Electronic contributions

Electronic Contribution. If a molecule has an electronic ground state of degeneracy go a first excited state at an energy Cei with a degeneratgrgj, then the electronic partition function will be 

Free radicals with an unpaired electron will have an electronic degeneracy of 2 arising from the resultant spin of 1/2. The same situation arises with NOg, and LA In 2 has therefore to be added to S T) and - (G (r) - H%Q) IT whereas C% X) and H T) - H (0) /r are unaffected. 

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The value of at zero temperature can be estimated from the electron density ( equation Al.3.26). Typical values of the Femii energy range from about 1.6 eV for Cs to 14.1 eV for Be. In temis of temperature (Jp = p//r), the range is approxunately 2000-16,000 K. As a consequence, the Femii energy is a very weak ftuiction of temperature under ambient conditions. The electronic contribution to the heat capacity, C, can be detemiined from... 

Figure A2.2.6. Electronic contribution to the heat capacity Cy of copper at low temperatures between 1 and 4 K. (From Corak et al [2]).

Tromp R M, Hamers R J and Demuth J E 1986 Atomic and electronic contributions to Si(111)-(7 7) scanning-tunnelling-microscopy images Rhys. Rev. B 34 1388... 

I lie electronic contribution arises from a continuous function of electron density and must be calculated using the appropriate operator ... 

The dipole moment operator is a sum of one-electron operators r , and as such the electronic conlribution to the dipole moment can be written as a sum of one-electron contributions. The eleclronic contribution can also be written in terms of the density matrix, P, as follows ... 

The electronic contribution to the dipole moment is thus determined from the density matrix and a series of one-electron integrals J dr< (-r)0. The dipole moment operator, r, h.)-components in the x, y and z directions, and so these one-electron integrals are divided into their appropriate components for example, the x component of the electronic contribution to the dipole moment would be determined using ... 

When you perform a single point semi-empirical or ab initio calculation, you obtain the energy and the first derivatives of the energy with respect to Cartesian displacement of the atoms. Since the wave function for the molecule is computed in the process, there are a number of other molecular properties that could be available to you. Molecular properties are basically an average over the wave function of certain operators describing the property. For example, the electronic dipole operator is basically just the operator for the position of an electron and the electronic contribution to the dipole moment is... 

The point r is the position of a positive probe charge. is the nuclear charge on atom Alocated at position R. The function p(r ) is the electronic density. In the above equation, the first term represents the contribution of the nuclei to the electrostatic potential and the second term is the electronic contribution. Substituting the electron density expression ... 

For measurements carried out closer to resonances, thermal contributions to the optical nonlinearity, given by the following formula, can compete with electronic contributions (54,55) ... 

For main group elements the number of framework electrons contributed is equal to (t + a — 2) where v is the number of valence shell electrons of that element, and x is the number of electrons from ligands, eg, for Ff, x = and for Lewis bases, x = 2. Examples of 2n + 2 electron count boranes and heteroboranes, and the number of framework electrons contributed by their skeletal atoms, ate given in Table 1. 

Number of atoms multipbed by, in parentheses, the number of electrons contributed to the framework gives the total electron contribution for the element. 

Table 2. Framework Electron Contributions for Metal Moieties ...

The relatively high mobilities of conducting electrons and electron holes contribute appreciably to electrical conductivity. In some cases, metallic levels of conductivity result ia others, the electronic contribution is extremely small. In all cases the electrical conductivity can be iaterpreted ia terms of carrier concentration and carrier mobiUties. Including all modes of conduction, the electronic and ionic conductivity is given by the general equation ... 

Computed values are plotted in Figure 6 against the number of methyl groups. Note that these components include the electronic contributions (net contribution for isolated mole-... 

One further breaks down the secondary electron contributions into three groups SEI, SEII and SEIII. SEIs result from the interaction of the incident beam with the sample at the point of entry. SEIIs are produced by BSE s on exiting the sample. SEIIIs are produced by BSEs which have exited the surface of the sample and further interact with components on the interior of the SEM usually not related to the sample. SEIIs and SEIIIs come from regions far outside that defined by the incident probe and can cause serious degradation of the resolution of the image. 

The spin multiplicity for a molecule is given by the equation 2S + 1, where S is the total spin for the molecule. Paired electrons contribute nothing to this quantity. They have a net spin of zero since an alpha electron has a spin of +Vi and a beta electron has a spin of -Vi. Each unpaired electron contributes +Vi to S. Thus, a singlet—a system with no unpaired electrons—has a spin multiplicity of 1, a doublet (one unpaired electron) has a spin multiplicity of 2, a triplet (two unpaired electrons of like spin) has a spin multiplicity of 3, and so on. 

As each B atom contributes 1 electron to its B-Ht bond and 2 electrons to the framework MOs, the (n + 1) framework bonding MOs are just filled by the 2n electrons from nB atoms and the 2 electrons from the anionic charge. Further, it is possible (conceptually) to remove a BHt group and replace it by 2 electrons to compensate for the 2 electrons contributed by the BHi group to the MOs. Electroneutrality can then be achieved by adding the appropriate number of protons this does not alter the number of electrons in the system and hence all bonding MOs remain just filled. 

Boys and Cook refer to these properties as primary properties because their electronic contributions can be obtained directly from the electronic wavefunction As a matter of interest, they also classified the electronic energy as a primary property. It can t be calculated as the expectation value of a sum of true one-electron operators, but the Hartree-Fock operator is sometimes written as a sum of pseudo one-electron operators, which include the average effects of the other electrons. 

The derivations given above related to a single particle in a constant magnetic induction. For a molecule within the Bom-Oppenheimer approximation, the derivation is similar except that we take the nuclei to be fixed in space. There is a nuclear and an electronic contribution to each property. 

In the BeF2 molecule, there are two electron-pair bonds. These electron pairs are located in the two sp hybrid orbitals. In each orbital, one electron is a valence electron contributed by beryllium the other electron comes from the fluorine atom. 

Studies of double carrier injection and transport in insulators and semiconductors (the so called bipolar current problem) date all the way back to the 1950s. A solution that relates to the operation of OLEDs was provided recently by Scott et al. [142], who extended the work of Parmenter and Ruppel [143] to include Lange-vin recombination. In order to obtain an analytic solution, diffusion was ignored and the electron and hole mobilities were taken to be electric field-independent. The current-voltage relation was derived and expressed in terms of two independent boundary conditions, the relative electron contributions to the current at the anode, jJfVj, and at the cathode, JKplJ. 

In many electron atoms the maximum contributions to the polarizability and to London forces arise from configurations with more than one electron contributing to the net dipole moment of the atom. But in such configurations the electronic repulsion is especially high. The physical meaning to be attributed to the Qkl terms is just the additional electron repulsive energy which these configurations require. 

The assumption that only the outermost subshell of electrons contributes to either a or EL. 

The P —alumina structures are remarkable not only for their ionic conductivities but also for the versatility in isomorphous replacement. There is little of the structure of (Na2S)l+Jt 11A1203 which cannot be substituted, at least in part, by an alternative ion. MgO and Li20 are preferred additives to P —alumina in order to obtain good ionic conduction with no electronic contribution. 

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