  The Quasi-harmonic (QH) approximation uses harmonic vibrational frequencies
omega(H,Q,V), computed at volumes V near the volume where the Born-Oppenheimer
(BO) energy is minimum. When this is used in the harmonic free energy, QH
approximation gives a good zeroth order theory of thermal expansion, and first
order theory of bulk modulus. Here, n-th order means smaller than the leading
term by n powers of epsilon, where epsilon is the ratio hbar omega(Q)/E(el) or
kT/E(el), and E(el) is an electronic energy scale, typically 2 to 10 eV.
Experiment often shows evidence for next order corrections. When such
corrections are needed, anharmonic interactions must be included. The most
accessible measure of anhamonicity is the quasiparticle (QP) energy,
omega(Q,V,T), seen experimentally by vibrational spectroscopy. However, this
cannot just be inserted into the harmonic free energy F(H). In this paper, a
free energy formula is found which corrects the double-counting of anharmonic
interactions that is made when F is approximated by F(H,omega(Q,V,T)). The term
"QP thermodynamics" is used for this way of treating anharmonicity. It enables
(n+1)-order corrections, if QH theory is accurate to order n. This procedure is
used to give corrections to specific heat and volume thermal expansion. The QH
formulas for isothermal and adiabatic bulk moduli are clarified, and the route
to higher order corrections is indicated.
