CCSD-CTOCD Static Dipole Shielding Polarizability forQuantification of the Chiral NMR Effects in OxaziridineDerivatives

Stefano Pelloni[a] and Inmaculada Garc�ıa Cuesta*[b]

Chiral discrimination by nuclear magnetic resonance (NMR)

spectroscopy might be achieved through the pseudo-scalar

derived from the dipole shielding polarizability tensor. Coupled

Cluster Singles and Doubles-Quadratic Response (CCSD-QR) cal-

culations inside the continuous translation of the origin of the

current density formalism have been carried out to determine

the effects of basis set, electron correlation, and gauge transla-

tion on the determination of this magnitude in oxaziridine deriva-

tives. Inclusion of electronic correlation is needed for adequately

describing the pseudo-scalar for the heavier nuclei, making CCSD

a rigorous and affordable method to compute these high order

properties in medium-sized molecules. The observable magni-

tudes for chiral discrimination (produced RF voltage and required

electric field) are calculated. Half of the considered molecules

show values of the observable magnitudes near the lower limit

for experimental detection. Nuclei 19F, 31P, and 79Br produce

the largest values of RF voltage (50–80 nV). Moreover, 31P and79Br are the nuclei requiring smallest electric fields (3 MVm21) to

separate the NMR signals, being then suitable for both the

techniques. VC 2014 Wiley Periodicals, Inc.

DOI: 10.1002/jcc.23689

Introduction

It is known that nuclear magnetic resonance (NMR) spectroscopy

has a “chiral blindness” as the two mirror images of a chiral mol-

ecule cannot be distinguished. Magnetic properties like nuclear

magnetic shieldings or coupling constants do not change as

they have the same sign and magnitude on inverting the coordi-

nate system, that is, passing from one enantiomer to another.

Conversely, resolution of racemic mixtures, recognition of enan-

tiomers, and absolute configuration assignment are central prob-

lem in synthetic biochemistry. Some years ago, Buckingham[1]

tried to give a solution with the introduction of a pseudo-scalar

related to the magnetoelectric third-rank tensor dipole shielding

polarizability (DSP). The DSP describes the interaction of a mag-

netic and electric field with matter, that is, an NMR experiment

with an external electric field that can be static or dynamic. The

important point is that this tensor changes sign passing from a

system to its mirror image and, therefore, makes it possible, at

least in principle, to recognize two enantiomers of a molecule.

Chiral discrimination via NMR spectroscopy could be a quick and

affordable method, but most of theoretical calculations of DSP pre-

dicts magnitudes too small to be experimentally detected and,

hence, to differentiate between two enantiomers of a molecule by

NMR measures was not feasible.[2–4] However, recent works have

encountered[5] at least two order of magnitude larger than those

reported so far; therefore, the experimental detection is starting to

be possible. In this context, an accurate theoretical description of

the magnitude is of fundamental importance to be susceptible of

being unequivocally confirmed by experiment.

The effects of an electric field on NMR spectral parameters

have been theoretically determined since some time ago,[3,6–14]

either in a purely analytically manner through quadratic response

(QR) methods or by numerical differentiation of analytical shield-

ing tensors. Even though there are predominate calculations at

the coupled Hartree–Fock (CHF) level, correlation contributions

may be important as the nuclear magnetic shieldings are, in

many molecules, strongly affected by electron correlation.[15–21]

Indeed, it is possible to find a variety of examples in the literature

of calculations of DSP with different correlated methods, includ-

ing Complete Active Space Self Consistent Field (CASSCF), MP2

and MP3, coupled cluster, and several parameterizations of the

density functional theory (DFT). Unfortunately, such studies have

mainly focused on small molecular models not necessarily pre-

senting chiral behavior. However, chiral molecules of chemical

interest are usually of larger size and contain several heavy atoms

often even from the third or fourth row of the periodic system.

Therefore, we find it convenient to carry out a systematic investi-

gation of DSP in some medium-sized chiral molecules account-

ing also for correlation effects.

As the nuclear magnetic shieldings are origin sensitive, it is

advantageous to use a gauge-independent method as the

continuous translation of the origin of the current density

(CTOCD).[22] We use in this work, a coupled cluster implemen-

tation of the CTOCD method in its diamagnetic zero (DZ)

[a] S. Pelloni

Dipartimento di Chimica Universit�a degli Studi di Modena e Reggio Emilia.

via Campi 183, 41100 Modena, Italy

[b] I. G. Cuesta

Instituto de Ciencia Molecular, Universidad de Valencia, P.O. Box 22085,

46071 Valencia, Spain

E-mail: garciain@uv.es

Contract grant sponsor: Spanish MICINN; Contract grant number:

CTQ2010–19738; Contract grant sponsor: Italian MIUR (Ministero

dell’Istruzione, dell’Universit�a e della Ricerca; PRIN 2009 scheme)

VC 2014 Wiley Periodicals, Inc.

Journal of Computational Chemistry 2014, 35, 1815–1823 1815

FULL PAPERWWW.C-CHEM.ORG

formulation, the only CTOCD method in which analytical QR is

available, while the others need numerical integration. A study

on (R)-N-fluor-C-methyl-oxa-aziridine with a hierarchy of

increasing basis sets and of coupled cluster methods has been

carried out to investigate for the first time the performance of

the CTOCD-DZ approach to compute DSP. Furthermore, differ-

ent gauge origins have been considered, namely the center of

mass (CM) or each one of the atoms themselves.

The choice of such aziridine derivative has three principal

reasons: (i) oxaziridine molecule is often used as substrate in a

wide range of enantioselective organic oxidation reactions,

and there is a lot of publications on the chemistry of oxaziri-

dine;[23,24] (ii) in a previous article,[4] the (2R)N-methyloxaziri-

dine shows the highest value of DSP found in the literature

(for oxygen atom) and so oxaziridine was a good starting

point to build a new molecule that may show interesting fea-

tures; (iii) this system contains C, O, N, and F atoms, that is,

the nonmetals of the second row of the periodic table, those

are the bricks to build the 90% of the known molecules.

From the conclusions of the previous assessment, the study

has been extended to include also a set of derivatives of the

(R)-N-fluor-C-methyl-oxa-aziridine obtained by substitution of

oxygen by sulfur, nitrogen by phosphorus, and/or fluorine by

chlorine or bromine. As we are mainly interested in the chiral

discrimination problem, we also computed the chiral NMR

effects for the whole set of considered molecules using two of

the strategies previously proposed by Buckingham:[1,2] the cal-

culation of the RF voltage produced by the rotating chiral

electric polarization induced by permanent magnetic nuclear

momentum and the required electric field to produce a differ-

ence of 0.01 ppm between the NMR chemical shifts of the

two enantiomers.

This article is organized as follows: in Calculation Method

section, we summarize the theory that leads to the definitions

of the response properties of interest as well as the technical

details of the reported calculations. The first part of Results

and Discussion section analyzes the effects of atomic basis,

gauge origin, and correlation level on the computed CTOCD-

DZ DSP of (R)-N-fluor-C-methyl-oxa-aziridine. The second part

of this section is devoted to the theoretical determination of

observable magnitudes for chiral discrimination for the whole

set of considered derivatives. Finally, some concluding remarks

are given in Conclusions section.

Calculation Method

Theoretical background

From a Taylor-series expansion of the total energy of a mole-

cule with nuclei with nuclear magnetic dipoles mI in the pres-

ence of external magnetic field B and electric field E (see for

instance, eq. (365) of Ref. [25]), it is trivial to identify the

nuclear magnetic shielding as

rIab5

@2W

@mIa@Bb

����mI;B!0

(1)

and the nuclear magnetic shielding polarizability as

rIabc5

@3W

@mIa@Bb@Ec

����mI;B!0;E!0

(2)

For an isotropic medium, it is expedient to define molecular

average properties by introducing the pseudo-scalar[26–28]

rð1ÞI5

1

6�abcr

Iabc (3)

where �abc is the skew-symmetric Levi-Civita tensor. Therefore,

the tensor rIabc has a chirally sensitive isotropic part rð1Þ

I. Fur-

thermore, the pseudo-scalar allows for compact expressions

for the magnetic field induced at nucleus I

BnI 52rIB2r 1ð Þ I

B3E (4)

the electric dipole induced in the electron cloud

M5 2r 1ð Þ ImI3B (5)

and the induced magnetic dipole

M 5 2rImI2r 1ð Þ IE3mI (6)

Being a third-order property, the shielding polarizability can

be expressed in terms of QR functions. Using the notation of

Bishop, Orr, and Ward[6–8] for the propagator,[29,30] eqs. (7) and

(8) give, respectively, the expressions for the frequency-

dependent paramagnetic ðrpIabcÞ and diamagnetic (rdI

abcÞ contri-

butions to the magnetic-shielding polarizability

rpIabc 2xr; x1;x2ð Þ52hhBn

Ia; mb; lciix1;x2

521

�h2

XP

Xj;k 6¼a

hajBn

Iajjih jjlc jki hkjmbjaixja2xr� �

xka2x1ð Þ(7)

rdIabc 2x; xð Þ52<hhrdI

ab; lciix

51

�h

Xj 6¼a

2xja

x2ja2x2

� �< hajrdIabjji hjjlcjai

� �(8)

where, in addition to the standard definitions of the electric and

magnetic dipole operators, we have used xr 5 x1 1 x2, over-

lined operators are defined, for instance, as la5la2hajlajaiand RP stands for the sum over all permutations of the pairs

Bn

Ia=2xr

� �, mb=x1

� �, lc=x2

� �. The operator for the diamag-

netic contribution is written as

rdIab5

e

2mec2

Xn

i51

ricEi

Icdab2riaEi

Ib

� �" #(9)

Ei

I is the operator for the electric field exerted on nucleus I by

electron i, such that the operator for the total field of n elec-

trons is

En

I 5Xn

i51

Ei

I5Xn

i51

1

4p�0e

ri2RI

jri2RIj3(10)

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1816 Journal of Computational Chemistry 2014, 35, 1815–1823 WWW.CHEMISTRYVIEWS.COM

while Bn

I is the operator of the magnetic field of n electrons

on nucleus I[25,27,31]

Bn

I 52l0

4pe

meM

n

I 52l0

4pe

me

Xn

i51

ri2RI

jri2RIj33pi (11)

In the CTOCD-DZ[25] scheme, the diamagnetic term is

replaced by a D contribution (rDIabcÞ:

rDIabc 2xr; x1;x2ð Þ5�bklhhPk; Rc; T

n

Ilaiix1;x2

5�bkl1

�h2

XP

Xj;k 6¼a

hajPkjjihjjRc jki hkjTn

Ilajaixja2xr� �

xka2x1ð Þ(12)

In eq. (12), P is the linear momentum operator and T is a

Hermitian operator needed in the CTOCD-DZ approach:

Tn

Iab51

2

Xn

i51

ria2r0a� �

Mi

Ib1Mi

Ib ria2r0a� �

(13)

An indirect measure of the pseudo-scalar rð1ÞI

can be

obtained by taking into account[1,2] that the precessing

nuclear magnetization MIx induces a rotating chiral electric

polarization PIy ,

PIy5rð1Þ

IMI

x Bz (14)

which generates a RF voltage in a capacitor with plates at 6 d2

on the y axis incorporated into the resonance circuit

VðIÞ5PI

y d

e21ð Þe05

NIjrð1ÞIj gIlNB0

z

� �2II II11ð Þ

3kT

d

e21ð Þe0(15)

where NI is the number of nuclei per unit volume, gI is the

g-factor of the nucleus II, mN is the nuclear magneton, B0z the

magnetic field, II is the nuclear spin, e is the dielectric constant

of the medium and e0 is the permitivity of free space.

An alternative strategy is based on the chiral chemical shift

(i.e., the difference of nuclear magnetic shieldings between the

two enantiomers) produced by the external electric

field.[2,4,5,32] The shift depends on the magnitude of the field

and on the pseudo-scalar rð1ÞI

[see eq. (4)]. Consequently, it is

possible to determine the electric field required to observe an

induced magnetic field equivalent to an experimentally detect-

able shift. Note, however, that the induced field would be per-

pendicular to the strong magnetic field B of an NMR

spectrometer and, therefore, with no effect on the chemical

shifts at first order. Experimental designs overpassing this

problem have been discussed in Ref. [2].

Computational details

All geometries were optimized using the Gaussian03 pro-

gram[33] at the B3LYP/aug-cc-pVTZ level. The paramagnetic

[eq. (7)] and D [eq. (12)] contributions to static DSP tensor

components were calculated using the Dalton[34] program via

QR for both correlated and uncorrelated calculations. As cus-

tomary in CC response theory, we use the unrelaxed CC

approach, although—obviously—full relaxation is included in

the CHF estimations. To reach the Hartree–Fock limit in the

CHF calculations, four basis set of increasing size were used:

aug-cc-pVDZ, aug-cc-pVTZ, aug-cc-pVQZ, and aug-cc-pV5Z.[35]

Likewise, Sadlej-pVTZ[36–40] developed for electric polarizabil-

ities was also checked for latter use in the correlated calcula-

tions. Electron correlation effects in the calculation of the

shielding polarizability pseudo-scalar are analyzed in the

CTOCD-DZ scheme at different coupled cluster levels (CCS,

CC2, CCSD) inside the QR formalism. In addition, the men-

tioned pseudo-scalar was also computed through finite field’s

(FF) calculations at the CC2 and CCSD levels using analytically

evaluated nuclear magnetic shieldings. According to previous

literature,[9,41,42] we have used electric fields of 0.001 au of

magnitude.

Results and Discussion

Calculation assessment

To validate the used theoretical approach, we start by carrying

out a preliminary study of the effects of basis sets, gauge ori-

gin, and correlation contributions in the parent molecule (R)-

N-fluor-C-methyl-oxa-aziridine (see Fig. 1). This system is a chi-

ral molecule containing four heavy atoms with an asymmetric

carbon in a three-member ring.

It is well known that the CTOCD-DZ approach requires in

many cases rather large basis to guarantee reliable results.

Conversely, the N6 scaling—N is the size of the atomic basis

Figure 1. (R)-N-fluor-C-methyl-oxa-aziridine.

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Journal of Computational Chemistry 2014, 35, 1815–1823 1817

set—of CCSD calculations makes it convenient from a compu-

tational point of view to reduce as much as possible the afore-

mentioned basis set size. Clearly, for size-enlarged and less-

symmetric molecules, the use of extended basis becomes

more and more problematic. It is, therefore, desirable to use

basis set of relatively small size but still able to reproduce the,

in principle, more accurate results form larger basis. Conse-

quently, we have analyzed the convergence of the calculated

value of the pseudo-scalar rð1ÞI

with a hierarchy of Dunning’s

correlation consistent basis sets for the considered (R)-N-fluor-

C-methyl-oxa-aziridine. In Table 1, our results of rð1ÞI

for heavy

nuclei 13C, 14N, 17O, and 19F are shown. We present the para-

magnetic, D, and total contributions, which have been com-

puted at the CHF-QR(CTOCD-DZ) level of theory. In addition,

the results corresponding to the complete basis set (CBS)

extrapolation,[44–46] as well as those using Sadlej’s TZV basis

are also presented. The former were calculated as

lim X!1 a1bexp 2c Xð Þf g.To calculate the CBS limit, the extrapolation was carried out

independently for each one of the QR functions in eqs. (7) and

(12) using Dunning’s aug-cc-pVXZ with X ranging from D to 5.

In most of the cases, a monotonic behavior was encountered

although in some instances the aug-cc-pVDZ basis results

break the monotonicity. At any rate, differences from including

or not the aug-cc-pVDZ basis results are not relevant, as they

appear only in the smallest contributions to the pseudo-scalar,

the dominant terms in it showing a monotonous trend. How-

ever, those small components—together with the opposite

sign of the large components—are responsible of the lack of

monotonicity of the overall pseudo-scalar. Only the nitrogen

atom behaves differently, because of a very slow convergence

of the XZY element of the paramagnetic DSP tensor, which is

the one dominating rð1ÞN

. To illustrate this fact, consider that

the CBS limit of rpIxzy varies from 231.0719 au to 229.9382 au

depending whether the aug-cc-pVDZ basis is included or not

in the extrapolation, in spite of the perfectly monotonic

behavior of the quadratic function with respect of basis set

size in this case.

For all atoms, the values of rð1ÞI

total are rather similar with

all basis sets; indeed, the computed value for the aug-cc-pV5Z

is nearly identical in all the cases to the limit value with the

exception mentioned above. Nonetheless, the paramagnetic

and D contributions vary a little bit more, even though the

paramagnetic contribution—that is, the leading one—shows a

fast convergence. The convergence of the D contribution is

slightly slower, but we note that it converges to zero in

accordance to the fact that the diamagnetic part of the DSP

tensor rIabc is symmetric in bc and, therefore, does not contrib-

ute to the pseudo-scalar if the origin of the coordinate system

is taken on the nucleus in question.[2]

As mentioned before, we have also tested Sadlej’s TZV

basis looking for a reduced-size atomic basis set. Having

been optimized for the computation of electric properties,

notably it shows a performance close to that of the aug-cc-

pVQZ for C1, C2, O, and F atoms as can be checked by

comparison to the limit values. Due to the mentioned slow

convergence of the large XZY element of the DSP tensor in

the case of N atom, Sadlej’s basis results are not so close

to those from aug-cc-pVQZ basis, as the overestimation of

the paramagnetic part is not cancelled by an adequate

compensation from the D contribution, as happens for the

fluorine atom. Remarkably, Sadlej’s basis gives for nitrogen

the results closest to the CBS limit. Moreover, Sadlej’s TZV

basis provides results that are inside a margin of 2 ppm

with respect to the values extrapolated to the complete

basis.

Even though the convergence of the basis at the CPHF level

is a necessary condition to achieve convergence also at CCSD,

it is obviously not sufficient, as correlated methods require

larger basis sets, especially for the QR functions that we calcu-

late. An estimation of the quality of Sadlej’s basis for CCSD-QR

calculations can be found by comparing the results using it to

those using aug-cc-pVQZ basis. We focus in the fluorine

atom, for it is the most sensitive to the basis set size in (R)-N-

fluor-C-methyl-oxa-aziridine. In particular, we obtained a value

of rð1ÞF5 226:8 ppm with Dunning’s and a slightly higher

value of rð1ÞF5 228:5 ppm with Sadlej’s basis set. Summariz-

ing, Sadlej’s TZV basis set represents a fair compromise

between accuracy and computational cost of the DPS pseudo-

scalar rð1ÞI.

Table 1. Basis set dependence of static dipole-shielding polarizability

pseudo-scalar rð1ÞI

for heavy atoms of (R)-N-fluor-C-methyl-oxa-aziridine

molecule in ppm au.

Atom Basis set Paramagnetic D Total

C1 aug-cc-pVDZ 222.346 20.531 222.876

aug-cc-pVTZ 224.218 0.110 224.108

aug-cc-pVQZ 224.001 0.146 223.855

aug-cc-pV5Z 224.254 0.072 224.182

CBS 224.79 0.04 224.75

Sadlej-VTZ 223.793 0.051 223.742

C2 aug-cc-pVDZ 27.848 0.766 27.082

aug-cc-pVTZ 28.360 0.313 28.048

aug-cc-pVQZ 28.338 0.070 28.268

aug-cc-pV5Z 28.355 0.010 28.344

CBS 28.41 20.17 28.57

Sadlej-VTZ 28.310 0.354 27.955

O aug-cc-pVDZ 230.792 1.362 232.154

aug-cc-pVTZ 231.994 21.478 230.516

aug-cc-pVQZ 233.090 20.798 232.292

aug-cc-pV5Z 234.111 20.286 233.825

CBS 234.09 20.26 233.83

Sadlej-VTZ 232.343 0.071 232.414

N aug-cc-pVDZ 243.706 1.746 241.961

aug-cc-pVTZ 238.402 0.639 237.762

aug-cc-pVQZ 237.956 0.166 237.790

aug-cc-pV5Z 237.976 0.140 237.835

CBS 244.09 20.05 244.14

Sadlej-VTZ 245.077 1.032 244.045

F aug-cc-pVDZ 223.280 6.364 216.916

aug-cc-pVTZ 221.097 1.247 219.849

aug-cc-pVQZ 220.865 0.442 220.423

aug-cc-pV5Z 220.823 20.024 220.847

CBS 220.14 21.68 221.82

Sadlej-VTZ 224.365 5.008 219.357

The conversion factors for nuclear magnetic shielding polarizability, from

atomic units to SI and cgs units, obtained from the CODATA recom-

mended values (Ref. [43]) are 1 ppm au 5 1026(ea0/Eh) 5 1.94469057 3

10218 m V21 5 5.83003566 3 10214 cm stat V21, respectively.

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To confirm the intrinsic gauge-origin independence of the

CTOCD-DZ approach, as well as to indirectly check the cor-

rectness of the variety of contractions of four index tensors

carried out, five CTOCD-DZ QR shifted origin calculations

have been performed for each atoms putting the origin of

the coordinate system in all heavy atoms. The results are

summarized in Table 2.

The gauge variation causes a change in the paramagnetic

contribution and an opposite, with equal magnitude, shift for

the D term, and as a result the sum does not vary signifi-

cantly. In fact, the maximum difference found because of the

displacement of the origin of gauge is approximately 0.4

ppm for the fluorine atom. Such a difference is very much

smaller than the deviations introduced for not taking a com-

plete (and clearly unfeasible) basis set. Again, it is convenient

to recheck the quality of our results when correlation is

switched on. The last section of Table 2 shows that gauge

origin invariance is maintained at the CCSD level. The

obtained results, then, prove the reliability of the chosen

method to calculate the DSP. Thus, the rest of reported calcu-

lations have been done using Sadlej’s TZV basis and CTOCD-

DZ approach.

We finish the study on (R)-N-fluor-C-methyl-oxa-aziridine

considering the effects of electron correlation on the CTOCD-

DZ estimations of the pseudo-scalar at different coupled clus-

ter levels, specifically CCS, CC2, and CCSD. As customary in

coupled cluster response theory, we use unrelaxed QR func-

tions, that is, Hartree–Fock orbitals are not allowed to adapt to

the perturbation in the post-Hartree–Fock part of the calcula-

tions. Anyway, relaxation effects are partially accounted for as

single amplitudes are treated to infinite order.[47,48] Further-

more, we recall that for static properties the unrelaxed

coupled cluster hierarchies converges toward the full-

configuration interaction limit, if the basis sets are perturba-

tion independent as we do here.[49] For the sake of complete-

ness, we have also performed CCS calculations, as they do

include neither correlation nor relaxation effects. Table 3

shows the computed values of rð1ÞI

for the heavy nuclei of

the studied aziridine at the different considered levels of

theory.

The same that in the case of CTOCD-DZ nuclear magnetic

shieldings,[16] the first point to remark on the numbers in

Table 3 is the absolute lack of accuracy of the CCS esti-

mates for rð1ÞI, which are consistently lower than the pre-

dicted by the other methods. Parallel again to the NMR

shieldings, the inadequacy of the CCS approach is caused

by not including at all the orbital relaxation contributions

as proven by the reasonable quality of the values com-

puted inside the CHF formalism, which implicitly includes

orbital relaxation.

Table 2. Origin dependence of static dipole-shielding polarizability

pseudo-scalar rð1ÞI

for heavy atoms of (R)-N-fluor-C-methyl-oxa-aziridine

molecule in ppm au.

Atom Gauge Paramagnetic D Total

CHF-QR

C1 C.M. 223.793 0.051 223.742

C1 223.777 0.028 223.748

C2 224.722 0.974 223.748

O 225.966 2.229 223.737

N 223.624 20.115 223.739

F 224.418 22.330 223.748

C2 C.M. 28.310 0.354 27.955

C1 28.292 0.341 27.952

C2 28.049 0.095 27.954

O 26.273 21.686 27.960

N 28.613 0.660 27.953

F 29.978 2.020 27.958

O C.M. 232.343 0.070 232.414

C1 231.776 0.692 232.467

C2 227.280 5.167 232.447

O 232.530 20.057 232.473

N 232.223 0.125 232.349

F 236.645 24.341 232.304

N C.M. 245.077 1.032 244.045

C1 245.173 1.159 244.014

C2 242.814 21.116 243.930

O 247.979 4.108 243.871

N 244.062 20.012 244.074

F 245.027 0.861 244.166

F C.M. 224.365 5.008 219.357

C1 229.555 10.173 219.382

C2 238.788 19.450 219.337

O 214.239 24.779 219.019

N 221.357 2.006 219.351

F 219.377 20.064 219.441

CCSD-QR

C1 C.M. 229.040 20.110 229.150

F 226.908 22.241 229.150

O C.M. 203.955 0.709 204.664

F 201.688 2.982 204.670

N C.M. 259.380 20.303 259.683

F 261.560 1.876 259.684

F C.M. 232.595 5.749 226.845

F 226.817 20.029 226.846

All calculations used the Sadlej-pVTZ basis set. CM is the center of

mass.

Table 3. CTOCD-DZ Coupled Cluster estimates of dipole-shielding polariz-

ability pseudo-scalar rð1ÞI

in ppm au for (R)-N-fluor-C-methyl-oxa-aziridine

molecule using the Sadlej-pVTZ basis set.

Atom Method Paramagnetic D Total

C1 CHF-QR 223.793 0.051 223.742

CCS-QR 219.338 0,715 218.624

CC2-QR(FF[a]) 233.728 0.347 233.382 (233.639)

CCSD-QR(FF) 229.040 20.110 229.150 (229.154)

C2 CHF-QR 28.310 0.354 27.955

CCS-QR 27.541 21.242 28.782

CC2-QR(FF) 213.370 0.900 212.470 (212.500)

CCSD-QR(FF) 211.227 20.132 211.359 (211.361)

O CHF-QR 232.343 0.070 232.414

CCS-QR 165.708 9.838 175.546

CC2-QR(FF) 245.984 23.804 242.180 (241.256)

CCSD-QR(FF) 203.955 0.709 204.664 (204.681)

N CHF-QR 245.077 1.032 244.045

CCS-QR 232.440 8.400 224.040

CC2-QR(FF) 281.002 20.581 281.583 (280.555)

CCSD-QR(FF) 259.380 20.303 259.683 (259.686)

F CHF-QR 224.365 5.008 219.357

CCS-QR 219.257 13.323 25.934

CC2-QR(FF) 244.977 5.279 239.698 (239.185)

CCSD-QR(FF) 232.595 5.749 226.845 (226.843)

[a] Finite Fields.

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Journal of Computational Chemistry 2014, 35, 1815–1823 1819

From the analysis above, it is evident that orbitals’ relaxation

effects are required to get a trustable picture of the electron

correlation contribution to the DSP pseudo-scalar. The lowest

level coupled cluster method is CC2 model, where as in CCSD,

the t1 amplitudes are treated to infinite order with the aim

of providing an approximate implicit description of orbital

relaxation, but some approximations are introduced in the

doubles part of CCSD equations. Moreover, in cases where

the correlation contributions are essential, the CC2 numbers

are much closer than CHF ones to the highly correlated

CCSD(T) results.[16,50] For the molecules considered in this

article, it is very likely that CC2 strongly overestimates corre-

lation effects, giving values that can exceed the CPH results

up to 50%.

This indicates the necessity of an improved treatment of

dynamical correlation using the CCSD method. CCSD is a rigor-

ous, correlated ab initio level of theory, which is, generally, a

good systematic level available for these high-order properties,

for which inclusion of linked triples would be extremely costly.

However, it must be realized that correlation may become more

important in chiral molecule with large p-electrons systems. In

these systems, an explicit treatment of triples excitation—at

least in a perturbative manner—could be fundamental.

Our best estimates for rð1ÞI

have been computed using

CCSD-QR. Comparing these results to the uncorrelated CHF

ones, it is observed that for C, N, and F atom correlation effects

increase the absolute value of the property from a minimum of

�19% (C1 atom) to a maximum of �30% (C2 atom). The oxygen

case behaves oppositely, as there is a decrease of �14% when

including electron correlation. At any instance, it is evident that

electron correlation effects are very important to get an

adequate description of the pseudo-scalar rð1ÞIin (R)-N-fluor-C-

methyl-oxa-aziridine.

The encountered concordance between the analytical CCQR

numbers and those from unrelaxed FF approach (see Table 3) in

the calculations of the shielding polarizability pseudo-scalar is

excellent for these systems, confirming previous results.[9,20,41,42]

Additionally, using an electric field with 0.001 au magnitude has

been proven to be adequate for FF procedure also in the study

of the rð1ÞI

pseudo-scalar. The good agreement indicates that

this approach is viable for the calculation of the DSP in systems

with larger size, where CCSD-QR calculations may be prohibitive.

Estimation of chiral-discrimination observables in oxaziridine

derivatives

DSP pseudo-scalar has been calculated for a set of derivatives

of the parent (R)-N-fluor-C-methyl-oxa-aziridine molecule by

systematically changing nitrogen, oxygen, and fluorine atoms

by other atoms of the same group in the periodic system

(CH3ACHAZAYAX, for X@F, Cl, and Br; Y@N and P; Z@O

and S). According to the results from the previous section, we

use a CCSD-QR approach and Sadlej’s TZV basis along the

whole set of calculations. The obtained results for the heavy

atoms (13C, 14N, 17O, 19F, 31P, 33S, 35Cl and 79Br) of the consid-

ered derivatives are collected in Table 4.

It can be observed in Table 3 that the computed values of

rð1ÞI

are very different for each atom in the parent molecule.

Thus, while for the asymmetric carbon the value of the

pseudo-scalar is only of 29 ppm, rð1ÞI

is duplicated for the

nitrogen bonded to it. Notably, the oxygen atom that com-

pletes the ring shows a pseudo-scalar magnitude that is

approximately seven times larger, to be precise 205 ppm. Con-

versely, the fluorine atom bonded to nitrogen present a mag-

nitude very similar to that of the asymmetric carbon. It is to

remark that the large value of rð1ÞI

that we have calculated for

the oxygen atom stands out against those found in similar

molecular systems (2R)-N-methyloxaziridine (40 ppm) or

(2R)22-methyloxirane (10 ppm).[4]

The change of heavy atom causes different consequences

on rð1ÞI

depending on the environment. In this way, the DSP

pseudo-scalars for nitrogen and phosphorus are analogous

when these atoms are bonded to oxygen, and both are very

sensitive to the change of either the chalcogen or the halo-

gen. We observe an enlargement of the pseudo-scalar when

going down in the group. Oppositely, oxygen and sulfur show

very dissimilar magnitudes of rð1ÞI, the largest values corre-

sponding to sulfur. For these two atoms, the change of the

neighbor nitrogen by phosphorus does not produce a signifi-

cant variation, and even a lesser influence has the substitution

of the fluorine atom, which is not directly joined to the refer-

ence atom. Finally, among the halogens, fluorine presents the

smaller magnitude of the DSP pseudo-scalar; additionally, fluo-

rine is also less sensitive to the substitution of nitrogen by

phosphorus. It is noteworthy, that the change of oxygen by

sulfur increases in a factor of �2.5 the value of rð1ÞF, even

though they are not directly bonded. The chlorine and bro-

mine atoms show the largest values of rð1ÞI

among the halo-

gens, both of them being also exceedingly affected by

changes in the environment in all the considered cases.

The highest values of rð1ÞI

that we have found are those

corresponding to sulfur and phosphorus in the compounds of

larger molecular weight, that is, CH3ACHASANBr and

CH3ACHASAPBr. For those cases, the value of the DSP

pseudo-scalar becomes as high as 800 ppm, a value sensibly

larger than most of the previously reported. We stress that the

presence of a sulfur nucleus significantly enhances the DSP

pseudo-scalar of the neighboring heavy atoms, especially in the

case of phosphorus in which it can become four times bigger.

Anyway, it is worthy to mention that larger values of the

Table 4. CCSD-QR(CTOCD-DZ) dipole-shielding polarizability pseudo-

scalar rð1ÞI

in ppm using the Sadlej-pVTZ basis set.

Molecule N/P O/S F/Cl/Br

CH3ACHAOANF 259.68 204.66 226.84

CH3ACHAOANCl 2171.10 224.71 227.23

CH3ACHAOANBr 2232.39 276.68 280.01

CH3ACHAOAPF 269.68 210.74 248.19

CH3ACHAOAPCl 2215.70 276.63 2156.12

CH3ACHAOAPBr 2282.02 340.94 2374.43

CH3ACHASANF 2184.33 623.01 290.65

CH3ACHASANCl 2392.93 724.12 2127.03

CH3ACHASANBr 2518.91 830.39 2266.65

CH3ACHASAPF 2287.40 547.15 2118.34

CH3ACHASAPCl 2612.04 694.81 2281.68

CH3ACHASAPBr 2771.10 802.56 2650.09

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1820 Journal of Computational Chemistry 2014, 35, 1815–1823 WWW.CHEMISTRYVIEWS.COM

pseudo-scalar do not imply per se higher chance of measurabil-

ity, as (i) the larger the spin, the lower the accuracy of an NMR

measurement and (ii) the larger the natural abundance, the

stronger the intensity of the NMR bands. Therefore, we empha-

size that the best candidate for chiral discrimination by NMR is

the fluorine in CH3ACHASPF and CH3ACHASNF followed by

the phosphorous in CH3ACHASPCl and CH3ACHASPBr.

The presence of bromine in some of the molecules consid-

ered makes it opportune to look for a gross estimation of the

effect of relativity on the computed values of the DSP pseudo-

scalar. As relativistic calculations require very large basis specif-

ically designed for them, CCSD-QR calculations are too costly

to be carried out and, therefore, relativistic effects have been

approximately evaluated at the DFT-KT2 level of theory in a

finite differences approach using triple-zeta Dyall’s basis for

bromine and Sadlej’s elsewhere. To this end, we have use

DIRAC[51] program to calculate rð1ÞBr

in CH3ACHAONBr and

CH3ACHAOPBr and compared fully relativistic values of the

pseudo-scalar to those at the nonrelativistic limit. Such calcula-

tions indicate that relativity increases around 7% the nonrela-

tivistic result.

Having available accurate values of the CCSD shielding polariz-

ability pseudo-scalar, we make an estimation of the values that

could be experimentally measured and used as chiral discrimina-

tors. As stated before, two methodologies previously discussed

by Buckingham and coworkers are used to this end. First, we

consider the RF voltage generated in an appropriated experi-

mental setup. According to eq. (15), in addition to intrinsic prop-

erties of the considered nucleus and some universal constants,

voltage depends on the density of the liquid, the temperature,

the applied magnetic field, and the characteristics of the capaci-

tor. In Table 5, we report the generated RF voltage assuming a

magnetic field with flux density B 5 23 T, a pure liquid of density

1.0 g cm23, a capacitor with ðd=ðe21ÞÞ50:016 m, an isotopic

abundance of 100% and room temperature (T 5 300 K).

The largest RF voltage we have found is 76 nV and it is induced

by the 79Br nucleus in CH3ACHASAPBr. Relativistic effects are pre-

sumed to increase this magnitude, and therefore, 76 nV is the mini-

mum expected value. Note, nonetheless, that 79Br has a spin of

I 5 3/2, which can cause problems in the precision of the measure.

Somewhat smaller values, around 50 nV, are induced by 19F and by31P in CH3ACHASAYAF and CH3ACHASAPAX molecules. Notice

that both 19F and 31P have spin I 5 1/2 and 100% of abundance,

what in principle favors the experimental determination. It is also

worth to mention that all the “measurable favorable” compounds

contain sulfur in the three-member ring and a pair of them phos-

phorus too. As previously reported by Monaco and Zanasi,[32] fluo-

rine has proven to be a good candidate for detecting the discussed

RF voltage as its high value of g-factor provides values of that mag-

nitude above average. Nevertheless, the elevate values of the RF

voltage encountered for phosphorus are due to a large rð1ÞPin the

studied molecules.

The detection of the RF voltage depends on the experimen-

tal conditions, contrary to the determination of the NMR

chemical shift displacement in the presence of an external

static electric field. Table 6 collects the magnitude of the exter-

nal electric field required to induce a magnetic field perpen-

dicular to that of the NMR spectrophotometer and equivalent

to a difference of shifts of 0.01 ppm between the two enan-

tiomers. Anyway, this technique can be applicable only to

nuclei of low spin because the larger the spin, the lower the

precision of NMR measurement, due to band broadening aris-

ing for nuclear quadrupolar interaction.

In the studied molecules, the electric fields required to pro-

duce detectable chiral shifts have intensities of the order of

MV/m, which is maybe affordable but still too high for routine

measurements. At any rate, our results provide information on

the characteristics of compounds and nuclei more adequate

for a first experimental confirmation. Again, the “best”—that is,

the smallest—values are found for the nuclei of sulfur, phos-

phorus, bromine, and nitrogen in CH3ACHASANX and

CH3ACHASAPX. Even though the study of sulfur and bromine

could need signal-separation techniques for their high nuclear

spin, phosphorus and nitrogen should not face this difficulty.

Conclusions

NMR spectroscopy can distinguish diastereoisomers through a

pseudo-scalar obtained from the static DSP tensor, which has

Table 5. rf-voltage, V(I)(nV), in a capacitor with plates at 6d/2 on a direc-

tion perpendicular to the spectrometer’s magnetic field, induced by the

precessing nuclei of the species I, at 300 K, assuming density of 1 g

cm23 and 100% of isotopic abundance.[a]

Molecule N/P O/S F/Cl/Br

CH3ACHAOANF 0.6 30.7 16.6

CH3ACHAOANCl 1.4 27.9 0.8

CH3ACHAOANBr 1.3 23.3 11.5

CH3ACHAOAPF 6.5 25.9 24.4

CH3ACHAOAPCl 17.3 29.0 3.7

CH3ACHAOAPBr 16.2 25.6 48.0

CH3ACHASANF 1.5 10.6 46.4

CH3ACHASANCl 2.7 10.6 3.0

CH3ACHASANBr 2.5 8.6 34.4

CH3ACHASAPF 23.1 7.9 51.3

CH3ACHASAPCl 42.9 8.8 5.8

CH3ACHASAPBr 40.1 7.5 75.5

[a] Natural abundance: 1.108% (13C), 99.635% (14N), 0.037% (17O), 100%

(19F), 100% (31P), 0.750% (33S), 75.50% (35Cl), 50.60% (79Br), 49.40%

(80Br). All calculations were carried out at the CCSD-QR(CTOCD-DZ) level

with the Sadlej-pVTZ basis set.

Table 6. Electric field in 106 (Vm21) to be applied to yield a 0.01 ppm dif-

ference between NMR chemical shift of two enantiomers.

Molecule N/P O/S F/Cl/Br

CH3ACHAOANF 43.08 12.56 95.79

CH3ACHAOANCl 15.03 11.44 94.41

CH3ACHAOANBr 11.06 9.29 32.14

CH3ACHAOAPF 36.90 12.20 53.35

CH3ACHAOAPCl 11.92 9.29 16.47

CH3ACHAOAPBr 9.12 7.54 6.87

CH3ACHASANF 13.93 4.13 28.36

CH3ACHASANCl 6.54 3.55 20.24

CH3ACHASANBr 4.95 3.10 9.64

CH3ACHASAPF 8.95 4.70 21.73

CH3ACHASAPCl 4.20 3.70 9.13

CH3ACHASAPBr 3.33 3.20 3.95

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Journal of Computational Chemistry 2014, 35, 1815–1823 1821

opposite sign in each enantiomer and is zero for achiral mole-

cules. A systematic study has been carried out for the first

time to investigate the ability of the CTOCD-DZ approximation

to determine the DSP pseudo-scalar.

Using the parent molecule (R)-N-fluor-C-methyl-oxa-aziridine

as reference, the dependence of the computed pseudo-scalar

on basis set, gauge origin, and electron correlation has been

extensively investigated. Furthermore, the CBS limit at the CHF

level of theory has been determined, showing that basis satu-

ration is achieved with Dunning’s aug-ccpV5Z basis set and

that a reasonable compromise between numerical perform-

ance and the number of basis function is obtained with

Sadlej’s TZ that yields results that are in satisfactory agreement

with aug-ccpV5Z and CBS basis. The gauge invariance of

CTOCD-DZ method was checked and confirmed.

The reported calculations show that correlation may cause

large changes in the property value—its inclusion being

required for an accurate picture of the DSP pseudo-scalar—

even though it does not change the order of magnitude of

the calculated property. It seems that CHF and CCS generally

underestimate DSP, while CC2 overestimate it, exaggerating

electron correlation effects in its typical way. Conversely, CCSD

appears as a rigorous and affordable method to theoretically

determine accurate DSP in medium-sized molecules where the

inclusion on linked triples excitation would be too expensive.

All the molecules considered in this study show large values

of the DSP pseudo-scalar in one or more nuclei. Moreover, in at

least half of the studied systems the RF voltage generated by

the rotating chiral electric polarization, induced by the perma-

nent magnetic dipole moment and the chiral chemical shift are

inside the experimentally detectable range, allowing for chiral

discrimination by NMR techniques. The largest computed DSP

pseudo-scalar corresponds in all the considered examples to

the sulfur atom, but this is not the case for the measurable chi-

ral effects. However, its presence in the ring substantially

increases the value of the pseudo-scalar on the nearby atoms

N, F, Br, and especially P, whose isotopic abundance and nuclear

spin make it very favorable for NMR spectroscopy. Summarizing,

among the studied molecular systems, we suggest the fluorine

in CH3ACHASPF and CH3ACHASNF as well as the phosphorous

in CH3ACHASPCl and CH3ACHASPBr as the best candidates

for experimental verification of NMR chiral discrimination.

Acknowledgment

The authors are grateful to Profs. P. Lazzeretti and A. S�anchez de

Mer�as for helpful discussions.

Keywords: shielding polarizability � CCSD � complete basis

set � chiral discrimination

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Received: 28 February 2014Revised: 23 June 2014Accepted: 7 July 2014Published online on 22 July 2014

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