In the previous chapter, the experimental techniques used to obtain the thermochemical and phase equilibrium data that were the inputs for the thermodynamic modeling of a system were summarized. However, experimental data are not always available. This is due to the fact that (i) the experiments are expensive, especially when they involve developing new materials, and (ii) the experiments cannot reliably access the non-stable phases in most cases. The alternative approach is to predict the thermochemical data by first-principles calculations. The prediction of material properties, without using phenomenological parameters, is the basic spirit of first-principles calculations. In particular, the steady increase of both computer power and the efficiency of computational methods have made the first-principles predictions of most thermodynamic properties possible, including both enthalpy and entropy as a function of temperature, volume, and/or pressure.

By definition, the term “first-principles” represents a philosophy that the prediction is to be based on a basic, fundamental proposition or assumption that cannot be deduced from any other proposition or assumption. This implies that the computational formulations are based on the most fundamental theory of quantum mechanics, the Schrödinger equation or density functional theory, and the inputs to the calculations must be based on well-defined physical constants – the nuclear and electronic masses and charges. In other words, once the atomic species of an assigned material are known, the theory should predict the energies of all possible crystalline structures, without invoking any phenomenological fitting parameters.

This chapter is organized in sequence from thermodynamic calculations to fundamental theory, to help those readers who are more interested in realistic calculations using existing computer codes. Detailed theoretical discussions follow the subsections on thermodynamic calculations for those readers who are also interested in the derivation of the formulations used in the thermodynamic calculations. The subsections are arranged accordingly in the order: (i) examples of the commonly adopted calculation procedures for thermodynamic properties using the elemental metal nickel as the main prototype; (ii) derivation of the Helmholtz energy expression under the first-principles framework; (iii) introduction of the solution to the electronic Schrödinger equation within two well-developed frameworks – the quantum chemistry approach and the density functional theory; (iv) detailed description of the procedure on how to solve the Schrödinger equation for the motions of atomic nuclei by means of lattice dynamics; and (v) First-principles approaches to disordered alloys.