1.) Try powering the amplifier from +/- 2.5V. It might be a rail to rail problem.
2.) Try decreasing the 1M resistor to 100K and the other one also. It could be an offset voltage due to bias and offset current.
It works finally!
After powering the op amp with +/- 2V5, it workes like a charm!
I used LM358 op amp to split the 5V voltage to +/- 2V5, but I noted something which was very strange!
After powering the AD8629 op amp, and before connecting anything to the non-inverted input, the voltage over -V and +V were -2V5 respectively +2V5, but after connecting to it, the voltage became 0V respectively +5V!!!
Is that normal?
Tommy