Spline representation. I. Linear spline bases for atomic calculations

Abstract

The use of a basis of cardinal splines for atomic calculations with the expansion method is proposed. The basis has the property that the coefficients C_p in the orbital expansion ${\mathrm{\phi \; (r)\; =\; \Sigma }}_{\text{p=−1}}^{\text{N+1}}{\mathrm{\hspace{0.17em}C}}_p{\chi }_p\mathrm{(r)}$ are given by C₋₁=φ′(r₀), C_p=φ(r_p), p=0,1, ···, N, and C_N+1=φ′(r_N), where r₀, r₁, ···,r_N is the mesh of knots on which the spline is defined. The basis functions χ_p(r) are continuous with continuous first and second derivatives, and may readily be calculated from the expansion ${\chi }_p{\mathrm{(r)\; =\; \Sigma }}_{\text{k=1}}^N{\mathrm{\hspace{0.17em}\Sigma }}_{\text{s=0}}^3{\mathrm{\hspace{0.17em}r}}^s{d}_k{\mathrm{(r)S}}_{\mathrm{ksp}}$ , where d_k(r)=1 if r_k−1 ≤ r<r_k and 0 otherwise and S_ksp is a matrix of coefficients which are uniquely determined by the mesh alone and may be constructed by simple algebraic operations, the most complex being the inversion of a single (N+1)×(N+1) matrix. Atomic Hartree‐Fock matrix elements are given by simple polynomial expressions. The number of two‐electron integrals increases only as the square of the number of basis functions and can be stored in factored form, so that the storage requirements for the two‐electron integrals are trivial. The largest two arrays (equal in size) are the density matrix and the matrix elements of the exchange operator. The results of numerical tests for the hydrogen atom using several different distributions of mesh points are presented. It is found that the number of mesh points required is smaller than for standard numerical methods by a factor of about 4. The accuracy of a linear spline representation with N + 1 knots is comparable to that of a Gaussian expansion with N basis functions.