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The hydrogen atom Consists of two particles, a positively charged nucleus and a negatively charged electron Hamiltonian has three terms Physical Chemistry Lecture 16 The Hydrogen Atom, a CentralForce Problem Center of mass, RCM Relative position, r Total mass, M Reduced mass, H Tn Te pn2 2mn Vne pe2 2me 1 e2 4 0 | rne | Simplification of the hydrogenatom problem Center of mass Hamiltonian as a function of nuclear and electronic coordinates is complex Define Nuclear kinetic energy Electronic kinetic energy Nuclear-electron Coulombic potential energy By substitution, the hydrogen-atom Hamiltonian becomes H m1 r1 M r1 R CM r r2 M m1 p12 2m1 H CM CM 2 PCM 2M 1 2 p 2 1 e2 4 0 r H rel Decompose into two problems m2 p22 2m2 1 2 PCM 2M H CM m2 r2 M m1m2 m1 m2 H rel rel CM rel p2 2 Center-of-mass problem A particle of mass, M, moving in no potential Like the particle-in-a-box problem Energies and wave functions known from that model Translational degrees of freedom of the hydrogen atom ECM CM Erel rel E ECM Erel The relative problem Consider relative motion of nucleus and electron, with only Coulombic interaction The dependence of 2 on angular variables simplifies the problem Eigenfunctions of L2 are well known H E 2 2 2 1 e2 E 4 0 r 1 2 2 1 2 L r 2 r 2 r r 2 r 2 E 1 e2 4 0 r (r , , ) R (r )Ym ( , ) 1 Hydrogenic energies and eigenfunctions The radial equation Use of the spherical harmonic functions gives an equation for the radial part 1 2 R r 2 r 2 r r 2 1 2 2 R r2 2 1 e R ER 4 0 r This can be put into dimensionless form by defining the radial distance in bohrs r a0 a0 En 1 e2 1 4 0 2a0 n 2 gn n2 1 hartree 27.2114 electron volts 1 rydberg 13.606 electron volts Energy level in which the electron and the nucleus are just pulled apart States are labeled by n and 0.5 hartree States of different given by letters s ( = 0) p ( = 1) d ( = 2) 1 electron volt 96.4853 kilojoules / mole Addition of spin The spatial part is incomplete One incorporates spin as a separate coordinate Wave function is a product n ,l ,m , I ,mI (r , , ) Ln(x) 1 2-x 1 3 – 2x + 2x2/9 4 – 2x/3 1 Energies of the state depend only on n Joules Ergs Hartree Rydberg Electron volt 0 0 1 0 1 2 Defined relative to the “vacuum level” Often convenient to use “atomic units” n 1 2 2 3 3 3 Hydrogen-atom energy levels Can give energies in “macroscopic units” Laguerre polynomials A Ln ( ) exp / n Energies Anm r / a0 Ln ( r / a0 ) exp(r / na0 )Ym ( , ) nm (r , , ) 0h2 me e 2 The dimensionless equation is related to Laguerre’s differential equation Rn ( ) Products of spherical harmonic functions with functions related to Laguerre polynomials A Rnl (r / a0 )Ylm ( , ) spin , S ,ms Energy is unchanged by addition of spin variables Increases degeneracy Electron spin S=½ Two states m = ½ or m = - ½ or Eigenstates of the spin angular momentum operators S2 = ½(½+1)2 S2 = ½(½+1)2 Sz = (/2) Sz = - (/2) 2 Summary Central-force (hydrogen-atom) problem separates Center of mass movement (translation) Relative motion Angular part is constant-angular-momentum problem Radial part is related to Laguerre’s equation Energy depends on principal quantum number, n, as predicted by Rydberg Degeneracy States with same n but different angular momentum quantum numbers, and m, have same energy gn = n2 3