[R-sig-ME] effect of adding a constant to predictor

[Don Cohen]

I expected that adding a constant to an input in a mixed
model would affect some of the coefficients, but not such basic
stuff as AIC (or loglik, from which it is derived).
But I seem to have been wrong about that:

> AIC(lmer(output ~ input1 + I(input1 * input2) + input2 + ((1 | group) + (0 + input2 | group)) , data=xxx, REML=F, control=contr))
[1] 1556.341
> xxx$input2 = xxx$input2 + 1
> AIC(lmer(output ~ input1 + I(input1 * input2) + input2 + ((1 | group) + (0 + input2 | group)) , data=xxx, REML=F, control=contr))
[1] 1551.005

Can someone explain why this change to input2 affects AIC (or loglik) ?

BTW, changing (0 + input2 | group) to (0 + input1 | group) 
produces the same AIC before and after the change to input2.