Maintaining Adiabaticity in Car-Parrinello Molecular Dynamics Nicholas Walker

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