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We present a generalization of the phaseless auxiliary-field quantum Monte Carlo (AFQMC) method to cavity quantum-electrodynamical (QED) matter systems. The method can be formulated in both the Coulomb and the dipole gauge. We verify its accuracy by benchmarking calculations on a set of small molecules against full configuration interaction and state-of-the-art QED coupled cluster (QED-CCSD) calculations. Our results show that (i) gauge invariance can be achieved within correlation-consistent Gaussian basis sets, (ii) the accuracy of QED-CCSD can be enhanced significantly by adding the standard perturbative triples correction without light-matter coupling, and (iii) there is a straightforward way to evaluate the differential expression for the photon occupation number that works in any gauge. The high accuracy and favorable computational scaling of our AFQMC approach will enable a broad range of applications. Besides polaritonic chemistry, the method opens a way to simulate extended QED matter systems., Comment: 10 pages, 7 figures

In recent years, digital games have become increasingly present in people's lives both as a leisure activity or in gamified activities of everyday life. Despite this growing presence, large-scale, data-driven analyses of video games remain a small fraction of the related literature. In this sense, the present work constitutes an investigation of patterns in popularity series of video games based on monthly popularity series, spanning eleven years, for close to six thousand games listed on the online platform Steam. Utilizing these series, after a preprocessing stage, we perform a clustering task in order to group the series solely based on their shape. Our results indicate the existence of five clusters of shape patterns named decreasing, hilly, increasing, valley, and bursty, with approximately half of the games showing a decreasing popularity pattern, 20.7% being hilly, 11.8% increasing, 11.0% bursty, and 9.1% valley. Finally, we have probed the prevalence and persistence of shape patterns by comparing the shapes of longer popularity series during their early stages and after completion. We have found the majority of games tend to maintain their pattern over time, except for a constant pattern that appears early in popularity series only to later originate hilly and bursty popularity series., Comment: 36 pages, 19 figures

We introduce an electron-photon exchange-correlation functional for quantum electrodynamical density-functional theory (QEDFT). The approach, photon MBD (pMBD), is inspired by the many-body dispersion (MBD) method for weak intermolecular interactions, which is generalized to include both electronic and photonic (electromagnetic) degrees of freedom on the same footing. We demonstrate that pMBD accurately captures effects that arise in the context of strong light-matter interactions, such as anisotropic electron-photon interactions, beyond single-photon effects, and cavity modulated van der Waals interactions. Moreover, we show that pMBD is computationally efficient and allows simulations of large complex systems coupled to optical cavities., Comment: 10 pages, 3 figures

Tasks of different nature and difficulty levels are a part of people's lives. In this context, there is a scientific interest in the relationship between the difficulty of the task and the persistence need to accomplish it. Despite the generality of this problem, some tasks can be simulated in the form of games. In this way, we employ data from a large online platform, called Steam, to analyze games and the performance of their players. More specifically, we investigated persistence in completing tasks based on the proportion of players who accomplished game achievements. Overall, we present five major findings. First, the probability distribution for the number of achievements is log-normal distribution. Second, the distribution of game players also follows a log-normal. Third, most games require neither a very high degree of persistence nor a very low one. Fourth, players also prefer games that demand a certain intermediate persistence. Fifth, the proportion of players as a function of the number of achievements declines approximately exponentially. As both the log-normal and the exponential functions are memoryless, they are mathematical forms that describe random effects arising from the nature of the system. Therefore our first two findings describe random processes of fragmenting achievements and players while the last three provide a quantitative measure of the human preference in the pursuit of challenging, achievable, and justifiable tasks., Comment: 17 pages, 4 figures

Despite significant efforts devoted to understanding the underlying complexity and emergence of collective movement in animal groups, the role of different external settings on this type of movement remains largely unexplored. Here, by combining time series analysis and complex network tools, we present an extensive investigation of the effects of shady environments on the behavior of a fish species (Silver Carp Hypophthalmichthys molitrix) within earthen ponds. We find that shade encourages fish residence during daylight hours, but the degree of preference for shade varies substantially among trials and ponds. Silver Carp are much slower and exhibit lower persistence in their speeds when under shade than out of it during daytime and nighttime, with fish displaying the highest persistence degree and speeds at night. Furthermore, our research shows that shade affects fish schooling behavior by reducing their polarization, number of interactions among individuals, and the stability among local neighbors; however, fish keep a higher local degree of order when under shade compared to nighttime positions., Comment: 16 pages, 4 figures, supplementary information; accepted for publication in Scientific Reports

The recently developed average-case analysis of optimization methods allows a more fine-grained and representative convergence analysis than usual worst-case results. In exchange, this analysis requires a more precise hypothesis over the data generating process, namely assuming knowledge of the expected spectral distribution (ESD) of the random matrix associated with the problem. This work shows that the concentration of eigenvalues near the edges of the ESD determines a problem's asymptotic average complexity. This a priori information on this concentration is a more grounded assumption than complete knowledge of the ESD. This approximate concentration is effectively a middle ground between the coarseness of the worst-case scenario convergence and the restrictive previous average-case analysis. We also introduce the Generalized Chebyshev method, asymptotically optimal under a hypothesis on this concentration and globally optimal when the ESD follows a Beta distribution. We compare its performance to classical optimization algorithms, such as gradient descent or Nesterov's scheme, and we show that, in the average-case context, Nesterov's method is universally nearly optimal asymptotically., Comment: To be published in ICML 2022

The particle-hole interaction problem is longstanding within time-dependent density functional theory (TDDFT) and leads to extreme errors in the prediction of K-edge X-ray absorption spectra (XAS). We derive a linear-response formalism that uses optimized orbitals of the n-1-electron system as reference, building orbital relaxation and a proper hole into the initial density. Our approach is an exact generalization of the static-exchange approximation that ameliorates particle-hole interaction error associated with the adiabatic approximation and reduces errors in TDDFT XAS by orders of magnitude. With a statistical performance of just 0.5 eV root-mean-square error and the same computational scaling as TDDFT under the core-valence separation approximation, we anticipate that this approach will be of great utility in XAS calculations of large systems., Comment: 11 pages (main text), 12 pages (supporting information), 9 figures

We investigate the use of orbital-optimized references in conjunction with single-reference coupled-cluster theory with single and double substitutions (CCSD) for the study of core excitations and ionizations of 18 small organic molecules, without any use of response theory or equation-of-motion formalisms. Three schemes are employed to successfully address the convergence difficulties associated with the coupled-cluster equations, and the spin contamination resulting from the use of a spin symmetry-broken reference, in the case of excitations. In order to gauge the inherent potential of the methods studied, an effort is made to provide reasonable basis set limit estimates for the transition energies. Overall, we find that the two best-performing schemes studied here for Delta-CCSD are capable of predicting excitation and ionization energies with errors comparable to experimental accuracies. The proposed Delta-CCSD schemes seem to fare better than the widely used equation-of-motion CCSD (EOM-CCSD) with core-valence separation protocol, with statistical errors being reduced by more than a factor of two when compared to FC-CVS-EOM-CCSD.

We systematically investigate the underlying relations among different levels of approximation for simulating electronic de-excitations, with a focus on modeling X-ray emission spectroscopy (XES). Using Fermi's golden rule and explicit modeling of the initial, core-excited state and the final, valence-hole state, we show that XES can be accurately modeled by using orbital optimization for the various final states within a Slater-determinant framework. However, in this paper, we introduce a much cheaper approach reliant only on a single self-consistent field for all the final states, and show that it is typically sufficient. Further approximations reveal that these fundamentally many-body transitions can be reasonably approximated by projections of ground state orbitals, but that the ground state alone is insufficient. Furthermore, except in cases where the core-ionization induces negligible changes in polarization, linear-response approaches within the adiabatic approximation will have difficulty in accurately modeling de-excitation to the core level. Therefore, change in the net dipole moment of the valence electrons can serve as a metric for the validity of the linear-response approximation.

Core-level spectra of 1s electrons of elements heavier than Ne show significant relativistic effects. We combine advances in orbital optimized DFT (OO-DFT) with the spin-free exact two-component (X2C) model for scalar relativistic effects, to study K-edge spectra of third period elements. OO-DFT/X2C is found to be quite accurate at predicting energies, yielding $\sim 0.5$ eV RMS error vs experiment with the modern SCAN (and related) functionals. This marks a significant improvement over the $>50$ eV deviations that are typical for the popular time-dependent DFT (TDDFT) approach. Consequently, experimental spectra are quite well reproduced by OO-DFT/X2C, sans empirical shifts for alignment. OO-DFT/X2C combines high accuracy with ground state DFT cost and is thus a promising route for computing core-level spectra of third period elements. We also explored K and L edges of 3d transition metals to identify limitations of the OO-DFT/X2C approach in modeling the spectra of heavier atoms.

Simulating solids with quantum chemistry methods and Gaussian-type orbitals (GTOs) has been gaining popularity. Nonetheless, there are few systematic studies that assess the basis set incompleteness error (BSIE) in these GTO-based simulations over a variety of solids. In this work, we report a GTO-based implementation for solids, and apply it to address the basis set convergence issue. We employ a simple strategy to generate large uncontracted (unc) GTO basis sets, that we call the unc-def2-GTH sets. These basis sets exhibit systematic improvement towards the basis set limit as well as good transferability based on application to a total of 43 simple semiconductors. Most notably, we found the BSIE of unc-def2-QZVP-GTH to be smaller than 0.7 m$E_h$ per atom in total energies and 20 meV in band gaps for all systems considered here. Using unc-def2-QZVP-GTH, we report band gap benchmarks of a combinatorially designed meta generalized gradient functional (mGGA), B97M-rV, and show that B97M-rV performs similarly (a root-mean-square-deviation (RMSD) of 1.18 eV) to other modern mGGA functionals, M06-L (1.26 eV), MN15-L (1.29 eV), and SCAN (1.20 eV). This represents a clear improvement over older pure functionals such as LDA (1.71 eV) and PBE (1.49 eV) though all these mGGAs are still far from being quantitatively accurate. We also provide several cautionary notes on the use of our uncontracted bases and on future research on GTO basis set development for solids.

L\"owdin's symmetry dilemma is an ubiquitous issue in approximate quantum chemistry. In the context of Hartree-Fock (HF) theory, the use of Slater determinants with some imposed constraints to preserve symmetries of the exact problem may lead to physically unreasonable potential energy surfaces. On the other hand, lifting these constraints leads to the so-called broken symmetry solutions that usually provide better energetics, at the cost of losing information about good quantum numbers that describe the state of the system. This behavior has been previously extensively studied in the context of bond dissociation. This paper studies the behavior of different classes of Hartree-Fock spin polarized solutions (restricted, unrestricted, generalized) in the context of ionization by strong static electric fields. We find that, for simple two-electron systems, UHF is able to provide a qualitatively good description of states involved during the ionization process (neutral, singly-ionized and doubly ionized states), whereas RHF fails to describe the singly ionized state. For more complex systems, even though UHF is able to capture some of the expected characteristics of the ionized states, it is constrained to a single $M_s$ (diabatic) manifold in the energy surface as a function of field intensity. In this case a better qualitative picture can be painted by GHF as it is able to explore different spin manifolds and follow the lowest solution due to lack of collinearity constraints on the spin quantization axis.