1. Maintaining Adiabaticity in Car-Parrinello Molecular Dynamics
Nicholas Walker

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. 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
Initial electronic system will reside on BO surface

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

7. 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

8. Adiabaticity 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

9. Controlling Adiabaticity
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

10. 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 parameters

11. Using Thermostats Metals do not have a band gap
Adiabatic separation can be maintained with an electron thermostat
Ensures the electrons stay “cold”
A separate thermostat can also be applied to the ions
Ensures the ions stay “hot”
Small energy transfer between the systems becomes a non-issue
Implemented with a frictional force that modifies kinetic energy
Maintains desired average temperature
Introduces energy fluctuations

12. Thermostat Parameters
The temperature of the ions is chosen at will
The appropriate fictitious electron kinetic energy cannot be known a priori
An exact value is not needed, so you only need a decent guess
The ionic thermostat frequencies must be chosen wisely
Excite vibrational modes
Period must be longer than interactions, shorter than simulation
The electron thermostat frequency is less important
Must lie far above ionic vibrational spectrum
Reduces thermal exchange

13. Examples

14. Conclusion
Car-Parrinello MD allows for faster ab initio simulations than BOMD
Care must be taken to choose parameters to maintain adiabatic separation of the “hot” ionic and “cold” electronic systems
Modifying the electron fictitious mass is sufficient for insulators
Metals need the addition of a thermostat to keep the electrons “cold”
The thermostat parameters cannot be known a priori
Exact values are not needed, just ballpark values
There is a lot of trial and error