Relativistic fourcomponent calculations of Buckingham birefringence using London atomic orbitals
Abstract
We present the first relativistic study of the electricfieldgradient induced birefringence (Buckingham birefringence), with application to the series of molecules CX_{2} (X = O, S, Se, Te). A recently developed atomicorbitaldriven scheme for the calculation of timedependent molecular properties using one, two and fourcomponent relativistic wave functions (Bast et al. in Chem Phys 356:177, 2009) is extended to firstorder frequencydependent magneticfield perturbations, using London atomic orbitals to ensure gaugeorigin independent results and to improve basisset convergence. Calculations are presented at the Hartree–Fock and Kohn–Sham levels of theory and results for CO_{2} and CS_{2} are compared with previous highlevel coupledcluster calculations. Except for the heaviest member of the series, relativistic effects are small—in particular for the temperatureindependent contribution to the birefringence. By contrast, the effects of electron correlation are significant. However, the reliability of standard exchangecorrelation functionals in describing Buckingham birefringence remains unclear based on the comparison with highlevel coupledcluster singlesanddoubles calculations.
Keywords
Relativistic effects Buckingham birefringence Nonlinear properties Response theory1 Introduction
More than half a century ago, Buckingham [1, 2] proposed that measurements of the birefringence induced by a gradient of an external electric field on linearly polarized radiation passing through a sample of molecules could be used as a direct method to measure molecular quadrupole moments [3]. Today, this idea has evolved into an important technique for determining the molecular quadrupole moment, which is the leading electric multipole in nonpolar molecules and as such plays a major role in determining the structural and spectroscopic behavior of matter [4, 5, 6, 7].
The electricfieldgradient induced (linear) birefringence, also known as the Buckingham effect or Buckingham birefringence, is measured by the optical retardation, proportional to the anisotropy n _{ x } − n _{ y }, generated in the real part of the complex refractive index when linearly polarized radiation travels in the z direction through a fluid immersed in an electric field gradient with components in the xy plane. The phase difference is directly proportional to the optical path length and inversely proportional to the wave length of the impinging radiation. As a result of the interactions between the radiation field, the electric field gradient and the sample, the beam exiting the sample cell is elliptically polarized.
To lowest order in a perturbation expansion involving fields and molecular multipoles at constant pressure, Buckingham birefringence has two contributions. The first accounts for the thermal orientational effect of the electric field gradient on the freely rotating molecules and is inversely proportional to the temperature for a measurement carried out at constant pressure. This contribution depends on the electric quadrupole moment and the electricdipole polarizability tensors; for polar molecules, the mixed electricdipole–electricquadrupole and electricdipole–magneticdipole polarizabilities contribute in addition to the static molecular dipole moment [4]. The second, temperatureindependent contribution to Buckingham birefringence arises from the response of the molecular electronic system to the external field and depends on higherorder mixed response properties. Although typically smaller than the temperaturedependent contribution, it is the only nonvanishing contribution to Buckingham birefringence in fluids consisting of atoms and spherically symmetric molecules.
In recent years, we have contributed to the study of Buckingham birefringence by developing and applying computational techniques for the ab initio prediction of the effect—see Refs. [8, 9, 10, 11]. Since our first investigation of Buckingham birefringence in 1998 [12], which focused on the nonpolar molecules H_{2}, N_{2}, C_{2}H_{2} and CH_{4} and led to a revised experimental quadrupole moment of N_{2} [13], we have extended our interest to larger and more complex systems, including polar [14, 15] as well as nonpolar [16, 17, 18, 19, 20, 21, 22] molecules, for which the effect of electron correlation and the choice of electronicstructure model have been analyzed in detail. In Ref. [23], the Buckingham birefringence of solvated molecules was investigated, thereby assessing the ability of our approach to simulate the effect of the environment. Our studies of carbon monoxide, nitrous oxide and carbonyl sulfide [14, 15] have resolved a controversy regarding the semiclassical theory of Buckingham birefringence [4, 24], supporting [25, 26, 27] the original derivation by Buckingham and LonguetHiggins [4]. The dependence of Buckingham birefringence on the density for helium, neon and argon was analyzed in Ref. [28]. Most recently, we presented an approach to the study of Buckingham birefringence with perturbationdependent basis sets that ensures invariance of the results to the origin of the multipole expansion [22].
An unexplored aspect of Buckingham birefringence is the influence of relativity. In this communication, we study relativistic effects on the Buckingham birefringence for the series of molecules CX_{2} (X = O, S, Se, Te). For CO_{2}, CS_{2} and CSe_{2}, we compare our results with previous nonrelativistic ab initio [17] data and with experiment [29, 30, 31, 32, 33, 34]. The atomicorbitaldriven (AOdriven) scheme recently introduced by Bast et al. [35] for calculating timedependent molecular properties with one, two and fourcomponent relativistic methods is extended to firstorder frequencydependent magnetic perturbations with London atomic orbitals (LAOs) [36, 37, 38, 39, 40], thereby ensuring gaugeorigin independence of the calculated results. The present work can also be considered an extension of our recent analytic implementation of Buckingham birefringence [22] to Kohn–Sham (KS) densityfunctional theory (DFT). Results are presented at the Hartree–Fock (HF) and KS levels of theory using nonrelativistic and fourcomponent relativistic (reference) wave functions.
2 Theory
We here review the theory underlying the Buckinghambirefringence calculations presented in this paper. First, the important quantities of Buckingham birefringence are reviewed in Sect. 2.1. Following a discussion of our treatment of relativity in Sect. 2.2, we outline the response theory used to calculate Buckingham birefringence at the relativistic HF and KS levels of theory in Sect. 2.3.
2.1 Buckingham birefringence
As shown by Buckingham and LonguetHiggins [4], for polar fluids, where the quadrupole moment is origin dependent, Eq. 3 must be generalized to include a correction to the temperaturedependent part involving the molecular dipole moment and the mixed electricdipole–magneticdipole and electricdipole–electricquadrupole polarizabilities. As we only study nonpolar molecules in this work, we do not consider this generalization here, referring the interested reader to Refs. [4, 14, 22] for more information.
2.2 Relativistic Hamiltonians and selfconsistentfield wave functions
2.3 Atomicorbitalbasis Kohn–Sham densityfunctional response theory
The framework used here to calculate the response functions contributing to Buckingham birefringence has been presented in Refs. [60, 61], which describe a selfconsistentfield AObased response theory for time and perturbationdependent basis sets. Being formulated in the AO basis, the approach is transparent to the explicit form of the molecular Hamiltonian and to the parametrization of the selfconsistentfield wave function. We have previously utilized this feature to extend the approach to two and fourcomponent relativistic wave functions for the calculation of higherorder molecular properties involving oneelectron operators [35]. Here, this approach is extended further to include KS exchange–correlation (XC) contributions using perturbationdependent basis sets to first order.
The evaluation of the XC functional derivatives has previously been described in the context of perturbationindependent basis sets by Sałek et al. [67] and, in the specific case of magneticfield perturbed densities, by Krykunov et al. [68] and by Kjærgaard et al. [69]. A general strategy for the evaluation of higherorder perturbed XC energies and functionals is given in Ref. [60]. Extensions to include spindensity contributions have furthermore been described [70], also in conjunction with the use of perturbationdependent basis sets [61].
To develop an implementation suitable for higherorder mixedfield XC response contributions, we have found it convenient to evaluate the XC terms in Eq. 30 in two steps. First, we calculate the magneticfield derivatives n ^{ eb }, ∇ n ^{ eb }, and \((\nabla n \cdot \nabla n^e)^b\) of the perturbed density variables n ^{ e }, ∇n ^{ e }, and \((\nabla n \cdot \nabla n^e)\), where e represents an electricfield perturbation. Next, this set of magneticfield derivatives is used as input variables to conventional (possibly LAOunaware) response modules that form the required matrix elements without the need for additional programming. The functional derivatives needed for constructing the XC matrix elements are obtained using automatic differentiation [71].
Whereas our expression for the quasienergy derivatives has been formulated in the AO basis, we do not solve the linear sets of equations in this basis. Instead, we transform the righthand side of Eq. 47 to the fourcomponent spinor basis and use the linear response solver of Saue and Jensen [72]. In this manner, we ensure that the electronic Hessian remains diagonally dominant, thereby improving convergence. The transformation to the spinor basis also allows us to utilize quaternion algebra in the solution of the linear equations in Eq. 47, as described by Saue and Jensen [72, 73]. The response vectors obtained are subsequently transformed back to the AO basis and converted into the homogeneous components of the perturbed density matrix, which is used in the calculation of the response functions that determine the Buckingham birefringence. For details, we refer to the literature describing the various aspects of the procedure [35, 60, 72].
3 Computational details
All Buckinghambirefringence results have been obtained using a development version of the DIRAC program package [74]. The relativistic calculations have been carried out employing the fourcomponent DC Hamiltonian; for the nonrelativistic reference values, we have used the LévyLeblond Hamiltonian [53].
In addition to the HF method, we have employed the KS method with the LDA (SVWN5) [75, 76], BLYP [77, 78, 79], B3LYP [80, 81], PBE [82], and PBE0 [83] XC functionals. These are nonrelativistic functionals which, with the DC Hamiltonian, have been evaluated using relativistic densities and density gradients. We have employed the full derivatives of the functionals provided by the XCFun library [71, 84]. Spindensity contributions [70] to XC matrix elements have been ignored.
Experimental equilibrium geometries from the compilation in Ref. [85] were used for CO_{2} (R _{CO} = 2.19169a _{0}) and CS_{2} (R _{CS} = 2.93391a _{0}). For CSe_{2}, we have used the experimental bond length of R _{CSe} = 3.19993a _{0} reported in Ref. [86]. For CTe_{2}, we have calculated the bond length of R _{CTe} = 3.60056a _{0} using the Gaussian 09 package [87] with the Ahlrichs def2TZVPP basis [88] for C and Te in combination with the Stuttgart/Dresden 28electron effective core pseudopotential [89] and the B3LYP XC functional.
We have used the uaugccpVDZ and uaugccpVTZ (“u” meaning uncontracted) basis sets of Dunning [90, 91] for C, O and S, and the augmented allelectron uDZ and uTZ basis sets of Dyall [92, 93] for Se and Te. The smallcomponent basis for the DC calculations has been generated using UKB, with RKB imposed in the canonical orthonormalization step [45]. In the selfconsistentfield and response calculations, the smallcomponent twoelectron Coulomb integrals (SSSS) have been approximated using a pointcharge model [94]. A Gaussian charge distribution has been chosen as the nuclear model in the relativistic and nonrelativistic calculations, using the recommended values in Ref. [95].
4 Results
4.1 The temperatureindependent part of the Buckingham birefringence
The value of b(ω) calculated using the DC Hamiltonian (atomic units)


 uaugDZ  uaugDZ  uaugTZ  uaugTZ  daugQZ 

CGO  LAO  CGO  LAO  CGO  
CO_{2}  HF  −34.7  −46.7  −43.6  −48.3  
λ = 632.8 nm  LDA  −40.5  −56.3  −51.4  −58.9  
BLYP  −41.7  −59.1  −52.4  −61.5  
B3LYP  −39.2  −54.9  −49.5  −57.1  
PBE  −41.3  −57.7  −51.5  −59.9  
PBE0  −38.6  −53.0  −48.3  −55.0  
CCSD^{a}  −54.48  
exp^{b}  −160 ± 80  
exp^{c}  −100 ± 200  
CS_{2}  HF  −380.4  −467.6  −462.8  −482.5  
λ = 632.8 nm  LDA  −353.3  −455.5  −435.8  −464.1  
BLYP  −352.5  −474.2  −433.8  −476.1  
B3LYP  −355.6  −462.0  −436.3  −468.3  
PBE  −356.3  −460.6  −428.7  −462.4  
PBE0  −361.7  −450.7  −432.3  −456.2  
CCSD^{a}  −410.91  
exp^{b}  −1,200 ± 800  
CSe_{2}  HF  −661.8  −766.4  −754.4  −781.5  
λ = 632.8 nm  LDA  −603.8  −722.3  −695.4  −728.9  
BLYP  −590.6  −737.6  −682.1  −731.4  
B3LYP  −606.9  −733.3  −699.1  −737.5  
PBE  −614.7  −738.7  −692.2  −733.5  
PBE0  −633.6  −737.1  −707.3  −738.1  
exp^{d}  ≈0  
CTe_{2}  HF  −1,294.1  −1,404.1  −1,798.4  −1,839.4  
λ = 632.8 nm  LDA  −981.6  −1,127.9  −1,283.4  −1,331.9  
BLYP  −757.1  −938.6  −1,126.0  −1,196.9  
B3LYP  −1,441.3  −1,599.3  −1,690.8  −1,745.8  
PBE  −982.3  −1,137.5  −1,299.1  −1,358.9  
PBE0  −1,495.3  −1,626.6  −1,697.9  −1,743.3  
CTe_{2}  HF  −1,379.6  −1,507.0  −1,419.8  −1,461.9  
λ = 694.3 nm  LDA  −1,076.5  −1,223.7  −1,395.0  −1,442.3  
BLYP  −917.6  −1,101.4  −1,330.0  −1,399.4  
B3LYP  −939.1  −1,098.9  −1,268.7  −1,324.1  
PBE  −1,070.6  −1,227.7  −1,401.8  −1,460.4  
PBE0  −1,075.1  −1,207.9  −1,307.2  −1,353.0 
The first thing to note from Table 1 is the importance of introducing field dependence in the AOs—in particular, in the smallest uaugDZ basis. For CO_{2} and CS_{2}, the LAO field dependence induces changes as large as 25–30%, largely independent of the choice of XC functional. Going down in the CX_{2} series, the effect of field dependence decreases, being only 10–15% for CTe_{2} in the uaugDZ basis. In the larger uaugTZ basis, the effect of LAOs is smaller, being on average 5–10% for the entire series, the largest effect being once again observed for the lightest members of the series. The changes observed in the LAO results when the basis is increased from uaugDZ to uaugTZ are small, about 5% for CO_{2} and less than 1–2% for CS_{2}, CSe_{2} and CTe_{2}. The observed importance of using LAOs for rapid basisset convergence corroborates the findings of our recent nonrelativistic study [22]. Please note that the b(ω) term for the studied series CX_{2} (X = O, S, Se, Te) is gaugeorigin independent by symmetry.
Given that the relativistic correction to the temperatureindependent part of Buckingham birefringence is negligible for CO_{2} (vide infra), we can compare our CO_{2} results directly with the highlevel coupledcluster singlesanddoubles (CCSD) results of Ref. [17]. As this study employed the large daugccpVQZ basis, these CCSD results are expected to be reasonably close to the basisset limit. Prior to this work, there have been three Buckinghambirefringence studies using KS theory, but only with fieldindependent basis sets [18, 19, 20]. Having established the importance of LAOs in Ref. [22] and in Table 1, the qualities of different XC functionals can now be more reliably assessed.
As seen from Table 1, the effect of electron correlation on b(ω) is significant, with changes from HF theory to CCSD theory of about 13% for CO_{2} and 15% for CS_{2}. Interestingly, whereas electron correlation increases the magnitude of the temperatureindependent contribution to the Buckingham birefringence of CO_{2}, the opposite happens for CS_{2}. Moreover, the CO_{2} and CS_{2} results obtained with different XC functionals do not lead to any clear conclusions regarding their ability to capture the effect of electron correlation on b(ω). For CO_{2}, all functionals overestimate the effect of correlation; the hybrid functionals B3LYP and PBE0 perform best, the PBE0 value being very close to the CCSD value. For CS_{2}, all XC functionals perform poorly, typically recovering less than one third of the total correlation effect as calculated using CCSD theory, the PBE0 functional again providing the best KS result.
Because of the very large experimental error bars, comparison with experimental results does not provide a stringent test on the different computational methods—all calculated values fall comfortably within three standard deviations from the center of the experimental distribution for CO_{2} and CS_{2}. As discussed for linear birefringences elsewhere [8, 9, 10, 11], this difficulty arises from the extreme sensitivity of the infinitetemperature extrapolation performed on the experimental data to estimate b(ω). From comparison with experimentally derived data, it is therefore not possible to draw definite conclusions regarding the reliability of the various XC functionals for the mixed hyperpolarizabilities that determine the temperatureindependent contribution to Buckingham birefringence. Note, however, that the b(ω) contribution for CSe_{2} is computed to be of the order of –700 a.u. We shall later return to the consequences that this computed value has for the estimate of the quadrupole moment of CSe_{2} made by Brereton and coworkers in Ref. [34].
The temperatureindependent contribution to Buckinghambirefringence, b(ω), and its individual components (isotropic averages, Einstein implicit summation implied) calculated using the uaugccpVTZ LAO basis and the DC Hamiltonian

 b(ω)  \(J^{\prime}_{\alpha,\beta,\gamma}\)  B _{αβ,αβ}  \({\fancyscript{B}_{\alpha,\alpha\beta,\beta}}\) 

CO_{2}  HF  −48.3 (−0.1)  5.1 (0.0)  −1,069.7 (−1.5)  −1,061.2 (−1.5) 
λ = 632.8 nm  LDA  −58.9 (−0.1)  6.2 (0.0)  −1,641.8 (−2.6)  −1,628.1 (−2.7) 
BLYP  −61.5 (−0.1)  6.5 (0.0)  −1,738.9 (−3.0)  −1,725.9 (−3.0)  
B3LYP  −57.1 (−0.1)  6.0 (0.0)  −1,519.2 (−2.5)  −1,507.1 (−2.5)  
PBE  −59.9 (−0.1)  6.3 (0.0)  −1,676.6 (−2.9)  −1,663.7 (−2.9)  
PBE0  −55.0 (−0.1)  5.8 (0.0)  −1,432.0 (−2.3)  −1,420.4 (−2.3)  
CS_{2}  HF  −482.5 (−1.9)  46.8 (0.2)  −11,867.3 (−61.8)  −11,497.1 (−60.8) 
λ = 632.8 nm  LDA  −464.1 (−2.2)  44.1 (0.2)  −14,335.1 (−92.3)  −13,914.8 (−91.8) 
BLYP  −476.1 (−2.2)  44.9 (0.2)  −15,307.2 (−106.7)  −14,857.0 (−106.3)  
B3LYP  −468.3 (−2.1)  44.6 (0.2)  −13,972.1 (−89.6)  −13,554.6 (−88.9)  
PBE  −462.4 (−2.1)  43.9 (0.2)  −14,321.7 (−93.0)  −13,904.1 (−92.5)  
PBE0  −456.2 (−2.0)  43.7 (0.2)  −12,984.2 (−76.9)  −12,598.8 (−76.1)  
CSe_{2}  HF  −781.5 (5.3)  71.9 (−1.0)  −22,358.3 (−407.3)  −21,493.6 (−376.4) 
λ = 632.8 nm  LDA  −728.9 (−0.7)  66.5 (0.2)  −26,054.2 (−571.8)  −25,205.4 (−580.6) 
BLYP  −731.4 (1.9)  66.0 (0.0)  −28,534.5 (−698.4)  −27,635.1 (−713.2)  
B3LYP  −737.5 (0.8)  67.1 (−0.0)  −25,948.1 (−570.3)  −25,078.0 (−574.4)  
PBE  −733.5 (−0.1)  66.9 (0.2)  −26,622.6 (−633.0)  −25,768.0 (−643.4)  
PBE0  −738.1 (−0.8)  67.8 (0.1)  −24,002.8 (−500.4)  −23,178.6 (−500.7)  
CTe_{2}  HF  −1,839.4 (−47.1)  −65.4 (−210.3)  −78,749.0 (−26,170.6)  −60,414.8 (−11,210.8) 
λ = 632.8 nm  LDA  −1,331.9 (245.5)  91.1 (−34.5)  −59,784.2 (−4,149.6)  −56,123.0 (−3,593.6) 
BLYP  −1,196.9 (359.1)  71.9 (−47.9)  −66,794.5 (−5,223.1)  −62,808.5 (−4,590.8)  
B3LYP  −1,745.8 (−141.2)  129.3 (2.9)  −61,634.2 (−4,832.6)  −57,516.6 (−3,973.1)  
PBE  −1,358.9 (237.1)  93.1 (−33.4)  −61,992.8 (−4,592.5)  −58,267.9 (−4,049.3)  
PBE0  −1,743.3 (−100.7)  131.5 (−1.6)  −57,060.6 (−4,374.7)  −53,114.7 (−3,510.6)  
CTe_{2}  HF  −1,461.9 (300.6)  42.6 (−95.8)  −62,159.3 (−11,635.2)  −54,436.8 (−6,591.7) 
λ = 694.3 nm  LDA  −1,442.3 (120.1)  105.9 (−15.5)  −56,883.4 (−3,393.2)  −54,136.4 (−3,112.1) 
BLYP  −1,399.4 (160.4)  98.9 (−19.8)  −63,238.4 (−4,218.2)  −6,0274.5 (−3,916.0)  
B3LYP  −1,324.1 (268.9)  89.1 (−33.7)  −58,556.7 (−4,017.4)  −55,417.2 (−3,463.5)  
PBE  −1,460.4 (126.1)  107.1 (−15.9)  −58,927.7 (−3,798.3)  −56,132.7 (−3,528.9)  
PBE0  −1,353.0 (266.2)  94.0 (−33.3)  −54,299.0 (−3,618.3)  −51,312.3 (−3,078.5) 
As expected, the relativistic correction to b(ω) is dominated by the correction to \(J^{\prime}_{\alpha,\beta,\gamma}. \) The correction is fairly small, however, even for CSe_{2}, thus leaving the resulting temperatureindependent Buckingham birefringence virtually unaffected by relativity for the three lightest members of the CX_{2} series. Further studies are needed to establish whether this is a general feature of b(ω), valid also for polar systems, for example, or whether this insensitivity to relativity is unique to the CX_{2} series. The relativistic corrections to B _{αβ,αβ} and \({\fancyscript{B}_{\alpha,\alpha\beta,\beta}}\) are small in relative terms, being only about 2% for CSe_{2}. By cancellation, the total relativistic correction from \({B_{\alpha\beta,\alpha\beta}\fancyscript{B}_{\alpha,\alpha\beta,\beta}}\) is even smaller, being less than one percent for all XC functionals.
The relativistic effects vary significantly with the choice of XC functional—in particular, for the heavier elements. It is noteworthy that the use of exact exchange (in HF and hybrid theories) gives a negative relativistic correction for CTe_{2} at λ = 632.8 nm, whereas pure KS theory gives a positive and much larger relativistic correction. For CTe_{2}, we note from Table 2 that the relativistic correction becomes substantial for b(ω) at λ = 632.8 nm, amounting to 30% for the BLYP functional.
The reason for the much larger relativistic corrections in CTe_{2} is the presence of a lowlying \(^3\Upsigma_u^+\) state (scalar relativistic notation), rather close in energy to the frequency of the applied field. Whereas the transition to this state is dipole forbidden in the nonrelativistic case, it is allowed in the fourcomponent relativistic case due to spin–orbit coupling. The fourcomponent relativistic calculations are thus much more dependent on the predicted excitation energy for this state than are the nonrelativistic calculations, as the approaching electronic resonance may affect the different XC functionals differently depending on how close the energy of the relevant \(^3\Upsigma_u^+\) state is to the applied laser frequency.
At λ = 694.3 nm, the relativistic correction in CTe_{2} is positive for all employed XC functionals, again with the hybrid functionals standing out and yielding very similar relativistic corrections.
4.2 The temperaturedependent contribution to the Buckingham birefringence
Results for \(\Updelta \alpha(\omega;\omega), \alpha_{\rm ave}(\omega;\omega), \Uptheta, b(\omega)\) and the temperaturedependent contribution to the Buckingham birefringence, F(ω) / T, for CO_{2},CS_{2}, and CSe_{2}, with \(F(\omega)=\frac{2\Uptheta \Updelta \alpha(\omega;\omega)}{15kT}, T = 298.15\,\hbox{K}\), and using the DC Hamiltonian

 \(\Updelta \alpha (\omega;\omega)\)  α_{ave} (− ω; ω)  \(\Uptheta\)  b(ω)  \(\frac{F(\omega)}{T}\)  _{m}Q(ω, T)  \(\frac{\Updelta n (\omega,T)}{\nabla E}\) 

a.u.  a.u.  a.u.  a.u. × 10^{2}  a.u. × 10^{4}  a.u. × 10^{28}  m^{2}V^{−1} × 10^{−23}  
CO_{2}  HF  12.0879 (−0.0016)  15.9517 (0.0067)  −3.8017 (0.0134)  −0.4835 (−0.0001)  −0.6489 (0.0024)  −1.65  −1.66 
λ = 632.8 nm  LDA  13.8954 (−0.0006)  17.8500 (0.0077)  −3.1004 (0.0144)  −0.5895 (−0.0001)  −0.6084 (0.0028)  −1.55  −1.56 
BLYP  13.9429 (−0.0005)  18.0041 (0.0083)  −3.2251 (0.0146)  −0.6150 (−0.0001)  −0.6350 (0.0029)  −1.62  −1.63  
B3LYP  13.5672 (−0.0008)  17.4243 (0.0077)  −3.3319 (0.0142)  −0.5713 (−0.0001)  −0.6384 (0.0028)  −1.62  −1.64  
PBE  13.8218 (−0.0006)  17.8480 (0.0081)  −3.0836 (0.0145)  −0.5989 (−0.0001)  −0.6019 (0.0029)  −1.53  −1.54  
PBE0  13.3737 (−0.0009)  17.1777 (0.0075)  −3.2292 (0.0142)  −0.5501 (−0.0001)  −0.6098 (0.0027)  −1.55  −1.56  
exp.  (14.25 ± 0.42)^{a}  (−3.18 ± 0.16)^{b,c}  (−1 ± 2)^{d}  (− 1.675 ± 0.023)^{e,f}  
exp.  (−3.19 ± 0.13)^{e}  (−1.6 ± 0.8)^{e}  
exp.  (−2.99 ± 0.09)^{g}  
exp.  (−3.339 ± 0.111)^{d}  
exp.  (−3.181 ± 0.136)^{h}  
calc.^{i}  14.90  (−3.18 ± 0.02)  −54.475  
calc.^{j}  −3.107 (−3.121)  
CS_{2}  HF  61.5719 (−0.0434)  57.8239 (0.0271)  2.3725 (0.0646)  −4.8246 (−0.0019)  2.0628 (0.0547)  5.08  5.12 
λ = 632.8 nm  LDA  60.9516 (−0.0190)  58.4256 (0.0476)  2.2808 (0.0681)  −4.6405 (−0.0022)  1.9631 (0.0580)  4.84  4.87 
BLYP  61.8957 (−0.0132)  59.1345 (0.0543)  1.9643 (0.0677)  −4.7608 (−0.0022)  1.7169 (0.0588)  4.21  4.24  
B3LYP  61.5490 (−0.0209)  58.3474 (0.0453)  2.1192 (0.0674)  −4.6832 (−0.0021)  1.8419 (0.0580)  4.53  4.56  
PBE  61.1159 (−0.0166)  58.1834 (0.0466)  2.1104 (0.0667)  −4.6240 (−0.0021)  1.8213 (0.0571)  4.48  4.51  
PBE0  60.7926 (−0.0251)  57.3998 (0.0374)  2.2776 (0.0663)  −4.5620 (−0.0020)  1.9553 (0.0562)  4.82  4.85  
exp.  (68.87 ± 1.88)^{a}  (2.67 ± 0.13)^{d}  (−12 ± 8)^{e}  (5.23 ± 0.12)^{e,k}  
exp.  (3.17 ± 0.25)^{l}  
exp.  (2.56 ± 0.11)^{e}  
exp.  (3.12 ± 0.67)^{a}  
calc.^{i}  70.63  2.338  −410.91  
calc.^{j}  2.288 (2.224)  
CSe_{2}  HF  88.4674 (−0.5702)  78.0131 (−0.2103)  4.2405 (0.2760)  −7.8148 (0.0053)  5.2976 (0.3128)  13.17  13.26 
λ = 632.8 nm  LDA  84.3227 (−0.6143)  77.2424 (−0.1047)  4.1771 (0.2999)  −7.2894 (−0.0007)  4.9739 (0.3234)  12.36  12.45 
BLYP  86.0432 (−0.5809)  79.0331 (−0.0577)  3.8547 (0.2977)  −7.3144 (0.0019)  4.6837 (0.3326)  11.63  11.71  
B3LYP  86.0481 (−0.6141)  77.9473 (−0.1201)  4.0218 (0.2961)  −7.3745 (0.0008)  4.8870 (0.3275)  12.14  12.23  
PBE  85.2390 (−0.5888)  77.9609 (−0.0761)  4.0601 (0.3033)  −7.3348 (−0.0001)  4.8872 (0.3338)  12.14  12.23  
PBE0  85.4450 (−0.6263)  76.8797 (−0.1468)  4.2283 (0.3002)  −7.3806 (−0.0008)  5.1019 (0.3274)  12.68  12.77  
exp.^{m}  (72.84 ± 1.46)  (3.43 ± 0.60)  0^{n}  (11.1 ± 1.9)  
exp.^{o}  (3.47 ± 0.60)  
calc.^{j}  4.234 (3.956) 
Results for \(\Updelta \alpha(\omega;\omega), \alpha_{\rm ave}(\omega;\omega), \Uptheta, b(\omega)\) and the temperaturedependent contribution to the Buckingham birefringence, F(ω)/T, for CTe_{2}, with \(F(\omega)=\frac{2\Uptheta \Updelta \alpha(\omega;\omega)}{15kT}, T = 298.15\,{\rm K}\), and using the DC Hamiltonian

 \(\Updelta \alpha (\omega;\omega)\)  α_{ave} (− ω; ω)  \(\Uptheta\)  b(ω)  \(\frac{F(\omega)}{T}\)  _{m}Q(ω, T)  \(\frac{\Updelta n (\omega,T)}{\nabla E}\) 

a.u.  a.u.  a.u.  a.u. × 10^{2}  a.u. × 10^{4}  a.u. × 10^{28}  m^{2}V^{−1} × 10^{−23}  
CTe_{2}  HF  198.6880 (38.1670)  138.8550 (12.0720)  6.8928 (0.7241)  −18.3937 (−0.0471)  19.3396 (5.3564)  48.32  48.67 
λ = 632.8 nm  LDA  143.0930 (−0.7840)  120.0180 (−0.3380)  6.4064 (0.7931)  −13.3186 (0.2455)  12.9452 (1.5404)  32.32  32.55 
BLYP  147.0880 (−0.3710)  123.7760 (−0.0240)  5.9893 (0.7847)  −11.9686 (0.3591)  12.4404 (1.6027)  31.08  31.30  
B3LYP  150.0150 (1.2220)  122.7310 (−0.0730)  6.2975 (0.7856)  −17.4583 (−1.4120)  13.3410 (1.7594)  33.21  33.45  
PBE  145.5880 (−0.7000)  122.1230 (−0.2420)  6.2772 (0.8112)  −17.4332 (0.1473)  12.9055 (1.6138)  32.21  32.44  
PBE0  149.3520 (0.9920)  121.3750 (−0.1840)  6.6075 (0.8092)  −13.5893 (−0.2837)  13.9358 (1.7879)  34.71  34.96  
Calc.^{a}  7.138 (6.341)  
CTe_{2}  HF  171.6980 (16.7520)  129.3050 (4.8730)  6.8928 (0.7241)  −14.6194 (3.0057)  16.7125 (13.0196)  41.79  42.09 
λ = 694.3 nm  LDA  137.6550 (−1.4420)  117.5910 (−0.6670)  6.4064 (0.7931)  −14.4233 (1.2006)  12.4533 (1.4274)  31.05  31.27 
BLYP  141.1180 (−1.1810)  121.0670 (−0.4540)  5.9893 (0.7847)  −13.9945 (1.6042)  11.9355 (1.4770)  29.75  29.97  
B3LYP  143.0660 (−0.6180)  120.1810 (−0.4010)  6.2975 (0.7856)  −13.2408 (2.6894)  12.7230 (1.5391)  31.76  31.99  
PBE  139.9380 (−1.3790)  119.6160 (−0.5680)  6.2772 (0.8112)  −14.6044 (1.2606)  12.4046 (1.4967)  30.92  31.14  
PBE0  142.6930 (−0.7110)  118.9280 (−0.4940)  6.6075 (0.8092)  −13.5296 (2.6623)  13.3144 (1.5723)  33.25  33.48 
Interestingly, the relativistic correction to the polarizability of CTe_{2} is largest in HF theory (Table 4) and positive for both frequencies, whereas the relativistic corrections to the polarizability are negligible for all studied XC functionals and negative except for the hybrid PBE0 and B3LYP functionals at λ = 632.8 nm.
By contrast, the relativistic corrections to the quadrupole moment are substantial, indicating that relativity leads to a significant restructuring of the electron density, increasing the quadrupole moment by 10–15%. Indeed, spin–free calculations confirm that the relativistic increase in the quadrupole moment is almost entirely a scalar relativistic effect. To understand this effect we have compared nonrelativistic and scalar relativistic orbital contributions to the quadrupole moment (data not shown). This analysis shows that the change in the quadrupole moment when including scalar relativity is due to a relativistic contraction of the valence σ orbitals related to a contraction of the participating s and porbitals. This decreases the electronic contribution to the quadrupole moment and increases the total (electronic + nuclear) quadrupole moment. Clearly, despite being a property that largely probes the outer part of the electron density, the quadrupole moment is strongly dependent on a proper relativistic treatment.
Whereas the isotropic polarizability is fairly insensitive to electron correlation, with the notable exception of CTe_{2}, electron correlation is moderately important for the polarizability anisotropy and quadrupole moment of the lighter members of the series (CO_{2}, CS_{2} and CSe_{2}), contributing 3–5%. Also in this case, CTe_{2} displays much larger dependence on correlation at λ = 632.8 nm (by almost 15% with the BLYP functional), possibly because of the lowlying \(^3\Upsigma_u^+\) state. Interestingly, the relativistic corrections show only a weak dependence on electron correlation. We also note that the polarizability anisotropy exhibits large correlation effects at λ = 632.8 nm, more than 25% at the relativistic fourcomponent level of theory for the LDA functional.
In Table 3, we report the available experimental reference data for most of the observables involved in Buckingham birefringence of the series of studied molecules. Whereas the electricdipole polarizability anisotropy of CO_{2} is reasonably well reproduced (albeit all XC functionals yield values below the center of the experimental distribution, with the PBE, LDA and BLYP functionals performing better than the hybrid PBE0 and B3LYP functionals), the disagreement with experiment is notable for CS_{2} (where we underestimate \(\Updelta \alpha\)) and CSe_{2} (where we overestimate \(\Updelta \alpha\)). For CO_{2} and CS_{2}, our DFT results lie on the opposite side of experiment relative to the highly accurate ab initio values of Coriani and coworkers in Ref. [17]. Note also the neglect of vibrational corrections, whose magnitude for the heavier members of the series may heavily affect the comparison.
With regard to the quadrupole moment, comparisons can again be made with experiment, with the CCSD(T) results obtained by Benkova and Sadlej [97] employing the HyPolX and HyPolX_dk basis sets and including scalar relativistic effects within the two–component Douglas–Kroll approximation, and, for CO_{2} and CS_{2}, with the results of Ref. [17]. Among the XC functionals, the BLYP and PBE0 functionals get closest to the nicely agreeing experimental and ab initio values (both aiming at a value of 3.18 a.u.) for the quadrupole moment of CO_{2}. Again, the reader should consult Ref. [17] for further details—for instance, on the effect of molecular vibrations.
There are several experimental values for the quadrupole moment of CS_{2}. Both ab initio and this relativistic DFT study are closer to the values in the range of 2.6–2.7 a.u. in Refs. [29, 30]. The experimental value for CSe_{2} was obtained by Brereton and coworkers [34] by infinitedilution extrapolation of data at 298 K for Buckingham birefringence observed in carbon tetrachloride, with a laser source at λ = 632.8 nm. In deriving the value of \(\Uptheta = 3.43 \pm 0.60\;\hbox{a.u.}\), the authors assumed a negligible b(ω) term, which we compute instead at a value of about −700 a.u. When, as done in other cases [8, 9, 10, 11], this value is inserted in the expression for _{m} Q(ω, T) and the quadrupole moment is reevaluated, assuming for the electricdipole polarizability anisotropy and the reactionfieldcorrection factor the same values as in Ref. [34], the experimental estimate of \(\Uptheta\) for CSe_{2} is revised to 3.47 ± 0.60 a.u., with a small but not negligible shift towards the calculated results.
4.3 Assessment of the calculated Buckingham birefringences
The last four columns of Tables 3 and 4 allow us to comment on the magnitude of the Buckingham birefringence in the CX_{2} (X = O, S, Se, Te) series. First, we observe that the temperatureindependent contribution to _{m} Q(ω, T) in CO_{2} never exceeds 1% of the temperaturedependent contribution in column F(ω)/T of Tables 3 and 4. This percentage rises to 2% for CSe_{2} and CTe_{2} and to 3% for CS_{2}. Still, the neglect of the temperatureindependent contribution to the Buckingham birefringence in the CX_{2} (X = O, S, Se, Te) series is a good approximation—for example, we have seen that, when b(ω) is taken into account in the derivation of the quadrupole moment of CSe_{2}, the quadrupole moment changes by only 1.2%.
Because of the large experimental error bars, our calculated value for the Buckingham constant _{m} Q(ω, T) at 632.8 nm and 298.15 K is in good agreement with the experimental value (at 298 K) for CSe_{2}, in spite of our overestimation of the quadrupole moment \(\Uptheta\) and the electricdipole polarizability anisotropy \(\Updelta \alpha (\omega;\omega)\).
The discrepancies between theory and experiment for these quantities are also responsible for our underestimation of the magnitude of _{m} Q(ω, T) for CO_{2} and CS_{2}, the HF values being closer to experiment than the DFT values. A reasonable value for the static electric field gradient in measurements of Buckingham birefringence is ∇E ≈ 1 × 10^{9} Vm^{−2}. With this value, we predict for the higher member of our series an anisotropy \(\Updelta n \approx 3\times 10^{13}\), which, for an optical path length of 1 m, yields a retardation of 3 × 10^{−6} rad at 632.8 nm, well above the current limits of detection.
5 Concluding remarks
We have presented the first fourcomponent relativistic study of Buckingham birefringence, using LAOs to ensure gaugeorigin independence of the calculated results and, more importantly for the small systems considered here, to improve basisset convergence. With the use of LAOs, the results obtained at the uaugDZ level of theory are within 5% of the estimated basisset limit for the Buckingham birefringence of the molecules studied in this work. Electron correlation has been described using KS theory.
We have investigated the importance of relativity and electron correlation for the description of the Buckingham birefringence of the four nonpolar molecules CO_{2}, CS_{2}, CSe_{2}, and CTe_{2}. Electron correlation is significant, leading to changes of 10–15% relative to the DC HF results. However, the ability of the standard XC functionals LDA, BLYP, B3LYP, PBE and PBE0 investigated here to recover the correlation contribution to Buckingham birefringence is doubtful, noting that the agreement with earlier CCSD values is good for CO_{2} but poor for CS_{2}, for which only one third of the electroncorrelation effects are recovered. Further studies on the adequacy of modern XC functionals in describing electroncorrelation contributions to Buckingham birefringence appear necessary.
In contrast to electron correlation, the effects of relativity on the temperatureindependent contribution to Buckingham birefringence are negligible. The only exception to this observation in the series is CTe_{2}, where the relativistic corrections amount to 20–30%. The importance of relativity for this molecule is due to lowlying, resonant states becoming dipole allowed when spinorbit interactions are included in the calculations.
Relativistic effects have been found to be more important for the temperaturedependent contribution to Buckingham birefringence, where the quadrupole moments display fairly large relativistic corrections considering that the property probes the outer regions of the electron density. These relativistic corrections are almost exclusively scalar in nature, and are found to be due to the relativistic contraction of σ valence orbitals. It would therefore be of interest to also investigate relativistic effects on polar molecules, as additional contributions to the temperaturedependent part enters the expression for the Buckingham birefringence in this case.
Notes
Acknowledgments
It is a pleasure for us to dedicate this work to Prof. Pekka Pyykkö, a pioneer in the development and understanding of relativistic effects in molecular properties. The authors thank Miroslav Iliaš, Małgorzata Olejniczak, Trond Saue and Andreas J. Thorvaldsen for helpful discussions. This work has received support from the Research Council of Norway through a Centre of Excellence Grant (Grant No. 179568/V30) and Grant No. 191251/V30. A grant of computer time from the Programme for Supercomputing is also gratefully acknowledged.
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