8.1 Functionalities of Escf and Egrad

escf and egrad are designed as efficient tools for response and excited state calculations on large molecules. escf serves to compute the following properties for HF and KS reference states:

• Eigenvalues of the electronic Hessian (stability analysis)

• Frequency-dependent polarizabilities and optical rotations

• Frequency-dependent electronic hyperpolarizabilities

• Vertical electronic excitation energies (TD-DFT)

• Vertical electronic excitation energies from the Bethe-Salpeter equation

• Transition moments, oscillator and rotatory strengths of electronic excitations
  \(\Rightarrow\) UV-VIS and CD spectra

• Two-photon transition moments
  \(\Rightarrow\) 2PA spectra

• Nuclear spin-spin coupling constants (SSCCs)

• Damped response TD-DFT and BSE

• Two-component TD-DFT and \(GW\)-BSE including spin-orbit coupling

• TD-DFT and \(GW\)-BSE in magnetic fields

• Berry curvature resulting from spin-orbit coupling or magnetic field

• Diagonal Born-Oppenheimer correction (DBOC)

Spin-restricted closed-shell and spin-unrestricted ground states (except for stability analysis) are supported. For 2c calculations, TD-DFT supports Kramers symmetric references, while \(GW\)-BSE supports all references. The RI-\(J\) approximation in conjunction with LDA, GGA, and meta-GGA (MGGA) functionals is implemented for all properties. The seminumerical semiJK algorithm is also available for all properties. Excitation energies and transition moments can be computed either within the full time-dependent HF (TDHF) or time-dependent DFT (TDDFT) formalisms or within the Tamm-Dancoff approximation (TDA).

Excited state first order properties can be evaluated analytically using egrad. They include:

• Gradients of the excited state energy with respect to nuclear positions
  \(\Rightarrow\) Excited state equilibrium structures (jobex), adiabatic excitation energies, emission spectra

• Exited state densities \(\Rightarrow\) Charge moments, population analysis

• Excited state force constants by numerical differentiation of gradients (using the script NumForce)

• First-order derivative couplings between the ground and an excited state as well as between two excited-states (state-to-state)

• Transition moments, oscillator strengths between two TDHF/TDDFT excited-states (state-to-state)

Moreover, analytical gradients of static and frequency-dependent polarizabilities are available from egrad. Together with vibrational normal modes from the aoforce or NumForce they are used to calculate vibrational Raman intensities. Excited state gradients for \(GW\)-BSE are presently unavailable.

Again, ground states may be spin-restricted closed-shell or spin-unrestricted, RI-\(J\) is available, and either full TDDFT/TDHF or the TDA can be used. For further details we refer to a recent review [217].