Transcript Document

Physics 250-06 “Advanced Electronic Structure”
Lecture 2. Density Functional Theory
Contents:
1. Thomas-Fermi Theory.
2. Density Functional Theory.
3. Local Density Approximation.
Supplement: Solving Kohn-Sham Equations
Thomas-Fermi Theory.
Thomas and Fermi (1927) proposed Total Energy as Density
Functional:
E[n]  T [n]  [n]
Let’s evaluate kinetic energy for homogeneous electron
gas:
T [n ]    k |  | k    k d k  Cn
2
k
kF
2
3
5/ 3
0
Hence we can write a simplest functional:
E[n]  C  n ( r )dV  
5/ 3
1 n( r )n( r ')
n( r )Vext ( r )  
2
|r  r'|
Thomas-Fermi Theory.
To find the minimum we constrain the density

n(r )dV  N
Let’s evaluate the variational derivative

 n( r )
[ E    n( r )dV ]  0
Obtain equation for the density:
5 2/3
n( r ')dV
Cn ( r )  Vext ( r )  
 0
3
|r  r' |
Thomas-Fermi Theory.
Solution of Thomas Fermi equation produces desired density.
The equation is remarkably simpler than full many-body
Equation. Once density is found one can evaluate total energy.
Corrections are easy to evaluate:
First, kinetic energy corrections (Weiszsacker, 1935)
1
[n ( r )]2
4
Second, exchange energy via homogeneous electron
Gas (Kohn-Sham, 1965)
i ( k k ')( r r ')
kF
1
e
Ex [n]   ij |
| ji   d 3kd 3k '  drdr '
 C2n 4 / 3
0
|r  r' |
|r  r' |
ij
Density Functional Theory, Hohenberg-Kohn, 1964
Many body wave function uniquely defines the density
n( r )    ( r, r2 ...rN )0 ( r, r2 ...rN )dr2 ...drN
*
0
H 0  E00
Hence, for a given external potetnial Vext(r) ground state
total energy can be viewed as density functional E[n]
Note that when density n is the true ground state density E[n]
becomes true ground state total energy. Away from the
minimum E[n] is not necessarily interpreted as the energy!
Density Functional Theory
Density becomes a functional of the external potential Vext(r)
Such densities are called V-representable.
Various extensions have been proposed:
Extensions to spin dependent systems (Barth, Hedin, 1972)
E[n , n ]
Extension to relativistic systems (Vignale, Kohn, 1988)
E[ j(r )]
Extension to finite temperatures
F [n]  E[n]  TS[n]
Time-Dependent DFT. (Runge, Gross, 1984)
Kohn-Sham Theory
Key assumption: represent density by a set of independent
particles moving in some effective field
n  | i |2
occ
Leads to writing down kinetic energy for independent particles
T    i |  | i 
2
occ
Kohn-Sham functional becomes
E[ i ]    i | 2 | i    nVext  EH  E xc
occ
Kohn-Sham Equations
Minimization subject to constrain

2
{E[ k ]   k  | k | dV }  0
*
 i
k
Leads to Kohn-Sham equations
(   Veff ( r )) i ( r )   i i ( r )
2
Veff ( r )  Vext ( r )  VH ( r )  Vxc ( r )
VH ( r )  
n( r ')
dr '
|r  r'|
Meaning of Eigenvalues
Unfortunately, DFT eigenvalues are not excitation energies since
E [n ]   f i  i
i
Therefore, Koopman’s theorem (valid in Hartree Fock) does not
hold in DFT.
However, it was proved by Janak (1977) that DFT eigenvalues
are derivatives with respect to one-electron occupation numbers
(Janak Theorem)
dE[n]
i 
dfi
Current puzzles: Fermi surface predictions, energy gaps,
Metal-insulator transition (e.g. metallization pressure)
Approximations for Exchange Correlation
Exchange Correlation Functional can be represented as
E xc [n ]  
n( r ) g xc ( r, r ')n( r ')
drdr '
|r  r' |
where gxc(r,r’) describes exchange correlation hole.
Approximations for gxc(r,r’) have led to Weighted Density
Approximation (WDA, Gunnarsson, 1979)
Implementation is not straightforward mainly because gxc(r,r’)
is highly non spherical but doable (David Singh et.al) !
Local Density Approximation
Exchange Correlation Functional can be represented as
Exc [n]   n( r ) xc [n( r )]dr
where xc[n(r)] is the energy density in homogeneous
electron gas. Exchange part is trivial:
i ( k k ')( r r ')
kF
1
e
Ex [n]   ij |
| ji   d 3kd 3k '  drdr '
 C2n 4 / 3
0
|r  r' |
|r  r' |
ij
Correlations are more complicated but doable as well
analytical forms:
Barth, Hedin, 1972, Gunnarsson, Lundquvisit, 1974
QMC simulations by Ceperley, Alder (1980)and their
parameterizations by Vosko, Wilk, Nussiar (1980).
Comparison with X-alpha method of Slater
Slater proposed an approximation similar to LDA. Let’s estimate
average Fock exchange potential for free-electrons:
 k | F (r, r ') | k   Cn
1/ 3
 Vx,Slater (r)
The constant C here is different from exchange potential found
in LDA by a factor 3/2 because LDA comes from the energy while
Slater found approximation for potential.
Slater proposed X-Alpha method:
VX  (r)  Cn
1/ 3
If =2/3 we come to Kohn-Sham exchange, if =1, we
come to Slater exchange.  can be treated as adjustable constant.
Important: Kohn Sham theory is variational!
LDA and it accuracy
40 years of experience: what LDA can do and what it cannot do.
Total energies, densities, P(V), elastic constants,
phonons, response functions.
Excitations, optical properties, transport – weakly correlated
systems vs strongly correlated systems.
Generalized Gradient Approximations
Perdew, Wang 1991, simplified version: Perdew, Wang, 1996
Corrections due to density gradients up to second order:
Exc [n, n, 2n]
GGA produces practically the same spectra as LDA.
GGA gives slight improvement of the ground state
properties such as lattice constants, bulk modulus, etc.
Solving Kohn-Sham Equations
Self-Consistent Cycles with respect to Density
Mixing Schemes:
Linear Mixing: Pratt Scheme
Non-linear Mixing: Broyden Scheme.