UV/vis absorption/emission

UV/vis absorption/emission#

Generalized eigenvalue equation#

The standard method to calculate UV/vis absorption and emission spectra is to solve the generalized eigenvalue equation [NRS18].

In the case of SCF theory, it is commonly referred to as the time-dependent density functional theory or Hartree–Fock (TDDFT or TDHF) approach. TDHF is also known as the Random Phase Approximation (RPA).

If electron de-excitations are ignored in the formation of the electronic Hessian, then one arrives at the Tamm–Dancoff approximation and which can be invoked with a keyword in the input file.

VeloxChem implements a reduced-space Davidson algorithm to solve the equation for the N lowest eigenvalues (bottom-up). Based on these eigenvalues, or transition frequencies, and the associated transition moments, the dimensionless oscillator strengths are calculated according to

\[ f_{n0} = \frac{2 m_\mathrm{e} \omega_{n0}}{3\hbar e^2} \sum_{\alpha = x,y,z} |\langle 0 | \hat{\mu}_\alpha | n \rangle |^2 \]

With oscillator strengths and transition frequencies, the linear absorption cross section can be determined from the expression [NRS18]

\[ \sigma(\omega) = \frac{2\pi^2 e^2 \omega}{(4\pi\varepsilon_0) m_\mathrm{e} c} \sum_{n > 0} f(\omega; \omega_{n0}, \gamma) \frac{ f_{n0} }{ \omega_{n0} } \]

where \(f\) is the Cauchy distribution.

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-uv-vis'
results = scf_drv.compute(molecule, basis)

rsp_drv = vlx.lreigensolver.LinearResponseEigenSolver()
rsp_drv.nstates = 10
rsp_drv.nto = True
rsp_results = rsp_drv.compute(molecule, basis, results)

Download a Python script type of input file to calculate the absorption of the ten lowest singlet excited states of the thiophene-quinoxaline molecule at the CAM-B3LYP/def2-svp level of theory.

Text file

Please refer to the keyword list for a complete set of options. Note that natural transition orbitals NTO can be saved using the nto keyword. By specifying the absorption property, both UV/vis and ECD spectra will be calculated.

@jobs
task: response
@end

@method settings
xcfun: cam-b3lyp
basis: def2-svp
@end

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

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

Download a text file type of input file to calculate the absorption of the ten lowest singlet excited states of the thiophene-quinoxaline molecule at the CAM-B3LYP/def2-svp level of theory.

Complex polarization propagator approach#

The linear absorption cross section can be determined directly from the imaginary part of the polarizability [NRS18]

\[ \sigma(\omega) = \frac{\omega}{\epsilon_0 c} \mathrm{Im}\left\{ \overline{\alpha}(-\omega;\omega) \right\} \]

where

\[ \overline{\alpha} = \frac{1}{3} \big( \alpha_{xx} + \alpha_{yy} + \alpha_{zz} \big) \]

and

\[ \alpha_{\alpha\beta}(-\omega;\omega) = - \langle \langle \hat{\mu}_\alpha ; \hat{\mu}_\beta \rangle \rangle^\gamma_\omega \]

The polarizability 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 \(\sigma(\omega)\) are presented in atomic units and can be converted to the SI unit of m\(^2\) by multiplying with a factor of \(a_0^2\).

The arbitrary frequency region is specified in the input file together with a requested frequency resolution.

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(0.1, 0.25, 0.0025)
cpp_drv.damping = 0.0045563
cpp_drv.cpp_flag = "absorption"
cpp_drv.filename = 'mol-cpp'

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

Download a Python script type of input file to calculate the absorption spectrum with CPP of the thiophene-quinoxaline molecule at the CAM-B3LYP/def2-svp level of theory.

Text file

Please refer to the keyword list for a complete set of options.

@jobs
task: response
@end

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

@response
property: absorption (cpp)
! frequency region (and resolution)
frequencies: 0.10-0.25 (0.0025)
damping: 0.0045563  ! this is the default value
@end

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

Download a text file] type of input file to calculate the absorption spectrum with CPP of the thiophene-quinoxaline molecule at the CAM-B3LYP/def2-svp level of theory.

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