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” one- electron atoms. Removing normalizing constants from the formulas for familiar orbitals reveals the underlying simplicity of their shapes. Chemistry 125: Lecture 9 Sept 21, 2009 More Dimensions: Chladni Figures and One-Electron Atoms For copyright notice see final page of this file

Reward for Finding  Knowledge of Everything e.g. Allowed Energies Structure Dynamics Bonding Reactivity

Single- vs. Double Minimum For Hooke's Law the Blue Energy is too Low and the Red Energy is too High. The Correct Lowest Energy must lie between these values. Single-Mimimum Actually this is a Double- Minimum. The Blue and Red  s are correct! What if the wells were further apart? Closer wells give lowered minimum energy and higher next energy Both are ~ same as single-minimum solution “ Splitting 1.3Å

in Ain B Wells far apart Wells far apart Total Energy of Particle "Mixing" localized   s for double minimum Wells close together in AB Antibonding Holds A & B together Black line is energy Blue line is  Bonding! Stabilization of Particle

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.

1.4 kcal/mole splitting  ~4  sec to get from well to well. Well-to-Well time  5  sec Splitting (kcal/mole) Assertion from time-dependent q. mech. Dynamics: Tunneling

Reward for Finding  Knowledge of Everything e.g. Allowed Energies Structure Dynamics Bonding Reactivity (coming later)

Morse Quantization "Erwin" can find  s for any complicated V(x) 7 Å and rank them by energy / "curvature" / # of nodes Don’t cross 0 in “forbidden” continuum. Don’t slope out and away from  = 0 in “forbidden” continuum. What’s wrong with this picture?

“Erwin” even handles Multiple Minima

“Erwin’s curve-tracing recipe won't work in more dimensions (e.g. 3N). 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. When there are many curvatures, it is not clear how to partition the kinetic energy among the different (d 2  / dx i 2 ) /  contributions.

E. F. F. Chladni ( ) Acoustics (1803) e.g. Chladni Figures in 2 Dimensions

Sand Collects in Nodes Touch in Different Places Bow in Different Places

dry ice Click for Short Chladni Movie (3MB) Click for Longer Chladni Movie (9.5MB) CO 2 brass plate

Crude Chladni Figures 3 Diameters / 1 Circle3 Circles 1 Diameter / 2 Circles 4 Diameters / 1 Circle from in-class demo

Chladni’s Nodal Figures for a Thin Disk Portion inside outer circular node Cf. (1,2)

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:" 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 8 Lines 4 Circles 2 Circles 4 Lines 3 Circles 2 Lines 1 Circle 6 Lines Number of Circles Number of Diameters 1 Circle 2 Lines

e.g. Daniel Bernoulli ( ) Great Mathematicians Worked on Chladni’s 2-D Problems:  2 /  x 2 +  2 /  x 2 +  2 /  x 2 ( abbreviated  2 ) the Laplacian Operator

 s for one-electron atoms involve Laplace’s “Spherical Harmonics” (3D-Analogues of Chladni Figures)

3-Dimensional H-Atom Wavefunctions   ( , ,  ) = R(r)   (   )   (   ) Adrien-Marie Legendre ( )  (  ) is the normalized “Associated Legendre Polynomial” Edmond Laguerre ( ) R(r) is the normalized “Associated Laguerre Function” with contributions from other old-time mathematicians

 Table for H-like Atoms V( x,y,z ) = sqrt(x 2 + y 2 + z 2 ) 1 simplifies V( r, ,  ) = r c Name  by quantum numbers (n > l ≥ m) or by nickname (1s, etc.)  = R nl (r)   lm (  )   m (  ) product of simple functions of only one variable each and  (x,y,z) is very complicated change coordinate system: x,y,z  r  x y z n e r  

 Table for H-like Atoms  = R(r)   (  )   (  ) 1s  r 2Z2Z na o Why  instead of r? Allows writing the same e  2 for any nuclear charge (Z) and any n. = K e -  /2 N.B. No surprise for Coulombic Potential x y z n e r   Note: all contain (Z / a o ) 3/2 Squaring gives a number, Z 3 per unit volume (units of probability density)

  r 2Z2Z na o exp -  r =  2Z2Z na o r 1H =  Å r 1C =  Å All-Purpose Curve shrunk by Z; expanded by n Å (1s H ) (0.26  Å ) Increasing nuclear charge sucks standard 1s function toward the nucleus 0.1 Å (1s C ) 0.2 (renormalization keeps total probability constant)  1/6  216 (0.044  Å) Å (1s C ) (0.044  Å) Different Å scales Common Å scale (1s H ~2 e/Å 3 )

H 1s C 1s Relative Electron Density Increasing nuclear charge sucks standard 1s function toward the nucleus (renormalization keeps total probability constant) Common Å scale Summary  r 2Z2Z na o What would the exponential part of……. look like? C2sC2s +5 Wrong scaling factor used. C should go not to 6, but to 216!

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 1s H with 2s C +5 in Energy 4) Do the 6 atomic orbital problems Click Here Click Here

2 2 2  Table for H-like Atoms 1s = K e -  /2 2s = K'(2-  ) e -  /2 Shape of H-like  = K'''(  cos(  )) e -  /2 2p z z Guess what 2p x and 2p y look like. Simpler (!) than Erwin 1-D Coulombic x y z n e r   r cos(  ) = z

End of Lecture 9 Sept 21, 2009 (see Lecture 10 for description of Atom-in-a-Box) Copyright © J. M. McBride 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).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