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Cite This: J. Phys. Chem. A XXXX, XXX, XXX−XXX

Performance of Property-Optimized Basis Sets for Optical Rotation with Coupled Cluster Theory J. Coleman Howard,† Shree Sowndarya S. V.,† Imaad M. Ansari,† Taylor J. Mach,‡ Angelika Baranowska-Łączkowska,¶ and T. Daniel Crawford*,† †

Department of Chemistry, Virginia Tech, Blacksburg, Virginia 24061, United States Concordia University, St. Paul, Minnesota 55104, United States ¶ Institute of Physics, Kazimierz Wielki University, Plac Weyssenhoffa 11, PL-85072 Bydgoszcz, Poland Downloaded via UNIV OF SUSSEX on July 4, 2018 at 16:31:22 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

S Supporting Information *

ABSTRACT: The effectiveness of the optical rotation prediction (ORP) basis set for computing specific rotations at the coupled cluster (CC) level has been evaluated for a test set of 14 chiral compounds. For this purpose, the ORP basis set has been developed for the second-row atoms present in the investigated systems (that is, for sulfur, phosphorus, and chlorine). The quality of the resulting set was preliminarily evaluated for seven molecules using time-dependent density-functional theory (TD-DFT). Rotations were calculated with the coupled cluster singles and doubles method (CCSD) as well as the second-order approximate coupled cluster singles and doubles method (CC2) with the correlation-consistent aug-cc-pVDZ and aug-ccpVTZ basis sets and extrapolated to estimate the complete basis-set (CBS) limit for comparison with the ORP basis set. In the compounds examined here, the ORP calculations on molecules containing only first-row atoms compare favorably with results from the larger aug-cc-pVTZ basis set, in some cases lying closer to the estimated CBS limit, while results for molecules containing second-row atoms indicate that larger correlation-consistent basis sets are necessary to obtain reliable estimates of the CBS limit. effective choice for estimating energies23−26 and selected properties (e.g., optical rotation values)3,27,28 near the complete basis-set (CBS) limit. Alternatively, one may improve the efficiency of basis sets by selecting a set of functions tailored to a particular type of calculation. That is, instead of applying large basis sets designed for general use in electronic structure calculations, such as the Dunning correlationconsistent basis sets,29−31 one may design the AO basis to target specific molecular properties. One approach to such property-specific basis sets is to explicitly include a dependence on the perturbation of interest in the definition of the basis functions themselves, such as the gauge-including atomic orbitals (GIAOs)32 commonly used in magnetic property calculations. Sadlej’s electric field variant (EFV) basis sets were similarly built with an electric field dependence for the purpose of streamlining polarizability calculations.33 In the case of the EFV basis sets, analysis of the basis functions’ dependence on the field eventually led to strategies for designing effective fieldindependent basis functions.34 The key to this method of basisset design is that there is no need for basis-set optimization, as the known dependence of the atomic orbitals on the field is

1. INTRODUCTION Coupled cluster theory offers the most reliable route to the accurate calculation of chiroptical properties.1 A sufficient degree of electron correlation, such as in the coupled cluster singles and doubles (CCSD) approach, is typically capable of reproducing gas-phase experimental optical rotatory dispersion curves for most systems studied to date, but the high-degree polynomial scaling associated with such accurate correlated wave function methods (e.g., [6(N 6)] for CCSD) is problematic for large molecules and/or cases of high sensitivity to the choice of the one-electron basis set.2−4 Indeed, large, flexible, atomic-orbital (AO) basis sets augmented with diffuse functions are required to obtain converged specific rotation values for many molecules. Furthermore, this high computational cost is exacerbated for flexible molecules where conformational sampling is needed5,6 and in cases where vibrational corrections are significant.7,8 While one may seek to ameliorate such costs by choosing less expensive electron correlation models (e.g., the second-order approximate coupled cluster singles and doubles method (CC2),9−13 which scales as [6(N 5)]) or by employing local-correlation or other reduced-scaling techniques,14−22 such methods still depend on the use of effective and efficient AO basis sets.4 With a series of calculations of systematically increased basis-set size, basis-set extrapolation has been shown to be an © XXXX American Chemical Society

Received: May 2, 2018 Revised: June 18, 2018 Published: June 20, 2018 A

DOI: 10.1021/acs.jpca.8b04183 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

workers43 was chosen as the source set, and the diffuse functions were obtained from anticipated geometric progression based on the two lowest orbital exponents. In the initial guess, values of the polarization functions’ exponents were taken in the range from 1.1 to 2.5 for the first function, from 0.2 to 1.0 for the second, and from 0.01 to 0.15 for the third. With utilization of steps equal to 0.2 for the first two polarization functions and 0.02 for the third, a total of thus 320 uncontracted basis sets per element were next employed in finite-field ROHF calculation of atomic polarizabilities, assuming the external electric field strength equal to 0.001 au. We considered as the optimal polarization functions’ exponents those minimizing the error in atomic polarizability values with respect to the reference values of Stiehler and Hinze.44 After recognizing the best values of exponents in the initial set, we improved them by testing another set of exponents around these optimal values. For this, we decreased the step to 0.05 for the first two polarization functions and to 0.01 for the third. Once the final polarization functions exponents were found, the basis set was contracted to the form [13s10p3d/ 6s5p3d] to decrease its size. While obviously this step delimits basis-set flexibility, and thus leads to some deterioration of results, it is also necessary to guarantee that the final set can be easily employed in specific rotation calculations in both smalland medium-sized organic systems. The resulting basis set is denoted as ORP and is presented in the Supporting Information. As a preliminary test of the new basis set, specific rotation calculations were carried out for seven chiral test systems using the TD-DFT method and the B3LYP exchange-correlation functional. Compounds 3, 4, 6, 7, 8, 9, and 13 were chosen for this purpose. Calculations were carried out at λ = 589.3 nm, using London atomic orbitals. Reference results were obtained in the aug-cc-pVXZ basis sets29−31 (denoted as aXZ in the following) with X up to 5 for molecule 3 and Q for the remaining systems. Specific rotation values were computed at four wavelengths, 355, 436, 589, and 633 nm, for the 14 molecules shown in Figure 1 utilizing the CC2 and CCSD methods. In this study, 10 of the 14 molecules were selected from the OR45 test set of Srebro et al.45 Four additional molecules were optimized at the same level of theory (the B3LYP functional46−48 with the 6311G(d,p) basis set). For comparison with the ORP basis set, specific rotation values for each molecule were computed with the aDZ and aTZ basis sets. All of the calculations employ the “modified velocity gauge” representation49 of the dipole operator to obtain origin-independent results. The “frozencore” approximation was adopted in every calculation. CC perturbed wave function equations were converged to at least an RMS of 10−6 in the perturbed amplitudes. To estimate CBS limit values, the aug-cc-pVDZ and aug-cc-pVTZ values are extrapolated according to eq 1

captured through the introduction of higher angular momentum polarization functions added to a source set. A series of polarized basis sets, viz. PolX,35,36 ZPolX,37−39 and LPol,40 were developed in this vein for the purpose of computing accurate molecular electric properties with more compact basis sets. The LPol basis sets, in particular, have been demonstrated to be very effective for calculating optical rotation spectra with density-functional theory (DFT) and coupled cluster (CC) methods, offering significant improvement in basis-set convergence and outperforming larger basis sets in some cases.4 Despite these successes, the LPol basis set can still be prohibitively large for some chiral molecules of interest, and by the nature of its design, will suffer from linear dependencies due to the unoptimized polarization functions sharing exponent values with valence-type functions. These issues motivated the creation of a new basis set, dubbed ORP, the optical rotation prediction basis set.41,42 In this case, the polarization function exponents are determined by an optimization procedure, minimizing the error in finitefield spin-restricted open-shell Hartree−Fock atomic polarizability calculations with respect to a set of reference values. The final ORP basis set is a contraction ([6s3p/4s3p] for hydrogen atoms and [11s7p3d/5s4p3d] for first-row atoms) intermediate in size between Dunning’s augmented correlation-consistent aug-cc-pVDZ and aug-cc-pVTZ basis sets.29−31 As with the LPol family basis sets, ORP has been shown to yield optical rotation values comparable to, and in some cases exceeding, the performance of much larger basis sets at the DFT level.41,42 In this work, we investigate the performance of the ORP basis set for the calculation of optical rotation at the CC level of theory. Utilizing the CC2 and CCSD methods, we compare the performance of ORP to augmented correlationconsistent basis sets aug-cc-pVDZ and aug-cc-pVTZ for a diverse test set consisting of 14 chiral compounds shown in Figure 1. In addition, the optical rotation values from these two

Figure 1. The 14 molecule test set used to test the basis sets considered in this work.

basis sets are used in an inverse power extrapolation introduced by Haghdani et al.28 to establish estimates of the complete basis-set (CBS) limit.

ORX = OR ∞ + AX −n

2. COMPUTATIONAL METHODS Within the present work, we carry out generation of the ORP basis set for phosphorus, sulfur, and chlorine, using the method proposed in reference 41. For this purpose, we have chosen a flexible but compact source set, augmented it with diffuse functions to increase its flexibility in the nucleus distant regions, and added three first-order uncontracted polarization functions. As previously, the VTZ basis set of Ahlrichs and co-

(1)

Here, X represents the cardinal number in an aug-cc-pVXZ basis set, and A is a parameter to be determined from a fixed exponent n and the results from two basis sets. The exponent n was fixed to n = 3, which was shown by Haghdani et al. to yield CC2 values in excellent agreement with aug-cc-pV6Z calculations. Core orbitals (1s for C, N, O, and F; 1s2s2p for P, S, and Cl) were kept frozen in all coupled cluster energy and response calculations. All CC2 and CCSD computations were B


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The Journal of Physical Chemistry A performed with the Psi4 software package.50 Atomic polarizability calculations were carried using the Molcas 7.8 package,51−53 and for TD-DFT OR calculations, the Gaussian 09 program was employed.54

the ORP set leads to a considerably worse agreement with reference values than does the aDZ basis set. Finally, in the case of molecule 13, both aDZ and ORP lead to similar error values calculated with respect to the aQZ value, both deviating from aQZ by approximately 3 deg dm−1 (g/mL)−1. Table 3 presents the results of optical rotation calculations for 14 chiral molecules at the CC2 and CCSD levels of theory. For each method, specific rotation values were computed at four wavelengths (λ): 355, 436, 533, and 633 nm. For each wavelength in the table, specific rotations are given in deg dm−1 (g/mL)−1 for the aug-cc-pVDZ and aug-cc-pVTZ basis sets (labeled aDZ and aTZ, respectively), the approximate CBS limit obtained using these two basis sets in conjunction with the two-point inverse power relationship in eq 1, and the ORP basis set, all computed using both the CC2 and CCSD methods. The size of the ORP basis set in each calculation is always intermediate between the aDZ and aTZ basis sets. For example, in the largest molecule considered here, 14 (1S,4Snorbornenone), the number of basis functions in the aDZ and aTZ basis sets are 256 and 552, respectively, while the ORP basis set contains 360 functions, when using their pure angular momentum forms. The first four rows of data in Table 3 present the results for 1, (S)-methyloxirane, which has drawn considerable interest as a target for high-accuracy optical rotation calculations.55−57 In progression from the aDZ basis set to aTZ, the CC2 specific rotation at 355 nm is reduced from −83.9 to −61.1 deg dm−1 (g/mL)−1, leading to a CBS limit estimate of −51.4 deg dm−1 (g/mL)−1. The corresponding value computed with the ORP basis set (given in the column following the CBS estimated values) is −46.0 deg dm−1 (g/mL)−1, in excellent agreement with the CBS estimate and, in fact, closer to the basis-set limit than the larger aTZ basis set. The CC2 values for the next three wavelengths show a similar trend for 1, with the ORP and aTZ computed values at 436, 589, and 633 nm all of smaller magnitude relative to aDZ, and the CC2 ORP predicted rotations are all closer to the CBS estimate than the aTZ values. The deviations between CC2 ORP and aTZ specific rotations decrease in both an absolute and a relative sense as the wavelength increases, differing by about 2 deg dm−1 (g/mL)−1 for the specific rotation computed at 633 nm. The last four columns of Table 3 give the analogous values computed at the CCSD level for 1. For each wavelength and basis set, the CCSD rotations are shifted to more positive values relative to CC2 computations. The basis-set convergence trends at the CCSD level follow the CC2 behavior, with both basis sets larger than aDZ, predicting specific rotation values of smaller magnitude. For the CCSD calculations of (S)-methyloxirane, the aTZ and the ORP rotations are in close agreement. Each of the four rotation values computed with the ORP basis set is within 4 deg dm−1 (g/mL)−1 of the aTZ value, and the specific rotation values at the two larger wavelengths are within 1 deg dm−1 (g/mL)−1. As with the CC2 results, the ORP values lie closer to the estimated basis-set limit than the aTZ specific rotations at each wavelength. (It should be noted that none of the reported specific rotations for 1 compare well to the positive experimental gas-phase values reported by Vaccaro and coworkers.58 Indeed, reconciliation between theory and experiment is obtained only after appropriate inclusion of higherorder electron-correlation effects and anharmonic vibrational corrections.59 However, the goal of this work is limited to exploring the performance of compact basis sets, not obtaining

3. RESULTS AND DISCUSSION Finite-field atomic polarizability values calculated employing the ORP basis set are presented in Table 1 (with associated Table 1. Atomic Polarizabilities Obtained within the FiniteField ROHF Approximationa system

ML

ORP

ref 44

P S

0 0 ±1 average 0 ±1 average

25.4480 21.4670 17.8950 19.0857 13.0498 14.7536 14.1857

25.469 21.277 18.058 19.131 13.233 14.740 14.238

Cl

a

All values in atomic units.

magnetic quantum numbers ML in the second column) and compared with the reference results by Stiehler and Hinze.44 The agreement between our average polarizabilities and the literature results is fair (errors do not exceed 1.5%, being in most cases lower than 1%) and should be sufficient for molecular electric property evaluation in organic molecules. Results of the TD-DFT OR calculations are shown in Table 2. It can be seen that the quality of the results depends strongly Table 2. TD-DFT Optical Rotation (deg dm−1 (g/mL)−1) Calculated at λ = 589.3 nm molecule

aDZ

aTZ

aQZ

a5Z

ORP

3 4 6 7 8 9 13

51.68 166.70 81.19 −26.74 88.08 −20.34 −41.73

26.45 137.11 78.60 −31.49 81.15 −25.33 −39.57

22.73 132.59 81.34 −28.78 80.11 −26.60 −38.46

21.81

27.22 132.56 84.73 −25.59 82.80 −25.75 −35.25

on the investigated system. In the case of molecule 3, significant improvement is observed on going from the aDZ (150 functions) to the ORP (210 basis functions) basis set, with the value obtained in the latter set being in a very good agreement with the aTZ (326 functions) result. Obviously, total timing of calculations does not depend exclusively on the basis-set size; however, it is worth noting that for the abovementioned basis sets, the following timing ratio was observed, aDZ/ORP/aTZ ≈ 1:3:16. Thus, in the case of molecule 3, the ORP basis set allows for a substantial improvement in the computed optical rotation values with respect to aDZ at a reasonable cost. The improvement over the aDZ results is encouraging also in the case of molecules 4 and 9, for which the ORP rotations are even closer to the aQZ values than those from the aTZ basis set. The magnitude of the ORP result obtained for molecule 8 is between that of the aDZ and aTZ results. The opposite situation is observed for molecules 6 and 7, for which already the aDZ basis set yields values very close to the reference. In these cases, despite the higher cost of calculations, C


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Table 3. Specific Rotation Values (deg dm−1 (g/mL)−1) Computed at the CC2 and CCSD Levels of Theory with aug-cc-pVXZ (X = D,T) and ORP Basis Sets CC2

CCSD

molecule

λ

aDZ

aTZ

CBS est.

ORP

aDZ

aTZ

CBS est.

ORP

1

355 436 589 633 355 436 589 633 355 436 589 633 355 436 589 633 355 436 589 633 355 436 589 633 355 436 589 633 355 436 589 633 355 436 589 633 355 436 589 633 355 436 589 633 355 436 589 633 355 436 589 633 355 436 589 633

−83.9 −75.8 −47.3 −41.5 9.0 12.7 9.3 8.4 110.5 112.4 68.8 60.1 504.2 326.7 172.4 148.2 409.3 247.6 125.5 107.4 296.6 151.9 71.4 60.6 2.4 −33.6 −24.4 −21.5 101.3 95.6 58.6 51.2 −375.8 −136.4 −43.6 −34.4 −37.5 −21.2 −10.2 −8.6 −508.4 −295.0 −144.5 −122.9 −367.9 −178.9 −75.8 −63.1 −41.4 −41.1 −26.4 −23.2 −5890.8 −2142.8 −801.8 −657.7

−61.1 −57.0 −36.1 −31.8 −5.7 1.9 3.1 2.9 52.7 70.3 45.4 39.9 399.6 265.1 141.1 121.4 355.5 215.2 109.2 93.4 289.7 150.2 71.2 60.5 −16.8 −45.0 −29.7 −26.1 156.4 126.4 73.5 64.0 −323.4 −113.2 −34.0 −26.5 −45.8 −26.4 −12.8 −10.9 −505.8 −293.1 −143.5 −122.0 −266.5 −124.0 −49.9 −41.0 −53.2 −44.5 −26.9 −23.5 −6032.5 −2190.6 −818.5 −671.2

−51.4 −49.1 −31.4 −27.6 −11.9 −2.6 0.5 0.7 28.4 52.6 35.6 31.4 355.6 239.1 127.9 110.1 332.8 201.5 102.3 87.5 286.8 149.4 71.1 60.4 −24.9 −49.8 −32.0 −28.0 179.6 139.4 79.9 69.3 −301.4 −103.4 −30.0 −23.2 −49.3 −28.5 −14.0 −11.9 −504.7 −292.4 −143.1 −121.7 −223.8 −100.9 −38.9 −31.8 −58.1 −45.9 −27.1 −23.6 −6092.1 −2210.6 −825.6 −677.0

−46.0 −50.6 −33.6 −29.7 −18.7 −5.5 −0.5 −0.2 32.8 62.3 42.5 37.4 416.7 269.8 142.4 122.4 357.4 215.8 109.3 93.5 372.6 190.9 89.7 76.0 62.0 −5.9 −12.0 −11.2 116.6 106.2 64.5 56.3 −396.4 −147.4 −48.7 −38.7 −47.5 −27.5 −13.5 −11.5 −491.6 −285.1 −139.6 −118.8 −279.1 −127.5 −50.5 −41.5 −21.2 −27.4 −19.0 −16.9 −5861.5 −2139.1 −801.9 −657.9

−60.0 −50.2 −30.9 −27.1 15.1 13.5 8.8 7.7 141.0 119.1 69.4 60.3 486.2 313.3 165.3 142.1 314.8 195.3 100.9 86.5 232.1 114.9 52.6 44.4 −94.9 −85.8 −49.7 −43.1 120.8 104.1 61.9 54.0 −355.4 −134.9 −45.6 −36.4 −26.4 −14.8 −7.0 −5.9 −463.8 −271.1 −133.6 −113.8 −268.5 −137.3 −60.3 −50.4 −99.7 −69.3 −38.6 −33.4 −3735.6 −1433.8 −554.3 −456.7

−32.1 −30.7 −20.0 −17.6 −4.4 0.4 1.6 1.5 91.0 83.0 49.3 42.9 406.9 265.9 140.9 121.1 273.6 170.2 88.1 75.6 217.9 108.9 50.3 42.5 −124.0 −102.9 −58.0 −50.2 172.1 134.3 77.1 67.0 −316.1 −117.1 −38.1 −30.2 −34.6 −19.8 −9.6 −8.2 −470.6 −274.7 −135.2 −115.1 −197.7 −97.6 −41.2 −34.2 −103.4 −69.5 −38.0 −32.8 −3666.5 −1420.0 −550.6 −453.8

−20.3 −22.4 −15.4 −13.6 −12.7 −5.1 −1.4 −1.1 69.9 67.8 40.9 35.7 373.6 246.0 130.6 112.3 256.3 159.6 82.8 71.0 212.0 106.3 49.3 41.7 −136.3 −110.0 −61.5 −53.2 193.7 147.0 83.5 72.4 −299.6 −109.6 −35.0 −27.6 −38.1 −22.0 −10.7 −9.1 −473.5 −276.2 −135.9 −115.7 −168.0 −80.9 −33.1 −27.3 −104.9 −69.6 −37.7 −32.5 −3637.3 −1414.2 −549.1 −452.6

−28.5 −29.4 −19.6 −17.4 −11.3 −3.7 −0.5 −0.3 77.2 77.0 47.0 41.1 419.5 268.2 141.0 121.2 276.0 171.4 88.6 76.0 291.0 145.4 66.9 56.6 −51.3 −65.2 −40.4 −35.3 139.2 116.4 68.6 59.7 −369.7 −142.5 −49.1 −39.4 −35.2 −20.3 −9.9 −8.4 −454.5 −265.6 −130.8 −111.4 −203.1 −99.1 −41.4 −34.2 −83.68 −58.84 −33.01 −28.64 −3684.2 −1420.3 −550.5 −453.8

2

3

4

5

6

7

8

9

10

11

12

13

14

D


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aziridine (6) seemingly exhibits quick convergence to the estimated CBS limit using the augmented correlationconsistent basis sets, with the aDZ basis set producing a specific rotation value already within 4% of the extrapolated basis-set limit for the largest rotation at 355 nm with the CC2 method. However, the ORP computed values differ significantly from the basis-set limit, consistently overestimating the specific rotation values with both methods and never closer to the CBS estimate than the aDZ values. To further investigate the discrepancies in molecule 6, we computed CC2 specific rotations using the doubly augmented form of the aDZ basis set (daDZ).60 Mach et al.4 previously employed the daDZ and daTZ basis sets in CC optical rotation calculations. In that work,4 the CC2/daDZ results were actually in better agreement with the computed daTZ values in comparison to the CC2/aTZ results for every molecule at each wavelength. For molecule 6, the CC2/daDZ values computed here lie between the aTZ and ORP results for each wavelength. The daDZ computed specific rotations (in order of increasing wavelength) are 337.8, 176.5, 83.8, and 71.1 deg dm−1 (g/ mL)−1. These values are in slightly better agreement with ORP than the aTZ computed rotations and suggest that the aDZ and aTZ basis sets are not large enough to reliably extrapolate to the CBS limit. Molecule 7, (2R,3R)-1-chloro-2,3-dimethylaziridine, does not display the fast convergence with correlation-consistent basis sets seen for molecule 6, but the ORP basis set once again appears to underperform relative to aDZ for this chlorine-containing compound. As seen for molecule 6, CC2/daDZ computations yielded specific rotation values in better agreement with the ORP basis set than the aTZ results, with daDZ values of 30.8, −17.4, −16.3, and −14.7 deg dm−1 (g/mL)−1, and basis sets larger than aTZ should be employed to establish reliable estimates for the CBS limit values. These two chlorinated aziridine compounds examined here also show a particular sensitivity to the level of electron correlation, with large deviations seen between the CC2 and CCSD CBS estimates, especially in the case of molecule 7. Replacement of the nitrogens in the aziridine compounds with phosphorus atoms gives the phosphirane molecules (8 and 9 in Table 1 and Figure 1). For (1S,2R)-1-chloro-2methylphosphirane (8), the ORP calculations are an improvement over the aDZ values but in no case are they closer to the estimated CBS limit than the aTZ values; they are always intermediate between the two correlation-consistent basis sets in this study. CC2 specific rotation values computed with the daDZ basis set for this molecule yield values comparable to those from the ORP and aDZ calculations. The average absolute deviation between daDZ and ORP values is just below 10 deg dm−1 (g/mL)−1, and these can be found in the Supporting Information. For the aDZ, aTZ, and ORP basis sets, CCSD results are very similar to the CC2 specific rotations. In the CC2 and CCSD rotation calculations of molecule 9, (2R,3R)-1-chloro-2,3-dimethylphosphirane, the ORP basis set does not appear to accelerate convergence to the CBS limit relative to the correlation-consistent basis sets for the large negative rotations seen for this molecule. In this case, daDZ rotation values are remarkably close to the ORP results, differing by less than 1 deg dm−1 (g/mL)−1 on average (see Supporting Information). Molecule 10, (S)-2-chloropropionitrile, provides a different structural motif compared to the previous examples. For this compound, Mach et al.4 observed a lack of significant basis-set

agreement with experiment. The vast majority of specific rotation data has been measured in solution, and the solvent environment often contributes significantly to the magnitude of the rotation, even shifting the sign in cases,58 making the comparison of computed specific rotations of isolated molecules to experimental measurements oftentimes inappropriate. The interested reader can refer to reference 45 for experimental values.) The next set of specific rotation values in Table 3 correspond to compound 2, (R)-fluorooxirane. Replacement of the methyl group with fluorine leads to much smaller rotations compared to those of methyloxirane. For the CC2 computations, the aDZ basis set predicts positive specific rotation values at every wavelength, while the aTZ calculations see each of these points on the ORD curve shift toward negative values. The extrapolated CBS estimate at 355 nm predicts a CC2 specific rotation at −11.9 deg dm−1 (g/mL)−1, and the computed rotation values at longer wavelength increase to small positive values through 589 and 633 nm a bisignate structure that arises due to two or more relatively low-lying excited states with opposite sign circular dichroism rotatory strengths. The results of the ORP calculations for this molecule are of comparable accuracy to the aTZ values relative to the extrapolated basis-set limit; although, the sign differs for the small rotations computed at 589 and 633 nm. For the corresponding CCSD results, the ORP specific rotations are in good agreement with the estimated CBS values, in this case predicting negative rotations at every wavelength and always within ca. 1.5 deg dm−1 (g/mL)−1 of the CBS limit estimate. On the other hand, the CCSD/aTZ rotation values for the three longest wavelengths are all small, positive values. The CC2 results for compound 3, (R)-methylthiirane, demonstrate very similar results to methyloxirane, with excellent performance by the ORP basis set, at each wavelength predicting a specific rotation value in between the aTZ and the estimated CBS limit values. The success of ORP is also seen in the CCSD calculations of (R)methylthiirane. Here, the importance of the level of electron correlation is apparent in the shorter wavelength regime of this molecule’s ORD spectrum, where the CC2 method predicts a specific rotation value at 355 nm near 28 deg dm−1 (g/mL)−1 compared to the CCSD result of around 70 deg dm−1 (g/ mL)−1 at the basis-set limit. The addition of a second methyl group in (2R,3R)-dimethylthiirane (molecule 4) leads to significantly larger rotation values at the CC2 and CCSD levels. In this case, the aTZ values fall closer to the CBS limit specific rotations for every wavelength at both levels of theory compared to the ORP basis set, although the ORP calculations do offer significant improvement relative to the aDZ values. For the largest rotation seen at 355 nm, the ORP and aTZ values differ by around 17 and 13 deg dm−1 (g/mL)−1 with the CC2 and CCSD methods, respectively, and the two basis sets provide very comparable values at longer wavelengths. Molecules 5, 6, and 7 represent another class of threemembered heterocyclic compounds as all are derivatives of aziridine. (2R,3R)-2,3-Dimethylaziridine (5) in Figure 1 displays excellent agreement between ORP rotation values and those computed with the aTZ basis set at both CC2 and CCSD levels of theory. The maximum deviation is only 2.4 deg dm−1 (g/mL)−1 from the CCSD specific rotation at 355 nm. The ORP and aTZ differences are, in fact, usually less than 1 deg dm−1 (g/mL)−1. For the chlorinated aziridines (6 and 7), the situation is much different. (1S,2R)-1-Chloro-2-methylE


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limit values. For most molecules examined here, the ORP rotation values are comparable to the larger aug-cc-pVTZ results, and in a few cases, the ORP results lie closer to the estimated CBS limit relative to aug-cc-pVTZ. However, the performance of the ORP basis set is less clear for some compounds containing second-row atoms, especially in some of the three-membered heterocyclic molecules containing chlorine atoms. In these cases, the ORP results lie farther from the extrapolated CBS limits than the specific rotations computed with the aug-cc-pVDZ basis set. This is also the case for one of the sulfur-containing compounds examined here, (R)-4-methylthiete 1,1,-dioxide. However, specific rotation values computed with an additional set of diffuse functions using the d-aug-cc-pVDZ basis set indicate that the aug-ccpVDZ and aug-cc-pVTZ results, in these cases, may lie far from convergence, and larger basis sets should be used for reliable extrapolations to the CBS limit. The results presented here suggest that the ORP basis set can be recommended as an economical alternative to larger augmented correlationconsistent basis sets, such as aug-cc-pVTZ, when calculating optical rotation spectra of molecules containing only first-row atoms at the coupled cluster level. For some molecules containing second-row atoms, larger basis sets beyond aug-ccpVTZ will be necessary to obtain reliable estimates of the basis-set limit and accurately assess the performance of the ORP basis set.

dependence for specific rotations and noted that the basis-set limit is nearly reached already with a doubly augmented correlation-consistent basis set. Unlike the aforementioned chlorine-containing compounds, no large discrepancies exist between aDZ and aTZ results and those obtained with the ORP basis set, as ORP performs unquestionably well for (S)-2chloropropionitrile, in excellent agreement with the CBS limit estimates, always within 2 deg dm−1 (g/mL)−1 at the CC2 level and within 3 deg dm−1 (g/mL)−1 for the slightly smaller (in magnitude) rotations from CCSD results. Molecules 11, 12, and 13 all include four-membered rings. In (S)-3-methylcyclobutene (11), the large negative rotations do not appear to be very sensitive to the choice of basis set or correlation level in these two CC approximations. The difference in aDZ and aTZ values are never larger than 7 deg dm−1 (g/mL)−1, seen for the CCSD specific rotation near −470 deg dm−1 (g/mL)−1 at 355 nm. Because the rotation values converge so quickly with respect to basis set, the ORP values are never closer to the basis-set limit when compared to aDZ, but they are always in good agreement with the CBS estimates. Both the correlation and basis-set effects are much more pronounced in molecule 12, (R)-2-methyloxetane. For the specific rotation at 355 nm, the difference in the correlation-consistent basis sets exceeds 100 deg dm−1 (g/ mL)−1, while the difference in CC2 and CCSD specific rotations at this wavelength is one of the largest considered in this work. With both CC2 and CCSD computations, the ORP results are in good agreement with the larger aTZ calculations, usually within a few deg dm−1 (g/mL)−1 and offering large improvement over the aDZ basis set. The ORP basis set, however, is less effective in the final four-membered-ring compound studied here, (R)-4-methylthiete 1,1,-dioxide (13). This molecule is another example of the aug-cc-pVXZ basis sets converging toward the CBS limit very efficiently, within 5 deg dm−1 (g/mL)−1 of the estimated basis-set limit, utilizing only aDZ in most cases. Here, the ORP basis set underestimates the magnitude of these negative rotations at each wavelength. The final compound in this study is (1S,4S)-norbornenone (14), which exhibits a very large-magnitude specific rotation relative to structurally similar molecules because of its optimal positioning of the carbonyl and alkenyl moieties.61,62 In an absolute sense, the deviations between ORP and the CBS limit appear to be significant, especially for the CC2 specific rotation of (1S,4S)-norbornenone computed at 355 nm exceeding −6000 deg dm−1 (g/mL)−1 at the CC2 CBS limit, although the deviations never represent more than a few percentage points. The closest values to the ORP rotations at the CC2 level are those from the aDZ basis set, while the smaller CCSD/ORP rotations are very close to the aTZ values, with the largest relative deviation between ORP and aTZ rotations less than 0.5%.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.8b04183.

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Optimized geometries of molecules 2, 8, and 9; CC2/daug-cc-pVDZ specific rotation values of molecules 6, 7, 8, and 9; and ORP basis set for P, S, and Cl atoms (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

J. Coleman Howard: 0000-0003-4296-1849 Angelika Baranowska-Łączkowska: 0000-0001-9285-4991 T. Daniel Crawford: 0000-0002-7961-7016 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by a grant (ACI-1450169) from the U.S. National Science Foundation and a HASI grant from the U.S. Department of Defense High Performance Computing Modernization Program as well as a grant (CHE-1465149) from the U.S. National Science Foundation. The authors also acknowledge Advanced Research Computing at Virginia Tech for providing computational resources and technical support that have contributed to the results reported within the paper. This work has been supported by the Foundation for Polish Science within the PARENT/BRIDGE program (Pomost/ 2013-7/1), cofinanced from European Regional Development Fund within Innovative Economy Operational Program.

4. CONCLUSIONS Specific rotations were computed for a test set of 14 compounds with the CC2 and CCSD methods to examine the basis-set convergence and evaluate the performance of the ORP basis set, designed specifically for the calculation of optical rotations. For each molecule, CC2 and CCSD specific rotations were computed with Dunning’s augmented correlation-consistent aug-cc-pVDZ and aug-cc-pVTZ basis sets. Basis-set extrapolations were performed using an inverse power relationship and a fixed exponent to estimate CBS F


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