?What irreducible representations do the four H1s orbitals of \(\mathrm{CH}_{4}\) span?

Identify and list four applications of character tables.

Explain how symmetry arguments are used to construct molecular orbitals.

Use symmetry properties to determine whether or not the integral \(\int \mathrm{p}_{x} z \mathrm{p}_{z} \mathrm{~d} \tau\) is necessarily zero in a molecule with symmetry \(C_{2 v}\). Text Transcription: intp_xzp_zdtau C_2v

Use symmetry properties to determine whether or not the integral \(\int \mathrm{p}_{x} z \mathrm{p}_{z} \mathrm{~d} \tau\) is necessarily zero in a molecule with symmetry \(D_{3 \mathrm{~h}}\). Text Transcription: Intp_xzp_zdtau D_3h

Is the transition \(\mathrm{A}_{1} \rightarrow \mathrm{A}_{2}\) forbidden for electric dipole transitions in a \(C_{3 v}\) molecule? Text Transcription: C_3v A_1A_2

Is the transition \(\mathrm{A}_{1 \mathrm{~g}} \rightarrow \mathrm{E}_{2 \mathrm{u}}\) forbidden for electric dipole transitions in a \(D_{6 \mathrm{~h}}\) molecule? Text Transcription: A_1gE_2u D_6h

Show that the function xy has symmetry species \(B_{1 \mathrm{~g}}\) in the group \(D_{2 \mathrm{~h}}\). Text Transcription: B_1g D_2h

Show that the function xyz has symmetry species \(\mathrm{A}_{\mathrm{u}}\) in the group \(D_{2 \mathrm{~h}}\). Text Transcription: A_u D_2h

Consider the \(C_{2 v}\) molecule \(\mathrm{OF}_{2}\); take the molecule to lie in the yz-plane, with z directed along the \(C_{2}\) axis; the mirror plane \(\sigma_{\mathrm{v}}^{\prime}\) is the yz-plane, and \(\sigma_{v}\) is the xz-plane. The combination \(\mathrm{p}_{z}(\mathrm{~A})+\mathrm{p}_{z}(\mathrm{~B})\) of the two F atoms spans \(\mathrm{A}_{1}\), and the combination\(\mathrm{p}_{z}(\mathrm{~A})+\mathrm{p}_{z}(\mathrm{~B})\) of the two F atoms spans \(\mathrm{B}_{2}\). Are there any valence orbitals of the central O atom that can have a non-zero overlap with these combinations of F orbitals? How would the situation be different in \(\mathrm{SF}_{2}\), where 3d orbitals might be available? Text Transcription: C_2v OF_2 C_2 sigma_v^prime Sigma_v p_zA+p_zB p_zA-p_zB A_1 B_2 SF_2

Consider the same situation as in Exercise E10C.4(a). Find the irreducible representations spanned by the combinations \(\mathrm{p}_{y}(\mathrm{~A})+\mathrm{p}_{y}(\mathrm{~B})\) and \(\mathrm{p}_{y}(\mathrm{~A})-\mathrm{p}_{y}(\mathrm{~B})\). Are there any valence orbitals of the central O atom that can have a non-zero overlap with these combinations of F orbitals? Text Transcription: p_yA+p_yB p_yA-p_yB

Consider the \(C_{2 v}\) molecule NO2. The combination \(\mathrm{p}_{x}(\mathrm{~A})-\mathrm{p}_{x}(\mathrm{~B})\) of the two O atoms (with x perpendicular to the plane) spans \(\mathrm{A}_{2}\). Is there any valence orbital of the central N atom that can have a non-zero overlap with that combination of O orbitals? What would be the case in \(\mathrm{SO}_{2}\), where 3d orbitals might be available? Text Transcription: C_2v p_xA-p_xB A_2 SO_2

Consider \(\mathrm{BF}_{3}\) (point group \(D_{3 \mathrm{~h}}\)). There are SALCs from the F valence orbitals which transform as \(\mathrm{A}_{2}^{\prime \prime}\) and \(\mathrm{E}^{\prime \prime}\). Are there any valence orbitals of the central B atom that can have a non-zero overlap with these SALCs? How would your conclusion differ for \(\mathrm{AlF}_{3}\), where 3d orbitals might be available? Text Transcription: BF_3 D_3h A_2^primeprime E^prime AIF_3

The ground state of \(\mathrm{NO}_{2}\) is \(A_{1}\) in the group \(C_{2 v}\). To what excited states may it be excited by electric dipole transitions, and what polarization of light is it necessary to use? Text Transcription: NO_2 A_1 C_2v

The ClO2 molecule (which belongs to the group \(C_{2 v}\)) was trapped in a solid. Its ground state is known to be \(\mathrm{B}_{1}\). Light polarized parallel to the y-axis (parallel to the OO separation) excited the molecule to an upper state. What is the symmetry species of that state? Text Transcription: C_2v B_1

A set of basis functions is found to span a reducible representation of the group \(C_{4 v}\) with characters 5,1,1,3,1 (in the order of operations in the character table in the Resource section). What irreducible representations does it span? Text Transcription: C_4v

A set of basis functions is found to span a reducible representation of the group \(D_{2}\) with characters 6,?2,0,0 (in the order of operations in the character table in the Resource section). What irreducible representations does it span? Text Transcription: D_2

A set of basis functions is found to span a reducible representation of the group \(D_{4 \mathrm{~h}}\) with characters 4,0,0,2,0,0,0,4,2,0 (in the order of operations in the character table in the Resource section). What irreducible representations does it span? Text Transcription: D_4h

A set of basis functions is found to span a reducible representation of the group \(O_{h}\) with characters 6,0,0,2,2,0,0,0,4,2 (in the order of operations in the character table in the Resource section). What irreducible representations does it span? Text Transcription: O_h

What states of (i) benzene, (ii) naphthalene may be reached by electric dipole transitions from their (totally symmetrical) ground states?

What states of (i) anthracene, (ii) coronene (7) may be reached by electric dipole transitions from their (totally symmetrical) ground states?

What irreducible representations do the four H1s orbitals of \(\mathrm{CH}_{4}\) span? Are there s and p orbitals of the central C atom that may form molecular orbitals with them? In \(\mathrm{SiH}_{4}\), where 3d orbitals might be available, could these orbitals play a role in forming molecular orbitals by overlapping with the H1s orbitals? Text Transcription: CH_4 SiH_4

Suppose that a methane molecule became distorted to (a) \(C_{3 v}\) symmetry by the lengthening of one bond, (b) \(C_{2 v}\) symmetry, by a kind of scissors action in which one bond angle opened and another closed slightly. Would more d orbitals on the carbon become available for bonding? Text Transcription: C_3v C_2v

Does the integral of the function \(3 x^{2}-1\) necessarily vanish when integrated over a symmetrical range in (a) a cube, (b) a tetrahedron, (c) a hexagonal prism, each centred on the origin? Text Transcription: 3x^2-1

In a spectroscopic study of \(\mathrm{C}_{60}\), Negri et al. (J. Phys. Chem. 100, 10849 (1996)) assigned peaks in the fluorescence spectrum. The molecule has icosahedral symmetry (\(I_{\mathrm{h}}\)). The ground electronic state is \(\mathbf{A}_{1 \mathrm{~g}}\), and the lowest-lying excited states are \(\mathrm{T}_{1 \mathrm{~g}}\) and \(\mathrm{G}_{\mathrm{g}}\). (a) Are photon-induced transitions allowed from the ground state to either of these excited states? Explain your answer. (b) What if the molecule is distorted slightly so as to remove its centre of inversion? Text Transcription: C_60 I_h A_1g T_1g G_g

P10C.5 In the square planar \(\mathrm{XeF}_{4}\) molecule, consider the symmetry-adapted linear combination \(p_{1}=\mathrm{p}_{\mathrm{A}}-\mathrm{p}_{\mathrm{B}}+\mathrm{p}_{\mathrm{C}}-\mathrm{p}_{\mathrm{D}}\), where \(\mathrm{P}_{A}\), \(\mathrm{P}_{B}\), \(\mathrm{P}_{C}\), and \(\mathrm{P}_{D}\) are \(2 p_{z}\) atomic orbitals on the fluorine atoms (clockwise labelling of the F atoms). Decide which of the various s, p, and d atomic orbitals on the central Xe atom can form molecular orbitals with \(P_{1}\). Text Transcription: XeF_4 p_1=p_A-p_B+p_C-p_D P_A P_B P_C P_D 2p_z P_1

The chlorophylls that participate in photosynthesis and the haem (heme) groups of cytochromes are derived from the porphine dianion group (8), which belongs to the \(D_{4 \mathrm{~h}}\) point group. The ground electronic state is \(A_{1 \mathrm{~g}}\) and the lowest-lying excited state is \(\mathrm{E}_{u}\). Is a photon-induced transition allowed from the ground state to the excited state? Explain your answer. Text Transcription: D_4h E_u A_1g

P10C.7 Consider the ethene molecule (point group \(D_{2 \mathrm{~h}}\)), and take it as lying in the xy-plane, with x directed along the C–C bond. By applying the projection formula to one of the hydrogen 1s orbitals generate SALCs which have symmetry \(A_{g}\), \(\mathrm{B}_{2 \mathrm{u}}\), \(\mathrm{B}_{3 \mathrm{u}}\), and \(\mathrm{B}_{1 \mathrm{g}}\). What happens when you try to project out a SALC with symmetry \(\mathrm{B}_{1 \mathrm{u}}\)? Text Transcription: D_2h A_g B_2u B_3u B_1g

Consider the molecule \(\mathrm{F}_{2} \mathrm{C}=\mathrm{CF}_{2}\) (point group \(D_{2 \mathrm{~h}}\)), and take it as lying in the xy-plane, with x directed along the C–C bond. (a) Consider a basis formed from the four \(2 \mathrm{p}_{z}\) orbitals from the fluorine atoms: show that the basis spans \(\mathrm{B}_{1 \mathrm{u}}\), \(\mathrm{B}_{2 \mathrm{g}}\), \(\mathrm{B}_{3 \mathrm{g}}\), and \(A_{u}\). (b) By applying the projection formula to one of the \(2 \mathrm{p}_{z}\)) orbitals, generate the SALCs with the indicated symmetries. (c) Repeat the process for a basis formed from four \(2 p_{x}\) orbitals (the symmetry species will be different from those for \(2 \mathrm{p}_{z}\)). Text Transcription: F_2C=CF_2 D_2h 2p_z B_1u B_2g A_u 2p_x 2p_z