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