1. Chemistry 301/ Mathematics 251 Chapter 4
Eigen value problems

2. Eigen Value Problems scalar Operator Matrix Differential equation
Not all values of x or l will work
l called Eigen value
x called Eigen vector (function)

3. Matrix Eigen Value Problems
Nontrivial solution

4. Matrix Eigen Value Problems
Will give 2 Eigen values
# of Eigen values is same as order of matrix

5. Matrix Eigen Value Problems

6. Eigen vectors?
Substitute each l into original Eigen value equation and determine vectors
For l = 2
For l = -3
Both equations give
Both equations give

7. Normalization
Find constants such that dot product of each Eigen vector with itself is 1
For l = 2
For l = -3

8. Orthogonal vectors
Consider the dot product of two different Eigen vectors
Eigen vectors are orthogonal

9. Transformation matrix
Form a transformation matrix where columns are normalized Eigen vectors
Diagonalizes matrix and elements of diagonal matrix are Eigen values

10. Back to Inertia Tensor (Chapter 2)
Water revisited

11. Back to Inertia Tensor (Chapter 2)
Eigen vectors

12. Differential Eigen value Problems
Schrödinger Eqn is an Eigen value problem

13. Systems of 1st order Linear Equations
Consider systems with constant coefficients
where
In matrix form

14. Systems of 1st order Linear Equations
Special case
xi must have the form
Why?

15. Systems of 1st order Linear Equations
Eigen value problem

16. Systems of 1st order Linear Equations

17. Systems of 1st order Linear Equations

18. Systems of 1st order Linear Equations

19. Systems of 1st order Linear Equations

20. Systems of 1st order Linear Equations

21. Systems of 1st order Linear Equations

22. Systems of 1st order Linear Equations

23. Systems of 1st order Linear Equations

24. Systems of 1st order Linear Equations

25. Systems of 1st order Linear Equations

26. Systems of 1st order Linear Equations
General Solution
At t=0

27. Lotka-Volterra Mechanism
System of nonlinear DE
Solve subject to A constant (replenished as needed)

28. Lotka-Volterra Mechanism
Approximate solution
Find critical points

29. Lotka-Volterra Mechanism
Close to the critical points

30. Lotka-Volterra Mechanism

31. Lotka-Volterra Mechanism

32. Lotka-Volterra Mechanism

33. Lotka-Volterra Mechanism

34. Lotka-Volterra Mechanism

35. Lotka-Volterra Mechanism

36. Molecular Orbital (MO) Theory
Linear Combination of Atomic Orbitals (LCAO)

37. Molecular Orbital (MO) Theory
Want to find the combination that gives the lowest energy
Gives
Eigen value problem with overlap

38. Eigen value problems with overlap

39. Eigen value problems with overlap
Diagonal matrix of Eigen values

40. Eigen value problems with overlap

41. Eigen value problems with overlap
Find Eigen values and vectors of S
Form a
Transform H into H
Find Eigen values and vectors of H
Determine C

42. Molecular Orbital (MO) Theory

43. Molecular Orbital (MO) Theory

44. Molecular Orbital (MO) Theory

45. Molecular Orbital (MO) Theory
Special case A = B

46. Molecular Orbital (MO) Theory
Special case A = B

47. Molecular Orbital (MO) Theory
Special case A = B

48. Molecular Orbital (MO) Theory
Special case no overlap

49. Molecular Orbital (MO) Theory
Special case HF without overlap

50. Molecular Orbital (MO) Theory
Special case HF without overlap

51. Molecular Orbital (MO) Theory
Homework
Find the allowed energies for the following