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.

@jobs
task: response
@end

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

@response
property: absorption
# tamm_dancoff: yes
nstates: 3
@end

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

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.

@jobs
task: response
@end

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

@response
property: absorption (cpp)
# frequency region (and resolution)
frequencies: 0.0-0.15 (0.0025)
damping: 0.0045563  # this is the default value
@end

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