Daniel H.
Friese
*a,
Christof
Hättig
b and
Antonio
Rizzo
c
aCentre for Theoretical and Computational Chemistry, University of Tromsø, Tromsø, Norway. E-mail: daniel.h.friese@uit.no
bLehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, Bochum, Germany
cConsiglio Nazionale delle Ricerche – CNR, Istituto per i Processi Chimico-Fisici, IPCF-CNR, Sede Secondaria di Pisa, Area della Ricerca, Via G. Moruzzi 1, I-56124 Pisa, Italy
First published on 3rd May 2016
We present the first origin-independent approach for the treatment of two-photon circular dichroism (TPCD) using coupled cluster methods. The approach is assessed concerning its behavior on the choice of the basis set and different coupled cluster methods. We also provide a comparison of results from CC2 with those from density functional theory using the CAM-B3LYP functional. Concerning the basis set we note that in most cases an augmented triple zeta basis or a doubly augmented double zeta basis is needed for reasonably converged results. In the comparison of different coupled cluster methods results from CCSD, CC3 and CC2 have been found to be quite similar in most cases, while CCS results differ remarkably from the results at the higher levels. However, this proof-of-principle study also shows that further benchmarking of DFT and CC2 against accurate coupled cluster reference values (e.g. CCSD or CC3) is needed.
The first theoretical description of TPCD dates back to the mid-seventies in work carried out almost simultaneously by Tinoco, by Power and by Andrews.1–3 First measurements have been reported in 1995 by Gunde and Richardson for chiral gadolinium complexes.9 The interest in the phenomenon was revived a decade ago, by the development of a computational protocol which allowed one to make reliable predictions of the intensity of two-photon circular dichroism,10–13 followed by the design of an experimental setup,12,14 leading, thirteen years after the first qualitative observation, to the flourishing of a new spectroscopic discipline, where the analysis of experimental TPCD spectra is carried out nowadays with the support of computed results.11
Two-photon spectroscopy methods are becoming increasingly popular, both from the experimental and computational aspects, because they have proven to be complementary to their one-photon analogues. Compared with the latter, the former allows exploring spectral regions using photons with increasing penetration depth, higher 3D confocality and reduced photobleaching effects. One-photon absorption often takes place in the far- and near-UV regions of the electromagnetic spectrum and overlaps in many cases with the response of standard aqueous buffer solutions or other common solvents. In these cases the lower laser frequencies used in two-photon spectrometers help getting access to information otherwise hidden. Another advantage of two-photon spectroscopy methods arises from the different transition rules. In particular for samples with an inversion point, transitions which are symmetry forbidden at the one-photon level are allowed in the corresponding two-photon spectroscopy. Although this aspect is less relevant for dichroic response, due to the reduced symmetry of chiral samples or environments, it is a fact that even for low-symmetry systems quite often states that are weak one-photon absorbers are active when probed by two photons.15
Different selection rules can also be observed when comparing absorptive and dichroic (differential absorptive) properties as they are related to different operators (magnetic dipole moment and electric quarupole moment) with different symmetry behaviour, therefore dichroic properties allow the detection of states which might be difficult to detect by absorptive properties.
In this respect it is worth noting that, when dealing with isotropic samples, TPCD is the lowest-order chiroptical property where electric quadrupole transitions play a role.11 Lower-order absorption properties, such as one-photon absorption and electronic circular dichroism (ECD), and two-photon absorption can be rationalized to the lowest order in perturbation theory by resorting to either only electric dipole (OPA, TPA) or magnetic and electric dipole transition moments (ECD). The contribution of electric quadrupole transitions, formally arising at the same order of perturbation theory as magnetic dipole transitions, averages out for these spectroscopy methods, whereas it is non-vanishing and often non-negligible in TPCD. In addition, TPCD is very sensitive to conformational changes in chiral molecules,11,16,17 and it can be used to gain insight into the conformational structure of a molecule.
In general the computational description of TPCD requires the calculation of two-photon transition strengths which correspond to the single residues of the cubic response function.18 The transition strengths can be decomposed in left and right transition moment tensors which can also be obtained from single residues of the quadratic response function.18,19 Implementations of two-photon absorption strengths are available today for SCF-based methods20,21 as well as at different levels of coupled cluster response theory which enables a description with a hierarchy of methods with systematically increasing accuracy with an approximate treatment of doubles,22 full doubles,23 approximate triples,24etc. These implementations can be used for TPCD calculations as long as the required perturbation operators (electric dipole moment, magnetic dipole moment and electric quadrupole moment, vide infra) are available in the corresponding code.
Since the computational protocol for TPCD has been outlined,10,25 quantum chemical calculations have consistently exploited a time-dependent density functional theory (TD-DFT) structure model, mainly due to the size of the systems investigated also by experimentalists. In this study we develop and test a novel approach where we transfer the computational protocol for TPCD to coupled cluster response theory. In contrast to TD-DFT calculations, which normally scale at most with the fourth power of the system size, coupled cluster methods have a steeper scaling behavior depending on the excitation level, but offer the possibility of systematic improvement in the hierarchy of coupled cluster methods.26 As for two-photon absorption,22,27 coupled cluster can therefore be considered to be a reference method which additionally does not have the known deficiencies of TD-DFT e.g. in the description of Rydberg and charge transfer states.28 It can therefore be used to validate TD-DFT and provides an alternative when TD-DFT is not applicable because it does not give correct excitation energies.
Since the rotatory strengths in TPCD spectra involve mixed electric dipole–magnetic dipole and electric dipole–electric quadrupole transition tensors, the problem of (magnetic and electric quadrupole gauge) origin dependence of the results arises, as it is also the case for electronic circular dichroic and optical rotation spectra. In methods that describe the response of the electronic wavefunction to the electro-magnetic field in an orbital-relaxed framework, this problem can be solved by using the so-called gauge-including atomic orbitals (GIAOs).29–34
This is, however, not easily possible for the standard coupled cluster response methods where unrelaxed orbitals are used for frequency-dependent properties and transition moments.35–37 Therefore other approaches are usually used to get origin-independent results for frequency-dependent magnetic properties with standard coupled cluster methods. For optical rotation and one-photon circular dichroism techniques have been established which express the electric dipole operator in velocity gauge and thereby achieve origin invariance.36–39 A similar velocity-gauge based technique, which is actually based on the original derivation of TPCD from ref. 1, has been introduced by one of us to obtain origin-independent results for TPCD using SCF-based methods (Hartree–Fock and TD-DFT) and has become standard for the computational treatment of TPCD.25 Very recently, a GIAO-based treatment of TPCD for SCF-based theory has also been realized by one of us.40 In the following we will adapt the velocity-gauge based approach to the requirements of coupled cluster theory.
The remainder of this article is organized as follows: in Section 2 we describe the basic theory of TPCD and its origin invariant description and show how it can be applied in connection with coupled cluster response methods. Section 3 is dedicated to technical information about the methods and molecules which were used. In Section 4 we present a general assessement of the approach where we compare results from different basis sets, coupled cluster models and density functional theory before we discuss our results in Section 5.
As all properties that involve the magnetic dipole and electric quadrupole operators TPCD also suffers in the length gauge formulation from the problem of origin dependence which occurs as long as the calculations are not carried out in a complete one-electron basis set.25 In CC theory also an untruncated cluster operator would be needed.43,44 However the approach presented by Rizzo and coworkers circumvents the problem of origin-dependence by treating the electric dipole operator completely in the velocity gauge and the electric quadrupole operator in a mixed length-velocity gauge. In the following we will very briefly recapitulate this approach before we present its generalization for coupled cluster response theory.
(1) |
The quantity fRTP is called TPCD rotatory strength. It is defined as
fRTP = −b11 − b22 − b33, | (2) |
(3) |
(4) |
(5) |
(6) |
(7) |
The tensor +,0fab is a special form of the electric-dipole–electric-quadrupole transition moment tensor which is defined as
(8) |
(9) |
Δma = −½εacdRcμpd, | (10) |
ΔT+ab = −μpaRb − μpbRa, | (11) |
(12) |
(13) |
(14) |
(15) |
(16) |
(17) |
Fig. 1 Small molecules in the test set for method and basis set behavior. The molecules were taken from a test set by Srebro et al.47 |
For the study of the basis set behavior and for a first comparison with DFT results we have used the CC2 method48 and some additional molecules that are larger and also feature larger π-systems so that they allow for the investigation of π→π*- and n→π*-excitations while the spectra of the smaller molecules are dominated by Rydberg states. They are, however, at the limit of what can be treated with the available CCSD implementation for two-photon transition strengths. The additional molecules for the basis set study are shown in Fig. 2.
Fig. 2 Molecules in the test set for the basis set behavior. All molecules are used in chiral conformations. No. 6 was studied in ref. 49. |
6 has been studied thoroughly by one of us using different conformers, basis sets and density functionals.49 In this paper a structure which was optimized using the B3LYP functional50–54 and the TZVP basis set.55 Structures 7, 8 and 9 in Fig. 2 are achiral, however all molecules are treated here in chiral, frozen conformations. For 7 and 8 these structures have also been optimized using the B3LYP density functional and the TZVP basis set, while for the optimization of 9 the MP2 model and the cc-pVTZ basis set have been used. The reason for this is that for 9 the presumably correct D2 point group symmetry is not correctly reproduced by many DFT functionals. Optimization in the D2 point group symmetry is achieved by MP2 in sufficiently large basis sets, as reported in several studies.56–58
Using both programs, the tensors p,0f, p,f0, p,0f, p,f0, +,0f and +,f0 have been calculated and combined according to eqn (15)–(17) to the intermediates CC1, CC2 and CC3 and the rotatory strength fRTP (eqn (2)).
The TD-DFT calculations have been carried out using the CAM-B3LYP density functional69 and the aug-cc-pVDZ basis set using the DALTON program.63,64
The factors forming the rotatory strength are set to b1 = 6.0, b2 = 2.0 and b3 = −2.0 following ref. 25. This choice corresponds to two circularly polarized photons which propagate parallel to each other.
To get the TPCD in Göppert-Mayer (GM) units, which are also used to report two-photon absorption cross-sections, we rewrite eqn (1) slightly following the lines of ref. 70 setting in the conversion factors for the speed of light in vacuo, the Bohr radius and time from atomic units to the cgs unit system and get
δTPCD = 4.87555 × 10−5ω2g(2ω)fRTP. | (18) |
The simulated TPCD spectra contain information on both the excitation energy and the TPCD strength for different singly and doubly augmented basis sets. The first important finding from this part of the study is that double augmentation is less important for the TPCD strength than for the excitation energy, where it is crucial if Rydberg states play a role. However, as this study focusses on the TPCD strength effects, the excitation energy will not be further discussed here. Therefore, the large and computationally demanding doubly augmented basis sets do not seem to be recommendable for an efficient description of TPCD.
The second important finding is that depending on the molecule, TPCD is about converged with the aug-cc-pVDZ (4–7) or aug-cc-pVTZ basis set (1, 2, 3, 8, and 9). In some cases (e.g.1, 2, 5 and 8) the basis set dependence for different states differs a lot so that we can conclude that basis set convergence of TPCD depends non-trivially on both the molecule and the character of the excited state. The results show that the basis set convergence can be expected already for aug-cc-pVDZ if the valence spectrum of larger molecules is studied (6, 7 or 8), while for smaller molecules, where Rydberg states are often important, aug-cc-pVTZ or even a larger basis set will be needed. The more difficult basis set behaviour of 9 might be strongly influenced by the π-stacking effects of the two aromatic rings, complicating the description of the electronic structure. These findings are consistent with the well-established finding that the basis set convergence is often faster for larger molecules, in particular if they have a three-dimensional structure.
1 | 2 | ||||||
---|---|---|---|---|---|---|---|
CCS | CC2 | CCSD | CC3 | CCS | CC2 | CCSD | CC3 |
140 (8.85) | 193 (6.41) | 173 (7.17) | 172 (7.20) | 228 (5.44) | 246 (5.03) | 249 (4.98) | 249 (4.94) |
137 (9.07) | 181 (6.86) | 168 (7.47) | 165 (7.51) | 189 (6.55) | 219 (5.66) | 213 (5.83) | 213 (5.79) |
133 (9.30) | 179 (6.94) | 164 (7.63) | 161 (7.67) | 182 (6.82) | 211 (5.89) | 209 (5.94) | 209 (5.92) |
133 (9.34) | 175 (7.07) | 159 (7.84) | 158 (7.86) | 170 (7.30) | 199 (6.23) | 194 (6.39) | 194 (6.36) |
132 (9.41) | 162 (7.67) | 155 (8.05) | 154 (8.05) | 164 (7.55) | 191 (6.48) | 187 (6.65) | 186 (6.61) |
3 | ||
---|---|---|
CCS | CC2 | CCSD |
176 (7.03) | 191 (6.50) | 188 (6.61) |
172 (7.21) | 188 (6.60) | 181 (6.83) |
161 (7.71) | 181 (6.84) | 174 (7.14) |
169 (7.33) | 183 (6.77) | 177 (7.01) |
155 (7.92) | 179 (6.92) | 171 (7.24) |
For the excitation energies we note that the results from CCS are strongly blue-shifted in all cases while CC2 shows a slight red shift compared to CCSD and CC3 which yield very similar excitation energies in all cases.
Simulated TPCD spectra are shown in Fig. 4. Regarding the excitation energies they show the expected behaviour for the different methods: CCS excitation energies are normally strongly blue-shifted compared to the others, while CC2 values are red-shifted for Rydberg states. For 1 we note that the spectra for CCSD and CC2 are in excellent agreement while CCS even fails to reproduce most of the TPCD signs. The plots for CCSD and CC3 are nearly on top of each other. The artificial spectra for CC2 and CCSD/CC3 would look even more similar if the CC2 values would be blue-shifted to the excitation energies from CCSD/CC3. For 2 the difference between CCSD and CC2 is a bit larger, however, all peaks can be identified in both spectra also with a good qualitative agreement in intensity. The agreement between CCSD and CC3 is much higher than the one between CCSD and CC2, however, we note that it is less pronounced than for 1. CCS is far off the higher-order methods. For 3 the difference between CC2 and CCSD is stronger than for 1 and 2, however, the dominating peaks can be identified in both spectra although they are a bit shifted. Anyway from the comparison of the peaks we can also see more similarities between CCS and CCSD compared to the other molecules. In general this small test series shows that CC2 and the higher order coupled cluster models are in quite good agreement when it comes to the description of TPCD spectra while CCS in general yields poorer results. This is also supported by the position of CC2 in the coupled cluster hierarchy where it is located between CCS and CCSD.26
In general time-dependent density functional theory is known for having deficiencies in the treatment of charge transfer and Rydberg states. Parts of these deficiencies are compensated by a flexible exact exchange contribution as it is realized in the so-called range-separated functionals such as the CAM-B3LYP functional which has been used for this study.
In Fig. 5 simulated spectra for the five lowest states of molecules 1–9 obtained using CC2 and CAM-B3LYP are shown. As in the previous parts of the study the spectra have been obtained by centering Gaussians with a unit width and a height corresponding to the TPCD strength at the excitation energy.
From these spectra we note that the agreement between DFT and CC2 is very different. A general tendency is that DFT/CAM-B3LYP excitation energies are to a certain extent blue-shifted. When evaluating the data plotted in Fig. 5 it has to be kept in mind that in both DFT and CC2 calculations only the five lowest excited states have been considered. This means that changes in the ordering of the states especially for S4 and S5 might occur.
For 1 we find a very good qualitative agreement between the spectra. The agreement for 2 is a bit poorer while for 3 we cannot really recognize a systematic agreement. For molecules 4, 5 and 6 we can at least identify the most intense peaks in both spectra. In particular for 7 and 8, this is at least the case for the two lowest states. In 9 it is possible to identify some peaks in both spectra, although this molecule, due to the strong influence of π-stacking effects on the structure, might be expected to be a problematic case in DFT treatment. Furthermore we have to recall that in 3 and 5 the chromophore is an isolated CC-double bond. Its TPCD spectrum is apparently described adaquately neither by CAM-B3LYP nor by CC2.
In general comparing the results from this part of the study we identify a moderate agreement between TD-DFT and CC2. Further benchmarking of TPCD values from TD-DFT against coupled cluster methods is needed. Note that some molecules of the test set we used here are relatively small, which give rise to low-lying Rydberg states. While these states are described well by coupled cluster, it is well known that their treatment is one of the strong deficiencies of TD-DFT with commonly used functionals.71
For this reason an upcoming benchmark study should focus especially on larger molecules where the problem of low-lying Rydberg states is less pronounced. However, such a benchmark requires the availability of efficient and parallelized CC3- and CCSD-implementations of two-photon transition tensors which is not yet the case. In case this is done, other density functionals should also be taken into account.
Regarding the basis set behavior we found that the aug-cc-pVDZ basis set from the correlation-consistent basis set family often does not yield satisfying results. For small molecules or when Rydberg states are important converged results require at least a basis set of aug-cc-pVTZ quality. For valence transitions in larger molecules the quality of the results obtained using the aug-cc-pVDZ basis set is, however, usually acceptable. In a comparison of different coupled cluster models we could find that there is a general excellent agreement between CCSD and CC3 and that in most cases also the agreement between CC2 and CCSD is quite good while CCS behaves poorly. For this reason we consider CC2 to be a promising candidate for a benchmark method for DFT calculations although further benchmarking of this is needed, especially for larger molecules. However, the availability of efficient and parallel CCSD and CC3 implementations of two-photon transition moments would be required for this.
In a comparison between CC2 and the CAM-B3LYP functional, however, we found a moderate agreement of the results. We find a strong deviation of excitation energies between CC2 and CAM-B3LYP, however, especially for larger TPCD values and molecules with an unproblematic electronic structure (no π-stacking as in paracyclophane), there is qualitative agreement between the artificial TPCD spectra for these two methods. Also, here a more thorough benchmarking is required which should not only involve higher-order coupled cluster models but also other density functionals.
It is important to note that the definitions and the labels of the operators differ between the different implementations. In the case of the electric dipole operator in velocity gauge multiplication with is already carried out prior to the printing of the results in the DFT code in DALTON, while in the coupled cluster codes in DALTON and TURBOMOLE this is not the case. Moreover the signs of the angular momentum operators differ between DALTON and TURBOMOLE. Table 2 gives an overview of the different operators, their labels and definitions. This information is, of course, subject to changes in the code and reflects the stage of the implementations when this work was written.
As can be seen from the table different implementations differ only by signs and prefactors. However this has important implications for the formation of the rotatory strengths. For the formation from the TD-DFT results in DALTON the following equation has to be used
(19) |
(20) |
(21) |
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