Accurate Scaling Functions of the Scaled Schrödinger Equation. II. Variational Examination of the Correct Scaling Functions with the Free Complement Theory Applied to the Helium Atom

In a previous paper [Phys. Rev. Lett. 2004, 93, 030403.], one of the authors introduced the scaled Schrödinger equation (SSE), g(H - E)ψ = 0 for atoms and molecules, where the scaling function g is the positive function of the electron-nuclear (e-n) and electron-electron (e-e) distances. The SSE is equivalent to the Schrödinger equation (SE), (H - E)ψ = 0, that governs the chemical world but does not have the divergence difficulty that occurs when we try to solve the SE to obtain the exact solution. The g function is essential not only to prevent this divergence difficulty but also to obtain the exact wave function of the SE or SSE. In paper I of this series [J. Chem. Phys. 2022, 156, 014113.], we introduced five analytical g functions that behave correctly at both the coalescence and asymptotic regions, but we examined them only for the e-e part. In this paper, we examine these correct g functions for both e-n and e-e parts by applying the free complement (complete-element) (FC) theory variationally to the He atom. However, even for the two-electron He atom, the analytical integral formulas were not obtained when we use the correct g functions for both e-n and e-e parts, except for g = 1 - exp(-γr), but we were able to perform variational FC calculations by employing numerical integration schemes. We examined not only the energy and wave function but also the H-square error (defined by eq 14 of the text), energy lower bound, and e-n and e-e cusp properties. For the energy lower bound, we applied our FC wave functions to the method proposed recently by Pollak, Martinazzo, and others and could obtain good results. With the use of the correct-group g functions, the convergence of the FC theory to the exact analytical solution of the SE or SSE became efficient, and the performance was particularly good with the g functions, r/(r + 1/γ), Ei, and 1 - exp(-γr) in this order. These results were always superior to those obtained with g = r.

Stochastic representation of many-body quantum states

The quantum many-body problem is ultimately a curse of dimensionality: the state of a system with many particles is determined by a function with many dimensions, which rapidly becomes difficult to efficiently store, evaluate and manipulate numerically. On the other hand, modern machine learning models like deep neural networks can express highly correlated functions in extremely large-dimensional spaces, including those describing quantum mechanical problems. We show that if one represents wavefunctions as a stochastically generated set of sample points, the problem of finding ground states can be reduced to one where the most technically challenging step is that of performing regression-a standard supervised learning task. In the stochastic representation the (anti)symmetric property of fermionic/bosonic wavefunction can be used for data augmentation and learned rather than explicitly enforced. We further demonstrate that propagation of an ansatz towards the ground state can then be performed in a more robust and computationally scalable fashion than traditional variational approaches allow.

Accurate Exponential Representations for the Ground State Wave Functions of the Collinear Two-Electron Atomic Systems

In the framework of the study of helium-like atomic systems possessing the collinear configuration, we propose a simple method for computing compact but very accurate wave functions describing the relevant S-state. It is worth noting that the considered states include the well-known states of the electron–nucleus and electron–electron coalescences as a particular case. The simplicity and compactness imply that the considered wave functions represent linear combinations of a few single exponentials. We have calculated such model wave functions for the ground state of helium and the two-electron ions with nucleus charge 1≤Z≤5. The parameters and the accompanying characteristics of these functions are presented in tables for number of exponential from 3 to 6. The accuracy of the resulting wave functions are confirmed graphically. The specific properties of the relevant codes by Wolfram Mathematica are discussed. An example of application of the compact wave functions under consideration is reported.

Exponentially correlated Hylleraas-configuration interaction non-relativistic energy of the 1S ground state of the helium atom

A generalization of the Hylleraas-Configuration Interaction method (Hy-CI) first proposed by Wang, et al., the Exponentially Correlated Hylleraas-Configuration Interaction method (E-Hy-CI) in which the single r _ij of an Hy-CI wave function is generalized to a form of the generic type r i j ν i j e - ω i j r i j , is explored. This type of correlation, suggested by Hirshfelder in 1960, has the right behavior both in the vicinity of the r_ij cusp as r_ij goes to 0 and as r_ij goes to infinity; this work explores whether wave functions containing both linear and exponential r _ij factors converge more rapidly than either one alone. The method of calculation of the two-electron E-Hy-CI kinetic energy and electron repulsion integrals in a stable and efficient way using recursion relations is discussed, and the relevant formulas are given. The convergence of the E-Hy-CI wave function expansion is compared with that of the Hy-CI wave function without exponential correlation factors, demonstrating both convergence acceleration and an improvement in the accuracy for the same basis. This makes the application of the E-Hy-CI method to systems with N > 4, for which this formalism with at most a single r i j ν i j e - ω i j r i j factor per term leads to solvable integrals, very promising. E-Hy-CI method variational calculations with up to 10080 expansion terms are reported for the ground ¹ S state of the neutral helium atom, with a resultant nonrelativistic energy of -2.9037 2437 7034 1195 9831 1084 hartree for the best expansion.

Precision spectroscopy of atomic helium

Helium is a prototype three-body system and has long been a model system for developing quantum mechanics theory and computational methods. The fine-structure splitting in the 23P state of helium is considered to be the most suitable for determining the fine-structure constant α in atoms. After more than 50 years of efforts by many theorists and experimentalists, we are now working toward a determination of α with an accuracy of a few parts per billion, which can be compared to the results obtained by entirely different methods to verify the self-consistency of quantum electrodynamics. Moreover, the precision spectroscopy of helium allows determination of the nuclear charge radius, and it is expected to help resolve the 'proton radius puzzle'. In this review, we introduce the latest developments in the precision spectroscopy of the helium atom, especially the discrepancies among theoretical and experimental results, and give an outlook on future progress.

The order of three lowest-energy states of the six-electron harmonium at small force constant

The order of low-energy states of six-electron harmonium is uncertain in the case of strong correlation, which is not a desired situation for the model system being considered for future testing of approximate methods of quantum chemistry. The computational study of these states has been carried out at the frequency parameter ω = 0.01, using the variational method with the basis of symmetry-projected, explicitly correlated Gaussian (ECG) lobe functions. It has revealed that the six-electron harmonium at this confinement strength is an octahedral Wigner molecule, whose order of states is different than in the strong confinement regime and does not agree with the earlier predictions. The results obtained for ω = 0.5 and 10 are consistent with the findings based on the Hund's rules for the s(2)p(4) electron configuration. Substantial part of the computations has been carried out on the graphical processing units and the efficiency of these devices in calculation of the integrals over ECG functions has been compared with traditional processors.

Singular analysis and coupled cluster theory

The primary motivation for systematic bases in first principles electronic structure simulations is to derive physical and chemical properties of molecules and solids with predetermined accuracy. This requires, however, a detailed asymptotic analysis of singularities.

The splitting of atomic orbitals with a common principal quantum number revisited: np vs. ns

Atomic orbitals with a common principal quantum number are degenerate, as in the hydrogen atom, in the absence of interelectronic repulsion. Due to the virial theorem, electrons in such orbitals experience equal nuclear attractions. Comparing states of several-electron atoms that differ by the occupation of orbitals with a common principal quantum number, such as 1s(2) 2s vs. 1s(2) 2p, we find that although the difference in energies, ΔE, is due to the interelectronic repulsion term in the Hamiltonian, the difference between the interelectronic repulsions, ΔC, makes a smaller contribution to ΔE than the corresponding difference between the nuclear attractions, ΔL. Analysis of spectroscopic data for atomic isoelectronic sequences allows an extensive investigation of these issues. In the low nuclear charge range of pertinent isoelectronic sequences, i.e., for neutral atoms and mildly positively charged ions, it is found that ΔC actually reverses its sign. About 96% of the nuclear attraction difference between the 6p (2)P and the 6s (2)S states of the Cs atom is cancelled by the corresponding interelectronic repulsion difference. From the monotonic increase of ΔE with Z it follows (via the Hellmann-Feynman theorem) that ΔL > 0. Upon increasing the nuclear charge along an atomic isoelectronic sequence with a single electron outside a closed shell from Z(c), the critical charge below which the outmost electron is not bound, to infinity, the ratio ΔC/ΔL increases monotonically from lim(Z→Z(c)(+))ΔC/ΔL=-1 to lim(Z→∞)ΔC/ΔL=1. These results should allow for a more nuanced discussion than is usually encountered of the crude electronic structure of many-electron atoms and the structure of the periodic table.

Confined helium atom low-lying S states analyzed through correlated Hylleraas wave functions and the Kohn-Sham model

Calculation including the electron correlation effects is reported for the ground 1 1S and lowest triplet 1 3S state energies of the confined helium atom placed at the center of an impenetrable spherical box. While the adopted wave-functional treatment involves optimization of three nonlinear parameters and 10, 20, and 40 linear coefficients contained in wave functions expressed in a generalized Hylleraas basis set that explicitly incorporates the interelectronic distance r12, via a Slater-type exponent and through polynomial terms entering the expansion, the Kohn-Sham model employed here uses the Perdew and Wang exchange-correlation functional in its spin-polarized version within the local-density approximation (LDA) with and without the self-interaction correction. All these calculations predict a systematic increase in the singlet-triplet energy splitting toward the high confinement regime, i.e., when the box radius is reduced. By using the variational results as benchmark, it is found that the LDA underestimates the singlet-triplet energy splitting, whereas the self-interaction correction overestimates such a quantity.

Highly Compact Wave Functions for He-Like Systems

Wave functions which are compact, but still quite accurate, are extremely valuable as tools for gaining understanding of quantum systems. This paper investigates the use for that purpose of functions that depend exponentially on all the interparticle distances of a few-body system, illustrated by a study of the ground electronic states of the He isoelectronic series (Z from 1 to 10). Using as few as 4 exponential basis functions, it is found that nonrelativistic energies are reproduced to within 38 microhartrees of the exact values, an error far less than for previously reported compact wave functions. Other properties are also well-represented.

Similarity-Transformed Hamiltonians by Means of Gaussian-Damped Interelectronic Distances

A method is presented to improve the description of electron correlation in configuration interaction (CI) calculations. In this method, the standard CI expansion ψ is multiplied by a correlation function ϕ = exp (F) with F = Σ(from m = 1 to M) cm Σ(for i < j) rij exp(-γmrij2). With this correlation function, the total wavefunction Ψ = ϕψ exhibits the right behavior when two electron coalesce while F vanishes for large interelectronic distances. The correlation function is implemented using the methodology of similarity-transformed Hamiltonians and is applied to two-electron systems. A generalization to many-electron systems is indicated. The new method yields more accurate results than standard CI calculations of the energy and interelectronic distance of the He atom. The H2 molecule was chosen to study the long-range behavior of the correlation function.

Estudo da série iso-eletrônica do átomo de hélio pelo método hiperesférico

Neste trabalho estudamos o espectro da série iso-eletrônica do átomo de hélio utilizando o método hiperesférico adiabático. Este método permite o estudo dos níveis de energia de sistemas atômicos por meio de um conjunto de curvas de potencial, de forma semelhante à aproximação de Born-Oppenheimer para sistemas moleculares. As curvas de potencial são definidas com relação a uma única variável radial, independentemente do número de elétrons do sistema. Desta forma a análise e classificação dos níveis de energia é realizada de forma simples e intuitiva, o que não se observa em métodos como o variacional e Hartree-Fock. O objetivo desta pesquisa é o de descrever o comportamento do estado fundamental de sistemas heliônicos com a variação da carga nuclear. Além do método hiperesférico simplificar muito a análise dos resultados, é um processo ab-initio, cujos erros são limitados apenas pelos truncamentos do número de equações acopladas. Já na sua aproximação mais simples, onde todos os acoplamentos radiais são desprezados, o erro obtido para a energia do estado fundamental é inferior a 1% e com a introdução do acoplamento diagonal o erro cai para cerca de 0.3%. Resultados de grande precisão são obtidos com os acoplamentos não diagonais, atingindo precisões da ordem de 10-3 %.