Vibrational (Infrared) Spectroscopy vibrational modes ← C≣≣O → equilibrium bond distance re can be changed by applying energy potential well for modified (Morse) potential classical vibrator well for a diatomic molecule   1. quantized – only certain energy levels may exist E = hω(u + 1/2)  u: vibrational quantum number w: vibrational frequency 1 k ω = ―― ―― 2p m m1 × m2 m = ――――― m1 + m2    2. too close – repulsion between nuclei and electrons too far apart – dissociation

ex. HCl u(HCl) = 2990 cm-1 DCl u(DCl) = 2145 cm-1 ex. u(NO) bond order NO+ 2273 cm-1 3 NO 1880 cm-1 2.5 NO- 1365 cm-1 2 NO2- 886 cm-1 1.5 number of vibrational modes a molecule consists of N atoms, there are 3N degrees of freedom translation rotation vibration nonlinear 3 3 3N – 6 linear 3 2 3N – 5 type of vibrational modes stretching mode u bending mode d IR active absorption Raman active absorption

frequencies for some commonly encountered groups, fragments, and linkages in inorganic and organic molecules

ex. W(CO)6 Mn(CO)5Br   compound u(CO) (cm-1) [Ti(CO)6]2- 1740 [V(CO)6]- 1860 Cr(CO)6 2000 [Mn(CO)6]+ 2095 stretching modes of CO and IR frequencies (a) terminal (b) doubly bridging (c) triply bridging    ex.

some ligands capable of forming linkage isomers     IR spectrum for nujol   salt plates NaCl 625 cm-1 KBr 400 cm-1 CsI 200 cm-1 721 1377 1462 2925 2855

symmetry of normal vibrations ex. CO32- 6 vibrational modes C3(u3a) = -1/2u3a + 1/2 u3b C3(u3b) = -3/2u3a - 1/2 u3b   determine the symmetry type of normal modes   E c = 12

C3     c = 0   C2 c = -2 c (sh) = 4 c (S3) = -2 ?? c (sv) = 2

G = A1’ + A2’ + 3E’ + 2A2” + E” 3 translatory modes: E’, A2” 3 rotational modes: A2’, E” genuine vibrational modes: Gg = A1’ +2E’ + A2” IR active: E’, A2” (3 bands) Raman active: A1’, E’ (3 bands) particular internal coordinates to normal modes C—O bonds E C3 C2 sh S3 sv G 3 0 1 3 0 1 GCO = A1’ + E’ in-plane stretching GOCO = A1’ + E’ in-plane bending A2” out-of-plane bending   ex. determine the number of IR active CO stretching bands for the following metal carbonyl compounds : M(CO)6 M(CO)5L cis-M(CO)4L2 trans-M(CO)4L2 fac-M(CO)3L3 mer-M(CO)3L3 M(CO)5 M(CO)4L M(CO)3L2 M(CO)4

(i) trans-M(CO)4L2 D4h E C4 C2 C2’ C2” i S4 sh sv sd OC CO OC CO ==> A1g + B1g + Eu L IR-active: Eu (ii) cis-M(CO)4L2 CO C2v E C2 s s’ OC L 4 0 2 2 OC L ==> 2A1 + B1 + B2 CO IR-active: 2A1, B1, B2   (iii) mer-M(CO)3L3 CO C2v E C2 s s’ L L 3 1 1 2 OC L ==> 2A1 + B1 OC IR-active: 2A1, B1 (iv) M(CO)5 D3h E C3 C2 sh S3 sv 5 2 1 3 0 3 ==> 2A1’ + A2” + E’ IR-active: A2”, E’

(v) M(CO)4L L D3v E C3 sv 4 1 2 ==> 2A1 + E IR-active: 2A1, E 4 1 2 ==> 2A1 + E IR-active: 2A1, E   D2v E C3 sv sv‘ L 4 0 2 2 ==> 2A1 + B1 + B2 IR-active: 2A1, B1, B2 (vi) M(CO)3L2 L D3h E C3 C2 sh S3 sv 3 0 1 3 0 1 ==> A1‘ + E’ L IR-active: E’ L Cs E sh 3 1 L ==> 2A‘ + A” IR-active: 2A’, A” (vii) M(CO)4 Td E C3 C2 S3 sd 4 1 0 0 2 ==> A1 + T2 IR-active: T2

number of CO stretching bands in IR spetcrum for metal carbonyl compounds

number of IR bands of some common geometric arrangements

calculation of force constants for diatomic molecule AB harmonic oscillator f • m-1 – l = 0 for polyatomic molecule Wilson’s method “The F and G matrix method” |FG – El| = 0 F: matrix of force constant (potential energy) G: matrix of masses and spatial relationship of atoms (kinetic energy) E: unit matrix  e.g. H2O Gg = 2A1 + B1 2 O-H distance Dd1, Dd2 A1 + B1 ∠HOH Dθ A1 using projection operator to obtain complete set of symmetry coordinates for vibrations A1 : S1 = Dθ S2 = 1/√2(Dd1 + Dd2) B1 : S3 = 1/√2(Dd1 - Dd2) F matrix 2V = Sfik si sk si, sk: change in internal coordinates for Dd1 Dd2 Dθ Dd1 fd fdd fdθ Dd2 fdd fd fdθ Dθ fdθ fdθ fθ

2V = fd(Dd1)2 + fd(Dd2)2 + fθ(Dθ)2 + 2 fdd(Dd1 Dd2) + 2 fdθ(Dd1 Dθ) + 2 fdθ(Dd2 Dθ) = [Dd1 Dd2 Dθ] fd fdd fdθ Dd1 fdd fd fdθ Dd2 fdθ fdθ fθ Dθ = s’f s   relationship between the internal coordinates and the symmetry coordinates S = U s U matrix Dq 0 0 1 Dd1 Dd1 + Dd2 = 1/√2 1/√2 0 Dd2 Dd1 - Dd2 1/√2 -1/√2 0 Dq S = U s s = U’ S s’ = (U’ S)’ = S’U s’fs = S’FS (S’U)f(U’S) = S’FS S’(UfU’)S = S’FS ==> F = UfU’ 0 0 1 fd fdd fdθ 0 1/√2 1/√2 F = 1/√2 1/√2 0 fdd fd fdθ 0 1/√2 -1/√2 1/√2 -1/√2 0 fdθ fdθ fθ 1 0 0 fθ √2 fdθ 0 = √2 fdθ fd + fdd 0 0 0 fd - fdd

G = 1/√2 1/√2 0 gdd gd gdθ 0 1/√2 -1/√2 G matrix G = UgU’ 0 0 1 gd gdd gdθ 0 1/√2 1/√2 G = 1/√2 1/√2 0 gdd gd gdθ 0 1/√2 -1/√2 1/√2 -1/√2 0 gdθ gdθ gθ 1 0 0 g33 √2 g13 0 = √2 g13 g11 + g12 0 0 0 g11 - g12 g11 = mH + mO g12 = mO cosθ g13 = -(mO/r) sinθ g33 = 2(mH + mO - mO cosθ)/r2 m : reciprocal of the mass 2(mH + mO - mO cosθ)/r2 -(√2mO/r) sinθ 0 G = -(√2mO/r) sinθ mH + mO (1+ cosθ) 0 0 0 mH + mO (1 - cosθ)   for H2O θ= 104.3o31’ r = 0.9580 Å 2.332 -0.0893 0 G = -0.0893 1.0390 0 0 0 1.0702 fθ √2 fdθ 2.332 -0.0893 l 0 A1: – = 0 √2 fdθ fd + fdd -0.0893 1.0390 0 l B1: 1.0702(fd - fdd) = l

elements of the g matrix mi: reciprocal mass of the ith atom rij: reciprocal of the distance between ith and jth

Raman spectroscopy light of energy less than that required to promote a molecule into an excited electronic state is absorbed by a molecule, a virtual excited state is created virtual state is very short lifetime, the majority of the light is re-emitted over 360oC, this is called Rayleigh scattering C. V. Raman found that the energy of a small proportion of re-emitted light differs from the incident radiation by energy gaps that correspond to some of the vibrational modes Stokes lines anti-Stokes line

schematic representation of Raman spectrometer selection rules for vibrational transitions • a fundamental will be infrared active if the normal mode which is excited belongs to the same representation as any one or several of the Cartesian coordinates • a fundamental will be Raman active if the normal mode involved belongs to the same representation as one or more of the components of the polarizability tensor of the molecule the exclusion rule – in centrosymmetric molecules, no Raman-active vibration is also IR-active and no IR-active vibration is also Raman-active only fundamentals of modes belonging to g representations can be Raman active and only fundamentals of modes belonging to u representations can be IR active

ex. Na2MoO4 dissolved in HCl exhibits Raman peaks at 964, 925, 392, 311, 246, 219 cm-1 925, 311 cm-1 being polarized what can be deduced from the spectrum? no n(Mo—H) and n(O—H) bands only M—Cl and M=O likely exist 964, 925 cm-1 Mo=O stretching bands 392 cm-1 Mo=O bending mode 311, 246, 219 cm-1 Mo—Cl stretching modes possible product:

normal vibrational modes for common structures