On the origin independence of the Verdet tensor<sup>†</sup>

Caputo, M. C.; Coriani, Sonia; Pelloni, Stefano; Lazzeretti, Paolo

doi:10.1080/00268976.2013.793845

Cited by 5 publications

(9 citation statements)

References 59 publications

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“…, is independent of the origin of the laboratory coordinate system, and it is the unique measurable quantity in a disordered medium containing freely tumbling optically active molecules. The separate components, diagonal and off‐diagonal, depend on the origin, 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 Eq.…”

Section: Translation Of Frequency‐dependent Response Tensorsmentioning

confidence: 99%

“…Within the dipole velocity‐angular momentum gauge ( P , L ), the mixed electric‐magnetic dipole polarizability (MEMDP) tensor is given by

κ_{α β}^{' (P L)} (ω) = - \frac{e^{2}}{2 m_{e}^{2} ℏ} \sum_{j updiagonalstrike = a} \frac{2 ω}{ω_{j a} true(ω_{j a}^{2} - ω^{2} true)} ℜ true(true〈 a | {\overset{P}{true}}_{α} | j true〉 true〈 j | {\overset{L}{true}}_{β} | a true〉 true),

and the equation for its change in a coordinate translation, Eq. , is

κ_{α β}^{' (P, L)} ({bold-italicr}^{''}) = κ_{α β}^{' (P, L)} ({bold-italicr}^{'}) - \frac{ω}{2} ε_{β γ δ} α_{α γ}^{(P, P)} d_{δ},

where the dynamic electric dipole polarizability within the dipole velocity gauge ( P , P ),

α_{α β}^{(P, P)} (ω) = \frac{e^{2}}{m_{e}^{2} ℏ} \sum_{j \neq a}^{} \frac{2}{ω_{j a} true(ω_{j a}^{2} - ω^{2} true)} ℜ true(true〈 a | {\overset{P}{true}}_{α} | j true〉 true〈 j

…”

Section: Translation Of Frequency‐dependent Response Tensorsmentioning

confidence: 99%

“…, have appeared . However, the first proof of the translational invariance the Verdet tensor has been reported only recently …”

Section: Translation Of Frequency‐dependent Response Tensorsmentioning

confidence: 99%

“…The origin independence of the Verdet propagator necessarily implies that, in the coordinate translation, Eqs. , , and in the resulting transformation

\begin{matrix} {〈 〈 {\hat{μ}}_{α} false(r^{''} false); {\hat{μ}}_{β} false(r^{''} false), {\hat{m}}_{γ} false(r^{''} false) 〉 〉}_{ω, 0} = {〈 〈 {\hat{μ}}_{α} false(r^{'} false); {\hat{μ}}_{β} false(r^{'} false), {\hat{m}}_{γ} false(r^{'} false) 〉 〉}_{ω, 0} \\ + \frac{e}{2 m_{e}} ε_{γ λ μ} d_{λ} {〈 〈 {\hat{μ}}_{α} false(r^{'} false); {\hat{μ}}_{β} false(r^{'} false), {\hat{P}}_{μ} 〉 〉}_{ω, 0}, \end{matrix}

the sum rule

{scriptZ}_{α β γ} (- ω; ω, 0) = - ℑ {true〈 true〈 {\overset{μ}{true}}_{α}; {\overset{μ}{true}}_{β}, {\overset{P}{true}}_{γ} true〉 true〉}_{ω, 0} = 0,

must be satisfied for any origin

{bold-italicr}^{'}

, as the polar tensor on the r.h.s.,

〈 〈 {\overset{μ}{true}}_{α}; {\overset{μ}{true}}_{β},

…”

Section: Translation Of Frequency‐dependent Response Tensorsmentioning

confidence: 99%

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Invariance of molecular response properties under a coordinate translation

Lazzeretti

2014

Int J of Quantum Chemistry

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The response of a molecule to static and dynamic electromagnetic fields and intramolecular perturbations is reviewed within the framework of the Rayleigh-Schr€ odinger perturbation theory. A semiclassical approach is adopted, using quantized form of electronic operators and the nonquantized description of the electromagnetic fields via the Maxwell equations. The Bloch multipolar gauge has been used to define operators suitable to describe the molecular interaction with nonhomogneous timedependent perturbations. It is shown that the quantum mechanical theory of magnetic properties can profitably be proposed in terms of electronic current densities induced by an external magnetic field and permanent magnetic dipole moments at the nuclei. Theoretical relationships are reported to evaluate magnetizability, nuclear magnetic shielding, and nuclear spin-spin coupling, proving that the whole theory can be reformulated via the equations of classical electromagnetism, provided that the current density is evaluated by quantum mechanical methods. Emphasis is placed on the invariance of response properties in a translation of the coordinate system as a basic requirement for measurability. The connections among translational invariance, gauge invariance, and electron charge conservation are outlined, showing that they can be illustrated via quantum mechanical sum rules. A number of relationships describing the change of static and dynamic electromagnetic properties in a translation of the reference frame are reported.

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Section: Translation Of Frequency‐dependent Response Tensorsmentioning

confidence: 99%

“…Within the dipole velocity‐angular momentum gauge ( P , L ), the mixed electric‐magnetic dipole polarizability (MEMDP) tensor is given by

κ_{α β}^{' (P L)} (ω) = - \frac{e^{2}}{2 m_{e}^{2} ℏ} \sum_{j updiagonalstrike = a} \frac{2 ω}{ω_{j a} true(ω_{j a}^{2} - ω^{2} true)} ℜ true(true〈 a | {\overset{P}{true}}_{α} | j true〉 true〈 j | {\overset{L}{true}}_{β} | a true〉 true),

and the equation for its change in a coordinate translation, Eq. , is

κ_{α β}^{' (P, L)} ({bold-italicr}^{''}) = κ_{α β}^{' (P, L)} ({bold-italicr}^{'}) - \frac{ω}{2} ε_{β γ δ} α_{α γ}^{(P, P)} d_{δ},

where the dynamic electric dipole polarizability within the dipole velocity gauge ( P , P ),

α_{α β}^{(P, P)} (ω) = \frac{e^{2}}{m_{e}^{2} ℏ} \sum_{j \neq a}^{} \frac{2}{ω_{j a} true(ω_{j a}^{2} - ω^{2} true)} ℜ true(true〈 a | {\overset{P}{true}}_{α} | j true〉 true〈 j

…”

Section: Translation Of Frequency‐dependent Response Tensorsmentioning

confidence: 99%

“…, have appeared . However, the first proof of the translational invariance the Verdet tensor has been reported only recently …”

Section: Translation Of Frequency‐dependent Response Tensorsmentioning

confidence: 99%

“…The origin independence of the Verdet propagator necessarily implies that, in the coordinate translation, Eqs. , , and in the resulting transformation

\begin{matrix} {〈 〈 {\hat{μ}}_{α} false(r^{''} false); {\hat{μ}}_{β} false(r^{''} false), {\hat{m}}_{γ} false(r^{''} false) 〉 〉}_{ω, 0} = {〈 〈 {\hat{μ}}_{α} false(r^{'} false); {\hat{μ}}_{β} false(r^{'} false), {\hat{m}}_{γ} false(r^{'} false) 〉 〉}_{ω, 0} \\ + \frac{e}{2 m_{e}} ε_{γ λ μ} d_{λ} {〈 〈 {\hat{μ}}_{α} false(r^{'} false); {\hat{μ}}_{β} false(r^{'} false), {\hat{P}}_{μ} 〉 〉}_{ω, 0}, \end{matrix}

the sum rule

{scriptZ}_{α β γ} (- ω; ω, 0) = - ℑ {true〈 true〈 {\overset{μ}{true}}_{α}; {\overset{μ}{true}}_{β}, {\overset{P}{true}}_{γ} true〉 true〉}_{ω, 0} = 0,

must be satisfied for any origin

{bold-italicr}^{'}

, as the polar tensor on the r.h.s.,

〈 〈 {\overset{μ}{true}}_{α}; {\overset{μ}{true}}_{β},

…”

Section: Translation Of Frequency‐dependent Response Tensorsmentioning

confidence: 99%

See 2 more Smart Citations

Invariance of molecular response properties under a coordinate translation

Lazzeretti

2014

Int J of Quantum Chemistry

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“…In oriented molecules,

ϕ

has been shown to depend also on

A_{α, β γ}

, the mixed electric dipole‐electric quadrupole polarizability (MEDQP), via an origin‐independent combination of terms which separately depend on the origin . At a given frequency ω of the incident monochromatic light, the change of the individual

{κ^{'}}_{α β} (- ω; ω)

and

A_{α, β γ} (- ω; ω)

components in a translation of the origin is described by equations involving the dynamic electric dipole polarizability (EDP),

α_{α β} (- ω; ω)

, at the same frequency . However, it was recently observed that the diagonal components of the MEMDP tensor become translationally invariant if it is referred to the coordinate frame of the eigenvectors of EDP .…”

Section: Introductionmentioning

confidence: 99%

Theoretical prediction of the optical rotation of chiral molecules in ordered media: A computational study of (R_a)‐1,3‐dimethylallene, (2R)‐2‐methyloxirane, and (2R)‐N‐methyloxaziridine

Caputo

Pelloni

Lazzeretti

2015

Int J of Quantum Chemistry

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A theoretical procedure has been developed and implemented to calculate the optical rotation of chiral molecules in ordered phase via origin-independent diagonal components j 0 xx ð2x; xÞ; j 0 yy ð2x; xÞ, j 0 zz ð2x; xÞ of the optical activity tensor and origin-independent components A a;bc ð2x; xÞ, for a 6 ¼ b 6 ¼ c, of the mixed electric dipole-electric quadrupole polarizability. Origin independence was achieved by referring these tensors to the principal axis system of the electric dipole dynamic polarizability a ab ð2x; xÞ at the same laser frequency x. The approach has been applied, allowing for alternative quantum mechanical methods based on different gauges, to estimate near Hartree-Fock values for three chiral molecules, (2R)-N-methyloxaziridine C 2 NOH 5 , (2R)-2-methyloxirane (also referred to as propylene oxide) C 3 OH 6 , and (R a )-1,3-dimethylallene C 5 H 8 , at two frequencies. The theoretical predictions can be useful for an attempt at measuring correspondent experimental values in crystal phase.

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