Electron representation

Bar code scanners, for example, exploit the directionality of a laser beam. The laser light reflects more strongly from white stripes than from black. Because of the directionality of the laser beam, the reflected laser beam faithfully mirrors the pattern of the bar code. A sensor reads these variations and converts the light pattern into an eiectronic representation of the bar code. The sensor transmits the electronic representation to a computer. 

Suppose now that A) and B) belong to an electronic representation I ,. Since H is totally symmetric, Eq. (6) implies that the matrix elements (A II TB) belong to the representation of symmetrized or anti-symmetrized products of the bras (A with the kets 7 A). However, the set TA) is, however, simply a reordering of the set ( A). Hence, the symmetry of the matrix elements in the even- and odd-electron cases is given, respectively, by the symmetrized [Ye x Te] and antisymmetrized Ff x I parts of the direct product of I , with itself. A final consideration is that coordinates belonging to the totally symmetric representation, To, cannot break any symmetry determined degeneracy. The symmetries of the Jahn-Teller active modes are therefore given by... 

Figure 2. (a) Electronic representation, (b) truth table, and (c) molecular implementation of YES (PASS) logic. 0 and 1 are digital representations of low and high signal levels respectively in all Figures. 

Figure 5. (a) Electronic representation, (b) truth table and (c) molecular implementation of XOR logic. 

Figure 5. One-electron representation of pir/dw orbitals involved in coupling in Creutz-Taube ion. Key a, 2b u (doubly filled) and 2bSg (half-filled) and b, localized orbitals 2"%(2b u . 2bsg) (21).

In this section, we introduce the model Hamiltonian pertaining to the molecular systems under consideration. As is well known, a curve-crossing problem can be formulated in the adiabatic as well as in a diabatic electronic representation. Depending on the system under consideration and on the specific method used, both representations have been employed in mixed quantum-classical approaches. While the diabatic representation is advantageous to model potential-energy surfaces in the vicinity of an intersection and has been used in mean-field type approaches, other mixed quantum-classical approaches such as the surfacehopping method usually employ the adiabatic representation. 

In order to introduce the classical-path approximation in the adiabatic electronic representation, we expand the total wave function in terms of... 

To apply the mapping formalism to vibronically coupled systems, we identify the / ) with electronic states and the h m with operators of the nuclear dynamics. Hereby, the adiabatic as well as a diabatic electronic representation may be employed. In a diabatic representation, we have [cf. Eq. (1)]... 

X + P )/4, which by construction varies between 0 (system is in /i)) and 1 (system is in /2)). Describing, as usual, the nuclear motion through the position X, the vibronic PO can then be drawn in the (A dia,v) plane. Here, the subscript dia emphasizes that we refer to the population of the diabatic states which are used to define the molecular Hamiltonian H. For inteipretational purposes, on the other hand, it is often advantageous to change to the adiabatic electronic representation. Introducing the adiabatic population A ad. where Nad = 0 corresponds to the lower and A ad = 1 to the upper adiabatic electronic state, the vibronic PO can be viewed in the (Nad,x) plane. Alternatively, one may represent the vibronic PO as a curve N d i + (1 — A ad)IFi between the... 

All of these chemical species have importance in the production of polymeric materials. There are several shorthand techniques for writing down the structures of polymers. The carbon-based polymer molecules using the stick representation are made up of atoms connected by covalent bonds (represented here by the straight lines between the carbon and the hydrogen and the carbon-to-carbon molecules), as shown in Fig. 2.6. To reiterate, carbon is always tetravalent, having four covalent bonds, and a schematic of the paired electrons for two of the incorporated carbon molecules can be seen in the bottom of Fig. 2.6. Thus each stick represents two electrons. For the two highlighted carbon atoms in the polyethylene molecule of Fig. 2.6, the electron representation is shown, where there are four covalent bonds associated with each carbon and each bond is made up of two shared electrons represented by the black dots. This polymer molecule is made up of only carbon and hydrogen with no double bonds, and it represents a linear form... 

Figure 2.6 Representations for PE resin a) stick method, and b) electron representation of the highlighted segment...

In this work we use an adiabatic electronic representation, and Jacobi nuclear coordinates are chosen r, the HE internuclear vector, and R, the vector joining the HE center-of-mass to the Li atom, in a body-fixed frame in which the three atoms lie on the a — body-fixed plane, with R being parallel to the body-fixed... 

Similar results have been found[58] in recent semiclassical simulations even when some variation was obtained depending on the particular method and on the electronic representation, either diabatic or adiabatic. The electronic representa-... 

Fig. 4. Concerning a one-electron representation of the/-electron energy in configuration/ ground-state. Numbers used in theoretical expressions appropriate to Ln .

V. I. Danilov and G. F. Kventsel, Electronic Representations in the Point Mutation Theory (in Russian). Naukova Dumka, Kiev, 1971. 

For one shell of equivalent electrons representation (14.65) has been found in which the irreducible tensorial products are constructed separately for creation operators and annihilation operators. Likewise, such a representation can also be found for two-shell configurations. In that case,... 

The enabled OR - another type of gate that requires a minimum of three inputs -has just become available. Figure 15 shows the equivalent electronic representation, with the appropriate connection of AND and OR gates. In this context we note that the INHIBIT gate discussed in Section 11.10 also involves an enabling/disabling input. One strand of Pina, Maestri, and Balzani s novel exploitation of flavylium derivative photochemistry was briefly mentioned in Section 11.4. This AND logic system used protons and ultraviolet photons as the two inputs, to create a visible... 

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Another one-input, one-output gate is a NOT gate. This gate can be considered an inverted YES gate, as can be seen by the electronic representation in Figure 1. The corresponding truth table for this logic situation can also be seen. 

The correlation function can be written in a basis set Ra) = R) a) chosen as the tensor product of the coordinate representation for the nuclear degrees of freedom, and a nuclear coordinate independent electronic basis. In this chapter we shall refer to this electronic representation as the diabatic basis. (We refer the interested reader to reference [40] for the development of this approach with a more general electronic representation). By inserting resolutions of the identity in this basis, one obtains... 

M. Baer and R. Englman, A study of the diabatic electronic representation within the Born-Oppenheimer approximation, Mol. Phys. 75 293 (1992). 

Comparison of the various ab initio basis sets has been made to resolve the question of the best basis set for a particular application. It was concluded that of the lower-level basis sets (STO-3G, 4-3IG, STO-3G ), STO-3G provides the best electronic representation of hypervalent phosphorus and sulphur species (Collins et al 1976). Of the higher-level basis sets (3-2IG, 3-21G( ), 6-31G ), it was concluded that the 6-31G and 3-21G( ) sets give good representation of such species (Pietro et al., 1982). A major conclusion of both these studies is that d-orbital functions are essential for calculations on species containing hypervalent second-row elements and bonds involving two heavy atoms. 

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