Transcript No Slide Title
Computational Chemistry
•
Molecular Mechanics/Dynamics F = Ma
•
Quantum Chemistry
Schr
Ö
dinger Equation
H
= E
Density-Functional Theory
SchrÖdinger Equation H
=
E
Wavefunction
Hamiltonian H =
- (
h 2 /2m e )
i
i 2 e 2 /r ij
+
i V(r i )
+
i
j
Energy
Text Book: Density-Functional Theory for Atoms and Molecules by Robert Parr & Weitao Yang
Hohenberg-Kohn Theorems 1 st Hohenberg-Kohn Theorem: The external potential V(r)
is determined, within a trivial additive constant, by the electron density
r
(r).
Implication: electron density determines every thing.
2 nd
r
Hohenberg-Kohn Theorem: For a trial density
(r), such that
r
(r)
0 and , Implication: Variation approach to determine ground state energy and density.
2 nd Hohenberg-Kohn Theorem: Application
Minimize
E
ν
[
ρ
] by varying ρ(
r
) : under constraint: Then, construct Euler-Langrage equation : (
N is number of electrons
) Minimize this Euler-Langrage equation: (chemical potential or Fermi energy)
Thomas-Fermi Theory
Ground state energy Constraint: number of electrons
Using :
Kohn-Sham Equations
In analogy with the Hohenberg-Kohn definition of the universal function
F
HK [ρ], Kohn and Sham invoked a corresponding
noninteracting reference system
, with the Hamiltonian in which there are no electron-electron repulsion terms, and for which
the ground state electron density is exactly ρ
. For this system there will be an exact determinantal ground-state wave function
/2
The kinetic energy is Ts[ρ]:
For the real system, the energy functional
/2
ν
eff
(
r
)
is the effective potential:
ν
xc
(
r
)
is exchange-correlation potential:
Density Matrix
One-electron density matrix: Two-electron density matrix:
Thomas-Fermi-Dirac Theory
where,
r s
is the radius of a sphere whose volume is the effective volume of an electron;
The correlation energy: At high density limit: At low density limit: where,
r s
is the radius of a sphere whose volume is the effective volume of an electron.
In general:
Xα method
If the correlation energy is neglected: we arrive at
Xα equation:
Finally
:
Further improvements
General Gradient Approximation (GGA): Exchange-correlation potential is viewed as the functional of density and the gradient of density: Meta-GGA: Exchange-correlation potential is viewed as the functional of density and the gradient of density and the second derivative of the density: Hyper-GGA: further improvement
The hybrid B3LYP method The exchange-correlation functional is expressed as: where, ,
Hu, Wang, Wong & Chen, J. Chem. Phys. (Comm) (2003) B3LYP/6-311+G(d,p) B3LYP/6-311+G(3df,2p) RMS=21.4 kcal/mol RMS=12.0 kcal/mol RMS=3.1 kcal/mol RMS=3.3 kcal/mol B3LYP/6-311+G(d,p)-NEURON & B3LYP/6-311+G(d,p)-NEURON: same accuracy
First-Principles Methods
Usage: interpret experimental results numerical experiments Goal: predictive tools Inherent Numerical Errors caused by Finite basis set Electron-electron correlation Exchange-correlation functional How to achieve chemical accuracy: 1~2 kcal/mol?
In Principle: DFT is exact for ground state TDDFT is exact for excited states
To find:
Accurate / Exact Exchange-Correlation Functionals Too Many Approximated Exchange-Correlation Functionals System-dependency of XC functional ???
ex E XC
[ r ] =
E XC
[ r ] +
E XC
[ r ]
E XC
[ r ] :
Existing Approx. XC functional
E ex XC
[ r ]
a
[ r ] = = ( 1 +
a
[
E XC
[ r ] / r ])
E XC
[
E XC
[ r ] r ]
When the exact XC functional is projected onto an existing XC functional, it should be system-dependent
ex E XC
[ r ] =
E XC
[ r ] +
E XC
[ r ]
E XC
[ r ] =
a
0 [ r ]
E X Slater
[ r ]
a
0 [ r ]
E X HF
[ r ] +
a X
[ r ]
E X Becke
[ r ] +
a C
[ r ]
E C LYP
[ r ]
a C
[ r ]
E C VMN
[ r ]
ex E XC
[ r ] =
a
0 [ r ]
E X Slater
[ r ] + ( 1 -
a
0 [ r ])
E X HF
[ r ] +
a X
[ r ]
E X Becke
[ r ] +
a C
[ r ]
E C LYP
[ r ] + ( 1 -
a C
[ r ])
E VMN C
[ r ]
E XC
[ r ] is system-dependent functional of r
Any hybrid exchange-correlation functional is system-dependent
Neural-Networks-based DFT exchange-correlation functional
a
[ r ] ?
Exp. Database
Descriptors must be functionals of electron density XC Functional
Neural Networks
v- and N-representability
We can minimize E[ρ] by varying density ρ, however, the variation can not be arbitrary because this ρ is not guaranteed to be ground state density. This is called the v-representable problem.
A density ρ ( ν (
r
).
r
) is said to be v-representable if ρ (
r
) is associated with the ground state wave function of Homiltonian Ĥ with some external potential
v- and N-representability
For more information about
N
-representable density, please refer to the following papers.
① . E.H. Lieb, Int. J. Quantum Chem. (1983), 24(3), p 243-277.
② . J. E. Hariman, Phys. Rev. A (1988), 24(2), p 680-682.
c c c c
Basis set of GTFs
STO-3G, 3-21G, 4-31G, 6-31G, 6-31G*, 6-31G** ------------------------------------------------------------------------------------ complexity & accuracy
Minimal basis set
: one STO for each atomic orbital (AO) STO-3G: 3 GTFs for each atomic orbital 3-21G: 3 GTFs for each inner shell AO 2 CGTFs (w/ 2 & 1 GTFs) for each valence AO 6-31G: 6 GTFs for each inner shell AO 2 CGTFs (w/ 3 & 1 GTFs) for each valence AO 6-31G*: adds a set of d orbitals to atoms in 2nd & 3rd rows 6-31G**: adds a set of d orbitals to atoms in 2nd & 3rd rows and a set of p functions to hydrogen
Polarization Function
Diffuse/Polarization Basis Sets:
For excited states and in anions where electronic density is more spread out, additional basis functions are needed.
Polarization functions to 6-31G basis set as follows:
6-31G* - adds a set of polarized d orbitals to atoms in 2 nd & 3 rd rows (Li - Cl). 6-31G** - adds a set of polarization d orbitals to atoms in 2 nd & 3 rd rows (Li- Cl) and a set of p functions to H
Diffuse functions + polarization functions:
6-31+G*, 6-31++G*, 6-31+G** and 6-31++G** basis sets.
Double-zeta (DZ) basis set:
two STO for each AO
6-31G for a carbon atom:
1s 2s
(10s12p) [3s6p]
2p i (i=x,y,z)
6GTFs 3GTFs 1GTF 3GTFs 1GTF 1CGTF 1CGTF 1CGTF 1CGTF 1CGTF (s) (s) (s) (p) (p)
Time-Dependent Density-Functional Theory (TDDFT) Runge-Gross Extension: Phys. Rev. Lett. 52, 997 (1984) Time-dependent system
r
(r,t)
Properties P (e.g. absorption) TDDFT equation: exact for excited states
To solve Time-dependent Kohn-Sham equation
i
r =
h
, r t = 0 : system is at ground state, r = r (0) t > 0 : r = r (0) + r
r = r
(0)
+ r
E(
t
)
i
r =
heff
, r ( 0 ) + r
Yokojima & Chen, Chem. Phys. Lett., 1998; Phys. Rev. B, 1999
We have performed the TDLDA-LDM calculations on several polyacetylene oligomers. Their excitation energies are presented in Table I. A 6-31G basis set is employed and no cutoff is used in the calculations.
E(t) is set parallel to the
molecules. The time step and total time of the simulation are 0.005 and 70 fs, respectively. We study the excitation to the optically allowed 1 1
Bu state. Compared with the available
experimental excitation energies for the oligomers, it is confirmed that our TDLDA-LDM code yields reasonable predictions for excitation energies.
First-principles method for isolated systems
Ground-state density functional theory (DFT) HK Theorem P. Hohenberg & W. Kohn, Phys. Rev.
136, B864 (1964) r(
r
)
all system properties
Time-dependent DFT for excited states (TDDFT) RG Theorem E. Runge & E. K. U. Gross, Phys. Rev. Lett.
52, 997 (1984) r(
r
,t )
Excited state properties
First-principles method for isolated systems
Ground-state density functional theory (DFT) Kohn-Sham Equation ( 1 2 2 +
V eff
)
i
=
i
i
Time-dependent DFT for excited states (TDDFT) Time-dependent Kohn-Sham equation
i
j
= ˆ
eff
j
= ( 1 2 2 +
V eff
)
j
Time-dependent density-functional theory for open systems
Open Systems
H = H
S
+ H
B
+ H
SB particle energy
First-principles method for open systems?
ρ
D
(
, t)
open system properties
Is the electron density function of any physical system a real analytical function ?
A real function is said to be analytic if it possesses derivatives of all orders and agrees with its Taylor series in the neighborhood of every point.
Analyticity of basis functions
r
(r)
•
Gaussian-type orbital
•
Slater-type orbital
D •
Plane wave
• Linearized augmented plane wave (LAPW)
Holographic electron density theorem for time independent systems
• • Riess and Munch, Theor. Chim. Acta. 58, 295(1981). Mezey, Mol. Phys. 96, 169 (1999) • • Fournais et.al. Ark. Mat. 42, 87(2004).
Jeko T., arXiv.0904.0221v7[math ph](2010).
D r
(r)
Analytical continuation r D
(r)
r
(r)
HK
system properties
Holographic electron density theorem for time dependent systems
It is difficult to prove the analyticity for r (r,t) rigorously!
r D
(r,t) v(r,t) system properties
Holographic electron density theorem r
(r,t)
D
X. Zheng and G.H. Chen, arXiv:physics /0502021 (2005); Yam, Zheng & Chen, J. Comput. Theor. Nanosci. 3, 857 (2006); Recent progress in computational sciences and engineering , Vol. 7A, 803 (2006); Zheng, Wang, Yam, Mo & Chen, PRB (2007).
The electron density distribution of the reduced system determines all physical properties or processes of the entire system!
Existence of a rigorous TDDFT for Open System
Auguries of Innocence
William Blake
To see a world in a grain of sand , And a heaven in a wild flower, Hold infinity in the palm of your hand, And eternity in an hour...
Time-Dependent Density-Functional Theory
Time –dependent Kohn-Sham equation:
i
i
t
=
h KS
i
= ( 1 2 2 +
v eff
(
t
))
i
EOM for density matrix:
i
= [
h
, ]
Q
describes the interactions between system and environment. According to the holographic DFT,
Q
is the functional of ρ
D
(
r
,t), the EOM can be recast into a self-closed form,
Time-Dependent DFT for Open Systems boundary condition
,m L ,m R Left electrode system to solve right electrode Dissipation functional Q (energy and particle exchange with the electrodes)
Poisson Equation
with boundary condition via potentials at S L and S R
Zheng, Wang, Yam, Mo & Chen, Phys. Rev. B 75, 195127 (2007)
Quantum Dissipation Theory (QDT): Louiville-von Neumann Equation
i
r =
h
, r +
Q
where
r
is the reduced N-electron density matrix of the system Our theory: rigorous one-electron QDT Quantum kinetic equation for transport (EOM for Wigner function)
(r,r’;t)=
(R,
;t)
Wigner function: f(R, k; t)
Fourier Transformation with R
= (
r+r’
)/2;
= r-r’ Our EOM:
First-principles quantum kinetic equation for transport i
= [
h
, ] +
Q
Very General Equation: Time-domain, O(N) & Open systems!
System: Sim. Box: (5,5) Carbon Nanotube w/ Al(001)-electrodes 60 Carbon atoms & 48x2 Aluminum atoms
Transient Current Density Distribution through Al-CNT-Al Structure Time dependent Density Func. Theory Color: Current Strength Yellow arrow: Local Current direction
Xiamen, 12/2009
Transient current (red lines) & applied bias voltage (green lines) for the Al CNT-Al system. (a) Bias voltage is turned on exponentially, V b = V 0 (1-e t/a ) with V 0 = 0.1 mV & a = 1 fs . Blue line in (a) is a fit to transient current, I 0 (1-e -t/τ ) with τ = 2.8 fs & I 0 =13.9 nA. (b) Bias voltage is sinusoidal with a period of T = 5 fs. The red line is for the current from the right electrode & squares are the current from the left electrode at different times.
V b V 0 = V 0 (1-e -t/a ) = 0.1 mV & a = 1 fs Switch-on time: ~ 10 fs
(a) Electrostatic potential energy distribution along the central axis at t = 0.02, 1 and 12 fs. (b) Charge distribution along Al-CNT Al at t = 4 fs. (c) Schematic diagram showing induced charge accumulation at two interfaces which forms an effective capacitor.
Dynamic conductance calculated from exponentially turn-on bias voltage (solid squares) and sinusoidal bias voltage (solid triangle). The red line are the fitted results. Upper ones are for the real part and lower ones are for the imaginary part of conductance.
L ~
h
R L
~
d h e
2 ≈ 18.8 pH R L 7.39 kΩ L 16.6 pH R c 6.45 kΩ (
0.5g
0 -
1 ) C 0.073 aF
g 0 =2e 2 /h
Q/ V = 0.052 aF
Buttiker, Thomas & Pretre, Phys. Lett. A 180, 364 (1993) Science 313, 499 (2006)