On the calculation of quantum mechanical electron transfer rates

Abstract

We present a simple interpolation formula for the rate of an electron transfer reaction as a function of the electronic coupling strength. The formula only requires the calculation of Fermi Golden Rule and Born-Oppenheimer rates and so can be combined with any methods that are able to calculate these rates. We first demonstrate the accuracy of the formula by applying it to a one dimensional scattering problem for which the exact quantum mechanical, Fermi Golden Rule, and Born-Oppenheimer rates are readily calculated. We then describe how the formula can be combined with the Wolynes theory approximation to the Golden Rule rate, and the ring polymer molecular dynamics (RPMD) approximation to the Born-Oppenheimer rate, and used to capture the effects of nuclear tunnelling, zero point energy, and solvent friction on condensed phase electron transfer reactions. Comparison with exact hierarchical equations of motion (HEOM) results for a demanding set of spin-boson models shows that the interpolation formula has an error comparable to that of RPMD rate theory in the adiabatic limit, and that of Wolynes theory in non-adiabatic limit, and is therefore as accurate as any method could possibly be that attempts to generalise these methods to arbitrary electronic coupling strengths. arXiv:1909.09882v1 [physics.chem-ph]

A general non-adiabatic quantum instanton approximation

“…When the Born-Oppenheimer rate is calculated using RPMD rate theory 9,10 and the golden rule rate using Wolynes theory, 13 the resulting interpolated rate is independent of the choice of position space dividing surface for all values of the electronic coupling strength. 32 However, the present NAQI approach clearly provides some advantages over the use of an interpolation formula. For example, one could imagine using its path integral implementation to obtain direct information about the imaginary time trajectories that are important in the reaction and how these trajectories change as a function of the electronic coupling.…”

“…We have recently suggested an alternative approach to calculating non-adiabatic reaction rates, which avoids the need to optimize the dividing surface. 32,33 The idea is to combine the Born-Oppenheimer and golden rule rates with an appropriate interpolation formula. When the Born-Oppenheimer rate is calculated using RPMD rate theory 9,10 and the golden rule rate using Wolynes theory, 13 the resulting interpolated rate is independent of the choice of position space dividing surface for all values of the electronic coupling strength.…”

“…While there now exist several such methods that are routinely used to study electronically adiabatic reactions, [6][7][8][9][10][11][12] as well as theories that can be applied to electronically non-adiabatic reactions in the golden rule limit, [13][14][15][16][17][18][19][20] the development of accurate path integral techniques for more general electronically non-adiabatic reactions with intermediate electronic coupling strengths is still a very active area of research. [21][22][23][24][25][26][27][28][29][30][31][32][33] When developing new path integral methods for studying nonadiabatic systems, a common approach has been to generalize a pre-existing adiabatic method. One of the most successful path integral techniques for electronically adiabatic reactions is ring polymer molecular dynamics 34,35 (RPMD) reaction rate theory, 9,10 and because of this, much of the work in this area has been focused on trying to extend it to treat non-adiabatic systems.…”

An improved path-integral method for golden-rule rates

“…[49][50][51][52][53][54][55][56][57][58][59][60] However, we note that LGR can already be used to calculate reaction rates in systems with arbitrary coupling strengths, by combining it with Born-Oppenheimer RPMD using a simple interpolation formula that interpolates between the golden-rule and Born-Oppenheimer limits. 61 Future work will look to investigate the accuracy of this approach in systems where Wolynes theory is known to break down in the golden-rule limit.…”

An analysis of isomorphic RPMD in the golden rule limit

“…14,33 Moreover one can connect Wolynes theory in the golden rule limit to conventional RPMD rate theory in the Born-Oppenheimer limit using a simple interpolation formula, which enables the calculation of accurate reaction rates for arbitrary electronic coupling strengths. 34,35 In order to apply iso-RPMD to such systems one would need to modify the isomorphic Hamiltonian or the dynamics to make it more closely related to an accurate approximation to the quantum rate in Eq. (13) in the golden rule limit.…”