<p>We present a detailed variational study of the helium atom using compact, generalized exponential basis functions (GEFs) that incorporate non-integer radial powers and adjustable exponential decay parameters. These trial wave functions, originally introduced by Koga and Kanayama, offer improved flexibility for describing electron behavior near the nucleus and at large distances. By optimizing the variational parameters a, b, and x, we construct wave functions that closely approximate the Hartree-Fock (HF) ground state using only a single basis term. We evaluate key expectation values, including <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\langle \hbox {r}\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mtext>r</mtext> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\langle 1/\textrm{r}\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mtext>r</mtext> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\langle \hbox {r}_{12}\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <msub> <mtext>r</mtext> <mn>12</mn> </msub> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\langle \hbox {r}_{<}\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <msub> <mtext>r</mtext> <mo><</mo> </msub> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\langle \hbox {r}_{>}\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <msub> <mtext>r</mtext> <mo>></mo> </msub> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>, and analyze the effects of radial power and decay parameters on kinetic, nuclear attraction, and electron-electron repulsion energies. Our results demonstrate that the total energy can be lowered to within 0.20 millihartree of the HF limit, matching the performance of larger Slater-type orbital expansions with far fewer parameters. We further investigate the influence of wave function parameters on the nuclear cusp and radial probability density. The findings highlight the utility of GEFs in compact atomic modeling, offering both computational efficiency and near-HF-limit accuracy, with significant pedagogical value for quantum chemistry instruction.</p>

“Analysis of Compact, Generalized Exponential Basis Functions for Helium”

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发表日期：

2025年10月7日

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Jesse W. Tye

摘要

We present a detailed variational study of the helium atom using compact, generalized exponential basis functions (GEFs) that incorporate non-integer radial powers and adjustable exponential decay parameters. These trial wave functions, originally introduced by Koga and Kanayama, offer improved flexibility for describing electron behavior near the nucleus and at large distances. By optimizing the variational parameters a, b, and x, we construct wave functions that closely approximate the Hartree-Fock (HF) ground state using only a single basis term. We evaluate key expectation values, including $\langle \hbox {r}\rangle $ $⟨ r ⟩$ , $\langle 1/\textrm{r}\rangle $ $⟨ 1 / r ⟩$ , $\langle \hbox {r}_{12}\rangle $ $⟨ r_{12} ⟩$ , $\langle \hbox {r}_{<}\rangle $ $⟨ r_{<} ⟩$ , and $\langle \hbox {r}_{>}\rangle $ $⟨ r_{>} ⟩$ , and analyze the effects of radial power and decay parameters on kinetic, nuclear attraction, and electron-electron repulsion energies. Our results demonstrate that the total energy can be lowered to within 0.20 millihartree of the HF limit, matching the performance of larger Slater-type orbital expansions with far fewer parameters. We further investigate the influence of wave function parameters on the nuclear cusp and radial probability density. The findings highlight the utility of GEFs in compact atomic modeling, offering both computational efficiency and near-HF-limit accuracy, with significant pedagogical value for quantum chemistry instruction.