Transcript BO approximation - University of Minnesota

Fundamentals of DFT
R. Wentzcovitch
U of Minnesota
VLab Tutorial
• Hohemberg-Kohn and Kohn-Sham theorems
• Self-consistency cycle
• Extensions of DFT
- Basic
equations for
interacting electrons and nuclei
BO approximation
Ions (RI ) + electrons (ri )
Hˆ ion
2
2
2
2
Z
Z
e
Z
e
1
e
2
2
I J
I







 i 2



I
2me i
r

R
2
M
r

r
i j i
i,I
I
i , I RI  RJ
i
I
I
j
2
Vˆint
Tˆe
Hˆ ion  
I
2
2M I
Vˆext
  Hˆ tot ( R)
2
I
Tˆion
R  RI 
r  rI 
Hˆ tot  Tˆ  Vˆext  Vˆint  Eionion  Hˆ  Eionion
Etot  R  
Eionion
 el Hˆ  el
 el |  el
This is the quantity calculated
by total energy codes.
 Eionion
Pseudopotentials
1.0
3s orbital of Si
rRl (r)
0.5
Pseudoatom
Real atom
0.0
-0.5
0
1
2
3
4
5
Radial distance (a.u.)
V(r)
Nucleus
Core electrons
Valence electrons
1/2 Bond length
Pseudopotential
Ion potential
r
BO
approximation
• Born-Oppenheimer approximation (1927)
Ions (RI ) + electrons (ri )
2


2
 Etot ( R)    R     R 
  I
2
2
2M I  RI


e2
Etot ( R) 
2

forces
FI  
Etot ( R)
RI
I J
R  RI 
ZI ZJ
 E  R
RI  RJ
phonons
stresses
lm  
Etot ( R)
lm
Molecular dynamics
det
 2 Etot ( R)
 2  0
M I M J RI RJ
1
Lattice dynamics
Electronic Density Functional Theory (DFT)
(T = 0 K)
Etot ( R) 
 el Hˆ  el
 el |  el
 Eionion  Tˆ  Vˆint   d 3rVext (r )n(r )  Eionion
• Hohemberg and Kohn (1964). Exact theory of many-body systems.
Theorem I: For any system of interacting particles in an external
DFT1 V (r) is determined
potential Vext(r), the potential
ext
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
E (1)  (1) Hˆ (1) (1)  (2) Hˆ (1) (2)
(2) Hˆ (1) (2)  (2) Hˆ (2) (2)  (2) Hˆ (1)  Hˆ (2) (2)
 E (2)   d 3 r Vext(1) (r )  Vext(2) (r ) n0 (r )
Then
and
Adding both
(1)
(2)
E (1)  E (2)   d 3 r Vext
(r )  Vext
(r ) n0 (r )
(1)
E (2)  E (1)   d 3 r Vext(2) (r )  Vext
(r ) n0 (r )
E (2)  E (1)  E (1)  E (2)
which is an absurd!
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.
EHK [n]  T [n]  Eint [n]   drVext (r )n(r )  Eion ion
 FHK [n]   drVext (r )n(r )  Eion ion
(1)
Consider Vext
(r )
E (1)  EHK n(1)    (1) Hˆ (1) (1)
(2)
(2)
V
and a different n (r) corresponding to a different ext (r )
It follows that
E (1)  (1) Hˆ (1) (1)  (2) Hˆ (1) (2)
Hohemberg and Kohn, Phys. Rev. 136, B864 (1964)
The Kohn-Sham Ansatz
Kohn and Sham, Phys. Rev. 140, A1133 (1965)
Hohemberg-Kohn functional:
E[n]  T  n   Eint  n    d 3rVext (r )n(r )
How to find n?
Replacing one problem with another…(auxiliary and tractable
non-interacting system)
• Kohn and Sham(1965)
n(r )   i (r ) i ( r )
i
1
2
T [n ] 
i p i

2m i
dr ' n(r ')n(r )
EHartree (r )  
r r'
E[n]  T [n]  EHartree [n]   drVext (r )n(r )  Exc [n]
Kohn-Sham equations: (one electron equation)
df•Minimizing
E[n] expressed in terms of the non-interacting
t2 system w.r.t. Ψs, while constraining Ψs to be orthogonal:
 i | j   i, j
 2 2

 
  Vext ( r )  VHartree ( r )  Vxc ( r )  i ( r )   i i ( r )
 2m

With εis as Lagrange multipliers associated with the
orthonormalization constraint and
EHartree [n]
dr' n( r )
VHartree ( r ) 

n( r )
r  r'
and Vxc (r ) 
E xc [n]
n(r )
• 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.
with E xc  n   drn(r ) xc ([ n], r ) and

 Exc [n]
 xc ([n], r )
Vxc (r ) 
  xc  n , r   n(r )
 n( r )
 n( r )
• Local density approximation (LDA) uses εxc[n] calculagted
exactly for the homogeneous electron system
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
dE
i 
dni
• 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
Self consistency cycle
nin0 (r)
V [nin0 ]
2

 out ( i )
2
in( i )
in( i )
out ( i )
  Vext (r )  VHartree (r )  Vxc (r )  i (r )   i i (r )

 2m

i
nout
(r )
i  i 1
until
i
V [nout
]
i
out
n (r ) n (r )
V [n ]  V [n ]  V [n ]
i 1
in
i
in
i
in
i
out
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 )
E  EHK [n, s]
2

 
2



 



V
(
r
,
s
)

V
(
r
,
s
)

V
(
r
)

(
r
)



 i
ext
Hartree
xc
i  i (r )
 2m

• 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.
n(r )   fi i (r ) i (r )
i
1
fi 
 i 
1  exp  

 k BTel 
F[n, T ]  EHK [n, T ]  Tel S
S  kB [ fi ln (1  fi ) ln fi ]
i
D. Mermim, Phys. Rev. 137, A1441 (1965)
Use of the Mermin functional is recommended in
the study of metals. Even at 300 K, states
above 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 conserved
when the energy levels are occupied
according to the Fermi-Dirac distribution is
the Mermin free energy, F[n,T].
Wentzcovitch, Martins, Allen, PRB 1991
Dissociation phase boundary
Umemoto, Wentzcovitch, Allen
Science, 2006
Few references:
-Theory of the Inhomogeneous electron gas, ed. by
S. 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…