1. 3: Density Functional Theory
The Nuts and Bolts of First-Principles Simulation
3: Density Functional Theory
CASTEP Developers’ Group with support from the ESF k Network
CASTEP Workshop, Durham University, 6 – 13 December 2001

2. Density functional theory Mike Gillan, University College London
Ground-state energetics of electrons in condensed matter
Energy as functional of density: the two fundamental theorems
Equivalence of the interacting electron system to a non-interacting system in an effective external potential
Kohn-sham equation
Local-density approximation for exchange-correlation energy
CASTEP Workshop, Durham University, 6 – 13 December 2001

3. CASTEP Workshop, Durham University, 6 – 13 December 2001
The problem
Hamiltonian H for system of interacting electrons acted on by electrostatic field of nuclei:
with T kinetic energy, U mutual interaction energy of electrons, V interaction energy with field of nuclei.
To develop theory, V will be interaction with an arbitrary external field:
with ri position of electron i.
Ground-state energy is impossible to calculate exactly, because of electron correlation. DFT includes correlation, but is still tractable because it has the form of a non-interacting electron theory.
CASTEP Workshop, Durham University, 6 – 13 December 2001

4. Energy as functional of density: the first theorem
For given external potential v(r), let many-body wavefunction be . Then ground-state energy Eg is:
and the electron density n(r) by:
where the density operator is defined as:
Theorem 1: It is impossible that two different potentials give rise to the same ground-state density distribution n(r).
Corollary: n(r) uniquely specifies the external potential v(r) and hence the many-body wavefunction
CASTEP Workshop, Durham University, 6 – 13 December 2001

5. Convexity of the energy (1)
Theorem 1 expresses convexity of the energy Eg as function of external potential.
Convexity means: For two external potentials and , go along linear path between them; if is ground-state energy for then:
Proof of follows from 2nd-order perturbation theory:
with and wavefns of ground and excited states, and their energies, and
CASTEP Workshop, Durham University, 6 – 13 December 2001

6. Convexity of the energy (2)
Theorem 1 is equivalent to saying that a change of external potential cannot give a vanishing change of density
This follows from convexity. Convexity implies that at is less than at But , so that:
so that:
Hence:
which demonstrates that , and this is Theorem 1.
CASTEP Workshop, Durham University, 6 – 13 December 2001

7. DFT variational principle the second theorem
Since ground-state energy Eg is uniquely specified by n(r), write it as Eg[n(r)]. It’s useful to separate out the interaction with the external field, and write:
Where F[n(r)] is ground-state expectation value of H0 when density is n(r).
Theorem 2 (variational principle): Ground-state energy for a given v(r) is obtained by minimising Eg[n(r)] with respect to n(r) for fixed v(r), and the n(r) that yields the minimum is the density in the ground state.
Proof: Let v(r) and v’(r) be two different external potentials, with ground-state energies Eg and Eg’ and ground-state wavefns and By Rayleigh-Ritz variational principle:
Where n’(r) is density associated with This proves the theorem. The usual assumptions of non-degenerate ground state is needed.
CASTEP Workshop, Durham University, 6 – 13 December 2001

8. CASTEP Workshop, Durham University, 6 – 13 December 2001
The Euler equation
Write F[n(r)] as:
where T[n] is kinetic energy of a system of non-interacting electrons whose density distribution is n(r). Then:
Variational principle:
subject to constraint:
Handle the constant-number constraint by Lagrange undetermined multiplier, and get:
with undetermined multiplier the chemical potential.
CASTEP Workshop, Durham University, 6 – 13 December 2001

9. CASTEP Workshop, Durham University, 6 – 13 December 2001
Kohn-Sham equation
Rewrite the Euler equation for interacting electrons:
by defining , so that:
But this is Euler equation for non-interacting electrons in potential veff(r), and must be exactly equivalent to Schroedinger equation:
with n(r) given by:
Then put n(r) back into G[n(r)] to get total energy:
CASTEP Workshop, Durham University, 6 – 13 December 2001

10. CASTEP Workshop, Durham University, 6 – 13 December 2001
Self consistency
How to do DFT in practice???
We don’t know G[n(r)], and probably never will, but suppose we know an adequate approximation to it.
Make an initial guess at n(r), calculate and hence
for this initial n(r).
Solve the Kohn-Sham equation with this veff(r) to get the KS orbitals
and hence calculate the new n(r):
The output n’(r) is not the same as the input n(r). So iterate to reduce residual:
The whole procedure is called ‘searching for self consistency’.
CASTEP Workshop, Durham University, 6 – 13 December 2001

11. Exchange-correlation energy
We have already split the total energy into pieces:
Now separate out the Hartree energy:
Then exchange-correlation energy Exc[n] is defined by:
So far, everything is formal and exact. If we knew the exact Exc[n], then we could calculate the exact ground-state energy of any system!
CASTEP Workshop, Durham University, 6 – 13 December 2001

12. Local density approximation
There is one extended system for which Exc is known rather precisely: the uniform electron gas (jellium). For this system, we know exchange-correlation energy per electron as a function of density n.
Local density approximation (LDA): assume the xc energy of an electron at point r is equal to for jellium, using the density n(r) at point r. Then total Exc for the whole system is:
Some kind of justification can be given for LDA (see xxxxxxx). But the main justification is that it works quite well in practice.
CASTEP Workshop, Durham University, 6 – 13 December 2001

13. Kohn-Sham potential in LDA
The effective Kohn-Sham effective potential in general is:
The Hartree potential is:
Exchange-correlation potential in LDA:
Where:
So in LDA, everything can be straightforwardly calculated!
CASTEP Workshop, Durham University, 6 – 13 December 2001

14. CASTEP Workshop, Durham University, 6 – 13 December 2001
Useful references
Here is a selection of references that contain more detail about DFT:
P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)
W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965)
N. D. Mermin, Phys. Rev. 137, A1441 (1965)
R. O. Jones and O. Gunnarsson, Rev. Mod. Phys., 61, 689 (1989)
M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias and J. D. Joannopoulos, Re. Mod. Phys., 64, 1045 (1992)
CASTEP Workshop, Durham University, 6 – 13 December 2001