The dimerized ground state

We seek a solution of H for arbitrary A . In Section 4.4 we discuss the Hellmann-Feynman theorem, which gives us a general solution for any eigenstate. For now, however, we describe the Peierls mechanism, which gives us the dimerized, broken-symmetry ground state. 

We calculate the equilibrium bond distortion as follows (Longuet-Higgins and Salem 1959). Let us suppose that there is a uniform staggered dimerization, 

We see that as a function of - for small 5 - the kinetic energy decreases more quickly than the increase in elastic energy. Thus, the chain spontaneously dimerizes to a finite value of 6. The energy as a function of 5 is shown in Fig. 4.1. Notice that the ground state is doubly degenerate, with 5 and —6 corresponding to the A and B phases, respectively. 

Minimizing Eq with respect to 6 we find that there is a saddle point at = 0 and stable minima at 

In the noninteracting limit, the band gap is directly proportional to the dimerization gap. In fact, this prediction is violated in traris-polyacteylene, indicating the importance of electron-electron interactions - as we describe in Chapter 7. 

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Auzinsh, M.P. (1990). Nonlinear phase resonance of quantum beats in the dimer ground state, Opt. Spectrosc. (USSR), 68, 750-752. 

Since RSA s for arbitrary wavelengths are not known, it is important to be able to design molecules which behave as a RSA at a desired wavelength. Briefly, the idea is to use a dimer molecule formed from monomers whose ground state absorption peaks near the wavelength of interest. If synthesis of the dimer with fixed orientation of the monomers at desired distances from one another is possible, and the dimer is sufficiently stable to withstand dissociation upon absorption of two quanta of radiation, and a number of other criteria are satisfied, the dimer may behave as a RSA at the desired wavelength. The dimer ground state absorption is to a state in which the excitation is spread over both monomeric units and the dimer excited state absorption commences from this state to the doubly excited electronic state in which both monomeric units are excited. 

Consider a dimer molecule synthesized with the monomers tied together so that their relative geometry is fixed. The shift of the dimer ground state absorption spectrum relative to that of the monomer is due mainly to (a) the splitting of the first excited state of the dimer resulting from the dipole-dipole interaction... 

The sign of the shift of the dimer ground state absorption spectrum relative to the monomer is controlled by the positioning of the orientation of the monomers within the dimer. The first excited dimer states undergo first-order interaction to give... 

We start with the alternating Huckel chain in Eq. (1), which simply consists of the bond order operators pd,s for nearest-neighbor hopping between sites 2n, 2n -1 and 2n, 2n + 1. Following SSH [10,11] and LHS [8], we discuss the dimerized ground state and write the linear e-ph Hamiltonian, Eq. (11), in terms of the symmetry coordinates in Eq. (16) for single- and double-bond stretches. In the = 0 subspace, we have... 

Fitting results for fs DFWM experiments of the formic acid dimer (ground state) rotational constants, CD constants, parametrized polarizability parameter cp, Ray s asymmetry parameter k and temperature T. ... 

EPR experiments on carbon-centred radicals with either a- or /J-boronic ester substituents have been reported.168 While the a-substituted radicals were modestly thermodynamically stable, the /J-substituted radicals underwent easy /J-climination. An EPR experiment on the photo-oxidation of phenolic compounds containing at least one free ortho position has indicated the formation of persistent secondary radicals derived from dimerization or polymerization from C-0 coupling.169 The structure of the succinimidyl radical has been re-examined using density functional theory with a variety of basis sets. The electronic ground state was found to be of cr-symmetry allowing for facile -scission. These conclusions were also predicted using MP2 but... 

The even-leg ladders show a spin-liquid ground state described by a Resonance Valence Bond (RVB) state introduced by P.W. Anderson m[25] such system may be efficiently described by using a direct-space method as the DMRG and RVA ones. In this note, we will consider only the simplest even-leg ladder, the two-legladder, and we will describe it with the simplest RVB state, the Dimer-RVB state where the elementary singlets are between two nearest-neighbour sites. This problem is easily solved by... 

Using (3) one can generate recursively the Dimer RVB state N) from previous states with lower length and estimate variationally the ground-state energy as,... 

When both 10ns of a Cr dimer are in the electronic ground state, the exchange interaction can be represented by the well-known Heisenberg Hamiltonian... 

For the reflection symmetric two-level electron-phonon models with linear coupling to one phonon mode (exciton, dimer) Shore et al. [4] introduced variational wave function in a form of linear combination of the harmonic oscillator wave functions related with two levels. Two asymmetric minima of elfective polaron potential turn coupled by a variational parameter (VP) respecting its anharmonism by assuming two-center variational phonon wave function. This approach was shown to yield the lowest ground state energy for the two-level models [4,5]. 

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