1. Car Parrinello Molecular Dynamics
Nicholas Walker
Jürg Hutter, “Introduction to Ab Initio Molecular Dynamics.” Physical Chemistry Institute, University of Zurich. Winterthurerstrasse 190
Dominik Marx and Jürg Hutter, “Ab initio molecular dynamics: Theory and Implementation.” Modern Methods and Algorithms of Quantum Chemistry, J. Grotendorst (Ed.), John von Neumann Institute for Computing, Jülich, NIC Series, Vol. 1, ISBN , pp , 2000.

2. Introduction – Ab Initio Molecular Dynamics
Chemically complex systems are not well-suited for classical MD
Many different types of atoms
Qualitative changes in electronic structure
Ab initio MD relies on DFT (Kohn-Sham)
Electronic variables explicitly considered
Not integrated out beforehand
Treated as active degrees of freedom
Emergent properties can be observed easily
Tracing back behavior to a specific mechanism is difficult

3. Introduction – Why Do We Care?
Ab initio molecular dynamics is accurate, but slow
Electronic structure problem is difficult
Smaller timesteps are used
Born-Oppenheimer method computationally complex
Recalculate electronic structure problem at every timestep
Car Parrinello method avoids recalculating the electronic structure
Considerable speedup is gained
But at what cost and with what challenges?

4. Relevant Kohn-Sham Equations
Kohn-Sham Energy
Charge Density
Exchange-Correlation Energy

5. Born-Oppenheimer MD
The BO approximation allows for the separation of the ionic and electronic systems
The ionic interaction energy is the same in BOMD as in Kohn-Sham
The Lagrangian of the system is thus expressed as

6. BOMD Scheme Verlet algorithm for dynamics v[:] = v[:]+dt/(2*m[:])*f[:]
r[:] = r[:]+dt*v[:]
Optimize Kohn-Sham orbitals
Calculate forces
dt – timestep
m – ion mass
r – ion position
v – ion velocity
f – force acting on ion

7. Car Parrinello MD
Exploit time-scale separation of fast electronic and slow nuclear motion
Classical mechanical adiabatic energy-scale separation
Map two component quantum/classical problem to two component classical problem
Separate energy scales
Lose explicit time-dependence of quantum subsystem

8. CPMD Lagrangian
Extended energy functional to introduce orthonormality constraint
Orbitals considered as classical fields in Lagrangian
Resulting equations of motion

9. CPMD Temperature
The nuclei evolve in time at an instantaneous physical temperature
Proportional to sum of nuclear kinetic energies (equipartition theorem)
The electrons evolve in time at a fictitious temperature
Proportional to sum of fictitious electronic kinetic energies (equipartition theorem)
Electrons are “cold” – close to instantaneous minimized energy (BO surface)
Ground state wavefunction optimized for initial configuration will stay close to the ground state during time evolution if it is at a sufficiently low temperature

10. CPMD Adiabacity Separate nuclear and electronic motion
Electronic subsystem must stay cold for a long time
Electronic subsystem must follow slow nuclear motion adiabatically
Nuclei still kept at higher temperature
Achieved through decoupling of the two subsystems and adiabatic time evolution
Power spectra of both dynamics must not have too much overlap in the frequency domain
Energy transfer between “hot” nuclei and “cold” electrons becomes practically impossible

11. CPMD Controlling Adiabacity
Adiabatic separation satisfied by large frequency gap
Frequency spectrum of orbital classical fields close to the minimum (ground state)
Both the nuclei frequency spectrum and the smallest energy gap are determined by the system
Only control parameter is the fictitious electronic mass
Decreasing the mass shifts the frequency spectrum up, but also stretches it

12. CPMD Timescale
The maximum possible frequency determined by the cutoff frequency is also shifted up by lowering the electronic mass
This imposes an arbitrary condition on the maximum possible molecular dynamics time step
Because of this, compromises must be made on the control parameter
One work-around is to increase the nuclear masses to depress the nuclear vibrational frequency (at the cost of renormalization)

13. CPMD Forces and Constraints
The forces on the orbital fields are calculated as the action of the Kohn-Sham Hamiltonian
The forces with respect to the nuclear positions are the same as in BOMD
The constraint forces arise from the extended energy functional

14. CPMD Integrator dt – timestep m – mass r – position
Constraint terms prohibit direct application of Verlet method
Assuming there is no nuclear position dependent overlap of the wavefunctions, only the orbital fields are constrained
Constraints are incorporated using the RATTLE algorithm
v[:] = v[:]+dt/(2*m[:])*f[:]
rp[:] = r[:]+dt*v[:]
Calculate Lagrange multiplier
r[:] = rp[:]+L*r[:]
Calculate forces
v[:] = v[:]+L*r[:]
dt – timestep
m – mass
r – position
rp – temporary position
v – velocity
f – force

15. CPMD Performance Comparison With BOMD
Simulation of 8 Si atoms for 8ps at K with energy cutoff of 10Ry at the gamma point of a simple cubic supercell and fictitious mass of 400au

16. Conclusions
CPMD is a useful method for running faster ab initio molecular dynamics simulations
Fictitious electron dynamics
Adiabatic energy-scale separation
Electrons kept near ground state to retain accurate ionic forces
Fictitious mass control parameter
BOMD can still be made as fast or faster than CPMD
Sacrifices energy conservation
CPMD provides far better energy conservation while still being fast
Small timesteps