Question

In the textbook, Theories of Molecular Reaction Dyanmics by Flemming Y. Hansen and Niels E. Henriksen, there is a derivation of the equation of motion for the adiabatic approximation (pgs. 5-7). The author starts by showing that the wavefunction can be written as a product state by separating the slow and fast components (i.e. nuclear and electronic) as shown below, Ψ (r,R,t) = χ (R,t) ψ (r;R) where χ (R,t) is the nuclear component and ψ (r;R) is the electronic component which depends parametrically on the nuclear position(R).

We take the following time-dependent Schrodinger equation, written as
iℏ ∂Ψ(r,R,t) / ∂t = (T^_Nuc + H^ₑ) Ψ (r,R,t)
and plug in the product state to get the following equation
iℏ ∂χ(R,t) / ∂t = [T^_Nuc + Eᵢ (R) + ⟨ψ|TNuc^|ψ⟩0] χ (R,t).
It is said that this term is equal to zero, leaving us with the equation of motion of the adiabatic approximation
iℏ ∂χ(R,t) / ∂t=[T^_Nuc + Eᵢ (R)] χ (R,t)
Questions
1. How do we get the ⟨ψ|TNuc^|ψ⟩0 term during the derivation? I am confused on how that arises.
2. What does this term mean? Is it just the coupling between the nuclear and electronic states?

Answer 1

The term ⟨ψ|TNuc^|ψ⟩0 represents the coupling between the nuclear and electronic states and is equal to zero in the adiabatic approximation.

Step-by-step explanation:

The term <strong><em><u>⟨ψ|TNuc^|ψ⟩0</u></em></strong> arises during the derivation when the time-dependent Schrödinger equation is plugged in with the product state wavefunction Ψ (r,R,t) = χ (R,t) ψ (r;R).

The term represents the coupling between the nuclear and electronic states and is equal to zero in the adiabatic approximation.