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L. Thøgersen, J. Olsen / Chemical Physics Letters 393 (2004) 36–43 41 A smaller than the FCI value, but the frequency is 6 cm 1 too large. The deviations of the various CC methods obtained here for CN are very similar to the previously obtained deviations for N 2 . Thus, it has previously been reported that the contribution from connected quadruple excitations to the harmonic frequency for this molecule [23,32] is 20 cm 1 . It is currently not feasible to obtain FCI energies for CN in the cc-pVTZ basis with an accuracy that is sufficient to obtain the frequency with an accuracy of 1cm 1 or less. One can instead estimate the convergence by examining the changes in the constants through the CC hierarchy. It is seen from Table 4 that the changes between the CCSDT and CCSDTQ results are very similar in the cc-pVDZ and cc-pVTZ basis sets. In both basis sets, the quadruple excitations increase the distance by 0.0020 A and reduce the harmonic frequency by about 20 cm 1 . This suggests that it may be feasible to obtain the quadruple corrections to these constants in rather small basis sets. It should be noted that although the quadruple corrections to the properties are rather constant, the quadruples corrections to the raw energies are very different in the two basis sets. The errors of the harmonic frequencies arise from two sources. First of all, the positive curvatures of the CC deviation curves around the equilibrium geometries lead to CC frequencies that are larger than the FCI frequency. Furthermore, as the third derivative of the energies with respect to the distance in general is large and negative, the somewhat shorter internuclear distances obtained with the CC methods than with FCI lead also to frequencies that are too large. These two sources of errors may be analyzed in the cc-pVDZ basis by evaluating the CC frequencies at the FCI equilibrium geometry. For the orbital based methods one then obtains the frequencies 2035, 2027 and 2022 cm 1 for the CCSD, CCSDT and CCSDTQ methods. Whereas, the CCSDT frequency evaluated at the optimized CCSDT distance deviates from the FCI frequency by 23 cm 1 , the CCSDT frequency evaluated at the FCI geometry thus deviates by only 7 cm 1 . Although the errors connected with the positive curvatures of the deviation curves are not vanishing, the major errors of the frequencies seem to arise from the errors of the equilibrium distances. The experimental values for the equilibrium distance and the harmonic frequency are 1.1718 A and 2069 cm 1 , respectively, [30]. Comparing the results obtained using the cc-pVTZ basis to the experimental values, it is observed that the CCSDT results are in better agreement with experiment than the CCSDTQ results. A better estimate of the importance of the quadruples corrections may be obtained using CCSDT results for large basis sets. Feller and Sordo [2] have calculated the CCSDT spectroscopic constants for CN using the augcc-pVQZ basis and obtained the equilibrium distance 1.1739 A and the harmonic frequency of 2082 cm 1 . Adding our quadruples correction to these CCSDT results gives an equilibrium geometry of 1.1759 A and a harmonic frequency of 2060 cm 1 . To obtain spectroscopic constants that are significantly more accurate than the CCSDT results, other corrections, most important core-correlation contributions, must be included together with the quadruple excitations. 3.4. Atomization energy It has previously been reported that quadruple and even quintuple excitations may be important to obtain atomization energies with high accuracy [3,4,12] In Table 5, we list the atomization energies using the CCSD, CCSDT, CCSDTQ, and FCI approaches with the cc-pVDZ basis and the CCSD, CCSDT, and CCSDTQ approaches with the cc-pVTZ basis set. All molecular calculations were carried out at the experimental equilibrium distance. It is again noticed that there are no significant difference between the results obtained using the CC(orb) and CC(spin–orb) approaches. The two approaches differ by only 0.1 kJ/mol at the CCSDT and CCSDTQ levels. The quadruple excitations change the atomization energy by 4 kJ/mol with both the cc-pVDZ and the ccpVTZ basis sets. These results are in agreement with previous calculations of the contributions from connected quadruple excitations [4]. From the difference between CCSDTQ and the FCI atomization energy, it is seen that the quintuple excitations contribute 0.5 kJ/mol to the atomization energy. The above contribution from quadruple and quintuple excitations are very similar to the results previously reported for N 2 [3]. The contribution from higher excitations to the atomization energy of CN has previously been studied by Feller and Sordo [2]. They obtained a significantly smaller contribution from quadruple excitations, 0.3 kcal/mol or 1.2 kJ/mol. There are several experimental measurements of the atomization energies, and Feller and Sordo [2] quotes Table 5 The electronic contribution to the dissociation energy (kJ/mol) for CN CCSD(orb) cc-pVDZ 631.6 CCSDT(orb) cc-pVDZ 663.0 CCSDTQ(orb) cc-pVDZ 666.5 CCSD(spin–orb) cc-pVDZ 629.2 CCSDT(spin–orb) cc-pVDZ 662.9 CCSDTQ(spin–orb) cc-pVDZ 666.4 FCI cc-pVDZ 667.0 CCSD(spin–orb) cc-pVTZ 674.2 CCSDT(spin–orb) cc-pVTZ 714.4 CCSDTQ(spin–orb) cc-pVTZ 718.5 D e
42 L. Thøgersen, J. Olsen / Chemical Physics Letters 393 (2004) 36–43 values in the range 745–762 kJ/mol for the experimental electronic contribution. Adding our estimate of quadruples correction to the estimated CCSDT limit of 748 kJ/mol result of Feller and Sordo gives a value of 752 kJ/ mol for the electronic atomization energy for CN. 3.5. The vertical electron affinity An FCI calculation for the CN anion using the aug 0 - cc-pVDZ basis was carried out at the experimental equilibrium geometry of the radical. The FCI calculation contains about 20 billion Slater determinants and sparsity of the CI-vectors was only used to reduce discstorage, not computation time. This FCI calculation represents one of the largest FCI calculations we hitherto have carried out. The FCI energy for the anion was obtained as )92.627391(2) E h . Combining this energy with the FCI energy of )92.497766 E h for the radical in the same basis set leads to an FCI value of 0.12962 E h for the vertical electron affinity. CC expansions using spin–orbital occupations for restrictions of excitations were also carried out for the radical and the anion in the aug 0 -cc-pVDZ basis and the resulting electron affinities are given in Table 6. As the differences between the CC calculations using orbital and spin–orbital restrictions already have been shown to be small, no orbital-restricted calculations were carried out. Already at the CCSD level, the calculated electron affinity differs from the FCI affinity by less than 1 mE h , and at the CCSDT level the calculated electron affinity differs from the FCI result by less than 0.1 mE h . The deviations of the CC energies from the FCI energies for the radical and the anion are also listed in Table 6. It is seen that the high accuracy of the CC affinities is caused by cancellation of the errors of the radical and anion – the deviation of the affinity is roughly an order of magnitude smaller than the deviation of the individual energies. It is also interesting to see that the electron affinity converges from above – the CC affinities are larger than the FCI affinity. As seen from the other columns of Table 6, the CC expansion converges slightly faster for the anion than for the radical. The faster convergence of the anion may seem surprising as the anion contains one more electron than the radical but is probably caused by CN being slightly more multiconfigurational than the anion. The electron affinity of CN calculated using CC calculations in large basis sets has been the subject of several recent studies [33,34]. These studies also found small contributions to the electron affinity from triple excitations. 4. Conclusion Full configuration interaction calculations using the cc-pVDZ basis and CC calculations using the cc-pVDZ and cc-pVTZ basis sets have been carried out for the CN radical at various geometries. Single reference configuration interaction calculations were also carried out using the cc-pVDZ basis at the experimental internuclear distance. At the CCSDT level, the energies differ from the FCI energy by 1.5 mE h , and at the CCSDTQ level, the energies are 0.2 mE h from the FCI energy. The CC energies converge toward the FCI energy in an approximately linear fashion with a decrease in the deviation by about a factor of 10 for each added excitation level. This is in contrast to an analysis based on perturbation theory, predicting that adding even orders give larger decreases in the deviations than adding odd orders. The observed convergence for CN in the ccpVDZ basis is very similar to the convergence previously reported for N 2 , indicating that the open-shell nature of CN does not affect the convergence. A comparison of the FCI and CC energies at various internuclear distances, reveals that the deviations of the CC approaches do not occur suddenly for large internuclear distances. The deviations are instead nearly linear functions of the internuclear distance. At the FCI level, the equilibrium geometry and harmonic frequency are obtained as 1.1969 A and 2020.1 cm 1 , respectively. The CCSDT and CCSDTQ frequencies are 25 and 5 cm 1 above the FCI value, respectively. The quadruple corrections to both the equilibrium distance and the harmonic frequency were found to be nearly identical in the cc-pVDZ and ccpVTZ basis sets. The major errors of the CC frequencies come from the errors of the distances where these are evaluated. For the electronic contribution to the atomization energy, a value of 667.0 kJ/mol is obtained at the FCI level using the cc-pVDZ basis set. The CCSDT and CCSDTQ atomization energies are 4 and 0.5 kJ/mol below the FCI atomization energy, respectively. The quadruple contributions in the cc-pVDZ and cc-pVTZ Table 6 The vertical electron affinity (E h ) of CN calculated in the aug 0 -cc-pVDZ basis EA EA EA FCI E CN EFCI CN E CN –EFCI CN CCSD(spin–orb) 0.13025 0.00063 0.01529 0.01466 CCSDT(spin–orb) 0.12977 0.00014 0.00154 0.00140 CCSDTQ(spin–orb) 0.12966 0.00003 0.00020 0.00016 FCI 0.12962
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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 Benchmarking for Radicals ar
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Part 3 Benchmarking for Radicals As
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Part 3 Benchmarking for Radicals R
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