Density functional theory
1
CHEM 6670
omputational Chemistry & Biophysics
Computational Chemistry & Biophysics
Density functional theory
January 29, 2008
Course instructor: Dr. Jana Khandogin
[email protected]
Course web site:
http://ccb.ou.edu/teaching.aspx
Topics
•
Review: Hartree
‐
Fock theory and electron correlation
ethods
methods
•
Reduced density matrix
•
Thomas
‐
Fermi
‐
Dirac model
•
Hohenberg
‐
Kohn theorem
•
Kohn
‐
Sham theory
2
Electronic structure methods
e
e
e
e
E
H
ψ
=
ˆ
Hartree
‐
Fock method
I MP CC
Trial wave function: Single determinant
Variational principle
Approximations of integrals
Adding more determinants
Systematic convergence to
exact solution
3
Semi
‐
empirical methods
CI, MP, CC
Density functional methods
(energy is a functional of electron density;
implicitly models electron exchange and correlation)
Function and functional
•
A function produces a number from a set of variables.
•
A functional produces a number from a function.
•
Wave function is a function of 3N coordinates (N electrons).
)
,
,
(
3
2
1
x
x
x
f
)]
,
,
(
[
3
2
1
x
x
x
f
F
aeuc
to
s auc
o 3
coo d ates (
eecto s)
•
Electron density is a function of 3 coordinates (one electron).
4
)
,...,
(
1
N
r
r
r
r
)
(
1
r
r
ρ
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2
Reduced density matrix
•
We define a density matrix as
•
First
‐
order reduced density matrix
•
econd
‐
rder reduced density matrix
N
N
N
r
d
r
d
r
r
r
r
r
r
N
r
r
r
r
r
r
r
r
r
r
r
r
⋅
⋅
⋅
=
∫
2
2
1
2
1
1
1
1
)
,...
,
'
(
)
,...
,
(
*
)
'
,
(
ψ
γ
)
'
'
,
'
(
)
,
(
*
)
,
'
'
(
2
1
2
1
1
1
1
N
N
N
N
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
⋅
⋅
⋅
⋅
⋅
⋅
=
⋅
⋅
⋅
⋅
⋅
⋅
Seco dode
educed de s ty
at
5
N
N
N
r
r
d
r
r
r
r
r
r
N
N
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
⋅
⋅
⋅
−
=
∫
3
2
1
2
1
2
1
2
1
1
)
,...
'
,
'
(
)
,...
,
(
*
2
)
1
(
)
'
,
'
,
,
(
Electron density function
•
Diagonal elements of the first
‐
order density matrix give the
lectron density function. It is the probability of finding
electron density function. It is the probability of finding
electron one at
∫
⋅
⋅
⋅
=
⋅
⋅
⋅
=
=
N
N
N
r
d
r
d
r
r
r
N
r
d
r
d
r
r
r
r
r
r
N
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
2
2
2
1
2
1
1
1
1
1
)
,...
,
(
)
,...
,
(
)
,...
,
(
*
)
,
(
)
(
ρ
1
r
r
•
The probability of finding electron one anywhere is,
6
...
)
...
(
)
(
2
1
2
1
1
1
N
r
d
r
d
r
d
r
r
N
r
d
r
N
N
=
=
∫
∫
r
r
r
r
r
r
r
∫
N
N
2
2
1
,
Total number of electrons
Electron pair density function
•
Probability of finding electron one at
and electron two at
iagonal elements of the second
‐
rder density matrix by
1
r
r
2
r
r
Diagonal elements of the second order density matrix by
setting
and
.
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 Spring '08
 janaKhandogin
 Chemistry, pH, Quantum Chemistry, Kinetic Energy, electron density, Computational Chemistry & Biophysics Computational Chemistry & Biophysics Density

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