## Final Report Summary - PROPAGATE (New Propagation Techniques for the simulation of dynamical processes in extended systems)

The scientific aim of PROPAGATE was to enable quantum mechanical methods for large-scale computer simulations. We have picked an approximation to density-functional theory (DFT), the density-functional based tight-binding (DFTB) method, as method of choice as we have been familiar with its merits and limitations, as it offers the highest possible efficiency among all methods that allow fully quantum-mechanical treatment of the electrons, and even though it requires parameterization, it is applicable to all elements of the periodic table. Depending on system size and level of DFT accuracy, DFTB calculations are typically up tp4 orders of magnitude faster than DFT calculations.

To reach PROPAGATE’s aim, two essential bottlenecks had to be removed – the first one concerns the memory usage, the other the computational cost. For the memory bottleneck, the standard DFTB implementation requires to store matrices describing electronic interactions and orbitals that scale quadratically with the number of electrons and that therefore quickly meet the limits even of the largest modern computers. In ADF, this bottleneck has been removed by employing sparse matrix technology. Here, the matrices are separated in sub-matrices, and sub-matrices containing only numbers below a threshold are omitted from storage (and calculation). The advantage of this approach is evident when looking at the second bottleneck, the computational cost. In DFTB, solving a generalised eigenvalue problem is the computationally most costly task, taking up typically more than 90% of the computer time and scaling cubically with the number of electrons. Reducing the scaling by employing sparse matrix techniques often hit another problem: highly efficient matrix algebra routines use dense matrices and reach CPU peak performance. Changing to a different format is sacrificing this advantage. Our matrix storage scheme now combines the advantages of sparse matrices with peak matrix performance, as instead of one operation on very large matrices, we have rewritten the computational driver such that it applies many operations on small, dense matrices, thus approaching peak performance, and still avoiding to store and manipulate submatrices that are empty.

A density matrix purification scheme has been developed and is available in ADF2017 and found to be superior to alternative propagation schemes. This is due to the fact that density propagation schemes require small time steps in molecular dynamics simulations, and they are difficult to apply in traditional calculations such as geometry optimizations. The density matrix purification scheme is independent on the dynamics of the nuclei and can thus be applied in all situations.

Time-dependent density-functional theory (TDDFT) is the workhorse for most excited state applications, e.g. in the calculation of UV/Vis spectra. The theory is well-established and used in many applications. However, if solved directly, the computer time and memory requirements are huge, much larger than in ground state DFT calculations, as a coupling matrix with dimension of the number of occupied times number of virtual orbitals needs to be calculated, and then solved for its eigenvalues and eigenvectors. Again, DFTB is an interesting alternative to DFT as it provides a similar performance boost as compared to the ground state. We have thus implemented time-dependent DFTB (TDDFTB) into ADF. Moreover, we have employed a scheme, originally developed for TDDFT, to TDDFTB, which only accounts for those excitation matrix elements that are significant, which allows the removal of a significant part of the excitation matrix and thus reduces memory and computational requirements, while calculated spectra remain essentially unaffected. The TDDFTB method further allows going beyond the state-of-the-art in UV/Vis spectroscopy: we have included the vibrational coupling in the TDDFTB calculations, where for each excitation the excited state geometry must be calculated. In a traditional DFT scheme this is only applicable for very small systems, but in TDDFTB it becomes feasible for large molecules. This has been made possible due to our new efficient implementation of excited-state gradients in DFTB context.

Finally, perhaps the most significant contribution to this Objective, we provide an alternative to TDDFT, where we use DFT orbitals and the TDDFT computational framework, but we calculate the coupling matrix elements using DFTB approximations. As no parameters are needed, this approach is directly applicable to the periodic table.

In order to run many independent or dependent calculations, which is needed in many situations, but in this case intentionally for hierarchical methods such as the QM/MM scheme, we have developed a parallel scripting environment, PLAMS, which is included in ADF since the 2014 release. PLAMS is currently adopted to develop an environment that allows for an alternative route to generate DFTB parameters.

The PLAMS library has already been picked up in a broader workflow project in The Netherlands [1] that supports complicated multi-step workflows for many quantum chemistry and materials science packages [2]. SCM has made PLAMS available as open source to the community [3].

Finally, we developed a new method to generate large molecular framework structures from its building blocks and integrated it with ADF methods (force fields, DFTB, DFT) into a graphical user interface as well as making it accessible via (python) scripts [4]. This is already being used and built upon by other groups that are studying MOFs, including an MCSA ITN network that SCM is currently participating in. These methods and tools developed within PROPAGATE will be made available to the community of materials modellers as part of the ADF2017 Modeling Suite [5], planned to be released in April. This will ensure the wide dissemination and continuity of PROPAGATE’s results.

[1] https://www.esciencecenter.nl/project/computational-chemistry-made-easy

[2] https://github.com/SCM-NV/qmworks

[3] https://github.com/SCM-NV/PLAMS

[4] https://www.scm.com/doc.trunk/Tutorials/GUI_overview/Building_Frameworks_and_Reticular_Compounds.html

[5] https://www.scm.com/adf-modeling-suite/

To reach PROPAGATE’s aim, two essential bottlenecks had to be removed – the first one concerns the memory usage, the other the computational cost. For the memory bottleneck, the standard DFTB implementation requires to store matrices describing electronic interactions and orbitals that scale quadratically with the number of electrons and that therefore quickly meet the limits even of the largest modern computers. In ADF, this bottleneck has been removed by employing sparse matrix technology. Here, the matrices are separated in sub-matrices, and sub-matrices containing only numbers below a threshold are omitted from storage (and calculation). The advantage of this approach is evident when looking at the second bottleneck, the computational cost. In DFTB, solving a generalised eigenvalue problem is the computationally most costly task, taking up typically more than 90% of the computer time and scaling cubically with the number of electrons. Reducing the scaling by employing sparse matrix techniques often hit another problem: highly efficient matrix algebra routines use dense matrices and reach CPU peak performance. Changing to a different format is sacrificing this advantage. Our matrix storage scheme now combines the advantages of sparse matrices with peak matrix performance, as instead of one operation on very large matrices, we have rewritten the computational driver such that it applies many operations on small, dense matrices, thus approaching peak performance, and still avoiding to store and manipulate submatrices that are empty.

A density matrix purification scheme has been developed and is available in ADF2017 and found to be superior to alternative propagation schemes. This is due to the fact that density propagation schemes require small time steps in molecular dynamics simulations, and they are difficult to apply in traditional calculations such as geometry optimizations. The density matrix purification scheme is independent on the dynamics of the nuclei and can thus be applied in all situations.

Time-dependent density-functional theory (TDDFT) is the workhorse for most excited state applications, e.g. in the calculation of UV/Vis spectra. The theory is well-established and used in many applications. However, if solved directly, the computer time and memory requirements are huge, much larger than in ground state DFT calculations, as a coupling matrix with dimension of the number of occupied times number of virtual orbitals needs to be calculated, and then solved for its eigenvalues and eigenvectors. Again, DFTB is an interesting alternative to DFT as it provides a similar performance boost as compared to the ground state. We have thus implemented time-dependent DFTB (TDDFTB) into ADF. Moreover, we have employed a scheme, originally developed for TDDFT, to TDDFTB, which only accounts for those excitation matrix elements that are significant, which allows the removal of a significant part of the excitation matrix and thus reduces memory and computational requirements, while calculated spectra remain essentially unaffected. The TDDFTB method further allows going beyond the state-of-the-art in UV/Vis spectroscopy: we have included the vibrational coupling in the TDDFTB calculations, where for each excitation the excited state geometry must be calculated. In a traditional DFT scheme this is only applicable for very small systems, but in TDDFTB it becomes feasible for large molecules. This has been made possible due to our new efficient implementation of excited-state gradients in DFTB context.

Finally, perhaps the most significant contribution to this Objective, we provide an alternative to TDDFT, where we use DFT orbitals and the TDDFT computational framework, but we calculate the coupling matrix elements using DFTB approximations. As no parameters are needed, this approach is directly applicable to the periodic table.

In order to run many independent or dependent calculations, which is needed in many situations, but in this case intentionally for hierarchical methods such as the QM/MM scheme, we have developed a parallel scripting environment, PLAMS, which is included in ADF since the 2014 release. PLAMS is currently adopted to develop an environment that allows for an alternative route to generate DFTB parameters.

The PLAMS library has already been picked up in a broader workflow project in The Netherlands [1] that supports complicated multi-step workflows for many quantum chemistry and materials science packages [2]. SCM has made PLAMS available as open source to the community [3].

Finally, we developed a new method to generate large molecular framework structures from its building blocks and integrated it with ADF methods (force fields, DFTB, DFT) into a graphical user interface as well as making it accessible via (python) scripts [4]. This is already being used and built upon by other groups that are studying MOFs, including an MCSA ITN network that SCM is currently participating in. These methods and tools developed within PROPAGATE will be made available to the community of materials modellers as part of the ADF2017 Modeling Suite [5], planned to be released in April. This will ensure the wide dissemination and continuity of PROPAGATE’s results.

[1] https://www.esciencecenter.nl/project/computational-chemistry-made-easy

[2] https://github.com/SCM-NV/qmworks

[3] https://github.com/SCM-NV/PLAMS

[4] https://www.scm.com/doc.trunk/Tutorials/GUI_overview/Building_Frameworks_and_Reticular_Compounds.html

[5] https://www.scm.com/adf-modeling-suite/