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42) If we suppose that the reference structure is determined by the σ bonding alone, as for example in polyethylene, and that distortions from this structure are small, then we may express Vnσ as a sum of harmonic springs, Vnσ = K 2 (ui+1 − ui )2 . 43) i K is therefore the spring constant associated with the σ-bonds. Now suppose that we consider the molecular structure arising from both the σ and π electrons, as for example in trans-polyacetylene. As we shall see shortly, the coupling of the π electrons to the lattice leads to both an overall reduction in the chain length, and to a regular distortion of the lattice.
30) the density-dependent hopping, V˜iii,i+1 , and the bond-charge repulsion, V˜i,i+1,i+1,i . These terms are smaller than the diagonal ones, as they are determined by the wave function overlap, φ∗i (r)φj (r). 5 Another simpliﬁcation is to assume that the two-centre integrals, tij , are only nonzero for electrons in the same orbital or on neighbouring orbitals. 33) i=j where Ni = σ Niσ counts the number of electrons in the orbital φi (r) and Vn−n is the screened nuclear-nuclear interaction. Since this Hamiltonian is widely used throughout this book, we now describe the physical meaning of each term.
4 State labels The many-body states are labelled as np X ± , where • n is the overall quantum number, • p = 1 for singlets and p = 3 for triplets SYMMETRIES AND QUANTUM NUMBERS 25 • X = Ag or Bu for even or odd inversion symmetry, respectively, • ± refers to the particle-hole symmetry eigenvalue being ±1. 9 In the noninteracting limit the singlet and triplet 1Bu states are degenerate, and lie below the 2Ag state. In large part, the aim of this book is to explain how electronic interactions and electron-phonon coupling determines the character and energetic ordering of these states.