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Quantum Chemistry 7th edition Ira N. Levine - Solutions

Give the possible values of the total-angular-momentum quantum number J that result from the addition of angular momenta with quantum numbers a. 3/2 and 4; b. 2, 3, and 1/2.
True or false? The angular-momentum addition rule (11.39) shows that the number of values of J obtained by adding j1 and j2 is always 2j< + 1, where j< is the smaller of j1 and j2 or is j1 if j1 = j2.
For each of the following symmetry operations, find the matrix representative in the x, y, z basis. (a) E; (b) σ (xy); (c) σ (yz); (d) C2(x); (e) S4(z); (f) C3(z).
(a) Use SF6 (Fig. 12.7) to verify that C2(x) Ïƒ (xy) = Ïƒ(xz).In figure 12.7(b) Write down the matrix representatives in the x, y, z basis of the three operations in part (a). Verify that these matrices multiply the same way the symmetry operations multiply.
(a) What are the eigenvalues of OC4? (b) Is this operator Hermitian?
Do the same as in Prob. 12.12 for OC2. (a) What are the eigenvalues of OC4? (b) Is this operator Hermitian?
To what function is a 2pz hydrogenlike orbital converted by (a) OC4(z); (b) OC4(zy?
It is common to use rotation-inversion axes (rather than rotation-reflection axes) to classify the symmetry of crystals. Any Sn axis is equivalent to a rotation-inversion axis (symbolized by p) whose order p may differ from n. A rotation-inversion operation consists of rotation by 2Ï€/p
State whether each of the following is a group. (a) All the real numbers with the rule of combination for forming the product of two elements being addition. (b) All positive integers with the rule of combination being multiplication. (c) All real numbers except zero, with the rule of
Does the set of all square matrices of order four form a group if the rule for forming the "product" of two elements is matrix addition?
Does the set of the nth roots of unity [Eq. (1.36)] with the rule of combination being ordinary multiplication form a group? Justify your answer.
Give the point group of each of the following molecules. (a) CH4; (b) CH3F; (c) CH2F2; (d) CHF3; (e) SF6; (f) SF5Br; (g) trans@SF4Br2; (h) CDH3.
Give all the symmetry elements of each of the following molecules: (a) H2S; (b) NH3; (c) CHF3; (d) HOCl; (e) 1,3,5-trichlorobenzene; (f) CH2F2; (g) CHFClBr.
Give the point group of (a) Benzene; (b) Fluorobenzene; (c) O-difluorobenzene; (d) M-difluorobenzene; (e) P-difluorobenzene; (f) 1, 3, 5-trifluorobenzene; (g) 1, 4-difluoro-2, 5-dibromobenzene; (h) Naphthalene; (i) 2-chloronaphthalene.
Give the point group of (a) HCN; (b) H2S; (c) CO; (d) CO; (e) C2H2; (f) CH3OH; (g) ND3; (h) OCS; (i) P4; (j) PCl3; (k) PCl5; (l) B12Cl2-12; (m) UF6; (n) Ar.
Give the point group of (a) FeF3-6; (b) IF5; (c) CH2 == C == CH2; (d) C8 H8, cubane; (e) C6H6Cr(CO)3; (f) B2H6; (g) XeF4; (h) F2O; (i) spiropentane.
Give the order of each of the following groups: (a) B3v; (b) Bs; (c) L∞v; (d) D3h.
The product of two members of a group must be a member of that group. (a) List the members (the symmetry operations) of the group 2v, using the x, y, and z axes to specify the axis or plane with respect to which each symmetry operation is performed. (b) For every possible product of two members of
State whether each of these groups is Abelian. (a) B3; (b) B3v.
What is the point group of the tris(ethylenediamine) cobalt(III) complex ion? (Each NH2CH2CH2NH2 group occupies two adjacent positions of the octahedral coordination sphere.)
List all the symmetry operations of each of the molecules in Prob. 12.2. In problem 12.2 (a) H2S; (b) NH3; (c) CHF3; (d) HOCl; (e) 1,3,5-trichlorobenzene; (f) CH2F2; (g) CHFClBr.
Give the point group of (a) A square-based pyramid; (b) A right circular cone; (c) A square lamina; (d) A square lamina with the top and bottom sides painted different colors; (e) A right circular cylinder; (f) A right circular cylinder with the two ends painted different colors; (g) A right
(a) What Platonic solid is dual to the regular tetrahedron? (b) How many vertices does a pentagonal dodecahedron have?
(a) For what values of n does the presence of an Sn axis imply the presence of a plane of symmetry? (b) For what values of n does the presence of an Sn axis imply the presence of a center of symmetry? (c) The group Dnd has an S2n axis. For what values of n does it have a center of symmetry?
For which point groups can a molecule have a dipole moment?
For which point groups can a molecule be optically active?
Two people play the following game. Each in turn places a penny on the surface of a large chessboard. The pennies can be put anywhere on the board, as long as they do not overlap previously placed pennies. A penny may overlap more than one square. Once placed, a penny cannot be moved. When one of
Consider the square-planar ion PtCl2-4. Suppose we interchange two Cl atoms that are cis to each other. Does this interchange meet the definition of a symmetry operation? If so, express it in terms of some combination of the four kinds of symmetry operations discussed.
What symmetry operation is each of the following products of operations equal to?
Use Fig. 12.7 to state what symmetry operation in SF6 each of the following products of symmetry operations is equal to. (a) C2(x)C4(z); b) C4(z)C2(x);
For SF6, which of the following pairs of operations commute? (a) C4(z), σ(xy); (b) C4(z), σ(yz); (c) C2(z), C2(x); (d) σ(xy), σ(yz); (e) i, σ (xy).
What information does symmetry give about the dipole moment of each of the molecules in Prob. 12.2? In problem 12.2 (a) H2S; (b) NH3; (c) CHF3; (d) HOCl; (e) 1,3,5-trichlorobenzene; (f) CH2F2; (g) CHFClBr.
(a) Does H2O2 (Fig. 12.12) have an Sn axis? (b) Is it optically active? Explain.
Analytical solution of the Schrödinger equation for the Morse-function potential energy gives Evib = (v + 1/2)hve - (v + ½)2h2v2e/4De. (The Morse-function U becomes infinite at R = - ∞ The expression given for Evib corresponds to the boundary conditions that c becomes zero at R = ∞ and at R =
Derive (13.60) for the overlap integral Sab.
(a) Show that the variational integral W1 for the ground state of H+2 can be written as W1 = k2F(t) + kG(t), where t K kR and where F and G are certain functions of t.(b) Show that the minimization condition ˆ‚W1/ˆ‚k = 0 leads toUsing this equation, we can find k for a given
Write a computer program that will calculate the optimum value of the orbital exponent k for the H+2 trial function (13.54) for a given R value. Have the program calculate W1 + 1/R [where W1 is given by (13.63)] for k ranging from 0 to 3 in steps of 0.001 and have the program pick the k value that
For the H2 ground electronic state, D0 = 4.4781 eV. Find ΔHo0 for H2(g) → 2H(g) in kJ/mol.
Write a computer program that will simultaneously find the values of k and R that minimize W1 + 1/R in equation. On the first run, have R range from 0.1 to 6 in steps of 0.01 and have k range from 0 to 3 in steps of 0.01; compute W1 + 1/R for all possible pairs of k and R values in these ranges and
(a) Use Mathcad to create an animation showing how contours and three-dimensional plots of the H+2 LCAO MOs ϕ1 and ϕ2 [Eqs. (13.57) and (13.58)] for a plane containing the nuclei change as R changes from 3.8 to 0.1 bohr. Proceed as follows. Define the function U1R, k2 by adding the inter-nuclear
Use a spreadsheet to calculate the k values asked for in Prob. 13.22 In Problem 22 (a) Use Mathcad to create an animation showing how contours and three-dimensional plots of the H+2 LCAO MOs ϕ1 and ϕ2 [Eqs. (13.57) and (13.58)] for a plane containing the nuclei change as R changes from 3.8 to 0.1
Verify Eq. (13.79) for a Ïƒh reflection.In Equation 13.79
Which species of each pair has the greater De? (a) Li2 or Li+2 ; (b) C2 or C+2; (c) O2 or O+2; (d) F2 or F+2.
How many independent electronic wave functions correspond to each of the following diatomic-molecule terms? (a) 1Σ-; (b) 3Σ+; (c) 3Π; (d) 1ϕ; (e) Δ
Give the levels belonging to each of the terms in Prob. 13.28. In problem 13.28 (a) 1Σ-; (b) 3Σ+; (c) 3Π; (d) 1ϕ; (e) Δ
Use the D0 value of H2 14.478 eV2 and the D0 value of H+2 (2.651 eV) to calculate the first ionization energy of H2 (that is, the energy needed to remove an electron from H2).
Show that the four functions of (13.89) have the indicated eigen values with respect to a Ïƒv(xz) reflection of electronic coordinates. Start by showing that this reflection converts Ï• to -Ï• and leaves ra and rb unchanged
Show that for a diatomic molecule [Lz, Oσ] ≠ 0.
The ground state of H2 has 1Σ+g symmetry. What restriction does this impose on the values of m, n, j, and k in the James and Coolidge wave function?
For the determinant (13.112), add column 1 to column 3 and add column 2 to column 4. Show that the result is equal toIn this determinant, subtract column 3 from column 1 and subtract column 4 from column 2. Then use column interchanges to show that the result equals (13.113). At each step, state
Write down abbreviated expressions for the remaining six determinants of the N2 VB function of Section 13.12. Use the rule given in that section to find the coefficient of each determinant in the wave function.
Use simple MO theory to predict the number of unpaired electrons and the ground term of each of the following: (a) BF; (b) BN; (c) BeS; (d) BO; (e) NO; (f) CF; (g) CP; (h) NBr; (i) ClO; ( j) BrCl. Compare your results with the experimentally observed ground terms: (a) 1Σ+; (b) 3Π; (c) 1Σ+; (d)
The infrared absorption spectrum of 1H35Cl has its strongest band at 8.65 x 1013Hz. For this molecule, D0 = 4.43 eV. (a) Find De for 1H35Cl. (b) Find D0 for 2H35Cl
In applying quantum chemistry to chemical reactions, which would be the more accurate approximation, the simple MO or the simple VB method?
In Section 13.14, an experimental value for U (Re) of F2 was given. Explain how this value is found from certain other data. Hint: See the paragraph after Eq. (13.1).In Equation 13.1
Use notation such as 1sa(1) to write down without consulting this chapter (a) The simple MO wave function for H2 including spin; (b) The simple VB wave function for H2
(a) Verify that if anharmonicity is taken into account by inclusion of the vexe term in the vibrational energy, then De = D0 + 1/2 hve - 1/4 hvexe. (b) The 7Li1H ground electronic state has D0 = 2.4287 eV, ve/c = 1405.65 cm-1, and ne xe/c = 23.20 cm-1, where c is the speed of light. (These last
Without consulting the text, write down the complete non-relativistic Hamiltonian operator for the H2 molecule. Then write down the purely electronic Hamiltonian operator for H2.
Verify the Taylor-series expansion (13.24).In Equation 13.24
(a) Use the Numerov method with the endpoints and interval recommended in the Section 13.2 example to find the lowest six Morse-function vibrational levels of H2. (b) Calculate (xr) and (R) for these six levels. (c) Find the Morse-function v = 6, 7, 8, and 9 H2 vibrational levels.
Derive (14.6) for rVB and rMO.In Equation 17.6
(a) From (14.5), show thatWhere ( is the electron probability density of an n-electron molecule. (b) Use the result of (a) and Eq. (14.43) to show that (c) Show that n = Tr(PS*), which becomes n = Tr(PS) Which become for real basis functions. Here, P and S are the density and overlap matrices.
Verify the equalities (14.47) for electron-repulsion integrals.
Use Eq. (9.124) of Prob. 9.14 to verify the expressions for the integrals (11| 22) and |22| 22) in the Section 14.3 exampleIn Eq. 9.14
For the He-atom SCF calculation in Section 14.3, find the initial estimate of c11/c21 given by the approximation Frs ≈ Hrscore
Derive (14.48) from the normalization condition for Ï•1.Where k = c11/c21
Verify the numerical results for P11, P12, P22, F11, F12, F22, e1, e2, c11, and c21 obtained on the last cycle of calculation in the Section 14.3 example
Show that rMO of (14.6) is greater than rVB of (14.6) at the midpoint of the line joining the nuclei.In Equation 14.6
(a) Write a computer program that will perform the helium-atom SCF calculation of Section 14.3. Have the input to the program be ζ1, ζ2, and the initial guess for c11/c21. Do not use the Section 14.3 values of the integrals, but have the program calculate all integrals from ζ1 and ζ2. Have the
Calculate r for the He SCF wave function of the Section 14.3 example at r = 0 and at r = 1 bohr.
Given thatWhere the x's functions are orthonormal, show that the orbitals Ï•i form an orthonormal set if the matrix C' of coefficients c'si is unitary.
Which of the following functions are homogeneous? Give the degree of homogeneity. (a) x + 3yz; (b) 179; (c) x2/yz3; (d) (ax3 + bxy2)1/2
A one-dimensional harmonic oscillator in a stationary state has (T) = 5.0 X 10-19 J. Find E and (V) for this state.
Show that the dipole moment (14.9) of a system of charges is independent of the choice of the coordinate origin, provided the system has no net charge.
The zero level of potential energy is arbitrary and we can always add a constant C to the potential energy function V. If we add C to V, state what happens to each of the following for a stationary state: (V), (T), E. Do these results contradict the virial theorem? Explain your answer.
Prove that ˆ‚(V) / ˆ‚R must be nonnegative at R = Re; that is, (V) cannot be increasing with decreasing R as we go through the minimum in the U® curve (Fig. 14.1). State and prove the corresponding theorem for (Tel).
Let ( be the complete wave function for a molecule, with the Born-Oppenheimer approximation ( = (el(N not necessarily holding. Is it true thatWhere Tel and TN are the kinetic-energy operators for the electrons and nuclei and V is the complete potential-energy operator? Justify your answer.
Given that De = 4.75 eV and Re = 0.741 Å for the ground electronic state of H2, find U(Re), (V)Re, (Vel)Re, and (Tel)Re for this state.
The Fues potential-energy function for nuclear vibration of a diatomic molecule is U(R) = U(∞) + De(-2Re/R + Re2/R2). Find the expressions for (Tel) and (V) predicted by this potential and comment on the results.
(a) Apply the generalized Hellmann-Feynman theorem with Z as the parameter to find (1/r) for the hydrogen like-atom bound states (nlm. (b) Since hydrogen like functions (nlm with the same n but different l or m have the same energy, we must be sure that the functions (nlm are the correct
(a) Explain why the permanent dipole moment of a many-electron atom in a stationary state is always zero. (b) Explain why the permanent electric dipole moment of H can be nonzero for certain excited states. (c) Show qualitatively that two of the four correct zeroth-order functions of Prob. 9.23
Use Fz,a = -ˆ‚U/ˆ‚za and to show that in Fig. 14.4, Fz,a = -(ˆ‚U/ˆ‚R)[(za- zb)/R]. Find a similar equation for Fz,b and verify that Fz,a = -Fz,b.
For NaCl, Re = 2.36 Å. The ionization energy of Na is 5.14 eV, and the electron affinity of Cl is 3.61 eV. Use the simple model of NaCl as a pair of spherical ions in contact to estimate De and the dipole moment of NaCl. Compare with the experimental values De = 4.25 eV and m = 9.0 D. [One debye
Explain the origin of the extra terms in the molecular Hartree-Fock operator (14.26) as compared with the atomic Hartree operator of (11.9) and (11.7).In Equation 14.26
Verify that the Coulomb and exchange integrals Jij and Kij can be written in terms of theCoulomb and exchange operators of Section 14.3 as
Verify Eq. (14.40) for the Kj integral over basis functions.In Equation 14.40
Verify that (15.3) are the possible symmetry species for Ï2v.
The fit in Prob. 15.9 gives S(r) ≈ Σ3i = 1 Ci Gi (r), where W(r) is a 1s STO with orbital exponent 1 and Gi is a 1s GTO with orbital exponent αi. Let S(r, ζ) = ζ3/2π-1/2e- ζr. Let Gi (r,ζ) function obtained by replacing αi in Gi(r) by αiζ2. Show that S(r, ζ) ≈ Σ3i = 1 Ci Gi (r, ζ).
Use the Basis Set Exchange (bse.pnl.gov) to find (a) The 6-31G** basis functions for H; (b) The 6-31G* basis functions for C.
Give an example of a molecule for which the HF/6-31G* and HF/6-31G** energies are the same.
Sketch the 1a1, 3a1, 1b1, 4a1, and 2b2 MOs of water.
Give the form of the normalization constants for the symmetry orbitals in Eq. (15.13).
Suppose a ground-state calculation gives us some virtual orbitals for the molecule M. In which one of the following species would an excited electron occupy an MO that was well approximated by a virtual orbital of ground-state M? (a) M; (b) M+; (c) M-. Explain.
Work out the possible symmetry species for
Suppose an SCF calculation that includes 3d orbitals on oxygen is done on H2O (Fig. 15.1). For each of the occupied ground-state H2O MOs, use symmetry-species arguments to decide which of the following 3d oxygen AOs will contribute to that MO: 3dz2, 3dxz, 3dyz, 3dxy, 3dx2-y2.In Figure 15.1
(a) As noted in the text, for the cc-pVnZ basis sets, it is best not to use the n = 2 energy value in extrapolations. Also, the n = 6 energy value is often not available. Use (15.23) and the H2O HF/cc-pVnZ energies for n = 3, 4, and 5 given just before (15.23) to find the values of the three
(a) Instead of (15.23), the following equation has been proposed for extrapolation from SCF aug-cc-pVQZ and aug-cc-pV5Z energies to the CBS limit [A. Karton and J. M. L. Martin, Theor. Chem. Acc., 115, 330 (2006)]:Where L is the largest orbital-angular momentum l value that appears in the basis
(a) Use (15.24) to show that(b) Use the result of (a) to verify that
(a) Verify the net populations given at the end of the Section 15.6 example. (b) Verify the H2O 2a1 and 1b2 interatomic overlap populations given in Section 15.6. (c) Verify the gross populations given in Section 15.6 for the H2O basis functions.
Use Coulomb's law to show that the electric potential ϕP at a point P a distance d from a point charge Q is ϕP = Q/4πε0d.

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