Basis Sets

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Basis Sets

• Patrick Briddon

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Contents

• What is a basis set? Why do we need them?
• Gaussian basis sets
• Uncontracted
• Contracted
• Accuracy a case study
• Some concluding thoughts

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What is a basis set?

• Solutions to the Schrödinger equation

are continuous functions, ?(x). ? not good for a
modern computer (discrete)
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Why a basis set?

• Idea
• write the solution in terms of a series of
  functions

• The function ? is then stored as a number of
  coefficients

5
A few questions

• What shall I choose for the functions?
• How many of them do I need?
• How do I work out what the correct coefficients
  are?

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Choosing Basis functions

• Try to imagine what the true wavefunction will be
  like

V
?
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Choosing Basis functions
?
Basis states
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The coefficients

• These are determined by using the variational
  principle of quantum mechanics.
• If we have a trial wave-function

• Choose the coefficients to minimise the energy.

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How many basis functions?

• The more the better (i.e. the more accurate).
• Energy always greater than true energy, but
  approaches it from above.
• The more you use, the slower the calculation!
• In fact time depends on number-cubed!
• The better they are, the fewer you need.

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Basis sets ad LCAO/MO

• There is a close relationship between chemistry
  ideas and basis sets.
• Think about the H2 molecule

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Basis sets and LCAO

• Physicists call this LCAO (linear combination of
  atomic orbitals)
• The basis functions are the atomic orbitals
• Chemists call this molecular orbital theory
• There is a big difference though
• In LCAO/MO the number of basis functions is equal
  to the number of MOs.
• There is no variational freedom.

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What about our basis functions?

• Atomic orbitals are fine, but they are
• Not well defined you cant push a button on a
  calculator and get one!
• Cumbersome to use on a computer
• AIMPRO used Gaussian orbitals
• It is called a Gaussian Orbital code.

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Gaussian Orbitals

• The idea

• There are thus three ingredients
• An exponent, a controls the width of the
  Gaussian.
• A centre R controls the location
• A coefficient varied to minimise the energy

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The Exponents

• Typically vary between 0.1 and 10
• Si 0.12 up to 4
• F 0.25 up to 10
• These are harder to find than coefficients.
• Small or large exponents are dangerous
• Fixed in a typical AIMPRO run
• determined for atom or reference solid.
• i.e. vary exponents to get the lowest energy for
  bulk Si
• Put into hgh-pots
• then keep them fixed when we look at other defect
  systems.

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The Positions/Coefficients

• Positions we put functions on all atoms
• In the past we put them on bond centres too
• Abandoned what if a bond disappears during a
  run?
• You cannot put two identical functions on the
  same atom the functions must all be different.
• That is why small exponents are dangerous.
• Coefficients AIMPRO does that for you!

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How good are Gaussians?

• Problems near the nucleus?
• True AE wave function was a cusp
• but the pseudo wave function does not!

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How good are Gaussians?

• Problems at large distance?
• True wave function decays exponentially exp-br
• Our function will decay more quickly exp-br2
• Not ideal, but is not usually important for
  chemical bonding.
• Could be important for VdW forces
• But DFT doesnt get them right anyway
• Only ever likely to be an issue for surfaces or
  molecules (our solution ghost orbitals)

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AIMPRO basis set

• We do not only use s-orbitals of course.
• Modify Gaussians to form Cartesian Gaussian
  functions

• Alongside the s orbital that will give 4
  independent functions for the exponent.

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What about ds?

• We continue, multiplying by 2 pre-factors

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What about ds?

• This introduces 6 further functions
• i.e. giving 10 including the s and ps
• Of these 6 functions, 5 are the d-orbitals
• One is an additional s-type orbital

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ddpp and all that

• We often label basis sets as ddpp.
• What does this mean?
• 4 letters means 4 different exponents.
• The first (smallest) has s/p/d functions (10)
• The next also has s/p/d functions (10)
• The last two (largest exponents) have s/p (4
  each)
• Total of 28 functions

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Can we do better?

• Add more d-functions
• dddd with 40 functions per atom
• this can be important if states high in the
  conduction band are needed (EELS).
• Clearly crucial for elements like Fe!
• Add more exponents
• ddppp
• Pddppp
• Put functions in extra places (bond centres)
• Not recommended

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How good is the energy?

• We can get the energy of an atom to 1 meV when
  the basis fitted.
• BUT larger errors encountered when transferring
  that basis set to a defect.
• The energy is not well converged.
• But energy differences can be converged.
• So
• ONLY SUBTRACT ENERGIES CALCULATED WITH THE SAME
  BASIS SET!

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Other properties

• Structure converges fastest with basis set
• Energy differences converge next fastest
• Conduction band converges more slowly
• Vibrational frequencies also require care.
• Important to be sure, the basis set you are using
  is good enough for the property that you are
  calculating!

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Contracted basis sets

• A way to reduce the number of functions whilst
  maintaining accuracy.
• Combine all four s-functions together to create a
  single combination

• The 0.1, 0.2, etc. are chosen to do the best for
  bulk Si.
• They are then frozen kept the same for large
  runs.
• Do the same for the p-orbitals.
• This gives 4 contracted orbitals

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The C4G basis

• These 4 orbitals provide a very small basis set.
• How much faster than ddpp?
• Answer (28/7)3 or 343 times!
• Sadly not good enough!
• You will probably never hear this spoken of!
• Chemistry equivalent STO-3G
• Also regarded as rubbish!

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The C44G basis

• Next step up choose two different s/p
  combinations
• We will now have 8 functions per atom.
• (8/4)3 or 8 times slower than C4G!
• (28/8)3 or 43 times faster than ddpp.
• Sadly still not good enough!

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The C44G basis

• Main shortcoming change of shape of s/p
  functions when solid is formed.
• Need d-type functions.
• Add 5 of these.
• Gives 13 functions
• What we call C44G (again PRB speak)
• Similar to chemists 6-31G

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The C44G basis

• 13 functions still (28/13)3 times faster than
  ddpp
• Diamond generally very good
• Si conduction band not converged various
  approaches (Jons article on Wiki)
• Chemists use 6-31G for much routine work.

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Results for Si (JPG)
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The way forwards?

• 13 functions still (28/13)3 times faster than
  ddpp
• 4 functions was (28/4)3 times faster.
• Idea at Nantes form combinations not just of
  functions on one atom.
• Be very careful how you do this.
• Accuracy can be as good as ddpp.

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Plane Waves

• Another common basis set is the set of plane
  waves recall the nearly free electron model.
• We can form simple ideas about the band structure
  of solids by considering free electrons.
• Plane waves are the equivalent to atomic
  orbitals for free electrons.

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Gaussians vs Plane Waves

• Number of Gaussians is very small
• Gaussians 20/atom
• Plane Waves 1000/atom
• Well written Gaussian codes are therefore faster.
• Plane waves are systematic no assumption as to
  true wave function
• Assumptions are dangerous (they can be wrong!)
• but they enable more work if they are faster

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Gaussians vs Plane Waves

• Plane waves can be increased until energy
  converges
• In reality it is not possible for large systems.
• Number of Gaussians cannot be increased
  indefinitely
• Gaussians good when we have a single difficult
  atom
• Carbon needs a lot of pane waves ? SLOW!
• 1 C atom in 512 atom Si cell as slow as diamond
• True for 2p elements (C, N, O, F) and 3d metals.
• Gaussians codes are much faster for these.

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In conclusion

• Basis set is fundamental to what we do.
• A quick look at the mysterious hgh-pots.
• Uncontracted and contracted Gaussian bases.
• Rate of convergence depends on property.
• A good publication will demonstrate that results
  are converged with respect to basis.