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9.5 Rotation and Vibration of Diatomic Molecules

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9.5 Rotation and Vibration of Diatomic Molecules 9.5. Rotation and Vibration of Diatomic Molecules 349 9.5 Rotation and Vibration of Diatomic Molecules Up to now we have discussed the electronic states of ri- gid molecules, where the nuclei are clamped to a fixed position. In this section we will improve our model of molecules and include the rotation and vibration of dia- tomic molecules. This means, that we have to take into account the kinetic energy of the nuclei in the Schrö- dinger equation Hψ Eψ, which has been omitted in ˆ the foregoing sections=. We then obtain the Hamiltonian 2 2 1 2 N H ! 2 ! 2 ˆ = − 2 M ∇k − 2m ∇i k 1 k e i 1 != != 2 e Z1 Z2 1 1 1 el Fig. 9.41. Energy En (R) of the rigid molecule and total + 4πε R + r − r + r energy E of the nonrigid vibrating and rotating molecule 0 i, j i, j i i1 i2 ! ! $ % Enucl Eel E0 T H (9.73) ˆkin kin pot ˆk ˆ0 = + + = + can be written as the product of the molecular wave where the first term represents the kinetic energy of the function χ(Rk) (which depends on the positions Rk of nuclei, the second term that of the electrons, the third the nuclei), and the electronic wave function Φ(ri , Rk) represents the potential energy of nuclear repulsion, the of the rigid molecule at arbitrary but fixed nuclear po- fourth that of the electron repulsion and the last term sitions Rk, where the electron coordinates ri are the the attraction between the electrons and the nuclei. variables and Rk can be regarded as a fixed parameter. This implies that nuclear motion and electronic motion 9.5.1 The Adiabatic Approximation are independent and the coupling between both is ne- glected. The total energy E is the sum of the energy Because of their much larger mass, the nuclei in a mol- el En (R) of the rigid molecule in the nth electronic state, ecule move much slower than electrons. This implies which is represented by the potential curve in Fig. 9.41 that the electrons can nearly immediately adjust their and the kinetic energy (Evib Erot) of the nuclei. positions to the new nuclear configuration when the + nuclei move. Although the electronic wave functions Note that the total energy is independent of R! ψ(r, R) depend parametrically on the internuclear di- stance R they are barely affected by the velocity of the Inserting this product into the Schrödinger equation moving nuclei. The kinetic energy of the nuclear motion (9.73) gives the two equations (see Problem 9.4) 1 2 Ekin 2 Mv is small compared to that of the electrons. = el (0) el We therefore write the total Hamiltonian H in (9.73) as H0Φ (r, Rk) E Φ (r, Rk) (9.75a) ˆ n = n · n the sum (0) Tk E χn,m(R) En,mχn,m(R) . (9.75b) ˆ + n = H H0 Tk = + The(first equat)ion describes the electronic wave func- of the Hamiltonian H0 of the rigid molecule and Tk of tion Φ of the rigid molecule in the electronic state (0) the kinetic energy of the nuclei. Since the latter is small (n, L, Λ) and En is the total electronic energy of this compared to the total energy of rigid molecule we can state without the kinetic energy Tk of the nuclei. regard Tk as a small perturbation of H. In this case the The second equation determines the motion of the total wave function nuclei in the potential (o) el ψ(ri , Rk) χ(Rk) Φ(ri , Rk) (9.74) E E Epot(ri , Rk) , (9.76) = · n = kin + * + 350 9. Diatomic Molecules which consists of the time average of the kinetic Inserting the product (9.80) into (9.79) gives, as energy of the electrons and the total potential energy has been already shown in Sect. 4.3.2, the following of electrons and nuclei. The total energy equation for the radial function S(R): E Enuc E(o) (9.77) 1 d dS n,k kin n R2 (9.80) = + R2 dR dR of the nonrigid molecule is the sum of the kinetic energy $ % 2M J(J 1)!2 of the nuclei and the total energy of the rigid molecule. E Epot(R) + S 0 . + 2 − − 2MR2 = Equation (9.75b) is explicitly written as ! , - 2 2 For the spherical surface harmonics Y(ϑ, ϕ) we obtain ! ! (n) − ∆A − ∆B Epot(Rk) χn,m(Rk) the Eq. (4.88), already treated in Sect. 4.4.2: 2MA − 2MB + ,$ % - 2 En,mχn,m(Rk) . (9.78) 1 ∂ ∂Y 1 ∂ Y = sin ϑ (9.81) sin ϑ ∂ϑ ∂ϑ + sin2 ϑ ∂ϕ2 In the center of the mass system this translates to $ % J(J 1)Y 0 . 2 + + = ! (n) − ∆ Epot(R) χn,m(R) En,mχn,m(R) While the first Eq. (9.80) describes the vibration of the 2M + = , - diatomic molecule, (9.81) determines its rotation. (9.79) where M M M /(M M ) is the reduced mass of A B A B 9.5.2 The Rigid Rotor the two nu=clei and the ind+ex m gives the mth quantum state of the nuclear movement (vibrational-rotational A diatomic molecule with the atomic masses MA and state). MB can rotate around any axis through the center The important result of this equation is: of mass with the angular velocity ω (Fig. 9.42). Its rotational energy is then The potential energy for the nuclear motion in 1 2 2 Erot Iω J /(2I) . (9.82) the electronic state (n, L, Λ) depends only on the = 2 = ϑ ϕ 2 2 2 nuclear distance R, not on the angles and , i.e., Here I MA R MB R MR where M MA MB/ = A + B = = it is independent of the orientation of the mol- (MA MB) is the moment of inertia of the molecule ecule in space. It is spherically symmetric. The with +respect to the rotational axis and J Iω is its wave functions χ χ(R, ϑ, ϕ), however, may still | | = = rotational angular momentum. Since the square of the depend on all three variables R, ϑ, and ϕ. angular momentum J 2 J(J 1)h2 Because of the spherically symmetric potential | | = + equation (9.77) is mathematically equivalent to the can take only discrete values that are determined by Schrödinger equation of the hydrogen atom. The diffe- the rotational quantum number J, the rotational ener- rence lies only in the different radial dependence of the gies of a molecule in its equilibrium position with an potential. Analogous to the treatment in Sect. 4.3.2 we internuclear distance Re are represented by a series of can separate the wave functions χ(R, ϑ, ϕ) into a radial part depending solely on R and an angular part, depen- ding solely on the angles ϑ and ϕ. We therefore try the R R product ansatz A B B χ(R, ϑ, ϕ) S(R) Y(ϑ, ϕ) . A = · The radial function S(R) depends on the radial form S M M of the potential, while the spherical surface harmo- A B nics Y(ϑ, ϕ) are solutions for all spherically symmetric R potentials, independent of their radial form. Fig. 9.42. Diatomic molecule as a rigid rotor 9.5. Rotation and Vibration of Diatomic Molecules 351 discrete values with the rotational constant 2 ! J(J 1) Be , (9.86) E ! . (9.83) = 4πcMR2 rot + 2 e = 2MRe which is determined by the reduced mass M and the The energy separation between the rotational levels J equilibrium nuclear distance Re. For historical reasons 1 1 and J 1 one writes Be in units of cm− instead of m− . + (J 1) 2 ∆E E (J 1) E (J) ! (9.84) EXAMPLES rot rot rot + 2 = + − = 2MRe 1. The H2 molecule has a reduced mass M 0.5MH increases linearly with J (Fig. 9.43). 28 = = This result can also be directly obtained from (9.80). 8.35 10− kg, and the equilibrium distance Re 0.742× 10 10 m I 4.60 10 48 kg m2. The ro=- For a fixed nuclear distance R the first term in (9.80) × − ⇒ = × − is zero. Therefore the second term must also be zero, tational energies are because the sum of the two terms is zero. The kinetic 21 Erot(J) 1.2 10− J(J 1) Joule energy of a rigid rotor, which does not vibrate, is Ekin = × + = 7J(J 1) meV . Erot E Epot, where E is the total energy. The bracket = + = − of the second term in (9.80) then becomes for R Re The rotational constant is B 60.80 cm 1. = e − equal to (9.83). 2. For the HCl molecule th=e figures are M 27 10 = In the spectroscopic literature, the rotational term 0.97 AMU 1.61 10− kg, Re 1.27 10− m = ×22 = × values F(J) E(J)/hc are used instead of the energies. Erot 2.1 10− J(J 1) Joule 1.21 J(J 1) = ⇒ = × 1 + = + Instead of (9.83) we write meV, Be 10.59 cm . = − J(J 1)!2 F (J) B J(J 1) (9.85) In Table 9.7 the equilibrium distances Re and the rot + 2 e = 2hcMRe = + rotational constants are listed for some diatomic mol- ecules. The figures show that the rotational energies are within the range of J F(J) (J + 1) h2 6 2 4 E Erot (10− 10− ) J(J 1) eV . ∆ rot = 2 MRe = − + For a rotational angular momentum J the rotational ν4 period becomes α 2πI/! Trot .
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