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UV/vis absorption/emission

VeloxChem provides UV/vis absorption and emission spectra through two complementary approaches—an eigenvalue‑based TDDFT solver that yields excitation energies and oscillator strengths, and the complex polarization propagator (CPP) approach that computes frequency‑dependent response functions relating directly to broadened, experimentally comparable spectra. For further theoretical details, see Norman et al. (2018).

Generalized eigenvalue equation

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

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

fn0=2meωn03ℏe2∑α=x,y,z∣⟨0∣μ^α∣n⟩∣2f_{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
(1)

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

σ(ω)=2π2e2ω(4πε0)mec∑n>0f(ω;ωn0,γ)fn0ωn0\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} }
(2)

where ff is the Cauchy distribution.

Python script

import veloxchem as vlx

molecule = vlx.Molecule.read_name("paranitroaniline")
basis = vlx.MolecularBasis.read(molecule, "def2-svp")

scf_drv = vlx.ScfRestrictedDriver()
scf_drv.xcfun = "cam-b3lyp"
scf_results = scf_drv.compute(molecule, basis)

rsp_drv = vlx.LinearResponseEigenSolver()
rsp_drv.nstates = 5
rsp_results = rsp_drv.compute(molecule, basis, scf_results)
<Figure size 800x500 with 2 Axes>

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

Complex polarization propagator approach

The linear absorption cross section can be determined directly from the imaginary part of the polarizability Norman et al. (2018)

σ(ω)=ωϵ0cIm{α‾(−ω;ω)}\sigma(\omega) = \frac{\omega}{\epsilon_0 c} \mathrm{Im}\left\{ \overline{\alpha}(-\omega;\omega) \right\}
(3)

where

α‾=13(αxx+αyy+αzz)\overline{\alpha} = \frac{1}{3} \big( \alpha_{xx} + \alpha_{yy} + \alpha_{zz} \big)
(4)

and

ααβ(−ω;ω)=−⟨⟨μ^α;μ^β⟩⟩ωγ\alpha_{\alpha\beta}(-\omega;\omega) = - \langle \langle \hat{\mu}_\alpha ; \hat{\mu}_\beta \rangle \rangle^\gamma_\omega
(5)

The polarizability is complex and calculated with a damping term, ℏγ\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 σ(ω)\sigma(\omega) are presented in atomic units and can be converted to the SI unit of m2^2 by multiplying with a factor of a02a_0^2.

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

Python script

import veloxchem as vlx
import numpy as np

molecule = vlx.Molecule.read_name("ethylene")
basis = vlx.MolecularBasis.read(molecule, "def2-svp")

scf_drv = vlx.ScfRestrictedDriver()
scf_drv.xcfun = "cam-b3lyp"
scf_results = scf_drv.compute(molecule, basis)

cpp_drv = vlx.ComplexResponse()
cpp_drv.frequencies = np.arange(0.2, 0.35, 0.0025)
cpp_drv.cpp_flag = "absorption"

cpp_results = cpp_drv.compute(molecule, basis, scf_results)
<Figure size 800x500 with 1 Axes>

Text file

@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

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

References
  1. Norman, P., Ruud, K., & Saue, T. (2018). Principles and practices of molecular properties. John Wiley & Sons, Ltd.
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