Basis Sets

1. Basis Sets Patrick Briddon

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

3. What is a basis set? Solutions to the Schrödinger equation: are continuous functions, ψ(x). → not good for a modern computer (discrete)

4. Why a basis set? • Idea: • write the solution in terms of a series of functions: • The function Ψ 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?

6. Choosing Basis functions • Try to imagine what the true wavefunction will be like: V ψ

7. Choosing Basis functions ψ Basis states

8. The coefficients • These are determined by using the variationalprinciple of quantum mechanics. • If we have a trial wave-function: • Choose the coefficients to minimise the energy.

9. 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.

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

11. 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”.

12. What about our basis functions? • Atomic orbitals are fine, but they are: • Not well defined – you can’t 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.

13. 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

14. 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.

15. 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!

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

17. 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 doesn’t get them right anyway • Only ever likely to be an issue for surfaces or molecules (our solution: ghost orbitals)

18. 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.

19. What about d’s? • We continue, multiplying by 2 pre-factors:

20. What about d’s? • This introduces 6 further functions • i.e. giving 10 including the s and p’s • Of these 6 functions, 5 are the d-orbitals • One is an additional s-type orbital:

21. 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

22. 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

23. 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!

24. 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!

25. 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

26. 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!

27. 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!

28. 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*

29. The C44G* basis • 13 functions still (28/13)3 times faster than ddpp • Diamond generally very good • Si: conduction band not converged – various approaches (Jon’s article on Wiki) • Chemists use 6-31G* for much routine work.

30. Results for Si (JPG)

31. 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.

32. 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.

33. 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

34. 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.

35. 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.