On the definition of molecular dynamic magnetizability

Lazzeretti, Paolo

doi:10.1007/s00214-015-1621-y

Cited by 5 publications

(5 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the shift of origin

{boldo}^{'} \to {boldo}^{''} = {boldo}^{'} + d

of the multipole expansion analogous to (32), using Equation (), one finds 7,63

Δ ⟨{\overset{m}{true}}_{α}^{'} (o^{''})⟩ = Δ ⟨{\overset{m}{true}}_{α}^{'} (o^{'})⟩ - \frac{1}{2} ε_{italicαβγ} d_{β} \frac{d}{italicdt} Δ ⟨{\overset{μ}{true}}_{γ} (o^{'})⟩,

where

Δ ⟨{\overset{μ}{true}}_{γ} (o^{''})⟩ = Δ ⟨{\overset{μ}{true}}_{γ} (o^{'})⟩

. This relation is consistent with () and (), which give

Δ ⟨{\overset{m}{true}}_{α}^{'} (o^{''})⟩ = Δ ⟨{\overset{m}{true}}_{α}^{'} (o^{'})⟩ - \frac{1}{2} ε_{italicαβγ} d_{β} \int J_{γ}^{((), 1)} (bold-italicr, ω, t) d^{3} r,

by employing Equation (102) of reference 54,

\int J_{γ}^{((), 1)} (bold-italicr, ω, t)

…”

Section: Quantum Mechanical Approach To Induced Electric and Magnetic...supporting

confidence: 67%

“…using either the polarization density (2.10) or definitions of response tensors reported previously. 55,63 The vector Δ b μ α h i, Equation (2.27), is invariant of origin of r α , which can be arbitrarily chosen, since To make these aspects clear also for dynamic magnetizabilities, consider, for instance the contribution ξ αβ B β to (2.31), using the definition of magnetizability in terms of CDT, 21,52,53…”

Section: Induced Electric and Magnetic Dipole Momentsmentioning

confidence: 99%

“…Two basic relations can be used to compute the induced electric dipole moment,

Δ ⟨{\overset{μ}{true}}_{α}⟩ = \int r_{α} ρ^{((), 1)} (r, ω, t) d^{3} r

\begin{matrix} newline= α_{αβ} E_{β} goodbreak+ α_{αβ}^{'} {\dot{E}}_{β} ω^{- 1} goodbreak+ α_{α, italicβγ} E_{γβ} goodbreak+ α_{α, italicβγ}^{'} {\dot{E}}_{γβ} ω^{- 1} \\ goodbreak+ κ_{αβ} B_{β} goodbreak+ κ_{αβ}^{'} {\dot{B}}_{β} ω^{- 1}, \end{matrix}

using either the polarization density () or definitions of response tensors reported previously 55,63 . The vector

Δ ⟨{\overset{truê}{μ}}_{α}⟩

, Equation (), is invariant of origin of

r_{α}

, which can be arbitrarily chosen, since

\int ρ^{((), 1)} (bold-italicr, ω, t) d^{3} r = 0,

owing to orthonormality of the exact electronic states

{normalΨ}_{j}^{((), 0)}

in Equations ().…”

Section: Quantum Mechanical Approach To Induced Electric and Magnetic...mentioning

confidence: 99%

“…The vector

Δ ⟨{\overset{truê}{μ}}_{α}⟩

, Equation (), is invariant of origin of

r_{α}

, which can be arbitrarily chosen, since

\int ρ^{((), 1)} (bold-italicr, ω, t) d^{3} r = 0,

owing to orthonormality of the exact electronic states

{normalΨ}_{j}^{((), 0)}

in Equations (). Nonetheless, terms in Equation (), for example, tensors

α_{α, βγ}

and

κ_{italicαβ}^{'}

, vary in a coordinate translation, in such a way that

Δ ⟨{\overset{truê}{μ}}_{α}⟩

remains the same: translational invariance of induced electric dipole, expressed via molecular response tensors of increasing rank coupled to fields and spatial field derivatives, Equation (), has been proven up to the electric octopole approximation (EOA), 63 allowing for the Maxwell equations and the hypervirial relations (). In actual calculations based on the algebraic approximation, 61 the degree of translational invariance obtained via () provides a test of basis set quality and completeness with respect to representation of relevant operators, for example, (), as well as fulfillment of the hypervirial theorem ().…”

Section: Quantum Mechanical Approach To Induced Electric and Magnetic...mentioning

confidence: 99%

See 3 more Smart Citations

On the computation of frequency‐dependent molecular magnetizabilities via dynamical charge and current electron densities

2023

Self Cite

View full text Add to dashboard Cite

In the computation of molecular dynamic magnetizabilities and magnetic dipole moments, three different reference points are required: (i) origin of coordinate system, (ii) origin of vector potential A, and (iii) origin of multipole expansion. This study shows that methods relying on continuous translation of origin of the current density bold-italicIbold-italicBbold-italicrωt induced by optical magnetic fields provide an effective solution to the problem of choices (i) and (ii), in that they yield origin independent bold-italicIbold-italicB within the algebraic approximation, for any basis set. Frequency‐dependent magnetizabilities are also invariant with respect to (iii), as a consequence of symmetry, for a number of molecular point groups. In molecules of lower symmetries, computed magnetizabilities depend on origin of the multipole expansion. Large basis set computations carried out for water, ammonia, methane, ethane, ethylene, boranylborane, and hydroxilamine, at the DFT level, have been reported to document these statements. A comparison is made for results obtained within the conventional common origin approach for static magnetic field. Sum rules for invariance of computed properties are discussed. Representations of streamlines and stagnation graphs of dynamical current density vector field induced in the water molecule by monochromatic waves of four frequencies are displayed.

show abstract

“…In the shift of origin

{boldo}^{'} \to {boldo}^{''} = {boldo}^{'} + d

of the multipole expansion analogous to (32), using Equation (), one finds 7,63

Δ ⟨{\overset{m}{true}}_{α}^{'} (o^{''})⟩ = Δ ⟨{\overset{m}{true}}_{α}^{'} (o^{'})⟩ - \frac{1}{2} ε_{italicαβγ} d_{β} \frac{d}{italicdt} Δ ⟨{\overset{μ}{true}}_{γ} (o^{'})⟩,

where

Δ ⟨{\overset{μ}{true}}_{γ} (o^{''})⟩ = Δ ⟨{\overset{μ}{true}}_{γ} (o^{'})⟩

. This relation is consistent with () and (), which give

Δ ⟨{\overset{m}{true}}_{α}^{'} (o^{''})⟩ = Δ ⟨{\overset{m}{true}}_{α}^{'} (o^{'})⟩ - \frac{1}{2} ε_{italicαβγ} d_{β} \int J_{γ}^{((), 1)} (bold-italicr, ω, t) d^{3} r,

by employing Equation (102) of reference 54,

\int J_{γ}^{((), 1)} (bold-italicr, ω, t)

…”

Section: Quantum Mechanical Approach To Induced Electric and Magnetic...supporting

confidence: 67%

Section: Induced Electric and Magnetic Dipole Momentsmentioning

confidence: 99%

“…Two basic relations can be used to compute the induced electric dipole moment,

Δ ⟨{\overset{μ}{true}}_{α}⟩ = \int r_{α} ρ^{((), 1)} (r, ω, t) d^{3} r

\begin{matrix} newline= α_{αβ} E_{β} goodbreak+ α_{αβ}^{'} {\dot{E}}_{β} ω^{- 1} goodbreak+ α_{α, italicβγ} E_{γβ} goodbreak+ α_{α, italicβγ}^{'} {\dot{E}}_{γβ} ω^{- 1} \\ goodbreak+ κ_{αβ} B_{β} goodbreak+ κ_{αβ}^{'} {\dot{B}}_{β} ω^{- 1}, \end{matrix}

using either the polarization density () or definitions of response tensors reported previously 55,63 . The vector

Δ ⟨{\overset{truê}{μ}}_{α}⟩

, Equation (), is invariant of origin of

r_{α}

, which can be arbitrarily chosen, since

\int ρ^{((), 1)} (bold-italicr, ω, t) d^{3} r = 0,

owing to orthonormality of the exact electronic states

{normalΨ}_{j}^{((), 0)}

in Equations ().…”

Section: Quantum Mechanical Approach To Induced Electric and Magnetic...mentioning

confidence: 99%

“…The vector

Δ ⟨{\overset{truê}{μ}}_{α}⟩

, Equation (), is invariant of origin of

r_{α}

, which can be arbitrarily chosen, since

\int ρ^{((), 1)} (bold-italicr, ω, t) d^{3} r = 0,

owing to orthonormality of the exact electronic states

{normalΨ}_{j}^{((), 0)}

in Equations (). Nonetheless, terms in Equation (), for example, tensors

α_{α, βγ}

and

κ_{italicαβ}^{'}

, vary in a coordinate translation, in such a way that

Δ ⟨{\overset{truê}{μ}}_{α}⟩

Section: Quantum Mechanical Approach To Induced Electric and Magnetic...mentioning

confidence: 99%

See 2 more Smart Citations

On the computation of frequency‐dependent molecular magnetizabilities via dynamical charge and current electron densities

2023

Self Cite

View full text Add to dashboard Cite

show abstract

“…, in Eq. , the induced magnetic dipole and quadrupole are formally the same,

\begin{matrix} Δ 〈 {\overset{m}{true}}_{α}^{'} 〉 \equiv M_{α} = χ_{α β} B_{β} + χ_{α, β γ} B_{γ β} + χ_{α, β γ δ} B_{δ γ β} + \dots \\ - \sum_{I = 1}^{N^{'}} σ_{β α}^{I} m_{I_{β}}, \end{matrix}

\begin{matrix} Δ 〈 {\overset{m}{true}}_{α β}^{'} 〉 \equiv M_{α β} = χ_{α β, γ} B_{γ} + χ_{α β, γ δ} B_{δ γ} + χ_{α β, γ δ ε} B_{ε δ γ} + \dots \\ - \sum_{I = 1}^{N^{'}} σ_{γ, α β}^{I} m_{I_{γ}} . \end{matrix}

…”

Section: Linear Response To a Static Nonhomogeneous Magnetic Field Anmentioning

confidence: 99%

Invariance of molecular response properties under a coordinate translation

Lazzeretti

2014

Int J of Quantum Chemistry

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Gauge invariance and origin independence of electronic charge density and current density induced by optical fields

Lazzeretti

2018

The Journal of Chemical Physics

View full text Add to dashboard Cite

Expressions for the first-order polarization charge density ρ(1) and current density J(1) induced in a molecule by a monochromatic plane wave, obtained by time-dependent quantum mechanical perturbation theory, have been investigated to assess their gauge invariance and independence of the coordinate system in passive and active translations. The conditions arrived at show that, within the (long wavelengths) dipole approximation, only the electric contributions to these densities are needed to rationalize the phenomenology. To the next higher quadrupole approximation, assuming that the magnetic field and the electric field gradient are uniform over the molecular dimensions, corresponding contributions to ρ(1) and J(1) are considered. It has been found that total densities are independent of the origin, whereas the contributions from electric and magnetic fields are not separately invariant. A magnetic contribution to J(1), which is by itself origin independent, can be defined by means of an approach based on continuous translation of the origin of the coordinate system.

show abstract

On the definition of molecular dynamic magnetizability

Cited by 5 publications

References 42 publications

On the computation of frequency‐dependent molecular magnetizabilities via dynamical charge and current electron densities

On the computation of frequency‐dependent molecular magnetizabilities via dynamical charge and current electron densities

Invariance of molecular response properties under a coordinate translation

Gauge invariance and origin independence of electronic charge density and current density induced by optical fields

Contact Info

Product

Resources

About