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.
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.
Structure optimization; example: Vanadium dimer
F
• Put atoms “anywhere”
F
F
• Calculate forces
F
• Forces will move atoms toward configuration with lowest energy (forces = 0)
Algorithms for structure optimization:
•Damped Newton dynamics (option: NEWT)
Δx n  mF n  lΔx n1
•Parameters l and m need to be optimized.
•Quasi-Newton structure optimization (BFGS scheme; option: PORT)
 
Δx  H
n
n 1
F
n
Hessian:
2E
H ij 
xi x j
•No parameters needed
•In principle, only one iteration needed if Hessian is known, and fully harmonic.
•In practise, a few iterations and Hessian updates are needed.
•Nevertheless, this is typically the recommended option.
My results for vanadium: lbond = 1.80 Å (experimental value: 1.77 Å)
1.80 Å
Bigger vanadium clusters: V8+
Different start geometries lead to very different structures
E=0.4 eV
E=0 eV
E=0.8 eV
E=1.8 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.
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)
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)
In - terminated
As - terminated
[110]
[110]
Which reconstruction ?
Use density-functional theory (DFT)
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
Possible structures
a(2x4)
b(2x4)
a2(2x4)
b2(2x4)
a3(2x4)
b3(2x4)
z(4x2)
We also considered the corresponding (4x2) structures
(which are rotated by 90o, and In and As atoms are interchanged)
Phase diagram for InAs(001)
a2 (2x4)
Typical experimental
regime
C. Ratsch et al., Phys. Rev. B 62, R7719 (2000).
Predictions confirmed by STM images
Low As pressure
a2(2x4)
High As pressure
b2(2x4)
Barvosa-Carter, Ross, Ratsch, Grosse, Owen, Zinck, Surf. Sci. 499, L129 (2002)
Phase transitions
Yesterday, we have learned how
to calculate the lattice constant,
by calculating E(V).
• Without pressure, structure B is stable
• With pressure, eventually structure A
becomes stable
Minimize Gibbs free energy (at T=0):
Etot
G  Etot  PV
Structure A
Pressure for phase transition is determined
by:
GA (V1 )  GB (V2 )
Structure B
V1
V2
lattice constant
volume
Etot, A (V1 )  PV1  Etot, B (V2 )  PV2
P
Etot, A (V1 )  Etot, B (V2 )
V2  V1
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)
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)
Ying and Cohen, PRL, 1980
Ying and Cohen, PRB, 1982
• Nevertheless, these calculations are considered one of the first successes
of (the predictive power) of DFT calculations.
Lattice vibrations: A 1-dimensional monatomic chain of atoms
n-2
n-1
n
n+1
n
u(na): displacement of atom n
..
Equation of Motion: M u (na)  K 2u(na)  u((n  1)a)  u((n  1)a)
Mass of atom
Spring constant
Assume solution of form: u (na, t )  ei ( knat )
Periodic boundary condition requires: k 
2 n
a N
Upon substitution, we get solution
 k   2
K
1
| sin ka |
M
2
Dispersion curve for a
monatomic chain
Lattice vibrations of a chain with 2 ions per primitive cell
n-1
n
n+1
n
u1(na): displacement of atom n,1
Spring K
Spring G
u2(na): displacement of atom n,2
Coupled equations of motion:
..
M1 u1 (na)   K u1(na)  u2 (na)  Gu1(na)  u2 ((n  1)a)
..
M 2 u2 (na)   K u2(na)  u1 (na)  Gu2(na)  u1 ((n  1)a)
Solution
 2 k  
K G
1

M eff
M eff
K 2  G 2  2KG cos ka
Dispersion relation for the diatomic linear chain
K G
1
 k  

M eff
M eff
2
K 2  G 2  2KG cos ka
There are N
values of k:
k
2 n
a N
•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
1

 k  2  k  h (k )

2
Optical branch
Because long wavelength modes can interact
with electromagnetic radiation
Acoustic branch
Because   ck for small k, which
is characteristic of sound waves
Lattice vibrations in 3D with p ions per unit cell
•Analysis essentially the same
•For each k, there are 3p normal modes
•The lowest 3 branches are acoustic
•The remaining 3(p-1) branches are
optical
•A “real” phonon spectrum might look
slightly different;
•The reason is that interactions beyond
nearest neighbors are not included, the
potential might not be harmonic, there
are electron-phonon coupling, etc.
•More in the talk by Claudia AmbroschDraxl
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
Frozen phonon calculations
•Choose a supercell that corresponds to
the inverse of wave vector k
•Calculate dynamical matrix D
•in principle, this is done by
displacing each atom in each
direction, and get the forces acting
on all other atoms:
F
Dij  i
x j
/3a /2a
/a
•eigenvalues l are the frequencies
i 
li
M

Dii( diag)
M
•in practise, one exploits the
symmetry of the system (need
group theory)
•Repeat for several k
•More details in the presentation by
Mahboubeh Hortamani
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)
Spectra for VxAry+
3
14
4
5
6
7
8
15
9
10
11
16
17
18
19
20
21
12
22
13
23
• 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
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
Tunable free electron
Laser (FELIX)
Example:
Excitation of V7Ar1 and V7Ar2 at
313 cm-1
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?
Structure determination for V8+
experiment
theory
E=0
E=0.4eV
E=0.8eV
E=1.8eV
Structure determination for V9+
experiment
theory
E=0
E=0.01eV
S=1
S=0
E=0.06eV
S=1
E=0.08eV
S=0
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.
Niobium 7
Experimental Spectra
Calculated Spectra of
lowest energy structure
neutral
neutral
cationic
cationic
Niobium 6
Experimental Spectra
Calculated Spectra of
lowest energy structure
neutral
neutral
cationic
cationic
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.
But there is the
timescale problem:
Even with big computers, we
can’t describe dynamics
beyond microseconds, for up
to ~ 106 atoms
10-13 s
More about molecular dynamics in the talk by Karsten Reuter.
One way to “speed up dynamics” is to use transition state theory (TST)
Transition state theory (TST) to calculate microscopic rate parameters
Transition state theory (Vineyard, 1957):
Energy barrier Ed  Etrans  Ead
Ed
Etrans
D  0 exp(  Ed / kT )
Attempt frequency
(using harmonic
approximations)
0

j 1 j
3 N 1 *
j
j 1

 j ,  *j normal mode frequencies at
adsorption and transition site
Ead
•Finding transition state is a big challenge
•Sometimes, intuition is enough
•Often, sophisticated schemes are needed
(nudged elastic band method, dimer
method, …. )



3N
 *j
j
Model system: Ag/Ag(111) and Ag/Pt(111)
(Brune et al, Phys. Rev. B 52, 14380 (1995))
Ag/Pt(111)
o
100 A
Ag/ 1ML Ag/Pt(111)
T = 65 K, Coverage = 0.12 ML
Nucleation Theory: N ~ (D/F)-1/3
System: Ag on
Pt(111)
Ag/Pt(111)
Ag(111)
Ed (meV)
157 (10)
60 (10)
97 (10)
0 (s-1)
1 x 1013 (0.4)
1 x 109 (0.6)
2 x 1011 (0.5)
Ag/Ag(111)
Results and comparison: Diffusion barrier
System: Ag on
Pt(111)
Ag(111)
1ML Ag
on Pt(111)
Ag with Pt
lattice const.
Experiment
(meV)
157
97
60
DFT
(meV)
150
81
65
60
• Tensile strain: less diffusion
• Compressive strain: increased
diffusion
Lowered diffusion barrier for Ag on
Ag/Pt(111) is mainly an effect of strain.
Ratsch et al., Phys. Rev. B 55, 6750 (1997)
How to calculate the prefactor
Attempt frequency 0



3N

j 1 j
3 N 1 *
j
j 1

• Calculate force constant matrix by displacing each atom in x,y,z-direction,
and by calculating the forces that act on all atoms
*
• Eigenvalues of this matrix are the normal mode frequencies  j ,  j
• Important question: how many degrees of freedom need to be included?
Convergence test for Ag/Ag(111)
only adatom
2x2 cell
adatom and top
layer, 2x2 cell
adatom and 2
layers, 4x4 cell
# degrees of freedom 3
15
99
Prefactor 0 (THz)
0.82
0.71
1.55
Ratsch et al, Phys. Rev. B 58, 13163 (1998)
Results and comparison: Prefactor
System: Ag on
Experiment
Ed (meV)
Experiment
0 (THz)
Pt
Ag (stretched)
Ag
Ag (compressed)
1 ML Ag/Pt
157
10
97
65
0.2
0.001
Theory
Ed (meV)
Theory
0 (THz)
106
81
60
63
0.25
0.82
1.3
~ 7.0
Compensation effect:
• Prefactor is O(1 THz) for all systems
Higher barrier
• Compensation effect is not confirmed
Higher prefactor
• Explanation for experimental result:
• Long range interactions are important for
systems with small barrier (Fichthorn and
Scheffler, PRL, 2000; Bogicevic et al., PRL, 2000),
• Simple nucleation theory does not apply any
longer but can be modified (Venables, Brune,
PRB 2002)
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.