Hydrogen-like centres

The symmetry of an isolated atom is that of the full rotation group R+ (3), whose irreducible representations (IRs) are D where j is an integer or half an odd integer. An application of the fundamental matrix element theorem [22] tells that the matrix element (5.1) is non-zero only if the IR DW of Wi is included in the direct product x of the IRs of ra and < f. The components of the electric dipole transform like the components of a polar vector, under the IR l)(V) of R+(3). Thus, when the initial and final atomic states are characterized by angular momenta Ji and J2, respectively, the electric dipole matrix element (5.1) is non-zero only if D(Jl) is contained in Dx D(j 2 ) = D(J2+1) + T)(J2) + )(J2-i) for j2 > 1 This condition is met for = J2 + 1, J2, or J2 — 1. However, it can be seen that a transition between two states with the same value of J is allowed only for J 0 as DW x D= D( D(°) is the unit IR of R+(3)). For a hydrogen-like centre, when an atomic state is defined by an orbital quantum number , this can be reduced to the Laporte selection rule A = 1. This is of course formal, as it will be shown that an impurity state is the weighted sum of different atomic-like states with different values of but with the same parity P = ( —1) These states are represented by an atomic spectroscopy notation, with lower case letters for the values of (0, 1, 2, 3, 4, 5, etc. correspond to s, p, d, f, g, h, etc.). The impurity states with P = 1 and -1 are called even- and odd-parity states, respectively. For the one-valley EM donor states, this quasi-atomic selection rule determines that the parity-allowed transitions from Is states are towards np (n > 2), n/ (n > 4), nh (n > 6), or nj (n > 8) states. For the acceptor states in cubic semiconductors, the even- and odd-parity states labelled by the double IRs T of Oh or Td are indexed by + or respectively, and the parity-allowed transition take place between Ti+ and... 

The present volume entitled Optical Absorption of Impurities and Defects in Semiconducting Crystals - Hydrogen-like Centres is by Dr. Bernard Pajot. He is an internationally recognized condensed matter experimenter. He has made numerous significant contributions to the field of donors and acceptors local vibrations of oxygen related complexes in Si and Ge and magneto-and piezo-spectroscopy of donors and acceptors. This volume contains an authoritative and clear presentation of the theory of donors and acceptors. The... 

A well known operator is the Hamiltonian of an electron in centre-of-mass coordinates of a hydrogen-like atom... 

We consider a hydrogen-like atom, with nuclear charge Z, enclosed in a spherical well, of radius R, with an impenetrable wall. The nucleus is assumed fixed at the centre of the well and we note that, for finite R, it is not therefore possible to separate out the translational motion of the centre of mass of the system. Pupyshev [18] proved that, for the ground state, the energy is a minimum when the nucleus is at r = 0. In a non-relativistic approximation the Schrodinger equation for the electronic motion is1... 

From the coupling of the spectroscopic results with electrical measurements has emerged a classification of the electrically-active foreign centres into hydrogen-like (H-like) centres on the one side, opposed to deep centres on the other side. This classification is somewhat abrupt as there exist centres, like those related to the transition metals, which can display properties related to one or to the other category. 

The 1-electron 1-centre eigenfunctions, which satisfy an equation of the form (1.2.9), are atomic orbitals (AOs) they are hydrogen-like AOs if we use the Hamiltonian (1.2.2a), corresponding to one electron in the field of the bare nucleus, but may have a somewhat different functional form if they are eigenfunctions of a more general Hamiltonian (1.1.2b) in whidi the potential V may allow partly for the presence of other electrons. In any case the AOs are essentially those of a central field, in w ich V depends only on the radial distance of the electron from the nucleus, and have characteristic forms which are well described in elementary textbooks on valency. For completeness, the forms of the Is, 2s, 2p,..., 4f orbitals and the significance of the classification are reviewed in Appendix 1. 

Point defects and complexes exliibit metastability when more than one configuration can be realized in a given charge state. For example, neutral interstitial hydrogen is metastable in many semiconductors one configuration has H at a relaxed bond-centred site, bound to the crystal, and the other has H atomic-like at the tetrahedral interstitial site. 

In the case of the retro Diels-Alder reaction, the nature of the activated complex plays a key role. In the activation process of this transformation, the reaction centre undergoes changes, mainly in the electron distributions, that cause a lowering of the chemical potential of the surrounding water molecules. Most likely, the latter is a consequence of an increased interaction between the reaction centre and the water molecules. Since the enforced hydrophobic effect is entropic in origin, this implies that the orientational constraints of the water molecules in the hydrophobic hydration shell are relieved in the activation process. Hence, it almost seems as if in the activated complex, the hydrocarbon part of the reaction centre is involved in hydrogen bonding interactions. Note that the... 

A more eflicient and general synthetic procedure is the Masamune reaction of aldehydes with boron enolates of chiral a-silyloxy ketones. A double asymmetric induction generates two new chiral centres with enantioselectivities > 99%. It is again explained by a chair-like six-centre transition state. The repulsive interactions of the bulky cyclohexyl group with the vinylic hydrogen and the boron ligands dictate the approach of the enolate to the aldehyde (S. Masamune, 1981 A). The fi-hydroxy-x-methyl ketones obtained are pure threo products (threo = threose- or threonine-like Fischer formula also termed syn" = planar zig-zag chain with substituents on one side), and the reaction has successfully been applied to macrolide syntheses (S. Masamune, 1981 B). Optically pure threo (= syn") 8-hydroxy-a-methyl carboxylic acids are obtained by desilylation and periodate oxidation (S. Masamune, 1981 A). Chiral 0-((S)-trans-2,5-dimethyl-l-borolanyl) ketene thioketals giving pure erythro (= anti ) diastereomers have also been developed by S. Masamune (1986). 

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