Could Electronic Anapolar Interactions Drive Enantioselective

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Quantum Electronic Structure

Could electronic anapolar interactions drive enantioselective syntheses in strongly nonuniform magnetic fields? A computational study Gabriel Ignacio Pagola, Marta B. Ferraro, Patricio Federico Provasi, Stefano Pelloni, and Paolo Lazzeretti J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b01002 • Publication Date (Web): 14 Jan 2019 Downloaded from http://pubs.acs.org on January 15, 2019

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Journal of Chemical Theory and Computation

Could electronic anapolar interactions drive enantioselective syntheses in strongly nonuniform magnetic fields? A computational study. G. I. Pagola, M. B. Ferraro Departamento de Física, Facultad de Ciencias Exactas y Naturales, and IFIBA, CONICET, Universidad de Buenos Aires, Ciudad Universitaria, Pab. I, (1428) Buenos Aires, Argentina; P. F. Provasi Departamento de Física, Northeastern University, Av. Libertad 5500, W3400 AAS, Corrientes, Argentina; S. Pelloni Istituto d’istruzione superiore Francesco Selmi, via Leonardo da Vinci, 300, 41126 Modena, Italy. and P. Lazzeretti∗ Istituto di Struttura della Materia, Consiglio Nazionale delle Ricerche, Via del Fosso del Cavaliere 100, 00133 Roma, Italy; ∗ e-mail: [email protected] January 13, 2019

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Abstract It is shown that the anapolar interaction of the electrons of a molecule with an external uniform magnetic field B and a uniform curl C = ∇ × B ′ determines different thermodynamical stabilization of the ground state for the enantiomers and diastereoisomers of a chiral molecule. A series of potential candidates for enantioselective syntheses has been investigated in a computational study at SCF-HF, B3LYP and various coupled cluster approaches to determine the energy difference between different enantiomers and diastereoisomers. The calculations show that these differences are very small for B and C presently available, but approximately 3 orders of magnitude bigger than those determined by parity violation effects. The chances that enantioselective synthesis may be attempted in the future are discussed. Recognition of anapolar interaction in chiral molecules via measurements of induced magnetic dipole moment in ordered phase may become possible in the presence of nonuniform magnetic field with strong gradient.

Keywords: magnetic response properties, molecules in a magnetic field with uniform gradient, anapole magnetizabilities, electron correlation effects, enatioselective syntheses

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1

Introduction

Assuming that the dissimmetry of the Universe is determined by some force akin to a magnetic field, and put on the wrong track by a deceiving resemblance between the natural optical activity and the magneto-optical rotation discovered by Faraday, 1 Pasteur asked Rhumkorff in Strasbourg to construct powerful magnets, 2 which would provide a proper influence on the reaction environment to drive enantioselective chemical syntheses. The fallacy of the analogy which misled Pasteur, evident to Faraday 3 and Lord Kelvin 4 , who decidedly reminded to keep away from a fictitious likeness, is even more discernible by the modern interpretation of molecular chirality in connection with the fundamental discrete symmetries of nature. 5,6 The intrinsic properties characterizing enantiomeric species are odd under parity, represented by an operator P which inverts the sign of the coordinates of all the particles, but even under the time-reversal operator T which inverts the direction of their motion. Instead, a spatially uniform magnetic field B is an axial vector, unaffected by P , i.e., P B = B, a character early recognized by Curie, who suggested to represent it by an arrow tangent to a circumference, 7 but odd under T , i.e., T B = −B. Therefore, in the presence of magnetic fields, a distinction between true and false chirality is needed. 5 The former is typical of natural optical activity, which is rationalized by a time-even pseudoscalar, the trace κ′ = κ′αα of a second-rank tensor, the mixed electric dipole/magnetic dipole polarizability κ′αβ . For instance, a beam of plane polarized light is rotated by a chiral species in disordered phase by an angle depending on κ′ . The Faraday effect, magnetic circular dichroism and magnetic optical activity are instead related to a time-odd axial vector with the same direction as the impinging radiation. 5 In spite of the recognized P -even and T -odd character of B, some unsuccessful efforts to bring about enantioselective syntheses in the presence of uniform magnetic field have been reported in the literature. 8 An egregious instance is that of alleged near-tocompletion Grignard reactions of aromatic aldehydes and ketones in the presence of B, 9 which was proved false 10,11,12 and eventually dismissed. 13 Notwithstanding, the existence of induced anapole moments in chiral molecules inter3



acting with nonuniform magnetic fields, 14,15,16,17,18,19,20,21,22 and their relevance in molecular physics, 23,24 unveils new possibilities. In nonordered media, the anapolar interaction energy is a pseudoscalar with opposite sign for the enantiomers of a chiral species in the presence of a magnetic field B and a curl C = ∇ × B ′ , which, at least in principle, would determine preferential stabilization of one over the other. 18,19 Calculations reported so far show that the energy difference between two enantiomers caused by anapolar interaction is far too small to be detectable for B and C available nowadays. 16,18,19 Nonetheless, suitable experimental conditions for synthesizing a preferred enantiomer can be studied and might possibly be implemented. Recent advancements in nuclear magnetic resonance (NMR) technology led to the construction of spectrometers operating at 1 Gigahertz, 25 with a magnetic field of 23.5 T. A groundbreaking 32 T all-superconducting magnet, successfully tested in 2017 in the National High Magnetic Field Laboratory, 26 reached a full field magnitude of 36 T. It is not the strongest continuous-field magnet in the world, which is MagLab’s 45 T hybrid magnet. It is, however, expected to become the strongest magnet in the world by far for NMR. 27 High-temperature superconducting tapes suitable for magnets at 50 T and beyond are forthcoming. 26 Presently available magnets capable of generating magnetic fields with high gradients have been recently reviewed. 28,29,30,31 Field gradient values up to 4.1 × 106 T/m were predicted for thermomagnetically patterned micromagnets 32 and micropatterned Nd-FeB hard magnetic films. 33 High field gradient magnetic flux sources can be constructed by micro-magnetic imprinting: simulations indicate magnetic field gradients of up to 5 × 105 T/m. 34 Further progress in magnet development can reasonably be expected for the foreseeable future, therefore grounds for believing that chemical reactions in the presence of strongly nonuniform magnetic fields may yield good rates of the energetically more stable enantiomer do not seem completely unsound. Accordingly, the present paper aims at predicting the average anapole magnetizability of some simple chiral molecules, which may be candidates for enantioselective syntheses, in the hope of arousing the interest of

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chemists and of stimulating their efforts to put our attempts into effect in their laboratory. An outline of the basic ideas and equations underlying the computations is given in Section 2, and results are reported in Section 3, determining, for each molecule, the enantiomer energetically stabilized by anapolar interaction in a fixed relative orientation of B and C. Conclusions and outlook are presented in Section 4.

2

Anapole moment and anapole magnetizabilities

Standard tensor formalism is employed throughout this article, e.g., the Einstein convention of implicit summation over two repeated Greek subscripts is assumed and ǫαβγ is the Levi-Civita pseudotensor. The SI units have been chosen. The notation is the same as in previous references. 14,15,16,18,35 Let us consider a diamagnetic molecule with n electrons and N nuclei responding to a time-independent, uniform magnetic field with flux density B, and to a nonuniform B ′ , arbitrarily oriented with respect to B and with spatially uniform gradient ∇B ′ . The symmetric components (1/2)(∇αBβ′ + ∇β Bα′ ) of the gradient are not taken into account as comparatively less interesting, retaining the antisymmetric components via the curl C = ∇ × B′. The perturbed Hamiltonian has a first-order part,

ˆ (1) = H ˆB + H ˆ C, H n e X B B ˆ (Aα pˆα + pˆα AB H = α )k 2me

(1)

k=1

ˆC H

= −m ˆ α Bα , n e X C (Aα pˆα + pˆα AC = α )k 2me k=1 = −âα Cα ,

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(2)

(3)


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where the relationships 1 ǫαβγ Bβ rγ , 2 1 = − (r 2 δαβ − rα rβ )Cβ 6

AB = α

(4)

AC α

(5)

define the contributions to the vector potential,

m ˆα = −

e ˆ Lα , 2me

(6)

is the operator for the electronic magnetic dipole moment related to the orbital angular ˆ α , and the Hermitian operator momentum L ˆC 1 ∂H = − ǫαβγ m ˆ βγ ∂Cα 2 n e X 2 [(r δαβ − rα rβ )ˆ pβ + pˆβ (r 2 δαβ − rα rβ )]k = 12me

a ˆα = −

=

e 6me

k=1 n X

[(r 2 δαβ − rα rβ )ˆ pβ + i~rα ]k ,

(7)

k=1

represents the anapole of the electron cloud. 14,15,19 It is connected with the antisymmetric part 8 of the magnetic quadrupole operator 14,15,19

m ˆ αβ

n e Xˆ = (lα rβ + rβ ˆlα )k . 6me k=1

(8)

The second-order part of the Hamiltonian contains a term bilinear in B and C, n

2 X ˆ BC = − e ǫαβγ Bβ Cγ (rα r 2 )k H 12me k=1

(9)

whereby one obtains a diamagnetic contribution to the anapole operator, defined as its second derivative, n

a ˆdαβ

X ˆ BC e2 ∂2H (r 2 rγ )k . = ǫαβγ =− ∂Cα ∂Bβ 12me k=1 6


(10)

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In these equations, contributions from electron spin are neglected. Allowing for these relationships, the linear response of a molecule to a nonuniform magnetic field can be rationalized via Rayleigh-Schrödinger perturbation theory (RSPT), 14,15,35,36 introducing the electronic wavefunction for the reference a singlet state in the form

Bα Cα Ψa = Ψ(0) a + Ψa Bα + Ψa Cα ,

(11)

where the first-order RSPT wavefunctions are 1 X −1 ω |jihj|m ˆ α|ai, ~ j6=a ja 1 X −1 α |ΨC ωja |jihj|âα|ai, a i = ~

α |ΨB a i =

(12) (13)

j6=a

(0)

(0)

where ωja = (Ej − Ea )/~ is a natural transition frequency. The anapole magnetizability can be obtained by the second derivative of the energy W BC , aαβ = −

∂Aα ∂Mβ ∂ 2 W BC = = , ∂Cα ∂Bβ ∂Bβ ∂Cα

(14)

and by the first derivative of either the induced anapole

Aα = aαβ Bβ

(15)

with respect to the components of B, or by the first derivative of the magnetic dipole

Mβ = aαβ Cα

(16)

with respect to the components of C. It is expedient to express the anapole magnetizability as a sum of paramagnetic and diamagnetic contributions,

aαβ = apαβ + adαβ , 7


(17)


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specified via the formulae

apαβ =

1X 2 ℜ(ha|âα |jihj|m ˆ β |ai), ~ ωja

(18)

j6=a

n X e2 ǫαβγ ha| (r 2 rγ )k |ai. = 12me k=1

adαβ

(19)

It is immediately verified from Eqs. (18) and (19), that the average anapole susceptibility a = (1/3)aαα ≡ (1/3)apαα

(20)

contains only the paramagnetic contribution, since the trace adαα vanishes. In a shift d of origin r ′ → r ′′ = r ′ + d, the change of the anapole magnetizability is given by 1 aγδ (r ′′ ) = aγδ (r ′ ) + ǫαβγ ξδα dβ , 2

(21)

p d where ξδα is a component of the magnetizability tensor ξαβ = ξαβ + ξαβ ,

1X 2 ℜ(ha|m ˆ α |jihj|m ˆ β |ai), ~ j6=a ωja + * n 2 X e 2 a (r δαβ − rα rβ )k a . = − 4me

p ξαβ =

d ξαβ

(22) (23)

k=1

Equation (21) shows that the diagonal components of the anapole magnetizability are independent of the origin in the principal axis system of ξδα . 16 The anapole magnetizability defined via equations (14)-(19) is a parity-odd, time-even second-rank tensor as the mixed electric dipole-magnetic dipole polarizability, 8

κ′αβ = −

1X 2ω ℑ(ha|ˆ µα|jihj|m ˆ β |ai). 2 ~ j6=a ωja − ω 2

8


(24)

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Therefore, the anapolar contribution

W BC = −aαβ Cα Bβ

(25)

to the interaction energy has the same magnitude, but opposite sign, for two enantiomeric molecules. 16,18 An alternative nonperturbative approach based on the definition (14) has been applied to evaluate a series of tensors, 17,20 employing a flexible and effective computational approach, designed to study linear and nonlinear atomic and molecular response to strong magnetic fields via basis sets of London orbitals. 37,38,39 Calculation efficiency and theoretical appeal of such a method, which appears in some instances preferable to RSPT techniques for nonlinear response to very strong magnetic field, 40,41,42 have been demonstrated. 43,44 Nonetheless, some nice features of the RSPT approach can justify its application, e.g., possibility of separate definition for diamagnetic and paramagnetic contributions according to (18) and (19), fulfillment of the Hellmann-Feynman theorem in the case of B-independent basis sets, definition of auxiliary sum rules, e.g., 8

X

ℜ(ha|âα |jihj|m ˆ α|ai) = 0,

(26)

j6=a

analogous to the Condon relationship for rotational strengths, 45

X j6=a

Raj =

X

ℑ(ha|ˆ µα|jihj|m ˆ α |ai = 0,

(27)

j6=a

conditions for origin independence of calculated magnetizabilities, etc., which can be useful for estimating limit values within a given variational procedure, e.g., self consistent field (SCF) approximation to the Hartree-Fock (HF) method, 14,15,36 and coupled cluster response functions. 46,47,48 Therefore, the symmetry properties of the aαβ anapole magnetizability under parity 9



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P and time reversal T are identical to those of κ′αβ , whose trace κ′αα is related to the optical rotatory power of chiral species. 49,50 Since aαβ is odd under parity and even under time reversal, its diagonal components (and trace) vanish in nonchiral molecules. However, it has the same magnitude but opposite sign for two enantiomers, and therefore can, in principle, be used for chiral discrimination. 16,18 In addition, in disordered phase, gas or solution, the orbital electronic anapole and the magnetic dipole

A = aB,

M = aC,

(28)

induced by an external magnetic field, have opposite direction in two enantiomeric molecules.

3

Calculations of anapole magnetizabilities for chiral molecules in a magnetic field with uniform curl

All the calculations of anapole magnetizability reported in the present study were carried out by the DALTON package. 51 For each molecule, the aαβ tensors reported in Tables 1-10 are relative to the principal axis system of the corresponding ξαβ magnetizability. As such, they are invariant of the origin of the coordinate system. 16 Molecular geometries were optimized at the B3LYP 52 /6-31G* and B3LYP/ 6-31+G(d,p) levels via the GAUSSIAN code. 53 A preliminary computational test was made on the H2 O2 molecule to check the difference between the definitions of anapole magnetizability adopted by two groups of authors. An exhaustive series of calculations at the SCF-HF level has been reported by Tellgren and Fliegl in Table II of Ref. 17 and by Sen and Tellgren in Table VII of Ref. 20 , for the dihedral angle of 120◦ , to document basis set convergence of calculated values. These results are very important because of the saddle character of the anapolar interaction energy as a function of variational parameters and basis set characteristics. Thus, at variance with the case of the diagonal components of, e.g., electric polarizability and paramagnetic contribution to magnetizability, whose accuracy can be ascertained by the

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criterion “the larger, the better" for basis sets of different quality and size owing to variational criteria, 54 convergence to limit values of anapole magnetizabilities for a given approximation can only be assessed by careful painstaking attempts. Other SCF predictions for H2 O2 have been reported for a slightly different equilibrium geometry. 22 By comparing the average anapole magnetizabilities from these papers, it is evident that values in Refs. 17 and 20 are approximately twice as big as in Ref. 22 . This is confirmed by SCF-HF calculations displayed in Table 1 for the B3LYP equilibrium geometry (via the TZ basis set - see basis set documentation hereafter -, dihedral angle 119.32), and for the dihedral angle of 120◦: the trace of the tensors in the table is nearly one half that of Refs. 17 and 20 . Actually, the factor 2 in the definition of the anapole operator, equation (14) of both papers, 17,20 does not appear in (7) here, which explains the difference. Thus the agreement between the results in Table 1 and those reported in Refs. 17 and 20 is actually very good. As a further matter, the theoretical predictions obtained via B3LYP and various coupled-cluster schemes for a of H2 O2 in Table 1 are quite close to HF, indicating that contributions from electron correlation are small for this molecule. Larger variations are observed on passing from TZ to MODENA1 basis sets for the tensor components in Table 1. An analogous trend had been previously discussed for other molecules, 18 and can also be noticed hereafter in Table 3. It depends on two factors: (i) the different features of the basis sets, which directly bias the quality of aαβ , (ii) the different response of aαβ and ξαβ . These factors determine a superposition of errors, since the former is, in each calculation, rotated to the principal axis system of the latter. Thus insufficient convergence of ξαβ further affects the convergence of aαβ . A second test was made using the fluoro-oxirane molecule, first synthesized by Hollenstein et al., 55 to select the basis set for a series of chiral compounds, and looking for a reasonable trade-off between size and quality of computed estimates. Extended correlation consistent basis sets of Gaussian functions, from the compilation by Dunning and coworkers, 56,57,58 were adopted. This choice was motivated by the features of the response properties studied and by the need to describe electron correlation systematically.

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Within the algebraic approximation, higher quality results are expected by carefully improving the basis set. In fact, it had previously been found that the basis sets from Dunning et al. 56,57,58 are useful to study convergence of related properties, ξαβ and ξα,βγ magnetizabilities, to limit values. 18,17 Therefore, we attempted to reach saturation by employing four Dunning basis sets 56,57,58 of increasing dimension and characteristics, referred to in Tables 2 and 3 as DZ, TZ, QZ and 5Z for brevity, to designate aug-cc-pCVDZ, aug-cc-pVTZ aug-cc-pVQZ and aug-ccpV5Z. The number of atomic Gaussian functions for each basis set is specified in the Tables 1-10. These tests were made to estimate the degree of convergence of calculated anapole magnetizabilities aαβ , Eqs. (17), and pseudoscalar a. On the other hand, the QZ and 5Z basis sets from Dunning become unpractical for big molecules, therefore the basis set MODENA1, an uncontracted (13s10p5d2f/8s4p1d) carrying optimized polarization functions for electric and magnetic properties, 16 smaller in size than the largest Dunning basis sets, was also employed to further document its quality in view of future applications to larger molecular systems. The SCF-HF results displayed in Tables 2 and 3 suggest that TZ and MODENA1 basis sets can be used with confidence in their quality. Increasing accuracy of computed results seems to characterize the B3LYP predictions in Table 3. Moreover, the contributions of electron correlation predicted by B3LYP and CCSD calculations are very close to one another. Therefore, the theoretical scheme outlined in Section 2 has been applied to evaluate aαβ anapole magnetizabilities using the TZ basis set for a series of chiral molecules, (S-C,S-N)-chloro-methyl-aziridine C3 H6 NCl, (S-N)-chloro-dimethyl-aziridine C4 H8 NCl, (R-C,R-N)-fluoro-methyl-aziridine C3 H6 NF, (R-C,S-N)-methoxy-methyl-aziridine C4 H9 NO, and R-glycidol C3 H6 O2 . R-amino-methyl-fluoro-methane CHFNH2 CH3 and R-fluoroamino-methanol CHFNH2 OH were studied employing both TZ and MODENA1 basis sets at various levels of electron correlation. The origin of the coordinate system is the center of mass (CM) in all cases. The convergence degree of the components of the aαβ tensor is difficult to assess.

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Change of sign of some components is observed for the HF and CCS results on passing from TZ to MODENA1 basis sets in Table 2. In spite of that, a, which is origin independent in the limit of complete basis set, 18 stabilizes at a nearly converged value with TZ in most cases for a given approximation and its magnitude can be considered reliably estimated in general, according to the basic motivation of our study, which remains that of indicating possible candidates for enantioselective reactions. Therefore, allowing for the results arrived at in a previous paper 18 and in the present investigations, we are confident that the MODENA1 basis set, developed ad hoc for near HF estimates of second-rank magnetizabilities, constitutes a satisfactory compromise between size and accuracy for a.

4

Concluding remarks and outlook

The present study is a follow-up to a previous research. 21 Building on the same basic theory of anapolar response of diamagnetic chiral molecules, it reports a further examination and goes beyond by using more reliable correlation methods and different sets of molecules. Only a few computational predictions of anapole magnetizability tensors have been reported so far, indicating that the tensor trace is generally very small. 17,18,19,20,21,22 Nonetheless, the results of the computations displayed in Tables 2-10 show that the off-diagonal components of the aαβ tensor are generally much bigger than the diagonal ones. In the same way as the mixed electric dipole-magnetic dipole polarizability κ′αβ , Eq. (24), the latter are characterized by different sign for all the molecules considered here, which determines the tiny value of the average a. In view of experimental efforts to detect, and possibly measure, the impact of anapolar interaction, one may tentatively look for chiral systems with the characteristics of potential candidates by extensive and systematic calculations. It can be expected that some molecules have larger anapole moments than others, and one should then focus on the molecular characteristics that will maximize the desired property. In the series of possible candidates for enantioselective syntheses examined in this

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study, S-fluoro-oxirane C2 H3 FO, (S-C,S-N)-chloro-methyl-aziridine C3 H6 NCl, (S-N)-chloro-dimethyl-aziridine C4 H8 NCl, (R-C,R-N) fluoro-methyl-aziridine C3 H6 NF, (R-C,S-N)-methoxy-methyl-aziridine C4 H9 NO, R-glycidol C3 H6 O2 , R-amino-methyl-fluoro-methane CHFNH2 CH3 , and R-fluoro-amino-methanol CHFNH2 OH, the largest values of a pseudoscalar were estimated via B3LYP calculations for the S-fluoro-oxirane molecule, ≈ 0.032 a.u., Table 3, and for R-fluoro-methyl-aziridine, ≈ 0.068 a.u., Table 6, which, allowing for the conversion factor reported in Table 2, correspond to ≈ 1 × 10−40 JT−2 m and ≈ 3 × 10−40 JT−2 m, respectively. Therefore, the energy difference ∆W BC stabilizing the S- enantiomer of fluoro-oxirane would be ≈ 2 × 10−40 J per molecule for |B| = 1 T and |C| = 1 T/m, in the presence of B k C. In these conditions, the R- enantiomer of fluoro-methyl-aziridine was calculated to be more stable than the S-enantiomer by ≈ 6 × 10−40 J per molecule. The same thermodynamical stabilization would favor formation of the S-enantiomer for B antiparallel to C. This behavior is illustrated qualitatively for R-glycidol in Figure 3. These values could increase by a factor as big as ≈ 107 − 108 allowing for the largest values of magnetic field 26 and magnetic field gradient presently available. 33,34 Since the R-enantiomer of fluoro-oxirane was calculated 59 to be more stable than the S-enantiomer by the amount of 2 × 10−12 J mol−1 , i.e., 3 × 10−36 J per molecule, owing to parity violation, the energy difference between the enantiomers of fluoro-oxirane estimated by the present calculations would be ≈ 3 orders of magnitude larger than that arising from parity violation. Eventually, according to our limited computational experience, we can conclude that fluoro-oxirane and fluorinated aziridine systems, see Tables 3 and 6, are characterized by values of diagonal tensor components comparatively larger than those estimated so far for other compounds. In addition, for C2 H3 FO and C3 H6 NF, aαα is one order of magnitude bigger than that reported in the Tables for other molecules. These features would seem to provide important indications for future computational investigations on, as well as the synthesis of, suitable chiral substances, e.g., one could study the result of systematic addition of substituents carrying other fluorine atoms to

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aziridine and fluoro-oxirane units.

It remains to be seen whether the tiny magnitude

of these predictions could prompt attempts to synthesize a thermodynamically stabilized enantiomer in laboratory for a given configuration of B and C. Anyway, further theoretical investigations are needed to determine the bias of nonuniform magnetic fields on the intermediate steps of enantioselective reactions, which may be much more important. More fruitful attempts at detecting the effects of electronic anapoles induced by the curl of nonuniform magnetic field may possibly be made employing physical methods. Experiments in transparent crystals provide important information on gyrotropic phenomena, 60 thus one may tentatively devise procedures for observing the change of magnetic dipole moment predicted by Eq. (16) for a chiral molecule in ordered phase, in the presence of strongly nonuniform magnetic field. The magnitude of such a change can be estimated allowing for the conversion factors reported in the footnote to Table 2, e.g., for Cz ≈ 1 × 106 Tm−1 and azz ≈ 1.8 × 10−39 JT−2 m, displayed for fluoro-methyl aziridine in Table 6, the change of magnetic dipole moment would be Mz = azz Cz ≈ 1.8 × 10−33 JT−1 . This can be compared with the magnetic dipole moment Mz = ξzz Bz ≈ 1.0 × 10−28 JT−1 induced by a magnetic field Bz of 0.1 T. Within the principal axis system of the ξαβ symmetric tensor, the eigenvalues, computed at the CCSD level for fluoro-methyl aziridine, are ξxx = −17.09, ξyy = −18.30, ξzz = −13.08 a.u., very close to those estimated by B3LYP and HF methods. Therefore the contribution to the magnetic dipole moment induced by the strongest field gradients available nowadays would determine a change as small as 10−4 − 10−5 JT−1 , which does not seem to be presently detectable. In spite of that, possible future advancement in magnetic-field technology may rise hope for recognition of anapolar interaction in chiral molecules via measurements of induced magnetic dipole moments.

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Acknowledgments PFP acknowledges financial support from CONICET and UNNE (PI: 15/F002 Res. 1017/15 C.S.). MBF and GIP acknowledge financial support from Universidad de Buenos Aires (20020170100456BA) and CONICET (PIP 11220130100377).

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References [1] Faraday, M. XLIX. Experimental Researches in Electricity. Nineteenth Series. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 1846, 28, 294–317. [2] Pasteur, 2–6,

L.

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Reprinted

Dissymétrie in

‘Oeuvres

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https://webext.pasteur.fr/biblio/ressources/histoire/pasteur.php1884 See also http://gallica.bnf.fr/ark:/12148/bpt6k2151022.image.r =revue+scientifique+1884.f5.langFR. [3] Faraday, M. Faraday’s Diary. Volume IV, Nov. 12, 1847,

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http://faradaysdiary.com/ws3/faraday.pdf. [4] Lord Kelvin, Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light; London: C. J. Clay and Sons, Cambridge University Press Warehouse, Ave Maria Lane, 1904; footnote p. 619. [5] Barron, L. D. Molecular Light Scattering and Optical Activity, 2nd ed.; Cambridge University Press: Cambridge, 2004. [6] Wagnière, G. H. On Chirality and the Universal Asymmetry: Reflections on Image and Mirror Image, 1st ed.; Wiley-VCH, 2007. [7] Curie, P. Sur la symétrie dans les phénomènes physiques - Symétrie d’un champ électrique et d’un champ magnétique; t. III, 393, (1894); Villars,

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pp. 113–141 (1908), Gordon and Breach

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24


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Page 25 of 36 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


Table 1: Anapole magnetizability aαβ of hydrogen peroxide H2 O2 via two basis sets at various levels of electron correlation.

Level/Basis set

Energy

aug-cc-pVTZ: 138 HF

−150.8383

B3LYP

−151.5506

CCS

−150.8383

CC2

−151.3844

CCSD

−151.3830

MODENA1: 268 HF

−150.8502

HF†

−150.8502

B3LYP

−151.5649

B3LYP†

−151.5649

CCS

−150.8502

CC2

−151.4973

CCSD

CCSD†

−151.4948

−151.4948

x

y

z

0.1396 0.0266 0.0000 0.1666 0.0543 0.0000 0.1434 0.0352 0.0000 0.1562 0.0847 0.0000 0.1524 0.0542 0.0000

0.0826 −0.0025 0.0000 0.1408 −0.0077 0.0000 0.0659 0.0083 0.0000 0.1439 −0.0065 0.0000 0.1241 −0.0052 0.0000

0.0000 0.0000 −0.1761 0.0000 0.0000 −0.2093 0.0000 0.0000 −0.1867 0.0000 0.0000 −0.1860 0.0000 0.0000 −0.1885

0.1691 −0.0136 0.0000 0.1676 −0.0906 0.0000 0.1997 0.0246 0.0000 0.1980 −0.0435 0.0000 0.1750 −0.0165 0.0000 0.1960 0.0491 0.0000 0.1882 0.0169 0.0000 0.1864 −0.0576 0.0000

0.1062 −0.0096 0.0000 0.2232 −0.0099 0.0000 0.1679 −0.0168 0.0000 0.2869 −0.0172 0.0000 0.0746 −0.0043 0.0000 0.1726 −0.0178 0.0000 0.1511 −0.0138 0.0000 0.2737 −0.0146 0.0000

0.0000 0.0000 −0.2017 0.0000 0.0000 −0.2005 0.0000 0.0000 −0.2403 0.0000 0.0000 −0.2390 0.0000 0.0000 −0.2085 0.0000 0.0000 −0.2221 0.0000 0.0000 −0.2216 0.0000 0.0000 −0.2197

†

a

−0.0130

−0.0168

−0.0117

−0.0121

−0.0138

−0.0141

−0.0142

−0.0191

−0.0194

−0.0126

−0.0146

−0.0157

−0.0160

For a dihedral angle of 120◦ , calculated within the principal axis system of the magnetizability tensor ξαβ , in which the diagonal components of aαβ are independent of the origin. All the other values were obtained for the dihedral angle 119.3◦ of the equilbrium geometry, Section 3. 25



Page 26 of 36

Table 2: Anapole magnetizability of S-fluoro-oxirane C2 H3 FO with two basis sets at several levels of electron correlation.†

Level/Basis set/#

Energy

aug-cc-pVTZ: 253 HF

−251.8266

CCS

−251.8266

CC2

−252.7371

CCSD

−252.7376

MODENA1: 479 HF

−251.8459

CCS

−251.8459

x

y

z

−0.0483 −1.1926 −0.3298 −0.0062 −1.4177 −0.4116 −0.0492 −1.3098 −0.3319 −0.0212 −1.2235 −0.3321

0.6968 −0.0650 0.3407 0.8838 −0.1372 0.5084 0.6582 −0.0810 0.4927 0.6773 −0.0932 0.4355

0.4709 0.1120 0.1802 0.4286 −0.1060 0.2240 0.5039 0.0736 0.2353 0.4532 0.1318 0.2103

0.0160 −1.3893 0.7394 0.0138 −1.8211 0.4050

1.0405 0.0866 0.2598 1.3452 −0.0583 0.4995

−0.4460 0.2654 −0.0387 −0.2741 0.0037 0.1226

†

a

0.0223

0.0269

0.0350

0.0320

0.0213

0.0260

In all the Tables the origin of the coordinate system is the center of mass. From the CODATA compilation, 61 the conversion factor from SI-a.u. to SI units per molecule for magnetizability ξαβ , Eqs. (22) and (23), is e2 a20 /me = 7.891 036 433 × 10−29 JT−2 , that for anapole magnetizability aαβ , Eq. (17), is e2 a30 /me = 4.175 756 62×10−39 JT−2 m. The conversion factor for the magnetic dipole moment is ~/me = 1.854 801 90 × 10−23 JT−1 , that for the anapole moment is eEh a30 /~ = 9.815 188 95 × 10−34 JT−1 m ≡ m3 A.


Page 27 of 36 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


Table 3: Anapole magnetizability of S-fluoro-oxirane C2 H3 FO via aug-cc-pVXZ (X = D,T Q and 5) basis sets at the B3LYP level of calculation.

Level

Energy

B3LYP/DZ

−252.9472

B3LYP/TZ

−253.0167

B3LYP/QZ

−253.0352

B3LYP/5Z

x

−253.0406

−0.0399 −0.3824 −1.1501 −0.0331 −1.1606 −0.3719 −0.0138 −1.3904 0.2619 −0.0025 −1.3935 0.4564

y −0.3933 −0.1791 0.3028 0.5448 −0.0757 0.4327 0.8684 0.0371 0.4459 0.8988 0.0776 0.4402


z 1.1146 0.1778 0.3175 0.5061 0.1406 0.2063 −0.0058 0.1934 0.0721 −0.1653 0.2028 0.0198

a

0.0329

0.0325

0.0318

0.0316


Page 28 of 36

Table 4: Anapole magnetizability of (S-C,S-N)-chloro-methyl-aziridine C3 H6 NCl via augcc-pVTZ basis sets at HF, B3LYP and CCSD levels of calculation.

Level

Energy

x

HF/TZ

−631.0400

B3LYP/TZ

−632.7304

CCSD/TZ

−632.1177

y

−0.4648 9.3947 2.6314 −0.4794 9.1192 2.5934 −0.5275 8.4795 2.8667

z

−10.4392 0.4132 −0.8586 −10.1219 0.4775 −0.9882 −9.4619 0.4351 −0.7068

a

−2.1499 1.2236 0.1158 −2.0449 1.3058 0.0897 −2.2496 1.1911 0.1363

0.0214

0.0293

0.0146

Table 5: Anapole magnetizability of (S-N)-chloro-dimethyl-aziridine C4 H8 NCl with augcc-pVTZ basis sets at HF, B3LYP and CCSD levels of calculation.

Level

Energy

x

Cl-dimethyl-aziridine : 464 HF/TZ

−670.0882

−0.0507 −14.2798 −3.2929 −0.0478 −13.9827 −3.3669 0.0070 −13.6446 −3.3648

B3LYP/TZ

CCSD/TZ

−672.0309

−671.3775

y 15.3067 −0.1878 −0.9904 14.9840 −0.1718 −1.2425 14.7596 −0.3008 −0.8221


z 2.8403 0.3156 0.2426 2.8098 0.4188 0.2269 2.8791 0.3514 0.2724

a

0.0014

0.0024

−0.0071

Page 29 of 36 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


Table 6: Anapole magnetizability of (R-C,R-N)-fluoro-methyl-aziridine via aug-cc-pVTZ basis sets at HF, B3LYP and CCSD level of calculation.

Level

Energy

x

HF/TZ

−271.0312

B3LYP/TZ

−272.4337

CCSD/TZ

−272.1317

−0.5384 −3.1752 0.0441 −0.6023 −3.0695 −0.0506 −0.4979 −3.4249 0.0698

y 1.9885 0.3206 −0.0394 1.6553 0.3741 −0.0084 2.0749 0.2890 0.0154

z 0.0572 −0.0744 0.3611 0.1355 −0.1040 0.4314 −0.0646 −0.0868 0.3867

a

0.0478

0.0677

0.0593

Table 7: Anapole magnetizability of (R-C,S-N)-methoxy-methyl-aziridine via aug-ccpVTZ basis sets at HF, B3LYP and CCSD level of calculation.

Level

Energy

x

HF/TZ

−286.0645

B3LYP/TZ

−287.6947

CCSD/TZ

−287.3646

−0.5825 −1.3337 −0.6233 −0.6461 −1.0952 −0.5717 −0.6014 −1.0744 −0.6577

y 0.2911 0.1369 −1.6998 0.2694 0.1031 −1.8209 0.1066 0.1421 −1.7643

z 1.2442 1.6280 0.4152 1.3904 1.5116 0.5033 1.3152 1.6355 0.4345

a

−0.0101

−0.0132

−0.0083

Table 8: Anapole magnetizability of R-glycidol C3 H6 O2 via aug-cc-pVTZ basis set at HF, B3LYP and CCSD levels of calculation.

Level HF/TZ

B3LYP/TZ

CCSD/TZ

Energy

x

−266.8649

−268.2822

−267.9780

−0.5537 0.1255 −1.3450 −0.6404 0.0416 −1.4707 −0.6035 0.1090 −1.4749

y 0.5838 0.5181 0.2629 0.6306 0.5981 0.2896 0.7214 0.5206 0.3136


z 0.6161 −1.1531 0.0101 0.5031 −1.2621 0.0092 0.4901 −1.2617 0.0542

a

−0.0085

−0.0110

−0.0096


Page 30 of 36

Table 9: Anapole magnetizability of R-amino-methyl-fluoro-methane CHFNH2 CH3 with two basis sets at various levels of electron correlation.

Level/Basis set augccpvtz: 322 HF

B3LYP

Energy −233.2142

−234.4022

CCS

−233.2142

CC2

−234.1289

CCSD

−234.1477

MODENA1: 635 HF

−233.2319

x

y

z

−0.0968 0.5043 −0.6498 −0.0247 0.3965 −1.6069 0.0525 0.6200 −2.2814 −0.0103 0.4343 −1.5300 −0.0482 0.5035 −1.1185

−0.3729 −0.0450 −1.4774 −0.5038 −0.1080 0.2665 −0.5781 −0.1422 −1.3487 −0.3602 −0.0723 −1.3406 −0.3671 −0.0815 −1.4112

0.6818 1.2026 0.1369 1.2919 −0.3459 0.1309 1.8869 0.8944 0.1035 1.3687 1.0918 0.0811 1.0365 1.0791 0.1277

−0.0300 0.7401 −1.2940

−0.6932 −0.0186 −0.9370

1.2174 0.6281 0.0440


a

−0.0017

−0.0006

0.0046

−0.0005

−0.0007

−0.0015

Page 31 of 36 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


Table 10: Anapole magnetizability of R-fluoro-amino-methanol CHFNH2 OH via two basis sets and various levels of electron correlation.

Level/Basis set augccpvtz: 276 HF

B3LYP

Energy −269.0613

−270.3280

CCS

−269.0613

CC2

−270.0360

CCSD

−270.0383

MODENA1: 531 HF

−269.0823

CC2

−270.0360

x

y

z

0.0201 0.5098 −0.8051 0.0255 0.4987 −0.7617 −0.1271 0.4105 −0.1771 0.0301 0.1978 5.6444 0.0301 0.1688 3.3902

−0.4867 −0.0776 0.4617 −0.5141 −0.0564 0.5063 −0.6716 0.0384 1.6151 −0.1933 −0.0213 −7.3228 −0.1697 −0.0739 −8.7471

0.8379 −0.7106 0.0731 0.8163 −0.7164 0.0569 −0.0442 −1.5038 0.0949 −5.4918 7.2432 0.0218 −3.3536 8.5384 0.0641

0.0373 0.3159 −0.8697 −0.0240 −0.1389 −0.7965

−0.3450 −0.0382 0.4289 −0.1953 −0.0378 1.5467

0.9593 −0.5949 0.0150 0.6924 −1.4012 0.0664


a

0.0052

0.0087

0.0021

0.0102

0.0068

0.0047

0.0015


a

b

c

d

Figure 1: a (S-C,S-N)-chloro-methyl-aziridine C3 H6 NCl, b (S-N)-chloro-dimethylaziridine C4 H8 NCl, c (R-C,R-N)-fluoro-methyl-aziridine C3 H6 NF, d (R-C,S-N)-methoxymethyl-aziridine C4 H9 NO. Here and in the following figure, dark red, red, cyan, and green dots denote carbon, oxygen, nitrogen, chlorine or fluorine atoms, respectively. Hydrogen atoms are represented by smaller grey dots. For each molecule, the origin of the coordinate system is in the center of mass.


Page 32 of 36

Page 33 of 36 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


e

f

g

h

Figure 2: e R-amino-methyl-fluoro-methane, f R-fluoro-amino-hydroxy-methane, g Rglycidol. h S-fluoro-oxirane



W

W

BC

∆W=2 a B C

C

B

S

1

BC

S

R

B

Page 34 of 36

∆W=2 a B C

C

R

2

Figure 3: (1) On the left: in a disordered medium, in the presence of parallel B and C, the S-enantiomer of glycidol is energetically more stable than the R-enantiomer. The absolute value of the energy difference is 2|aB · C|, with a the average anapole magnetizability. (2) On the right: the R-enantiomer is more stable than the S-enantiomer in antiparallel B and C.


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W

W

BC

S

R

B

∆W=2 a B C

C

B

∆W=2 a B C

S

1

BC

R

2

Figure 4: Graphical TOC Entry


C

1


W

W

BC

ΔW=2 a B C

BC

S

R

B

Page 36 of 36

C

B

ΔW=2 a B C

S

R


2

C

Could Electronic Anapolar Interactions Drive Enantioselective

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