1. Jacob Buchanan Department of Chemistry Central Washington University
Using a Spreadsheet to Solve the Schrödinger Equations for H2 in the Ground and Excited States
Jacob Buchanan
Department of Chemistry
Central Washington University

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

3. Goals Better 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.

4. Prior Knowledge Eigen functions of an operator.
Average is figured out this way when there the function is not an eigen function.

5. 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. 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
Exercises 1 and 2 introduced in lecture but can be introduced in lab.
Ex. 1-2 help students better understand abstract QM concepts and gain experience with spreadsheet calculations.
Exercise 3 done in lab.

6. Ex. 1. Visualization of tunneling effect in a H atom
−1 2 𝛻 2 𝜓− 1 𝑟 𝜓 =𝐸𝜓
Students can verify that 𝜓= 𝑒 −𝑟 is an eigenfunction with 𝐸=− 1 2 au.
Radius
Total E
Potential E
Kinetic E
(r)
= -1/2
= -1/r
=1/r – 1/2 1
-0.5
-1.0
0.5 4
-0.25 …
Use exponential function instead of gaussian functions.
At radius of 2, electrons kinetic energy becomes negative which violates classical mechanics.
Talk about graph a little more in detail with pointer.

7. Ex. 2. Visualization of particle-like properties of waves
Double-slit diffraction patterns of electrons visually illustrate the wave-like properties of particles.
Wavefunction created by averaging cosine functions.

8. Ex. 2. Visualization of particle-like properties of waves
Photoelectric effects illustrate (less visually) the particle-like property of light.
Wavefunction created by averaging cosine functions.

9. Ex. 2. Visualization of particle-like properties of waves
Visualization of the particle-like property of waves on a spreadsheet:
Ψ =[ cos θ + cos 2θ + cos 3θ +cos⁡(4θ) ] / 4 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
-0.15
-0.06 -1
0.54
-0.38
0.14
1.00 1 2 3
Ψ 2
Wavefunction created by averaging cosine functions.

10. Combination of many waves exhibit particle-like property
Ψ = 𝑛=1 𝑁 cos 𝑛θ /𝑁
𝑁=1
𝑁=10
𝑁=100
Ψ 2
Waves can exhibit properties of particles.
Can identify the magnitude of the momentum (cosine is not an eigenfunction and has two possible values).
Introduction of Heisenberg uncertainty principle.
wave particle
# of values the momentum (p = h/λ) may have:
Uncertainty of momentum
Uncertainty of position

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

12. Integral calculation on a spreadsheet
Approximations used to be able to do math on a spreadsheet.
Fast decaying function allows us to focus on the volume of interest.

13. H2+ −∞ ∞ Ψ 𝑂 Ψ𝑑τ ≅𝑉 𝑖=1 𝑁 Ψ 𝜏 𝑖 𝑂 Ψ 𝜏 𝑖 𝑁 6 6 7.5 rA rB HA HB
e- (x, y, z) 6 rA rB HA HB
RAB = 1.5
Box of interest.
Nuclei distant fixed at 1.5 atomic units.
Coordinates of electron randomly generated.
Tow I is the coordinates which is sampled several thousand times. 6
7.5
−∞ ∞ Ψ 𝑂 Ψ𝑑τ ≅𝑉 𝑖=1 𝑁 Ψ 𝜏 𝑖 𝑂 Ψ 𝜏 𝑖 𝑁

14. − 1 2 𝛻 2 ( 𝑒 −𝑟 ) = ( 1 𝑟 − 1 2 ) 𝑒 −𝑟 Random coordinates 𝑒 − 𝑟 𝐴
− 1 2 𝛻 2 ( 𝑒 −𝑟 ) = ( 1 𝑟 − 1 2 ) 𝑒 −𝑟
Random
coordinates
𝑒 − 𝑟 𝐴
𝑒 − 𝑟 𝐵
φA+φB
Attraction
Kinetic E 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
0.089
0.178
0.032
0.013
-0.003
0.7
1.0
1.4
1.9
0.152
0.305
0.093
0.049
0.003
1.2
-1.8
2.2
0.115
0.229
0.053
0.024
-0.002
0.8
0.154
0.309
0.095
0.051
-0.4
-0.8
3.0
3.1
0.045
0.091
0.008
-0.001
-0.7
2.0
0.139
0.278
0.077
0.039
0.001
-0.6
0.213
0.425
0.181
0.117
0.026
-2.2
0.046
0.009
0.2
0.4
0.192
0.384
0.147
0.016
1.8
2.9
3.7
0.052
-1.7
0.142
0.285
0.081
0.042
0.0
-1.2
-1.9
2.3
0.104
0.208
0.043
0.019 …
Laplacian is figured out before hand so that simplified equation can be put into spreadsheet.

15. H2
e1- 6 HA HB
r12
e2- 6
7.5
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.

16. 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
Randomly generated coordinates of electron 1.
Randomly generated coordinates of electron 2.
Distance of electrons from nuclei.
Atomic orbitals.
Molecular orbitals.
Hartree product wave function.
Probability Density.
Blue = energy of electron-nuclei attraction.
Purple = energy of electron-electron repulsion.
Red = KE of electrons (Laplacian contains both electrons).
-ψ2/r1A
-ψ2/r1B
-ψ2/r2A
-ψ2/r2B
ψ2/r12
- 1 2 ψ 𝛻 2 ψ
Attraction
Repulsion
Kinetic Energy

17. Average of 10 runs (each contains 8000 sampling)
Kinetic E
e-e repulsion
Attraction

18. Range and standard deviation of 10 runs
H2+ H2
Kinetic E
e-e repulsion
Total E
Potential E
Total E
Attraction
Potential E
Top to bottom is the range.
Length wise is one standard deviation.
Attraction

19. Comparison of H2 energy calculated using spreadsheet and using HF theory with various basis sets
HF/cc-pV6Z around 100 gaussian functions provides HartreeFock limit

20. Ex. 4. Exchange energy in H2. : first singlet vs
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. This reduction of e-e repulsion is called exchange energy.
The e-e repulsion for the S1 state and T1 state:
< 1 𝑟 12 > = ψ 1 𝑟 12 ψd𝜏 ψ ψd𝜏 ≅ ψ2/r ψ2
ψ 𝑆1 = (σ1σ∗2 + σ2σ∗1) ψ 𝑇1 = (σ1σ∗2- σ2σ∗1)
Exchange energy can only occur when wave functions are antisymmetric.
Tell students that exchange energy only involves electron-electron repulsion.
Hartree product does not include exchange energy.
Determinant wave function does include exchange energy.
Spin integral gets cancelled out.

21. Ex. 5. Correlation energy in H2: ψ 𝑔𝑠 + C 𝑒𝑥 ψ 𝑒𝑥
Hartree-Fock energy (1 determinant) E
Correlation energy
Exact energy (infinite # of determinants)
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.
Correlation energy is the difference between the Hartree-Fock limit energy and the exact or true energy.
Hartree-Fock uses one determinant to describe wave function.
Lower energy by using a linear combination of determinants.
Students compare the lowest energy 2-determinant wave function with the one determinant wave function (when Cex = 0).

22. Ex. 5. Correlation energy in H2: ψ 𝑔𝑠 + C 𝑒𝑥 ψ 𝑒𝑥
Hartree-Fock energy (1 determinant) E
Correlation energy from double excitation
CID energy (2 determinants)
Using Hartree product functions to calculate correlation energy approximately for a H2 molecule:
ψ 𝑔𝑠 =σ1σ w/o correlation
ψ 𝑐𝑜𝑟𝑟 =σ1σ2 + Cexσ∗1σ∗ w/ correlation
Simplify problem for students by using the Hartree product to create the wave functions instead of determinants.
Correlation energy is the repulsion energy with correlation minus the repulsion energy without correlation.
Ecorr ≅ ( ψ 𝑐𝑜𝑟𝑟 2/r12) ψ 𝑐𝑜𝑟𝑟 2 − ( ψ 𝑔𝑠 2/r12) ψ 𝑔𝑠 2

23. Acknowledgements
Benjamin Livingston (CWU Alumnus pursuing PhD in Math at Oregon State University)
Dr. Robert Rittenhouse
CWU COTS Summer Writing Grant & Faculty Development Fund (YG)
CWU Office of Graduate Studies & Research Faculty Travel Fund (YG)
CWU Department of Chemistry

24. Questions?

25. Difference between S1 and T1 excited states:σ* σ
σα1σ*β1 (S1) σα1σ*α1 (T1)
w/o exchange w/ exchange
Use Hartree product ψ 𝐻𝑃 = σ1σ*2 Use determinantal ψ 𝑑𝑒𝑡
Visual representation to help students understand what is going on.
Singlet excited state, no exchange, used hartree product.
Triplet excited state, includes exchange, wave function created by using a determinant.
Attraction potential energy, repulsion potential energy, kinetic energy.

26. Exchange energy is the essentially the difference between repulsion of electrons with same spin vs. repulsion of electrons with opposite spin.
ψ 𝐻𝑃
ψ 𝑑𝑒𝑡
ψ 𝐻𝑃 2
ψ 𝑑𝑒𝑡 2
ψ 𝐻𝑃 2/r12
ψ 𝑑𝑒𝑡 2/r12
σ1σ*2
σ1σ∗2 - σ2σ∗1 …
Basic explanation of exchange energy.
Hartree product wave function, determinant wave function, squared modulus, repulsion energies.
Exchange energy is the energy of the triplet excited state minus the energy of the singlet excited state.
Exchange E ≅ ψ 𝑑𝑒𝑡 2/r ψ 𝑑𝑒𝑡 2 − ψ 𝐻𝑃 2/r ψ 𝐻𝑃 2

27. −1 2 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝜕𝑟 + 1 𝑠𝑖𝑛𝜃 𝜕 𝜕𝜃 𝑠𝑖𝑛𝜃 𝜕 𝜕𝜃 + 1 𝑠𝑖𝑛 2 𝜃 𝜕 2 𝜕 𝜙 2 𝜓− 1 𝑟 𝜓 =𝐸𝜓
−1 2 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝜕𝑟 + 1 𝑠𝑖𝑛𝜃 𝜕 𝜕𝜃 𝑠𝑖𝑛𝜃 𝜕 𝜕𝜃 𝑠𝑖𝑛 2 𝜃 𝜕 2 𝜕 𝜙 𝑒 −𝑟
= −1 2 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝜕𝑟 𝑒 −𝑟
= 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝑒 −𝑟
= 1 2 𝑟 2 (2𝑟 𝑒 −𝑟 − 𝑟 2 𝑒 −𝑟 )
= ( 1 𝑟 − 1 2 ) 𝑒 −𝑟

28. H atom calculations Familiarize students with spreadsheet format.
Calculation of kinetic energy and potential energy.
Comparison to analytical solution.

29. Compilation
Results from each student compiled using Google spreadsheet
Results averaged and compared to HF/STO-3G calculations and experimental data.