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The isomeric fulminate ion, CNO (Figure 3.5), can be drawn with three similar structures, but the resulting formal charges have larger magnitudes than in OCN. Because the order of electronegativities is C < N < O, none of these are ideal structures, and it is not surprising that this ion is unstable. The only common fu... | {
"Header 1": "3.1.2 Higher Electron Counts",
"Header 3": "CNO-",
"token_count": 318,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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A few molecules—such as BeF<sub>2</sub>, BeCl<sub>2</sub>, and BF<sub>3</sub>—seem to require multiple bonds to satisfy the octet rule for Be and B, even though multiple bonds for F and Cl are not generally expected on the basis of the high electronegativities of these halogens. Structures minimizing formal charges for... | {
"Header 1": "3.1.4 Multiple Bonds in Be and B Compounds",
"token_count": 2032,
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BeCl<sub>2</sub> dimerizes to a 3-coordinate structure in the vapor phase, but the linear monomer is formed at high temperatures. This monomeric structure is unstable due to the electronic deficiency at Be; in the dimer and the network formed in the solid-state, the halogen atoms share lone pairs with the Be atom in an... | {
"Header 1": "3.1.4 Multiple Bonds in Be and B Compounds",
"token_count": 361,
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**Valence shell electron-pair repulsion (VSEPR)** is an approach that provides a method for predicting the shape of molecules based on the electron-pair electrostatic repulsion described by Sidgwick and Powell4 in 1940 and further developed by Gillespie and Nyholm5 in 1957 and in the succeeding decades. Despite this me... | {
"Header 1": "3.2 **[Valence Shell Electron-Pair Repulsion](#page-4-0)**",
"token_count": 1216,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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Bonding models are useful only if their explanations are consistent with experimental data. New theories are continually being suggested and tested. Because we are working with such a wide variety of atoms and molecular structures, a single approach will unlikely work for all of them. Although the fundamental ideas of ... | {
"Header 1": "3.2.1 Lone-Pair Repulsion",
"token_count": 227,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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The isoelectronic molecules CH<sub>4</sub>, NH<sub>3</sub>, and H<sub>2</sub>O (Figure 3.10) illustrate the effect of lone pairs on molecular shape. Methane has four identical bonds between carbon and each of the hydrogens. When the four pairs of electrons are arranged as far from each other as possible, the result is ... | {
"Header 1": "3.2.1 Lone-Pair Repulsion",
"Header 3": "**Steric Number = 4**",
"token_count": 482,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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For the trigonal bipyramidal geometry, there are two unique locations for electron pairs, axial and equatorial. If there is a single lone pair, for example in SF<sub>4</sub>, the lone pair occupies an equatorial position. This position provides the lone pair with the most space and minimizes the interactions between th... | {
"Header 1": "3.2.1 Lone-Pair Repulsion",
"Header 3": "**Steric Number = 5**",
"token_count": 1040,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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In octahedral structures, all six positions are equivalent. When a single lone pair is present, it typically repels adjacent bonding pairs, reducing bond angles accordingly, as for IF<sub>5</sub> in Figure 3.13. In octahedron-based structures with two lone pairs, lone pair–lone pair repulsion is minimized if these pair... | {
"Header 1": "3.2.1 Lone-Pair Repulsion",
"Header 3": "Steric Numbers = 6 and 7",
"token_count": 520,
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SbF<sub>4</sub> has a single lone pair on Sb. Its structure is therefore similar to SF<sub>4</sub>, with a lone pair occupying an equatorial position. This lone pair causes considerable distortion, giving an F—Sb—F (axial positions) angle of 155° and an F—Sb—F (equatorial) angle of 90°.
SF<sub>5</sub><sup>-</sup> has... | {
"Header 1": "3.2.1 Lone-Pair Repulsion",
"Header 3": "**EXAMPLE 3.4**",
"token_count": 246,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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The VSEPR model considers double and triple bonds to have slightly greater repulsive effects than single bonds because of the repulsive effect of $\pi$ electrons that increase the electron density between the bonded atoms beyond that present in a $\sigma$ bond. For example, the $H_3C-C-CH_3$ angle in $(CH_3)_2C=... | {
"Header 1": "3.2.2 Multiple Bonds",
"token_count": 483,
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**HCP**, like HCN, is linear, with a triple bond: H - C = P:
**IOF**<sub>4</sub><sup>-</sup> has a single lone pair on the side opposite the oxygen. The lone pair has a slightly greater repulsive effect than the double bond to oxygen, as shown by the average O-I-F angle of 89°. (The extra repulsive character of the I... | {
"Header 1": "3.2.2 Multiple Bonds",
"Header 3": "**EXAMPLE 3.5**",
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Linus Pauling introduced the concept of electronegativity in the 1930s as a means of describing bond energies. Pauling recognized that polar bonds have higher bond energies than nonpolar bonds formed from the same elements. For example, he observed that the bond energy of HCl, 432 kJ/mol, was much higher than the avera... | {
"Header 1": "3.2.2 Multiple Bonds",
"Header 3": "**Electronegativity Scales**",
"token_count": 2041,
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By the VSEPR approach, trends in many bond angles can be explained by electronegativity. Consider the bond angles in the following molecules:
| Molecule | X–P–X Angle (°) | Molecule | X–S–X Angle (°) |
|----------|-----------------|----------|-----------------|
| PF3 | 97.8 | OSF2 | 92.3 ... | {
"Header 1": "3.2.2 Multiple Bonds",
"Header 3": "**Electronegativity and Bond Angles**",
"token_count": 1595,
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Which molecule has the smallest bond angle in each series?
OSeBr<sub>2</sub> (halogen–Se–halogen angle) **a.** OSeF<sub>2</sub> OSeCl<sub>2</sub>
**b.** SbCl<sub>3</sub> SbBr<sub>3</sub> c. $PI_3$ $AsI_3$ SbI<sub>3</sub>
#### **Effects of Size**
In the examples considered so far, the most electronegative atom... | {
"Header 1": "3.2.2 Multiple Bonds",
"Header 3": "EXERCISE 3.4",
"token_count": 445,
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For main group atoms having a steric number of 5, it is instructive to consider the relative bond lengths for axial and equatorial positions. For example, in PCl<sub>5</sub>, SF<sub>4</sub>, and ClF<sub>3</sub>, the central atom-axial distances are longer than the distances to equatorial atoms, as shown in Figure 3.17.... | {
"Header 1": "3.2.2 Multiple Bonds",
"Header 3": "**Molecules Having Steric Number = 5**",
"token_count": 1062,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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As in the case of individual atoms, numerous approaches have been taken to estimate electron-attracting abilities of groups such as CH<sub>3</sub>, CF<sub>3</sub>, and OH which may be bonded to central atoms. For example, a CF<sub>3</sub> group would be expected to attract electrons more strongly than a CH<sub>3</sub> ... | {
"Header 1": "3.2.2 Multiple Bonds",
"Header 3": "**Group Electronegativities**",
"token_count": 772,
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O
$$F_3C$$
$F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $F_3C$ $... | {
"Header 1": "3.2.2 Multiple Bonds",
"Header 3": "**Group Electronegativities**",
"token_count": 2064,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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The **ligand close-packing (LCP)** model developed by Gillespie<sup>23</sup> uses the distances between outer atoms in molecules as a guide to molecular shapes. For a series of molecules having the same central atom, the *non*bonded distances\* between the outer atoms are consistent, but the bond angles and bond length... | {
"Header 1": "3.2.4 Ligand Close Packing",
"token_count": 1063,
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The LCP model predicts that nonbonded atom-atom distances in molecules remain approximately the same, even if the bond angles around the central atom are changed. For example, the fluorine-fluorine distances in NF<sub>4</sub><sup>+</sup> and NF<sub>3</sub> are both 212 pm, even though the F—N—F bond angles are signific... | {
"Header 1": "3.2.4 Ligand Close Packing",
"Header 3": "**Ligand Close Packing and Bond Distances**",
"token_count": 546,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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In $PF_4^+$ the $F \cdots F$ and P - F distances are 238 pm and 145.7 pm, respectively. Predict the P—F distance in POF<sub>3</sub>, which has an F—P—F angle of 101.1°.
#### **SOLUTION**
The LCP model predicts that the $F \cdots F$ distances should be approximately the same in both structures. Sketches simila... | {
"Header 1": "3.2.4 Ligand Close Packing",
"Header 3": "**EXAMPLE 3.6**",
"token_count": 555,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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When atoms with different electronegativities are bonded, the resulting molecule has polar bonds, with the electrons of the bond concentrated, perhaps very slightly, on the more electronegative atom; the greater the difference in electronegativity, the more polar the bond. As a result, the bonds are dipolar, with relat... | {
"Header 1": "3.3 **[Molecular Polarity](#page-4-0)**",
"token_count": 792,
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| Molecule | Experimental (D) | Calculated from Vectors (D) |
|----------|------------------|-----------------------------|
| CH3Cl | 1.90 | 1.77 |
| CH2Cl2 | 1.60 | 2.008 |
| CHCl3 | 1.04 | 1.82 |
*Source*... | {
"Header 1": "3.3 **[Molecular Polarity](#page-4-0)**",
"Header 3": "**TABLE 3.8 Dipole Moments of Chloromethanes**",
"token_count": 591,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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Ammonia, water, and hydrogen fluoride all have much higher boiling points than other similar molecules, as shown in Figure 3.26. These high boiling points are caused by hydrogen bonds, in which hydrogen atoms bonded to nitrogen, oxygen, or fluorine also form weaker bonds to a lone pair of electrons on another nitrogen,... | {
"Header 1": "3.4 Hydrogen Bonding",
"token_count": 1304,
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- **1.** G. N. Lewis, *J. Am. Chem. Soc* ., **1916** , *38* , 762; *Valence and the Structure of Atoms and Molecules* , Chemical Catalogue Co., New York, 1923.
- **2.** L. Suidan, J. K. Badenhoop, E. D. Glendening, F. Weinhold, *J. Chem. Educ* ., **1995** , *72* , 583; J. Cioslowski, S. T. Mixon, *Inorg. Chem* ., **199... | {
"Header 1": "3.4 Hydrogen Bonding",
"Header 3": "**References**",
"token_count": 2006,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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Good sources for bond lengths and bond angles are the works of Wells, Greenwood and Earnshaw, and Cotton and Wilkinson cited in Chapter 1 . Reviews of electron-dot diagrams and formal charges can be found in most general chemistry texts. One of the best VSEPR references is still the early paper by R. J. Gillespie and R... | {
"Header 1": "3.4 Hydrogen Bonding",
"Header 3": "**[General References](#page-4-0)**",
"token_count": 527,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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**3.1** The dimethyldithiocarbamate ion, $[S_2CN(CH_3)_2]^-$ , has the following skeletal structure:
$$\begin{array}{cccccccccccccccccccccccccccccccccccc$$
- a. Give the important resonance structures of this ion, including any formal charges where necessary. Select the resonance structure likely to provide the be... | {
"Header 1": "3.4 Hydrogen Bonding",
"Header 3": "**Problems**",
"token_count": 1880,
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- a. PCl<sub>5</sub> is a stable molecule, but NCl<sub>5</sub> is not.
- **b.** $SF_4$ and $SF_6$ are known, but $OF_4$ and $OF_6$ are not.
- **3.18** X-ray crystal structures of ClOF<sub>3</sub> and BrOF<sub>3</sub> have been determined.
- **a.** Would you expect the lone pair on the central halogen to be axia... | {
"Header 1": "3.4 Hydrogen Bonding",
"Header 3": "**3.17** Explain the following:",
"token_count": 2006,
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and C-Te-C angles of 90.86(18)° and 91.73(18)°, respectively (Aboulkacem, S.; Naumann, D.; Tyrra, W.; Pantenburg, I. Organometallics, 2012, 31, 1559).
$$\begin{array}{cccccccccccccccccccccccccccccccccccc$$
- a. Explain why the angles are more acute for the Te compound relative to the Se compound.
- **b.** These ang... | {
"Header 1": "3.4 Hydrogen Bonding",
"Header 3": "**3.17** Explain the following:",
"token_count": 1856,
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Symmetry is a phenomenon of the natural world, as well as the world of human invention (**Figure 4.1**). In nature, many flowers and plants, snowflakes, insects, certain fruits and vegetables, and a wide variety of microscopic plants and animals exhibit characteristic symmetry. Many engin... | {
"Header 1": "[Symmetry and Group](#page-4-0) Theory",
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All molecules can be described in terms of their symmetry, even if it is only to say they have none. Molecules or any other objects may contain **symmetry elements** such as mirror planes, axes of rotation, and inversion centers. The actual reflection, rotation, or inversion is called a **symmetry operation**. To conta... | {
"Header 1": "4.1 **[Symmetry Elements and Operations](#page-4-0)**",
"token_count": 747,
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Structure of the lowest energy isomer of gas phase
$$TaB_{10}$$
$C_1$
$C_2$
$C_3$
$C_4$
$C_5$
$C_6$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C_7$
$C... | {
"Header 1": "4.1 **[Symmetry Elements and Operations](#page-4-0)**",
"token_count": 2035,
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Inversion results in two hydrogen atoms in the horizontal plane on the right and two hydrogen atoms in the vertical plane on the left. Inversion is therefore *not* a symmetry operation of methane, because the orientation of the molecule following the *i* operation differs from the original orientation.
![](_page_91_P... | {
"Header 1": "4.1 **[Symmetry Elements and Operations](#page-4-0)**",
"token_count": 1323,
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Staggered ethane has three mirror planes, each containing the CiC bond axis and passing through two hydrogens on opposite ends of the molecule. It has a *C*3 axis collinear with the carbon–carbon bond and three *C*2 axes bisecting the angles between the mirror planes. (Use of a model is helpful for viewing the *C*2 axe... | {
"Header 1": "4.1 **[Symmetry Elements and Operations](#page-4-0)**",
"Header 3": "**Ethane (staggered conformation)**",
"token_count": 210,
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Each molecule has a set of symmetry operations that describes the molecule's overall symmetry. This set of symmetry operations is called the **point group** of the molecule. **Group theory** , the mathematical treatment of the properties of groups, can be used to determine the molecular orbitals, vibrations, and other ... | {
"Header 1": "4.2 **[Point Groups](#page-4-0)**",
"token_count": 269,
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- **4.** Does the molecule have a mirror plane $(\sigma_h)$ perpendicular to the principal $C_n$ axis? If so, it is classified as $C_{nh}$ or $D_{nh}$ . If not, continue with Step 5.
- **5.** Does the molecule have any mirror planes that contain the... | {
"Header 1": "4.2 **[Point Groups](#page-4-0)**",
"Header 3": "H H O",
"token_count": 356,
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**1.** Determine whether the molecule belongs to one of the special cases of low or high symmetry.
Inspection of the molecule will determine if it fits one of the low-symmetry cases. These groups have few or no symmetry operations and are described in Table 4.2.
**TABLE 4.2 Groups of Low Symmetry**
| Group | Symm... | {
"Header 1": "4.2.1 Groups of Low and High Symmetry",
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Molecules with many symmetry operations may fit one of the high-symmetry cases of linear, tetrahedral, octahedral, or icosahedral symmetry with the characteristics described in Table 4.3. Molecules with very high symmetry are of two types, linear and polyhedral. Linear molecules having a center of inversion have $D_{\... | {
"Header 1": "4.2.1 Groups of Low and High Symmetry",
"Header 3": "**High Symmetry**",
"token_count": 556,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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<sup>3</sup><sup>+</sup> NH3, 1,5-dibromonaphthalene, H2O2, 1,3,5,7-tetrafl uorocyclooctatetraene
> Molecules with no perpendicular *C*<sup>2</sup> axes are in one of the groups designated by the letters *C* or *S* .
While point group assignments have not yet been made, the molecules are now divided into two major ... | {
"Header 1": "4.2.1 Groups of Low and High Symmetry",
"Header 3": "**Yes:** *D Groups* **No:** *C or S Groups*",
"token_count": 1926,
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No
$$\overline{D_n}$$
No $C_n$ or $S_{2n}$ [Co(en)<sub>3</sub>]<sup>3+</sup> is $D_3$ H<sub>2</sub>O<sub>2</sub>, 1,3,5,7-tetrafluorocyclooctatetraene
These molecules are in the simpler rotation groups *Dn*, *Cn*, and *S*2*n* because they do not have any mirror planes. *Dn* and *Cn* point groups have *only Cn... | {
"Header 1": "4.2.1 Groups of Low and High Symmetry",
"Header 3": "**Yes:** *D Groups* **No:** *C or S Groups*",
"token_count": 248,
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**Yes** *S*2*<sup>n</sup>* 1,3,5,7-tetrafl uorocyclooctatetraene is *S*<sup>4</sup>
**No** *Cn* H2O2 is C2
We have only one example in our list that falls into the *S*2*<sup>n</sup>* groups, as seen in Figure 4.12 . A flowchart that summarizes this point group assignment method is given in Figure 4.7 , and more exa... | {
"Header 1": "4.2.1 Groups of Low and High Symmetry",
"Header 3": "*D* **Groups** *C* **and** *S* **Groups**",
"token_count": 882,
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All molecules having these classifications must have a $C_n$ axis. If more than one $C_n$ axis is found, the highest order (principal) axis (largest value of n) is used as the reference axis. It is generally useful to orient this axis vertically.
| ... | {
"Header 1": "4.2.1 Groups of Low and High Symmetry",
"Header 3": "C Versus D Point Group Classifications",
"token_count": 390,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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The high-symmetry point groups $I_h$ , $O_h$ , and $T_d$ are ubiquitous in chemistry and are represented by the classic molecules C<sub>60</sub>, SF<sub>6</sub>, and CH<sub>4</sub>. For each of these point groups, there is a purely rotational subgroup (I, O, and T, respectively) in which the only symmetry operation... | {
"Header 1": "4.2.1 Groups of Low and High Symmetry",
"Header 3": "Groups Related to $I_h$ , $O_h$ , and $T_d$ Groups",
"token_count": 864,
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#### **Examples from Point Group**
$C_{3\nu}$ molecules (and *all* molecules) contain the identity operation E.
- 2. Each operation must have an **inverse** that, when combined with the operation, yields the identity operation (sometimes a symmetry operation may be its own inverse). Note: By convention, we perform... | {
"Header 1": "4.3 Properties and Representations of Groups",
"Header 3": "1. Each group must contain an **identity** operation that commutes (in other words, EA = AE) with all other members of the group and leaves them unchanged (EA = AE = A).",
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Important information about the symmetry aspects of point groups is summarized in character tables, described later in this chapter. To understand the construction and use of character tables, we must consider the properties of matrices, which are the basis for the tables.\*
A matrix is an ordered array of numbers, s... | {
"Header 1": "4.3.1 Matrices",
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$$k \qquad j \qquad j \qquad j \\ i \begin{bmatrix} 1 & 5 \\ 2 & 6 \end{bmatrix} \times \begin{bmatrix} 7 & 3 \\ 4 & 8 \end{bmatrix} k = \begin{bmatrix} (1)(7) + (5)(4) & (1)(3) + (5)(8) \\ (2)(7) + (6)(4) & (2)(3) + (6)(8) \end{bmatrix} i = \begin{bmatrix} 27 & 43 \\ 38 & 54 \end{bmatrix} i$$
This example has two ro... | {
"Header 1": "4.3.1 Matrices",
"Header 3": "**EXAMPLE 4.3**",
"token_count": 854,
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We will now consider how the *C*2*<sup>v</sup>* point group symmetry operations transform a set of *x* , *y* , and *z* coordinates. The water molecule possesses *C*2*<sup>v</sup>* symmetry. It has a *C*2 axis through the oxygen and in the plane of the molecule, no perpendicular *C*2 axes, and no horizontal mirror plane... | {
"Header 1": "4.3.2 **[Representations of Point Groups](#page-4-0)**",
"Header 3": "**Symmetry Operations: Matrix Representations**",
"token_count": 1203,
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Verify the transformation matrices for the E and $\sigma_{\nu}'(yz)$ operations of the $C_{2\nu}$ point
This set of matrices satisfies the properties of a mathematical group. We call this a matrix **representation** of the $C_{2y}$ point group. This representation is a set of matrices, each corresponding to an ... | {
"Header 1": "4.3.2 **[Representations of Point Groups](#page-4-0)**",
"Header 3": "**EXERCISE 4.5**",
"token_count": 398,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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Each transformation matrix in the *C*2*<sup>v</sup>* set can be "block diagonalized"; that is, it can be broken down into smaller matrices along the diagonal, with all other matrix elements equal to zero:
$$E: \begin{bmatrix} [1] & 0 & 0 \\ 0 & [1] & 0 \\ 0 & 0 & [1] \end{bmatrix} C_2: \begin{bmatrix} [-1] & 0 & 0 \\... | {
"Header 1": "4.3.2 **[Representations of Point Groups](#page-4-0)**",
"Header 3": "**Reducible and Irreducible Representations**",
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Three of the representations for $C_{2\nu}$ , labeled $A_1, B_1$ , and $B_2$ , have now been determined. The fourth, called $A_2$ , can be found by using the group properties described in **Table 4.7**. A complete set of irreducible representations for a point group is called the character table for that group. The... | {
"Header 1": "4.3.3 Character Tables",
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Full descriptions of the matrices for the operations in this group will not be given, but the characters can be found by using the properties of a group. Consider the $C_3$ rotation shown in Figure 4.16. Counterclockwise rotation of $120^{\circ}$ results in a new x' and y' as shown, which can be described in terms ... | {
"Header 1": "4.3.3 Character Tables",
"Header 3": "**Another Example:** $C_{3v}(NH_3)$",
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$$E: \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
$$C_2: \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
$$\sigma_{v}(xz): \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
$$E : \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \qqua... | {
"Header 1": "4.3.3 Character Tables",
"Header 3": "**Reducible Matrix Representations**",
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$$\begin{bmatrix} [1] & 0 & 0 \\ 0 & [1] & 0 \\ 0 & 0 & [1] \end{bmatrix}$$
$$\begin{bmatrix} [1] & 0 & 0 \\ 0 & [1] & 0 \\ 0 & 0 & [1] \end{bmatrix} \qquad \begin{bmatrix} [-1] & 0 & 0 \\ 0 & [-1] & 0 \\ 0 & 0 & [1] \end{bmatrix} \qquad \begin{bmatrix} [1] & 0 & 0 \\ 0 & [-1] & 0 \\ 0 & 0 & [1] \end{bmatrix} \qquad ... | {
"Header 1": "4.3.3 Character Tables",
"Header 3": "**Block Diagonalized Matrices**",
"token_count": 307,
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| $C_{2v}$ | E | $C_2$ | $\sigma_v(xz)$ | $\sigma_{v}{}'(yz)$ | Matching Functions | |
|----------|---|-------|----------------|---------------------|--------------------|-----------------|
| $A_1$ | 1 | 1 | 1 | 1 | z | $x^2, y^2, z^2$ |
| $A_2$ ... | {
"Header 1": "4.3.3 Character Tables",
"Header 3": "**Character Table**",
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"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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Number of irreducible representations | $3(A_1, A_2, E)$ ... | {
"Header 1": "4.3.3 Character Tables",
"Header 3": "**Character Table**",
"token_count": 966,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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Which point groups are possible for chiral molecules? (Hint: refer to the character tables in Appendix C.)
Air blowing past the stationary propellers in Figure 4.18 will be rotated in either a clockwise or counterclockwise direction. By analogy, plane-polarized light will be rotated on passing through chiral molecule... | {
"Header 1": "4.4.1 Chirality",
"Header 3": "**EXERCISE 4.7**",
"token_count": 323,
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Because the study of vibrations is the study of motion of the atoms in a molecule, we must first attach a set of x, y, and z coordinates to each atom. For convenience, we assign the z axes parallel to the $C_2$ axis of the molecule, the x axes in the plane of the molecule, and the y axes perpendicular to the plane (F... | {
"Header 1": "4.4.1 Chirality",
"Header 3": "Water (C<sub>2v</sub> Symmetry)",
"token_count": 1211,
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Write the corresponding $9 \times 9$ transformation matrices for the $\sigma(xz)$ and $\sigma(yz)$ operations in $C_{2\nu}$ symmetry.
Because all nine direction vectors are included in this representation, it represents all the motions of the molecule: three translations, three rotations, and (by difference) ... | {
"Header 1": "4.4.1 Chirality",
"Header 3": "**EXERCISE 4.8**",
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The next step is to determine how the irreducible representations sum to give the reducible representation. This requires another property of groups. The number of times any irreducible representation contributes to a reducible representation is equal to the sum of the products of the characters of the reducible and ir... | {
"Header 1": "4.4.1 Chirality",
"Header 3": "**Reducing Representations to Irreducible Representations**",
"token_count": 1176,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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Using the x, y, and z coordinates for each atom in $XeF_4$ , determine the reducible representation for all molecular motions; reduce this representation to its irreducible components; and classify these representations into translational, rotational, and vibrational modes.
First, it is useful to assign x, y, and z ... | {
"Header 1": "4.4.1 Chirality",
"Header 3": "**EXAMPLE 4.4**",
"token_count": 1492,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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$$n_{A_g} = \frac{1}{4}[(1)(4)(1) + (1)(0)(1) + (1)(2)(1) + (1)(2)(1)] = 2$$
$$n_{B_g} = \frac{1}{4}[(1)(4)(1) + (1)(0)(-1) + (1)(2)(1) + (1)(2)(-1)] = 1$$
$$n_{A_u} = \frac{1}{4}[(1)(4)(1) + (1)(0)(1) + (1)(2)(-1) + (1)(2)(-1)] = 0$$
$$n_{B_u} = \frac{1}{4}[(1)(4)(1) + (1)(0)(-1) + (1)(2)(-1) + (1)(2)(1)] = 1$$ ... | {
"Header 1": "4.4.1 Chirality",
"Header 3": "**SOLUTION**",
"token_count": 702,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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It is often useful to consider a particular vibrational mode for a compound. For example, useful information often can be obtained from the CiO stretching bands in infrared spectra of metal complexes containing CO (carbonyl) ligands. The following example of *cis* - and *trans* -dicarbonyl square planar complexes shows... | {
"Header 1": "4.4.1 Chirality",
"Header 3": "**Selected Vibrational Modes**",
"token_count": 1367,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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Absorbance 1.0 0.8 0.6 0.4 0.2 -0.0
2400
Determine the number of IR-active CO stretching modes for fac-Mo(CO)<sub>3</sub>(CH<sub>3</sub>CH<sub>2</sub>CN)<sub>3</sub>, as shown in the margin.
This molecule has $C_{3\nu}$ symmetry. The operations to be considered are E, $C_3$ , and $\sigma_{\nu}$ . E leaves the... | {
"Header 1": "4.4.1 Chirality",
"Header 3": "**EXAMPLE 4.6**",
"token_count": 486,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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This spectroscopic method uses a different approach to observe molecular vibrations. Rather than directly observing absorption of infrared radiation as in IR spectroscopy, in Raman spectroscopy higher energy radiation, ordinarily from a laser, excites molecules to higher electronic states, envisioned as short-lived "vi... | {
"Header 1": "4.4.1 Chirality",
"Header 3": "Raman Spectroscopy",
"token_count": 786,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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There are several helpful books on molecular symmetry and its applications. Good examples are D. J. Willock, Molecular Symmetry, John Wiley & Sons, Chichester, UK, 2009; F. A. Cotton, Chemical Applications of Group Theory, 3rd ed., John Wiley & Sons, New York, 1990; S. F. A. Kettle, Symmetry and Structure: Readable Gro... | {
"Header 1": "4.4.1 Chirality",
"Header 3": "**General References**",
"token_count": 231,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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- **4.1** Determine the point groups for
- a. Ethane (staggered conformation)
- **b.** Ethane (eclipsed conformation)
- **c.** Chloroethane (staggered conformation)
- **d.** 1,2-Dichloroethane (staggered *anti* conformation)
- **4.2** Determine the point groups for
- a. Ethylene
- **b.** Chloroethylene
- **c.** The pos... | {
"Header 1": "4.4.1 Chirality",
"Header 3": "**Problems**",
"token_count": 1835,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
} |
- **e.** Central design on the Ethiopian flag:
- **f.** Turkey
- **g.** Japan
- **h.** Switzerland
- **i.** United Kingdom (be careful!)
- **4.20** For *trans*-1,2-dichloroethylene, which has *C*2*h* symmetry,
- **a.** List all the symmetry operations ... | {
"Header 1": "4.4.1 Chirality",
"Header 3": "**d.** Field of stars in flag of Micronesia",
"token_count": 1962,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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McIndoe, *Dalton Trans.*, 2006, 4570.) Can these data be used to unambiguously establish whether the PPh<sub>3</sub> ligand is bound in either an equatorial or axial site in this trigonal bipyramidal complex? Support your decision by determining the number of IR-active CO stretching modes for these isomers.
- 4.34 Disu... | {
"Header 1": "4.4.1 Chirality",
"Header 3": "**d.** Field of stars in flag of Micronesia",
"token_count": 1925,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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Molecular orbital theory uses group theory to describe the bonding in molecules; it complements and extends the introductory bonding models in Chapter 3. In molecular orbital theory the symmetry properties and relative energies of atomic orbitals determine... | {
"Header 1": "[Molecular Orbitals](#page-5-0)",
"token_count": 319,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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As with atomic orbitals, Schrödinger equations can be written for electrons in molecules. Approximate solutions to these molecular Schrödinger equations can be constructed from **linear combinations of atomic orbitals (LCAO)**, the sums and differences of the atomic wave functions. For diatomic molecules such as H2, su... | {
"Header 1": "5.1 **[Formation of Molecular Orbitals from Atomic Orbitals](#page-5-0)**",
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Consider the interactions between two *s* orbitals, as in H2. For convenience, we label the atoms of a diatomic molecule *a* and *b*, so the atomic orbital wave functions are c(1*sa*) and c(1*sb*). We can visualize the two atoms approaching each other, until their electron clouds overlap and merge into larger molecular... | {
"Header 1": "5.1.1 **[Molecular Orbitals from](#page-5-0)** *s* **Orbitals**",
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Molecular orbitals formed from p orbitals are more complex since each p orbital contains separate regions with opposite signs of the wave function. When two orbitals overlap, and the overlapping regions have the same sign, the sum of the two orbitals has an increased electron probability in the overlap region. When two... | {
"Header 1": "5.1.1 **[Molecular Orbitals from](#page-5-0)** *s* **Orbitals**",
"Header 3": "5.1.2 Molecular Orbitals from p Orbitals",
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In the heavier elements, particularly the transition metals, *d* orbitals can be involved in bonding. **Figure 5.3** shows the possible combinations. When the *z* axes are collinear, two *dz* <sup>2</sup> orbitals can combine end-on for s bonding. The *dxz* and *dyz* orbitals form p orbitals. When atomic orbitals meet ... | {
"Header 1": "5.1.3 **[Molecular Orbitals from](#page-5-0)** *d* **Orbitals**",
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Sketch the overlap regions of the following combination of orbitals, all with collinear *z* axes, and classify the interactions.
**EXERCISE 5.1** Repeat the process in the preceding example for the following orbital combinations, again using collinear *z* axes.
$$p_x$$
and $d_{xz}$ ... | {
"Header 1": "5.1.3 **[Molecular Orbitals from](#page-5-0)** *d* **Orbitals**",
"Header 3": "**E X A M P L E 5 .1**",
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As mentioned previously, nonbonding molecular orbitals have energies essentially equal to that of atomic orbitals. These can form in larger molecules, for example when there are three atomic orbitals of the same symmetry and similar energies, a situation that requires the formation of three molecular orbitals. Most com... | {
"Header 1": "5.1.4 **[Nonbonding Orbitals and Other Factors](#page-5-0)**",
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Although apparently satisfactory Lewis electron-dot structures of N2, O2, and F2 can be drawn, the same is not true with Li2, Be2, B2, and C2, which violate the octet rule. In addition, the Lewis structure of O2 predicts a double-bonded, diamagnetic (all electrons paired) molecule ( OO ), but experiment has shown O2 to... | {
"Header 1": "5.2.1 **[Molecular Orbitals](#page-5-0)**",
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In Figure 5.5, we only considered interactions between atomic orbitals of identical energy. However, atomic orbitals with similar, but unequal, energies can interact if they have appropriate symmetries. We now outline two approaches to analyzing this phenomenon, one in which we first consider the atomic orbitals that c... | {
"Header 1": "5.2.2 Orbital Mixing",
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Before proceeding with examples of homonuclear diatomic molecules, we must define two types of magnetic behavior, **paramagnetic** and **diamagnetic** . Paramagnetic compounds are attracted by an external magnetic field. This attraction is a consequence of one or more unpaired electrons behaving as tiny magnets. Diamag... | {
"Header 1": "5.2.3 **[Diatomic Molecules of the First and Second Periods](#page-5-0)**",
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Here is an example in which the MO model has a distinct advantage over the Lewis dot model. B<sub>2</sub> is a gas-phase species; solid boron exists in several forms with complex bonding, primarily involving B<sub>12</sub> icosahedra.
B<sub>2</sub> is paramagnetic. This behavior can be explained if its two highest en... | {
"Header 1": "$B_2[\\pi_u^{-1}\\pi_u^{-1}(2p)]$",
"token_count": 592,
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The MO model of C<sub>2</sub> predicts a doubly bonded molecule, with all electrons paired, but with both highest occupied molecular orbitals (HOMOs) having $\pi$ symmetry. C<sub>2</sub> is unusual because it has two $\pi$ bonds and no $\sigma$ bond. Although $C_2$ is a rarely encountered allotrope of carbon (c... | {
"Header 1": "$B_2[\\pi_u^{-1}\\pi_u^{-1}(2p)]$",
"Header 3": "$C_{2}[\\pi_{u}^{2}\\pi_{u}^{2}(2p)]$",
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N<sub>2</sub> has a triple bond according to both the Lewis and the molecular orbital models. This agrees with its very short N-N distance (109.8 pm) and extremely high bonddissociation energy (942 kJ/mol). Atomic orbitals decrease in energy with increasing nuclear charge Z as discussed in Section 2.2.4, and further de... | {
"Header 1": "$B_2[\\pi_u^{-1}\\pi_u^{-1}(2p)]$",
"Header 3": "$N_2[\\pi_u^2\\pi_u^2\\sigma_g^2(2p)]$",
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O<sub>2</sub> is paramagnetic. As for B<sub>2</sub>, this property cannot be explained by the Lewis dot structure :0=0:, but it is evident from the MO picture, which assigns two electrons to the degenerate $\pi_g^{\ *}$ orbitals. The paramagnetism can be demonstrated by pouring liquid O2 between the poles of a strong... | {
"Header 1": "$O_2[\\sigma_q^2 \\pi_u^2 \\pi_u^2 \\pi_q^{*1} \\pi_q^{*1}(2p)]$",
"token_count": 888,
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**Figure 5.8** shows the variation of bond distance with the number of valence electrons in second-period *p* -block homonuclear diatomic molecules having 6 to 14 valence electrons. Beginning at the left, as the number of electrons increases the number in bonding orbitals also increases; the bond strength becomes great... | {
"Header 1": "$O_2[\\sigma_q^2 \\pi_u^2 \\pi_u^2 \\pi_q^{*1} \\pi_q^{*1}(2p)]$",
"Header 3": "**Bond Lengths in Homonuclear Diatomic Molecules**",
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In addition to data on bond distances and energies, specific information about the energies of electrons in orbitals can be determined from photoelectron spectroscopy.<sup>6</sup> In this technique, ultraviolet (UV) light or X-rays eject electrons from molecules:
$$O_2 + h\nu(photons) \rightarrow O_2^+ + e^-$$
The ... | {
"Header 1": "5.2.4 Photoelectron Spectroscopy",
"token_count": 1639,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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The application of molecular orbital theory to heteronuclear diatomic molecules is similar to its application to homonuclear diatomics, but the different nuclear charges of the atoms require that interactions occur between orbitals of unequal energies and shifts the resulting molecular orbital energies. In dealing with... | {
"Header 1": "5.3.1 **[Polar Bonds](#page-5-0)**",
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The most efficient approach to bonding in heteronuclear diatomic molecules employs the same strategy as for homonuclear diatomics with one exception: the more electronegative element has atomic orbitals at lower potential energies than the less electronegative element. Carbon monoxide, shown in Figure 5.13 , shows this... | {
"Header 1": "5.3.1 **[Polar Bonds](#page-5-0)**",
"Header 3": "**Carbon Monoxide**",
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Molecular orbitals for HF can be found by using the approach used for CO. The 2*s* orbital of the fluorine atom is more than 26 eV lower than that of the hydrogen 1*s*, so there is very little interaction between these orbitals. The fluorine 2*pz* orbital (-18.65 eV) and the hydrogen 1*s* (-13.61 eV), on the other hand... | {
"Header 1": "5.3.1 **[Polar Bonds](#page-5-0)**",
"Header 3": "**E X A M P L E 5 . 3**",
"token_count": 847,
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Ionic compounds can be considered the limiting form of polarity in heteronuclear diatomic molecules. As mentioned previously, as the atoms forming bonds differ more in electronegativity, the energy gap between the interacting atomic orbitals also increases, and the concentration of electron density is increasingly bias... | {
"Header 1": "5.3.2 **[Ionic Compounds and Molecular Orbitals](#page-5-0)**",
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The linear ion FHF-, an example of very strong hydrogen bonding that can be described as a covalent interaction, 9 provides a convenient introduction to the concept of **group orbitals** , collections of matching orbitals on outer atoms. To generate a set of group orbitals, we will use the valence orbitals of the fluor... | {
"Header 1": "[5.4.1](#page-5-0) **FHF** -",
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**FIGURE 5.16** Interaction of Fluorine Group Orbitals with the Hydrogen 1 s Orbital.
The central hydrogen atom in FHF-, with only its 1*s* orbital available for bonding, is only eligible on the basis of its symmetry to interact with group orbitals 1 and ... | {
"Header 1": "[5.4.1](#page-5-0) **FHF** -",
"Header 3": "Atomic Orbitals Used Group Orbitals",
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The approach used so far can be applied to other linear species—such as CO2, N3 -, and BeH2 —to consider how molecular orbitals can be constructed on the basis of interactions of group orbitals with central atom orbitals. However, we also need a method to understand the bonding in more complex molecules. We will first ... | {
"Header 1": "[5.4.2](#page-5-0) **CO2**",
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Using the *D*2*<sup>h</sup>* character table shown, verify that the group orbitals in Figure 5.18 match their irreducible representations.
| $D_{2h}$ | Ε | $C_2(z)$ | $C_2(y)$ | $C_2(x)$ | i | $\sigma(xy)$ | $\sigma(xz)$ | $\sigma(yz)$ | | |
|----------|---|----------|----------|----------|----... | {
"Header 1": "[5.4.2](#page-5-0) **CO2**",
"Header 3": "**E X E R C I S E 5 . 5**",
"token_count": 1307,
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Using orbital potential energies, show that group orbital 4 is more likely than group orbital 2 to interact strongly with the $2p_z$ orbital of carbon.
The $2p_y$ orbital of carbon has $B_{2u}$ symmetry and interacts with group orbital 5 (Figure 5.23). The result is the formation of two $\pi$ molecular orbita... | {
"Header 1": "[5.4.2](#page-5-0) **CO2**",
"Header 3": "**EXERCISE 5.6**",
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Molecular orbitals of nonlinear molecules can be determined by similar procedures. Water is a useful example:
- 1. Water is a bent molecule with a $C_2$ axis through the oxygen and two mirror planes that intersect along this axis (**Figure 5.26**). The point group is $C_{2\nu}$ .
- **2.** The $C_2$ axis is chose... | {
"Header 1": "5.4.3 **H<sub>2</sub>O**",
"token_count": 363,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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| $C_{2v}$ | E | $C_2$ | $\sigma_{v}(xz)$ | $\sigma_{v}'(yz)$ | | |
|----------|---|-------|------------------|-------------------|----------|-----------------|
| $A_1$ | 1 | 1 | 1 | 1 | z | $x^2, y^2, z^2$ |
| $A_2$ | 1 | 1 | -1 ... | {
"Header 1": "5.4.3 **H<sub>2</sub>O**",
"Header 3": "$C_{2\\nu}$ Character Table",
"token_count": 600,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
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| | r | | 11- | | |
|----------|---|-------|------------------|---------------------|---|
| $C_{2v}$ | Ε | $C_2$ | $\sigma_{v}(xz)$ | $\sigma_{v}{}'(yz)$ | |
| Γ | 2 | 0 | 2 | 0 | |
| $A_1$ | 1 | 1 | 1 ... | {
"Header 1": "5.4.3 **H<sub>2</sub>O**",
"Header 3": "The reducible representation $\\Gamma = A_1 + B_1$ :",
"token_count": 2035,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
} |
The VSEPR approach describes ammonia as a pyramidal molecule with a lone pair of electrons and $C_{3\nu}$ symmetry. To obtain a molecular orbital description of $NH_3$ , it is convenient to view this molecule down the $C_3$ , or z, axis and with the yz plane passing through one of the hydrogen atoms, as shown in **... | {
"Header 1": "5.4.4 **NH**<sub>3</sub>",
"token_count": 1248,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
} |
The same general SALCs are obtained regardless of the initial atomic orbital examined. Show that if hydrogen 1s orbital $H_b$ is chosen as the basis (instead of $H_a$ ), the resulting $A_1$ and $A_2$ linear combinations would be identical to those shown previously, and the E linear combination would feature the ... | {
"Header 1": "5.4.4 **NH**<sub>3</sub>",
"Header 3": "**EXERCISE 5.9**",
"token_count": 2025,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
} |
The 1s orbital energies (-13.61 eV) of the hydrogen atoms match well with the energies of the nitrogen 2p orbitals (-13.18 eV), resulting in large differences between the bonding and antibonding orbital energies. The nitrogen 2s has such a sufficiently low energy (-25.56 eV) that its interaction with the hydrogen orbit... | {
"Header 1": "5.4.4 **NH**<sub>3</sub>",
"Header 3": "**EXERCISE 5.9**",
"token_count": 283,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
} |
Section 5.4.2 outlines the process for determining group orbitals in the linear case where the outer atoms employ both s and p orbitals; Figure 5.18 illustrates the group orbitals comprised of $2p_x$ , $2p_y$ , $2p_z$ , and 2s orbitals, respectively. While these orbitals can be deduced by
![](_page_169_Figure_10.j... | {
"Header 1": "5.4.5 CO<sub>2</sub> Revisited with Projection Operators",
"token_count": 1711,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
} |
| Original Orbital | E | C2(z)<br>- | C2(y)<br>- | C2(x) | i<br>- | s(xy) | s(xz) | s(yz)<br>- |
|---------------------|--------------|-------------------------|--------------|-------------------|-----------... | {
"Header 1": "5.4.5 CO<sub>2</sub> Revisited with Projection Operators",
"token_count": 472,
"source_pdf": "datasets/websources/biochem/inorganic-chemistry-g-l-miessler-2014.pdf"
} |
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