The magnitude of the curvature of a wave function relates to the kinetic energy of the system, and the square of the wave function relates to probability density. The requirement that the wavefunction not diverge in areas of negative kinetic energy can constrain total energies to certain values, a property which is explored for the harmonic oscillator, the Morse potential, and the Columbic potential. Consideration of the influence of mass reveals an “isotope effect” on dynamics and on the energy, vibration frequency, and length of bonds. Introducing the double minimum potential leads to the study of bonding. Chemistry 125: Lecture 8 Sept 17, 2010 One-Dimensional Wave Functions For copyright notice see final page of this file

Rearranging Schrödinger to give a formula for curve tracing. C  Curvature of  m + V = E C  Curvature of  m (V - E) = Curves away from 0 for V>E; toward 0 for V<E. Since m, C, V(x) are given, this curvature recipe allows stepwise tracing of  (x), from initial  (0) [= 1], with initial slope [0], and a guessed E. C m V 1 etc.

100 kcal/mole 2.5Å 0 Too Cold Too Hot Just Right! kcal/mole Guess 21 kcal/mole Guess 20 kcal/mole Danger Negative Kinetic Energy (Curve Away from Baseline) Danger Negative Kinetic Energy (Curve Away from Baseline) Infinitesimal  denominator allows large negative KE without significant curvature. Thus initial slope  0

100 kcal/mole 2.5Å 0 Could there be a lower-energy  ? Could there be an energy between? kcal/mole 4.15 kcal/mole kcal/mole NODES 0 because of sign change More Energy  More Curvature  More Nodes

Finding Solutions is Much Harder with Many Particles. Is it worth our effort?

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

Harmonic Spacing Even Energy Spacing for Hooke’s Law E = k (n + ) 1 2

“We only wish that we could glean an inkling of what  could mean.”

Structure:  2  Probability Density Max Born (June 25, 1926) If one wishes to translate this result into physical terms, only one interpretation is possible,  signifies the probability [of the structure] 1 ) Correction in proof: more careful consideration shows that the probability is proportional to the square of the size of . 1)1) Oops!

Structure:  2  Probability Density Aber eine innere Stimme sagt mir, dass das doch nicht der wahre Jakob ist. Die Theorie liefert viel, aber dem Geheimnis des Alten bringt sie uns kaum näher. Jedenfalls bin ich überzeugt, dass der nicht würfelt. But an inner voice tells me, that this is not the real thing. The theory yields a great deal, but it brings us no nearer to the secret of the Old One. Anyway I am convinced that He does not play dice. Albert Einstein to Max Born December 4, 1926

Probability Density Suppose the total mass in the flask is 1 kg. How much (or what fraction) is exactly 1 cm from the bottom? Multiply density by volume for mass (or fraction, or probability). 0 !

“Normalization” Scale  so that total (integral of)   2  volume = 1

Harmonic Probability Ultimately Probability Builds Up at the Extremes 1.5 Å (not normalized!) Probability Penetrates the Classically ‘Forbidden’ Region

Morse Quantization Morse Potential : Quantized; Probability Spreads to Right Large on right because low kinetic energy means low curvature 7 Å ~ Exponential Decay (e -x ) (~ constant negative kinetic energy) Total Potential Kinetic

Morse Quantization Morse Potential : Quantized; Probability Spreads to Right Energies not evenly spaced (not like Hooke’s Law) 7 Å As the energy increases, along with the number of nodes, the well widens more than it would for a Hooke’s Law parabola. Thus wavelengths become longer, and energies lower, than expected for Hooke’s Law.

Morse Continuum Morse Potential : Not Quantized above Dissociation Limit ~ sin(x) (~ constant positive kinetic energy) Total Potential Kinetic

Coulombic Spacing 50 Å One-dimensional e - in Coulombic Potential of a Proton ! High Curvature (Erwin Program is approximate)

Coulomb Three E = k n2n2 50 Å Higher levels spread way out

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

Change mass Single-bonded H Increase mass from 1 to 14 (H to C 14 ) C  Curvature of  m Greater mass means more curvature for the same energy.

Change mass Single-bonded H Increase mass from 1 to 14 (H to C 14 ) Need to lower energy (curvature) for m=14 C  Curvature of  m Greater mass means more curvature for the same energy.

Mass Effect What about U 235 or a marble? m = 1 m = 14

Mass Effect and Vibration C H Higher-energy H shifted to right in unsymmetrical Morse Potential. half-maximum probability density

Mass Effect and Vibration H C

H C H C ±0.05Å (3% of X-C) ±0.1Å (9% of X-H)

Dunitz et al. (1981) Typically vibrating by ±0.050 Å in the crystal

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 ~ same as single-minimum solution “Splitting”

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! lower Kinetic Energy! Stabilization of Particle reduced curvature increased curvature

End of Lecture 8 Sept 17, 2010 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