Locating multiple self-consistent field solutions: an approach inspired by metadynamics

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.

Orbital-Optimized Versus Time-Dependent Density Functional Calculations of Intramolecular Charge Transfer Excited States

The performance of time-independent, orbital-optimized calculations of excited states is assessed with respect to charge transfer excitations in organic molecules in comparison to the linear-response time-dependent density functional theory (TD-DFT) approach. A direct optimization method to converge on saddle points of the electronic energy surface is used to carry out calculations with the local density approximation (LDA) and the generalized gradient approximation (GGA) functionals PBE and BLYP for a set of 27 excitations in 15 molecules. The time-independent approach is fully variational and provides a relaxed excited state electron density from which the extent of charge transfer is quantified. The TD-DFT calculations are generally found to provide larger charge transfer distances compared to the orbital-optimized calculations, even when including orbital relaxation effects with the Z-vector method. While the error on the excitation energy relative to theoretical best estimates is found to increase with the extent of charge transfer up to ca. -2 eV for TD-DFT, no correlation is observed for the orbital-optimized approach. The orbital-optimized calculations with the LDA and the GGA functionals provide a mean absolute error of ∼0.7 eV, outperforming TD-DFT with both local and global hybrid functionals for excitations with a long-range charge transfer character. Orbital-optimized calculations with the global hybrid functional B3LYP and the range-separated hybrid functional CAM-B3LYP on a selection of states with short- and long-range charge transfer indicate that inclusion of exact exchange has a small effect on the charge transfer distance, while it significantly improves the excitation energy, with the best-performing functional CAM-B3LYP providing an absolute error typically around 0.15 eV.

QSym2: A Quantum Symbolic Symmetry Analysis Program for Electronic Structure

Symmetry provides a powerful machinery to classify, interpret, and understand quantum-mechanical theories and results. However, most contemporary quantum chemistry packages lack the ability to handle degeneracy and symmetry breaking effects, especially in non-Abelian groups, and they are not able to characterize symmetry in the presence of external magnetic or electric fields. In this article, a program written in Rust entitled QSym2 that makes use of group and representation theories to provide symmetry analysis for a wide range of quantum-chemical calculations is introduced. With its ability to generate character tables symbolically on-the-fly and by making use of a generic symmetry-orbit-based representation analysis method formulated in this work, QSym2 is able to address all of these shortcomings. To illustrate these capabilities of QSym2, four sets of case studies are examined in detail in this article: (i) high-symmetry C84H64, C60, and B9- to demonstrate the analysis of degenerate molecular orbitals (MOs); (ii) octahedral Fe(CN)63- to demonstrate the analysis of symmetry-broken determinants and MOs; (iii) linear hydrogen fluoride in a magnetic field to demonstrate the analysis of magnetic symmetry; and (iv) equilateral H3+ to demonstrate the analysis of density symmetries.

Rydberg electron stabilizes the charge localized state of the diamine cation

A previous controversial discussion regarding the interpretation of Rydberg spectra of gaseous dimethylpiperazine (DMP) as showing the co-existence of a localized and delocalized mixed-valent DMP+ radical cation is revisited. Here we show by high-level quantum-chemical calculations that an apparent barrier separating localized and delocalized DMP+ minima in previous multi-reference configuration-interaction (MRCI) calculations and in some other previous computations were due to unphysical curve crossings of the reference wave functions. These discontinuities on the surface are removed in state-averaged MRCI calculations and with some other, orthogonal high-level approaches, which do not provide a barrier and thus no localized minimum. We then proceed to show that in the actually observed Rydberg state of neutral DMP the 3s-type Rydberg electron binds more strongly to a localized positive charge distribution, generating a localized DMP* Rydberg-state minimum, which is absent for the DMP+ cation. This work presents a case where interactions of a Rydberg electron with the underlying cationic core alter molecular structure in a fundamental way.

Nonorthogonal Multireference Wave Function Description of Triplet-Triplet Energy Transfer Couplings

In this study, the use of self-consistent field quasi-diabats is investigated for calculation of triplet energy transfer diabatic coupling elements. It is proposed that self-consistent field quasi-diabats are particularly useful for studying energy transfer (EnT) processes because orbital relaxation in response to changes in electron configuration is implicitly built into the model. The conceptual model that is developed allows for the simultaneous evaluation of direct and charge-transfer mechanisms to establish the importance of the different possible EnT mechanisms. The method's performance is evaluated using two model systems: the ethylene dimer and ethylene with the methaniminium cation. While states that mediate the charge-transfer mechanism were found to be higher in energy than the states involved in the direct mechanism, the coupling elements that control the kinetics were found to be significantly larger in the charge-transfer mechanism. Subsequently, we discuss the advantage of the approach in the context of practical difficulties with the use of established approaches.

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.

Target State Optimized Density Functional Theory for Electronic Excited and Diabatic States

A flexible self-consistent field method, called target state optimization (TSO), is presented for exploring electronic excited configurations and localized diabatic states. The key idea is to partition molecular orbitals into different subspaces according to the excitation or localization pattern for a target state. Because of the orbital-subspace constraint, orbitals belonging to different subspaces do not mix. Furthermore, the determinant wave function for such excited or diabatic configurations can be variationally optimized as a ground state procedure, unlike conventional ΔSCF methods, without the possibility of collapsing back to the ground state or other lower-energy configurations. The TSO method can be applied both in Hartree-Fock theory and in Kohn-Sham density functional theory (DFT). The density projection procedure and the working equations for implementing the TSO method are described along with several illustrative applications. For valence excited states of organic compounds, it was found that the computed excitation energies from TSO-DFT and time-dependent density functional theory (TD-DFT) are of similar quality with average errors of 0.5 and 0.4 eV, respectively. For core excitation, doubly excited states and charge-transfer states, the performance of TSO-DFT is clearly superior to that from conventional TD-DFT calculations. It is shown that variationally optimized charge-localized diabatic states can be defined using TSO-DFT in energy decomposition analysis to gain both qualitative and quantitative insights on intermolecular interactions. Alternatively, the variational diabatic states may be used in molecular dynamics simulation of charge transfer processes. The TSO method can also be used to define basis states in multistate density functional theory for excited states through nonorthogonal state interaction calculations. The software implementing TSO-DFT can be accessed from the authors.

Nonorthogonal Active Space Decomposition of Wave Functions with Multiple Correlation Mechanisms

The nonorthogonal active space decomposition (NO-ASD) methodology is proposed for describing systems containing multiple correlation mechanisms. NO-ASD partitions the wave function by a correlation mechanism, such that the interactions between different correlation mechanisms are treated with an effective Hamiltonian approach, while interactions between correlated orbitals in the same correlation mechanism are treated explicitly. As a result, the determinant expansion scales polynomially with the number of correlation mechanisms rather than exponentially, which significantly reduces the factorial scaling associated with the size of the correlated orbital space. Despite the nonorthogonal framework of NO-ASD, the approach can take advantage of computational efficient matrix element evaluation when performing nonorthogonal coupling of orthogonal determinant expansions. In this work, we introduce and examine the NO-ASD approach in comparison to complete active space methods to establish how the NO-ASD approach reduces the problem dimensionality and the extent to which it affects the amount of correlation energy recovered. Calculations are performed on ozone, nickel-acetylene, and isomers of μ-oxo dicopper ammonia.

The Role of Local DFT+U Minima in the First-Principles Modeling of the Metal-Insulator Transition in Vanadium Dioxide

The DFT+U method is frequently employed to improve the first-principles description of strongly correlated materials. However, it is prone to deliver metastable electronic minima. While these local minima of the DFT+U method are often considered to be computational artifacts, their physical meaning and relationship to true excited states remains unclear. In this work, the possibility of theoretically modeling transformations in the solid state that require thermal or optical excitations of electrons is explored, taking into account the metastable states of the computationally undemanding DFT+U formalism. For this purpose, we choose to examine the example of the VO₂ metal-insulator transition. Metastable states that are located on different electronic potential energy surfaces are found to correspond to experimentally observed VO₂ phases. The identified metastable electronic states can be used to model the collapse of the VO₂ band gap at elevated temperatures and upon photoexcitation as well as other monoclinic-monoclinic phase transformations. The results suggest that local DFT+U minima can indeed carry physical meaning, while they remain under-reported in theoretical literature on transition metal oxides like VO₂.

Relativistic nonorthogonal configuration interaction: application to L2,3-edge X-ray spectroscopy

In this article, we develop a relativistic exact-two-component nonorthogonal configuration interaction (X2C-NOCI) for computing L-edge X-ray spectra. This article to our knowledge is the first time NOCI has been used for relativistic wave functions. A set of molecular complexes, including SF₆, SiCl₄ and [FeCl₆]^3-, are used to demonstrate the accuracy and computational scaling of the X2C-NOCI method. Our results suggest that X2C-NOCI is able to satisfactorily capture the main features of the L_2,3-edge X-ray absorption spectra. Excitations from the core require a large amount of orbital relaxation to yield reasonable energies and X2C-NOCI allows us to treat orbital optimization explicitly. However, the cost of computing the nonorthogonal coupling is higher than in conventional CI. Here, we propose an improved integral screening using overlap-scaled density combined with a continuous measure of the generalized Slater-Condon rules that allows us to estimate if an element is zero before attempting a two-electron integral contraction.

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.

Localized Spin Rotations: A Size-Consistent Approach to Nonorthogonal Configuration Interaction

Current nonorthogonal configuration interaction (NOCI) methods often use a set of self-consistent field (SCF) states selected based on chemical intuition. However, it may be challenging to track these SCF states across a dissociation profile and the NOCI states recovered may be spin contaminated. In this Article, we propose a method of applying spin rotation on symmetry broken unrestricted Hartree-Fock (sb-UHF) states to generate a basis for NOCI. The dissociation of ethene was examined by localizing spin rotation on each resulting carbene fragment. We show that this gives a size-consistent description of its dissociation and results in spin-pure states at all geometries. The dissociation was also studied with different orbitals, namely, canonical UHF and absolutely localized molecular orbitals (ALMO). Furthermore, we demonstrate that the method can be used to restore spin symmetry of symmetry broken SCF wave functions for molecules of various sizes, marking an improvement over existing NOCI methods.

Variational coupled cluster for ground and excited states

In single-reference coupled-cluster (CC) methods, one has to solve a set of non-linear polynomial equations in order to determine the so-called amplitudes that are then used to compute the energy and other properties. Although it is of common practice to converge to the (lowest-energy) ground-state solution, it is also possible, thanks to tailored algorithms, to access higher-energy roots of these equations that may or may not correspond to genuine excited states. Here, we explore the structure of the energy landscape of variational CC and we compare it with its (projected) traditional version in the case where the excitation operator is restricted to paired double excitations (pCCD). By investigating two model systems (the symmetric stretching of the linear H₄ molecule and the continuous deformation of the square H₄ molecule into a rectangular arrangement) in the presence of weak and strong correlations, the performance of variational pCCD (VpCCD) and traditional pCCD is gauged against their configuration interaction (CI) equivalent, known as doubly occupied CI, for reference Slater determinants made of ground- or excited-state Hartree-Fock orbitals or state-specific orbitals optimized directly at the VpCCD level. The influence of spatial symmetry breaking is also investigated.

Orbital optimization in nonorthogonal multiconfigurational self-consistent field applied to the study of conical intersections and avoided crossings

Nonorthogonal approaches to electronic structure methods have recently received renewed attention, with the hope that new forms of nonorthogonal wavefunction Ansätze may circumvent the computational bottleneck of orthogonal-based methods. The basis in which nonorthogonal configuration interaction is performed defines the compactness of the wavefunction description and hence the efficiency of the method. Within a molecular orbital approach, nonorthogonal configuration interaction is defined by a "different orbitals for different configurations" picture, with different methods being defined by their choice of determinant basis functions. However, identification of a suitable determinant basis is complicated, in practice, by (i) exponential scaling of the determinant space from which a suitable basis must be extracted, (ii) possible linear dependencies in the determinant basis, and (iii) inconsistent behavior in the determinant basis, such as disappearing or coalescing solutions, as a result of external perturbations, such as geometry change. An approach that avoids the aforementioned issues is to allow for basis determinant optimization starting from an arbitrarily constructed initial determinant set. In this work, we derive the equations required for performing such an optimization, extending previous work by accounting for changes in the orthogonality level (defined as the dimension of the orbital overlap kernel between two determinants) as a result of orbital perturbations. The performance of the resulting wavefunction for studying avoided crossings and conical intersections where strong correlation plays an important role is examined.

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.

An Overview of Self-Consistent Field Calculations Within Finite Basis Sets

A uniform derivation of the self-consistent field equations in a finite basis set is presented. Both restricted and unrestricted Hartree-Fock (HF) theory as well as various density functional approximations are considered. The unitary invariance of the HF and density functional models is discussed, paving the way for the use of localized molecular orbitals. The self-consistent field equations are derived in a non-orthogonal basis set, and their solution is discussed also in the presence of linear dependencies in the basis. It is argued why iterative diagonalization of the Kohn-Sham-Fock matrix leads to the minimization of the total energy. Alternative methods for the solution of the self-consistent field equations via direct minimization as well as stability analysis are briefly discussed. Explicit expressions are given for the contributions to the Kohn-Sham-Fock matrix up to meta-GGA functionals. Range-separated hybrids and non-local correlation functionals are summarily reviewed.

GronOR: Massively parallel and GPU-accelerated non-orthogonal configuration interaction for large molecular systems

GronOR is a program package for non-orthogonal configuration interaction calculations for an electronic wave function built in terms of anti-symmetrized products of multi-configuration molecular fragment wave functions. The two-electron integrals that have to be processed may be expressed in terms of atomic orbitals or in terms of an orbital basis determined from the molecular orbitals of the fragments. The code has been specifically designed for execution on distributed memory massively parallel and Graphics Processing Unit (GPU)-accelerated computer architectures, using an MPI+OpenACC/OpenMP programming approach. The task-based execution model used in the implementation allows for linear scaling with the number of nodes on the largest pre-exascale architectures available, provides hardware fault resiliency, and enables effective execution on systems with distinct central processing unit-only and GPU-accelerated partitions. The code interfaces with existing multi-configuration electronic structure codes that provide optimized molecular fragment orbitals, configuration interaction coefficients, and the required integrals. Algorithm and implementation details, parallel and accelerated performance benchmarks, and an analysis of the sensitivity of the accuracy of results and computational performance to thresholds used in the calculations are presented.

Excited State Orbital Optimization via Minimizing the Square of the Gradient: General Approach and Application to Singly and Doubly Excited States via Density Functional Theory

We present a general approach to converge excited state solutions to any quantum chemistry orbital optimization process, without the risk of variational collapse. The resulting square gradient minimization (SGM) approach only requires analytic energy/Lagrangian orbital gradients and merely costs 3 times as much as ground state orbital optimization (per iteration), when implemented via a finite difference approach. SGM is applied to both single determinant ΔSCF and spin-purified restricted open-shell Kohn-Sham (ROKS) approaches to study the accuracy of orbital optimized DFT excited states. It is found that SGM can converge challenging states where the maximum overlap method (MOM) or analogues either collapse to the ground state or fail to converge. We also report that ΔSCF/ROKS predict highly accurate excitation energies for doubly excited states (which are inaccessible via TDDFT). Singly excited states obtained via ROKS are also found to be quite accurate, especially for Rydberg states that frustrate (semi)local TDDFT. Our results suggest that orbital optimized excited state DFT methods can be used to push past the limitations of TDDFT to doubly excited, charge-transfer, or Rydberg states, making them a useful tool for the practical quantum chemist's toolbox for studying excited states in large systems.

Atom-Based Bootstrap Embedding For Molecules

Recent developments in quantum embedding have offered an attractive approach to describing electron correlation in molecules. However, previous methods such as density matrix embedding theory (DMET) require rigid partitioning of the system into fragments, which creates significant ambiguity for molecules. Bootstrap embedding (BE) is more flexible because it allows overlapping fragments, but when done on an orbital-by-orbital basis, BE introduces ambiguity in defining the connectivity of the orbitals. In this Letter, we present an atom-based fragment definition that significantly augments BE's performance in molecules. The resulting method, which we term atom-based BE, is very effective at recovering valence electron correlation in moderate-sized bases and delivers near-chemical-accuracy results using extrapolation. We anticipate atom-based BE may lead to a low-scaling and highly accurate approach to electron correlation in large molecules.

Half-Projected σ Self-Consistent Field For Electronic Excited States

Fully self-consistent mean-field solutions of electronic excited states have been much less accessible compared to ground state solutions (e.g., Hartree-Fock). The main reason for this is that most excited states are energy saddle points, and hence energy-based optimization methods such as Δ-SCF often collapse to the ground state. Recently, our research group has developed a new method, σ-SCF [ J. Chem. Phys. 2017 , 147 , 214104 ], that successfully solves the "variational collapse" problem of energy-based methods. Despite the success, σ-SCF solutions are often spin-contaminated for open-shell states due to the single-determinant nature; unphysical behaviors such as disappearing solutions and discontinuous first-order energy derivatives are also observed along with the spontaneous breaking of spin or spatial symmetries. In this work, we tackle these problems by partially restoring the broken spin-symmetry of a σ-SCF solution through an approximate spin-projection scheme called half-projection. Orbitals of the projected wave function are optimized in a variation-after-projection (VAP) manner. The resulting theory, which we term half-projected (HP) σ-SCF, brings substantial improvement to the description of singlet and triplet excitations of the original σ-SCF method. Numerical simulations on small molecules suggest that HP σ-SCF delivers high-quality excited-state solutions that exist in a wide range of geometries with smooth potential energy surfaces. We also show that local excitations in HP σ-SCF can be size-intensive.

σ-SCF: A direct energy-targeting method to mean-field excited states

The mean-field solutions of electronic excited states are much less accessible than ground state (e.g., Hartree-Fock) solutions. Energy-based optimization methods for excited states, like Δ-SCF (self-consistent field), tend to fall into the lowest solution consistent with a given symmetry-a problem known as "variational collapse." In this work, we combine the ideas of direct energy-targeting and variance-based optimization in order to describe excited states at the mean-field level. The resulting method, σ-SCF, has several advantages. First, it allows one to target any desired excited state by specifying a single parameter: a guess of the energy of that state. It can therefore, in principle, find all excited states. Second, it avoids variational collapse by using a variance-based, unconstrained local minimization. As a consequence, all states-ground or excited-are treated on an equal footing. Third, it provides an alternate approach to locate Δ-SCF solutions that are otherwise hardly accessible by the usual non-aufbau configuration initial guess. We present results for this new method for small atoms (He, Be) and molecules (H₂, HF). We find that σ-SCF is very effective at locating excited states, including individual, high energy excitations within a dense manifold of excited states. Like all single determinant methods, σ-SCF shows prominent spin-symmetry breaking for open shell states and our results suggest that this method could be further improved with spin projection.

Spatial and Spin Symmetry Breaking in Semidefinite-Programming-Based Hartree-Fock Theory

The Hartree-Fock problem was recently recast as a semidefinite optimization over the space of rank-constrained two-body reduced-density matrices (RDMs) [ Phys. Rev. A 2014 , 89 , 010502(R) ]. This formulation of the problem transfers the nonconvexity of the Hartree-Fock energy functional to the rank constraint on the two-body RDM. We consider an equivalent optimization over the space of positive semidefinite one-electron RDMs (1-RDMs) that retains the nonconvexity of the Hartree-Fock energy expression. The optimized 1-RDM satisfies ensemble N-representability conditions, and ensemble spin-state conditions may be imposed as well. The spin-state conditions place additional linear and nonlinear constraints on the 1-RDM. We apply this RDM-based approach to several molecular systems and explore its spatial (point group) and spin ( Ŝ² and Ŝ₃) symmetry breaking properties. When imposing Ŝ² and Ŝ₃ symmetry but relaxing point group symmetry, the procedure often locates spatial-symmetry-broken solutions that are difficult to identify using standard algorithms. For example, the RDM-based approach yields a smooth, spatial-symmetry-broken potential energy curve for the well-known Be-H₂ insertion pathway. We also demonstrate numerically that, upon relaxation of Ŝ² and Ŝ₃ symmetry constraints, the RDM-based approach is equivalent to real-valued generalized Hartree-Fock theory.

Gentlest ascent dynamics for calculating first excited state and exploring energy landscape of Kohn-Sham density functionals

We develop the gentlest ascent dynamics for Kohn-Sham density functional theory to search for the index-1 saddle points on the energy landscape of the Kohn-Sham density functionals. These stationary solutions correspond to excited states in the ground state functionals. As shown by various examples, the first excited states of many chemical systems are given by these index-1 saddle points. Our novel approach provides an alternative, more robust way to obtain these excited states, compared with the widely used ΔSCF approach. The method can be easily generalized to target higher index saddle points. Our results also reveal the physical interest and relevance of studying the Kohn-Sham energy landscape.

Energy landscape of the finite-size mean-field 2-spin spherical model and topology trivialization

Motivated by the recently observed phenomenon of topology trivialization of potential energy landscapes (PELs) for several statistical mechanics models, we perform a numerical study of the finite-size 2-spin spherical model using both numerical polynomial homotopy continuation and a reformulation via non-Hermitian matrices. The continuation approach computes all of the complex stationary points of this model while the matrix approach computes the real stationary points. Using these methods, we compute the average number of stationary points while changing the topology of the PEL as well as the variance. Histograms of these stationary points are presented along with an analysis regarding the complex stationary points. This work connects topology trivialization to two different branches of mathematics: algebraic geometry and catastrophe theory, which is fertile ground for further interdisciplinary research.

Certification and the potential energy landscape

Typically, there is no guarantee that a numerical approximation obtained using standard nonlinear equation solvers is indeed an actual solution, meaning that it lies in the quadratic convergence basin. Instead, it may lie only in the linear convergence basin, or even in a chaotic region, and hence not converge to the corresponding stationary point when further optimization is attempted. In some cases, these non-solutions could be misleading. Proving that a numerical approximation will quadratically converge to a stationary point is termed certification. In this report, we provide details of how Smale's α-theory can be used to certify numerically obtained stationary points of a potential energy landscape, providing a mathematical proof that the numerical approximation does indeed correspond to an actual stationary point, independent of the precision employed.