Stability and Reactivity of TiO2n$$ {\left({\mathrm{TiO}}_2\right)}_n $$, <i>n</i> = 1–10, Clusters and Their Interactions With CO2$$ {\mathrm{CO}}_2 $$

Small titanium dioxide clusters <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mfenced><mml:msub><mml:mi>TiO</mml:mi> <mml:mn>2</mml:mn></mml:msub> </mml:mfenced> <mml:mi>n</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {\left({\mathrm{TiO}}_2\right)}_n $$</mml:annotation></mml:semantics> </mml:math> (with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>n</mml:mi></mml:mrow> <mml:annotation>$$ n $$</mml:annotation></mml:semantics> </mml:math> = 1-10) are promising photocatalysts for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>CO</mml:mi> <mml:mn>2</mml:mn></mml:msub> </mml:mrow> <mml:annotation>$$ {\mathrm{CO}}_2 $$</mml:annotation></mml:semantics> </mml:math> conversion; however, their size-dependent stability and reactivity are not fully characterized. This study uses density functional theory (M06/def2-TZVP) and global and local reactivity descriptors to identify "magic number" clusters that exhibit high stability. The stability function ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msup><mml:mi>ε</mml:mi> <mml:mn>3</mml:mn></mml:msup> </mml:mrow> <mml:annotation>$$ {\varepsilon}^3 $$</mml:annotation></mml:semantics> </mml:math> ), reveals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>n</mml:mi></mml:mrow> <mml:annotation>$$ n $$</mml:annotation></mml:semantics> </mml:math> = 2, 4, and 8 as magic numbers. Electrophilicity analysis ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>Δ</mml:mi> <mml:mi>ω</mml:mi></mml:mrow> <mml:annotation>$$ \Delta \omega $$</mml:annotation></mml:semantics> </mml:math> ) shows moderate electrophilicity for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>n</mml:mi></mml:mrow> <mml:annotation>$$ n $$</mml:annotation></mml:semantics> </mml:math> = 1-5 and strong electrophilicity for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>n</mml:mi></mml:mrow> <mml:annotation>$$ n $$</mml:annotation></mml:semantics> </mml:math> = 7-10, while the magic numbers display reduced reactivity. Fukui functions and fractional occupation number-weighted density ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>N</mml:mi> <mml:mi>FOD</mml:mi></mml:msub> </mml:mrow> <mml:annotation>$$ {N}_{FOD} $$</mml:annotation></mml:semantics> </mml:math> ) highlight localized reactivity. Notably, they reveal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>n</mml:mi></mml:mrow> <mml:annotation>$$ n $$</mml:annotation></mml:semantics> </mml:math> = 6 to be highly electrophilic, with distinct "hot" electron sites. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>CO</mml:mi> <mml:mn>2</mml:mn></mml:msub> </mml:mrow> <mml:annotation>$$ {\mathrm{CO}}_2 $$</mml:annotation></mml:semantics> </mml:math> interaction energies inversely correlate with cluster stability: unstable clusters ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>n</mml:mi></mml:mrow> <mml:annotation>$$ n $$</mml:annotation></mml:semantics> </mml:math> = 3, 5, and 9) strongly bind <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>CO</mml:mi> <mml:mn>2</mml:mn></mml:msub> </mml:mrow> <mml:annotation>$$ {\mathrm{CO}}_2 $$</mml:annotation></mml:semantics> </mml:math> (up to 0.72 eV), while magic numbers weakly physisorb it (e.g., 0.45 eV for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>n</mml:mi></mml:mrow> <mml:annotation>$$ n $$</mml:annotation></mml:semantics> </mml:math> = 8). Non-covalent interaction (NCI) analysis confirms Ti-OCO attraction and C-repulsive sites. Together, these results establish design principles for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>TiO</mml:mi> <mml:mn>2</mml:mn></mml:msub> </mml:mrow> <mml:annotation>$$ {\mathrm{TiO}}_2 $$</mml:annotation></mml:semantics> </mml:math> cluster catalysts that balance stability with tailored reactivity for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>CO</mml:mi> <mml:mn>2</mml:mn></mml:msub> </mml:mrow> <mml:annotation>$$ {\mathrm{CO}}_2 $$</mml:annotation></mml:semantics> </mml:math> activation.