Rationale for the extrapolation procedure in selected configuration interaction.

Selected configuration interaction (SCI) methods have emerged as state-of-the-art methodologies for achieving high accuracy and generating benchmark reference data for ground and excited states in small molecular systems. However, their precision relies heavily on extrapolation procedures to produce a final estimate of the exact result. Using the structure of the exact electronic energy landscape, we provide a rationale for the common linear extrapolation of the variational energy as a function of the second-order perturbative correction. In particular, we demonstrate that the energy gap and the coupling between the so-called internal and external spaces are the key factors determining the rate at which the linear regime is reached. Starting from the first principles, we also derive a new non-linear extrapolation formula that improves the post-processing of data generated from SCI methods and can be applied to both ground- and excited-state energies.

Can GW handle multireference systems?

Due to the infinite summation of bubble diagrams, the GW approximation of Green's function perturbation theory has proven particularly effective in the weak correlation regime, where this family of Feynman diagrams is important. However, the performance of GW in multireference molecular systems, characterized by strong electron correlation, remains relatively unexplored. In the present study, we investigate the ability of GW to handle closed-shell multireference systems in their singlet ground state by examining four paradigmatic scenarios. First, we analyze a prototypical example of a chemical reaction involving strong correlation: the potential energy curve of BeH2 during the insertion of a beryllium atom into a hydrogen molecule. Second, we compute the electron detachment and attachment energies of a set of molecules that exhibit a variable degree of multireference character at their respective equilibrium geometries: LiF, BeO, BN, C2, B2, and O3. Third, we consider a H6 cluster with a triangular arrangement, which features a notable degree of spin frustration. Finally, the dissociation curve of the HF molecule is studied as an example of single bond breaking. These investigations highlight a nuanced perspective on the performance of GW for strong correlation depending on the level of self-consistency, the choice of initial guess, and the presence of spin-symmetry breaking at the Hartree-Fock level.

Excited States, Symmetry Breaking, and Unphysical Solutions in State-Specific CASSCF Theory.

State-specific electronic structure theory provides a route toward balanced excited-state wave functions by exploiting higher-energy stationary points of the electronic energy. Multiconfigurational wave function approximations can describe both closed- and open-shell excited states and avoid the issues associated with state-averaged approaches. We investigate the existence of higher-energy solutions in complete active space self-consistent field (CASSCF) theory and characterize their topological properties. We demonstrate that state-specific approximations can provide accurate higher-energy excited states in H2 (6-31G) with more compact active spaces than would be required in a state-averaged formalism. We then elucidate the unphysical stationary points, demonstrating that they arise from redundant orbitals when the active space is too large or symmetry breaking when the active space is too small. Furthermore, we investigate the singlet-triplet crossing in CH2 (6-31G) and the avoided crossing in LiF (6-31G), revealing the severity of root flipping and demonstrating that state-specific solutions can behave quasi-diabatically or adiabatically. These results elucidate the complexity of the CASSCF energy landscape, highlighting the advantages and challenges of practical state-specific calculations.

Generalized nonorthogonal matrix elements. II: Extension to arbitrary excitations.

Electronic structure methods that exploit nonorthogonal Slater determinants face the challenge of efficiently computing nonorthogonal matrix elements. In a recent publication [H. G. A. Burton, J. Chem. Phys. 154, 144109 (2021)], I introduced a generalized extension to the nonorthogonal Wick's theorem that allows matrix elements to be derived between excited configurations from a pair of reference determinants with a singular nonorthogonal orbital overlap matrix. However, that work only provided explicit expressions for one- and two-body matrix elements between singly- or doubly-excited configurations. Here, this framework is extended to compute generalized nonorthogonal matrix elements between higher-order excitations. Pre-computing and storing intermediate values allows one- and two-body matrix elements to be evaluated with an O(1) scaling relative to the system size, and the LIBGNME computational library is introduced to achieve this in practice. These advances make the evaluation of all nonorthogonal matrix elements almost as easy as their orthogonal counterparts, facilitating a new phase of development in nonorthogonal electronic structure theory.

Complex analysis of divergent perturbation theory at finite temperature.

We investigate the convergence properties of finite-temperature perturbation theory by considering the mathematical structure of thermodynamic potentials using complex analysis. We discover that zeros of the partition function lead to poles in the internal energy and logarithmic singularities in the Helmholtz free energy that create divergent expansions in the canonical ensemble. Analyzing these zeros reveals that the radius of convergence increases at higher temperatures. In contrast, when the reference state is degenerate, these poles in the internal energy create a zero radius of convergence in the zero-temperature limit. Finally, by showing that the poles in the internal energy reduce to exceptional points in the zero-temperature limit, we unify the two main mathematical representations of quantum phase transitions.

Generalized nonorthogonal matrix elements: Unifying Wick's theorem and the Slater-Condon rules.

Matrix elements between nonorthogonal Slater determinants represent an essential component of many emerging electronic structure methods. However, evaluating nonorthogonal matrix elements is conceptually and computationally harder than their orthogonal counterparts. While several different approaches have been developed, these are predominantly derived from the first-quantized generalized Slater-Condon rules and usually require biorthogonal occupied orbitals to be computed for each matrix element. For coupling terms between nonorthogonal excited configurations, a second-quantized approach such as the nonorthogonal Wick's theorem is more desirable, but this fails when the two reference determinants have a zero many-body overlap. In this contribution, we derive an entirely generalized extension to the nonorthogonal Wick's theorem that is applicable to all pairs of determinants with nonorthogonal orbitals. Our approach creates a universal methodology for evaluating any nonorthogonal matrix element and allows Wick's theorem and the generalized Slater-Condon rules to be unified for the first time. Furthermore, we present a simple well-defined protocol for deriving arbitrary coupling terms between nonorthogonal excited configurations. In the case of overlap and one-body operators, this protocol recovers efficient formulas with reduced scaling, promising significant computational acceleration for methods that rely on such terms.

Hartree-Fock critical nuclear charge in two-electron atoms.

Electron correlation effects play a key role in stabilizing two-electron atoms near the critical nuclear charge, representing the smallest charge required to bind two electrons. However, deciphering the importance of these effects relies on fully understanding the uncorrelated Hartree-Fock description. We investigate the properties of the ground state wave function in the small nuclear charge limit using various symmetry-restricted Hartree-Fock formalisms. We identify the nuclear charge where spin-symmetry breaking occurs to give an unrestricted wave function that predicts an inner and outer electron. We also identify closed-shell and unrestricted critical nuclear charges where the highest occupied orbital energy becomes zero and the electron density detaches from the nucleus. Finally, we identify the importance of fractional spin errors and static correlation for small nuclear charges.

Variations of the Hartree-Fock fractional-spin error for one electron.

Fractional-spin errors are inherent in all current approximate density functionals, including Hartree-Fock theory, and their origin has been related to strong static correlation effects. The conventional way to encode fractional-spin calculations is to construct an ensemble density that scales between the high-spin and low-spin densities. In this article, we explore the variation of the Hartree-Fock fractional-spin (or ghost-interaction) error in one-electron systems using restricted and unrestricted ensemble densities and the exact generalized Hartree-Fock representation. By considering the hydrogen atom and H+ 2 cation, we analyze how the unrestricted and generalized Hartree-Fock schemes minimize this error by localizing the electrons or rotating the spin coordinates. We also reveal a clear similarity between the Coulomb hole of He-like ions and the density depletion near the nucleus induced by the fractional-spin error in the unpolarized hydrogen atom. Finally, we analyze the effect of the fractional-spin error on the Møller-Plesset adiabatic connection, excited states, and functional- and density-driven errors.

Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package.

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.

Complex adiabatic connection: A hidden non-Hermitian path from ground to excited states.

Processes related to electronically excited states are central in many areas of science; however, accurately determining excited-state energies remains a major challenge in theoretical chemistry. Recently, higher energy stationary states of non-linear methods have themselves been proposed as approximations to excited states, although the general understanding of the nature of these solutions remains surprisingly limited. In this letter, we present an entirely novel approach for exploring and obtaining excited stationary states by exploiting the properties of non-Hermitian Hamiltonians. Our key idea centres on performing analytic continuations of conventional quantum chemistry methods. Considering Hartree-Fock theory as an example, we analytically continue the electron-electron interaction to expose a hidden connectivity of multiple solutions across the complex plane, revealing a close resemblance between Coulson-Fischer points and non-Hermitian degeneracies. Finally, we demonstrate how a ground-state wave function can be morphed naturally into an excited-state wave function by constructing a well-defined complex adiabatic connection.

Microscopic Marangoni Flows Cannot Be Predicted on the Basis of Pressure Gradients.

A concentration gradient along a fluid-fluid interface can cause flow. On a microscopic level, this so-called Marangoni effect can be viewed as being caused by a gradient in the pressures acting on the fluid elements or as the chemical-potential gradients acting on the excess densities of different species at the interface. If the interface thickness can be ignored, all approaches should result in the same flow profile away from the interface. However, on a more microscopic scale, the different expressions result in different flow profiles, only one of which can be correct. Here we compare the results of direct nonequilibrium molecular dynamics simulations with the flows that are generated by pressure and chemical-potential gradients. We find that the approach based on the chemical-potential gradients agrees with the direct simulations, whereas the calculations based on the pressure gradients do not.

Excited State-Specific CASSCF Theory for the Torsion of Ethylene.

State-specific complete active space self-consistent field (SS-CASSCF) theory has emerged as a promising route to accurately predict electronically excited energy surfaces away from molecular equilibria. However, its accuracy and practicality for chemical systems of photochemical interest have yet to be fully determined. We investigate the performance of the SS-CASSCF theory for the low-lying ground and excited states in the double bond rotation of ethylene. We show that state-specific approximations with a minimal (2e,2o) active space provide comparable accuracy to state-averaged calculations with much larger active spaces, while optimizing the orbitals for each excited state significantly improves the spatial diffusivity of the wave function. However, the incorrect ordering of state-specific solutions causes excited state solutions to coalesce and disappear, creating unphysical discontinuities in the potential energy surface. Our findings highlight the theoretical challenges that must be overcome to realize practical applications of state-specific electronic structure theory for computational photochemistry.

Energy Landscape of State-Specific Electronic Structure Theory.

State-specific approximations can provide a more accurate representation of challenging electronic excitations by enabling relaxation of the electron density. While state-specific wave functions are known to be local minima or saddle points of the approximate energy, the global structure of the exact electronic energy remains largely unexplored. In this contribution, a geometric perspective on the exact electronic energy landscape is introduced. On the exact energy landscape, ground and excited states form stationary points constrained to the surface of a hypersphere, and the corresponding Hessian index increases at each excitation level. The connectivity between exact stationary points is investigated, and the square-magnitude of the exact energy gradient is shown to be directly proportional to the Hamiltonian variance. The minimal basis Hartree-Fock and excited-state mean-field representations of singlet H2 (STO-3G) are then used to explore how the exact energy landscape controls the existence and properties of state-specific approximations. In particular, approximate excited states correspond to constrained stationary points on the exact energy landscape, and their Hessian index also increases for higher energies. Finally, the properties of the exact energy are used to derive the structure of the variance optimization landscape and elucidate the challenges faced by variance optimization algorithms, including the presence of unphysical saddle points or maxima of the variance.

Energy Landscapes for Electronic Structure.

Orbital-optimized multiple self-consistent-field (SCF) solutions are increasingly being interpreted as mean-field approximations of diabatic or excited electronic states. However, surprisingly little is known about the topology of the electronic energy landscape from which these multiple solutions emerge. In this contribution, we extend energy landscape methods, developed for investigating molecular potential energy surfaces, to investigate and understand the structure of the electronic SCF energy surface. Using analytic gradients and Hessians, we systematically identify every real SCF minimum for the prototypical H4 molecule with the 3-21G basis set, and the index-1 saddles that connect these minima. The resulting SCF energy landscape has a double-funnel structure, with no high-energy local minima. The effect of molecular symmetry on the pathways is analyzed, and we demonstrate how the SCF energy landscape changes with the basis set, SCF potential, molecular structure, and spin state. These results provide guiding principles for the future development of algorithms to systematically identify multiple SCF solutions from an orbital optimization perspective.

Perturbation theory in the complex plane: exceptional points and where to find them.

We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree-Fock approximation and Rayleigh-Schrödinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within Møller-Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Padé and quadratic approximants) that can improve the overall accuracy of the Møller-Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.

Reaching Full Correlation through Nonorthogonal Configuration Interaction: A Second-Order Perturbative Approach.

Nonorthogonal multireference methods can predict statically correlated adiabatic energies while providing chemical insight through the combination of diabatic reference states. However, reaching quantitative accuracy using nonorthogonal multireference expansions remains a significant challenge. In this work, we present the first rigorous perturbative correction to nonorthogonal configuration interaction, allowing the remaining dynamic correlation to be reliably computed. Our second-order "NOCI-PT2" theory exploits a zeroth-order generalized Fock Hamiltonian and builds the first-order interacting space using single and double excitations from each reference determinant. This approach therefore defines the rigorous nonorthogonal extension to conventional multireference perturbation theories. We find that NOCI-PT2 can quantitatively predict multireference potential energy surfaces and provides state-specific ground and excited states for adiabatic avoided crossings. Furthermore, we introduce an explicit imaginary-shift formalism requiring shift values that are an order of magnitude smaller than those used in conventional multireference perturbation theories.

Towards a Holomorphic Density Functional Theory.

Self-consistent-field (SCF) approximations formulated using Hartree-Fock (HF) or Kohn-Sham density-functional theory (KS-DFT) have the potential to yield multiple solutions. However, the formal relationship between multiple solutions identified using HF or KS-DFT remains generally unknown. We investigate the connection between multiple SCF solutions for HF or KS-DFT by introducing a parameterized functional that scales between the two representations. Using the hydrogen molecule and a model of electron transfer, we continuously map multiple solutions from the HF potential to a KS-DFT description. We discover that multiple solutions can coalesce and vanish as the functional changes, forming a direct analogy with the disappearance of real HF solutions along a change in molecular structure. To overcome this disappearance of solutions, we develop a complex-analytic extension of DFT-the "holomorphic DFT" approach-that allows every SCF stationary state to be analytically continued across all molecular structures and exchange-correlation functionals.

General Approach for Multireference Ground and Excited States Using Nonorthogonal Configuration Interaction.

A balanced description of ground and excited states is essential for the description of many chemical processes. However, few methods can handle cases where static correlation is present, and often these scale very unfavorably with system size. Recently, multiple Hartree-Fock (HF) solutions have been proposed as a basis for nonorthogonal configuration interaction (NOCI) to provide multireference ground- and excited-state energies, although applications across multiple geometries have been limited by the coalescence of HF solutions. Holomorphic HF (h-HF) theory allows solutions to be analytically continued beyond the Coulson-Fischer points at which they vanish, but, until now, this has only been demonstrated for small model systems. In this work, we propose a general protocol for computing NOCI ground- and excited-state energies using multiple HF solutions. To do so, we outline an active space variation of SCF metadynamics that allows a chemically relevant set of HF states to be identified and describe how these states can be routinely traced across all molecular geometries by exploiting the topology of h-HF solutions in the complex plane. Finally, we illustrate our approach using the dissociation of the fluorine dimer and the pseudo-Jahn-Teller distortion of cyclobutadiene, demonstrating its applicability for multireference ground and excited states.

Parity-Time Symmetry in Hartree-Fock Theory.

PT-symmetry-invariance with respect to combined space reflection P and time reversal T-provides a weaker condition than (Dirac) Hermiticity for ensuring a real energy spectrum of a general non-Hermitian Hamiltonian. PT-symmetric Hamiltonians therefore form an intermediate class between Hermitian and non-Hermitian Hamiltonians. In this work, we derive the conditions for PT-symmetry in the context of electronic structure theory and, specifically, within the Hartree-Fock (HF) approximation. We show that the HF orbitals are symmetric with respect to the PT operator if and only if the effective Fock Hamiltonian is PT-symmetric, and vice versa. By extension, if an optimal self-consistent solution is invariant under PT, then its eigenvalues and corresponding HF energy must be real. Moreover, we demonstrate how one can construct explicitly PT-symmetric Slater determinants by forming PT-doublets (i.e., pairing each occupied orbital with its PT-transformed analogue), allowing PT-symmetry to be conserved throughout the self-consistent process. Finally, considering the H2 molecule as an illustrative example, we observe PT-symmetry in the HF energy landscape and find that the spatially symmetry-broken unrestricted HF wave functions (i.e., diradical configurations) are PT-symmetric, while the spatially symmetry-broken restricted HF wave functions (i.e., ionic configurations) break PT-symmetry.

Holomorphic Hartree-Fock Theory: The Nature of Two-Electron Problems.

We explore the existence and behavior of holomorphic restricted Hartree-Fock (h-RHF) solutions for two-electron problems. Through algebraic geometry, the exact number of solutions with n basis functions is rigorously identified as 1/2(3n - 1), proving that states must exist for all molecular geometries. A detailed study on the h-RHF states of HZ (STO-3G) then demonstrates both the conservation of holomorphic solutions as geometry or atomic charges are varied and the emergence of complex h-RHF solutions at coalescence points. Using catastrophe theory, the nature of these coalescence points is described, highlighting the influence of molecular symmetry. The h-RHF states of HHeH2+ and HHeH (STO-3G) are then compared, illustrating the isomorphism between systems with two electrons and two electron holes. Finally, we explore the h-RHF states of ethene (STO-3G) by considering the π electrons as a two-electron problem and employ NOCI to identify a crossing of the lowest energy singlet and triplet states at the perpendicular geometry.