# Vertiefungsfach Theoretische Chemie / Computerchemie Praktikum ...

Vertiefungsfach Theoretische Chemie / Computerchemie Praktikum ...

FSU Jena, Institut für Physikalische Chemie, VF 7, L. González, D. Bender, SS2010 1Vertiefungsfach Theoretische Chemie / ComputerchemiePraktikum im Sommersemester 2010Versuch 2T:2010-04-15SCF calculation ”by hand“ Do an SCF calculation for the helium-atom ground stateusing a basis set of two 1s STOs with orbital exponents ζ 1 = 1.45 and ζ 2 = 2.91.1. χ 1 = 2ζ 3/21 e −ζ 1r Y 00 , 2. χ 2 = 2ζ 3/22 e −ζ 2r Y 00 ,with ζ 1 = 1.45, ζ 2 = 2.91Procedure:Use Mathematica to solve the Roothaan-Hall equationsevaluating the integralsb∑c ni (F mn − ε i S mn ) = 0, m = 1, 2, . . . b (1)n=1F mn ≡〈χ m | ˆF〉|χ nThe integrals F mn are given by the following expressionF mn = H mn +F mn = H mn +b∑(2)S mn ≡ 〈χ m |χ n 〉 . (3)n/2 b∑ ∑c ∗ tic uj [2 (mn|tu) − (mu|tn)] (4)t=1 u=1 j=1b∑b∑t=1 u=1P tu[(rs|tu) − 1 2 (ru|ts) ]n/2∑P tu ≡ 2 c ∗ tjc uj , t = 1, 2, . . . b, u = 1, 2, . . . b (6)H mn ≡j=1〈〉χ m (1)|ĥ0(1)|χ n (1)ĥ 0 (1) ≡ − 1 2 ∇2 1 − Z r 1(8)Many of the electron repulsion integrals (mn|tu) are equal to one another. Which ones?To start the calculation, an initial guess for the ground-state AO expansion coefficients c niin equation (1) is needed. The normalization condition∫|φ 1 | 2 dτ = 1 (9)(5)(7)

FSU Jena, Institut für Physikalische Chemie, VF 7, L. González, D. Bender, SS2010 2gives for real coefficientsc ni = (1 + k 2 + 2kS mn ) −1/2 , (10)where k ≡ c in /c ni .Start with a ratio of k = 2.How does the He ground state SCF AO for this basis set looks like?What is the energy you obtain?The Hartree-Fock limit (found with five basis functions) is -2.8616799 hartrees (C. Roettiand E. Clementi, J. Chem. Phys. 60, 4725 (1974)). How does the energy found here compareto the Hartee-Fock limit?How does the convergence rate depend on the initial ratio k of the coefficients? Redo thecalculations, setting k = 100, 10, 1, −1, −10.Calculate the density ρ for the He SCF wave function at r = 0 and r = 1 bohr.Store the program in your home directory, we might need it again ;-)

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