where H o is the ground-state Hamiltonian, the linear term vanishes (as the ground state is found by minimizing the energy in terms of the ionic position), and Γ Γ i j , α α β β and Ψ Ψ i j k , α α β β γ γ are the second- and third-order force constants between atoms i, j, and k when moved in along coordinate α α , β β , and γ γ . The second order term is primarily responsible for the phonon dispersion, while the anharmonic (third order and higher) terms are responsible for thermal expansion as well as the phonon up- (multiple low-energy optical phonons combine to form a high-energy phonon) and downconversion (a high-energy phonon splits into multiple low-energy phonons).