Transcript No Slide Title

Computational Chemistry
• Molecular Mechanics/Dynamics
F = Ma
• Quantum Chemistry
SchrÖdinger Equation
H = E
Beginning of Computational Chemistry
In 1929, Dirac declared, “The underlying physical
laws necessary for the mathematical theory of ...the
whole of chemistry are thus completely know, and
the difficulty is only that the exact application of
these laws leads to equations much too complicated
to be soluble.”
Dirac
SchrÖdinger Equation
H=E
Wavefunction
Hamiltonian
H = (-h2/2m)2 - (h2/2me)ii2
+  ZZe2/r - i  Ze2/ri
+ i j e2/rij
Energy
Hartree-Fock Equation:
(f + J  K) f = e f
Fock Operator:
Ff+JK
Hydrogen Molecule H2
e
+
+
e
The Pauli principle
two electrons cannot be in the same state.
Hartree-Fock equation
(f+J)f=ef
f(1) = Te(1)+VeN(1)
J(1) = dt2 f*(2) e2/r12 f(2)
one electron operator
two electron Coulomb
operator
LCAO-MO:
f = c11 + c22
Multiple 1 from the left and then integrate :
c1F11 + c2F12 = e (c1 + S c2)
Multiple 2 from the left and then integrate :
c1F12 + c2F22 = e (S c1 + c2)
where,
Fij =  dt i* ( f + J ) j = Hij +  dt i* J j
S =  dt 1 2
(F11 - e) c1 + (F12 - S e) c2 = 0
(F12 - S e) c1 + (F22 - e) c2 = 0
Secular Equation:
F11 - e F12 - S e
F12 - Se F22 - e
bonding orbital:
= 0
e1 = (F11+F12) / (1+S)
f1 = (1+2) / 2(1+S)1/2
antibonding orbital: e2 = (F11-F12) / (1-S )
f2 = (1-2) / 2(1-S)1/2
Hartree-Fock Method
1. Many-Body Wave Function is approximated
by Slater Determinant
2. Hartree-Fock Equation
F fi = ei fi
F Fock operator
fi the i-th Hartree-Fock orbital
ei the energy of the i-th Hartree-Fock orbital
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
Polarization
and a set of p functions to hydrogen
Function
3. Roothaan Method (introduction of Basis functions)
fi = k cki k LCAO-MO
{ k } is a set of atomic orbitals (or basis functions)
4. Hartree-Fock-Roothaan equation
j ( Fij - ei Sij ) cji = 0
Fij  < i| F | j >
Sij  < i| j >
5. Solve the Hartree-Fock-Roothaan equation
self-consistently
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 2nd & 3rd rows (Li - Cl).
6-31G** - adds a set of polarization d orbitals to atoms in
2nd & 3rd 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
(10s12p)  [3s6p]
6-31G for a carbon atom:
1s
2s
2pi (i=x,y,z)
6GTFs
3GTFs
1GTF
3GTFs
1GTF
1CGTF
(s)
1CGTF
(s)
1CGTF
(s)
1CGTF
(p)
1CGTF
(p)
Density-Functional Theory
SchrÖdinger Equation
H=E
Wavefunction
Hamiltonian
H = - (h2/2me)ii2 + i V(ri) + i j e2/rij
Energy
Text Book:
Density-Functional Theory for Atoms and Molecules
by Robert Parr & Weitao Yang
Hohenberg-Kohn Theorems
1st 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.
2nd Hohenberg-Kohn Theorem: For a trial density r’(r),
such that r’(r) 0 and  r’(r) dr = N,
E0  Ev[r’(r)]
Implication: Variation approach to determine ground
state energy and density.
Thomas-Fermi Theory
Kohn-Sham Equations
/2
/2
Density Matrix
Thomas-Fermi-Dirac Theory
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
The National Science and Technology Council (NSTC) was established by Executive Order
12881 on November 23, 1993. This Cabinet-level Council is the principal means within the
executive branch to coordinate science and technology policy across the diverse entities that
make up the federal research and development enterprise. Chaired by the President, the NSTC
is made up of the Vice President, the Director of the Office of Science and Technology Policy,
Cabinet Secretaries and Agency Heads with significant science and technology
responsibilities, and other White House officials.
This new focused initiative will better leverage existing Federal
investments through the use of computational capabilities, data
management, and an integrated approach to materials science and
engineering.
At present, the time frame for incorporating new classes of materials
into applications is remarkably long, typically about 10 to 20 years
from initial research to first use.
The lengthy time frame for materials to move from discovery to
market is due in part to the continued reliance of materials research
and development programs on scientific intuition and trial and error
experimentation. Much of the design and testing of materials is
currently performed through time-consuming and repetitive
experiment and characterization loops. Some of these experiments
could potentially be performed virtually with powerful and accurate
computational tools, but that level of accuracy in such simulations
does not yet exist.
The Materials Genome Initiative will develop the toolsets necessary
for a new research paradigm in which powerful computational
analysis will decrease the reliance on physical experimentation.
This new integrated design continuum — incorporating greater use
of computing and information technologies coupled with advances
in characterization and experiment — will significantly accelerate
the time and number of materials deployed by replacing lengthy and
costly empirical studies with mathematical models and
computational simulations.
Now is the ideal time to enact this initiative; the computing capacity
necessary to achieve these advances exists and related technologies
such as nanotechnology and bio- technology have matured to enable
us to make great progress in reducing time to market at a very low
cost.
Integrating materials computational tools and information with
sophisticated computational and analytical tools already in use in
engineering fields... [promises] to shorten the materials development
cycle from its current 10-20 years to 2 or 3 years. --- National Research
Council of the National Academies of Sciences, in its report on
Integrated Computational Materials Engineering
Computational Tools
The ultimate goal is to generate computational tools that enable realworld materials development, that optimize or minimize traditional
experimental testing, and that predict materials performance under
diverse product conditions.
Achieving these objectives requires a focus in three necessary areas:
(1) creating accurate models of materials performance and validating
model predictions from theories and empirical data;
(2) implementing an open- platform framework to ensure that all code
is easily used and maintained by all those involved in materials
innovation and deployment, from academia to industry; and
(3) creating software that is modular and user- friendly in order to
extend the benefits to broad user communities.
Experimental Tools
The emphasis of the Initiative is on developing and improving
computational capabilities, but it is essential to ensure that these new
tools both complement and fully leverage existing experimental
research on advanced materials. Effective models of materials
behavior can only be developed from accurate and extensive sets of
data on materials properties. Experimental data is required to create
models as well as to validate their key results.
Experimental outputs will additionally be used to provide model
parameters, validate key predictions, and supplement and extend the
range of validity and reliability of the models.
Data
Data — whether derived from computation or experiment — are the
basis of the information that drives the materials development
continuum. Data inform and verify the computational models that
will streamline the development process.
This initiative will emphasize accuracy and verifiability of models
and experimental tools being developed and support informatics
research to enable the most effective retrieval and analysis of
materials data in this new paradigm.
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
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 ???
E [r ] = EXC[r ] + EXC[r ]
ex
XC
EXC [ r ] :
Existing Approx. XC functional
E [ r ] = (1 + a[ r ])E XC [ r ]
ex
XC
a[ r ] = EXC [ r ] / EXC [ r ]
When the exact XC functional is projected onto an
existing XC functional, it should be system-dependent
ex
EXC
[r ] = EXC[r ] + EXC[r ]
E XC [ r ] = a0 [ r ]E XSlater[ r ] - a0 [ r ]E XHF [ r ] + a X [ r ]E XBecke[ r ]
+ aC [ r ]ECLYP [ r ] - aC [ r ]ECVMN [ r ]
ex
EXC
[ r ] = a0[ r ]EXSlater[ r ] + (1 - a0[ r ])EXHF [ r ] + aX [ r ]EXBecke[ r ]
+ aC [ r ]ECLYP [ r ] + (1 - aC [ r ])ECVMN [ r ]
EXC[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
XC Functional
Descriptors must be
functionals of electron density
Neural Networks
v- and N-representability
c
c
c
c
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
r=
(0)
r
+



i

r
=
h
,
r
r
E(t)
Yokojima & Chen, Chem. Phys. Lett., 1998; Phys. Rev. B, 1999
 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
Time-dependent density-functional theory for open systems
Open Systems
H = HS + HB + HSB
particle
energy
 First-principles method for open systems?

ρD (r , 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
•
•
Plane wave
Linearized augmented plane wave (LAPW)
D
 Holographic electron density theorem for timeindependent systems
•
Riess and Munch (1981)
•
Mezey (1999)
•
Fournais (2004)
r(r)
D
Analytical
continuatio
r n(r)
D
r(r)
HK
system properties
 Holographic electron density theorem for timedependent systems
It is difficult to prove the analyticity for r(r,t) rigorously!
rD(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
1 2
i
= hKS i = (-  + veff (t )) i
t
2
EOM for density matrix:
i = [h,  ]
 Time-Dependent DFT for Open Systems
boundary condition
,mL
Left electrode
,mR
right electrode
system to solve
Dissipation functional Q
(energy and particle exchange
with the electrodes)
Poisson Equation with boundary condition via potentials at SL and SR
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 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 AlCNT-Al system. (a) Bias voltage is
turned on exponentially, Vb = V0 (1-et/a) with V = 0.1 mV & a = 1 fs. Blue
0
line in (a) is a fit to transient current,
I0(1-e-t/τ) with τ = 2.8 fs & I0 =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.
Vb = V0 (1-e-t/a)
V0 = 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-CNTAl at t = 4 fs. (c) Schematic
diagram showing induced
charge accumulation at two
interfaces which forms an
effective capacitor.