  Splitting schemes are numerical integrators for Hamiltonian problems that may
advantageously replace the St\"ormer-Verlet method within Hamiltonian Monte
Carlo (HMC) methodology. However, HMC performance is very sensitive to the step
size parameter; in this paper we propose a new method in the one-parameter
family of second-order of splitting procedures that uses a well-fitting
parameter that nullifies the expectation of the energy error for univariate and
multivariate Gaussian distributions, taken as a problem-guide for more
realistic situations; we also provide a new algorithm that through an adaptive
choice of the $b$ parameter and the step-size ensures high sampling performance
of HMC. For similar methods introduced in recent literature, by using the
proposed step size selection, the splitting integration within HMC method never
rejects a sample when applied to univariate and multivariate Gaussian
distributions. For more general non Gaussian target distributions the proposed
approach exceeds the principal especially when the adaptive choice is used. The
effectiveness of the proposed is firstly tested on some benchmarks examples
taken from literature. Then, we conduct experiments by considering as target
distribution, the Log-Gaussian Cox process and Bayesian Logistic Regression.
