Symmetry, bonding, and wannier function construction in the PF6− molecule: a first-principles case study

We present a comprehensive analysis of the electronic structure of the PF <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mn>6</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> anion, a prototypical octahedral molecular system with high <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>O</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>h</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> symmetry. Using symmetry-adapted linear combinations of atomic orbitals and group theoretical techniques, we construct molecular orbitals and provide a systematic classification according to irreducible representations of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>O</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>h</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> point group. The role of the P-centered <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m5"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>s</mml:mi> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m6"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m7"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> orbitals, together with the symmetry-adapted 2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m8"> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> </mml:math> and 2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m9"> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> orbitals of the six surrounding fluorine atoms, is explicitly analyzed. The electronic structure is described both within the theory of molecular orbitals and with the picture based on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m10"> <mml:mrow> <mml:mi>s</mml:mi> <mml:msup> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> hybridization. Maximally localized Wannier functions derived from first-principles density functional theory calculations using the siesta and wannier90 codes are computed. The constructed Wannier functions accurately reflect the expected molecular symmetries and provide a natural minimal basis for tight-binding and second-principles modeling. A detailed comparison is made between bonding, nonbonding, and antibonding orbitals, as well as their energetic ordering. Our results demonstrate the interplay between symmetry, bonding, and electronic structure in molecular systems with high cubic symmetry and set the stage for the development of accurate minimal models for such systems.