Noninteracting electrons

The dimerized chain is the simplest model of semiconducting polymers, and is applied in particular to trans-polyacetylene. The noninteracting electronic structure of conjugated polymers with more complex unit cells, such as poly(para-phenylene), will be discussed in their relevant chapters. We emphasize that the noninteracting model is a simple model. It is not a realistic description of the electronic states of conjugated polymers, as it neglects two key physical phenomena electron-phonon coupling and electron-electron interactions. Despite these deficiencies it does provide a useful framework for the more complex descriptions to be described in later chapters. 

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In the discussion above, scattering from molecules is treated as a superposition of noninteracting electron... 

The Vext operator is equal to Vne for A = 1, for intermediate A values, however, it is assumed that the external potential Vext(A) is adjusted so that the same density is obtained for both A = 1 (the real system) and A = 0 (a hypothetical system with noninteracting electrons). For the A = 0 case the exact solution to the Schrddinger equation is given as a Slater determinant composed of (molecular) orbitals, for which the... 

Thus the interacting multi-electron system can be simulated by the noninteracting electrons under the influence of the effective potential l eff(r)- Kohn and Sham [51] took advantage of the fact that the case of non-interacting electrons allows an exact computation of the particle density and kinetic energy as... 

The simplest approach is to describe the valence electrons in the solid as a free noninteracting electron gas in a box with the volume V, as we did in Chapter 3. We have to find the ground state for the Schrodinger equation... 

The connection to HF theory has been accomplished in a rather ingenious way by Kohn and Sham (KS) by referring to a fictitious reference system of noninteracting electrons. Such a system is evidently exactly described by a single Slater determinant but, in the KS method, is constrained to share the same electron density with the real interacting system. It is then straightforward to show that the orbitals of the fictitious system fulfil equations that very much resemble the HF equations ... 

Exc accounts for the energy of exchange interactions, correlation effects, and the difference between the exact kinetic energy and that of the reference system of noninteracting electrons with the density p(r). 

The third term on the right hand side of this expression is the single-particle kinetic energy of the noninteracting electrons whereas the functional Exc[p contains the additional contribution to the energy that is needed to make Eq. (8) equal to Eq. (6). 

The inherent problems associated with the computation of the properties of solids have been reduced by a computational technique called Density Functional Theory. This approach to the calculation of the properties of solids again stems from solid-state physics. In Hartree-Fock equations the N electrons need to be specified by 3/V variables, indicating the position of each electron in space. The density functional theory replaces these with just the electron density at a point, specified by just three variables. In the commonest formalism of the theory, due to Kohn and Sham, called the local density approximation (LDA), noninteracting electrons move in an effective potential that is described in terms of a uniform electron gas. Density functional theory is now widely used for many chemical calculations, including the stabilities and bulk properties of solids, as well as defect formation energies and configurations in materials such as silicon, GaN, and Agl. At present, the excited states of solids are not well treated in this way. 

The problem of T[p] is cleverly dealt with by mapping the interacting many-electron system on to a system of noninteracting electrons. For a determinantal wave function of a system of N noninteracting electrons, each electron occupying a normalized orbital >p, (r), the Hamiltonian is given by... 

The basic variable in density functional theory (DFT)22 is the electron density n(r). In the usual implementation of DFT, the density is calculated from the occupied single-particle wave functions (r) of an auxiliary system of noninteracting electrons... 

The reason for calling I [n] the noninteracting kinetic energy functional is recognized immediately when the GS problem of a system of N noninteracting electrons, moving in an external potential rs(r), is considered. In analogy with Eq. (9) we write... 

Thus, the minimization in Eq. (64) can be performed alternatively as a solution of an equivalent noninteracting electron problem, leading to equations... 

By applying all above considerations to the HF method posed as the DFT in Section 2.4, where the equivalent noninteracting electron problem leads to the HF-KS equations (70), we obtain from Eqs. (174) and (175)... 

The significance of the electrochemical potential is apparent when related to the concepts of the usual stati.stical model of free electrons in a body where there are a large number of quantum states e populated by noninteracting electrons. If the electronic energy is measured from zero for electrons at rest at infinity, the Fermi-Dirac distribution determines the probability P(e) that an electron occupies a state of energy e given by... 

Operators corresponding to physical quantities, in second-quantization representation, are written in a very simple form. In the quantum mechanics of identical particles we normally have to deal with two types of operators symmetric in the coordinates of all particles. The first type includes N-particle operators that are the sum of one-particle operators. An example of such an operator is the Hamiltonian of a system of noninteracting electrons (e.g. the first two terms in (1.15)). The second type are iV-particle operators that are the sum of two-particle operators (e.g. the energy operator for the electrostatic interaction of electrons - the last term in (1.15)). In conventional representations these operators are... 

The activational temperature dependence Eq. (16) holds for all T -C 7), because for noninteracting electrons at T = 0 the product Tj uj)Aq(k, uj) is exactly zero [11] at k < mSv. If electrons interact, some overlap of A and A2 exists even at small k C mSv. This yields a further contribution to r, that has a power-law temperature dependence. We will evaluate this contribution for weak intrawire interaction. 

The conductance for a spacing of 2 /xm between gates g and g2 is shown in Fig. 2. The measured bright and dark curves in the plot can be interpreted as spectral peaks tracing out the dispersions of the elementary excitations in the wires. [3] In the case of noninteracting electrons, the curves are expected to map out parabolas defining the continua of electron-hole excitations across... 

Fig. 2. Measurement of G(V, B) for a 2 pm junction. Light shows positive and dark negative differential conductance. A smoothed background has been subtracted to emphasize the spectral peaks and the finite-size oscillations. The solid black lines are the expected dispersions of noninteracting electrons at the same electron densities as the lowest ID bands of the wires, ui) and li). The white lines are generated in a similar way but after rescaling the GaAs band-structure mass, and correspondingly the low-voltage slopes, by a factor of 0.7. Only the fines labeled by a, b, c, and d in the plot are found to trace out the visible peaks in G(V,B), with the fine d following the measured peak only at V > —10 mV.

The introduction in 1965 by Kohn and Sham7 of a practical computational scheme may, therefore, be considered to be the next major milestone in the development of formal DFT. The essential ingredient in this approach is the postulation of a reference system of N noninteracting electrons, moving in an effective external potential vs(r), the so-called Kohn-Sham potential, instead of the electrostatic potential v(r) of the nuclei ... 

For noninteracting electrons the transition rates are determined by the single-electron tunneling rates, and are nonzero only for the transitions between the states with the number of electrons different by one. For example,... 

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