On the determination of the diagonal components of the optical activity tensor in chiral molecules Stefano Pelloni and Paolo Lazzeretti Citation: The Journal of Chemical Physics 140, 074105 (2014); doi: 10.1063/1.4865229 View online: http://dx.doi.org/10.1063/1.4865229 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Atomic partition of the optical rotatory power of methylhydroperoxide J. Chem. Phys. 128, 064318 (2008); 10.1063/1.2826351 Calculation of origin-independent optical rotation tensor components in approximate time-dependent density functional theory J. Chem. Phys. 125, 034102 (2006); 10.1063/1.2210474 Ab initio calculations of nonlinear optical rotation by several small chiral molecules and by uridine stereoisomers J. Chem. Phys. 124, 184305 (2006); 10.1063/1.2196880 On the effect of a radiation field in modifying the intermolecular interaction between two chiral molecules J. Chem. Phys. 124, 014302 (2006); 10.1063/1.2140000 Density-functional theory calculations of optical rotatory dispersion in the nonresonant and resonant frequency regions J. Chem. Phys. 120, 5027 (2004); 10.1063/1.1647515
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THE JOURNAL OF CHEMICAL PHYSICS 140, 074105 (2014)
On the determination of the diagonal components of the optical activity tensor in chiral molecules Stefano Pelloni and Paolo Lazzerettia) Dipartimento di Scienze Chimiche e Geologiche, Università degli Studi di Modena e Reggio Emilia, Via Campi 183, 41100 Modena, Italy
(Received 28 December 2013; accepted 29 January 2014; published online 19 February 2014) It is shown that the diagonal components of the mixed electric-magnetic dipole polarizability tensor, used to rationalize the optical rotatory power of chiral molecules, are origin independent, if they are referred to the coordinate system defined by the eigenvectors of the dynamic electric dipole polarizability, for a given value ω of the frequency of a monochromatic wave impinging on an ordered sample. Within this reference frame, the individual diagonal components of the mixed electric-magnetic dipole polarizability are separately measurable properties. The theoretical method is applied via a test calculation to the cyclic 1,2-M enantiomer of the dioxin molecule, using a large Gaussian basis set to estimate near Hartree-Fock values within a series of dipole length, velocity, and acceleration representations. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4865229] I. INTRODUCTION
The optical rotatory power (ORP) is the property of chiral molecules in an isotropic medium to rotate the plane of polarization of a beam of linearly polarized light.1, 2 The rotation angle is rationalized via a pseudoscalar, that is, the mean value (one third the trace) of the optical gyration polarizability,3–5 a non-symmetric second-rank axial tensor, also referred to as optical activity tensor, and “mixed electric-magnetic dipole polarizability” (MEMDP),6–8 odd under parity P and even under time reversal T. The components of the MEMDP tensor, and the related pseudoscalar, have the same magnitude but opposite sign in the enantiomers of a chiral species. This pseudoscalar is in general smaller than any of the three diagonal components, as they have different sign. It vanishes in optically inactive molecules, which nonetheless may be characterized by individual non-zero off-diagonal components of the MEMDP tensor. Only the trace of this tensor is independent of the origin of the laboratory coordinate system, and it is the sole measurable quantity in a disordered medium containing freely tumbling optically active molecules. The separate components, diagonal and off-diagonal, depend on the origin,9–15 and consequently they could not, in principle, be experimentally determined in a molecular crystal. However, diagonal components invariant under a translation of coordinate system can be formally defined allowing for the results of previous papers.12–15 The scope of the present work is to show that, for any frequency ω of the incident plane polarized light, the diagonal components of the optical gyration tensor referred to the principal axis system of the electric dipole polarizability (EDP) are independent of the origin. Therefore, they can be calculated as sharply defined properties of optically active molecules, and they are by all means measurable in ordered phase.
A compact self-contained account of the basic theory needed for a sensible discussion of this point is presented hereafter in Sec. II. The results of calculations carried out for C4 H4 O2 , the cyclic 1,2-M enantiomer of the dioxin molecule, are given in Sec. III, and a summary is reported in the concluding section, Sec. IV.
II. THE MEMDP TENSOR WITHIN DIPOLE LENGTH, VELOCITY, AND ACCELERATION GAUGES
We use the notation adopted in previous papers.12–15 For a molecule with n electrons and N nuclei, charge, mass, position with respect to an arbitrary origin, canonical, and angular momentum of the ith electron are indicated by −e, me , r i , pˆ i , lˆi = r i × pˆ i , i = 1, 2 . . . n. Corresponding quantities for nucleus I are indicated by ZI e, MI , RI . Capital letters denote total electronic operators, e.g., ˆ = R
Pˆ =
ri,
i=1
n
pˆ i ,
Lˆ =
i=1
n
lˆi ,
i=1
etc. Throughout this article we use SI units and the Einstein convention of summing over two repeated Greek subscripts. The operators for the electric and magnetic dipole moment of electrons are μˆ α = −eRˆ α
(1)
and ˆα = − m
e ˆ Lα , 2me
(2)
respectively. The multiplicative operator for the electric field of electron i on nucleus I is defined by ˆ iI = E
a) E-mail: [email protected]
0021-9606/2014/140(7)/074105/6/$30.00
n
140, 074105-1
r i − RI e , 4π 0 |r i − RI |3
(3)
© 2014 AIP Publishing LLC
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Allowing for the hypervirial relationships,12, 17
and the operator for the total n-electron field by ˆ nI = E
n
ˆ iI . E
(4)
a|Rˆ α |j =
i=1
(R,L) καβ (ω)
2ω 1 ˆ β |a), (5) =− (a|μˆ α |j j |m ¯ j =a ωj2a − ω2
where takes the imaginary part of the term within brackets, ωj a =
1 (0) Ej − Ea(0) ¯
(6)
are the natural transition frequencies of the molecule in the reference state |a(0) ≡ |a, with energy eigenvalue Ea(0) , excited state energies Ej(0) , determined by solving the Schrödinger equation for the unperturbed Hamiltonian Hˆ (0) , and ω is the angular frequency of a monochromatic wave impinging on the molecule. ˆ and m ˆ are, respectively, a polar and an axial vecAs μ tor, the trace of the tensor defined by Eq. (5), which is related to quantities experimentally accessible,16 is a pseudoscalar, changing sign under inversion of the coordinate system. Equation (5) defines a second-order property in the length(RL) (ω) ≡ καβ (ω). At any angular momentum picture, i.e., καβ rate, owing to the peculiar features of quantum mechanics, one is free to choose different representations, or gauges, via canonical transformations of the Hamiltonian. Alternative definitions for the tensor introduced via expression (5) can be immediately arrived at by Ehrenfest off-diagonal hypervirial relations,17 allowing for velocity, force (or acceleration), and torque gauges.12 In an unperturbed molecule, the electronic operators for ˆN ˆN the total force, F n , and torque, K n (r 0 ) about the origin r 0 , acting on the electron cloud, contain only contributions from the nuclei, as total electron-electron force and torque cancel out by action equals reaction. Therefore, 1 ˆ i ˆ (0) ˆ P, [H , R] = ¯ me N n i ˆ (0) ˆ e2 r i − R I ˆN [H , P] = F = − Z , I n ¯ 4π 0 I =1 |r i − RI |3 i=1
(7)
(8)
i ˆ (0) ˆ N [H , L(r 0 )] = Kˆ n (r 0 ) ¯ =
N n e2 r i − R I ZI × (RI − r 0 ). 4π 0 I =1 |r i − RI |3 i=1
(9) The operators for total force and torque can also be conveniently rewritten12 in terms of the electric field, Eq. (3).
N e −2 ωj a ZI a|Eˆ Inα |j , me I =1
(10)
ˆN a|Lˆ α |j = iωj−1 a a|Knα |j ,
(11)
=
The ORP of a chiral molecule is customarily accounted for via the MEMDP tensor within dipole length-angular momentum formalism (R, L),12
i −1 1 N ωj a a|Pˆα |j = − ωj−2 a|Fˆnα |j me me a
five alternative expressions are found for the MEMDP tensor,12–14, 18 (RK) καβ (ω) =
2ω e2 N 2 a|Rˆ α |j j |Kˆ nβ |a , 2 2me ¯ j =a ωj a ωj a −ω (12)
(P L) καβ (ω) = −
2ω e2 2 a|Pˆα |j j |Lˆ β |a , 2 2 2me ¯ j =a ωj a ωj a −ω (13)
(P K) καβ (ω) = −
2ω e2 N 2 a|Pˆα |j j |Kˆ nβ |a , 2 2 2me ¯ j =a ωj a ωj a −ω2 (14)
(F L) καβ (ω) =
2ω e2 N a|Fˆnα |j j |Lˆ β |a , 2m2e ¯ j =a ωj2a ωj2a −ω2 (15)
(F K) καβ (ω) = −
2ω e2 N N a|Fˆnα |j j |Kˆ nβ |a . 2m2e ¯ j =a ωj3a ωj2a −ω2 (16)
(ω) is unafOf course the axial tensor character of καβ N N fected by the change of picture, as Fˆnα and Kˆ nα are, respectively, polar and axial vectors. All these definitions are equivalent in quantum mechanics, as a consequence of the invariance of the theory in a canonical transformation. However, the estimates of MEMDP are numerically the same only if the eigenfunctions |a and |j are the exact eigenstates to a model Hamiltonian, or optimal variational wavefunctions,17 a condition that is hardly met in practice. In a calculation based on the algebraic approximation, values arrived at by relationships (5) and (12)–(16) may be appreciably different: their numerical agreement gives a benchmark of basis set completeness and an a priori quality criterion of computational accuracy. In other words, if MEMDP values calculated via Eqs. (5) and (12)–(16) are close to one another, one can reasonably assume that they specify the limit for a given approximation. In a translation of origin of the coordinate frame,
r = r + d,
(17)
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the MEMDP tensor changes according to a relationship involving α, the dynamic EDP,12 καβ (r ) = καβ (r ) −
ω βγ δ ααγ dδ , 2
Tr[κ (r )] = Tr[κ (r )]. (18)
These equations would be exactly fulfilled in ideal calculations, e.g., using exact wavefunctions. In approaches employing truncated basis sets, the change of the components for different pictures, Eqs. (5) and (12)–(16), is obtained by (R,L) (R,L) καβ (r ) = καβ (r ) −
ω (R,P ) βγ δ ααγ dδ , 2
(19)
(R,K) (R,K) καβ (r ) = καβ (r ) −
ω (R,F ) βγ δ ααγ dδ , 2
(20)
(P ,L) (P ,L) (r ) = καβ (r ) − καβ
ω (P ,P ) βγ δ ααγ dδ , 2
(21)
(P ,K) (P ,K) καβ (r ) = καβ (r ) −
ω (P ,F ) βγ δ ααγ dδ , 2
(22)
(F,L) (F,L) (r ) = καβ (r ) − καβ
ω (F,P ) βγ δ ααγ dδ , 2
(23)
(F,K) (F,K) (r ) = καβ (r ) − καβ
ω (F,F ) βγ δ ααγ dδ , 2
(24)
where the frequency-dependent EDPs in different pictures are e2 2ωj a (a|Rˆ α |j j |Rˆ β |a), ¯ j =a ωj2a − ω2
(R,R) ααβ (ω) =
(25)
2 e2 (a|Rˆ α |j j |Pˆβ |a), (26) 2 me ¯ j =a ωj a − ω2
(R,P ) ααβ (ω) =
(R,F ) ααβ (ω) = −
2 e2 N 2 (a|Rˆ α |j j |Fˆnβ |a), me ¯ j =a ωj a ωj a −ω2 (27)
(P ,P ) ααβ (ω) =
2 e2 (a|Pˆα |j j |Pˆβ |a), m2e ¯ j =a ωj a ωj2a −ω2 (28)
(P ,F ) (ω) = ααβ
2 e2 N (a|Pˆα |j j |Fˆnβ |a), m2e ¯ j =a ωj2a ωj2a −ω2 (29)
(F,F ) (ω) = ααβ
2 e2 N N 2 (a|Fˆnα |j j |Fˆnβ |a). 3 2 me ¯ j =a ωj a ωj a −ω2 (30)
According to Eqs. (21) and (24), the trace of the κ (P ,L) and κ (F,K) tensors calculated via gaugeless basis sets is invariant in a translation of coordinate system, that is, T r[κ (P ,L) (r )] = T r[κ (P ,L) (r )], T r[κ (F,K) (r )] = T r[κ (F,K) (r )].
(31)
If the hypervirial theorems, Eqs. (10) and (11), are obeyed, for instance, in the case of complete basis sets, corresponding to that of optimal variational electronic wavefunctions within the algebraic approximation,17 all these definitions of dynamic polarizability would provide the same numerical response. Thus, within the true coupled Hartree-Fock (CHF) approach,17 equivalent to the random-phase approximation (RPA),19 the definitions (5) and (12)–(16), and (25)– (30) would give identical results. In actual self-consistent field (SCF) calculations adopting truncated basis sets, the closeness of the values obtained in different pictures will give a measure of basis set completeness and ability to represent a given operator in different gauges. In particular, in SCF calculations, the (R, R) formalism for static EDP yields diagonal components approximating the exact Hartree-Fock values from below.20 Therefore, for a given molecular geometry, the largest results are the best ones. In quantum chemical calculations, London atomic orbitals (LAO),21 also referred to as gauge including atomic orbitals (GIAO),22 are currently used6–8 to ensure origin independence of theoretical estimates of Tr[κ ]. At any rate, the individual components, diagonal and off-diagonal, of the MEMDP tensor computed via a LAO basis set are origin tensor in the dependent.8 Nonetheless, the trace of the καβ dipole velocity/angular momentum gauge is invariant in any approximate calculation, which makes the (P, L) formalism useful if only gaugeless basis sets are available.13, 14 Such a procedure provides an interesting alternative to using LAO basis sets.23 In contrast to the Hartree-Fock, multi-configurational self consistent field and density functional theory methodologies, the inclusion of London orbitals does not eliminate the problem of origin dependence in coupled cluster calculations.24 Equations (19)–(24) and (25)–(30) provide a clue to define origin-independent diagonal components of MEMDP in computations using finite basis sets of gaugeless functions. The electric polarizabilities calculated within the algebraic approximation allowing for the dipole velocity, Eq. (28), and dipole acceleration, Eq. (30), are symmetric tensors with real eigenvalues and orthogonal eigenvectors. Therefore, within the principal axis system of either α (P ,P ) or α (F,F ) , the second addendum on the rhs of Eqs. (21) or (24), respectively, vanishes. The same statement applies to the ideal case of complete basis sets, optimal variational wavefunctions,17 and, all the more, exact wavefunctions. Consistently with this quantum-mechanical result, an experimental determination of the diagonal components of the MEMDP tensor becomes possible within the principal axis system of the EDP, for a given frequency ω. The intrinsic symmetry α αβ = α βα may be violated in finite basis set calculations of polarizabilities in mixed gauges, Eqs. (26), (27), and (29). Accordingly, the eigenvectors of
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the polarizability matrix computed within mixed pictures may be non-orthogonal, so that translationally invariant diagonal components of MEMDP tensors (12), (14), and (15) cannot in principle be adequately defined. III. CALCULATIONS
The theoretical procedure outlined in Sec. II was applied to calculate origin independent diagonal components of MEMDP in the 1,2-M enantiomer of the C4 H4 O2 dioxin molecule, at the RPA level of accuracy, using the SYSMO code.25 The optimized molecular geometry was taken from previous papers.26, 27 An extended basis set of 662 freely varying Gaussian functions was employed to calculate near Hartree-Fock EDPs in different gauges, according to Eqs. (25)–(30). The 13s/8p substrata of the basis sets for C and O, and 8s for H, from the van Duijneveldt compilation,28 were used in the present study. The polarization functions consist of two sets of 2p’s with exponents 1512.9 and 355.1, plus 5 sets of Cartesian 3d’s and 2 sets of 4f functions from Dunning29 for carbon. The exponents of the 2p functions for oxygen are 3694.545 and 859.598, those for the 5 sets of Cartesian 3d’s and 2 sets of 4f functions are from Dunning.29 The sets of four 2p and one 3d functions for hydrogen are also from Dunning.29 These basis sets are especially meant to describe effects taking place in the vicinity of the nuclei, as well as in medium and tail regions of the molecular domain.30 Therefore, they are expected to be quite good for MEMDP too, but possibly less suitable to predict response properties within formalisms involving the force and torque operators, Eqs. (12), (14)–(16), (27), (29), and (30). ˆN ˆN Since F n , Eq. (8), and K n (r 0 ), Eq. (9), weigh the electron distribution in regions close to the nuclei, polarized basis sets constructed by different criteria, e.g., to predict infrared intensities from nuclear electric shielding tensors,31–33 may be adopted for EDP and MEMDP within formalisms involving force and torque operators. The Sadlej compilations34–36 could alternatively be used to predict EDP and MEMDP within dipole length and velocity pictures. On the other hand, attempts at using the recipes recommended by Wolinski et al.32 for the dioxin molecule were frustrated by the problem of quasi-linear dependence of the polarized basis set. Other numerical tests have shown that polarized basis sets32 cannot really represent the MEMDP tensor in the velocity gauge, in agreement with previous findings (see Table 2 of Pedersen et al.23 ). At any rate, investigating these aspects lies far beyond the aims of the present research. The 3 × 3 eigenvalue equation for the dynamic EDP in the (P, P) picture, Eq. (28), for a given frequency ω, α (P ,P ) U = Uλα ,
(32)
is solved, then the similarity transformation (P ,L) κD = U † κ (P ,L) U
(33)
of MEMDP within the (P, L) gauge, Eq. (13), is performed.37 We recall that, within such a gauge, the trace of the κ (P ,L) tensor calculated via gaugeless basis sets, left invariant by
TABLE I. Dipole polarizability tensor of the cyclic 1,2-M enantiomer of the dioxin molecule, C4 H4 O2 , in a.u.a Formalism/ Comp. (R, R) (P, R) (P, P) (F, R) (F, P) (F, F)
xx
yy
zz
xy
yx
av
38.065 38.022 37.981 37.962 37.920 37.890
66.045 66.034 66.025 66.183 66.174 66.332
58.321 58.295 58.270 58.301 58.276 58.283
1.7432 1.7321 1.7316 1.9319 1.9317 1.9140
1.7432 1.7430 1.7316 1.7182 1.7066 1.9140
54.144 54.117 54.092 54.149 54.123 54.168
The conversion factor to SI units is 1 a.u. of electric dipole polarizability e2 a02 /Eh = 1.648 777 274 × 10−41 C2 m2 J−1 ≡ F m2 , from CODATA recommended values of the fundamental physical constants.43 ω = 0.0345439 a.u. a
TABLE II. Optical activity tensor of the cyclic 1,2-M enantiomer of the dioxin molecule, C4 H4 O2 , in a.u.a Formalism/ Comp. xx × 10−1 yy × 10−2 zz × 10−2 xy × 10−1 (R, L) (P, L) (F, L) (R, K) (P, K) (F, K)
− 1.7670 − 1.7909 − 0.8511 − 1.7633 − 1.7871 − 0.8492
3.4798 3.4958 2.5914 4.1289 4.1462 3.2627
− 9.5388 − 9.5706 − 9.5828 − 9.5890 − 9.6208 − 9.6316
− 5.5933 − 5.5780 − 5.7234 − 5.5912 − 5.5763 − 5.7130
yx
av × 10−2
1.6663 1.6668 1.7327 1.6630 1.6636 1.7292
− 7.9095 − 7.9945 − 5.1674 − 7.6977 − 7.7820 − 4.9535
a
The conversion factor to SI units is 1 a.u. of mixed electric-magnetic dipole polarizability e2 a03 /¯ = 3.607 015 64 × 10−35 m C T−1 ≡ m kg−1 s C2 , from CODATA recommended values of the fundamental physical constants.43 ω = 0.0345439 a.u.
TABLE III. Eigenvectors of the dipole polarizability tensor within the (P, P) gauge.a
x y z a
x
y
z
0.9981135 − 0.0613962 0.0
− 0.0613962 − 0.9981135 0.0
0.0 0.0 1.0000000
ω = 0.0345439 a.u.
TABLE IV. Eigenvectors of the dipole polarizability tensor within the (F, F) gauge.a
x y z a
x
y
z
0.9977635 − 0.0668434 0.0
− 0.0668434 − 0.9977635 0.0
0.0 0.0 1.0000000
ω = 0.0345439 a.u.
TABLE V. Dipole polarizability tensor within the (P, P) and (F, F) gauges of the cyclic 1,2-M enantiomer of the dioxin molecule, C4 H4 O2 , in the corresponding principal axes systems, in a.u.a Comp.
(P, P)
(F, F)
xx yy zz xy yx av
37.874 66.131 58.270 0.0 0.0 54.092
37.762 66.460 58.283 0.0 0.0 54.168
a
ω = 0.0345439 a.u.
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TABLE VI. Optical activity tensors of the cyclic 1,2-M enantiomer of the dioxin molecule, C4 H4 O2 , within the (P, L) and (F, K) representations, in the corresponding principal axes systems of the (P, P) and (F, F) dipole polarizabilities, in a.u.a Comp. xx yy zz xy yx av a
(P, L)
(F, K)
− 0.246240 0.102111 − 0.095707 0.575101 − 1.649507 − 0.079945
− 0.161616 0.109325 − 0.096316 0.584316 − 1.716174 − 0.049535
The diagonal components are origin independent. ω = 0.0345439 a.u.
Eq. (33), is origin independent according to Eqs. (21) and (31). The principal axis system is defined by the eigenvectors U, that is, the new coordinates of nucleus I are RI,D = U RI .
(34)
(P ,L) components of the κ D (P ,P )
The diagonal tensor, referred to , are origin independent acthe principal axis system of α cording to Eq. (21). An analogous viable procedure is applied within the acceleration/torque formalism, employing κ (F,K) , Eq. (16), α (F,F ) , Eq. (30), and relationships analogous to Eqs. (32)–(34). The eigenvectors used to define the principal axes system of EDP displayed in Tables III and IV for the (P, L) and (F, K) pictures are satisfactorily close to one another. The results obtained, reported in Tables I–VI, document the practicality of the approach discussed above. In particular, the overall agreement among results from length, velocity, and acceleration pictures gives a benchmark of accuracy of the present calculations. IV. CONCLUDING REMARKS
The results reported in Tables I–VI show that the large basis sets used for C, O, and H are virtually complete, at least as regards accurate representation of the electric and magnetic dipole operators. In fact, they provide numerical predictions useful to estimate near Hartree-Fock EDP and MEMDP for the 1,2-M enantiomer of the dioxin molecule, C4 H4 O2 . The agreement among values calculated within different gauges is remarkable, some discrepancies being observed only for MEMDP within the acceleration picture. In fact, see Table II, the largest diagonal element of the MEMDP tensor is reduced by a factor of two in the (F, L) and the (F, K) pictures compared to the other representations. Analogously, the originindependent (F, K) results in the third column of Table VI are comparatively less accurate than the (P, L). Therefore, the diagonal components of the MEMDP tensor reported in the (P, L) column of Table VI can be considered origin-independent measurable properties. Although the numerical test is limited to a single molecule, we can reasonably argue that the theoretical procedure proposed in the present study can be applied to define translationally invariant diagonal components of the optical activity tensor for any chiral molecule. An experimental estimate of these diagonal components can actually be arrived at, for any value of the an-
gular frequency ω, within the reference frame defined by the principal axis system of the dynamic electric polarizability at the same frequency. As regards the experimental determination of the diagonal components of the κ tensor, the most promising approach may be that of differential Rayleigh scattering of left- and right-circularly polarized light, see Bogaard et al.,38 and references therein. In fact, nematic liquid crystal solvents, successfully used to investigate the effect of orientation on nuclear magnetic resonance spectral parameters, may be unsuitable for chiral species, since optically active solutes convert nematic mesophases into the cholesteric form.38 The experimental electric polarizabilities at the same frequency can be obtained by well-documented methods. Average values are measurable from the refractive index of a low-pressure gas.39, 40 A combination of average values with light scattering data38 yields the anisotropy and the principal components of the dynamic polarizability for molecules with a threefold or higher symmetry axis. The optical activity of uniaxial crystals or of oriented molecules can also be considered.38, 41, 42 ACKNOWLEDGMENTS
Financial support to the present research from the Italian MIUR (Ministero dell’Istruzione, Università e Ricerca), via PRIN 2009 funds, is gratefully acknowledged. 1 E.
U. Condon, Rev. Mod. Phys. 9, 432 (1937). J. Caldwell and H. Eyring, Adv. Quantum Chem. 6, 143 (1972). 3 A. D. Buckingham, Adv. Chem. Phys. 12, 107 (1967). 4 A. D. Buckingham and P. J. Stiles, Acc. Chem. Res. 7, 258 (1974). 5 The names “gyration,” used by Landau and Lifshitz,41 and “gyration polarizability,” used by Buckingham and Stiles4 and adopted here for brevity, are not universally accepted and may cause confusion. In physics, the gyration tensor is also a tensor that describes the second moments of position of a collection of particles (see W. Mattice and U. Suter, Conformational Theory of Large Molecules (Wiley Interscience, New York, 1994)). The denomination “mixed electric-magnetic dipole polarizability” (MEMDP)6–8 may possibly be preferable. 6 J. R. Cheeseman, M. J. Frisch, F. J. Devlin, and P. J. Stephens, J. Phys. Chem. A 104, 1039 (2000). 7 P. J. Stephens, F. J. Devlin, J. R. Cheeseman, and M. J. Frisch, J. Phys. Chem. A 105, 5356 (2001). 8 K. Ruud and T. Helgaker, Chem. Phys. Lett. 352, 533 (2002). 9 W. Moffitt, J. Chem. Phys. 25, 467 (1956). 10 A. Moscowitz, Modern Quantum Chemistry, edited by O. Sinanoglu (Academic Press, New York, 1965), Vol. 3. 11 A. E. Hansen and T. D. Bouman, Adv. Chem. Phys. 44, 545 (1980). 12 P. Lazzeretti, Adv. Chem. Phys. 75, 507 (1989). 13 P. Lazzeretti, “Electric and magnetic properties of molecules,” in Handbook of Molecular Physics and Quantum Chemistry. Part 1 (John Wiley and Sons, Ltd., Chichester, 2003), Vol. 3, Chapter 3, pp. 53–145. See Eqs. (857), (866), (943), and the discussion after Eq. (946). 14 A. Ligabue, P. Lazzeretti, M. P. B. Varela, and M. B. Ferraro, J. Chem. Phys. 116, 6427 (2002). 15 M. C. Caputo, S. Coriani, S. Pelloni, and P. Lazzeretti, Mol. Phys. 111, 1405 (2013). 16 L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University Press, Cambridge, 1982). 17 S. T. Epstein, The Variation Method in Quantum Chemistry (Academic Press, New York, 1974). 18 P. Lazzeretti and R. Zanasi, J. Chem. Phys. 87, 472 (1987). 19 P. Jørgensen and J. Simons, Second Quantization-Based Method in Quantum Chemistry (Academic Press, New York, 1981). 20 R. Moccia, Chem. Phys. Lett. 5, 265 (1970). 21 F. London, J. Phys. Radium 8, 397 (1937). 22 A. E. Hansen and T. D. Bouman, J. Chem. Phys. 82, 5035 (1985), this is the first reference in which the reinterpretation of the GIAO acronym 2 D.
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for gauge-including-atomic-orbitals has been proposed, see footnote 6, p. 5047. 23 T. B. Pedersen, H. Koch, L. Boman, and A. M. Sánchez de Merás, Chem. Phys. Lett. 393, 319 (2004). 24 K. Ruud, P. J. Stephens, F. J. Devlin, P. R. Taylor, J. R. Cheeseman, and M. J. Frisch, Chem. Phys. Lett. 373, 606 (2003). 25 P. Lazzeretti, M. Malagoli, and R. Zanasi, Technical report on project “Sistemi informatici e calcolo parallelo,” Research Report 1/67, CNR, 1991. 26 S. Pelloni, F. Faglioni, and P. Lazzeretti, Rendiconti Lincei 24, 283 (2013). 27 S. Pelloni and P. Lazzeretti, Mol. Phys. 111, 2387 (2013). 28 F. B. van Duijneveldt, “Gaussian basis sets for the atoms H-Ne for use in molecular calculations,” Research Report RJ 945, IBM, 1971. 29 T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 (1989). 30 S. Pelloni and P. Lazzeretti, Int. J. Quantum Chem. 111, 356 (2011). 31 B. O. Roos and A. J. Sadlej, Chem. Phys. 94, 43 (1985). 32 K. Wolinski, B. O. Roos, and A. J. Sadlej, Theor. Chim. Acta 68, 431 (1985). 33 P. Lazzeretti and R. Zanasi, J. Chem. Phys. 83, 1218 (1985). 34 A. J. Sadlej, Collect. Czech. Chem. Commun. 53, 1995 (1988).
J. Chem. Phys. 140, 074105 (2014) 35 A.
Sadlej, Theor. Chim. Acta 79, 123 (1991). Sadlej, Theor. Chim. Acta 81, 45 (1991). 37 The unitary eigenvectors of Hermitian matrices are defined apart from a unitary transformation, those of symmetric matrices, like EDP, except for a ± sign. Therefore, in the most general case, the U matrix corresponds to a rotoreflection, then the sign of its columns should be adjusted to describe a proper rotation. 38 M. P. Bogaard, A. D. Buckingham, R. K. Pierens, and A. H. White, J. Chem. Soc., Faraday Trans. 1 74, 3008 (1978). 39 H. E. Watson and K. L. Ramaswamy, Proc. R. Soc. London 156, 144 (1936). 40 M. P. Bogaard and B. J. Orr, “Electric dipole polarisabilities of atoms and molecules,” in International Review of Science. Molecular Structure and Properties, Physical Chemistry Series Two Vol. 2, edited by A. D. Buckingham (Butterworths, London, 1975), pp. 149–194. 41 L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, Oxford, 1981). 42 A. D. Buckingham and M. B. Dunn, J. Chem. Soc. A 1971, 1988. 43 P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 77, 1 (2005). 36 A.
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THE JOURNAL OF CHEMICAL PHYSICS 140, 074105 (2014)
On the determination of the diagonal components of the optical activity tensor in chiral molecules Stefano Pelloni and Paolo Lazzerettia) Dipartimento di Scienze Chimiche e Geologiche, Università degli Studi di Modena e Reggio Emilia, Via Campi 183, 41100 Modena, Italy
(Received 28 December 2013; accepted 29 January 2014; published online 19 February 2014) It is shown that the diagonal components of the mixed electric-magnetic dipole polarizability tensor, used to rationalize the optical rotatory power of chiral molecules, are origin independent, if they are referred to the coordinate system defined by the eigenvectors of the dynamic electric dipole polarizability, for a given value ω of the frequency of a monochromatic wave impinging on an ordered sample. Within this reference frame, the individual diagonal components of the mixed electric-magnetic dipole polarizability are separately measurable properties. The theoretical method is applied via a test calculation to the cyclic 1,2-M enantiomer of the dioxin molecule, using a large Gaussian basis set to estimate near Hartree-Fock values within a series of dipole length, velocity, and acceleration representations. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4865229] I. INTRODUCTION
The optical rotatory power (ORP) is the property of chiral molecules in an isotropic medium to rotate the plane of polarization of a beam of linearly polarized light.1, 2 The rotation angle is rationalized via a pseudoscalar, that is, the mean value (one third the trace) of the optical gyration polarizability,3–5 a non-symmetric second-rank axial tensor, also referred to as optical activity tensor, and “mixed electric-magnetic dipole polarizability” (MEMDP),6–8 odd under parity P and even under time reversal T. The components of the MEMDP tensor, and the related pseudoscalar, have the same magnitude but opposite sign in the enantiomers of a chiral species. This pseudoscalar is in general smaller than any of the three diagonal components, as they have different sign. It vanishes in optically inactive molecules, which nonetheless may be characterized by individual non-zero off-diagonal components of the MEMDP tensor. Only the trace of this tensor is independent of the origin of the laboratory coordinate system, and it is the sole measurable quantity in a disordered medium containing freely tumbling optically active molecules. The separate components, diagonal and off-diagonal, depend on the origin,9–15 and consequently they could not, in principle, be experimentally determined in a molecular crystal. However, diagonal components invariant under a translation of coordinate system can be formally defined allowing for the results of previous papers.12–15 The scope of the present work is to show that, for any frequency ω of the incident plane polarized light, the diagonal components of the optical gyration tensor referred to the principal axis system of the electric dipole polarizability (EDP) are independent of the origin. Therefore, they can be calculated as sharply defined properties of optically active molecules, and they are by all means measurable in ordered phase.
A compact self-contained account of the basic theory needed for a sensible discussion of this point is presented hereafter in Sec. II. The results of calculations carried out for C4 H4 O2 , the cyclic 1,2-M enantiomer of the dioxin molecule, are given in Sec. III, and a summary is reported in the concluding section, Sec. IV.
II. THE MEMDP TENSOR WITHIN DIPOLE LENGTH, VELOCITY, AND ACCELERATION GAUGES
We use the notation adopted in previous papers.12–15 For a molecule with n electrons and N nuclei, charge, mass, position with respect to an arbitrary origin, canonical, and angular momentum of the ith electron are indicated by −e, me , r i , pˆ i , lˆi = r i × pˆ i , i = 1, 2 . . . n. Corresponding quantities for nucleus I are indicated by ZI e, MI , RI . Capital letters denote total electronic operators, e.g., ˆ = R
Pˆ =
ri,
i=1
n
pˆ i ,
Lˆ =
i=1
n
lˆi ,
i=1
etc. Throughout this article we use SI units and the Einstein convention of summing over two repeated Greek subscripts. The operators for the electric and magnetic dipole moment of electrons are μˆ α = −eRˆ α
(1)
and ˆα = − m
e ˆ Lα , 2me
(2)
respectively. The multiplicative operator for the electric field of electron i on nucleus I is defined by ˆ iI = E
a) E-mail: [email protected]
0021-9606/2014/140(7)/074105/6/$30.00
n
140, 074105-1
r i − RI e , 4π 0 |r i − RI |3
(3)
© 2014 AIP Publishing LLC
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Allowing for the hypervirial relationships,12, 17
and the operator for the total n-electron field by ˆ nI = E
n
ˆ iI . E
(4)
a|Rˆ α |j =
i=1
(R,L) καβ (ω)
2ω 1 ˆ β |a), (5) =− (a|μˆ α |j j |m ¯ j =a ωj2a − ω2
where takes the imaginary part of the term within brackets, ωj a =
1 (0) Ej − Ea(0) ¯
(6)
are the natural transition frequencies of the molecule in the reference state |a(0) ≡ |a, with energy eigenvalue Ea(0) , excited state energies Ej(0) , determined by solving the Schrödinger equation for the unperturbed Hamiltonian Hˆ (0) , and ω is the angular frequency of a monochromatic wave impinging on the molecule. ˆ and m ˆ are, respectively, a polar and an axial vecAs μ tor, the trace of the tensor defined by Eq. (5), which is related to quantities experimentally accessible,16 is a pseudoscalar, changing sign under inversion of the coordinate system. Equation (5) defines a second-order property in the length(RL) (ω) ≡ καβ (ω). At any angular momentum picture, i.e., καβ rate, owing to the peculiar features of quantum mechanics, one is free to choose different representations, or gauges, via canonical transformations of the Hamiltonian. Alternative definitions for the tensor introduced via expression (5) can be immediately arrived at by Ehrenfest off-diagonal hypervirial relations,17 allowing for velocity, force (or acceleration), and torque gauges.12 In an unperturbed molecule, the electronic operators for ˆN ˆN the total force, F n , and torque, K n (r 0 ) about the origin r 0 , acting on the electron cloud, contain only contributions from the nuclei, as total electron-electron force and torque cancel out by action equals reaction. Therefore, 1 ˆ i ˆ (0) ˆ P, [H , R] = ¯ me N n i ˆ (0) ˆ e2 r i − R I ˆN [H , P] = F = − Z , I n ¯ 4π 0 I =1 |r i − RI |3 i=1
(7)
(8)
i ˆ (0) ˆ N [H , L(r 0 )] = Kˆ n (r 0 ) ¯ =
N n e2 r i − R I ZI × (RI − r 0 ). 4π 0 I =1 |r i − RI |3 i=1
(9) The operators for total force and torque can also be conveniently rewritten12 in terms of the electric field, Eq. (3).
N e −2 ωj a ZI a|Eˆ Inα |j , me I =1
(10)
ˆN a|Lˆ α |j = iωj−1 a a|Knα |j ,
(11)
=
The ORP of a chiral molecule is customarily accounted for via the MEMDP tensor within dipole length-angular momentum formalism (R, L),12
i −1 1 N ωj a a|Pˆα |j = − ωj−2 a|Fˆnα |j me me a
five alternative expressions are found for the MEMDP tensor,12–14, 18 (RK) καβ (ω) =
2ω e2 N 2 a|Rˆ α |j j |Kˆ nβ |a , 2 2me ¯ j =a ωj a ωj a −ω (12)
(P L) καβ (ω) = −
2ω e2 2 a|Pˆα |j j |Lˆ β |a , 2 2 2me ¯ j =a ωj a ωj a −ω (13)
(P K) καβ (ω) = −
2ω e2 N 2 a|Pˆα |j j |Kˆ nβ |a , 2 2 2me ¯ j =a ωj a ωj a −ω2 (14)
(F L) καβ (ω) =
2ω e2 N a|Fˆnα |j j |Lˆ β |a , 2m2e ¯ j =a ωj2a ωj2a −ω2 (15)
(F K) καβ (ω) = −
2ω e2 N N a|Fˆnα |j j |Kˆ nβ |a . 2m2e ¯ j =a ωj3a ωj2a −ω2 (16)
(ω) is unafOf course the axial tensor character of καβ N N fected by the change of picture, as Fˆnα and Kˆ nα are, respectively, polar and axial vectors. All these definitions are equivalent in quantum mechanics, as a consequence of the invariance of the theory in a canonical transformation. However, the estimates of MEMDP are numerically the same only if the eigenfunctions |a and |j are the exact eigenstates to a model Hamiltonian, or optimal variational wavefunctions,17 a condition that is hardly met in practice. In a calculation based on the algebraic approximation, values arrived at by relationships (5) and (12)–(16) may be appreciably different: their numerical agreement gives a benchmark of basis set completeness and an a priori quality criterion of computational accuracy. In other words, if MEMDP values calculated via Eqs. (5) and (12)–(16) are close to one another, one can reasonably assume that they specify the limit for a given approximation. In a translation of origin of the coordinate frame,
r = r + d,
(17)
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the MEMDP tensor changes according to a relationship involving α, the dynamic EDP,12 καβ (r ) = καβ (r ) −
ω βγ δ ααγ dδ , 2
Tr[κ (r )] = Tr[κ (r )]. (18)
These equations would be exactly fulfilled in ideal calculations, e.g., using exact wavefunctions. In approaches employing truncated basis sets, the change of the components for different pictures, Eqs. (5) and (12)–(16), is obtained by (R,L) (R,L) καβ (r ) = καβ (r ) −
ω (R,P ) βγ δ ααγ dδ , 2
(19)
(R,K) (R,K) καβ (r ) = καβ (r ) −
ω (R,F ) βγ δ ααγ dδ , 2
(20)
(P ,L) (P ,L) (r ) = καβ (r ) − καβ
ω (P ,P ) βγ δ ααγ dδ , 2
(21)
(P ,K) (P ,K) καβ (r ) = καβ (r ) −
ω (P ,F ) βγ δ ααγ dδ , 2
(22)
(F,L) (F,L) (r ) = καβ (r ) − καβ
ω (F,P ) βγ δ ααγ dδ , 2
(23)
(F,K) (F,K) (r ) = καβ (r ) − καβ
ω (F,F ) βγ δ ααγ dδ , 2
(24)
where the frequency-dependent EDPs in different pictures are e2 2ωj a (a|Rˆ α |j j |Rˆ β |a), ¯ j =a ωj2a − ω2
(R,R) ααβ (ω) =
(25)
2 e2 (a|Rˆ α |j j |Pˆβ |a), (26) 2 me ¯ j =a ωj a − ω2
(R,P ) ααβ (ω) =
(R,F ) ααβ (ω) = −
2 e2 N 2 (a|Rˆ α |j j |Fˆnβ |a), me ¯ j =a ωj a ωj a −ω2 (27)
(P ,P ) ααβ (ω) =
2 e2 (a|Pˆα |j j |Pˆβ |a), m2e ¯ j =a ωj a ωj2a −ω2 (28)
(P ,F ) (ω) = ααβ
2 e2 N (a|Pˆα |j j |Fˆnβ |a), m2e ¯ j =a ωj2a ωj2a −ω2 (29)
(F,F ) (ω) = ααβ
2 e2 N N 2 (a|Fˆnα |j j |Fˆnβ |a). 3 2 me ¯ j =a ωj a ωj a −ω2 (30)
According to Eqs. (21) and (24), the trace of the κ (P ,L) and κ (F,K) tensors calculated via gaugeless basis sets is invariant in a translation of coordinate system, that is, T r[κ (P ,L) (r )] = T r[κ (P ,L) (r )], T r[κ (F,K) (r )] = T r[κ (F,K) (r )].
(31)
If the hypervirial theorems, Eqs. (10) and (11), are obeyed, for instance, in the case of complete basis sets, corresponding to that of optimal variational electronic wavefunctions within the algebraic approximation,17 all these definitions of dynamic polarizability would provide the same numerical response. Thus, within the true coupled Hartree-Fock (CHF) approach,17 equivalent to the random-phase approximation (RPA),19 the definitions (5) and (12)–(16), and (25)– (30) would give identical results. In actual self-consistent field (SCF) calculations adopting truncated basis sets, the closeness of the values obtained in different pictures will give a measure of basis set completeness and ability to represent a given operator in different gauges. In particular, in SCF calculations, the (R, R) formalism for static EDP yields diagonal components approximating the exact Hartree-Fock values from below.20 Therefore, for a given molecular geometry, the largest results are the best ones. In quantum chemical calculations, London atomic orbitals (LAO),21 also referred to as gauge including atomic orbitals (GIAO),22 are currently used6–8 to ensure origin independence of theoretical estimates of Tr[κ ]. At any rate, the individual components, diagonal and off-diagonal, of the MEMDP tensor computed via a LAO basis set are origin tensor in the dependent.8 Nonetheless, the trace of the καβ dipole velocity/angular momentum gauge is invariant in any approximate calculation, which makes the (P, L) formalism useful if only gaugeless basis sets are available.13, 14 Such a procedure provides an interesting alternative to using LAO basis sets.23 In contrast to the Hartree-Fock, multi-configurational self consistent field and density functional theory methodologies, the inclusion of London orbitals does not eliminate the problem of origin dependence in coupled cluster calculations.24 Equations (19)–(24) and (25)–(30) provide a clue to define origin-independent diagonal components of MEMDP in computations using finite basis sets of gaugeless functions. The electric polarizabilities calculated within the algebraic approximation allowing for the dipole velocity, Eq. (28), and dipole acceleration, Eq. (30), are symmetric tensors with real eigenvalues and orthogonal eigenvectors. Therefore, within the principal axis system of either α (P ,P ) or α (F,F ) , the second addendum on the rhs of Eqs. (21) or (24), respectively, vanishes. The same statement applies to the ideal case of complete basis sets, optimal variational wavefunctions,17 and, all the more, exact wavefunctions. Consistently with this quantum-mechanical result, an experimental determination of the diagonal components of the MEMDP tensor becomes possible within the principal axis system of the EDP, for a given frequency ω. The intrinsic symmetry α αβ = α βα may be violated in finite basis set calculations of polarizabilities in mixed gauges, Eqs. (26), (27), and (29). Accordingly, the eigenvectors of
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the polarizability matrix computed within mixed pictures may be non-orthogonal, so that translationally invariant diagonal components of MEMDP tensors (12), (14), and (15) cannot in principle be adequately defined. III. CALCULATIONS
The theoretical procedure outlined in Sec. II was applied to calculate origin independent diagonal components of MEMDP in the 1,2-M enantiomer of the C4 H4 O2 dioxin molecule, at the RPA level of accuracy, using the SYSMO code.25 The optimized molecular geometry was taken from previous papers.26, 27 An extended basis set of 662 freely varying Gaussian functions was employed to calculate near Hartree-Fock EDPs in different gauges, according to Eqs. (25)–(30). The 13s/8p substrata of the basis sets for C and O, and 8s for H, from the van Duijneveldt compilation,28 were used in the present study. The polarization functions consist of two sets of 2p’s with exponents 1512.9 and 355.1, plus 5 sets of Cartesian 3d’s and 2 sets of 4f functions from Dunning29 for carbon. The exponents of the 2p functions for oxygen are 3694.545 and 859.598, those for the 5 sets of Cartesian 3d’s and 2 sets of 4f functions are from Dunning.29 The sets of four 2p and one 3d functions for hydrogen are also from Dunning.29 These basis sets are especially meant to describe effects taking place in the vicinity of the nuclei, as well as in medium and tail regions of the molecular domain.30 Therefore, they are expected to be quite good for MEMDP too, but possibly less suitable to predict response properties within formalisms involving the force and torque operators, Eqs. (12), (14)–(16), (27), (29), and (30). ˆN ˆN Since F n , Eq. (8), and K n (r 0 ), Eq. (9), weigh the electron distribution in regions close to the nuclei, polarized basis sets constructed by different criteria, e.g., to predict infrared intensities from nuclear electric shielding tensors,31–33 may be adopted for EDP and MEMDP within formalisms involving force and torque operators. The Sadlej compilations34–36 could alternatively be used to predict EDP and MEMDP within dipole length and velocity pictures. On the other hand, attempts at using the recipes recommended by Wolinski et al.32 for the dioxin molecule were frustrated by the problem of quasi-linear dependence of the polarized basis set. Other numerical tests have shown that polarized basis sets32 cannot really represent the MEMDP tensor in the velocity gauge, in agreement with previous findings (see Table 2 of Pedersen et al.23 ). At any rate, investigating these aspects lies far beyond the aims of the present research. The 3 × 3 eigenvalue equation for the dynamic EDP in the (P, P) picture, Eq. (28), for a given frequency ω, α (P ,P ) U = Uλα ,
(32)
is solved, then the similarity transformation (P ,L) κD = U † κ (P ,L) U
(33)
of MEMDP within the (P, L) gauge, Eq. (13), is performed.37 We recall that, within such a gauge, the trace of the κ (P ,L) tensor calculated via gaugeless basis sets, left invariant by
TABLE I. Dipole polarizability tensor of the cyclic 1,2-M enantiomer of the dioxin molecule, C4 H4 O2 , in a.u.a Formalism/ Comp. (R, R) (P, R) (P, P) (F, R) (F, P) (F, F)
xx
yy
zz
xy
yx
av
38.065 38.022 37.981 37.962 37.920 37.890
66.045 66.034 66.025 66.183 66.174 66.332
58.321 58.295 58.270 58.301 58.276 58.283
1.7432 1.7321 1.7316 1.9319 1.9317 1.9140
1.7432 1.7430 1.7316 1.7182 1.7066 1.9140
54.144 54.117 54.092 54.149 54.123 54.168
The conversion factor to SI units is 1 a.u. of electric dipole polarizability e2 a02 /Eh = 1.648 777 274 × 10−41 C2 m2 J−1 ≡ F m2 , from CODATA recommended values of the fundamental physical constants.43 ω = 0.0345439 a.u. a
TABLE II. Optical activity tensor of the cyclic 1,2-M enantiomer of the dioxin molecule, C4 H4 O2 , in a.u.a Formalism/ Comp. xx × 10−1 yy × 10−2 zz × 10−2 xy × 10−1 (R, L) (P, L) (F, L) (R, K) (P, K) (F, K)
− 1.7670 − 1.7909 − 0.8511 − 1.7633 − 1.7871 − 0.8492
3.4798 3.4958 2.5914 4.1289 4.1462 3.2627
− 9.5388 − 9.5706 − 9.5828 − 9.5890 − 9.6208 − 9.6316
− 5.5933 − 5.5780 − 5.7234 − 5.5912 − 5.5763 − 5.7130
yx
av × 10−2
1.6663 1.6668 1.7327 1.6630 1.6636 1.7292
− 7.9095 − 7.9945 − 5.1674 − 7.6977 − 7.7820 − 4.9535
a
The conversion factor to SI units is 1 a.u. of mixed electric-magnetic dipole polarizability e2 a03 /¯ = 3.607 015 64 × 10−35 m C T−1 ≡ m kg−1 s C2 , from CODATA recommended values of the fundamental physical constants.43 ω = 0.0345439 a.u.
TABLE III. Eigenvectors of the dipole polarizability tensor within the (P, P) gauge.a
x y z a
x
y
z
0.9981135 − 0.0613962 0.0
− 0.0613962 − 0.9981135 0.0
0.0 0.0 1.0000000
ω = 0.0345439 a.u.
TABLE IV. Eigenvectors of the dipole polarizability tensor within the (F, F) gauge.a
x y z a
x
y
z
0.9977635 − 0.0668434 0.0
− 0.0668434 − 0.9977635 0.0
0.0 0.0 1.0000000
ω = 0.0345439 a.u.
TABLE V. Dipole polarizability tensor within the (P, P) and (F, F) gauges of the cyclic 1,2-M enantiomer of the dioxin molecule, C4 H4 O2 , in the corresponding principal axes systems, in a.u.a Comp.
(P, P)
(F, F)
xx yy zz xy yx av
37.874 66.131 58.270 0.0 0.0 54.092
37.762 66.460 58.283 0.0 0.0 54.168
a
ω = 0.0345439 a.u.
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074105-5
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J. Chem. Phys. 140, 074105 (2014)
TABLE VI. Optical activity tensors of the cyclic 1,2-M enantiomer of the dioxin molecule, C4 H4 O2 , within the (P, L) and (F, K) representations, in the corresponding principal axes systems of the (P, P) and (F, F) dipole polarizabilities, in a.u.a Comp. xx yy zz xy yx av a
(P, L)
(F, K)
− 0.246240 0.102111 − 0.095707 0.575101 − 1.649507 − 0.079945
− 0.161616 0.109325 − 0.096316 0.584316 − 1.716174 − 0.049535
The diagonal components are origin independent. ω = 0.0345439 a.u.
Eq. (33), is origin independent according to Eqs. (21) and (31). The principal axis system is defined by the eigenvectors U, that is, the new coordinates of nucleus I are RI,D = U RI .
(34)
(P ,L) components of the κ D (P ,P )
The diagonal tensor, referred to , are origin independent acthe principal axis system of α cording to Eq. (21). An analogous viable procedure is applied within the acceleration/torque formalism, employing κ (F,K) , Eq. (16), α (F,F ) , Eq. (30), and relationships analogous to Eqs. (32)–(34). The eigenvectors used to define the principal axes system of EDP displayed in Tables III and IV for the (P, L) and (F, K) pictures are satisfactorily close to one another. The results obtained, reported in Tables I–VI, document the practicality of the approach discussed above. In particular, the overall agreement among results from length, velocity, and acceleration pictures gives a benchmark of accuracy of the present calculations. IV. CONCLUDING REMARKS
The results reported in Tables I–VI show that the large basis sets used for C, O, and H are virtually complete, at least as regards accurate representation of the electric and magnetic dipole operators. In fact, they provide numerical predictions useful to estimate near Hartree-Fock EDP and MEMDP for the 1,2-M enantiomer of the dioxin molecule, C4 H4 O2 . The agreement among values calculated within different gauges is remarkable, some discrepancies being observed only for MEMDP within the acceleration picture. In fact, see Table II, the largest diagonal element of the MEMDP tensor is reduced by a factor of two in the (F, L) and the (F, K) pictures compared to the other representations. Analogously, the originindependent (F, K) results in the third column of Table VI are comparatively less accurate than the (P, L). Therefore, the diagonal components of the MEMDP tensor reported in the (P, L) column of Table VI can be considered origin-independent measurable properties. Although the numerical test is limited to a single molecule, we can reasonably argue that the theoretical procedure proposed in the present study can be applied to define translationally invariant diagonal components of the optical activity tensor for any chiral molecule. An experimental estimate of these diagonal components can actually be arrived at, for any value of the an-
gular frequency ω, within the reference frame defined by the principal axis system of the dynamic electric polarizability at the same frequency. As regards the experimental determination of the diagonal components of the κ tensor, the most promising approach may be that of differential Rayleigh scattering of left- and right-circularly polarized light, see Bogaard et al.,38 and references therein. In fact, nematic liquid crystal solvents, successfully used to investigate the effect of orientation on nuclear magnetic resonance spectral parameters, may be unsuitable for chiral species, since optically active solutes convert nematic mesophases into the cholesteric form.38 The experimental electric polarizabilities at the same frequency can be obtained by well-documented methods. Average values are measurable from the refractive index of a low-pressure gas.39, 40 A combination of average values with light scattering data38 yields the anisotropy and the principal components of the dynamic polarizability for molecules with a threefold or higher symmetry axis. The optical activity of uniaxial crystals or of oriented molecules can also be considered.38, 41, 42 ACKNOWLEDGMENTS
Financial support to the present research from the Italian MIUR (Ministero dell’Istruzione, Università e Ricerca), via PRIN 2009 funds, is gratefully acknowledged. 1 E.
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