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Fig. 21 Logarithm of bond order versus bond length for ozone molecule. Colors of data points indicate singlet (red), cation doublet (green), or triplet (blue) spin states. Shapes of data points indicate the exchange-—correlation theory. Left: Bonds between the middle and outer atoms. Right: Bond between the two outer atoms. As shown in Fig. 23, the DDEC6 serial computation required ~20 seconds per atom. The OpenMP code ran slightly faster on one processor than the serial (non-OpenMP) code. The paral- lelization efficiency dropped substantially when increasing the number of parallel processors. For 16 processors, the paralle- lization efficiency dropped to ~11%. As shown in Fig. 23, this was because the time required for setting up density grids did not decrease significantly with increasing number of proces- sors. This was because the input file reading was not paral- lelized. When the time required for setting up the density grids is omitted, parallelization efficiencies increase to = ~60%. Although one could potentially improve the parallelization efficiency of setting up the density grids by parallelizing the input file reading, we chose not to do so at this time to keep the Every C atom in the Ceo molecule is equivalent, being shared by two C, rings and one C; ring. There are two types of bonds: (a) a bond shared between a C, and a C; ring, and (b) a bond shared between two C, rings. The C NAC in N@Cgo was nearly zero (—0.007 to 0.003). The bond order type (a) in N@Cgo was 1.12, while the bond order type (b) was 1.30. The SBO for each C atom in N@C¢o was 3.93. This SBO is in line with the chemical expectation that each C atom shares four valence electrons.

Figure 21 Logarithm of bond order versus bond length for ozone molecule. Colors of data points indicate singlet (red), cation doublet (green), or triplet (blue) spin states. Shapes of data points indicate the exchange-—correlation theory. Left: Bonds between the middle and outer atoms. Right: Bond between the two outer atoms. As shown in Fig. 23, the DDEC6 serial computation required ~20 seconds per atom. The OpenMP code ran slightly faster on one processor than the serial (non-OpenMP) code. The paral- lelization efficiency dropped substantially when increasing the number of parallel processors. For 16 processors, the paralle- lization efficiency dropped to ~11%. As shown in Fig. 23, this was because the time required for setting up density grids did not decrease significantly with increasing number of proces- sors. This was because the input file reading was not paral- lelized. When the time required for setting up the density grids is omitted, parallelization efficiencies increase to = ~60%. Although one could potentially improve the parallelization efficiency of setting up the density grids by parallelizing the input file reading, we chose not to do so at this time to keep the Every C atom in the Ceo molecule is equivalent, being shared by two C, rings and one C; ring. There are two types of bonds: (a) a bond shared between a C, and a C; ring, and (b) a bond shared between two C, rings. The C NAC in N@Cgo was nearly zero (—0.007 to 0.003). The bond order type (a) in N@Cgo was 1.12, while the bond order type (b) was 1.30. The SBO for each C atom in N@C¢o was 3.93. This SBO is in line with the chemical expectation that each C atom shares four valence electrons.

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Fig.1 The triple role of AIM methods. AIM methods provide atomistic descriptors that can be used (i) to understand the chemical properties of materials, (ii) to parameterize force fields used in classical atomistic simulations, and (iii) to provide dispersion interactions in some DFT + dispersion methods or to produce localized electron distributions for use in QC methods.
Fig.1 The triple role of AIM methods. AIM methods provide atomistic descriptors that can be used (i) to understand the chemical properties of materials, (ii) to parameterize force fields used in classical atomistic simulations, and (iii) to provide dispersion interactions in some DFT + dispersion methods or to produce localized electron distributions for use in QC methods.
Fig. 2 Six pillars of great performance for an AIM method. conformational transferability. An AIM method should prefer ably work well for a broad range of material types includins small and large molecules, ions, organometallic complexes porous and dense solids, solid surfaces, nanostructures, botl magnetic and non-magnetic materials, both heavy and ligh chemical elements, both organic and inorganic materials, botl conducting and insulating materials, etc. In order to be used a: a default atomic population analysis method (APAM) in OC
Fig. 2 Six pillars of great performance for an AIM method. conformational transferability. An AIM method should prefer ably work well for a broad range of material types includins small and large molecules, ions, organometallic complexes porous and dense solids, solid surfaces, nanostructures, botl magnetic and non-magnetic materials, both heavy and ligh chemical elements, both organic and inorganic materials, botl conducting and insulating materials, etc. In order to be used a: a default atomic population analysis method (APAM) in OC
Fig. 3 Flow diagram of the DDEC6 method as implemented in the CHARGEMOL program. The dotted line indicates the path that would be followed if using atom-centered integration grids. Read input files. The program reads the system's informa- tion from input file(s) containing geometry, electron density, and spin density (for magnetic materials). Our program reads the input information in chunks so that it does not overflow the read buffer set by the operating system. This allows the program to read large input files without generating an overflow error. Currently, the program can read several input file formats: VASP,”*”’ wfx, xsf, and cube files. CHARGEMOL identifies the type of input file and reads it using an appropriate module. At Check grid spacing and number of valence electrons. Before starting charge partitioning, the program performs the
Fig. 3 Flow diagram of the DDEC6 method as implemented in the CHARGEMOL program. The dotted line indicates the path that would be followed if using atom-centered integration grids. Read input files. The program reads the system's informa- tion from input file(s) containing geometry, electron density, and spin density (for magnetic materials). Our program reads the input information in chunks so that it does not overflow the read buffer set by the operating system. This allows the program to read large input files without generating an overflow error. Currently, the program can read several input file formats: VASP,”*”’ wfx, xsf, and cube files. CHARGEMOL identifies the type of input file and reads it using an appropriate module. At Check grid spacing and number of valence electrons. Before starting charge partitioning, the program performs the
Table 2 DDEC6 charge partitioning minimizes a series of 14 Lagrangians arranged in seven charge partitioning steps 2.4 Spin partitioning details
Table 2 DDEC6 charge partitioning minimizes a series of 14 Lagrangians arranged in seven charge partitioning steps 2.4 Spin partitioning details
quantifies the maximum ASM component error on the first spin cycle (i.e., proportional spin partitioning) compared to the final converged result. The next to last spin cycle is the first spin cycle for which max_ASM_change < spin_convergence_tolerance. A final spin cycle is required to confirm convergence has reached this level. The first +1 in eqn (14) accounts for the first spin cycle, and the second +1 in eqn (14) accounts for the last spin Therefore, the number of spin_cycles required to achieve convergence of all ASM components to within spin_conver- gence_tolerance follows the equation’””
quantifies the maximum ASM component error on the first spin cycle (i.e., proportional spin partitioning) compared to the final converged result. The next to last spin cycle is the first spin cycle for which max_ASM_change < spin_convergence_tolerance. A final spin cycle is required to confirm convergence has reached this level. The first +1 in eqn (14) accounts for the first spin cycle, and the second +1 in eqn (14) accounts for the last spin Therefore, the number of spin_cycles required to achieve convergence of all ASM components to within spin_conver- gence_tolerance follows the equation’””
Table 5 Comparison of correction schemes for core electron density grid and total electron density grid
Table 5 Comparison of correction schemes for core electron density grid and total electron density grid
Fig.6 Elements for achieving an efficient shared memory parallelization of the DDEC6 method
Fig.6 Elements for achieving an efficient shared memory parallelization of the DDEC6 method
Table 6 Minimum memory required (in MB) to complete the DDEC6 calculation a single thread should consecutively run over adjacent values of the inner index, and threads should be parallelized over one of the outer indices. This means that a should be the inner (fast) loop and c should be the outer (slow) loop when assigning values to my_array(a, b, c) using a Fortran Do loop. The correct organization of the loops in the program is represented in Fig. 8. When using the correct loop order, each thread will update several consecutive array values before requiring a fetch from main memory (or outer cache) to inner cache of the next several array values. Since the different parallel threads are working on array elements corresponding to different cache lines, each thread will not slow down the performance of the other threads. according to the distance from the atomic nucleus and a second index depends on the atom number: my_array_1(radial_shell, atom_number). Let us define another array that only depends on the atom number: my_array_2(atom_number). In the first case, loops were set so radial_shell was the fast index. We tested the efficiency of parallelizing loops in the DDEC6_valence_it- erator module that iterate over the number of atoms. The system tested was an ice crystal with 6144 atoms. We used 16 processors. Because of false sharing where different processors try to update adjacent array values in the same cache line, a parallel loop containing variables of the type my_array_2 took almost three times longer when parallelized than in the serial mode. In contrast, the parallelization efficiency was 98% when the loop containing variables of the type my_array_1 was par- allelized over atom_number. False sharing is minimized for this kind of loop because radial_shell is the fast index and different processors work on different values of the slow index (atom_- number). This is why we parallelized over the number of atoms some loops that contained variables of the type my_array_1 but not my_array_2.
Table 6 Minimum memory required (in MB) to complete the DDEC6 calculation a single thread should consecutively run over adjacent values of the inner index, and threads should be parallelized over one of the outer indices. This means that a should be the inner (fast) loop and c should be the outer (slow) loop when assigning values to my_array(a, b, c) using a Fortran Do loop. The correct organization of the loops in the program is represented in Fig. 8. When using the correct loop order, each thread will update several consecutive array values before requiring a fetch from main memory (or outer cache) to inner cache of the next several array values. Since the different parallel threads are working on array elements corresponding to different cache lines, each thread will not slow down the performance of the other threads. according to the distance from the atomic nucleus and a second index depends on the atom number: my_array_1(radial_shell, atom_number). Let us define another array that only depends on the atom number: my_array_2(atom_number). In the first case, loops were set so radial_shell was the fast index. We tested the efficiency of parallelizing loops in the DDEC6_valence_it- erator module that iterate over the number of atoms. The system tested was an ice crystal with 6144 atoms. We used 16 processors. Because of false sharing where different processors try to update adjacent array values in the same cache line, a parallel loop containing variables of the type my_array_2 took almost three times longer when parallelized than in the serial mode. In contrast, the parallelization efficiency was 98% when the loop containing variables of the type my_array_1 was par- allelized over atom_number. False sharing is minimized for this kind of loop because radial_shell is the fast index and different processors work on different values of the slow index (atom_- number). This is why we parallelized over the number of atoms some loops that contained variables of the type my_array_1 but not my_array_2.
ig.8 Example of the arrangement of loop and matrix indices to maximize cache efficiency. This structure was used throughout the program To test cache performance, we generated a code with the correct (“fast code”) and incorrect (“slow code”) order of loop indices when writing updated values to the big arrays. Computational tests for the Mnj,,-acetate single molecule magnet using the PBE/planewave densities are shown in Table 7. Three runs were performed and the average and standard deviation are reported. The slow code took longer to run than the fast code. On 8 and 16 processors, spin partitioning took We tested the efficiency of parallelizing over the number of atoms on two different types of arrays. Let us define one array where the first index (corresponding to the radial shell) varies
ig.8 Example of the arrangement of loop and matrix indices to maximize cache efficiency. This structure was used throughout the program To test cache performance, we generated a code with the correct (“fast code”) and incorrect (“slow code”) order of loop indices when writing updated values to the big arrays. Computational tests for the Mnj,,-acetate single molecule magnet using the PBE/planewave densities are shown in Table 7. Three runs were performed and the average and standard deviation are reported. The slow code took longer to run than the fast code. On 8 and 16 processors, spin partitioning took We tested the efficiency of parallelizing over the number of atoms on two different types of arrays. Let us define one array where the first index (corresponding to the radial shell) varies
Table 7 Ratio of slow to fast code times required to perform DDEC6 calculations. These tests were performed on the Mnj2-acetate single molecule magnet (PBE/planewave)
Table 7 Ratio of slow to fast code times required to perform DDEC6 calculations. These tests were performed on the Mnj2-acetate single molecule magnet (PBE/planewave)
Table 8 Time per atom (seconds) to perform DDEC6 calculations on Mny2-acetate SMM (PBE/planewave) using different thread configurations. The averages and standard deviations for three runs are shown. The calculations with 32 threads used two threads per processor, while the calculations with 16 threads used one thread per processor. We examined the effects of setting OMP_PROC_BIND to true, false, and not specifying the variable's value
Table 8 Time per atom (seconds) to perform DDEC6 calculations on Mny2-acetate SMM (PBE/planewave) using different thread configurations. The averages and standard deviations for three runs are shown. The calculations with 32 threads used two threads per processor, while the calculations with 16 threads used one thread per processor. We examined the effects of setting OMP_PROC_BIND to true, false, and not specifying the variable's value
“ Several additional sets of grid points were also considered for this material as described in Sections 4.5 and 4.6. ’ Additional calculations performed for the same molecule with CASSCF/AUG-cc-pVTZ, CCSD/AUG-cc-pVTZ, PW91/6-311+G*, and SAC-CI/AUG-cc-pVTZ levels of theory with identical total number of grid points and volume per grid point. ° Additional calculations performed for the same molecule with CCSD/ AUG-cc-pVTZ and PW91/6-311+G* levels of theory with identical total number of grid points and volume per grid point. “ Endohedral complexes with X = Am", Cs, Eu", Li, N, and Xe. Table 9 Summary of the systems studied for computational timing and memory tests
“ Several additional sets of grid points were also considered for this material as described in Sections 4.5 and 4.6. ’ Additional calculations performed for the same molecule with CASSCF/AUG-cc-pVTZ, CCSD/AUG-cc-pVTZ, PW91/6-311+G*, and SAC-CI/AUG-cc-pVTZ levels of theory with identical total number of grid points and volume per grid point. ° Additional calculations performed for the same molecule with CCSD/ AUG-cc-pVTZ and PW91/6-311+G* levels of theory with identical total number of grid points and volume per grid point. “ Endohedral complexes with X = Am", Cs, Eu", Li, N, and Xe. Table 9 Summary of the systems studied for computational timing and memory tests
Fig.10 Parallelization timing and efficiency results for Ni bulk metal (1 atom per unit cell, PBE/planewave method). Three trials were per- formed to compute error bars for the computational times and par- allelization efficiencies. Table 10 also lists the DDEC6 SBO for each atom type. The standard deviation was =0.10. SBOs for C atom types ranging from 3.84 to 4.25, which agrees with a chemically expected value of ~4. The H atoms have DDEC6 SBOs of nearly 1, which is the chemically expected value. SBOs for N atoms were between 3 and 4. The O SBO is expected to be ~2; however, we obtained O
Fig.10 Parallelization timing and efficiency results for Ni bulk metal (1 atom per unit cell, PBE/planewave method). Three trials were per- formed to compute error bars for the computational times and par- allelization efficiencies. Table 10 also lists the DDEC6 SBO for each atom type. The standard deviation was =0.10. SBOs for C atom types ranging from 3.84 to 4.25, which agrees with a chemically expected value of ~4. The H atoms have DDEC6 SBOs of nearly 1, which is the chemically expected value. SBOs for N atoms were between 3 and 4. The O SBO is expected to be ~2; however, we obtained O
Fig. 11 Parallelization timing and efficiency results for the B-DNA decamer (733 atoms per unit cell, PBE/planewave method). The lines mark the unit cell boundaries. We now discuss computation: molecule magnets that demon tational approach. Fig. 13 disp single molecule magnet using al performance for two single strate versatility of our compu- ays results for the Mn,,-acetate both planewave (left panel) and Gaussian (middle panel) type basis sets. Mn,-acetate has been one of the most widely studied shown in Fig. 13, computationa Gaussian basis set coefficients single molecule magnets.*”* As times were much higher for the input than for the density grid input derived from the planewave calculation. The program must explicitly compute the e ectron and spin density grids from the Gaussian basis set coefficients input. Parallelization
Fig. 11 Parallelization timing and efficiency results for the B-DNA decamer (733 atoms per unit cell, PBE/planewave method). The lines mark the unit cell boundaries. We now discuss computation: molecule magnets that demon tational approach. Fig. 13 disp single molecule magnet using al performance for two single strate versatility of our compu- ays results for the Mn,,-acetate both planewave (left panel) and Gaussian (middle panel) type basis sets. Mn,-acetate has been one of the most widely studied shown in Fig. 13, computationa Gaussian basis set coefficients single molecule magnets.*”* As times were much higher for the input than for the density grid input derived from the planewave calculation. The program must explicitly compute the e ectron and spin density grids from the Gaussian basis set coefficients input. Parallelization
Table 10 DDEC6 NACs and SBOs with standard deviation and maximum and minimum values for the B-DNA decamer (CCATTAATGQG)po. CHARMM27 and AMBER 4.1 NACs shown for comparison @ CHARMM27 NACs from ref. 47. 7 AMBER 4.1 RESP HF/6-31G* NACs from ref. 48.
Table 10 DDEC6 NACs and SBOs with standard deviation and maximum and minimum values for the B-DNA decamer (CCATTAATGQG)po. CHARMM27 and AMBER 4.1 NACs shown for comparison @ CHARMM27 NACs from ref. 47. 7 AMBER 4.1 RESP HF/6-31G* NACs from ref. 48.
Fig. 12 Computed DDEC6 bond orders in the guanine-cytosine and adenine-thymine base pairs. The hydrogen bonds are shown as dotted rec lines.
Fig. 12 Computed DDEC6 bond orders in the guanine-cytosine and adenine-thymine base pairs. The hydrogen bonds are shown as dotted rec lines.
Fig.13 Parallelization timing and efficiency results for the Mn,2-acetate single molecule magnet (148 atoms per unit cell) that exhibits collinear magnetism. Left: PBE/planewave results. Middle: PBE/LANL2DZ results. Right: Chemical structure with Mn atoms colored by type: Mn type 1 (blue), Mn type 2 (red), Mn type 3 (yellow). The PBE/LANL2DZ computed ASMs were —2.56 (Mn type 1), 3.63 (Mn type 2), 3.57 (Mn type 3), and =0.077 in magnitude on all other atoms."®
Fig.13 Parallelization timing and efficiency results for the Mn,2-acetate single molecule magnet (148 atoms per unit cell) that exhibits collinear magnetism. Left: PBE/planewave results. Middle: PBE/LANL2DZ results. Right: Chemical structure with Mn atoms colored by type: Mn type 1 (blue), Mn type 2 (red), Mn type 3 (yellow). The PBE/LANL2DZ computed ASMs were —2.56 (Mn type 1), 3.63 (Mn type 2), 3.57 (Mn type 3), and =0.077 in magnitude on all other atoms."®
Fig.14 Computed bond orders (blue) and NACs (black) for the Mn,2-acetate single molecule magnet using the PBE/LANL2DZ electron and spin densities. The atoms are colored by element: Mn type 1 (blue), Mn type 2 (green), Mn type 3 (yellow), O (red), C (grey), H (pink). All of the atoms (i.e., the full chemical structure) were included the DDEC6 calculation, but for display purposes only a portion of the atoms are shown here. The fragments shown here were chosen so that together they include all of the symmetry unique atoms and bonds. 96, 324, 768, 1500, 2592, 4116, 6144, and 8748 atoms in the unit cell. As described in another paper, these structures were con- structed as n x n x n times the primitive cell containing 4 water molecules (12 atoms), for n = 1 to 9.° These unit cells were used as inputs for VASP calculations to compute the electron distribu- tions.’® As shown in Fig. 18, the total memory required is almost independent of the number of processors and scales linearly with increasing system size. This indicates an efficient memory management. Fig. 18 also shows the serial and parallel compu- tational times scaled linearly with increasing system size and decreased with increasing number of parallel processors. Table 11 shows how the data from Fig. 18 fits to lines of the form y = ax” for the time and memory needed to complete the CHARGEMOL program. The fitted exponents b are 0.8723 to 1.0389, which
Fig.14 Computed bond orders (blue) and NACs (black) for the Mn,2-acetate single molecule magnet using the PBE/LANL2DZ electron and spin densities. The atoms are colored by element: Mn type 1 (blue), Mn type 2 (green), Mn type 3 (yellow), O (red), C (grey), H (pink). All of the atoms (i.e., the full chemical structure) were included the DDEC6 calculation, but for display purposes only a portion of the atoms are shown here. The fragments shown here were chosen so that together they include all of the symmetry unique atoms and bonds. 96, 324, 768, 1500, 2592, 4116, 6144, and 8748 atoms in the unit cell. As described in another paper, these structures were con- structed as n x n x n times the primitive cell containing 4 water molecules (12 atoms), for n = 1 to 9.° These unit cells were used as inputs for VASP calculations to compute the electron distribu- tions.’® As shown in Fig. 18, the total memory required is almost independent of the number of processors and scales linearly with increasing system size. This indicates an efficient memory management. Fig. 18 also shows the serial and parallel compu- tational times scaled linearly with increasing system size and decreased with increasing number of parallel processors. Table 11 shows how the data from Fig. 18 fits to lines of the form y = ax” for the time and memory needed to complete the CHARGEMOL program. The fitted exponents b are 0.8723 to 1.0389, which
Fig. 15 Parallelization timing and efficiency results for the Fe4O,2N4C4oHs2 single molecule magnet (112 atoms per unit cell, PW91/planewave method) that exhibits non-collinear magnetism. Left: Chemical structure reproduced with permission of ref. 16 (© The Royal Society of Chemistry 2017). The atoms and spin magnetization vectors are colored by: Fe (orange), O (red), N (blue), C (gray), H (white). The Fe atoms exhibited DDEC6 ASM magnitudes of 2.33, and the ASM magnitudes were negligible on the other atoms.*® Right: Parallelization timing and efficiency results. For periodic materials, calculations using a planewave basis set have several advantages.”°*° First, the basis set functions are orthonormal, which greatly simplifies matrix element calcula- tions. Second, the planewave coefficients are related to a Fourier transform.”®** Fast Fourier transform algorithms allow func- tions to be quickly transformed between real space and recip- rocal space. Because the standard fast Fourier transform algorithm requires uniformly spaced sampling points, uniformly spaced integration grids are the natural choice for planewave QC calculations. Integrations and other mat hemat- ical manipulations can thus be performed in whichever space (real or reciprocal) is most convenient. Third, the Because the cell periodic functions {u,(k,7)} have the same periodicity as the unit cell, they can be expanded in terms of
Fig. 15 Parallelization timing and efficiency results for the Fe4O,2N4C4oHs2 single molecule magnet (112 atoms per unit cell, PW91/planewave method) that exhibits non-collinear magnetism. Left: Chemical structure reproduced with permission of ref. 16 (© The Royal Society of Chemistry 2017). The atoms and spin magnetization vectors are colored by: Fe (orange), O (red), N (blue), C (gray), H (white). The Fe atoms exhibited DDEC6 ASM magnitudes of 2.33, and the ASM magnitudes were negligible on the other atoms.*® Right: Parallelization timing and efficiency results. For periodic materials, calculations using a planewave basis set have several advantages.”°*° First, the basis set functions are orthonormal, which greatly simplifies matrix element calcula- tions. Second, the planewave coefficients are related to a Fourier transform.”®** Fast Fourier transform algorithms allow func- tions to be quickly transformed between real space and recip- rocal space. Because the standard fast Fourier transform algorithm requires uniformly spaced sampling points, uniformly spaced integration grids are the natural choice for planewave QC calculations. Integrations and other mat hemat- ical manipulations can thus be performed in whichever space (real or reciprocal) is most convenient. Third, the Because the cell periodic functions {u,(k,7)} have the same periodicity as the unit cell, they can be expanded in terms of
Fig. 16 Computed bond orders (blue) and NACs (black) for the Fe,- OyN4C4oHs2 single molecule magnet that exhibits non-collinear magnetism. The atoms are colored by element: Fe (yellow), O (red), C (grey), N (blue), H (pink). The distorted cuboidal Fe,O, core is shown together with one adsorbed methanol molecule and one of the organic ligands. The dashed blue line illustrates interaction between the methanol lone pair and the adjacent Fe atom (i.e., Lewis acid—base interaction). The other three adsorbed methanol molecules and three organic ligands were included in the calculation but are not shown here for display purposes.
Fig. 16 Computed bond orders (blue) and NACs (black) for the Fe,- OyN4C4oHs2 single molecule magnet that exhibits non-collinear magnetism. The atoms are colored by element: Fe (yellow), O (red), C (grey), N (blue), H (pink). The distorted cuboidal Fe,O, core is shown together with one adsorbed methanol molecule and one of the organic ligands. The dashed blue line illustrates interaction between the methanol lone pair and the adjacent Fe atom (i.e., Lewis acid—base interaction). The other three adsorbed methanol molecules and three organic ligands were included in the calculation but are not shown here for display purposes.
Fig.17 The convergence rate of spin partitioning is highly predictable. A plot of the logarithm of the max_ASM_change versus iteration number is linear with a slope equal to In(fpin) = In(1 — \/1/2) = In(0.29) = 1.23 for both collinear and non-collinear magnetism.
Fig.17 The convergence rate of spin partitioning is highly predictable. A plot of the logarithm of the max_ASM_change versus iteration number is linear with a slope equal to In(fpin) = In(1 — \/1/2) = In(0.29) = 1.23 for both collinear and non-collinear magnetism.
Fig. 18 DDEC6 results for ice crystal using 12, 96, 324, 768, 1500, 2592, 4116, 6144, and 8748 atoms per unit cell (PBE/planewave method). These results show both the computational tine and memory required scale linearly with increasing system size. Left: The computational time scales linearly with the number of atoms and decreases with increasing number of processors. Right: Total RAM required is almost independent of the number of processors. The predicted memory required is from eqn (17). times the real-space unit cell volume constant. This in turn corresponds to The VASP POTCAR file (which contains the PAW potential) lists the ‘ENMAX’ value; this is the planewave cutoff energy beyond which results are considered well-converged. A reason- able choice is to set the planewave cutoff energy (ENCUT) to the greater of 400 eV and the POTCAR-listed ENMAX value:
Fig. 18 DDEC6 results for ice crystal using 12, 96, 324, 768, 1500, 2592, 4116, 6144, and 8748 atoms per unit cell (PBE/planewave method). These results show both the computational tine and memory required scale linearly with increasing system size. Left: The computational time scales linearly with the number of atoms and decreases with increasing number of processors. Right: Total RAM required is almost independent of the number of processors. The predicted memory required is from eqn (17). times the real-space unit cell volume constant. This in turn corresponds to The VASP POTCAR file (which contains the PAW potential) lists the ‘ENMAX’ value; this is the planewave cutoff energy beyond which results are considered well-converged. A reason- able choice is to set the planewave cutoff energy (ENCUT) to the greater of 400 eV and the POTCAR-listed ENMAX value:
“ The memory tests were performed on the Comet cluster at the San Diego Supercomputing Center (SDSC) with the serial code and parallel code using one and eight cores. Table 11 Data from Fig. 18 fitted to lines of the form y = ax” for the time and memory required to complete the CHARGEMOL calculation. The last column lists the parallelization efficiency on the ice crystal containing 8748 atoms per unit cell
“ The memory tests were performed on the Comet cluster at the San Diego Supercomputing Center (SDSC) with the serial code and parallel code using one and eight cores. Table 11 Data from Fig. 18 fitted to lines of the form y = ax” for the time and memory required to complete the CHARGEMOL calculation. The last column lists the parallelization efficiency on the ice crystal containing 8748 atoms per unit cell
Fig.19 Time and memory required to complete DDEC6 analysis for a 324 atom ice unit cell with 12 700 800, 35 123 200, and 81 285 120 grid points. These calculations were run in serial mode on a single processor. Both the computational time and memory required scaled linearly with increasing number of grid points and were independent of the planewave cutoff energy.
Fig.19 Time and memory required to complete DDEC6 analysis for a 324 atom ice unit cell with 12 700 800, 35 123 200, and 81 285 120 grid points. These calculations were run in serial mode on a single processor. Both the computational time and memory required scaled linearly with increasing number of grid points and were independent of the planewave cutoff energy.
able 12 +Computational tests performed on ice supercell having 324 atoms. The planewave cutoff energy, volume per grid point, and number o' ‘points were varied. The mean absolute deviation (MAD) quantifies the difference between symmetry equivalent atoms in the same set. The lean absolute deviation of mean (MADM) quantifies the difference in average value for symmetry equivalent atoms in the set compared to the gh precision set 6
able 12 +Computational tests performed on ice supercell having 324 atoms. The planewave cutoff energy, volume per grid point, and number o' ‘points were varied. The mean absolute deviation (MAD) quantifies the difference between symmetry equivalent atoms in the same set. The lean absolute deviation of mean (MADM) quantifies the difference in average value for symmetry equivalent atoms in the set compared to the gh precision set 6
Fig. 20 Parallelization timing and efficiency results for ozone singlet, +1 doublet, and triplet at different levels of theory: BSLYP/6-311+G*, CASSCF/AUG-cc-pVTZ, CCSD/AUG-cc-pVTZ, PW91/6-311+G*, and SAC-Cl/AUG-cc-pVTZ. Serial and 16 parallel processors calculations are compared. The cation spin doublet calculation at the CCSD/AUG-cc- pVTZ level of theory took the longest time for DDEC6 analysis. To verify this was statistically significant, we performed three serial runs each for the cation doublet and neutral triplet at CCSD/AUG-cc-pVTZ. The average and error bars (i.e., standard deviation) for each part of the calculation are shown in Fig. 20 but are too small to be visible. These results showed that indeed A series of endohedral complexes was studied: [AmM@Cgo]"’; Cs@Cgo, [EU@Ceo]*, Li@Ceo, N@Coo, and Xe@Cgo. These contain light and heavy elements from main group, lanthanide, and actinide elements. Previously, we demonstrated chemical consistency between the computed DDEC6 NACs and ASMs for these endohedral complexes.” Fig. 22 compares results for serial computation to 16 parallel processors. The parallelization
Fig. 20 Parallelization timing and efficiency results for ozone singlet, +1 doublet, and triplet at different levels of theory: BSLYP/6-311+G*, CASSCF/AUG-cc-pVTZ, CCSD/AUG-cc-pVTZ, PW91/6-311+G*, and SAC-Cl/AUG-cc-pVTZ. Serial and 16 parallel processors calculations are compared. The cation spin doublet calculation at the CCSD/AUG-cc- pVTZ level of theory took the longest time for DDEC6 analysis. To verify this was statistically significant, we performed three serial runs each for the cation doublet and neutral triplet at CCSD/AUG-cc-pVTZ. The average and error bars (i.e., standard deviation) for each part of the calculation are shown in Fig. 20 but are too small to be visible. These results showed that indeed A series of endohedral complexes was studied: [AmM@Cgo]"’; Cs@Cgo, [EU@Ceo]*, Li@Ceo, N@Coo, and Xe@Cgo. These contain light and heavy elements from main group, lanthanide, and actinide elements. Previously, we demonstrated chemical consistency between the computed DDEC6 NACs and ASMs for these endohedral complexes.” Fig. 22 compares results for serial computation to 16 parallel processors. The parallelization
Table 13. Summary of NACs for ozone singlet, +1 doublet, and triplet. The range in each cell is the minimum and the maximum value across the different levels of theory for the underlined atoms 4.9 The effects of large vacuum space in unit cell: water dimer
Table 13. Summary of NACs for ozone singlet, +1 doublet, and triplet. The range in each cell is the minimum and the maximum value across the different levels of theory for the underlined atoms 4.9 The effects of large vacuum space in unit cell: water dimer
“ Using 46 frozen Cs core electrons. ? Using 54 simulated frozen Cs core electrons. Table 14 Summary of DDEC6 bond orders and NACs for the different X@Cgo endohedral complexes studied
“ Using 46 frozen Cs core electrons. ? Using 54 simulated frozen Cs core electrons. Table 14 Summary of DDEC6 bond orders and NACs for the different X@Cgo endohedral complexes studied
Fig.22 Parallelization timing and efficiency results for: [Am@Ceol**, Cs@Cé0 (46 frozen Cs core electrons), Cs@C¢o (54 simulated frozen Cs core electrons), [EU@C¢o1**, Li@Ceo, N@Ceéo, and Xe@Cgo. These calculations have 61 atoms per unit cell and were computed using PBE/planewave. We summarized theory, flow diagrams, main features, and computational parameters for DDEC6 analysis. A key advantage is that the DDEC6 method has predictably rapid and robust convergence. DDEC6 charge partitioning requires seven charge partitioning steps, and this corresponds to solving a series of 14 Lagrangians in order. DDEC6 ASMs are computed using the DDEC spin partitioning algorithm of Manz and Sholl” applied using the DDEC6 {pa(7,)}..7"* These ASMs_ converge DDEC6 NACs and bond orders are also shown in Fig. 23. The H-O distances are 2.01 A for the hydrogen bond and 0.96 A for the covalent bond. The computed DDEC6 H-O bond orders are 0.84-0.90 for the covalent bond and 0.10 for the hydrogen bond. The computed NACs indicate that the H-O covalent bond in water is polar-covalent with electrons drawn from the hydrogen towards the oxygen. As discussed in our previous publication, the DDEC6 H,O NACs are in good agreement with common 3- site water models typically used in classical atomistic molecular
Fig.22 Parallelization timing and efficiency results for: [Am@Ceol**, Cs@Cé0 (46 frozen Cs core electrons), Cs@C¢o (54 simulated frozen Cs core electrons), [EU@C¢o1**, Li@Ceo, N@Ceéo, and Xe@Cgo. These calculations have 61 atoms per unit cell and were computed using PBE/planewave. We summarized theory, flow diagrams, main features, and computational parameters for DDEC6 analysis. A key advantage is that the DDEC6 method has predictably rapid and robust convergence. DDEC6 charge partitioning requires seven charge partitioning steps, and this corresponds to solving a series of 14 Lagrangians in order. DDEC6 ASMs are computed using the DDEC spin partitioning algorithm of Manz and Sholl” applied using the DDEC6 {pa(7,)}..7"* These ASMs_ converge DDEC6 NACs and bond orders are also shown in Fig. 23. The H-O distances are 2.01 A for the hydrogen bond and 0.96 A for the covalent bond. The computed DDEC6 H-O bond orders are 0.84-0.90 for the covalent bond and 0.10 for the hydrogen bond. The computed NACs indicate that the H-O covalent bond in water is polar-covalent with electrons drawn from the hydrogen towards the oxygen. As discussed in our previous publication, the DDEC6 H,O NACs are in good agreement with common 3- site water models typically used in classical atomistic molecular
Fig.23 Left: NACs (black) and bond orders (blue) of water dimer. Right: Parallelization timing and efficiency results for water dimer (6 atoms per unit cell, PBE/planewave method). The solid blue line is the overall parallelization efficiency. The dashed blue line is the parallelization efficiency excluding setting up density grids.
Fig.23 Left: NACs (black) and bond orders (blue) of water dimer. Right: Parallelization timing and efficiency results for water dimer (6 atoms per unit cell, PBE/planewave method). The solid blue line is the overall parallelization efficiency. The dashed blue line is the parallelization efficiency excluding setting up density grids.
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Quantum PhysicsComputational ChemistryQuantum ChemistryHigh Performance Scientific ComputingDFT calculationComputational Quantum ChemistryDensity functional theory calculationsComputational & Theoretical ChemistryElectronic Structure TheoryPhysical, Theoretical and Computational Chemistry
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