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L. Thøgersen, J. Olsen / Chemical Physics Letters 393 (2004) 36–43 39 From the above comparisons, it may be concluded, that the open-shell nature of CN does not lead to slower convergence of the CC hierarchy than previously observed for N 2 . However, it should be noted, that the convergence of the CC hierarchy for N 2 is rather slow compared to the convergence observed for e.g., H 2 O [19] and F 2 . 3.2. The potential curve for CN FCI calculations were carried out at a number of internuclear distances. To obtain accurate spectroscopic constants, the CC energies were converged to 10 9 E h Table 2 FCI energies (E h ) as a function of internuclear distance ( A) for CN using the cc-pVDZ basis R E R E 0.9 )92.169732 1.2118 )92.494065103 1.0 )92.384032 1.2169 )92.493765608 1.0918 )92.469313943 1.2369 )92.491837833 1.1318 )92.485677652 1.2518 )92.489691918 1.1518 )92.490432414 1.30 )92.479361 1.1569 )92.491327096 1.40 )92.447147 1.1718 )92.493262415 1.50 )92.408657 1.1769 )92.493704946 1.60 )92.370048 1.1869 )92.494267979 1.7577 )92.316388 1.1918 )92.494402963 2.05065 )92.255688 1.1919 )92.494404785 2.3436 )92.241450 1.1969 )92.494449358 2.9295 )92.240346 1.2019 )92.494404774 3.5154 )92.239697 1.2069 )92.494274026 for internuclear distances close to the experimental value. For the remaining geometries the energy was converged to 10 6 E h . The obtained FCI energies are listed in Table 2. The graph of the FCI potential curve is given in Fig. 2. To associate the various internuclear distances with a degree of bond-breaking it is useful to examine the coefficient of the Hartree–Fock determinant in the FCI wave-function. Around the equilibrium geometry, the weight of the HF-determinant is about 0.92. Increasing the internuclear distance leads to a steady lowering of this weight and at 1.3 and 1.8 A, the weights are 0.79 and 0.57, respectively. From 1.8 to 2.0 A the weight drops sharply so the weight at 2.0 A is 0.25 and at 2.5 A less than 0.04. We may therefore say that the bond is half broken at 1.8 A and broken at 2.5 A. In addition to FCI calculations, CCSD(orb), CCSDT(orb) and CCSDTQ(orb) calculations were performed at the various internuclear distances up to 1.8 A. Although it is possible to converge the CC equations for larger distances, we find this of less interest, due to the breakdown of the single-reference approximation. In Fig. 3, we plot the deviations of the CCSDT and CCSDTQ energies from the FCI energy, and in Table 3, we list the non-parallelity error (NPE), i.e., the difference between the largest and smallest deviation from the FCI energy. At the equilibrium distance, both deviation curves in Fig. 3 have a positive curvature. For internuclear distances larger than the equilibrium distance, both the CCSDT and CCSDTQ deviation curves are nearly -92.15 -92.20 -92.25 FCI energy -92.30 -92.35 -92.40 -92.45 -92.50 0.5 1 1.5 2 2.5 3 3.5 4 Internuclear distance Fig. 2. The FCI potential curve for CN using the cc-pVDZ basis. The energies are in Hartrees and the inter-nuclear distances are in A.
40 L. Thøgersen, J. Olsen / Chemical Physics Letters 393 (2004) 36–43 0.007 0.006 CCSDT CCSDTQ Deviation from FCI energy 0.005 0.004 0.003 0.002 0.001 0 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Internuclear distance Fig. 3. The difference between the CCSDT and CCSDTQ energies and the FCI energy for CN using the cc-pVDZ basis. The energies are in Hartrees and the inter-nuclear distances are in A. Table 3 Non-parallelity error (NPE) (E h ) for CCSD, CCSDT, and CCSDTQ Method NPE CCSD 0.042326 CCSDT 0.006355 CCSDTQ 0.001742 linear functions of the internuclear distance. Actually, the slope of the CCSDT deviation is smaller for larger internuclear distances than for the equilibrium distance. The analogous CCSDT- and CCSDTQ-curves for the nitrogen molecule exhibit maxima for an internuclear distance around 1.5 A, (3 au) [23]. 3.3. Spectroscopical constants for CN Equilibrium geometries and harmonic frequencies were obtained for the CCSD, CCSDT, CCSDTQ and FCI methods using quartic interpolation of the energies. The harmonic frequency for a given method was evaluated at the equilibrium geometry of this method. In Table 4 we list the obtained equilibrium distances and frequencies. In addition, the table contains the CCSD, CCSDT and CCSDTQ results for the cc-pVTZ basis. We will first discuss the results obtained using the ccpVDZ basis. The CC calculations using orbital-based excitation spaces are slightly more accurate than those using spin–orbital-based excitation spaces, but the differences are small compared to the size of the deviations. We will therefore, discuss only the spin–orbital based Table 4 Equilibrium distance ( A) and harmonic frequency (cm 1 ) for CN CCSD(orb) cc-pVDZ 1.1860 2111 CCSDT(orb) cc-pVDZ 1.1946 2043 CCSDTQ(orb) cc-pVDZ 1.1964 2025 CCSD(spin–orb) cc-pVDZ 1.1855 2114 CCSDT(spin–orb) cc-pVDZ 1.1944 2046 CCSDTQ(spin–orb) cc-pVDZ 1.1964 2026 FCI cc-pVDZ 1.1969 2020.1 CCSD(spin–orb) cc-pVTZ 1.1688 2136 CCSDT(spin–orb) cc-pVTZ 1.1783 2067 CCSDTQ(spin–orb) cc-pVTZ 1.1804 2045 Expt. 1.1718 2069 excitation spaces. Since the deviation curves for the CC energies are increasing functions, the CC equilibrium distances are necessarily shorter than the FCI equilibrium distance. The causes of the errors of the harmonic frequencies will be discussed in detail below. At the CCSD level, the distance is 0.01 A shorter than the FCI value and the harmonic frequency is about 90 cm 1 larger than the FCI value, stressing the inaccuracy of this method for predicting equilibrium properties. The errors are significantly reduced by the CCSDT method with errors of 0.0025 A and 26 cm 1 for the equilibrium distance and frequency, respectively. The errors are further reduced by about a factor of five by using the CCSDTQ instead of the CCSDT method. At the CCSDTQ level, the equilibrium geometry is only 0.0005 R eq x e
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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 2 Atomic Orbital Based Respons
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Part 3 Benchmarking for Radicals ar
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Part 3 Benchmarking for Radicals le
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Part 3 Benchmarking for Radicals As
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Part 3 Benchmarking for Radicals R
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Summary The developments in compute
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