Optical activity and dichroism

Rotatory strengths¶

The strength of an ECD band is given by the anisotropy of the decadic molar extinction coefficient Norman et al. (2018)

\Delta\epsilon(\omega) = \frac{ 16\pi N_\mathrm{A} }{ \ln\left(10\right) \left(4\pi\varepsilon_0\right) c^2 } \frac{\pi}{3 \hbar} \sum_{n>0} f(\omega; \omega_{n0},\gamma)\, \omega_{n0} R_{n0}

(1)

where $N_\mathrm{A}$ is Avogadro’s constant, $f$ is the Cauchy distribution, and $R_{n0}$ is the rotatory strength defined as

R_{n0} = \sum_{\alpha = x,y,z} \Im \langle 0 | \hat{\mu}_\alpha | n \rangle \langle n | \hat{m}_\alpha | 0\rangle = \sum_{\alpha = x,y,z} \frac{-e}{m_\mathrm{e} \omega_{n0}} \langle 0 | \hat{p}_\alpha | n \rangle \langle n | \hat{m}_\alpha | 0\rangle

(2)

In VeloxChem, the rotatory strength is evaluated in the velocity gauge as given in the second expression, and it is thereby gauge-origin independent.

Python script

import veloxchem as vlx

xyz="""
...
"""

molecule = vlx.Molecule.read_xyz_string(xyz)
basis = vlx.MolecularBasis.read(molecule, 'def2-svp')

scf_drv = vlx.ScfRestrictedDriver()
scf_drv.xcfun = 'cam-b3lyp'
scf_drv.filename = 'mol-ecd'
results = scf_drv.compute(molecule, basis)

rsp_drv = vlx.lreigensolver.LinearResponseEigenSolver()
rsp_drv.nstates=10
rsp_drv.nto = True
rsp_drv.filename = 'mol-ecd'
results = rsp_drv.compute(molecule, basis, results)

Text file

@jobs
task: response
@end

@method settings
basis: def2-svpd
xcfun: b3lyp
@end

@response
property: ecd
nstates: 10
nto: yes
@end

@molecule
charge: 0
multiplicity: 1
xyz:
...
@end

Extinction coefficient¶

The anisotropy of the decadic molar extinction coefficient can be determined directly from the complex polarization propagator evaluated for mixed electric- and magnetic-dipole operators Jiemchooroj & Norman (2007)

\Delta\epsilon(\omega) = \frac{ 16 \pi N_\mathrm{A} \omega^2 }{ \ln(10) \left(4\pi\varepsilon_0\right) c^2 } \, \beta(\omega)

(3)

where the molecular response property, $\beta(\omega)$ , is defined as

\beta(\omega) = -\frac{1}{3 \omega} (G_{xx} + G_{yy} + G_{zz})

(4)

and

G_{\alpha\beta} = - \Re\langle\langle\hat{\mu}_\alpha;\hat{m}_\beta \rangle\rangle_\omega^\gamma = - \frac{e}{\omega m_e} \Im \langle\langle\hat{p}_\alpha; \hat{m}_\beta \rangle\rangle_\omega^\gamma

(5)

The mixed electric–magnetic dipole tensor, $G$ , is evaluated in the velocity gauge as given in the second expression. Furthermore, it is complex and calculated with a damping term, $\hbar \gamma$ , associated with the inverse finite lifetime of the excited states. The default program setting for this parameter is 0.124 eV (or 0.004556 a.u.).

The resulting values for $\Delta \epsilon(\omega)$ are converted from atomic units to units of L mol $^{-1}$ cm $^{-1}$ by multiplying with a factor of $10\, a_0^2$ .

Python script

import veloxchem as vlx

xyz="""
....
"""

molecule = vlx.Molecule.read_xyz_string(xyz)
basis = vlx.MolecularBasis.read(molecule, 'def2-svp')

scf_drv = vlx.ScfRestrictedDriver()
scf_drv.xcfun = 'cam-b3lyp'
scf_drv.filename = 'mol-cpp'
scf_results = scf_drv.compute(molecule, basis)

cpp_drv = vlx.ComplexResponse()
cpp_drv.frequencies = np.arange(0.2, 0.35, 0.0025)
cpp_drv.damping = 0.0045563
cpp_drv.cpp_flag = "ecd"
cpp_drv.filename = 'mol-cpp'

cpp_results = cpp_drv.compute(molecule, basis, scf_results)

Text file

@jobs
task: response
@end

@method settings
basis: def2-svpd
xcfun: b3lyp
@end

@response
property: ecd (cpp)
! frequency region (and resolution)
frequencies: 0.05-0.15 (0.0025)
damping: 0.0045563  ! this is the default value
@end

@molecule
charge: 0
multiplicity: 1
xyz:
...
@end

Exciton coupling model¶

VeloxChem implements the exciton coupling model to determine circular dichroism spectra.

Python script

to be added.

Text file

@jobs
task: exciton
@end

@method settings
xcfun: b3lyp
basis: cc-pvdz
@end

@exciton
fragments: 40
atoms_per_fragment: 55
nstates: 5
ct_nocc: 0
ct_nvir: 0
@end

@molecule
charge: 0
multiplicity: 1
xyz:
...
! XYZ coordinates for 40 x 55 atoms
...
@end

References¶

Norman, P., Ruud, K., & Saue, T. (2018). Principles and practices of molecular properties. John Wiley & Sons, Ltd.
Jiemchooroj, A., & Norman, P. (2007). Electronic circular dichroism spectra from the complex polarization propagator. J. Chem. Phys., 126(13), 134102. 10.1063/1.2716660