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Quantum Chemistry 7th edition Ira N. Levine - Solutions

(a) The infrared absorption spectrum of 1H35Cl has its strongest band at 8.65 × 1013 Hz. Calculate the force constant of the bond in this molecule. (b) Find the approximate zero-point vibrational energy of 1H35Cl. (c) Predict the frequency of the strongest infrared band of 2H35Cl.
The v = 0 → 1 and v = 0 → 2 bands of 1H35Cl occur at 2885.98 cm-1 and 5667.98 cm-1. (a) Calculate ve/c and ne vexe/c for this molecule. (b) Predict the wavenumber of the v = 0 → 3 band of 1H35Cl.
(a) The v = 0 → 1 band of LiH occurs at 1359 cm-1. Calculate the ratio of the v = 1 to v = 0 populations at 25°C and at 200°C. (b) Do the same as in (a) for ICl, whose strongest infrared band occurs at 381 cm-1.
(a) Verify (4.62). (b) Find the corresponding equation for the v = 0 → v2 transition.
Show that if one expands U(R) in Fig. 4.6 in a Taylor series about R = Re and neglects terms containing (R - Re)3 and higher powers (these terms are small for R near Re), then one obtains a harmonic-oscillator potential with k = d2/dR2| R = Re.
The Morse function U® = De[1 - e-a(R-Re)]2 is often used to approximate the U(R) curve of a diatomic molecule, where the molecule's equilibrium dissociation energy De is De K U(∞) - U(Re). (a) Verify that this equation for De is satisfied by the Morse function. (b) Show that a = (ke/2De)1/2.
(a) Find the Taylor-series expansion about x = 0 for ex. (b) Use the Taylor series (about x = 0) of sin x, cos x, and ex to verify that eiθ = cos θ + i sin θ [Eq. (1.28)].
Use the Numerov method to find the lowest three stationary-state energies for a particle in a one-dimensional box of length l with walls of infinite height. use either a program similar to that in Table 4.1, a spreadsheet, or a computer-algebra system such as Mathcad. If negative eigenvalues are
(a) Use the Numerov method to find all the bound-state eigenvalues for a particle in a rectangular well (Section 2.4) of length l with V0 = 20h2/ml2. Note that V is different in different regions and ψ ≠ 0 at the walls. (b) Repeat (a) for V0 = 50h2/ml2. (c) Check your results using the automatic
Use the Numerov method to find the lowest three energy eigenvalues for a one-particle system with V = cx4, where c is a constant. Use either a program similar to that in Table 4.1, a spreadsheet, or a computer-algebra system such as Mathcad. If negative eigenvalues are being sought using Excel
Use the Numerov method to find the lowest three energy eigenvalues for a one-particle system with V = ax8, where a is a constant. Use either a program similar to that in Table 4.1, a spreadsheet, or a computer-algebra system such as Mathcad. If negative eigenvalues are being sought using Excel
Use the Numerov method to find the lowest four energy eigenvalues for a one-particle system with V = ∞ for x ≤ 0 and V = bx for x > 0, where b is a positive constant. (For b = mg, this is a particle in a gravitational field.) Use either a program similar to that in Table 4.1, a spreadsheet, or
Consider a one-particle system with V = - 31.5 (h2/ma2) / (ex/a + e-x/a)2, where a is a positive constant. (a) Find Vr. (b) Use a spreadsheet, graphing calculator, or Mathcad to graph Vr versus xr. (See Prob. 4.39.) (c) Use the Numerov method to find all bound-state eigenvalues less than - 0.1.
Consider a one-particle system with V = 1/4 b2/c - bx2 + cx4, where b and c are positive constants. If we use h, m, and b to find A and B in Er = E/A and xr = x/B, we will get the same results as for the harmonic oscillator, except that k is replaced by b. Thus, Eq. (4.74) gives B = m-1/4b-1/4h1/2.
A one-dimensional double-well potential has V = ∞ for x < -1/2 l, V = 0 for -1/2 l ≤ x ≤ -1/4 l, V = V0 for -1/4 l < x < 1/4 l, V = 0 for 1/4 l ≤ x ≤ 1/2 l, and V = ∞ for x > 12 l, where l and V0 are positive constants. Sketch V. Use the Numerov method to find the lowest four
(a) For the harmonic-oscillator Numerov example, we went from - 5 to 5 in steps of 0.1 and found 0.4999996 as the lowest eigenvalue. For this choice of xr, 0 and sr, find all eigenvalues with Er < 6; then find the eigenvalue that lies between 11 and 12 and explain why the result is not accurate.
Spreadsheets contain pitfalls for the unwary. (a) If cell A1 contains the value 5, what would you expect the formula = −A1^2+A1^2 to give? (b) Using Excel, enter 5 in cell A1, enter = −A1^2+A1^2 in cell A2, and enter = +A1^2−A1^2 in A3. What results do you get?
Derive (4.29) for E of a classical oscillator.
Use (4.67) to show that if one multiplies c1 in the Numerov method by a constant c, then ψ2, ψ3, care all multiplied by c, so the entire wave function is multiplied by c, which does not affect the eigenvalues we find.
Use the normalized Numerov-method harmonic-oscillator wave functions found by going from -5 to 5 in steps of 0.1 to estimate the probability of being in the classically forbidden region for the v = 0 and v = 1 states.
In the Taylor series (4.85) of Prob. 4.1, let the point x = a be called xn (that is, xn ‰¡ a) and let s ‰¡ x - a = x - xn, so x = xn + s.(a) Use this notation to write (4.85) as f(xn + s) equal to a power series in s and evaluate all terms through s5.(b) In the result of part
Use dimensional analysis to verify (4.74) for B.
In applying the Numerov method to count the nodes in ψr, we assumed that ψ changes sign as it goes through a node. However, there are functions that do not have opposite signs on each side of a node. For example, the functions y = x2 and y = x4 are positive on both sides of the node at x = 0. For
Suppose V = cx8, where c is a positive constant, and we want all eigenvalues with Er < 10. (a) Show that Vr = x8r and that for Er = 10 the boundaries of the classically allowed region are at xr = ± 1.33. (b) Set up a spreadsheet and verify that if we take xr, 0 = - 3, xr,max = 3, and sr = 0.05, ψ
Modify the program of Table 4.1 to find the normalized wave function.
Rewrite the program of Table 4.1 to eliminate all array variables.
Spreadsheets and computer-algebra systems can easily be used to solve equations of the form f(x) = 0. For example, suppose we want to solve ex = 2 - x2. A computer-made graph shows that the function ex - 2 + x2 equals zero at only two points, one positive and one negative. In Excel, enter an
(a) Find the recursion relation for the coefficients cn in the power-series solution of (1 - x2) y"(x) - 2xy - (x) + 3y (x) = 0. (b) Express c4 in terms of c0 and c5 in terms of c1.
(a) Show that if ki and fi are eigenvalues and eigenfunctions of the linear operator A, then cki and fi are eigenvalues and eigenfunctions of cA. (b) Give an operator whose eigenvalues are 1/2, 3/2, 5/2, . . . . (c) Give an operator whose eigenvalues are 1, 2, 3,c.
(a) A certain system in a certain stationary state has ψ = Ne-ax4 (N is the normalization constant.) Find the system's potential-energy function V(x) and its energy E. (b) Sketch V(x). (c) Is this the ground-state ψ? Explain.
True or false? (a) In the classically forbidden region, E > V for a stationary state. (b) If the harmonic-oscillator wave function ψv is an even function, then ψv+1 is an odd function. (c) For harmonic-oscillator wave functions, ∫∞-∞ ψ*v(x) ψv+1 (x) dx = 0. (d) At a node in a bound
Which of the following are even functions? odd functions? (a) sin x; (b) cos x; (c) tan x; (d) ex; (e) 13; (f) x cos x; (g) 2 - 2x; (h) (3 + x)(3 - x).
Prove the statements made after Eq. (4.51) about products of even and odd functions.
(a) If f(x) is an even function that is everywhere differentiable, prove that f -(x) is an odd function. Do not assume that f(x) can be expanded in a Taylor series. (b) Prove that the derivative of an everywhere-differentiable odd function is an even function. (c) If f(x) is an even function that
For the ground state of the one-dimensional harmonic oscillator, find the average value of the kinetic energy and of the potential energy; verify that (T) = (V) in this case.
Let A have the components (3, - 2, 6); let B have the components (-1, 4, 4). Find |A|, |B|, A + B, A - B, A B, A × B. Find the angle between A and B.
Use the vector dot product to find the obtuse angle between two diagonals of a cube. What is the chemical significance of this angle?
(a) Use the vector dot product to show that in HCBr3, cos (∠BrCBr) = 1 - 1.5 sin2(∠HCBr). (b) In HCBr3, HCBr = 107.2°. Find BrCBr in HCBr3.
Let f = 2x2 - 5xyz + z2 - 1. Find grad f. Find ∇2f.
The divergence of a vector function A is a scalar function defined by(a) Verify that div [grad g(x, y, z)] ‰¡ ˆ‡ ˆ™ ˆ‡g = ˆ‚2g/ˆ‚x2 + ˆ‚2g/ˆ‚y2 + ˆ‚2g/ˆ‚z2. This is the origin of the notation
For the vector (3, -2, 0, 1) in four-dimensional space, find (a) The length; (b) The direction angles.
Derive Eq. (5.68) for L2 from Eqs. (5.65)-(5.67).
Find the spherical coordinates for points with the following (x, y, z) coordinates: (a) (1, 2, 0); (b) (-1, 0, 3); (c) (3, 1, -2); (d) (-1, -1, -1).
Verify the commutator identities (5.1)-(5.5).
Find the (x, y, z) coordinates of the points with the following spherical coordinates: (a) r = 1, u = p/2, f = p; (b) r = 2, u = p/4, f = 0.
Give the shape of a surface on which (a) r is constant; (b) u is constant; (c) f is constant.
By integrating the spherical-coordinates differential volume element dτ over appropriate limits, verify the formula 4/3 πR3 for the volume of a sphere of radius R.
Calculate the possible angles between L and the z axis for l = 2.
(a) Show that for the orbital-angular-momentum eigenfunctions, the smallest possible value for the angle between L and the z axis obeys the relation cos2 θ = l/ (l + 1). (b) As l increases, does this smallest-possible angle increase or decrease?
Substitute d/dθ = - (1 - w2)1/2(d/dw) [which follows from (5.85)] and Eq. (5.85) into d2S/dθ2 = (d/dθ) (dS/dθ) to verify Eq. (5.86).
Derive the formula for S2,0 (Table 5.1) in two ways: (a) by using (5.146) in Prob. 5.34; (b) by using the recursion relation and normalization.
Use the recursion relation (5.98) and normalization to find (a) Y03; (b) Y13.
Apply the L2 operator (5.68) to Y02 and verify that the eigenvalue equation (5.104) is obeyed.
(a) If we measure Lz of a particle that has angular-momentum quantum number l = 2, what are the possible outcomes of the measurement? (b) If we measure Lz of a particle whose state function is an eigenfunction of L2 with eigenvalue 12h2, what are the possible outcomes of the measurement?
If we measure Ly of a particle that has angular-momentum quantum number l = 1, what are the possible outcomes of the measurement?
At a certain instant of time t', a particle has the state function ψ = Ne-ar2Y12 (θ, ϕ), where N and a are constants. (a) If L2 of this particle were to be measured at time t', what would be the outcome? Give a numerical answer. (b) If Lz of this particle were to be measured at t', what would be
The associated Legendre functions Pl|m| (w) are defined byVerify that P00(w) = 1, P01(w) = w, P11(w) = (1 - w2)1/2 and find P02(w), P12(w), and P22(w). It can be shown that (Pauling and Wilson, page 129) Equations (5.146) and (5.145) give the explicit formula for the normalized theta factor in the
Apply the lowering operator L- three times in succession to Y11 (θ, ϕ) and verify that we obtain functions that are proportional to Y01, Y1-1, and zero.
True or false? (a) The L2 eigenvalues are degenerate except for l = 0. (b) Since L2Ylm = l(l + 1)h2Ylm, it follows that L2 = l(l + 1)h2. (c) L2 commutes with Lx. (d) Y00 is a constant. (e) Y00 is an eigenfunction of L2, Lx, Ly, and Lz. (f) If A and B do not commute, it is impossible for an
For the ground state of the one-dimensional harmonic oscillator, compute the standard deviations Δx and Δpx and check that the uncertainty principle is obeyed. Use the results of Prob. 4.9 to save time.
At a certain instant of time, a particle in a one-dimensional box of length l (Fig. 2.1) is in a nonstationary state with ψ = (105 / l7)1/2x2(l - x) inside the box. For this state, find Δx and Δpx and verify that the uncertainty principle Δx Δpx ≥ 1/2 h is obeyed.
Show that the standard deviation ΔA is 0 when ψ is an eigenfunction of A.
Derive (ΔA)2 = 〈A2〉 - 〈A〉2 [Eq. (5.11)].
Let w be the variable defined as the number of heads that show when two coins are tossed simultaneously. Find 〈w〉 and σw.
Classify each of these as a scalar or vector. (a) 3B; (b) C: B; (c) C B; (d) B; (e) Velocity; (f) Potential energy.
The J = 2 to 3 rotational transition in a certain diatomic molecule occurs at 126.4 GHz, where 1 GHz ≡ 109 Hz. Find the frequency of the J = 5 to 6 absorption in this molecule.
The J = 7 to 8 rotational transition in gas-phase 23Na35Cl occurs at 104189.7 MHz. The relative atomic mass of 23Na is 22.989770. Find the bond distance in 23Na35Cl.
For a certain diatomic molecule, two of the pure-rotational absorption lines are at 806.65 GHz and 921.84 GHz, where 1 GHz ≡ 109 Hz, and there are no pure-rotational lines between these two lines. Find the initial J value for each of these transitions and find the molecular rotational constant B.
(a) For 12C16O in the v = 0 vibrational level, the J = 0 to 1 absorption frequency is 115271.20 MHz and the J = 4 to 5 absorption frequency is 576267.92 MHz. Calculate the centrifugal distortion constant D for this molecule. (b) For 12C16O in the v = 1 level, the J = 0 to 1 absorption is at
Verify Eq. (6.51) for I of a two-particle rotor. Begin by multiplying and dividing the right side of (6.50) by m1m2 / (m1 + m2). Then use (6.49).
Calculate the ratio of the electrical and gravitational forces between a proton and an electron. Is neglect of the gravitational force justified?
(a) Explain why the degree of degeneracy of an H-atom energy level is given by(b) Break this sum into two sums. Evaluate the first sum using the fact that Show that the degree of degeneracy of the H-atom levels is n2 (spin omitted). (c) Prove that by adding corresponding terms of the two series 1,
(a) Calculate the wavelength and frequency for the spectral line that arises from an n = 6 to n = 3 transition in the hydrogen atom. (b) Repeat the calculations for He+; neglect the change in reduced mass from H to He+.
Assign each of the following observed vacuum wavelengths to a transition between two hydrogen- atom levels: 656.47 nm, 486.27 nm, 434.17 nm, 410.29 nm.........(Balmer series) Predict the wavelengths of the next two lines in this series and the wavelength of the series limit. (Balmer was a Swiss
Each hydrogen-atom line of Prob. 6.18 shows a very weak nearby satellite line. Two of the satellites occur at the vacuum wavelength 656.29 nm and 486.14 nm. (a) Explain their origin. (The person who first answered this question got a Nobel Prize.) (b) Calculate the other two satellite wavelengths.
The particle in a spherical box has V = 0 for r ≤ b and V = ∞ for r > b. For this system: (a) Explain why ψ = R(r)f(θ, ϕ), where R(r) satisfies (6.17). What is the function f(θ, ϕ)? (b) Solve (6.17) for R® for the l = 0 states. The substitution R(r) = g(r)/r reduces (6.17) to an easily
Verify that for large values of j, the ratio bj+1/bj in (6.88) is the same as the ratio of the coefficient of rj+1 to that of r j in the power series for e2Cr.
For the particle in a box with infinitely high walls and for the harmonic oscillator, there are no continuum eigenfunctions, whereas for the hydrogen atom we do have continuum functions. Explain this in terms of the nature of the potential-energy function for each problem.
The positron has charge + e and mass equal to the electron mass. Calculate in electronvolts the ground-state energy of positronium-an "atom" that consists of a positron and an electron.
For the ground state of the hydrogenlike atom, show that 〈r〉 = 3a/2Z.
Find 〈r〉 for the 2p0 state of the hydrogenlike atom.
Find 〈r2〉 for the 2p1 state of the hydrogenlike atom.
For a hydrogenlike atom in a stationary state with quantum numbers n, l, and m, prove that
Derive the 2s and 2p radial hydrogenlike functions.
What is the value of the angular-momentum quantum number l for a t orbital?
If the three force constants in Prob. 4.20 are all equal, we have a three-dimensional isotropic harmonic oscillator. (a) State why the wave functions for this case can be written as ψ = f(r)G(θ, ϕ). (b) What is the function G? (c) Write a differential equation satisfied by f(r). (d) Use the
If we were to ignore the interelectronic repulsion in helium, what would be its ground-state energy and wave function? (See Section 6.2.) Compute the percent error in the energy; the experimental He ground-state energy is - 79.0 eV.
For the ground state of the hydrogenlike atom, find the most probable value of r.
(a) For the hydrogen-atom ground state, find the probability of finding the electron farther than 2a from the nucleus. (b) For the H-atom ground state, find the probability of finding the electron in the classically forbidden region.
A stationary-state wave function is an eigenfunction of the Hamiltonian operator H = T + V. Students sometimes erroneously believe that c is an eigenfunction of T and of V. For the ground state of the hydrogen atom, verify directly that c is not an eigenfunction of T or of V, but is an
For the hydrogen-atom ground state, (a) find 〈V〉; (b) use the results of (a) and 6.35 to find 〈T〉; then find 〈T〉 > 〈V〉; (c) use 〈T〉 to calculate the root-mean-square speed 〈v2〉1/2 of the electron; then find the numerical value of 〈v2〉1/2/c, where c is the speed of light.
The hydrogenlike wave functions 2p1, 2p0, and 2p-1 can be characterized as those 2p functions that are eigenfunctions of Lz. What operators can we use to characterize the functions 2px, 2py, and 2pz, and what are the corresponding eigenvalues?
Given that Af = af and Ag = bg, where f and g are functions and a and b are constants, under what condition(s) is the linear combination c1f + c2g an eigenfunction of the linear operator A?
Verify Eq. (6.6) for the Laplacian in spherical coordinates. (This is a long, tedious problem, and you probably have better things to spend your time on.)
State which of the three operators L2, Lz, and the H-atom H each of the following functions is an eigenfunction of: (a) 2pz; (b) 2px; (c) 2p1.
For the real hydrogenlike functions: (a) What is the shape of the n - l - 1 nodal surfaces for which the radial factor is zero? (b) The nodal surfaces for which the ϕ factor vanishes are of the form ϕ = constant. Thus they are planes perpendicular to the xy plane. How many such planes are there?
Verify the orthogonality of the 2px, 2py, and 2pz functions.
Find the radius of the sphere defining the 1s hydrogen orbital using the 95% probability definition.
Show that the maximum value for 2py [Eq. (6.123)] is k3/2 π-1/2e-1. Use Eq. (6.123) to plot the 2py contour for which ψ = 0.316ψmax.
Sketch rough contours of constant |ψ| for each of the following states of a particle in a twodimensional square box: nxny = 11; 12; 21; 22. What are you reminded of?

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