Metropolis-Hastings algorithm for harmonic oscillator

Question

I am exploring the use of the Metropolis-Hastings algorithm and I use a simple harmonic oscillator (with $x_0=1$) as model system. The usual Metropolis algorithm with the move $x'=x+\epsilon$, where $\epsilon\in[-\Delta,\Delta]$ is a random number is trivial: we only need to compute the Boltzmann factor of the energy difference $E(x')-E(x)$. A simple simulation gives $<x>=1$ as expected. However I have problems to obtain the correct result with the Metropolis-Hastings algorithm with the following move: $x'=x+\epsilon$ where with probability 1/2 we have $\epsilon\in[0,\Delta]$ while with probability 1/2 we have $\epsilon\in[-\Delta/2,0]$. How can I derive the correct Metropolis-Hastings algorithm for this move?