Practical approach for beryllium atomic clusters: TD-DFT potential energy surfaces from equilibrium to dissociation for excited states of 2s → 2p

To shed light on the spectroscopic energy-smearing mechanism of the recently developed technique for multi-element analysis (e.g., forensic tests on artworks), which is called plume laser-induced fluorescence, practically computer simulations need to be accurate and fast enough for generating wide-range excited-state potential energy surfaces (PESs) from equilibrium to dissociation of large (non-equilibrated) atomic clusters. Currently, all the very high-level post-Hartree–Fock theories (e.g., coupled-cluster theory) are computationally too expensive for this purpose (e.g., taking at least ~ 23 years). After considering the extensive applications of time-dependent density-functional theory (TD-DFT) in computing the optical properties of many-body systems, we wonder how TD-DFT performs in the wide-range-PES generations because no such TD-DFT application is found by us in the literature (perhaps due to its insufficient static/non-dynamical correlation energy). In this work, we carried out TD-DFT calculations on beryllium atomic clusters from Be₁ to Be₅₀ (in contrast to open-shell H₂ or Li₂ system, we think the individual closed-shell Be systems at the dissociation limit could help TD-DFT to describe the static/non-dynamical correlation. For example, it is usually considered that Be₂ has zero non-dynamical correlation energy in the literature) with and without Tamm–Dancoff approximation (TDA). Electronic excited-state properties (e.g., PES, equilibrium position, dissociation energy, fork intersection) associated with 2s → 2p are systematically investigated at a total of 48 levels of TD-DFT. Our calculations are thoroughly benchmarked against experimental results and (very) high-level theoretical values. In short, we find that for accurate simulations, it is necessary to use diffuse basis sets and the TDA approximation. TD-DFT with the small basis set 3-21+G* took ~ 2 min to finish a single-point excitation energy calculation on a linear Be₅₀ (a toy model), while the large basis set 6-311+G(2df,2p) took 30 min. By contrast, a couple-cluster theory EOM-CCSD with 6-311+G(2df,2p) took 2.5 h to do the same but merely for a smaller linear Be₁₀. In general, we suggest that TD-DFT:TDA with 3-21+G* should be practically fast and accurate enough for emulating our desired wide-range excited-state PESs of large Be atomic clusters (including conical intersection points of close energies) in the future studies. Nevertheless, benchmarking work on (much) larger sizes of Be clusters will always be more desirable whenever its computational costs at very high level of theories becomes more tractable.