Half-filled

For the transition metals it is often impossible to reach a noble gas structure except in covalent compounds (see effective atomic number rule) and it is found that relative stability is given by having the sub-shells (d or f) filled, half-filled or empty. 

The half-filled moleeular eonduetion band in gives rise to a maximized density of states at tire Fenni... 

The Bolir radius is very large, 3-5 nm, and tlie shallow impurity wavefunction extends over a large portion of the crystal. Doping up to tlie Tnetallic limit consists in implanting a sufficiently high concentration of donors so tliat tlie shallow-donor wavefunctions overlap, creating a half-filled impurity band in which tlie electrons move freely. 

The teodeocy to aitaia either a half filled or fully filled set of d orbitals at the expense of the outer s orbital is shown by both chromium and copper and should be noted. This apparent irregularity will be discussed in more detail in Chapter 13. 

Table 2.6 shows the electron affinities, for the addition of one electron to elements in Periods 2 and 3. Energy is evolved by many atoms when they accept electrons. In the cases in which energy is absorbed it will be noted that the new electron enters either a previously unoccupied orbital or a half-filled orbital thus in beryllium or magnesium the new electron enters the p orbital, and in nitrogen electron-pairing in the p orbitals is necessary. 

In the older form of the periodic table, chromium was placed in Group VI, and there are some similarities to the chemistry of this group (Chapter 10). The outer electron configuration, 3d 4s. indicates the stability of the half-filled d level. 3d 4s being more stable than the expected 3d 4s for the free atom. Like vanadium and titanium, chromium can lose all its outer electrons, giving chromium)VI) however, the latter is strongly oxidising and is... 

In the actinides, the element curium, Cm, is probably the one which has its inner sub-shell half-filled and in the great majority of its compounds curium is tripositive, whereas the preceding elements up to americium, exhibit many oxidation states, for example -1-2, -1-3. -1-4, -1-5 and + 6, and berkelium, after curium, exhibits states of -1- 3 and -E 4. Here then is another resemblance of the two series. 

Acetylene is obtained from a cylinder (at ground level outside the fum chamber) and is freed from acetone by passing through two 500 ml. wash bottles, half filled with concentrated sulphuric acid, at the rate of 2-3 litres per minute when the acid in the second wash bottle becomes discoloured, the wash bottles should be recharged with fresh acid. The... 

One important question is how many orbitals are available at any given energy level. This is shown using a density of states (DOS) diagram as in Figure 34.2. It is typical to include the Fermi level as denoted by the dotted line in this figure. A material with a half-filled energy band is a conductor, but it may be a... 

The Fermi energy is the energy of the highest-energy filled orbital, analogous to a HOMO energy. If the orbital is half-filled, its energy will be found at a collection of points in /c-space, called the Fermi surface. 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

The characteristic feature of valence bond theory is that it pictures a covalent bond between two atoms in terms of an m phase overlap of a half filled orbital of one atom with a half filled orbital of the other illustrated for the case of H2 m Figure 2 3 Two hydrogen atoms each containing an electron m a Is orbital combine so that their orbitals overlap to give a new orbital associated with both of them In phase orbital overlap (con structive interference) increases the probability of finding an electron m the region between the two nuclei where it feels the attractive force of both of them... 

A vexing puzzle m the early days of valence bond theory concerned the fact that methane is CH4 and that the four bonds to carbon are directed toward the corners of a tetrahedron Valence bond theory is based on the overlap of half filled orbitals of the connected atoms but with an electron configuration of s 2s 2p 2py carbon has only two half filled orbitals (Figure 2 8a) How can it have bonds to four hydrogens ... 

All four sp orbitals are of equal energy Therefore according to Hund s rule (Sec tion 1 1) the four valence electrons of carbon are distributed equally among them making four half filled orbitals available for bonding... 

The axes of the sp orbitals point toward the corners of a tetrahedron Therefore sp hybridization of carbon is consistent with the tetrahedral structure of methane Each C—H bond is a ct bond m which a half filled Is orbital of hydrogen over laps with a half filled sp orbital of carbon along a line drawn between them... 

FIGURE 2 9 Each half filled sp orbital overlaps with a half filled hydrogen Is or bital along a line between them giving a tetrahedral arrangement of four ct bonds Only the major lobe of each sp orbital is shown Each orbital contains a smaller back lobe which has been omitted for clarity... 

FIGURE 2 10 The C—C ct bond in ethane pictured as an overlap of a half filled sp orbital of one carbon with a half filled sp hybrid orbital of the other... 

Each carbon atom still has at this point an unhybridized 2p orbital available for bonding These two half filled 2p orbitals have their axes perpendicular to the frame work of CT bonds of the molecule and overlap m a side by side manner to give what is... 

In valence bond theory a covalent bond is described m terms of m phase overlap of a half filled orbital of one atom with a half filled orbital of another When applied to bonding m H2 the orbitals involved are the Is orbitals of two hydrogen atoms and the bond is a ct bond... 

Section 2 6 Bonding m methane is most often described by an orbital hybridization model which is a modified form of valence bond theory Four equiva lent sp hybrid orbitals of carbon are generated by mixing the 2s 2p 2py and 2p orbitals Overlap of each half filled sp hybrid orbital with a half filled hydrogen Is orbital gives a ct bond... 

Section 2 21 Carbon is sp hybridized m acetylene and the triple bond is of the ct + Tt + Tt type The 2s orbital and one of the 2p orbitals combine to give two equivalent sp orbitals that have their axes m a straight line A ct bond between the two carbons is supplemented by two tr bonds formed by overlap of the remaining half filled p orbitals... 

In addition to its three sp hybrid orbitals each carbon has a half filled 2p orbital that can participate m tt bonding Figure >b shows the continuous rr system that encompasses all of the carbons that result from overlap of these 2p orbitals The six tt electrons of benzene are delocalized over all six carbons... 

Each carbon in propane is bonded to four atoms and is sp hybridized The C—C bonds are a bonds involving overlap of a half filled sp hybrid orbital of one carbon with a half filled sp hybrid orbital of the other The C—H bonds are a bonds involving overlap of a half filled sp hybrid onbital of carbon with a half filled hydrogen li orbital... 

Hund s rule (Section 1 1) When two orbitals are of equal en ergy they are populated by electrons so that each is half filled before either one is doubly occupied Hybrid orbital (Section 2 6) An atomic orbital represented as a mixture of vanous contributions of that atom ss p d etc orbitals... 

Valence bond theory (Section 2 3) Theory of chemical bond mg based on overlap of half filled atomic orbitals between two atoms Orbital hybridization is an important element of valence bond theory... 

Atoms wifh a ground configuration in which an orbifal is exacfly half-filled, as for example in N(2/> ), Mn(3ti ) and Eu(4/ ), always have an S ground sfafe. Since such sfafes have only one componenf fhe problem of a normal or inverted mulfiplef does nof arise. Table 7.1 gives fhe ground sfafes of all atoms in fhe periodic fable. 

A common example of the Peieds distortion is the linear polyene, polyacetylene. A simple molecular orbital approach would predict S hybddization at each carbon and metallic behavior as a result of a half-filled delocalized TT-orbital along the chain. Uniform bond lengths would be expected (as in benzene) as a result of the delocalization. However, a Peieds distortion leads to alternating single and double bonds (Fig. 3) and the opening up of a band gap. As a result, undoped polyacetylene is a semiconductor. 

As the vessel is only about half filled with slurry, the disks become coated with the cake when immersed, the cake is dewatered when the disks emerge from the slurry, and scraped or blown off, by reverse blow, into the central conveyor which takes the cake to one end of the vessel. The planetary action and the slow movement of the disks through the feed slurry ensure exceptionally good homogeneity of the cake which is critically important for good dewatering characteristics the typical speed of rotation of the planetary system of shafts is from 0.8 to 1 rpm. 

The initial bond formation between the -> ir excited carbonyl compound and an alkene can occur by interaction of the half-filled n -orbital of the [I CO] with the ir-system of the alkene, in a sense transferring a tt-electron to the -orbital and making a bond between an alkene carbon and the carbonyl oxygen. In this process (common for electron rich olefins) the plane formed by the alkene carbons and their four substituents is perpendicular to the plane of the carbonyl groups and its two substituents (Figure 1). In the... 

Figure 4.20. Gruneisen parameter versus pressure for different regimes are indicated. Pluses indicate properties of stishovite phase, half-filled circles and closed circles indicate properties of high-density molten material, whereas open triangles and open circles and upper branch indicate behavior of coesitelike phase (Simakov and Trunin, 1990).

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