Using a Spreadsheet to Solve the Schrödinger Equations for Small Molecules: An Undergraduate

Quantum Chemistry LabJacob Buchanan, Benjamin Livingston, Yingbin Ge*

Department of ChemistryCentral Washington University

“I think you should be more explicit here in Step Two.”By Sidney Harris, Copyright 2007, The New Yorker

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GoalsBetter understanding of quantum chemistry • by visualizing quantum phenomena on a spreadsheet• by solving Schrödinger equation for molecules by

hand• by breaking the total energy into pieces– Potential energy: N-N repulsion, e-e repulsion, e-N

attraction.– Kinetic energy of each electron.

• by hand calculating exchange and correlation energy on a spreadsheet.

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Prior Knowledge

• Multivariable differential and integral calculus• Quantum mechanical postulates:

ψ = oψ<o> = (when is a real function.)

• Linear algebra and differential equations are not required

• Experience with Excel or Google spreadsheet

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Quantum chemistry exercises on a spreadsheet

Ex. 1. Visualization of tunneling effect in a H atom

Ex. 2. Visualization of particle-like properties of waves

Ex. 1-2 help students better understand abstract QM concepts and gain experience with spreadsheet calculations.

Ex. 3. Solving the Schrödinger equation for H2+ and H2

Ex. 4. Exchange energy in H2*: first singlet vs triplet excited states

Ex. 5. Correlation energy in H2: a 2-determinant wave function

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Ex. 1. Visualization of tunneling effect in a H atom

−12𝛻2𝜓−

1𝑟𝜓=𝐸𝜓

Students can easily verify that is an eigenfunction with .

0.5 1.0 1.5 2.0 2.5 3.0 3.5-2.0-1.5-1.0-0.50.00.51.01.5 Total E

Potential EKinetic E

Radius

Radius Total E Potential E Kinetic E(r) = -1/2 = -1/r =1/r – 1/2

0.5 -0.5 -2.0 1.50.6 -0.5 -1.7 1.20.7 -0.5 -1.4 0.9

… … … …

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Ex. 2. Visualization of particle-like properties of waves

Double-slit diffraction patterns of electrons visually illustrate the wave-like properties of particles.

Photoelectric effects illustrate (less visually) the particle-like property of light. Visualization of the particle-like property of waves on a spreadsheet:

θ cos(θ) cos(2θ) cos(3θ) cos(4θ) Ψ Ψ2

-3 -0.99 0.96 -0.91 0.84 -0.02 0.00-2 -0.42 -0.65 0.96 -0.15 -0.06 0.00-1 0.54 -0.42 -0.99 -0.65 -0.38 0.140 1.00 1.00 1.00 1.00 1.00 1.001 0.54 -0.42 -0.99 -0.65 -0.38 0.142 -0.42 -0.65 0.96 -0.15 -0.06 0.003 -0.99 0.96 -0.91 0.84 -0.02 0.00 -3 0 3

0

1

] / 4

Ψ 2

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Combination of many waves exhibit particle-like property

Ψ  =∑𝑛=1

𝑁

cos (𝑛θ )/𝑁

𝑁=1 𝑁=100𝑁=10

-4 0 4 -4 0 4-4 0 4

Ψ 2

wave particle

# of values the momentum (p = h/λ) may have:1 10 100

Uncertainty of momentum

Uncertainty of position

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H2+

• Introduction to solving problems without analytical solutions.

• Nucleus positions fixed with a bond length of 1.5 a.u.• Calculate every potential and kinetic energy

component approximately.• Need to evaluate integrals on a spreadsheet.

Ex. 3. Solving the Schrödinger equation for H2+ and H2

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Integral calculation on a spreadsheet

• Using a spreadsheet to estimate the value of an integral:

• For a fast decaying function f(x,y,z), the function value at several thousand randomly selected data points are calculated within a given volume.

• The product of the volume and the average function value is approximately the integral.

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HA HB

e- (x, y, z)

Z

X

YrA rB

RAB = 1.5

∫−∞

∞

Ψ �̂�Ψ 𝑑 τ≅ 𝑉∑𝑖=1

𝑁

Ψ (𝜏 𝑖 ) �̂�Ψ (𝜏 𝑖 )

𝑁

H2+

x y z rA rB φA φB σ σ2 -σ2/rA -σ2/rB σ[(1/rA-1/2)φA+(1/rB-1/2)φB]0.3 -0.5 -2.4 1.5 2.4 0.226 0.088 0.314 0.099 0.067 0.041 0.010

-1.6 1.7 0.5 2.4 2.4 0.089 0.089 0.178 0.032 0.013 0.013 -0.0030.7 1.0 1.4 1.9 1.9 0.152 0.152 0.305 0.093 0.049 0.049 0.003

-0.5 1.2 -1.8 2.2 2.2 0.115 0.115 0.229 0.053 0.024 0.024 -0.002-0.5 -1.6 0.8 1.9 1.9 0.154 0.154 0.309 0.095 0.051 0.051 0.003-0.4 -0.8 3.0 3.1 3.1 0.045 0.045 0.091 0.008 0.003 0.003 -0.0011.5 1.2 -0.7 2.0 2.0 0.139 0.139 0.278 0.077 0.039 0.039 0.001

-0.6 1.2 0.8 1.5 1.5 0.213 0.213 0.425 0.181 0.117 0.117 0.0261.2 -1.8 -2.2 3.1 3.1 0.046 0.046 0.093 0.009 0.003 0.003 -0.002

-1.6 0.2 0.4 1.7 1.7 0.192 0.192 0.384 0.147 0.089 0.089 0.0161.8 1.4 2.9 3.7 3.7 0.026 0.026 0.052 0.003 0.001 0.001 -0.001

-1.7 -0.8 0.5 2.0 2.0 0.142 0.142 0.285 0.081 0.042 0.042 0.0010.0 -1.2 -1.9 2.3 2.3 0.104 0.104 0.208 0.043 0.019 0.019 -0.003

… … … … … … … … … … … …

Kinetic EAttractionRandomcoordinates 𝑒−𝑟 𝐴 φA+φB

() = ()

𝑒−𝑟 𝐵

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HA HB

e1-

Z

X

Y

e2-

r12

Nucleus positions fixed with a bond length of 1.5 a.u.No exchange energy between electrons with opposite spin.Hartree product wave function ψ = σ(1)σ(2) is used.

H2

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There are more columns but calculations can still be done conveniently on a spreadsheet.

x1 y1 z1 x2 y2 z2 r1A r1B r2A r2B r12

… … … … … … … … … … …

φ1A φ1B φ2A φ2B σ1 σ2 Ψ Ψ2

… … … … … … σ1σ2 …

-ψ2/r1A -ψ2/r1B -ψ2/r2A -ψ2/r2B ψ2/r12 -ψψ… … … … … …

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-4

-2

0

2

SpreadsheetHF/STO-3G

a. u

.

Potential E

Kinetic E

Total E

Attraction

Potential E

Total E

Attraction

e-e repulsion

Kinetic EH2+ H2

Average of 10 runs (each contains 8000 sampling)

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-4

-2

0

2

a. u

.

Kinetic E

Attraction

Potential E

Total E

H2+ H2

e-e repulsionKinetic E

Total E

Potential E

Attraction

Range and standard deviation of 10 runs

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Ex. 4. Exchange energy in H2*: first singlet vs. triplet excited states

No 2 electrons can have the same set of quantum numbers.

Electrons with same spin can better avoid each other than electrons with opposite spin, which reduces the e-e repulsion.

Hartree product and determinantal wave functions can be used to estimate the e-e repulsion for the S1 (w/o exchange) state and T1 state (w/exchange).

<> =

(S1) (- ) (T1)

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σ

σ*

σα1σ*β1 (S1) σα1σ*α1 (T1)w/o exchange w/ exchangeUse Hartree product = σ1σ*2 Use determinantal

Difference between S1 and T1 excited states:

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2 2 2/r122/r12

σ1σ*2 - … … … …

Exchange E

Exchange energy is the essentially the difference between repulsion of electrons with same spin vs. repulsion of electrons with opposite spin.

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Ex. 5. Correlation energy in H2: +

Correlation energy

• 2-determinant wave function that includes a parametric constant Cex for the excited state determinant.

• Variational method to minimize Ecorr.• Minimum energy using the 2-determinant wave function

compared to the HF energy with Cex = 0.

Exact energy (infinite # of determinants)

Hartree-Fock energy (1 determinant)

E

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Ex. 5. Correlation energy in H2: +

Using Hartree product functions to calculate correlation energy approximately for a H2 molecule:

w/o correlation w/ correlation

Hartree-Fock energy (1 determinant)

CID energy (2 determinants)

Correlation energy from double excitation

Ecorr

E

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Acknowledgements

• Central Washington University (CWU) College of the Sciences Summer Writing Grant & Faculty Development Fund

• CWU Office of Graduate Studies & Research Faculty Travel Fund

• CWU Department of Chemistry• Dr. Robert Rittenhouse for helpful discussion

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Questions, comments, & suggestions?

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−12𝑟 2 [ 𝜕𝜕𝑟 (𝑟2 𝜕

𝜕𝑟 )+ 1𝑠𝑖𝑛𝜃

𝜕𝜕 𝜃 (𝑠𝑖𝑛𝜃 𝜕

𝜕𝜃 )+ 1𝑠𝑖𝑛2𝜃 ( 𝜕2

𝜕𝜙2 )]𝜓 − 1𝑟 𝜓=𝐸𝜓

)

( )

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H atom calculations

• Familiarize students with spreadsheet format.• Calculation of kinetic energy and potential

energy.• Comparison to analytical solution.

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Compilation

• Results from each student compiled using Google spreadsheet

• Results averaged and compared to HF/STO-3G calculations and experimental data.