The Nuts and Bolts of First-Principles Simulation

The Nuts and Bolts of First-Principles Simulation 13: Gradient corrections and other functionals Durham, 6th-13th December 2001 CASTEP Developers’ Groupwith support from the ESF k Network

Outline • Introduction • Exchange and correlation interactions • What are GGA’s • Implementation • Some examples • When should they be used • Future developments • Closing statements 12: Gradient corrections

Exact XC interaction is unknown. Introduction Within DFT we can write the XC interaction as This relation defines the XC energy. It is simply the Coulomb interaction between an electron an r and the value of its XC hole nxc(r,r’) at r’. 12: Gradient corrections

The XC hole • From the Pauli exclusion principle we have for a given r Which is known as the sum rule. 12: Gradient corrections

Non-local Nature of XC Hole 12: Gradient corrections

Approximations to Exc As we have already seen a simple, but effective approximation to the exchange-correlation interaction is The generalised gradient approximation contains the next term in a derivative expansion if the charge density: 12: Gradient corrections

Why the GGA? • LDA depends only on one variable. • GGA’s require knowledge of 2 variables (the density and its gradient). • In principle one can continue with this expansion. • If quickly convergent, it would characterise a class of many-body systems with increasing accuracy by functions of 1,2,6,…variables. • How fruitful is this? As yet, unknown. 12: Gradient corrections

Some properties of Exc • However we fit the XC contribution, there are some properties that should hold: if we scale the density Then, for example …and many more. However we fit Exc, we don’t want to break what we know is correct. There are many GGA’s that don’t obey known rules. 12: Gradient corrections

Commonly used GGA’s • PW91: J. P. Perdew and Y. Wang, “Accurate and simple analytic representation of the electron-gas correlation energy”, Phys. Rev. B 45 13244 (1992). • PBE: J. P. Perdew, K. Burke and M Ernzerhof, “Generalised gradient approximation made simple”, Phys. Rev. Lett.77 3865 (1996). • RPBE: B. Hammer, L. B. Hansen and J. K. Norskov, “Improved adsorption energies within DFT using revised PBE functionals”, Phys. Rev. B 59 7413 (1999). 12: Gradient corrections

Parameterisations of GGA’s • They are usually split into two easier parts: • Exc=Ex+Ec : Exchange and correlation are treated independently • Commonly used form for the exchange interaction is: where 12: Gradient corrections

Electron exchange • The function generates the gradient expansion for the exchange energy. • It is found that the non-localities given by the gradient term for exchange and correlation tend to oppose one another (see later). • They nearly cancel at low metallic densities. • At higher densities exchange this exchange dominates and its full non-locality is unveiled. 12: Gradient corrections

Gradient Corrections for Exchange • In the previous expression the function F(s) is parameterised by the scaled density gradient and F(s) is…. 12: Gradient corrections

Gradient Corrected Correlation • Ec has a similar form Where ec contains terms in the charge density and its gradient. Accurate analytic representations for ec are available. They all take fairly complicated functional forms. The main problem with fitting GGA’s is not the data to which it’s fitted but getting a simple functional form that correctly represents that data. 12: Gradient corrections

Different Fits for the GGA • PW91 is the first reasonable GGA that can be reliably used over a very wide range of materials. • PW91 contains much of the known correct physics of the exchange and correlation interactions. • PBE is based on PW91 containing the correct features of LDA but the correct (but hopefully not important!) features of PW91 that are ignored are: • (1) Correct 2nd order gradient coefficients of ex and Ec in the slowly varying limit. • (2) Correct non-uniform scaling of ex in the limits where s tend to infinity. 12: Gradient corrections

GGA’s (Continued) • The revised PBE (known as RPBE) only differs from PBE functional in the choice of mathematical form for ex. • It was chosen only for its simplicity (a simpler F(s) than stated previously). • PBE and RPBE contain the same essential physics and also leave out the same (non-essential?) Physics. 12: Gradient corrections

Atomic Energies (PW91) 12: Gradient corrections

Lattice Parameters Lattice parameters of various semiconducting materials (in Angstroms) for the GGA functionals considered here. 12: Gradient corrections

Bulk Modulus Bulk modulus of various semiconducting materials (in GPA) for the GGA functionals considered here. 12: Gradient corrections

Charge Differences (LDA-PW91) 12: Gradient corrections

Non-local XC Functionals Exact XC expression involves an integration over the XC hole nxc(r,r’) surrounding an electron: Where the XC hole is determined from the pair correlation function: Subject to the condition that nxc contains one electron 12: Gradient corrections

Pair-correlation function is unknown. We have approximated it by: We have fitted gxc of this form to quantum Monte Carlo data and also such that it obeys the correct XC scaling rules and cusp conditions For example, the simplest form would be to fit to the exchange in a homogeneous electron gas: where 12: Gradient corrections

Some pictures: XC hole for a simple cosine potential 12: Gradient corrections

Exchange-correlation holes in silicon: 12: Gradient corrections

Closing Statements • GGA’s improve on LDA results in general (but not always!) – Often better lattice parameters, binding energies, etc. • Computationally more complicated that LDA so will be slightly slower to run. • Should be used in particular for transition metal and molecular systems. • It’s not a fix for everything (band gaps underestimated). 12: Gradient corrections

The Nuts and Bolts of First-Principles Simulation