Perturbatively corrected ring-polymer instanton theory for accurate tunneling splittings

Abstract

We introduce an approach for calculating perturbative corrections to the ring-polymer instanton approximation to tunneling splittings (RPI+PC) by computing higher-order terms in the asymptotic expansion in ℏ. The resulting method goes beyond standard instanton theory by using information on the third and fourth derivatives of the potential along the tunneling path to include additional anharmonic effects. This leads to significant improvements both in systems with low barriers and in systems with anharmonic modes. We demonstrate the applicability of RPI+PC to molecular systems by computing the tunneling splitting in full-dimensional malonaldehyde and a deuterated derivative. Comparing to both experiment and recent quantum mechanical benchmark results, we find that our perturbative correction reduces the error from −11% to 2% for hydrogen transfer and performs even better for the deuterated case. This makes our approach more accurate than previous calculations using diffusion Monte Carlo and path-integral molecular dynamics while being more computationally efficient.

Numerical Accuracy Matters: Applications of Machine Learned Potential Energy Surfaces

“…The corrections c CF to the RPI splittings for two different step sizes in the numerical differentiation and for both PESs are reported in Table . For NN F64 the correction is c CF = 0.91 irrespective of step size and yields a corrected tunneling splitting of 22.1 cm –1 which is in excellent agreement with the experimental value of 21.6 cm –1 . , As a comparison, using a semiglobal PES, very recent RPI+PC simulations also found 22.1 cm –1 . Contrary to using NN F64 , the “correction”, which is expected to be ∼1, ranges from 8.9 to 21.7 depending on the step size when using NN F32 .…”

“…33 Recently, a perturbative correction for RPI simulations was presented which determines a multiplicative factor c CF to account for anharmonic effects and intermode coupling. 35 Computation of c CF requires third-and fourth-order derivatives of the total energy with respect to Cartesian coordinates which need to be carried out numerically, similar to the situation for VPT2 calculations. The corrections c CF to the RPI splittings for two different step sizes in the numerical differentiation and for both PESs are reported in Table 3.…”

“…31,34 As a comparison, using a semiglobal PES, 36 very recent RPI+PC simulations also found 22.1 cm −1 . 35 Contrary to using NN F64 , the "correction", which is expected to be ∼1, ranges from 8.9 to 21.7 depending on the step size when using NN F32 . Hence, the roughness of NN F32 (see below) has a detrimental effect for higher-order derivatives.…”

Proton-Tunneling Dynamics along Low-Barrier Hydrogen Bonds: A Full-Dimensional Instanton Study of 6-Hydroxy-2-formylfulvene

“…Alternatively, the perturbatively corrected ring-polymer instanton theory recently introduced by Richardson and coworkers affords a novel approach for incorporating higherorder terms beyond those of the quadratic expansion. 51 It will be interesting to apply such methods to HFF in future studies.…”

Smolyak Scheme for solving the Schrödinger equation: Application to Malonaldehyde in Full Dimensionality

“…[47] From a theoretical point of view this molecule has been intensively studied. In particular several simulations have been done: Reaction Surface Hamiltonian (RSH) [48][49] instanton, [50][51][52][53] DMC, [3,4] MCTDH [54][55][56] approaches. In particular and due to the availability of recent full dimensionality potential energy surfaces, [3,4] the DMC and MCTDH lead to tunneling splitting values in good agreement with the experimental ones (see Table 1).…”