Chemistry 301/ Mathematics 251 Chapter 4

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Presentation transcript:

Chemistry 301/ Mathematics 251 Chapter 4 Eigen value problems

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)

Matrix Eigen Value Problems Nontrivial solution

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

Matrix Eigen Value Problems

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

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

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

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

Back to Inertia Tensor (Chapter 2) Water revisited

Back to Inertia Tensor (Chapter 2) Eigen vectors

Differential Eigen value Problems Schrödinger Eqn is an Eigen value problem

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

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

Systems of 1st order Linear Equations Eigen value problem

Systems of 1st order Linear Equations

Systems of 1st order Linear Equations

Systems of 1st order Linear Equations

Systems of 1st order Linear Equations

Systems of 1st order Linear Equations

Systems of 1st order Linear Equations

Systems of 1st order Linear Equations

Systems of 1st order Linear Equations

Systems of 1st order Linear Equations

Systems of 1st order Linear Equations

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

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

Lotka-Volterra Mechanism Approximate solution Find critical points

Lotka-Volterra Mechanism Close to the critical points

Lotka-Volterra Mechanism

Lotka-Volterra Mechanism

Lotka-Volterra Mechanism

Lotka-Volterra Mechanism

Lotka-Volterra Mechanism

Lotka-Volterra Mechanism

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

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

Eigen value problems with overlap

Eigen value problems with overlap Diagonal matrix of Eigen values

Eigen value problems with overlap

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

Molecular Orbital (MO) Theory

Molecular Orbital (MO) Theory

Molecular Orbital (MO) Theory

Molecular Orbital (MO) Theory Special case A = B

Molecular Orbital (MO) Theory Special case A = B

Molecular Orbital (MO) Theory Special case A = B

Molecular Orbital (MO) Theory Special case no overlap

Molecular Orbital (MO) Theory Special case HF without overlap

Molecular Orbital (MO) Theory Special case HF without overlap

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