Velocity Gauge for Oscillator Strength in $Δ$SCF theory

Abstract

Delta self-consistent-field ($Δ$SCF) theory is widely used for electronic excitation energy calculations. However, calculating the corresponding oscillator strengths is challenging. The corresponding many-electron wavefunctions are not directly accessible. Both the ground-state and the excited-state wave functions from $Δ$SCF are described by reference Kohn-Sham (KS) single-determinant wavefunctions for the fictitious non-interacting systems. The non-orthogonality between the ground and excited Kohn-Sham determinants from two different SCF calculations leads to unphysically origin-dependent transition properties, such as transition dipole moment and length-gauge oscillator strength. Including nuclei contribution in the perturbation is theoretically rigorous, but its effectiveness is only limited to neutral systems, as we show theoretically and numerically. While several other practical approaches have been proposed to tackle the non-orthogonality problem and yield reasonable results, inevitably the determinant of the ground state or the excited state is changed, as well as the density matrix. In this work, we explore the use of the velocity gauge to compute oscillator strength within $Δ$SCF theory. We demonstrate that the velocity gauge is capable of naturally accounting for the non-orthogonality of $Δ$SCF KS wavefunctions and offering origin-independent predictions without any additional correction schemes to the KS wavefunctions. Compared to the length-gauge results obtained via symmetric orthogonalization, velocity gauge can offer comparable results. Furthermore, the adoption of spin-purified singlet excitation energy in the velocity-gauge transition dipole moment significantly enhances the overall performance of the velocity gauge for $Δ$SCF oscillator strength predictions on conjugated chromophores.

Figures (4)

• Figure 1: The error in oscillator strength of the lowest singlet states. The labels "$\Delta\text{E}_{\text{exci}}^{\text{purified}}$" and "$\Delta\text{E}_{\text{exci}}^{\text{unpurified}}$" indicate the spin-purified singlet excitation energy or the unpurified singlet excitation energy, which are used to calculate the corresponding transition dipole moment (TDM), i.e. the denominator of $\frac{\langle\Phi_{0}|\hat{P}|\Phi_{m}\rangle}{\Delta E_{\text{exci}}}$. The unpurified singlet excitation energy is defined as difference between the ground-state and excited-state SCF ($\Delta$SCF) total energy. The spin purification is performed according to Eq. \ref{['eq:AP_wave']}.. No additional correction on the wave function, e.g. symmetric orthogonalization or projection, is adopted for the velocity gauge. All calculations are done at the same DFT level (CAM-B3LYP/aug-cc-pVTZ), and the length-gauge $\Delta$SCF results bourne_worster_reliable_2021 are based on symmetric orthogonalization. The TDDFT results are in length gauge. Connection lines are just guide to eyes.
• Figure 2: The origin dependence of length-gauge $\Delta$SCF transition dipole moment with nuclei contribution and the origin-independence of velocity-gauge $\Delta$SCF transition dipole moment, for a charged system, $\text{CH}_{3}^{-}$ anion. The blue and green points represents the modulus of transition dipole moment in length gauge and velocity gauge, respectively. The red line is a linear fit of the length-gauge transition dipole moment, while the black line is for the velocity gauge. Since the transition dipole moment is irrelevant to the origin in velocity gauge, only the standard deviation of transition dipole moment in velocity gauge is given, instead of the R$^{2}$.
• Figure 3: The excitation energies of the chromophore lowest singlet states, predicted by $\Delta\text{SCF}$ and TDDFT. The $\Delta$SCF calculation is performed at the same DFT level (PBE0/def2-SVP) as the TDDFT reference bourne_worster_reliable_2021. Connection lines are just guide to eyes.
• Figure 4: A comparison between the $\Delta$SCF transition properties computed with different gauges and singlet excitation energies, at the DFT level of PBE0/def2-SVP. The labels "$\Delta E_{\text{exci}}^{\text{purified}}$" and "$\Delta E_{\text{exci}}^{\text{unpurified}}$" indicate the spin-purified singlet excitation energy or the unpurified singlet excitation energy, which are used to calculate the corresponding transition dipole moment (TDM), i.e. the denominator of $\frac{\langle\Phi_{0}|\hat{P}|\Phi_{m}\rangle}{\Delta E_{\text{exci}}}$. The unpurified singlet excitation energy is defined as difference between the ground-state and excited-state SCF ($\Delta$SCF) total energy. The spin purification is performed according to Eq. \ref{['eq:AP_wave']}. The results in length gauge adopting symmetric orthogonalization are from bourne_worster_reliable_2021. The black dashed line ($y=0.0$) is an indication of the agreement with the TDDFT oscillator strength.