The Nuts and Bolts of First-Principles Simulation Lecture 16: DFT for Metallic Systems Durham, 6th- 13th December 2001 CASTEP Developers Group with support from the ESF ψ k Network
Overview of talk What is a metal? Problems with metals Finite temperature DFT Density mixing Ensemble DFT Conclusions Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 2
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 Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 3
2. Problems with metals Band crossings at Fermi level Charge sloshing These manifest themselves in several different ways Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 4
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 Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 5
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 Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 6
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 Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 7
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 n(r) = f i Ψ i 2 i Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 8
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 Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 9
The new line search Since the density is now a function of the occupancies, as well as the wavefunctions, we need to modify our search algorithm. Ψ i + λφ i As before, we take a trial step We now recalculate the occupancies as well as the density Fit parabola and move to parabolic minimum Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 10
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. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 11
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 F = E σ n S Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 12 ε σ n E F
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 Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 13
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 Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 14
Smearing Schemes Gaussian Fermi- Dirac Cold Smearing (Methfessel-Paxton) Methfessel and Paxton expanded the deltafunction 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. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 15
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: J = 1 χu Where χ is the susceptibility and U is given by: G U G = δ GG 4 π G e 2 2 Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 16
Thus we have: V H n(g) G 2 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 Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 17
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 n ( i+ 1) ( r ) = n ( i) ( r ) +α R[ n ( i ) ( r)] Where R is the density residual, defined as: R [ ( )] ( i ) 2 ( i ) n r = f Ψ n ( r ) n Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 18 n n
We can use a more general mixing scheme: n i ( i + 1) ( i+ 1) ( r ) = n ( r) + α j j = 1 R [ ] ( j ) n 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 Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 19
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 Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 20
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 Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 21
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 Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 22
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. Nuts and Bolts 2001 Lecture 16: DFT for Metallic Systems 23