Norm Conserving Pseudopotentials and The Hartree Fock Method

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Norm Conserving Pseudopotentialsand The Hartree
Fock Method

• Eric Neuscamman
• Mechanical and Aerospace Engineering 715
• May 7, 2007

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Goals

• Produce a stand alone Hartree Fock code using C
• Apply code to generate pseudopotentials

3
Goals

• Produce a stand alone Hartree Fock code using C
• Apply code to generate pseudopotentials

Success!
4
Goals

• Produce a stand alone Hartree Fock code using C
• Apply code to generate pseudopotentials

Success!
Very limited success.
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Basic Idea
Pseudopotentials simplify calculations

• 1. By removing core electrons

Carbon n 6 n3 216
nv 4 nv3 64
Silicon n 14 n3 2744
2. By creating nodeless valence orbitals
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Creating Pseudopotentials

• Step 1 Solve atomic system exactly.
• Step 2 Construct the target valence orbital.
  It should be exact for large r.
• Step 3 Invert the Schrodinger eq. for the V(r)
  that produces your valence orbital

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The Hartree Fock Method

• Apply the variational principal to a Slater
  determinant.
• Result

This differential equation has an infinite number
of solutions.
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Solving the Fock Equation

• Introduce a basis

Restricted Hartree Fock
Assumes all orbitals are doubly occupied
Unrestricted Hartree Fock
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Solving the Fock Equation

• The basis gives a matrix eigenvalue problem

Diagonal Eigenvalue Matrix
Fock Matrix
Orbital Coefficients
Overlap Matrix
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Solving the Fock Equation

• Structure of the Fock operator

One Electron Integrals
Two Electron Integrals
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Solving the Fock Equation

• Finding the integrals is the hardest part!

One Electron Integrals
Two Electron Integrals
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Basis Sets

• To simplify integration, choose linear
  combinations of
• gaussian basis functions.
• This gives us

We have reduced a 6-dimensional integral to a
2-dimensional integral
An analytic integral
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The SCF Algorithm
Solution Found
No
Converged?
Yes
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Target Valence Orbitals
Now that we know the exact valence orbital
functions, we may construct new valence orbitals
that lack nodes.

• Questions

• How do we ensure they match in the valence
  region?
• How do we enforce normalization? (Norm
  conservation)

Answer Method of Hamman, Schluter, and Chiang
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Introduce a cutoff function
Holding the core electrons fixed, re-solve the
Schrodinger (Fock) equation, but with a modified
potential
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Introduce a cutoff function
To generate our target valence orbital, we
repeatedly solve the modified Fock equation,
adjusting c until the eigenvalue of the nodeless
solution is the same as for the exact orbital.
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A Potential Problem
Incorporating in the Hartree Fock
method means modifying the one and two electron
integrals.
One Electron Not a problem
Two Electron Breaks symmetry!
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Solution Only modify the OEI
Rather than modifying the nuclear, coulomb, and
exchange potentials, only modify the nuclear
attraction.
Then only the one electron integrals need
modification and the method is tractable again.
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Norm Conservation
Due to the homogeneity of the Schrodinger
equation, the modified and exact valence
functions may differ by a constant multiple.
This is easily fixed by scaling the modified
function to match the exact function in the
valence region.
Afterwards, however, our function is no longer
normalized!
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Norm Conservation
To re-normalize our function without altering its
valence behavior, we must change its form in the
core. This is achieved by using a cutoff function
again.
The normalization condition is then
Of the two roots for d, choosing the smaller one
will produce a smoother wavefunction.
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The Pseudopotential
We have now generated a target valence orbital
that is normalized and matches the exact orbital
outside the core.
Our pseudopotential is then whatever potential
generates our target orbital. To find it, we
invert the Fock equation.
Solve for me!
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Easier Said Than Done
The integral form of the coulomb and exchange
operators, coupled with the fact that not all the
core electrons will be in spherically symmetric
orbitals, make inverting this equation cumbersome.
Currently, my code can only calculate Vpseudo
when all of the occupied orbitals are spherically
symmetric.
This limits me to Lithium and Berillium. Yuck!
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Results
Hartree Fock code correctly predicts orbital
occupations for 2nd row elements (need to check
3rd row).
Small modification would allow d-orbital basis
functions to be employed, permitting modeling of
transition metals. However, accuracy will
degrade as relativistic effects grow.
Pseudopotential method successful for both Li and
Be. Results reported here employ the 6-31G basis
set.
Substantial work needed to allow calculation of
PPs for atoms with p electrons.
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Results for Lithium (rc 2.0)
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Results for Lithium (rc 2.0)
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Results for Lithium (rc 2.0)
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Results for Beryllium (rc 0.9)
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Results for Beryllium (rc 0.9)
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Results for Beryllium (rc 0.9)
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Conclusion
Although implementing it was an excellent
educational tool, the Hartree Fock is ill-suited
for pseudopotentials
Even if my codes current shortcomings could be
removed, relavistic effects would prevent the
method from applying to heavier atoms where
pseudopotentials can greatly reduce the number of
electrons.
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Questions
?