The instanton theory is reformulated with use of the path integral approach and the Wentzel–Kramers–Brillouin approximation to the Schrödinger equation. Both approaches are shown to provide the same results. A new practically useful semiclassical formula is derived for the tunneling splitting of the ground state, which can be implemented for high-dimensional systems. The theory is applicable to systems of arbitrary Riemannian metric and is also supplemented by a practical numerical recipe to evaluate the instanton trajectory, i.e., periodic orbit, in multidimensional space. Numerical examples are presented for three-dimensional (3D) and 21D systems of HO2 and malonaldehyde, respectively.

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A practical and accurate semiclassical method for calculating the tunneling splitting of the ground state in polyatomic molecules is presented based on a recent version of the instanton theory [J. Chem. Phys. 115, 6881 (2001)]. The method uses ab initio quantum chemical data for the potential energy surface without any concomitant extrapolation and requires only a small number of ab initio data points to get convergence even for large molecules. This enables one to use an advanced level of electronic structure theory and achieve a high accuracy of the result. The method is applied to the 9-atomic malonaldehyde molecule by making use of the potential energy surface at the level of CCSD(T) with the hybrid basis set of aug-cc-pVTZ (for oxygen atoms and the transferred hydrogen atom) and cc-pVTZ (for other atoms).

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We report an accurate and efficient full dimensional semiclassical ab initio method for calculation of energy level splitting due to tunneling in polyatomic system. The method is applied to 21-dimensional 9-atomic malonaldehyde molecule. The tunneling splittings obtained are ΔE(H)=21.2 cm−1 for hydrogen atom transfer and ΔE(D)=3.0 cm−1 for deuterium atom transfer, which are in excellent agreement with the experimental values of 21.6 cm−1 and, 2.9 cm−1 respectively. We believe that the present analysis gives the final solution to the longstanding problem.

We present a generalized formulation of the trajectory surface hopping method applicable to a general multidimensional system. The method is based on the Zhu-Nakamura theory of a nonadiabatic transition and therefore includes the treatment of classically forbidden hops. The method uses a generalized recipe for the conservation of angular momentum after forbidden hops and an approximation for determining a nonadiabatic transition direction which is crucial when the coupling vector is unavailable. This method also eliminates the need for a rigorous location of the seam surface, thereby ensuring its applicability to a wide class of chemical systems. In a test calculation, we implement the method for the DH(2) (+) system, and it shows a remarkable agreement with the previous results of C. Zhu, H. Kamisaka, and H. Nakamura, [J. Chem. Phys. 116, 3234 (2002)]. We then apply it to a diatomic-in-molecule model system with a conical intersection, and the results compare well with exact quantum calculations. The successful application to the conical intersection system confirms the possibility of directly extending the present method to an arbitrary potential of general topology.

We develop the instanton theory for calculating the tunneling splitting of excited states. For the case of low vibrational quantum states we derive a canonically invariant formula which is applicable to a multidimensional system of arbitrary Riemannian metric. The effect of multidimensionality in relation to the vibrational excitation is explained in terms of the effective frequencies along the instanton trajectory. The theory is demonstrated to work well by taking HO2 molecule as an example.

The previously developed instanton theory [G. V. Mil'nikov and H. Nakamura, J. Chem. Phys. 122, 124311 (2005)] is applied to the calculation of vibrationally assisted tunneling splitting of the deuterated formic acid dimer (DCOOH)2 with all the degrees of freedom taken into account. The ground-state tunnel splitting is determined by the density-functional theory combined with coupled cluster level of quantum chemistry to be 0.0038 cm(-1) which is comparable to the experimental value of 0.0029 cm(-1). Further, the tunnel splittings of fundamental excitations are estimated for frequencies below 300 cm(-1). In this energy range it is found that the excitation modes may either enhance or suppress tunneling as compared to the ground state. For the higher-frequency modes a rapid growth of the tunnel splitting is observed. At frequencies above 1000 cm(-1) the semiclassical solution becomes unstable and no reliable tunneling splittings can be obtained. This is in vast contrast to the adiabatic approximation to the instanton theory in which the tunnel splittings can be retrieved up to 3000 cm(-1). We discuss this disparity from the viewpoint of the multidimensional character of tunneling in hydrogen bonds and the adiabatic approximation is concluded to be inaccurate.