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Singlet-triplet gaps in diradicals by the spin-flip approach: A benchmark study
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73.This formula has been proposed for the extrapolation of correlation energy. However, it is unclear how to define correlation energy in the EOM methods. One possibility is to define correlation energy as a difference between the total energy of the final state and the Hartree–Fock energy of the (triplet) reference state. We have found that in this case the resulting values differ from the ones obtained by extrapolating the total energies by no more than 0.001 eV.
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80.J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, and R. J. Bartlett, ACES II, 1993. The package also contains modified versions of the MOLECULE Gaussian integral program of J. Almlöf and P. R. Taylor, the ABACUS integral derivative program written by T. U. Helgaker, H. J. Aa. Jensen, P. Jørgensen and P. R. Taylor, and the PROPS property evaluation integral code of P. R. Taylor.
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82.Basis sets were obtained from the Extensible Computational Chemistry Environment Basis Set Database, Version, as developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory which is part of the Pacific Northwest Laboratory, P.O. Box 999, Richland, Washington 99352, and funded by the U.S. Department of Energy. The Pacific Northwest Laboratory is a multiprogram laboratory operated by Battelle Memorial Institute for the U.S. Department of Energy under Contract No. DE-AC06-76RLO 1830. Contact David Feller or Karen Schuchardt for further information.
83.The DZP basis for carbon and oxygen is defined by Dunning’s double-ζ contraction [T. H. Dunning, J. Chem. Phys. 53, 2823 (1970)] of Huzinaga’s primitive Gaussian functions
83.[ S. Huzinaga, J. Phys. Chem. 42, 1293 (1965)] and a single set of polarization functions [ ], the contraction scheme being The triple-ζ (TZ) Huzinaga-Dunning bases [Ref. 15;
83.T. H. Dunning, J. Chem. Phys. 55, 716 (1971)] [contraction schemes are for carbon and oxygen, and for silicon] are augmented by: (i) two sets of polarization functions (TZ2P) [ 0.375; 0.4250; 0.25]; (ii) three sets of polarization functions (TZ3P) 0.75,0.1875; 0.85,0.2125]. Then additional sets of functions are added: (i) one set of functions for TZ2P (TZ2PF) [ ]; (ii) two sets of functions for TZ3P (TZ3P2F) [ 0.40; 0.70]. In addition, TZ2P and TZ2PF bases augmented by one set of diffuse functions ( and respectively) are considered [ ]. Cartesian and functions are used for carbon and oxygen, and pure angular momentum for silicon. In addition, series of correlation consistent basis sets (cc-pVDZ, cc-pVTZ, and cc-pVQZ [Ref. 90; D. Wooh and T. H. Dunning (to be published)]) are used (these employ pure angular momentum and -functions).
84.A. D. Walsh, J. Chem. Soc. 1953, 2260.
85.The TZ bases for carbon and hydrogen are from Refs. 33 and 60; for nitrogen from Ref. 34. The TZ bases for silicon and phosphorus are from Refs. 35 and 36, respectively. All carbon bases are the same as in the previous section. However, some labels are slightly different: i.e., and are equivalent to TZ2PF and TZ3P2F from Sec. III A. The TZ-quality basis sets for hydrogen and nitrogen are derived from the Huzinaga–Dunning basis [S. Huzinaga, J. Chem. Phys. 42, 1293 (1965) and
85.T. H. Dunning, J. Chem. Phys. 55, 716 (1971)],
85.the contraction scheme being for N, and for H. The TZ-quality bases for silicon and phosphorus are defined by the McLean-Chandler TZ contraction [A. D. McLean and G. S. Chandler, J. Chem. Phys. 72, 5639 (1980)] of the Huzinaga’s primitive Gaussian functions [S. Huzinaga, Approximate Atomic Wave Functions, Vol. II, University of Alberta, Alberta, (1971)]: the contraction scheme is for Si and P. We augment these basis sets by: (i) two sets of polarization functions (TZ2P) [ 0.375; 0.40; 0.25 and 0.30]; (ii) three sets of polarization function, [ 0.75, 0.1875].
85.The orbital exponents of higher angular momentum functions are: (i) and for a single set of higher angular momentum functions bases] and (ii) for a double set of higher angular momentum functions In addition, the TZ2P and bases are augmented by one set of diffuse functions [ and respectively], with the exponent (diffuse functions on heavy atoms are the same as in the previous section). Lastly, correlation-consistent basis sets have been employed, i.e., cc-pVDZ, cc-pVTZ, and cc-pVQZ [Refs. 90 and D. Wooh and T. H. Dunning (to be published)]. Pure angular momentum and functions are used in all the calculations reported in this section.
86.To calculate adiabatic energy separations in methylene, FCI/TZ2P optimized geometries (Ref. 60) are used ( in in degree): 1.0775/133.29, 1.1089/101.89, 1.0748/141.56, and 1.0678/170.08 for the and states, respectively. The equilibrium geometries for are optimized geometries (Ref. 34): 1.0295/150.88, 1.0459/107.96, 1.0293/161.47, and 1.0315/180.00 for the and states, respectively. For and we employ the optimized geometries from Refs. 35 and 36, respectively. The equilibrium structures of the and states of are 1.5145/92.68, 1.4770/118.26, 1.4830/122.65, and 1.4573/162.34, respectively. For geometries of these states are: 1.4178/93.06, 1.4056/121.77, 1.4194/124.84, and 1.4118/159.62.
87.H. Petek, D. J. Nesbitt, D. C. Darwin, P. R. Ogilby, C. B. Moore, and D. A. Ramsay, J. Chem. Phys. 91, 6566 (1989).
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89.In variational methods, the energy of an approximate wave function can be lowered by breaking its spin or point group symmetry. Thus, for any variational model (e.g., Hartree-Fock or CI), energies of spin-contaminated solutions are always lower than those of the spin-pure ones. Rigorously, this may not be the case for non-variational methods such as CCSD or B-CCD. Practically, however, spin-contaminated energies are usually lower than spin-pure ones for non-variational methods as well. Spin-contamination of coupled-cluster wave functions has been recently discussed by A. I. Krylov, J. Chem. Phys. 113, 6052 (2000).
90.T. H. Dunning, J. Chem. Phys. 90, 1007 (1989).
91.The following equilibrium bond lengths for the and states are used (Ref. 67): 1.034 and 1.036 Å for NH; 1.308 and 1.317 Å for NF; 1.215 and 1.207 Å for For the experimental geometry of the state is used for both states [theoretical calculations (Ref. 56) predict that the difference in the of these states is very small, i.e. ].
92.A. I. Krylov, Y. Shao, and M. Head-Gordon (unpublished).
93.We used the functional composed of the equal mixture of the following exchange and correlation parts: for exchange, and for correlation.
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96.The DZP basis for carbon is the same as in Sec. III A. The DZP basis for hydrogen is a Huzinaga–Dunning [S. Huzinaga, J. Chem. Phys. 42, 1293 (1965)
96.and T. H. Dunning, J. Chem. Phys. 53, 2823 (1970)] basis augmented by a single set of polarization functions Pure angular momentum functions are used.
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99.This is similar to the diradical at linear geometry where the and states become degenerate as two components of the state.
100.Note that the PES of the open-shell singlet has no minimum at the planar structure. Likewise, the PES of the triplet state has no minimum at the twisted structure. This can be demonstrated by the vibrational analysis which yields one imaginary frequency corresponding to the rotation of methylene group for the structures obtained by the constrained optimization. We do not discuss these saddle points, even though these structures have been previously reported (Ref. 45).
101.The scaling of the spin-orbital implementation of the SF-CIS model is (due to integral transformation); however, it can easily be reduced to approximately by implementing the direct algorithm for the CIS procedure (Ref. 78). The scaling of the SF-CIS(D) model is and the step is noniterative. Similar to their non SF counterparts, the SF-CISD, SF-CCSD, and SF-OD models scale as The scaling of MR models is factorial due to the MCSCF part. The scaling bottleneck restricts full valence space MCSCF calculations to systems of just two or three heavy atoms. For example, the full valence active space for TMM consists of 16 orbitals and contains 16 electrons. It is not yet possible to carry out such calculations with today’s hardware and software tools. As far as timing is considered, the comparison between MR and SF models is more difficult since the absolute timing strongly depends upon implementation.
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