Atomic contributions

Referring to figure Bl.8.5 the radii of the tliree circles are the magnitudes of the observed structure amplitudes of a reflection from the native protein, and of the same reflection from two heavy-atom derivatives, dl and d2- We assume that we have been able to detemiine the heavy-atom positions in the derivatives and hl and h2 are the calculated heavy-atom contributions to the structure amplitudes of the derivatives. The centres of the derivative circles are at points - hl and - h2 in the complex plane, and the three circles intersect at one point, which is therefore the complex value of The phases for as many reflections as possible can then be... 

Figure Bl.8.5. Pp Pdl and Fdl are the measured stnicture amplitudes of a reflection from a native protein and from two heavy-atom derivatives. and are the heavy atom contributions. The pomt at which the tliree circles intersect is the complex value of F. ...

Figure C2.12.1. Origin of ion exchange capacity in zeolites. Since every oxygen atom contributes one negative charge to the tetrahedron incoriDorated in the framework, the silicon tetrahedron carries no net charge while the aluminium tetrahedron carries a net charge of-1 which is compensated by cations M.

The most stable nuclear configuration of this system is a pair of H2 molecules. There are three possible spin coupling combinations for H4 corresponding to three distinct stable product H2 pairs H1 H2 with H3 H4, H1 H3 with H2 H4, and H1 H4 with H2 H3. Each H atom contributes one electron, the dot diagrams indicate spin pairing. The three combinations are designated as Hfl), HOT), and H(III), respectively. They may be interconverted via square transition states, Figure 2. 

Miller [10] later revised these atomic contributions, O , somewhat, on the basis of new experimental data. 

The accuracy of an additivity scheme can be increased by going from atomic contributions through bond contributions to group contributions. 

Fhe van der Waals and electrostatic interactions between atoms separated by three bonds (i.c. the 1,4 atoms) are often treated differently from other non-bonded interactions. The interaction between such atoms contributes to the rotational barrier about the central bond, in conjunction with the torsional potential. These 1,4 non-bonded interactions are often scaled down by an empirical factor for example, a factor of 2.0 is suggested for both the electrostatic and van der Waals terms in the 1984 AMBER force field (a scale factor of 1/1.2 is used for the electrostatic terms in the 1995 AMBER force field). There are several reasons why one would wish to scale the 1,4 interactions. The error associated wilh the use of an repulsion term (which is too steep compared with the more correct exponential term) would be most significant for 1,4 atoms. In addition, when two 1,4... 

These absorptions are ascribed to n-n transitions, that is, transitions of an electron from the highest occupied n molecular orbital (HOMO) to the lowest unoccupied n molecular orbital (LUMO). One can decide which orbitals are the HOMO and LUMO by filling electrons into the molecular energy level diagram from the bottom up, two electrons to each molecular orbital. The number of electrons is the number of sp carbon atoms contributing to the n system of a neuhal polyalkene, two for each double bond. In ethylene, there is only one occupied MO and one unoccupied MO. The occupied orbital in ethylene is p below the energy level represented by ot, and the unoccupied orbital is p above it. The separation between the only possibilities for the HOMO and LUMO is 2.00p. 

Recent progress in this field has been made in predicting individual atoms contribution to optical activity. This is done using a wave-functioning, partitioning technique roughly analogous to Mulliken population analysis. 

CHEOPS is based on the method of atomic constants, which uses atom contributions and an anharmonic oscillator model. Unlike other similar programs, this allows the prediction of polymer network and copolymer properties. A list of 39 properties could be computed. These include permeability, solubility, thermodynamic, microscopic, physical and optical properties. It also predicts the temperature dependence of some of the properties. The program supports common organic functionality as well as halides. As, B, P, Pb, S, Si, and Sn. Files can be saved with individual structures or a database of structures. 

The impurity atoms used to form the p—n junction form well-defined energy levels within the band gap. These levels are shallow in the sense that the donor levels He close to the conduction band (Fig. lb) and the acceptor levels are close to the valence band (Fig. Ic). The thermal energy at room temperature is large enough for most of the dopant atoms contributing to the impurity levels to become ionized. Thus, in the -type region, some electrons in the valence band have sufficient thermal energy to be excited into the acceptor level and leave mobile holes in the valence band. Similar excitation occurs for electrons from the donor to conduction bands of the n-ty e material. The electrons in the conduction band of the n-ty e semiconductor and the holes in the valence band of the -type semiconductor are called majority carriers. Likewise, holes in the -type, and electrons in the -type semiconductor are called minority carriers. 

There are two processes where nuclear and atomic contributions are iaterrelated. These are the emission of electrons from the atomic shells as an alternative to the emission of a photon and the emission of bremsstrahlung photons ia the P decay process. 

The molecular formula is C4H10O. Using the atomic contribution values from Table 2-386. 

Perhaps the most notable difference between S-N and N-O compounds is the existence of a wide range of cyclic compounds for the former. As indicated by the examples illustrated below, these range from four- to ten-membered ring systems and include cations and anions as well as neutral systems (1.14-1.18) (Sections 5.2-5.4). Interestingly, the most stable systems conform to the well known Htickel (4n -1- 2) r-electron rule. By using a simple electron-counting procedure (each S atom contributes two electrons and each N atom provides one electron to the r-system in these planar rings) it can be seen that stable entities include species with n = 1, 2 and 3. 

Election counting and orbital bookkeeping can easily be checked in these diagrams as each B has 4 valency orbitals (s -F 3p) there should be 4 lines emanating from each open circle likewise, as each B atom contributes 3 electrons in all and each H atom contributes 1 electron, the total... 

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. 

One way to investigate the electrophilic properties of these molecules is to examine the orbital that each uses to accept electrons from a nucleophile. This orbital is the lowest-unoccupied molecular orbital (LUMO). Examine the LUMO for methyl acetate (Z=OCH3), acetaldehyde (Z=H), N,N-dimethylacetamide (Z=N(CH3)2) and acetyl chloride (Z=C1) (acetaldehyde is not a carboxylic acid derivative, but is included here for comparison). What is the shape of the LUMO in the region of the carbonyl group Is it a o or 7U orbital Is it bonding or antibonding What other atoms contribute to the LUMO Which bonds, if any, would be weakened when a nucleophile transfers its electrons into the LUMO ... 

Aceording to Frontier Molecular Orbital (FMO) theory, thiophene s most reactive site can be identified by examining the shape of its highest-occupied molecular orbital (HOMO). Which atoms contribute to thiophene s HOMO Which atom(s) contributes the most Which nitration product should form preferentially ... 

In addition to electrophilic attack on the pyrrole ring in indole, there is the possibility for additions to the fused benzene ring. First examine the highest-occupied molecular orbital (HOMO) of indole. Which atoms contribute the most What should be the favored position for electrophilic attack Next, compare the energies of the various protonated forms of indole (C protonated only). These serve as models for adducts formed upon electrophilic addition. Which carbon on the pyrrole ring (C2 or C3) is favored for protonation Is this the same as the preference in pyrrole itself (see Chapter 15, Problem 2)1 If not, try to explain why not. Which of the carbons on the benzene ring is most susceptible to protonation Rationalize your result based on what you know about the reactivity of substituted benzenes toward electrophiles. Are any of the benzene carbons as reactive as the most reactive pyrrole carbon Explain. 

For the minute, imagine an HF-LCAO treatment of just the jr-electrons in ethene where each carbon atom contributes just one electron and one atomic orbital of the correct symmetry to the conjugated system. Without any particular justification except chemical intuition, we make the following assumptions. 

In the PPP model, each first-row atom such as carbon and nitrogen contributes a single basis functiqn to the n system. Just as in Huckel theory, the orbitals x, m e not rigorously defined but we can visualize them as 2p j atomic orbitals. Each first-row atom contributes a certain number of ar-electrons—in the pyridine case, one electron per atom just as in Huckel 7r-electron theory. 

In the TT-electron theories, each first-row atom contributes a single basis function. For the all valence electron models there is now an additional complication in at some of the basis functions could be on the same atomic centre. So how should we treat integrals involving basis functions all on the same atomic centre such as... 

To interpret the strong dependence of the conductivity from composition, we also evaluated the electronic density-of-states and analyzed its specific atomic contributions. For this discussion and for comparison we also calculated the electrical conductivities and the electronic densitity-of-states using a simplified density-functional (DFT)- based LCAO scheme [12]. 

This strong dependence of the conductivity on the composition can be understood qualitatively considering the densities-of-states. The DOS s from the LCAO calculation, splitted into their atomic contributions (partial DOS s), are given in Fig. 1 and can be discussed as follows ... 

Nitrogen, Sulfur and Oxygen Compounds. These are usually abbreviated as NSO compounds and sometimes referred to as asphaltics. Although present in small amounts, the N, S, and O atoms contribute greatly to the nonhydrocarbon fraction of a crude oil by their incorporation into hydrocarbon molecules. The residuum contains a high percentage of NSO compounds. 

Pyrrole (two r s, one /) and imidazole are /ive-membered heterocycles, yet both have six tt electrons and are aromatic. In pyrrole, each of the four. sp2-hybridized carbons contributes one tt electron, and the sp2-hybridized nitrogen atom contributes the two from its lone pair, which occupies a p orbital (Figure 15.9). Imidazole, also shown in Figure 15.9, is an analog of pyrrole that has two nitrogen atoms in a five-membered, unsaturated ring. Both nitrogens are sp2-hybridized, but one is in a double bond and contributes only one electron to the aromatic tt system, while the other is not in a double bond and contributes two from its lone pair. 

---