PCCP PAPER
Cite this: Phys. Chem. Chem. Phys., 2016, 18, 13267
A complex-polarization-propagator protocol for magneto-chiral axial dichroism and birefringence dispersion† Janusz Cukras,ab Joanna Kauczor,c Patrick Norman,c Antonio Rizzo,d Geert L. J. A. Rikkene and Sonia Coriani*af A computational protocol for magneto-chiral dichroism and magneto-chiral birefringence dispersion is presented within the framework of damped response theory, also known as complex polarization
Received 2nd March 2016, Accepted 12th April 2016 DOI: 10.1039/c6cp01465h
propagator theory, at the level of time-dependent Hartree–Fock and time-dependent density functional theory. Magneto-chiral dichroism and magneto-chiral birefringence spectra in the (resonant) frequency region below the first ionization threshold of R-methyloxirane and L-alanine are presented and compared with the corresponding results obtained for both the electronic circular dichroism and the magnetic circular dichroism. The additional information content yielded by the magneto-chiral phenomena, as well
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as their potential experimental detectability for the selected species, is discussed.
1 Introduction Magneto-chiral dichroism (MChD)1–10 is related to the difference in the absorption coefficients of a chiral molecule in a magnetic field parallel and antiparallel to the incident light beam. In other words, if one shines light (in any polarization state) on a chiral species in the presence of a static magnetic field with a component parallel to the direction of propagation of light, the absorption coefficient n 0 mm for B (magnetic field) and k (direction of propagation) parallel to each other differs from the one measured for B and k antiparallel, n 0 mk. The same occurs for the refractive indices, and the resulting birefringence, corresponding to an anisotropy of the phase of the beams, is then known as magneto-chiral birefringence (MChB).1–9,11 Moreover, the two effects will be oppositely signed for the two mirror-image enantiomers, or reversing the relative directions of the magnetic field and the propagation vector. a
` degli Studi di Dipartimento di Scienze Chimiche e Farmaceutiche, Universita Trieste, via Giorgieri 1, 34127 Trieste, Italy. E-mail:
[email protected] b Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warszawa, Poland c Department of Physics, Chemistry and Biology, Linko¨ping University, S-58183, ¨ping, Sweden Linko d Consiglio Nazionale delle Ricerche – CNR, Istituto per i Processi Chimico-Fisici, IPCF-CNR, Sede Secondaria di Pisa, Area della Ricerca, Via G. Moruzzi 1, I-56124 Pisa, Italy e Laboratoire National des Champs Magne´tiques Intenses, UPR3228 CNRS/INSA/ UJF/UPS, Toulouse & Grenoble, France f Aarhus Institute of Advanced Studies, Aarhus University, DK-8000 Aarhus C, Denmark † Electronic supplementary information (ESI) available. See DOI: 10.1039/ c6cp01465h
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Magneto-chiral dichroism has long been considered a plausible candidate for explaining the homochirality of life, in addition to the Earth’s rotational motion and circularly polarized light induced asymmetric photochemical reactions.5,7,12 Yet, its experimental detection in compounds of relevance in life sciences, like amino acids, is elusive. The only experimental reports of MChD spectra of organic molecules concern a porphyrin complex13 and an artificial light-harvesting antenna.14 Recently, there was a report on the detection of strong magneto-chiral dichroism in a paramagnetic molecular helix observed by hard X-rays.15 This is clearly a case where computational simulations can be of fundamental help. However, no computational protocol for MChD has been proposed so far, despite the fact that a molecular theory of the effect has been available for more than thirty years,4 and it bears strong similarities with the theory of magnetic circular dichroism (MCD).16–18 Building upon our expertise on the development of computational methods for magnetic circular dichroism,19–23 and magneto-chiral birefringence,24,25 we present here the definition and implementation of a computational protocol for magneto-chiral dichroism and birefringence dispersion in the framework of the complex polarization propagation approach (CPP), also known as damped response theory.26–30 In the CPP approach, an empirical linewidth parameter is introduced in the response functions to account for the finite lifetime of the excited states. The real and imaginary components of the resulting complex response function are well-behaved even in the resonant regions of the spectrum. Depending on the perturbation operators involved, dispersion and absorption contributions to a given physical phenomenon are identified by the real and
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imaginary components. These are almost straightforwardly identified with the molecular property tensors emerging within the semi-classical formalism when considering the oscillating electric and magnetic multipole moments induced by the incident light wave.1,31 Thus, for instance, the imaginary part of the electric dipole–electric dipole linear response function relates to the one-photon absorption spectrum;26 the real part of the electric dipole–magnetic dipole linear response function yields the electronic circular dichroism (ECD) spectrum;32 the real part of the electric dipole–electric dipole–magnetic dipole quadratic response function connects to the MCD spectrum; its imaginary counterpart gives the dispersion of the Faraday rotation (angle).21 Our ultimate purpose is to use the proposed protocol to estimate the effect on molecular systems important in life sciences, in an attempt to identify which compounds may have a response above the detectability limit of the current experimental setups. Indeed, in the experiment performed on chromium trisoxalate33 it was possible to detect a magnetochiral dichroism signature corresponding to a dissymmetry factor (essentially a dichroism to total absorption ratio, see below) of a few times 10!6 T!1. Here, we will present pilot results for R-methyloxirane and L-alanine.
2 Theory The expression of the molecular quantities describing magnetochiral birefringence (MChB) and magneto-chiral dichroism (MChD) can be derived either within the refractive index approach or within the refringent scattering approach, similarly as for onephoton absorption (OPA), optical rotation dispersion (ORD), electronic circular dichroism (ECD), magnetic optical rotatory dispersion (MORD) and magnetic circular dichroism (MCD). Both approaches have been extensively discussed in the literature, see e.g. Buckingham31 and Barron.1 Here we follow closely ref. 4. Within the refractive index approach, the refractive and absorption indices of incident light of circular frequency o passing through a diluted molecular medium of number density N are expressed in terms of appropriate combinations of dynamical molecular tensors. For light linearly polarized along x and propagating in the z direction, we have the refraction (n) and absorption (n 0 ) indices, respectively, given by n"1þ
N ½axx ð f Þ þ zxxz ð f Þ . . .' 4e0
(1)
N ½axx ðgÞ þ zxxz ðgÞ . . .' (2) 4e0 where e0 is the electric constant. The molecular tensors on the right-hand side of the above expressions are components of the complex dynamic dipole polarizability tensor (Barron’s notation1) ~ aab = aab ! iaab0 and the optical rotatory tensor n0 "
~zabl ¼ zabl ! iz 0 abl
zabl
# $ " 1 1 ! 0 0 o Aa;bl þ Ab;al þ edla Gbd þ edlb Gad ¼ c0 3
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0
zabl ¼ !
# $ & 1 1 % 0 0 o Aa;bl ! Ab;al þ edla Gbd ! edlb Gad c0 3
where c0 is the speed of light in vacuo and eabl is the alternating (Levi–Civita) tensor. Each molecular tensor is further decomposed into dispersive (i.e., a dispersion line-shape function is associated with the tensor) and absorptive components, as indicated by the ( f ) and ( g) notations, respectively. For instance, aab = aab( f ) + iaab( g).
(6)
Explicit sum-over-state expressions ! " for the electric dipole– 0 magnetic dipole G~ab ¼ Gab ! iGab and electric dipole–electric ! " 0 dynamic molecular tensors quadrupole A~a;bg ¼ Aa;bg ! iA a;bg
can be found, e.g. in ref. 1 and 31, see also ref. 4. For circularly polarized light, the equations are4
N' axx ð f Þ þ ayy ð f Þ þ zxxz ð f Þ þ zyyz ð f Þ 4e0 h 0 i o 0 ) 2 axy ð f Þ þ zxyz ð f Þ þ * * * ;
nR=L " 1 þ
0
n R=L "
N' axx ðgÞ þ ayy ðgÞ þ zxxz ðgÞ þ zyyz ðgÞ 4e0 h 0 i o 0 ) 2 axy ðgÞ þ zxyz ðgÞ þ * * *
(7)
(8)
where superscripts R and L indicate right and left circular polarization, respectively. For unpolarized light (regarded as incoherent superposition of right and left circularly polarized light), we have4 n"1þ n0 "
) N( axx ð f Þ þ ayy ð f Þ þ zxxz ð f Þ þ zyyz ð f Þ 4e0
) N( axx ðgÞ þ ayy ðgÞ þ zxxz ðgÞ þ zyyz ðgÞ 4e0
(9)
(10)
The quantities axx( g) + ayy( g) and zxxz( g) + zyyz( g) are independent of the polarization state of incident light and, while canceling out for natural optical activity, they are responsible for conventional absorption and magneto-chiral dichroism, respectively. The polarization-dependent terms axx( g) ! ayy( g) and axy( g) are responsible for linear dichroism, zxxz( g) ! zyyz( g) 0
and zxyz(g) for gyrotropic dichroism, axy ðgÞ for magnetic circular 0
dichroism, and zxyz ðgÞ for natural circular dichroism.1,4
When a static magnetic field B aligned with the direction of propagation z is applied to the diluted medium, within a perturbative approach including only terms linear in Bz, the electric dipole polarizability ~ aab and the molecular tensors entering eqn (4) and (5) are modified as follows: aab(Bz) = aab + a(m) ab,zBz 0
0
0 ðmÞ
aab ðBz Þ ¼ aab þ aab;z Bz
(3) 0
(4)
(5)
0
0 ðmÞ
(11) (12)
aa;bl ðBz Þ ¼ Aa;bl þ Aa;bl;z Bz
(13)
Gab(Bz) = Gab + G(m) ab,zBz
(14)
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MChB and MChD anisotropies arise when the effect of the static perturbing magnetic field is included in the expressions of the refractive indices for unpolarized light (eqn (9) and (10), and similarly for linearly polarized light) for two different reciprocal orientations (parallel, mm, and antiparallel, mk) of the direction of propagation and of the external magnetic field. A classical Boltzmann average over all orientations taken by the molecules of the medium in a fluid then yields the following expressions noh 0 i o 0 ðmÞ N ðmÞ ðmÞ Bz 3Aa;ab;b ð f Þ ! Aa;bb;a ð f Þ þ eabl Gab;l ð f Þ 3e0 c0 15 n o N 0 Bz BA ð f Þ þ BG ð f Þ + 3e0 c0 (15)
n"" ! n"# "
0
0
Z¼
" ol ! 0 L 0 n !nR 2c0
(22)
In experiments, the quantities of relevance are the specific or molar rotations ([a] and [f], respectively) and ellipticities ([c] and [Y], respectively), defined as optical rotations (ellipticities) per unit path length (given in decimeter for specific observables and in centimeter for molar quantities) divided by the density d of the sample (given in grams per liter for the specific observables and in 10!2 moles per liter for molar quantities). Equations for specific properties are given in the Appendix. Here we report the explicit expression for molar quantities. 0 0 In NOA the polarization-dependent zxyz ð f Þ and zxyz ðgÞ tensors are of relevance. Tumble averaging further reduces the 0
anisotropies to the contribution from the Gab tensor alone
noh 0 i o 0 ðmÞ N ðmÞ ðmÞ 3Aa;ab;b ðgÞ ! Aa;bb;a ðgÞ þ eabl Gab;l ðgÞ Bz 3e0 c0 15 n o N 0 Bz BA ðgÞ þ BG ðgÞ + 3e0 c0 (16)
Dn " !
n "" ! n "# "
2N 0 G ðfÞ 3e0 c0 aa
(23)
2N 0 G ðgÞ 3e0 c0 aa
(24)
Dn0 " !
In terms of molar observables Before proceeding we give here some relevant definitions and establish the notation used later when comparing our results regarding MChD and MChB with other closely related dichroisms and birefringences. In conventional OPA, the quantity of interest is the absorption coefficient n 0 for unpolarized light, and the molecular tensor quantity responsible for it is then axx(g) + ayy(g). For a sample of randomly oriented (achiral) molecules, in the absence of external magnetic fields, tumble-averaging yields n0 "
N aaa ðgÞ 6e0
(17)
In the following, we will report the OPA spectra as absorption cross-section s in a.u. (the SI unit is the barn = 10!28 m2) o aaa ðgÞ sðoÞ ¼ 3e0 c0
(18)
(19)
s(o) [barn] E 8.55965 , 105 , o [a.u.]aaa(g) [a.u.]
(20)
The last two equations yield the observable in atomic units and SI units, respectively, when the circular frequency o and the molecular tensor aaa(g) are given in atomic units. For circularly differential refractive effects, as those yielding natural optical activity (NOA) and magnetic-field-induced optical activity (MOA), we deal with anisotropies of the refractive and absorptive indices for light polarized in the L and R directions. This yields specifically a rotation Dy or an ellipticity Z, in radians, which can (always) be written as (l is the optical path length)1 & ol % L n ! nR ; 2c0
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½Y'ECD " !
100oNa 0 100 Lnð10Þ 0 De G ðgÞ ¼ ! 4 3e0 c02 aa
(21)
(25)
(26)
In the last equation, we introduced the molar absorptivity e 0 (SI unit m3 mol!1; usual unit cm!1 mol!1 dm3). Na is Avogadro’s constant. The molar rotation (ellipticity) is obtained in the commonly used units of deg dm3 cm!1 mol!1 via the expressions (the conversion from radians to degrees implies the multiplication by the factor 180/p) ½f'OR ½deg dm3 cm!1 mol!1 ' 0
s(o) [a.u.] E 3.05671 , 10!2 , o [a.u.]aaa(g) [a.u.]
Dy ¼
100oNa 0 G ðfÞ ½f'OR " ! 3e0 c02 aa
" !2:15524 , 104 , o ½a:u:' , Gaa ð f Þ ½a:u:' ½Y'ECD ½deg dm3 cm!1 mol!1 ' 0
" !2:15524 , 104 , o ½a:u:' , Gaa ðgÞ ½a:u:'
(27)
(28)
[Y]ECD [deg dm3 cm!1 mol!1] E !3.29821 , 103 , De 0 [cm!1 mol!1 dm3]
(29)
where, as for eqn (19) and (20) above and in other instances below, we specify within square brackets the units to be employed for the quantities on the right hand side to obtain the observable in the units given on the left hand side of the equations. In MORD and MCD, manifestations of MOA, the effect of a static magnetic field on the tensors entering the indices for the R and L circularly polarized components of plane-polarized light (again eqn (7) and (8)) to first order in the magnetic field is 0 ðmÞ
considered. Only the first-order response aabl of the antisymmetric
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polarizability aab , see eqn (12), survives, yielding a non-vanishing contribution to the anisotropies. After averaging, we obtain Dn ¼ !
0 ðmÞ N Bz eabl aabl ð f Þ 6e0
Dn0 ¼ !
(30)
0 ðmÞ N Bz eabl aabl ðgÞ 6e0
(31)
In terms of molar observables ½f'MOR " !
0 ðmÞ 25oNa Bz eabl aabl ð f Þ 3e0 c0
(32)
0 ðmÞ 25oNa Bz eabl aabl ðgÞ 3e0 c0
(33)
½Y'MCD " !
( ) ½f'MOR deg dm3 cm!1 mol!1
0 ðmÞ
( ) ½Y'MCD deg dm3 cm!1 mol!1
0 ðmÞ
Although the MChB and MChD are not circular differential properties, for the sake of comparing quantities with the same units, we generalize eqn (21) and (22) to axial anisotropies and define molar observables as
½Y'
MChD
o & 50oNa n A0 ol % "" G n ! n"# " B B ðf Þ þ B ðf Þ (36) z 2c0 3e0 c02 ¼
2c0
n
0 ""
!n
0 "#
"
0 ðmÞ ( 0 ) 0 2 n L ðBÞ ! n L ð!BÞ 1 Bz eabl aabl ¼! gMCD ðBÞ ¼ 0 L 2 aaa ðgÞ ½n ðBÞ þ n 0 L ð!BÞ'
n
o
50oNa 0 " Bz BA ðgÞ þ BG ðgÞ 3e0 c02 (37)
[f]MChB [deg dm3 cm!1 mol!1] 0
E 4.58460 , 10!2 , o [a.u.] , Bz [T] , {BA ( f ) + BG( f )} [a.u.] (38)
Connection to (damped) response functions
The molecular quantity that yields the OPA spectrum is the imaginary component of the damped electric-dipole–electric dipole linear response function26 aab( g) = !Imhhma;mbiigo
E 4.58460 , 10!2 , o [a.u.] , Bz [T] , {B (g) + BG( g)} [a.u.] (39) Note that temperature-dependent terms have been omitted in the above expressions, since they involve the expectation-value of the magnetic dipole operator in the ground state, which is quenched for closed shell systems as those we consider in this study. A criterion to decide on whether or not magneto-chiral dichroism in a particular absorption band is measurable for a given instrumental sensitivity is given by the so-called magneto-chiral dissymmetry (or asymmetry) factor gMChA or gMChD (also known as d),4,7 ( 0 ) 0 0 Bz BA ðgÞ þ BG ðgÞ n "" ! n "# (40) gMChD ðBÞ ¼ ¼2 1 0 "" c0 * aaa ðgÞ ðn þ n 0 "# Þ 2
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(44)
where ma indicates a Cartesian component of the electric dipole operator, g is the damping factor and o is the damped (complex) frequency. For ORD and ECD, the imaginary and real components of the damped magnetic dipole–electric dipole linear response function propagator are required, respectively32,35 ** ++g 0 Gab ð f Þ ¼ Im ma ; mb o (45) ** ++g 0 Gab ðgÞ ¼ Re ma ; mb o
(46)
where ma is the magnetic dipole moment operator component. For MOR and MCD, the relevant complex propagator is the quadratic response function involving two electric and one magnetic dipole operator 21,22 ** ++g 0 ðmÞ aabl ðf Þ ¼ Im ma ; mb ; ml o;0 (47) ** ++g 0 ðmÞ aabl ðgÞ ¼ !Re ma ; mb ; ml o;0
[Y]MChD [deg dm3 cm!1 mol!1] A0
(43)
The above expressions are used to evaluate the empirical relation gMChD = gMCD*gECD.34
(35)
" !3:14128 , o ½a:u:' , Bz ½T' , eabl , aabl ðgÞ ½a:u:'
ol !
which relates to the intensity of the OPA band.34 Similar dissymmetry (or asymmetry) factors are defined for MCD and ECD % 0 0 & 0 2 nL!nR 2Gaa ðgÞ gECD ¼ ¼! (42) 0L 0R ðn þ n Þ c0 * aaa ðgÞ
2.1 (34)
" !3:14128 , o ½a:u:' , Bz ½T' , eabl , aabl ð f Þ ½a:u:'
½f'MChB ¼
The denominator in the magneto-chiral dissymmetry factor is given by " 1! 0 "" N 0 n þ n "# ¼ aaa ðgÞ (41) 2 6e0
(48)
Generalizing the results of ref. 24, for MChB, the molecular 0 ðmÞ
tensors Aa;br;l ð f Þ and G(m) ab,l( f ) in eqn (15) correspond to the quadratic response functions
g G(m) ab,l(!o;o,0)( f ) = Rehhma;mb,mliio,0 ** ++g 0 ðmÞ Aa;br;l ð!o; o; 0Þð f Þ ¼ !Im ma ; Ybr ; ml o;0
(49) (50)
where (Ybr) is the traceless quadrupole moment operator component. When the frequency of the incident light approaches the absorbing region of the sample, the MChD emerges. Similar to what we did in the case of the MCD,21,22 we consider the absorptive counterparts of the quadratic response functions describing the MChB, that is
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Fig. 1 R-Methyloxirane. Upper panels: One photon absorption (OPA) cross sections, electronic circular dichroism (ECD), magnetic circular dichroism (MCD) and magneto-chiral dichroism (MChD) calculated at the B3LYP/t-aug-cc-pVDZ (left) and CAMB3LYP/t-aug-cc-pVDZ (right) levels. Lower panels: Natural optical rotatory (OR), magnetic optical rotation (MOR) and magneto-chiral birefringence (MChB) dispersions calculated at the B3LYP/t-aug-cc-pVDZ (left) and CAMB3LYP/t-aug-cc-pVDZ (right) levels. Absorption cross sections are reported in atomic units. All other quantities are given in deg dm3 cm!1 mol!1. MCD, MChD, MOR and MChB spectra were computed at 1 T of magnetic field intensity.
g G(m) ab,l(!o;o,0)( g) = Imhhma;mb,mliio,0
** ++g 0 ðmÞ Aa;br;l ð!o; o; 0ÞðgÞ ¼ !Re ma ; Ybr ; ml o;0 :
(51) (52)
The fact that the equations above involve the electric dipole moment, the magnetic dipole moment and the traceless quadrupole moment operators raises the issue of origin invariance, which in the case of magneto-chiral birefringence has been discussed in detail by Coriani and co-workers.24 These authors proved the independence of the choice of the magnetic gauge
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for the expression of eqn (15) describing magnetochiral birefringence in the ‘‘exact wave function’’ case, assuming in other words that the Ehrenfest theorem is valid and that the basis set used for the expansion of the wavefunction is complete. In the ‘‘real’’ case, implying the use of finite basis sets, an analytic expression was given for the magnetic gauge origin dependence of the birefringence, in terms of combinations of quadratic response functions involving, besides the operators mentioned above, also the velocity operator components mpa , see ref. 24 for further details. This is the case in actual computational
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simulations. We add that for variational models, as the Hartree–Fock approximation or Kohn–Sham density functional theory, where Ehrenfest theorem holds and the hypervirial relationship relating the velocity operator to the commutator of the Hamiltonian and the electric dipole operator is satisfied, convergence towards full magnetic gauge origin independence of the results obtained via standard response theory is usually guaranteed. A detailed study of the matter for another property (magnetic field induced second harmonic generation, MFISHG, in chiral samples36) involving nonlinear response functions, and where the problem of magnetic gauge origin dependence of the results also arises, was studied recently and proved that origin dependence becomes less severe as the size and complexity of the molecule increase. Within the CPP framework used in the present study, the analysis becomes more complicated, with couplings between real and imaginary terms, as illustrated, for instance, in ref. 37, and we have not performed it here, limiting ourselves to test the effect of adopting different gauge origins in our calculations. Fig. S5 in the ESI,† shows how the MChD (and MCD) B3LYP spectra of R-methyloxirane change when computed at three different gauge origins, namely the center of mass, the chiral center, and one given H atom. As it can be appreciated from the figure, the differences are negligible, so we will only discuss center-of-mass results in the following.
3 Computational details To start the investigation we chose the R-methyloxirane molecule as the smallest organic chiral system. Next, in order to stay in the context of biologically relevant species, we investigated a chiral proteinogenic amino acid, L-alanine. The calculations of R-methyloxirane were performed on the optimized structure of the (R) enantiomer taken from ref. 24,
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where it was obtained at the MP2/cc-pVTZ level of theory. For the natural amino-acid L-alanine, we considered the equilibrium structures of the two lowest lying gas-phase conformers Ala-I and Ala-IIA fully optimized at the coupled-cluster [CCSD(T)/cc-pVTZ] level of theory by Jaeger et al.38 The authors of ref. 38 computed a relative energy of 2.11 kJ mol!1 between these two conformers, corresponding to Boltzmann weights of E0.7 and E0.3, for Ala-I and Ala-IIA, respectively. The same authors discuss a third conformer (labelled Ala-IIB) very close in energy to Ala-IIA and which they claimed as still unobserved when the paper was published. We did not include this conformer in our analysis. OPA, ECD, MCD and MOR as well as MChB and MChD spectra were computed both at the Hartree–Fock and at the TD-DFT level (B3LYP and CAMB3LYP functionals) in the (multiply-)augmented correlation consistent basis sets aug-cc-pVDZ, d-aug-cc-pVDZ, and t-aug-cc-pVDZ. The convergence threshold of the response calculations was set to 10!5. In the CPP calculations, we used the empirical linewidth g equal to 1000 cm!1 (0.00456 a.u.) and the step between successive frequencies in the transition region equal to 0.0025 a.u. Obviously, the choice of g affects the calculated spectra, a larger value yielding broader spectral features with lower peak intensities. As we here deal with pure electronic structure calculations where various factors (such as vibronic couplings) affecting the experimental linewidths are not considered, the value of g can be empirically adjusted to approximately match the observed spectra. So far there exist no experimental spectra of MChD for the chosen compounds, and we rely in our selection of g in previous application of the CPP approach for MCD in systems of composition comparable to the present ones.39,40 The code to evaluate the MChD and MChB properties was implemented within the DALTON program.41,42 The plots, obtained by spline fitting the computed observables, were done using the libraries Python Scipy43 and Matplotlib44. The code was first tested against the MChB results obtained in ref. 24
Fig. 2 R-Methyloxirane. The gECD, gMCD and gMChD dissimmetry factors, calculated at the B3LYP/t-aug-cc-pVDZ (left) and CAMB3LYP/t-aug-cc-pVDZ (right) levels. Note that the MCD and MChD dissymmetry factors are given for 1 T of magnetic field intensity.
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Fig. 3 L-Alanine, conformer I. Upper panels: One photon absorption (OPA) cross sections, electronic circular dichroism (ECD), magnetic circular dichroism (MCD) and magneto-chiral dichroism (MChD), calculated at the B3LYP/t-aug-cc-pVDZ (left) and CAMB3LYP/t-aug-cc-pVDZ (right) levels. Lower panels: Natural optical rotatory (OR), magnetic optical rotation (MOR) and magneto-chiral birefringence (MChB) dispersions, calculated at B3LYP/ t-aug-cc-pVDZ (left) and CAMB3LYP/t-aug-cc-pVDZ (right) levels. Absorption cross sections are reported in atomic units. All other quantities are given in deg dm3 cm!1 mol!1. MCD, MChD, MOR and MChB spectra were computed at 1 T of magnetic field intensity.
and it was found that the values obtained therein could be reproduced, in particular, the values of B(A 0 ) and B(G) for H2O2 and R-methyloxirane.
4 Results and discussion As mentioned above, we have computed the dispersion and absorption spectra both at TD-Hartree–Fock and TD-DFT levels, the latter with two different functionals, and three basis sets,
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aug-cc-pVDZ, d-aug-cc-pVDZ and t-aug-cc-pVDZ. Spectra were simulated in the wavelength region including the lowest ten electronic excitations. The full set of results is collected in the ESI,† and we refrain from a very detailed discussion in the following, limiting ourselves to summarizing here the main points. The results reported and discussed in the following are those obtained in the largest (t-aug-cc-pVDZ) basis set. For R-methyloxirane all three basis sets yield, for each individual electronic structure model, rather similar spectral profiles, the differences mainly emerging at higher frequencies
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Fig. 4 L-Alanine, conformer IIA. Upper panels: One photon absorption (OPA) cross section, electronic circular dichroism (ECD), magnetic circular dichroism (MCD) and magneto-chiral dichroism (MChD), calculated at B3LYP/t-aug-cc-pVDZ (left) and CAMB3LYP/t-aug-cc-pVDZ (right) levels of theory. Lower panels: Natural optical rotatory (OR), magnetic optical rotation (MOR) and magneto-chiral birefringence (MChB) dispersions, calculated at B3LYP/t-aug-cc-pVDZ (left) and CAMB3LYP/t-aug-cc-pVDZ (right) levels of theory. Absorption cross sections are reported in atomic units. All other quantities are given in deg dm3 cm!1 mol!1. MCD, MChD, MOR and MChB spectra were computed at 1 T of magnetic field intensity.
(shorter wavelengths) consistent with the increasing number of excited states when approaching the ionization limit. We have collected in Table S1 of the ESI† the excitation energies, the OPA oscillator strengths and the ECD rotatory strengths of the first ten excitations calculated for all three methods in the larger basis set. Much more pronounced are the differences, in the same basis sets, between Hartree–Fock and DFT results, and, to a lower extent, between the two sets of DFT results. As commonly seen, Hartree–Fock spectra are shifted towards shorter wavelengths (first excitation at 8.83 eV in the t-aug-cc-pVDZ basis),
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followed by the CAMB3LYP (first bright excitation at 7.01 eV) and then by the B3LYP (6.44 eV) ones. Note that the first ionization limit has been estimated by Koopmans’ theorem at 11.8 eV for Hartree–Fock, at 9.26 eV for CAMB3LYP and 7.43 eV for B3LYP. R-Methyloxirane has been a reference system for studies of chiroptical properties. In previous studies,45 where the vertical electronic excitation manifold and the related rotatory strengths of R-methyloxirane, including the first eight excited states, were studied in greater detail, with also an analysis of basis set behavior, it was shown that CAMB3LYP yields results closer to
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Fig. 5 L-Alanine. Boltzmann averages. Upper panels: One photon absorption (OPA) cross sections, electronic circular dichroism (ECD), magnetic circular dichroism (MCD) and magneto-chiral dichroism (MChD), calculated at the B3LYP/t-aug-cc-pVDZ (left) and CAMB3LYP/t-aug-cc-pVDZ (right) levels. Lower panels: Natural optical rotatory (OR), magnetic optical rotation (MOR) and magneto-chiral birefringence (MChB) dispersions, calculated at B3LYP/t-aug-cc-pVDZ (left) and CAMB3LYP/t-aug-cc-pVDZ (right) levels. Absorption cross sections are reported in atomic units. All other quantities are given in deg dm3 cm!1 mol!1. MCD, MChD, MOR and MChB spectra were computed at 1 T of magnetic field intensity.
experiment, in particular for ECD rotatory strengths computed with sufficiently large basis sets. Indeed, the authors of ref. 46 concluded that CAMB3LYP is the recommended functional for excitation energy calculations, since it yields a balanced description of local, Rydberg, and charge transfer excitations. We note on one hand that the comparison with experiment requires accounting for vibrational and, when applicable, solvent effects,
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both of which are neglected in our analysis. On the other hand, we have shown, see in particular ref. 47, that for higher order properties no clear cut conclusion can be drawn in principle on the superiority of CAMB3LYP over B3LYP, and that the degree of agreement between theory and experiment depends on several parameters, including the region of the spectrum under investigation.
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Despite the shift in the spectra and the underlying different composition of the bands in terms of individual excited states, both DFT functionals show qualitatively similar band features in the CPP spectra, see the upper panels of Fig. 1. The ECD spectrum has a +/! band structure in both cases corresponding to the first five excitations. The MCD spectrum exhibits a weak negative peak around the first excitation in the CAMB3LYP case, not evident in the B3LYP spectrum, otherwise both functionals yield a positive band in the region around the second to fourth excitation. The MChD spectra have a !/+ pattern for both functionals. Remarkably, all three dichroisms show different sign patters, indicating that three effects yield complementary information. Inspection of the figures also shows that the different spectroscopic effects have very different intensities (yet the two functionals give comparable intensities). The MCD signal is about one order of magnitude weaker (per unit of magnetic field) than the ECD one. The MChD signal is two to three orders of magnitude weaker than the MCD one. The dispersion curves, see Fig. 1 (lower panels), also show some qualitatively similar features for the two functionals, and give an indication of the origin of the resonances in the MCD and MChD CPP spectra, as they correspond to inflection points on the dispersion profiles. A detailed investigation of pure and cascaded magnetochiral dichroism has led Rikken and Raupach34 to the conclusion that in diamagnetic media, between the magnitude of the anisotropy factor for the pure magneto-chiral effect and that for natural and magnetic circular dichroism, the following relation should hold approximately: |gMChD| E |gECD||gMCD|
(53)
For R-methyloxirane, the dissymmetry factors of ECD, MCD and MChD, together with their absolute values and the product of the ECD and MCD factors, are plotted in Fig. 2. The comparison shows that the curves yielded by the approximate relation above are rather different from those computed directly from the simulated MChD and OPA results, yet the order of magnitude is roughly the same. In the specific case the factor obtained from the ECD and MCD product is lower than the one computed directly for MChD. Turning our attention to the two conformers of L-alanine, see Fig. 3 and 4 and ESI,† we note that, like in R-methyloxirane, the basis set effects on the spectra are contained, despite significant differences in the underlying composition of the spectral bands, as also appreciated from the results of the excitation energies and oscillator and rotatory strengths tabulated in Tables S2 and S3 of the ESI.† For Ala-I, a remarkable difference between B3LYP and CAMB3LYP is observed in the intermediate region of the ECD spectrum, which can be attributed to the different rotatory strengths of the second and third excited states. The MCD spectra, on the other hand, are qualitatively similar in terms of band structure (+/!/+). An interesting feature in comparing the MCD and MChD spectra is the fact that no MChD band emerges around the second excited state, whereas it clearly does so in the MCD spectrum. As for R-methyloxirane, the predicted MChD intensities are roughly the same for both functionals, and significantly smaller than the MCD (and ECD) ones. In Ala-IIA the sign pattern of the ECD bands corresponding to the lowest excited states is similar for the two functionals. The MCD signal is almost quenched in correspondence to the two first excited states (at least in the given ordinate scale), whereas a positive band emerges for the first excited state in the MChD spectrum. The larger separation between the first and
Fig. 6 L-Alanine. The gECD, gMCD and gMChD dissimmetry factors, calculated at the B3LYP/t-aug-cc-pVDZ (left) and CAMB3LYP/t-aug-cc-pVDZ (right) level of theory from the Boltzmann averaged spectra. Note that the MCD and MChD dissymmetry factors were computed at 1 T of magnetic field intensity.
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the second excited states probably accounts for the appearance of a weak band in the CAMB3LYP spectrum in correspondence to the second excited state. For the higher states, the MCD and MChD signals have the same sign pattern, yet with a different composition in terms of contributions from the excited states underlying the various bands. Once again, the spectral intensities are comparable for the two functionals. The dispersion profiles for OR, MOR and MChB have also been included in Fig. 3 and 4. The two conformers have rather different spectra and dispersion profiles. The curves have been combined according to their Boltzmann weights and are shown in Fig. 5. The curves corresponding to the Ala-I and Ala-IIA conformers were included in the total spectrum with the factors of 0.7 and 0.3, respectively. The corresponding dissymmetry factor curves are shown in Fig. 6.
using the expressions ( ) ½a' deg cm3 dm!1 g!1
" !2:15524 , 106 ,
( ) ½c' deg cm3 dm!1 g!1 " !2:15524 , 106 ,
A protocol for the calculation of magneto-chiral dichroism and magneto-chiral birefringence has been proposed and implemented within the complex polarization propagator formalism of Hartree–Fock and density functional response theory. Pilot results have been presented for R-methyloxirane and L-alanine. Despite the differences in the fine spectral details, both B3LYP and CAMB3LYP yield similar results (in terms of intensity) for the MChD, which is computed two to three orders of magnitude smaller than the corresponding MCD signal. The MChD dissymmetry factor curves are different from the product of the dissymmetry factors for MCD and ECD, but roughly of the same order of magnitude. In a previous experiment on chromium tris-oxalate,33 a resolution of g of a few times 10!6 T!1 had been reached. Our computational predictions are at the limit of the current experimental resolution, obtained in the visible wavelength range, and should stimulate the development of a similar resolution in the UV and deep UV, in view of the ongoing discussion on the role of magneto-chiral dichroism in the homochirality of life. The rational search for organic molecules of biological relevance with a detectable magneto-chiral dichroism has just begun.
Appendix: more definitions and unit conversions In terms of specific observables, rotations and ellipticities, for NOA, are ½a' " !
oNa 0 G ðfÞ 3e0 c02 M aa
(54)
½c' " !
oNa 0 G ðgÞ 3e0 c02 M aa
(55)
o ½a:u:' 0 , Gaa ðgÞ ½a:u:' M ½g mol!1 '
(57)
0 ðmÞ oNa Bz eabg aabg ð f Þ 12e0 c0 M
(58)
0 ðmÞ oNa ½c' " ! Bz eabg aabg ðgÞ 12e0 c0 M
(59)
½a' " !
" !3:14128 , 102 ,
o ½a:u:' M ½g mol!1 ' 0 ðmÞ
, Bz ½T' , eabg , aabg ð f Þ ½a:u:'
(60)
( ) ½c' deg cm3 dm!1 g!1
" !3:14128 , 102 ,
o ½a:u:' M ½g mol!1 ' 0 ðmÞ
, Bz ½T' , eabg , aabg ðgÞ ½a:u:'
(61)
and for MChD and MChB, ½a' " !
n 0 o oNa Bz BA ð f Þ þ BG ð f Þ 2 6e0 c0 M
½c' " !
n 0 o oNa Bz BA ðgÞ þ BG ðgÞ 2 6e0 c0 M
(62) (63)
( ) ½a' deg cm3 dm!1 g!1 " !4:58460 ,
n 0 o o ½a:u:' , Bz ½T' , BA ð f Þ þ BG ð f Þ ½a:u:' !1 M ½g mol ' (64)
( ) ½c' deg cm3 dm!1 g!1 " !4:58460 ,
n 0 o o ½a:u:' , Bz ½T' , BA ðgÞ þ BG ðgÞ ½a:u:' !1 M ½g mol ' (65)
Here we also give a compilation of relevant conversion units (same units for primed and unprimed tensors): - e0 = 8.85419 , 10!12 F m!1 = 7.95775 , 10!2 a0 me e2 h !!2; h a0!1 me!1; - c0 = 299 792 459 m s!1 = 1.37036 , 102 !
where M is the molar mass. The specific rotation (ellipticity) is obtained in the commonly used units of deg cm3 dm!1 g!1
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(56)
For MOA,
( ) ½a' deg cm3 dm!1 g!1
5 Conclusions
o ½a:u:' 0 , Gaa ð f Þ ½a:u:' M ½g mol!1 '
- Na = 6.02214 , 1023 mol!1; h a0!2 me!1; - [o] - s!1 = 2.4188 , 10!17 !
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- [Bz] - T = 4.25438 , 10!6 !h e!1 a0!2; - [N] - m!3 = 1.48185 , 10!31 a03; - [M] - g mol!1 = 1.09777 , 1027 me mol!1; - [aab] - C2 m2 J!1 = 6.06510 , 1040 e2 a04 me h!!2; - [Gab] - C2 m3 J!1 s!1 = 2.77238 , 1034 h!!1 a03 e2; - [Aa,bg] - C2 m3 J!1 = 1.14614 , 1051 h!!2 a05 e2 me;
h 0 i ðmÞ - aa;b;l ! C3 m4 s!1 J!2 ¼ 1:42561 , 1046 e3 a06 !h!3 me ; 3 5 !2 !2 - [G(m) s = 6.51652 , 1039 e3 a05 h !!2; ab,g] - C m J 3 5 !2 !1 - [A(m) s = 2.69402 , 1056 e3 a07 me h !!3; a,bg,d] - C m J
Acknowledgements This research work was carried out with support from the University of Trieste (Grant CHIM02-Ricerca, J. C. and S. C.), the PRIN2009 funding scheme (Project No. 2009C28YBF_001, S. C.), the AIAS-COFUND Marie-Curie program (Grant Agreement No. 609033, S. C.), the COST-CMTS Action CM1002 COnvergent Distributed Environment for Computational Spectroscopy (CODECS), the Interdisciplinary Centre for Mathematical and Computational Modelling (ICM), and the University of Warsaw (Grant No. G56-20, J. C.). Computer time from CINECA-UniTS ¨ping is and the National Supercomputer Centre (NSC) in Linko also acknowledged.
References 1 L. D. Barron, Molecular light scattering and optical activity, Cambridge University Press, Cambridge, 2004. `re and A. Meyer, Chem. Phys. Lett., 1982, 93, 2 G. H. Wagnie 78–81. `re, Chem. Phys. Lett., 1984, 110, 546–551. 3 G. H. Wagnie 4 L. Barron and J. Vrbancich, Mol. Phys., 1984, 51, 715–730. 5 G. L. Rikken and E. Raupach, Nature, 1997, 390, 493–494. `re, Chem. Phys., 1999, 245, 165–173. 6 G. Wagnie 7 G. L. Rikken and E. Raupach, Nature, 2000, 405, 932–935. `re and G. L. Rikken, Chem. Phys. Lett., 2009, 8 G. H. Wagnie 481, 166–168. `re, On Chirality and the Universal Asymmetry. 9 G. H. Wagnie Reflections on Image and Mirror Image, Wiley-VCH, Zurich, 2007. 10 L. Barron, Nature, 2000, 405, 895. 11 N. B. Baranova, Y. V. Bogdanov and B. Y. Zel’dovich, Opt. Commun., 1977, 22, 243–247. `re and A. Meier, Experimentia, 1983, 39, 1090. 12 G. H. Wagnie 13 Y. Kitagawa, H. Segawa and K. Ishii, Angew. Chem., Int. Ed., 2011, 50, 9133.
13278 | Phys. Chem. Chem. Phys., 2016, 18, 13267--13279
14 Y. Kitagawa, T. Miyatake and K. Ishii, Chem. Commun., 2012, 48, 5091–5093. 15 R. Sessoli, M.-E. Boulon, A. Caneschi, M. Mannini, L. Poggini, F. Wilhelm and A. Rogalev, Nat. Phys., 2015, 11, 69–74. 16 A. D. Buckingham and P. J. Stephens, Annu. Rev. Phys. Chem., 1966, 399–432. 17 W. R. Mason, A Practical guide to Magnetic Circular Dichroism Spectroscopy, Wiley, New York, 2007. 18 T. Kjærgaard, S. Coriani and K. Ruud, WIREs Comput. Mol. Sci., 2012, 2, 443–455. 19 S. Coriani, P. Jørgensen, A. Rizzo, K. Ruud and J. Olsen, Chem. Phys. Lett., 1999, 300, 61–68. ¨ttig, P. Jørgensen and T. Helgaker, J. Chem. 20 S. Coriani, C. Ha Phys., 2000, 113, 3561. 21 H. Solheim, K. Ruud, S. Coriani and P. Norman, J. Chem. Phys., 2008, 128, 094103. 22 T. Kjærgaard, K. Kristensen, J. Kauczor, P. Jørgensen, S. Coriani and A. Thorvaldsen, J. Chem. Phys., 2011, 135, 024112. 23 T. Fahleson, J. Kauczor, P. Norman and S. Coriani, Mol. Phys., 2013, 111, 1401–1404. 24 S. Coriani, M. Pecul, A. Rizzo, P. Jørgensen and M. Jaszunski, J. Chem. Phys., 2002, 117, 6417. 25 B. Jansı´k, A. Rizzo, L. Frediani, K. Ruud and S. Coriani, J. Chem. Phys., 2006, 125, 234105. 26 P. Norman, D. Bishop, H. J. Aa. Jensen and J. Oddershede, J. Chem. Phys., 2001, 115, 10323. 27 P. Norman, D. Bishop, H. J. Aa. Jensen and J. Oddershede, J. Chem. Phys., 2005, 123, 194103. 28 K. Kristensen, J. Kauczor, T. Kjærgaard and P. Jørgensen, J. Chem. Phys., 2009, 131, 044112. 29 P. Norman, Phys. Chem. Chem. Phys., 2011, 13, 20519. 30 J. Kauczor, P. Jørgensen and P. Norman, J. Chem. Theory Comput., 2011, 7, 1610. 31 A. D. Buckingham, Adv. Chem. Phys., 1967, 12, 107. 32 A. Jiemchooroj and P. Norman, J. Chem. Phys., 2007, 126, 134102. 33 E. Raupach, G. Rikken, C. Train and B. Malezieux, Chem. Phys., 2000, 261, 373–380. 34 G. L. Rikken and E. Raupach, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1998, 58, 5081–5084. 35 P. Norman, K. Ruud and T. Helgaker, J. Chem. Phys., 2004, 120, 5027. ´vet, Phys. Chem. 36 A. Rizzo, G. L. J. A. Rikken and R. Mathe Chem. Phys., 2016, 18, 1846–1858. 37 N. H. List, J. Kauczor, T. Saue, H. J. A. Jensen and P. Norman, J. Chem. Phys., 2015, 142, 244111. ´sza ´r 38 H. M. Jaeger, H. F. Schaefer III, J. Demaison, A. G. Csa and W. D. Allen, J. Chem. Theory Comput., 2010, 6, 3066–3078. 39 F. Santoro, R. Improta, T. Fahleson, J. Kauczor, P. Norman and S. Coriani, J. Phys. Chem. Lett., 2014, 5, 1806–1811. 40 T. Fahleson, J. Kauczor, P. Norman, F. Santoro, R. Improta and S. Coriani, J. Phys. Chem. A, 2015, 119, 5476–5489. 41 K. Aidas, C. Angeli, K. L. Bak, V. Bakken, R. Bast, L. Boman, O. Christiansen, R. Cimiraglia, S. Coriani, P. Dahle,
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¨m, T. Enevoldsen, J. J. Eriksen, E. K. Dalskov, U. Ekstro ´ndez, L. Ferrighi, H. Fliegl, P. Ettenhuber, B. Ferna ¨ttig, H. Heiberg, L. Frediani, K. Hald, A. Halkier, C. Ha T. Helgaker, A. C. Hennum, H. Hettema, E. Hjertenæs, S. Høst, I.-M. Høyvik, M. F. Iozzi, B. Jansı´k, H. J. Aa. Jensen, D. Jonsson, P. Jørgensen, J. Kauczor, S. Kirpekar, T. Kjærgaard, W. Klopper, S. Knecht, R. Kobayashi, H. Koch, J. Kongsted, A. Krapp, K. Kristensen, A. Ligabue, O. B. Lutnæs, J. I. Melo, K. V. Mikkelsen, R. H. Myhre, C. Neiss, C. B. Nielsen, P. Norman, J. Olsen, J. M. H. Olsen, A. Osted, M. J. Packer, F. Pawlowski, T. B. Pedersen, P. F. Provasi, S. Reine, Z. Rinkevicius, T. A. Ruden, K. Ruud, V. V. Rybkin, P. Sałek, C. C. M. Samson, A. S. ´s, T. Saue, S. P. A. Sauer, B. Schimmelpfennig, de Mera K. Sneskov, A. H. Steindal, K. O. Sylvester-Hvid, P. R. Taylor, A. M. Teale, E. I. Tellgren, D. P. Tew, A. J.
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Paper
42 43
44 45 46 47
Thorvaldsen, L. Thøgersen, O. Vahtras, M. A. Watson, D. J. D. Wilson, M. Ziolkowski and H. Ågren, WIREs Comput. Mol. Sci., 2014, 4, 269–284. Dalton, A molecular electronic structure program, Release Dalton2015, 2015, see http://daltonprogram.org/. E. Jones, T. Oliphant and P. Peterson, et al., SciPy: Open source scientific tools for Python, 2001, http://www.scipy.org/, online; accessed 2016-02-06. J. D. Hunter, Comput. Sci. Eng., 2007, 9, 90–95. J. Bloino, M. Biczysko, F. Santoro and V. Barone, J. Chem. Theory Comput., 2010, 6, 1256–1274. M. J. G. Peach, P. Benfield, T. Helgaker and D. J. Tozer, J. Chem. Phys., 2008, 128, 044118. N. Lin, F. Santoro, X. Zhao, C. Toro, L. D. Boni, F. E. ´ndez and A. Rizzo, J. Phys. Chem. B, 2011, 115, Herna 811–824.
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