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Part 3 Benchmarking for Radicals As expected, the cc-pVDZ basis set does not provide accurate geometries and frequencies, and the cc-pVTZ numbers are clearly more in the range of the experimental data than the cc-pVDZ numbers. CCSD displays its insufficiency for prediction of equilibrium properties by differing from the FCI values by 0.01Å in the geometry, 90 cm -1 in the frequency, and 35 kJ/mol in the atomization energy. The errors in R eq and ω e are reduced by a factor of four going to the CCSDT level and a factor of five going from the CCSDT to the CCSDTQ level. The error in the atomization energy is reduced by a factor of nine going to the CCSDT level and a factor of eight going from the CCSDT to the CCSDTQ level, but while the equilibrium geometry on the CCSDTQ level is only 0.0005Å from the FCI value, the harmonic frequency is still about 5 cm -1 too high. Both the equilibrium geometry and the harmonic frequency are apparently better approximated by the CCSDT method than the CCSDTQ. This is due to a favorable cancellation in errors for CCSDT calculations in small basis sets. By extrapolation to the larger aug-cc-pVQZ basis, 67,75 we get an equilibrium distance of 1.1759Å and a harmonic frequency of 2060cm -1 at the CCSDTQ level. 3.3.4 The Vertical Electron Affinity of CN Calculations on CN - and CN were carried out in the aug´-cc-pVDZ basis at the experimental equilibrium geometry for CN. The FCI calculation on CN - is one of the largest FCI calculations carried out so far containing about 20 billion Slater determinants. The vertical electron affinity (EA) was found and is displayed in Table 3-2. Again only CC(spin-orb) calculations have been carried out because of the rather small difference in performance of CC(spin-orb) and CC(orb). Table 3-2 The vertical electron affinity of CN. EA / E h EA - EA FCI CCSD(spin-orb) aug’-cc-pVDZ 0.13025 0.00063 CCSDT(spin-orb) aug’-cc-pVDZ 0.12977 0.00014 CCSDTQ(spin-orb) aug’-cc-pVDZ 0.12966 0.00003 FCI aug’-cc-pVDZ 0.12962 --- The convergence is remarkable; already at the CCSD level we are down to an error of 0.5% of the FCI value, on the CCSDT level it is 0.1% and on the CCSDTQ level 0.02%. The reason for the excellent convergence is found in a cancellation of errors that influence the result. The deviations of the individual energies are always roughly an order of magnitude larger than the deviation of the affinity, 75 but the errors cancel when the CN and CN - energies are subtracted. That the convergence is from above is also noteworthy. This is because the CC hierarchy converges faster for CN - than for 82
Numerical Results CN. This seems surprising since CN - contains one more electron than CN, but it could be explained by CN - being more one-configurational than CN. 3.3.5 The Equilibrium Geometry of CCH The equilibrium geometry of CCH found from FCI/cc-pVDZ calculations is used in ref. 76 to calibrate coupled cluster calculations in larger basis sets. The FCI correction is assumed to be independent of basis set. To optimize for the two variables R(CC) and R(CH), the CCH radical is assumed linear and the CC and CH bonds are then distorted in step-lengths of δ = 0.01Å from an initial geometry making a grid of single-point calculations around the equilibrium geometry with R(CC) on the one axis and R(CH) on the other. The initial geometry is taken from a CCSDT cc-pVDZ study 76 , the geometry being R CCSDT (CC) = 1.23448Å and R CCSDT (CH) = 1.07924Å. The resulting potential energy surface is seen in Fig. 3.5. -76.4020 -76.4024 E FCI / E h -76.4028 -76.4032 -76.4036 1.09924 1.08924 1.07924 1.06924 R (C-H)/Å 1.21448 1.22448 1.23448 R (C-C)/Å 1.24448 1.25448 1.05924 Fig. 3.5 The potential energy surface of CCH. From finite-difference expressions with the error being of the order δ 4 , the gradient and Hessian are found for the initial geometry and a Newton step is taken giving an improved guess for the equilibrium geometry. The FCI equilibrium geometry is thus found as FCI CCSDT −1 R = R −H G, (3.1) where G is the gradient, H the Hessian, and R CCSDT the CCSDT geometry. The equilibrium geometry at the FCI level is found to be 83
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PhD Thesis Optimization of Densitie
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Contents Preface ..................
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Summary............................
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List of Publications This thesis in
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Part 1 Improving Self-consistent Fi
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