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Unified Formulae for Augmented Near Orthonormalized STO‐mG Basis Sets via Atomic Orbital Energy Fit: Fitting Algorithm and Tables.

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Title: Unified Formulae for Augmented Near Orthonormalized STO‐mG Basis Sets via Atomic Orbital Energy Fit: Fitting Algorithm and Tables.
Authors: Kristyan, Sandor1 (AUTHOR) kristyan.sandor@ttk.hu
Source: International Journal of Quantum Chemistry. 2/15/2026, Vol. 126 Issue 4, p1-15. 15p.
Subjects: Atomic orbitals, Gaussian function, Quantum chemistry
Abstract: STO‐mG type basis functions for 1s to 4f hydrogen‐like orbitals by "energy fit" are reported as simple functions of running parameter atomic number Z and quantum numbers to utilize the basis set from these functions in molecular electronic structure and energy calculations. We mimic the accurate solution of Slater‐type atomic orbitals (STO, here called HTO) as close as possible (in shape and orbital energy) with a linear combination of m Gaussian functions; the use of Gaussians is vital for the analytical evaluation of molecular integrals. We analyze how they reproduce the one‐electron atomic wave function shapes and energy values (−Z2/(2n2)) as an obvious primary claim, as well as we compare it to the literature. The direct wave function (shape) fit to the exp.(−Z r/n) part of Hydrogen‐like orbitals with a linear combination of Gaussians, ∑i=1mAiexp(−ai r2), is the basic way to create STO‐mG basis sets in literature, yielding a huge number of tables for different atoms, that is, listed separately for different atomic numbers Z in the periodic table. Our fit is based on three devices: (1) Instead of ∑i Aiexp(−ai r2), the P(Z,r) ∑iAiexp(−ai (Z r/n)2) with proper polynomial P is used (for the atomic radial part), allowing the optimization for running Z as a parameter, that is, a common basis set has been reported that can be used for any atoms; only the value for Z has to be substituted. (2) The polynomial part, P, takes care of nodes, taking on the role of the alternating sign of Gaussians (as in literature), as well as it is designed to produce even powers finally for r (in the product of atomic auxiliary and radial parts), necessary for analytical molecular integral evaluations in practice. Even more, the Z drops from integral < P(Z,r) ∑iAiexp(−ai (Z r/n)2) | P(Z,r) ∑iAiexp(−ai (Z r/n)2) > for norm. (3) Not "radial (shape) fit" by overlap integral is used in the optimization as the basic guide in literature, but "energy fit," that is, minimizing energy integral < P(Z,r) ∑iAiexp(−ai (Z r/n)2) |[h]| P(Z,r) ∑iAiexp(−ai (Z r/n)2) > ≈ − Z2(2n2), which mimics the shape and energy of the true wave function with the help of the one‐electron Hamiltonian operator, [h]. For this, a "variation‐like" property is also discussed for excited states (2s, 2p, 3s, ...); that is, besides relation "≈" the relation "≥" also holds in the previous expression. The optimizations have been done by using least squares fits via the Lagrange multiplier method for energy with a constraint for normalization. This algorithm to create STO‐mG basis functions allows for a compact list. All these STO‐mG basis functions are normalized exactly to one in our tables, and the virial theorem holds approximately at least between −2 ± 0.01. A general problem of STO‐mG(3 and 4du∧2) and STO‐mG(4fu∧3 and 4fuv∧2) basis functions among the six technical 3duv as well as the 10 technical 4f used in the practice of molecular structure calculations is commented on, and a better strategy is suggested. Some special features of the seemingly trivial normalization are detailed in our production of the STO‐mG basis set along with recursive formulae for overlap molecular integrals. [ABSTRACT FROM AUTHOR]
Copyright of International Journal of Quantum Chemistry is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Unified Formulae for Augmented Near Orthonormalized STO‐mG Basis Sets via Atomic Orbital Energy Fit: Fitting Algorithm and Tables.
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  Data: STO‐mG type basis functions for 1s to 4f hydrogen‐like orbitals by &quot;energy fit&quot; are reported as simple functions of running parameter atomic number Z and quantum numbers to utilize the basis set from these functions in molecular electronic structure and energy calculations. We mimic the accurate solution of Slater‐type atomic orbitals (STO, here called HTO) as close as possible (in shape and orbital energy) with a linear combination of m Gaussian functions; the use of Gaussians is vital for the analytical evaluation of molecular integrals. We analyze how they reproduce the one‐electron atomic wave function shapes and energy values (−Z2/(2n2)) as an obvious primary claim, as well as we compare it to the literature. The direct wave function (shape) fit to the exp.(−Z r/n) part of Hydrogen‐like orbitals with a linear combination of Gaussians, ∑i=1mAiexp(−ai r2), is the basic way to create STO‐mG basis sets in literature, yielding a huge number of tables for different atoms, that is, listed separately for different atomic numbers Z in the periodic table. Our fit is based on three devices: (1) Instead of ∑i Aiexp(−ai r2), the P(Z,r) ∑iAiexp(−ai (Z r/n)2) with proper polynomial P is used (for the atomic radial part), allowing the optimization for running Z as a parameter, that is, a common basis set has been reported that can be used for any atoms; only the value for Z has to be substituted. (2) The polynomial part, P, takes care of nodes, taking on the role of the alternating sign of Gaussians (as in literature), as well as it is designed to produce even powers finally for r (in the product of atomic auxiliary and radial parts), necessary for analytical molecular integral evaluations in practice. Even more, the Z drops from integral &lt; P(Z,r) ∑iAiexp(−ai (Z r/n)2) | P(Z,r) ∑iAiexp(−ai (Z r/n)2) &gt; for norm. (3) Not &quot;radial (shape) fit&quot; by overlap integral&lt;exp.(−Z r/n) − ∑i=1mAiexp(−ai r2) | exp.(−Z r/n) − ∑i=1mAiexp(−ai r2) &gt; is used in the optimization as the basic guide in literature, but &quot;energy fit,&quot; that is, minimizing energy integral &lt; P(Z,r) ∑iAiexp(−ai (Z r/n)2) |[h]| P(Z,r) ∑iAiexp(−ai (Z r/n)2) &gt; ≈ − Z2(2n2), which mimics the shape and energy of the true wave function with the help of the one‐electron Hamiltonian operator, [h]. For this, a &quot;variation‐like&quot; property is also discussed for excited states (2s, 2p, 3s, ...); that is, besides relation &quot;≈&quot; the relation &quot;≥&quot; also holds in the previous expression. The optimizations have been done by using least squares fits via the Lagrange multiplier method for energy with a constraint for normalization. This algorithm to create STO‐mG basis functions allows for a compact list. All these STO‐mG basis functions are normalized exactly to one in our tables, and the virial theorem holds approximately at least between −2 &#177; 0.01. A general problem of STO‐mG(3 and 4du∧2) and STO‐mG(4fu∧3 and 4fuv∧2) basis functions among the six technical 3duv as well as the 10 technical 4f used in the practice of molecular structure calculations is commented on, and a better strategy is suggested. Some special features of the seemingly trivial normalization are detailed in our production of the STO‐mG basis set along with recursive formulae for overlap molecular integrals. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: &lt;i&gt;Copyright of International Journal of Quantum Chemistry is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder&#39;s express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.&lt;/i&gt; (Copyright applies to all Abstracts.)
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        Value: 10.1002/qua.70152
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      – Code: eng
        Text: English
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        PageCount: 15
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    Subjects:
      – SubjectFull: Atomic orbitals
        Type: general
      – SubjectFull: Gaussian function
        Type: general
      – SubjectFull: Quantum chemistry
        Type: general
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      – TitleFull: Unified Formulae for Augmented Near Orthonormalized STO‐mG Basis Sets via Atomic Orbital Energy Fit: Fitting Algorithm and Tables.
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            NameFull: Kristyan, Sandor
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              M: 02
              Text: 2/15/2026
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              Y: 2026
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              Value: 126
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