Structure and spectra of polyatomic molecules

Transcription

1 Chapter 5 Structure and spectra of polyatomc molecules 5.1 Structure of polyatomc molecules The same approxmatons can be used for the statonary states of a polyatomc molecule as for a datomc molecule, n partcular the approxmate separablty of the motons and the Born-Oppenhemer approxmaton dscussed n Chapter Rotatonal Structure The rotatonal energy level structure of a molecule can be determned by analyzng the rotatonal moton classcally and then, by usng the correspondence prncple, to determne the Hamlton operator Ĥrot and fnally, by determnng the egenfunctons and egenvalues of Ĥ rot. Angular velocty, angular momentum and nertal tensor Let us consder a rgd rotor. Each partcle of the rotor has the same angular velocty ω around an axs passng through the center of mass (CM) as shown n Fgure 5.1. Fgure 5.1: Angular velocty ω. 135

2 136 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES The angular momentum s J = N m ( r v )= =1 N m [ r ( ω r )]. (5.1) Usng v 1 ( v 2 v 3 )=( v 1 v 3 ) v 2 ( v 1 v 2 ) v 3, Equaton (5.1) can be rewrtten as J = N =1 =1 m [ ω r 2 r ( r ω)] (5.2) wth r ω = x ω x + y ω y + z ω z.thus: [ ] [ ] [ ] J x = m (r 2 x 2 ) ω x m x y ω y m x z ω z, (5.3) [ ] [ ] [ ] J y = m x y ω x + m (r 2 y 2 ) ω y m y z ω z, (5.4) [ ] [ ] [ ] J z = m x z ω x m y z ω y + m (r 2 z 2 ) ω z. (5.5) One can express Equatons (5.3) (5.5) n matrx notaton: J x J y J z J = I ω (5.6) I xx I xy I xz = I yx I yy I yz I zx I zy I zz ω x ω y ω z (5.7) wth I αα = m (r 2 α 2 ) I αβ = m α β, (5.8) whereby α, β = x, y, z. ThematrxI s real and symmetrc. A untary transformaton can brng I n dagonal form wth dagonal elements I a, I b, I c where a, b, c are the prncpal axes of the rgd rotor. I a, I b and I c are the roots of the determnantal equaton: I xx λ I xy I xz I yx I yy λ I yz =0. (5.9) I zx I zy I zz λ Conventon: The prncpal axes are labelled a, b and c so that I a I b I c. (5.10)

3 5.1. STRUCTURE OF POLYATOMIC MOLECULES 137 The angular momentum components n the prncpal axs system are J a = I a ω a, J b = I b ω b, (5.11) J c = I c ω c. Classfcaton of rotors Molecules can be classfed nto dfferent types of rotors dependng on the relaton between the three dagonal components of the moment of nerta. I a =0,I b = I c : lnear molecule I a <I b = I c : prolate symmetrc top wth the shape of a rugby ball, see Fgure 5.2 (example: CH 3 Cl) I a = I b <I c : oblate symmetrc top wth the shape of a dsc, see Fgure 5.2 (example: benzene) I a = I b = I c = I : sphercal top (examples: CH 4,C 60 ) I a <I b <I c : asymmetrc top (example: H 2 O) Fgure 5.2: Scheme of oblate and prolate symmetrc top rotors. Moreover, for planar molecules: I a + I b = I c (5.12) Rgd rotor energy levels The classcal knetc energy of a rotor s: T = 1 m v 2 = 1 m v ( ω r )= ω m ( r v )= 1 ω J 2 = 1 2 [ ω 2 a I a + ω 2 b I b + ω 2 c I c ] = J 2 a 2I a + J 2 b 2I b + J 2 c 2I c. (5.13)

4 138 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES Accordng to the correspondence prncple, the Hamltonan for a rgd rotor s therefore Ĥ rot = Ĵ 2 a 2I a + Ĵ 2 b 2I b + Ĵ 2 c 2I c. (5.14) The quantum mechancal descrpton of the rotatonal moton of a rotor requres three quantum numbers because there are three degrees of freedom assocated wth the rotaton n free space. We use the followng quantum numbers: J the rotatonal quantum number, M the quantum number assocated wth the projecton of J onto the space-fxed Z axs and K the quantum number assocated wth the projecton of J onto the z axs of the molecule-fxed reference frame. Wth these notatons, the wave functons J KM are egenfunctons of Ĵ 2 and Ĵz. The rotatonal constants A, B and C are defned as 2 2I a = hca, 2 2I b = hcb, and (5.15) 2 2I c = hcc. Wth ths defnton the rotatonal constants are expressed n cm 1. In the followng we wll smplfy the expresson of the rotatonal energy, when possble, for each specfc case of rotors (see also lecture notes Physcal Chemstry III). Sphercal top: I a = I b = I c Equaton (5.14) becomes Ĥ rot = Ĵ 2 2I, (5.16) Ĥ rot JKM = E J JKM 1 2I Ĵ 2 JKM = 2 J(J +1) J KM. (5.17) } 2I {{} Egenvalue The egenvalues for the rotatonal moton of a sphercal top are E J = 2 J(J +1). (5.18) 2I The spectroscopc usage s to wrte E J = hcbj(j +1)wthB = 2 2hcI. The energy does not depend on K or M and each level has a degeneracy factor g =(2J + 1)(2J +1).

5 5.1. STRUCTURE OF POLYATOMIC MOLECULES 139 Oblate symmetrc top: I a = I b <I c. The axes wll be chosen as follows: a x, b y, c z. Equaton (5.14) becomes [ 1 Ĥ rot JKM = (Ĵ x 2 + 2I Ĵ y 2 )+ 1 ] Ĵz 2 J KM b 2I c [ 1 = (Ĵ 2 2I Ĵ z 2 )+ 1 ] Ĵz 2 J KM b 2I c = hc { BJ(J +1)+(C B)K 2} J KM (5.19) }{{} Egenvalue wth B>C. The degeneracy s g J,0 =2J +1forK = 0 and g J,K =2(2J +1)forK 0. Prolate symmetrc top: I a <I b = I c The axes wll be chosen as follows: a z, b x, c y. Equaton (5.14) becomes Ĥ rot JKM = [ 1 (Ĵ 2 2I Ĵ z 2 )+ 1 ] Ĵz 2 J KM = hc { BJ(J +1)+(A B)K 2} JKM b 2I a (5.20) wth A>B. The degeneracy s g J,0 =2J +1forK = 0 and g J,K =2(2J +1)forK 0. Datomc molecule: The case of datomc molecules (already seen n Chapter 3) can be consdered as a specal case of prolate top wth A =. Hence K =0. E JKM = hcbj(j + 1) (5.21) wth a degeneracy of g J =2J + 1. Fgure 5.3 summarzes the rotatonal level structure of the four specfc rotors dscussed. Fgure 5.3: Schematc dagram representng the rotatonal level structure of dfferent rotors. The dagram also ncludes the degeneracy g of the rotatonal levels. Levels wth K J do not exst.

6 140 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES Asymmetrc top: I a I b I c The J KM functons are not egenfunctons of Ĥrot. However, they form a complete set of bass functons and the wave functons J τm of an asymmetrc top can be represented as a lnear combnaton of these bass functons: a J τm = J K= J c τ,k JKM (5.22) To obtan the energy levels of an asymmetrc top, the followng procedure can be followed: one constructs Ĥrot n matrx form and obtans the energes by lookng for the egenvalues of Ĥ rot (after dagonalzaton of the matrx). Ĥ rot = 1 2I a Ĵ 2 a + 1 2I b Ĵ 2 b + 1 2I c Ĵ 2 c (5.23) Example: constructon of the matrx representaton of Ĵz and Ĵ z 2 n the { J K(M) } bass for J =1: { } Ĵ z Ĵ 2 z The matrx elements are calculated n the followng way: (J z ) 11 = 11 Ĵz 11 = 11 K 11 = K = K = (J z ) 12 = 11 Ĵz 10 = = =0 0=0 }{{} 0 (J z ) 22 = 10 Ĵz 10 = =0 (J z ) 33 = 1 1 Ĵz 1 1 = = = a Note that τ s not a good quantum number but smply an ndex runnng from J to J wth the assocated energy levels E Jτ arranged n ascendng order. In the lterature, the double ndex K a,k c s used nstead of τ, wherek a and K c represent K n the prolate top and oblate top lmtng case, respectvely; τ = K a K c.

7 5.1. STRUCTURE OF POLYATOMIC MOLECULES Vbratonal Structure The number of vbratonal degrees of freedom s f =3N 6 for a nonlnear molecule, and f =3N 5 for a lnear molecule. In the treatment of the vbratonal moton of a polyatomc molecule, one makes use of the near-separablty of the nuclear Schrödnger equaton n the normal modes. pont s Ĥ = 3N =1 One ntroduces mass-weghted dsplacement coordnates so that The startng ˆp 2 2m + V (x 1,...,z N ). (5.24) q = m Δx = m (x x eq ), (5.25) ˆT = 1 2 3N =1 q 2. (5.26) The potental s then expanded around the equlbrum poston (eq) n a Taylor seres: Harmonc vbratons 3N ( ) V V = V eq + q + 1 q =1 eq 2 }{{} 0 at the mnmum 3N 3N =1 j ( 2 V q q j ) q q j +... (5.27) eq Settng V eq = 0 and makng the harmonc approxmaton, Equatons (5.24) and (5.27) smplfy to Ĥ = ˆT N 3N =1 j k j q q j, (5.28) wth ( 2 ) V k j =. (5.29) q q j eq By a sutable bass transformaton and separaton of the rotatonal and translatonal moton one obtans the normal modes Q k as lnear combnaton of the dsplacement coordnates: Q k = 3N =1 L k q k =1, 2,..., 3N 6, (5.30) where L k represent the expanson coeffcents. The normal modes form an orthogonal set and Ĥ can be wrtten as a sum of terms actng only on the coordnate Q k Ĥ = 3N 6 =1 ( 1 2 Q ) 2 λ Q 2. (5.31)

8 142 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES Because Ĥ s separable n the normal modes, one can wrte the soluton of the Schrödnger equaton as E v1,v 2,...,v 3N 6 = 3N 6 =1 ( hν v + 1 ) 2 (5.32) Ψ v1,v 2,...,v 3N 6 = Ψ v1 (Q 1 ) Ψ v2 (Q 2 )... Ψ v3n 6 (Q 3N 6 ). (5.33) Example: tratomc molecules (N =3) The number of degrees of freedom s 3 for SO 2 and 4 for CO 2. In CO 2 the bendng mode s doubly degenerate (see Fgure 5.4). The exctaton of a degenerate vbratonal mode leads to a vbratonal angular momentum l. In CO 2 ths angular momentum s parallel to the molecular axs (see Fgure 5.5) and corresponds to a superposton of moton n the two degenerate bendng modes. l = l z = l wth l = v 2,v 2 +2,..., v 2. The total angular momentum s J = J R + l (5.34) and thus E rot = hcbj R (J R +1)=hcB[J(J +1) l 2 ] (5.35) Anharmonc vbratons Takng hgher terms n Equaton (5.27) one gets: V = V N 3N =1 j k j q q j + 1 3! 3N 3N 3N =1 j k j k jk q q j q k +... (5.36) and T (v 1,..., v 3N 6 ) = E v 1,...,v 3N 6 = hc + degenerate modes l ( ω v + d ) + 2 j ( x j v + d )( v j + d ) g j l l j (5.37) where d s the degree of degeneracy of the vbratonal mode. The constants ω and x j are gvenncm 1 n Equaton (5.37). Example: CO 2

9 5.1. STRUCTURE OF POLYATOMIC MOLECULES 143 Fgure 5.4: Schematc representaton of the normal modes of CO 2 (top) and SO 2 (bottom). Fgure 5.5: Vbratonal angular momentum l of CO 2 ω 1 = cm 1, ω 2 = cm 1, ω 3 = cm 1, x 11 = 3.1 cm 1, x 22 = 1.59 cm 1,

10 144 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES x 33 = 12.5 cm 1, x 12 = 5.37 cm 1, x 13 = cm 1, x 23 = cm 1, g 22 = 0.62 cm 1. ( T (v 1,v 2,v 3 ) = ω 1 v ) ( + ω 2 (v 2 +1)+ω 3 v ) 2 2 ( +x 11 v 1 + 2) 1 2 ( ) 2 + x 22 (v 2 +1) 2 + x 33 +x 12 ( v ) (v 2 +1)+x 13 ( v v )( v ) ( + x 23 (v 2 +1) v ) 2 +g 22 l 2 2 (5.38) The zero pont vbratonal energy contans a contrbuton from the anharmoncty terms (l =0for v 2 =0) T (0, 0, 0) = ω ω 2 + ω x x 22 + x x x x (5.39) Molecular orbtals, electronc confguratons, and electronc states The electronc structure of polyatomc molecules can be descrbed usng the same prncples as those ntroduced for datomc molecules n chapter 3. However, the same polyatomc molecule can have dfferent geometres and belongs to dfferent pont groups dependng on ts electronc state. The varety of possble electronc states and molecular structures s so large that t s mpossble to gve a complete overvew n ths chapter. We therefore restrct the dscusson to only a few representatve molecular systems: Molecules of the form HAH as prototypcal small molecules, the cyclopentadenyl caton and benzene as typcal hghly symmetrcal molecules and adenne as example of a nonsymmetrcal large molecule. The prncples that we descrbe are easly generalzed to arbtrary molecules. Small polyatomc molecules wth the example of HAH molecules Molecules possessng the chemcal formula HAH (A desgnates an atom, e. g., Be,B,C,N,O, etc.) are ether lnear and belong to the D h pont group (see character table n Chapter 4), or bent and belong to the C 2v pont group (see character table 5.1). Molecular orbtals are therefore classfed ether n the D h or the C 2v pont group (see character tables n chapter 4). For smplcty we consder here only valence states of HAH molecules wth A beng an atom from the second or thrd row of the perodc table, so that l 2 atomc orbtals can be gnored n the dscusson of the electronc structure. The determnaton of the molecular orbtals may proceed along the followng scheme:

11 5.1. STRUCTURE OF POLYATOMIC MOLECULES 145 C 2v E C z 2 σ xz v σ yz v A T z α xx,α yy,α zz A R z α xy B T x,r y α xz B T y,r x α yz Table 5.1: Character table of the C 2v pont group. 1. Identfcaton of all atomc orbtals partcpatng n the formaton of molecular orbtals. Symmetry restrcts the number of these orbtals. In the case of HAH molecules, the requred orbtals are the two 1s orbtals of the hydrogen atoms and the ns andnp valence orbtals of the central atom A, where n 2 represents the row of the perodc system of elements to whch A belongs. Orbtals belongng to nner shells of the central atom usually le so deep n energy and are so strongly localzed on the nucleus that they hardly contrbute to molecular bonds. 2. Formaton of symmetry-adapted molecular orbtals from the set of atomc orbtals determned under (a) ( = 6 n the case of HAH molecules). Symmetry-adapted molecular orbtals transform as rreducble representatons of the correspondng pont group. In molecules such as HAH n whch two or more atoms are equvalent, t s convenent to frst buld symmetry-adapted lnear combnatons of the orbtals of these equvalent atoms,. e., of the H 1s orbtals n the case of HAH molecules (see Fgure 5.6). z g u a 1 b 2 (a) (b) Fgure 5.6: Symmetry adapted lnear combnatons of 1s orbtals that partcpate n the constructon of molecular orbtals of (a) lnear and (b) bent HAH molecules. These orbtals are then used to form molecular orbtals wth the orbtals of the central atom A havng the correspondng symmetry, as shown n Fgure 5.7 for the pont group D h. The ns (σ g )andnp z (σ u ) orbtals of the central atom can be combned wth the symmetryadapted orbtals of the H atoms n two ways each, resultng n four molecular orbtals of σ symmetry. The energetc orderng of these molecular orbtals can be derved from the num-

12 146 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES 1s 2s 1s 1s 2p z 1s 1s 2s 1s 1s 2p z 1s 2 g 1 u 3 g 2 u 1 u Fgure 5.7: Lnear combnatons of atomc orbtals n lnear HAH molecules ber of nodal planes of the wave functons. The 2σ g (zero nodal plane) and 1σ u (one nodal plane) orbtals n Fgure 5.7 are bondng, whereas the 3σ g (two nodal planes) and 2σ u (three nodal planes) orbtals are antbondng. For symmetry reasons, the 2p x and 2p y orbtals of the central atom (both of π u symmetry) cannot combne wth the 1s orbtals of the H atoms and are therefore nonbondng orbtals. The energetc orderng of these molecular orbtals s gven on the rght-hand sde of Fgure 5.8 whch also shows how the energes of the molecular orbtals change as the molecule s progressvely bent from the lnear D h structure ( (HAH)= 180 )towardthec 2v structure wth (HAH)= 90. The orbtals of the bent molecules dsplayed on the left-hand sde of Fgure 5.8 are gven symmetry labels of the C 2v pont group accordng to ther transformaton propertes (see character table). The two lowest-lyng orbtals are nvarant under rotatons around the C2 z axs, and also under reflecton n both the σ xz and σ yz planes, and are therefore totally symmetrc (a 1 ). The next hgher-lyng orbtal s antsymmetrc under C 2 rotaton and under σ xz reflecton, but symmetrc under σ yz reflecton,andsthusofb 2 symmetry. As n the lnear geometry, the energetc orderng essentally follows the number of nodal planes. The degeneracy of the two nonbondng π u orbtals s lfted as the molecule bends. The molecular orbtal correspondng to the p orbtal n the molecular plane becomes bondng and correlates wth the 3a 1 molecular orbtal of the bent molecule. The other molecular orbtal, whch s perpendcular to the molecular plane, remans a nonbondng orbtal and correlates wth the 1b 1 orbtal of the bent molecule. The angle dependence of the 1π u 3a 1 orbtal energy s of partcular mportance, because ths orbtal s the only one that becomes sgnf-

13 5.1. STRUCTURE OF POLYATOMIC MOLECULES 147 cantly more stable n the nonlnear geometry. All other molecular orbtals are destablzed when the HAH angle s decreased. The occupaton of ths orbtal wth one or two electrons can result n a bent equlbrum structure of the molecule. Correlaton dagrams as the one shown n Fgure 5.8 are known as Walsh dagrams. As n the case of atoms and datomc molecules, the electronc confguratons of polyatomc molecules are obtaned by fllng the molecular orbtals wth a maxmum of two electrons. Whether a molecule of the form HAH s lnear or bent depends on the occupaton of the orbtals, especally of the 3a 1 orbtal, as dscussed above. The symmetres of the electronc states that result from a gven confguraton are obtaned from the drect product of the rreducble representatons of the occuped molecular orbtals (see Secton 4.3.4). Fnally, the multplctes (2S + 1) are derved followng exactly the same procedure as dscussed for atoms and datomc molecules n chapters 2 and 3 (see also Secton below).

14 148 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES Fgure 5.8: Walsh dagram for HAH molecules. The symmetry labels on the left-hand sde correspond to C 2v pont-group symmetry, those on the rght-hand sde to D h symmetry. Examples: BeH 2 :...(2σ g ) 2 (1σ u ) 2 The dependence of the energes of the occuped orbtals favors a lnear structure (see Fgure 5.8) and the ground electronc state s therefore the X 1 Σ + g state. BeH 2 :...(2a 1 ) 2 (1b 2 ) 1 (3a 1 ) 1 The frst excted confguraton leads to a bent structure because the 3a 1 1π u orbtal s occuped. The electronc confguraton s thus gven usng C 2v symmetry labels. The two electronc states resultng from ths confguraton are of 3 B 2 and 1 B 2 symmetry. H 2 O:...(2a 1 ) 2 (1b 2 ) 2 (3a 1 ) 2 (1b 1 ) 2 Because the 3a 1 orbtal n H 2 O s doubly occuped, the electronc ground state s also bent. The ground electronc state s thus the X 1 A 1 state.

15 5.1. STRUCTURE OF POLYATOMIC MOLECULES 149 H 2 O + :...(2a 1 ) 2 (1b 2 ) 1 (3a 1 ) 2 (1b 1 ) 1 X+ 2 B 1. Walsh dagrams such as that dsplayed n Fgure 5.8 are also useful n the dscusson of vbronc nteractons because they enable one to see how the degeneracy of π orbtals and of the Π, Δ, Φ,... electronc states are lfted, and how the electronc character changes, when the molecules bend out of ther lnear structures. Larger symmetrc molecules To determne the molecular orbtals of larger polyatomc molecules that have a hgh symmetry, t s useful to follow the systematc approach ntroduced n chapter 4; the symmetrzed lnear combnaton of atomc orbtals (LCAO) are determned usng projecton operators ˆP Γ (Equaton (4.11)) that are appled onto one of the atomc orbtals of the set of dentcal atoms. To llustrate the applcaton of the projecton formula, we use t to derve the system of π molecular orbtals of benzene n the D 6h pont group by buldng symmetry-adapted lnear combnatons Φ (s) 6 Γ = c Γ, φ (5.40) =1 of the carbon 2p z orbtals ϕ ( =1 6). From the sx 2p z atomc orbtals nvolved n the π orbtal system, whch form a sx-dmensonal reducble representaton of the D 6h pont group, a total of 6 orthogonal molecular orbtals can be formed. The reducble representaton Γ of the carbon 2p z orbtals can be constructed usng the character table of the D 6h pont group presented n Table 5.2. Under the group operatons of D 6h,the2p z orbtals ϕ have the same symmetry propertes as the components z of the nuclear dsplacement vectors of the carbon atoms. The 2p z orbtals are mapped onto each other by the symmetry operatons of the group. From the propertes of the representaton matrces, t can be easly establshed that each orbtal that s left unchanged by a symmetry operaton Ô adds 1 to the character χ(ô), each orbtal that s nverted adds -1 to the character, each orbtal that s mapped onto another orbtal gves no contrbuton to the character.

16 150 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES D 6h E 2C 6 2C 3 C 2 3C 2 3C 2 2S 3 2S 6 σ h 3σ d 3σ v A 1g A 2g R z B 1g B 2g E 1g R x, R y E 2g A 1u A 2u z B 1u B 2u E 1u x, y E 2u Table 5.2: Character table of the D 6h pont group. Thus, the characters of the reducble representaton Γ of the sx 2p z dfferent classes of symmetry operatons Ô are orbtals under the D 6h E 2C 6 2C 3 C 2 3C 2 3C 2 2S 3 2S 6 σ h 3σ d 3σ v Γ Reducng Γ usng Equaton (4.10) yelds the symmetres of the sx π molecular orbtals: Γ=B 2g A 2u E 1g E 2u, (5.41) each of the two e rreducble representatons beng two-dmensonal. The orthonormal set of symmetry-adapted bass functons {Φ (s) } s constructed by projectng the ϕ on ther rreducble components usng Equaton (4.11) (projecton operator) and Table 5.2: Φ (s) A 2u = 1 N ˆP A 2u φ 1 = 1 ( ) φ1 + φ 2 + φ 3 + φ 4 + φ 5 + φ 6, (5.42) 6 where N s a normalzaton constant. The same result s obtaned by applyng the projector to any other φ, 1. Φ (s) B 2g = 1 N ˆP B 2g φ 1 = 1 6 ( φ1 φ 2 + φ 3 φ 4 + φ 5 φ 6 ). (5.43)

17 5.1. STRUCTURE OF POLYATOMIC MOLECULES 151 For mult-dmensonal subspaces, the projecton technque usually yelds nonorthogonal lnear combnatons of the orgnal bass vectors. To construct the symmetry-adapted bass, t s suffcent to determne d lnearly ndependent vectors and then choose sutable orthogonal lnear combnatons of them: Φ (s) E 1g,1 = 1 N ˆP E 1g φ 1 = 1 12 ( 2φ1 + φ 2 φ 3 2φ 4 φ 5 + φ 6 ), (5.44) Φ (s) E 1g,2 = 1 N ˆP E 1g φ 2 = 1 12 ( φ1 +2φ 2 + φ 3 φ 4 2φ 5 φ 6 ). (5.45) The set of vectors {Φ (s) E 1g,1, Φ(s) E 1g,2} s lnearly ndependent, but not orthogonal. A set of orthogonal bass vectors can be obtaned by usng the Schmdt orthogonalzaton algorthm: f whch s orthog- ψ 1,ψ 2 are non-orthogonal, normalzed bass vectors, then a bass vector ψ2 onal to ψ 1 can be constructed usng Equaton (5.46): ψ 2 = ψ 2 ψ 2 ψ 1 ψ1, (5.46) where.. denotes the scalar product. Thus: After normalzaton, one obtans Φ (s), E 1g,2 = Φ(s) E 1g,2 Φ (s) E 1g,2 Φ (s) E 1g,1 Φ (s) E 1g,1 = Φ (s) E 1g,2 1 2 Φ(s) E 1g,1. (5.47) Φ (s), E 1g,2 = 1 2( φ2 + φ 3 φ 5 φ 6 ). (5.48) The set of symmetry-adapted bass vectors {Φ (s) E 1g,a, Φ(s) E 1g,b } = {Φ(s) E 1g,1, Φ(s), E 1g,2 } for the E 1g subspace s thus: Φ (s) E 1g,a = 1 12 ( 2φ1 + φ 2 φ 3 2φ 4 φ 5 + φ 6 ), (5.49) Φ (s) E 1g,b = 1 2( φ2 + φ 3 φ 5 φ 6 ). (5.50) Smlarly, one obtans an orthonormal set of symmetry-adapted bass vectors for the E 2u subspace: Φ (s) E 2u,a = 1 12 ( 2φ1 φ 2 φ 3 +2φ 4 φ 5 φ 6 ), (5.51) Φ (s) E 2u,b = 1 2( φ2 φ 3 + φ 5 φ 6 ). (5.52) These molecular orbtals are depcted n Fgure 5.9. Ther energetc orderng can be determned from the number of nodal planes. The a 2u orbtal must be the most stable because

18 152 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES t possesses a sngle nodal plane (the plane contanng the carbon atoms). The e 1g orbtal possesses two nodal planes and has therefore the second lowest energy. The e 2u and b 2g orbtals possess three and four nodal planes and are thus the orbtals of second hghest and hghest energy, respectvely. A more quanttatve estmate of the relatve energes of the molecular orbtals can be acheved usng the Hückel molecular orbtal (HMO) model. The HMO model represents a smple sememprcal method to calculate the electronc energy level structure of molecules that exhbt conjugated π molecular orbtals such as polyenes and aromatc molecules. The model s useful to gan a sem-quanttatve descrpton of the π molecular orbtals and ther relatve energes and s wdely used n physcal-organc chemstry. Wthn the framework of the HMO model, the π molecular orbtals are constructed by lnear combnatons of orthogonal 2p z atomc orbtals centered on the carbon atoms. The energes E k of the π molecular orbtals are obtaned by solvng the secular determnant det(h j E k S j )=0, (5.53) where H j are the matrx elements of a formal Hamltonan operator Ĥ descrbng the π electron system (the Hückel operator ) and S j denotes the overlap ntegral between the p z orbtals of atoms and j. The expanson coeffcents c (k) of the molecular orbtal Φ k n the bass of the atomc 2p z orbtals {ϕ } are obtaned by solvng the set of secular equatons The followng approxmatons are ntroduced: c (k) ( ) Hj E k S j =0. (5.54) All overlap ntegrals vansh (S j = 0) unless = j, n whch case S =1. All dagonal elements of H are the same: H = α. All off-dagonal elements of H are set to zero, except those between neghborng atoms, whch are H j = β. β s usually negatve (β <0). α and β are treated as effectve parameters that can n prncple be estmated from calormetrc data. The matrx representaton of the Hückel operator Ĥ descrbng the π molecular orbtal system

19 5.1. STRUCTURE OF POLYATOMIC MOLECULES 153 can be derved n the bass of the carbon 2p z atomc orbtals {φ },ands α β β β α β β α β 0 0 H =. (5.55) 0 0 β α β β α β β β α The egenvectors and egenvalues of the matrx (5.55) represent the molecular orbtals and ther energes, respectvely. Alternatvely, the Hückel operator can be expressed n the bass of symmetry-adapted bass functon {Φ (s) } by evaluatng the matrx elements accordng to H (s) j = Φ (s) Ĥ Φ (s) j (5.56) usng the bass functons gven n Equatons (5.42), (5.43), (5.49), (5.50), (5.51), (5.52). For the Φ (s) A 2u orbtal (Equaton (5.42)), one fnds, for nstance: (s) Φ Ĥ A 2u Φ (s) 1 A 2u = φ1 + φ 2 + φ 3 + φ 4 + φ 5 + φ 6 Ĥ φ 1 + φ 2 + φ 3 + φ 4 + φ 5 + φ 6 6 = α +2β. (5.57) Matrx elements between functons of dfferent symmetry and matrx elements between orthogonal bass functons wthn the E 1g and E 2u subspaces are zero because H s totally symmetrc, so that one obtans the followng Hückel matrx α +2β α 2β H (s) 0 0 α + β =, (5.58) α + β α β α β whch s already n dagonal form. The symmetry-adapted orthonormal bass functons {Φ (s) } are thus the egenvectors of the Hückel operator and represent the π molecular orbtals of benzene depcted n Fgure 5.9.

20 154 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES Fgure 5.9: Energy level dagram and schematc representaton of the π molecular orbtals of benzene. The sze of the crcles represents the weght of the atomc orbtal n the molecular orbtal wave functon. The two grey tones of the shadng ndcate the relatve sgn of the 2p z orbtals whch form the molecular orbtal. The energes of the molecular orbtals ncrease wth the number of nodal planes and are expressed, on the rght-hand sde of the fgure, as a functon of the Hückel parameters α and β. The arrows represent schematcally the occupaton of the molecular orbtals n the ground-state confguraton of benzene. One should note that a group theoretcal treatment normally only dvdes the Hamltonan matrx n as many dagonal blocks as there are rreducble representatons,. e., n the present case, n two 1 1 dagonal blocks correspondng to A 2u and A 2g and two 2 2 dagonal blocks correspondng to E 1g and E 2u. The fact that the Hamltonan matrx s fully dagonal n Equaton (5.58) s a consequence of the partcular choce made durng the Schmdt orthogonalzaton procedure. The lowest energy confguraton of π electrons n benzene can thus be wrtten (a 2u ) 2 (e 1g ) 4, gvng rse to a sngle electronc state of symmetry 1 A 1g. The frst excted electronc confguraton of benzene s (a 2u ) 2 (e 1g ) 3 (e 2u ) 1. Ths confguraton gves rse to several electronc states as wll be dscussed n Secton The drect product of the partally occuped orbtals s E 1g E 2u =B 1u B 2u E 1u. Snce two dfferent spatal orbtals are partally occuped, there s no restrcton on the total electron spn mposed by the generalzed Paul prncple (see Secton 3.4.3), and all electronc states contaned n the drect product can exst

21 5.1. STRUCTURE OF POLYATOMIC MOLECULES 155 as ether snglet or trplet states. The confguraton (a 2u ) 2 (e 1g ) 3 (e 2u ) 1 thus gves rse to the electronc states 3 B 1u, 1 B 1u, 3 B 2u, 1 B 2u, 3 E 1u and 1 E 1u. To llustrate the case where each orbtal of a degenerate par of orbtals s sngly occuped, we now present the group theoretcal and HMO treatments of the electronc structure of the cyclopentadenyl caton C 5 H + 5 presented n Table 5.3. usng the D 5h pont group, the character table of whch s D 5h E 2C 5 2C 2 5 5C 2 σ h 2S 5 2S 2 5 5σ d A A R z E 1 2 ω 2 ω ω 2 ω 1 0 x, y E 2 2 ω 1 ω ω 1 ω 2 0 A A z E 1 2 ω 2 ω ω 2 -ω 1 0 R x,r y E 2 2 ω 1 ω ω 1 -ω 2 0 Table 5.3: Character table of the D 5h pont group wth ω 1 =2 cos(4π/5) and ω 2 =2 cos(2π/5). The matrx representaton of the Hückel operator can be determned n analogy to benzene and takes the form α β 0 0 β β α β 0 0 H = 0 β α β 0. (5.59) 0 0 β α β β 0 0 β α Dagonalzaton of ths matrx gves rse to the fve egenvalues α +2β, α + ω 1 β and α + ω 2 β, where ω 1 and ω 2 are defned n Table 5.3. The applcaton of the reducton formula of Equaton (4.10) to the fve-dmensonal reducble representaton of the fve p z orbtals and of the projecton operators gves rse to fve egenvectors of symmetres A 2,E 1 and E 2. The energetc orderng of the correspondng orbtals s depcted n Fgure 5.10a n the form of a so-called Frost-Musuln dagram. A Frost-Musuln dagram s derved by drawng a regular

22 156 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES polygone representng the cyclc polyene nto a crcle, placng one vertex on the lowest pont of the crcle. The vertces of the polygone then provde the energes of the π orbtals of the polyene. Such dagrams provde an elegant graphcal method to determne the sequence and degeneracy of the HMO of cyclc polyenes. Lookng back at Fgure 5.9, t becomes apparent that a Frost-Musuln dagram ndeed adequately descrbes the HMOs of benzene. The most stable electronc confguraton of C 5 H + 5 s (a 2 )2 (e 1 )2, as depcted n Fgure 5.10a. The drect product of the rreducble representatons of the partally occuped orbtals s E 1 E 1 =A 2 E 2 A 1. In ths case, the two components of a degenerate orbtal may both be sngly occuped. The total electronc wave functon must be antsymmetrc under the exchange of the two electrons n the e 1 orbtals, whch restrcts the number of possble states, as wll be dscussed n general terms n Secton The snglet states have an antsymmetrc electron spn wave functon (wth respect to the permutaton of the electrons). By consequence, ths electron spn wave functon must be combned wth the symmetrc (rather than the drect) product of the rreducble representatons [E 1 E 1 ]=E 2 A 1, resultng n a 1 E 2 and a 1 A 1 state. Correspondngly, the trplet state has a symmetrc electron spn wave functon (wth respect to the permutaton of the electrons). By consequence, ths electron spn wave functon must be combned wth the antsymmetrc product {E 1 E 1 } =A 2,resultng n a sngle 3 A 2 state (see the analogous dscusson of the electronc structure of O 2 n Subsecton 3.4.3). The energetc orderng of the three states 3 A 2, 1 E 2 and 1 A 1 s gven n Fgure 5.10b. The Hartree-Fock energes of these three states are 2h + J 23 K 23,2h + J 23 + K 23 and 2h + J 22 + K 23, respectvely, where h, J j and K j represent the one-electron orbtal energy, the Coulomb and the exchange ntegrals, respectvely, and the ndces desgnate the π molecular orbtals n order of ncreasng energy (by symmetry, J 22 J 23 =2K 23, see e. g. W. T. Borden, Dradcals, John Wley and Sons, New York (1982)). Usng the Hückel molecular orbtal approach, one can determne the one-electron energy to be h = α + ω 1 β. Large polyatomc molecules Unlke small polyatomc molecules, most large molecules have a low symmetry, and the classfcaton of electronc states by ther rreducble representatons loses ts relevance. When the molecule possesses no symmetry elements, all electronc transtons nvolvng nomnally a sngle electron are allowed by symmetry. Consequently, a dfferent nomenclature s used to label both the electronc states and the electronc transtons of large molecules, as already mentoned n the ntroducton.

23 5.1. STRUCTURE OF POLYATOMIC MOLECULES 157 Fgure 5.10: a) Frost-Musuln dagram of the π molecular orbtals of the cyclopentadenyl caton. The arrows ndcate the occupaton correspondng to the lowest-lyng electronc confguraton. b) Energetc orderng of the correspondng electronc states n D 5h symmetry (rght-hand sde). K 23 represents the exchange ntegral (adapted from H. J. Wörner and F. Merkt, J. Chem. Phys., 127, (2007)). The electronc states are desgnated by a captal letter representng ther spn multplcty: S for snglets, D for doublets, T for trplets etc. A numercal subscrpt s used to ndcate thegroundstate(e. g. S 0 ) and the hgher-lyng excted states of the same multplcty (S 1, S 2 etc...). States of another multplcty of the same molecule are also labelled n order of ncreasng energes but startng wth the subscrpt 1 rather than 0 (e. g. T 1,T 2 etc.). Electronc transtons n polyatomc molecules are often labeled accordng to the type of molecular orbtals nvolved. One dstngushes between bondng orbtals of σ or π type, the correspondng ant-bondng orbtals (σ or π ) and nonbondng orbtals (n). Ths nomenclature has the advantage that t hghlghts the nature of the electronc transton, from whch qualtatve predctons of ther ntenstes can be made: Transtons nvolvng the exctaton of an electron from a bondng to the correspondng antbondng orbtal (σ σ or π π ) are usually assocated wth a large oscllator strength, whereas transtons from nonbondng to ant-bondng orbtals (n σ or n π ) are weak. The nomenclature outlned above s often used n the dscusson of the photochemstry and photophyscs of larger molecules, lke the DNA bases. Although the solated DNA bases absorb strongly n the ultravolet ( nm), they hardly show any fragmentaton, unlke many other molecules. Ths property may be of mportance n preservng the genetc nfor-

24 158 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES maton (see e. g. A. L. Sobolewsk and W. Domcke, Eur. Phys. J. D, 20, (2002).) and arses from the ablty of the molecules to convert the energy of the photon to vbratonal energy. Fgure 5.11: Resonance-enhanced two-photon onzaton spectrum of adenne n the gas phase (adapted from N. J. Km et al., J. Chem. Phys., 113, (2000)). The bands A and D are assgned to the orgns of the n π and π π exctatons, respectvely. The bands B, C and E are vbronc levels of mxed electronc character. The rght-hand sde of the fgure shows an energy level dagram of the three lowest electronc snglet states of adenne. Adenne also represents a good example llustratng the dffculty assocated wth all nomenclature systems for large molecules. The sequence of snglet states conssts of the S 0 ground state wth the confguraton...(π) 2 (n) 2 (π ) 0, followed by two electronc states of domnant confguratons,...(π) 2 (n) 1 (π ) 1 and...(π) 1 (n) 2 (π ) 1, respectvely. One can only ndcate the domnant confguratons for these two electronc states because they le energetcally very close, and confguraton nteracton between them s mportant. Snce the energetc orderng of these states has been debated n the lterature, t s dffcult to apply the usual labels S 1 and S 2 to these electronc states. To avod ths dffculty, the recent lterature uses the desgnaton 1 nπ and 1 ππ for these two states, the 1 superscrpt desgnatng the spn multplcty. Fgure 5.11 shows a resonant two-photon onzaton spectrum of adenne n a supersonc expanson and a dagram of the electronc energy levels as derved from the spectrum. The band labeled A was assgned to the orgn of the 1 nπ state, whereas band D was assgned to the orgn of the 1 ππ state. The wave number scale on top of Fgure 5.11 s gven wth respect to band A. The energy-ntegrated absorpton of the 1 ππ state s strong, and the

25 5.1. STRUCTURE OF POLYATOMIC MOLECULES 159 band turns nto a broad absorpton band above cm Spn multplcty As n the treatment of datomc molecules n Subsecton 3.4.3, we wll only consder twoelectron wave functons. Because of the Paul prncple, the two-electron wave functon must ether have a symmetrc spatal part (Ψ R (s) (q )) and an antsymmetrc spn part (Ψ S (a) (m )) or vce versa (Ψ R (a) (q )) and (Ψ S (a) (m )), see Tables 2.2 and 2.3. The stuaton s slghtly dfferent from the case of datomc molecules, because the components of degenerate orbtals can no longer be classfed accordng to a good quantum number (λ n the case of datomc molecules). However, group theory provdes a smple approach to determnng the exstng multplctes. Two cases can be dstngushed: 1. The two electrons are located n dfferent molecular-orbtal shells. Both the symmetrc and the antsymmetrc spatal parts of the wave functons are nonzero n ths case. No restrctons result from the Paul prncple: The electronc states are gven by the drect product of the representatons of the partally occuped orbtals, and all terms contaned n the drect product exst as both snglet and trplet states. Ths stuaton arses n the frst excted states of BeH 2 arsng from the confguraton...(2a 1 ) 2 (1b 2 ) 1 (3a 1 ) 1.Snce b 2 a 1 =b 2, the two electronc states 3 B 2 and 1 B 2 are obtaned. The same apples to the (a 2u ) 2 (e 1g ) 3 (e 2u ) 1 confguraton of benzene dscussed n Secton 5.1.3b, gvng rse to the electronc states 3 B 1u, 1 B 1u, 3 B 2u, 1 B 2u, 3 E 1u and 1 E 1u. 2. The two electrons are located n the same molecular-orbtal shell. If the molecularorbtal shell s nondegenerate, the spatal part of the wave functon s necessarly symmetrc. The spn part must therefore be antsymmetrc, resultng n a totally symmetrc (A 1 ) snglet state. If the molecular-orbtal shell s degenerate, the spatal part has both symmetrc and ant-symmetrc components. The symmetry propertes of these components s determned by the symmetrc and antsymmetrc parts, respectvely, of the drect product of the orbtal symmetry wth tself. Ths stuaton arses n the (a 2 )2 (e 1 )2 confguraton of C 5 H + 5 dscussed n Secton 5.1.3b. The symmetrc spatal part of the wave functon s gven by [E 1 E 1 ]=E 2 A 1, resultng n a 1 E 2 and a 1 A 1 state. Correspondngly, the trplet state s obtaned from the antsymmetrc product {E 1 E 1 } =A 2, resultng n a sngle 3 A 2 state.

26 160 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES 5.2 Spectra of polyatomc molecules Rovbratonal Transtons As dscussed n Secton 4.3.6, a vbratonal transton s allowed when the matrx element φ vb φ el μ α φ el φ vb s non-zero,. e. when the rreducble representatons of the vbratonal levels and the dpole operator fulfll the followng relaton Γ vb Γ α Γ vb A 1. (5.60) When vbratonal transtons are measured at hgh resoluton, they reveal ther rotatonal structure whch follows the followng selecton rules: symmetrc tops: α = z, parallel transton, ΔK =0,ΔJ = ±1 fork =0andΔJ =0, ±1 fork 0. α = x, y, perpendcular transton, ΔK = ±1, ΔJ =0, ±1. asymmetrc tops: α = a, a-type transton, ΔK a even, ΔK c odd, ΔJ = ±1 fork a =0 and ΔJ =0, ±1 fork a 0. α = b, b-type transton, ΔK a odd, ΔK c odd, ΔJ = ±1. α = c, c-type transton, ΔK a odd, ΔK c even, ΔJ = ±1 fork c =0 and ΔJ =0, ±1 fork c 0. Fgure 5.12 shows a hgh-resoluton Fourer-transform nfrared spectrum of the ν 1 fundamental band of CHD 2 I, whch s an asymmetrc top. The panel n the center shows an overvew of the spectrum n whch both a- and c-type transtons are vsble. The top panel shows an enlarged vew of the low-wavenumber-sde of the spectrum n whch c-type transtons wth ΔK a = 1 domnate. The bottom panel shows an enlarged vew of the regon n whch a-type transtons wth ΔJ = +1 domnate (R-branch).

27 5.2. SPECTRA OF POLYATOMIC MOLECULES 161 Fgure 5.12: Fourer transform nfrared spectrum of the ν 1 fundamental of CHD 2 I (data from S. Albert, C. Manca Tanner, and M. Quack, Molecular Physcs, 108, (2010)) Electronc Transtons An electronc transton s allowed f the matrx element φ el μ α φ el s non-zero,. e. when the rreducble representatons of the electronc states and the dpole operator fulfll the followng relaton Γ el Γ α Γ el A 1. (5.61) When an electronc transton s allowed, the vbratonal structure of the transton can be predcted by factorzng the matrx element φ vb φ el μ α φ el φ vb = φ vb φ vb φ el μ α φ el. (5.62) Consequently, a transton to a fnal vbratonal state φ vb wll only be observable f the rreducble representatons of the two vbratonal states fulfl the relaton: Γ vb Γ vb A 1. (5.63) Just as n datomc molecules, the quantty φ vb φ vb 2 determnes the relatve ntenstes of the vbratonal bands n an electronc transton and s called the Franck-Condon factor (see

28 162 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES Chapter 3 and Fgure 3.12). When an electronc transton s forbdden, t can sometmes nevertheless be weakly observed through couplng of the electronc and vbratonal degrees of freedom. The mechansm by whch such transtons are observed s the Herzberg-Teller ntensty borrowng mechansm mentoned n the secton enttled Electroncally forbdden but vbroncally allowed transtons. Snce the electronc wave functons depend parametrcally on the nuclear coordnates Q, the factorzaton used n Equaton (5.62) s not adequate when the electronc wave functons change sgnfcantly wth the molecular geometry. Ths s however often the case when electronc states are close to each other and especally for molecular geometres devatng from the hghest symmetry. In such cases, electroncally forbdden transtons can become observable f the followng relaton s fulflled: Γ vb Γ el Γ α Γ el Γ vb A 1. (5.64) Example: Let us consder the electroncally forbdden transton between the X 1 A 1 vbratonless ground state and the Ỹ 1 A 2 electronc state of a C 2v molecule. Exctaton of a B 1 vbraton n the upper electronc state results n an excted state of vbronc symmetry Γ ev =A 2 B 1 =B 2. A transton to ths state orgnatng n the A 1 state s vbroncally allowed. However, the transton only carres sgnfcant ntensty f the B 2 vbronc state nteracts wth a close-lyng electronc state Z of electronc symmetry B 2. The ntensty of the transton s borrowed from the Z X bythe Herzberg-Teller effect. Electroncally allowed transtons In polyatomc molecules, an electronc transton can be nduced by any of the three Cartesan components of the transton dpole moment. When an electronc transton s allowed, the relatve ntenstes of the transtons to dfferent vbratonal levels of the electroncally excted state approxmately correspond to Franck-Condon factors, and the vbratonal structure of an electroncally allowed transton contans nformaton on the relatve equlbrum geometres of the two electronc states connected through the transton. An llustratve example of an electroncally allowed transton s the absorpton spectrum of ammona, whch s dsplayed n Fgure The electronc ground state of ammona has an equlbrum structure of pyramdal C 3v pont-group symmetry. However, two pyramdal

29 5.2. SPECTRA OF POLYATOMIC MOLECULES 163 confguratons are separated by a low barrer along the symmetrc bendng (umbrella) mode, whch leads to nverson of the molecule through tunnelng on the pcosecond tme scale and to a tunnelng splttng of 0.8 cm 1. When ths tunnelng splttng s resolved, the approprate pont group to treat the energy level structure s D 3h, the character table of whch s gven n Table 5.4. D 3h I 2C 3 3C 2 σ h 2S 3 3σ v A A R z E x, y A A z E R x,r y Table 5.4: Character table of the D 3h pont group. The vbratonal wave functons are nevertheless manly localzed at the mnma of the potental energy surfaces correspondng to a C 3v geometry. In the C 3v pont group, the electronc confguraton of ammona n the X ground electronc state s (1a 1 ) 2 (2a 1 ) 2 (1e) 4 (3a 1 ) 2. The lowest-lyng electroncally allowed transton corresponds to the exctaton of an electron from the 3a 1 orbtal, whch s a nonbondng orbtal (lone par) of the ntrogen atom, nto the dffuse 3s Rydberg orbtal wth a NH + 3 planar on core. In the electroncally excted state, the molecule has a planar structure wth D 3h pont-group symmetry. In ths pont group, the excted electronc confguraton and the electronc state are labeled (1a 1 )2 (2a 1 )2 (1e ) 4 (1a 2 )1 (3sa 1 )1 à 1 A 2. The absorpton spectrum of the à X transton recorded usng a room-temperature sample n whch only the ground vbratonal level of the X state s sgnfcantly populated s dsplayed n Fgure The spectrum conssts of a sngle progresson n the out-of-plane bendng (umbrella) mode ν 2 of the à state. The orgn band s labelled as 00 0 and the members of the progresson as 2 n 0, ndcatng that the electronc transton orgnates n the vbratonal ground state of the X state of ammona and ends n the v 2 = n vbratonally excted level of the à state. The weak band observed at lower wave numbers than the orgn band s the hot band The very long progresson, extendng beyond n = 15, s characterstc of a large

30 164 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES change n equlbrum geometry between the two electronc states nvolved n the transton. Fgure 5.13: Absorpton spectrum of the à X transton of ammona (data from B.-M. Cheng et al., Astrophys. J., 647, 1535 (2006)). The orgn of the electronc transton s labeled as and the domnant progresson n the out-of-plane bendng mode ν 2 s labelled 2 n 0. The members of ths progresson orgnate n the vbratonal ground state of the X state and end n the v 2 = n level of the à excted state. The smplest way to understand the fact that transtons to both even and odd vbratonal levels are observed wthout notceable ntensty alternatons between even and odd levels s to evaluate the selecton rules n the C 3v pont group (see table n Secton 4.2.2). In ths group, the electronc transton s allowed, and the umbrella mode s totally symmetrc, so that the vbratonal selecton rule and ntensty dstrbuton can be descrbed by the vbratonal selecton rule (5.63) and the Franck-Condon factors, respectvely. The vbratonal ntensty dstrbuton can also be explaned n the D 3h pont group. However, n ths group, the umbrella mode ν 2 s not totally symmetrc, but of a 2 symmetry. Consequently, one would predct on the bass of Equaton (5.63) that the vbratonal bands correspondng to odd values of the vbratonal quantum number v 2 of the umbrella mode should be mssng n an absorpton spectrum from the ground vbratonal level. The reason why transtons to vbratonal levels wth odd values of v 2 are observed s that they orgnate from the upper tunnelng component of the ground state whch has A 2 vbronc symmetry (and thus may be regarded as the frst excted vbratonal level of the ground state).

31 5.2. SPECTRA OF POLYATOMIC MOLECULES 165 Transtons to vbratonal levels wth even values of v 2 orgnate from the lower tunnelng component of the ground state, whch has A 1 symmetry. The two tunnelng components are almost equally populated under the expermental condtons used to record the spectrum dsplayed n Fgure 5.13, so that no ntensty alternatons n the ν 2 progresson are observed. Ths example also served the purpose of llustratng some of the dffcultes one encounters n nterpretng electronc states wth equlbrum structures correspondng to dfferent pont groups. Electroncally forbdden but vbroncally allowed transtons Electroncally forbdden transtons may gan ntensty from an allowed transton through vbronc couplng medated by a non-totally-symmetrc mode (the Herzberg-Teller effect), as dscussed above. A prototypcal example of ths stuaton s the electroncally forbdden à 1 B 2u X 1 A 1g transton of benzene (C 6 H 6 ). Ths transton s also referred to as the S 1 S 0 transton, accordng to the nomenclature ntroduced n Secton The excted electronc state arses from the electronc confguraton (a 2u ) 2 (e 1g ) 3 (e 2u ) 1 (showng the π molecular orbtals only). The drect product of the rreducble representatons of the partally occuped orbtals s E 1g E 2u =B 1u B 2u E 1u, gvng rse to the electronc states 3 B 1u, 1 B 1u, 3 B 2u, 1 B 2u, 3 E 1u,and 1 E 1u. Inboththeà 1 B 2u and the X 1 A 1g state, the benzene molecule has D 6h pont-group symmetry. The dpole moment operator transforms as A 2u E 1u and thus the only allowed electronc transtons orgnatng from the ground electronc state end n states of electronc symmetry A 2u or E 1u. The à 1 B 2u X 1 A 1g transton n benzene s thus forbdden, whle the C 1 E 1u X 1 A 1g transton s allowed (see Table 5.2). However, vbratonal modes of symmetry B 2u E 1u =E 2g nduce vbronc couplng between the Ãand C electronc states. Fgure 5.14 shows a low-resoluton overvew spectrum of benzene n the regon of cm 1, whch was frst analyzed n J. H. Callomon, T. M. Dunn and I. M. Mlls, Phl. Trans. Roy. Soc. A, 259, (1966). The spectrum s domnated by a strong regular progresson of absorpton bands connectng the ground vbratonal level of the X state to vbratonally excted levels of the à state. The nomenclature 1n ndcates that the lower level of the transton has the quantum numbers v 1 = v 6 =0,. e. both ν 1 and ν 6 are unexcted, whereas the upper level of the transton has v 1 = n and v 6 = 1. The orgn of the band system, desgnated as 0 0 0, does not carry ntensty, as expected for an electroncally forbdden

32 166 CHAPTER 5. STRUCTURE AND SPECTRA OF POLYATOMIC MOLECULES transton. The ν 1 and ν 6 vbratonal modes have A 1g and E 2g symmetry, respectvely, n both electronc states. The vbronc symmetry of the upper levels of the observed transton s thus Γ e Γ v =B 2u [A 1g ] n E 2g =E 1u whch can be accessed from the ground vbronc state through the E 1u component of the electrc-dpole-moment operator (see Table 5.2). Fgure 5.14: Low-resoluton absorpton spectrum of benzene n the cm 1 regon (data from T. Etzkorn et al., Atmos. Envron., 33, (1999)). The transtons labelled 1 n orgnate from the vbratonal ground state of the X 1 A 1g state and end n the (v 1 = n,v 6 = 1) vbratonal states of the à 1 B 2u electroncally excted state. The orgn of the band (marked as ) does not carry ntensty. The vbroncally allowed hot band 60 1 below the orgn band. s observed All strong transtons n ths band system end n v 6 = 1 levels, whch ndcates that, among all vbratonal modes of benzene, ν 6 s the mode prmarly nvolved n medatng the vbronc nteracton. Below the orgn of the band, the weak transton labelled as orgnates from the thermally populated v 6 = 1 vbratonally excted level of the ground electronc state and ends n the vbratonal ground state of the à 1 B 2u state. Such a transton s a hot band and s not observed when the vbratonal temperature of the molecule s suffcently low. One should note that the band s vbroncally allowed, whch explans why t s observed, whereas transtons from other thermally populated excted vbratonal levels of the ground state are not detected.