Part III: Methods, this and that

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Part III Methods, this and that
Hai-Ping Cheng Department of Physics and the
Quantum Theory Project University of
Florida Beijing, China, June 2 Collaborators C.
Zhang, P. Jiang, J.L. Krause, A.E. Roitberg

• Time-dependent DFT
• Finite bias DFT
• Quantum chemistry hierarchy
• Molecular dynamics and self-assembled alkanethiol
  monolayer

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Quantum method hierarchy
DFT Traditional physics approach -- now used
extensively by QM community
Traditional QM approach -- now scientists are
more educated than ever before
Modern quantum chemistry A. Szabo and N.S.
Ostlund, (Dover) Molecular modeling Principles
and application A. R. Leach second edition
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A brief sketch of TDDFT
In TDDFT, one start with,
The action is defined as,
?(r,t) is a function of V(r,t). From
With adiabatic approximation and linear response
theory,
Optical response function
where
The response function has poles that correspond
to excitation energy. Equivalently, one solves an
eigenvalue problem
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Important references on TDDFT
Gross, E.K.U. and W. Kohn, Time-dependent density
functional theory. Advances in quantum chemistry,
1990. 21 p. 255. Runge, E. and E.K.U. Gross,
Density-Functional Theory for Time-Dependent
Systems. Physical Review Letters, 1984. 52(12)
p. 997-1000. Casida, M.E., Time-dependent density
functional response theory for molecules, in
Recent advances in density functional methods,
D.P. Chong, Editor. 1995, World Scientific
Singapore. p. 155. Jamorski, C., M.E. Casida, and
D.R. Salahub, Dynamic polarizabilities and
excitation spectra from a molecular
implementation of time-dependent
density-functional response theory N-2 as a case
study. Journal of Chemical Physics, 1996.
104(13) p. 5134-5147. Bauernschmitt, R. and R.
Ahlrichs, Treatment of electronic excitations
within the adiabatic approximation of time
dependent density functional theory. Chemical
Physics Letters, 1996. 256(4-5) p.
454-464. Petersilka, M., U.J. Gossmann, and
E.K.U. Gross, Excitation energies from
time-dependent density-functional theory.
Physical Review Letters, 1996. 76(8) p.
1212-1215. Onida, G., L. Reining, and A. Rubio,
Electronic excitations density-functional versus
many-body Green's-function approaches. Reviews of
Modern Physics, 2002. 74(2) p. 601-659. Daniel,
C., Electronic spectroscopy and photoreactivity
in transition metal complexes. Coordination
Chemistry Reviews, 2003. 238 p. 143-166.
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Berry-adiabatic approximation A feasible
approach?
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Introduction
The non-equilibrium self-consistent procedure
based on DFT
Bound states
?
functional
Single-particle potential
Scattering states
(Non-equilibrium distribution functions)
Is the electron density the basic variable of the
system?
or
Can electron density determine the transport
properties of the system under a finite bias?
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Hohenberg-Kohn Theorem
Model
Current I
Left lead
Scattering Region
Right lead
Left and right reservoirs are at infinity and
maintain their own Fermi energies, ?l and ?r. The
bias voltage is defined as Vb ?l- ?r.
General assumptions
Two leads are in equilibrium with two reservoirs.
The temperature of two leads is zero.
The goal of the research is to relate the current
to the bias voltage.
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Hohenberg-Kohn Theorem
?
Obviously, I is a function of Vb.
According to H-K theorem for ground state, ?l is
determined by ?l(r) to an arbitrary constant Vl
and ?r is determined by ?r(r) to another constant
Vr.
If only given the electron density in two leads,
the bias voltage is an arbitrary number.
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Hohenberg-Kohn Theorem
The adiabatic time-dependent process
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Hohenberg-Kohn Theorem
The voltage-dependence of functional
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Hohenberg-Kohn Theorem
The voltage-dependence of functional
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Hohenberg-Kohn Theorem
The non-equilibrium electron density

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Hohenberg-Kohn Theorem
The non-equilibrium electron density
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Hohenberg-Kohn Theorem
The self-consistent procedure
Single-particle states lower than µr

Single-particle states higher than µr
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The Uniform Electron Gas
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The Uniform Electron Gas
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The Uniform Electron Gas
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First Principles Approach Based on DFT
Problems
Functional-dependence Basis-set-dependence
K-S orbital is not real
Efficiency
Truncated Hamiltonian changes channel
Current-induced force
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Hartree-Fock Equation
Hamiltonian of a N-electron, M-nuclei system
Schrödinger Equation
Born-Oppenheimer Approximation Hartree product
Slater determinant
Note There are 2K spin orbitals
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For a two-electron system
The probability of finding the system in such
state is
If the two electrons have different spin and the
probability of finding 1 at r1 and 2 at r2
is i.e. the probability is un-correlated if same
spin,
The probability is correlated! Especially, for
r1r2, P(r1, r1) 0, a fermi hole is said to
exist around an electron!
Introducing notation
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Unrestricted determinants
Use unrestricted determinants as wave
function, one obtain spin-unrestricted
Hartee-Fock (UHF) equation. Unrestricted
determinants are not an eigen- -state of total
spin S2. They cannot be spin-adaptive by
combining a finite number of unrestricted determin
ants (it can be done for spin-restricted Hartree-F
ock (RHF), i.e., UHF recovers some electron
correlation since Ecorr. ? Etrue - ERHF .
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Linear combination of molecular orbitals (LACO)
The choice of basis sets is a matter of
art! numerical atomic wave-function,
all-electron Gaussian functions Slater
orbitals Wanier function pseudo-potential wavele
ts Planwaves pseudo potential atomic
wave-function pseudo-potential ..
Both local functions and planewaves have been
used for finite and crystalline systems!
A basis set is a basis set, a basis set!
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For a polyatomic system,
Each term contains
complicated but
tractable! Define
Kinetic and potential energy term of a electron
in the field of nuclei
Coulomb interaction between electrons
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negative
Exchange energy. This term is zero only if two
electrons have the same spin. In short hand,
Let use a close-shell system as an example, i.e.
total spin 0. In this case, the wave function
clapses to RHF, N electrons, N/2 orbitals.
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Variation principle
For one particular electron, the equation becomes,
Note that
The operator Kj is defined as which is an awkward
operator. We now have,
The famous Fock operator is
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This form of HF equations do not have an unique
solution. One can manipulate the equations such
that ?ij?ij?ij, and then
which can be solved self-consistently! (note the
solutions are NOT more correctly than other
solutions but they can provide some chemical
picture. If use LCAO, the
variation principle leads to
we obtain the Roothan-Hall equation
(close-shell)
FCSCE
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Self-consistent procedure

• 0. diagonalize S--gt UTSU
• Cal. F integrals
• Cal. S
• Diagonalize S
• Form S-1/2
• Initial guess of P
• Form F using integral and P
• Form FS-1/2 F S-1/2 
• Solve eqn. F-EI0 to get E and C by
  diagonalizing F
• Cal CS-1/2 C
• Cal new P from C
• Check convergence

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Semi-empirical methods going down
Start with
1. Zero differential overlap (ZDO)
Complete neglect of differential overlap a
flavor of ZDO Parameterize (????) as ?AB and
?AA A, B different atoms, AA same atom
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Intermediate neglect differential overlap
(INDO) Keep (????) if ??, ?? are from the same
atom --gt keep mono- atomic differential overlap
for one-center integrals. This includes some spin
effects. Neglect diatomic differential overlap
(NDDO) Keep all one-center integrals, all two
center integrals of ?? are on the same atom (A)
and ?? are also on the same atom (B) MINDO/3
parameterization of INDO ZINDO/(s) Zerners
parameterization of INDO (s) spectroscopy MNDO
modified NDDO AM1 improved MNDO PM3 third
parameterization of MNDO MOPACK SAM
semi-ab initio model 1 Hukel, extended Hukel
simplest model
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Hartree-Fock-Slater equation
This equation approximate the exchange K in
f-operator by A simple local operator the
precursor of DFT Slater (1951)
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Ab initio Method

• 0. Spin-unrestricted HF
• Configuration interaction CI
• ?C0?0C1?1 C2?2
• ?o ground state
• ?1 excited state with one electron in upper
  (virtual) orbital
• ?2 two
• The Ci are obtained using variation method.
• Single CI ?C0?C1?
• Active space orbitals involved in constructing
  ?0 ?1

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Many-body perturbation theory
Moller and Plesset (1934) HHoV, to approach
H gradually, let HHo?V 0lt?lt1
.. MP2, MP4 second and 4th order
of perturbation theory
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Molecular dynamics
R Nuclei coordinates r Electron coordinates
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Background

• Self-assembled monolayers (SAMs) are ordered
  molecular assemblies that are formed
  spontaneously by the adsorption of a surfactant
  with a specific affinity of its headgroup to a
  substrate.
• Alkanethiol CH3(CH2) n-1SH SAMs on Au(111)
  Since its discovery in the early 1980s, this
  system has received enormous attention because of
  1) structural simplicity 2) well-defined order
  3) stable
• An ideal system for understanding the
  fundamental interactions in determining the
  self-assembled structure
• Many applications protective coatings (corrosion
  inhibition), wetting control (changing the
  endgroup, hydrophilic vs. hydrophobic),
  bio-related applications and now, molecular
  electronics!
• Experimental work on alkanethiol monolayers
• Low energy helium diffraction (J. Chem. Phys.
  91, 1 ,1989)
• Grazing incidence X-ray diffraction (GIXD)
  (Phys. Rev. Lett. 70, 2447, 1993 Science, 266,
  1216, 1994)
• Scanning tunneling microscopy (STM)
  (Langmuir 10, 2853, 1994)
• Atomic force microscopy (AFM) (J. Chem.
  Phys. 111, 9797, 1999)

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1 J. Chen, et al., Appl. Phys. Lett., 75,624
(1999) 2 Z. J. Donhause, et al., Science,
292,2303 (2001) 3 G. K. Ramachandran, et al.,
Science, 300, 1413 (2003) 4 T. Hugel et al.,
Science, 296, 1103 (2002) 5 S. Yasuda et al.,
J. Am. Chem. Soc., 125, 16430 (2003)
Azobenzene molecules embedded in a alkanethiol
monolayer
A. Ulman, Chem. Rev., 96, 1533 (1996) F.
Schreiber, Prog. Surf. Sci., 65, 151(2000)
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• Azo-benzene molecule as a host molecule because
  of its unique structural response to the laser
  excitation, which provides hope for future
  nano-electronics.
• The structure and dynamics can potentially be
  altered by the properties of the matrix
  monilayer.

trans
cis
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Simulation model
Dreiding force field

• Set parameters

--- Au-S LJ potential (from harmonic) ---
Au-S-C truncated harmonic
potential --- Au C, H, S vdW interaction LJ
12-6 form
Au (111)
Sutton-Chen Potential
1 S.L. Mayo, et al., J. Phys. Chem., 94,8897
(1990) 2 Sutton, A. P., Chen. J, Philos. Mag.,
61, 139 (1990) 3 Finnis, M. W., Sinclair. J.
E., Philos. Mag. A, 50,45 (1984)
Software DL-Poly
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• 0K fully relaxed
• --- 90 molecules, SCH3(CH2)12 (C13)
• --- hcp hollow site, ( )R30
• --- 9 layers with 270 Au atoms each
• layer
• --- PBC and slab model

---- S atom (without chain)
---- 1st layer Au atom
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Caloric curve and heat capacity

• Caloric Curve (Heating) and Heat Capacity
• Low temperature (50K) high
  temperature (800K)

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Snapshots Structure at various T

• Structure

50K
300K
500K
cis
trans
C12
Snapshots of configurations of molecules on a
Au(111) surface at 50K, 300K and 800K
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Software

• Band Structure
• VASP, PWSCF, ABINIT, SIESTA, CRYSTAL
• Finite systems
• Gaussian, NWchem, DMOL, ACES, HyperChem
• Quantum MD
• CPMD, BO-LSD-MD, FHI98MD, Endyne
• Semi-empirical MOPACK
• Classical MD AMBER, TINKER, DL_POLY

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Monolayer Matrix

• The alkanethiol monolayer matrix has been used
  as a host to contain, support and isolate guest
  molecules to study their electrical properties,
  molecular electronic and structural properties,
  which were determined using a STM tip
• An azobenzene molecule was chosen as the guest
  molecule. This molecule has been proposed as a
  light-driven molecular switch.

S. Yasuda et al., J. Am. Chem. Soc., 125, 16430
(2003) C. Zhang et al., Phys. Rev. Lett, 92,
158301 (2004)
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