First-principles theories for anharmonic lattice vibrations.

Size-extensive generalizations of the vibrational self-consistent field (VSCF), vibrational Moller-Plesset perturbation (VMP), and vibrational coupled-cluster (VCC) methods are made to anharmonic lattice vibrations of extended periodic systems on the basis of a quartic force field (QFF) in delocalized normal coordinates. Copious terms in the formalisms of VSCF that have nonphysical size dependence are identified algebraically and eliminated, leading to compact and strictly size-extensive equations. This "quartic" VSCF method (qVSCF) thus defined has no contributions from cubic force constants and alters only the transition energies of the underlying harmonic-oscillator reference from a subset of quartic force constants. It also provides a way to evaluate an anharmonic correction to the lattice structure due to cubic force constants of a certain type. The second-order VMP and VCC methods in the QFF based on the qVSCF reference are shown to account for anharmonic effects due to all cubic and quartic force constants in a size-extensive fashion. These methods can be readily extended to a higher-order truncated Taylor expansion of a potential energy surface in normal coordinates. An algebraic proof of the lack of size-extensivity in the vibrational configuration-interaction method is also presented.