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Part 3 Benchmarking for Radicals level closer to the FCI solution. Describing the convergence pattern of the CI and CC hierarchies through orders of Møller-Plesset perturbation theory (MPPT), 73 the form of the curves can be predicted. Because also disconnected products of excitations are included in the ansatz of CC, the order of its error grows continually in the order of MPPT. Going from uneven to even excitation levels, both methods have an increase in the order of error in energy of two orders of MPPT, thus, the graphs are parallel. Going from even to uneven excitation levels, the CC error increases one order, whereas the CI error remains unchanged, giving a greater slope for the CC curve. This explains the parallel behavior going from uneven to even excitation levels and the smoother convergence of the CC hierarchy compared to the CI hierarchy. The stepwise convergence predicted by MPPT, which should be significant for CI and noticeably for CC, is not apparent though. The reason could be that CN is not strictly mono-configurational. The convergence patterns for CI and CC are very similar to the convergence patterns previously reported for N 2 . 74 Therefore, 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. 3.3.2 The Potential Curve for CN The potential curve for CN was determined from single-point calculations at the FCI level with basis set cc-pVDZ. Close to equilibrium the energies were converged to 10 -9 E h making the determination of accurate spectroscopic constants possible. The result is displayed in Fig. 3.3. E FCI / E h -92.15 -92.20 -92.25 -92.30 -92.35 -92.40 -92.45 -92.50 0.5 1.5 R / Å 2.5 3.5 Fig. 3.3 The potential curve for CN found from FCI cc-pVDZ calculations. E dev / E h 0.03 0.02 0.01 0.00 CCSD CCSDT CCSDTQ 0.9 1.2 R / Å 1.5 1.8 Fig. 3.4 E dev for the CC potential curves. E dev (R) = E(R) – E FCI (R). The potential curve was also created with the methods CCSD(orb), CCSDT(orb) and CCSDTQ(orb) in the basis set cc-pVDZ. Since the weight of the reference HF- determinant decreases as the internuclear distance increases, we examine the HF-coefficients from the FCI calculations and discover that it is irrelevant to make single-reference CC calculations beyond R = 1.8Å, since the weight of the reference has already dropped to 0.57 at that point. Fig. 3.4 displays the differences of the CC 80
Numerical Results potential curves compared to the FCI curve. At a given inter-nuclear distance, the FCI energy has been subtracted from the CC energy. The decreasing weight of the reference ground state with increasing atomic distance is reflected in the quality of the CC wave functions. The correlation in the wave function compensates partially for the lack of a single dominant configuration; the higher the correlation level, the better the compensation. This is illustrated by the slopes of the curves in Fig. 3.4. Furthermore, it should be noticed how the deviation energy is nearly linear in R, with a slightly positive curvature around the equilibrium geometry. 3.3.3 Spectroscopic Constants and Atomization Energy for CN The equilibrium geometry and harmonic frequency for CN were found from single-point calculations using quartic interpolation. The atomization energy was found at the experimental equilibrium distance. The results are displayed in Table 3-1. Table 3-1 Equilibrium geometry, harmonic frequency, and atomization energy for CN. R eq / Å ω e / cm -1 D e / kJ/mol CCSD(spin-orb) cc-pVDZ 1.1855 2114 629.2 CCSD(orb) cc-pVDZ 1.1860 2111 631.6 CCSDT(spin-orb) cc-pVDZ 1.1944 2046 662.9 CCSDT(orb) cc-pVDZ 1.1946 2043 663.0 CCSDTQ(spin-orb) cc-pVDZ 1.1964 2026 666.4 CCSDTQ(orb) cc-pVDZ 1.1964 2025 666.5 FCI cc-pVDZ 1.1969 2020 667.0 CCSD(spin-orb) cc-pVTZ 1.1688 2136 674.2 CCSDT(spin-orb) cc-pVTZ 1.1783 2067 714.4 CCSDTQ(spin-orb) cc-pVTZ 1.1804 2045 718.5 Experimental 72 1.1718 2069 --- As mentioned in Section 3.2, it is not feasible to carry out FCI calculations at the cc-pVTZ level. Still, the convergence of the CC hierarchy can be estimated by examining the changes in the constants. Since the difference in accuracy between the models CC(orb) and CC(spin-orb) is negligible compared to the deviation from FCI, only the CC(spin-orb) results are discussed from now on and only the CC(spin-orb) numbers are found at the cc-pVTZ level. The deviation curves for the coupled cluster energies (see Fig. 3.4) are increasing functions, and thus the coupled cluster equilibrium bond lengths are shorter than the one found from FCI. Furthermore, the positive curvature of the deviation-curves around the equilibrium leads to coupled cluster frequencies that are higher than the FCI frequency. 81
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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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