Quantum mechanics 2 - Lecture 10 Igor Luka cevi...
Transcript of Quantum mechanics 2 - Lecture 10 Igor Luka cevi...
The electronic structure of materials 3 - QMDQuantum mechanics 2 - Lecture 10
Igor Lukacevic
UJJS, Dept. of Physics, Osijek
January 9, 2013
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 1 / 11
Contents
1 Quantum molecular dynamics (QMD)
2 Literature
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 2 / 11
Quantum molecular dynamics (QMD)
Contents
1 Quantum molecular dynamics (QMD)
2 Literature
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 3 / 11
Quantum molecular dynamics (QMD)
Basic concepts of MD
Molecular dynamics
B. Alder & T. Wainwright(1950s)
A. Rahman (1960s)
Rubber balls and sticks model.
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 4 / 11
Quantum molecular dynamics (QMD)
Basic concepts of MD
Molecular dynamics
B. Alder & T. Wainwright(1950s)
A. Rahman (1960s)
Rubber balls and sticks model.
A question
If we, for example, seek the atomicequilibrium positions, which conditionshould the forces satisfy? What wouldbe our stopping criterion?
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 4 / 11
Quantum molecular dynamics (QMD)
Basic concepts of MD
Molecular dynamics - requires apotential:
empirical
semi-empirical
pair potentials(Lennard-Jones)
many-body potentials
polarizable...
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 4 / 11
Quantum molecular dynamics (QMD)
Basic concepts of MD
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 4 / 11
Quantum molecular dynamics (QMD)
Basic concepts of MD
A question
Molecular dynamics treats quantum constituents as classical objects. When do you thinkis this treatment justified?Hint: Instead of matter waves, try to think about the objects’ probability densities.
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 4 / 11
Quantum molecular dynamics (QMD)
Basic concepts of MD
A question
Molecular dynamics treats quantum constituents as classical objects. When do you thinkis this treatment justified?
√2mkB T
~�(
N
V
)1/3
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 4 / 11
Quantum molecular dynamics (QMD)
Basic concepts of MD
Classical objects Newton’s equations of motion
mx(t) = F (x)
x(t) - molecular trajectories
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 4 / 11
Quantum molecular dynamics (QMD)
Lennard-Jones potential
U(x1, . . . , xN ) =∑
ij
UL−J (rij ) , rij = |xi − xj |
UL−J (rij ) = 4ε
[(σ
rij
)p
−(σ
rij
)q]99K Lennard-Jones potential
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 5 / 11
Quantum molecular dynamics (QMD)
Lennard-Jones potential
U(x1, . . . , xN ) =∑
ij
UL−J (rij )
UL−J (rij ) = 4ε
[(σ
rij
)p
−(σ
rij
)q]99K Lennard-Jones potential
rij = |xi − xj |
Most common choice:p = 12q = 6
}=⇒ 12-6 L-J potential
Questions
1 If energy minimum is found at r0 = 21/6 · σ, what is the value of UL−J (r0)? What doyou think is the physical meaning of UL−J (r0)?
2 What is the value of UL−J (σ)?
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 5 / 11
Quantum molecular dynamics (QMD)
Lennard-Jones potential
U(x1, . . . , xN ) =∑
ij
UL−J (rij )
UL−J (rij ) = 4ε
[(σ
rij
)p
−(σ
rij
)q]99K Lennard-Jones potential
rij = |xi − xj |
Most common choice:p = 12q = 6
}=⇒ 12-6 L-J potential
Questions
1 If energy minimum is found at r0 = 21/6 · σ, what is the value of UL−J (r0)? What doyou think is the physical meaning of UL−J (r0)? UL−J (r0) = −ε
2 What is the value of UL−J (σ)? UL−J (σ) = 0
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 5 / 11
Quantum molecular dynamics (QMD)
Lennard-Jones potential
Forces:
Fi = −dU
dxi= −∂U(rij )
∂rij· drij
xi
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 5 / 11
Quantum molecular dynamics (QMD)
Lennard-Jones potential
Forces:
Fi = −dU
dxi= −∂U(rij )
∂rij· drij
xi
∂U(rij )
∂rij=
4ε
σ
[− p
ρp+1+
q
ρq+1
], ρ =
rij
σ
drij
xi=
xi − xj
rij
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 5 / 11
Quantum molecular dynamics (QMD)
Time evolution (trajectories)
Particle trajectory (analytical):
t =
∫ x(t)
0
dx√2
m
{E − 4ε
[(σx
)12
−(σ
x
)6]}
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 6 / 11
Quantum molecular dynamics (QMD)
Time evolution (trajectories)
Particle trajectory (analytical):
t =
∫ x(t)
0
dx√2
m
{E − 4ε
[(σx
)12
−(σ
x
)6]}
⇒ Discretize the derivatives
x(t) =x(t + dt)− x(t)
dt
x(t) =x(t+dt)−x(t)
dt− x(t)−x(t−dt)
dt
dt=
x(t + dt)− 2x(t) + x(t − dt)
dt2
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 6 / 11
Quantum molecular dynamics (QMD)
Time evolution (trajectories)
t → t + dt:
x(t + dt) = 2x(t)− x(t − dt) +1
mF [x(t)]dt2
Problem: error accumulation
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 6 / 11
Quantum molecular dynamics (QMD)
Timestep
t < molecular phenomena timescale
[t] =[EM−1L−2
]E ≈ 1 eV = 1.6 · 10−19 J
M ≈ 1/6.02 · 10−6 kg
L ≈ 1 A = 10−10 m
t ≈ 1.014 · 10−14 ≈ 1 fs
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 7 / 11
Quantum molecular dynamics (QMD)
QMD
forces from
MD various effective potentialsQMD ab initio from electrons
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 8 / 11
Quantum molecular dynamics (QMD)
QMD
forces from
MD various effective potentialsQMD ab initio from electrons
Hellmann-Feynman theorem
FI = − ∂E
∂RI= −
⟨Ψ
∣∣∣∣ ∂H
∂RI
∣∣∣∣Ψ⟩
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 8 / 11
Quantum molecular dynamics (QMD)
QMD
Hellmann-Feynman theorem
FI = − ∂E
∂RI= −
⟨Ψ
∣∣∣∣ ∂H
∂RI
∣∣∣∣Ψ⟩
E from DFT(−1
2∆ + Veff (r)− εj
)ϕj (r) = 0 ,
n (r) =N∑
j=1
|ϕj (r)|2 ,
Veff = Vext(r) + VHartree(r) + Vxc (r) ,
FI = − ∂E
∂RI
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 8 / 11
Quantum molecular dynamics (QMD)
QMD
A question
Consider as system of interacting ions and electrons. Which variables will the totalenergy E depend on if we know how to use DFT?
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 8 / 11
Quantum molecular dynamics (QMD)
QMD
A question
Consider as system of interacting ions and electrons. Which variables will the totalenergy E depend on if we know how to use DFT?
E = E [{ψi}, {RI}]
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 8 / 11
Quantum molecular dynamics (QMD)
QMD
A question
Consider as system of interacting ions and electrons. Which variables will the totalenergy E depend on if we know how to use DFT?
E = E [{ψi}, {RI}]
DFT Car-Parrinello MD (QMD)
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 8 / 11
Quantum molecular dynamics (QMD)
QMD
Car-Parrinello approach one unified problem
{motion of the nucleiKS equations for electrons
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 8 / 11
Quantum molecular dynamics (QMD)
QMD
Car-Parrinello approach one unified problem
{motion of the nucleiKS equations for electrons
Lagrangian
L =N∑
i=1
1
2(2µ)
∫|ψi (r)|2dr +
∑i
1
2MI R
2I − E [ψi ,RI ]
+∑
ij
Λij
[∫ψ∗
i (r)ψj (r)dr − δij
]
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 8 / 11
Quantum molecular dynamics (QMD)
QMD
Lagrangian
L =N∑
i=1
1
2(2µ)
∫|ψi (r)|2dr +
∑i
1
2MI R
2I − E [ψi ,RI ]
+∑
ij
Λij
[∫ψ∗
i (r)ψj (r)dr − δij
]
Equations of motion
µψi (r, t) = − δE
δψ∗i (r)
+∑
k
Λikψk (r, t)
= −Hψi (r, t) +∑
k
Λikψk (r, t)
MI RI = FI = − ∂E
∂RI
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 8 / 11
Quantum molecular dynamics (QMD)
Application examples
HT/HP phase diagram of carbon.
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 9 / 11
Quantum molecular dynamics (QMD)
Application examples
Liquid carbon at p ≈ 0 and T = 5000 K. Left: CPMD and tight-binding method results (describes well the 2-, 3- and 4-fold coordinated C structures) for
g(r). Right: Snapshot of typical low-pressure liquid structure.
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 9 / 11
Quantum molecular dynamics (QMD)
Application examples
Electronic properties of liquid carbon at p ≈ 0 and T = 5000 K. Left: Time averaged DOS close to free-electron parabola (dashed line). Right:
Conductivity σ(ω) averaged over configurations and similar to Drude form, expected for a metal.
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 9 / 11
Quantum molecular dynamics (QMD)
Application examples
HP/HT (core-mantle boundary) radial density of liquid Fe. The integral under the first peak is ≈ 12 atoms, indicating a close packed liquid at all pressures.
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 9 / 11
Quantum molecular dynamics (QMD)
Application examples
Ziegler-Nata reaction during the polymer forming at Ti catalytic sites on MgCl2. Simulating the insertion of a second ethylene molecule offers an insight
into the chain growth process and the stereochemical character of the polymer.
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 9 / 11
Quantum molecular dynamics (QMD)
Application examples
Up: Models of Si(100) surface including clusters of different size. Down: The slab geometry.
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 9 / 11
Quantum molecular dynamics (QMD)
Application examples
Atomic positions in competing Sin nanoclusters (n = 9, 10, 12 atoms).
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 9 / 11
Quantum molecular dynamics (QMD)
Application examples
Equilibrium structures and magnetic moments of Fe clusters with predicted non-collinear spin states. These simulations are essential for treating bulk
magnetism at finite temperature.
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 9 / 11
Literature
Contents
1 Quantum molecular dynamics (QMD)
2 Literature
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 10 / 11
Literature
Literature
1 R. M. Martin, “Electronic Structure - Basic Theory and Practical Methods”,Cambridge University Press, Cambridge, 2004.
2 A. Mattoni, “Introduction to MPMD”, School on Numerical Methods for MaterialsScience Related to Renewable Energy Applications, ICTP, November 2012.
3 http://www.youtube.com/watch?v=NQhjAtCKghE
4 http://www.youtube.com/watch?v=RIW65QLWsjE
5 http://www.youtube.com/watch?v=v4T2Qu2Qtig
Igor Lukacevic (UJJS, Dept. of Physics, Osijek) The electronic structure of materials 3 - QMD January 9, 2013 11 / 11