VISUALIZINGEigenfunctions, Wave Functions and

Orbitals

Excel to understand what we are plotting3D (GeoWall) to explore the plots

Outline

• Background (the math, physics, and chemistry perspectives on Ψ)

• Some questions– connections btwn chem and phys views?

• Getting at some answers Using Excel Using 3D visualization

What are we visualizing

• Math: Eigenfunctions to the Separable Partial Differential Equation.

• Physics: Wavefunctions of the Hydrogen Atom with Definite Energy, Angular Momentum, and z-component of Angular Momentum.

• Chemistry: Orbitals of Hydrogen-like Atoms – key to bonding.

What are we visualizing?

The probability that an electron will be found at a particular location.

The electron is bound to a nucleus (charge Z). Assume no effect from other electrons! (and this is ok!)

The wave function

The wave function

Ψ=R(r)Θ(θ)Φ(φ)

| Ψ|2 (= Ψ Ψ*) tells us probability

Rn,l(r) tells us how it varies with distance (constant direction)

Θl,m (θ) tells us how it varies with angle to z axis along an arc

Φm (φ) tells us how it varies with angle along a circle centered on z axis.

y

x

z

θ

φ

Specific case: What do the angular solutions for n=3, l=2, m=1 look like?

• Math and physics say:

• Chemistry says:

From Mathematica

From Orbital Viewer

WHAT ARE WE PLOTTING???

Using Excel to visualize the solutions

• The Excel spreadsheet has macros for creating the n=3, l=2, m=1 solutions.

Using Excel to understand the plots

• Use ctrl-s to calculate the values of Ψ at points in the y-z plane for (nlm)=(3,2,1)

• The size of the bubbles is proportional to the value of Ψ

• Use ctrl-b to highlight the points along an arc or a radius.• Observe the shape of the plot of Ψ vs. r for various angles,

or vs. θ for various r.

• Can you see what R, Θ, Φ are doing, individually? What does “separable” mean?

What do these mean? Where do they come from?

• Now, you need to know what the functions are!

• You can find the PDE (Schrodinger’s equation, with the Coulomb potential) HERE

• YIKES!!!

solutions

• They worked out the solutions for some values of n, l, and m.

• Don’t worry about the constants. Do note:– R is exponentials times Laguerre polynomials

(Mathematica knows about these)– Θ is Legendre Polynomials of cos(θ) (Mathematica

knows about these, too)

– Φm (φ)=eimφ which has magnitude 1!

N=3, l=2, m=1

• Can you tell from the mathematical solutions how each part (R, Θ, Φ) behaves?

• Does the Excel plot match your understanding?

• Can you explain in physics terms what the picture show?

• Alt-F11 opens the macro window.– Manually change the function for Ψ. We have n=3,

l=2, m=1. What does n=3, l=2, m=-1 look like?

• What should these look like in 3D?

• {Mathematica or MathCad might be nicer!}– These know about complex numbers (a+ib).– Try plotting | Ψ Ψ*| using the math equations.– Create and plot linear combinations.

Orbitals vs. Wavefunctions

• The orbital viewer (http://www.orbitals.com/) orbitals don’t look like that. Why????

• The solutions you have seen are complex.• Orbital Viewer plots a real function that is a

linear superposition of two complex solutions.• What does that mean mathematically?• What does that mean physically?

m

• Are the physics m’s the same as the chemistry m’s? (NO!)

• What is the relationship???

• What does m tell us in physics? What does it tell us in chemistry?

3D visualization• Now that we understand something about what the

plots represent and• We have some questions• Lets explore some 3D pictures!• Revisit the questions:

– What does m tell us in physics? How does that show up in the pictures?

– What does m tell us in Chemistry? How does that show up in the pictures?

– How can we construct an orbital picture (chem) in Excel or the 3D visualization using the math/physics wave function data?

Questions to explore

• Math: What is an eigen-function? What is an eigenvalue? What are the properties of nth-order polynomials? What is linear superposition?

• Physics: What do the solutions look like? How does the physics of angular momentum relate to the plots? What is linear superposition?

• Chemistry: What do the orbitals mean??? What about more than one atom?

Useful linkshttp://chemviz.ncsa.uiuc.edu/content/demo-waltz3.htmlhttp://www.chm.davidson.edu/ChemistryApplets/AtomicOrbitals/d-orbitals.html has the fuzzy cloud pictures of

orbitalshttp://mathworld.wolfram.com/SphericalHarmonic.html has the mathematica functions and output for

eigenfunctions of the angular PDEhttp://www.orbitals.com/ Has a nice orbital rendering program for download. It draws the orbitals (as used in

chemistry) for one or more atoms. The orbital is the region where the probability is greater than a specified value. Phase (+ or -) is indicated by color.

Useful features: Can choose any n<30, and any l, m for these n’sCan visualize wavefunction for superposition of more than one atom. By placing two atoms in same location,

can create superpositions of the wavefunction.3D features:Can set transparency of rendering so that can “look through” the drawing for 3D informationCan watch as orbital is constructed to “see inside” before the surface gets coveredCan set probability for outer surface, and can animate sequence for drawing surfaces of decreasing probability

(“slice in probability space”)Can render drawings for stereoscopic viewingCaveat:The orbitals are NOT the eigenfunctions of the Lz operator. The orbital for “±m” is actually the superposition

for the two real-valued linear superpositions of the +m and –m solutions.http://www.orbitals.com/orb/ov.pdf pdf file explaining orbial drawing software.

Non sequitur

Is there quasidiffusion in biological systems?

Quasidiffusion = the random walk, but time dependent mean free path.

In physics (e.g. phonons in crystal) mean free path and life time both depend on energy. As phonons decay, step size gets longer, but rate of decay also slows down.

In biology… ?