Momentum distributions and Compton profiles from two-electron atomic wavefunctions containing exponential correlation terms
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Fourier integral techniques are used to generate non-diagonal first-order density matrices (1-matrices) from two-electron atomic wavefunctions containing exponential correlation terms. Although the techniques presented here can be applied to larger systems, the lengthy amounts of algebra limit their usefulness. The 1-matrices generated from two-electron correlated wavefunctions have been transformed to momentum space. The Compton profile obtained from a He wavefunction yielding over 99% of the correlation energy is presented.Keywords:
Momentum (technical analysis)
Electronic correlation
A recently developed method (the GF method) which is equivalent to optimizing the orbitals of a Slater determinant after spin projection has been applied to H−, He, Li+, Be++, Li, Be+,B++, Li−, Be, B+ and C++. These wavefunctions, which can be given an independent particle interpretation, yield better energies than those of the Hartree-Fock method. For example, H− and Li− are correctly predicted to be stable in contradistinction with the Hartree-Fock results. The new correlation energies are tabulated and compared to the Hartree-Fock values. In the case of the two-electron systems the new wavefunctions are nearly at the radial limit, accounting for 93% to 97% of the radial correlation error present in the Hartree-Fock description.
Electronic correlation
Slater determinant
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A series of increasingly accurate wavefunctions for H-, and He built from Gaussian geminals is presented. Such wavefunctions are found to be much more compact than configuration interaction wavefunctions of similar accuracy. Moreover, comparison with nearly exact values shows that they predict one- and two-electron moments in position space quite well but perform less satisfactorily with respect to point properties. The wavefunctions can be Fourier-Dirac transformed in closed form and are used to predict values of one- and two-electron momentum space properties including Compton profiles. Most of these momentum properties are expected to be of benchmark quality.
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Abstract ‘This chapter sketches how the electron correlation is treated in post-Hartree-Fock (HF) wavefunction methods. The distinction between static and dynamic correlation is explained. A configuration interaction (CI) wavefunction is a linear combination of several or many Slater determinants (SDs). Following a HF calculation, different SDs can be constructed by replacing 1, 2, 3, … occupied orbitals in the HF wavefunction with 1, 2, 3,… unoccupied or virtual orbitals, leading to pseudo-excited electron configurations at the singles, doubles, triples, … (S, D, T, …) level. The virtual orbitals are usually available as a by-product of the HF calculation in a basis set. Full CI (FCI) considers all possible substitutions, up to N-fold for an N-electron system. FCI is impractical for all but the smallest molecules. CI truncated at a lower level, e.g. S and D, suffers from lack of size extensitivity. Truncated coupled-cluster (CC) is size extensive. Open-shell systems generally require a multi-reference treatment. The chapter concludes with a treatment of the static correlation in the bond breaking of H2.
Electronic correlation
Slater determinant
Coupled cluster
Open shell
Electron configuration
Basis (linear algebra)
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The excitation transitions considered are 11S to 21P and 11S to 31P. Electron correlation is introduced into the description of the 11S ground state in a systematic way by using the natural expansion of a configuration-interaction (CI) wavefunction developed by Weiss. By truncating this expansion to the first X terms and renormalizing a series of wavefunctions ranging from a Hartree-Fock equivalent (X=1) to the full CI function (X=15) are obtained. Operating within the first Born approximation, three standard formulations for evaluating f(K) are used and their sensitivity with respect to correlation effects is discussed and a comparison of the present results is made with those of other workers. Although the inclusion of the first correlation configuration in the 11S wavefunction did not necessarily guarantee an improvement in f(K), the introduction of the second correlation term in the natural expansion gave rise to a significant improvement in f(K) for all K regardless of the mode of calculation.
Electronic correlation
Full configuration interaction
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The implications of the repulsive Coulombic interaction between two electrons are discussed. They lead to a general correlation property of the wavefunction when the two electrons are close to each other. The importance of this property is illustrated by including a simple correlation function in two pedagogically useful model wavefunctions for the ground state and (2p)2 3P state of He and isoelectronic ions. It leads to a significant improvement in the predicted values for the energies and other properties.
Electronic correlation
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Quantum defect theory is applied to the calculation of non-hydrogenic radial wavefunctions in the momentum space representation, given in the form of Hankel transform integrals of the position space functions in the Coulomb approximation. It is shown that, even for low excited states, the momentum space functions for important values of the momentum are insensitive to the inner-core region of the atom. Analytical expressions are presented and illustrative examples are given for some Rydberg states in sodium, magnesium and caesium.
Momentum (technical analysis)
Hydrogen-like atom
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ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTNumerical Methods for Finding Momentum Space DistributionsFrank Rioux View Author Information Department of Chemistry, Saint John''s University , Collegeville, MN 56321Cite this: J. Chem. Educ. 1997, 74, 5, 605Publication Date (Web):May 1, 1997Publication History Received3 August 2009Published online1 May 1997Published inissue 1 May 1997https://pubs.acs.org/doi/10.1021/ed074p605https://doi.org/10.1021/ed074p605research-articleACS PublicationsRequest reuse permissionsArticle Views242Altmetric-Citations2LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access optionsGet e-Alertsclose SUBJECTS:Fourier transforms,Schrodinger equation Get e-Alerts
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Electronic correlation
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The hydride ion is one of the simplest systems possessing electrons with antiparallel spins and, therefore, it is ideal for a study of electron correlation. Five wavefunctions for H− have been analysed and compared. Two functions were based on the single determinant independent-particle model and three were correlated wavefunctions. The present article is devoted, in the main, to a discussion of radial correlation, hence, graphs are presented for the two-particle density ρ(r1, r2) and the radial density D(r) for each treatment. Values are determined for the coherent x-ray scattering contribution f00(X), the three-dimensional Dirac delta functions 〈δ3(r1)〉 and 〈δ3(r12)〉, and also for 〈rn〉 when −2 ≤ n ≤ 4. In this way, the effect of electron correlation on various physical properties could be examined. In particular, by comparing the results with those obtained from a more accurate wavefunction, it was possible to assess the usefulness of a limited configuration-interaction treatment which employed `floating' orbitals. On the whole, this treatment of H− proved to be good. Finally, it was observed that the inclusion of electron correlation effects within a wavefunction for H− causes the electron density to become more diffuse, the inner regions around the nucleus resembling, fairly closely, the density within an isolated hydrogen atom.
Electronic correlation
Slater determinant
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The authors report results on the wavefunction representation of doubly excited states of He and on their autoionisation probability amplitudes in momentum space. These are presented in terms of conditional probability density and differential transition amplitude plots. The correlated momentum wavefunctions are obtained by Fourier-transforming position wavefunctions computed by multiconfigurational Hartree-Fock and variational procedures. Special techniques are developed for the Fourier transform of the numerical Hartree-Fock scattering orbitals. The differential transition amplitude plots lead to the same conclusions regarding the mechanism of autoionisation as did the previous work in position space.
Position (finance)
Momentum (technical analysis)
Configuration space
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