Kinetic energy finite difference representation for electronic structure calculations. Domenico Ninno and Giovanni Cantele CNR-INFM and Universit` a di Napoli Federico II, Dip. di Scienze Fisiche, Via Cintia, I-80126 Napoli, Italy. The finite difference method in a electronic structure calculation[1] is based on the following representation of the kinetic energy ˆ Tψ ≈−  2 2µ M ∑ m=-M C (M) m ψ(x n+m ) (1) where M is the representation order controlling the accuracy, x n = na,n = 0, 1 ...N is a uniform grid (the extension to 3D is straightforward) and the coefficients C (M) m can be calculated by means of specific algorithms[2]. With this contribution we address two basic points: (i) what is the basis set be- hind Eq(1)? and (ii) why is it that in an electronic structure calculation the energy levels converge from below? Using as unique starting point the canonical commutation [  x,  p]= i  I we show that the basis is made of discrete coordinate eigenkets |x n ⟩. Using this basis set we obtain a new expression for the second derivative weights C (M) m = 1 a 2    1 m 2 Ω(M,m) if m ̸=0 − M ∑ l=1 2 l 2 Ω(M,l) if m =0 (2) Ω(M,m)= M ∏ l=1 f (l,m), f (l,m)= { 1 if l = m 1 − ( m l ) 2 if l ̸= m (3) The simple structure of Eqs (2) and (3) allows us to show, using periodic boundary conditions, that for very small grid spacing (N →∞) the kinetic energy eigenvalues can be written as ϵ =  2 2µ ( 2πν L ) 2 [ 1 − ( 2πν N ) 2M Δ+ ··· ] ,Δ= 2    ∑ M m=1 m 2M Ω(M,m)    (2M + 2)! (4) 1