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Accurate quantum-mechanical simulations have significantly extended the current picture of the Earth and hold a great promise for the future of the Earth and planetary sciences. Studies of phase transitions, equations of state, elasticity and thermoelastic properties of the Earth-forming minerals are essential to geophysics. This chapter gives a basic background of the physics of the deep Earth and outlines the theory of phase transitions, equations of state, elasticity and thermoelastic properties. A particular emphasis is put on the principles of quantum-mechanical simulations and some recent results relevant to geophysics.

The importance of quantum-mechanical simulations is reflected by the award of the 1998 Nobel Prize in Chemistry to W. Kohn and J. Pople, who were among the pioneers of this field. Areas of application of such simulations are extremely diverse and include studies of the electronic structure, reactivity, catalysis, bulk and surface structure, prediction of materials structures and properties, especially at extreme conditions, calculation of phase diagrams and studies of phase transitions etc.

One of the most exciting areas of application of such simulations is the study of the Earth- (and planet-) forming minerals at the extreme conditions of the Earth's interior. One can accurately predict the structures, properties, and behaviour of minerals. This often reveals new aspects of mineral crystal chemistry and allows one to explain geophysical measurements and understand better how the Earth works as a planet. This chapter consists of five major parts – Part I: Brief geophysical introduction, Part II: Thermoelastic properties, Part III: Phase transitions, Part IV:

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