Specific heats of quantum double-well systems

Specific heats of quantum systems with symmetric and asymmetric double-well potentials have been calculated. In numerical calculations of their specific heats, we have adopted the combined method which takes into account not only eigenvalues of $\epsilon_n$ for $0 \leq n \leq N_m$ obtained by the energy-matrix diagonalization but also their extrapolated ones for $N_m+1 \leq n < \infty$ ($N_m=20$ or 30). Calculated specific heats are shown to be rather different from counterparts of a harmonic oscillator. In particular, specific heats of symmetric double-well systems at very low temperatures have the Schottky-type anomaly, which is rooted to a small energy gap in low-lying two-level eigenstates induced by a tunneling through the potential barrier. The Schottky-type anomaly is removed when an asymmetry is introduced into the double-well potential. It has been pointed out that the specific-heat calculation of a double-well system reported by Feranchuk, Ulyanenkov and Kuz'min [Chem. Phys. 157, 61 (1991)] is misleading because the zeroth-order operator method they adopted neglects crucially important off-diagonal contributions.