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Computational Chemistry
• Molecular Mechanics/Dynamics
F = Ma
• Quantum Chemistry
SchrÖdinger Equation
H = E
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
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 Equation:
(f + J  K) f = e f
Fock Operator:
Ff+JK
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.
/2
Kohn-Sham Equations
/2
DFT Method
1. Many-Body Wave Function is approximated
by Slater Determinant
2. Kohn-Sham 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)
Thomas-Fermi Theory
Density Matrix
Thomas-Fermi-Dirac Theory
MAD=22.6 kcal/mol
Hu, Wang, Wong & Chen, J. Chem. Phys. (Comm) (2003)
B3LYP/6-311+G(d,p)
MAD=22.6 kcal/mol
MAD=1.59 kcal/mol
B3LYP/6-311+G(3df,2p)
MAD=11.6 kcal/mol
MAD=1.45 kcal/mol
Table 3. Mean absolute deviations (MAD) and root-mean-square (RMS) for 539
molecules obtained from data set in this study (all data are in the units of kcal mol-1).
Figure 5. Experimental ΔfH298Ɵ versus the calculated ΔfH298Ɵ for 90 molecules in the testing set. Lines
in the bottom indicate the distribution of training set along the X-axis. The error bars on the figures are
obtained from bootstrapping approach. All values are in the units of kcal mol-1.
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
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)