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Chapter 18 Electronic Structure of the Hydrogen Atom

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Chapter 18 Electronic Structure of the Hydrogen Atom Chapter 18 Electronic Structure of the Hydrogen Atom P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Electronic Structure of the Hydrogen Atom Hydrogen atom is simplest atomic system where Schrödinger equation can be solved analytically and compared to experimental measurements. Analytical solution serve as basis for obtaining approximate solutions for multi-electron atoms and molecules, where no analytical solution exists. Warning: Working through analytical solution of H-atom may cause drowsiness. Do not operate heavy machinery during this derivation. P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Two Particle Problem ⃗ ⃗ 2 particles with mass mp and me at rp and re. z Total mass, center of mass, and inter-particle distance vector is ⃗ ⃗ mprp + mere M = m + m ; R⃗ = ; and ⃗r = ⃗r * ⃗r p e M e p y ⃗ ⃗ ⃗ ⃗ Express rp and re in terms of M, R and r as m mp ⃗r = R⃗ * e ⃗r and ⃗r = R⃗ + ⃗r x p M e M Express individual momenta of 2 particles as ⃗ drp d⃗r p⃗ = m and p⃗ = m e p p dt e e dt then in terms of M, R⃗ and ⃗r as 0 1 0 1 ⃗ ⃗ dR m d⃗r dR mp d⃗r p⃗ = m * e and p⃗ = m + p p dt M dt e e dt M dt P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Two Particle Problem Next, consider total energy ⃗ 2 2 pp p⃗ E = + e + V.⃗r/ 2mp 2me V.⃗r/ depends only on distance between 2 particles. Write energy in terms of M, R⃗ and ⃗r, and obtain 0 1 0 1 2 2 1 dR⃗ 1 d⃗r E = M + 휇 + V.⃗r/ 2 dt 2 dt 휇 is reduced mass, given by 1 1 1 휇 = + mp me Define 2 new momenta associated with center of mass and reduced mass, dR⃗ d⃗r p⃗ = M ; and p⃗ = 휇 R dt r dt P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Two Particle Problem Total energy becomes p⃗ 2 p⃗ 2 E = R + r + V.⃗r/ 2M 2휇 1st term is translational energy of center of mass 2nd term is kinetic energy due to relative motion of 2 particles Translating to quantum mechanics, we write time independent Schrödinger equation for 2 particle system as L M 4 5 p⃗ 2 p⃗ 2 `2 `2 R + r + V.⃗r/ f .R⃗; ⃗r/ = * (⃗ 2 * (⃗ 2 + V.⃗r/ f .R⃗; ⃗r/ = Ef .R⃗; ⃗r/ 2M 2휇 2M R 2휇 r P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Two Particle Problem 4 5 `2 `2 * (⃗ 2 * (⃗ 2 + V.⃗r/ f .R⃗; ⃗r/ = Ef .R⃗; ⃗r/ 2M R 2휇 r Wave function can be separated into product of two wave functions f .R⃗; ⃗r/ = 휒.R⃗/ .⃗r/ 휒.R⃗/ depending only on center of mass .⃗r/ depending only on relative motion of 2 particles Substitute product into Schrödinger Eq above, we obtain 2 wave equations 4 5 `2 `2 * (⃗ 2 휒.R⃗/ = E 휒.R⃗/ and * (⃗ 2 + V.⃗r/ .⃗r/ = E .⃗r/ 2M R R 2휇 r r where E = ER + Er On left is wave equation for translational motion of free particle of mass M On right is wave equation for particle with mass 휇 in potential V.⃗r/ For electron bound positively charged nucleus, we focus on PDE for .⃗r/ P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Schrödinger Equation in Spherical Coordinates P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Schrödinger Equation in Spherical Coordinates Focus on PDE for .⃗r/: Electron bound to positively charged nucleus 4 5 `2 * (2 + V̂ .r/ .⃗r/ = E .⃗r/ 2휇 Coulomb potential is 2 ̂ Zqe V.r/ = * 휋휖 4 0r Central potential with 1_r dependence. V → −∞ as r → 0 V → 0 as r → ∞ P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Schrödinger Equation in Spherical Coordinates Since V̂ .r/ only depends on r, we adopt spherical coordinates 4 0 ( ) 15 `2 )2 2 ) 1 1 ) ) 1 )2 * + + sin 휃 + + V̂ .r/ = E 2휇 )r2 r )r r2 sin 휃 )휃 )휃 sin2 휃 )휙2 «­­­­­­­­­­­­­­­­­­­­­­­­­­¯­­­­­­­­­­­­­­­­­­­­­­­­­­¬ *L̂ 2_`2 «­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­¯­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­¬ (2 Take term in parentheses as *L̂ 2_`2, rearrange, and simplify to )2 .r; 휃; 휙/ ) .r; 휃; 휙/ 2휇r2 ( ) 1 r2 + 2r + E * V̂ .r/ .r; 휃; 휙/ = L̂ 2 .r; 휃; 휙/ )r2 )r `2 `2 To solve PDE use separation of variables .r; 휃; 휙/ = R.r/Y.휃; 휙/ P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Schrödinger Equation in Spherical Coordinates Substitute .r; 휃; 휙/ into PDE, and dividing both sides by .r; 휃; 휙/ r2 )2R.r/ 2r )R.r/ 2휇r2 ( ) 1 1 + + E * V̂ .r/ = L̂ 2Y.휃; 휙/ R.r/ )r2 R.r/ )r `2 Y.휃; 휙/ `2 Left side depends only on r Right side depends only on 휃 and 휙. We have turned one PDE into two ODEs. ̂ 1 We know eigenfunctions and eigenvalues of L2 on right side, ̂ 2 휃; 휙 l l `2 휃; 휙 L Yl;m. / = . + 1/ Yl;m. / 2 Define l.l + 1/ as separation constant and obtain ODE for radial part r2 d2R.r/ 2r dR.r/ 2휇r2 ( ) + + E * V̂ .r/ = l.l + 1/ R.r/ dr2 R.r/ dr `2 P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Schrödinger Equation in Spherical Coordinates Expanding and rearranging ODE for radial part, we obtain 4 0 1 5 `2 d2 2 d `2 l.l + 1/ * + + + V̂ .r/ R.r/ = ER.r/ 2휇 )r2 r dr 2휇 r2 «­­­­­­­­­­­¯­­­­­­­­­­­¬ Veff.r/ Looks like 1D time independent Schrödinger Eq with effective potential `2 l.l + 1/ Zq2 V̂ .r/ = * e eff 2휇 r2 4휋휖 r «­­­­¯­­­­¬ 0 Centrifugal Term ù Angular momentum, l.l + 1/`, creates centrifugal force that pushes electron away from nucleus. ̂ Notice 2 terms in Veff.r/ always have opposite signs. P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom l Plot of Veff for = 0, 1, 2, and 3. 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 0 2 4 6 8 10 12 14 Note: potential minimum shifts to higher radii with increasing angular momentum (i.e., centrifugal force). P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom ̂ Veff.r/: Effective Potential of H-Like Atom Switching to Atomic Units: Convenient units for atomic physics Atomic unit of length, also called Bohr radius, is defined as 4휋휖 `2 a ≡ 0 = 52:9177210526763 pm 0 2 qeme Atomic unit of energy is `2 q2 1E ≡ = e = 27:21138602818051 eV h 2 4휋휖 a mea0 0 0 P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Wave Equation for the Radial Part L 0 1 M 2 `2 d2 2 d `2 l.l + 1/ Zq * + + * e R.r/ = ER.r/ 휇 2 휇 2 휋휖 2 )r r dr 2 r 4 0r pour yourself a fifth cup of coffee... P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Radial Wave Equation To obtain radial wave function, R.r/, we must solve L 0 1 M 2 `2 d2 2 d `2 l.l + 1/ Zq * + + * e R.r/ = ER.r/ 휇 2 휇 2 휋휖 2 )r r dr 2 r 4 0r Begin by dividing and both sides by E, and obtain 4 5 H I 2 `2 d2 2 d `2 l.l + 1/ Zq * + R.r/ + * e * 1 R.r/ = 0 휇 2 휇 2 휋휖 2 E )r r dr 2 E r 4 0rE Define 2휇E `2휅2 휅2 ≡ * or E = * `2 2휇 휅 is in wave numbers. Rearrange potential energy expression to 2 2 2 Zq Zq 2휇 Zq 휇 1 2휌 * e = e = e = 0 휋휖 휋휖 `2휅2 휋휖 `2휅 휅 휅 4 0rE 4 0r 2 0 r r Zq2휇 휌 is a dimensionless quantity: 휌 ≡ e 0 0 휋휖 `2휅 2 0 P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Radial Wave Equation Substitute expression for potential energy into ODE 4 5 0 1 1 d2 2 d l.l + 1/ 2휌 + R.r/* 1 + + 0 R.r/ = 0 휅2 )r2 r dr .휅r/2 휅r Define dimensionless variable: 휌 ≡ 2휅r Rewrite radial part as function of 휌 as 4 5 0 1 d2 2 d l.l + 1/ 2휌 + R.휌/* 1 + + 0 R.휌/ = 0 d휌2 휌 d휌 휌2 휌 Further simplify ODE by defining du.휌/ dR.휌/ d2u.휌/ d2R.휌/ dR.휌/ u.휌/ = 휌R.휌/; = 휌 + R.휌/; = 휌 + 2 d휌 d휌 d휌2 d휌2 d휌 Re-express ODE as 0 1 d2u.휌/ 1 휌 l.l + 1/ * * 0 + u.휌/ = 0 d휌2 4 2휌 휌2 휌 varies from 0 to ∞. 1st look at asymptotic solutions: (1) 휌 → ∞ and (2) 휌 → 0 P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Radial Wave Equation, 휌 → ∞ Starting with 0 1 d2u.휌/ 1 휌 l.l + 1/ * * 0 + u.휌/ = 0 d휌2 4 2휌 휌2 (1) 휌 → ∞. At large values of 휌 approximate ODE as d2u.휌/ u.휌/ * ù 0 d휌2 4 Solutions to this ODE have form u.휌/ ∼ Ae*휌_2 + Be휌_2 Reject positive exponent since require that solution be finite everywhere, u.휌/ ∼ Ae*휌_2 in limit that 휌 → ∞ P.
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