Molecular crystals Hamiltonian

In order to demonstrate the NDCPA a model of a system of excitons strongly coupled to phonons in a crystal with one molecule per unit cell is chosen. This model is called here the molecular crystal model. The Hamiltonian of... 

The special case where only rotators are present, Np = 0, is of particular interest for the analysis of molecular crystals and will be studied below. Here we note that in the other limit, where only spherical particles are present, Vf = 0, and where only symmetrical box elongations are considered with boxes of side length S, the corresponding measure in the partition function (X Qxp[—/3Ep S, r )], involving the random variable S, can be simplified considerably, resulting in the effective Hamiltonian... 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

The hope of understanding the concept of molecular structure quantum-mechanically would obviously be at its most realistic for the smallest of molecules at the absolute zero of temperature. However, under these conditions completely different pictures emerge for the molecule in, either total isolation, or in a macroscopic sample. In the latter case the molecule appears embedded in a crystal, which is quantum-mechanically described by a crystal hamiltonian with the symmetry of the crystal lattice. The isolated molecule has a spherically symmetrical hamiltonian. The two models can obviously not define the same quantum molecule. 

In previous chapters we have seen that the Hamiltonian describing a nuclear spin system is considerably simplified when molecules tumble rapidly and randomly, as in the liquid state. However, that simplicity masks some fundamental properties of spins that help us to understand their behavior and that can be applied to problems of chemical interest. We turn now to the solid state, where these properties often dominate the appearance of the spectra. Our treatment is limited to substances such as molecular crystals, polymers, and glasses, that is, solids in which there are well-defined individual molecules. We do not treat metals, ionic crystals, semiconductors, superconductors, or other systems in which delocalization of electrons is of critical importance. 

The interaction with optical phonons is at the core of the molecular crystal (MC) Hamiltonian [20]. In one dimension this becomes... 

The Hamiltonians of relevance here are the molecular crystal [Eq. (4)] and the SSH Hamiltonians [Eq. (5)]. The electron-phonon interactions have the following forms [45] ... 

All familiar molecular structures have been identified in the crystalline state. To describe such molecules quantum-mechanically requires specification of a crystal Hamiltonian. This procedure is never attempted in practice. Instead, history is taken for granted by assuming a specific connectivity among nuclei and the crystal environment is assumed to generate well-defined conformational features characteristic of all molecules. Although these decisions may not always be taken consciously, the conventional approach knows no other route from wave equation to molecular conformation. 

The problem with the theory of electronic transport in molecular crystals has been to deduce the transport, given a model Hamiltonian containing what one considers to be the essential physical interactions. Since several interactions may be comparable in size, simple perturbative methods fail. The method (12) adopted here yields a rather direct solution to the problem. 

The harmonic approximation consists of expanding the potential up to second order in the atomic or molecular displacements around some local minimum and then diagonalizing the quadratic Hamiltonian. In the case of molecular crystals the rotational part of the kinetic energy, expressed in Euler angles, must be approximated, too. The angular momentum operators that occur in Eq. (26) are given by... 

Just as the perturbation theory described in the previous section, the self-consistent phonon (SCP) method applies only in the case of small oscillations around some equilibrium configuration. The SCP method was originally formulated (Werthamer, 1976) for atomic, rare gas, crystals. It can be directly applied to the translational vibrations in molecular crystals and, with some modification, to the librations. The essential idea is to look for an effective harmonic Hamiltonian H0, which approximates the exact crystal Hamiltonian as closely as possible, in the sense that it minimizes the free energy Avar. This minimization rests on the thermodynamic variation principle ... 

A scheme as described here is indispensable for a quantum dynamical treatment of strongly delocalized systems, such as solid hydrogen (van Kranendonk, 1983) or the plastic phases of other molecular crystals. We have shown, however (Jansen et al., 1984), that it is also very suitable to treat the anharmonic librations in ordered phases. Moreover, the RPA method yields the exact result in the limit of a harmonic crystal Hamiltonian, which makes it appropriate to describe the weakly anharmonic translational vibrations, too. We have extended the theory (Briels et al., 1984) in order to include these translational motions, as well as the coupled rotational-translational lattice vibrations. In this section, we outline the general theory and present the relevant formulas for the coupled... 

The Hamiltonian for Exciton-Phonon Coupling in Molecular Crystals. 

EM was quite extensively and successfully applied to model optical spectra of molecular crystals and aggregates. Extensions were discussed [18] to account for disorder, whose effects are particularly important in aggregates, and to include the coupling between electronic degrees of freedom and molecular vibrations [48], needed to properly describe the absorption and emission bandshapes. However, as it was already recognized in original papers [7, 46], other terms enter the excitonic Hamiltonian. Electrostatic interactions between local excitations can in fact be introduced as ... 

Crystal states, where one molecule is ionized, and the conduction band contains one electron, can be used as those intermediate states, as has been shown in (31) (the role of such intermediate states in theory of photoconductivity of molecular crystals has been discussed by Lyons (32)). The use of intermediate states becomes indispensable when the second-order perturbation theory is applied in the case of a degenerate term. According to (33), correct linear combinations of crystal states, containing one molecule in a triplet state and all remaining in the ground state, can be found by perturbation theory when in the corresponding secular equation the following effective Hamiltonian is used... 

While the Sternlicht and McConnel paper (26) was originally devoted to resonance phenomenon, by justifying the exciton spin Hamiltonian, its main influence has been on the large number of experiments on the magnetic field dependence of delayed fluorescence from molecular crystals. A review of fluorescence studies can be found in the paper by Swenberg and Giacintov (46). 

To demonstrate the idea of the calculations we consider below the case of a molecular crystal with one two-level molecule in the unit cell (Subsection 3.4.1). For this model the second-quantized Hamiltonian of the crystal has the form... 

The operator II for a molecular crystal with fixed molecules is defined by the expression (3.1) and the elementary excitation spectrum in the excitonic energy region corresponding to this Hamiltonian was discussed in Chs. 2 and 3. 

To obtain Hamiltonian (6.11) we proceeded above from the model of a molecular crystal. Actually, its range of application includes nonmolecular crystals as well, provided we are concerned with optical phonons whose bandwidth is much narrower than the phonon frequency. In these spectral regions the vibrations of the atoms inside the unit cell are similar to intramolecular vibrations in molecular crystals, since the comparatively narrow phonon bandwidth is indicative of the weakness of the interaction between the vibrations of atoms located in different unit cells. 

For the discussion of the excitonic spectrum in a one-dimensional molecular crystal (with one molecule per unit cell) we use the following Hamiltonian ... 

In Ch. 3 we have applied the second quantization for investigation of exciton states. The first step was to express the crystal Hamiltonian in terms of creation and annihilation Pauli operators Pjf and P/ of single-molecule excited states, where the index s indicates the lattice points where the molecule is placed, and / labels the molecular excited states. When taking into account only one fth excited molecular state, the operators P%f and P/ satisfy the following commutation rules (see eqn 3.28)... 

We assume that the crystal consists of N unit cells. We will limit ourselves here to crystals with no more than two molecules in the unit cell. The Hamiltonian H of the pure molecular crystal is given by... 

Crystal energy matrix, 307 Crystal Hamiltonian, 294, 307 Crystals, organic molecular, 286, 327 Cube, Cn axes of, 19 commutation of, 23 symmetry operations of, 23 symmetry planes of, 20, 21 Cubic point groups, 66ff Current density, 109 Cyclobutadiene, v molecular orbitals of, 178-179... 

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