Applications of Density-Functional Theory:

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Applications of Density-Functional
Theory Structure Optimization, Phase
Transitions, and Phonons
Christian Ratsch UCLA, Department of Mathematics
In previous talks, we have learned how to
calculate the ground state energy, and the forces
between atoms. Now, we will discuss some
important concepts and applications of what we
can do with this.
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Outline

• Structure Optimization
• Optimize bond length
• Optimize atomic structure of a cluster or
  molecule
• Optimize structure of a surface
• Phase transitions
• Phonons
• Structural and vibrational properties of metal
  clusters
• Dynamics on surfaces
• Molecular dynamics
• Use transition state theory DFT can be used to
  calculate energy barriers, prefactors.

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Structure optimization example Vanadium dimer

• Put atoms anywhere
• Calculate forces
• Forces will move atoms toward configuration with
  lowest energy (forces 0)

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Bigger vanadium clusters V8
Different start geometries lead to very different
structures
E0 eV
E1.8 eV

• Each structure is in an energetic local minimum
  (i.e., forces are zero).
• But which one is the global minimum?
• Finding the global minimum is a challenging task
• Sometimes, good intuition is all we need
• But even for O(10) atoms, intuition is often
  not good enough.
• There are many strategies to find global minima.

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Surface relaxation on a clean Al(111) surface
Relaxation obtained with DFT (DMol3 code) (Jörg
Behler, Ph.D. thesis)
D12
D23

• Top layer relaxes outward
• Second layer relaxes inward (maybe)

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Things are more complicated on semiconductor
surfaces

• Semiconductor surfaces reconstruct. Example
  InAs(100)
• Surface reconstruction is important for
  evolution of surface morphology which influences
  device properties
• RHEED experiments show transition of symmetry
  from (4x2) to (2x4)

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Computation details of the DFT calculations

• Computer code used fhi98md
• Norm-conserving pseudopotentials
• Plane-wave basis set with Ecut 12 Ryd
• k summation 64 k points per (1x1) cell
• Local-density approximation (LDA) for
  exchange-correlation
• Supercell with surface on one side,
  pseudo-hydrogen on the other side
• Damped Newton dynamics to optimize atomic
  structure

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Possible structures
b(2x4)
a(2x4)
z(4x2)
b2(2x4)
a2(2x4)
a3(2x4)
b3(2x4)
We also considered the corresponding (4x2)
structures (which are rotated by 90o, and In and
As atoms are interchanged)
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Phase diagram for InAs(001)
C. Ratsch et al., Phys. Rev. B 62, R7719 (2000).
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Predictions confirmed by STM images
Barvosa-Carter, Ross, Ratsch, Grosse, Owen,
Zinck, Surf. Sci. 499, L129 (2002)
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Phase transitions
Yesterday, we have learned how to calculate the
lattice constant, by calculating E(V).
Etot
Structure A
lattice constant volume
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Classical example Phase transition of silicon
Si has a (cubic) diamond structure, which is
semiconducting
The pressure of phase transition has been
computed from DFT to be 99 kbar (experimental
value 125 kbar)
Under pressure, there is a phase transition to
the tetragonal b-tin structure, which is metalic
M.T. Yin, and M.L. Cohen, PRL 45, 1004 (1980)
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Historical remark

• In the original paper, the energies for bcc and
  fcc were not fully converged
• Luckily, this did not matter (for the phase
  transition)

• Nevertheless, these calculations are considered
  one of the first successes of (the predictive
  power) of DFT calculations.

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Lattice vibrations A 1-dimensional monatomic
chain of atoms
Periodic boundary condition requires
Upon substitution, we get solution
Dispersion curve for a monatomic chain
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Lattice vibrations of a chain with 2 ions per
primitive cell
Coupled equations of motion
Solution
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Dispersion relation for the diatomic linear chain
There are N values of k

• For each k, there are 2 solutions, leading to a
  total of 2N normal modes.
• The normal modes are also called phonons, in
  analogy to the term photons, since the energy
  of the N elastic modes are quantized as

Optical branch
Because long wavelength modes can interact with
electromagnetic radiation
Acoustic branch
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Lattice vibrations in 3D with p ions per unit
cell
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How can we calculate phonon spectrum?

• Frozen phonon calculation

• This is what you will do this afternoon.

• DFT perturbation theory

• Molecular Dynamics

• Do an MD simulation for a sufficiently long time
• Measure the time of vibrations for example for
  dimer, this is obvious
• For bigger systems, one needs to do Fourier
  analysis to do this
• Very expensive

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Frozen phonon calculations

• Choose a supercell that corresponds to the
  inverse of wave vector k

• More details in the presentation by Mahboubeh
  Hortamani

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Structural and vibrational properties of small
vanadium clusters

• Why do we care about small metal clusters?
• Many catalytic converters are based on clusters
• Clusters will play a role in nano-electronics
  (quantum dots)
• Importance in Bio-Chemistry

• Small clusters (consisting of a few atoms) are
  the smallest nano-particles!

This work was also motivated by interesting
experimental results by A. Fielicke, G.v.-Helden,
and G. Meijers (all FHI Berlin)
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Spectra for VxAry

• Each cluster has an individual signature
• V13 is the only structure with peaks that are
  beyond 400 cm-1
• Beginning at size 20, the spectra look
  similar. This suggests a bulk-like structure

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Experimental setup using a tunable free electron
laser
Laser Beam clusters are formed, Ar attaches
Mass-Spectrometer
Gas flow (1 Ar in He)
metal-rod
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DFT calculations for small metal clusters

• Computer Code used DMol3
• GGA for Exchange-Correlation (PBE) but we also
  tested and compared LDA, RPBE
• We tested a large number of possible atomic
  structures, and spin states.
• All atomic structures are fully relaxed.
• Determine the energetically most preferred
  structures
• Calculate the vibrational spectra with DFT (by
  diagonalizing force constant matrix, which is
  obtained by displacing each atom in all
  directions)
• Calculate the IR intensities from derivative of
  the dipol moment

• What can we learn from these calculations?
• Confirm the observed spectra
• Determine the structure of the clusters
• Is the spectrum the result of one or several
  isomers?

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Structure determination for V8
experiment
theory
E0
E0.4eV
E0.8eV
E1.8eV
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Structure determination for V9
experiment
theory
S1
E0
E0.01eV
S0
E0.06eV
S1
E0.08eV
S0
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Niobium

• Open issues
• Sometimes neutral and cationic niobium have
  similar spectrum, sometimes they are very
  different
• Cationic Nb is sometimes like cationic V,
  sometimes different.

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Niobium 7
Experimental Spectra
neutral
cationic
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Niobium 6
Experimental Spectra
neutral
cationic
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Molecular dynamics
Once we have the forces, we can solve the
equation of motion for a large number of atoms,
describing the dynamics of a system of interest.
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Transition state theory (TST) to calculate
microscopic rate parameters
Transition state theory (Vineyard, 1957)
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Model system Ag/Ag(111) and Ag/Pt(111) (Brune et
al, Phys. Rev. B 52, 14380 (1995))
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Results and comparison Diffusion barrier
Lowered diffusion barrier for Ag on Ag/Pt(111) is
mainly an effect of strain.
Ratsch et al., Phys. Rev. B 55, 6750 (1997)
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How to calculate the prefactor
Ratsch et al, Phys. Rev. B 58, 13163 (1998)
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Results and comparison Prefactor
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Conclusion and summary

• DFT calculations can be used to optimize the
  atomic structure of a system
• DFT calculations can be used to calculate the
  pressure of a phase transition. This will be part
  of the exercises this afternoon!
• DFT calculations can be used to calculate a
  phonon spectrum. This will also be part of the
  exercises this afternoon.
• DFT calculations can be used to obtain structural
  and vibrational properties of clusters
• DFT calculations can be used to obtain the
  relevant microscopic parameters that describe the
  dynamics on surfaces.