Electronic circular dichroism spectra using the algebraic diagrammatic construction schemes of the polarization propagator up to third order

Expressions for the calculation of rotatory strengths using the algebraic diagrammatic construction (ADC) scheme of the polarization propagator in both length and velocity gauges have been implemented. This enables the simulation of electronic circular dichroism (ECD) spectra at the ADC level up to third order of perturbation theory. The ADC(n) methods produce rotatory strengths of comparable accuracy to those obtained with coupled cluster methods of corresponding approximation levels, as evaluated for methyloxirane, methylthiirane, dimethyloxi-rane, dimethylthiirane, hydrogen peroxide, and dihydrogen disulfide. ECD spectra of (1 R )-camphor, (1 R )-norcamphor, and (1 R )-fenchone computed at the third order ADC(3) level of theory are shown to agree very favorably with experimental gas phase spectra, demonstrating the usefulness of ADC for the calculation of chiro-optical properties of organic molecules. ADC(2) in combination with the polarizable continuum model is shown to successfully reproduce the ECD spectrum


I. INTRODUCTION
Electronic circular dichroism (ECD) is a widely used linear optical effect to gain information about the stereochemistry of molecular systems. 1 Different enantiomers of a chiral molecule preferentially interact with one of the two circularly polarized components of plane polarized light. This results in a rotation of the plane of polarization, as observed in optical rotation measurements, and differential absorption, as detected by ECD spectroscopy. As a spectroscopic tool, ECD has found use in a wide range of scientific fields ranging from biochemistry, where it is commonly employed to detect the secondary structures of proteins, 2 to pharmaceutical synthesis, where the enantiomeric purity is of importance. 3 Especially important in ECD spectroscopy is the sign of the measured signal as enantiomers produce mirror-like ECD spectra. This allows for quick and straightforward quality control of synthesized chiral molecules once accurate theoretical predictions have been carried out. Furthermore, achiral systems interacting with chiral systems, e.g., a solvent arranging itself around a solute in a structurally chiral manner, can give rise to a CD signal, which is a process referred to as induced circular dichroism (ICD). Thus, ICD provides information not only on the absolute configuration of the chiral system but also on the orientation of its achiral (e.g., solvent) counterpart. 4 For the prediction of ECD spectra, many theoretical methods have been developed over the last decades with methods suited for different molecular sizes and environments. Progress has been made, and highly accurate ECD spectra can be simulated and compared directly with experiment. 1 For proteins and macro-molecules, semi-empirical methods have been shown to accurately reproduce ECD spectra of low-lying excitations. 5,6 Similarly, for molecules on the order of hundreds of atoms, time-dependent Hartree-Fock (TD-HF) [7][8][9] and time-dependent density functional theory (TD-DFT) [10][11][12][13][14] have proven successful. For ECD spectra of small chiral molecules, on the order of tens of atoms, configuration interaction (CI) 15 and linear response based on a complete active space self-consistent field (CAS-SCF) reference 16 can be applied. Furthermore, the well-known equation of motion coupled cluster (EOM-CC) 17,18 and the similar linear response coupled cluster (LR-CC) approach [19][20][21][22] have been applied to the calculation of ECD spectra and proven to be highly accurate.
The algebraic diagrammatic construction (ADC) 23 scheme for the polarization propagator is an ab initio method, which shares many features with CC excited-state methods. ADC requires the solution of a Hermitian eigenvalue equation and provides a hierarchy of approximations. In comparison with CC methods, this has the advantage of not requiring the solution of both the left-and right-hand side equations. It has been employed to predict several linear and non-linear properties, e.g., two-photon absorption (TPA) 24 and resonant inelastic x-ray scattering (RIXS). 25 Furthermore, static properties such as static polarizabilities and C 6 dispersion coefficients 26,27 have recently been investigated. The computational effort required for ADC(n) schemes scales as n 4 for ADC(1), n 5 for ADC (2), and n 6 for ADC(2)-x and ADC(3/2) with n being the number of basis functions. For comparison, the corresponding CC schemes, CC2, coupled-cluster singles and doubles (CCSD), and CC3, formally scale as n 5 , n 6 , and n 7 , respectively, and, moreover, require an iterative solution for the corresponding ground state as well.
Here, we report the first implementation of ADC-based methods to simulate chiro-optical properties such as ECD.
We choose methyloxirane and methylthiirane molecules as well as their dimethylated derivatives to investigate the performance of the ADC methods for the calculation of ECD spectra. For these molecules, highly resolved gas phase spectra exist for comparison. Furthermore, we include the inherently chiral hydrogen peroxide H 2 O 2 and dihydrogen disulfide H 2 S 2 for their small size and well studied chiro-optical properties. Beside these relatively small molecules, we evaluate the ECD spectra of camphor, norcamphor, and fenchone using a third order ADC scheme in comparison with experimental gas phase spectra. [28][29][30][31][32] The latter molecules have previously been investigated using TD-DFT but not yet using a third order ab initio method. We also include epinephrine, a common medical drug and hormone, 33 for which there are aqueous ECD solution spectra reported. 34 The molecular structures of the investigated systems are shown in Fig. 1.
A complication in the calculation of ECD spectra is the lack of gauge invariance of approximate solutions of the Schrödinger equation, which can lead to origin dependent rotatory strengths in the length gauge. This is circumvented in one of the two ways: (i) using magnetic-field dependent "gauge-including" atomic orbitals (GIAOs, also referred to as London orbitals) 11,[35][36][37][38][39][40][41][42] or alternatively (ii) by a gauge transformation to the velocity gauge, which is formally origin invariant. 19,[43][44][45][46][47] The latter strategy is referred to as the velocity gauge formalism or gradient form of the rotatory strength, and this method will be used throughout this work. For the sake of completeness, the length gauge formulation of the rotatory strength is also briefly presented.
Experimentally, an ECD spectrum is recorded by measuring the difference in absorption between the left and right circularly polarized light, where ϵL(ω) and ϵR(ω) are the absorption coefficients, at incident frequency ω, for the left and right circularly polarized light, respectively. The measured signal may then be related to the underlying rotatory strength, governing the excitation by integration over a frequency range [ω 1 , ω 2 ], 48 as where the photon energy ω is in eV and Δϵ(ω) is given in L mol −1 cm −1 . The rotatory strength, R, is in the usually reported units of 10 −40 esu cm erg/G or 10 −40 c.g.s. units. 49 The crucial assumption of Eq. (2) is that the integrated peak results from one electronic transition only. This is not necessarily the case and not known a priori, which emphasizes the need for robust theoretical models assisting in the interpretation of the experimental ECD spectra.

II. THEORY AND METHODOLOGY
A. Optical rotatory and oscillator strengths in the length-and velocity-gauge formalisms within the intermediate state representation The fundamental property governing the preferential absorption of circularly polarized light is the rotatory strength, Rn 0 . It is given by the well-known Rosenfeld equation 50 as the scalar product of the electric and magnetic transition dipole moments between the ground state |Ψ 0 ⟩ and an excited state |Ψn ⟩, whereμ andm are the electric and magnetic dipole operators, respectively. From the Rosenfeld equation, we can see that the electric transition dipole moment and the magnetic transition dipole moment must be non-orthogonal for a transition to be active in ECD. Using the electric dipole operator,μ = −er, and the magnetic dipole operator,m = ie ̵ h 2m e cr ×∇, 43 within Eq. (3) yields the rotatory strength in its length gauge formulation, where e is the elementary charge, me is the electron mass, c is the speed of light, and ̵ h is the reduced Planck constant. In the sumover-states expression, this form of the rotatory strength sums up to zero, which results in zero rotatory strengths far away from resonance, 51 but lacks gauge-origin independence. 52 Hence, a translation of the origin of the position operator alters the calculated rotatory strength. As mentioned previously, we circumvent the origin dependence of the rotatory strengths in the length gauge by employing the velocity gauge. By using the fact that the position operator commutes with every term in the electronic molecular Hamiltonian, except the kinetic energy operator, 47,53 and applying this to Eq. (4) yields the rotatory strength in the velocity (gradient) form The corresponding oscillator strength of the same transition is computed in the length gauge as 43 which in analogy to the rotatory strength can be transformed to yield the oscillator strength in the velocity gauge, The two forms of the oscillator strength are origin independent, and they are usually computed together as a measure of basis sets completeness.
In the context of the algebraic diagrammatic construction scheme for the polarization propagator, the excitation energies and transition moments required in order to evaluate Eq. (6) are obtained by finding the eigenvectors and eigenvalues of the Hermitian ADC matrix M, which is a representation of the electronic HamiltonianĤ in the so-called intermediate states |ΨI⟩ shifted by the exact ground-state energy, E 0 . 54,55 The ADC matrix is connected to the diagonal representation of the shifted Hamiltonian in exact states, by unitary transformation with the eigenvector matrix X. Ground-to-excited state transition moments for a given operatorÔ can be evaluated by contracting an eigenvector x n with the vector of so-called modified transition moments F(Ô), which correspond to the transition moments between the ground state and the intermediate states, To efficiently evaluate transition moments for different operators, the modified transition moments are not constructed explicitly, but rather the transition density ρ 0n is calculated, where c † p and cq are creation and annihilation operators in second quantization. The transition moments are then evaluated as with the property integrals opq = ⟨ϕp|Ô|ϕq⟩ in the molecular orbital basis {ϕp}. By consistently constructing the intermediate states up to a given order n of perturbation theory using the Møller-Plesset Hamiltonian partitioning, one arrives at the corresponding ADC(n) scheme. Using the eigenvalues and eigenvectors of the third-order ADC matrix with the second-order modified transition moments is referred to as ADC(3/2). This mixed third and second order scheme is used throughout this paper and called ADC(3) for brevity. 56,57 Full-ADC is formally gauge invariant. However, this is not true for truncated ADC schemes. The remaining difference between the length and velocity forms of the oscillator and rotatory strengths occurs in the highest-order contributions of their perturbational expansion or, in other words, in O (n) for ADC(n). Consequently, the gauge-variance of ADC(3/2) is of the same order as ADC(2), i.e., the differences occur in the O (2) contributions. For further theoretical details regarding the length and velocity gauge behavior of ADC(n) methods, see Ref. 58.

B. Computational details
To simulate the ECD spectra, the computed rotatory strengths are converted from a.u. to 10 −40 c.g.s. using the conversion factor 471.44 and subsequently broadened by a Lorentzian function, where Δϵn is the peak intensity of the nth transition given in L mol −1 cm −1 , Rn 0 is the rotatory strength in 10 −40 c.g.s., γn is the Lorentzian broadening factor, and ω and ωn 0 are the incident optical frequency and excitation energy in eV. A broadening factor corresponding to full width at half maximum of γn = 1000 cm −1 is used throughout this work. 59 Calculations of oscillator and rotatory strengths in both length and velocity gauges using ADC were performed using a development version of Q-Chem 5.2. 60 For the CC calculations of the same properties, Dalton2018 was used. 61 Dunning basis sets were employed throughout the paper. 62,63 The origin for the position operator was set to the hetero-atom for the oxiranes, thiiranes, and camphor and its derivatives and to one of the oxygen/sulfur atoms in the symmetrical H 2 O 2 and H 2 S 2 molecules. Geometries of methyloxirane, dimethyloxirane, methylthiirane, and dimethylthiirane were optimized using MP2/aug-cc-pVTZ and those of H 2 O 2 and H 2 S 2 at the MP2/d-aug-cc-pVTZ level. For camphor, norcamphor, and fenchone, DFT/CAM-B3LYP 64 and aug-cc-pVTZ were used for the geometry optimization.

III. RESULTS AND DISCUSSION
A. ECD spectra of methyloxirane and methylthiirane Methyloxirane has been widely used for the evaluation of quantum chemical methods for the simulation of ECD spectroscopy, primarily due to its small size as well as its available gas phase circular dichroism spectrum. 65,66 Furthermore, there is extensive literature available, both experimental and theoretical work, covering electronic structure, 67 excited state properties, 68 and two-photon absorption ECD. 69 CC methods have been proven successful in reproducing the gas phase ECD spectrum. 70 The inclusion of molecular dynamics has addressed the water solution spectrum, 71 and the use of a polarizable embedding model has also addressed the spectrum in various solvents. [72][73][74] It has been shown that the inclusion of diffuse functions is important to ensure correct signs of the optical rotatory strength of methyloxirane. 75 Pople basis sets seem inadequate even with diffuse functions, while Dunning basis sets have been proven more capable. 11 Therefore, we performed an extensive basis set study for ADC(2), ADC(3), CC2, and CCSD using Dunning basis sets and comparing to experimentally measured values of methyloxirane. The excitation energies and oscillator and rotatory strengths calculated at the ADC(2), ADC(3), CC2, and CCSD levels of theory for the nine lowest excited states of (R)-methyloxirane and (R)methylthiirane are collected in Table I (see Tables S1-S4 in the supplementary material for all excited states computed). The simulated ECD and UV/VIS spectra at the same levels of theory of the 15 energetically lowest excited states are shown in Fig. 2.

Basis set impact on methyloxirane
Starting with ADC(2) and CC2, the smallest basis of the cc-pVXZ series (X: D, T, Q), cc-pVDZ, yields nearly identical excitation energies for ADC (2) and CC2 with the first vertically excited state calculated at 8.24 eV at the ADC(2) level and 8.35 eV at the CC2 level, which should be compared with the experimental value of 7.08 eV. 65 Increasing the basis set size from double-to quadruplezeta lowers the excitation energy to 7.51 eV and 7.54 eV for ADC (2) and CC2, respectively. However, the rotatory strength of the first transition is largely unaffected and is roughly overestimated by a factor of 2 compared to experiment. However, the reported value in Ref. 65 is based on Eq. (2) with the limits of integration determined by a low-level CI calculation possibly impairing the analysis. In general, careful analysis is always needed when comparing computed and experimental rotatory strengths.
Turning to the singly augmented, aug-cc-pVXZ series (X: D, T), aug-cc-pVDZ yields similar excitation energies for the first vertically excited state at the ADC(2) and CC2 level with 6.23 eV and 6.37 eV, respectively. Increasing the basis from double-to triplezeta improves the excitation energies to 6.52 eV and 6.55 eV for ADC(2) and CC2, respectively. In general, the rotatory strengths are much more sensitive to augmentation of the basis set than the excitation energies. Even with aug-cc-pVDZ, the rotatory strength of the first transition is now within 20% of experiment and within 15% for aug-cc-pVTZ, which has been overestimated by roughly 100% using non-augmented basis sets.
Moving to the doubly augmented, d-aug-cc-pVXZ series (X: D, T), the rotatory strength of the first excited state is within 10% of experiment for both ADC(2) and CC2 using d-aug-cc-pVDZ, but the excitation energy (<0.04 eV) is hardly affected. Going to d-augcc-pVTZ, the rotatory strength is largely unaffected, while the excitation energy is increased, as was the case for the singly augmented aug-cc-pVTZ basis. However, double augmentation has a substantial impact on the rotatory strengths of higher-lying excited states. For cc-pVXZ and aug-cc-pVXZ basis sets, the second vertically excited state has an incorrect positive sign of the rotatory strength at ADC(2) and CC2 levels of theory but is correctly described as negative using d-aug-cc-pVXZ basis sets. Higher order methods such as ADC (3), CCSD, or CC3 reproduce the correct sign regardless of which basis set is employed. This illustrates the importance of diffuse functions as well as the degree of electron correlation included for a correct assignment of higher excited states.
Overall, the differences between the length and velocity forms of the rotatory strength are relatively small for the first electronic transition, i.e., on the order of ∼10-20% for cc-pVXZ, ∼8-10% for aug-cc-pVXZ, and ∼7-10% for d-aug-cc-pVXZ at the ADC(2) and CC2 level of theory. The difference is on the same order of magnitude for energetically higher-lying excited states with minor exceptions (to see the effect on a shift in the gauge-origin on the length gauge rotatory strength, refer Fig. S5 of the supplementary material). The difference between the length and velocity forms of the oscillator strength is very similar to the one observed for the rotatory strengths.

(R)-methyloxirane
A comparison of the performance of the individual methods in the calculation of the ECD spectrum of methyloxirane when the daug-cc-pVDZ basis set is used reveals ADC(2) and CC2 to yield very similar spectra, as obtained from the first 15 excitations calculated ( Table S3 in the supplementary material). All spectra are shifted for the first state to correspond to the experimental value of 7.10 eV. The shifts required are +0.92 eV for ADC(2) and +0.78 eV for CC2, respectively. Two prominent features of the experimental spectrum, A and B, are well reproduced by ADC(2) and CC2 (Fig. 2). Furthermore, at these levels of theory, A-C arise from oxygen lone-pair excitations to Rydberg orbitals. According to our calculations, A is a convolution of two states of negative amplitude and B is a convolution of three states with an overall positive amplitude. Finally, C corresponds to a convolution of three states and is incorrectly predicted as positive by both methods.   Methylthiirane   The spectra calculated at the ADC(3) and CCSD level of theory are again shifted by −0.74 eV and −0.02 eV, respectively, for the first vertical excitation energy to match the experimental value. At these levels of theory, in addition to A and B, C is also correctly predicted as negative albeit red-shifted by ∼0.5 eV for ADC(3).

(R)-methylthiirane
Methylthiirane is the sulfur containing analog of methyloxirane, and again a high-quality experimental gas phase ECD spectrum, 65,66 as well as experimental vibrational circular dichroism (VCD) 76 and Raman optical activity (ROA) 77 spectra, is available. The ECD spectrum and the optical rotatory dispersion have also been addressed in theoretical studies. 78,79 The experimental ECD spectrum of methylthiirane possesses three distinct features, A, B, and C in analogy to methyloxirane. Using the d-aug-cc-pVDZ basis set, ADC(2) and CC2 yield very similar transition energies and rotatory strengths for the energetically lowest 15 computed states (Table S4 in the supplementary material). All excited states correspond to electronic transitions from sulfur lone-pair orbitals into Rydberg orbitals. ADC(2) and CC2 yield ECD spectra correctly predicting the signs of A, B, and C. The excitation energies are redshifted by roughly 0.4 eV in comparison to experiment (Fig. 2). The first peak arises from a weakly absorbing state with a negative rotatory strength, which has also been experimentally observed but was not shown in the experimental spectrum as it was barely visible. The first significant feature of the experimental spectrum, A, is a convolution of two peaks, which are correctly predicted as negative, but the amplitude is underestimated by a factor of 2 at these levels of theory. B and C correspond to convolutions of multiple states, the signs of which are correctly predicted using ADC(2) and CC2.
Proceeding to ADC (3), the excitation energies require no shift and transition amplitudes agree very well with experiment. The two electronic states constituting the spectral feature A, a 1 and a 2 , lie within ∼0.02 eV of the experimental spectrum at the ADC(3) level, but the rotatory strengths of these states appear to be swapped compared to experiment. However, it cannot be excluded that these arise from vibrational effects. The improvement is of equal quality for CCSD, although slightly red-shifted compared to experiment. B consists of four vibronic peaks in the experimental spectrum. At the ADC(3) level, two excited electronic states are computed, exhibiting the correct excitation energies and signs of their rotatory strengths. C is a convolution of several peaks in the experimental spectrum and is incorrectly described as negative by both ADC(3) and CCSD.

B. ECD spectra of dimethyloxirane and dimethylthiirane
In addition, for dimethyloxirane and dimethylthiirane, highquality experimental gas phase ECD spectra are available 65,66 and show features similar to the ones of methyloxirane and methylthiirane. The excitation energies and oscillator and rotatory strengths calculated at the ADC(2), ADC(3), CC2, and CCSD levels of theory for the nine lowest excited states of (R,R)-dimethyloxirane and (R,R)-dimethylthiirane are collected in Table II (see Tables S5-S8 in the supplementary material for all excited states computed). The simulated ECD and UV/VIS spectra at the same levels of theory of the 15 energetically lowest excited states are shown in Fig. 3.

(R,R)-dimethyloxirane
At the ADC(2) and CC2 levels of theory using the d-aug-cc-pVDZ basis set, both methods produce very similar spectra for the energetically lowest 15 states computed with the ADC(2) excitation energies red-shifted by ∼0.15 eV-0.20 eV compared to CC2. All spectra are shifted for the first state to correspond to the experimental value of 7.0 eV. The shifts required are +0.92 eV and +0.78 eV for ADC(2) and CC2, respectively. At these levels of theory, the vertical excited states arise exclusively from lone-pair orbital excitations from the oxygen atom into Rydberg orbitals. Furthermore, the spectral feature A (Fig. 3) arises solely from the lowest transition, the excitation energy of which is red-shifted by ∼0.8 eV from experiment, reproduced as 6.08 eV and 6.22 eV for ADC(2) and CC2, respectively. The rotatory strength, however, is in reasonable agreement with experiment, although it is roughly overestimated by 40% for both ADC(2) and CC2. Furthermore, ADC(2) and CC2 show B arising solely from the second excited state for which the excitation energy is calculated at 6.38 eV and 6.53 eV for ADC (2) and CC2, respectively, which is again red-shifted by ∼0.9 eV compared to experiment. The rotatory strength is overestimated by an order of magnitude. The band C at these levels of theory arises from the third and fourth excited states, which in comparison with experiment are red-shifted by roughly 1 eV for both ADC(2) and CC2. D arises from the fifth and sixth excited states, which are again too low in energy by roughly 1.1 eV.
The spectra of the higher order methods ADC(3) and CCSD are also shifted to correspond to experiment. The shifts required are −0.81 eV and −0.07 eV for ADC(3) and CCSD, respectively. ADC(3) again yields higher excitation energies, as was already the case for methyloxirane. The CCSD excitation energies on the other hand agree to within ∼0.10 eV with experiment. At the ADC(3) level, the spectral feature A arises from two states with negative rotatory strengths, separated by 0.10 eV. These peaks appear in agreement with the experimental ECD spectrum. We note that other studies have assigned these peaks to vibronic structures originating from up to two electronic states. 81 Furthermore, these two peaks are not reproduced in the CCSD spectrum, which, alike ADC(2) and CC2, shows A arising from a single excited electronic state, whose width closely agrees with the experimental one. At the ADC(3) level of theory, B is a convolution of the third and fourth excited state, the first sharply positive and latter negative. This results in a relatively weak positive feature, in reasonably good agreement with experiment. A similar convolution is observed at the CCSD level, however, of the second and third excited state. The larger energetic separation between the two states in CCSD results in a more intense positive Bband feature, whereas the negative component merges with the negative C band. C is again a convolution of many states at the ADC(3) level resulting in a broad positive band, not present in experiment, followed by a blue-shifted negative peak. CCSD predicts the overall negative band of C, which is originating from the fourth and fifth excited states (with a shoulder due to the third one), in better agreement with experiment. D is correctly predicted as positive and arising from the tenth excited state using ADC(3) and a convolution of many states using CCSD.

(R,R)-dimethylthiirane
The three bands, A-C, of dimethylthiirane are again investigated using the d-aug-cc-pVDZ basis set and are characterized by lone-pair excitations into Rydberg orbitals by ADC(2) and CC2. The excitation energy of the first excited state is calculated as 5.03 eV and 5.08 eV by ADC(2) and CC2, respectively, which are blue-shifted by roughly 0.5 eV, compared to the reported experimental value of 4.5 eV. The first rotatory strength exhibits an error of more than an order of magnitude, 5.7 (×10 −40 c.g.s.) and 4.3 (×10 −40 c.g.s.) for ADC(2) and CC2, respectively, compared to the experimental value of 0.1 (×10 −40 c.g.s.). 80 However, this is a weakly active state and as such experimental limitations may also play a role.
The first distinct feature A in the experimental spectrum is a convolution of the second to fifth excited states at these levels of theory. The second feature, B, is a convolution of the sixth to eighth excited state, and the third band, C, arises from higher-lying excited states above the eighth state. It is interesting to recognize that the calculated spectra at ADC(2) and CC2 levels could be mistaken as   that of red-shifted (S,S)-dimethylthiirane; however, this is not the case for the higher order methods. Using ADC(3) and CCSD with the d-aug-cc-pVDZ basis set, the lowest excitation energy remains mostly unaltered at 4.97 eV and 5.04 eV, respectively. Unlike ADC(2), ADC(3) yields a rotatory strength for the first weak peak of similar magnitude to that of the experiment, deviating by ∼10%, however, still overestimated by an order of magnitude for CCSD. Here, 14 excited states are considered, and for the first and second band feature A and B, ADC(3) results agree well with experiment, although the rotatory strength of A is still overestimated by a factor of 2. Significantly, the last band feature, C, is correctly predicted as negative at the ADC(3) level resulting from a convolution of several excited states but incorrectly predicted as positive using CCSD. We note, however, that more excited states may be needed at the CCSD level to reproduce this spectral region. of Chemical Physics ARTICLE scitation.org/journal/jcp

C. ECD spectra of H 2 O 2 and H 2 S 2
The chiro-optical properties of H 2 O 2 and H 2 S 2 have been the subject of benchmark calculations due to their small size and complex electronic structure. 14,82,83 The dependence of the optical rotatory strength on the dihedral angle of H 2 S 2 has been studied at the CIS level. It was discovered that H 2 S 2 has degenerate excited states at a dihedral angle of ∼90 ○ , which are poorly described using a HF wavefunction. Later, the performance of various density functionals was benchmarked against MRCI and CC2 calculations. 84 As can be expected from a sulfur containing chromophore, low-lying excitations are characterized by Rydberg to a certain extent. 84 Fewer studies have been carried out on H 2 O 2 , and for neither molecule, the ECD spectra have been simulated using a third order ab initio method. The excitation energies and oscillator and rotatory strengths calculated at the ADC(2), ADC(3), CC2, CCSD, and CC3 levels of theory for the nine lowest excited states of H 2 O 2 and H 2 S 2 are collected in Table III (see Tables S9-S12  ADC (2) and CC2 yield very similar ECD spectra for the energetically lowest 15 singlet states. The first feature, A, at the ADC(2) and CC2 levels of theory arises from the first electronic excited state, and the excitation energies and rotatory strengths are calculated to within 0.03 eV and ∼15% from the corresponding CC3 excitation energy and rotatory strength, respectively. The second spectral feature, B, arises from the second excited state and likewise agrees well between ADC(2) and CC2, but it is red-shifted for both methods by ∼0.5 eV and the rotatory strengths are within ∼10% with respect to CC3 energies and rotatory strengths. C corresponds to the third excited state, and ADC(2) and CC2 energies are blue-shifted by ∼0.04 eV and rotatory strengths are within ∼10% to those at the CC3 level; however, due to the larger energetic gap between the third and fourth excited state at the CC3 level, the convoluted spectra at ADC(2) and CC2 levels show C overestimated by a factor of 2. D corresponds to the fourth excited state and is red-shifted by roughly 0.4 eV at ADC(2) and CC2 levels, with rotational strengths calculated to within ∼10%, compared with CC3 energies and rotatory strengths. Finally, E is caused by the fifth excitation and is red-shifted by ∼0.5 eV with rotatory strengths underestimated by ∼50% and ∼30% for ADC (2) and CC2, respectively, as compared with CC3. It is noteworthy that E could be mistaken as positive at the ADC(2) and CC2 levels, caused by the sixth excitation.
For the higher order methods, ADC(3), CCSD, and CC3, the first excited state responsible for the spectral feature A has an excitation energy of 5.95 eV, 6.14 eV, and 6.08 eV, respectively. For the second band, B, ADC(3) and CCSD are blue-shifted by 0.5 eV, compared to ADC(2) and CC2; however, at the ADC(3) level, the sign of the transition is positive. This reversal can be explained by the dependence of the rotatory strength on the H 2 O 2 structural parameters, specifically the length of the oxygen-oxygen bond. Contraction of this bond by 1.8 pm from its MP2 equilibrium geometry (1.452 Å) interchanges the second and third excitations, i.e., B and C, at the ADC(3) level, which then fall directly into reasonable agreement with the CCSD and CC3 results (Fig. 4). An identical contraction leaves CCSD and CC3 excited states invariant, showing a unique dependence on the structural parameters for ADC (3). D and E correspond to the same excited states as those calculated using ADC (2) and CC2 and agree well between ADC(3) and CC3 after contraction of the oxygen-oxygen bond.

H 2 S 2
The optimized geometrical parameters for H 2

D. ECD spectrum of solvated epinephrine
Only the D-enantiomer of epinephrine (commonly known as adrenaline) is biologically relevant, yet organic synthesis produces a racemic mixture of both enantiomers. 85 An efficient method to obtain enantiopure D-epinephrine is via HPLC separation in combination with ECD spectroscopy. 34 To demonstrate the general ability of our implementation of rotatory strengths within ADC to simulate the ECD spectra of solvated molecules, the solution spectrum of L-epinephrine in water is computed using a conductor-like polarizable continuum model (PCM) for water. 86,87 It is clear that for a more thorough investigation, explicit water epinephrine interactions need to be considered; however, for a qualitative agreement and a proof-of-concept, the PCM approach suffices. The geometry of the L-epinephrine enantiomer was optimized using CAM-B3LYP/cc-pVTZ with and without PCM (the converged geometry in both cases corresponds to the AG1a conformer using the nomenclature adopted in Ref. 88). The ECD spectra of these molecular systems were then computed at the ADC(2)/aug-cc-pVDZ/PCM(water) and of Chemical Physics ARTICLE scitation.org/journal/jcp   ADC(2)/aug-cc-pVDZ levels of theory for the five energetically lowest excited states, as shown in Fig. 5 (see Table S13 for numerical values). All states essentially correspond to ππ * excitations with contributions from the oxygen/nitrogen lone pair into π * -orbitals of the phenyl ring. Two bands are seen in the experimental spectrum: one weakly positive peak centered around 270 nm and one broad negative peak centered around 230 nm. The band at 270 nm is, however, not unambiguously assignable in the experiment and may well correspond to noise. Using the gas-phase geometry at the ADC(2)/aug-cc-pVDZ level, the second band of the ECD spectrum is clearly reproduced as a convolution of the third to fifth vertically excited states. The first band is a convolution of the first two excited states, producing an overall negative band.
Including the PCM for water solvation for the calculation of the spectrum only leads to a blue-shift of the center of the second band and drastically alters the computed rotatory strengths, which falls in line with the experimental spectrum, however, with the first of Chemical Physics band again predicted as a weakly absorbing negative band. Using the PCM for both geometry optimization and the calculation of the ECD spectrum yields a very similar spectrum to that obtained using the gas-phase geometry, showing that the optimized geometry is only negligibly influenced by solvation. The difference between the length and velocity gauges for the energetically lowest excited states is as large as a factor of 2 but small in absolute terms, owing to the weak absorption, which can already originate from the incompleteness of the one-particle basis set. For energetically higher states with higher absorption, the gauges agree acceptably well with deviations of ∼15%. For all states, the sign of the rotatory strength remains consistent.

E. ECD spectra of camphor, norcamphor, and fenchone
Camphor and the structurally related molecules, norcamphor and fenchone, have long been used as molecular standards for ECD calibrations, [89][90][91][92][93] and their characteristic ECD spectra as well as VCD spectra have been extensively explored. 28,[94][95][96][97] To the best of our knowledge, the ECD spectra of these molecules have not been calculated using high-level ab initio methods. The excitation energies and oscillator and rotatory strengths calculated at the ADC(2) and ADC(3) levels of theory for the nine lowest excited states of camphor, norcamphor, and fenchone are collected in Table IV. The simulated ECD and UV/VIS spectra at the same levels of theory of the ten energetically lowest excited states of the same molecules are shown in Fig. 6.

(1R)-camphor
As seen in Fig. 6, the ECD spectrum of (1R)-camphor in the gas phase is fully reproduced using ADC(3)/aug-cc-pVDZ with the ten lowest lying excited states computed. The signs of the three bands (A-C) are all correctly predicted, and the simulated ECD spectral amplitudes are in good agreement with experiment. We shift the simulated ECD spectrum −0.32 eV to correspond with experiment. The first peak A is caused by the energetically lowest excited state. Although it is a dark state in the UV/VIS spectrum, it becomes visible in the ECD spectrum due to its large magnetic transition dipole moment. The rotatory strength of this band matches well with that of experiment. The second band B appears as a vibrationally resolved peak in the experimental spectrum and is calculated as a convolution of the second to fifth excited electronic state, which matches closely in amplitude. C arises from the sixth to ninth excitation. All excited states correspond to n-Rydberg excitations.
At the ADC(2)/aug-cc-pVDZ level, A, B, and C arise from the same states as those computed at the ADC(3) level with identical signs. We shift the spectrum −0.08 eV to correspond with experiment. The rotatory strength of A, at this level of theory, matches well with experiment and roughly overestimates ADC(3) by a factor of 2. B and C are both red-shifted with respect to experiment by ∼0.8 eV and ∼1 eV, respectively. The rotatory strength of B and C is calculated to roughly half and roughly one quarter, respectively, as those at the ADC(3) level. Table IV. Transition energy ω (eV), optical rotatory strength R (10 −40 c.g.s.), and oscillator strengths (×10 3 ) of the ten lowest electronic transitions of camphor, norcamphor, and fenchone calculated at the ADC(3) and ADC(2) levels of theory using the aug-cc-pVDZ basis set for camphor and fenchone and the d-aug-cc-pVDZ basis set for norcamphor. Noticeably, the first band feature A is weakly positive (+0.3 × 10 −40 c.g.s.) in the length gauge and weakly negative (−0.5 × 10 −40 c.g.s.) in the velocity gauge at the ADC(3)/d-aug-cc-pVDZ level of theory. This can be attributed to basis set incompleteness. A arises from the first excited state, which is again characterized by the n-Rydberg character, as all other computed higher-lying states of norcamphor as well. The second band of the experimental ECD spectrum, B, is caused by the second vertical excited state and again aligns excellently with the experimental reference. The sharply positive third band, C, arises from a convolution of the third and fourth excited states with a vibrational fine structure appearing in the experimental spectrum. D arises from convolutions of the fifth to seventh excited states, which are hard to separate. At the ADC(2)/d-aug-cc-pVDZ level, A, B, and C are consistently reproduced as those at the ADC(3) level. A shift of −0.14 eV is applied to agree with experiment. B and C are both red-shifted by ∼1.0 eV with rotatory strengths roughly half as those calculated at the ADC(3) level. D is not reproduced at the ADC(2) level from the lowest ten electronic excited states.

(1R)-fenchone
Finally, we turn to (1R)-fenchone, which is a constitutional isomer of (1R)-camphor. Not surprisingly, all computed excited states of fenchone correspond again to transitions from lone-pair oxygen n-orbitals to Rydberg orbitals. The first band feature A is a dark state in the UV/VIS spectrum as was the case for (1R)-camphor, yet becomes visible in the ECD spectrum, again arising from the energetically lowest excited state, which is in good agreement with experiment. The second band structure B appears again as a vibrationally resolved peak with a slightly red-shifted shoulder, as was the case for camphor. At the ADC(3) level, it arises from a convolution of the second to fifth excited states, which are further blue-shifted by ∼0.4 eV from experiment. The Lorentzian broadened spectral amplitude is in excellent agreement with experiment. C arises from a convolution of the eighth excitation and upwards.
At the ADC(2)/aug-cc-pVDZ level, the spectrum is shifted by +0.04 eV to correspond to the first bright state. At this level, A, B, and C are consistently reproduced as those calculated of Chemical Physics

IV. SUMMARY AND CONCLUSIONS
Expressions for the calculation of rotatory strength in the length and velocity gauges have been implemented for the algebraic diagrammatic construction scheme for the polarization propagator up to third order. The accuracy of the simulated ECD spectra using the ADC(2) and ADC(3) schemes has been assessed by comparison to the corresponding coupled cluster schemes CC2, CCSD, and (for the smaller systems) also CC3. For that objective, the gas phase ECD spectra of the R-enantiomers of methyloxirane, methylthiirane, dimethyloxirane, dimethylthiirane, hydrogen peroxide, hydrogen disulfide, camphor, norcamphor, and fenchone have been computed. Furthermore, addressing the spectra of molecules in solution, the water spectrum of epinephrine was evaluated using ADC (2) in combination with the polarizable continuum model.
A strong basis set dependence of the computed rotatory strengths has been observed, in particular, with respect to the inclusion of diffuse orbital functions. However, this owes to the n-Rydberg excitation character of all transitions studied here and can be expected to be less prominent when other chiral organic molecules involving less diffuse n-π * and π-π * electronic transitions are investigated.
Not surprisingly, ADC(2) yields rotatory strengths and ECD spectra very similar to those obtained at the CC2 level. ADC(3) produces spectra similar in quality to CCSD, which are both comparable to CC3. ADC(3) spectra are in better agreement with the experimental ones for the larger systems studied here, i.e., camphor, norcamphor, and fenchone, than for the smaller molecules. For the purpose of assigning an ECD spectrum to one specific enantiomer, ADC(2) and ADC(3) are clearly sufficiently accurate. Together they represent an excellent complementary toolbox for the simulation of ECD spectra, which holds a promise that the same will also apply of Chemical Physics ARTICLE scitation.org/journal/jcp for other chiro-optical properties, which will be the subject of future studies. An important aspect in the simulation of experimental ECD spectra of medium-sized to large molecules is the influence of molecular environments, as has been shown for the solution spectrum of epinephrine where the application of PCM shifted the simulated spectrum. Within the ADC framework, further different solvation models are available, for example, frozen density embedding, 98,99 effective fragment potentials, 100 or polarizable embedding. 101,102 The suitability of these environment models in combination with ADC for the simulation of ECD spectra will be investigated in the future.

SUPPLEMENTARY MATERIAL
Optimized geometries, excitation energies, and rotatory and oscillator strengths of all excited states computed are provided in the supplementary material. The impact of the oxygen-oxygen bond contraction on the rotatory strengths of H 2 O 2 computed at the ADC(3) level and the basis set impact on excitation energies and rotatory strengths of methyloxirane are also included.

ACKNOWLEDGMENTS
We acknowledge financial support from the Marie Skłodowska-Curie European Training Network COSINE, Grant Nos. 765739 and VR 2018-4343. Computational resources from the Swedish National Infrastructure for Computing (SNIC) are also acknowledged.

DATA AVAILABILITY
The data that support the findings of this study are available within the article and its supplementary material.