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The ORP Basis Set Designed for Optical Rotation
Calculations
Angelika Baranowska-Ła˛czkowska,*[a] and Krzysztof Z. Ła˛czkowski[b]
Details of generation of the optical rotation prediction (ORP)
basis set developed for accurate optical rotation (OR) calculations are presented. Specific rotation calculations carried out
at the density functional theory (DFT) level for model chiral
methane molecule, fluorooxirane, methyloxirane, and dimethylmethylenecyclopropane reveal that the ORP set outperforms
larger basis sets, among them the aug-cc-pVTZ basis set of
Dunning (J. Chem. Phys. 1989, 90, 1007) and the aug-pc-2
basis set of Jensen (J. Chem. Phys. 2002, 117, 9234; J. Chem.
Theory Comput. 2008, 4, 719). It is shown to be an attractive
choice also in the case of larger systems, namely
norbornanone, b-pinene, trans-pinane, and nopinone. The ORP
basis set is further used in OR calculations for 24 other systems, and the results are compared to the aug-cc-pVDZ values.
Whenever large discrepancies of results are observed, the ORP
values are in an excellent agreement with the aug-cc-pVTZ
results. The ORP basis set enables accurate specific rotation
calculations at a reduced cost and thus can be recommended
for routine DFT OR calculations, also for large and conformaC 2013 Wiley Periodicals, Inc.
tionally flexible molecules. V
Introduction
may lead to a serious loss of accuracy.[28,32] Unfortunately,
already the aVTZ basis set leads to a significant increase in the
calculation cost, making OR calculations for larger systems—in
particular those conformationally flexible—unfeasible despite
the fast development of computational resources.
A solution to this problem are basis sets designed for calculation of property of given type. An example of such basis sets
are the LPol-n sets[33] belonging to the Pol family of basis
sets,[34–42] and designed for calculations of linear and nonlinear
electric properties. Despite the small size, already the LPol-ds
basis set—the smallest among the LPol-n sets—gives electric
property results very close to those yielded by traditional basis
sets of larger size, thus enabling accurate calculations for systems larger than previously. Excellent performance of the LPoln sets is reported for linear and nonlinear electric properties of
isolated systems, as well as interaction-induced electric properties in hydrogen-bonded systems and van der Waals complexes.[33,43–51]
Recently, we have carried out extensive studies on the basis
set dependence of the calculated specific rotation of an artificial model system, the chiral methane molecule,[32] within the
time-dependent Hartree–Fock, and density functional approximations, using the naVXZ basis sets of Dunning and coworkers
Determination of the spatial arrangement of atoms forming
chiral molecules, that is, the absolute configuration (AC) of chiral compounds, is a crucial step in asymmetric synthesis and in
drug development. It can be achieved using experimental
methods such as x-ray crystallography or chemical correlation.
However, experimental methods sometimes encounter difficulties and thus an alternative way of AC determination is highly
desired. Over the years, attempts were made to assign AC of
compounds by computing optical rotation (OR) of one of the
two enantiomers and comparing the result with experimental
value.[1–28] Although often successful, theoretical calculations
are not always able to determine AC correctly. It turns out that
the results depend strongly on a number of parameters, such
as the choice of the level of theory or accounting for solvent
effects. One of these parameters is the choice of basis set
used in the calculation. Since correct theoretical description of
the system interacting with external electromagnetic field
needs a basis set including polarization and diffuse functions,
the use of large and flexible basis sets is crucial for obtaining
accurate values of optical properties. However, in the case of
OR calculations, the—usually large—size of investigated system forces the use of relatively small basis sets. Basis set
choice is thus to a large extent the result of a compromise
between the desired accuracy of calculations and their
feasibility.
The aug-cc-pVDZ basis set of Dunning and coworkers[29–31]
(the n-aug-cc-pVXZ basis sets are abbreviated as naVXZ
throughout) is well-recognized to be a reasonable choice for
density functional theory (DFT) B3LYP OR calculations for large
molecules.[5,24,26] It has been shown, however, that in some
cases the use of basis set of at least the aVTZ (or, preferably,
the aVQZ or daVTZ) set quality is mandatory to obtain results
close to the basis set limit, and that the use of the aVDZ set
2006
Journal of Computational Chemistry 2013, 34, 2006–2013
DOI: 10.1002/jcc.23347
[a] A. Baranowska-Ła˛czkowska
Institute of Physics, Kazimierz Wielki University, Plac Weyssenhoffa 11,
PL-85072 Bydgoszcz, Poland
E-mail: [email protected].
[b] K. Z. Ła˛czkowski
Department of Chemical Technology and Pharmaceuticals, Faculty of
Pharmacy, Nicolaus Copernicus University, 2 Jurasz St., PL-85089
Bydgoszcz, Poland
Contract grant sponsor: Polish Ministry of Science and Higher Education
within the Iuventus Plus programme; Contract grant number: IP2010
051070
C 2013 Wiley Periodicals, Inc.
V
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Figure 1. Molecules 1–31 for which specific rotation calculations are carried out.
and the LPol-n (n 5 ds, dl, fs, fl) basis sets. The chiral methane
is a model system obtained from the real nonchiral methane
molecule by appropriate distortion of its tetrahedral geometry.[32] Among the conclusions of that work is that the LPol-n
sets are competitive to the larger naVXZ basis sets in the OR
calculations. We have also carried out specific rotation calculations for (S)-methyloxirane and (S)-fluorooxirane molecules
using the LPol-ds basis set and have shown that for these two
systems the LPol-ds set performs comparably to the over twice
larger aVQZ basis set.[32] This makes the LPol-ds basis set very
useful in accurate OR calculations for larger molecules. Moreover, Mach and Crawford[52] have shown for a set of test molecules that the LPol-n sets outperform the commonly used
correlation-consistent basis sets of comparable size also in the
coupled cluster (CC) OR calculations.
The LPol-n sets—and especially the smallest among them,
the LPol-ds set—can be strongly recommended for the OR calculations because they allow for computing time savings while
yielding results of the aVQZ basis set quality. However, even
the relatively small LPol-n sets soon become too large to be
used in regular ab initio or DFT calculations as the system size
increases. Another problem encountered while using the LPoln sets is the numerical linear dependence of orbitals: polarization functions in the LPol-n sets have the same orbital exponents as the p-type functions (s-type functions in the case of
hydrogen), soon leading to linear dependencies. The number
of functions removed from the total LPol-n set due to their
near linear dependence is substantial even in relatively small
molecules. This step decreases the size of the basis set in a
not fully controlled manner, which in turn may influence accuracy of the description of the investigated system.
The aim of the present project is to develop basis set possibly smaller than the LPol-n and more resistant to near-linear
dependencies, but yielding OR results of similar quality as the
LPol-n sets. The new set presented in this work is referred to
as ORP (acronym of the optical rotation prediction). Systematic
generation of basis set for five elements commonly occurring
in chiral molecules—hydrogen, carbon, nitrogen, oxygen, and
fluorine—is carried out. The new set is tested in the B3LYP OR
calculations for the model chiral methane molecule, and seven
rigid test systems (see Fig. 1), namely (R)-fluorooxirane (1), (S)methyloxirane (2), (2 R,3 R)-dimethylmethylenecyclopropane
(3), norbornanone (4), b-pinene (5), trans-pinane (6), and nopinone (7). To judge the performance of the new set, the OR of
these systems is calculated also using possibly large Dunning
(d)aVXZ basis sets.
Additionally, we test the performance of other medium-size
basis sets, namely the aug-pc-1, aug-pc-2, aug-pcS-1, and augpcS-2 (abbreviated as apc1, apc2, apcS1, and apcS2, respectively) basis sets of Jensen,[53,54] and the SVPD and TZVPD sets
of Rappoport and Furche.[55] The apcS1 basis set has been
recently reported to yield at wavelength k 5 589.3 nm OR
Journal of Computational Chemistry 2013, 34, 2006–2013
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values better than or similar to those obtained with the aVDZ
and aVTZ basis sets, while it is only marginally larger than the
aVDZ set.[28] At 355.0 nm, the authors recommended the use
of the larger apcS2 set which is competitive to the aVTZ basis
set.
The ORP set is further used for evaluation of specific rotation of 20 other rigid systems, namely molecules 8 through 27
(see Fig. 1). These systems where chosen from the set of molecules investigated recently by Srebro et al.[56] Results obtained
for systems 8–27 are compared to the aVDZ values, and in the
cases where large discrepancy is observed between the two
results, also to the aVTZ values.
Additionally, we use the new basis set in the B3LYP OR calculations carried out for a set of four flexible molecules,
namely b-amino alcohols derived from (1)-3- and (1)-2-carene
(molecules 28 through 31). Specific rotation of these systems
was previously calculated within the B3LYP/aVDZ approximation.[27] We compare the B3LYP/ORP results with the available
experimental data,[57] and the aVDZ values.
In the following section, we outline details of the ORP basis
set generation and OR calculations, and in section Results and
Discussion, we discuss the results. In the last section, we summarize and conclude.
Methodology
Optical rotation ½ak of conformationally flexible enantiomer A
at wavelength k is defined as[58]
½ak 5
N
eeðeA Þ X
Xi ½aki
100 i51
(1)
with eeðeA Þ being the enantiomeric excess of enantiomer A in
the mixture, ½aki denoting the OR of conformer i, and Xi—the
fractional population of conformer i defined as[58]
Xi 5
exp ð2Di E=kTÞ
:
N
X
exp ð2Dj E=kTÞ
(2)
j51
In the above, Di E is the relative energy of conformer i, k is the
Boltzmann constant, T is the temperature, and the summation
runs over all N stable conformers of enantiomer A.
Because estimation of the rotation angle of polarization
plane requires calculation of the trace of mixed electric
dipole–magnetic dipole polarizability tensor, basis sets optimized with respect to linear electric properties should be able
to describe OR correctly. Indeed, as shown in our earlier investigation,[32] the LPol-n basis sets perform very well in OR calculation of test systems and are competitive to the larger allpurpose basis sets of Dunning and coworkers. On the basis of
this conclusion, the shape of our new basis set is optimized
with respect to atomic polarizabilities, similarly as in the case
of the LPol-n sets.
The ORP set is developed for hydrogen, carbon, nitrogen,
oxygen, and fluorine by adding three uncontracted first-order
polarization functions to some initial set of functions. This
2008
Journal of Computational Chemistry 2013, 34, 2006–2013
initial set is obtained from the uncontracted VTZ basis set of
Ahlrichs and coworkers[59] augmented with diffuse functions
to further improve the description of regions distant from the
nucleus. One s-type function is added in the case of hydrogen,
and one s- and one p-type functions in the case of other elements. Analogously to the LPol-n basis set generation,[33]
orbital exponents of these additional functions are determined
from the anticipated geometric progression based on the two
lowest exponents.
Next, three uncontracted first-order polarization functions
are added to the initial set. In the initial guess, their orbital
exponent values lay between 1.10 and 2.50 for the first function, 0.20 and 1.00 for the second, and 0.01 and 0.15 for the
third, with the step equal to 0.20 for the first two functions,
and 0.02 for the third. Thus, the total of 320 uncontracted
basis sets is tested for each of the investigated elements. Finite
field restricted open-shell hartree-fock (ROHF) calculations of
atomic polarizabilities are carried out to find the set of orbital
exponents minimizing the error in atomic polarizability values
with respect to the reference values taken from the work of
Stiehler and Hinze.[60] For hydrogen, the exact value equal to
4.5 a.u. is used as reference. The external electric field strength
is assumed equal to 0.001 a.u. Once the best set of orbital
exponents is found from the initial guess, additional iteration
with smaller steps (0.05 for the first two functions and 0.01 for
the third) is carried out in the vicinity of their values, leading
to optimal values of exponents.
The new basis set is of the form (6s3p) for hydrogen and
(11s7p3d) for other atoms. To decrease its size and thus the cost
of calculations, the basis set is next contracted. Contraction of
the set is to a large extent arbitrary providing that the resulting
set remains flexible enough. Here, contraction coefficients are
determined from atomic ROHF calculations carried out using the
initial set, and polarized basis set is contracted to the form
[6s3p/4s3p] for hydrogen and [11s7p3d/5s4p3d] for other atoms,
with polarization functions remaining uncontracted. The resulting set is referred to as ORP. It contains 13 functions for hydrogen and 32 for other elements, and thus is slightly smaller than
the smallest of the LPol-n sets (the LPol-ds containing 13 functions for hydrogen and 36 for other elements), and considerably
smaller than the aVTZ set (23 and 46 functions, respectively).
The ORP set is further tested in the DFT/B3LYP calculations
of the specific rotation of chiral methane molecule previously
studied in Ref. [32], and test molecules 1 through 7. Calculations for chiral methane are carried out at wavelength k 5
227.82 nm, and for molecules 1–7 at k 5 633.0, 589.3, and
355.0 nm. Reference results are obtained in the aVXZ basis
sets of Dunning and coworkers (with X up to six for chiral
methane, five for molecule 1, Q for molecules 2–4, and T for
systems 5–7) and in the daVXZ basis sets (with X up to five
for chiral methane and molecule 1, Q for molecule 2, and T
for systems 3–7). We also test the performance of other
medium-size basis sets, namely the apc1, apc2, apcS1, and
apcS2 basis sets of Jensen,[53,54] and the SVPD and TZVPD sets
of Rappoport and Furche.[55]
Next, we use the ORP basis set in the evaluation of specific
rotation of 20 other rigid molecules examined recently by
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Srebro et al.,[56] namely molecules 8–27 (see Fig. 1). Calculations are carried out at wavelength k 5 589.3 nm. Results are
compared with the aVDZ values, and whenever large discrepancies are observed also with the aVTZ values. As the last test,
the ORP set is used in the DFT/B3LYP OR calculations performed at k 5 589.3 nm for four conformationally flexible bamino alcohols (molecules 28 through 31) investigated by us
previously using the aVDZ basis set.[27]
Geometrical parameters of chiral methane molecule and systems 2, 5, and 6 are taken from Ref. [32], those of molecule 1
from the work of Pedersen et al.,[61] molecules 3 and 8
through 27 from the recent work of Srebro et al.,[56] and of bamino alcohols 28–31 from Ref. [27]. Geometries of systems 4
and 7 are optimized within the B3LYP/aVDZ approximation.
For these two molecules, vibrational frequency calculations are
carried out to confirm that optimized structures correspond to
the real minima on potential energy surface.
All atomic calculations are carried using the MOLCAS 7.8
package.[62–64] Geometry optimization, vibrational frequency,
and OR calculations are performed using the GAUSSIAN 09
program.[65] Although for some of the investigated systems OR
values obtained in the basis sets of Dunning and Jensen are
available in literature, we repeat the calculations here to avoid
differences in results caused by differences in the computing
codes or in the geometrical parameters.
Atomic polarizability results are given in atomic units. The
OR values are reported in 1021deg cm2 g21 referred to as the
OR-units throughout. The VTZ, daVXZ, apc(S)n, SVPD, and
TZVPD basis sets are taken from the EMSL Basis Set
Library.[66,67]
Results and Discussion
Atomic polarizability results obtained in the ORP basis set are
given in Table 1, together with the LPol-ds values.[33] Reference results are taken from the work of Stiehler and Hinze,[60]
and are among the most accurate results available. For hydrogen, the exact value of polarizability is used as reference. From
Table 1, it follows that the ORP basis set yields results of practically the same accuracy as does the LPol-ds set. They are
slightly closer to the reference than the LPol-ds values in the
Table 1. The ROHF finite field atomic static electric dipole polarizabilities.
System
H
C
N
O
F
ML
ORP
LPol-ds[33]
Ref. [60]
0
0
61
Average
0
0
61
Average
0
61
Average
4.5000
10.1024
12.9862
12.0249
7.3546
5.1076
4.5434
4.7315
3.0972
3.3736
3.2815
4.4996
10.1058
12.9868
12.0265
7.3546
5.1258
4.5488
4.7411
3.0976
3.3816
3.2869
4.5000
10.112
12.994
12.033
7.3581
5.0690
4.5658
4.7335
3.1174
3.3670
3.2838
External electric field strength equal to 0.001 au. Polarizabilities in
atomic units.
case of hydrogen, oxygen, and fluorine. Differences between
the new results and literature values are negligible taking into
account that the ORP set is designed for molecular linear
property calculations.
Optical rotation results for the chiral methane molecule are
gathered in Table 2. All LPol-n as well as the ORP basis sets
yield results very close to the reference values obtained in the
aV6Z basis set. Differences are in the order of at most 0.24
OR-units, with the only exception being the LPol-fs value
obtained for geometry 2 (difference equal to 0.32 OR-units).
The apcS1 basis set performs very well in the case of geometry 1, but slightly worse for geometry 2. Similar situation is
observed in the case of the TZVPD set, whereas the opposite
for the apc1, apc2, and apcS2 sets. The SVPD basis set performs satisfactorily for geometry 1, especially taking into
account its exceptionally small size; however, it has problems
with correct description of OR for geometry 2. On the basis of
results obtained for the chiral methane molecule, the ORP and
LPol-ds sets can be recommended for OR calculations due to
their small size and the accuracy of results. They are both
competitive to the much larger aVTZ basis set, which use in
OR calculations is restricted to small and medium size molecules. Although among the investigated non-Dunning sets, the
LPol-fl basis set yields results closest to the reference, one has
to keep in mind that it contains significantly more basis functions than does the aVTZ set, and thus its use in the case of
large organic molecules is practically not feasible.
We now turn our attention to results obtained for molecules
1 through 7, presented in Tables (3–9). For system 1, we
observe excellent performance of the LPol-ds basis set. Among
even smaller basis sets, the SVPD and ORP sets prove to be
very attractive choices. These three sets contain significantly
less functions than the aVTZ basis set while introduce much
Table 2. Optical rotation of the chiral methane molecule.
Basis set
N
Geometry 1
Geometry 2
aVDZ
aVTZ
aVQZ
aV5Z
aV6Z
daVDZ
daVTZ
daVQZ
daV5Z
apc1
apc2
apcS1
apcS2
SVPD
TZVPD
LPol-ds
LPol-dl
LPol-fs
LPol-fl
ORP
59
138
264
447
697 (5)
84
190
353
583 (6)
59
138
62
141
52
73
88
113
142 (3)
194 (3)
84
27.67
28.92
29.14
29.04
29.10
29.91
29.28
29.22
29.08
29.82
28.50
29.15
28.58
28.90
28.70
29.22
28.86
29.16
29.14
28.90
17.46
17.04
17.13
17.18
17.20
17.37
17.32
17.25
17.23
17.10
17.05
16.65
17.01
13.33
16.06
17.14
17.22
16.88
17.26
17.25
Symbol N denotes the total number of basis functions. The number of
functions removed due to near-linear dependence, if any, given in
parentheses. All results in OR-units.
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Table 3. Optical rotation of molecule 1.
Basis set
N
aVDZ
aVTZ
aVQZ
aV5Z
daVDZ
daVTZ
daVQZ
daV5Z
apc1
apc2
apcS1
apcS2
SVPD
TZVPD
LPol-ds
LPol-dl
LPol-fs
LPol-fl
ORP
119
253
458
748 (2)
167
344 (1)
606 (7)
967 (15)
119
253
131
265
110
181
183
231 (1)
269 (3)
360 (9)
167
Table 5. Optical rotation of molecule 3.
633.0
589.3
355.0
Basis set
N
633.0
589.3
355.0
23.96
211.45
212.52
212.72
29.55
212.63
212.77
212.78
24.10
212.05
24.47
212.00
212.62
213.38
212.79
212.05
213.11
212.58
212.51
24.93
213.63
214.86
215.09
211.42
215.00
215.16
215.17
25.09
214.32
25.52
214.26
215.03
215.86
215.18
214.32
215.55
214.94
214.84
230.81
256.85
260.55
261.14
250.34
260.95
261.28
261.34
231.44
258.67
232.67
258.53
262.52
263.35
261.43
258.61
262.41
260.63
259.98
aVDZ
aVTZ
aVQZ
daVDZ
daVTZ
apc1
apc2
apcS1
apcS2
SVPD
TZVPD
LPol-ds
LPol-dl
LPol-fs
LPol-fl
ORP
228
506
940 (2)
322 (8)
692 (25)
228
506 (3)
246
524 (3)
200
312
346 (4)
440 (10)
530 (29)
716 (63)
322
10.25
22.75
24.64
20.56
25.70
20.20
28.54
21.28
28.30
22.37
24.56
27.16
26.32
26.41
26.80
26.11
11.90
23.38
25.63
20.84
26.88
20.38
210.19
21.64
29.91
23.10
25.61
28.60
27.61
27.72
28.18
27.36
22.18
232.90
242.48
224.82
246.89
220.99
257.45
224.63
256.22
238.12
246.19
252.79
249.07
250.03
251.19
247.40
Symbol N denotes the number of basis functions. The number of functions removed due to near-linear dependence, if any, given in parentheses. All results in OR-units, wavelengths in nm.
Symbol N denotes the number of basis functions. The number of functions removed due to near-linear dependence, if any, given in parentheses. All results in OR-units, wavelengths in nm.
Table 6. Optical rotation of molecule 4.
smaller error with respect to the reference values obtained in
the daV5Z set. Considering molecule 2, again the LPol-ds and
the ORP sets perform more than satisfactorily, outperforming
larger basis sets. For molecule 3, correct description of OR is
more difficult and it should be noted that the results obtained
in the largest Dunning’s basis sets used for this system may be
still somewhat distant from the basis set limit. The ORP, the
SVPD and TZVPD, and the LPol-n basis sets perform very well,
most of them outperforming both the aVDZ and aVTZ sets.
The investigated apc(S)n basis sets yield for molecules 1–3
results in general in a worse agreement with the reference values than the ORP and the LPol-n sets. In particular, the LPol-fs
N
aVDZ
aVTZ
aVQZ
daVDZ
daVTZ
daVQZ
apc1
apc2
apcS1
apcS2
SVPD
TZVPD
LPol-ds
LPol-dl
LPol-fs
LPol-fl
ORP
146
322
596
206
440 (7)
792 (21)
146
322
158
334
131
205
222 (1)
282 (3)
338 (10)
456 (24)
206
633.0
589.3
355.0
214.15
27.55
26.91
210.49
26.82
26.82
29.17
25.40
29.31
25.33
28.72
28.38
27.25
26.31
26.59
26.63
27.27
215.50
27.90
27.15
211.24
27.04
27.04
29.74
25.39
29.92
25.31
29.40
28.91
27.52
26.42
26.76
26.81
27.54
6.15
25.55
27.82
19.05
28.90
28.50
21.74
33.28
20.29
33.46
9.70
17.89
28.29
31.66
30.32
29.50
28.71
Symbol N denotes the number of basis functions. The number of functions removed due to near-linear dependence, if any, given in parentheses. All results in OR-units, wavelengths in nm.
2010
N
633.0
589.3
355.0
aVDZ
aVTZ
aVQZ
daVDZ
daVTZ
apc1
apc2
apcS1
apcS2
SVPD
TZVPD
LPol-ds
LPol-dl
ORP
274
598
1100 (6)
386 (13)
816 (37)
274
598 (9)
298
622 (9)
243
389
418 (10)
530 (18)
386 (1)
28.86
29.13
28.24
28.65
27.87
29.09
28.54
28.39
28.69
27.99
28.70
27.42
28.10
27.15
211.40
211.67
210.63
211.15
210.19
211.73
210.98
210.91
211.15
210.52
211.21
29.65
210.47
29.31
2132.23
2129.87
2127.15
2132.67
2123.56
2139.01
2128.06
2137.02
2128.68
2159.08
2137.64
2122.12
2126.62
2118.20
Symbol N denotes the number of basis functions. The number of functions removed due to near-linear dependence, if any, given in parentheses. All results in OR-units, wavelengths in nm.
Table 4. Optical rotation of molecule 2.
Basis set
Basis set
Journal of Computational Chemistry 2013, 34, 2006–2013
Table 7. Optical rotation of molecule 5.
Basis set
N
633.0
589.3
355.0
aVDZ
aVTZ
daVDZ
daVTZ
apc1
apc2
apcS1
apcS2
SVPD
TZVPD
LPol-ds
LPol-dl
ORP
374
828 (2)
528 (30)
1132 (79)
374
828 (25)
404
858 (25)
328
514 (1)
568 (19)
722 (37)
528 (5)
18.58
16.37
17.12
15.71
16.30
15.75
16.68
15.55
9.03
15.16
15.82
15.19
16.13
24.37
21.76
22.66
21.00
21.69
21.02
22.12
20.78
12.59
20.21
21.10
20.38
21.47
251.86
241.13
245.95
238.55
240.59
236.76
240.55
235.63
160.83
220.84
237.51
236.08
239.15
Symbol N denotes the number of basis functions. The number of functions removed due to near-linear dependence, if any, given in parentheses. All results in OR-units, wavelengths in nm.
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Table 8. Optical rotation of molecule 6.
Table 10. Optical rotation of molecules 8–27 calculated in aVDZ and ORP
basis sets for k 5 589.3 nm.
Basis set
N
633.0
589.3
355.0
aVDZ
aVTZ
daVDZ
daVTZ
apc1
apc2
apcS1
apcS2
SVPD
TZVPD
LPol-ds
LPol-dl
ORP
392
874 (3)
554 (30)
1196 (87)
392
874 (28)
422
904 (29)
344
532 (1)
594 (20)
756 (39)
554 (5)
2.53
1.80
2.38
1.45
3.24
1.43
2.71
1.32
20.56
0.64
1.59
1.41
1.34
3.15
2.29
2.96
1.88
3.97
1.86
3.35
1.73
20.49
0.93
2.05
1.84
1.76
20.28
17.72
19.51
16.36
22.80
16.25
20.67
15.81
6.81
12.41
16.77
15.98
15.89
Symbol N denotes the number of basis functions. The number of functions removed due to near-linear dependence, if any, given in parentheses. All results in OR-units, wavelengths in nm.
basis set yields results significantly closer to the reference than
any of the apc(S)n sets while its size is only marginally larger
than that of the apcS2 basis set.
For molecules 4 through 7, the LPol-ds and ORP basis sets
are the most attractive choices among the smaller basis sets,
and the LPol-dl and the apc2 among those larger. It should be
stressed that the ORP basis set is the smallest set performing
satisfactorily in all seven investigated organic molecules and at
the three wavelengths, which is very encouraging. Also the
slightly larger LPol-ds basis set yields OR results very close to
the reference values. The performance of the SVPD and TZVPD
depends on the system, for example, it is excellent for molecule 1 but rather poor for molecule 2. Nevertheless, the TZVPD
could be an attractive choice in the case of some systems due
to its small size, and further studies of its performance carried
out for a larger group of systems are highly desirable.
Results of another test of the ORP basis set, carried out for
a set of 20 molecules 8–27, are presented in Table 10. The
ORP values are compared to the aVDZ values obtained in the
present study. Corresponding differences are in the order of a
Table 9. Optical rotation of molecule 7.
Basis set
N
aVDZ
aVTZ
daVDZ
daVTZ
apc1
apc2
apcS1
apcS2
SVPD
TZVPD
LPol-ds
LPol-dl
ORP
356
782 (1)
502 (24)
1068 (66)
356
782 (20)
386
812 (20)
315
499 (1)
542 (16)
688 (30)
502 (3)
633.0
589.3
355.0
210.75
211.09
210.26
210.61
210.14
211.35
210.15
211.39
25.09
211.00
210.01
210.69
210.60
211.09
211.55
210.51
210.97
210.40
211.83
210.39
211.88
24.48
211.39
210.27
211.06
210.96
111.63
102.43
116.25
105.82
108.94
103.70
110.37
103.07
129.09
107.33
108.97
106.66
106.46
Symbol N denotes the number of basis functions. The number of functions removed due to near-linear dependence, if any, given in parentheses. All results in OR-units, wavelengths in nm.
aVDZ
System
[a]
8
9[b]
10[c]
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
ORP
½a589:3
Ndel
½a589:3
Ndel
75.87
58.41
101.97
2169.23
265.48
129.59
21192.73
58.14
266.97
94.13
116.23
84.09
79.54
43.44
29.88
149.80
111.53
46.16
2254.31
408.13
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
61.66
49.02
91.33
2162.33
260.69
133.77
21174.35
58.41
263.77
91.15
113.32
83.79
77.42
42.76
29.71
145.71
108.25
45.11
2250.33
408.76
0
0
0
0
0
0
1
6
5
0
1
0
2
2
2
0
1
5
5
0
[a] The aVTZ result: 62.70 OR-units. [b] The aVTZ result: 52.64 OR-units.
[c] The aVTZ result: 93.28 OR-units. Symbol Ndel denotes the number of
basis functions removed due to near-linear dependence. All results in
OR-units.
few up to 20 percent of the total OR value. In the case of molecules 8, 9, and 10, for which observed differences are the
largest, calculations carried out using the aVTZ basis set (see
Table 10) have shown that the use of the ORP basis set allows
for significant improvement of accuracy of the results.
Table 11 presents OR values obtained in the aVDZ and ORP
basis sets for test flexible systems. In general, good agreement
of the aVDZ and ORP results is observed, with the only exception being the conformer 30A for which the two basis sets
yield opposite sign of OR. Test calculation carried out for this
conformer in the aVTZ basis set gave an agreement of sign
with the ORP set. This suggests that in some cases the use of
ORP basis set might help to avoid errors in the calculated OR
sign with respect to the basis set limit while keeping the computing cost reasonable. Agreement of the ORP results with the
experimental values is in general similar as in the case of the
aVDZ set (slightly improved for systems 29 and 31, and worse
for system 30). The use of the ORP set did not improve the
agreement of theoretical OR sign with the experimental one in
the difficult case of molecule 28. Most probably other factors,
for example, solvent effects, need to be taken into account to
correct the OR sign for this molecule.
We finally comment on the near-linear dependence of basis
functions in the investigated basis sets. The number of functions removed from each basis set due to near-linear dependence is shown in Tables (2–10). Although the number of
linearly dependent functions increases rapidly with the
increase of system size for the LPol-n basis sets, as previously
reported by Srebro et al.,[56] the ORP set behaves similarly to
other medium-size basis sets and its use is thus not likely to
lead to numerical problems.
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2011
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Table 11. Optical rotation of flexible test molecules.
aVDZ
Di E
Xi ½%
Xi ½%½a589:3
i
232.49
32.42
246.50
44.36
228.45
79.51
12.31
4.09
2.15
1.93
(theor.)
(exp.)
0.0000
1.1053
1.7574
2.1392
2.2026
223.34
12.5
247.12
293.10
231.54
218.79
13.42
22.21
(theor.)
(exp.)
0.0000
0.0745
0.8841
1.3646
1.7659
2.9072
260.64
229.9
(theor.)
(exp.)
0.0000
1.7879
2.1899
2.5417
2.7982
26.38
28.4
(theor.)
(exp.)
0.0000
0.3783
0.6841
0.7220
1.0409
2.3715
50.67
36.4
Conformer
28
A
B
C
D
E
½a589:3
½a589:3
29
A
B
C
D
E
F
½a589:3
½a589:3
30
A[a]
B
C
D
E
½a589:3
½a589:3
31
A
B
C
D
E
F
½a589:3
½a589:3
ORP
½a589:3
i
Di E
½a589:3
i
Xi ½%
Xi ½%½a589:3
i
225.83
3.99
21.90
0.95
20.55
0.0000
1.1676
1.8372
2.1779
2.2404
225.62
12.5
234.34
32.09
247.31
43.67
229.88
81.14
11.31
3.65
2.05
1.85
227.86
3.63
21.73
0.90
20.55
44.15
38.94
9.93
4.41
2.24
0.33
220.81
236.25
23.13
20.83
0.30
0.07
0.0000
0.2127
1.0157
1.4557
1.8004
2.9426
257.26
229.9
243.79
293.01
232.56
220.20
12.17
21.16
49.53
34.59
8.92
4.24
2.37
0.35
221.69
232.17
22.90
20.86
0.29
0.07
22.46
264.27
24.46
233.00
292.54
91.21
4.46
2.26
1.25
0.81
22.24
22.87
20.10
20.41
20.75
0.0000
1.9171
2.3149
2.6110
2.8694
21.40
28.4
2.04
261.08
22.70
231.81
289.45
92.63
3.64
1.86
1.13
0.73
1.89
22.23
20.05
20.36
20.65
73.78
217.68
85.17
68.35
43.61
252.56
42.92
22.67
13.53
12.69
7.41
0.78
31.67
24.01
11.52
8.67
3.23
20.41
0.0000
0.2423
0.5210
0.7409
1.0140
2.3009
49.59
36.4
75.29
216.69
86.68
69.50
45.41
248.69
38.96
25.88
16.17
11.16
7.04
0.80
29.33
24.32
14.02
7.75
3.20
20.39
[a] The aVTZ result for A conformer: 0.86 OR-units. Relative energy of conformer i, Di E, in kcal/mol, specific rotation in OR-units.
On the basis of the presented results, we recommend the
ORP basis set as an efficient tool for DFT/B3LYP OR calculations due to its overall satisfactory performance and small size.
Other basis sets which are attractive choices are the LPol-ds
and -dl, and whenever the size of basis set does not need to
be kept very small, also the apc(S)2 and LPol-fs basis sets. The
ORP basis set gives results of quality at least comparable to
that obtained using the LPol-ds and -dl sets, enabling
approaching the basis set limit at a reduced cost. Its competitiveness to other basis sets of similar and larger size is particularly evident in the case of Tables (2–4) where the results
obtained in up to the aV6Z, daV5Z, and daVQZ basis set,
respectively, are available. Because the ORP set is only slightly
larger than the aVDZ set, it may be routinely used in the DFT
OR calculations for medium size and large molecules, including
conformationally flexible systems. The ORP set is thus a valuable tool for accurate calculations of specific rotation at a
reduced cost. Its use obviously does not ensure that the
results will be in excellent agreement with the experimental
data, especially in the case of measurements carried out in solutions. Most probably accounting for the solvent effects and
resorting to the CC methods will be necessary in trouble
cases.
2012
Journal of Computational Chemistry 2013, 34, 2006–2013
Summary and Conclusions
We have presented details of generation of the ORP basis set
developed for accurate OR calculations and the results of test
calculations performed for the total of 32 test systems. Despite
its relatively small size, the ORP basis set yields OR values very
close to those obtained in the largest basis sets of Dunning
and coworkers used in the study. This gives possibility of carrying more accurate DFT-specific rotation calculations at a
reduced cost, enabling them for systems larger than previously, also for large and conformationally flexible molecules.
Presently, we plan to use the ORP basis set in the OR calculations carried out within the CC approximation.
Other attractive choices among the investigated basis
sets are the LPol-ds and -dl sets. In the case of small and
medium-size molecules, also the larger apc(S)2 and LPol-fs
basis sets can be recommended. The TZVPD set yields attractive results for some of the investigated molecules and we
plan to investigate its performance in more details in close
future.
Keywords: absolute configuration optical rotation density
functional theory LPol-n ORP basis set
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How to cite this article: A. Baranowska-Ła˛czkowska, K. Z.
Ła˛czkowski, J. Comput. Chem. 2013, 34, 2006–2013. DOI:
10.1002/jcc.23347
Additional Supporting Information may be found in the
online version of this article.
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Received: 8 January 2013
Revised: 19 April 2013
Accepted: 12 May 2013
Published online on 5 June 2013
Journal of Computational Chemistry 2013, 34, 2006–2013
2013