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

In 1929, Dirac declared, “The underlying physical laws necessary for the mathematical theory of...the whole of chemistry are thus completely know, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.” Beginning of Computational Chemistry Dirac

H  E  Schr Ö dinger Equation Hamiltonian H =   (  h 2 /2m      h 2 /2m e )  i  i 2 +     Z  Z  e   r   i     e 2 /r i   i  j  e 2 /r ij Wavefunction Energy

 f + J  – K  F  f + J  – K Hartree-Fock Equation: Fock Operator:

e + + e two electrons cannot be in the same state. Hydrogen Molecule H 2 The Pauli principle

 f + J  f(1) = T e (1)+V eN (1) one electron operator J(1) =   d      e 2 /r 12  two electron Coulomb operator Hartree-Fock equation LCAO-MO:  c 1  1 + c 2  2 Multiple  1 from the left and then integrate : c 1 F 11 + c 2 F 12 =  (c 1 + S c 2 )

Multiple  2 from the left and then integrate : c 1 F 12 + c 2 F 22 =  (S c 1 + c 2 ) where, F ij =  d  i * ( f + J )  j = H ij +  d  i * J  j S =  d  1  2 (F 11 -  ) c 1 + (F 12 - S  ) c 2 = 0 (F 12 - S  ) c 1 + (F 22 -  ) c 2 = 0

Secular Equation: F 11 -  F 12 - S   F 12 - S  F 22 -  bonding orbital:  1 = (F 11 +F 12 ) / (1+S)    = (     ) /  2(1+S) 1/2 antibonding orbital:  2 = (F 11 -F 12 ) / (1-S )    = (     ) /  2(1-S) 1/2

1. Many-Body Wave Function is approximated by Slater Determinant 2. Hartree-Fock Equation F  i =  i  i F Fock operator  i the i-th Hartree-Fock orbital  i the energy of the i-th Hartree-Fock orbital Hartree-Fock Method

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

3. Roothaan Method (introduction of Basis functions)  i =  k c ki  k LCAO-MO {  k } is a set of atomic orbitals (or basis functions) 4. Hartree-Fock-Roothaan equation  j ( F ij -  i S ij ) c ji = 0 F ij  i  F  j  S ij  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 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)

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   ’(r) dr = N, E 0  E v [  ’(r)] Implication: Variation approach to determine ground state energy and density.

Thomas-Fermi Theory

Kohn-Sham Equations /2

Density Matrix

Thomas-Fermi-Dirac Theory

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)

The National Science and Technology Council (NSTC) was established by Executive Order on November 23, This Cabinet-level Council is the principal means within the executive branch to coordinate science and technology policy across the diverse entities that make up the federal research and development enterprise. Chaired by the President, the NSTC is made up of the Vice President, the Director of the Office of Science and Technology Policy, Cabinet Secretaries and Agency Heads with significant science and technology responsibilities, and other White House officials.

This new focused initiative will better leverage existing Federal investments through the use of computational capabilities, data management, and an integrated approach to materials science and engineering. At present, the time frame for incorporating new classes of materials into applications is remarkably long, typically about 10 to 20 years from initial research to first use. The lengthy time frame for materials to move from discovery to market is due in part to the continued reliance of materials research and development programs on scientific intuition and trial and error experimentation. Much of the design and testing of materials is currently performed through time-consuming and repetitive experiment and characterization loops. Some of these experiments could potentially be performed virtually with powerful and accurate computational tools, but that level of accuracy in such simulations does not yet exist.

The Materials Genome Initiative will develop the toolsets necessary for a new research paradigm in which powerful computational analysis will decrease the reliance on physical experimentation. This new integrated design continuum — incorporating greater use of computing and information technologies coupled with advances in characterization and experiment — will significantly accelerate the time and number of materials deployed by replacing lengthy and costly empirical studies with mathematical models and computational simulations. Now is the ideal time to enact this initiative; the computing capacity necessary to achieve these advances exists and related technologies such as nanotechnology and bio- technology have matured to enable us to make great progress in reducing time to market at a very low cost.

Integrating materials computational tools and information with sophisticated computational and analytical tools already in use in engineering fields... [promises] to shorten the materials development cycle from its current years to 2 or 3 years. --- National Research Council of the National Academies of Sciences, in its report on Integrated Computational Materials Engineering

Computational Tools The ultimate goal is to generate computational tools that enable real- world materials development, that optimize or minimize traditional experimental testing, and that predict materials performance under diverse product conditions. Achieving these objectives requires a focus in three necessary areas: (1) creating accurate models of materials performance and validating model predictions from theories and empirical data; (2) implementing an open- platform framework to ensure that all code is easily used and maintained by all those involved in materials innovation and deployment, from academia to industry; and (3) creating software that is modular and user- friendly in order to extend the benefits to broad user communities.

Experimental Tools The emphasis of the Initiative is on developing and improving computational capabilities, but it is essential to ensure that these new tools both complement and fully leverage existing experimental research on advanced materials. Effective models of materials behavior can only be developed from accurate and extensive sets of data on materials properties. Experimental data is required to create models as well as to validate their key results. Experimental outputs will additionally be used to provide model parameters, validate key predictions, and supplement and extend the range of validity and reliability of the models.

Data Data — whether derived from computation or experiment — are the basis of the information that drives the materials development continuum. Data inform and verify the computational models that will streamline the development process. This initiative will emphasize accuracy and verifiability of models and experimental tools being developed and support informatics research to enable the most effective retrieval and analysis of materials data in this new paradigm.

Usage: interpret experimental results numerical experiments Goal: predictive tools Inherent Numerical Errors caused by Finite basis set Electron-electron correlation Exchange-correlation functional 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

c cc c

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.