Analytical Excited-State Gradients and Derivative Couplings in TDDFT with Minimal Auxiliary Basis Set Approximation and GPU Acceleration

TL;DR

This work develops analytical excited-state gradients and derivative couplings within the TDDFT-ris framework, which uses a minimal auxiliary-basis RI approximation and GPU acceleration. The authors adapt the Lagrangian/Z-vector formalism to the RIS scheme and implement the approach in GPU4PySCF, enabling faster calculations of gradients and derivative couplings. Benchmarking on medium-sized molecules shows a two- to threefold speedup for gradients and derivative couplings with TDDFT-ris, while emission energies and excited-state geometries remain accurate relative to standard TDDFT; however, derivative couplings between nearly degenerate states can be less reliable, necessitating system-specific validation. The method significantly extends the practical reach of excited-state dynamics simulations, with future work focusing on speeding the Z-vector solve and expanding to spin-flip TDDFT and related techniques.

Abstract

Calculating excited-state gradients and derivative couplings using time-dependent density functional theory (TDDFT) remains a computationally demanding task. An efficient variant, TDDFT with resolution of the identity and a minimal auxiliary basis (TDDFT-ris), has been developed to accelerate excitation energy calculations. However, the formulation and implementation of analytical derivatives for this method have not yet been reported. In this work, we present an implementation of analytical excited-state gradients and derivative couplings within the TDDFT-ris framework. Benchmark calculations on medium-sized organic molecules demonstrate a two- to three-fold speedup for both gradients and derivative couplings compared to standard TDDFT. The accuracy of the TDDFT-ris approach is assessed for gradient-dependent applications, including geometry optimizations, emission energy calculations, and the localization of minimum-energy crossing points. Overall, the TDDFT-ris method provides reliable approximations for most cases, with noticeable errors mainly occurring in derivative couplings between nearly degenerate states.

Figures (10)

• Figure 1: Performance comparison of the TDA-ris and standard TDA methods for calculating the SCF energy, excitation energy, derivative couplings ($\boldsymbol{g}_{01}^{\xi}$ and $\boldsymbol{g}_{12}^{\xi}$), and excited-state gradient based on the total computational time for the molecules listed in Table \ref{['tab:atoms_orbitals']}. The two-electron integrals in the ground state and $Z$-vector equations are evaluated with (a) the exact algorithm and (b) density fitting approximation.
• Figure 2: Scaling of the computational cost for the different stages in a TDA-ris calculation with respect to system size (number of basis functions). The density fitting approximation is employed for the ground state and $Z$-vector computations.
• Figure 3: Vertical emission energies (in eV) computed with the TDA-ris method versus the TDA method. For each data point, both the geometry optimization and the energy calculation are performed at the corresponding level of theory. The complete emission energies are presented in Table \ref{['tab:emission']}. The dashed line indicates perfect agreement ($y=x$), and the solid red line represents the linear regression fit.
• Figure 4: Derivative couplings for azulene. The vectors derived from the TDA and TDA-ris methods are depicted as magenta and cyan arrows, respectively, illustrating the coupling between the (a) $S_0/S_1$ ($\boldsymbol{g}_{01}^{\xi}$) and (b) $S_1/S_2$ ($\boldsymbol{g}_{12}^{\xi}$) electronic states.
• Figure 5: The geometry of the $S_1/S_2$ minimum energy crossing point of furan using TDA (pink atoms) and TDA-ris (red atoms).