The Nuts and Bolts of First-Principles Simulation

Lecture 16: DFT for Metallic Systems

CASTEP Developers’ Groupwith support from the ESF k Network

Durham, 6th-13th December 2001

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

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

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

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

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

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

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, {fi}, which are 0 if the band is unoccupied and 1 if occupied

n(r) f i i

2

i

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

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

i i

Since the density is now a function of the occupancies, as well as the wavefunctions, we need to modify oursearch algorithm.

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.

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

n

Fn ESEF

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

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

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.

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:

UJ 1

Where is the susceptibility and U is given by:

2

24

G

eGUG GG

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

VH n(G)

G2

Thus we have:

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

)]([)()( )()()1( rnRrnrn iii

Where R is the density residual, defined as:

rnfrnR i

nnn

i )(2)(

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

i

j

jj

ii nRrnrn1

)()1()1( )()(

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

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

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

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.