1. Consider a diatomic molecule with two electronic states, 1 and 2 and a bond length R. The Hamiltonian for this system in the diabatic basis is given by: ()()( ) ( )( )2ˆ21ˆ22)ˆ(11ˆ1ˆ22221211211221RVRVRVRVHRR+∇−++++∇−= where the relevant interactions are given by ()( )( ) ( )2206.211228.2211015.ˆˆ1.05.ˆ0ˆRReRVRVeRVRV−−==−==. Plot the associated adiabatic potentials and non-adiabatic coupling. Describe what you see. The adiabatic electronic states diagonalize the potential matrix: =2221120VVVV and the adiabatic energies are the eigenvalues of the adiabatic states. A little work with Mathematica gives the eigenvalues 2124222222VVEV+±=± and eigenvectors ( ∆≡222V, Γ≡+2124222VV): ( )Γ±∆−+Γ∆=±1221221VV∓c . Thus, the non-adiabatic coupling between the two states is: ( ) ( )( )( )( )( )412222121212122122212211VVVVVVVdRdT+Γ−∆′+Γ′+∆′Γ−∆=′Γ′−∆′−Γ+∆−+Γ−∆+Γ+∆=⋅−+ccIf we plot the eigenenergies (along with the diabatic states), we find: Thus, there are two crossings in the diabatic picture (around R=±1.5) which turn into avoided crossings in the adiabatic picture. Meanwhile, the diabatic coupling looks like: Clearly, the diabatic coupling is large near the two avoided crossings, indicating that the adiabatic states change rapidly around these points. E+ E- E2 E12. A Jahn-Teller distortion occurs when a molecule that would have a degenerate electronic configuration if the nuclei were arranged symmetrically instead distorts so that the electronic degeneracy is lifted and the energy is lowered. This problem concerns a model of Jahn-Teller distortion. Consider a system with two low-lying diabatic electronic states, 1 and 2 , and two vibrational modes, x and y. The Hamiltonian for this molecule is: ( )( ) ( )2,21,22),(11,1ˆ222221222121121122212221+∂∂−∂∂−++++∂∂−∂∂−=yxVyxyxVyxVyxVyxH where the electronic matrix elements are ()()()( ) ( ) ( )iyxkyxVyxVyxyxVyxV+==+==*21122222211,,,,ω. In this model, the origin (0==yx) corresponds to the symmetric configuration and the x and y modes correspond to vibrational motions that distort the molecule away from the symmetric geometry. Note that the first and second diabatic states are degenerate, as advertised. a) Determine the adiabatic electronic energies of this potential. Express the result in terms of polar coordinates θρcos→x and θρsin→y. Does this model properly describe a distorted molecule? If so what is the magnitude of the distortion (in terms of ω and k)? What is the energy lowering due to the distortion? What happens to the two surfaces at the symmetric geometry (0=ρ)? The adiabatic electronic states diagonalize the potential matrix: =22211211VVVVVand the adiabatic energies are the eigenvalues of the adiabatic states. A little work with Mathematica gives the eigenvalues ()ωωωωρρρ222222kkkV −±=±=± Hence, the effect of the coupling is to take the two degenerate diabatic states and split them into two shifted harmonic oscillators. Pictorially, the two surfaces look like: Note that the surfaces are rotationally symmetric (the potential just depends on ρρρρ). Since ρρρρ=0 is the symmetric geometry, the shifted oscillators do, indeed describe a distorted molecule, at least for the lower surface (which is what the Jahn-Teller theorem applies to). The magnitude of the distortion is the distance the oscillator minimum is from the symmetric geometry (k/ωωωω) and the energetic stabilization is the energy at the minimum of the lower surface (-k2/2ωωωω). At the symmetric geometry, the picture shows that there is a conical intersection, and the two surfaces cross (i.e. they are degenerate). b) Determine the adiabatic electronic states. Use these states to compute the non-adiabatic coupling. Make sure to choose the relative phase of the two states so that the diagonal part is zero: 0=∇=∇upperupperlowerlowerψψψψ. c) What happens to the nonadiabatic coupling at the symmetric geometry? Note that the physical state of the molecule is the same at the polar points ()θρ, and ()πθρ2,+ since this just corresponds to a 360 degreerotation. Show that the adiabatic wavefunctions at ()θρ, and ()πθρ2,+ are not the same. Could this have any experimental consequences? If you ask Mathematica to give you the eigenvectors of the potential matrix (and are careful to make sure they are normalized), you get the two states: −==−+121121θθψψiiee Unfortunately, an equivalent set of eigenvectors is obtained if you multiply the above vectors by 2/θie−: −==−−−+2/2/2/2/2121θθθθψψiiiieeee Both expressions are correct. However, the second set of states is the traditional choice because it makes the diagonal elements of the non-adiabatic coupling zero: ( ) ( )( )002/2/2/2/2/2/2/2/2/2/2/2/=−=−∂∂−∝∇=−=−=∂∂∝∇−−−−−−−−++iieeeeiiieieeeeeeeiiiiiiiiiiiiθθθθθθθθθθθθθψψθψψThe first states are single-valued for all θθθθ while the second set acquires a “-“ sign on going from θθθθ to θ+2πθ+2πθ+2πθ+2π. This minus sign problem occurs anytime there is a conical intersection. In order to have a single valued wavefunction overall, the extra phase in the adiabatic electronic wavefunction is exactly cancelled by an extra phase (of the opposite sign) in the nuclear part – the so-called “Berry’s phase”. The extra phase required is different on the two different surfaces, and so for non-adiabatic processes the phase shows up as an experimentally observable interference pattern. To calculate the non-adiabatic coupling, we need+−∇ψψ. This can be computed in one of two ways; we can either re-write the eigenvectors in terms of x and y and then computethe gradient in those coordinates, or we can use the expression for ∇ in polar coordinates (which can be found in the back of CTDL): θρρθρ∂∂+∂∂=∇1ee . Taking the second route and remembering to take the conjugate transpose of the bra vector, ( ) (