In this paper, we describe an automated integration-free path-integral (AIF-PI) method [Wong, K.-Y.; Gao, J. J. Chem. Phys. 2007, 127, 211103], based on Kleinert's variational perturbation (KP) theory, to treat internuclear quantum-statistical effects in molecular systems. We have developed an analytical method to obtain the centroid potential as a function of the variational parameter in the KP theory, which avoids numerical difficulties in path-integral Monte Carlo or molecular dynamics simulations, especially at the limit of zero-temperature. Consequently, the variational calculations using the KP theory can be efficiently carried out beyond the first order, i.e., the Giachetti-Tognetti-Feynman-Kleinert variational approach, for realistic chemical applications. By making use of the approximation of independent instantaneous normal modes (INM), the AIF-PI method can be applied to many-body systems, and it was shown previously that the AIF-PI method is accurate for computing the quantum effects including a water molecule and the collinear H3 reaction. In this work, the accuracy and properties of the KP theory are further investigated by using the first three-order perturbations on an asymmetric double-well potential, the bond vibrations of H2, HF, and HCI represented by the Morse potential, and a proton-transfer barrier modeled by the Eckart potential. The zero-point energy, quantum partition function, and tunneling factor for these systems have been determined and are found to be in excellent agreement with the exact quantum results. Using our new analytical results at the zero-temperature limit, we show that the minimum value of the computed centroid potential in the KP theory is in excellent agreement with the ground-state energy (zero-point energy), and the position of the centroid potential minimum is the expectation value of particle position in wave mechanics. The fast convergent property of the KP theory is further examined in comparison with results from the traditional Rayleigh-Ritz variational approach and Rayleigh-Schrödinger perturbation theory in wave mechanics. The present method can be used for thermodynamic and quantum dynamic calculations, including systematically determining the exact value of zero-point energy and studying kinetic isotope effects for chemical reactions in solution and in enzymes.