and stay close to the BO surface. - Optimization method instead of variational equation. - Equation of motion instead of
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)
