Quantum Molecular Dynamics

Pham Tuan Anh Condensed Matter Physics The Abdus Salam International Center for Theoretical Physics

Content ●

Overview of DFT and Ab Initio Calculation

●

Born-Oppenheimer Molecular Dynamics

●

Car-Parrinello Molecular Dynamics

●

Conclusion

Overview of DFT and Ab initio Calculation Many particle problem

 ℏ2 1 2  ∑i − 2m ∆i − Ze r i − R 

 2 1 e Ψ + ∑   Ψ = EΨ 2 i≠ j r i − r j  

Hartree: Smooth distribution of negative charge

U

el



 1    ' ' r = −e ∫ dr ρ r 
 ; ρ r = −e∑ Ψ i r i r − r'

()

()

()

()

2

Seft-consistent equation



    ℏ − ∆Ψ i r + U ion r Ψ i r + e 2  ∑ ∫ dr ' Ψ j r '  2m  2

()

() ()

()

2

   1  
'  Ψ i r = εi Ψ i r r−r  

()

But the wave function does not obey the Pauli principle

()

Hartree-Fock equation


 Ψ r1 ,..., rN = Slater

(

)


 e 2

    ' * ' ' H Hartree Ψ i r − ∑ dr 
 Ψ j r Ψ i r Ψ j r δsis j = εi Ψ i r j r − r'

()

( ) ( ) ()

()

- Include the Exchange term - More complexity since the exchange term contains the integral operator

The rise of DFT Hohenberg-Kohn: - The ground state density determines all the properties of system - The Potential can be described in term of ground state density

Kohn-Sham energy   ℏ  2 3 ion ∇ Ψ + E = ∑ ∫ Ψi  − d r V i  ∫ 2m i   2

+ E XC


  '  2 n r n r
   3 e  3 3 ' r n r d r+ d rd r 
 ∫ ' 2  − r r 

( ) ( ) 

() ()

 n r + E ion ( R I )

( ( ))

Kohn-Sham equations        ℏ 2  2 ion  −  ∇ + V r + VH r + VXC r  Ψ i r = εi Ψ i r  2m  
  ' n r
   δE XC r 2 3 ' VH r = e ∫ 
 d r , VXC r =  δn r r − r'

()

()

()

()

()

()

()

()

() ()

To calculate the Exchange-Correlation term: Local-Density approximation

E XC = ∫ ε XC

  3 r n r d r,

() ()

ε XC

  hom  r = ε XC n r   

()

()

 

Calculation Procedure

Born-Oppenheimer MD 1. From the ionic configuration at time t we compute the minimal orbitals and minimal energy 2. Then we get the force:


 .. ∂E 0 FI ( t ) = − , MR I = FI ( t ) ∂R I 3. Advance ionic configuration R(t) to R(t+dt) by solving Newton's equation of motion (using Verlet algorithm) 4. Back to 1

Drawbacks - A lot of effort for diagonalization - Need to calculate all the eigenvalues of Kohn-Sham Equation

Car-Parrinello MD Car-Parrinello Lagrangian 2 . . 2   * 1 1 3 3 L = ∑ µ ∫ d r Ψ i + ∑ M I R I − E [ ψ i , R I ] + ∑ Λ ij  ∫ d rΨ i Ψ j − δij    i 2 i 2 i

Equation of Motion

 µ Ψ i r, t = − ..

( )

 δE  + ∑ Λ ik Ψ k r, t * Ψ i r, t k  = − Hψ i + ∑ Λ ik Ψ k r, t

( )

( )

( )

k ..

M I R I = −∇ R I E Using Verlet algorithm to describe both the motion of ions and electrons

Remark Equilibrium state   Ψ i r, t = 0 → −Hψ i + ∑ Λ ik Ψ k r, t = 0 ..

( )

( )

k

- Reduces to the Kohn-Sham equation - The orbitals are approximately minimal and stay close to the BO surface - Optimization method instead of variational equation - Equation of motion instead of matrix diagonalization

Be careful - Fictitious mass of electrons - Constraint conditions (using SHAKE)

Conclusions CP MD

BO MD ●

Exactly on BO surface, more accurate in principle

●

Always slightly off BO surface, less accurate

●

dt = Ionic time scales

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dt << Ionic time scales

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Diagonalization or minimization expensive at every time step

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Only orthogonalization, far less expensive time steps

●

Not stable against deviation from BO surface

●

Stable

References ● M. Payne et al, Rev. Mod. Physics 64, 1045 (1992) ● Kohn, Rev. Mod. Physics 71, 1253 (1999) ● M. Martin, Electronic structure