In pair space

The particle current density in pair space ja/ (2)(r1,r2) is given by the six-component vector... 

To evaluate qv we must express ja i<2) as a function of the local velocities. In order to do this we shall postulate the validity in pair space of quasi-phenomenological relations strictly analogous... 

We now postulate that the following equation in pair space, analogous to Eq. IV.9 in singlet space, is valid ... 

Equation IV. 18 splits the heat of transport into two terms, the first of which is quasi-thermodynamic in that it involves only averages over equilibrium ensembles, and the second of which arises from the deviation of the distribution function in pair space from the equilibrium distribution function. Qn corresponds to the expressions for the heat of transport found by previous authors who have neglected the nonequilibrium perturbation to the distribution function. 

The theory and results given here determine and illustrate the effect of Pauli exclusion principle on the distribution of electronic charge in real space. It has been demonstrated that the localization of charge in real space is determined by the localization of the Fermi correlation in pair space and that the extent of pairing and localization proceed hand in hand. 

All Np states belonging to the Pth sub-space interact strongly with each other in the sense that each pair of consecutive states have at least one point where they form a Landau-Zener type interaction. In other words, all j = I,... At/> — I form at least at one point in configuration space, a conical (parabolical) intersection. 

Spin orbitals arc grouped in pairs for an KHF ealetilation, Haeti mem her of ih e pair dilTcrs in its spin function (one alpha and one beta), hilt both must share the same space function. For X electrons, X/2 different in olecu lar orbitals (space function s larc doubly occupied, with one alpha (spin up) and one beta (spin down) electron forming a pair. 

One way to describe the conformation of a molecule other than by Cartesian or intern coordinates is in terms of the distances between all pairs of atoms. There are N(N - )/ interatomic distances in a molecule, which are most conveniently represented using a N X N S5munetric matrix. In such a matrix, the elements (i, j) and (j, i) contain the distant between atoms i and and the diagonal elements are all zero. Distance geometry explort conformational space by randomly generating many distance matrices, which are the converted into conformations in Cartesian space. The crucial feature about distance geometi (and the reason why it works) is that it is not possible to arbitrarily assign values to ti... 

Ethylene—Dicarboxylic Acid Copolymers. Partial neutralization of copolymers containing carboxyls in pairs on adjacent carbons, eg, ethylene—maleic acid, has been described (11). Surprisingly, there is no increase in stiffness related to neutralization. Salts with divalent metal cations are not melt processible. The close spacing of the paired carboxyl groups has resulted in ionic cluster morphology which is distinct from that of the commercial ionomer family. 

The primary photochemical act, subsequent to near-uv light (wavelengths <400 nm) absorption by Ti02 particles, is generation of electron—hole pairs where the separation (eq. 3) into conduction band electrons (e g ) and valence band holes (/lyB ) faciUtated by the electric field gradient in the space charge region. Chemically, the hole associated with valence band levels is constrained at... 

In most metals the electron behaves as a particle having approximately the same mass as the electron in free space. In the Group IV semiconductors, dris is usually not the case, and the effective mass of electrons can be substantially different from that of the electron in free space. The electronic sUmcture of Si and Ge utilizes hybrid orbitals for all of the valence elecU ons and all electron spins are paired within this structure. Electrons may be drermally separated from the elecU on population in dris bond structure, which is given the name the valence band, and become conduction elecU ons, creating at dre same time... 

In all the equations above we have omitted the dependencies on r the symbol denotes convolution in r-space. The symmetry of the correlation functions implies that Similarly to the total pair corre-... 

In a normal gear set, each of the sidebands are spaced by exactly the lx running speed of the input shaft and the entire gear mesh is symmetrical as seen in Figure 44.50. In addition, the sidebands always occur in pairs, one below and one above the gear mesh frequency, and the amplitude of each pair is identical (Figure 44.51). 

The structure factor for the 104-atom complex with almost perfect icosahedral symmetry determines the intensities of the diffraction maxima, in correspondence with the inverse relationship between intensity in reciprocal space and the atom-pair vectors in real space that was discovered fifty years ago by Patterson.27 The icosahedral nature of the clusters in the cubic crystal explains the appearance of the Fibonacci numbers and the golden ratio. 

Thallous halides offer a unique possibility of studying the stereochemistry of the (chemically) inert electron pair, since their structures and their pressure and temperature-dependent phase transitions have been well established. Thallium (1) fluoride under ambient conditions, adopts an orthorhombic structure in the space group Pbcm which can be regarded as a distorted rocksalt structure (Fig. 2.4). In contrast to TIF, the thallium halides with heavier halogens, TlCl, TlBr and Til, adopt the highly symmetric cubic CsCl structure type under ambient conditions [46]. Both TlCl and TlBr, at lower temperatures, undergo phase transitions to the NaCl type of structure [47]. 

The equations to be fulfilled by momentum space orbitals contain convolution integrals which give rise to momentum orbitals ( )(p-q) shifted in momentum space. The so-called form factor F and the interaction terms Wij defined in terms of current momentum coordinates are the momentum space counterparts of the core potentials and Coulomb and/or exchange operators in position space. The nuclear field potential transfers a momentum to electron i, while the interelectronic interaction produces a momentum transfer between each pair of electrons in turn. Nevertheless, the total momentum of the whole molecule remains invariant thanks to the contribution of the nuclear momenta [7]. 

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