Fundamentals of DFT R. Wentzcovitch U of Minnesota VLab Tutorial
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Transcript of Fundamentals of DFT R. Wentzcovitch U of Minnesota VLab Tutorial
Fundamentals of DFTR. WentzcovitchU of MinnesotaVLab Tutorial
• Hohemberg-Kohn and Kohn-Sham theorems
• Self-consistency cycle
• Extensions of DFT
BO approximation
Ir r
intVeT
- Basic equations for interacting electrons and nuclei Ions (RI ) + electrons (ri )
222 2 22 2
, ,
1ˆ2 2 2
I JIion i I
i i j i I I i Ie i I I I Ji j
Z Z eZ eeH
m r R M R Rr r
intˆ ˆ ˆ ˆ ˆtot ext ion ion ion ionH T V V E H E
IRR
ionTextV ion ionE
22ˆ ˆ ( )
2ion I totI I
H H RM
ˆ
|
el el
tot ion ionel el
HE R E
This is the quantity calculatedby total energy codes.
Pseudopotentials
NucleusCore electronsValence electrons
V(r)
1.0
0.5
0.0
-0.5
0
Radial distance (a.u.)
rRl (
r)
1 2 3 4 5
3s orbital of Si
Real atom
Pseudoatom
r
Ion potential
Pseudopotential
1/2 Bond length
BO approximation• Born-Oppenheimer approximation (1927) Ions (RI ) + electrons (ri )
2 2
2 2( )
2 totII I
E R R RM R
2
( )2
I Jtot I J
I J
Z ZeE R E R
R R
( )totI
I
E RF
R
( )tot
lmlm
E R
22( )1
det 0tot
I JI J
E R
R RM M
IRR
Molecular dynamics Lattice dynamics
forces stresses phonons
Electronic Density Functional Theory (DFT) (T = 0 K)
• Hohemberg and Kohn (1964). Exact theory of many-body systems.
3int
ˆˆ ˆ( ) ( ) ( )
|
el el
tot ion ion ext ion ionel el
HE R E T V d rV r n r E
DFT1Theorem I: For any system of interacting particles in an external potential Vext(r), the potential Vext(r) is determined uniquely, except for a constant, by the ground state electronic density n0(r).Theorem II: A universal functional for the energy E[n] in terms of the density n(r) can be defined, valid for any external potential Vext(r). For any particular Vext(r), the exact ground state energy is the global minimum value of this functional, and the density n(r), that minimizes the functional is the ground state density n0(r).
• Proof of theorem I
Assume Vext(1)(r) and Vext
(2)(r) differ by more than a constant and produce the same n(r). Vext
(1)(r) and Vext(2)(r) produce H(1) and H(2) ,
which have different ground state wavefunctions, Ψ(1) and Ψ(2)
which are hypothesized to have the same charge density n(r). It follows that
Then
and
Adding both which is an absurd!
(1) (1) (1) (1) (2) (1) (2)ˆ ˆE H H
(2) (1) (2) (2) (2) (2) (2) (1) (2) (2)ˆ ˆ ˆ ˆH H H H
(2) 3 (1) (2)0( ) ( ) ( )ext extE d r V r V r n r
(1) (2) 3 (1) (2)0( ) ( ) ( )ext extE E d r V r V r n r
(2) (1) 3 (2) (1)0( ) ( ) ( )ext extE E d r V r V r n r
(2) (1) (1) (2)E E E E Hohemberg and Kohn, Phys. Rev. 136, B864 (1964)
• Proof of theorem II
Each Vext(r) has its Ψ(R) and n(r). Therefore the energy Eel(r) can be viewed as a functional of the density.
Consider
and a different n(2)(r) corresponding to a different
It follows that (1) (1) (1) (1) (2) (1) (2)ˆ ˆE H H
(1) (1) (1) (1) (1)ˆHKE E n H
int[ ] [ ] [ ] ( ) ( )HK ext ion ionE n T n E n drV r n r E
[ ] ( ) ( )HK ext ion ionF n drV r n r E (1) ( )extV r
(2) ( )extV r
Hohemberg and Kohn, Phys. Rev. 136, B864 (1964)
The Kohn-Sham Ansatz
3int[ ] ( ) ( )extE n T n E n d rV r n r
[ ] [ ] [ ] ( ) ( ) [ ]Hartree ext xcE n T n E n drV r n r E n
Replacing one problem with another…(auxiliary and tractable non-interacting system)
• Kohn and Sham(1965)
Hohemberg-Kohn functional:
How to find n?
i
ii pmnT 2
2
1][
)()()( rrrni
ii
' ( ') ( )( )
'Hartree
dr n r n rE r
r r
Kohn and Sham, Phys. Rev. 140, A1133 (1965)
• Kohn-Sham equations: (one electron equation)
)()()()()(2
22
rrrVrVrVm iiixcHartreeext
'
)('
)(
][)(
rr
rndr
rn
nErV Hartree
Hartree
)(
][)(
rn
nErV xc
xc
With εis as Lagrange multipliers associated with the orthonormalization constraint and
and
dft2
Minimizing E[n] expressed in terms of the non-interacting system w.r.t. Ψs, while constraining Ψs to be orthogonal:
,|i j i j
• Exchange correlation energy and potential: By separating out the independent particle kinetic energy and the long range Hartree term, the remaining exchange correlation functional Exc[n] can reasonably be approximated as a local or nearly local functional of the density.
[ ] ([ ], )( ) , ( )
( ) ( )xc xc
xc xc
E n n rV r n r n r
n r n r
with and
• Local density approximation (LDA) uses εxc[n] calculagted exactly for the homogeneous electron system
( ) ([ ], )xc xcE n drn r n r
Quantum Monte Carlo by Ceperley and Alder, 1980
• Generalized gradient approximation (GGA) includes density gradients in εxc[n,n’]
• Meaning of the eigenvalues and eigenfunctions:• Eigenvalues and eigenfunctions have only mathematical meaning
in the KS approach. However, they are useful quantities and often have good correspondence to experimental excitation energies and real charge densities. There is, however, one important formal identity
• These eigenvalues and eigenfunctions are used for more accurate
calculations of total energies and excitation energy.
• The Hohemberg-Kohn-Sham functional concerns only ground state
properties.
• The Kohn-Sham equations must be solved self-consistently
ii
dE
dn
Self consistency cycle
0 ( )inn r
0[ ]inV n
( ) ( ) ( ) ( )2
2 ( ) ( ) ( ) ( ) ( )2
i i i iin in out outext Hartree xc i i iV r V r V r r r
m
( )ioutn r
[ ]ioutV n
1[ ] [ ] [ ]i i iin in outV n V n V n
1i i ( ) ( )i i
out inn r n runtil
Extensions of the HKS functional• Spin density functional theory
The HK theorem can be generalized to several types of particles. The most important example is given by spin polarized systems.
( ) ( ) ( )n r n r n r ( ) ( ) ( )s r n r n r
[ , ]HKE E n s2
2 ( , ) ( , ) ( ) ( ) ( )2 ext Hartree xc i i iV r s V r s V r r rm
• Finite T and ensemble density functional theory
The HK theorem has been generalized to finite temperatures.
This is the Mermin functional. This is an even stronger generalization of density functional.
[ , ] [ , ]HK elF n T E n T T S
[ ln (1 ) ln ]B i i ii
S k f f f
1
1 expi
i
B el
f
k T
( ) ( ) ( )ii i
i
n r f r r
D. Mermim, Phys. Rev. 137, A1441 (1965)
Wentzcovitch, Martins, Allen, PRB 1991
Use of the Mermin functional is recommended in the study of metals. Even at 300 K, statesabove the Fermi level are partially occupied.It helps tremendously one to achieve self-consistency. (It stops electrons from “jumping” from occupied to empty states in one step of the cycle to the next.)
This was a simulation of liquid metallic Li at P=0 GPa. The quantity that is conservedwhen the energy levels are occupied according to the Fermi-Dirac distribution is the Mermin free energy, F[n,T].
Dissociation phase boundary
Umemoto, Wentzcovitch, AllenScience, 2006
Few references:-Theory of the Inhomogeneous electron gas, ed. byS. Lundquist and N. March, Plenum (1983).- Density-Functional Theory of Atoms and Molecules, R. Parr and W. Yang, International Series of Monographs on Chemistry, Oxford Press (1989).- A Chemist’s Guide to Density Fucntional Theory, W. Koch, M. C. Holthause, Wiley-VCH (2002).
Much more ahead…