Canonically modified Nosé-Hoover equation with explicit inclusion of the virial

The Nos\'e-Hoover equation has recently been introduced to simulate, in a deterministic and reversible way, the equilibrium properties of a system at constant temperature. However, for many one-dimensional potentials, such as the harmonic oscillator, the Nos\'e-Hoover scheme is not adequate since the dynamics is not sufficiently ergodic. We present modifications of the Nos\'e-Hoover equation in which the kinetic energy and the virial are treated in an equivalent manner and which explicitly include the virial within a canonical framework. We show that these modifications can yield an adequate statistical description for one-dimensional potentials such as the double-well and harmonic oscillator.