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Chemistry 125: Lecture 9 Sept 21, 2009 More Dimensions: Chladni Figures and One-Electron Atoms Double-minimum potentials generate one-dimensional bonding, A different technique is needed to address multi-dimensional problems. Solving Schroedinger’s three-dimensional differential equation might have been daunting, but it was not, because the necessary formulas had been worked out more than a century earlier in connection with acoustics. Acoustical “Chladni” figures show how nodal patterns relate to frequencies. The analogy is pursued by studying the form of wave functions for “hydrogen-like” oneelectron atoms. Removing normalizing constants from the formulas for familiar orbitals reveals the underlying simplicity of their shapes. For copyright notice see final page of this file Reward for Finding Y Knowledge of Everything e.g. Allowed Energies Structure Dynamics Bonding Reactivity @ 1.3Å @ 0.6Å For Hooke's Law Actually Singlevs. Double Minimum the Blue Energy this is a is too Low DoubleBoth are and Minimum. ~ same as single-minimum the Red Energy solution is too High. The Blue and Red Ys are correct! Single-Mimimum “Splitting” The Correct Lowest What if wells give Closer Energy must lie thelowered wells were minimum energy between values. further apart?these and higher next energy Black line is energy Blue line is Y Antibonding in A Stabilization of Particle Total Energy of Particle "Mixing" localized Y s for double minimum in B Bonding! Wells far apart in AB Wells far apart Holds Wells close A & B together together Dynamics: Tunneling Dynamics: Tunneling The word "Tunneling" is one of my pet peeves: It is misleading and mischievous because it suggests that there is something weird about the potential energy in a double minimum. . In fact it simply involves the same negative kinetic energy that one sees in the tails of EVERY bounded wavefunction. The word reveals naiveté about quantum mechanics. Dynamics: Tunneling 510-14 sec Well-to-Well time Assertion Splitting (kcal/mole) from time-dependent q. mech. 1.4 kcal/mole ~410-14 sec to get from well to well. splitting Reward for Finding Y Knowledge of Everything e.g. Allowed Energies Structure Dynamics Bonding Reactivity (coming later) "Erwin" Don’t slope out and away from can Y find Y s for any complicated = 0 in “forbidden” continuum. V(x) Morse Quantization and rank them by energy / "curvature" / # of nodes What’s wrong with this picture? Don’t cross 0 in “forbidden” continuum. 7Å “Erwin” even handles Multiple Minima “Erwin’s curve-tracing recipe won't work in more dimensions (e.g. 3N). When there are many curvatures, it is not clear how to partition the kinetic energy among the different (d2Y / dxi2) / Y contributions. But Schrödinger had no trouble finding solutions for the 3-dimensional H atom, because they were familiar from a long tradition of physicists studying waves. Acoustics (1803) e.g. Chladni Figures E. F. F. Chladni in 2 Dimensions (1756-1827) Sand Collects in Nodes Touch in Different Places Bow in Different Places dry ice brass plate CO2 Click for Short Chladni Movie (3MB) Click for Longer Chladni Movie (9.5MB) 3 Circles 3 Diameters / 1 Circle Crude Chladni Figures 1 Diameter / 2 Circles 4 Diameters / 1 Circle from in-class demo Chladni’s Nodal Figures for a Thin Disk (1,2) Portion inside outer circular node Cf. http://www.kettering.edu/~drussell/Demos/MembraneCircle/Circle.html Chladni’s Nodal Figures for a Thin Disk Number of Diametrical Nodes Number of Circular Nodes PITCH 47 Patterns! "These pitch relationships agree approximately with the squares of the following numbers:" 1 Circle Number of Circles 4 Circles 2 Lines Number of Diameters 3 Circles 2 Lines 2 Circles 4 Lines 1 Circle 6 Lines 8 Lines Frequency ≈ (Diametrical Nodes + 2 Circular Nodes) 2 Note: Increasing number of ways to get a higher frequency by mixing different numbers of circles and lines Great Mathematicians Worked on Chladni’s 2-D Problems: e.g. Daniel Bernoulli (1700-1782) 2/x2 + 2/x2 + 2/x2 (abbreviated 2) the Laplacian Operator Ys for one-electron atoms involve Laplace’s “Spherical Harmonics” (3D-Analogues of Chladni Figures) 3-Dimensional H-Atom Wavefunctions Y (,,) = R(r) ( ) ( ) R(r) is the normalized “Associated Laguerre Function” () is the normalized “Associated Legendre Polynomial” with contributions from other old-time mathematicians Edmond Laguerre Adrien-Marie Legendre (1834-1886) (1752 -1833) Y Table for H-like Atoms 1 V(x,y,z) = sqrt(x2 + y2 + z2) Y(x,y,z) is very complicated z change coordinate system: r e x,y,z r n x y c simplifies V(r,,) = r and Y = Rnl(r) lm() m() product of simple functions of only one variable each Name Y by quantum numbers (n > l ≥ m) or by nickname (1s, etc.) Y Table for H-like Atoms Y = R(r) () () z r n )3/2 Note: all contain (Z / ao Squaring gives a number, N.B. Nounit surprise for Z 3 per volume Coulombic Potential (units of probability density) x e y -/2 = K e 1s Why instead of r? 2Z r nao Allows writing the same e-/2 for any nuclear charge (Z) and any n. (1sH ~2 e/Å3) All-Purpose Curve - exp shrunk by Z; expanded by n Increasing nuclear charge sucks standard 1s function toward the nucleus 1/6 (renormalization keeps total probability constant) nao r= 2Z 0.53Å r1H = 2 (0.26 Å) 0.53Å r1C = 12 (0.044 Å) 216 0.1(1s0.2 Å (1sC) Å C) Different Å scale scales Common 2Z r nao 0.5 0.1 1.0 0.2 Å (1sH) Relative Electron Density Wrong scaling factor used. C should go not to 6, but to 216! Summary Increasing nuclear charge sucks standard 1s function toward the nucleus +5 C1s (renormalization keeps total probability constant) 2Z r nao What would the exponential part +5 of……. C2s look like? H1s 0.1 0.2 Common Å scale 0.5 1.0 For Wednesday: 1) Why are there no Chladni Figures with an odd number of radial nodes? (e.g. 3 or 5 radii) 2) Why are the first two cells [(0,0) and (1,0)] in Chladni's table vacant? 3) Compare 1sH with 2sC+5 in Energy 4) Do the 6 atomic orbital problems Click Here Y Table for H-like Atoms Shape of H-like Y z r r cos() = z n x e y -/2 = K e 1s 2 Simpler (!) than Erwin 1-D Coulombic 2s = K'(2-) e-/2 2 2 -/2 2pz = K'''( cos( z )) e Guess what 2px and 2py look like. End of Lecture 9 Sept 21, 2009 (see Lecture 10 for description of Atom-in-a-Box) Copyright © J. M. McBride 2009. Some rights reserved. Except for cited third-party materials, and those used by visiting speakers, all content is licensed under a Creative Commons License (Attribution-NonCommercial-ShareAlike 3.0). Use of this content constitutes your acceptance of the noted license and the terms and conditions of use. Materials from Wikimedia Commons are denoted by the symbol . Third party materials may be subject to additional intellectual property notices, information, or restrictions. The following attribution may be used when reusing material that is not identified as third-party content: J. M. McBride, Chem 125. License: Creative Commons BY-NC-SA 3.0