Bond Valence Energy Landscape (BVEL)

• BVEL is a method that integrates covalent and ionic descriptors to estimate oxygen vacancy migration barriers in rutile 3d transition-metal dioxides.
• It employs exponential distance fits of ICOHP for covalent bonds and Madelung energy for ionic interactions, bypassing expensive NEB calculations.
• The approach achieves a mean absolute error of ~0.09 eV, making it a valuable tool for high-throughput screening in materials design.

The Bond Valence Energy Landscape (BVEL) provides a physically motivated, rapid estimation of migration barriers for oxygen vacancies in rutile-type $3d$ transition-metal dioxides. Developed by Kim and Choi, the BVEL methodology integrates covalent and ionic bond strength descriptors—quantified by the integrated crystal orbital Hamilton population (ICOHP) and Madelung energy, respectively—to yield an empirical predictor for the vacancy migration barrier $E_B$ using only bulk electronic structure data and simple exponential distance fits (Kim et al., 8 Apr 2026). This approach enables estimation of $E_B$ without the need for computationally intensive nudged elastic band (NEB) calculations and is parameterized specifically for TM$^{4+}$–O$^{2-}$ bonds in rutile oxides.

1. Theoretical Foundations: Covalent and Ionic Descriptors

The BVEL framework decomposes local atomic interactions at the migrating oxygen site (O*) into distinct covalent and ionic contributions. The covalent bond-strength descriptor $S_c$ is computed within the crystal orbital Hamilton population (COHP) formalism:

$$\mathrm{ICOHP}_{ij} = \int_{-\infty}^{E_F} -\mathrm{COHP}_{ij}(E)\, dE$$

The contribution for O* is defined as

$$S_c = \sum_{j : r_{\mathrm{O^*}j} \leq 7\, \text{\AA}} \left( -\mathrm{ICOHP}_{\mathrm{O^*},j} \right)$$

where $j$ indexes TM and O atoms within a 7 Å cutoff. This sum accounts for all TM–O* and O–O* bonding interactions, yielding $S_c$ (in eV), which becomes positive for stronger covalent bonds due to the sign convention.

The ionic descriptor $E_B$0 is constructed from the Madelung energy:

$E_B$1

where $E_B$2 is a partial atomic charge (typically Mulliken or Löwdin), and $E_B$3 the corresponding electrostatic potential. The O*-localized ionic bond strength is then

$E_B$4

Both $E_B$5 and $E_B$6 may be evaluated along the migration path to capture the energy landscape relevant to vacancy hopping.

2. Linkage to Vacancy Migration Barriers

For rutile oxides, neither $E_B$7 nor $E_B$8 alone provides a sufficient predictor of the first-principles vacancy migration barrier $E_B$9. BVEL adopts a physically motivated empirical combination: the arithmetic average of their maximal values along the migration coordinate,

$E_B$0

where the superscripts denote the partial charge scheme. Kim and Choi found, for rutile 3$E_B$1-TM dioxides, a near one-to-one linear relationship

$E_B$2

with a mean absolute error (MAE) of approximately 0.09 eV across TiO$E_B$3, VO$E_B$4, CrO$E_B$5, ..., CuO$E_B$6 (Kim et al., 8 Apr 2026).

3. Exponential Fitting Parameters and Model Generalization

Inspired by the classical bond-valence model (BVM), BVEL expresses the distance-dependence of $E_B$7 with an exponential form:

$E_B$8

where $E_B$9 (in Å) is the nominal bond length for $^{4+}$0 eV, and $^{4+}$1 (in Å) characterizes bond strength decay with distance. These parameters were fitted to a dataset of $^{4+}$2800 oxides, with the following values for rutile TM$^{4+}$3–O$^{4+}$4 pairs:

TM$^{4+}$5–O$^{4+}$6	$^{4+}$7 (Å)	$^{4+}$8 (Å)
Ti	2.438	0.352
V	2.357	0.340
Cr	2.361	0.361
Mn	2.324	0.377
Fe	2.331	0.394
Co	2.241	0.369
Ni	2.202	0.372

As in BVM, $^{4+}$9 approximately scales linearly with cation–anion ionic radii sum; $^{2-}$0 encodes overlap decay. Additional parameters for TM–TM and O–O interactions are available for more comprehensive modeling (see Table VI in (Kim et al., 8 Apr 2026)).

4. Application Workflow for Predicting $^{2-}$1

BVEL enables rapid evaluation of $^{2-}$2 in rutile oxides via a sequence:

1. Input structure: Start from the relaxed rutile structure (experiment or DFT relaxation).
2. Site identification: Select the migrating O* and its six nearest TM neighbors.
3. Covalent strength calculation: Use the parameterized exponential (ICOHP-based BVM) for each TM–O* bond:

$^{2-}$3

Sum over the six contributions for $^{2-}$4.

1. Ionic strength calculation: Obtain $^{2-}$5 and $^{2-}$6 (e.g., Mulliken/Löwdin), then compute $^{2-}$7.
2. Barrier estimation: Form $^{2-}$8 and estimate $^{2-}$9, bypassing NEB.

5. Empirical Accuracy and Domain of Applicability

Across eight rutile 3$S_c$0-TM dioxides, BVEL achieves MAE $S_c$1 0.09 eV for $S_c$2 prediction (maximum observed error $S_c$3 0.2 eV). The covalent-only or ionic-only metrics yield systematically inferior accuracy (e.g., $S_c$4 MAE $S_c$5 0.26 eV). This underscores the necessity of including both bonding characters for reliable prediction.

BVEL is most accurate for TiO$S_c$6, CrO$S_c$7, CoO$S_c$8, and NiO$S_c$9; slight degradation is observed for CuO$$\mathrm{ICOHP}_{ij} = \int_{-\infty}^{E_F} -\mathrm{COHP}_{ij}(E)\, dE$$0 due to less transferable Cu$$\mathrm{ICOHP}_{ij} = \int_{-\infty}^{E_F} -\mathrm{COHP}_{ij}(E)\, dE$$1–O$$\mathrm{ICOHP}_{ij} = \int_{-\infty}^{E_F} -\mathrm{COHP}_{ij}(E)\, dE$$2 parameters. The applicability is contingent on reliable partial charges and valid comparison of bond lengths at equilibrium and saddle-point configurations. Extensions to non-rutile phases or TM valences outside $$\mathrm{ICOHP}_{ij} = \int_{-\infty}^{E_F} -\mathrm{COHP}_{ij}(E)\, dE$$3 require additional parameterization.

6. Limitations and Prospects

BVEL relies on Mulliken or Löwdin population analyses, which introduces a known basis dependency in $$\mathrm{ICOHP}_{ij} = \int_{-\infty}^{E_F} -\mathrm{COHP}_{ij}(E)\, dE$$4. The method presupposes that the saddle-point geometry along the vacancy pathway does not deviate substantially from the equilibrium bond environment—a condition potentially violated in highly distorted lattices. Broader generalization (e.g., to non-rutile structures or different transition-metal oxides) requires recomputation of exponential-fit parameters against a representative data set.

A plausible implication is that BVEL provides a physically transparent alternative to empirical barriers for the targeted family of materials, with scope for expansion to more complex systems given suitable parameterization. The approach's speed and moderate accuracy position it as a viable screening tool within high-throughput computational materials design workflows for ionic and electronic conductors.