A multiscale approach to coupled nuclear and electronic dynamics. II. Exact and approximated evaluation of nonradiative transition rates

Thisworkfollowsacompanionarticle,whichwillbereferredtoasPaperI[Campeggioetal.,J.Chem.Phys.158,244104(2023)]inwhicha quantum-stochasticLiouvilleequationforthedescriptionofthequantum–classicaldynamicsofamoleculeinadissipativebathhasbeenformulatedincurvilinearinternalcoordinates.Insuchanapproach,thecoordinatesofthesystemareseparatedintothreesubsets:thequantum coordinates,theclassicalrelevantnucleardegreesoffreedom,andtheclassicalirrelevant(bath)coordinates.Theequationhasbeenderived innaturalinternalcoordinates,whicharebondlengths,bondangles,anddihedralangles.Theresultingequationneedstobeparameterized.Inparticular,oneneedstocomputethepotentialenergysurfaces,thefrictiontensor,andtherateconstantsforthenonradiativejumps amongthequantumstates.Whilestandardmethodsexistforthecalculationofenergyanddissipativeproperties,anefficientevaluationof thetransitionratesneedstobedeveloped.Inthispaper,anapproximatedtreatmentisintroduced,whichleadstoasimpleexplicitformula withasingleadjustableparameter.Suchanapproximatedexpressioniscomparedwiththeexactcalculationoftransitionratesobtainedvia moleculardynamicssimulations.Tomakesuchacomparisonpossible,asimplesandboxsystemhasbeenused,withtwoquantumstates andasingleinternalcoordinate(togetherwithitsconjugatemomentum).Resultsshowthattheadjustableparameter,whichisaneffective decoherencetime,canbeparameterizedfromtheeffectiverelaxationtimesoftheautocorrelationfunctionsoftheconjugatedmomentaof therelevantnuclearcoordinates.