear response method which together with range-separated density functionals like ωB97X-D3 produces ECD spectra in very good agreement with experiment. problems that can occur in TD-DFT,13,14 but it is well-known that the sum rules for the oscillator and rotatory strengths are no longer fulfilled by TDA-DFT.15,16 Oscillator strengths are unsigned quantities; therefore, only their magnitude changes when going from TD-DFT to TDA-DFT. The resulting discrepancy between TD-DFT and TDA-DFT is often beyond the accuracy necessary for comparison of experimental and theoretical UV spectra. For rotatory strengths, which are the signed quantities required for electronic circular dichroism (ECD) spectra, the consequence of the Tamm−Dancoff approximation (TDA) may be more severe. Thus, TDA-DFT is generally not used to compute ECD spectra, as long as a TDDFT calculation is still feasible. For an overview on current state-of-the-art computation of ECD spectra, see refs 15 and 17−19. To our knowledge, it has not been studied where TDA-DFT might still be applicable or how bad its performance actually is for so-called “worst cases”. Similar considerations also hold for the configuration interaction singles (CIS) approach which is
1. INTRODUCTION Kohn−Sham density functional theory (KS-DFT) is now the most widely used method for electronic ground-state calculations of molecules and solids. Because of the moderate computational cost and reasonable accuracy, time-dependent density functional theory (TD-DFT)1−4 based on a KS-DFT reference has become the leading method used to calculate excited-state properties and electronic spectra (see refs 5−9 for reviews; for TD-DFT calculations of ground-state properties like dispersion coefficients, see refs 10 and 11). It is routinely applicable to large systems (about a few hundred atoms) in which correlated wave-function-based methods of similar accuracy are not feasible. In very large systems with about 2000 atoms, the calculation of a few excited states by TD-DFT applying special software on graphics processing units has also been reported.12 However, the theoretical treatment of an entire electronic spectrum in a typical excitation energy range (e.g., up to 7 eV) for systems with several hundred atoms is still too demanding, in particular if hybrid TD-DFT is used. Applying the Tamm− Dancoff approximation to TD-DFT (TDA-DFT) generally reduces the computational cost but sacrifices the gaugeinvariance of TD-DFT. TDA-DFT is more robust to instability © 2015 American Chemical Society
Received: February 18, 2015 Revised: March 23, 2015 Published: March 23, 2015 3653
DOI: 10.1021/acs.jpca.5b01680 J. Phys. Chem. A 2015, 119, 3653−3662
Article
The Journal of Physical Chemistry A
spectra from TD-DFT while the earlier reported simplified TDA-DFT variant (sTDA)44 behaves like its nonsimplified counterpart TDA-DFT.43 Because of its good and representative performance and the possibility of combining it with common hybrid density functionals, this method will be used in the present work. For another approach to facilitate the computation of entire absorption spectra, see ref 45. 1.2. Systems under Consideration. In the present work, we study the effect of the Tamm−Dancoff approximation on the computed ECD spectra of three inherently chiral and structurally rigid chromophores. The first is the D2 symmetric C76 fullerene; the second is a cutout from a the chiral (11,7) carbon nanotube. The smallest chromophore studied is the [11]helicene. All systems are highly conjugated π-systems with delocalized orbitals. We first noticed a discrepancy between ECD spectra from sTDA and sTD-DFT for a substituted, chiral C60 derivative in our original paper on the sTD-DFT method.43 Therein, the spectra (obtained from the velocity form, vide infra) computed by both approaches differed in sign over a wide energy range. Therefore, a further study of this phenomenon seems appropriate. In the present study, C76 was chosen because, due to its rigidity, we can ignore conformational effects on the ECD spectrum. In that respect, substituted chiral C 60 derivatives (see e.g., ref 46.) are less suited. Furthermore, we may exploit its symmetry allowing us to additionally perform standard TD-DFT/TDA-DFT and TD-HF/CIS calculations on this system. According to the isolated pentagon rule,47 there is only one chiral isomer of the C76 fullerene allowing direct comparison to the experimental ECD spectrum based on a single structure. This is a big advantage compared to the previously studied C84 fullerene for which the occurrence of other isomers might have caused differences in the calculated and experimental ECD intensities.48 The second model system is a cutout from a chiral (11,7) carbon nanotube which has a diameter of about 13 Å.49 It contains 180 carbon atoms, and the edges are capped by hydrogens (18 on each edge); we will denote it as (11,7)hCNTh. This system is different compared to fullerenes because the π-system is noncontinuous in one direction. The second difference is the connectivity within the π-system. Fullerenes contain hexagonic as well as pentagonic structural motifs while carbon nanotubes contain only hexagons. [11]Helicene is the third inherently chiral chromophore to be studied here. An experimental ECD spectrum was presented a few years ago.50 In our previous work, we have shown that most of the ECD bands are reproduced correctly by either sTDA with a range-separated hybrid (RSH)51 or sTD-DFT with a global hybrid (BHLYP).43 Even though the negative bands in the range of 280-340 nm were qualitatively reproduced, the ECD at 250−280 nm was improperly described. In the present study, we investigate this system by combining a gauge-invariant method (i.e., sTD-DFT) with an RSH functional. Another conceptually different case to be studied is a polypeptide. Our model system is Ac(Ala)19Me, i.e., an Nterminus acylated, C-terminus methyl-aminated poly-Ala chain (in a right-handed α-helix conformation) containing in total 20 peptide chromophores. Here, the chromophores are not conjugated and instead separated by the α-carbons. The lowlying excitations may therefore be constructed from coupled, localized states.52 This system with rather high-lying excitations
equivalent to Tamm−Dancoff approximated time-dependent Hartree−Fock (TD-HF).7 It is often used in combination with semiempirical variants of Hartree−Fock (HF). Typically applied semiempirical approaches employ the zero differential overlap (ZDO) approximation20,21 and have been particularly parametrized for the calculation of electronic spectra such as CNDO/S,22−24 INDO/S,25,26 or MSINDO-sCIS.27−29 For a very recent variant (dubbed INDO/X) particularly designed for electronic excited states, see ref 30. The CIS formalism has been used in previous studies to calculate the ECD spectra of various systems either in combination with an independent chromophore approach (then often termed “matrix method”)31−35 or in combination with the above-mentioned semiempirical methods.36−38 However, the limited applicability of these approaches to particular systems due to the use of the TDA/CIS approximation is rarely discussed. It is shown here that TDA/CIS methods can lead to computed ECD spectra of reasonable shape but with an almost mirror-image relation to the correct spectra, which would lead to incorrect absolute configuration assignments. 1.1. Suitable Methods for Computing Full Range ECD Spectra of Large Systems. There are only a few gaugeinvariant approaches that are capable of computing a whole spectrum up to e.g., 7 eV, in particular for large molecules with several hundred or even more than a thousand atoms. The fastest variant is the time-dependent density functional tight binding (TD-DFTB) approach of Elstner and co-workers.39 In terms of the computational cost, it is comparable to the semiempirical ZDO approaches because of the use of a minimal basis set as well as integral approximations. In contrast to the HF-based semiempirical CIS approaches, the linear-response time-dependent (LR-TD) eigenvalue problem is solved instead of the Tamm−Dancoff approximated one, retaining the gaugeinvariance. The benefit from using a DFT-derived (i.e., from density functional tight-binding) reference is the diagonal nature of the (A − B) matrix (see below), considerably reducing the computational cost. Some significant drawbacks are, however, the small basis set (as in most semiempirical methods) and particularly the lack of exact Fock-exchange. The semilocal exchange results in a wrong description of chargetransfer (CT) and Rydberg states. For a discussion of the same problem in semilocal TD-DFT, see refs 40 and 41. Attempts have been made to remove these spurious “ghost states” by removing configuration state functions (CSFs) with small electric transition dipole moments.42 Such an approach may lead to removal of electrically forbidden, yet physically important CSFs as well (e.g., nπ* transitions). It should also be noted that such a criterion is not applicable for ECD as the rotatory strength depends on both the electric and magnetic transition dipole moment. As for most semiempirical methods, the parametrization of (TD-)DFTB is nontrivial and is thus restricted to a certain small set of elements. Another method suitable for computing entire ECD spectra of large systems is the simplified TD-DFT (sTD-DFT) approach that has recently been proposed by our group.43 Similar to regular TD-DFT, it uses a (hybrid) KS-DFT reference but is orders of magnitude faster than TD-DFT in the post-SCF procedure because of a drastic simplification of the two-electron integrals as well as a massive reduction of the single excitation configuration space (vide infra). The computational bottleneck is the preceding KS-DFT procedure to obtain the ground-state orbitals. It could be shown that this method produces electronic excitation spectra of similar quality as 3654
DOI: 10.1021/acs.jpca.5b01680 J. Phys. Chem. A 2015, 119, 3653−3662
Article
The Journal of Physical Chemistry A (A − B)1/2 (A + B)(A − B)1/2 Z = ω 2 Z
(200 nm, 200 nm,
