Single-particle electronic state

Because the lattice is two-dimensional, all translations commute with reflection in the plane of the lattice, so any electron (or vibrational) eigenstate can be chosen to be either even or odd under this reflection. For this reason, the single-particle electron states are rigorously separated into two classes, called a and 7t. The even a states are derived from carbon s and p, py orbitals (that is, their hybridized sp orbitals), while the odd Jt states are derived from carbon p orbitals. These latter are cylindrically symmetric in the x—y plane, lie near the Fermi level (half-filled) and are the electrically active states of interest in low energy experimental probing of graphene. 

In Dirac notation we may represent a single-particle electronic state as the ket f). Suppose that the single-particle states f) form an orthonormal basis. The projection of i) onto the coordinate representation, r), (where r) is an eigenstate of the position operator, f) gives the single-particle wave function (or... 

For a perturbative calculation of the spectrum of (10.4.1) we assume that electron 1 is in the single-particle SSE state m) and electron 2 is in a highly excited state, much more weakly bound than electron 1. In this situation electron 2 spends most of its time in regions with X2 xi. Therefore, we assume that the eigenfunctions of (10.4.1) are approximately of the form... 

For open-shell systems, therefore, the model space is no longer simple and sometimes not even known before the computations really start. For closed-shell systems, in contrast, the model space can be formed by a single many-electron state, which is taken as the reference state and which is sufficient in order to classify the single-electron states into particle (unoccupied) and hole (occupied or core) states, respectively. In open-shell systems, instead, the valence shells are neither empty nor completely filled. To facilitate the handling of such open shells, we shall provide (and discuss) a orbital notation in Subsection 3.4 which is appropriate for the derivation of perturbation expansions. 

One practical way to implement the calculation of all the coherent amplitudes needed in the IPM is the FHBS method. We form many-electron states from suitably antisymmetrised single particle pseudo-states, we shall call these states Xn-... 

The corresponding fiinctions i-, Xj etc. then define what are known as the normal coordinates of vibration, and the Hamiltonian can be written in tenns of these in precisely the fonn given by equation (AT 1.69). witli the caveat that each tenn refers not to the coordinates of a single particle, but rather to independent coordinates that involve the collective motion of many particles. An additional distinction is that treatment of the vibrational problem does not involve the complications of antisymmetry associated with identical fennions and the Pauli exclusion prmciple. Products of the nonnal coordinate fiinctions neveitlieless describe all vibrational states of the molecule (both ground and excited) in very much the same way that the product states of single-electron fiinctions describe the electronic states, although it must be emphasized that one model is based on independent motion and the other on collective motion, which are qualitatively very different. Neither model faithfully represents reality, but each serves as an extremely usefiil conceptual model and a basis for more accurate calculations. 

Parker [55] studied the IN properties of MEH-PPV sandwiched between various low-and high work-function materials. He proposed a model for such photodiodes, where the charge carriers are transported in a rigid band model. Electrons and holes can tunnel into or leave the polymer when the applied field tilts the polymer bands so that the tunnel barriers can be overcome. It must be noted that a rigid band model is only appropriate for very low intrinsic carrier concentrations in MEH-PPV. Capacitance-voltage measurements for these devices indicated an upper limit for the dark carrier concentration of 1014 cm"3. Further measurements of the built in fields of MEH-PPV sandwiched between metal electrodes are in agreement with the results found by Parker. Electro absorption measurements [56, 57] showed that various metals did not introduce interface states in the single-particle gap of the polymer that pins the Schottky contact. Of course this does not imply that the metal and the polymer do not interact [58, 59] but these interactions do not pin the Schottky barrier. 

The continuum model with the Hamiltonian equal to the sum of Eq. (3.10) and Eq. (3.12), describing the interaction of electrons close to the Fermi surface with the optical phonons, is called the Takayama-Lin-Liu-Maki (TLM) model [5, 6], The Hamiltonian of the continuum model retains the important symmetries of the discrete Hamiltonian Eq. (3.2). In particular, the spectrum of the single-particle states of the TLM model is a symmetric function of energy. 

We may express the single-particle wave function tpniqd fhe product of a spatial wave function 0n(r,) and a spin function % i). For a fermion with spin such as an electron, there are just two spin states, which we designate by a(i) for m = and f i) for Therefore, for two particles there are three... 

However, it is indeed fortunate that the IV-representability problem for the electron density p(r) greatly simplifies itself. In fact, the necessary and sufficient conditions that a given p(r) be /V-representable are actually given by Equation 4.5 above. Nevertheless, question remains Can the single-particle density contain all information about a many-electron system, at least in its ground state An affirmative answer to this question can be given from Kato s cusp condition for a nuclear site in the ground state of any atom, molecule, or solid, viz.,... 

The conclusion that it may be possible to formulate the quantum mechanics of many-electron systems solely in terms of the single-particle density was put on a firm foundation by the two Hohenberg-Kohn theorems (1964), which are stated below, without proof. 

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