To expand the QUEST database of highly accurate vertical transition energies, we consider a series of large organic chromogens ubiquitous in dye chemistry, such as anthraquinone, azobenzene, BODIPY, and naphthalimide. We compute, at the CC3 level of theory, the singlet and triplet vertical transition energies associated with the low-lying excited states. This leads to a collection of more than 120 new highly accurate excitation energies. For several singlet transitions, we have been able to determine CCSDT transition energies with a compact basis set, finding minimal deviations from the CC3 values for most states. Subsequently, we employ these reference values to benchmark a series of lower-order wave function approaches, including the popular ADC(2) and CC2 schemes, as well as time-dependent density-functional theory (TD-DFT), both with and without applying the Tamm–Dancoff approximation (TDA). At the TD-DFT level, we evaluate a large panel of global, range-separated, local, and double hybrid functionals. Additionally, we assess the performance of the Bethe–Salpeter equation (BSE) formalism relying on both G0W0 and evGW quasiparticle energies evaluated from various starting points. It turns out that CC2 and ADC(2.5) are the most accurate models among those with respective O(N5) and O(N6) scalings with system size. In contrast, CCSD does not outperform CC2. The best performing exchange–correlation functionals include BMK, M06–2X, M06-SX, CAM-B3LYP, ωB97X-D, and LH20t, with average deviations of approximately 0.20 eV or slightly below. Errors on vertical excitation energies can be further reduced by considering double hybrids. Both SOS-ωB88PP86 and SOS-ωPBEPP86 exhibit particularly attractive performances with overall quality on par with CC2, whereas PBE0-DH and PBE-QIDH are only slightly less efficient. BSE/evGW calculations based on Kohn–Sham starting points have been found to be particularly effective for singlet transitions, but much less for their triplet counterparts.

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In this paper we examine the numerical approximation of the limiting invariant measure associated with Feynman-Kac formulae. These are expressed in a discrete time formulation and are associated with a Markov chain and a potential function. The typical application considered here is the computation of eigenvalues associated with non-negative operators as found, for example, in physics or particle simulation of rare-events. We focus on a novel lagged approximation of this invariant measure, based upon the introduction of a ratio of time-averaged Feynman-Kac marginals associated with a positive operator iterated l ∈ N times; a lagged Feynman-Kac formula. This estimator and its approximation using Diffusion Monte Carlo (DMC) have been extensively employed in the physics literature. In short, DMC is an iterative algorithm involving N ∈ N particles or walkers simulated in parallel, that undergo sampling and resampling operations. In this work, it is shown that for the DMC approximation of the lagged Feynman-Kac formula, one has an almost sure characterization of the L1-error as the time parameter (iteration) goes to infinity and this is at most of O(exp{-κl}/N ), for κ > 0. In addition a non-asymptotic in time, and time uniform L1-bound is proved which is O(l/ √ N ). We also prove a novel central limit theorem to give a characterization of the exact asymptotic in time variance. This analysis demonstrates that the strategy used in physics, namely, to run DMC with N and l small and, for long time enough, is mathematically justified. Our results also suggest how one should choose N and l in practice. We emphasize that these results are not restricted to physical applications; they have broad relevance to the general problem of particle simulation of the Feynman-Kac formula, which is utilized in a great variety of scientific and engineering fields.

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In a recent work we have proposed an original analytic expression for the partition function of the quartic oscillator. This partition function, which has a simple and compact form with {\it no adjustable parameters}, reproduces some key mathematical properties of the exact partition function and provides free energies accurate to a few percent over a wide range of temperatures and coupling constants. In this work, we present the derivation of the energy spectrum of this model. We also generalize our previous study limited to the quartic oscillator to the case of a general anharmonic oscillator. Numerical application for a potential of the form $V(x)=\frac{\omega^2}{2} x^2 + g x^{2m}$ show that the energy levels are obtained with a relative error of about a few percent, a precision which we consider to be quite satisfactory given the simplicity of the model, the absence of adjustable parameters, and the negligible computational cost.

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The era of exascale computing presents both exciting opportunities and unique challenges for quantum mechanical simulations. While the transition from petaflops to exascale computing has been marked by a steady increase in computational power, the shift towards heterogeneous architectures, particularly the dominant role of graphical processing units (GPUs), demands a fundamental shift in software development strategies. This review examines the changing landscape of hardware and software for exascale computing, highlighting the limitations of traditional algorithms and software implementations in light of the increasing use of heterogeneous architectures in high-end systems. We discuss the challenges of adapting quantum chemistry software to these new architectures, including the fragmentation of the software stack, the need for more efficient algorithms (including reduced precision versions) tailored for GPUs, and the importance of developing standardized libraries and programming models.

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