The field of quantum chemistry is concerned with the modelling and simulation of the behaviour of molecular systems on the basis of the fundamental equations of quantum mechanics. Since these equations exhibit an extreme case of the curse of dimensionality (the Schrödinger equation for N electrons being a partial differential equation on \mathbb R^{3N} ), the quantum-chemical simulation of even moderate-size molecules already requires highly sophisticated model-reduction, approximation, and simulation techniques. The workshop brought together selected quantum chemists and physicists, and the growing community of mathematicians working in the area, to report and discuss recent advances on topics such as coupled-cluster theory, direct approximation schemes in full configuration-interaction (FCI) theory, interacting Green’s functions, foundations and computational aspects of density-functional theory (DFT), low-rank tensor methods, quantum chemistry in the presence of a strong magnetic field, and multiscale coupling of quantum simulations.
The field of quantum chemistry is concerned with the modelling and simulation of the behaviour of molecular systems on the basis of the fundamental equations of quantum mechanics. Since these equations exhibit an extreme case of the curse of dimensionality (the Schrödinger equation for N electrons being a partial differential equation on \mathbb R^{3N} ), the quantum-chemical simulation of even moderate-size molecules already requires highly sophisticated model-reduction, approximation, and simulation techniques. The workshop brought together selected quantum chemists and physicists, and the growing community of mathematicians working in the area, to report and discuss recent advances on topics such as coupled-cluster theory, direct approximation schemes in full configuration-interaction (FCI) theory, interacting Green’s functions, foundations and computational aspects of density-functional theory (DFT), low-rank tensor methods, quantum chemistry in the presence of a strong magnetic field, and multiscale coupling of quantum simulations.
Quantum chemistry focuses on modelling and simulating the behaviour of molecular systems using the fundamental principles of quantum mechanics. However, the underlying equations, such as the Schrödinger equation for computing the ground state of N electrons, which is a partial differential equation defined on R3N, suffer from the curse of dimensionality. As a result, simulating even moderately sized molecules demands advanced techniques in analysis, approximation and reduced-order modelling for overcoming the naive computational complexity. In this workshop, the rapidly growing mathematical community working in the field together with several quantum chemists and physicists were gathered to discuss recent advances in areas such as (1) the mathematical and numerical analysis of standard models used in the field, including Density Functional Theory, Coupled Cluster, and the Density Matrix Renormalization Group (2) the development of efficient numerical methods, relying e.g. on the development of error bounds and (3) the opportunities of recent data-driven methods towards the approximation of wavefunctions or density matrices.
Quantum chemistry focuses on modelling and simulating the behaviour of molecular systems using the fundamental principles of quantum mechanics. However, the underlying equations, such as the Schrödinger equation for computing the ground state of N electrons, which is a partial differential equation defined on \mathbb{R}^{3N} , suffer from the curse of dimensionality. As a result, simulating even moderately sized molecules demands advanced techniques in analysis, approximation and reduced-order modelling for overcoming the naive computational complexity. In this workshop, the rapidly growing mathematical community working in the field together with several quantum chemists and physicists were gathered to discuss recent advances in areas such as (1) the mathematical and numerical analysis of standard models used in the field, including Density Functional Theory, Coupled Cluster, and the Density Matrix Renormalization Group (2) the development of efficient numerical methods, relying e.g. on the development of error bounds and (3) the opportunities of recent data-driven methods towards the approximation of wavefunctions or density matrices.
The conventional numerical methods when applied to multidimensional problems suffer from the so-called "curse of dimensionality", that cannot be eliminated by using parallel architectures and high performance computing. The novel tensor numerical methods are based on a "smart" rank-structured tensor representation of the multivariate functions and operators discretized on Cartesian grids thus reducing solution of the multidimensional integral-differential equations to 1D calculations. We explain basic tensor formats and algorithms and show how the orthogonal Tucker tensor decomposition originating from chemometrics made a revolution in numerical analysis, relying on rigorous results from approximation theory. Benefits of tensor approach are demonstrated in ab-initio electronic structure calculations. Computation of the 3D convolution integrals for functions with multiple singularities is replaced by a sequence of 1D operations, thus enabling accurate MATLAB calculations on a laptop using 3D uniform tensor grids of the size up to 10 15 . Fast tensor-based Hartree-Fock solver, incorporating the grid-based low-rank factorization of the two-electron integrals, serves as a prerequisite for economical calculation of the excitation energies of molecules. Tensor approach suggests efficient grid-based numerical treatment of the long-range electrostatic potentials on large 3D finite lattices with defects.The novel range-separated tensor format applies to interaction potentials of multi-particle systems of general type opening the new prospects for tensor methods in scientific computing. This research monograph presenting the modern tensor techniques applied to problems in quantum chemistry may be interesting for a wide audience of students and scientists working in computational chemistry, material science and scientific computing.
Mathematical Chemistry // Editors: W. I. Hong // Book description: Mathematical chemistry is the area of research engaged in the novel applications of mathematics to chemistry ; it concerns itself principally with the mathematical modeling of chemical phenomena. Mathematical chemistry has also sometimes been called computer chemistry, but should not be confused with computational chemistry. // Major areas of research in mathematical chemistry include chemical graph theory, which deals with topics such as the mathematical study of isomerism and the development of topological descriptors or indices which find application in quantitative structure-property relationships ; chemical aspects of group theory, which finds applications in stereochemistry and quantum chemistry ; and topological aspects of chemistry. // Table of Contents: // Preface // Exact Treatment of Finite Rank Modifications of Finite-Dimensional and Infinite-Dimensional Quantum Systems // (Tomislav P. Živkovic, Ruđer Boskovic Institute, Zagreb, Croatia) // Molecular Symmetry and Fuzzy Symmetry // (Xuezhuang Zhao, Department of Chemistry, Nankai University, Tianjin, P.R.China) // Information Perspective on Molecular Electronic Structure // (Roman F. Nalewajski, Faculty of Chemistry, Jagiellonian University, Cracow, Poland) // The Chemistry and Mathematics of DNA Polyhedra // (Wen-Yuan Qiu, Ze Wang, Guang Hu, State Key Laboratory of Applied Organic Chemistry, Department of Chemistry, Lanzhou University, Lanzhou, People’s Republic of China) // High-Gain Observers for Estimation of Kinetics in Batch and Continuous Bioreactors // (Dan Seli^(o)teanu, Emil Petre, Dorin ^(a)endrescu, Monica Roman, Department of Automatic Control, University of Craiova, Craiova, Romania) // Remarks on Novel Exact Solution Methodologies in Wavefunction Analysis // (Valentino A. Simpao, Mathematical Consultant Services, Greenville, Kentucky, USA) // On The Enumeration of Various Networks // (Michael J. Bucknum, Eduardo A. Castro, Consejo de Investigaciones Cientificas y Tecnicas (CONICET), Buenos Aires, Argentina, and others) // Chemical Physics of Phonons and Superconductivity: A Heuristic Approach // (Michael J. Bucknum, Eduardo A. Castro, INIFTA, Division Quimica Teorica Universidad Nacional de La Plata, La Plata, Buenos Aires, Argentina) // An Intensity-Enhanced 2D Graphical Representation of DNA and RNA Sequence // (Teik-Cheng Lim, School of Science and Technology, SIM University (UniSIM), Republic of Singapore) // Laguerre Functions with Applications from Mathematical Chemistry // (Damian Trif, Department of Applied Mathematics, Babes-Bolyai University of Cluj-Napoca, Romania) // Stability of Chemical and Phase Equilibrium: Alternative Forms of Equations for Themodynamic Analysis // (A.M. Toikka, Saint-Petersburg State University, Department of Chemical Thermodynamics & Kinetics, Saint-Petersburg, Russia) // Limitations in Using the Sanderson's Principle to Estimate Global Reactivity // (Jorge Martinez, Sociedad de Innovacion Cientifica--Tecnologica InnovaSciTech Ltda., Santiago, Chile) // A General Method for Central Potentials in Quantum Mechanics // (Amlan K. Roy, Division of Chemical Sciences, Indian Institute of Science Education and Research (IISER), Kolkata, India) // A Review of Reduction Methods for Multiple Time Scale Stochastic Reaction Networks // (Chang Hyeong Lee, Roger Liu, School of Mechanical and Advanced Materials Engineering, Ulsan Nacional Institute of Science and Technology (UNIST), South Korea) // Index
Studying complex molecular systems, whether in the gas or a condensed phase, requires highly accurate quantum chemistry methods whose computational effort remains tractable when treating realistic systems. Commonly used ab initio methods such as Coupled Cluster (CC) or Density Functional Theory (DFT) generally provide reliable results when the underlying approximations and parameterizations are appropriate for the systems being studied. Among the less popular quantum chemistry methods, Valence Bond (VB) theory and Quantum Monte Carlo (QMC) methods, each provides different unique advantages and disadvantages. Ab initio VB, while usually only qualitative for energetic properties such as enthalpies of formation, has the distinct advantage of providing very compact wave functions that can be readily mapped into the intuitive Lewis structures that chemists very often use when qualitatively describing the valence electronic structure of atoms and molecules. QMC using trial wave functions based on multireference ab initio quantum methods can give some of the most accurate energies and related properties, however, this is often at a great computational cost. Furthermore, the large wave function expansions in the trial function are often difficult to qualitatively interpret. In this lecture, we will present the Valence-Bond Quantum Monte Carlo methods (VB-QMC) which retain the interpretability of VB and the high accuracy of QMC. We will show a few applications which demonstrate their accuracy for molecules in the gas phase and we will examine areas where innovations could be made.
The field of quantum chemistry has long been defined by the central computational challenge of the quantum many-body fermionic system problem, main the electronic Schrödinger equation. This exponential scaling makes it computationally intractable to solve, the exact governing electronic Schrödinger equation. Approximation methods have been developed, ranging from the widely used Density Functional Theory (DFT) to Coupled Cluster (CC) theory. While powerful, these conventional approaches often involve controlled truncations which caused a trade-off between accuracy, leaving many systems—particularly those with strong electron correlation—beyond the reach of ab initio simulation. Furthermore, the famous fermionic sign oscillation problem is particularly severe in stochastic methods like Quantum Monte Carlo (QMC). 1 Traditional Ansätze are also frequently engineered specifically for particular systems, which results in a fragmentation of results and fundamentally limits the transferability. Recent research demonstrates that with carefully designed architectures, ANNs can not only represent ground states of model systems with high precision but also tackle the sign problem, offering a promising path forward for strongly correlated electronic structure calculations.2,3 This review charts the paradigm shift driven by Neural Network Quantum States (NNQS), which leverage the representational power of deep learning to overcome these barriers. We detail the architectural evolution from Restricted Boltzmann Machines to autoregressive models like RNNs and Transformers, which enable exact, uncorrelated sampling and bypass critical bottlenecks.2,4,5Furthermore, we explore the geometric foundation of high-efficiency optimization via the neural networks, which decouples optimization cost from model complexity.6 The frontier of the field is marked by the integration of operator learning and hybrid quantum-classical frameworks, such as Reinforcement Learning for contracted quantum eigen-solvers and Quantum-Enhanced Neural Networks, which leverage quantum processors to enhance expressivity for hardware-efficient Ansätze.7,8 By framing the problem through the lens of computational resource allocation—specifically, trading exponential memory demands for polynomialcomplexity optimization and sampling—this review elucidates how recent advances at the intersection of artificial neural networks and quantum computing have evolved. The emergence of those methods not merely progress into a powerful simulation-tools, but also a hybrid computational interface.9 This convergence offers a versatile and systematically improvable framework for potentially addressing some of the most challenging problems in quantum physics and chemistry.