Quantum cell model

S. Ramasesha, Z. G. Soos, in Valence Bond Theory, D. L. Cooper, Ed., Elsevier, Amsterdam, The Netherlands, 2002, pp. 635-697. Valence Bond Theory of Quantum Cell Models. 

Valence Bond Theory of Quantum Cell Models... 

The shared features of quantum cell models are specified orbitals, matrix elements and spin conservation. As emphasized by Hubbard[5] for d-electron metals and by Soos and Klein [11] for organic crystals of 7r-donors or 7r-acceptors, the operators o+, and apa in (1), (3) and (4) can rigorously be identified with exact many-electron states of atoms or molecules. The provisos are to restrict the solid-state basis to four states per site (empty, doubly occupied, spin a and spin / ) and to stop associating the matrix elements with specific integrals. The relaxation of core electrons is formally taken into account. Such generalizations increase the plausibility of the models and account for their successes, without affecting their solution or interpretation. 

Since the many-electron basis of quantum cell models is finite and complete, we have... 

We see that additional orbitals or spins can readily be introduced in quantum cell models. The real constraint is the total number of orbitals, which governs the... 

Amsterdam, The Netherlands, 2002, pp. 635-697. Valence Bond Theory of Quantum Cell Models. 

Quantum-chemical calculations on conjugated hydrocarbons support the spectroscopic estimate, (3 Rq) = -2.40eV, and all-electron descriptions are appealing as soon as they become feasible. There are too many levels of theory to enumerate here, but quantitative ones are not yet applicable to conjugated polymers. Moreover, we are interested in excited states, which remain challenging even in molecules. The rationale for ct-tt separability, for the Coulomb potential V(R), and for the zero differential overlap (ZDO) approximation were discussed [1] in connection with the PPP model. Hubbard [33] considered the same issues for d electrons in transition metals. Quantum cell models [12,13,34] for frontier orbitals of any kind implicitly invoke ZDO to obtain two-center interactions. In many cases, the relevant transfer integrals t, Hubbard repulsion U, and intersite interactions V(R) are small and hence difficult to evaluate in... 

Quantum cell models are introduced phenomenologically in solid-state physics to describe frontier orbitals in extended systems. They are not derived in the tt-electron sense, although ZDO and related approximations are readily discerned. Models pose well-defined theoretical problems at the expense of direct contact with actual systems. Such contact is clearly vital for spectroscopic studies, however, of real solids or polymers. The problem is to identify contributions due to n electrons or frontier orbitals w ithout detailed knowledge of the full system. For example, the oriented gas model of organic molecular crystals [36] describes small vibrational or electronic shifts and splitting relative to the gas phase. The microscopic parameters of quantum cell models can then be directly related to vibrational or electronic spectra. 

The electronic response x in Eq. (13) is a static susceptibility involving even-parity singlets of polymers with centrosymmetric backbones. The p in Eq. (1) indicate to be a bond-order/bond-order polarizability [20]. Since the excited states F> in the sum depend on V R), different s are found in Hubbard or PPP models. The large NLO responses of conjugated polymers are also due to V electrons. The dipole operator in the ZDO approximation of quantum cell models is [12]... 

V. CORRELATED EXCITED STATES OF QUANTUM CELL MODELS... 

The vibrational consequences of 7r-electron fluctuations discussed in Sections II and III drew on both molecular spectroscopy and solid-state physics. The analysis of NLO and EA spectra in Section IV combined PPP models for molecules with quantum cell models of alternating chains. We proposed at the outset to relate the conjugated polymers in Fig. 6.2 to alternating Fliickel or PPP chains and have so far discussed vibrational and optical implications of 7r-electron models rather than the Hamiltonian, Eq. (7), or its mathematical properties. The analysis holds for any H(8) with appropriate vibrational or optical susceptibilities. Equation (7) is sufficiently general to encompass Hiickel, Hubbard, extended Hubbard, PPP, and other models with suitable choices of U and Vp,. This generality is an extremely useful feature of solid-state models. 

Accurate correlated excited states pose major theoretical challenges for extended systems, even at the tt-electron level. We consider quantum cell models from several perspectives in this section, starting with symmetries and a many-electron basis. In addition to the total spin 5, Eq. (7) has electron-hole (e-h) symmetry [12,117-119] for arbitrary intersite interactions V, in systems with one electron per site. The correlated singlets G and A have e-h index J = 1, while Bu singlets... 

The interaction terms of Eq. (7) are completely specified by number operators rip and thus commute with /x in Eq. (28). Quantum cell models satisfy a sum rule for oscillator strengths based on the identity [143,144]... 

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