Maintaining Adiabaticity in CarParrinello Molecular Dynamics Nicholas Walker
Maintaining Adiabaticity in Car-Parrinello Molecular Dynamics Nicholas Walker Supported by the US National Science Foundation through co-operative agreement OIA-1541079
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
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?
Relevant Kohn-Sham Equations • Kohn-Sham Energy • Charge Density • Exchange-Correlation Energy
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
Lagrangian • Extended energy functional to introduce orthonormality constraint • Orbitals considered as classical fields in Lagrangian • Resulting equations of motion
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
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
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
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
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
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
Examples
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
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