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 . (9.87) 3 ∆E h2 = √J(J 1) tgα ∝ rot = + J + 1 MR2 e Depending on the rotational constant Be they range from 14 10 1 ν3 T 10 s to 10 s. For B 1 cm one obtains 0 1 2 3 4 5 6 J rot − − e − b) = 11 = Trot 1.6 10 /√J(J 1) s. If an electro-magnetic = × − + 2 wave falls onto a sample of molecules it can be ab- I(ν) sorbed on rotational transitions J J 1 resulting in ν2 absorption lines with frequencies → + 1 νrot(J) E(J 1) E(J) /! (9.88a) ν1 = [ + ] − ] 0 1 ν1 ν2 ν3 ν4 νrot or, in wavenumber units cm− , a) c) νrot(J) 2Be(J 1) . (9.88b) Fig. 9.43. (a) Energy levels of the rigid rotor (b) Separati- = + ons ∆Erot Erot(J 1) Erot(J) (c) Schematic rotational The rotational transitions between levels J and spectrum = + − J 1 fall into the spectral range with frequencies + 352 9. Diatomic Molecules
Table 9.7. Equilibrium di- Molecule R /pm B D α ω ω x stances and rotational and e e e e e e e 2 vibrational constants in H2 74.16 60.8 1.6 10− 3.06 4401 121.3 1 × 6 units of cm− for some Li2 267.3 0.673 9.9 10− 0.007 351.4 2.6 × 6 diatomic molecules N2 109.4 2.01 5.8 10− 0.017 2359.0 14.3 × 6 O2 120.7 1.45 4.8 10− 0.016 1580.0 12.0 × 9 I2 266.6 0.037 4.2 10− 0.0001 214 0.61 35 × 4 H Cl 127.4 10.59 5.3 10− 0.31 2990 52.8 35 × 4 D Cl 127.4 5.45 1.4 10− 0.11 2145 27.2 × 8 ICl 232.1 0.114 4.0 10− 0.0005 384 1.50 × 6 CO 112.8 1.931 6 10− 0.017 2170 13.29 NO 115.1 1.705 0.×5 10 6 0.017 1904 14.08 × −
ν 109 1013 Hz, i.e., in the Gigahertz–Terrahertz E = − 5 1 Epot(R) range with wavelengths between λ 10− 10− m. This spectral region is called the micro=wave r−ange.
In Sect. 9.6.2 we will see, that only molecules with a permanent electric dipole moment can absorb or → → emit radiation on rotational transitions (except for Fr Fz very weak quadrupole transitions). Therefore ho- monuclear diatomic molecules show no rotational absorption or emission spectra!
Re R Fig. 9.44. Compensation of centrifugal and restoring force in the nonrigid rotating molecule 9.5.3 Centrifugal Distortion A real molecule is not rigid. When it rotates, the centrifugal force acts on the atoms and the inter- energy Epot(R) is, for R > Re, larger than Epot(Re) we nuclear distance widens to a value R where this force have to include the additional energy ∆E 1 k(R 2 pot 2 Fc Mω R is compensated by the restoring force 2 = − = − Re) in the rotational energy of the nonrigid rotor. The Fr dEpot(R)/dR holding the two atoms together, total energy of the nonrigid rotor is then wh=ich−depends on the slope of the potential energy J(J 1)!2 1 function Epot(R) (Fig. 9.44). E k(R R )2 . (9.91) rot + 2 e In the vicinity of the equilibrium distance Re the = 2MR + 2 − potential can be approximated by a parabolic function If we express R on the right side of (9.90) by Re and k (see Sect. 9.4.4). This leads to a linear restoring force with the help of (9.89) we obtain J(J 1)!2 Fr k(R Re)R . (9.89) R R 1 R (1 x) = − − ˆ e + 4 e = + MkRe = + From the relation J2 I 2ω2 M2 R4ω2 we obtain: $ % = = with x 1. This allows us to expand 1/R2 into the 2 J(J 1) ! ' Mω2 R ! k(R R ) power series + 3 e = MR = − 2 2 1 1 2J(J 1)! J(J 1)! 1 + (9.92) R Re + , (9.90) R2 = R2 − MkR4 ⇒ = + MkR3 e , e 3J2(J 1)2!4 which means that the internuclear distance R is wi- + . . . + M2kR8 ∓ dened by the molecular rotation. Since the potential e - 9.5. Rotation and Vibration of Diatomic Molecules 353 and the rotational energy becomes spin S precesses independently around the z-axis with a projection J(J 1)!2 J2(J 1)2!4 Erot + + (9.93) = 2MR2 − 2M2kR6 Sz Msh . (9.96b) e e ) * = 3J3(J 1)3 6 ! . . . . Both projections add to the total value 3+2 10 + 2M k Re ± Ωh (Λ Ms)h . (9.96c) = + In the case of strong spin-orbit coupling L and S couple For a given value of the rotational quantum to Jel L S with the projection number J the centrifugal widening makes the mo- = + el ment of inertia larger and therefore the rotational J Ω h z = × energy smaller. This effect overcompensates for (see* Sec+ t. 9.3.3). the increase in potential energy. The total angular momentum J of the rotating mol- ecule is now composed of the angular momentum N of Using the term-values instead of the energies, (9.94) the molecular rotation and the projection Λh or Ωh. For becomes Ω 0 the total angular momentum J of the molecule is n+=o longer perpendicular to the z-axis (Fig. 9.45). 2 2 Frot(J) Be J(J 1) De J (J 1) = 3+ − 3 + He J (J 1) . . . Since the total angular momentum of a free mol- + + − ecule without external fields is constant in time, (9.94) the molecule rotates around the space-fixed direc- tion of J and for Λ 0 the rotational axis is no with the rotational constants longer perpendicular+=to the molecular z-axis. 3 B ! , D ! , (9.95) e 2 e 2 6 = 4πcMRe = 4πckM Re In a simple model, the whole electron shell can be re- 3 5 garded as a rigid charge distribution that rotates around H ! . e 2 3 10 the z-axis. The rotating molecule can then be described = 4πck M Re as a symmetric top with two different moments of in- The spectroscopic accuracy is nowadays sufficiently ertia: 1.) The moment I1 of the electron shell rotating high to measure even the higher order constant H. around the z-axis and 2.) the moment I2 of the molecule When fitting spectroscopic data by (9.95) this constant, therefore, has to be taken into account.
→ N → 9.5.4 The Influence of the Electron Motion J → Up to now we have neglected the influence of the elec- L tron motion on the rotation of molecules. In the axial symmetric electrostatic field of the two nuclei in the nonrotating molecule, the electrons precess around the Λ space-fixed molecular z-axis. The angular momentum L(R) Σli (R) of the electron shell, which depends on = the separation R of the nuclei, has, however, a constant A Λh B z projection R
Lz Λh (9.96a) ) * = independent of R. For molecular states with electron Fig. 9.45. Angular momenta of the rotating molecule spin S 0 in atoms with weak spin-orbit coupling the including the electronic contribution += 354 9. Diatomic Molecules
(nuclei and electrons) rotating around an axis perpendi- depend on the integer vibrational quantum number v cular to the z-axis. Because the electron masses are very 0, 1, 2, . . . . = small compared with the nuclear masses, it follows that They are equally spaced by ∆E !ω. The = I1 I2. frequency ω √kr/M depends on the constant ' 2 = 2 The rotational energy of this symmetric top is kr (d Epot/dR )R in the parabolic potential and on = e 2 2 2 the reduced mass M of the molecule. The lowest vibra- J J J 1 x y z tional level is not E 0 but E !ω. The solutions Erot (9.97) = = 2 = 2Ix + 2Iy + 2Iz of (9.80) with a parabolic potential are the vibrational with Ix Iy I1 Iz I2 . eigenfunctions = = += = πMω/hR From Fig. 9.45 the following relations can be obtained: S(R) ψvib(R, v) e− Hv(R) (9.102) = = · 2 2 2 where the functions Hv(R) are the Hermitian polyno- Jz Ω ! 2 2 = 2 2 2 2 mials. Some of these vibrational eigenfunctions of the J J N ! J J (9.98) x + y = = − z harmonic oscillator are compiled in Table 4.1 and are 2 2 J(J 1) Ω ! . illustrated in Fig. 4.20. = + − Although the real potential of a diatomic molecule Inserting this i.nto (9.97) gives/ the term values F(J) = can be well approximated by a parabolic potential in the Erot(J)/hc of the rotational levels vicinity of the potential minimum at R Re, it deviates = more and more for larger R Re (see Fig. 9.38). This 2 2 | − | F(J, Ω) Be J(J 1) Ω AΩ (9.99) figure also illustrates that the Morse potential is a much = + − + better approximation. Inserting the Morse potential . / with the two rotational constants a(R Re) 2 Epot(R) ED 1 e− − (9.103) ! ! = − A Be . (9.100) into the radial part (9.80) of the Schrödinger equation = 4πcI , = 4πcI . / 1 2 allows its exact analytical solution (see Problem 9.5). The term AΩ2, which does not depend on J, is generally The energy eigenvalues are now: added to the electronic energy Te of the molecular state, 2 2 2 1 ! ω0 1 since it is constant for all rotational levels of a given Evib(v) !ω0 v v = + 2 − 4E + 2 electronic state with quantum number Ω. It is therefore $ % D $ % also not influenced by the centrifugal distortion. (9.104) The ground states of the majority of diatomic mol- with energy separations ecules are 1Σ-states with Λ Ω 0. For these cases = = ∆E(v) Evib(v 1) Evib(v) (9.105a) A 0 and (9.99) is identical to (9.94). = + − = !ω !ω 1 (v 1) , = − 2E + 9.5.5 Vibrations of Diatomic Molecules , D - where ED is the dissociation energy of the rigid mol- For a nonrotating molecule, the rotational quantum ecule. The vibrational levels are no longer equidistant number J in (9.80) is zero. The solutions S(R) of (9.80) but separations decrease with increasing vibrational are then the vibrational wave functions of the diatomic quantum number v, in accordance with experimental molecule. For J 0 they solely depend on the radial observations. form of the poten=tial energy E (R). For a parabolic pot The term-values Tv Ev/hc are potential, the vibrating molecule is a harmonic oscilla- = 1 1 Tvib(v) ωe(v ) ωexe(v ) (9.105b) tor, which has been already treated in Sect. 4.2.5. The = + 2 − + 2 result obtained there was the quantization of the energy with the vibrational constants levels. ω ω2 hc The energy levels of the harmonic oscillator 0 ! 0 2 ωe , ωexe ωe . = 2πc = 8πcED = 4ED E(v) (v 1 )hω (9.101) (9.105c) = + 2 9.5. Rotation and Vibration of Diatomic Molecules 355
The vibrational frequency Fig. 9.46. Vibrating rotor
ω0 a 2ED/M (9.106) R(t) = correspond0s to that of a classical oscillator with the S 2 restoring force constant kr 2a ED. From measure- Re = ments of kr (for instance from the centrifugal distortion of rotational levels) and the dissociation energy ED the constant a in the Morse potential can be determined. With the more general expansion of the potential n 1 ∂ Epot n frequency is higher than the rotational frequency by Epot(R) (R Re) (9.107) = n ∂Rn − one to two orders of magnitude, the molecule un- n ! $ %Re ! dergoes many vibrations (typically 5 100) during the Schrödinger equation can only be solved numeri- one rotational period (Fig. 9.46). This m−eans that the cally. We will, however, see in Sect. 9.5.7 that the real nuclear distance changes periodically during one full potential can be very accurately determined from the rotation. measured term values of the rotational and vibrational levels. EXAMPLES
Note: 14 1 1. For the H2 molecule, ωe 1.3 10 s− Tvib 14 = × 13 ⇒ = 4.8 10− s, while Trot 2.7 10− √J(J 1) s. The distance between vibrational levels decreases × = × 12 1 + 2. For the Na molecule, ωe 4.5 10 s− Tvib • with increasing v, but stays finite up to the disso- 12 = × 10 ⇒ = 1.4 10− s, while Trot 1.1 10− √J(J 1) s. ciation energy. This means that only a finite number × = × + of vibrational levels fit into the potential well of Since the total angular momentum J I ω of a bound molecular state. This is in contrast to the = · infinite number of electronic states in an atom such a freely rotating molecule has to be constant in time, as the H atom. Here the distance between Ryd- but the moment of inertia I periodically changes, berg levels converges with n towards zero the rotational frequency ω has to change accor- (see (3.88)). This different beha→vio∞r stems from the dingly with a period Tvib. Therefore the rotational different radial dependence of the potentials in the energy two cases. J(J 1)!2 One has to distinguish between the experimen- Erot + 2 • exp = 2m R tally determined dissociation energy ED , where the molecule is dissociated from its lowest vibration also varies periodically with a period Tvib. level, and the binding energy EB of the potential well, which is measured from the minimum of the Because the total energy E Erot Evib Epot potential (Fig. 9.41). The difference is has to be constant, there is a p=eriodic+excha+nge of exp 1 rotational, vibrational and potential energy in the E EB !ω . D = − 2 vibrating rotor (Fig. 9.47).
The rotational energy, considered separately, is 9.5.6 Interaction Between Rotation the time average over a vibrational period. This time and Vibration average can be calculated as follows: Up to now we have looked at the rotation of a non- The probability to find the nuclei within the interval vibrating molecule and the vibration of a nonrotating dR around the distance R is molecule. Of course a real molecule can simul- 2 P(R)dR ψvib(R) dR . taneously rotate and vibrate. Since the vibrational = | | 356 9. Diatomic Molecules
E always equal to Re (Fig. 9.48). Therefore, the rotatio- nal constant Bv of the rotating harmonic oscillator also Erot depends on v. For the more realistic anharmonic poten- tial, both R as well as 1/R2 change with v. While E vib R increa)ses* 1/R2 dec)reases*with increasing v. ) * ) * In order to express the rotational term values by a rotational constant similar to (9.86) or (9.95) we introduce, instead of Be, the rotational constant Epot 1 ! ψ (v, ) ψ (v, ) Bv vib∗ R 2 vib R dR t = 4πcM R 1 (9.110a) Fig. 9.47. Exchange between rotational, vibrational and potential energy during a vibrational period averaged over the vibrational motion. It depends on the vibrational quantum number v. For a Morse potential we then obtain The quantum mechanical expectation values of R and 2 1 1/R are then Bv Be αe(v ) (9.111a) = − + 2
R ψ∗ Rψvib dR , and (9.108) where αe Be. In a similar way an average centrifugal ) * = vib ' 1 constant 2 1 1/R ψ∗ ψvib dR . 3 = vib R2 ! 1 1 Dv ψvib∗ ψvib dR (9.110b) * + = 4πckM2 R6 This gives the mean rotational energy, averaged over 1 one vibrational period can be defined, which is related to De by 2 J(J 1)! 1 1 Erot(v) + ψ∗ (v) ψvib(v) dR . Dv De βe(v ) with βe De . (9.111b) ) * = 2M vib R2 = − + 2 ' 1 (9.109) For a general potential, higher order constants have to Note: be introduced and one writes
Even for a harmonic potential the expectation value of B β α (v 1 ) γ (v 1 )2 . . . (9.112a) 1/R2 depends on the vibrational quantum number v. It v e e 2 e 2 = − + 1 + + 1 2+ increases with v although R is independent of v and Dv De βe(v 2 ) δe(v 2 ) . . . . (9.112b) ) * = + + + + +
1 B = 〈R〉 R2 v R e 2 = = 〈 〉 R Be v = 6 v = 4 5 4 3 3 2 b a 2
1 1
0 0 Fig. 9.48. Mean internuclear distance R 0.8 1 1.2 2 ) * and rotational constant Bv 1/R for the a) Re R b) Re R harmonic (a)and anharmon∝ic)(b)po*tential 9.5. Rotation and Vibration of Diatomic Molecules 357
The term value of a rotational-vibrational level can then and also to the coefficients an in the general potential be expressed as the power series expansion (9.114) [9.10].
1 1 2 T(v, J) Te ωe(v 2 ) ωexe(v 2 ) = + +1 3 − +1 4 9.5.8 Rotational Barrier ωe.ye(v 2 ) ωeze(v 2 ) . . . + + + 2 + 2 + The effective potential for a rotating molecule (see Bv J(J 1) Dv J (J 1) / + 3 + −3 + (9.80)) .Hv J (J 1) . . . . (9.113a) 2 + + ∓ (v) J(J 1)! Eeff (R) E (R) + (9.117) For a Morse potential this series is r/educed to pot = pot + 2MR2 includes, besides the potential E (R) of the nonrota- T Morse(v, J) T ω (v 1 ) (9.113b) pot e e 2 ting molecule, a centrifugal term that depends on the = + +1 2 ωexe(v 2 ) Bv J(J 1) rotational quantum number J and falls of with R as − 2 + 2+ + 2 Dv J (J 1) 1/R (Fig. 9.49). For a bound electronic state this leads − + eff to a maximum of Epot(R) at a distance Rm, which can where only five constants describe the energies of all be obtained by setting the first derivative of (9.117) to levels (v, J) up to energies where the Morse potential zero. This distance is still a good approximation. 1/3 J(J 1)!2 Rm + (9.118) = M(dE /dR) 9.5.7 The Dunham Expansion , pot - depends on the rotational quantum number J and on the In order to also reproduce the rotational-vibrational slope of the rotationless potential. term values T(v, J) of a rotating molecule for a more The minimum of the potential is shifted by the ro- general potential (9.107) tation of the molecule from Re to larger distances and the dissociation energy becomes smaller. E (R) a (R R )n , (9.114) pot n e Energy levels E(v, J) above the dissociation energy = n − ! ED can be still stable, if they are below the maximum of with 1 ∂n E a pot . n n E −1 = n ∂R / cm J 275 $ %Re hc = ! Predissociation Dunham introduced the expansion 9,000 J = 250 1 i 2 k 8,000 T(v, J) Yik(v ) J (J 1) Λ = + 2 · + − i k 7,000 ! ! . (9./115) 6,000 where the Dunham coefficients Yik are fit parameters J = 200 chosen such that the term values T(v, J) best repro- 5,000 duce the measured term values of rotational levels in v = 10, J = 200 J = 150 vibrational states of the molecule. 4,000 v = 25, J = 150 With (9.115) the energies of all vibrational- D = 6,000 cm−1 rotational levels of a molecule can be described by 3,000 a set of molecular constants. These constants are re- 2,000 lated to the coefficients in the expansion (9.113a) by J = 120 the relations 1,000 J = 0
Y10 ωe , Y20 ωexe , Y30 ωe ye 0 ≈ ≈ − ≈ 2 3 4 5 6 7 8 9 R / Å Y01 Be , Y02 De , Y03 He (9.116) ≈ ≈ ≈ Fig. 9.49. Effective potential curves of the rotating Na2 Y11 αe , Y12 βe , Y21 γe ≈ − ≈ ≈ molecule for different rotational quantum number J 358 9. Diatomic Molecules
E variety of molecular states, with energies depending on E (A + B) v, J kin the electronic, the rotational and vibrational structure of the molecule, the matrix elements of molecules are more complicated than those of atoms. In this section En(R) ED we will discuss their structure and the molecular spectra derived from them. For spontaneous emission (fluorescence spectra) the emission probability of a transition i k is given | * → | * R by the Einstein coefficient Aik. According to (7.17) Aik is related to the transition dipole matrix element MiK by Fig. 9.50. Predissociation of a molecule through the rotational barrier 2 ω3 ik 2 . Aik 3 Mik (9.119a) = 3 ε0c ! | | the potential barrier. However, due to the tunnel effect For the absorption or stimulated emission of radiation (Sect. 4.2.3) molecules in these levels can dissociate by the transition probability Pik Bikw(νik) is proportio- tunneling through the barrier (Fig. 9.50). This effect is nal to the spectral energy den=sity w(ν) of the radiation called predissociation by tunneling. The tunnel proba- field. In Sect. 7.2 it was shown that Pik is given by bility depends exponentially on the width of the barrier π and on the energy gap between the maximum of the 2 2 Pik E0 ψk∗ε pψi dτ (9.119b) barrier and the level energy. = 2!2 · 2 2 The predissociation rate can be determined by mea- where ε E/ E 2is the unit ve2ctor in the direction of suring the width δE h/τ of levels with a lifetime τ. If the electr=ic fiel|d E| of the electromagnetic wave, incident = the predissociation rate is large compared to the radia- onto the molecules. The transition probability therefore tive decay of a level, the lifetime τ is mainly determined depends on the scalar product ε p of electric field vector by predissociation. Measuring τ(v) for all levels above and electric transition dipole of· the molecule. the dissociation limit gives information on the form and heights of the potential barrier. The dissociating fragments have a kinetic energy 9.6.1 Transition Matrix Elements The dipole matrix element for a transition between two Ekin E(v, J) ED(J 0) , = − = molecular states with wave functions ψi and ψk is which is shared by the two fragments according to their masses. Mik ψ∗ pψk dτel dτN . (9.119c) = i 11
9.6 Spectra of Diatomic Molecules
When a molecule undergoes a transition
Ei (ni , Λi , vi , Ji ) Ek(nk, Λk, vk, Jk) ↔ between two molecular states i and k , electromagne- tic radiation can be absorbed or| e*mitte|d *with a frequency ν ∆E/h. Whether this transition really occurs de- pe=nds on its transition probability, which is proportional to the absolute square of the dipole matrix element Mik (see Sect. 7.1). The relative intensities of spectral lines can therefore be determined if the matrix elements of Fig. 9.51. Illustration of nuclear and electronic contributions the transitions can be calculated. Because of the larger to the molecular dipole moment 9.6. Spectra of Diatomic Molecules 359
The integration extends over all 3(ZA ZB) electronic function χ(R); which only depends on the nuclear coor- coordinates and over the six nuclear co+ordinates. Often dinates. Inserting (9.120, 9.121) into (9.119) the matrix only one of the electrons is involved in the transition. In elements is written as thiscasetheintegrationover dτel only needs to beperfor- med over the coordinates of this electron. The vector p Mik Φ∗ χ∗ (pel pN)Φkχ∗ dτel dτN . = i N,i + N,k is the dipole operator, which depends on the coordina- 11 (9.122a) tes of the electrons, involved in the transition and on the nuclear coordinates. In Fig. 9.51 it can be seen that Rearranging the different terms gives
p e ri e(ZA RA ZB RB) pel pN = − + + = + Mik χ∗ Φ∗ pelΦk dτel χk dτN (9.122b) i = i i ! (9.120) 1 ,1 - where p is the contribution of the electrons and p χ∗ pN Φ∗Φk dτel χk dτN . el N + 1 i that of the nuclei. 1 ,1 - We distinguish between two different cases (Fig. 9.52): Note that for homonuclear molecules ZA ZB but = The two levels i and k belong to the same elec- RA RB. Therefore pN 0! • | * | * = − = tronic state (Φi Φk). This means that the dipole = Within the adiabatic approximation we can separate transition occurs between two vibrational-rotational the total wave function ψ(r, R) into the product levels in the same electronic state Φi . In this case the first term in the sum (9.122b) is zero because the ψ(r, R) Φ(r, R) χN(R) (9.121) 2 = × integrand r Φi is an ungerade function of the elec- of electronic wave function Φ(r, R) of the rigid mol- tron coordi|nate|s r x, y, z . The integration from ecule at a fixed nuclear distance R and the nuclear wave to theref=ore{ gives }zero. −∞ +∞
En
el E2 (R)
Rotational levels E'(J')
v'= 3 ED' v'= 2 Vibrational v'= 1 states E'(v') v'= 0
Re' el E1 (R)
Rotational levels E''(J'') el el E2 − E1 v''= 3 ED'' v'' 2 = Vibrational v''= 1 states E''(v'') v''= 0 Fig. 9.52. Rotational and vibrational le- vels in two different electronic states of Re'' Re' R a diatomic molecule 360 9. Diatomic Molecules
Since the electronic wave functions Φ are Homonuclear diatomic molecules have no dipole- orthonormal, i.e., allowed vibrational-rotational spectra. This means they do not absorb or emit radiation on Φ∗φk dτel δik (9.123) i = transitions within the same electronic state. They 1 the integral over the electronic coordinates in the may have very weak quadrupole transitions. second term in the sum (9.122b) is equal to one. The matrix element then becomes Note: The molecules N and O , which represent the major Mik χi,N pNχk,N dτN . (9.124) 2 2 = constituents of our atmosphere, cannot absorb the in- 1 frared radiation emitted by the earth. Other molecules, The integrand solely depends on the nuclear such as CO2, H2O, NH3 and CH4 do have an electric coordinates, not on the electronic coordinates! dipole moment and absorb infrared radiation on their Transitions between levels in two different electro- numerous vibrational-rotational transitions. Although • nic states. In this case the integral over the electronic they are present in our atmosphere only in small concen- coordinates in the second term of (9.122b) is zero be- trations they can seriously perturb the delicate energy cause the Φi , Φk are orthonormal. The second term balance between absorbed incident sun radiation and is therefore zero and the matrix element becomes the energy radiated back into space by the earth (green- house effect). If their concentration is increased by only small amounts this can increase the temperature of the Mik χ∗ Φ∗ pelΦk dτel χk dτN atmosphere at the earth’s surface (greenhouse effect). = i i 1 el,1 - χ∗ M (R)χk dτN . (9.125) The structure of the vibration-rotation-spectrum and = i ik the pure rotation spectrum can be determined as follows. Since the interaction potential between the two We will now discuss both cases separately. atoms is spherically symmetric, we choose spherical coordinates for the description of the nuclear wave function χN(R, ϑ, ϕ). If the interaction between rotation and vibration is 9.6.2 Vibrational-Rotational Transitions sufficiently weak we can write χN as the product All allowed transitions (vi , Ji ) (vk, Jk) between two M χN(R, ϑ, ϕ) S(R)Y (ϑ, ϕ) (9.127) rotational-vibrational levels in t↔he same electronic state = J form for vi vk the vibrational-rotational spectrum of of the vibrational wave function S(R) in (9.102) and the += M the molecule in the infrared spectral region between rotational wave function YJ (ϑ, ϕ) for a rotational level λ 2 20 µm. For v v we have pure rotational with angular momentum J and its projection M ! onto i k · tra=nsit−ions between rota=tional levels within the same the quantization axis, which is a preferential direction vibrational state, which form the rotational spectrum in the laboratory coordinate system. For absorbing tran- in the microwave region with wavelengths in the range sition the quantization axis is, for instance, the direction 0.1 10cm. of the incident electromagnetic wave, or the direction −The dipole matrix element for these transitions is of its E-vector. according to (9.120) and (9.124) With R RA RB and RA/RB MB/MA (Figs. 9.42 and=9|.51)−and p| p/ p the dipo=le moment rot vib ˆ = | | M e χ∗(ZA RA ZB RB)χk dτN . (9.126) can be written as ik = i + 1 MB ZA MA ZB pN p pN e · − · R p For homonuclear diatomic molecules with ZA ZB = = ˆ · | | = MA MB · · ˆ and MA MB is RA RB and therefore the integrand + is zero = Mrot vib =0. − CRp . (9.128) ⇒ ik = = ˆ 9.6. Spectra of Diatomic Molecules 361
The volume element in spherical coordinates is z Fig. 9.53. Orienta- 2 tion of molecular dτN R dR sin ϑ dϑ dϕ . = dipole moment p in This gives the matrix element a space-fixed coor- pz dinate system 3 Mik C S∗ (R)Sv (R)R dR (9.129) = · vi k 1 → R θ p Y Mi Y Mk p sin ϑ dϑ dϕ . × Ji Jk ˆ ϑ1,ϕ φ px The first factor describes the vibrational transition vi ↔ py vk. If the harmonic oscillator functions are used for the vibrational functions S(R) the calculations of the x y integral shows that the integral is zero, unless
∆v vi vk 0 or 1 . (9.130) = − = ± The sign stands for absorbing, the minus sign for which gives emitt+ing transitions. Transitions with ∆v 0 are pure = rotational transitions within the same vibrational level. ε p (9.131c) This selection rule means that for the harmo- ˆ · = 4π 0 εx iεy 1 εx iεy 1 nic oscillator only transitions between neighboring p εzY − + Y + Y − . 3 1 + √ 1 + √ 1 vibrational levels are allowed. 3 $ 2 2 % For anharmonic potentials, such as the Morse po- Inserting this into the second integral in (9.129) and tential, higher order transitions with ∆v 2, 3, . . . extracting the components of the space fixed unit vec- = ± ± are also observed. Such overtone-transitions are, howe- tor ε out of the integral gives for the angular part of the ver, much weaker than the fundamental transitions with transition probability integrals of the form ∆v 1. = ± M M The second integral in (9.129) describes the rota- Y i Y ∆MY k dΩ with ∆M 0, 1 Ji 1 Jk = ± tional transitions. It depends on the orientation of the 1 molecular dipole moment p in space. with the result that these integrals are always zero, The amplitude of the radiation emitted into the di- except for ∆J Ji Jk 1. rection k in space is proportional to the scalar product This select=ion r−ule is=r±eadily understandable, be- of k p and the intensity is the square of this amplitude. cause the absorbed or emitted photon has the spin For a·bsorbing transitions it is proportional to the scalar s 1h and the total angular momentum of the system product E p of electric field amplitude and molecular ph=ot±on molecule has to be conserved. dipole mom· ent p. For t+he projection quantum number M the selection With the orientation angles Θ and φ of p p/ p rules are analogue to that for atoms: against the space-fixed axis X Y Z we oˆb=tain t|he| ∆M 0 for linear polarization of the radiation and relation (Fig. 9.53) ; ; ∆M =1 for circular polarization. = ± ε p p(εx sin Θ cos φ εy sin Θ sin φ εz cos Θ) ˆ · = + + (9.131a) Note: where εi is the ith component of ε E/ E against The angle ϑ is measured against the molecular axis in the space fixed axis i X, Y, Z. Tˆh=e ang|les| can be the molecular coordinate system, while Θ and φ are the = M angles between the molecular dipole moment and the expressed by the spherical surface harmonics YJ : space fixed quantization axis (see above). 4π 0 8π 1 iφ Y1 cos Θ Y1± sin Θ e± 3 3 = ; 3 3 = ∓ · In order to save indices in spectroscopic literature (9.131b) the upper state (vk, Jk) is always labeled with a prime as 362 9. Diatomic Molecules
(v2, J2), whereas the lower state (vi , Ji ) is labeled with J'' a double prime as (v22, J22). Transitions with 10
∆J J2 J22 1 8 = − = + are called R-transitions, those with 6
∆J J2 J22 1 P branch R branch = − = − 4 are P-transitions. 2 All allowed rotational transitions appear in the spec- trum as absorption- or emission lines (Fig. 9.54). All rotational lines of a vibrational transition form a vi- ν0 ν brational band. Its rotational structure is given by the Fig. 9.55. Fortrat diagram of the P- and R-branch of wavenumbers of all rotational lines vibrational-rotational transitions
ν(v2, J v22, J22) (9.132) ↔ 2 2 ν0 Bv2 J2(J2 1) Dv2 J2 (J2 1) = + + − 2 + 2 B22 J22(J22 1) D22 J22 (J22 1) − v + − v + where ν is the band origin. It gives the position of 0. / 2,700 2,800 2,900 3,000 a fictious Q-line with J J 0. Since this line does 2 = 22 = Fig. 9.56. Vibrational-rotational absorption of the H35Cl not exist in rotational-vibrational spectras of diatomic and H37Cl isotopomers in the infrared region between molecules, there is a missing line at ν ν0 (Fig. 9.54). λ 3.3 3.7 µm = 1 Since the rotational constant Bv Be αe(v ) = − = − + 2 generally decreases with increasing v (αe > 0 for most molecules) it follows that Bv2 < Bv22. Plotting ν(J J22) the P lines are on the low frequency side. In Fig. 9.56 for P- and R transitions as a function of ν giv=es the the vibration-rotation spectrum of HCl is shown with Fortrat-diagram shown in Fig. 9.55. The R-lines are on the P- and R-branch. The lines are split into two the high frequency side of the band origin ν0 while components, because the absorbing gas was a mixture of the two isotopomers of HCl with the two atomic isotopes 35Cl and 37Cl. Since the rotational and vibra- tional constants depend on the reduced mass M the J' v'= 1 lines of different isotopomers are shifted against each 4 other. R(3) 3 E 9.6.3 The Structure of Electronic Transitions P(4) R(2) 2 We will now evaluate the matrix element (9.125) for P(3) R(1) 1 el electronic transitions. The electronic part Mik(R) de- 0 pends on the internuclear distance R, because the electronic wave functions Φ depend parametrically J''
P(1) v''= 0 P(2) 3 on R. In many cases the dependence on R is weak el and we can expand Mik(R) into a Taylor series 2 el 1 el dMik M (R) Mik(Re) (R Re) . . . . 0 ik = + dR − + $ %Re P R Fig. 9.54. P and R rota- (9.133) tional transitions between the vibrational levels v22 In a first approximation only the first term, independent ν0 ν 0 and v 0 = of R, is considered, which can be regarded as an average 2 = 9.6. Spectra of Diatomic Molecules 363
el of Mik(R) over the range of R-values covered by the Mik /Debeye vibrating molecule. In this case the constant Mik(Re) can be put before the integral over the nuclear coor- 1 + 1 + dinates. Using the normalized nuclear wave functions 10 Σu − Σg χN S(R) Y(ϑ, ϕ) and the vibrational wave functions = · ψvib R S(R) the matrix element becomes 3 + 3 + = · Σg − Σu el 9.5 Mik M ψ∗ (vi )ψvib(vk) dR (9.134) = ik vib 1 Mi Mk YJ YJ sin ϑ dϑ dϕ · i k Atomic value 1 where Mh is the projection of the rotational angular 9 1 1 + momentum J onto a selected axis (for instance, in the Πu− Σg direction of the E-vector or the k-vector of the incident electromagnetic wave for absorbing transitions, or in the 0.15 3 6 9 R / nm direction from the emitting molecule to the observer for Fig. 9.57. Dependence of electronic transition dipole on in- fluorescent transitions). ternuclear distance R for several transitions of the Na2 molecule Note: This approximation of an electric transition dipole mo- depends on the rotational angular momenta and their ment independent of R is, for many molecules with orientation in space. This factor determines the spatial a strong dependence Mel(R), too crude (Fig. 9.57). In ik distribution of the emitted radiation. such cases the second term in the expansion (9.133) has An electric dipole transition in fluorescence can only to be taken into account. take place if none of these three factors is zero. The probability of absorbing transitions depends Since the probability of spontaneous emission is according to (9.119b) on the scalar product of the proportional to the square M 2, the intensity of ik electric field vector E and the dipole moment p a spectral emission line | | 2 Pik E Mik . el 2 I(ni , vi , Ji nk, vk, Jk) M (9.135) ∝ | · | ↔ ∝ ik Since only the last factor in (9.135) depends on the FCF(vi , vk) 2HL(2Ji , Jk) orientation of the molecule in space, i.e., the direc- · ·2 2 is determined by three factors. tion of Mik against the electric field vector E, only the el 2 Hönl–London factor differs for spontaneous emission The electronic part Mik gives the probability of 2 | | and absorbing transitions. For the intensity I ε0 E an electron jump from the electronic state i to k . It = depends on the overlap of the electronic wav|e*func|ti*ons of the incident electromagnetic wave we obtain with E ε E the transition probability Φi and Φk and their symmetries. = · | | 2 The Franck–Condon factor 2 el 2 vi vk Pik ε0 E M (Re) ψ ψ dR 2 = · ik × vib · vib · 21 2 FCF(vi , vk) ψvib(vi ) ψvib(vk) dR (9.136) 2 2 2 2 2 = · 2 Mi 2Mk 2 2 21 2 Y ε pY sin2 ϑ dϑ dϕ . (92.138) 2 2 × Ji ˆ · ˆ Jk is determined by th2e square of the overlap2integral of 21 2 2 2 2 2 the vibrational wave functions ψvib(vi ) and ψvib(vk) in 2 2 the upper and lower electronic state. a) The Gen2eral Structure 2 The Hönl–London factor of Electronic Transitions 2 Mi Mk HL(Ji , Jk) Y Y sin ϑ dϑ dϕ (9.137) Molecular electronic spectra have structures as shown = Ji Jk 21 2 in (Fig. 9.58). 2 2 2 2 2 2 364 9. Diatomic Molecules
excitation mechanism. Generally, the energy of the up- E per electronic state is for T 300 K large compared to the thermal energy kT. The=refore the thermal popula- tion is negligible. Optical pumping with lasers allows the population of single selected levels. In this case the Ei fluorescence spectrum becomes very simple because it (v', J') is emitted from a single upper level. In gas discharges, Emission many upper levels are excited by electron impact and Electronic Absorption the number of lines in the emission spectrum becomes transitions (UV / vis) very large. The absorption spectrum consists of all allowed R transitions from populated lower levels. Their intensity, as given in Sect. 7.2, is given by abs I gi Ni w(ν)Bik . (9.139b) ik = EK At thermal equilibrium the population distribution (v'', J'') follows a Boltzmann distribution Rotational-vibrational Pure rotational Ei /kT transitions transitions Ni gi e− . (9.140) = Fig. 9.58. Schematic representation of the structure of molecular transitions b) The Rotational Structure of Electronic Transitions All allowed transitions J J between the ro- i22 ←→ k2 tational levels Jk2 of a given vibrational level v2 in the The wavenumnber of a rotational line in the electro- upper electronic state and Ji22 of v22 in the lower elec- nic spectrum of a diatomic molecule corresponding to tronic state form a band. In absorption or fluorescence a transition (ni , vi , Ji ) (nk, vk, Jk) is ↔ spectra such a band consists of many rotational lines. νik (T 2 T 22) Tvib(v2) Tvib(v22) (9.141) Transitions with ∆J 0 form the Q-branch, those = e − e + − = Trot(J2) Trot(J22) with ∆J Jk2 Ji22 1 the R-branch and with + −4 5 ∆J 1 =the P−-bran=ch+. Q-branches are only present where Te g4ives the minimum5of the potential curves in tr=ans−itions where the electronic angular momentum Epot(R) of the electronic states i or k , Tvib is the changes by 1h, (e.g., for Σ Π transitions) in order to | * | * ↔ term value of the vibrational state for J 0 and Trot(J) compensate for the spin of the absorbed or emitted pho- the pure rotational term value. = ton. Electronic transitions with ∆Λ 0 (e.g., between = The rotational structure of a vibrational band is two Σ-states) have only P and R branches. then (similarly to the situation for vibrational–rotational The total system of all vibrational bands of this transitions within the same electronic state) given by electronic transition is called a band system. The total number of lines in such a band system depends not only νik ν0(ni , nk, vi , vk) Bv2 J2(J2 1) (9.142) = 2 2+ + on the transition probabilities but also on the number of Dv2 J2 (J2 1) populated levels in the lower or upper electronic state. − + 2 2 B22 J22(J22 1) D22 J22 (J22 1) . The intensities of the lines in the emission spectrum − v + − v + are proportional to the population of the emitting upper In contrast t.o (9.132), the rotational constant/Bv2 in the levels and to the transition probability Aik: upper state can now either be larger or smaller than Bv22 in em the lower electronic state. This depends on the binding Iik gk Nk Aik (9.139a) = energies and the equilibrium distances Re in the two where gk (2Jk 1) is the statistical weight of the states. The Fortrat-Diagrams shown in Fig. 9.59 has level. The=numbe+r of emitting levels depends on the a different structure for each of the two cases. 9.6. Spectra of Diatomic Molecules 365
At those J-values where the curve ν(J) becomes vertical, the density of rotational lines within a given 1GHz spectral interval has a maximum. The derivative dν/dJ changes its sign. For the case Bv22 > Bv2 the positions ν(J) of the rotational lines increase for R-lines before the maximum and then decrease again (Fig. 9.59a). The po- sition νh of this line pileup is called the band head. For Bv22 > Bv2 the R-lines show a band head at the high fre- quency side of the band, while for Bv22 < Bv2 the P-lines accumulate in a band head at the low frequency side (Fig. 9.59b). The line density may become so high, that even with very high spectral resolution the different li- nes cannot be resolved. This is illustrated by Fig. 9.60, which shows the rotational lines in the electronic tran- sition of the Cs2 molecule around the band head, taken with sub-Doppler resolution. In molecular electronic spectra taken with photogra- Fig. 9.60. Band head of the vibrational band v2 9 v22 1 1 = ← = phic detection and medium resolution where only part 14 of the electronic transition C Πu X Σ+ of the Cs2 ← g of the rotational lines are resolved, a sudden jump of the molecule, recorded with sub-Doppler-resolution blackening on the photoplate appears at the band head while the line density gradually decreases with incre- asing distance from the band head. The band appears shadowed (Fig. 9.61) to the opposite frequency side of the band head. For Bv22 > Bv2 the band is red-shadowed and the band head is on the blue side of the band, while for Bv22 < Bv2 the band is blue-shadowed and the band head appears on the red side. In cases where the electronic transition allows Q- lines, their spectral density is higher than that of the P- and R-lines. For B B all Q-lines Q(J) have the v2 = v22 same position. For Bv2 > Bv22 their positions ν(J) increase with increasing J (Fig. 9.59a) while for Bv2 < Bv22 they decrease (Fig. 9.59b).
2,977 2,820 3,805 3,577 3,371 3,159
v'' 2 1 0 0 0 0 v' 0 0 0 1 2 3 Fig. 9.61. Photographic recording of the band structure in the electronic transition 3Πg 3Πu of the N2 molecule. The wavelengths of the band h←eads are given in Å 0.1 nm above the spectrum (with the kind permission of=the late Fig. 9.59a,b. Fortrat-diagram for P, Q and R branches in Prof. G. Herzberg [G. Herzberg: Molecular Spectra and electronic transitions: (a) Bv22 > Bv2 (b) Bv22 < Bv2 Molecular Structure Vol. I (van Nostrand, New York, 1964)]) 366 9. Diatomic Molecules
c) The Vibrational Structure it follows that the electronic transition takes place at and the Franck–Condon Principle a nuclear distance R∗ where the kinetic energies of the vibrating nuclei in the upper and lower state are equal, i.e., E (R ) E (R ). This can be graphically The vibrational structure of electronic transitions is go- kin2 ∗ = kin22 ∗ verned by the Franck–Condon factor (9.136), which in illustrated by the difference potential turn depends on the overlap of the vibrational wave U(R) E22 (R) E2 (R) E(v2) (9.144) functions in the two electronic states. In a classical = pot − pot + model, which gives intuitive insight into electronic tran- introduced by Mulliken (Fig. 9.63). The electron jump sitions, the absorption or emission of a photon occurs from one electronic state into the other takes place at within a time interval that is short compared to the vi- such a value R∗, where Mulliken’s difference potential brational period Tvib of the molecule. In a potential intersects the horizontal energy line E E(v22), where diagram (Fig. 9.62) the electronic transitions between = U(R∗) E(v22) . the two states can be then represented by vertical ar- = rows. This means, that the internuclear distance R is In the quantum mechanical model, the probability the same for the starting point and the final point of the for a transition v2 v22 is given by the Franck–Condon transition. Since the momentum p hν/c of the absor- factor (9.136). Th↔e ratio bed or emitted photon is very small=compared to that of the vibrating nuclei, the momentum p of the nuclei is ψ2 (R)ψ22 (R) dR P (R) dR vib vib (9.145) conserved during the electronic transition. Also, the ki- = ψvib2 (R)ψvib22 (R) dR 2 netic energy Ekin p /2M does not change. From the energy balance = gives the probabi6lity that the transition takes place in the interval dR around R. It has a maximum for R R . = ∗ hv E2(v2) E22(v22) If the two potential curves Ep2 ot(R) and Ep22ot(R) = − have a similar R-dependence and equilibrium distan- Ep2 ot(R) Ekin2 (R) Ep22ot(R) Ekin22 (R) = + − [ + ] ces Re2 Re22 the FCF for transitions with ∆v 0 are Ep2 ot(R∗) Ep22ot(R∗) (9.143) maximu≈m and for ∆v 0 they are small (Fig.=9.62a). = − +=
'
∆v > 0 U(R) difference potential
''
Fig. 9.62. Illustration of the Franck-Condon principle for vertical transitions with ∆v 0 (a) and ∆v > 0 in case of Fig. 9.63. Illustration of the Mulliken-difference potential potential curves with R R=and R > R V(R) E (R) E (R) E(v ) e22 = e2 e2 e22 = p22ot − p2 ot + 2 9.6. Spectra of Diatomic Molecules 367
The larger the shift ∆R Re2 Re22 the larger becomes transferred to the Rydberg electron, which then gains the difference ∆v for ma=ximu−m FCF (Fig. 9.62b). sufficient energy to leave the molecule (Fig. 9.65). The situation is similar to that in doubly excited Rydberg atoms where the energy can be transferred from one ex- 9.6.4 Continuous Spectra cited electron to the Rydberg electron (see Sect. 6.6.2). If absorption transitions lead to energies in the upper However, while this process in atoms takes place wi- 13 15 electronic state above its dissociation energy, unbound thin 10− 10− s, due to the strong electron-electron states are reached with non-quantized energies. The ab- interaction−, in molecules it is generally very slow (bet- 6 10 sorption spectrum then no longer consists of discrete ween 10− 10− s), because the coupling between lines but shows a continuous intensity distribution I(ν). the motion−of the nuclei and the electron is weak. A similar situation arises, if the energy of the upper In fact, within the adiabatic approximation it would state is above the ionization energy of the molecule, be zero! The vibrational or rotational autoionization similarly to atoms (see Sect. 7.6). of molecules represent a breakdown of the Born– In the molecular spectra the ionization continuum is, Oppenheimer approximation. The decay of these levels however, superimposed by many discrete lines that cor- by autoionization is slow and the lines appear sharp. respond to transitions into higher vibrational-rotational In Fig. 9.65 an example of the excitation scheme of levels of bound Rydberg states in the neutral elec- autoionizing Rydberg levels is shown. The Rydberg le- tron. Although the electronic energy of these Rydberg vels are generally excited in a two-step process from states is still below the ionization limit, the additio- the ground state g to level i by absorption of a pho- nal vibrational-rotational energy brings the total energy ton from a laser a|n*d the furth| e*r excitation i k by above the ionization energy of the non-vibrating and a photon from another laser. The autoioniz|a*ti→on |of*the non-rotating molecule (Fig. 9.64). Rydberg level k is monitored by observation of the Such states can decay by autoionization into a lower resultant molec|ul*ar ions. A section of the autoioniza- state of the molecular ion, where part of the kinetic tion spectrum of the Li2-molecule with sharp lines and energy of the vibrating and rotating molecular core is a weak continuous background, caused by direct pho- toionization, is shown in Fig. 9.66. The lines have an asymmetric line profile called a Fano-profile [9.11]. The reason for this asymmetry is an interference effect between two possible excitation paths to the energy E∗ in the ionization continuum, as illustrated in Fig. 9.67:
Auto ionization M* |k〉 M+ |f〉 neutral ion + − M*(k) → M (f) + Ekin(e )
|i〉
|g〉 Fig. 9.65. Two-step excitation of a molecular Rydberg le- Fig. 9.64. Excitation (1) of a bound Rydberg level in the vel k , which transfers by auto ionization into a lower level f neutral molecule and (2) of a bound level in the molecular of th| e*molecular ion. The difference energy is given to the f|ree* ion M+ electron 368 9. Diatomic Molecules
Fig. 9.66. Section of the auto ioniza- tion spectrum of the Li2 molecule 8,000
6,000 Γ = 0.02 cm−1 ⇒ τeff = 1.3 ns Γ = 0.034 cm−1 4,000 ⇒ τeff = 0.76 ns Γ = 0.027 cm−1
Ion signal ⇒ τeff = 0.96 ns 2,000
0 42,136.6 42,136.8 42,137.0 42,137.2 42,137.4 42,137.6 42,137.8 Wavenumber (cm − 1 )
1. The excitation of the Rydberg level k of the neu- sonance destructive on the other constructive, resulting tral molecule from level i with t|he* probability in an asymmetric line profile. | * amplitude D1 with subsequent autoionization, Continuous spectra can also appear in emission, if 2. The direct photoionization from level i with the a bound upper level is excited that emits fluorescence | * probability amplitude D2. into a repulsive lower state. Such a situation is seen in excimers, which have stable excited upper states but an When the frequency of the excitation lasers is tu- unstable ground state (Fig. 9.68). For illustration, the ned, the phase of the transition matrix element does not emission spectrum of an excited state of the NaK alkali change much for path 2, but much more for path 1, be- molecule is shown in Fig. 9.69. This state is a mixture of cause the frequency is tuned over the narrow resonance of a discrete transition. The total transition probability
2 Pif D1 D2 = | + | E D1Π therefore changes with the frequency of the excitation 3 Π laser because the interference is on one side of the re- Continuous fluorescence |k 〉 E* Discrete D 12 σ lines
D2 D1 1 2
3 Σ+ R Laser excitation σd |i〉 X1Σ a) b) 1/q −q ε Fig. 9.67. (a) Interference of two possible excitation pathways to the energy E∗ in the ionization continuum (b) Resultant Fano-profile with asymmetric line shape. σd is the absorption Fig. 9.68. Level scheme of the NaK molecule with excitation crosssection for direct photoionization and discrete and continuous emission spectrum 9.6. Spectra of Diatomic Molecules 369 a) E b) E'pot(R)
v'= 7
2 I ∝ ∫ ψv'ψv''dr Airy function 630 625 v'' Discrete emission spectrum U(R)
v'' Continuous emission spectrum
v'' E'' (R) 0 pot ν / cm−1 R λ / nm 695 675 655 635 Fig. 9.69. (a) Vibrational overlap and Franck-Condon factor for the continuous emission (b) Measured emission spectrum of the NaK molecule
a singlet and a triplet state, due to strong spin-orbit coup- overlap integral between the vibrational wave function ling. Therefore transitions from this mixed state into of the bound level in the upper electronic state with lower singlet as well as triplet states becomes allowed. the function of the unstable level in the repulsive po- While the emission into the stable singlet ground state tential above the dissociation energy of the lower state X 1Σ shows discrete lines, the emission into the weakly which can be described by an Airy function. The fre- 3 bound lowest triplet state 3 Σ shows, on the short wave- quency ν E2(R) E22(R) and the wavelength λ c/ν length side, a section of discrete lines terminating at of the em=ission de−pends on the internuclear dista=nce R bound vibrational-rotational levels in the shallow poten- because the emission terminates on the Mulliken po- tial well of the a 3Σ state and, on the long wavelength tential of the repulsive lower state (dashed blue curve side, a modulated continuum terminating on energies in Fig. 9.69). The number q v2 1 of nodes in the above the dissociation limit of the a 3Σ state. The inten- fluorescence spectrum gives =the v−ibrational quantum sity modulation reflects the FCF, i.e., the square of the number v2 of the emitting level. 370 9. Diatomic Molecules
S U M M A R Y For the simplified model of a rigid diatomic monotonically decreases with increasing R the • molecule, the electronic wave functions ψ(r, R) state is unstable and it dissociates. and the energy eigenvalues E(R) can be ap- The vibration of a diatomic molecule can be de- proximately calculated as a function of the • scribed as the oscillation of one particle with internuclear distance R. The wave functions are reduced mass M MA MB/(MA MB) in the = + written as a linear combination of atomic orbitals potential Epot(R). In the vicinity of Re the po- (LCAO approximation) or of other suitable basis tential is nearly parabolic and the vibrations can functions. be well-approximated by a harmonic oscillator. In a rotating and vibrating molecule the kinetic The allowed energy eigenvalues, defined by the • energy of the nuclei is generally small compa- vibrational quantum number v, are equidistant red to the total energy of a molecular state. This with a separation ∆E !ω. For higher vibratio- allows the separation of the total wave func- nal energies the molecu=lar potential deviates from el tion ψ(r, R) χN(R)Φ (r, R) into a product of a harmonic potential. The distances between vi- a nuclear wa=ve function χ(R) and an electro- brational levels decrease with increasing energy. nic function Φel(r, R), which depends on the A good approximation to the real potential is the electronic coordinates r and only contains R as Morse-potential, where ∆Evib decreases linearly a free parameter. This approximation, called the with energy. Each bound electronic state has only adiabatic or Born–Oppenheimer approximation, a finite number of vibrational levels. neglects the coupling between nuclear and elec- The rotational energy of a diatomic molecule • 2 tron motion. The potential equals that of the rigid Erot J(J 1)h /2I is characterized by the ro- molecule and the vibration and rotation takes tation=al qu+antum number J and the moment place in this potential. of inertia I MR2. Due to the centrifugal Within this approximation the total energy of force F the= distance R increases slightly • c a molecular level can be written as the sum with J until Fc is compensated by the resto- E Eel Evib Erot of electronic, vibrational ring force Fr dEpot/dR and the rotational and=rotat+ional e+nergy. This sum is independent energy becom=es−smaller than that of a rigid of the nuclear distance R. molecule. The electronic state of a diatomic molecule is The absorption or emission spectra of a diatomic • characterized by its symmetry properties, its to- • molecule consists of: tal energy E and by the angular momentum and a) Pure rotational transitions within the same vi- spin quantum numbers. For one-electron systems brational level in the microwave region these are the quantum numbers λ lz/h and b) Vibrational-rotational transitions within the = σ sz/h of the projections lz of the electronic same electronic state in the infrared region = orbital angular momentum and sz of the spin s c) Electronic transitions in the visible and UV onto the internuclear z-axis. For multi-electron region systems L Σli , S Σsi , Λ Lz/h Σλi , and The intensity of a spectral line is proportional to = = = = • 2 Ms Σσi = Sz/h. Although the vector L might the product N Mik of the population density N = ·| | depend on R, the projection Lz does not. in the absorbing or emitting level and the square The potential curves E (R) are the sum of mean of the matrix element M . • pot ik kinetic energy Ekin of the electrons, their po- Homonuclear diatomic molecules have neither tential energy )and t*he potential energy of the • a pure rotational spectrum nor a vibrational- nuclear repulsion. If these potential curves have rotational spectrum. They therefore do not absorb a minimum at R Re, the molecular state is sta- in the microwave and the mid-infrared region, un- ble. The molecule=vibrates around the equilibrium less transitions between close electronic states fall distance Re. If Epot(R) has no minimum, but into this region. ! Problems 371
The electronic spectrum consists of a system equal to the square of the vibrational overlap • of vibrational bands. Each vibrational band integral. includes many rotational lines. Only rotatio- Continuous absorption spectra arise for transi- nal transitions with ∆J 0; 1 are allowed. • tions into energy states above the dissociation The intensity of a rota=tiona±l transition de- energy or above the ionization energy. Continuous pends on the Ho¨nl-London factor and those emission spectra are observed for transitions of the different vibrational bands are determi- from bound upper states into a lower state with ned by the Franck-Condon factors, which are a repulsive potential.
P R O B L E M S 1. How large is the Coulomb repulsion of the nuclei 5. Show that the energy eigenvalues (9.104) are in the H2+ ion and the potential energy of the obtained when the Morse potential (9.103) is electron with wave function Φ+(r, R) at the inserted into the Schro¨dinger equation (9.80). equilibrium distance Re 2a0? First calculate 6. What is the ionization energy of the H2 mol- = the overlap integral SAB(R) in (9.13) with the ecule when the binding energies of H2 and H2+ wave function (9.9). What is the mean kinetic are EB(H2) 4.48 eV and EB(H2+) 2.65 eV energy of the electron, if the binding energy is and the io=ni−zation energy of th=e −H atom Epot(Re) 2.65 eV? Compare the results with EIo 13.6 eV? the corres=po−nding quantities for the H atom. 7. Calc=ulate the frequencies and wavelengths for 2. How large is the electronic energy of the H2 mol- the rotational transition J 0 J 1 and = → = ecule (without nuclear repulsion) for R Re and J 4 J 5 for the HCl molecule. The in- = = → = for the limiting case R 0 of the united atom? ternuclear distance is Re 0.12745 nm. What is = = 3. a) Calculate the total electronic energy of the H2 the frequency shift between the two isotopomers molecule as the sum of the atomic energies of H35Cl and H37Cl for the two transitions? What is the two H atoms minus the binding energy of H2. the rotational energy for J 5? = b) Compare the vibrational and rotational energy 8. If the potential of the HCl molecule around Re of H2 at a temperature T 300 K with the energy is approximated by a parabolic potential Epot = 2 = of the first excited electronic state of H2. k(R Re) a vibrational frequency ν0 9 13− 1 = × 4. Prove that the two separated equations (9.75) 10 s− is obtained. What is the restoring force are obtained when the product ansatz (9.74) is constant k? How large is the vibrational amplitude inserted into the Schro¨dinger equation (9.73). for v 1? =