The Nuts and Bolts of First-Principles Simulation

1. The Nuts and Bolts of First-Principles Simulation 25: DFT Plane Wave Pseudopotential versus Other Approaches Durham, 6th-13th December 2001 CASTEP Developers’ Groupwith support from the ESF k Network

2. Outline • Basis states • Plane waves • The disadvantages of plane waves • The advantages of plane waves • Some myths and half-truths about plane waves • Ionic potentials • Pseudopotentials • The disadvantages of pseudopotentials • The advantages of pseudopotentials • Some myths and half truths about pseudopotentials Lecture 25: DFT PW PP v Other Approaches

3. Outline • Empirical and semi-empirical approaches, quantum mechanical techniques • Density functional theory • Disadvantages of DFT • Advantages of DFT • Myths and half truths about DFT Lecture 25: DFT PW PP v Other Approaches

4. Basis sets • The purpose of a basis set is to allow any arbitrary function y to be expressed as a sum of the basis functions {fi}. • The basis set is said to be COMPLETE if this representation of y is accurate. • If everyone used complete basis sets the only argument would be about the cost of the calculation. • No one uses a complete basis set. Lecture 25: DFT PW PP v Other Approaches

5. Basis sets • Questions: • Can a sufficient accuracy be achieved using an incomplete basis set? • Can you test convergence of predicted properties with the number of basis states? • What is the computational cost of achieving converged properties? Lecture 25: DFT PW PP v Other Approaches

6. Disadvantages of Plane Waves • You need lots of them! • You are forced to use periodic unit cells. • They provide the same accuracy at all points in space even if there is no electronic density there. Lecture 25: DFT PW PP v Other Approaches

7. Advantages of plane waves • There are very efficient algorithms for performing Fourier transforms allowing calculations to be performed in the most efficient space - real space or reciprocal space • Convergence of physical properties is controlled by a single parameter, the cutoff energy, and can be tested. Lecture 25: DFT PW PP v Other Approaches

8. Advantages of plane waves • They provide the same accuracy at all points in space. • The simplicity of plane waves makes it easy to add new functionality to codes and to try alternative algorithms. Lecture 25: DFT PW PP v Other Approaches

9. Some myths and half-truths about plane waves • You cannot use a Coulomb potential for the ions and are forced to use pseudopotentials • At one time it was common to use a Coulomb potential for hydrogen rather than a pseudopotential. • Chris Pickard has performed all electron calculations for carbon systems using a plane wave basis set (which was admittedly rather large!!). Lecture 25: DFT PW PP v Other Approaches

10. Some myths and half-truths about plane waves • You are forced to use periodic supercells if you use plane waves. • It is possible to use a plane wave basis set for non-periodic systems. • Plane waves are totally delocalised functions (which cannot be used in linear scaling approaches) • There are localised basis functions which are very closely related to plane waves. These give identical results to plane wave calculations (and offer better scaling). Lecture 25: DFT PW PP v Other Approaches

11. Ionic potentials • Actually the question is not about the ionic potential which is a Coulomb potential, it is about the core electrons and the way you handle them. • The core electrons have enormous energies - the energy of the 1s electrons is -13.6 Z2 eV, where Z is the atomic number • The core electrons have very small orbits - the radius of the orbitals of the 1s electrons is 1/Z Bohr radii. • Very small errors in the description of the core electrons can produce errors in energies that would completely dominate calculations. Lecture 25: DFT PW PP v Other Approaches

12. Ionic potentials • In most calculations the core electron wavefunctions are not allowed to vary during the calculation - this is the frozen core approximation. • The pseudopotential approximation builds on this idea by removing these core orbitals and includes the orthogonality constraint (to the core orbitals) with the Coulomb potential to produce an effective potential - the pseudopotential - that acts on the valence electrons. Lecture 25: DFT PW PP v Other Approaches

13. Disadvantages of pseudopotentials • The pseudopotential is not unique. • It is rather easy to generate pseudopotentials that do not work (ghost states, etc,). • Local and non-local pseudopotentials use a linear approximation for the exchange-correlation energy due to the addition of valence and core charges. This approximation can fail in which case you need to apply non-linear core corrections. • Ultra-soft pseudopotentials automatically include non-linear core corrections. Lecture 25: DFT PW PP v Other Approaches

14. Some myths and half-truths about pseudopotentials • You cannot describe properties which have significant contribution from the core region (eg. nuclear magnetic resonance). • Augmentation procedures can be used to obtain information in the core. • You cannot allow the core electrons to relax in response to changes in the valence electron system • A fully self-consistent pseudopotential scheme has been implemented. Lecture 25: DFT PW PP v Other Approaches

15. Empirical and semi-empirical approaches v QM. More atoms • Require parameterisation • Empirical (interatomic potentials) • Semi-empirical (tight binding) • Parameter free • Density functional theory • Correlated methods - MP2, MP4, coupled clusters, configuration interaction • Quantum Monte Carlo Higher accuracy Lecture 25: DFT PW PP v Other Approaches

16. Disadvantages of DFT • It only applies to the electronic groundstate (or an electronic system in thermal equilibrium). • Have to use approximations to the true density functional. • Not possible to predict error in the value of any particular property. • Not possible to systematically improve accuracy of calculation. Lecture 25: DFT PW PP v Other Approaches

17. Advantages of DFT • Kohn was awarded the Nobel prize for DFT - this has endowed DFT with a prestige that makes it hard to criticise! • It offers very good scaling of computational cost with system size. • It allows calculations to be performed on large and complex systems. • Given the very large number of DFT calculations the likely accuracy of property prediction for many properties/systems is known. Lecture 25: DFT PW PP v Other Approaches

18. Some myths and half-truths about DFT • DFT tells you nothing about the excited states. • Although the bandgap is wrong, the wavefunctions of excited states are meaningful. • Density functional theory can fail. • The true density functional gets all groundstate energies and densities correct, but it only manages this by psychopathic behaviour. Infinitely small changes in the electron density produce large changes in the XC potential infinitely far away. Lecture 25: DFT PW PP v Other Approaches