Computational Chemistry Molecular Mechanics/Dynamics F = Ma Quantum Chemistry Schr Ö dinger Equation H  = E 

H  E  SchrÖdinger Equation Hamiltonian H =  h 2 /2m e )  i  i 2  i V(r i )  i  j e 2 /r ij Wavefunction Energy Density-Functional Theory Text Book: Density-Functional Theory for Atoms and Molecules by Robert Parr & Weitao Yang

Hohenberg-Kohn Theorems 1 st 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.

2 nd Hohenberg-Kohn Theorem: For a trial density  (r), such that   (r)  0 and, Implication: Variation approach to determine ground state energy and density.

2 nd Hohenberg-Kohn Theorem: Application Minimize E ν [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 F HK [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

ν eff (r) is the effective potential: ν xc (r) is exchange-correlation potential:

Density Matrix One-electron density matrix: Two-electron density matrix:

Thomas-Fermi-Dirac Theory

where, r s 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, r s 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 method The exchange-correlation functional is expressed as: where,,

B3LYP/6-311+G(d,p)B3LYP/6-311+G(3df,2p) RMS=21.4 kcal/molRMS=12.0 kcal/mol RMS=3.1 kcal/molRMS=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 set Electron-electron correlation Exchange-correlation functional How to achieve chemical accuracy: 1~2 kcal/mol? First-Principles Methods

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 ???

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

E XC [  ] is system-dependent functional of  Any hybrid exchange-correlation functional is system-dependent

XC Functional Exp. Database Neural Networks Neural-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 ②. J. E. Hariman, Phys. Rev. A (1988), 24(2), p

c cc c

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 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 2 nd & 3 rd rows (Li - Cl). 6-31G** - adds a set of polarization d orbitals to atoms in 2 nd & 3 rd rows (Li- Cl) and a set of p functions to H Diffuse functions + polarization functions: 6-31+G*, G*, 6-31+G** and G** basis sets. Double-zeta (DZ) basis set: two STO for each AO

6-31G for a carbon atom:(10s12p)  [3s6p] 1s2s2p i (i=x,y,z) 6GTFs 3GTFs 1GTF3GTFs 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    

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,t  Excited state properties  r  all  system properties

Open Systems particle energy H = H S + H B + H SB Time-dependent density-functional theory for open systems

 First-principles method for open systems?

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)(r)

 Holographic electron density theorem for time- independent systems Fournais (2004) Mezey (1999) Riess and Munch (1981)  D (r)  (r)  system properties Analytical continuatio n D (r)(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/ (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: Time–dependent Kohn-Sham equation:

 Time-Dependent DFT for Open Systems Left electroderight electrode system to solve boundary condition Poisson Equation with boundary condition via potentials at S L and S R  L  R Dissipation functional Q (energy and particle exchange with the electrodes) Zheng, Wang, Yam, Mo & Chen, Phys. Rev. B 75, (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 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

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

Xiamen, 12/2009 Color: Current Strength Yellow 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, V b = V 0 (1-e - t/a ) with V 0 = 0.1 mV & a = 1 fs. Blue line in (a) is a fit to transient current, I 0 (1-e -t/τ ) with τ = 2.8 fs & I 0 =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. V b = V 0 (1-e -t/a ) V 0 = 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.

R L 7.39 kΩ L 16.6 pH R c 6.45 kΩ (0.5g 0 -1 ) C aF g 0 =2e 2 /h ≈ 18.8 pH L ~ Q/  V = aF ~

Buttiker, Thomas & Pretre, Phys. Lett. A 180, 364 (1993) Science 313, 499 (2006)