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Computational Chemistry

- Molecular Mechanics/Dynamics

F = Ma

- Quantum Chemistry

SchrÖdinger Equation

H = E

Hy = Ey

SchrÖdinger Equation

Wavefunction

Hamiltonian

H = - (h2/2me)ii2+ iV(ri) + ije2/rij

Energy

Text Book:

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.

2nd Hohenberg-Kohn Theorem: For a trial density r(r),

such that r (r) 0 and ,

Implication: Variation approach to determine ground state energy and density.

2nd Hohenberg-Kohn Theorem: Application

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)

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]:

/2

νeff(r) is the effective potential:

νxc(r) is exchange-correlation potential:

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.

In general:

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

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

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?

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 ofXC functional ???

Existing Approx. XC functional

When the exact XC functional is projected onto an

existing XC functional, it should be system-dependent

EXC[r] is system-dependent functional of r

Any hybrid exchange-correlation functional is system-dependent

Neural-Networks-based DFT exchange-correlation functional

Exp. Database

XC Functional

Neural Networks

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

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 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

6-31G for a carbon atom: (10s12p) [3s6p]

1s 2s 2pi (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,t) Properties P (e.g. absorption)

TDDFT equation: exact for excited states

r(r,t) Excited state properties

- First-principles method for isolated systems

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)

Time-dependent density-functional theory for open systems

Open Systems

H = HS + HB + HSB

particle

energy

- First-principles method for open systems?

Analyticity of basis functions

- Gaussian-type orbital

D

- Slater-type orbital

- Plane wave

- Linearized augmented plane wave (LAPW)

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.

Holographic electron density theorem for time-independent systems

- Riess and Munch (1981)

- Mezey (1999)

r(r)

- Fournais (2004)

rD(r) r(r)system properties

D

Analytical continuation

rD(r,t) v(r,t)system properties

Holographic electron density theorem

r(r,t)

D

- Holographic electron density theorem for time-dependent systems

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 DFT for Open Systems

boundary condition

Left electrode

right electrode

system to solve

,mL

,mR

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)

Quantum Dissipation Theory (QDT): Louiville-von Neumann Equation

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)

Fourier Transformation

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!

System: (5,5) Carbon Nanotube w/ Al(001)-electrodes

Sim. Box: 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

Al Crystal

Carbon Nanotube

Al Crystal

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, 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.

Buttiker, Thomas & Pretre, Phys. Lett. A 180, 364 (1993)

Science 313, 499 (2006)

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