The E⊗e dynamic Jahn-Teller problem: A new insight from the strong coupling limit

Abstract

Correct boundary conditions for the E x e dynamic Jahn-Teller problem are considered explicitly for the first time to obtain approximate analytical solutions in the strong coupling limit. Numerical solutions for the decoupled equations using the finite difference method are also presented. The numerical solutions for the decoupled equations exhibit avoided crossings in the weak coupling region, which explains the oscillating behavior of the solutions obtained by Longuet-Higgins et al. for the coupled equations. The obtained analytical energy expressions show improved agreement with the numerical calculations as compared with the previous treatment in which the potentials were assumed to be harmonic. We demonstrate that the pseudorotational energy j(2)/(2g(2)), where g is the dimensionless vibronic coupling constant, and j total angular momentum: j=+/-1/2,+/-3/2,..., in the conventional strong coupling expression for the vibronic levels of the lower sheet is exact. Non-Hermitian first-order perturbation theory gives the energy which is correct up to 1/g(4). The asymptotic behavior of the wave function at the origin does not influence the corrected energy up to order of 1/g(4). At the same time the treatment of the upper sheet with correct boundary conditions gives solutions which are entirely different from the corresponding Slonczewski's solutions. Besides, the correct boundary conditions enable us to evaluate the nonadiabatic coupling between the lower and upper potential sheets. The energy correction due to the nonadiabatic coupling is estimated to be of order 1/g(6).

Analytical solutions for the E⊗e dynamic Jahn–Teller problem in the strong coupling limit

“…In the term of the 1/g 4 order, we find a dependence on a. However, the first-order perturbations exactly cancel the a-dependence in the 1/g 4 order [13].…”

“…where P(q) and Q(q) is a function of a, j, g, and q. Since the neighborhood of q = q ±,0 is the most important region in the strong coupling limit, expanding P(q) and Q(q) around q = q ±,0 , and keeping terms up to a linear term in P(q) and a quadratic term in Q(q), we obtain an analytically solvable equation [13],…”

“…Recently we have derived analytical solutions with the correct boundary condition of the dynamic E e problem in the strong coupling limit [13]. In this paper, we summarize the analytical solutions in the strong coupling limit.…”

Near-infrared photoabsorption by C60 dianions in a storage ring

“…A possible explanation of the strong absorption at high energies is transition to so-called cone states, which are JT states with the opposite sign of the deformation amplitude and positive JT energy. [33][34][35] In the JT ground state the magnitude of the deformation amplitude q is determined by a balance between the JT force driving the deformation and the harmonic restoring force, but for cone states with the opposite sign of q these forces act together to reduce the magnitude of q. The deformation is instead stabilized by the centrifugal force associated with pseudorotations.…”

The Jahn–Teller Effect in Chemistry

“…Along this path, the three nuclei rotate around their symmetry positions in a concerted way. 4 The waveform of the lowest root is expressed as:…”