Optical Rotation from Coupled Cluster and Density Functional Theory

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Optical rotation from coupled cluster and density functional theory: the role of basis set convergence Shokouh Haghdani, Per-Olof Åstrand, and Henrik Koch J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.5b00721 • Publication Date (Web): 16 Dec 2015 Downloaded from http://pubs.acs.org on December 21, 2015

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Journal of Chemical Theory and Computation

Optical Rotation from Coupled Cluster and Density Functional Theory: The Role of Basis Set Convergence Shokouh Haghdani, Per-Olof ˚ Astrand, and Henrik Koch∗ Department of Chemistry, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway E-mail: [email protected]

∗

To whom correspondence should be addressed

1

ACS Paragon Plus Environment


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Abstract We have calculated the electronic optical rotation of seven molecules using coupled cluster singles-doubles (CCSD) and the second-order approximation (CC2) employing the aug-cc-pVXZ (X=D, T or Q) basis sets. We have also compared to time-dependent density functional theory (TDDFT) by utilizing two functionals B3LYP and CAMB3LYP and the same basis sets. Using relative and absolute error schemes, our calculations demonstrate that the CAM-B3LYP functional predicts optical rotation with the minimum deviations compared to CCSD at λ = 355 and 589.3 nm. Furthermore, our results illustrate that the aug-cc-pVDZ basis set provides the optical rotation in a good agreement with the larger basis sets for molecules not possessing small-angle optical rotation at λ = 589.3 nm. We have also performed several two-point inverse power extrapolations for the basis set convergence, i.e. OR∞ + AX −n , using the CC2 model at λ = 355 and 589.3 nm. Our results reveal that a two-point inverse power extrapolation with the aug-cc-pVTZ and aug-cc-pVQZ basis sets at n = 5 provides optical rotation deviations similar to those of aug-cc-pV5Z with respect to the basis limit.

Keywords: chirality, optical activity, coupled cluster method, density functional theory, two-point extrapolation

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I. Introduction Optical activity inherent to chiral molecules has been of great interest to a broad range of the scientific community during the past decades due to its fundamental importance in theoretical and experimental chemistry as well as potential applications in medical chemistry, biochemistry, and industry. 1–9 One of the most valuable practical applications of optical activity is the determination of absolute configuration (AC) of chiral molecules. The importance of determination the AC manifests itself, for example, in medical chemistry where one of the enantiomers may be undesirable and even harmful. 10 Two enantiomers of chiral molecules can be distinguished spectroscopically by investigating optical responses in refraction, absorption, or scattering. 11 One of the most well-known optical properties of chiral molecules is the optical rotation which is defined by the ability of a molecule to rotate the plane of an incident linearly polarized electromagnetic wave. 11 The optical rotation is wavelength dependent and equal in magnitude but opposite in sign for two enantiomers of a chiral compound. The AC of an enantiomer is realized solely by the sign of the optical rotation if the experiment agrees with a theoretical calculation for a given enantiomer. Several studies have demonstrated the promise of this approach employing both theoretical and experimental methods. 12–20 Hence, a tool for prediction of the AC is important in this context. The determination of AC has motivated considerable theoretical efforts. First, empirical and semi-empirical approaches were used as practical methods. 21–25 However, quantum chemical methods have been developed extensively for predicting accurate responses of chiral molecules where the pioneering work utilized Hartree-Fock (HF) theory. 26–32 To include electron correlation, density functional theory (DFT) 20,33–44 and coupled cluster (CC) 37,39,42,43,45–52 approaches have been employed. The CC methods provide the most accurate predictions of optical rotations and have drastically enhanced the accuracy of theoretical predictions compared to gas-phase experiments. 49,53,54 With the various methods, a diverse group of basis sets have been investigated. Basis sets with diffuse functions have 3



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demonstrated good efficiency for calculating the optical rotation. 33,34,38 Among these basis sets, the basis sets augmented with diffuse functions, aug-cc-pVXZ (X=D, T and Q), 55–58 have been employed extensively. 41–44 Also, the large polarized (LPol) basis sets 59 provide comparable optical rotation results to the aug-cc-pVXZ basis sets. 40–42,44 Basis set convergence is crucial for any level of theory and especially for correlated methods where convergence to the complete basis set limit is slower than for the HF and DFT methods. 60,61 From basis set studies one can find extrapolation schemes and estimate properties in the basis set limit on large systems where large basis sets cannot be employed. 60–63 It has been demonstrated that the energy converges by an inverse power scheme for correlated methods 60,61 while the HF energy converges exponentially. 62,63 Also, the basis set convergence of the optical rotation has been investigated at the DFT level using inverse power and exponential fits. 64,65 This paper consists of two parts. In the first part, we perform electronic optical rotation calculations for seven rigid chiral molecules using HF, DFT, and CC models in the gas-phase. Vibrational contributions (both harmonic and anharmonic), 66,67 solvent effects 68–74 and the temperature dependence 75–77 should be included for a detailed comparison between calculated and experimental optical rotations. However, in this work we focus on the electronic optical rotation using different methods with a hierarchy of augmented basis sets at three different wavelengths. We choose CCSD as the reference method and suggest a method including a DFT functional and basis set which predicts the optical rotation closest to CCSD. In the second part, we present systematic optical rotation calculations to investigate the basis set convergence and extrapolation to the aug-cc-pV6Z basis set at the CC2 level using the modified dipole-velocity gauge (MVG). 47 The inverse power scheme is examined for different exponents to estimate the optical rotation values in the basis set limit (aug-cc-pV6Z). Finally, we suggest an appropriate basis set for computing optical rotations which is crucial especially to large molecules where large basis sets are computationally challenging to use. The paper is organized as follows. In section II, we discuss the chosen computational

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methods. In section III, the results are presented and discussed. Finally, in section IV, we summarize our results and give concluding remarks.

II. Computational Details The specific rotation in unit of deg[dm g/cm3 ]−1 is defined by the total optical rotation normalized through path length (dm) and concentration of system (g/cm−3 ) as 78,79

[α] = 28800π

2

˜2 β ′ (ω)

ν NA a40

(1)

M

where NA is Avogadro’s number, a0 is the Bohr radius in cm, ν˜ is the frequency of the incident light in cm−1 (ω is the frequency in atomic units), M is the molecular weight in ′

g/mol, and β (in atomic units, a40 ) denotes to the Cartesian trace of the frequency-dependent electric dipole−magnetic dipole polarizability.

O

O

O

N

F (2S)-(1)

(2S)-(2)

(2S,3S)-(3)

(3R)-(4)

O

F

O (S)-(5)

(2R)-(6)

(1R,5S)-(7)

Figure 1: The molecule structures (1)-(7) considered for the specific rotation calculations In this paper, we study the following chiral molecules: (1) (2S)-2-fluorooxirane, (2) (2S)2-methyloxirane, (3) (2S,3S)-2,3-dimethyloxaziridine, (4) (3R)-3-methylcyclobutene, (5) (S)-1,3-dimethylallene, (6) (2R)-2-fluoroheptane, and (7) (1R,5S)-6,8-dioxabicyclo[3.2.1]octane, shown in Figure 1. The test set contains molecular structures with various sizes, chemical functionalities and enantiomers which cause a wide variety of optical rotation values. In 5



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previous studies, the smaller molecules (1), (2) and (5) have been investigated using both DFT and CC methods 39,42,43,48 while only the DFT method has been employed for studying molecules (3), (4) and (7). 34,41,44,64 Although (2R)-1-fluoroheptane was studied using the CC model, 80,81 molecule (6) has not been investigated before. The optical rotation calculations are carried out using a development version of DALTON 82 using time-dependent Hartree-Fock (TDHF), time-dependent density functional theory (TDDFT), coupled cluster singles and doubles (CCSD), 83 and second-order approximate coupled cluster singles and doubles (CC2). 84 The B3LYP functional 85–87 and the long-range corrected CAM-B3LYP functional 88 are employed for the TDDFT calculations. The core 1s orbitals for C, O, and N, are held frozen in the CC calculations. We have optimized the geometries at the DFT level using the B3LYP functional and the cc-pVTZ basis set 89 by the NWChem software. 90 The basis sets for computing the electronic optical rotations are augmented with diffuse functions, aug-cc-pVXZ (X=D, T, or Q). 55–58 We also investigate different gauges: dipole-length gauge (LG), gauge-including atomic orbitals (GIAOs), 91–93 dipole-velocity gauge (VG), and modified dipole-velocity gauge (MVG). 47 Origin-independent results for TDHF and TDDFT are obtained using GIAOs while for CCSD and CC2 methods this can be achieved by employing the MVG. We consider three wavelengths for the incident light: λ = 355, 589.3 (canonical sodium D line excitation) and 633 nm. The optical rotations show the largest variations with respect to the basis sets and different methods at λ = 355 nm but, on the other hand, experimental measurements of specific rotations are mostly done at λ = 589.3 nm. The advantage of the sodium D line wavelength is that experimentally, most organic molecules absorb less incident photons and therefore, the theoretical analysis of the optical properties at the sodium D line wavelength would provide the most important information closest to the experimental measurements. 11 The Cholesky decomposition of the two-electron integrals is employed for the B3LYP and CC2 calculations with a decomposition threshold of 10−8 which is sufficient to obtain conventional results within negligible deviations. 46,94 We note that for large-size molecules, the optical rotations for the CCSD method with the

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aug-cc-pVTZ and aug-cc-pVQZ basis sets are not available in Tables 3-7 solely because of high demands on the computational resources.

III. Results and Discussion III. A. Comparison Between Different Methods The optical rotation (OR) of molecules (1) and (2) (see Figure 1 and Tables 1-2) (2S)-2fluorooxirane and (2S)-2-methyloxirane, respectively, have already been studied extensively theoretically 34,35,38,39,41,47,48,69 and experimentally. 68,95,96 These two molecules have smallangle optical rotations, and are therefore challenging to predict for both DFT and CC with respect to both sign and magnitude. It has been demonstrated that even CC models give the wrong sign and magnitude of the optical rotation for molecule (2) also by including triple excitations (CC3 model 97,98 ) at λ = 355 nm. 70 However, including zero-point vibrational effects calculated at the CCSD level predicts correct optical rotations both in magnitude and sign at the three wavelengths, 75 whereas zero-point vibrational corrections calculated at DFT/B3LYP level to the CC3 model overestimate the magnitude compared to the gas-phase experiment. 99 Therefore, for small optical rotations such as for molecules (1) and (2), the vibrational contribution to the optical rotation is essential when comparing to experiments. Table 1: Specific rotation of molecule (1), (2S)-2-fluorooxirane, (deg[dm g/cm3 ]−1 ) Basis set λ = 355.0nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 589.3nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 633.0nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ

HF GIAOs LG

DFT/B3LYP GIAOs LG MVG

DFT/CAM-B3LYP GIAOs LG MVG

LG

CC2 VG

MVG

LG

CCSD VG MVG

26.9 43.8 48.2

21.9 42.0 46.9

31.8 57.8 61.6

24.3 55.5 60.4

36.4 55.9 60.4

14.8 40.4 44.7

8.3 37.9 43.4

14.5 36.9 42.7

-31.2 -11.9 -8.3

-411.5 -100.6 -41.9

-12.2 2.3 5.9

-24.9 -3.0 2.6

-470.9 -167.6 -111.6

-19.0 0.4 6.5

8.0 13.6 15.1

6.3 13.0 14.7

5.1 13.7 15.0

2.9 12.9 14.6

5.9 12.5 14.2

1.7 10.1 11.6

-0.2 9.2 11.1

0.9 8.5 10.6

-15.8 -8.4 -6.8

-409.6 -107.1 -50.2

-10.3 -4.1 -2.4

-11.1 -3.4 -1.3

-461.9 -170.8 -118.6

-9.9 -2.8 -0.5

6.7 11.6 12.9

5.2 11.0 12.5

4.1 11.6 12.7

2.2 10.8 12.3

5.0 10.8 12.3

1.3 8.5 9.8

-0.4 7.7 9.4

0.9 7.4 9.2

-13.9 -7.5 -6.0

-408.5 -106.8 -50.1

-9.2 -3.8 -2.3

-9.7 -3.1 -1.2

-460.7 -170.6 -118.6

-8.7 -2.6 -0.6

7



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Table 2: Specific rotation of molecule (2), (2S)-2-methyloxirane, (deg[dm g/cm3 ]−1 ) Basis set λ = 355.0nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 589.3nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 633.0nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ

HF GIAOs LG



LG

CC2 VG

MVG

LG

CCSD VG MVG

-19.8 -5.8 -4.4

-24.1 -6.9 -4.7

11.0 29.6 31.7

-0.6 26.5 30.6

1.5 24.6 31.4

-16.2 3.6 5.5

-27.0 0.5 4.7

-27.9 -3.0 3.8

-46.3 -27.7 -26.8

-580.5 -360.2 -337.0

-76.1 -53.9 -48.9

-50.1 -27.4 -24.3

-318.0 -112.1 -91.2

-54.9 -27.5 -21.3

-11.2 -6.1 -5.5

-12.5 -6.4 -5.6

-14.6 -7.2 -6.5

-17.9 -8.2 -6.8

-18.3 -9.3 -7.0

-16.7 -9.3 -8.7

-19.8 -10.3 -8.9

-20.6 -11.7 -9.5

-35.4 -25.8 -24.5

-549.2 -340.1 -319.2

-44.8 -33.8 -31.1

-26.9 -17.9 -16.5

-292.4 -103.1 -85.9

-29.2 -18.4 -15.9

-9.9 -5.5 -5.1

-11.0 -5.8 -5.0

-13.4 -6.9 -6.3

-16.2 -7.8 -6.6

-15.9 -8.4 -6.5

-14.8 -8.5 -7.9

-17.5 -9.3 -8.1

-18.2 -10.4 -8.4

-31.4 -23.0 -21.8

-543.9 -336.1 -315.5

-39.4 -29.8 -27.4

-23.7 -15.9 -14.6

-288.9 -101.0 -84.1

-25.7 -16.3 -14.2

Table 3: Specific rotation of molecule (3), (2S,3S)-2,3-dimethyloxaziridine, (deg[dm g/cm3 ]−1 ) Basis set λ = 355.0 aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 589.3 aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 633.0 aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ

HF GIAOs LG



-572.2 -546.3 -543.2

-579.8 -547.6 -542.9

-631.5 -591.6 -584.3

-643.1 -670.9 -596.1 -608.8 -585.2 -586.6

-583.6 -550.3 -544.1

-593.9 -613.7 -692.7 -1779.2 -715.1 -541.5 -1039.9 -553.4 -553.9 -562.8 -639.3 -1532.2 -654.1 -508.3 -874.3 -527.9 -544.7 -545.3 -625.1 -1491.3 -630.9

-180.7 -172.4 -171.4

-183.1 -172.9 -171.3

-191.4 -180.0 -177.9

-195.6 -205.1 -181.6 -186.6 -178.3 -180.2

-180.9 -170.9 -169.1

-184.4 -192.5 -209.8 -1279.6 -215.5 -168.0 -172.1 -177.0 -194.7 -1076.2 -198.1 -157.9 -169.4 -169.4 -190.9 -1052.3 -191.9

-664.5 -516.9

-178.1 -170.5

-155.2 -148.1 -147.2

-157.2 -148.5 -147.1

-164.0 -154.2 -152.6

-167.5 -174.4 -155.6 -158.5 -152.8 -153.1

-155.1 -146.5 -145.0

-158.1 -163.2 -179.7 -1248.6 -184.5 -144.1 -147.6 -149.7 -166.8 -1047.7 -169.6 -135.4 -145.2 -145.2 -163.6 -1024.8 -164.4

-639.5 -493.0

-153.0 -146.5

LG

CC2 VG

MVG

LG

CCSD VG

MVG

Table 4: Specific rotation of molecule (4), (3R)-3-methylcyclobutene, (deg[dm g/cm3 ]−1 ) Basis set λ = 355.0nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 589.3nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 633.0nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ

HF GIAOs LG



LG

CC2 VG

MVG

LG

CCSD VG MVG

444.3 442.2 444.1

446.2 443.9 444.1

599.2 574.9 577.8

588.4 577.0 577.9

589.1 579.5 578.0

570.6 550.4 552.8

561.7 552.2 552.9

558.0 552.9 552.1

536.0 532.0 529.4

689.4 675.6 670.1

508.1 504.7 501.7

445.5 447.3

547.3 551.7

462.5 468.5

126.2 125.5 126.0

126.8 126.0 126.0

169.4 161.1 162.0

166.0 161.8 162.1

166.7 165.4 161.6

159.8 153.3 154.1

157.0 153.9 154.1

158.6 155.5 155.4

152.5 151.2 150.6

325.7 314.1 310.8

144.4 143.2 142.5

127.7 127.9

218.0 217.8

133.2 134.6

107.6 106.9 107.4

108.2 107.4 107.4

144.2 137.2 137.9

141.3 137.8 138.0

141.3 138.4 138.0

136.2 130.4 131.1

133.7 130.9 131.1

132.9 131.2 131.0

129.9 128.8 128.3

304.3 292.8 289.7

123.0 121.9 121.4

108.8 109.0

198.4 197.9

113.5 114.7

Tables 1 and 2 summarize our results for the molecules (1) and (2) using various methods and basis sets. We confirm that CC and DFT models give contradicting signs and different magnitudes for the small-angle OR of molecule (1) at all the wavelengths λ = 355, 589.3 and 8


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Table 5: Specific rotation of molecule (5), (S)-1,3-dimethylallene, (deg[dm g/cm3 ]−1 ) Basis set λ = 355.0nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 589.3nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 633.0nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ

HF GIAOs LG



LG

CC2 VG

MVG

LG

CCSD VG MVG

319.3 322.3 323.7

315.8 321.2 324.4

265.6 280.8 285.9

243.8 278.9 286.0

254.7 288.4 288.2

311.3 326.2 328.8

292.5 324.4 329.2

307.7 333.4 333.8

223.8 270.5 284.1

865.1 940.4 947.7

310.6 358.6 367.4

178.6 206.6 213.4

556.5 630.2 638.1

370.2 428.2 436.4

111.1 111.8 112.0

110.2 111.5 112.2

125.1 128.3 129.4

119.6 127.9 129.5

121.1 130.5 130.3

121.2 124.6 125.2

116.1 124.2 125.3

115.5 125.0 124.5

97.1 106.2 108.6

675.6 713.3 713.1

120.6 131.5 132.8

74.3 79.7 80.7

319.3 350.4 351.7

133.0 148.4 150.1

95.5 96.2 96.4

94.8 95.9 96.6

108.6 111.2 112.2

103.9 110.9 112.3

105.6 113.1 112.9

104.6 107.5 107.9

100.3 107.1 108.1

103.1 109.0 108.9

84.2 91.8 93.8

659.3 695.3 694.8

104.2 113.8 114.5

64.3 68.9 69.7

301.0 329.8 330.8

114.7 127.8 129.2

Table 6: Specific rotation of molecule (6), (2R)-2-fluoroheptane, (deg[dm g/cm3 ]−1 ) Basis set λ = 355.0nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 589.3nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 633.0nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ

HF GIAOs LG



LG

CC2 VG

MVG

LG

CCSD VG

MVG

-85.7 -86.2 -85.7

-76.7 -83.5 -85.2

-125.9 -124.3 -123.2

-117.7 -120.9 -122.7

-123.4 -122.4 -123.1

-117.4 -116.1 -114.9

-107.9 -112.5 -114.3

-113.7 -115.5 -117.7

-105.1 -110.4 -111.6

-193.2 -192.2 -191.4

-112.0 -113.9 -113.9

-100.8

-162.7

-109.2

-28.4 -28.5 -28.4

-25.1 -27.6 -28.2

-40.0 -39.5 -39.1

-36.8 -38.3 -39.9

-39.1 -38.9 -39.1

-37.7 -37.3 -36.9

-34.2 -36.0 -36.7

-36.5 -37.1 -37.8

-32.7 -35.0 -35.5

-116.9 -114.7 -114.0

-35.8 -36.5 -36.5

-32.0

-88.8

-35.3

-24.4 -24.5 -24.4

-21.6 -23.7 -24.3

-34.3 -33.8 -33.5

-31.5 -32.8 -33.4

-33.6 -33.3 -33.5

-32.4 -32.0 -31.7

-29.3 -30.9 -31.5

-31.3 -31.8 -32.5

-28.1 -30.1 -30.5

-111.9 -109.6 -108.9

-30.7 -31.3 -31.4

-27.5

-83.8

-30.3

Table 7: Specific rotation of molecule (7), (1R,5S)-6,8-dioxabicyclo[3.2.1]octane, (deg[dm g/cm3 ]−1 ) Basis set λ = 355.0nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 589.3nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ λ = 633.0nm aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ

HF GIAOs LG



LG

CC2 VG

MVG

LG

CCSD VG MVG

205.5 199.6 199.3

202.8 200.3 199.6

284.6 274.8 273.9

279.7 276.4 274.7

283.8 278.9 275.1

264.4 254.8 253.9

259.3 256.1 254.3

261.9 257.2 253.6

300.8 295.5 292.9

453.6 387.2 380.4

288.4 281.9 279.1

260.2

408.0

263.5

67.0 65.1 64.9

66.1 65.3 65.0

93.4 90.1 89.7

91.8 90.6 90.0

93.1 91.1 90.1

85.3 82.2 82.0

83.6 82.6 82.2

85.6 83.1 82.3

98.3 95.8 94.8

259.5 196.7 191.6

94.3 91.3 90.3

84.0

230.0

85.5

57.6 55.9 55.8

56.8 56.1 55.9

80.3 77.4 77.1

78.9 77.8 77.3

80.1 78.4 77.4

73.2 70.5 70.3

71.8 70.9 70.4

72.7 71.2 70.6

84.4 82.3 81.5

246.2 183.7 180.2

81.0 78.4 77.6

72.2

218.0

73.5

633 nm. To our knowledge, there are no experimental data available for molecule (1). The gas-phase experimental results for molecule (2) are reported as 7.5 and −8.4 deg[dm g/cm3 ]−1 for λ = 355 and 633 nm, respectively. 96 Although the DFT calculations on molecule (2) 9



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

give a sign consistent with the experiments at λ = 355 nm, the most accurate calculations are provided by the CC models 53,54,75 and the success of DFT in this case stems from a cancellation of errors in obtaining the lowest electronic excitation energies. 48 The excitation energies for molecules (1) and (2) are far from the selected λ wavelengths. We calculate the two lowest excitation energies using the CC2 method and the aug-cc-pVTZ basis set for molecule (1) to 163.8 nm and 158.6 nm which are close to the previously reported values 163.9 nm and 158.7 nm. 39 For molecule (2), the two lowest excitations are at 188.8 nm and 177.4 nm which are comparable to the reported values 48 as well as the corresponding experimental values, 174.1 nm (7.12 eV) and 160.0 nm (7.75 eV). 100 The more accurate CC models with higher orders of correlations was suggested for resolving the sign discrepancy. 39,70 Thus, we have expanded our calculations to the CC3 level for molecules (1) and (2) with the aug-cc-pVDZ basis set at λ = 355 and 589.3 nm. For molecule (1), the CC3(MVG) method results in -20.1 deg[dm g/cm3 ]−1 at λ = 355 nm and -9.5 deg[dm g/cm3 ]−1 at λ = 589.3 nm while CCSD gives -19.0 and -9.9 deg[dm g/cm3 ]−1 . At λ = 355 and 589.3 nm, the results of CC3(MVG) for molecule (2) are -50.2 and -27.7 deg[dm g/cm3 ]−1 while CCSD finds -54.9 and -29.2 deg[dm g/cm3 ]−1 . Comparing to CCSD, the CC3 level of calculations only introduces small corrections to the optical rotation magnitudes in agreement with previous results for molecule (2). 70 Therefore, the additional correlation is also unable to obtain the correct sign for the optical rotation. To compare the CC2, CCSD and CC3 methods for molecules (1) and (2), we analyze the optical rotation values of LG and MVG using the aug-cc-pVDZ basis set at λ = 589.3 nm since the difference between LG and MVG results indicates the deviation of a method from the exact theory in the limit of a complete basis set. 101–105 For molecule (1), the CC2(LG) and CC2(MVG) methods give -15.8 and -10.3 deg[dm g/cm3 ]−1 while the CCSD(LG) and CCSD(MVG) results are -11.1 and -9.9 deg[dm g/cm3 ]−1 . The CC3 method results in -8.6 and -9.5 deg[dm g/cm3 ]−1 using LG and MVG, thus the difference between two gauges is reduced only 0.2 deg[dm g/cm3 ]−1 compared to CCSD but still in good agreement with each

10


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other. For molecule (2), the CC2(LG) and CC2(MVG) methods give -35.4 and -44.8 deg[dm g/cm3 ]−1 whereas CCSD(LG) and CCSD(MVG) predict -26.9 and -29.2 deg[dm g/cm3 ]−1 . The CC3(LG) and CC3(MVG) results are -24.3 and -27.7 deg[dm g/cm3 ]−1 so, the difference of LG and MVG using the CCSD method is close to the CC3(LG) and CC3(MVG) variation. On the other hand, the differences of the LG and MVG results using CC2 are larger than the two other methods which indicates that the CCSD method can be essential for investigating molecules with small-angle optical rotation. We now turn to the deviations between the GIAOs and MVG results for the DFT methods while for the CC models, the differences of LG and MVG are considered with the aug-ccpVXZ (X=D, T, Q) basis sets at λ = 589.3 nm. Using the results of molecule (1) shown in Table 1, the differences between B3LYP(GIAOs) and B3LYP(MVG) are given as 0.8, 1.2 and 0.8 deg[dm g/cm3 ]−1 while the deviations of CAM-B3LYP(GIAOs) and CAM-B3LYP(MVG) are 0.8, 1.6 and 1.0 deg[dm g/cm3 ]−1 . The variations between the CC2 results using LG and MVG are given as 5.5, 4.3 and 4.4 deg[dm g/cm3 ]−1 whereas the LG and MVG deviations are obtained 1.2, 0.6 and 0.8 deg[dm g/cm3 ]−1 employing the CCSD method. For the findings of molecule (2) given in Table 2, the variations between GIAOs and MVG for both B3LYP and CAM-B3LYP functionals are reduced with increasing the basis set size. The differences of B3LYP(GIAOs) and B3LYP(MVG) are 3.7, 2.1 and 0.5 deg[dm g/cm3 ]−1 while the deviations between GIAOs and MVG using CAM-B3LYP change as 3.9, 2.4 and 0.8 deg[dm g/cm3 ]−1 . Also, the CC2(LG) and CC2(MVG) differences are reduced to 9.4, 8.0 and 6.6 deg[dm g/cm3 ]−1 whereas the CCSD(LG) and CCSD(MVG) show smaller variations 2.3, 0.5 and 0.6 deg[dm g/cm3 ]−1 to the size of the basis sets. For molecules (1) and (2), the gauge variations are reduced using B3LYP and CAM-B3LYP which show the convergence between GIAOs and MVG with increasing the size of the basis sets. The CC2(LG) and CC2(MVG) deviations are large while the CCSD differences are reduced with the basis set size. Considering different methods and basis sets, our findings are consistent with the previously reported results for molecules (1) and (2). To be concise and restricting the discussion

11



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to the aug-cc-pVDZ basis set, we note that for molecule (1) we obtain 31.8, -12.2 and -19.0 deg[dm g/cm3 ]−1 using B3LYP(GIAOs), CC2(MVG) and CCSD(MVG) at λ = 355 nm which are close to those reported in the literature i.e. 31.9, -12.1 and -18.9 deg[dm g/cm3 ]−1 , respectively. 39 B3LYP(GIAOs) with the LPol-ds basis set gives 62.6 deg[dm g/cm3 ]−1 40 which is comparable to 61.6 deg[dm g/cm3 ]−1 obtained using the aug-cc-pVQZ basis set at λ = 355 nm. For molecule (2), B3LYP(GIAOs), CC2(MVG) and CCSD(MVG) give 11.0, -76.1 and -54.9 deg[dm g/cm3 ]−1 at λ = 355 nm which are in good agreement with 12.1, -74.6 and -56.4 deg[dm g/cm3 ]−1 . 42 We also obtain the molar rotation of molecule (2) as -8.5 and -9.9 deg[mol dm/cm3 ]−1 using B3LYP(GIAOs) and CAM-B3LYP(MVG) at λ = 589.3 nm which are comparable to the reported values, -9.9 and -12.0 deg[mol dm/cm3 ]−1 . 41 At λ = 355 nm, the reported values are 34.1, -34.6 and -21.7 deg[dm g/cm3 ]−1 using B3LYP(GIAOs), CC2(MVG) and CCSD(MVG) for the LPol-ds basis set. 42 The optical rotation calculations for molecules (3)-(7) are presented in Tables 3-7. For these molecules, all the different methods predict the same signs at all wavelengths. Also, the magnitude of the ORs follow similar trends, namely the values reduce for molecules (3), (4), (6) and (7) while increases for molecule (5) with increasing size of the basis set. We note that the OR at λ = 633 nm exhibits similar behaviour as compared to λ = 589.3 nm. Therefore the discussion of the results is limited to λ = 355 and 589.3 nm where significant variations can be seen at λ = 355 nm while the optical rotation magnitudes do not show large variations for different methods and basis sets at λ = 589.3 nm. We compare the origin-independent results using the GIAOs for the HF and DFT calculations and the MVG for the CC methods. We further note that the differences of GIAOs and MVG results for the DFT methods as well as the deviations between LG and MVG findings for the CC models via increasing the basis set at λ = 589.3 nm. As for molecules (1) and (2), the two lowest excitation energies are calculated by employing the CC2 method and the aug-cc-pVTZ basis set for molecules (3)-(5) while the aug-cc-pVDZ basis set is used for molecules (6) and (7) because of their relatively large sizes.

12


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In Table 3, we report the results for molecule (3). The largest basis set for CCSD is aug-cc-pVTZ and therefore this basis set is considered for comparison between the different methods. For λ = 355 nm, HF, B3LYP and CAM-B3LYP give -546.3, -591.6 and -550.3 deg[dm g/cm3 ]−1 while CC2 and CCSD give -654.1 and -527.9 deg[dm g/cm3 ]−1 , respectively. All the methods overestimate the CCSD result with the largest deviation for the CC2 method. The HF, B3LYP and CAM-B3LYP methods give the ORs -172.4, -180.0 and -170.9 deg[dm g/cm3 ]−1 whereas the CC2 and CCSD results are -198.1 and -170.5 deg[dm g/cm3 ]−1 at λ = 589.3 nm. As seen, all the methods are in good agreement except CC2 which overestimates the optical rotation compared to CCSD with 27.6 deg[dm g/cm3 ]−1 deviation. In particular, the CAM-B3LYP functional gives results similar to CCSD. At λ = 589.3 nm, the differences of B3LYP(GIAOs) and B3LYP(MVG) are given as 13.7, 6.6 and 2.3 deg[dm g/cm3 ]−1 with the size of basis set while the deviations of CAM-B3LYP results using GIAOs and MVG are reduced as 11.6, 6.1 and 0.3 deg[dm g/cm3 ]−1 . For the DFT methods using the aug-cc-pVQZ basis set, the small deviation between GIAOs and MVG indicates that the result is close to the optical rotation in the basis set limit. The deviations between LG and MVG are 5.7, 3.4 and 1.0 deg[dm g/cm3 ]−1 using the CC2 method whereas the differences between CCSD(LG) and CCSD(MVG) are 10.1 and 12.6 deg[dm g/cm3 ]−1 for the aug-cc-pVDZ and aug-cc-pVTZ basis sets. The optical rotation of molecule (3) is obtained as -191.4, -180.0 and -177.9 deg[dm g/cm3 ]−1 using the B3LYP(GIAOs) method with the aug-cc-pVXZ (X=D, T an Q) basis sets which are in good agreement with the literature data, -190.2, -180.3 and -177.6 deg[dm g/cm3 ]−1 . 64 The two lowest excitation energies are calculated as 205.6 nm and 201.2 nm for molecule (3) that confirm the optical rotations are calculated far from the excitation energies. Table 4 presents the results for molecule (4). Also for this molecule, we compare the different methods for the aug-cc-pVTZ basis set. For λ = 355 nm, the obtained optical rotations by the HF, B3LYP and CAM-B3LYP models are 442.2, 574.9 and 550.4 deg[dm g/cm3 ]−1 while the CC2 and CCSD methods give 504.7 and 468.5 deg[dm g/cm3 ]−1 . As for

13



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

molecule (3), all the methods overestimate the optical rotation as compared to CCSD apart from HF. At λ = 589.3 nm, the HF, B3LYP and CAM-B3LYP methods give 125.5, 161.1 and 153.3 deg[dm g/cm3 ]−1 whereas the ORs for CC2 and CCSD are 143.2 and 134.6 deg[dm g/cm3 ]−1 . The smallest and largest deviations come from CC2 and B3LYP as compared to the CCSD, 8.6 and 26.5 deg[dm g/cm3 ]−1 , respectively. At λ = 589.3 nm, the variations between B3LYP(GIAOs) and B3LYP(MVG) show the convergence of gauges as 2.7, 4.3 and 0.4 deg[dm g/cm3 ]−1 while the differences of CAM-B3LYP(GIAOs) and CAM-B3LYP(MVG) with increasing the basis set are obtained 1.2, 2.2 and 1.3 deg[dm g/cm3 ]−1 . Using the CC2 method, LG and MVG show unchanged deviations as 8.1, 8.0 and 8.1 deg[dm g/cm3 ]−1 . Although the differences of CCSD(LG) and CCSD(MVG) increase slightly, 5.5 and 6.7 deg[dm g/cm3 ]−1 with basis set, these deviations are smaller than the corresponding differences using the CC2 method which demonstrate the role of additional correlations. For molecule (4), the molar rotation is obtained 115.2 and 107.9 deg[mol dm/cm3 ]−1 using B3LYP(GIAOs) and CAM-B3LYP(MVG) by the aug-cc-pVDZ basis set at λ = 589.3 nm which are close to the reported values, 114.5 and 106.4 deg[mol dm/cm3 ]−1 . 41 Also, 111.2 deg[mol dm/cm3 ]−1 is given using B3LYP(GIAOs) and the LPol-ds(ZPolX) basis set which is comparable to the aug-cc-pVDZ result. 41 For molecule (4), the two lowest excitation energies are calculated at 187.9 nm and 177.0 nm which demonstrate that the selected wavelengths are far from the resonances. The optical rotations for molecule (5) are shown in Table 5. The OR magnitudes increase with the basis set from aug-cc-pVDZ to aug-cc-pVQZ. The aug-cc-pVQZ results are available for all methods for molecule (5) and considered for comparison between the different methods. At λ = 355 nm, the HF, B3LYP and CAM-B3LYP models give optical rotations equal to 323.7, 285.9 and 328.8 deg[dm g/cm3 ]−1 while the results of the CC2 and CCSD methods are 367.4 and 436.4 deg[dm g/cm3 ]−1 , respectively. As already discussed, there are large variations between the different methods at λ = 355 nm and CC2 and B3LYP give the smallest and largest differences, 69 and 150.5 deg[dm g/cm3 ]−1 as compared to CCSD.

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For λ = 589.3 nm, HF, B3LYP and CAM-B3LYP predict 112.0, 129.4 and 125.2 deg[dm g/cm3 ]−1 while CC2 and CCSD give 132.8 and 150.1 deg[dm g/cm3 ]−1 . All the methods underestimate OR compared to CCSD and the CC2 method shows the smallest difference of 17.3 deg[dm g/cm3 ]−1 . At λ = 589.3 nm, the deviations of GIAOs and MVG using the B3LYP method are reduced to 4, 2.2 and 0.9 deg[dm g/cm3 ]−1 with the size of the basis sets while the deviations of CAM-B3LYP(GIAOs) and CAM-B3LYP(MVG) are 5.7, 0.4 and 0.7 deg[dm g/cm3 ]−1 . The CC2(LG) and CC2(MVG) differences are given by 23.5, 25.3 and 24.2 deg[dm g/cm3 ]−1 whereas the CCSD(LG) and CCSD(MVG) findings show larger variations 58.7, 68.7 and 69.4 deg[dm g/cm3 ]−1 . The large variations between LG and MVG using the CC methods indicate that the methods with higher orders of correlations are necessary for investigating molecule (5). The optical rotations of molecule (5) are given by 265.6 and 370.2 deg[dm g/cm3 ]−1 , employing the B3LYP(GIAOs) and CCSD(MVG) methods with the augcc-pVDZ basis set at λ = 355 nm, compared to the literature data, 273.8 and 370.3 deg[dm g/cm3 ]−1 . 43 Also B3LYP(GIAOs) with augD-STO-3G and augD-3-21G (minimal basis sets augmented with diffuse functions) obtains 243.7 and 232.1 deg[dm g/cm3 ]−1 at λ = 355 nm 43 while the CCSD(MVG) method with the same basis sets finds 296.5 and 321.7 deg[dm g/cm3 ]−1 , respectively. 43 Using B3LYP(GIAOs) and CAM-B3LYP(MVG) with the aug-ccpVDZ basis set, the molar rotations of molecule (5) are 85.1 and 78.6 deg[mol dm/cm3 ]−1 at λ = 589.3 nm which are close to the reported values, 89.3 and 83.9 deg[mol dm/cm3 ]−1 . 41 Also B3LYP(GIAOs) with the LPol-ds(ZPolX) basis set finds 94.9 deg[mol dm/cm3 ]−1 at λ = 589.3 nm. 41 The gas-phase experimental values for molecule (5) are reported as 407.8 and 127.8 deg[dm g/cm3 ]−1 at λ = 355 and 633 nm 71 which are close to the optical rotations produced using the CCSD(MVG) method and the aug-cc-pVTZ basis set, 428.2 and 127.8 deg[dm g/cm3 ]−1 . We have also calculated the two lowest excitation energies of molecule (5) and found to be at 200.0 nm and 187.7 nm. Our results for molecule (6) are shown in Table 6. Since this molecule is larger, CCSD calculations have only been carried out for the aug-cc-pVDZ basis set. Thus, we compare

15



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the different methods using this basis set. At λ = 355 nm, the obtained optical rotations using the HF, B3LYP and CAM-B3LYP models are -85.7, -125.9 and -117.4 deg[dm g/cm3 ]−1 whereas CC2 and CCSD give -112.0 and -109.2 deg[dm g/cm3 ]−1 . The CC2 and HF methods show the smallest (2.8 deg[dm g/cm3 ]−1 ) and largest (23.9 deg[dm g/cm3 ]−1 ) deviations with respect to the CCSD results. At λ = 589.3 nm, the HF, B3LYP and CAM-B3LYP methods give optical rotations equal to -28.4, -40.0 and -37.7 deg[dm g/cm3 ]−1 as compared using CC2 and CCSD methods, ORs are -35.8 and -35.3 deg[dm g/cm3 ]−1 . As for λ = 355 nm, the CC2 and HF methods show the smallest and largest differences in OR predictions as compared to CCSD results, 0.5 and 6.9 deg[dm g/cm3 ]−1 . At λ = 589.3 nm, for molecule (6), the GIAOs and MVG results are relatively close using the DFT methods which indicate the reliability of the calculated optical rotations for all basis sets. The differences of GIAOs and MVG are 0.9, 0.6 and 0 deg[dm g/cm3 ]−1 employing B3LYP as well as 1.2, 0.2 and 0.9 deg[dm g/cm3 ]−1 using CAM-B3LYP. The variations of CC2(LG) and CC2(MVG) are reduced to 3.1, 1.5 and 1 deg[dm g/cm3 ]−1 whereas the CCSD(LG) and CCSD(MVG) difference is 3.3 deg[dm g/cm3 ]−1 using the aug-cc-pVDZ basis set. The two lowest excitation energies of this molecule are calculated at 158.8 nm and 151.8 nm which are relatively far from the wavelengths considered in our computations. Table 7 shows the optical rotations for molecule (7). As for molecule (6), CCSD calculations using only the aug-cc-pVDZ basis set have been performed due to the size of the molecule. At λ = 355 nm, the HF, B3LYP and CAM-B3LYP models give 205.5, 284.6 and 264.4 deg[dm g/cm3 ]−1 while the CC2 and CCSD methods predict 288.4 and 263.5 deg[dm g/cm3 ]−1 , respectively. CAM-B3LYP is closest to the CCSD result. At λ = 589.3 nm, the optical rotations predicted by the HF, B3LYP and CAM-B3LYP methods are 67.0, 93.4 and 85.3 deg[dm g/cm3 ]−1 whereas, the CC2 and CCSD methods give ORs equal to 94.3 and 85.5 deg[dm g/cm3 ]−1 . As for λ = 355 nm, the CAM-B3LYP result has the smallest deviation as compared to CCSD. At λ = 589.3 nm, the B3LYP(GIAOs) and B3LYP(MVG) results converge properly as we see from their differences, 0.3, 1.0 and 0.4 deg[dm g/cm3 ]−1 . This

16


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trend appears even using the CAM-B3LYP method, 0.3, 0.9 and 0.3 deg[dm g/cm3 ]−1 . The difference of CC2(LG) and CC2(MVG) are close with the size of basis set, 4, 4.5 and 4.5 deg[dm g/cm3 ]−1 . The CCSD(LG) and CCSD(MVG) difference is 1.3 deg[dm g/cm3 ]−1 using the aug-cc-pVDZ basis set. This difference is smaller than the CC2(LG) and CC2(MVG) deviations which demonstrates the role of additional correlations. The molar rotation of molecule (7) is obtained 107.3 and 98.4 deg[mol dm/cm3 ]−1 using B3LYP(GIAOs) and CAM-B3LYP(MVG) with the aug-cc-pVDZ basis set at λ = 589.3 nm which are in good agreement with data reported in the literature i.e. 107.3 and 97.2 deg[mol dm/cm3 ]−1 . 41 Also, B3LYP(GIAOs) with the LPol-ds(ZPolX) basis set obtains 101.2 deg[mol dm/cm3 ]−1 for the molar rotation of molecule (7). 41 The two lowest excitation energies of this molecule are at 205.3 nm and 197.5 nm, respectively.

11 3.6 -5.8

31.8 26.9 14.8 5.9 2.3

-16.2

-48.9 -53.9

570.6

Different Methods

40.4

-550.3

Different Methods

48.2

Different Methods

Different Methods

599.2

29.6

61.6 57.8

-572.2 -583.6

-631.5 -654.1

550.4

508.1

444.3 -715.1

-12.2 -76.1 -19

CCSD-MVG 0.4

6.5

-54.9

(a) molecule (1)

CCSD-MVG

-553.4 CCSD-MVG -527.9

-27.5 -21.3

(b) molecule (2)

367.4 358.6

(c) molecule (3)

285.9

Different Methods

328.8 319.3 310.6

-112

-117.4

265.6

-125.9 370.2

CCSD-MVG

428.2

(e) molecule (5)

CCSD-MVG

λ=355nm

205.5 263.5

CCSD-MVG

CCSD-MVG

(g) molecule (7)

(f) molecule (6)

Figure 2: Electronic optical rotations: differing methods (origin-independent gauges) vs CCSD(MVG). The points shown by squares, circles and triangles show the results using the aug-cc-pVDZ, aug-cc-pVTZ and aug-cc-pVQZ basis sets, respectively. The purple features correspond to the HF(GIAOs), the green and red features correspond to DFT(GIAOs) with B3LYP and CAM-B3LYP functionals, respectively, whereas the blue ones are related to the CC2(MVG) predictions at λ = 355 nm. The panels pertain to each molecule data set given in the Tables 1-7.

17


468.5

(d) molecule (4)

264.4

-109.2 436.4

462.5

288.4 284.6

-85.7

Different Methods

Different Methods

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



-2.4 -4.1 -10.3

-181.2

-191.7 -198.1

-31.1 -33.8

-44.8 -9.94

CCSD-MVG -2.81

Different Methods

1.7

169.7

-171.1

-6.2 -9.3 -11.2 -14.6 -16.6

Different Methods

10.1 7.9 5.1

Different Methods

Different Methods

13.8

-0.48

CCSD-MVG -18.4 -15.9

-178.1 CCSD-MVG

(b) molecule (2)

160.2 153.5 144.4

126.5

-215.5 -29.2

(a) molecule (1)

-170.5

(c) molecule (3)

133.2

CCSD-MVG

94.3

Different Methods

125.3 124.6 121.3 120.6

Different Methods

93.5

128.4

-35.8 -37.7

85.4

λ=589.3nm

-40 67.1

111.1 133

CCSD-MVG 148.4

(e) molecule (5)

85.5

-35.3 150.1

CCSD-MVG

CCSD-MVG

(f) molecule (6)

(g) molecule (7)

Figure 3: The same presentation as in Figure 2 but at λ = 589.3 nm. Tables 1-7 show the convergence of basis sets for the different molecules. We see that the optical rotation differences for the basis sets aug-cc-pVTZ and aug-cc-pVQZ as expected are smaller than the deviation between the aug-cc-pVDZ and aug-cc-pVTZ basis sets. This trend is found for all molecules and methods which demonstrates the basis set convergence. We have summarized the tables in Figures 2 and 3 for λ = 355 and 389.3 nm to illustrate the basis set dependence for each method plotted against CCSD. The OR at λ = 355 nm shows the largest variations for all methods and basis sets and therefore the basis set convergence is slower than for the other wavelengths. Also, the relatively large difference compared to CCSD(MVG) at this wavelength means that the different methods lead to considerably different results. The optical rotation is well converged when using the aug-cc-pVQZ basis set irrespective of wavelength. Nonetheless, considering the time-consuming computations and small improvements of the aug-cc-pVQZ basis set compared to aug-cc-pVTZ, the triplezeta basis set is a reasonable approximation for obtaining the optical rotation. Also, Tables 1-7 demonstrate that for molecules (3)-(7) (not possessing small-angle optical rotations) at large wavelengths such as λ = 589.3 nm, the aug-cc-pVDZ basis set provides the optical 18


134.6

(d) molecule (4)

-28.4

132.8 131.5

Different Methods

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

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rotation in a good agreement with the larger basis sets using all methods. Therefore, our results suggest that the aug-cc-pVDZ basis set is a reliable choice for investigating the optical rotation, especially for large molecules where larger basis sets lead to considerable computational costs. The signs predicted by all the methods considered in this paper are in agreement for all the molecules except molecules (1) and (2) shown in panels (a) and (b) in Figure 2 (λ = 355 nm) and molecule (1) in panel (a) in Figure 3 (λ = 589.3 nm) because of the small-angle optical rotations of these two molecules as discussed above. Based on the data presented in Tables 1-7, we would like to find a method and basis set which reproduces optical rotations of CCSD(MVG), and at the same time reduces the computational time substantially. We have therefore calculated the arithmetic mean and median relative as well as absolute differences, denoted by ∆r [α], δ r [α], ∆[α], δ[α], respectively, which are presented in Tables 8-11 at λ = 355 and 589.3 nm. We find similar trends for λ = 589.3 and 633 nm, and therefore we do not include results for λ = 633 nm in the tables. For investigating the basis set effect on the deviation between the DFT and CC method, we also consider the CC2(MVG) model as reference data in Tables 8 and 9. Since the optical rotations calculated by CCSD(MVG) and the aug-cc-pVQZ basis set are available only for molecules (1), (2) and (5), the deviations using this basis set are not calculated in Tables 10 and 11. In general, large variations with respect to the reference data, CC2(MVG) in Tables 8 and 9 and CCSD(MVG) in Tables 10 and 11 are found for λ = 355 nm. Also, the deviations between the different methods and reference data increase with the size of the basis set, in particular when CCSD is used as the reference data. We first use the CC2(MVG) results as reference data for λ = 355 and 589.3 nm given in Tables 8 and 9. At λ = 355 nm, CCSD(MVG) has the smallest mean relative deviations with respect to the CC2 values for both the aug-cc-pVDZ and aug-cc-pVTZ basis sets while the CAM-B3LYP(GIAOs) method provides the minimum mean absolut deviations using all basis sets. The deviations calculated for the B3LYP(GIAOs) and CAM-B3LYP(GIAOs) methods have differences, in

19



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particular when increasing the basis set. At λ = 589.3 nm, the CCSD(MVG) model predicts OR within the smallest deviations using the aug-cc-pVDZ basis set. For larger basis sets, B3LYP(GIAOs), CAM-B3LYP(GIAOs) and CCSD methods predict OR with approximately similar deviations from the CC2 predictions. Generally, CAM-B3LYP(GIAOs) and B3LYP(GIAOs) have similar deviations for all basis sets. Comparing these results with those of CC2(MVG), we conclude that both CAM-B3LYP(GIAOs) and B3LYP(GIAOs) functionals produce sufficiently reliable results at λ = 589.3 nm with reasonable computational times. For the small wavelength λ = 355 nm, deviations between the DFT and CC2 predictions are larger and demonstrate that the CC level of calculation is more desirable than DFT. Table 8: Comparison of mean (∆r [α]) and median (δ r [α]) relative deviations of calculated data with respect to the CC2(MVG) results (in percent %). Also, the mean and median absolute deviations are shown by ∆[α] and δ[α] (in deg[dm g/cm3 ]−1 ). The wavelength of incident light is λ = 355 nm. Basis sets/Methods aug-cc-pVDZ HF(GIAOs) B3LYP(GIAOs) CAM-B3LYP(GIAOs) CCSD(MVG) aug-cc-pVTZ HF(GIAOs) B3LYP(GIAOs) CAM-B3LYP(GIAOs) CCSD(MVG) aug-cc-pVQZ HF(GIAOs) B3LYP(GIAOs) CAM-B3LYP(GIAOs)

∆r [α]

δ r [α]

∆[α]

δ[α]

26.9±16.3 28.7±28.6 20.4±19.4 14.9±8.2

21.7± 15.1 13.4±20.2 10.3±16.0 14.1±8.2

63.5±33.0 54.1±33.2 47.3±37.3 52.6±38.6

60.0±33.0 64.3±33.2 41.9±37.3 35.2±36.3

30.3±19.6 35.3±39.9 25.3±27.1 23.7±12.6

20.4±17.3 11.7±28.2 9.3±18.7 19.3±10.5

60.8±23.4 51.9±28.8 44.9±24.2 64.6±33.3

55.3±23.4 66.3±25.2 39.0±24.2 52.9±33.3

26.6±21.4 41.0±41.3 29.3±32.8

12.9± 16.6 18.7±30.7 10.5±22.8

45.9±17.7 59.1±20.7 46.4±21.2

44.1±17.3 68.2±20.3 51.1±20.3

Secondly, we use CCSD(MVG) as reference data and perform the same analysis to determine a preferred method considering the computational cost. The results are presented in Tables 10 and 11 at λ = 355 and 589.3 nm, respectively. Comparing to the reference data, B3LYP(GIAOs) produces optical rotations with larger differences with respect to CAM-B3LYP(GIAOs) deviations for both the aug-cc-pVDZ and the aug-cc-pVTZ basis sets. Higher efficiencies are found for CAM-B3LYP(GIAOs) with lesser deviations. This might be explained by noting the fact that the CAM-B3LYP functional separates the exchange contributions into long- and short-range parts which enables to properly describe properties 20


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Table 9: Comparison of mean (∆r [α]) and median (δ r [α]) relative deviations of calculated data with respect to the CC2(MVG) results (in percent %). Also, the mean and median absolute deviations are shown by ∆[α] and δ[α] (in deg[dm g/cm3 ]−1 ). The wavelength of incident light is λ = 589.3 nm. Basis sets/Methods aug-cc-pVDZ HF(GIAOs) B3LYP(GIAOs) CAM-B3LYP(GIAOs) CCSD(MVG) aug-cc-pVTZ HF(GIAOs) B3LYP(GIAOs) CAM-B3LYP(GIAOs) CCSD(MVG) aug-cc-pVQZ HF(GIAOs) B3LYP(GIAOs) CAM-B3LYP(GIAOs)

∆r [α]

δ r [α]

∆[α]

δ[α]

26.8±16.7 18.7±16.2 17.4±15.1 13.5±8.4

18.4± 14.6 11.4± 13.4 10.1±12.3 9.8±7.3

21.8±10.1 14.8±11.6 14.9±11.1 14.3±8.1

22.7±10.1 14.3±11.6 12.2±11.1 11.8±7.5

28.8±17.7 18.7±20.0 18.4±18.0 19.6±13.0

18.4±15.4 8.6± 14.7 8.5±13.6 13.4±10.1

20.8±5.7 11.6±9.2 13.1±8.5 17.1±5.2

22.7±5.7 10.5±9.2 9.6±7.5 16.1±5.1

25.4±18.9 22.3± 19.6 18.7±17.7

13.6± 14.7 10.5± 16.8 10.0±28.2

16.4±5.8 13.6±7.1 12.4±6.7

18.5±5.8 15.8±6.9 10.7±6.5

Table 10: Comparison of mean (∆r [α]) and median (δ r [α]) relative deviations of calculated data with respect to the CCSD(MVG) results (in percent %). Also, the mean and median absolute deviations are shown by ∆[α] and δ[α] (in deg[dm g/cm3 ]−1 ). The wavelength of incident light is λ = 355 nm. Basis sets/Methods aug-cc-pVDZ HF(GIAOs) B3LYP(GIAOs) CAM-B3LYP(GIAOs) CC2(MVG) aug-cc-pVTZ HF(GIAOs) B3LYP(GIAOs) CAM-B3LYP(GIAOs) CC2(MVG)

∆r [α] 21.4± 35.8± 20.5± 17.6±

δ r [α]

∆[α]

δ[α]

14.4 28.0 17.6 10.8

17.6± 21.8± 11.7± 12.9±

14.4 23.4 16.1 10.3

34.1± 13.9 70.5 ± 35.9 40.8± 28.4 52.6± 38.7

29.3± 72.0± 34.4± 35.2±

13.9 35.9 27.7 36.3

28.2± 25.3 69.2± 69.2 39.6± 36.7 35.9± 30

15.2± 28.5± 20.6± 20.1±

23.6 51.8 28.8 23.9

43.1± 31.4 93.6±33.2 59.3± 32.6 64.6± 33.3

24.0± 85.0± 56.5± 52.9±

23.0 33.2 32.6 33.3

Table 11: Comparison of mean (∆r [α]) and median (δ r [α]) relative deviations of calculated data with respect to the CCSD(MVG) results (in percent %). Also, the mean and median absolute deviations are shown by ∆[α] and δ[α] (in deg[dm g/cm3 ]−1 ). The wavelength of incident light is λ = 589.3 nm. Basis sets/Methods aug-cc-pVDZ HF(GIAOs) B3LYP(GIAOs) CAM-B3LYP(GIAOs) CC2(MVG) aug-cc-pVTZ HF(GIAOs) B3LYP(GIAOs) CAM-B3LYP(GIAOs) CC2(MVG)

∆r [α]

δ r [α]

∆[α]

δ[α]

13.7 13.1 12.0 13.2

18.0± 13.3 11.3± 11.3 7.8±10.5 9.8± 10.9

12.4± 6.9 14.1 ± 7.5 9.4± 7.5 14.3± 8.1

12.5±6.9 10.6± 7.2 7.3± 7.6 11.8± 7.5

24.8± 21.0 24.9±17.9 19.9± 14.7 29.4±27.1

15.7± 20.9 16.6± 15.3 14.9±12.8 13.8± 20.5

14.9±10.7 16.8±6.5 13.0± 8.2 17.1± 5.2

10.8±9.5 15.6± 6.5 13.9± 8.2 16.1± 5.1

21.0± 18.8± 13.4± 17.3±

21



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involving the long-range component of exchange energy. For example, this functional can provide reasonable results to the excitation energies, 106 electronic circular dichroism, 107 and optical rotation 108 where the long-range component has considerable influences. Considering the CAM-B3LYP(GIAOs) deviations for both the aug-cc-pVDZ and aug-cc-pVTZ basis sets at λ = 589.3 nm (see Table 11), we find that the former basis sets has smallest deviations, mean relative and absolute deviations 13.4 (in percent %) and 9.4 (in deg[dm g/cm3 ]−1 ) while the mean relative and absolute deviations of the latter are 19.9 (in percent %) and 13.0 (in deg[dm g/cm3 ]−1 ), respectively. With the same behaviour at λ = 355 nm, therefore, our results suggest that CAM-B3LYP(GIAOs) method may be best approximation to CCSD(MVG) level of theory using both the aug-cc-pVDZ and aug-cc-pVTZ basis sets and reproduces its results with less computational expenses. It should be mentioned that all deviations in Tables 8-11 are calculated by excluding molecule (1) because of unreliable relative deviations due to its small OR.

III. B. Extrapolation and Optical Rotation at Basis Set Limit To estimate optical rotations in the basis set limit, several extrapolation schemes can be used. While an exponential scheme was suggested for the convergence of the energy at the Hartree-Fock level of theory, 62,63 an inverse power fitting scheme was analytically derived for the extrapolation to the basis set limit where the correlation energies converge slowly with respect to the increment of basis set size. 60,61 In recent studies, both methods have been used for extrapolation of optical rotation at the DFT level of theory. 64,65 To the best of our knowledge, extrapolation to the basis set limit for optical rotation has not been addressed at the CC level. Here, we examine the inverse power scheme of extrapolation for optical rotation in the CC2(MVG) level at λ = 355 and 589.3 nm. The series of aug-cc-pVXZ basis sets, X=D, T, Q, 5 and 6, are employed for calculating the optical rotation of all the molecules in Figure 1 except molecule (6) (because of its large size). We consider the following inverse

22


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power extrapolation form for the optical rotation which allows a three-point investigations

ORX = OR∞ + AX −n ,

(2)

where X is the highest angular function of a basis set (X = 2 − 6 for aug-cc-pVDZ to augcc-pV6Z and we label the basis sets with these numbers), OR∞ is the optical rotation at the basis set limit, and n is the exponent of the extrapolation scheme. It has been demonstrated that such an inverse power fitting with n = 3 gives the basis set extrapolation for correlation energies. 60,61 By simply considering the exponent n in eq 2 as a fitting parameter, three points with nonlinear variations are required for obtaining three fitting parameters in this scheme. Such three-point extrapolations result in a wide range of solutions to the exponent n, satisfying eq 2. To avoid this, we have assumed that n is fixed and examined n = 2, 3, 4, 5 and 6 which permits two-point extrapolations. It has been shown that two-point fits by the two highest basis sets i.e. X = 5 and 6 provide the correlation energy at the basis set limit. 61 Thus the optical rotation at the basis set limit can be obtained by combining X = 5 and 6 in eq 2 in line with ref 61. However, here we consider the aug-cc-pV6Z basis set as the basis limit to all exponents. First, we evaluate the CC2 method through investigating the CC2(MVG) and CC2(LG) convergence with size of the basis sets as differences diminish for the exact theory. For this purpose, we plot CC(MVG)−CC2(LG) (differences between CC2(MVG) and CC2(LG)) with respect to the basis sets for all the molecules at λ = 589.3 nm in Figure 4. Figure 4 illustrates that the differences are reduced for all the molecules (except molecules (4) and (7) which show relatively constant differences between CC2(MVG) and CC2(LG) with size of the basis set) by employing the aug-cc-pVQZ basis set. For larger basis sets it saturates although the reduction of differences seems to be dependent on the molecules and even is 0.5 deg[dm g/cm3 ]−1 for molecule (3). While molecule (5) exhibits the largest differences between CC2(MVG) and CC2(LG) (24.0 deg[dm g/cm3 ]−1 ), the other molecules show small

23



11

7 6 5 4 3 2

2

3

4

5

10 9 8 7 6 5

6

6

CC2(MVG) - CC2(LG)

CC2(MVG) - CC2(LG)

CC2(MVG) - CC2(LG)

8

2

3

4

X

(a) molecule (1)

7 6 4

2 1 2

3

5

6

5

6

(c) molecule (3) 7

26 25 24 23 22 21

4

X

CC2(MVG) - CC2(LG)

CC2(MVG) - CC2(LG)

8

3

3

(b) molecule (2)

9

2

4

6

27

10

5

5

5

X

11

CC2(MVG) - CC2(LG)

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

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2

3

4

5

X

X

(d) molecule (4)

(e) molecule (5)

6

6 5 4 3 2 1

2

3

4

5

6

X

(f) molecule (7)

Figure 4: Differences of optical rotations between the CC2(MVG) and CC2(LG) methods vs basis sets (indicated with their highest angular function, X) for each molecule, separately, at λ = 589.3 nm. variations between CC2(LG) and CC2(MVG) at the largest basis set. The results of optical rotation extrapolations are summarized in Tables 12 and 13 together with the results of the series of aug-cc-pVXZ basis sets at λ = 355 and 589.3 nm, respectively. Since the optical rotation shows large variations with respect to the basis sets at λ = 355 nm, the investigation of extrapolation can provide useful information at this wavelength. As seen, the optical rotation of molecules (1)-(5) are obtained up to the largest basis set (augcc-pV6Z) while the OR of molecule (7) is given to the aug-cc-pV5Z basis set (because of its size) and therefore aug-cc-pV5Z will be considered as the basis set limit for molecule (7). The OR predictions of the aug-cc-pVDZ basis set exhibit large variations when moving to its next basis set, aug-cc-pVTZ, for each molecules as seen in Tables 12 and 13. These variations become smaller by moving from the aug-cc-pVTZ basis set to the aug-cc-pVQZ basis set. We note that for molecule (1), the CC2 method predicts bisignate nonresonant optical rotations (namely, predicts different signs for the optical rotations at λ = 355 and 589.3 nm) while molecules (2)-(7) exhibit monosignate nonresonant optical rotatory dispersions using CC2 since their optical rotation signs are identical at all wavelengths. For molecule (1), 24


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Table 12: CC2(MVG) specific rotations using aug-cc-pVXZ (X=2−6) and two-point extrapolation ORX = OR∞ + AX −n , (X,X+1), with different n at λ = 355 nm. MAD between calculated ORs and aug-cc-pV6Z basis set. n

X 2 3 4 5 6

(1) -12.19 2.33 5.87 6.44 6.59

(2) -76.14 -53.89 -48.91 -50.12 -50.13

Molecules (3) (4) -715.10 508.09 -654.13 504.75 -630.97 501.75 -625.99 500.42 -623.07 499.39

(5) 310.61 358.63 367.40 368.96 369.87

(7) 288.42 281.88 279.09 278.32 -

MAD 35.81 9.87 2.57 0.83 -

2 2 2 2

(2,3) (3,4) (4,5) (5,6)

13.94 10.42 7.45 6.93

-36.09 -42.51 -52.27 -50.15

-605.35 -601.19 -617.13 -616.43

502.08 497.89 498.05 497.05

397.04 378.67 371.73 371.94

276.65 275.50 276.96 -

11.77 7.74 2.25 -

3 3 3 3

(2,3) (3,4) (4,5) (5,6)

8.44 8.45 7.09 6.79

-44.52 -45.27 -51.39 -50.14

-628.46 -614.07 -620.76 -619.06

503.34 499.56 499.02 497.97

378.85 373.80 370.59 371.12

279.12 277.05 277.52 -

4.43 3.51 0.98 -

4 4 4 4

(2,3) (3,4) (4,5) (5,6)

5.90 7.51 6.83 6.73

-48.41 -46.60 -50.96 -50.14

-639.12 -620.25 -622.53 -620.35

503.93 500.36 499.50 498.43

370.45 371.46 370.04 370.72

280.27 277.80 277.79 -

4.25 1.72 0.40 -

5 5 5 5

(2,3) (3,4) (4,5) (5,6)

4.53 6.97 6.78 6.69

-50.51 -47.36 -50.71 -50.13

-644.88 -623.76 -623.56 -621.11

504.24 500.81 499.77 498.70

365.91 370.13 369.72 370.48

280.89 278.22 277.95 -

5.94 0.94 0.35 -

6 6 6 6

(2,3) (3,4) (4,5) (5,6)

3.73 6.63 6.64 6.66

-51.75 -47.83 -50.55 -50.13

-648.26 -625.95 -624.22 -621.60

504.43 501.10 499.95 498.87

363.25 369.30 369.51 370.33

281.25 278.48 278.05 -

7.37 1.28 0.47 -

the rotatory strengths of the two first low-lying transitions with different signs contribute significantly to the optical rotation 42 which result in the bisignate optical rotations. 109,110 From Tables 12 and 13, for molecule (1), we find that all the basis sets predict positive sign for the optical rotation at λ = 355 nm (except aug-cc-pVDZ) whereas negative sign is predicted at λ = 589.3 nm. Thus, the bisignate optical rotation of molecule (1) is predicted using all the basis sets employed. Also, Tables 12 and 13 demonstrate the same extrapolation trends for molecule (1) as the other molecules. Also, to evaluate an optimal exponent based on the two-point fits, we calculate the mean absolute deviation (MAD) of unsigned optical rotations for all the basis sets along with two-point extrapolations with respect to the optical rotation in the aug-cc-pV6Z basis set (the aug-cc-pV5Z basis set for molecule (7)) at the CC2 level in Tables 12 and 13. We here 25



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Table 13: CC2(MVG) specific rotations using aug-cc-pVXZ (X=2−6) and two-point extrapolation ORX = OR∞ + AX −n , (X,X+1), with different n at λ = 589.3 nm. MAD between calculated ORs and aug-cc-pV6Z basis set. n

X 2 3 4 5 6

(1) -10.35 -4.15 -2.40 -1.98 -1.83

(2) -44.82 -33.79 -31.11 -31.05 -30.87

Molecules (3) (4) -215.50 144.42 -198.08 143.22 -191.92 142.51 -190.80 142.24 -190.11 141.99

(5) 120.64 131.56 132.78 132.85 132.94

(7) 94.31 91.35 90.27 89.97 -

MAD 11.15 2.86 0.60 0.22 -

2 2 2 2

(2,3) (3,4) (4,5) (5,6)

0.79 -0.15 -1.24 -1.49

-24.95 -27.66 -30.93 -30.46

-184.14 -184.01 -188.80 -188.55

142.27 141.59 141.75 141.44

140.29 134.36 132.97 133.13

88.98 88.88 89.45 -

3.59 2.32 0.46 -

3 3 3 3

(2,3) (3,4) (4,5) (5,6)

-1.55 -1.13 -1.55 -1.62

-29.14 -29.15 -30.98 -30.62

-190.74 -187.43 -189.61 -189.16

142.72 141.99 141.95 141.66

136.15 133.68 132.92 133.05

90.10 89.48 89.67 -

1.12 1.06 0.21 -

4 4 4 4

(2,3) (3,4) (4,5) (5,6)

-2.63 -1.59 -1.69 -1.69

-31.07 -29.87 -31.00 -30.70

-193.79 -189.07 -190.02 -189.47

142.93 142.18 142.05 141.77

134.24 133.35 132.90 133.02

90.62 89.77 89.77 -

1.26 0.51 0.11 -

5 5 5 5

(2,3) (3,4) (4,5) (5,6)

-3.21 -1.86 -1.78 -1.73

-32.11 -30.27 -31.01 -30.75

-195.43 -190.00 -190.25 -189.65

143.04 142.29 142.10 141.83

133.21 133.16 132.88 132.99

90.90 89.93 89.83 -

1.70 0.21 0.10 -

6 6 6 6

(2,3) (3,4) (4,5) (5,6)

-3.56 -2.02 -1.84 -1.76

-32.72 -30.53 -31.02 -30.78

-196.40 -190.59 -190.40 -189.76

143.11 142.35 142.14 141.87

132.61 133.05 132.88 132.98

91.06 90.03 89.87 -

2.06 0.25 0.12 -

examine different exponents: n = 2, 3, 4, 5 and 6 as well. The smallest basis set i.e. augcc-pVDZ has the largest deviation which is in agreement with previous work. 60,61 Generally, MAD reduces when the exponent n in eq 2 increases. If we consider two-point results in this table, we see that, for example, the errors calculated by points (2, 3) are smaller than those obtained using the aug-cc-pVTZ basis set (except for n = 2). The reduction of errors can be observed for the other two-point predictions. These results demonstrate the accuracy of extrapolation schemes to find optical rotations close to the basis set limit. Tables 12 and 13 show that a two-point inverse power extrapolation scheme with n = 5 results in reliable predictions for extrapolation of OR at the CC2 level of theory for both wavelengths. We find that if the results of aug-cc-pVQZ calculations are available, extrapolation for points (3, 4), with n = 5, provides the best predictions through comparing MAD obtained by the aug-cc26


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pV5Z basis set with respect to the basis set limits. For larger system where the aug-cc-pVQZ data may not be available, one can rely on the predictions of inverse power extrapolation by points (2, 3) with n = 3 or 4 which lead to similar and rather small deviations for CC2. We note that the MAD values are reliable criteria for selecting the exponent and two-point extrapolation as different molecules have similar contributions to MAD even if the optical rotation values are different. For example, the absolute deviations between points (3, 4) with n = 5 and the aug-cc-pV6Z basis set (the aug-cc-pV5Z basis set for molecule (7)) are 0.03, 0.60, 0.11, 0.30, 0.22 and 0.04 deg[dm g/cm3 ]−1 for molecules (1)-(5) and (7) at λ = 589.3 nm.

IV. Conclusions We have computed the optical rotation of seven molecules by the HF, DFT/B3LYP and CAM-B3LYP, CC2 and CCSD methods, with the augmented basis sets aug-cc-pVXZ at the wavelengths λ = 355, 589.3 and 633 nm. To confirm the validity of the considered wavelengths and thus the obtained results, we have also calculated corresponding wavelengths to the two first excitation energies in the energy spectrum of all the molecules using the CC2 method. We show that the two lowest excitation wavelengths are sufficiently far from those we employed in our calculations. To evaluate the different methods, we have compared the deviations of GIAOs and MVG for the DFT methods as well as the deviations between LG and MVG for the CC methods with the size of basis sets. Our results show that the gauge differences reduce by increasing the basis set sizes using both B3LYP and CAM-B3LYP for all molecules. Also, the deviations between LG and MVG using the CC2 method are reduced for all molecules except molecules (4) and (7) which show unchanged variations. While the CCSD(LG) and CCSD(MVG) differences for molecules (3) and (5) are increased with the basis set size, the variations are reduced for other molecules which are smaller than the corresponding CC2 deviations. We have used CCSD as a reference point and compared

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with DFT to recommend a more feasible method at the DFT level. Using various deviation schemes, we find that the CAM-B3LYP functional reproduces the CCSD results with the least deviation. Furthermore, the basis set convergence for the different methods suggests that the aug-cc-pVTZ basis set is a desirable choice to produce reliable results. Also, our results demonstrate that the aug-cc-pVDZ basis set can be a favourable option for large molecules because this basis set predicts the optical rotation in a good agreement with the larger basis sets for λ = 589.3 nm and, at the same time, reduce the computational costs considerably. We have also examined a two-point inverse power extrapolation scheme for the optical rotation at the CC2 level. We have first evaluated the CC2 method compared to the exact theory through investigating the deviations between CC2(MVG) and CC2(LG) with increasing basis set sizes for all the molecules at λ = 589.3 nm. We have found that the deviations properly reduce for molecules (1)-(3) and (5) while the differences are relatively constant for molecules (4) and (7). Although molecule (5) shows the largest deviation between CC2(MVG) and CC2(LG) (24.0 deg[dm g/cm3 ]−1 ), the differences are small (even 0.5 deg[dm g/cm3 ]−1 for molecule (3)) at the aug-cc-pV6Z basis set. Our calculations show that a two-point inverse power extrapolation of the aug-cc-pVTZ and aug-cc-pVQZ basis set with n = 5 as exponent predicts optical rotation of the aug-cc-pV6Z basis set with small deviations at both λ = 355 and 589.3 nm. For larger molecules, where the aug-cc-pVQZ basis set may not be available, one may employ a two-point inverse power extrapolation between the aug-cc-pVDZ and aug-cc-pVTZ basis set with n = 3 or 4.

Acknowledgement A grant of computer time is acknowledged from the NOTUR project (account 2920k) at the Norwegian Research Council. HK acknowledges financial support from the FP7-PEOPLE2013-IOF funding scheme (project No. 625321).

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Graphical TOC Entry

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Optical Rotation from Coupled Cluster and Density Functional Theory

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