Spin alternant

The Longuet-Higgins theory based on Huckel MO theory and Hund s rule was pioneering work in the area of molecular design of high-spin alternant hydrocarbons and has been appreciated for its predictability. 

Two of the determinants, labeled c and Oq( are unique, and as depicted in 4 and 5 in Scheme 5.1, the electrons are arranged in these determinants in an alternant manner, spin-up—spin-down, and so on. In contrast, in the other determinants for example, fl",-, there are pairs of identical spins on adjacent carbon atoms. The spin-alternant determinants, flqc and Oqc, were earlier called the antiferromagnetic determinants, the Neel state (16), or the (QC) state (17). The corresponding wave functions in Equations 5.8 and 5.9 use the QC shorthand notation for these determinants ... 

These spin-alternant determinants are the only ones that are common to the two Kekule structures in Equations 5.6 and 5.7. Moreover, they represent the... 

The role of ionic structures is crucial, and it can be appreciated based on a simple symmetry analysis. Indeed, as shown recently (24), the mixing of all ionic structure types into the spin-alternant determinant in benzene mediates the circular electronic motion (between the two determinants) and induce the diamagnetic ring current in a magnetic field. By contrast, the exclusion of the diagonal ionics in CBD excludes this motion. Exercise 5.4 demonstrates qualitatively this dichotomy of ring currents. 

It is experimentally known that benzene exhibits diamagnetic ring currents, while cyclobutadiene (CBD) does not. In this exercise, we will consider the wave functions of benzene and CBD as primarily made of their spin-alternant determinants HqC and I1qC, and we will model ring currents by the passage from one spin-alternant determinant to the other (HqC Hqc), which is associated with a collective circular flow of electrons. 

The first term has a triplet CO moiety, while the second exhibits spin alternation on the carbonyl fragment. Since each interaction between two electrons of the same spin account for one Pauli repulsion, there are two Pauli repulsion in the first term of P and only one in the second (the interaction between the spins on CH3 and O has a longer distance and is discounted). Hyperconjugation would have the effect of delocalizing the pseudo-orbital of methyl on the neighboring carbon, or, equivalently of adding the following two VB structures to P ... 

It is seen that the coefficient of the abc determinant, which is the spin-alternant determinant [or the quasiclassical (QC) determinant discussed in Chapters 3—5], is doubled compared with the other two. As such, using it once with abc we account for pairing of the electrons in b—c, namely, a double bond on the right-hand side, while using it the second time with abc gives rise to the double bond across a—b, on the left-hand side. This is the delocalized... 

Thus, as we did for the allyl radical case, here too the bonding characteristics of the two covalent states of benzene can be deduced from the respective wave functions. As discussed in Chapter 5, each Kekule structure can be generated by a product of the corresponding bond wave functions, each having a spin factor a(l)(3(2)- 3(l)a(2). Each Kekule structure possesses the two spin alternant QC determinants, which are related to each other by a cyclic permutation of the spins over the ring, and are shown in Fig. 7.4d. To illustrate clearly the building blocks of the two Kekule structures, we express them as a sum of all the permutations of the QC determinants, as follows ... 

Here, the first QC determinant, f2QC, is the spin-alternant determinant starting with spin a on position 1 and ending with (S on atom 6, while Qqc is the second QC determinant the two being related by a cyclic permutation of the spins, as... 

Express the VB wave functions for the ground state ( B,) and first covalent excited state (2A2) of the pentadienyl radical in terms of Kekule structures. Deduce the qualitative spin distribution change upon excitation. Hints For the excited-state case, you will need to express the Kekule structures in terms of determinants. For the ground state, you may express the wave function as the spin alternant determinant, plus some minor contributions of the two determinants that exhibit a single spin frustration (two identical neighboring spins). You may consult ChemPhysChem. 5, 515 (2004). 

The aim of this exercise is to compare the bonding features of the ground versus first excited states of benzene, with a method different from that of the preceding exercise. The bond index for an r—s bond will be taken as the probability of finding a spin alternation from r to s in the wave function. Thus, the r—s bond index can be estimated by summing up the squares of the coefficients of the determinants displaying an r-s spin alternation, in the wave function of the state under consideration. 

Use the semiempirical theory in Chapter 3 to obtain quantitative expressions for the energies and wave functions of the l Ag and 2 Ag states of butadiene. Hint Express the energies of the two Rumer structures relative to the QC determinant (the spin alternant determinant). Deduce the matrix element between the structures keeping only the close neighbor 2(35 term (for simplicity define X = —2(35). Neglect overlap in the normalization constant. 

FIGURE 7.Ans.l (a) Resonance structures of pentadienyl radical and their symmetry properties, (b) The VB mixing of the resonance structures, (c) The quasiclassical (spin alternant) determinant that dominates the ground-state wave function, and the corresponding secondary determinants, and the resulting spin density distribution (p) in the ground state, (d) The spin distribution in the covalent excited state. 

The values of the coefficients A and jjl in Equation 7.26 depend on the geometry of the hexatriene (the length of the various C—C bonds in the molecule). Thus, since the coefficients A and p are not known precisely, the easiest way is to deduce the spin density pictorially from the spin alternant determinant, DqC in part (c) of the figure. It is seen that DQC predicts the mode of spin polarization in the ground state. Addition of the determinants with a single spin frustration, Dr and DL, which have smaller weights due to the Pauli repulsion in the spin frustrated sites, will increase the spin density in the middle carbon and will decrease it on all other carbons. 

Since Ri possesses two bonds and a nonbonding interaction between the bonds, its energy relative to the energy of the spin alternant determinant, taken as the... 

The VB structure R2 has only one bond and two nonbonding interactions, leading to the same energy as the spin-alternant determinant ... 

Remembering that the energy of the spin-alternant is taken as zero, and that determinants interact only if they differ by only one spin permutation between adjacent atoms (in which case their matrix element is X), the Hamiltonian matrix element between the two Rumer structure is... 

As already reasoned throughout this book, the lowest energy determinants are those possessing maximum spin-alternation. In a linear polyene or any... 

FIGURE 8.1 Energies of the singlet ( Fo) and triplet excited ( P ) states of ethylene relative to the most spin-alternant determinant (MSAD). 

A simple principle of the spin-Hamiltonian VB theory, first formulated by Ovchinnikov (13), applies to alternant conjugated molecules, that is, those molecules that possess fully spin-alternant determinants. The rule is stated as follows ... 

These two inputs will yield the energy of a spin-alternant determinant of benzene. 

The second input uses the spin-orbitals saved in the %chk file as a guess function, but the tt MOs of this guess function are replaced by one-centered AOs of alternate spins so as to represent a spin-alternant determinant. These AOs are all the same and can be taken, as done here, as the constituting AOs of an ethylenic tt MO. Alternatively, the tt AOs can arise from a separate calculation of planar CH3. 

After this second calculation is completed, the energy of the spin-alternant determinant is read in the output as the UHF energy at iteration zero. 

SAD Spin-alternant determinant. The VB determinant with one electron per site and with alternating spins. Other terms describing the same determinant are the quasiclassical (QC) state, and the antiferromagnetic (AF) state. In nonalternant hydrocarbons, where compete spin alternation is impossible, the determinant is called MS AD, namely, the maximum spin-alternating determinant. The SAD MSAD are the leading terms in the wave function of molecules with one electron per site, for example, conjugated hydrocarbons. In radicals (e.g., allyl radical) the SAD is the root cause of spin polarization (i.e., negative spin densities flanked by positive ones). See Chapters 7 and 8. 

Denotes the spin-alternant quasi-classical determinant. 

The energy matrix entries can be evaluated by simple rules. For a given determinant, the diagonal entry equals the negative value of the number of spin-alternations for nearest-neighbor pairs of carbon atoms in the determinant, i.e.,... 

Sanchez Marcos and coworkers107,110 have reported on more refined analyses of the electron distributions in vinylamine, nitroethylene and cis- and trans- l-amino-2-nitroethene. The populations of ionic and spin alternant electron pairs are analyzed in... 

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