Effective non-adiabatic Hamiltonians for the quantum nuclear motion over coupled electronic states

Abstract

The quantum mechanical motion of the atomic nuclei is considered over a single-or a multidimensional subspace of electronic states which is separated by a gap from the rest of the electronic spectrum over the relevant range of nuclear configurations. The electron-nucleus Hamiltonian is block-diagonalized up to O(ε n+1 ) through a unitary transformation of the electronic subspace and the corresponding nth-order effective Hamiltonian is derived for the quantum nuclear motion.Explicit but general formulae are given for the second-and the third-order corrections. As a special case, the second-order Hamiltonian corresponding to an isolated electronic state is recovered which contains the coordinate-dependent mass-correction terms in the nuclear kinetic energy operator. For a multi-dimensional, explicitly coupled electronic band, the second-order Hamiltonian contains the usual BO terms and non-adiabatic corrections but generalized mass-correction terms appear as well.These, earlier neglected terms, perturbatively account for the outlying (discrete and continuous) electronic states not included in the explicitly coupled electronic subspace. * Electronic address: matyus@chem.elte.hu † Electronic address: stefan.teufel@uni-tuebingen.de

Non-adiabatic mass correction for excited states of molecular hydrogen: Improvement for the outer-well HH¯ 1Σg+ term values

“…For this purpose, one can either look for higher-order corrections-the third-order correction formulae can be found in Ref. [1]-, or for explicit coupling with the near-lying perturber state(s), in this case GK (and other states), and to use the effective non-adiabatic Hamiltonian of Ref. [1] for a multi-dimensional electronic subspace.…”

“…[1]-, or for explicit coupling with the near-lying perturber state(s), in this case GK (and other states), and to use the effective non-adiabatic Hamiltonian of Ref. [1] for a multi-dimensional electronic subspace. Note that in the single-state treatment only the EF state is projected out from the resolvent, Eq.…”

“…Note that in the single-state treatment only the EF state is projected out from the resolvent, Eq. (1), whereas in a multi-state treatment the full explicitly coupled subspace will be projected out [1], which will result in smaller corrections from electronic states better separated in energy.…”

“…Although their computed values are off by 10-35 cm −1 from experiment, probably due to the fact that they used less accurate potential energy curves, their results seem to underline the general observation that the many-state Born-Oppenheimer (BO) expansion converges relatively slowly when one aims to achieve spectroscopic accuracy.Because of the slow convergence of the BO expansion, it is important to think about the truncation error. Direct truncation of the electronic space introduces an error of O(ε) in the rovibronic energies, where ε = (m e /m nuc ) 1/2 is the square root of the electron-to-nucleus mass ratio[1,2,22]. This truncation error can be made lower order in ε by using adiabatic perturbation theory[2,22].…”

“…For an isolated electronic state, the first-order corrections can be made to vanish. The second-order non-adiabatic effective Hamiltonian, used in the present work, reproduces eigenvalues of the full electron-nucleus Hamiltonian with an error of O(ε 3 ), but it contains corrections both to the potential energy as well as to the kinetic energy of the atomic nuclei[1], which gives rise to effective coordinate-dependent masses to the different types of motions.…”