Accurate Wave Functions Research Articles

Modelling the non-local thermodynamic equilibrium spectra of silylene (SiH2).

This paper sets out a robust methodology for modelling spectra of polyatomic molecules produced in reactive or dissociative environments, with vibrational populations outside local thermal equilibrium (LTE). The methodology is based on accurate, extensive ro-vibrational line lists containing transitions with high vibrational excitations and relies on the detailed ro-vibrational assignments. The developed methodology is applied to model non-LTE IR and visible spectra of silylene (SiH2) produced in a decomposition of disilane (Si2H6), a reaction of technological importance. Two approaches for non-LTE vibrational populations of the product SiH2 are introduced: a simplistic 1D approach based on the Harmonic approximation and a full 3D model incorporating accurate vibrational wavefunctions of SiH2 computed variationally with the TROVE (Theoretical ROVibrational Energy) program. We show how their non-LTE spectral signatures can be used to trace different reaction channels of molecular dissociations.

Ultra‐compact accurate wave functions for He‐like and Li‐like iso‐electronic sequences and variational calculus: I. Ground state

AbstractSeveral ultra‐compact accurate wave functions in the form of generalized Hylleraas‐Kinoshita functions and Guevara‐Harris‐Turbiner functions, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC), for the ground state energies (4‐5 significant digits [s.d.]) of He‐like and Li‐like iso‐electronic sequences in the static approximation with point‐like, infinitely heavy nuclei are constructed. It is shown that for both sequences the obtained parameters can be fitted in Z by simple smooth functions: in general, these parameters differ from the ones emerging in variational calculations. For the He‐like two‐electron sequence the approximate expression for the ground state function, which provides absolute accuracy for the energy ∼10−3 a.u. and the same relative accuracies ∼10−2 to 10−3 for both the cusp parameters and the six expectation values, is found. For the Li‐like three‐electron sequence the most accurate ultra‐compact function taken as the variational trial function provides absolute accuracy for energy ∼10−3 a.u., 2 to 3 s.d. for the electron‐nuclear cusp parameter for Z ≤ 20 and 3 s.d. for the two expectation values for Z = 3.

Valence Bond Alternative Yielding Compact and Accurate Wave Functions for Challenging Excited States. Application to Ozone and Sulfur Dioxide.

A novel state-averaged version of ab initio nonorthogonal valence bond method is described, for the sake of accurate theoretical studies of excited states in the valence bond framework. With respect to standard calculations in the molecular orbital framework, the state-averaged breathing-orbital valence bond (BOVB) method has the advantage to be free from the penalizing constraint for the ground and excited state(s) to share the same unique set of orbitals. The ability of the BOVB method to faithfully describe excited states and to compute accurate transition energies from the ground state is tested on the five lowest-lying singlet electronic states of ozone and sulfur dioxide, among which 11B2 and 21A1 are the challenging ones. As the 11A2, 11B1, and 11B2 states are of different symmetries than the ground state, they can be calculated at the state-specific BOVB level. On the other hand, the 21A1 states and the 11A1 ground states, which are of like symmetry, are calculated with the state-averaged BOVB technique. In all cases, the calculated vertical energies are close to the experimental values when available, and at par with the most sophisticated calculations in the molecular framework, despite the extreme compactness of the BOVB wave functions, made of no more than 5-9 valence bond structures in all cases. The features that allow the combination of compactness and accuracy in challenging cases are analyzed. For the "ionic" 11B2 states, which are the site of important charge fluctuations, it is because of the built-in dynamic correlation inherent to the BOVB method. For the 21A1 ones, this is the fact that these states have the degree of freedom of having different orbitals than the ground states, even though they are of like symmetry and calculated simultaneously using the newly implemented state-average BOVB algorithm. Finally, the description of the excited states in terms of Lewis structures is insightful, rationalizing the fast ring closure for the 21A1 state of ozone and predicting some diradical character in the so-called "ionic" 11B2 states.

Combination of tensor network states and Green's function Monte Carlo

We propose an approach to study the ground state of quantum many-body systems in which Tensor Network States (TNS), specifically Projected Entangled Pair States (PEPS), and Green's function Monte Carlo (GFMC) are combined. PEPS, by design, encode the area law which governs the scaling of entanglement entropy in quantum systems with short range interactions but are hindered by the high computational complexity scaling with bond dimension (D). GFMC is a highly efficient method, but it usually suffers from the infamous negative sign problem which can be avoided by the fixed node approximation in which a guiding wave function is utilized to modify the sampling process. The trade-off for the absence of negative sign problem is the introduction of systematic error by guiding wave function. In this work, we combine these two methods, PEPS and GFMC, to take advantage of both of them. PEPS are very accurate variational wave functions, while at the same time, only contractions of single-layer tensor network are necessary in GFMC, which reduces the cost substantially. Moreover, energy obtained in GFMC is guaranteed to be variational and lower than the variational energy of the guiding PEPS wave function. Benchmark results of $J_1$-$J_2$ Heisenberg model on square lattice are provided.

Exploring Hilbert space on a budget: Novel benchmark set and performance metric for testing electronic structure methods in the regime of strong correlation.

This work explores the ability of classical electronic structure methods to efficiently represent (compress) the information content of full configuration interaction (FCI) wave functions. We introduce a benchmark set of four hydrogen model systems of different dimensionalities and distinctive electronic structures: a 1D chain, a 1D ring, a 2D triangular lattice, and a 3D close-packed pyramid. To assess the ability of a computational method to produce accurate and compact wave functions, we introduce the accuracy volume, a metric that measures the number of variational parameters necessary to achieve a target energy error. Using this metric and the hydrogen models, we examine the performance of three classical deterministic methods: (i) selected configuration interaction (sCI) realized both via an a posteriori (ap-sCI) and variational selection of the most important determinants, (ii) an a posteriori singular value decomposition (SVD) of the FCI tensor (SVD-FCI), and (iii) the matrix product state representation obtained via the density matrix renormalization group (DMRG). We find that the DMRG generally gives the most efficient wave function representation for all systems, particularly in the 1D chain with a localized basis. For the 2D and 3D systems, all methods (except DMRG) perform best with a delocalized basis, and the efficiency of sCI and SVD-FCI is closer to that of DMRG. For larger analogs of the models, the DMRG consistently requires the fewest parameters but still scales exponentially in 2D and 3D systems, and the performance of SVD-FCI is essentially equivalent to that of ap-sCI.

Variational ansatz for p-wave fermions confined in a one-dimensional harmonic trap

We propose a very accurate and efficient variational scheme for the ground state of the system of p-wave attractively interacting fermions confined in a one-dimensional harmonic trap. By the construction, the method takes the non-analytical part of interactions exactly into account and thus it approximates the true ground-state wave function in a whole range of interactions very accurately. Within the method, we determine different properties of the system for a different number of particles and different interactions. In this way, we explore how the system and its features transit from the ideal non-interacting Fermi gas to the system of infinitely strong attractions. Additionally, we demonstrate that the ansatz may also be used on a repulsive branch of interactions where other numerical methods break down. The presented method of including zero-range interactions is very universal and may be easily generalized to other one-dimensional confinements.

Small recoil momenta double ionization of He and two-electron ions by high energy photons

We calculate various differential and double differential characteristics of ionization by a single photon for H−, He and for the two-electron ions with Z = 3, 4, 5 in the region of the so-called quasi-free mechanism (QFM) domination. We employ highly accurate wave functions at the electron-electron coalescence line where coordinates of both ionized electrons coincide. We trace the Z dependence of the double differential distributions. For all considered targets we discuss the dependence of the photoelectron energy distribution on the photon energy. Our calculation demonstrated the rapid decrease of QFM contribution with increase of the difference in energy of two outgoing electrons, and with decrease of the angle between two outgoing momenta. As a general feature, we observe the decrease of QFM contribution with nuclear charge growth.

Water mediated electron attachment to nucleobases: Surface-bound vs bulk solvated electrons.

We have presented a mechanism for electron attachment to solvated nucleobases using accurate wave-function based hybrid quantum/classical (QM/MM) simulations and uracil as a test case. The initial electron attached state is found to be localized in the bulk water, and this water-bound state acts as a doorway to the formation of the final nucleobase bound state. The electron transfer from water to uracil takes place because of the mixing of electronic and nuclear degrees of freedom. The water molecules around the uracil stabilize the uracil-bound anion by creating an extensive hydrogen-bonding network and accelerate the rate of electron attachment to uracil. The complete transfer of the electron from water to the uracil occurs in a picosecond time scale, which is consistent with the experimentally observed rate of reduction of nucleobases in the presence of water. The degree of solvation of the aqueous electron can lead to a difference in the initial stabilization of the uracil-bound anion. However, the anions formed due to the attachment of both surface-bound and bulk-solvated electrons behave similarly to each other at a longer time scale.

Simulating disordered quantum Ising chains via dense and sparse restricted Boltzmann machines.

In recent years, generative artificial neural networks based on restricted Boltzmann machines (RBMs) have been successfully employed as accurate and flexible variational wave functions for clean quantum many-body systems. In this article, we explore their use in simulations of disordered quantum Ising chains. The standard dense RBM with all-to-all interlayer connectivity is not particularly appropriate for large disordered systems, since in such systems one cannot exploit translational invariance to reduce the amount of parameters to be optimized. To circumvent this problem, we implement sparse RBMs, whereby the visible spins are connected only to a subset of local hidden neurons, thus reducing the amount of parameters. We assess the performance of sparse RBMs as a function of the range of the allowed connections, and we compare it with that of dense RBMs. Benchmark results are provided for two sign-problem-free Hamiltonians, namely pure and random quantum Ising chains. The RBM Ansätzes are trained using the unsupervised learning scheme based on projective quantum Monte Carlo (PQMC) algorithms. We find that the sparse connectivity facilitates the training process and allows sparse RBMs to outperform their dense counterparts. Furthermore, the use of sparse RBMs as guiding functions for PQMC simulations allows us to perform PQMC simulations at a reduced computational cost, avoiding possible biases due to finite random-walker populations. We obtain unbiased predictions for the ground-state energies and the magnetization profiles with fixed boundary conditions, at the ferromagnetic quantum critical point. The magnetization profiles agree with the Fisher-de Gennes scaling relation for conformally invariant systems, including the scaling dimension predicted by the renormalization-group analysis.

Variational Localized Active Space Self-Consistent Field Method.

Fragmentation methods applied to multireference wave functions constitute a road toward the application of highly accurate ab initio wave function calculations to large molecules and solids. However, it is important for reproducibility and transferability that a fragmentation scheme be well defined with minimal dependence on initial orbital guesses or user-designed ad hoc fragmentation schemes. One way to improve this sort of robustness is to ensure that the energy obeys a variational principle; that is, the active orbitals and active space wave functions minimize the electronic energy in a certain ansatz for the molecular wave function. We extended the theory of the localized active space self-consistent field, LASSCF, method (JCTC 2019, 15, 972) to fully minimize the energy with respect to all orbital rotations, rendering it truly variational. The new method, called vLASSCF, substantially improves the robustness and reproducibility of the LAS wave function compared to LASSCF. We analyze the storage and operation cost scaling of vLASSCF compared to orbital optimization using a standard complete active space self-consistent field (CASSCF) approach, and we show results of vLASSCF calculations on some simple test systems. We show that vLASSCF is energetically equivalent to CASSCF in the limit of one active subspace and that vLASSCF significantly improves upon the reliability of LASSCF energy differences, allowing for more meaningful and subtle analysis of potential energy curves of dissociating molecules. We also show that all forms of LASSCF have a lower operation cost scaling than the orbital optimization part of CASSCF.

Computer program ATOM-MOL-nonBO for performing calculations of ground and excited states of atoms and molecules without assuming the Born-Oppenheimer approximation using all-particle complex explicitly correlated Gaussian functions.

In this work, we describe a computer program called ATOM-MOL-nonBO for performing bound state calculations of small atoms and molecules without assuming the Born-Oppenheimer approximation. All particles forming the systems, electrons and nuclei, are treated on equal footing. The wave functions of the bound states are expanded in terms of all-particle one-center complex explicitly correlated Gaussian functions multiplied by Cartesian angular factors. As these Gaussian functions are eigenfunctions of the operator representing the square of the total angular momentum of the system, the problem separates and calculations of states corresponding to different values of the total rotational quantum number can be solved independently from each other. Due to thorough variational optimization of the Gaussian exponential parameters, the method allows us to generate very accurate wave functions. The optimization is aided by analytically calculated energy gradient determined with respect to the parameters. Three examples of calculations performed for diatomic and triatomic molecules are shown as an illustration of calculations that can be performed with this program. Finally, we discuss the limitations, applicability range, and bottlenecks of the program.

Normal ordering of the su(1, 1) ladder operators for the quasi-number states of the Morse oscillator

The Morse oscillator is an adequate zero-order model for describing the highly excited vibrational states and large-amplitude vibrational motion. The corresponding Schrödinger equations can be conveniently solved by algebraic methods using the so-called quasi-number states (QNS) resembling the true wave functions of the Morse oscillator. The associated QNS ladder operators are occupation number, creation and annihilation operators K0,K+,K− that satisfy the commutation relations [K0,K±]=±K± and [K+,K−]=−2K0 of the Lie algebra su(1, 1). The efficient applications of the corresponding algebraic techniques within the variational or perturbative approximations require reduction of arbitrary multiplicative combinations of the su(1,1) ladder operators to the linear combinations of the normalized products K0aK+bK−c. The closed form solution of this problem for a typical complex form of the operator product is derived and proven mathematically. The validity and the numerical stability of the derived expressions are demonstrated by the systematic numerical tests for the product of six-term multiplicative operators.

The Molpro quantum chemistry package.

Molpro is a general purpose quantum chemistry software package with a long development history. It was originally focused on accurate wavefunction calculations for small molecules but now has many additional distinctive capabilities that include, inter alia, local correlation approximations combined with explicit correlation, highly efficient implementations of single-reference correlation methods, robust and efficient multireference methods for large molecules, projection embedding, and anharmonic vibrational spectra. In addition to conventional input-file specification of calculations, Molpro calculations can now be specified and analyzed via a new graphical user interface and through a Python framework.

Electron correlation effects in isomers of C20

The C20 molecule exists as three low-energy isomers: the monocyclic ring, a corannulene-like bowl structure, and the cage-the smallest possible fullerene. The curious structures of these isomers, along with the valuable properties and possible applications of fullerenes more generally, mean that C20 has attracted interest both experimentally and computationally. Unfortunately, previous theoretical studies have found these C20 isomers present a significant computational challenge, and widely used methods such as Density Functional Theory with different functionals have been unable to agree on even the relative ordering of the isomer energies. Even accurate high-level methods such as coupled-cluster with singles, doubles, and perturbative triples (CCSD(T)) and Diffusion Monte Carlo (DMC) have previously disagreed on the energetic ordering of these isomers. Here we re-examine the ring, bowl, and cage isomers of C20 using a more accurate DMC technique. We employ a novel method to go beyond the single-determinant DMC approaches previously used, and instead use more accurate multi-determinant trial wavefunctions. Our results show that the fullerene cage exhibits different electron correlation effects than the ring and bowl, which when taken into account leads to a reordering of their relative energies. This finally establishes agreement between DMC and relatively recent complete-basis CCSD(T) results, thereby resolving a long-standing disparity between these two high-level descriptions of the C20 isomers. The approach we use is generalisable, and could be used to provide insight into even larger systems in future.

Deep Quantum Geometry of Matrices

We employ machine learning techniques to provide accurate variational wavefunctions for matrix quantum mechanics, with multiple bosonic and fermionic matrices. Variational quantum Monte Carlo is implemented with deep generative flows to search for gauge invariant low energy states. The ground state, and also long-lived metastable states, of an $\mathrm{SU}(N)$ matrix quantum mechanics with three bosonic matrices, as well as its supersymmetric `mini-BMN' extension, are studied as a function of coupling and $N$. Known semiclassical fuzzy sphere states are recovered, and the collapse of these geometries in more strongly quantum regimes is probed using the variational wavefunction. We then describe a factorization of the quantum mechanical Hilbert space that corresponds to a spatial partition of the emergent geometry. Under this partition, the fuzzy sphere states show a boundary-law entanglement entropy in the large $N$ limit.

Multicomponent Orbital-Optimized Perturbation Theory Methods: Approaching Coupled Cluster Accuracy at Lower Cost

Multicomponent quantum chemistry methods such as the nuclear-electronic orbital (NEO) method allow the consistent quantum mechanical treatment of electrons and nuclei. The development of computationally practical, accurate, and robust multicomponent wave function methods is challenging because of the importance of orbital relaxation effects. Herein the variational orbital-optimized coupled cluster with doubles (NEO-OOCCD) method and the orbital-optimized second-order Møller-Plesset perturbation theory (NEO-OOMP2) method with scaled-opposite-spin (SOS) versions are developed and applied to molecular systems in which a proton and all electrons are treated quantum mechanically. The results highlight the importance of orbital relaxation in multicomponent wave function methods. The NEO-SOS'-OOMP2 method, which scales the electron-proton correlation energy as well as the opposite-spin and same-spin components of the electronic correlation energy, is found to achieve nearly the same level of accuracy as the NEO-OOCCD method for proton densities, proton affinities, and optimized geometries. An advantage of the NEO-SOS'-OOMP2 method is that it can be implemented with N4 scaling, where N is a measure of the system size. This method will enable future multicomponent wave function calculations of structures, energies, reaction paths, and dynamics for substantially larger chemical systems.

Tuning spatial entanglement in interacting two-electron quantum dots

Confined geometries such as semiconductor quantum dots are promising candidates for fabricating quantum computing devices. When several quantum dots are in proximity, spatial correlation between electrons in the system becomes significant. In this article, we develop a fully variational action integral formulation for calculating accurate few-electron wavefunctions in configuration space, irrespective of potential geometry. To evaluate the Coulomb integrals with high accuracy, a novel numerical integration method using multiple Gauss quadratures is proposed. Using this approach, we investigate the confinement of two electrons in double quantum dots, and evaluate the spatial entanglement. We investigate the dependence of spatial entanglement on various geometrical parameters. We derive the two-particle wavefunctions in the asymptotic limit of the separation distance between quantum dots, and obtain universal saturation values for the spatial entanglement. Resonances in the entanglement values due to avoided level-crossings of states are observed. We also demonstrate the formation of electron clusters, and show that the entanglement value is a good indicator for the formation of such clusters. Further, we show that a precise tuning of the entanglement values is feasible with applied external electric fields.

Semianalytical wavefunctions and Kohn–Sham exchange-correlation potentials for two-electron atomic systems in two-dimensions

We propose accurate semianalytical wavefunctions for two-electron atomic systems in two-dimensions. These wavefunctions take into account the cusp conditions arising due to the Coulombic nature of external potential and electron–electron interaction, and the screening effects. The first wavefunction is motivated by the Le Sech wavefunction for two-electron atoms and ions in three-dimensions while the second one is constructed to give more variational freedom to the Le Sech wavefunction and is obtained by solving a Schrödinger like equation self-consistently. The results for energies, densities and other quantities are highly accurate and therefore may play an essential role in developing and testing density functional methods for two-dimensional systems.

Calculations of long-range three-body interactions for Li(2 2S )-Li(22 S )-Li+(1 1S)

SynopsisUsing perturbation theory for energies, we evaluate the additive and nonadditive interaction co-efficients C4, C6 , C7 , C8 and C9 for the Li(22S)-Li(22S)-Li+(11S) system. The obtained coefficients Cn are evaluated with highly accurate variationally-generated nonrelativistic wave functions in Hylleraas coordinates. The nonadditive interaction coefficients depend on the geometrical configurations of this three-body system and on the different position of the ion for each configuration. Our calculations may be of interest for the study of three-body recombination and for constructing precise potential energy surfaces.

Convergent close-coupling calculations of positron collisions with the hydrogen negative ion

SynopsisConvergent close-coupling method has been applied to positron scattering by the hydrogen negative ion. Internal consistency of the method and convergence of the cross sections are achieved with use of both the single- and two-centre approaches. Calculations were performed using accurate target wave functions obtained with the multi-core approach.