Abstract
<jats:p> The many-body Schrödinger equation is a fundamental framework for describing the behaviors of electrons in molecular systems based on quantum mechanics. It largely forms the basis for quantum-chemistry-based energy calculation and has become the core concept of modern electronic structure theory. <jats:sup>1</jats:sup> However, its complexity increases exponentially with the number of interacting particles, and its exact solution remains intractable in most cases. To bridge this gap, various approximation strategies have been developed, forming the basis of modern quantum chemistry. <jats:sup>1</jats:sup> Accurate approximation methods for the many-body Schrödinger equation enable the solution of complex, otherwise intractable problems with enhanced accuracy and balanced computational costs. This review focuses on the conceptual foundations of these approximation approaches, ranging from mean-field theories, such as Hartree–Fock, through post–Hartree–Fock correlation methods, including configuration interaction, perturbation theory, and coupled-cluster techniques, to the widely adopted density functional theory and semi-empirical models; newly emerging methods such as quantum Monte Carlo, machine learning-augmented strategies, hybrid between classical and quantum computational methods, and the theoretical frontier of holographic principle which applies Ads/CFT correspondence to approximate quantum many body system inspired by higher energy physics would also be addressed. <jats:sup>2-8</jats:sup> The accuracy and efficiency will also be briefly mentioned for those approximation methods, highlighting the trade-offs between theoretical rigor and computational feasibility. These approximation strategies for the many-body Schrödinger equation continue to be central to quantum chemistry, enabling increasingly reliable predictions of molecular structure, energetics, and dynamics with reduced computational costs. </jats:p>