How to calculate the transition rate

1. Nastenka

1
1. The problem statement, all variables and given/known data
I have the usual equation for the transition amplitude:
A=< k|exp(-iHt)| j>,

while my Hamiltonian in Dirac notation looks like:

H=\sum E_a|a><a|+\sum_b (E_b|b><b|+V_1|0><b|+ V_2|b><0|)

In order to find the transition rate I should take a derivative as:
dA/dt, so that I will get something like:

A=< k|-iH*exp(-iHt)| j>

2. Relevant questions

Now, my question is:

how to treat it further?

3. The attempt at a solution

I know I should expand it with series for the exponent, but then I obtain in the middle this:
-iH(1+H)=-iH-iH2 that confuses me. I feel I am stucked in such an easy task. But the proble is that I cannot expand it with eigenvalues and eigenvectors which would simplify a lot my task....