Contents
1 Introduction 1
2 Theory 4
2.1 Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Adiabatic and Born-Oppenheimer approximation . . . . 5
2.2 Density Functional Theory: An Introduction . . . . . . . . . . 6
2.2.1 Local (Spin) Density Approximation (L(S)DA) and Generalized
Gradient Approximation (GGA) . . . . . . . . 7
2.3 Calculation of Bandgaps with DFT . . . . . . . . . . . . . . . . 8
2.3.1 A Brief Introduction to the Hybrid Functionals . . . . . 9
2.4 Implementation of DFT in the GPAW (Grid-based Projector
Augmented Wave) code . . . . . . . . . . . . . . . . . . . . . . 11
2.4.1 A Brief Introduction to the PAW Method . . . . . . . . 11
3 Heats of Formation of the Solids 13
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Calculation of the heats of formation without the fitting . . . . 14
3.3 Calculation of the heats of formation with the fitting . . . . . . 15
3.4 Outliers in the different predictions . . . . . . . . . . . . . . . . 17
3.5 True versus predicted error in the mBEEF functional . . . . . . 20
3.6 Cross validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Hydrogen Evolution from Two-Dimensional Materials 25
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Details of the atomic structure . . . . . . . . . . . . . . . . . . 26
4.3 Stability with respect to the standard states . . . . . . . . . . . 29
4.4 Adsorption of hydrogen on the basal planes . . . . . . . . . . . 29
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