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Part 3 Benchmarking for Radicals R FCI (CC) = 1.2367Å and R FCI (CH) = 1.0802Å. The error in the resulting geometry is a sum of the error from the finite difference approximations and the error from the Newton step. The gradient and Hessian carry an error of O(δ 4 ) where δ = 0.01Å, this is an error in the order of 10 -8 Å. The Newton step has an error of O((H -1 G) 2 ), in this case H -1 G is of the size 10 -3 Å and so the error is in the order of 10 -6 Å. The error in total is thus in the order of 10 -6 Å. The gradient for the FCI equilibrium geometry has been found as above, making single-point calculations at the FCI geometry and at geometries distorted in steps of 0.01Å from the FCI geometry. The same finite-difference expressions as before are used. The gradient is found to be ⎡ FCI 1.8593 10 E ⎢ ;3.0661 10 ⎣ Å ⎤ Å⎥⎦ G −5 h −5 = − ⋅ ⋅ h , (3.2) thus verifying the correctness of the FCI geometry. Since the geometry was determined at the CCSDT level to be R CCSDT (CC) = 1.23448Å and R CCSDT (CH) = 1.07924Å, the error due to truncation of the many-electron basis in CCSDT is in the order of 10 -3 Å. This is similar to the results obtained for CN. This also suggests that the quadruples correction to the equilibrium geometry is in the order of 0.001-0.002Å. 3.4 Conclusion Full configuration interaction (FCI) and coupled cluster (CC) calculations have been carried out on CN using the cc-pVDZ and cc-pVTZ basis sets. The equilibrium bond distance, harmonic frequency, atomization energy, and vertical electron affinity have been evaluated on the various levels of theory. As expected, the cc-pVDZ basis set does not provide accurate geometries and frequencies and CCSD is insufficient for prediction of equilibrium properties. Apparently, the CCSDT method is a better approximation than CCSDTQ for obtaining the equilibrium geometry and the harmonic frequency. This is due to a favorable cancellation of errors for CCSDT calculations in small basis sets. Also the vertical electron affinities are affected by cancellation of errors, and already at the CCSD level, the error is less than 1mE h compared to the FCI value. The convergence patterns for the CI and CC hierarchies are studied for CN and it is found similar to the convergence patterns previously reported for N 2 . 74 Thus, it does not seem that the open-shell nature of CN leads to slow convergence of the CI and CC hierarchies compared to closed shell cases. E 84
Conclusion For a number of the CC calculations, the excitation levels have been defined by spin-orbital excitations instead of orbital excitations. Certain internal excitations are thereby omitted, but it is seen that this does not affect the accuracy in any significant way. For a given excitation level, the energies obtained in the orbital formalism are in all cases closer to the FCI energy than the ones obtained in the spin-orbital formalism. However, the difference is negligible. The equilibrium geometry of CCH has been found at the FCI level in the cc-pVDZ basis set to be R FCI (CC) = 1.2367Å and R FCI (CH) = 1.0802Å. The correction found to the initial CCSDT geometry is in the order of 10 -3 Å. The FCI correction to the CCSDT equilibrium geometry of CN was of the same order. 85
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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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Part 3 A Coupled Cluster and Full C
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