Quantum mechanics 2 - Lecture 10 Igor Luka cevi cfizika.unios.hr/~ilukacevic/.../qm2/...structure_of_materials_3.pdf · The electronic structure of materials 3 - QMD Quantum mechanics

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


Quantum molecular dynamics (QMD)

Contents


2 Literature



Basic concepts of MD

Molecular dynamics

B. Alder & T. Wainwright(1950s)

A. Rahman (1960s)

Rubber balls and sticks model.




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?




Molecular dynamics - requires apotential:

empirical

semi-empirical

pair potentials(Lennard-Jones)

many-body potentials

polarizable...







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.




A question

Molecular dynamics treats quantum constituents as classical objects. When do you thinkis this treatment justified?

√2mkB T

~�(

N

V

)1/3




Classical objects Newton’s equations of motion

mx(t) = F (x)

x(t) - molecular trajectories



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




U(x1, . . . , xN ) =∑

ij

UL−J (rij )

UL−J (rij ) = 4ε

[(σ

rij

)p

−(σ

rij


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)?




U(x1, . . . , xN ) =∑

ij

UL−J (rij )

UL−J (rij ) = 4ε

[(σ

rij

)p

−(σ

rij


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




Forces:

Fi = −dU

dxi= −∂U(rij )

∂rij· drij

xi




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



Time evolution (trajectories)

Particle trajectory (analytical):

t =

∫ x(t)

0

dx√2

m

{E − 4ε

[(σx

)12

−(σ

x

)6]}




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




t → t + dt:

x(t + dt) = 2x(t)− x(t − dt) +1

mF [x(t)]dt2

Problem: error accumulation



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



QMD

forces from

MD various effective potentialsQMD ab initio from electrons



QMD

forces from

MD various effective potentialsQMD ab initio from electrons

Hellmann-Feynman theorem

FI = − ∂E

∂RI= −

⟨Ψ

∣∣∣∣ ∂H

∂RI

∣∣∣∣Ψ⟩



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



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?



QMD

A question


E = E [{ψi}, {RI}]



QMD

A question


E = E [{ψi}, {RI}]

DFT Car-Parrinello MD (QMD)



QMD

Car-Parrinello approach one unified problem

{motion of the nucleiKS equations for electrons



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

]



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



Application examples

HT/HP phase diagram of carbon.




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.




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.




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.




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.




Up: Models of Si(100) surface including clusters of different size. Down: The slab geometry.




Atomic positions in competing Sin nanoclusters (n = 9, 10, 12 atoms).




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.


Literature

Contents


2 Literature


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


http://www.youtube.com/watch?v=NQhjAtCKghE

http://www.youtube.com/watch?v=RIW65QLWsjE

http://www.youtube.com/watch?v=v4T2Qu2Qtig

Documents

Quantum mechanics 2 - Lecture 10 Igor Luka cevi cfizika.unios.hr/~ilukacevic/.../qm2/...structure_of_materials_3.pdf · The electronic structure of materials 3 - QMD Quantum mechanics