1. The Nuts and Bolts of First-Principles Simulation Lecture 16: DFT for Metallic Systems CASTEP Developers’ Group with support from the ESF  k Network Durham, 6th-13th December 2001

2. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 2 Overview of talk  What is a metal?  Problems with metals  Finite temperature DFT  Density mixing  Ensemble DFT  Conclusions

3. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 3 1. What is a metal?  For our purposes a metal is any system with unoccupied states very close to the Fermi level  This means that several bands may cross over near the Fermi surface

4. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 4 2. Problems with metals  Band crossings at Fermi level  Charge sloshing These manifest themselves in several different ways

5. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 5 Review of Orthogonalisation  We use a Gram-Schmidt method to orthogonalise each band of the search direction to all bands of the wavefunction  We do not orthogonalise with respect to bands above the valence band  For most systems these bands are much higher in energy, so the energy minimisation will remove them from the top band

6. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 6 Orthogonalisation Problems  For metals there are bands very close in energy to the valence band, so the energy minimisation will take a long time to remove them from our trial wavefunction  By including higher, unoccupied bands we can ensure they are orthogonal to the lower bands

7. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 7 Unoccupied Bands  We want to include some unoccupied bands in our calculation  The cost of our calculation scales quadratically with bands, so we need to keep these to a minimum  Unfortunately we cannot determine beforehand which bands are unoccupied  We must run the calculation, and then check that the top band is unoccupied

8. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 8 Band Occupancies  Since we’re including bands above the Fermi level, we need to know which bands are important when constructing the density  Introduce the concept of band occupancies, {f i }, which are 0 if the band is unoccupied and 1 if occupied

9. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 9 Assigning Occupancies  Our energy minimisation algorithm gives us wavefunctions which are orthogonal mixtures of the Kohn-Sham eigenstates  In order to determine which states are occupied or not, we need the true eigenstates of the Kohn-Sham Hamiltonian  We diagonalise the Hamiltonian in the subspace of the bands to get the true eigenstates and eigenenergies

10. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 10 The new line search  As before, we take a trial step  We now recalculate the occupancies as well as the density  Fit parabola and move to parabolic minimum Since the density is now a function of the occupancies, as well as the wavefunctions, we need to modify our search algorithm.

11. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 11 Unfortunately the occupancies are discontinuous and so even very small steps can completely change which bands are important for the density. The result is that this algorithm is often unstable.

12. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 12 3. Finite Temperature DFT  We know that at nonzero temperatures the occupancies are no longer discontinuous  Mermin extended the use of LDA DFT to systems with finite temperatures, in which case the bands are smeared in energy  We now need to minimise the free energy of the system

13. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 13 Partial Occupancies  We can use the same idea for our metallic calculations to improve the conditioning  We smear the bands in energy so that the occupancies become continuous  There is an additional entropic contribution to the energy which must be calculated. In general we have to approximate this term

14. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 14 A Smooth Operator  Because we’re only using the smearing as a way to improve the conditioning of our problem, we’re not restricted to physical smearing schemes  We can use any smooth operator to smear our bands, and if we can accurately calculate the entropic contribution then we can always recover the zero-temperature result

15. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 15 Smearing Schemes  Gaussian  Fermi-Dirac  Cold Smearing (Methfessel-Paxton) Methfessel and Paxton expanded the delta- function in Hermite polynomials. The entropic contribution due to this smearing can be calculated accurately, allowing good zero-temperature energies to be obtained using large smearing widths.

16. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 16 Charge Sloshing  Restrict our attention to the Hartree potential (usually the most important)  Response of the system to a perturbation is given by the dielectric matrix, J: Where  is the susceptibility and U is given by:

17. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 17  Sloshing instabilities arise for large simulation cells, where |G| is small  For metals, degenerate states at the Fermi level can also lead to instabilities, since macroscopic changes in the density can occur for little change in energy Thus we have:

18. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 18 4. Density Mixing  Wavefunctions found for fixed Hamiltonian  Trial density calculated from wavefunctions  The density is then mixed with previous densities, and the mixed density used to construct the new Hamiltonian Where R is the density residual, defined as:

19. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 19  We can use a more general mixing scheme:  Density evolution is now decoupled from that of the wavefunctions  The wavefunction and density searches can be preconditioned separately  The scheme is no longer variational

20. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 20 Density Evolution  By mixing the density with past densities, we damp out charge oscillations  If the damping is small, charge sloshing may still occur  If the damping is large, the system will not converge rapidly to the groundstate  Various mixing schemes exist which attempt to optimally mix the density at each step

21. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 21 5. Ensemble DFT  The instabilities arise because we do not properly account for the self-consistent variation of the density with our trial step  We can avoid these instabilities if we take the trial step not for a fixed Hamiltonian, but for fixed occupancies  Once we have found the optimum wavefunctions we minimise the energy again, this time with respect to the occupancies

22. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 22 The EDFT Algorithm  We have split our minimisation problem into a search over the wavefunctions followed by a search over the occupancies  We need to perform an occupancy search every time the wavefunctions are updated  Each time we update the occupancies we must reapply the Hamiltonian  Scheme is fully variational

23. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 23 6. Conclusions  For metallic systems we must include unoccupied bands  The bands must be smeared with a typical smearing width O(0.1)eV  Sloshing instabilities can arise, and must be quenched using density mixing, or circumvented using ensemble DFT  Metallic calculations are more expensive than for semiconductors, but good convergence can be achieved.