Triplet instabilities

Calculations of the vicinal F-F coupling constant for c/v-l,2-difluoropropene (5) suffer from triplet instabilities, when the aug-cc-pVTZ-J basis set is used... 

The triplet instability of both RB3LYP and RHF solutions is responsible for the difference in the symmetry of the ethylene-peroxynitrous acid transition structure. In general, an unsymmetrical transition structure with alkene epoxidation when an alternative symmetrical structure is possible, appears to be associated with an unstable wave function. 

The essential role of the size-consistency in molecular applications is strikingly conspicuous already in the SR case. Indeed, the SR CCSD method is manifestly size-extensive, yet it fails when breaking genuine chemical bonds, as the well-known examples illustrate [82,83]. This breakdown is of course most prominent when multiple bonds are involved, as the example of the CCSD PEC for N2, shown in Fig. 1, clearly illustrates [83]. Note that even when we employ the UHF reference, we will not generate a smooth PEC in view of the presence of the triplet instability (see, e.g., [84, 85] and references therein), whose onset occurs at an intermediately stretched geometry [86]. 

Recently, there have been several successful attempts at the calculation of spin-spin coupling constants at the density functional theory (DFT) level. " In view of the success of the DFT methodology, it would in particular be interesting to see how well DFT performs with respect to the calculation of spin-spin coupling constants. In particular, recent results demonstrated that DFT does not suffer from the triplet-instability problems that have plagued the application of Hartree-Fock theory to the calculation of spin-spin... 

The forms (4.25) and (4.32-4.34) of the second variation refer to the spin orbital level In investigating possible instabilities we always choose a particular set - The corresponding set Fukutome class as the set i/ M of the reference determinant or to a different one. In either case this type of information can be used to simplify the second variation further.17 The most well-known examples are the singlet and triplet instabilities of Paldus and Cizek.24 In the first case the set M is doubly filled and real and therefore belongs to the Fukutome class TICS like the set. As we have seen in (4.33), the fact that the spin orbitals are real reduces the matrix T from 2N x 2N to N x N. The fact that the orbitals are doubly filled reduces it further to (N/2) x (N/2). A similar reduction (but for another reason) occurs when we go from RHF to the Fukutome class ASDW.17 This provides an example of a triplet (or rather a nonsinglet) instability. 

It is well known that for singlet states, the UHF solutions with p pp are really possible when electron correlations become sufficiently strong. More exactly, the spin-polarized HF determinant < ) appear only under the non-singlet (triplet) instability which was defined by Cizek and Paldus in [63]. At the same time, solutions of the spin-projected variational HF method (the Lowdin s extended HF scheme) always exist [19]. The wave functions of this type will be signified by 4> ). This is usually defined by (apart from a normalization factor)... 

A strongly bound ( bind = 24.4 kcal mol M06-2X/6-31G level of theory) van-der-Waals dimer (7-complex in Fig. 18) exists that presumably forms without barrier [53]. The barrier for formation of the covalent dimer (7Mi in Fig. 18) from the van-der-Waals dimer (via transition structure 7Ti in Fig. 18) is 12.3 kcal mol at M06-2X/6-31G [53]. This is below the energy of two separated heptacene molecules. Hence, the energy gained by complex formation is enough to surmount the barrier for dimerization. One should note that the description of dispersion interactions in the anthracene dimer is problematic with density functional methods, even if empirical dispersion corrections are included [54], The M06-2X functional has been shown to perform well for a number of systems, but how reliable the data are for heptacene dimerization is not clear at this time. Another problem for the reliable computation of heptacene dimerizatiOTi may arise from the triplet instability of the spin-restricted Kohn-Sham (RKS) description of the heptacene molecule [55]. 

Auer and Gauss showed that for the CC calculations of the SSCCs with relaxation of the reference orbitals in the presence of the perturbations, unphysical results are obtained over a wide range of the potential curve. The effect of triplet instability in the relaxed methods was most dramatic for perturbative approaches like CCSD(T). CC calculations using the unperturbed HF orbitals (6 /0a = 0), i.e., so-called unrelaxed calculations, did not show any triplet instability. 

The triplet instability of the RHF solutions is a necessary, but insufficient, condition for the conclusion as to the biradical character or the triplet ground state of a given system, which would be important for an analysis of the internal mechanism of a number of reactions (see Sect. 5.1). Usually, a reliable result may be achieved in such cases by passing to the UHF approximation. 

Clearly, the rupture of the bond in XI and the transition to the biradical form XIa proceeds continuously, rather than in a jumpwise fashion. The sharpest changes during this continuous process occur in the region of instability of the HF solutions. It is for this reason that the analysis of stability of the HF values is a convenient technique for determining the part of the configurational space in which the PES contains biradical structures. The appearance of triplet instability may be regarded as an indication of the transformation of structures with closed electron shell into those with two separated radical centers or biradicals. 

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