Transcript Density Functionals: Basic DFT Theory
Density Functionals: Basic DFT Theory
Sergio Aragon San Francisco State University CalTech PASI January 4-16, 2004
DFT Publications
Outline
• Wavefunction methods: Hartree-Fock • To know ground state E, what else do we need to know? Electron Density.
• Basic DFT Theorems • Kohn-Sham Equations • Exchange Correlation Functionals • Excited States
Molecular Schroedinger Equation
H
Y (
x
1 ,
x
2 ,…,
x
N ;
R
1 ,…,
R
Nn ) = E Y (
x
1 ,
x
2 ,…,
x
N ;
R
1 ,…,
R
Nn )
H = Te +Ven +Vee +Vnn Te
= 2 2
m e
N
1 2
Ven
=
N N e
1
k n
1 4 0
Z
|
k
e r
2
R k
|
Vee
=
N
1
N e
e
1 4 0 |
e
2
r
r
| Born-Oppenheimer Aproximation
Vnn
=
j N N n
1
k n
1 4 0
Z k
|
Z
R j j e
2
R k
|
Hartree-Fock Theory
Y HF = 1
N
!
1 1 ( (
x
1
x
2 ) ) .
.
1 (
x N
) 2 (
x
1 ) .
.
.
.
.
.
.
.
.
.
.
.
.
.
N
N
(
x
1 ) (
x
2 ) .
.
N
(
x N
) Fictitious system of non interacting electrons.
h
1 2 2
k N
n
1
Z k r
k
E HF =< Y HF |
H
| Y HF
N
1 |
h
| 1 2
N N
1 1 [ 1 |
r
12 | 1 |
r
12 | ]
Vnn
{
h
V HF
(
x
)}
Correlation Energy
•
Exchange Correlation
: Fermions of the same spin cannot move independently of one another. Y is zero for two Fermions of the same spin located at the same point, regardless of their electric charge.
• HF Theory achieves exact Exchange correlation in a non-local manner (the Exchange operator acting on a given orbital requires information everywhere in space of that orbital.) •
Coulomb Correlation
: Electrons repel each other and therefore do not move independently even if they have the different spin state.
• HF Theory misses 100% of the Coulomb Correlation. Chemical accuracy requires post-Hartree Fock methods such as MP2 or CI.
Information: Basic Problem
Benzene N = 42 electrons Number of terms in Y : N! = 1.4 10 51 Number of Cartesian dimensions in which Y lives: 3N = 126 Y is an extremely complex object and contains more information that we need!
Can we make do with less??
The one-electron density!
r (
r
1 ) = N | Y
x
1
x
2 ...
x
N
R
1 ...
R
Nn
| 2
d
1
d x
2 ...
d x
N
This object is merely 3-dimensional, regardless of the number of electrons or the size of the molecule.
Electron Density of Water
Hohenberg-Kohn Theorems
Theorem 1
: The external potential, Ven is determined, up to an additive constant, by the electron density. Since r (
r
1 ) determines the number of electrons N, then the density also determines the ground state wavefunction and
all other electronic properties of the system.
Theorem 2
: There exists a Variational Principle for the density.
The correct density is the one that produces the minimum energy.
Hohenberg, P. & Kohn, W. “The inhomogeneous electron gas”, Phys. Rev.
136
, B864 (1964).
HK Proof I
Prove by contradiction. Assume that there exist two different external potentials V=Ven +Vnn and V’=Ven’ +Vnn’ which both give the same electron density r (
r
1 ). Then we have two Hamiltonians, and
H
’ with the same ground state density and different Y and Y ’. Now we use the variational principle, taking Y ’ as a trial function for the
H
Hamiltonian, to obtain: E 0
<
< Y ’|
H
| Y ’> = < Y ’|
H'
| Y ’> + < Y ’|(
H-H'
)| Y ’> = E 0 ' + < Y ’|(
V-V'
)| Y ’> In addition, we can take the Y as a trial function for the
H’
Hamiltonian, to obtain: E 0 ’
<
< Y |
H'
| Y > = < Y |
H
| Y > + < Y |(
H’-H
)| Y > = E 0 + < Y |(
V’-V
)| Y > Now we recognize that the expectation value of the difference in the external potentials differ only in sign because we assumed that the electron density is the same. When we add the two equations, we then obtain the contradiction: E 0 + E 0 ’ < E 0 ’ + E 0 Thus we conclude there there exists a unique map between the external potential V and the ground state density. This implies that all the energies, including the total energy, is a
functional
of the density. We write this as E = E[ r ]. The density determines the Hamiltonian, and thereby, the wavefunction.
HK Proof II
Suppose we have a trial density r ’. Then this density defines its own wavefunction Y ’, and the expectation value of the true Hamiltonian satisfies the variational principle: < Y ’ | H |Y ’ T[r] Vee[ r] r (
r
)
V
(
r
)
d r
E[r ’] E 0 [ r 0 ] = < Y 0 |
H
| Y 0 >.
Thus, the correct density is the one that produces the minimum energy.
What is a Functional ?
We give an intuitive definition by comparing the situation to the more familiar notion of
function
.
A function f: R R is a map (or assignment) between one number of the set R (say the real numbers) to another number of the set R.
We usually denote this assignment by y = f(x), and we agree that to each value of x there corresponds only one value of y. On the other hand, a
functional
bers; F: {f(x)} is a map between a set of functions and a set of num R, and we denote this by F[f(x)] = y. A simple example is the definite integration operator:
F
[
g
]
a
b g
(
x
)
dx
The specification of the integrable function g(x) produces a number defined in terms of the constants a and b.
Thomas-Fermi-Dirac Model
The uniform electron gas consists of N=Z electrons in a volume which is uniformily positive so as to keep the entire system electrically neutral. For this system, Thomas and Fermi (1927) computed the kinetic energy, and Dirac (1930) computed the Exchange correlation energy: E TFD [ r ] = T[ r ] + Ven[ r ] + Vee[ r ] + Vx[ r ] =
C F
r 5 / 3 (
r
)
d r
Z
r (
r r
)
d r
1 2 r (
r
1 ) r
r
12 (
r
2 )
d r
1
d r
2
C X
r 4 / 3 (
r
)
d r
Coulomb correlation for the electron gas was computed by Monte Carlo methods by Ceperly & Alder, Phys. Rev. Lett. 45, 566 (1980).
r 2 r r
TFD Atomic Density
HF TFD Kinetic Energy poorly represented!
1.0
Sqrt[r] a.u.
2.0
Kohn-Sham Reference System
Introduce a
fictitious
reference system of non-interacting particles with density r S . Then, a single Slater determinant gives the exact solution to the fictitious system, and,
T S
[ r
S
] 1 2
j N
1
j
| 2
j
|
j
r
S
(
r
)
j N
1 |
j
(
r
, ) | 2 Adjust the potential that these particles move under so that they have a density r = r S and write: E[ r ] = T S [ r ] + J[ r ] + Ven+Vnn + Exc[ r ] 1 2 r (
r
1 ) r
r
12 (
r
2 )
d r
1
d r
Exc[ r ] = T[ r ] - T S [ r ] + Eee[ r ] - J[ r ].
All unknowns here!
Kohn-Sham Equations
E
[ r ] 1 2
j N
1
j
| 2
j
|
j
1 2
j
1
i N N
1
j
i
1 |
r
12 |
j
i
Exc
[ r ]
a
1
j N Nn
1
a
j
1 |
r a
1 |
j
a
1
b Nn Nn
1
a
b r ab
EXACT EQUATION!
Variation principle { 1 2 1 2
a Nn
1
Z a r aj
V KS
(
r
1 )}
j
j
j V KS
(
r
1 ) r (
r
2 )
d
r
2
r
12
Vxc
(
r
1 )
HF and DFT Compared
• HF is never exact. DFT could be exact.
• HF formally scales as N Coulomb & Exchange.
4 while DFT scales as N • Both DFT & HF have kinetic energy error.
• HF can be systematically improved.
• DFT can be improved mostly by guessing!
• HF scaling increases dramatically as improved.
• DFT can do transition metals, HF can’t.
3 .
• HF misses Coulomb correlation. DFT includes both • Scaling in DFT is independent of Vxc improvements.
• DFT performs as HF//MP2 or better, with less effort.
HF/KS Orbitals Compared
HF • Koopman’s theorem applies to all HF orbital energies.
• Orbitals yield approx. wavefunction.
• Orbital energies lack correlation, probably not physical.
• HF orbitals do not yield correct density.
DFT • The DFT maximum filled orbital energy is minus the Ionization energy. • Orbitals do not yield wavefunction.
• Orbital energies are probably physical.
• KS orbitals yield the correct density.
Exchange-Correlation Functionals
• Uniform electron gas basis: LDA • Separate the Exchange and Coulomb correlation parts.
• Add terms depending explicitly on the gradient: GGA • Make hybrid functionals containing a percentage of HF exchange: B3LYP
The Local Density Approximation
The LDA Approximation (Koch & Holthausen)
LDA: Quantitatively
LDA E xc
[ r
xc
] ( r (
r
)) r
x
(
r
) ( r ( r
xc
(
r
)) (
r
))
d r
c
( r (
r
))
x
( r (
r
)) 3 4 3 r (
r
) 1 / 3 Coulomb correlation interpolated from Ceperly & Alder.
Generalized Gradient Approximation
Coulomb Correlation: Ec LYP : Lee, Parr &Yang (1988) PW91: Perdew & Wang (1991) P86: Perdew (1986)
F B
Becke (1988): 1 6
s
s
Sinh
1 (
s
) Exchange Correlation: Ex
E x CGA
Becke & Perdew/Wang: BPW91
E x LDA
F
(
s
) r 4 / 3 (
r
)
d r
s
| r r 4 / 3 | Becke & Lee/Parr/Yang: BLYP
F
86 1 Perdew (1986) : 1 .
296
x
2 14
x
4 0 .
2
x
6 1 / 15
x
( 24
s
2 ) 1 / 3
Hybrid Functionals
Add a percentage of HF Exchange since the systematic errors of HF and DFT are in opposite directions.
The famous B3LYP functional:
B
3
LYP E xc
( 1
a
)
E x LSD
aE
0
xc
bE c B
88
cE c LYP
( 1
c
)
E c LSD
The constants a =.2, b=.8, c=.7 are obtained from fits to a reference set of atoms & compounds. Semi-empirical?
Excited States
• The variational principle will always yield the lowest energy of a manifold of states of the same symmetry. Any excited states which have a different symmetry from the ground state can also be found.
• In general, the application of Time Dependent DFT is the most promising for excited states. This technique uses the idea of linear response and evaluates the excited state in terms of the ground state density. TDDFT is implemented in G98.
H
2