Wave functions butadiene

For the allowed reaction between ethylene and butadiene, D+A can mix into the transition state wave-function, since now and orbitals do overlap (p. 131). As a consequence the barrier is lowered and the transition state acquires some charge transfer character (107). Thus the role of symmetry in... 

In simple conjugated hydrocarbons, carbon utilizes sp2 hybrid orbitals to form a-bonds and the pure px orbital to give the it-MOs. Since the c-skeleton of the hydrocarbon is perpendicular to the wave functions of it-MO, only px AOs need be considered for the formation of it-MOs of interest for photochemists. Let us consider the case of butadiene with px AO contributed by 4 carbon atoms. The possible combinations are given in Figure 2.18. The energy increases with the number of nodes so that Et < < E3 < Et. 

We have tried to apply this method to several cases, notably those of irans-butadiene "5 and benzene, using antisymmetrized Slater orbitals to form the initial wave function. The function in fl>a, the excited state, is then supposed to be of the following form ... 

Rules for writing VB wave functions in the polyelectronic case are just extensions of the rules for the 2e/2c case above. First, let us consider butadiene 5 in Scheme 3.2, and restrict the description to the tt system. 

Denoting the tt AOs of the C1-C4 carbons by a,b,c, and d, respectively, the fully covalent VB wave function for the tt system of butadiene displays two singlet couplings one between a and b, and one between c and d. It follows that the wave function can be expressed in the form of Equation 3.8, as a product of the bond wave functions. 

An Example The Hartree—Fock Wave Function of Butadiene... 

The MOs of butadiene are listed in Table 4.1. At the Hartree—Fock level, only the two first MOs are occupied, and the wave function reads ... 

TABLE 4.2 Coefficients of the AO-Half-Determinants in the Expansion of the Hartree Fock Wave Function of Butadiene... 

Butadiene and Hexatriene The two covalent VB structures for butadiene are depicted in Fig. 7.9a as Ri and R2. Under the symmetry operations of the point group, the two structures transform as Ag (since both are unchanged by i), and will therefore mix to give rise to two states of the same symmetry. The corresponding wave functions are written in Equations 7.12a and b, with normalization constants that neglect all overlap terms between orbitals on atoms a—d. 

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. 

Exercise 7.6 The two covalent VB structures for butadiene are depicted in Fig. 7.9a as Ri and R2. The corresponding normalized wave functions are written below ... 

Classifying the roots obtained from these polynomials by spin multiplicity and point group Cs, we have eigenvalues of butadiene in the increased energy order, A (-3-31/2), 3A"(-2-2W2), 3A (-2), A (-3+31/2), 3A"(-2+2W2), and 5A (0). For these eigenvalues, one can also derive corresponding wave functions without difficulty. 

The next step up in complexity comes with four p orbitals conjugated together, with butadiene 1.5 as the parent member. There is a a framework 1.6 with 36 electrons and four p orbitals to house the remaining four. Using the electron in the box with four p orbitals, we can construct Fig. 1.31, which shows the four wave functions, inside which the p orbitals are placed at the appropriate regular intervals. We get a new set of orbitals, ipi, 3, and ip4, each described by Equation 1.11 with four terms. 

However, the HOMO of butadiene is 2 (in the ground state), in which the orbital lobes (terminal) that overlap to make the new a-bond have the opposite phase (sign of the wave function). Thus, in this case, the new a-bond between these two terminal orbital lobes cannot be formed by the disrotation. Thus, disrotatory ring closing of a diene to cyclobutene is thermally forbidden. If the terminal orbital lobes of the HOMO of butadiene were to close, it would be in a conrotatory fashion (Fig. 8.47). 

For jr—excitation energies the agreement between the calculated and experimentally observed data is more difficult to achieve. This is probably related to the intricacies of the correlation of a with Ji electrons in states with a highly zwitterionic character in the n part of the wave function. (Cf. Malmqvist and Roos, 1992.) Some of the difficulty is due to the limited availability of definitive and accurate experimental data. As an example, some data for butadiene are collected in Table 1.6. 

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