Understanding the properties of electronically excited states is a challenging task that becomes increasingly important for numerous applications in chemistry, molecular physics, molecular biology, and materials science. A substantial impact is exerted by the fascinating progress in time-resolved spectroscopy, which leads to a strongly growing demand for theoretical methods to describe the characteristic features of excited states accurately. Whereas for electronic ground state problems of stable molecules the quantum chemical methodology is now so well developed that informed nonexperts can use it efficiently, the situation is entirely different concerning the investigation of excited states. This review is devoted to a specific class of approaches, usually denoted as multireference (MR) methods, the generality of which is needed for solving many spectroscopic or photodynamical problems. However, the understanding and proper application of these MR methods is often found to be difficult due to their complexity and their computational cost. The purpose of this review is to provide an overview of the most important facts about the different theoretical approaches available and to present by means of a collection of characteristic examples useful information, which can guide the reader in performing their own applications.

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We prove the existence of nonequilibrium phases of matter in the prethermal regime of periodically driven, long-range interacting systems, with power-law exponent α > d, where d is the dimensionality of the system. In this context, we predict the existence of a disorder-free, prethermal discrete time crystal in one dimension-a phase strictly forbidden in the absence of long-range interactions. Finally, using a combination of analytic and numerical methods, we highlight key experimentally observable differences between such a prethermal time crystal and its many-body localized counterpart.

Semiempirical theories of bonding in molecules containing transition metals are often merely interpretive. That is, given the experimental fact that a molecule exists and has a particular shape, theory is used to provide a plausible ad hoc justification. Much interpretation of this type uses molecular orbital theory1 and ascribes bonding effects to orbital mixing. Earlier approaches such as ligand-field theory have been more or less abandoned along with the alternative insights they provided. Modern ab initio methods, on the other hand, have the possibility of being quantitatively predictive. They can be used to study the relative energies of the observed structure and nonobserved plausible alternatives. They can also be used to examine * Author to whom correspondence should be addressed. i Deceased, March 28, 1991.

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Pressure alters the physical, chemical, and electronic properties of matter. The diamond anvil cell enables tabletop experiments to investigate a diverse landscape of high-pressure phenomena. Here, we introduce and use a nanoscale sensing platform that integrates nitrogen-vacancy (NV) color centers directly into the culet of diamond anvils. We demonstrate the versatility of this platform by performing diffraction-limited imaging of both stress fields and magnetism as a function of pressure and temperature. We quantify all normal and shear stress components and demonstrate vector magnetic field imaging, enabling measurement of the pressure-driven a ↔ D phase transition in iron and the complex pressure-temperature phase diagram of gadolinium. A complementary NV-sensing modality using noise spectroscopy enables the characterization of phase transitions even in the absence of static magnetic signatures.3 of 6 Fig. 2. Full tensorial reconstruction of the stresses in a (111)-cut diamond anvil. (A) Spatially resolved maps of the loading stress (left) and mean lateral stress (right), s ⊥ ¼ 1 2 ðs XX þ s YY Þ, across the culet surface.In the inner region, where the culet surface contacts the pressure-transmitting medium (16:3:1 methanol/ ethanol/water), the loading stress is spatially uniform, whereas the lateral stress is concentrated toward the center; this qualitative difference is highlighted by a linecut (taken along the white-dashed line) of the two stresses (below), and reconstructed by finite-element analysis (orange and purple dashed lines). The black pixels indicate where the NV spectrum was obfuscated by the ruby microsphere. (B) Comparison of all stress tensor components in the fluid-contact region at P ¼ 4:9 GPa and P ¼ 13:6 GPa. At P ¼ 13:6 GPa, the pressure-transmitting medium has entered its glassy phase, and we observe a spatial gradient in the loading stress s ZZ (inset).

Extending the framework of statistical physics to the nonequilibrium setting has led to the discovery of previously unidentified phases of matter, often catalyzed by periodic driving. However, preventing the runaway heating that is associated with driving a strongly interacting quantum system remains a challenge in the investigation of these newly discovered phases. In this work, we utilize a trapped-ion quantum simulator to observe the signatures of a nonequilibrium driven phase without disorder—the prethermal discrete time crystal. Here, the heating problem is circumvented not by disorder-induced many-body localization, but rather by high-frequency driving, which leads to an expansive time window where nonequilibrium phases can emerge. Floquet prethermalization is thus presented as a general strategy for creating, stabilizing, and studying intrinsically out-of-equilibrium phases of matter.

We analyze the dynamics of periodically-driven (Floquet) Hamiltonians with short-and long-range interactions, finding clear evidence for a thermalization time, τ * , that increases exponentially with the drive frequency. We observe this behavior, both in systems with short-ranged interactions, where our results are consistent with rigorous bounds, and in systems with long-range interactions, where such bounds do not exist at present. Using a combination of heating and entanglement dynamics, we explicitly extract the effective energy scale controlling the rate of thermalization. Finally, we demonstrate that for times shorter than τ * , the dynamics of the system is well-approximated by evolution under a time-independent Hamiltonian D eff , for both short-and long-range interacting systems.