The Schrödinger equation is the master equation of quantum chemistry. The founders of quantum mechanics realised how this equation underpins essentially the whole of chemistry. However, they recognised that its exact application was much too complicated to be solvable at the time. More than two generations of researchers were left to work out how to achieve this ambitious goal for molecular systems of ever-increasing size. This book focuses on non-mainstream methods to solve the molecular electronic Schrödinger equation. Each method is based on a set of core ideas and this volume aims to explain these ideas clearly so that they become more accessible. By bringing together these non-standard methods, the book intends to inspire graduate students, postdoctoral researchers and academics to think of novel approaches. Is there a method out there that we have not thought of yet? Can we design a new method that combines the best of all worlds?Contents:Intracule Functional Theory (Deborah L Crittenden and Peter M W Gill)Explicitly Correlated Electronic Structure Theory (Frederick R Manby)Solving Problems with Strong Correlation Using the Density Matrix Renormalization Group (DMRG) (Garnet Kin-Lic Chan and Sandeep Sharma)Reduced-Density-Matrix Theory for Many-electron Correlation (David A Mazziotti)Finite Size Scaling for Criticality of the Schrödinger Equation (Sabre Kais)The Generalized Sturmian Method (James Avery and John Avery)Slater-Type Orbital Basis Sets: Reliable and Rapid Solution of the Schrödinger Equation for Accurate Molecular Properties (Philip E Hoggan)Modern Ab Initio Valence Bond Methods (Philippe C Hiberty & Sason Shaik)Quantum Monte Carlo Approaches for Tackling Electronic Correlation (Massimo Mella and Gabriele Morosi)Solving the Schrödinger Equation on Real-Space Grids and with Random Walks (Thomas L Beck and Joel H Dedrick)Changes in Dense Linear Algebra Kernels: Decades-Long Perspective (Piotr Luszczek, Jakub Kurzak, and Jack Dongarra)Readership: Graduate students, postdoctoral researchers and academics in the fields of computational chemistry, theoretical chemistry, super computing, molecular physics and solid state physics.