Wave Function Theory Research Articles

Solution of Schrödinger Equation for Simple Diatomic Molecules Using One‐Parameter 1s Slater‐Type Orbitals Wave Function

ABSTRACTThe Schrödinger equation for simple homonuclear and heteronuclear diatomic molecules is analytically solved using one‐parameter Slater‐type orbitals (STOs) to approximate the electronic wave function within a molecular orbital (MO)‐like approach. The resulting total energies, equilibrium bond lengths, potential curves, and electron densities are presented in detail. Calculations using a selected orbital exponent accurately reproduce results from standard methods. Furthermore, the optimization of the orbital exponent allows for a more accurate representation of the electronic wave function, leading to the improved results of the total energy and the equilibrium bond length, as well as minimal computational cost. Seen in the heteronuclear diatomic molecule, the use of the one‐parameter STOs allows the transformation of the heteronuclear problem into the homonuclear one, revealing the electron repulsion effect on the orbital exponent parameter.

Local Electronic Correlation in Multicomponent Møller-Plesset Perturbation Theory.

We present in this contribution the first application of local correlation in the context of multicomponent methods. Multicomponent approaches allow for the targeted simulation of electrons together with other Fermions (most commonly protons) as quantum particles. These methods have become increasingly popular over the last years, particularly for the description of nuclear quantum effects (in strong hydrogen bonds, proton tunneling, and many more). However, most implementations are still based on canonical formulations of wave function theory, which we know for decades to be computationally inefficient for capturing dynamical correlation effects. Local correlation approaches, particularly with the use of pair natural orbitals (PNOs), enable asymptotically linear scaling of computational costs with very little impact on the overall accuracy. In this context, the efficient use of density fitting approximations in the integral calculation proves essential. We start by discussing our implementation of density-fitted NEO-MP2 and NEO-PNO-LMP2, upgrading the electronic correlation treatment up to PNO local coupled cluster level of theory. Several challenging examples are provided to benchmark the method in terms of accuracy as well as computational cost scaling. Following appropriate protocols, anharmonic corrections to localized X-H stretches can be applied routinely with little computational overhead.

Unsupervised representation learning of Kohn–Sham states and consequences for downstream predictions of many-body effects

Representation learning for the electronic structure problem is a major challenge of machine learning in computational condensed matter and materials physics. Within quantum mechanical first principles approaches, density functional theory (DFT) is the preeminent tool for understanding electronic structure, and the high-dimensional DFT wavefunctions serve as building blocks for downstream calculations of correlated many-body excitations and related physical observables. Here, we use variational autoencoders (VAE) for the unsupervised learning of DFT wavefunctions and show that these wavefunctions lie in a low-dimensional manifold within latent space. Our model autonomously determines the optimal representation of the electronic structure, avoiding limitations due to manual feature engineering. To demonstrate the utility of the latent space representation of the DFT wavefunction, we use it for the supervised training of neural networks (NN) for downstream prediction of quasiparticle bandstructures within the GW formalism. The GW prediction achieves a low error of 0.11 eV for a combined test set of two-dimensional metals and semiconductors, suggesting that the latent space representation captures key physical information from the original data. Finally, we explore the generative ability and interpretability of the VAE representation.

Ground and Excited Electronic Structure Analysis of FeH with Correlated Wave Function Theory and Density Functional Approximations.

FeH is one of the most challenging diatomic molecules to study under electronic structure theory. Here, we have successfully studied 22 electronic states of FeH using ab initio multireference configuration interaction (MRCI), Davidson-corrected MRCI (MRCI+Q), and coupled cluster singles, doubles, and perturbative triples [CCSD(T)] levels of theory. We report their potential energy curves (PECs), excitation energies, dissociation energies, equilibrium electronic configurations, and a series of spectroscopic constants with the use of augmented triple-ζ, quadruple-ζ, and quintuple-ζ quality correlation consistent basis sets. The scalar relativistic effects and active space and core electron correlation contribution on the properties of FeH are also explored. The use of a large CASSCF active space that includes 4s, 4p, 3d, and 4d orbitals of Fe and the 1s of H is critical for producing accurate full PECs with proper dissociations and predicting the exact order of the electronic states. Our findings are in harmony with the experimental results available in the literature and will serve as reference values for future studies of FeH. Furthermore, with the use of PECs, the total internal partition function sum (TIPS) of FeH was calculated across a range of temperatures. Finally, we exploited the single-reference nature of the a6Δ of FeH and its ionized product FeH+ (X5Δ) to evaluate the associated density functional theory (DFT) errors on their dissociation energies and spectroscopic parameters.

Optical Gaps of Ionic Materials from GW/BSE-in-DFT and CC2-in-DFT.

This work presents a density functional theory (DFT)-based embedding technique for the calculation of optical gaps in ionic solids. The approach partitions the supercell of the ionic solid and embeds a small molecule-like cluster in a periodic environment using a cluster-in-periodic embedding method. The environment is treated with DFT, and its influence on the cluster is captured by a DFT-based embedding potential. The optical gap is estimated as the lowest singlet excitation energy of the embedded cluster, obtained using a wave function theory method: second-order approximate coupled-cluster singles and doubles (CC2), and a many-body perturbation theory method: GW approximation combined with the Bethe-Salpeter equation (GW/BSE). The calculated excitation energies are benchmarked against the periodic GW/BSE values, equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) results, and experiments. Both CC2-in-DFT and GW/BSE-in-DFT deliver excitation energies that are in good agreement with experimental values for several ionic solids (MgO, CaO, LiF, NaF, KF, and LiCl) while incurring negligible computational costs. Notably, GW/BSE-in-DFT exhibits remarkable accuracy with a mean absolute error (MAE) of just 0.38 eV with respect to experiments, demonstrating the effectiveness of the embedding strategy. In addition, the versatility of the method is highlighted by investigating the optical gap of a 2D MgCl2 system and the excitation energy of an oxygen vacancy in MgO, with results in good agreement with reported values.

Determinant- and derivative-free quantum Monte Carlo within the stochastic representation of wavefunctions

Abstract Describing the ground states of continuous, real-space quantum many-body systems, like atoms and molecules, is a significant computational challenge with applications throughout the physical sciences. Recent progress was made by variational methods based on machine learning (ML) ansatzes. However, since these approaches are based on energy minimization, ansatzes must be twice differentiable. This (a) precludes the use of many powerful classes of ML models; and (b) makes the enforcement of bosonic, fermionic, and other symmetries costly. Furthermore, (c) the optimization procedure is often unstable unless it is done by imaginary time propagation, which is often impractically expensive in modern ML models with many parameters. The stochastic representation of wavefunctions (SRW), introduced in (Atanasova et al 2023 Nat. Commun. 14 3601), is a recent approach to overcoming (c). SRW enables imaginary time propagation at scale, and makes some headway towards the solution of problem (b), but remains limited by problem (a). Here, we argue that combining SRW with path integral techniques leads to a new formulation that overcomes all three problems simultaneously. As a demonstration, we apply the approach to generalized ‘Hooke’s atoms’: interacting particles in harmonic wells. We benchmark our results against state-of-the-art data where possible, and use it to investigate the crossover between the Fermi liquid and the Wigner molecule within closed-shell systems. Our results shed new light on the competition between interaction-driven symmetry breaking and kinetic-energy-driven delocalization.

An Improved Quantum Particle Swarm Optimization Algorithm for Target Tracking Deployment in Spatial Sensor Networks

This paper presents a comprehensive review of the current research on spatial sensor networks and the node deployment methods employed for mobile target tracking. The study introduces particle swarm optimization (PSO) and its quantum behavior extension, detailing concepts such as the quantum state wave function and particle position representation. Subsequently, an improved Quantum Particle Swarm Optimization (QPSO) algorithm is proposed. This enhanced algorithm increases population diversity by incorporating quantum rotation gates and quantum mutation mechanisms, expands the search space through the superposition state and interference principles of quantum mechanics, and dynamically adjusts algorithm parameters to balance global exploration and local search. These modifications aim to improve both the convergence speed and accuracy of the algorithm. Simulation results demonstrate that the improved QPSO algorithm surpasses traditional mobile tracking deployment algorithms and the standard quantum behavior particle swarm optimization algorithm in terms of target tracking deployment within spatial sensor networks. Notably, it significantly enhances the tracking success rate and reduces tracking errors.

High-Throughput On-Demand Design Platform for Plasmonic Nanocavities: A Wavefunction Theory Approach.

Surface plasmon polaritons from plasmonic nanocavity have aroused great interest due to their applications in various fields, in which on-demand design is hindered by the lack of theoretical frameworks. Herein, based on its wave nature, we developed a wavefunction theory to explicitly describe individual surface plasmon polaritons and the resultant near-field and far-field behaviors, which serves as an efficient platform for high-throughput on-demand design of nanocavities. We found an applicative wavefunction form and proposed a two-body interaction function and a "shell" model for many-body interactions in surface plasmon polaritons' coupling. The wavefunction of individual surface plasmon polaritons and resultant near-field and far-field behaviors can be given explicitly and precisely. The theory provides a fundamental and quantitative understanding of surface plasmon polaritons and enables highly efficient on-demand design of plasmonic metamaterials and devices, leading to further methodological applications in numerous aspects.

Toward Reliable Conformational Energies of Amino Acids and Dipeptides─The DipCONFS Benchmark and DipCONL Datasets.

Simulating peptides and proteins is becoming increasingly important, leading to a growing need for efficient computational methods. These are typically semiempirical quantum mechanical (SQM) methods, force fields (FFs), or machine-learned interatomic potentials (MLIPs), all of which require a large amount of accurate data for robust training and evaluation. To assess potential reference methods and complement the available data, we introduce two sets, DipCONFL and DipCONFS, which cover large parts of the conformational space of 17 amino acids and their 289 possible dipeptides in aqueous solution. The conformers were selected from the exhaustive PeptideCS dataset by Andris et al. [ J. Phys. Chem. B 2022, 126, 5949-5958]. The structures, originally generated with GFN2-xTB, were reoptimized using the accurate r2SCAN-3c density functional theory (DFT) composite method including the implicit CPCM water solvation model. The DipCONFS benchmark set contains 918 conformers and is one of the largest sets with highly accurate coupled cluster conformational energies so far. It is employed to evaluate various DFT and wave function theory (WFT) methods, especially regarding whether they are accurate enough to be used as reliable reference methods for larger datasets intended for training and testing more approximated SQM, FF, and MLIP methods. The results reveal that the originally provided BP86-D3(BJ)/DGauss-DZVP conformational energies are not sufficiently accurate. Among the DFT methods tested as an alternative reference level, the revDSD-PBEP86-D4 double hybrid performs best with a mean absolute error (MAD) of 0.2 kcal mol-1 compared with the PNO-LCCSD(T)-F12b reference. The very efficient r2SCAN-3c composite method also shows excellent results, with an MAD of 0.3 kcal mol-1, similar to the best-tested hybrid ωB97M-D4. With these findings, we compiled the large DipCONFL set, which includes over 29,000 realistic conformers in solution with reasonably accurate r2SCAN-3c reference conformational energies, gradients, and further properties potentially relevant for training MLIP methods. This set, also in comparison to DipCONFS, is used to assess the performance of various SQM, FF, and MLIP methods robustly and can complement training sets for those.

Based on experiment and quantum chemical calculations: a study of the co-pyrolysis mechanism of polyesterimide enameled wires with polyvinyl chloride and the catalytic effect of endogenous metal Cu

Pyrolysis technology is a green and efficient method for recycling enameled wires. However, since waste enameled wires are typically recovered from electronic waste, they often contain small amounts of wires and cables. Therefore, during the pyrolysis process of waste enameled wires, it is inevitable for the paint film and the cable sheath to undergo co-pyrolysis. Polyesterimide enameled wires (EPEsI) and polyvinyl chloride (PVC) were chosen as represent enameled wires and cable sheath materials, respectively. Using thermogravimetric analysis with various pyrolysis kinetic analysis methods, the pyrolysis characteristics and kinetics of EPEsI and Mixture (mixture of EPEsI and PVC) were studied. Through synergy analysis and pyrolysis-gas chromatography/mass spectrometry analysis, the influence of PVC on the pyrolysis of EPEsI was elucidated from aspects such as pyrolysis characteristics and product distribution. Based on density functional theory calculations and wave function analysis, the role of endogenous metal Cu in EPEsI on the pyrolysis processes of PEsI and PVC, as well as the mechanism of HCl from PVC on the pyrolysis of PEsI, were clarified.

Quark spin-orbit correlations in spin-0 and spin-1 mesons using the light-front quark model

We investigate the spin-orbital angular momentum correlations for the active quark inside the light and heavy mesons for both the spin-0 and spin-1 cases. These correlations can be derived from the generalized transverse-momentum-dependent distributions as well as the generalized parton distributions. We employ the overlap representation of light-front wave functions in the light-front quark model to calculate our analytical results. The dependence of spin-orbit correlations (SOCs) on the longitudinal momentum fraction x as well as the transverse-momentum-dependence k⊥ is graphically presented. Even though the SOCs have already been studied for the spin-0 pions and kaons in other approaches, no calculations for the other light and heavy spin-0 mesons have been reported in the literature. Further, the correlations for any of the light and heavy spin-1 mesons are studied for the first time in the present work. Published by the American Physical Society 2024

Toward Efficient and Unified Treatment of Static and Dynamic Correlations in Generalized Kohn-Sham Density Functional Theory.

Accurate description of the static correlation poses a persistent challenge in electronic structure theory, particularly when it has to be concurrently considered with the dynamic correlation. We develop here a method in the generalized Kohn-Sham density functional theory (DFT) framework, named R-xDH7-SCC15, which achieves an unprecedented accuracy in capturing the static correlation, while maintaining a good description of the dynamic correlation on par with the state-of-the-art DFT and wave function theory methods, all grounded in the same single-reference black-box methodology. Central to R-xDH7-SCC15 is a general-purpose static correlation correction (SCC) model applied to the renormalized XYG3-type doubly hybrid method (R-xDH7). The SCC model development involves a hybrid machine learning strategy that integrates symbolic regression with nonlinear parameter optimization, aiming to achieve a balance between generalization capability, numerical accuracy, and interpretability. Extensive benchmark studies confirm the robustness and broad applicability of R-xDH7-SCC15 across a diverse array of main-group chemical scenarios. Notably, it displays exceptional aptitude in accurately characterizing intricate reaction kinetics and dynamic processes in regions distant from equilibrium, where the influence of static correlation is most profound. Its capability to consistently and efficiently predict the whole energy profiles, activation barriers, and reaction pathways within a user-friendly "black-box" framework represents an important advance in the field of electronic structure theory.

Combined Multireference-Multiscale Approach to the Description of Photosynthetic Reaction Centers.

A first-principles description of the primary photochemical processes that drive photosynthesis and sustain life on our planet remains one of the grand challenges of modern science. Recent research established that explicit incorporation of protein electrostatics in excited-state calculations of photosynthetic pigments, achieved for example with quantum-mechanics/molecular-mechanics (QM/MM) approaches, is essential for a meaningful description of the properties and function of pigment-protein complexes. Although time-dependent density functional theory has been used productively so far in QM/MM approaches for the study of such systems, this methodology has limitations. Here we pursue for the first time a QM/MM description of the reaction center in the principal enzyme of oxygenic photosynthesis, Photosystem II, using multireference wave function theory for the high-level QM region. We identify best practices and establish guidelines regarding the rational choice of active space and appropriate state-averaging for the efficient and reliable use of complete active space self-consistent field (CASSCF) and the N-electron valence state perturbation theory (NEVPT2) in the prediction of low-lying excited states of chlorophyll and pheophytin pigments. Given that the Gouterman orbitals are inadequate as a minimal active space, we define specific minimal and extended active spaces for the NEVPT2 description of electronic states that fall within the Q and B bands. Subsequently, we apply our multireference-QM/MM protocol to the description of all pigments in the reaction center of Photosystem II. The calculations reproduce the electrochromic shifts induced by the protein matrix and the ordering of site energies consistent with the identity of the primary donor (ChlD1) and the experimentally known asymmetric and directional electron transfer. The optimized protocol sets the stage for future multireference treatments of multiple pigments, and hence for multireference studies of charge separation, while it is transferable to the study of any photoactive embedded tetrapyrrole system.

Deep learning of many-body observables and quantum information scrambling

Machine learning has shown significant breakthroughs in quantum science, where in particular deep neural networks exhibited remarkable power in modeling quantum many-body systems. Here, we explore how the capacity of data-driven deep neural networks in learning the dynamics of physical observables is correlated with the scrambling of quantum information. We train a neural network to find a mapping from the parameters of a model to the evolution of observables in random quantum circuits for various regimes of quantum scrambling and test its generalization and extrapolation capabilities in applying it to unseen circuits. Our results show that a particular type of recurrent neural network is extremely powerful in generalizing its predictions within the system size and time window that it has been trained on for both, localized and scrambled regimes. These include regimes where classical learning approaches are known to fail in sampling from a representation of the full wave function. Moreover, the considered neural network succeeds in extrapolating its predictions beyond the time window and system size that it has been trained on for models that show localization, but not in scrambled regimes.

Strategies to Obtain Reliable Energy Landscapes from Embedded Multireference Correlated Wavefunction Methods for Surface Reactions.

Embedded correlated wavefunction (ECW) theory is a powerful tool for studying ground- and excited-state reaction mechanisms and associated energetics in heterogeneous catalysis. Several factors are important to obtaining reliable ECW energies, critically the construction of consistent active spaces (ASs) along reaction pathways when using a multireference correlated wavefunction (CW) method that relies on a subset of orbital spaces in the configuration interaction expansion to account for static electron correlation, e.g., complete AS self-consistent field theory, in addition to the adequate partitioning of the system into a cluster and environment, as well as the choice of a suitable basis set and number of states included in excited-state simulations. Here, we conducted a series of systematic studies to develop best-practice guidelines for ground- and excited-state ECW theory simulations, utilizing the decomposition of NH3 on Pd(111) as an example. We determine that ECW theory results are relatively insensitive to cluster size, the aug-cc-pVDZ basis set provides an adequate compromise between computational complexity and accuracy, and that a fixed-clean-surface approximation holds well for the derivation of the embedding potential. Additionally, we demonstrate that a merging approach, which involves generating ASs from the molecular fragments at each configuration, is preferable to a creeping approach, which utilizes ASs from adjacent structures as an initial guess, for the generation of consistent potential energy curves involving open-d-shell metal surfaces, and, finally, we show that it is essential to include bands of excited states in their entirety when simulating excited-state reaction pathways.

Modeling bicarbonate formation in an alkaline solution with multi-level quantum mechanics/molecular dynamics simulations

Understanding carbonate speciation and how it may be modulated is essential for the advancement of carbon dioxide (CO2) capture and storage technologies, which often rely on the transformation of CO2 into carbonate, e.g. via the formation of carbonate minerals. To date, few atomic-level, quantum-mechanics-based simulations have been carried out to characterize how carbonic acid (H2CO3) and bicarbonate ( HCO 3 − ) form in aqueous solution, and how pH affects this process. Recently, Martirez and Carter utilized rare-event sampling density functional theory molecular dynamics simulations in combination with multi-level embedded correlated wavefunction theory, thus accounting for both solvent dynamics and electron correlation accurately, to elucidate the mechanism of H2CO3 formation in neutral solution (J. Am. Chem. Soc., 145, 12561, 2023). Here, we perform a complementary simulation using the same method to map out the energetics of HCO 3 − formation from dissolved CO2 in basic solution. We find that, as in H2CO3 formation, including water dynamics is important to obtain an accurate prediction of the energetics for the aforementioned reaction. Furthermore, only with MD did we identify the correct pathway for the reaction, in which water – not hydroxide – acts as the initial nucleophile and only at the transition state does it lose a proton.

Coupling molecular density functional theory with converged selected configuration interaction methods to study excited states in aqueous solution.

This paper presents the first implementation of a coupling between advanced wavefunction theories and molecular density functional theory (MDFT). This method enables the modeling of solvent effect into quantum mechanical (QM) calculations by incorporating an electrostatic potential generated by solvent charges into the electronic Hamiltonian. Solvent charges are deduced from the spatially and angularly dependent solvent particle density. Such a density is obtained through the minimization of the functional associated with the molecular mechanics (MM) Hamiltonian describing the interaction between the fluid particles. The introduced QM/MDFT framework belongs to QM/MM family of methods, but its originality lies in the use of MDFT as the MM solver, offering two main advantages. First, its functional formulation makes it competitive with respect to sampling-based molecular mechanics. Second, it preserves a molecular-level description lost in macroscopic continuum approaches. The excited state properties of water and formaldehyde molecules solvated into water have been computed at the selected configuration interaction (SCI) level. The excitation energies and dipole moments have been compared with experimental data and previous theoretical work. A key finding is that using the Hartree-Fock method to describe the solute allows for predicting the solvent charge around the ground state with sufficient precision for the subsequent SCI calculations of excited states. This significantly reduces the computational cost of the described procedure, paving the way for the study of more complex molecules.

Unscrambling of single-particle wave functions in systems localized through disorder and monitoring

In systems undergoing localization-delocalization quantum phase transitions due to disorder or monitoring, there is a crucial need for robust methods capable of distinguishing phases and uncovering their intrinsic properties. In this paper, we develop a process of finding a Slater determinant representation of free-fermion wave functions that accurately characterizes localized particles, a procedure we dub “unscrambling.” The central idea is to minimize the overlap between envelopes of single-particle wave functions or, equivalently, to maximize the inverse participation ratio of each orbital. This numerically efficient methodology can differentiate between distinct types of wave functions: exponentially localized, power-law localized, and conformal critical, also revealing the underlying physics of these states. The method is readily extendable to systems in higher dimensions. Furthermore, we apply this approach to a more challenging problem involving disordered monitored free fermions in one dimension, where the unscrambling process unveils the presence of a conformal critical phase and a localized area-law quantum Zeno phase. Importantly, our method can also be extended to free fermion systems without particle number conservation, which we demonstrate by estimating the phase diagram of Z2-symmetric disordered monitored free fermions. Our results unlock the potential of utilizing single-particle wave functions to gain valuable insights into the localization transition properties in systems such as monitored free fermions and disordered models. Published by the American Physical Society 2024

Quantum mechanics of composite fermions

We establish the quantum mechanics of composite fermions based on the dipole picture initially proposed by Read. It comprises three complimentary components: a wave equation for determining the wave functions of a composite fermion in ideal fractional quantum Hall states and when subjected to external perturbations, a wave-function for mapping a many-body wave function of composite fermions to a physical wave function of electrons, and a microscopic approach for determining the effective Hamiltonian of the composite fermion. The wave equation resembles the ordinary Schrödinger equation but has drift velocity corrections that are not present in the Halperin-Lee-Read theory. The wave-function constructs a physical wave function of electrons by projecting a state of composite fermions onto a half-filled bosonic Laughlin state of vortices. Remarkably, Jain's wave-function can be reinterpreted as the new in an alternative wave-function representation of composite fermions. The dipole picture and the effective Hamiltonian can be derived from the microscopic model of interacting electrons confined in a Landau level, with all parameters determined. In this framework, we can construct the physical wave function of a fractional quantum Hall state deductively by solving the wave equation and applying the wave-function , based on the effective Hamiltonian derived from first principles, rather than relying on intuition or educated guesses. For ideal fractional quantum Hall states in the lowest Landau level, the approach reproduces the well-established results of the standard theory of composite fermions. We further demonstrate that the reformulated theory of composite fermions can be easily generalized for flat Chern bands. Published by the American Physical Society 2024

A non-hierarchical multi-layer multi-configurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces.

The correlation discrete variable representation (CDVR) enables multi-layer multi-configurational time-dependent Hartree (MCTDH) quantum dynamics simulations on general potential energy surfaces. In a recent study [R. Ellerbrock and U. Manthe, J. Chem. Phys. 156, 134107 (2022)], an improved CDVR that can account for the symmetry properties of a tree-shaped wavefunction representation has been introduced. This non-hierarchical CDVR drastically reduces the number of grid points required in the time-dependent quadrature used to evaluate all potential energy matrix elements. While the first studies on the non-hierarchical CDVR approach have been restricted to single-layer calculations, here the complete theory required for the implementation of the non-hierarchical CDVR approach in the multi-layer MCTDH context will be presented. Detailed equations facilitating the efficient recursive computation of all matrix elements are derived, and a new notation adapted to the symmetry properties of the tree-shaped representation is introduced. Calculations studying the non-adiabatic quantum dynamics of photoexcited pyrazine in 24 dimensions illustrate the properties of the non-hierarchical multi-layer CDVR.