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
@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
@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