Local and semilocal representation of Laplace- $\Delta$

The finite differencing scheme (FD) considers the derivative only by a local approximation such that in one dimension

$$\frac{d^{2}\psi (x_{i})}{dx^{2}}=\frac{\psi (x_{i+1})-2\psi (x_{i})+\psi (x_{i-1})}{(\Delta x)^{2}}.$$

Higher order differencing schemes involving more interpolation points $x_{j}$ are called semilocal. All of these approximations, however, are not very well-suited for application with quantum-mechanical problems as the consideration of wavefunctions is intrinsically global.