Understanding the electronic structure of molecules with high symmetry is crucial for rationalizing their stability, reactivity, and spectroscopic properties. The hexafluorophosphate anion, PF 6 − , serves as a paradigmatic system in this regard. Its octahedral geometry, with six fluorine atoms located at the vertices of a regular octahedron around a central phosphorus atom, leads to an ideal O h point group symmetry. This high symmetry allows for a rigorous analysis of its electronic structure using group-theoretical techniques, providing valuable insights into orbital interactions and chemical bonding.

The PF 6 − anion has attracted significant attention due to its widespread use in electrochemical and energy storage applications. Its high thermal stability, low nucleophilicity, and chemically inert character make it a common component in electrolytes for lithium-ion and lithium-air batteries, as well as in ionic liquids and superacid chemistry. At electrode interfaces, PF 6 − modulates local electronic environments, influencing O–O bond activity and improving kinetics in Li/O₂ battery systems [1].

The choice of anion plays a critical role in determining battery performance, with PF 6 − offering an optimal balance of ionic conductivity and electrochemical stability compared to ClO 4 − and other salts [2]. Luan et al. [3] reported that PF 6 − decomposition at the graphite/electrolyte interface affects solid electrolyte interphase stability, a key factor in long-term cell performance. In lithium-ion electrolytes, PF 6 − is valued for its combined conductivity, stability, and compatibility with electrodes. Beyond ionic transport, PF 6 − influences solvation-sheath structure and interfacial chemistry [4].

Recent investigations indicate that PF 6 − actively participates in interfacial reactions rather than acting as a passive anion. Spotte-Smith et al. [5] revealed LiPF 6 decomposition through P–F bond cleavage, while Ruan et al. [6] and Qi et al. [7] demonstrated that PF 6 − governs solvation-sheath composition and charge-transfer kinetics. Wang et al. [8] showed that optimized co-solvents can stabilize PF 6 − under high-voltage operation.

At the cathode interface, Peiris et al. [9] and Xing et al. [10] highlighted that anion identity dictates oxygen redox activity and stability. Moreover, PF 6 − adsorption or substitutional interaction with graphene surfaces has been reported to reduce Li 2 O 2 accumulation by altering charge distribution and promoting toroidal Li 2 O 2 growth, thereby mitigating cathode passivation and enhancing cycle reversibility [7, 9].

However, most theoretical treatments still idealize PF 6 − as a uniform anion, neglecting its localized electronic structure and symmetry-dependent orbital interactions. A deeper understanding of these features is essential for accurate modeling of electrochemical behavior.

Simulating an entire electrochemical cell at the first-principles level remains computationally prohibitive due to its size and complexity. Hence, reduced-order frameworks such as second-principles methods [11] based on tight-binding Hamiltonians constructed from maximally localized Wannier functions (MLWFs) enable efficient yet chemically accurate modeling of such systems. First steps in this direction have been taken in our group, where Li 2 O 2 (a key discharge product in non-aqueous lithium-air systems) has been extensively studied [12]. In particular, the coupling between lattice distortions and electronic states was characterized.

In this work, we present a comprehensive study of the isolated PF 6 − anion, focusing on the construction of symmetry-adapted MOs and a compact, chemically intuitive MLWF basis. The MOs are analyzed using both symmetry-adapted linear combination (SALC) methods and first-principles density functional theory calculations, implemented with siesta and wannier90. We emphasize the hybridization between phosphorus 3 s , 3 p , 3 d and fluorine 2 s , 2 p orbitals under O h symmetry. The resulting MLWFs bridge molecular orbital and real-space bonding pictures, providing a transparent foundation for building low-energy Hamiltonians. All pseudopotentials, basis sets, and parameters are made openly available for future second-principles modeling.

The rest of this paper is organized as follows. Section 2 outlines the computational methodology. Section 3 describes the relaxed atomic structure. In Section 4, we analyze the electronic states and their symmetry classification. Section 5 discusses the construction and properties of the MLWFs, along with the resulting tight-binding Hamiltonian. The paper concludes with a summary of the main findings and perspectives for future work.

Our calculations were carried out within the framework of density functional theory (DFT) [13], employing the generalized gradient approximation (GGA) for the treatment of exchange and correlation effects. The simulations relied on a numerical atomic orbital (NAO) basis, as implemented in the siesta code [14, 15]. We adopted the Perdew-Burke-Ernzerhof (PBE) GGA functional [16] to describe exchange-correlation interactions, utilizing the implementation provided by the libxc library [17, 18]. Given that semilocal GGA functionals suffer from self-interaction errors and deviations from piecewise linearity (thereby violating the Koopmans condition) we additionally performed calculations using the range-separated HSE06 hybrid functional [19], which mitigates these shortcomings and provides a more reliable estimate of the HOMO–LUMO gap by exploiting the newly implemented hybrid-functional capabilities of siesta.

Core electrons were modeled using norm-conserving pseudopotentials in the fully separable Kleinman-Bylander form [20]. In particular, we employed the optimized norm-conserving Vanderbilt pseudopotentials developed by Hamann [21], provided in the psml format [22], as distributed by the Pseudo-Dojo project [23, 24]. For fluorine and phosphorus atoms, the valence electrons included the 2 s and 2 p orbitals (F) and the 3 s and 3 p orbitals (P), respectively.

The electronic states were expanded using a basis set of strictly confined numerical atomic orbitals [25, 26], obtained as eigenfunctions of the isolated atoms subjected to a soft confinement potential [27]. The basis quality was varied by choosing between double- ζ and triple- ζ sets, corresponding to two or three radial components per valence orbital (i.e., 2 s , 2 p for F and 3 s , 3 p for P). To improve the angular flexibility of the basis, we added polarization functions with 3 d character on both species, including either one or two radial functions per angular shell, depending on the desired level of polarization (single or double). All basis set parameters were variationally optimized for the relaxed molecular geometry, using the procedure described in Ref. [27] and reference data obtained from fully converged plane-wave calculations.

All real-space integrals–such as the electronic density, the Hartree potential, and the exchange-correlation contributions–were evaluated on a uniform grid. The grid spacing was chosen to be equivalent to a plane-wave cutoff energy of 600 Ry, ensuring sufficient numerical accuracy [14].

To simulate the PF 6 − anion in isolation, we placed the molecule in a cubic supercell with 20 Å of side, sufficiently large to suppress interactions between periodic images. The system was given a net charge of − 1 e to account for the additional electron. For charged cells under periodic boundary conditions, electrostatic artifacts arise from the interaction between the charge and its periodic images. These were corrected by subtracting the leading-order Madelung energy, following the approach of Ref. [28].

Geometrical optimizations were performed using the conjugate gradient method, and the atomic coordinates were relaxed until all residual forces were below 10 meV/Å.

To validate the completeness of the NAO basis sets, we benchmarked our results against those from the abinit package [29–31], which uses a plane-wave basis. The comparison was designed to be as consistent as possible. Both siesta and abinit utilized the same pseudopotentials in psml format, incorporating an identical separation between the local potential and the nonlocal Kleinman-Bylander projectors. The same exchange-correlation functional was used in both codes, sourced from a common version of libxc. The primary difference lies in the basis representation: while siesta employed numerical atomic orbitals, abinit used a plane-wave basis with a cutoff energy of 60 Ha, sufficient to ensure convergence of total energies and forces.

The isolated PF 6 is octahedral in shape. Each fluorine atom sits on one of the six vertices of a regular octahedron with the phosphorus atom in the centre. Therefore, it exhibits O h symmetry in its ideal isolated form. The only internal degree of freedom is the distance x between the P and the F atoms. Numerous experimental and theoretical studies have reported values for the P–F bond distance. X-ray diffraction measurements on crystalline salts such as [N( CH 3 )₄ ] PF 6 [32] indicate a slightly distorted octahedral environment, with axial P–F distances ranging from 1.585 to 1.592Å, and equatorial bonds of 1.568 Å. The associated F–P–F bond angles were found to be 90 ° and 180 ° within experimental uncertainty. In other compounds, such as P( C 6 H 5 )₄ PF 6 , the measured P–F distances are slightly shorter, in the range of 1.555–1.556 Å. It is important to note that direct experimental observation of the PF 6 − anion in the gas phase or in solution is not feasible with current techniques. As such, experimental bond lengths obtained from crystals should be viewed as reference values for comparison. Computational investigations of the isolated PF 6 − molecule have also been reported. Xuan et al. [33] conducted a detailed analysis using basis sets of Gaussian-type orbitals and varying levels of theory. The computed P–F bond length spans a range from 1.5897 Å at the Hartree-Fock level to 1.6292 Å using the hybrid B3LYP functional, and 1.6348 Å within the MP2 formalism. In siesta, using a TZDP basis set within the HSE06 range-separated hybrid functional, the equilibrium distance amounts to 1.625 Å.

Table 1 presents the convergence behavior of numerical atomic orbital (NAO) basis sets for the PF 6 − anion in the gas phase. Results obtained using progressively larger, variationally optimized basis sets are compared against a reference calculation performed with a plane-wave (PW) basis set at a cutoff energy of 60 Ha, which serves as the converged-basis limit. All other computational parameters, including the pseudopotentials and exchange-correlation functional, were kept identical across all simulations to ensure consistency. It is important to emphasize that this reference PW result differs from lower-cutoff plane-wave calculations frequently employed in the literature, which may not fully eliminate basis-set incompleteness errors. A clear and systematic convergence trend is observed as the NAO basis size increases. The fully converged plane-wave result, where remaining errors are primarily associated with the choice of exchange-correlation functional and the pseudopotentials, yields a total energy marginally lower (by approximately 0.4%) than the most accurate NAO calculation tried here. As previously discussed, direct comparison with experimental bond lengths is not straightforward due to the lack of measurements for the isolated anion in the gas phase. Nevertheless, the P–F distance obtained with the siesta calculations overestimates the experimental values by roughly 4%, a deviation that falls within the expected range for simulations based on the PBE functional.

The PF 6 − anion adopts a highly symmetric octahedral geometry, described by the O h point group as illustrated in Figure 1. Its electronic structure can be efficiently analyzed by decomposing the valence atomic orbitals of phosphorus and fluorine into SALC that transform as irreducible representations of O h . This group-theoretical framework allows a systematic construction of MOs classified by symmetry, and highlights the role of bonding, nonbonding, and antibonding interactions in the stability of the anion.

Localized MOs, such as those described in Section 4, have a long-standing tradition in the chemistry literature as a powerful and intuitive framework for analyzing chemical bonding in molecular systems. These orbitals offer not only an insightful picture of bonding characteristics, coordination, and hybridization but also serve as a compact and efficient basis for high-accuracy electronic structure calculations. Maximally localized Wannier functions (MLWFs) [34, 35] represent a natural generalization of this concept to extended systems, providing a localized and orthonormal representation of the electronic structure in crystals and molecules alike. MLWFs retain much of the chemical interpretability of localized MOs while offering additional advantages: they preserve crystal symmetry, reflect the spatial distribution and coordination of electronic states, and enable the identification of transferable chemical building blocks. In this work, starting from the Kohn–Sham eigenstates computed with the siesta code, we construct MLWFs using the wannier 90 package [36].

The electronic structure of the PF 6 − anion has been extensively discussed in the literature using traditional molecular orbital (MO) theory and population-based methods, including Mulliken charges, NBO analysis, and energy-level diagrams. While these approaches offer chemically intuitive insights into bonding, they lack both real-space localization and symmetry-enforced orbital orthogonality. As such, they are not readily transferable to multiscale models or tight-binding Hamiltonians.

In contrast, the present work employs both symmetry-constrained and globally localized maximally localized Wannier functions (MLWFs), derived from first-principles DFT calculations. The symmetry-constrained MLWFs reproduce the expected irreducible representations of the Oh point group, providing a rigorous foundation for classifying molecular orbitals. More significantly, the global Wannierization procedure allows for full mixing across symmetry manifolds, producing a minimal and chemically insightful basis localized around each fluorine atom.

Each ligand-centered MLWF naturally decomposes into bonding, antibonding, or π -character, as revealed by their spatial shape and nodal structure. These MLWFs exhibit significantly reduced spatial spreads (down to 0.37 Ų), far below those of symmetry-constrained orbitals (2.3–3.2 Ų). Moreover, the resulting tight-binding Hamiltonian–constructed from the MLWF overlaps–reproduces the full Kohn–Sham eigenvalue spectrum with high fidelity, enabling physically grounded coarse-grained modeling.

To the best of our knowledge, no previous study has implemented a global Wannierization of PF 6 − to derive a transferable minimal basis suitable for model Hamiltonians. This approach thus bridges the gap between chemical intuition and first-principles electronic structure theory, and positions PF 6 − as a prototypical system for symmetry-aware multiscale simulation frameworks.

In this work, we have analyzed in detail the electronic structure of the PF 6 − anion, combining group theory, chemical intuition, and first-principles calculations. The high O h symmetry of the molecule permits the construction of SALCs of the fluorine orbitals, which form a natural basis for building MOs with well-defined symmetry properties. We have examined the contributions of the phosphorus 3 s , 3 p , and 3 d orbitals and their interaction with the F 2 s and 2 p SALCs, identifying the resulting bonding, nonbonding, and antibonding combinations. The comparison between the traditional sp 3 d 2 hybridization model and the MO picture derived from symmetry provides complementary insights into the bonding nature of the system. By projecting the Kohn–Sham states onto MLWFs, we obtained a minimal orthogonal basis that reproduces the symmetry and bonding character of the system and is suitable for tight-binding and model Hamiltonian construction. The obtained MLWFs confirm the expected irreducible representations and spatial localizations. Overall, the methodology employed here–combining symmetry analysis, localized orbital construction, and first-principles simulations–offers a robust and general approach to studying the electronic structure of molecular and extended systems with high symmetry, and lays the groundwork for efficient multiscale modeling strategies. The previous results on the structural and electronic properties of PF 6 − validates the combined use of the optimized norm-conserving Vanderbilt pseudopotentials proposed in Ref. [21] in combination with a basis of numerical atomic orbitals [37].