# Basis Sets

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12-Jan-2016Category

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### Transcript of Basis Sets

Basis SetsPatrick Briddon

ContentsWhat is a basis set? Why do we need them?Gaussian basis setsUncontractedContractedAccuracy: a case studySome concluding thoughts

What is a basis set?Solutions to the Schrdinger equation:

are continuous functions, (x).

not good for a modern computer (discrete)

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:

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?

Choosing Basis functionsTry to imagine what the true wavefunction will be like:V

Choosing Basis functionsBasis states

The coefficientsThese are determined by using the variational principle of quantum mechanics.If we have a trial wave-function:Choose the coefficients to minimise the energy.

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.

Basis sets ad LCAO/MOThere is a close relationship between chemistry ideas and basis sets.Think about the H2 molecule:

Basis sets and LCAOPhysicists call this LCAO (linear combination of atomic orbitals)The basis functions are the atomic orbitalsChemists call this molecular orbital theoryThere 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.

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 computerAIMPRO used Gaussian orbitalsIt is called a Gaussian Orbital code.

Gaussian OrbitalsThe idea:There are thus three ingredients:An exponent, a controls the width of the Gaussian.A centre R controls the locationA coefficient varied to minimise the energy

The ExponentsTypically vary between 0.1 and 10Si: 0.12 up to 4; F: 0.25 up to 10These 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-potsthen keep them fixed when we look at other defect systems.

The Positions/CoefficientsPositions: we put functions on all atomsIn the past we put them on bond centres tooAbandoned 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!

How good are Gaussians?Problems near the nucleus?True AE wave function was a cusp but the pseudo wave function does not!

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 forcesBut DFT doesnt get them right anywayOnly ever likely to be an issue for surfaces or molecules (our solution: ghost orbitals)

AIMPRO basis setWe 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.

What about ds?We continue, multiplying by 2 pre-factors:

What about ds?This introduces 6 further functions i.e. giving 10 including the s and psOf these 6 functions, 5 are the d-orbitalsOne is an additional s-type orbital:

ddpp and all thatWe 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

Can we do better?Add more d-functions: dddd with 40 functions per atomthis can be important if states high in the conduction band are needed (EELS). Clearly crucial for elements like Fe!

Add more exponentsddpppPddppp

Put functions in extra places (bond centres)Not recommended

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!

Other propertiesStructure converges fastest with basis setEnergy differences converge next fastestConduction band converges more slowlyVibrational frequencies also require care.

Important to be sure, the basis set you are using is good enough for the property that you are calculating!

Contracted basis setsA 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

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-3GAlso regarded as rubbish!

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!

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 functionsWhat we call C44G* (again PRB speak)Similar to chemists 6-31G*

The C44G* basis 13 functions still (28/13)3 times faster than ddppDiamond generally very goodSi: conduction band not converged various approaches (Jons article on Wiki)Chemists use 6-31G* for much routine work.

Results for Si (JPG)

Basis Num Etot/at(Ha) Erel/at(eV)a0 (au)B0 (GPa)Eg (eV)Time (s)Expt 10.26397.91.17216512dddd 40 -3.966670.00010.175 95.70.4725339ddpp 28 -3.964310.06410.195 96.90.52834827173C44G* 13 -3.963500.08610.192 98.50.7411494085Si-C4G 4 -3.942710.65210.390 92.12.28107411

The way forwards? 13 functions still (28/13)3 times faster than ddpp4 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.

Plane WavesAnother 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.

Gaussians vs Plane WavesNumber of Gaussians is very small Gaussians: 20/atomPlane Waves: 1000/atomWell written Gaussian codes are therefore faster.Plane waves are systematic: no assumption as to true wave functionAssumptions are dangerous (they can be wrong!) but they enable more work if they are faster

Gaussians vs Plane WavesPlane waves can be increased until energy convergesIn 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 diamondTrue for 2p elements (C, N, O, F) and 3d metals.Gaussians codes are much faster for these.

In conclusionBasis 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.