Matrix Elements of One Dimensional Explicitly Correlated Gaussian Basis Functions

Zaklama, Timothy; Zhang, David; Rowan, Keefer; Schatzki, Louis; Suzuki, Y.; Varga, K.

doi:10.1007/s00601-019-1539-3

Cited by 8 publications

(5 citation statements)

References 60 publications

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“…Equation (8) tells us that the expectation value obtained from |Ψ Trial is an upper bound to the exact ground-state energy.…”

Section: B Upper Bounds In Quantum Mechanicsmentioning

confidence: 99%

Integrals for lower bounds to the exact energy

Ireland

2021

Preprint

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Methods for calculating lower bounds to the exact energy using the variance of the upper bound energy are discussed and explored. All the matrix elements of the Hamiltonian squared are collected and considered, and those for which no known solutions could be found in the literature are derived for an explicitly correlated Gaussian (ECG) basis set. Analytical Solutions are determined for twoelectron, mono-nuclear systems, in addition to a one-dimensional integral expression which has use in polyatomic calculations. The newly derived integral expressions have been implemented in the integral library of the QUANTEN computer program. I. INTRODUCTIONSince the successful application of Schrödinger equation to the hydrogen atom in the mid 1920's, few-body systems have been of great interest to physicists and chemists. The ability to describe few-particle systems such as small atoms and molecules however becomes problematic due to a lack of analytical solutions. As such, quantum chemistry requires the use of approximations in order to simplify quantum problems, allowing for approximate solutions to be obtained for the physical properties of the systems in question.In atomic and molecular physics, one property of great interest is the energy of different eigenstates. The energy of a particular state can be calculated by approximating the wave function through linear and non-linear parameterisation. This approximate wave function is called the trial wave function for the system, and it can be shown that the energy expectation value associated with this trial wave function serves as an upper bound to the exact energy [1,2]. Calculating 'tight' upper bounds to the exact energy is well known and occurs frequently in modern quantum chemistry. However, the upper-bound value itself does not give information on how close it is to the exact energy. If a lower bound could also be calculated then this, along with the upper-bound value, would give an interval within which the exact energy can be found. Of course, the lower-bound value is only meaningful if the lower bound is of the same quality as the upper bound.As is true with upper bounds, improving the basis set with respect to the variational principle improves the lowerbound value, though the convergence of lower bounds is slower. Nevertheless, convergence of both upper and lower bounds leads to ever-tighter intervals, within which the exact energy lies. Two such methods for calculating lower bounds are the Weinstein criterion [3] and Temple's lower bound [4], presented by D. Weinstein and G. Temple respectively. Weinstein's bound tends to give lower bounds that are of too low a quality compared to the upper bounds, whereas Temple's bound is regarded to give better quality lower bounds [5,6]. Both the Weinstein criterion and Temple's bound make use of the variance of the upper bound energy, σ E 2 , which requires the matrix elements of the Hamiltonian operator squared, Ĥ2 .The trial wave function can be written as a linear combination of basis functions in the many-particle sta...

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“…Equation (8) tells us that the expectation value obtained from |Ψ Trial is an upper bound to the exact ground-state energy.…”

Section: B Upper Bounds In Quantum Mechanicsmentioning

confidence: 99%

Integrals for lower bounds to the exact energy

Ireland

2021

Preprint

View full text Add to dashboard Cite

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“…Another approach to calculating matrix elements of non-spherical ECGs is to calculate the matrix elements of the 1D case analytically 71 , and then generalize to 2D or 3D using tensor products if needed. The advantage of this approach is that the ECGs parameters can be different in different directions and problems with nonspherical potentials can be solved.…”

Section: Introductionmentioning

confidence: 99%

Deformed Explicitly Correlated Gaussians

Beutel,

Ahrens,

Huang

et al. 2021

Preprint

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Deformed correlated Gaussian basis functions are introduced and their matrix elements are calculated. These basis functions can be used to solve problems with nonspherical potentials. One example of such potential is the dipole self-interaction term in the Pauli-Fierz Hamiltonian. Examples are presented showing the accuracy and necessity of deformed Gaussian basis functions to accurately solve light-matter coupled systems in cavity QED.

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“…The basis parameters have been optimized using the stochastic variational approach (SVM) 36 . The advantage of the approach is that the matrix elements are analytically available 6,36,37 and it produces very accurate energies and wave functions 35 . This method has been used to describe excitonic complexes 8,35,[38][39][40][41] and two and three-dimensional quantum dots 42,43 .…”

Section: Introductionmentioning

confidence: 99%

Energy spectrum and structure of one-dimensional few-electron Wigner crystals with and without coupling to light in cavity

Huang,

Pitagora,

Zaklama

et al. 2021

Preprint

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Explicitly Correlated Gaussian basis is used to calculate the energies and wave functions of one dimensional few-electron systems in confinement potentials created by external potentials or coupling to light in cavity. The appearance and properties of Wigner crystal-like structures are discussed. It is shown that one dimensional Wigner crystals can be formed by coupling electrons to light due to the dipole self-interaction term in the light-matter Hamiltonian, provided an additional extremely weak confining potential is present.

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Matrix Elements of One Dimensional Explicitly Correlated Gaussian Basis Functions

Cited by 8 publications

References 60 publications

Integrals for lower bounds to the exact energy

Integrals for lower bounds to the exact energy

Deformed Explicitly Correlated Gaussians

Energy spectrum and structure of one-dimensional few-electron Wigner crystals with and without coupling to light in cavity

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