The accuracy of large-scale ab initio calculations is always desired, but they are unfortunately not usually practical in many fields, such as studies of heterogeneous catalysis, photosynthesis, and functional mechanisms of biomolecules. DFT serves as a prevailing method for large calculations due to its compromise between cost and accuracy, but it has a number of failure cases, like bond breaking. On the other hand, the simple theoretical nature of force-field simulations, which often work well, reminds us that large systems are not necessarily exponentially complicated. A recurring theme in the electronic structure of large systems is therefore the use of fragment-based approaches. Existing fragment approaches typically treat the most powerful electron correlations perturbatively, on the same footing with dispersion forces (if at all). In our approach, electrons are not treated individually in the global calculation, but subunits will be chosen. (e.g., amino acid residues or individual water molecules). The most important local states are selected to account for inter-subunit interactions. Most familiar wavefunction methods (SCF, PT, coupled-cluster) can be applied directly to this internally correlated Hamiltonian, and the Hamiltonian itself is systematically improvable to exactitude. Furthermore, different parts of a system can be treated at different accuracy. Our initial investigations focus on the correlation structure of groups of harmonic oscillators, where the coupling inside of groups is stronger than between groups. This mimics discrete molecules while providing us with exact analytical solutions for comparison. Correlations between grouped oscillators will be characterized by both numerical and analytical methods, in order to shed light on building an algorithm for real molecules and systems.
Abstract ................................................................................................. 359 4.1 Introduction .................................................................................. 359 4.2 Feynman’s Path Integral Quantum Formalism ............................ 364 4.2.1 Construction of Path Integral ........................................... 364 4.2.2 Propagator’s Equation ...................................................... 370 4.2.3 Recovering Wave Function’s Equation ............................ 373 4.3 Path Integrals for Basic Matter’s Structures ................................ 377 4.3.1 General Path Integral’s Properties ................................... 377 4.3.2 Path Integral for the Free Particle .................................... 384 4.3.3 Path Integral for Motion as the Harmonic Oscillator ...... 387 4.3.4 Path Integral Representation for the Bohr’s Atom ........... 394 4.3.5 Path Integral for Motion in the Quantum Well ................ 398 4.4 Density Matrix Approach Linking Path Integral Formalism ....... 402 4.4.1 On Mono-, Many-, and Reduced-Electronic Density Matrices .............................................................. 402 4.4.2 Canonical Density, Bloch Equation, and the Need of Path Integral ....................................................... 412 4.5 Roots of Self-Consistent Methods in Quantum Chemistry .......... 419 4.5.1 Molecular Orbital Approach ............................................ 420 4.5.2 Roothaan Approach .......................................................... 423 4.5.3 Introducing Semi-Empirical Approximations .................. 428 4.5.3.1 NDO Methods ................................................... 429 4.5.3.2 NDDO Methods ................................................ 432 4.5.4 Ab initio Methods: The Hartree-Fock Approach ............. 436 4.5.4.1 Hartree-Fock Orbital Energy ............................ 438 4.5.4.2 About Correlation Energy ................................. 442 4.5.4.3 Koopman’s Orbital Theorem with Hartree-Fock Picture ......................................... 444 4.5.4.4 Chemical Reactivity Indices in Orbital Energy Representation ...................................... 451 4.5.4.5 Testing Koopmans Theorem by Chemical Harness Reactivity Index .................. 458 4.6 Density Functional Theory: Observable Quantum Chemistry ..... 468 4.6.1 Hohenberg-Kohn theorems .............................................. 470 4.6.2 Optimized Energy-Electronegativity Connection ............ 475 4.6.3 Popular Energetic Density Functionals ............................ 480 4.6.3.1 Density Functionals of Kinetic Energy ............. 481 4.6.3.2 Density Functionals of Exchange Energy ......... 484 4.6.3.3 Density Functionals of Correlation Energy ...... 490 4.6.3.4 Density Functionals of Exchange-Correlation Energy ........................... 496 4.7 Observable Quantum Chemistry: Extending Heisenberg’s Uncertainty ............................................................. 502 4.7.1 Periodic Path Integrals ..................................................... 502 4.7.1.1 Effective Partition Function .............................. 502 4.7.1.2 Periodic Quantum Paths .................................... 505 4.7.2 Heisenberg Uncertainty Reloaded ................................... 508 4.7.3 Extending Heisenberg Uncertainty ...................................511 4.7.3.1 Averaging Quantum Fluctuations ......................511 4.7.3.2 Observed Quantum Evolution .......................... 514 4.7.3.3 Free Quantum Evolution ................................... 515 4.7.3.4 Free vs. Observed Quantum Evolution ............. 518 4.7.3.5 Spectral-IQ Method .......................................... 523 4.7.3.6 Spectral-IQ Results on Silica Sol-Gel-Based Mesosystems ..................................................... 527 4.8 Conclusion ................................................................................... 533 Keywords .............................................................................................. 535 References ............................................................................................. 536 Author’s Main References ........................................................... 536 Specific References ...................................................................... 537 Further Readings .......................................................................... 549 ABSTRACT Basing on the first principles of Quantum mechanics as exposed in the previous chapters and sections, special chapters of quantum theory are here unfolded in order to further extend and caching the quantum information from free to observed evolution within the matter systems with constraints (boundaries). As such, the Feynman path integral formalism is firstly exposed and then applied to atomic, quantum barrier and quantum harmonically vibration, followed by density matrix approach, opening the Hartree-Fock and Density Functional pictures of many-electronic systems, with a worthy perspective of electronic occupancies via Koopmans theorem, while ending with a further generalization of the Heisenberg observability and of its first application to mesosystems.
Quantum chemistry models atoms and molecules using quantum mechanics, with the electronic structure problem at its core. This problem, which determines the energy and properties of electrons in a molecule, is exponentially hard to solve exactly on classical computers. To approximate solutions, effective models introduce layers of simplifications. However, these approximations can fail for strongly correlated systems, necessitating alternative approaches. Quantum computers, capable of processing quantum information natively, offer a promising path to simulate molecular systems beyond classical reach, with potential applications in catalysis, materials science, and photochemistry. Despite rapid advancements in quantum hardware, practical quantum computations remain challenging due to limited resources. This thesis optimizes quantum resource usage in computational chemistry, introducing novel embedding techniques and exploring applications beyond ground-state energies. Chapter 1 establishes the foundation by reviewing quantum chemistry methods, from Hartree-Fock and Density Functional Theory (DFT) to active space approaches. It also provides an introduction to quantum computing for chemistry, focusing on near-term algorithms. The following chapters present novel methods aimed at reducing quantum resource requirements and tackling problems classical methods struggle with. Chapter 2 introduces a technique to reduce the
λ-norm of the electronic structure Hamiltonian, a key measure of quantum resource demand. By transforming molecular orbitals, this method enhances sampling efficiency in Noisy Intermediate-Scale Quantum (NISQ) algorithms and reduces runtime in Fault-Tolerant Quantum Computing (FTQC). Localized orbitals outperform canonical ones, and an orbital-optimization scheme is proposed to minimize the
λ-norm. Benchmarks on hydrogen chains, FeMoco, and Ruthenium complexes demonstrate its effectiveness. Chapter 3 presents the FragPT2 framework for simulating large systems with multiple interacting fragments. This method embeds each fragment in the mean field of others and applies multi-reference perturbation theory to capture inter-fragment correlations such as dispersion and charge transfer efficiently. The approach reduces computational overhead and enables scalable simulations, as shown in applications to \ce{N2} dimers, aromatic dimers, and butadiene. Chapter 4 focuses on photochemical systems, where multiple electronic states must be treated equally. The state-averaged orbital-optimized variational quantum eigensolver (SAOOVQE) is extended with analytical energy gradients and non-adiabatic couplings, enabling geometry optimizations and simulations of non-adiabatic dynamics on quantum hardware. This is demonstrated by locating the conical intersection in formaldimine, highlighting SAOOVQE’s potential for studying photoisomerization in biomolecules. Building on this, Chapter 5 develops a quantum algorithm to detect conical intersections, where the Born-Oppenheimer approximation breaks down. These intersections are characterized by a discrete Berry phase (0 or
π), representing their topological nature. The algorithm extracts this phase as a single bit of information, ensuring resilience to noise. Analytical proof and numerical validation using a toy molecular model illustrate its effectiveness for studying light-perception biochemistry and related photochemical processes. Finally, Chapter 6 explores machine learning density functionals trained on noisy quantum data. While DFT is computationally efficient, its accuracy depends on approximated exchange-correlation functionals. This work proposes a neural network-based approach to learn the exact functional from quantum data, showing that training on noisy outputs from NISQ devices can mitigate sampling errors. Applications to the Fermi-Hubbard model and other systems demonstrate its potential to generalize and provide accurate results under noise. This thesis advances quantum computational chemistry by optimizing quantum resource usage, developing new embedding strategies, and expanding the scope of quantum simulations beyond ground-state properties. These contributions pave the way for practical quantum algorithms that could revolutionize molecular simulations in the near future.
This dissertation is focused around enabling large-scale quantum chemistry calculations by developing novel algorithms on new heterogeneous hardware architectures. Section 2 contains a study on assessing the importance of many-body dispersion phenomena in molecular clusters and understanding the need to incorporate the phenomena into existing quantum chemistry methods to accurately capture long-range interactions. It is known that highly accurate quantum chemistry methods are computationally expensive; therefore, there is a need for new computer algorithms that will be able to vastly improve the time to solution and enable correlated quantum chemistry calculations on large molecular systems. Sections 3 and 4 focus on the development and implementation of various algorithms that enable quantum chemistry calculations on molecules of unprecedent sizes. Specifically, Section 3 describes a novel CUDA-based algorithm that enables GPU-acceleration of the main parts of the Self-Consistent Field (SCF) cycle, resulting in the fastest SCF code to date. Section 4 introduces a novel OpenMP-based implementation of the Fock build formation, which enables GPU-accelerated calculations on all existing GPU and CPU hardware. Section 5 expands upon the OpenMP work and describes a new GPU-accelerated library named LibERI for evaluating molecular integrals – often the main computational bottleneck in many quantum chemistry methods. LibERI is intended to serve as a library for fast quantum chemistry calculations that can be interfaced with any existing quantum chemistry software packages through the MolSSI Driver Interface (MDI). MDI is a standard interface between scientific computational codes. One of LibERI’s objectives is to seamlessly integrate with a Molecular Mechanics (MM) package using MDI to enable fast QM/MM calculations. This integration would allow for efficient GPU-accelerated treatment of large QM regions via LibERI. Lastly, Section 6 concludes this dissertation with a study that applies existing quantum chemistry methods to unravel the mechanism of a Pd-catalyzed carboacylation reaction involving the activation of ester C-O bonds.
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.
Computational chemistry has made remarkable progress in the last couple of decades due to the availability of better and more powerful computers. However, there is definitely more room for improvement. There are problems in material science and biology that require 102 to 104 atoms to be considered and the importance of such systems cannot be over-emphasized. Addressing such systems with ab-initio methods is still a quantum chemist’s dream. ¨ Brute force solutions of the Schrodinger equation are not possible for reasonably interesting and important systems due to the large number of Slater determinants that have to be stored in a computer’s memory. Thus electronic structure theorists have made chemically and mathematically intuitive approximations to simplify the problems. A particularly difficult class of systems that is very interesting to the theoretical and experimental chemists is the strongly correlated systems. High Tc superconductors, nanotubes, graphene sheets, transition metal complexes and photosynthetic materials all fall under this category. These systems are too large to be treated by traditional quantum chemical methods. Another reason why such systems have not been studied sufficiently by ab-initio method is that they comprise of complicated strong electron-electron correlation (and therefore the name strongly correlated), thus, making them impossible to be qualitatively defined by a molecular orbital picture, which has been a quantum chemist’s fa- vorite tool for a long time now. Therefore, in this thesis, we have tried to develop, improve and apply methods for treating such strongly correlated systems. The thesis is broadly divided into two parts. In the first part, we discuss the orbital optimized DMRG method developed to treat static correlation in large strongly correlated systems. In the next part we discuss the two methods that can treat dynamic correlation in such complicated systems.