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Write a detailed review of the following paper with expansion on the quantum mechanical aspects of the paper.

The review should provide a decent summary of the background and application, but should then focus on the computations and quantum mechanical aspects of the paper. The computational and quantum mechanical portions should be expanded upon and explained (i.e. talk about the Schrodinger equation, wavefunctions, Hartree-Fock, perturbation method, variational method, etc.)

The review and expansion must be as detailed as possible and understandable from a basic graduate level course in quantum mechanics.

Everything should be typed and pictures provided as necessary.

Paper: Theoretical Investigation of Water Oxidation Processes on Small MnₓTi₂₋ₓO₄ (x = 0−2) Complexes,
by Choongkeun Lee and Christine M. Aikens

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1. Paper summary

This paper is the theoretical study of the water oxidation reaction catalyzed with small metal oxide complexes. This area of research is very important since understanding these simplest architectures is essential for the bottom up design of commercially viable systems for catalytic conversion of solar energy into chemical energy, and, in a broader perspective, design of the clean energy sources able to replace fossil based fuels. After announcement of first electrochemical photocatalytical water oxidation reaction on TiO2, many metal oxide based photocatalysts have been investigated both theoretically and experimentally. The problem with TiO2 based systems used for used for water oxidation is that they require high overpotentials13 and possess high band gap (absorption in UV region, accounting for less than 5% of the full EM spectrum).
The paper represents the analysis of the Mn doping effect on water oxidation on a small TiO2 complex, MnxTi2−xO4 (x = 0−2).
Four possible catalytic reaction pathways are considered in detail, with the water oxidation process as the same outcome (the conversion of water into molecular oxygen, protons, and electrons). All four reaction pathways are analyzed with model catalyst consisting of oxides with two metal centers (Ti-Ti, Mn-Ti and Mn-Mn) connected by oxygen bridge, thus describing manganese doping of TiO2 with manganese. All examined pathways possess a diversity of electronically and structurally complicated intermediates where the coordination number, oxidation state and spin state of the metals are subject to alteration following the proposed catalytic mechanism....
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