<p>This work presents a physically consistent framework in which, in accordance with Gauss’s flux theorem, the partial charge densities are interpreted as source terms, each generating the corresponding conservative electronic force field. A new population constraint is formulated for the pseudoatomic basins <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega_{P}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ω</mi> <mi>P</mi> </msub> </math></EquationSource> </InlineEquation> defined in the total kinetic force field <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbf{F}_{k}(\mathbf{r})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">F</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>: in any many-electron multinuclear system, the integral populations of the von Weizsäcker and Pauli charge densities within the same closed <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega_{P}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ω</mi> <mi>P</mi> </msub> </math></EquationSource> </InlineEquation> basin exactly cancel each other. Under equilibrium conditions, this constraint can be combined with the previously established constraint for the equivalent pseudoatomic basins <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Omega_{P}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ω</mi> <mi>P</mi> </msub> </math></EquationSource> </InlineEquation> defined in the total static force field <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal{F}(\mathbf{r})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi mathvariant="bold">r</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This yields a comprehensive and unified condition that governs both local charge (re)distribution and force-field pseudoatomic charges and simultaneously reflects the balance between kinetic and static effects in the charge organization of matter. This framework is further applied to the hexagonal and cubic boron nitride (BN) crystals. Their electronic structures are analyzed in terms of the superposition of the chemically bonded atoms identified in the electron density gradient <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\nabla \rho(\mathbf{r})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">∇</mi> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi mathvariant="bold">r</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the pseudoatoms defined in the electrostatic force field <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbf{F}_{\text{es}}(\mathbf{r})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">F</mi> <mtext>es</mtext> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and those defined in the local-density-approximated force field <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbf{F}_{k}(\mathbf{r})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">F</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal{F}(\mathbf{r})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi mathvariant="bold">r</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, whose electron populations are subject to the corresponding charge constraints. This perspective elucidates both the interatomic electron transfer and the responsive sharing of the transferred density. The quantum chemical responses associated with the latter are nearly spatially complete within the graphene-like layers in hexagonal BN but are markedly attenuated between them. This contrast is further supported by the distribution of the local-density-approximated exchange charge density <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$q_x(\mathbf{r})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>q</mi> <mi>x</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="bold">r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which reveals a pronounced condensation of electron-pair density in the internuclear binding regions within the layers and a significant depletion in the binding regions between the layers. Concurrently, the B–N bond in cubic BN exhibits a comparably complete quantum chemical response. The differences in the spatial superposition patterns of the vector fields account for the distinct mechanical properties of the two polymorphs, despite the similar atomic and pseudoatomic charges within them. Additionally, in a closely related context, nonpolar interatomic interactions in graphite and diamond are briefly discussed.</p>

Pseudoatomic charge constraints governing the distribution of the partial charge densities. The atoms and force-field pseudoatoms in the hexagonal and cubic boron nitride crystals

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发表日期：

2026年2月4日

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Sergey V. Kartashov,
Robert R. Fayzullin

摘要

This work presents a physically consistent framework in which, in accordance with Gauss’s flux theorem, the partial charge densities are interpreted as source terms, each generating the corresponding conservative electronic force field. A new population constraint is formulated for the pseudoatomic basins $\Omega_{P}$ $Ω_{P}$ defined in the total kinetic force field $\mathbf{F}_{k}(\mathbf{r})$ $F_{k} (r)$ : in any many-electron multinuclear system, the integral populations of the von Weizsäcker and Pauli charge densities within the same closed $\Omega_{P}$ $Ω_{P}$ basin exactly cancel each other. Under equilibrium conditions, this constraint can be combined with the previously established constraint for the equivalent pseudoatomic basins $\Omega_{P}$ $Ω_{P}$ defined in the total static force field $\mathcal{F}(\mathbf{r})$ $F (r)$ . This yields a comprehensive and unified condition that governs both local charge (re)distribution and force-field pseudoatomic charges and simultaneously reflects the balance between kinetic and static effects in the charge organization of matter. This framework is further applied to the hexagonal and cubic boron nitride (BN) crystals. Their electronic structures are analyzed in terms of the superposition of the chemically bonded atoms identified in the electron density gradient $\nabla \rho(\mathbf{r})$ $\nabla ρ (r)$ , the pseudoatoms defined in the electrostatic force field $\mathbf{F}_{\text{es}}(\mathbf{r})$ $F_{es} (r)$ , and those defined in the local-density-approximated force field $\mathbf{F}_{k}(\mathbf{r})$ $F_{k} (r)$ or $\mathcal{F}(\mathbf{r})$ $F (r)$ , whose electron populations are subject to the corresponding charge constraints. This perspective elucidates both the interatomic electron transfer and the responsive sharing of the transferred density. The quantum chemical responses associated with the latter are nearly spatially complete within the graphene-like layers in hexagonal BN but are markedly attenuated between them. This contrast is further supported by the distribution of the local-density-approximated exchange charge density $q_x(\mathbf{r})$ $q_{x} (r)$ , which reveals a pronounced condensation of electron-pair density in the internuclear binding regions within the layers and a significant depletion in the binding regions between the layers. Concurrently, the B–N bond in cubic BN exhibits a comparably complete quantum chemical response. The differences in the spatial superposition patterns of the vector fields account for the distinct mechanical properties of the two polymorphs, despite the similar atomic and pseudoatomic charges within them. Additionally, in a closely related context, nonpolar interatomic interactions in graphite and diamond are briefly discussed.