Computational Chemistry

• Molecular Mechanics/DynamicsF = Ma

• Quantum Chemistry SchrÖdinger Equation

H = E

H E SchrÖdinger Equation

HamiltonianH = h2/2me)ii

2 i V(ri)ije2/rij

Wavefunction

Energy

Density-Functional Theory

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).

Implication: electron density determines every thing.

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

such that (r)0 and ,

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

2nd Hohenberg-Kohn Theorem: Application

Minimize Eν[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)

Thomas-Fermi Theory

Ground state energy

Constraint: number of electrons

Using

:

Kohn-Sham Equations

/2

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

For the real system, the energy functional

νeff(r) is the effective potential:

νxc(r) is exchange-correlation potential:

Density MatrixOne-electron density matrix:

Two-electron density matrix:

Thomas-Fermi-Dirac Theory

where, rs 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, rs 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 methodThe exchange-correlation functional is expressed as:

where,

,

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

Hu, Wang, Wong & Chen, J. Chem. Phys. (Comm) (2003)

Usage: interpret experimental results numerical experiments

Goal: predictive tools

Inherent Numerical Errors caused by

Finite basis setElectron-electron correlationExchange-correlation functional

How to achieve chemical accuracy: 1~2 kcal/mol?

First-Principles Methods

In Principle:DFT is exact for ground stateTDDFT is exact for excited states

To find:Accurate / Exact Exchange-Correlation Functionals

Too Many Approximated Exchange-Correlation Functionals

System-dependency of XC functional ???

][][][ XCXCexXC EEE

When the exact XC functional is projected onto an existing XC functional, it should be system-dependent

][/][][

][])[1(][

XCXC

XCexXC

EEa

EaE

:][XCE Existing Approx. XC functional

EXC[] is system-dependent functional of

Any hybrid exchange-correlation functional is system-dependent

][][][ XCXCexXC EEE

][][][][

][][][][][][][ 00

VMNCC

LYPCC

BeckeXX

HFX

SlaterXXC

EaEa

EaEaEaE

][])[1(][][

][][][])[1(][][][ 00

VMNCC

LYPCC

BeckeXX

HFX

SlaterX

exXC

EaEa

EaEaEaE

XC Functional

Exp. Database

Neural Networks

?][aNeural-Networks-based DFT exchange-correlation functional

Descriptors must be

functionals of electron density

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 (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).

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.

cc cc

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 orbital3-21G: 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 rows6-31G**: 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 densityis 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

Yokojima & Chen, Chem. Phys. Lett., 1998; Phys. Rev. B, 1999

t

,hi

HK Theorem P. Hohenberg & W. Kohn, Phys. Rev. 136, B864 (1964)

Ground-state density functional theory (DFT)

First-principles method for isolated systems

Time-dependent DFT for excited states (TDDFT)

RG Theorem E. Runge & E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984)

r,tExcited state properties

rallsystem properties

Open Systems

particle

energy

H = HS + HB + HSB

Time-dependent density-functional theory for open systems

First-principles method for open systems?

properties system opent),r(ρD

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

• Plane wave

• Slater-type orbital

• Gaussian-type orbital

• Linearized augmented plane wave (LAPW)

Is the electron density function of any physical system a real analytical function ?

D

(r)

Holographic electron density theorem for time-independent systems

• Fournais (2004)

• Mezey (1999)

• Riess and Munch (1981)

D(r) (r)system properties

Analytical continuati

on HK

D

(r)

Holographic electron density theorem for time-dependent systems

It is difficult to prove the analyticity for (r,t) rigorously!

D

(r,t)

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

Holographic electron density

theorem

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).

Existence of a rigorous TDDFT for Open System

The electron density distribution of the reduced system determines all physical properties or processes of the entire 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

EOM for density matrix:

],[ hi

Time–dependent Kohn-Sham equation:

ieffiKSi tvh

ti

))(2

1( 2

Time-Dependent DFT for Open Time-Dependent DFT for Open SystemsSystems

Left electrode right electrode

system to solve

boundary condition

Poisson Equation with boundary condition via potentials at SL and SR

L R

Dissipation functional Q

(energy and particle exchange with the

electrodes)

Zheng, Wang, Yam, Mo & Chen, Phys. Rev. B 75, 195127 (2007)

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

Qhi

,

Very General Equation:Time-domain, O(N) & Open systems!

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

where is the reduced density matrix of the system

Our theory: rigorous one-electron QDT

Qhi ],[

System: (5,5) Carbon Nanotube w/ Al(001)-electrodesSim. Box: 60 Carbon atoms & 48x2 Aluminum atoms

Xiamen, 12/2009

Color: Current StrengthYellow arrow: Local Current direction

Transient Current Density Distribution through Al-CNT-Al Structure

Carbon Nanotube

Al Crystal

Time dependent Density Func. Theory

Al Crystal

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.

RL 7.39 kΩL 16.6 pHRc 6.45 kΩ (0.5g0

-1)C 0.073 aF

g0=2e2/h

≈ 18.8 pH LRh

L ~Q/V = 0.052 aF

2e

hd~

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

Science 313, 499 (2006)