Computational Chemistry. Molecular Mechanics/Dynamics F = Ma Quantum Chemistry Schr Ö dinger Equation H = E . Density-Functional Theory. H y = E y. Schr Ö dinger Equation. Wavefunction. Hamiltonian H = - ( h 2 /2m e ) i i 2 + i V(r i ) + i j e 2 /r ij. Energy.
F = Ma
H = E
Hy = Ey
H = - (h2/2me)ii2+ iV(ri) + ije2/rij
Density-Functional Theory for Atoms and Molecules
by Robert Parr & Weitao Yang
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.
such that r (r) 0 and ,
Implication: Variation approach to determine ground state energy and density.
MinimizeEν[ρ] by varying ρ(r):
under constraint: (N is number of electrons)
Then, construct Euler-Langrage equation:
Minimize this Euler-Langrage equation:
(chemical potential or Fermi energy)
number of electrons
In analogy with the Hohenberg-Kohn definition of the universal function FHK[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
The kinetic energy is Ts[ρ]:
νxc(r) is exchange-correlation potential:
One-electron density matrix:
Two-electron density matrix:
where, rs is the radius of a sphere whose volume is the effective volume of an electron;
At high density limit:
At low density limit:
where, rs is the radius of a sphere whose volume is the effective volume of an electron.
If the correlation energy is neglected:
we arrive at Xα equation:
General Gradient Approximation (GGA):
Exchange-correlation potential is viewed as the functional of density and the gradient of density:
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 exchange-correlation functional is expressed as:
B3LYP/6-311+G(d,p)-NEURON & B3LYP/6-311+G(d,p)-NEURON: same accuracy
Usage: interpret experimental results
Goal: predictive tools
Inherent Numerical Errors caused by
Finite basis set
How to achieve chemical accuracy: 1~2 kcal/mol?
DFT is exact for ground state
TDDFT is exact for excited states
Accurate / Exact Exchange-Correlation Functionals
Too Many Approximated Exchange-Correlation Functionals
System-dependency ofXC functional ???
When the exact XC functional is projected onto an
existing XC functional, it should be system-dependent
Any hybrid exchange-correlation functional is system-dependent
Descriptors must be
functionals of electron density
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 (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 ν(r).
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.
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
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
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
1s 2s 2pi (i=x,y,z)
6GTFs 3GTFs 1GTF 3GTFs 1GTF
1CGTF 1CGTF 1CGTF 1CGTF 1CGTF (s) (s) (s) (p) (p)
Runge-Gross Extension: Phys. Rev. Lett. 52, 997 (1984)
(r,t) Properties P (e.g. absorption)
TDDFT equation: exact for excited states
Yokojima & Chen, Chem. Phys. Lett., 1998; Phys. Rev. B, 1999
r(r,t) Excited state properties
Ground-state density functional theory (DFT)
HK Theorem P. Hohenberg & W. Kohn, Phys. Rev.136, B864 (1964)
Time-dependent DFT for excited states (TDDFT)
RG Theorem E. Runge & E. K. U. Gross, Phys. Rev. Lett.52, 997 (1984)
H = HS + HB + HSB
Analyticity of basis functions
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.
rD(r) r(r)system properties
Holographic electron density theorem
It is difficult to prove the analyticity for r(r,t) rigorously!
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
William BlakeTo 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 Kohn-Sham equation:
EOM for density matrix:
system to solve
Dissipation functional Q
(energy and particle exchange with the electrodes)
Poisson Equationwith boundary condition via potentials at SL and SR
Zheng, Wang, Yam, Mo & Chen, Phys. Rev. B 75, 195127 (2007)
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)
s(r,r’;t)=s(R,D;t)Wigner function: f(R, k; t)
with R = (r+r’)/2; D= r-r’
Our EOM:First-principles quantum kinetic equation for transport
Very General Equation:
Time-domain, O(N) & Open systems!
Sim. Box: 60 Carbon atoms & 48x2 Aluminum atoms
Time dependent Density Func. Theory
Color: Current Strength
Yellow arrow: Local Current direction
Transient current (red lines) & applied bias voltage (green lines) for the Al-CNT-Al system. (a) Bias voltage is turned on exponentially, Vb = V0 (1-e-t/a) with V0 = 0.1 mV &a = 1 fs. Blue 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-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 16.6 pH
Rc 6.45 kΩ (0.5g0-1)
C 0.073 aF
Q/DV = 0.052 aF
≈ 18.8 pH
Science 313, 499 (2006)