Operator equivalents

For the nanotubes, then, the appropriate symmetries for an allowed band crossing are only present for the serpentine ([ , ]) and the sawtooth ([ ,0]) conformations, which will both have C point group symmetries that will allow band crossings, and with rotation groups generated by the operations equivalent by conformal mapping to the lattice translations Rj -t- R2 and Ri, respectively. However, examination of the graphene model shows that only the serpentine nanotubes will have states of the correct symmetry (i.e., different parities under the reflection operation) at the K point where the bands can cross. Consider the K point at (K — K2)/3. The serpentine case always sat-... 

A fully relativistic treatment of more than one particle has not yet been developed. For many particle systems it is assumed that each electron can be described by a Dirac operator (ca ir + p mc ) and the many-electron operator is a sum of such terms, in analogy with the kinetic energy in non-relativistic theory. Furthermore, potential energy operators are added to form a total operator equivalent to the Hamilton operator in non-relativistic theory. Since this approach gives results which agree with experiments, the assumptions appear justified. 

C. P. Slichter s textbook on magnetic resonance [4] may be recommended for further reading. It presents a very educational introduction into this issue of operator equivalence. A comprehensive, elaborate article on quadrupole interaction in Mossbauer spectroscopy is provided by H. Spiering in [5],... 

Stevens, K.W.H. 1952. Matrix elements and operator equivalents connected with the magnetic properties of rare earth ions. Proceedings of the Physical Society 65 209-215. 

It is worth noting here that negative q values correspond to complex operators, while Stevens parameters are always real [19]. The forms of the operator equivalents are reported in Table 1.3 [20], and the corresponding matrix elements are found tabulated in books by Abragam and Bleaney and by Altshuler... 

Since electrophilic and charge-transfer nitrations are both initiated via the same EDA complex and finally lead to the same array of nitration products, we infer that they share the intermediate stages in common. The strength of this inference rests on the variety of aromatic substrates (with widely differing reactivities and distinctive products) to establish the mechanistic criteria by which the identity of the two pathways are exhaustively tested. On this basis, electrophilic nitration is operationally equivalent to charge-transfer nitration in which electron-transfer activation is the obligatory first step. The extent to which the reactive triad in (90) is subject to intermolecu-lar interactions in the first interval (a few picoseconds) following electron transfer will, it is hoped, further define the mechanistic nuances of dissociative electron transfer in adiabatic and vertical systems (Shaik, 1991 Andrieux et al., 1992), especially when inner-sphere pathways are considered (Kochi, 1992). 

S improper rotation of 2n/n radians (n e N) improper rotations are regular rotations followed by a reflection in the plane perpendicular to the axis of rotation i inversion operator (equivalent to 2) a mirror plane... 

This elegant result has the consequence that the operator equivalence = (cr p) is reproduced exactly in the matrix form [18], [19, Equation (28)]... 

A complete tabulation of operator equivalents for crystals fields of various symmetries can be found in books by Low (5) and by Al tshuler and Kozyrev (9). 

When the matrix elements are calculated for states built from /-electron configurations it is always found that the constants A% (these quantities are related to the strength of crystal field) always occur with (the sharp brackets denote integration with respect to 4/ radial function). A parameters play an important role in crystal field calculations and can be used as parameters in describing the crystal field. For the lowest L S J state they can easily be determined by using the operator equivalent technique of Elliott and Stevens [545—547] and with the help of existing tables of matrix elements. Wybotjbne [548], however, feels that a better approach is to expand Vc in terms of the tensor operators,, as... 

The present procedure, based on the controlled lithiation of aliens, produces the operational equivalent of a propargyl dianion 1 (C3H2L12), and provides a convenient single-step route to propargylated derivatives. Lithiation of allene is... 

In the first term, the rate constant is multiplied by cHe / K% + cH ), the fraction of total enol present in neutral form E, and in the second term, k A is multiplied by K /(K + cH ), the fraction of total enol present in basic form Ee [Equation (4)]. The observed coefficient for general base catalysis is now seen to arise from the pre-equilibrium reaction shown in the second line of Scheme 3. Replacing (A /ch )cha by (K /K A)cAe we find k e=kf A / A. Thus, pre-equilibrium deprotonation of the enol by the general base followed by carbon protonation of the ensuing enol anion is operationally equivalent to general base catalysis. 

In the above formula operator equivalents 0 occur preceded by the potential constants B (here k is the tensor rank and m its component). The most... 

The total energy of a mechanical system is also given by the sum of its kinetic and potential energies, E = p2/2m + V. The operator equivalent of this classical expression, known as the Hamiltonian operator, is obtained by substituting the momentum operator into this equation, i.e. 

Product Operator. For multiple-quantum coherences, the pure zero-quantum and double-quantum states are shown with their Cartesian product operator equivalents. 

In this way, once the results for any given LS state have been obtained via the operator equivalent technique, those for any of the J sub-levels thereof may easily be derived. This procedure could of course be extended to evaluate matrix elements between states of different J, arising from the same LS state, but for calculations, whether in the LSMlMs) or the LSJMj) basis, in which it is required to incorporate mixing of different LS states the resulting cross-product matrix elements cannot be found by the operator equivalent method but must be determined directly from the wave functions. 

Stevens [4] developed an operator-equivalent method for evaluating crystal field matrix elements based on the Wigner-Eckart theorem. It was shown that within a particular J (or L) manifold all operators of the same rank have matrix elements which are proportional to one another. The matrix elements of these operators along with proportionality constants for the ground terms of f" ions have been tabulated [5]. 

Kutzelnigg/76/ and Kutzelnigg and Koch/77/ emphasized that the classification of the operators into the categories C, A, B and 0 is essentially a Fock space concept, so that the condition that L, vanishes will automativally generate operator equivalent of the eq. (7.1.3). L, is then the Fock-space effective hamiltonian H, and appropriate... 

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