Chemistry 301 Mathematics 251 Chapter 4

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Chemistry 301/ Mathematics 251Chapter 4

• Eigen value problems

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Eigen Value Problems
scalar
Operator
Differential equation
Matrix
Not all values of x or l will work l called Eigen
value x called Eigen vector (function)
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Matrix Eigen Value Problems
Nontrivial solution
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Matrix Eigen Value Problems
Will give 2 Eigen values of Eigen values is
same as order of matrix
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Matrix Eigen Value Problems
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Eigen vectors?
Substitute each l into original Eigen value
equation and determine vectors
For l -3
For l 2
Both equations give
Both equations give
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Normalization
Find constants such that dot product of each
Eigen vector with itself is 1
For l -3
For l 2
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Orthogonal vectors
Consider the dot product of two different Eigen
vectors
Eigen vectors are orthogonal
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Transformation matrix
Form a transformation matrix where columns are
normalized Eigen vectors
Diagonalizes matrix and elements of diagonal
matrix are Eigen values
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Back to Inertia Tensor (Chapter 2)
Water revisited
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Back to Inertia Tensor (Chapter 2)
Eigen vectors
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Differential Eigen value Problems
Schrödinger Eqn is an Eigen value problem
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Systems of 1st order Linear Equations
Consider systems with constant coefficients
where
In matrix form
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Systems of 1st order Linear Equations
Special case
xi must have the form
Why?
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Systems of 1st order Linear Equations
Eigen value problem
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Systems of 1st order Linear Equations
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Systems of 1st order Linear Equations
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Systems of 1st order Linear Equations
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Systems of 1st order Linear Equations
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Systems of 1st order Linear Equations
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Systems of 1st order Linear Equations
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Systems of 1st order Linear Equations
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Systems of 1st order Linear Equations
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Systems of 1st order Linear Equations
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Systems of 1st order Linear Equations
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Systems of 1st order Linear Equations
General Solution
At t0
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Lotka-Volterra Mechanism
System of nonlinear DE Solve subject to A
constant (replenished as needed)
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Lotka-Volterra Mechanism
Approximate solution Find critical points
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Lotka-Volterra Mechanism
Close to the critical points
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Lotka-Volterra Mechanism
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Lotka-Volterra Mechanism
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Lotka-Volterra Mechanism
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Lotka-Volterra Mechanism
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Lotka-Volterra Mechanism
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Lotka-Volterra Mechanism
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Molecular Orbital (MO) Theory
Linear Combination of Atomic Orbitals (LCAO)
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Molecular Orbital (MO) Theory
Want to find the combination that gives the
lowest energy
Gives
Eigen value problem with overlap
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Eigen value problems with overlap
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Eigen value problems with overlap
Diagonal matrix of Eigen values
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Eigen value problems with overlap
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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

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Molecular Orbital (MO) Theory
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Molecular Orbital (MO) Theory
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Molecular Orbital (MO) Theory
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Molecular Orbital (MO) Theory
Special case A B
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Molecular Orbital (MO) Theory
Special case A B
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Molecular Orbital (MO) Theory
Special case A B
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Molecular Orbital (MO) Theory
Special case no overlap
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Molecular Orbital (MO) Theory
Special case HF without overlap
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Molecular Orbital (MO) Theory
Special case HF without overlap
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Molecular Orbital (MO) Theory
Homework Find the allowed energies for the
following