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Atomic term symbols are sometimes called spectroscopic term symbols because atomic spectral lines can be assigned to transitions between states that are described by atomic term symbols. For example, the first few electronic states of atomic hydrogen are given in Table 8.5. The electron configuration 1s gives the term ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–8.** A Term Symbol Gives a Detailed Description of an Electron Configuration",
"Header 3": "8-11. Atomic Term Symbols Are Used to Describe Atomic Spectra",
"token_count": 1973,
"source_pdf": "datasets/websources/bioc... |
(The rule *!1L* = ±1
follows from the principle of conservation of angular momentum because a photon has a spin angular momentum of 1i.)
The selection rules given in Equations 8.57 tell us that 2 P --+ <sup>2</sup> S transitions are allowed, but that <sup>2</sup> S --+ <sup>2</sup> S transitions are not allowed bec... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–8.** A Term Symbol Gives a Detailed Description of an Electron Configuration",
"Header 3": "8-11. Atomic Term Symbols Are Used to Describe Atomic Spectra",
"token_count": 2003,
"source_pdf": "datasets/websources/bioc... |
63 | 5 <i>q</i> | 5g 2G | { 3½<br>4½ | 37060, 2 | | |
| 5 <b>s</b> | 58 2S | 3/2 | 33200. 696 | | _ | 6p 2P° | 1 | 37296, 51 | | |
| 4 <i>d</i> | 4d 2D ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**8–8.** A Term Symbol Gives a Detailed Description of an Electron Configuration",
"Header 3": "8-11. Atomic Term Symbols Are Used to Describe Atomic Spectra",
"token_count": 1294,
"source_pdf": "datasets/websources/bioc... |
8-1. Show that the atomic unit of energy can be written as
$$E_{\rm h} = \frac{\hbar^2}{m_{\rm e} a_0^2} = \frac{e^2}{4\pi\,\varepsilon_0 a_0} = \frac{m_{\rm e} e^4}{16\pi^2 \varepsilon_0^2 \hbar^2}$$
- **8-2.** Show that the energy of a helium ion in atomic units is *-2Eh.*
- 8-3. The electric potential at a dista... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
"token_count": 2020,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Argue that the normalization constant of an N x N Slater determinant of orthonormal spin orbitals is 1 I ../NT.
- 8-22. The total *z* component of the spin angular momentum operator for an N -electron system is <sup>N</sup>
$$\hat{S}_{z,\text{total}} = \sum_{i=1}^{N} \hat{S}_{zj}$$
Show that both
$$\psi = \frac{1... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
"token_count": 2043,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Show that
$$E_0 = -\frac{5}{2}E_{\rm h}$$
Explain why J is called an atomic Coulombic integral and K is called an atomic exchange integral.
Even though the above secular determinant is $4 \times 4$ and appears to give a fourth-degree polynomial in E, note that it really consists of two $1 \times 1$ blocks and... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
"token_count": 1968,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
(We have essentially done this problem in Problem 7–30.)
$$V^{
m eff}({f r}_1) = rac{Z^3}{\pi} \int d{f r}_2 rac{e^{-2Zr_2}}{r_{12}}$$
As in Problem 7–30, we use the law of cosines to write
$$r_{12} = (r_1^2 + r_2^2 - 2r_1r_2\cos\theta)^{1/2}$$
and so $V^{\text{eff}}$ becomes
$$V^{\text{eff}}(r_1) = \frac{Z... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
"token_count": 1690,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
**8-49.** The spin operators, *Sx, SY,* and S,, like all angular momentum operators, obey the commutation relations (Problem 6-13)
$$[\hat{S}_x, \hat{S}_y] = i\hbar \hat{S}_z \quad [\hat{S}_y, \hat{S}_z] = i\hbar \hat{S}_x \quad [\hat{S}_z, \hat{S}_x] = i\hbar \hat{S}_y$$
Define the (non-Hermitian) operators
$$... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
"token_count": 1609,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Because $\hat{S}_{\text{total}} = \hat{S}_1 + \hat{S}_2$ , we have
$$\begin{split} \hat{S}_{\text{total}}^2 &= (\hat{\mathbf{S}}_1 + \hat{\mathbf{S}}_2) \cdot (\hat{\mathbf{S}}_1 + \hat{\mathbf{S}}_2) = \hat{S}_1^2 + \hat{S}_2^2 + 2\hat{\mathbf{S}}_1 \cdot \hat{\mathbf{S}}_2 \\ &= \hat{S}_1^2 + \hat{S}_2^2 + 2(\hat{... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**Problems**",
"token_count": 1559,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
For simplicity, let's consider the simplest neutral molecule, H2 • The Hamiltonian operator for a hydrogen molecule is given by
$$\begin{split} \hat{H} &= -\frac{\hbar^2}{2M} (\nabla_{\rm A}^2 + \nabla_{\rm B}^2) - \frac{\hbar^2}{2m_{\rm e}} (\nabla_{\rm 1}^2 + \nabla_{\rm 2}^2) - \frac{e^2}{4\pi\varepsilon_0 r_{\rm ... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**9-1.** The Born-Oppenheimer Approximation Simplifies the Schrodinger Equation for Molecules",
"token_count": 1032,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The method we will use to describe the bonding properties of molecules is called *molecular-orbital theory.* Molecular-orbital theory was developed in the early 1930s and is now the most commonly used method to calculate molecular properties. In molecular-orbital theory, we construct molecular wave functions in a manne... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "9-2. Hi Is the Prototypical Species of Molecular-Orbital Theory",
"token_count": 1056,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Let's use 1/1 + given in Equation 9.6 to calculate the energy of Hi as a function of the internuclear separation, *R.* (Problem 9-5 has you calculate the energy associated with 1/1\_.) To determine *E* +, the energy associated with 1/J +, we start with Equation 9.5:
$$\hat{H}\psi_{+}(\mathbf{r};R) = E_{+}\psi_{+}(\ma... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "9-2. Hi Is the Prototypical Species of Molecular-Orbital Theory",
"Header 3": "**9-3.** The Overlap Integral Is a Quantitative Measure of the Overlap of Atomic Orbitals Situated on Different Atoms",
"token_count": 1486,
... |
So far, we have evaluated the denominator of Equation 9.7. The evaluation of the numerator is more complicated. Using Equation 9.4 for the Hamiltonian operator, we obtain
$$\int d\mathbf{r} \psi_{+}^{*} \hat{H} \psi_{+} = \int d\mathbf{r} (1s_{A}^{*} + 1s_{B}^{*}) \hat{H} (1s_{A} + 1s_{B})$$
$$= \int d\mathbf{r} (1... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "9-2. Hi Is the Prototypical Species of Molecular-Orbital Theory",
"Header 3": "**9-4.** The Stability of a Chemical Bond Is a Quantum-Mechanical Effect",
"token_count": 1918,
"source_pdf": "datasets/websources/biochem/F8... |
The integrals, J and K, can be evaluated analytically (Problem 9-6) and the results are
$$J = e^{-2R} \left( 1 + \frac{1}{R} \right) \tag{9.23}$$
and
$$K = \frac{S}{R} - e^{-R}(1+R) \tag{9.24}$$
Figure 9.6 shows a plot of the energy!),.£+= E+- <sup>E</sup>*<sup>1</sup> <sup>s</sup>*of Hi as a function of the in... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "9-2. Hi Is the Prototypical Species of Molecular-Orbital Theory",
"Header 3": "**9-4.** The Stability of a Chemical Bond Is a Quantum-Mechanical Effect",
"token_count": 603,
"source_pdf": "datasets/websources/biochem/F81... |
The two molecular orbitals $\psi_+$ and $\psi_-$ describe quite different states. The orbital $\psi_+$ describes a state that exhibits a stable chemical bond and is called a *bonding orbital*. The other possible linear combination of the two 1s atomic orbitals is
$$\psi_{-} = c_1 1 s_{\rm A} - c_2 1 s_{\rm B} \... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**9–5.** The Simplest Molecular Orbital Treatment of H<sub>2</sub><sup>+</sup> Yields a Bonding Orbital and an Antibonding Orbital",
"token_count": 1842,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
... |
Because $\psi_b$ is the molecular orbital corresponding to the ground-state energy of $H_2^+$ , we can describe the ground state of $H_2$ by placing two electrons with opposite spins in $\psi_b$ , just as we place two electrons in a 1s atomic orbital to describe the helium atom. The Slater determinant correspondi... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**9–6.** A Simple Molecular-Orbital Treatment of H<sub>2</sub> Places Both Electrons in a Bonding Orbital",
"token_count": 835,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
In this section, we will construct a set of of molecular orbitals and assign electrons to them in accord with the Pauli Exclusion Principle. This procedure will generate electron configurations for molecules similar to those discussed for atoms in Chapter 8. We will
The ground-state ene... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**9–6.** A Simple Molecular-Orbital Treatment of H<sub>2</sub> Places Both Electrons in a Bonding Orbital",
"Header 3": "9-7. Molecular Orbitals Can Be Ordered According to Their Energies",
"token_count": 2026,
"source_p... |
Because the *2p* and *2p* orbitals U*y g y* X *y* have identical energy and the resulting molecular orbitals differ only in their spatial orientation, the pairs of orbitals, *n: 2p* , *n: 2p* and *n: 2p* , *n: 2p* , are degenerate. Note *U X U y* g *X* g *<sup>y</sup>* :.· that unlike the bonding *a* orbitals, the bon... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**9–6.** A Simple Molecular-Orbital Treatment of H<sub>2</sub> Places Both Electrons in a Bonding Orbital",
"Header 3": "9-7. Molecular Orbitals Can Be Ordered According to Their Energies",
"token_count": 521,
"source_pd... |
For H2 through He2, we need to consider only the *a8 1s* and *a)s* orbitals, the two molecular orbitals of lowest energy. Consider the ground-state electron configuration of H2 • According to the Pauli Exclusion Principle, two electrons of opposite spin are <sup>p</sup>laced in the als orbital. The electron configurati... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**9–6.** A Simple Molecular-Orbital Treatment of H<sub>2</sub> Places Both Electrons in a Bonding Orbital",
"Header 3": "**9-8.** Molecular-Orbital Theory Predicts that a Stable Diatomic Helium Molecule Does Not Exist",
"t... |
Consider the homonuclear diatomic molecules $\text{Li}_2$ through $\text{Ne}_2$ . Each lithium atom has three electrons, so the ground-state electron configuration for $\text{Li}_2$ is $(\sigma_g 1s)^2 (\sigma_u 1s)^2 (\sigma_g 2s)^2$ , and the bond order is one. We predict that the diatomic lithium molecule is s... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**9–9.** Electrons Are Placed into Molecular Orbitals in Accord with the Pauli Exclusion Principle",
"token_count": 987,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The prediction of the correct electron configuration of an oxygen molecule is one of the most impressive successes of molecular-orbital theory. Oxygen molecules are paramagnetic; experimental measurements indicate that the net spin of the oxygen molecule corresponds to two unpaired electrons of the same spin. Let's see... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**9–10.** Molecular-Orbital Theory Correctly Predicts that Oxygen Molecules Are Paramagnetic",
"token_count": 1388,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
| Species | Ground-state electron configuration | Bond<br>order | Bond<br>length/pm | Bond energy/<br>kJ·mol <sup>-1</sup> |
|---... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**9–10.** Molecular-Orbital Theory Correctly Predicts that Oxygen Molecules Are Paramagnetic",
"token_count": 893,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The idea of atomic orbitals and molecular orbitals is rather abstract and sometimes appears far removed from reality. It so happens, however, that the electron configurations of molecules can be demonstrated experimentally. The approach used is very similar to the photoelectric effect discussed in Chapter 1. If high en... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**9–11.** Photoelectron Spectra Support the Existence of Molecular Orbitals",
"token_count": 309,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The molecular-orbital theory we have developed can be extended to *heteronuclear diatomic molecules*, that is, diatomic molecules in which the two nuclei are different. It is important to realize that the energies of the atomic orbitals on the two atoms from
**FIGURE 9.16** The photoel... | {
"Header 1": "Some Postulates and General Principles of Quantum Mechanics",
"Header 2": "**9–12.** Molecular–Orbital Theory Also Applies to Heteronuclear Diatomic Molecules",
"token_count": 1737,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
The molecular-orbital scheme we have presented thus far is the simplest possible molecular-orbital treatment. Each of the molecular orbitals in Figures 9.10 through 9.12 is formed from just one atomic orbital on each nucleus. In analogy with the atomic case, we can obtain better molecular orbitals by forming linear com... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"token_count": 1708,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
**FIGURE 9.20** Contour maps of the various molecular orbitals and the total electron density of the homonuclear diatomic molecules H2through F<sup>2</sup> •
**TABLE 9.4** A demonstration of the convergence to the Hartree-Fock limit for H<sup>2</sup> •... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"token_count": 870,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
In Section 8-8, the electronic states of atoms were designated by atomic term symbols. The electronic states of molecules are also designated by term symbols. Molecular term symbols happen to be easier to deduce than atomic term symbols. To determine molecular term symbols, we first calculate the possible values for th... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 3": "**9-14.** Electronic States of Molecules Are Designated by Molecular Term Symbols",
"token_count": 1889,
"source... |
This leaves
| | $M_{S}$ | | |
|-----------|---------|--------------|----|
| | 1 | 0 | -1 |
| 2 | - | 1+, 1- | |
| $M_L = 0$ | | $1^-, -1^+$ | |
| -2 | | $-1^+, -1^-$ | |
Two of the remaining terms in the column... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 3": "**9-14.** Electronic States of Molecules Are Designated by Molecular Term Symbols",
"token_count": 747,
"source_... |
Term symbols are also used to denote symmetry properties of a molecular wave function. (We will study the symmetry properties of molecules in detail in Chapter 12.) For homonuclear diatomic molecules, inversion through the point midway between the two nuclei leaves the nuclear configuration of the molecule unchanged. T... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**9–15.** Molecular Term Symbols Designate the Symmetry Properties of Molecular Wave Functions",
"token_count": 12... |
| Molecule | Electron configuration ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**9–15.** Molecular Term Symbols Designate the Symmetry Properties of Molecular Wave Functions",
"token_count": 28... |
So far we have considered only the ground electronic states of diatomic molecules. In this section we will consider some of the excited electronic states of molecular hydrogen. As we saw in Section 9–8, the electron configuration of the ground electronic state of $H_2$ is $(1\sigma_g)^2$ , whose molecular term symbo... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**9–15.** Molecular Term Symbols Designate the Symmetry Properties of Molecular Wave Functions",
"Header 3": "9-16... |
- 9-1. Express the Hamiltonian operator for a hydrogen molecule in atomic units.
- **9-2.** Plot the product $1s_A 1s_B$ along the internuclear axis for several values of R.
- **9-3.** The overlap integral, Equation 9.10, and other integrals that arise in two-center systems like H<sub>2</sub> are called *two-center i... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**9–15.** Molecular Term Symbols Designate the Symmetry Properties of Molecular Wave Functions",
"Header 3": "**Pr... |
| A<br>t<br>o<br>m | l<br>b<br>i<br>l<br>V<br>t<br>a<br>e<br>n<br>c<br>e<br>o<br>r<br>a | i<br>i<br>j<br>M<br>J<br>l<br>'<br>I<br>t<br>o<br>n<br>z<br>a<br>o<br>n<br>e<br>n<br>e<br>r<br>g<br>y<br>m<br>o<br>·<br>- |
|------------------|--------------------------------------------------------------------|---------------... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**9–15.** Molecular Term Symbols Designate the Symmetry Properties of Molecular Wave Functions",
"Header 3": "**Pr... |
Doing this, show that
$$E_{\pm} = \frac{\alpha_{\rm A} + \alpha_{\rm B} \pm [(\alpha_{\rm A} - \alpha_{\rm B})^2 + 4\beta^2]^{1/2}}{2}$$
Now if $\chi_{\rm A}$ and $\chi_{\rm B}$ have the same energy, show that $\alpha_{\rm A}=\alpha_{\rm B}=\alpha$ and that
$$E_{\pm} = \alpha \pm \beta$$
giving one level ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**9–15.** Molecular Term Symbols Designate the Symmetry Properties of Molecular Wave Functions",
"Header 3": "**Pr... |
Let
$$\begin{split} I &= -\int \frac{d\mathbf{r} 1 s_{\rm A}^* 1 s_{\rm A}}{r_{\rm B}} = -\frac{1}{\pi} \int d\mathbf{r} \frac{e^{-2r_{\rm A}}}{(r_{\rm A}^2 + R^2 - 2r_{\rm A}R\cos\theta)^{1/2}} \\ &= -\frac{1}{\pi} \int_0^\infty dr_{\rm A} r_{\rm A}^2 e^{-2r_{\rm A}} \int_0^{2\pi} d\phi \int_0^\pi \frac{d\theta \sin... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**9–15.** Molecular Term Symbols Designate the Symmetry Properties of Molecular Wave Functions",
"Header 3": "**Pr... |
The ground-state electron configuration of a carbon atom, ls 2s 2p!2p~, does not seem to lead to the tetrahedral bonding in methane and other saturated hydrocarbons. In fact, the electron configuration seems to imply that carbon should be divalent instead of tetravalent. You may have learned in general chemistry and or... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**10-1.** Hybrid Orbitals Account for Molecular Shape",
"token_count": 1983,
"source_pdf": "datasets/websources/... |
S 0 L U T I 0 N : If two orbitals, 1/11 and *1/1*<sup>2</sup> , are orthogonal, then
$$\int d\tau \psi_1^* \psi_2 = 0$$
Substituting the first two *sp*<sup>2</sup>hybrid orbitals (Equations 10.3 and 10.4) into this orthogonality integral gives
$$\int d\tau \psi_1^* \psi_2 = \int d\tau \left( \frac{1}{\sqrt{3}} ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**10-1.** Hybrid Orbitals Account for Molecular Shape",
"token_count": 1957,
"source_pdf": "datasets/websources/... |
The valence electrons on the central atoms of the molecules discussed in the last section (beryllium for BeH2 , boron for BH3 , and carbon for CH4 ) all occupy hybrid orbitals involved in bonding to a hydrogen atom. In this section, we consider the description of molecules in which the central atom has lone pairs of el... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**10-2.** Different Hybrid Orbitals Are Used for the Bonding Electrons and the Lone Pair Electrons in Water",
"tok... |
We have seen that BeH<sub>2</sub> is linear and H<sub>2</sub>O is bent. Although introducing hybrid orbitals can provide us an explanation of the observed geometry, the physical origin of this difference in molecular structure is not accounted for. The major difference between BeH<sub>2</sub> and H<sub>2</sub>O is the ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**10–3.** Why is BeH<sub>2</sub> Linear and H<sub>2</sub>O Bent?",
"token_count": 2043,
"source_pdf": "datasets/... |
The solid lines tell us how the energies of the molecular orbitals depend upon H–A–H bond angles between 90° and 180°.
valence electrons. For a linear structure, this would correspond to the electron configuration of $(2\sigma_g)^2(1\sigma_u)^2$ . A bent structure would have an electron configuration of $(2a_1)^2(1... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**10–3.** Why is BeH<sub>2</sub> Linear and H<sub>2</sub>O Bent?",
"token_count": 1203,
"source_pdf": "datasets/... |
We discussed photoelectron spectroscopy in Chapter 9, where we showed photoelectron spectra of $N_2$ and CO. Photoelectron spectroscopy can also be used for polyatomic molecules. Figure 10.14 shows the photoelectron spectrum of $H_2O$ vapor. The electron configuration $(2a_1)^2(1b_2)^2(3a_1)^2(1b_1)^2$ suggests t... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**10–4.** Photoelectron Spectroscopy Can Be Used to Study Molecular Orbitals",
"token_count": 1349,
"source_pdf"... |
In this section, we will discuss a well-known theory of bonding in unsaturated hydrocarbons. The simplest unsaturated hydrocarbon is ethene, C2H4 • Ethene is a planar unsaturated hydrocarbon, all of whose bond angles are approximately 120o. We can describe the structure of ethene by saying that the carbon atoms forms *... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**10-5.** Conjugated Hydrocarbons and Aromatic Hydrocarbons Can Be Treated by a :rr-Eiectron Approximation",
"toke... |
The case of butadiene is more interesting than that of ethene. Although butadiene exists in both the *cis* and *trans* configurations, we will ignore that and picture the butadiene molecule as simply a linear sequence of four carbon atoms, each of which contributes a 2p2 orbital to then-electron orbital (Figure 10.21).... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**10-6.** Butadiene Is Stabilized by a Delocalization Energy",
"token_count": 2007,
"source_pdf": "datasets/webs... |
#### **F** I G U **R E 1 0.23**
A schematic representation of the *n* molecular orbitals of butadiene. Note that the corresponding energy increases with the number of nodes.
#### **EXAMPLE 10-7**
Show that 1/f1 in Equation 10.26 is normalized and that it is orthogonal to 1/f2•
... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**10-6.** Butadiene Is Stabilized by a Delocalization Energy",
"token_count": 1766,
"source_pdf": "datasets/webs... |
- **10-1.** Show that *1/Jsp* = Jz (2s ± *2p)* is normalized.
- **10-2.** Show that the three *sp<sup>2</sup>*hybrid orbitals given by Equations 10.3 through 10.5 are normalized.
- **10-3.** Prove that the three *sp*<sup>2</sup>hybrid orbitals given by Equations 10.3 through 10.5 are directed at angles of 120° with res... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**Problems**",
"token_count": 1975,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Starting with the results of Problem 10-12, show that the third *s* p <sup>2</sup>hybrid orbital is given by
$$\psi_3 = 0.77 \cdot 2s - 0.64 \cdot 2p_z$$
At this point the lone pair orbitals are given by 1/!3 and the oxygen *2px* orbital. Construct two equivalent lone pair orbitals by taking the appropriate linear ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**Problems**",
"token_count": 2009,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
**10-32.** Calculate the $\pi$ -electron energy levels and the total $\pi$ -electron energy of bicyclobutadiene:
- **10-33.** Show that the Hückel molecular orbitals of benzene given in Equations 10.31 are orthonormal.
- **10-34.** Set up, but do not try to solve, the Hückel molec... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**Problems**",
"token_count": 2035,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Which molecule has a greater delocalization energy? Why?
- **10-44.** The problem of a linear conjugated polyene of N carbon atoms can be solved in general. The energies $E_j$ and the coefficients of the atomic orbitals in the jth molecular orbital are given by
$$E_j = \alpha + 2\beta \cos \frac{j\pi}{N+1}$$
$j = 1... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**Problems**",
"token_count": 893,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
At one time, quantum-chemical calculations were the domain of professional quantum chemists using large, powerful mainframe computers. Over the years, however, computer programs have become readily available that can be used by nonexperts to calculate reliable values of molecular properties such as geometries and energ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**Computational Quantum Chemistry**",
"token_count": 296,
"source_pdf": "datasets/websources/biochem/F814BC59158... |
Contemporary molecular-orbital theory calculations of the properties of polyatomic molecules are done using computers. We will examine both how such calculations are carried out and the accuracy of the methods used in predicting the properties of molecules. In analogy to the discussion of multielectron atoms in Chapter... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**11-1.** Gaussian Basis Sets Are Often Used in Modern Computational Chemistry",
"token_count": 1967,
"source_pd... |
#### **EXAMPLE 11-1**
Show that $\phi_{1s} = \phi_{1s}^{STO}(0, 1.24) = 0.779$ and that $\phi_{1s} = \phi_{1s}^{GF}(0, 0.4166) = 0.370$ as shown in Figure 11.1.
SOLUTION: From Equation 11.6,
$$\phi_{1s}^{\text{STO}}(r,\zeta) = \left(\frac{\zeta^3}{\pi}\right)^{1/2} e^{-\zeta r}$$
Setting r = 0 and $\zeta... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**11-1.** Gaussian Basis Sets Are Often Used in Modern Computational Chemistry",
"token_count": 1888,
"source_pd... |
| d | ! |
|-----------------------|----------------------------|
| ! | i |
| i | a |
| s | s |
| 0 | 0 ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**11-1.** Gaussian Basis Sets Are Often Used in Modern Computational Chemistry",
"token_count": 659,
"source_pdf... |
While the STO-NG (N = 1, 2, 3, ... ) basis sets were popular in the 1980s, they are not widely used today. Using a finite sum of Gaussian functions to describe an atomic orbital results in several inadequacies that affect the accuracy of the calculations. Here we consider one of the major limitations, and then we will ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**11-2.** Extended Basis Sets Accurately Account for the Size and Shape of Molecular Charge Distributions",
"token... |
(The z-axis lies along the bonds.) The standard orbital exponent in the STO-NG data set for the *2p* orbital on a carbon atom is 1.72 (Table 11.1). Thus, the contraction of the 2 *p z* orbitals and expansion of the 2 *p x* and 2 *p Y* orbitals of the carbon atom in a HCN molecule correspond to roughly a 20% change in t... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**11-2.** Extended Basis Sets Accurately Account for the Size and Shape of Molecular Charge Distributions",
"token... |
Consider the formation of a simple $\sigma 1s$ molecular orbital in $H_2$ . This orbital is formed from a 1s orbital on each hydrogen atom. Surely, however, the electron distribution about each hydrogen atom does not remain spherically symmetric as the two atoms approach each other. We can take this effect into acco... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**11–3.** Asterisks in the Designation of a Basis Set Denote Orbital Polarization Terms",
"token_count": 1573,
"... |
| Atom | $\alpha_{_{1si}}$ | $d_{1si}$ | $\alpha_{2si} = \alpha_{2pi}$ | $d_{2si}$ | $d_{2pi}$ | $\alpha_{2s}^{'}=\alpha_{2p}^{'}$ |
|------|-------------------|-----------|-------------------------------|-----------|-----------|-----------------------------------|
| C | 3.0475 | 1.8347 | 7.8683 ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**11–3.** Asterisks in the Designation of a Basis Set Denote Orbital Polarization Terms",
"token_count": 1676,
"... |
For our first calculation, let's return to a discussion of the simplest diatomic molecule, H2 • In Chapter 9, molecular orbitals for H2 were generated using the LCAO-MO approach. When only the ls orbitals on the two bonded atoms are used to generate the *lag* molecular orbital, an energy of -1.099 *Eh* is obtained. In ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**11-4.** The Ground-State Energy of H2Can Be Calculated Essentially Exactly",
"token_count": 1907,
"source_pdf"... |
We have said that a number of commercially available computer programs can be used by nonexperts to calculate molecular properties. GAUSSIAN 94, GAMESS, and SPARTAN are three of the most widely used. In this section, we will discuss
GAUSSIAN 94 and examine some of the actual input and output of GAUSSIAN 94 using wate... | {
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"Header 2": "**11–5.** GAUSSIAN 94 Calculations Provide Accurate Information About Molecules",
"token_count": 2045,
"source_p... |
A more accurate value can be obtained using configuration interaction.
Hartree-Fock calculations have been carried out on a large number of molecules and in many cases, excellent agreement between theory and experiment is observed. In Table 11.1 0, optimized structural information is displayed for various molecules u... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**11–5.** GAUSSIAN 94 Calculations Provide Accurate Information About Molecules",
"token_count": 1632,
"source_p... |
|
|------------------|-----------------------|--------|-------|--------|--------|---------|-------|
| Н, | r(HH) | 71.2 | 73.5 | 73.5 | 73.0 | 73.2 | 74.2 |
| LiH | r(LiH) | 151.0 | 164.0 | 164.0 | 163.6 | 162.3 | 159.6 |
| $CH_{4}$ | r(C... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**11–5.** GAUSSIAN 94 Calculations Provide Accurate Information About Molecules",
"token_count": 905,
"source_pd... |
| B<br>d<br>o<br>n | | | B<br>o | d<br>l<br>h<br>/<br>t<br>n<br>e<br>n<br>g<br>p | m | |
|------------------|-------------------------------------... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**11–5.** GAUSSIAN 94 Calculations Provide Accurate Information About Molecules",
"token_count": 3327,
"source_p... |
- **11-1.** Show that a three-dimensional Gaussian function centered at **<sup>r</sup> <sup>0</sup>**= *x0*i + *y0* <sup>j</sup>+ z<sup>0</sup> k is a product of three one-dimensional Gaussian functions centered on *x0 , y0 ,* and *z0•*
- **11-2.** Show that
$$\int_{-\infty}^{\infty} e^{-(x-x_0)^2} dx = \int_{-\infty... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**Problems**",
"token_count": 2004,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
- **11-12.** Compare $\phi_{1s}^{STO}(r, 1.00)$ and $\phi_{1s}^{GF}(r, 0.27095)$ graphically by plotting them on the same graph.
- **11-13.** In Problems 11–11 and 11–12, we discussed a one-term Gaussian fit to a 1s Slater orbital $\phi_{1s}^{STO}(r, 1.00)$ . Can we use the result of Problem 11–11 to find the op... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**Problems**",
"token_count": 2002,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
What would be the sum of the Mullikan Populations for the molecules $H_2CO$ , $CO_3^{2-}$ , and $NH_4^{+}$ ?
- 11-26. In this problem, we show that the Mullikan Populations (Problem 11-25) can be used to calculate the molecular dipole moment. Consider the formaldehyde molecule, $H_2CO$ . The calculated bond lengths... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**Problems**",
"token_count": 1514,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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Many physical operations such as magnification, rotation, and reflection through a plane can be represented mathematically by quantities called matrices. A matrix is simply a two-dimensional array that obeys a certain set of rules called matrix algebra. Even if matrices are entirely new to you, they are so convenient t... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**MATRICES**",
"token_count": 1704,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Substitute Equations F.4 into F.6 to obtain
$$x_3 = b_{11}(a_{11}x_1 + a_{12}y_1) + b_{12}(a_{21}x_1 + a_{22}y_1) y_3 = b_{21}(a_{11}x_1 + a_{12}y_1) + b_{22}(a_{21}x_1 + a_{22}y_1)$$
(F.10)
or
$$x_3 = (b_{11}a_{11} + b_{12}a_{21})x_1 + (b_{11}a_{12} + b_{12}a_{22})y_1$$
$$y_3 = (b_{21}a_{11} + b_{22}a_{21})x_1... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**MATRICES**",
"token_count": 1925,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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If it does happen that AB = BA, then A and B are said to *commute.*
#### **EXAMPLE F-4**
Do the matrices A and B commute if
$$A = \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$
I
SOLUTION:
$$AB = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmat... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**MATRICES**",
"token_count": 1461,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
#### **F-1** . Given the two matrices
$$A = \begin{pmatrix} 1 & 0 & -1 \\ -1 & 2 & 0 \\ 0 & 1 & 1 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} -1 & 1 & 0 \\ 3 & 0 & 2 \\ 1 & 1 & 1 \end{pmatrix}$$
form the matrices C = 2A- 3B and D = 6B- A.
#### F-2. Given the three matrices
$$A = \frac{1}{2} \begin{... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**Problems**",
"token_count": 1956,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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Do you see how this procedure generalizes to any number of simultaneous equations?
**F-12.** Solve the following simultaneous algebraic equations by the matrix inverse method developed in Problem F–11:
$$x + y - z = 1$$
$$2x - 2y + z = 6$$
$$x + 3z = 0$$
First show that
$$A^{-1} = \frac{1}{13} \begin{pmatrix} 6... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**Problems**",
"token_count": 562,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
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Group theory is perhaps the best example of a subject developed in pure mathematics and subsequently found to have wide application in the physical sciences. Many molecules have a certain degree of symmetry: methane is a tetrahedral molecule; benzene is hexagonal; sulfur hexafluoride and many inorganic ions are octahed... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**Group Theory: The Exploitation of Symmetry**",
"token_count": 229,
"source_pdf": "datasets/websources/biochem/... |
In Chapter 10, we applied Hiickel molecular-orbital theory to benzene. We used the 2 *p z* orbitals on each carbon atom as our atomic orbitals and found the 6 x 6 secular determinant
$$\begin{vmatrix} x & 1 & 0 & 0 & 0 & 1 \\ 1 & x & 1 & 0 & 0 & 0 \\ 0 & 1 & x & 1 & 0 & 0 \\ 0 & 0 & 1 & x & 1 & 0 \\ 0 & 0 & 0 & 1 & x... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12-1.** The Exploitation of the Symmetry of a Molecule Can Be Used to Significantly Simplify Numerical Calculation... |
The symmetry of a molecule can be described in terms of its *symmetry elements.* For example, a water molecule has the symmetry elements shown in Figure 12.1. The element C2 is an *axis of symmetry* and *<Y"* and *<Y:* are *planes of symmetry,* or *mirror <sup>p</sup>lanes.* Because the hydrogen atoms are indistinguish... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12-2.** The Symmetry of Molecules Can Be Described by a Set of Symmetry Elements",
"token_count": 1054,
"sourc... |
Illustrate the various symmetry elements for ammonia and ethene.
S 0 L UTI 0 N: The symmetry elements are illustrated in Figure 12.4.
#### **FIGURE 12.4**
(a) The symmetry elements of ammonia. Each mirror plane contains an N-H bond and bisects the opposite H-N-H bond angle. The thre... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12-2.** The Symmetry of Molecules Can Be Described by a Set of Symmetry Elements",
"Header 3": "**EXAMPLE 12-1**... |
A group is a set of entities that satisfy certain requirements. Specifically, the set $A, B, C, \ldots$ is said to form a group if
- 1. there is a rule for combining any two members of the group, and moreover, the result is a member of the group. This combining rule is commonly called multiplication and is denoted ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12-2.** The Symmetry of Molecules Can Be Described by a Set of Symmetry Elements",
"Header 3": "12-3. The Symmet... |
Thus, the four symmetry operations of C2" satisfy the conditions of being a group and are collectively referred to as the point group *czv·*
The other point group we will consider is **C3 ",** and we will use NH3as the example of a molecule whose symmetry properties are described by the C*3<sup>v</sup>*point group. I... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12-2.** The Symmetry of Molecules Can Be Described by a Set of Symmetry Elements",
"Header 3": "12-3. The Symmet... |
Let's consider $H_2O$ , which we have shown belongs to the $C_{2v}$ point group. We once again construct a set of Cartesian coordinates centered on the oxygen atom (Figure 12.6) and follow the effect of each symmetry operation on an arbitrary vector, $\mathbf{u}$ , where $\mathbf{u} = u_x \mathbf{i} + u_y \mathbf{... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "12-4. Symmetry Operations Can Be Represented by Matrices",
"token_count": 217,
"source_pdf": "datasets/websource... |
|
| | |
| | First operation | ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "12-4. Symmetry Operations Can Be Represented by Matrices",
"token_count": 2339,
"source_pdf": "datasets/websourc... |
We will denote a matrix that represents an operator by its corresponding sans serif symbol, without the caret. Similarly, $\hat{\sigma}_v$ changes $u_y$ to $-u_y$ and $\hat{\sigma}_v'$ changes $u_x$ to $-u_x$ , so the reflection operations can be represented by
$$\sigma_{v} = \begin{pmatrix} 1 & 0 & 0 \\ 0... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "12-4. Symmetry Operations Can Be Represented by Matrices",
"token_count": 2018,
"source_pdf": "datasets/websourc... |
The irreducible representations of the $C_{3v}$ point group are given in Table 12.6. Note how the number and dimensions of these irreducible representations satisfy Equation 12.7. Note also that $C_{3v}$ has a totally symmetric irreducible representation $A_1$ , which can be deduced by applying the six operations ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12–5.** The $C_{3v}$ Point Group Has a Two-Dimensional Irreducible Representation",
"token_count": 1706,
"sour... |
We write $\mathbf{u}_1$ as
$$\mathbf{u}_1 = u_{1x}\mathbf{i} + u_{1y}\mathbf{j} = (\cos\alpha)\mathbf{i} + (\sin\alpha)\mathbf{j}$$
Figure 12.8 shows that a reflection through $\sigma_v''$ sends $\alpha$ to $240^\circ - \alpha$ , so $\mathbf{u}_1$ becomes the vector $\mathbf{u}_2$ where
$$\mathbf{u}_{2... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12–5.** The $C_{3v}$ Point Group Has a Two-Dimensional Irreducible Representation",
"token_count": 882,
"sourc... |
Tables 12.5 and 12.6 show the irreducible representations of the point groups *C2 <sup>v</sup>*and C*3v,* respectively. It turns out that for almost all the applications of group theory we do not need the complete matrices, only the sum of the diagonal elements. The sum of the diagonal elements of a matrix is called it... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12-6.** The Most Important Summary of the Properties of a Point Group Is Its Character Table",
"token_count": 20... |
Note that Table 12.14 has triply degenerate representations, which are designated by the letter *T.* The point group *C*<sup>2</sup> h, which describes a trans-dichloroethene molecule, has a center of inversion, i, and the character table is given in Table 12.10. Irreducible representations that are symmetric under an ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12-6.** The Most Important Summary of the Properties of a Point Group Is Its Character Table",
"token_count": 18... |
In this section, we will give without proof a number of the mathematical properties associated with character tables. As we said in the previous section, character tables are square; in other words, the number of irreducible representations equals the number of classes. Furthermore, because an identity operation is rep... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12–7.** Several Mathematical Relations Involve the Characters of Irreducible Representations",
"token_count": 20... |
Therefore, if u and v are perpendicular (orthogonal), then cos *e* = 0 and
$$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 = 0 \tag{12.12}$$
Although we wrote u and vas three-dimensional vectors in Equations 12.11 and 12.12, they can ben-dimensional vectors, in which case Equation 12.11 becomes
$$\mat... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12–7.** Several Mathematical Relations Involve the Characters of Irreducible Representations",
"token_count": 20... |
These coefficients will tell us how many times each irreducible representation is
contained in r. Determining the a/s using the orthogonality re~ation given in Equation 12.20 is actually fairly easy. Multiply Equation 12.21 by *x.* (R) and sum both sides A I over R:
$$\sum_{\hat{R}} \chi(\hat{R}) \chi_i(\hat{R}) = ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12–7.** Several Mathematical Relations Involve the Characters of Irreducible Representations",
"token_count": 10... |
Recall from Chapters 9 and 10 that we encountered molecular integrals of the type
$$H_{ij} = \int \phi_i^* \hat{H} \phi_j d\tau$$
and $S_{ij} = \int \phi_i^* \phi_j d\tau$ (12.24)
We will now show that integrals like these will be equal to zero if we choose $\phi_i^*$ and $\phi_j$ such that they belong to dif... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12–8.** We Use Symmetry Arguments to Predict Which Elements in a Secular Determinant Equal Zero",
"token_count":... |
A counterclockwise rotation by 60° about the principal axis gives (remember that $\psi_j$ stands for a $2p_j$ orbital on carbon atom j)
$$\hat{C}_6 \phi_1 = \hat{C}_6 \psi_1 + \hat{C}_6 \psi_2 + \hat{C}_6 \psi_3 + \hat{C}_6 \psi_4 + \hat{C}_6 \psi_5 + \hat{C}_6 \psi_6$$
$$= \psi_6 + \psi_1 + \psi_2 + \psi_3 + \ps... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12–8.** We Use Symmetry Arguments to Predict Which Elements in a Secular Determinant Equal Zero",
"token_count":... |
There is a straightforward procedure to find linear combinations of atomic orbitals that are bases for the irreducible representations. It involves a quantity called a *generating operator*, whose formula we give without proof. The generating operator for the *j*th irreducible representation is
$$\hat{P}_j = \frac{d_... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12–9.** Generating Operators Are Used to Find Linear Combinations of Atomic Orbitals That Are Bases for Irreducibl... |
These results tell us that the four $2p_z$ orbitals belong to the reducible representation
$$\begin{array}{c|ccccccccccccccccccccccccccccccccccc$$
We can even write $\Gamma$ without writing out all the matrices. Note that there is a 1 on a diagonal in a representation if the $2p_z$ orbital is unchanged, a -1 ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12–9.** Generating Operators Are Used to Find Linear Combinations of Atomic Orbitals That Are Bases for Irreducibl... |
Using Equation 12.32 for $A_{2u}$ gives (use Figures 12.8 and 12.9)
$$\hat{P}_{A_{2u}}\psi_{1} = \frac{1}{24}(\underbrace{\psi_{1}}_{\hat{E}} + \underbrace{\psi_{2} + \psi_{6}}_{2\hat{C}_{6}} + \underbrace{\psi_{3} + \psi_{5}}_{2\hat{C}_{3}} + \underbrace{\psi_{4}}_{\hat{C}_{2}} + \underbrace{\psi_{1} + \psi_{3} + ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12–9.** Generating Operators Are Used to Find Linear Combinations of Atomic Orbitals That Are Bases for Irreducibl... |
- **12-1.** Neglecting overlap, show that ¢ <sup>1</sup>and ¢2 given by Equations 12.3 are orthonormal to the other four molecular orbitals.
- **12-2.** Using the six molecular orbitals given by Equations 12.3, verify that H11 *=a+ 2{3, H22 =a* - *2{3,* H12 = *H13* = H14 = H15 = *H16* = 0 (see Equation 12.4).
- **12-3.... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12–9.** Generating Operators Are Used to Find Linear Combinations of Atomic Orbitals That Are Bases for Irreducibl... |
What does your answer tell you about the expected Htickel secular determinant?
- **12-30.** Show that if we used a *2pz* orbital on each carbon atom as the basis for a (reducible) representation for cyclobutadiene (D4h), then r = 4 0 0 0 - 2 0 0 - 4 0 2. Reducer into its component irreducible representations. What does... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12–9.** Generating Operators Are Used to Find Linear Combinations of Atomic Orbitals That Are Bases for Irreducibl... |
Now argue that hybrid orbitals with $\mathbf{D}_{4h}$ symmetry can be formed from a s orbital, a $d_{x^2-y^2}$ orbital, and the $p_x$ and $p_y$ orbitals to give $sdp^2$ hybrid orbitals.
- **12-43.** Consider a trigoanl bipyramidal molecule XY<sub>5</sub> whose point group is $\mathbf{D}_{3h}$ . Using the pro... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**12–9.** Generating Operators Are Used to Find Linear Combinations of Atomic Orbitals That Are Bases for Irreducibl... |
To this point, we have mainly focused on the theoretical description of atomic and molecular orbitals and molecular structure. The interaction of electromagnetic radiation /". ------- - -- -·----·- with atoms and molecules, or spectroscopy, is one of the most important experimental pr~§\_JQ~\_studying atomic and molecu... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**Molecular Spectroscopy**",
"token_count": 367,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.... |
Molecular spectroscopy is the study of the interaction of electromagnetic radiation \_ .withlllQ~S-. Electromagnetic radiation is customarily divided into diffe(ent ~;;~ regions reflecting the different types of molecular processes that can be caused by 495 such radiation. The classifications we focus on in this chapte... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**13-1.** Different Regions of the Electromagnetic Spectrum Are Used to Investigate Different Molecular Processes",
... |
The quantum-mechanical properties of a diatomic harmonic oscillator were described in Section 5–4. Recall that the allowed energies of a harmonic oscillator are given by
$$E_v = (v + \frac{1}{2})hv$$
$v = 0, 1, 2, \dots$ (13.2)
where
$$\nu = \frac{1}{2\pi} \left(\frac{k}{\mu}\right)^{1/2} \tag{13.3}$$
is the f... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**13-1.** Different Regions of the Electromagnetic Spectrum Are Used to Investigate Different Molecular Processes",
... |
| M<br>l<br>l<br>o<br>e<br>c<br>u<br>e | 1<br>f<br>B<br>e<br>c<br>m<br>- | 1<br>f<br>a<br>e<br>c<br>m<br>- | 1<br>D<br>j<br>c<br>m<br>- | 1<br>f<br>v<br>e<br>c<br>m<br>- | 1<br>f<br>x<br>e<br>v<br>e<br>c<br>m<br>- | 0<br>)<br>/<br>R<br>(<br>e<br>=<br>p<br>m<br>v | 1<br>D<br>/<br>k<br>l·<br>l<br>... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**13-1.** Different Regions of the Electromagnetic Spectrum Are Used to Investigate Different Molecular Processes",
... |
When a molecule absorbs infrared radiation, the vibrational transition is accompanied by a rotational transition. The selection rules for absorption of infrared radiation in the rigid rotator-harmonic oscillator approximation are
$$\Delta v = +1$$
(absorption) (13.11)
$\Delta J = \pm 1$
For the case $\Delta J = ... | {
"Header 1": "**9–13.** An SCF–LCAO–MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently",
"Header 2": "**13-1.** Different Regions of the Electromagnetic Spectrum Are Used to Investigate Different Molecular Processes",
... |
The energies of a rigid rotator-harmonic oscillator are given by (Equation 13.10)
$$\tilde{E}_{v,J} = G(v) + F(J) = \tilde{v}(v + \frac{1}{2}) + \tilde{B}J(J+1)$$
where $\tilde{B}=h/8\pi^2c\mu\,R_e^2$ . Because the vibrational amplitude increases with the vibrational state (cf. Figure 13.1), we expect that $R_e$ ... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"token_count": 1572,
"source_pdf": "datasets/websources/biochem/F814BC5915875384820.pdf"
} |
Table 13 .3lists some of the observed lines in the pure rotational spectrum (no vibrational transitions) of H35Cl. The differences listed in the third column clearly show that the lines are not exactly equally spaced as the rigid rotator approximation predicts. The discrepancy can be resolved by realizing that a chemic... | {
"Header 1": "**13–3.** Vibration–Rotation Interaction Accounts for the Unequal Spacing of the Lines in the *P* and *R* Branches of a Vibration-Rotation Spectrum",
"Header 2": "**13-4.** The Lines in a Pure Rotational Spectrum Are Not Equally Spaced",
"token_count": 1034,
"source_pdf": "datasets/websources/bio... |
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