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Baskerville AL, Gray C, Cox H. Correlated energy from radial density-energy relations. ROYAL SOCIETY OPEN SCIENCE 2023; 10:221402. [PMID: 36938537 PMCID: PMC10014244 DOI: 10.1098/rsos.221402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/30/2022] [Accepted: 02/13/2023] [Indexed: 06/18/2023]
Abstract
Here, we demonstrate that the radial distribution function can be mapped into a radial density-energy space and the relationship between the radial density and radial energy is linear for the ground and excited states of helium-like systems; the gradient of the resulting straight line delivers the energy of the state considered. To utilize this finding, a simple analytical expression for the total energy in terms of the density at the most probable nucleus-electron distance of the systems considered is derived using a fitting procedure.
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Affiliation(s)
- Adam L. Baskerville
- Department of Chemistry, School of Life Sciences, University of Sussex, Falmer, Brighton BN1 9QJ, UK
| | - Conor Gray
- Department of Chemistry, School of Life Sciences, University of Sussex, Falmer, Brighton BN1 9QJ, UK
| | - Hazel Cox
- Department of Chemistry, School of Life Sciences, University of Sussex, Falmer, Brighton BN1 9QJ, UK
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Baskerville AL, Cox PU, Cox H. Consequences of approximating electron correlation effects. Mol Phys 2022. [DOI: 10.1080/00268976.2022.2146540] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
Affiliation(s)
| | | | - Hazel Cox
- Department of Chemistry, University of Sussex, Brighton, UK
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Baskerville AL, Targema M, Cox H. Reparametrization of the Colle-Salvetti formula. ROYAL SOCIETY OPEN SCIENCE 2022; 9:211333. [PMID: 35003756 PMCID: PMC8728172 DOI: 10.1098/rsos.211333] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/15/2021] [Accepted: 11/23/2021] [Indexed: 06/14/2023]
Abstract
We investigate the Colle-Salvetti (CS) formula, the basis of the Lee, Yang and Parr (LYP) correlation functional used in approximate density functional theory. The CS formula is reparametrized using high-accuracy Hartree-Fock (HF) wavefunctions to determine the accuracy of the formula to calculate anions. Fitting to the hydride ion or the two-electron system just prior to electron detachment at the HF level of theory does not, in general, improve the calculated correlation energies using the parameters derived from the CS/LYP method. An analysis of the CS parameters used in the popular LYP functional demonstrates the ingenuity and perhaps fortuitousness of the original formulation by CS.
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Affiliation(s)
- Adam L. Baskerville
- Department of Chemistry, School of Life Sciences, University of Sussex, Falmer, Brighton BN1 9QJ, UK
| | - Msugh Targema
- Department of Chemistry, School of Life Sciences, University of Sussex, Falmer, Brighton BN1 9QJ, UK
| | - Hazel Cox
- Department of Chemistry, School of Life Sciences, University of Sussex, Falmer, Brighton BN1 9QJ, UK
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Montgomery Jr. H, Sen K, Katriel J. Correlation effects close to the ground state critical charge of the two-electron atom. Chem Phys Lett 2021. [DOI: 10.1016/j.cplett.2021.139030] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Martínez-Sánchez MA, Martínez-Flores C, Vargas R, Garza J, Cabrera-Trujillo R, Sen KD. Ionization of many-electron atoms by the action of two plasma models. Phys Rev E 2021; 103:043202. [PMID: 34005965 DOI: 10.1103/physreve.103.043202] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2020] [Accepted: 03/22/2021] [Indexed: 11/07/2022]
Abstract
The Hartree-Fock equations for many-electron atoms embedded in a plasma medium are solved using two different plasma models: (a) Debye-Hückel screening (DHS) potential and (b) exponential cosine screened Coulomb (ECSC) potential. Roothaan's approach is implemented for these models after solving the inherent difficulties to evaluate integrals where screening appears explicitly. A corresponding computer code was developed using the method of global basis sets (GBS). The reliability of this approach was verified by solving the Hartree-Fock equations through implementation of the finite-differences and finite-element grid methods and applied to two-electron atoms, yielding excellent agreement with the Roothaan-GBS (RGBS) method. The RGBS method was used to study the energy evolution and ionization threshold of several closed- and open-shell many-electron atoms embedded either in weak or strong DHS or ECSC plasma conditions. In all cases, a critical value of the screening length is obtained for which ionization is achieved, being systematically larger for DHS conditions, indicating the effect of a more repulsive ECSC potential. For He-like atoms in the ground state, we report a comprehensive set of accurate total energy data as a function of the screening constant using the Lagrange mesh method, which includes the electron correlation effects. The electron correlation energy is estimated using this data with reference to the RGBS estimates of energy as the Hartree-Fock energy. The variation of correlation energy as a function of screening constant under the different plasma potentials is rationalized in terms of a conjectured comparison theorem. Finally, a discussion on the effect of plasma strength on localization or delocalization of the electronic density derived from the RGBS method is presented in terms of changes in the Shannon entropy, yielding consistent results for delocalization close to the ionization threshold.
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Affiliation(s)
- Michael-Adán Martínez-Sánchez
- División de Ciencias Básicas e Ingeniería, Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, 09340 Iztapalapa, México City, México
| | - César Martínez-Flores
- División de Ciencias Básicas e Ingeniería, Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, 09340 Iztapalapa, México City, México
| | - Rubicelia Vargas
- División de Ciencias Básicas e Ingeniería, Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, 09340 Iztapalapa, México City, México
| | - Jorge Garza
- División de Ciencias Básicas e Ingeniería, Departamento de Química, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, 09340 Iztapalapa, México City, México
| | - Remigio Cabrera-Trujillo
- Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Avenida Universidad S/N, Cuernavaca, Morelos 62210, México
| | - K D Sen
- School of Chemistry, University of Hyderabad, Hyderabad 500 046, India
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Abstract
Electron correlation effects play a key role in stabilizing two-electron atoms near the critical nuclear charge, representing the smallest charge required to bind two electrons. However, deciphering the importance of these effects relies on fully understanding the uncorrelated Hartree-Fock description. We investigate the properties of the ground state wave function in the small nuclear charge limit using various symmetry-restricted Hartree-Fock formalisms. We identify the nuclear charge where spin-symmetry breaking occurs to give an unrestricted wave function that predicts an inner and outer electron. We also identify closed-shell and unrestricted critical nuclear charges where the highest occupied orbital energy becomes zero and the electron density detaches from the nucleus. Finally, we identify the importance of fractional spin errors and static correlation for small nuclear charges.
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Affiliation(s)
- Hugh G A Burton
- Physical and Theoretical Chemistry Laboratory, Department of Chemistry, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom
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Cioslowski J, Prątnicki F. Uniform description of the helium isoelectronic series down to the critical nuclear charge with explicitly correlated basis sets derived from regularized Krylov sequences. J Chem Phys 2020; 153:224106. [PMID: 33317275 DOI: 10.1063/5.0029053] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
An efficient computational scheme for the calculation of highly accurate ground-state electronic properties of the helium isoelectronic series, permitting uniform description of its members down to the critical nuclear charge Zc, is described. It is based upon explicitly correlated basis functions derived from the regularized Krylov sequences (which constitute the core of the free iterative CI/free complement method of Nakatsuji) involving a term that introduces split length scales. For the nuclear charge Z approaching Zc, the inclusion of this term greatly reduces the error in the variational estimate for the ground-state energy, restores the correct large-r asymptotics of the one-electron density ρ(Z; r), and dramatically alters the manifold of the pertinent natural amplitudes and natural orbitals. The advantages of this scheme are illustrated with test calculations for Z = 1 and Z = Zc carried out with a moderate-size 12th-generation basis set of 2354 functions. For Z = Zc, the augmentation is found to produce a ca. 5000-fold improvement in the accuracy of the approximate ground-state energy, yielding values of various electronic properties with between seven and eleven significant digits. Some of these values, such as those of the norms of the partial-wave contributions to the wavefunction and the Hill constant, have not been reported in the literature thus far. The same is true for the natural amplitudes at Z = Zc, whereas the published data for those at Z = 1 are revealed by the present calculations to be grossly inaccurate. Approximants that yield correctly normalized ρ(1; r) and ρ(Zc; r) conforming to their asymptotics at both r → 0 and r → ∞ are constructed.
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Affiliation(s)
- Jerzy Cioslowski
- Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
| | - Filip Prątnicki
- Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
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