Questions and Answers of Molecular Spectroscopy

In this problem you will calculate the Stark effect on the rotational spectrum of a symmetric top molecule. The perturbation operator for a dipole o in an electric field E is H'=-μË. Take the
Derive Equations 5.41 and 5.42 for the spectra of a classical free rotor, and plot them using reduced units: ω* = (I/kBT)1/2ω. Do not worry about finding a closed form expression for the TCF of
Find the difference between the ground- and first excited-state translational energies of a helium atom in a 1 mm3 cubic box. Compare this to room temperature energy kBT at 300 K.
Find all the possible angles with respect to the z-axis for an angular momentum vector of magnitude √6h.
Show that the uncertainty as defined in Equation 1.15 vanishes when ψ is an eigenfunction of the operator Â. ΔΑ= (A2)-(A)
Can a harmonic oscillator ever be dissociated?
The classical harmonic oscillator obeys the equation of motion: F = −kx = md2x/dt2. Use this equation to verify Equation 1.29.Equation 1.29 Yf(r;Q)=y, (r;Q)x(Q)=yg (r;Q)x, (Q₁) X₂ (Q₂)...
The blackbody energy density given in Equation 2.75 is the energy density per unit frequency interval. In other words, du = ρ(ν)dν is the energy per unit volume between ν and ν + dν. Convert
The momentum of a photon is given by the de Broglie relation: p = h/λ = ħk. Suppose that a flat surface 1.00 m2 in area completely absorbs 1000 W of 400 nm light. Calculate the radiation pressure.
Convert the limits of the visible spectrum, 400–700 nm, to kJ/mol, eV, and cm−1.
The total intensity of sunlight, for all visible wavelengths striking the planet, is about 1000 W/m2. In a wavelength interval of about 1 nm in the visible range, the intensity is on the order of 2
Hydrides typically have rotational energies that are too widely spaced to be able to use Equation 1.99 at room temperature. Find zrot for HF at 300 K by summing the energies directly. Try to obtain a
Use the partition functions in Equations 1.98, 1.99, and 1.100 to find the translational, rotational, and vibrational contributions to the average energy of a diatomic molecule. Compare each result
Calculate the translational partition function for a mole of N2 at 1 atm and 300 K. Verify that the effective number of available states greatly exceeds the number of molecules.
Calculate the fraction of molecules in each of the first ten rotational levels of CO at 300 K, assuming that it is a rigid rotor with a bond length of 1.13 Â.
Show that if |n〉 is an eigenfunction of the operator Â with eigenvalue an, then Expand the exponential in a series (μ)² = (1/³ U u
Derive Equation 1.89.Equation 1.89 E =kBT²| دها alnz ar N,V
The vibrational frequency of 1H35Cl is ν0 = 8.97 × 1013 s−1. Find the force constant for the H–Cl bond.
Verify the conversion factor in front of the integral in Equation 6.26. Derive the conversion factor for finding the oscillator strength from the integrated molar absorptivity according to Equation
Graph a Gaussian and Lorentzian function with the same width at half the maximum intensity and comment on the essential difference between the two. What is the second moment in each case?
Given that the temperature of the sun is about 5700 K, use your result from Problem 4 to estimate the wavelength of the maximum in the solar emission spectrum.Problem 4The blackbody energy density
Are two-electron transitions permitted by electric dipole selection rules? To arrive at your answer, first consider the wavefunction to be a simple product of spin–orbitals. (This is called a
Refer to the energy level diagram for Ar+ given in Figure 7.6 and find the spin–orbit coupling constant A for the ground state 2P and excited state 4D terms. Predict the wavelength of the
The 2P →2S transition of Li is split by 0.34 cm−1. Find spin orbit coupling constant for the 2P multiplet and compare to the values found in the previous problem for Ar+.Data from previous
This problem illustrates the separation of internal and external motions for a one-dimensional rigid rotor. Consider a diatomic molecule consisting of masses m1 and m2 at positions x1 and x2. Using
Show that the inverse Fourier transform of ω2I(ω) gives the correlation functionYou can use the fact that an equilibrium average is independent of the origin of time: (À(0) · Â(t)).
Derive the expression given in Equation 5.71 for the nth spectral moment.Equation 5.71 M₁ = (-i)". - (A· A(t)) | ₁-0 dn dt" t=0
Verify that the imaginary part of the susceptibility χ′′ given by Equation 5.36 is equivalent to the expression of Equation 5.15 in the long-wavelength limit.Equation 5.56Equation 5.15 =
Prove that the free-rotor correlation function is Gaussian at short times.
Consider a single molecule with an electronic transition at 530 nm that is exposed to a source having an energy density of 10−19 J/m3 Hz at that wavelength. The transition dipole moment of the
Calculate the oscillator strength for the 1s → 2pz transition of the hydrogen atom. The wavefunctions arewhere a0 = 0.0529 nm is the Bohr radius. By symmetry, only the z component of the dipole
Derive the Thomas–Reiche–Kuhn sum rule for the oscillator strength:to a more useful quantity. The resolution of the identity is also helpful on this problem. it j fj =1
Treat a one-electron atom according to the Lorentz model and calculate the oscillator strength for the v = 0 → v = 1 transition. The relevant wavefunctions arewhere α ≡ (mω/ħ).
Derive the E1 selection rules for an electron in a one-dimensional box of length L. In other words, for what changes in the quantum number n will the transition dipole μnn' be nonzero? The
The molar absorptivity of the dye rhodamine 6G is about 105 L/mol cm at 540 nm. Estimate ε″ for a 10−3 M solution of this dye in ethanol.
Show that the classical Hamiltonian in Equation 2.54 is consistent with the Lorentz forceTo proceed, calculate the force in the x direction, Fx = mẍ, the help of the following expressions from
Use the formula for the vector potential of the quantized radiation fieldto derive the radiation Hamiltonian:See Section for an outline of the necessary steps. 1 A(r,t)=, - Νεον k qu(t)ük(r)
The work function of Cs is 2.14 eV. Find the maximum kinetic energy of electrons ejected by light of wavelength 400 nm.
An argon ion laser emits green light at 514.5 nm with a power of 1 W concentrated in a beam of cross section 0.01 cm2. Calculate(a) The electric field amplitude.(b) The photon flux.
Consider an electron moving parallel to an electric field of 100 V/m and perpendicular to a magnetic field of 1 T. How fast would the electron have to travel for the electric and magnetic forces to
Solve the Heisenberg equations of motion for the momentum and position of a harmonic oscillator:Verify that the expectation values evolve in time like those for a classical harmonic oscillator; that
Show thatInsert the resolution of the identity,between the operators y and ∂/∂z. (ƒ|yôlôz\i)=-mwf/ħ{f|yz|i).
The molar polarization PM of methanol decreases from about 36 to 33 cm3/mol on increasing the temperature from 298 to 333 K. Use the Debye equation to estimate the polarizability and dipole moment of
Below what temperature does the Debye equation predict ferroelectric behavior for water? Use μ0 = 1.86 D and α/4πε0 = 1.5 × 10−24 cm3.
The refractive index of CCl4 is nD = 1.4601 and the density is 1.594 g/ml. Use the Lorenz Lorentz equation to estimate the polarizability of CCl4. Compare to the literature value α/4πε0 = 1.25 ×
Estimate the polarizability of a ground state hydrogen atom from its size, in MKS and cgs units. Compare to the literature value, α/4πε0 = 0.667 Å3.
Consider four charges, equal in magnitude. Two positive charges are located at (x, y) = (1,1) and (−1,−1) and two negative charges are at (1,−1) and (−1,1). Calculate the nonzero components
Show that the dipole moment of a collection of charges is independent of origin, provided that the net charge is zero. Also, show that the quadrupole moment is independent of origin if the dipole
Use the data given in Table 9.2 to find the dissociation energies D̃e and D̃0 (in units of cm−1) for 2H35Cl. Table 9.2 Spectroscopic constants for diatomics in the ground electronic state V,
The vibrational frequency of 1H79Br is ṽe=−2649.7cm1 and the anharmonicity is xeṽe =−452.cm1. Find the frequencies of the fundamental, first overtone and second overtone for 1H79Br,
The rotational constants for HF are(a) Find the initial J value and the frequency for the most intense rotational transition taking place in the ground vibrational state at room temperature.(b) What
The rotational constant Be for 12C16O is 1.93127 cm−1 in the ground electronic state and 1.3099 cm−1 in the excited triplet electronic state. Calculate the bond length of CO in both the ground
Show thatfor ħω small compared to kBT. M2n-1 (h/2kgT) M 2n ≈
The inertia of a diatomic molecule is Iz = μR2. Calculate the rotational second moment of CO in cm−2, taking the bond distance to be 1.13 Å.
Show that the Debye model for the imaginary part of the permittivity, Equation 5.49, leads to a Lorentzian lineshape for the intensity I(ω).Equation 5.49 (Es-E..)@TD 1+ @o²t &(@)=- 2_2 D
Consider an electronic absorption band having an oscillator strength of 0.5 and an absorption maximum at 500 nm. Find(a) The transition dipole moment(b) The Einstein B coefficient(c) The radiative
Consider a molecule having a vibrational frequency of 2000 cm−1 and a transition dipole of 0.1 D. Calculate the radiative lifetime of the v = 1 state. What does this lifetime tell you about the
Compare the radiative linewidth obtained in the previous problem to the Doppler width that would be observed at room temperature. Assume a molecular mass equal to that of a CO molecule.
Find expressions for the Kubo lineshape formula ϕ(t) in the limits ATc¹.
Estimate the ratio of the rate of stimulated to spontaneous emission at three wavelengths:(a) 10 cm, in the microwave(b) 10−2 cm, in the far-IR(c) 10−5 cm, in the near-infrared. Assume a
Compute the transition dipole and oscillator strength for the following hydrogen atom transitions 1s ↔ 3p0 and 1s ↔ 4p0.
Explain why hydrogen atom E1 transitions having Δl = 0 are forbidden by symmetry.
Compute the three lowest energy transitions in the Lyman series for deuterium and compare to the same lines in hydrogen.
Figure out the term symbols associated with the following configurations:(a) s1p5(b) p3(c) p1d1(d) 2p13p2. For each case, decide the order of the term energies, and sketch a diagram showing how
For each transition in Figure 7.6 state whether or not it is permitted by E1, M1, and E2 selection rules.Figure 7.6 2P12 2
Compute the Landé g-factor for the 2P3/2 state of Figure 7.8 and complete the sketch of the field-induced splittings. Calculate the splittings, in wavenumbers, for a magnetic field of 1.50
Sketch the effect of a 10 kG magnetic field on the transition responsible for the 514.5 nm line of the argon ion laser. Compute the wavelengths of the allowed transitions.
Compute the effective temperature of Ne atoms emitting at 632.8 nm if the Doppler broadening is 1500 MHz.
Calculate the rotational constant for 13C16O in its ground electronic state.
The barrier to rotation about the C–C bond in CH3CH2Cl is about 15 to 20 kJ/mol. Estimate the torsional frequency and predict how it could be observed experimentally. Consult a table of bond
Assign the H2 pure rotational transitions in Figure 8.8, using data from Table 9.2 to calculate the predicted frequencies. Assuming the spectra were taken at room temperature, account for the
Prove that the moments of inertia Ia and Ib (see Figure 8.3) are equal for the benzene molecule. You do not need to know the bond distances, just invoke the hexagonal symmetry, and for simplicity,
The rotational constants B0 have been determined for three isotopic derivatives of chloroacetylene:Assume that the carbon isotope is 12C in all cases. Calculate the three bond distances in
Use the data given in Table 9.2 for 1H35Cl to estimate the cubic and quartic force constants: V(3) and V(4) Table 9.2 Spectroscopic constants for diatomics in the ground electronic state V, cm-¹x,V,
The vibrational frequency of NO is 1890 cm−1, the bond length is 1.1508 Å, and the dissociation energy is D0 = 6.5 eV. Use this information to find the Morse function for NO. Make a sketch of the
Use the harmonic oscillator raising and lowering operators to derive the selection rules for vibrational transitions that result from electrical anharmonicity in the form (²μ/dq²)oq².
According to Equation 9.30, the vibration–rotation constant for a harmonic oscillator is not zero and is in fact negative. Consider the rotational constant for the v = 0 state of a harmonic
Use the data in Table 9.2 to assign the Q branch lines in Figure 9.6. Account for the relative intensities.Figure 9.6 Table 9.2 Spectroscopic constants for diatomics in the ground electronic
Calculate the first-order correction to the energy of a harmonic oscillator, Ev(1), using the perturbation operator Ĥ' = bq2 + dq4 as defined. Compare your result to that obtained using
Using data from Table 9.2, find the force constants k for O2, I2, and N2. Is there a physical explanation for the relative magnitudes of these values? Table 9.2 Spectroscopic constants for diatomics
Estimate the radiative lifetime of a diatomic molecule in the v = 1 vibrational state, given a dipole derivative (∂μ/∂q)0 of about 10 D/Å and a vibrational frequency of 2000 cm−1.
The compound shown below has λmax = 736 nm in hexane and 553 nm in water. Explain the basis for the solvent shift. H3C H3C Z NO
The ground electronic state of Cr3+ in an octahedral field is 4A2g, which derives from a configuration. Is this state subject to Jahn–Teller distortion? 3 28
The SO2 molecule, analogous to H2O, has a 1A1 ground electronic state with the valence electronic configuration …(1a2)2(4b2)2(6a1)2(2b1)0 and low-lying excited singlet states of symmetry 1B1, 1A2,
Assign the bands in the benzene spectra displayed in Figures 10.12 and 10.13.Figure 10.12Figure 10.13 Absorbance benzene Qu m. 2000 Frequency, v (cm-¹) toluene 1000 3000
The absorption spectrum of I2 has E00/hc = 15,677 cm−1, and the onset of the continuum is at 19,735 cm−1. The excited state dissociates to I(2P3/2) + I(2P1/2), which is 7589 cm−1 above the
The bond length of CO is 1.128 Å in the ground electronic state and 1.370 Å in the first excited state. The vibrational frequency ṽe is 2170 cm−1 in the ground state and 1172 cm−1 in the
A general harmonic potential function for water isThe second line contains off-diagonal force constants, while the first three terms are diagonal. In matrix form, this can be expressed as is the
Find the state symmetries that derive from the ground and first excited electronic configurations of N2, F2, and CO. What transitions are possible between states?
Use the data in Table 10.4 to calculate the frequencies of the transitions (000) → (003) and (000) → (201) for water. Compare to the values given in Section. 10.7. Table 10.4 Vibrational
Evaluate the upper state symmetry for each of the following combination bands in benzene: ν16 + ν2, ν10 + ν13, ν2 + ν16 + ν18, and 2ν2 + ν18. In each case, what operator could result in
Find the symmetries of the states associated with the second overtone of the bending mode of CO2.
Explain how vibrational spectra can distinguish between cis or trans dichloroethylene.
Find the D matrix for CO2.
Find the symmetries of the normal modes of a planar AB4 molecule (D4h point group). Predict the Raman and IR activities of each fundamental. Repeat the analysis for the case where the molecule is
Use group theoretical arguments to predict the number of fundamentals observed in the Raman and IR spectra of a triatomic molecule AB2 for each of the possible structures: linear symmetric, linear
Using perturbation theory, derive the following expression for the matrix element pertinent to radiationless relaxation.How does this result compare to the vibronic coupling responsible for
Explain each of the following observations using the sum-over-states formalism (KHD equation), transform theory, and wave packet theory(a) Low intensity of low-frequency modes(b) Effect of
Show that the second moment of the Kubo lineshape is Δ2. How can the second moment remain constant as the motional narrowing limit is approached?
Show that Equation 12.2 leads to a symmetric tensor, αρσ = ασρ, in the limit ω0 << ωeg. 3 = (αpo) if 12 [{f|ºr|u}{u|ºr|t) (ƒ|ºr|u}{u|ºr|!)_ Wo +@nf + in =(i\â pof) @o-@ni-in

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