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“Enthalpy (Refs. 57 and 58) difference of 68.97(71) kJ/mol, corrected with AZPE (Ref. 59) 5.84 kcal/mol. TABLE VIII. Cycloaddition: 2C;H,—C,Hg. We use the same ethylene geometry as previously, and ou cyclobutane geometry is Rec=1.545 029 A, Renax= 1-089 404 A, Reyeg=1.0877 A, and Oycy= 109.18. Be- low the solid horizontal line are inconsistent calculations, where the number of GVB pairs for the two states are in the parentheses. We highlighted the result which meets our selection criteria.

Table 8 “Enthalpy (Refs. 57 and 58) difference of 68.97(71) kJ/mol, corrected with AZPE (Ref. 59) 5.84 kcal/mol. TABLE VIII. Cycloaddition: 2C;H,—C,Hg. We use the same ethylene geometry as previously, and ou cyclobutane geometry is Rec=1.545 029 A, Renax= 1-089 404 A, Reyeg=1.0877 A, and Oycy= 109.18. Be- low the solid horizontal line are inconsistent calculations, where the number of GVB pairs for the two states are in the parentheses. We highlighted the result which meets our selection criteria.

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FIG. 1. Typical ground state Jastrow functions used in this study for the aug-tz basis set. Based on our experience, we do not believe that any Jas- trows would look significantly different than these. In our optimization, we ignored minor flaws in the Jastrows, such as wiggles in the electron-H Jastrows, or the brief crossing of the exp(u,))=1 line in opposite spin Jastrows.
FIG. 1. Typical ground state Jastrow functions used in this study for the aug-tz basis set. Based on our experience, we do not believe that any Jas- trows would look significantly different than these. In our optimization, we ignored minor flaws in the Jastrows, such as wiggles in the electron-H Jastrows, or the brief crossing of the exp(u,))=1 line in opposite spin Jastrows.
Ill. RESULTS FIG. 2. Cancellation of time step error between triplet to singlet energies in methylene, using three pair delocalized GVB wave function. For this plot, individual calculations were stopped when they reached a statistical error of exactly 0.065 kcal/mol, corresponding to an error of 0.092 kcal/mol for the difference. We plot the differences here against the amount of simulated time, iterations X time step X average move acceptance probability.
Ill. RESULTS FIG. 2. Cancellation of time step error between triplet to singlet energies in methylene, using three pair delocalized GVB wave function. For this plot, individual calculations were stopped when they reached a statistical error of exactly 0.065 kcal/mol, corresponding to an error of 0.092 kcal/mol for the difference. We plot the differences here against the amount of simulated time, iterations X time step X average move acceptance probability.
FIG. 3. The final energies from Fig. 2, plotted against their time step. All appear to be producing approximately the same result, so we choose a 0.0075 time step for the rest of our calculations, in lieu of extrapolating to zero time step. The length of wallclock time for each calculation varied with many factors, but ranged from about 40 h on methylene to about 100 h on ethylene to about 400 h on cyclobutane. However, for these same calculations, each processor only
FIG. 3. The final energies from Fig. 2, plotted against their time step. All appear to be producing approximately the same result, so we choose a 0.0075 time step for the rest of our calculations, in lieu of extrapolating to zero time step. The length of wallclock time for each calculation varied with many factors, but ranged from about 40 h on methylene to about 100 h on ethylene to about 400 h on cyclobutane. However, for these same calculations, each processor only
“Experimental T,=Ty+AZPE, where Ty=3147+5 cm! (Ref. 36) and AZPE =128+18 cm! (Ref. 38). Optimizing basis functions and orbitals, from Ref. 39. ‘From Ref. 37. “From Ref. 54. TABLE I. Methylene excitations: 'A,—3B, and 'B,—%B,. For °B,, [Roy, Oycy]=[1.0753 A, 133.93] from experiment (Ref. 36), for 'A, [1.107 A, 102.4] from experiment (Ref. 53), and for 'B, [1.0723 A, 142.44] from theory (Ref. 54). By “B,” we are indicating our basis, by “O” we are indicating, where it matters, whether our GVB pairs are localized or delocalized, and by “J” we are indicating whether we are using two or three particle Jastrows. We highlighted the results which meets our selection criteria.
“Experimental T,=Ty+AZPE, where Ty=3147+5 cm! (Ref. 36) and AZPE =128+18 cm! (Ref. 38). Optimizing basis functions and orbitals, from Ref. 39. ‘From Ref. 37. “From Ref. 54. TABLE I. Methylene excitations: 'A,—3B, and 'B,—%B,. For °B,, [Roy, Oycy]=[1.0753 A, 133.93] from experiment (Ref. 36), for 'A, [1.107 A, 102.4] from experiment (Ref. 53), and for 'B, [1.0723 A, 142.44] from theory (Ref. 54). By “B,” we are indicating our basis, by “O” we are indicating, where it matters, whether our GVB pairs are localized or delocalized, and by “J” we are indicating whether we are using two or three particle Jastrows. We highlighted the results which meets our selection criteria.
TABLE II. The VMC energies for 'A,—*B, corresponding to the DMC calculations from Table I. The labels have the same meaning. We include the rms fluctuations for each calculation. We highlighted the result which meets our selection criteria. quicker, demonstrating their value in variance reduction. This comparison encourages their use, but this conclusion changes for cyclobutane. We wanted to discover the effect of optimizing different parts of the wave function. We pursued this by choosing some standard state for each atom for the Jastrows, and then selectively optimizing parts of the wave function, and com- paring these results to the comparable result from Table I. Our results, shown in Table II], tell us that optimizing the CI coefficients was worth 0.5 kcal/mol, and that optimizing the electron-nuclear Jastrows was worth another 0.4 kcal/mol. Of course, in the limit of zero time step Jastrows should not matter, so much of the error here can be called time step error. But it appears to be crucial that we optimize the CI coefficients, even if we are not optimizing the orbitals. Re- turning to the question of the CAS discrepancy, we tried optimizing all the Jastrows, while keeping the original CI coefficients. This produced 15.80+0.11 kcal/mol as the DMC energy splitting, the worst of all the results we ob- tained. Since this represents the comparison of the SCF func- tions directly without worrying about whether the VMC op- timization is falling into local minima, we can see that, given our methodology, a better SCF wave function does not al- ways improve accuracy unless everything is optimized.*° On the other hand, in separate investigations,” we found that CAS wave functions were the simplest accurate representa- tions we tried.
TABLE II. The VMC energies for 'A,—*B, corresponding to the DMC calculations from Table I. The labels have the same meaning. We include the rms fluctuations for each calculation. We highlighted the result which meets our selection criteria. quicker, demonstrating their value in variance reduction. This comparison encourages their use, but this conclusion changes for cyclobutane. We wanted to discover the effect of optimizing different parts of the wave function. We pursued this by choosing some standard state for each atom for the Jastrows, and then selectively optimizing parts of the wave function, and com- paring these results to the comparable result from Table I. Our results, shown in Table II], tell us that optimizing the CI coefficients was worth 0.5 kcal/mol, and that optimizing the electron-nuclear Jastrows was worth another 0.4 kcal/mol. Of course, in the limit of zero time step Jastrows should not matter, so much of the error here can be called time step error. But it appears to be crucial that we optimize the CI coefficients, even if we are not optimizing the orbitals. Re- turning to the question of the CAS discrepancy, we tried optimizing all the Jastrows, while keeping the original CI coefficients. This produced 15.80+0.11 kcal/mol as the DMC energy splitting, the worst of all the results we ob- tained. Since this represents the comparison of the SCF func- tions directly without worrying about whether the VMC op- timization is falling into local minima, we can see that, given our methodology, a better SCF wave function does not al- ways improve accuracy unless everything is optimized.*° On the other hand, in separate investigations,” we found that CAS wave functions were the simplest accurate representa- tions we tried.
TABLE III. The effect of optimizing different parts of our aug-tz GVB-3 delocalized methylene wave functions, with two particle Jastrows, and all calculations run at 0.0075 time step. The starting point for these calculations is the CI coefficients from GVB, electron-carbon and electron-electron Jas- trows from optimized carbon GVB-1 atom, and electron-hydrogen Jastrows from an optimized GVB-1 H, molecule. Each row corresponds to a different set of parameters which were optimized, where EN stands for electron- nuclear and EE for electron-electron.
TABLE III. The effect of optimizing different parts of our aug-tz GVB-3 delocalized methylene wave functions, with two particle Jastrows, and all calculations run at 0.0075 time step. The starting point for these calculations is the CI coefficients from GVB, electron-carbon and electron-electron Jas- trows from optimized carbon GVB-1 atom, and electron-hydrogen Jastrows from an optimized GVB-1 H, molecule. Each row corresponds to a different set of parameters which were optimized, where EN stands for electron- nuclear and EE for electron-electron.
TABLE IV. Methylene excitations using single determinant wave functions. Using our aug-tz basis set, we obtained orbitals from RHF or DFT, and added two particle Jastrows. determinant representations of the trial wave function. Here we can see that none of these wave functions are capable of addressing the missing angular correlation, and that perhaps DFT is worse than RHF. This demonstrates that the fixed node approximation is associated with static correlation, which DFT cannot do much about.
TABLE IV. Methylene excitations using single determinant wave functions. Using our aug-tz basis set, we obtained orbitals from RHF or DFT, and added two particle Jastrows. determinant representations of the trial wave function. Here we can see that none of these wave functions are capable of addressing the missing angular correlation, and that perhaps DFT is worse than RHF. This demonstrates that the fixed node approximation is associated with static correlation, which DFT cannot do much about.
“Energy-loss spectra, from Ref. 41. *Single determinant from CASSCF(4,8), using pseudopotentials, from Ref. 43 “Using pseudopotentials and their custom basis set, from Ref. 44. “These DMC results use VMC optimized orbitals. “Only determinants with coefficients >0.01 were included. ‘Computed value from Ref. 45. Only determinants with coefficients >0.001 were included. "Optical spectra, from Ref. 42. ‘Adsorption spectra, from Ref. 55. TABLE V. Vertical ethylene: *B,,,<— "Ay and 'B,,<'A,. For *B,,. For all calculations, we used Rec =1.339 A, Roy=1.086 A, and Oycy=117.6, in order for our results to be directly comparable with Schautz (Ref. 44). The entries below the horizontal line are inconsistent with the number of GVB pairs indicated in parentheses for each state individually. We highlighted the result which meets our selection criteria.
“Energy-loss spectra, from Ref. 41. *Single determinant from CASSCF(4,8), using pseudopotentials, from Ref. 43 “Using pseudopotentials and their custom basis set, from Ref. 44. “These DMC results use VMC optimized orbitals. “Only determinants with coefficients >0.01 were included. ‘Computed value from Ref. 45. Only determinants with coefficients >0.001 were included. "Optical spectra, from Ref. 42. ‘Adsorption spectra, from Ref. 55. TABLE V. Vertical ethylene: *B,,,<— "Ay and 'B,,<'A,. For *B,,. For all calculations, we used Rec =1.339 A, Roy=1.086 A, and Oycy=117.6, in order for our results to be directly comparable with Schautz (Ref. 44). The entries below the horizontal line are inconsistent with the number of GVB pairs indicated in parentheses for each state individually. We highlighted the result which meets our selection criteria.
“Only determinants with coefficients >0.001 were used. TABLE VI. Vertical ethylene: 2 Bis "Ag and 'B,,<'A,. For 3B,,. For these calculations, we used MP2 optimized Rec=1.331 046 A, Rey=1. 080. "564 A, and @ycy=121.35. We highlighted the result which meets our selection criteria.
“Only determinants with coefficients >0.001 were used. TABLE VI. Vertical ethylene: 2 Bis "Ag and 'B,,<'A,. For 3B,,. For these calculations, we used MP2 optimized Rec=1.331 046 A, Rey=1. 080. "564 A, and @ycy=121.35. We highlighted the result which meets our selection criteria.
“We “uncorrect’” the experimental value from Ref. 56 of 58(3) kcal/mol and all the computed results from Ri 45 by AZPE=3.2 kcal/mol, so that we can directly compare calculations. *Single determinant from CASSCF(4,8), using pseudopotentials, computed value from Ref. 43. “Our own CASSCF(12,12) calculation. TABLE VII. Adiabatic ethylene: *B,,<—'A,. For *B,,, we use Rcc=1.449 148 A, Roy=1.080 469 A, and @yycy= 121.5, and we use the same geometry as previously for 'A,. The entries below the horizontal line are unbalanced in terms of the number of orbitals. We highlighted the result which meets our selection criteria.
“We “uncorrect’” the experimental value from Ref. 56 of 58(3) kcal/mol and all the computed results from Ri 45 by AZPE=3.2 kcal/mol, so that we can directly compare calculations. *Single determinant from CASSCF(4,8), using pseudopotentials, computed value from Ref. 43. “Our own CASSCF(12,12) calculation. TABLE VII. Adiabatic ethylene: *B,,<—'A,. For *B,,, we use Rcc=1.449 148 A, Roy=1.080 469 A, and @yycy= 121.5, and we use the same geometry as previously for 'A,. The entries below the horizontal line are unbalanced in terms of the number of orbitals. We highlighted the result which meets our selection criteria.
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