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\title{Your high-impact title}

\author{Student T} \affiliation{Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213}

\author{John R. Kitchin} \email{[email protected]} \affiliation{Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213}

\date{\today}

\pacs{} \keywords{DFT+U, transition metal oxides, oxidation energy} \maketitle

Transition metal oxides (TMOs) have broad applications in heterogeneous catalysis, electrochemistry, photocatalysis, and sensors cite:Weaver2013,Doyle2013,Gouma2011. Standard exchange correlation functionals (LDA and GGA) in density functional theory (DFT) often fail to calculate accurate thermodynamic properties of TMOs, which hinders our ability to discover new TMOs for these applications. This failure has been partially attributed to a lack of cancellation of the self-interaction error cite:Cohen2008,PhysRevB.73.195107o,Franchini2007.

\begin{equation} Δ E = ∫_{\text{\textbf{V}}} \frac{dE}{d\textbf{\text{V}}} = ∫_{\text{\textbf{V}}} \left( \frac{∂ E}{∂\text{\textbf{V}}} + \frac{∂ E}{∂ U} \frac{dU}{d\text{\textbf{V}}} \right) d\text{\textbf{V}}. \label{dftuv-full} \end{equation}

\noindent Because $E$ depends on both $\textbf{V}$ and $U$, the total derivative contains changes in the total energy produced by both changes in $\textbf{V}$ $\left(\frac{∂ E}{∂\text{\textbf{V}}}\right)$ and changes in $U$ $\left(\frac{∂ E}{∂ U} \frac{dU}{d\text{\textbf{V}}}\right)$. Note, this derivation implies some $U(\textrm{\textbf{V}})$ relationship. If we assume that $E$ is continuous with respect to $U$ and $\textbf{V}$, the entire integral is path independent, but the contributions of each differential are not path independent.

\begin{align} Δ H_rxn& = E_{\mathrm{MO_y}}^U=b - E_{\mathrm{MO_x}}^U=a - \frac{y-x}{2} E_{\mathrm{O_2}}\label{derivation1}
Δ H_rxn& = (E_{\mathrm{MO_y}}^U=b - E_{\mathrm{M_{atom}}}^U=0 - \frac{y}{2}E_{\mathrm{O_2}}) - (E_{\mathrm{MO_x}}^U=a - E_{\mathrm{M_{atom}}}^U=0 - \frac{y}{2}E_{\mathrm{O_2}}) - \frac{y-x}{2} E_{\mathrm{O_2}}\label{derivation2}\ Δ H_rxn& = (E_{\mathrm{MO_y}}^U=b - E_{\mathrm{M_{atom}}}^U=0 - \frac{y}{2}E_{\mathrm{O_2}}) - (E_{\mathrm{MO_x}}^U=a - E_{\mathrm{M_{atom}}}^U=0 - \frac{x}{2}E_{\mathrm{O_2}})\label{derivation3}\ Δ H_rxn& = Δ E_\mathrm{DFT+U(\mathrm{\mathbf{V}}),\mathrm{MO_y}} - Δ E_\mathrm{DFT+U(\mathrm{\mathbf{V}}),\mathrm{MO_x}} \label{derivation4} \end{align}

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\begin{acknowledgments} J.R.K. gratefully acknowledges support from the U.S. Department of Energy (DOE) Office of Science, Early Career Research Program (DESC0004031). \end{acknowledgments}

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