This chapter begins with an introduction to how full quantum study benefits understanding the dynamic equilibrium of material composition in space and the earth environment. It then delves into the microscopic structure and dynamics of water as influenced by full quantum effects under natural conditions (i.e., near room temperature and atmospheric pressure), aiming at exploring their impact on the environment. Specific topics include the absorption and diffusion of water molecules on carbon and metal surfaces, anomalous static and dynamical isotope phenomena in bulk water, the structure and dynamic behavior of water clusters (e.g., H5O2+, H3O2−), and ion hydrates (e.g., Cl-•nH2O). It also discusses fluorine–water reaction in water and some aspects of atmospheric chemistry, including ozone formation and stability, the reaction of HOSO with NO2, and the reaction of HOSO + NO2 with H2O. In these scenarios, nuclear quantum effects influence hydrogen bond strength, water adsorption energy, proton-permeation configuration, liquid water volume, water molecule collision rate, water tunneling diffusion pathways, and inter-atomic distances, which results in unexpected properties. Regarding atmospheric chemical reactions, the chapter highlights the significant impact of nuclear quantum effects on the reaction potential energy profile, forming the basis for understanding microscopic reactions underlying air, ocean, and soil pollution.

This chapter provides readers who are encountering full quantum effects for the first time, with a comprehensive conceptual understanding of the condensed matter physics problems, and prepares them for further exploration of the intrinsic connection between specific macroscopic phenomena and their underlying microscopic mechanisms. It can introduce, in light-element systems, significant nuclear quantum effects (NQEs) and nonadiabatic effects, including quantum tunneling, quantum fluctuations, zero-point energy, and electron–phonon coupling. These effects have significant influence on the atomic structure (e.g., liquid state of helium), electronic structure (e.g., electron bandgap), thermodynamics (e.g., ultra-low thermal conductivity), optical properties (e.g., absorption spectra and formation of polaron), and quantum paraelectricity. As a prominent example, the superfluid and supersolid in helium systems originate from the NQEs. And nonadiabatic effects affect atomic vibrations (e.g., shift of phonon frequency) and collisions, leading to the breaking and reorganization of chemical bonds. These processes significantly affect ultrafast electron transfer (e.g., novel transfer rate dependence outside Fermi’s golden rule), electron–phonon coupling strength, the formation of charge density waves, phase transition dynamics, Berry phase, and superconductivity.

Chapters 1–4 give a comprehensive and detailed description of the physical-geometric optics method (PGOM). As the full name of PGOM implies, the method combines the theories and techniques of geometric optics and physical optics. The development of PGOM is inspired by previous research efforts on geometric optics and endeavors to improve the accuracy of geometric optics methods in light scattering computations by incorporating the effects of physical optics. Chapter 5 first presents a summary of PGOM from the perspectives of theory, technique, and applications. Then, we present our view of future efforts toward improving PGOM and enhancing its downstream applications.

In Chapter 3, the improved geometric optics method based on electromagnetic integral relations is introduced. Both the surface/volume integral equations linking the near field on the particle surface/volume internal field to the far field are derived. A proof is presented to show the equivalence of the surface and volume integral equations. The surface (or volume) integral equation is then employed to map the near field computed by the geometric optics method to the far field. To improve the computational efficiency of the mapping of the near field to the far field as well as ray tracing, a broad-beam method is presented. A beam-splitting technique based on computer graphics is presented to facilitate efficient beam tracing processes. The performance of PGOM is evaluated via comparing the PGOM simulations and the benchmarks provided by the invariant-imbedding T-matrix method (IITM). Furthermore, a simplified physical-geometric optics method, which considers the interference of emerging waves through the “ray-spreading effect,” is illustrated. A number of examples for the physical-geometric optics method and its simplified version are presented. Finally, this chapter presents a synergistically unified method based on a combination of IITM for small-to-moderate size parameters and PGOM for moderate-to-large size parameters.

This chapter delves into theoretical methods of electronic structure, which is the basis to simulate a real condensed matter system accurately at the quantum-mechanical level. It begins with an overview of the history of full quantum theory, which describes the quantum properties of nucleus, electron and their quantum coupling. Beyond the Born–Oppenheimer approximation, Born–Huang expansion is introduced which expands the system as a sum over electron–nuclear wavefunction products. It is a rigorous approach describing electron–phonon coupling and is believed to become the dominant paradigm in condensed matter physics with the rapid improvement of simulation and high-resolved experimental techniques. History, theoretical framework, and challenges of the mainstream electronic structure calculation methods are introduced. These include wavefunction method (Hartree method, Hartree–Fock method, and post-Hartree–Fock method), density functional theory (DFT), and Quantum Monte Carlo Method (QMC). The accuracy of post-Hartree–Fock methods (consider dynamical correlation of electron) makes them remain important while DFT is the most common method in physics, chemistry and material, being important foundation of full-quantum calculation. The impact of QMC keeps increasing in computational physics due to its adaptation to supercomputers. Then, the method for excited electronic state calculation is introduced, which is also needed in full quantum calculation. The GW Method is recommended because it can treat the quasiparticle–electron excitation properties accurately. Finally, the chapter introduces current realizations of electronic-structure calculation and their challenges (e.g., balance between accuracy and efficiency). It discusses key concepts such as basis set, pseudopotential, and boundary condition. The associated programs are summarized.

A key goal of condensed matter physics is to describe the properties of a quantum many-body interacting system in an accurate manner. The Born–Oppenheimer (BO) approximation, which was proposed right after the establishment of the Schrödinger Equation, has been the foundation of researches on polyatomic systems including condensed matter for about 100 years. For theoreticians, it is the starting point for the vast majority of simulations; for experimentalists, it is the theoretical tool on which most researchers rely when interpreting observations. Based on the BO approximation, the electronic structures of condensed matter can be described using a specific spatial configuration of the nuclei. Upon this, the vibrations of the lattice were treated using the concept of phonon. The electron–phonon interactions exist, and they can be addressed in terms of the many-body perturbation theory. In recent years, however, there has been a growing interest in studies of condensed matter beyond this scenario, driven by advancements in high-precision experiments and theoretical simulations. This situation prompts the studies of full quantum effects (FQEs), including those related to the nuclear quantum effects beyond the harmonic approximation and those related to the nonadiabatic effects. The Born–Huang expansion forms the basis of these studies. Various experiments introduced here were pushed to the limits of ultrahigh resolution in order to highlight observational details reflecting FQEs rather than the classical ones. Using water, the most ubiquitous substance on earth as an example, we show that FQEs can, both qualitatively and quantitatively, affect the water hydrogen bond structure, energy, and dynamics. Although FQEs are found in the past mostly prominent under extreme condition (e.g., low temperature, ultrafast timescale, single-molecule level) or in the elementary process (e.g., electron excitation, proton transfer, vibration), the unveiled law is expected to extend to the wide condensed physical and chemistry systems. In these systems, some of the basic interactions in the Schrödinger Equation, such as electron–electron and electron–nucleus interactions as well as nucleus character energy, are comparable to each other, which makes quantization of nuclei indispensable for accurate simulations and measurements. As such, simulation methods for nuclear quantum dynamics, nonadiabatic dynamics with electron excitation under external fields, as well as combination of the two, are the particular challenge in developing computational methods for FQE research. Experiments, in the meantime, are moving in a direction that combines ultrafast laser technology with traditional spectroscopy, while characterization techniques and the material quality need optimization in special environments to allow effective observations of FQE at the single-atom or single-bond level.

Chapter 2 discusses the concept of rays as localized plane waves and elucidates the criteria for the validity of defining a ray. This chapter also presents the conventional ray-tracing technique for light scattering by a nonabsorptive particle. In particular, the incident rays are specified with the Monte Carlo method or in a deterministic form. The ray directions in the ray-tracing process are specified in a closed set of equations and a vector form without referring to specific coordinate systems. Furthermore, the contributions of emerging (scattered) rays and diffraction to the amplitude scattering matrix are explicitly derived, followed by the formulas for averaging the single-scattering properties over particle orientations with respect to three Euler angles. For randomly oriented particles, a simplified ray-tracing method based on Stokes parameters is presented. The remaining portion of this chapter focuses on the ray-tracing process involving an absorptive particle, within which the electromagnetic waves may be inhomogeneous. Furthermore, the scattering of light by a particle with surface roughness is also discussed. This chapter ends with summarizing the inherent shortcomings of the conventional ray-tracing technique.

This chapter focuses on research progress related to full quantum effects in dense hydrogen, hydride compounds, and proton transport through 2D materials and bulk materials where hydrogen does not dominate in content. For dense hydrogen, this chapter describes nuclear quantum effects in high-pressure phase diagram, phase transition, and physical property. For the most common hydride compound, water, this chapter introduces nuclear quantum effects in water hydrogen bond property, water phase diagram, and physical and chemical property of confined water and ice. For other hydride compound, this chapter introduces full quantum effects in hydride superconductivity and nonlinear optical property of mineral hydride. These studies reveal that, full quantum effect not only has significant contribution to existing properties, such as hydrogen bond strength and phase diagram boundary probed by isotope substitution, but also causes novel states and properties, such as room temperature, high-pressure superconductivity of hydrides (e.g., sulfur hydride) and liquid forms of metallic and superconducting hydrogen. After these, this chapter introduces theory of proton tunneling rate, which is important from physical to life sciences. Theory of proton transport in perovskite-type oxides is also illustrated as typical example of nuclear quantum effects on hydrogen conduction in bulk materials. Finally, the chapter introduces work demonstrating importance of proton tunneling in DNA mutation.

This chapter explores, for condensed matter formed by other elements, the novel physical and chemical properties that are dominated by full quantum effects. This includes elementary substances including relatively light elements (e.g., helium (He), lithium (Li), carbon (C), and boron (B)) as well as relatively heavy elements (e.g., oxygen (O) and silicon (Si)), and their compounds. The objects studied include liquid and solid helium, lithium bulk metal and clusters, bulk boron, borophene, boron nitride nanotubes, hexagonal boron nitride, magnesium diboride, diamond, graphene, carbon atoms in organic molecules, strontium titanate, barium ferrite, silicon semiconductors, etc. The related physical and chemical properties include superfluidity, supersolidity, elastic or plastic deformation, heat capacity, high-pressure phase diagram, bonding lengths, diffusion, crystal structure, electron–phonon coupling, bandgap renormalization, thermal expansion coefficient, thermal conductivity, phonon frequency distribution, superconducting temperature and light absorption, etc. Through a variety of fruitful aspects readers would overview the situations where full quantum effect is pronounced and even critical.

This chapter explores the significant influence of full quantum effects on chemical reactions, surpassing the framework of classical transition-state theory (TST). For light element systems (e.g., H2, O2), quantum tunneling, vibration, and electron–nucleus quantum coupling have decisive impacts on the reaction process. Advancements in reaction rate theory incorporate tunneling and zero-point energy into TST, while the field waits for the development of a more comprehensive TST that precisely treats full quantum effects. Nuclear quantum effect is critical in hydrogen dynamics at room temperatures (even above 500 K), ultracold reactions, penning discharge, nonclassical product formation in organic reactions, performance improvement in electrolysis utilizing kinetic isotope effects (e.g., in aqueous batteries), and biochemical reactions. Quantum tunneling is likely to facilitate the origin of life at its early stage by increasing the rate of forming organic molecules. Nonadiabatic effects (NAEs) manifest in altering electronic eigenstate population (e.g., relaxation dynamics) and dynamic nuclear transfer between split potential energy surfaces (e.g., chlorine–hydrogen collision, water photodissociation), demonstrated or indicated by experiment or theoretical calculation.