On the origin independence of the Verdet tensor^†

Abstract

The condition for invariance under a translation of the coordinate system of the Verdet tensor and the Verdet constant, calculated via quantum chemical methods using gaugeless basis sets, is expressed by a vanishing sum rule involving a third-rank polar tensor. The sum rule is, in principle, satisfied only in the ideal case of optimal variational electronic wavefunctions. In general, it is not fulfilled in non-variational calculations and variational calculations allowing for the algebraic approximation, but it can be satisfied for reasons of molecular symmetry. Group-theoretical procedures have been used to determine (i) the total number of non-vanishing components and (ii) the unique components of both the polar tensor appearing in the sum rule and the axial Verdet tensor, for a series of symmetry groups. Test calculations at the random-phase approximation level of accuracy for water, hydrogen peroxide and ammonia molecules, using basis sets of increasing quality, show a smooth convergence to zero of the sum rule. Verdet tensor components calculated for the same molecules converge to limit values, estimated via large basis sets of gaugeless Gaussian functions and London orbitals.

Keywords:

• Faraday effect
• Verdet tensor
• Verdet constant
• sum rule for origin independence
• symmetry unique components

Acknowledgements

Financial support to the present research from the Italian MIUR (Ministero dell’Istruzione, Università e Ricerca), via PRIN 2009 funds, from CONICET(PIP0369) and UBACYT(W197) is gratefully acknowledged.

Notes

^aThe coordinate system is defined according to the Mulliken conventions [65,66]

^a The coordinate system is defined according to the Mulliken conventions [65,66].

^aIn au. The basis sets are specified in the text. Here and in the following tables, the entries between parentheses specify the number of basis functions. ω = 0.0345439 au.

^aIn au. Here and in Tables 6 and 8, the conversion factor from au to SI units is 1 v=ea ₀/ℏ=8.039 61763×10⁴ rad m⁻¹ T ⁻¹ from the CODATA compilation [67]. In au, the constant in Equation (11) is . For an ideal gas at 273.15 K, , from Ref. [32]. For each set of components, results from gaugeless basis sets are given in the first line and corresponding LAO results in the second line. The scalar and the Verdet constant v are defined in Equations (10) and (12) respectively. The origin of the coordinate system coincides with the centre of mass. Nuclear coordinates: O (0.0, 0.0, 0.124144), H₁ (0.0, 1.431530, −0.985266), in bohr. ω = 0.0345439 au.

^a In au, ω = 0.0345439 au.

^aIn au. See footnote to Table 3 for the conventions used. The origin of the coordinate system coincides with the centre of mass. Nuclear coordinates:

O₁ (0.0, 1.3824839, −0.06604896),

H₁ (1.4219547, 1.6931360, 1.0482451), in bohr.

ω = 0.0345439 au.

^aIn au, ω = 0.0345439 au.

^aIn au. See footnote to Table 3 for the conventions used. The origin of the coordinate system coincides with the centre of mass. Nuclear coordinates: N (0.0, 0.0, 0.11898517),

H₁ (1.7624493, 0.0, −0.55107391), in bohr. ω = 0.0345439 au.