Influence of dimensionality on deep tunneling rates: A study based on the hydrogen-nickel system Roi Baer Department of Physical Chemistry and the Fritz Haber Research Center, the Hebrew University, Jerusalem 91904, Israel Yehuda Zeiri Department of Chemistry, Nuclear Research Center-Negev Beer-Sheva 84190, P.O. Box 9001, Israel Ronnie Kosloff Department of Physical Chemistry and the Fritz Haber Research Center, the Hebrew University, Jerusalem 91904, Israel Received 26 April 1996 The tunneling of subsurface hydrogen into a surface site of a nickel crystal is used to investigate deep tunneling phenomena. A method to calculate tunneling lifetimes based on an energy and time filter is devel- oped, enabling converged lifetimes differing by 14 orders of magnitude. It is found that the reduced dimen- sional approximation always overestimates the tunneling rate. The vibrational adiabatic correction improves dramatically the one-dimensional calculation but nevertheless overestimates the cases of deep tunneling. The isotope effect is studied, pointing to experimental implications. S0163-18299651332-9 I. INTRODUCTION The tunneling motion of hydrogen atoms in metals is an important process with a range of practical applications, such as hydrogen embrittlement, catalysis, and fuel storage. 1 Moreover, tunneling draws fundamental interest since it ex- hibits dominant quantum effects at low temperatures. 2,3 Ex- perimentally, for tunneling processes the Arrhenius rate law breaks down, exhibiting a sharp crossover to a temperature- independent rate, 6 as predicted by Wigner in the 1930’s. 4,5 In molecular dynamics, tunneling phenomena are extreme examples where classical treatments fails. 2,3 Since quantum calculations scale exponentially with dimension, approxima- tions for the tunneling motion based on a reduced dimension- ality description have been developed. It has been established 6 that defining a ‘‘tunneling coordinate’’ and then freezing the perpendicular degrees of freedom causes severe errors in the estimates of tunneling rates. Instead, various studies have employed a reaction path approach, 7–10 which is essentially an adiabatic approximation suitable when the per- pendicular degrees of freedom can be treated as ‘‘fast’’ and the reaction, tunneling coordinate as ‘‘slow.’’ The approxi- mation amounts to adding the ground-state energy of the perpendicular oscillators to the reaction path potential, and will be denoted as the vibrationally adiabatic approximation VAA. Schatz et al. 11 have published a comparison of full quan- tum three-dimensional 3D calculations of proton transfer reactions vs VAA-type transition-state theory with tunneling treated semiclassicaly VAA/ST. They find large departures of the calculated reaction rates, up to a factor of 2.5 at room temperature. Garrett et al. 12 claim that this discrepancy is not necessarily indicative of the quality of the approximation and can also be attributed to differences in potential-energy sur- faces or to errors resulting from the numerical implementa- tion of the quantum method. For more conclusive results a simpler solution is necessary. Motivated by new experiments on the chemistry of bulk hydrogen, 13–16 the dynamics of a hydrogen atom in a nickel fcc crystal is studied. The atom is located in a subsurface interstitial site from which it tunnels, through a 0.65-eV bar- rier, to a threefold hollow site on the 111 metal surface. In this system, degrees of freedom can be divided into two types: those directly belonging to the tunneling atom, and those of the heavy metal atoms — which comprise the crystal vibrational heat bath. The masses associated with each class of degrees of freedom DOF are very different, and it is therefore assumed that the hydrogen atom is only weakly coupled to the motion of the crystal. The full 3D hydrogen quantum results are compared with reduced dimen- sional 2D and 1D calculations, as well as with the 1D vibra- tional adiabatic approximation 1D-VAA. The treatment of the crystal DOF’s as well as the possibility of exciting the metal electrons will be addressed in a future publication. II. METHOD OF CALCULATION The tunneling rate is related to the imaginary component of the complex eigenvalue of the Hamiltonian belonging to an asymptotically outgoing-only eigenstate. The tunneling eigenvalue spectrum is discrete and com- plex:  n =E n -i  n /2, n =0,1,2, . . . , where E n is the energy of the state and  n the tunneling rate. The tunneling states are improper eigenstates of the Hamiltonian, since the nega- tive imaginary part of the eigenvalue imposes a correspond- ing imaginary value of the outgoing momentum in the as- ymptotic channel, leading to the divergence of the eigenstate at infinitely large distances. In order to regularize these states, several methods have been developed, such as the complex scaling method 17,18 and others. 19 An alternative regularization method is achieved by a lo- calized negative imaginary potential NIP in the asymptotes. 20 The NIP’s are used to impose an outgoing boundary condition on the eigenstate calculations. By aug- PHYSICAL REVIEW B 15 AUGUST 1996-II VOLUME 54, NUMBER 8 54 0163-1829/96/548/52874/$10.00 R5287 © 1996 The American Physical Society