Lecture 2002 Ch316 Note 2

Lecture 2002 Ch316 Note 2 - Classes The members of a group...

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Unformatted text preview: Classes The members of a group can be divided into classes Two members of a group, P and R, belong to the same class if another member Z can be found such that P = Z-1RZ ; P and R are said to conjugate to each other and they form a class Êx 0ª ½* member class . ;  Ê =   = Ê for any  -1 -1 ½ Ĉ3x 0 Â-1 Ĉ3 “ 0ª ½ * group member ∧ σ v’ , σ v’ ’ , σ ∧ ’ -1 ’ ½ ½ σ v ˆ 0  σ v Âx 0ª v ’’’ Ĉ3, Ĉ3-1x 0 classX 0 ½ ½ ª ¨ 0 classˆ 0 ½ ½ ª . . Subgroup Group subset0 ß ¨ * ª group .¨ ß ½ NH3ˆ 0 ª ½ point group C3v¨ ß ½ ª rotational subgroup { Ê, Ĉ3, Ĉ3-1} .  Problem 2-3 Chap 3. Matrix Representation of Groups º¢ùˆ *  m ¬Î symmetry operation• ! ª º ù symmetry operator homomorphic→ 4 matrix O A, B vector cross product ˆˆ ij C = A  B = Ax AY B X By Complex conjugate ˆ k Az Bz : (A*)ij = (Aij)* t transpose : (A )ij = A ji Hermitian conjugate : A = (A t)* Inverse : AA -1 = 1 Trace : n ∑A ij i =1 Determinant special matrices A symmetric matrix : A t = A A Hermitian matrix : A = A -¬ Õ 1 è O .ß ¨ ª ½* w . .→ 4 O O@ point group symmetry operation group representationˆ ß !ª ½ . An orthogonal matrix : A t = A -1 A unitary matrix : A = A -1 Unitary (or orthogonal) transformation of vectors do nor alter the magnitude of a vector, but change its direction Matrices can be used in inverting a set of linear equations Ãx = -1 c → c =A c inverse matrix A-1…  O r @. M 11 Mn1 1 A-1 = A M 1n Mnn M ij A cofactor Matrix Eigenvalue Problem -> very important! N * N matrix àn eigenvalue n eigenvectorø D ª ½ . λi xi ( i = 1, , n) à xi =  eigenvector ÃØ Â n eigenvector n D ½ transformOO g á W 96 linear homogeneous eq. → λ ĩ) x = 0 ø X x = 0  trivial soln ½ ! ƒ → ˆ !ª . (à − Ö  λĩ → n | n→ →Õ *ª ¹ w Î n nontrivial solnÈ ª X ½ = 0 … secular determinant or secular equation eigenvectorØ ª X ½ ˆ˜ ª ( ~ X = x1, x 2, xn ) . ~ ~ ~~ A X = XΛ ---------- @ λ1 0 0 ~ 0 λ2 0 Λ = 0 0 λn ~→ Ø X!ª ´ ð !ª » . Λ ~ −1 ø D ª ~ −1 ~ ~ ~  @ X ½ X AX = Λ ~  eigenvector ˆ ~ A, A X A  diagonalize ~ ˆ½ < > È ª ½ Q ª A  eigenvector Xu X . ~ 4 1 A= 2 3 ˆ ˆ ( ) A X = λ X  X ë| ~¹ X Oª ( w¬ Î- . . 4 1 x1 x1 λ 0 x1 2 3 x 2 = λ x 2 = 0 λ x 2 1 x1 4 − λ = 0 b ∴ 2 3 − λ x 2 x1 , x 2  H secular determinant 4−λ 1 =0 2 3−λ λ2 − 7λ + 10 = 0 λ 1 = 2, λ 2 = 5 λ 1, λ 2 Å ª ½ b # ª ½* “ λ1 = 2 2 1 x1 2 1 x 2 = 0 ∴ 2 x1 + x 2 = 0 normalization 5 x1 = 1 2 λ2 = 5 − 1 1 x1 2 − 2 x 2 = 0 1 ~5 ∴X = −2 5 x1 = 1 2 , x2 = − 5 5 ∴ x1 = x 2,2 x1 2 = 1 x1 = 1 1 , x2 = 2 2 1 2 1 2 ½ secular eq.u !ª N a11 x1 + a12 x 2 + + a1nxn = λx1 a 21 x1 + a 22 x 2 + + a 2 nxn = λx 2 ----------------------------------- an1 x1 + an 2 x 2 + annxn = λxn ~ ¸b ò  AX = λ X nontrivial soln a11 − λ a 21 an1 a12 a 22 − λ an 2 . ¸ sp ò a1n a 2n ann − λ Similarity Transformation matrix ~ ~ A  matrix Z … . ~ ~~ ~ Z −1 A Z = B r O g O Similarity transformation ~ ~ ~ Z is a unitary matrix, then A and B are related via a unitary transformation ~~~ ~ ~ 2) If A, B , Z were matrix representation of symmetry operations, then A and B would 1) If be in the same class 3) Similarity transformatio eigenvalueè ª B ½ () ~| B !ª ¸ ê ~ ~| Z  A !ª ¸ ê 4) 5) Hermitian matrix unitary transformation Î-¬ Õ ~ ~ A  eigenvalue B Î-¬ óx Symmetry Operations and Position Vectors 1) Reflection X – ZªBx½µ ˆ σ 13 reflection x1 x1 x2 = − x2 x3 x3 ∴σ 13 ( do !ª . 1 0 0 D( σ 13 ) = 0 − 1 0 0 0 1 Bؽµª 1 0 0 −1 0 0 ˆ ˆ D( σ 12 ) = 0 1 0 , D( σ 23 ) = 0 1 0 0 0 − 1 0 0 1 2) rotation Z€ ! ª ¸ þ - ¬ ÕÎ w x′1 x1 ˆ C x 2 = x ′2 x3 x′3 θ ) ~~ ~ A = B , tr A = tr ( B )  . (eigenvector θ (radian) óx . óx . cosθ ˆ θ ) = − sin θ D(C 0 sin θ cosθ 0 0 0 1 A 6 W J Å R ˆ 2O g . x1 = r cosφ , x 2 = r sin φ x′1 = r cos(φ − θ ), x′2 = r sin(φ − θ ) A A cos(−θ ) sin(−θ ) 0 cosθ ˆ θ −1 ) = − sin( −θ ) cos(−θ ) 0 = sin θ D(C 0 0 1 0 [ ˆ ˆT D(Cθ −1 ) = D(Cθ ) ˆ D(Cθ ) A orthogonal matrix 3) inversion −1 0 0 A D (i ) = 0 − 1 0 0 0 − 1 4) Identitiy 1 0 0 ˆ ) = 0 1 0 D( E 0 0 1 <example> H2Oø N ½ 4 symmetry operatorx ª ½ N ˆˆ {E, C ,σˆ ,σˆ } 2 xz yz − sin θ cosθ 0 . 0 0 1  matrix representation¨ N ª½ −1 0 0 ˆ D(C 2) = 0 − 1 0 0 0 1 1 0 0 ˆ D( E ) = 0 1 0 0 0 1 −1 0 0 ˆ D(σyz ) = 0 1 0 0 0 1 1 0 0 ˆ D(σxz ) = 0 − 1 0 0 0 1 XZd È N !ª reflection YZ  reflection -> ä Xà ¦ *multiplication tableÈ Y y u !ª -¬ Õ X, N È Z d !ª w matrix→ l , !ª . Symmetry Operators and Basic vectors ˆ Cθ  » Z ² → 4 r O ä¦ À P( ¦À , N ½ y 4 @ Ĉ O r4 representation→ 4  row vectorØ ª D ½ p Ø ª½ O O .c ˆ /… 3*3 . ˆ Cθ basis vector ˆ Cθ  basis vector . . O. (point P ) O g O ! ½ª . basis vector column basis vector matrix representation }p .  p71ˆ ½ Nª  matrix¨ !ª / ¹ e1e 2e3e 4e5e6e7 e8e9e10e11e12 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ H1 H2 H3 N ˆ ½ ! ª½ σ ′V ¨ N NxÈ N H1X¨ !ª ½ N H2x H3x NH3 Â Õ ¬ 12 w . Ny . H1y basis vector . basis . Nz . H1Z , H2y H3y . . ‚ ˆ D(σ ′V ) H2z H3z N hh !ª . matrix _r â , 0 −1 00 00 00 00 10 0 1 0 0 0 0 ˆV NH3 σ ′  representh h ª½ 0 0 0 ( H2x, H2y, H2z, H3x, H3y, H3z) −1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 = (-H3x, H3y, H3z, -H2x, H2y, H2z) → i¸ !ª p71u 8 ª ½ Symmetry Operators and Basis functions vector¸ ½ (* ª Symmetry Operator Function Space vector space( i) scalar product : function H Ù * ª » ù å . . f i f j = ∫ f i * f j dτ ii) linear independence N l inearly independent functions are said to span the function space of dimension n. It is always possible to find a set of n orthogonal basis functions which span the space. Functon Spaces are commonly used in solving the Schrodinger eq. A set of degenerate wave functions associated with a single energy forms a function space. Gram-Schmidt Procedure n l inearly independent (but not orthogonal) basis function ¼ * ª \ functon , n orthogonal basis à *Representations of Groups A representation of a group : a set of matrices, each corresponding to a single operation in the group, that can be combined among themselves in a manner parallel to the way in which the group elements combine. C2 Symmetry group ` _* ¼ ª Õ ¬ > Symmetry operation, h h C2’ ( ß »* ½ ª C2 * ¼ \ ª product . how many representations can be found for any particular group? ( )¾ ¼. ª @ ä E C2 σ v σ v’ symmetry operation 1 C2v group – 1` _* ¼ ª Õ> Î representation . C2’ 11 1 −1 1 1 1 −1 1 −1 −1 1 1 1 −1 −1 C2v group multiplication table( ½ ª I σv E E C2 σ v C2 C2 E σ v ' σv σv σv' E σ v ' σ v ' σ v C2 E 4 σv' σv' σv C2 E C2 representation8 ½ I 8I ½ ? . higher order8 ½ I * group •¤ Ò representation8 . 7@ @ 1 0 0 −1 0 0 1 0 0 −1 0 0 E : 0 1 0 C 2 : 0 − 1 0 σ v : 0 − 1 0 σ v ': 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 ½ 9× 9, CH2Cl2 ˜ I ª ½ H2O? 8 8I ? * ª group è s ª * ª Õ ¬ 15× 15 8 Iª ½ Î > representation irreducible representation ° ? . . representation½ 8 I ? similarity transformation@ @ á ~ L È s J6 W ~~ A  A' s ¨ group representation mº * ~ ˜ A' A block-factored matrix3 0 A1 ' A2 ' ~ ~−1 ~ ~ A' = L A L = A3 ' 0 A4 ' A ~ A' 8? ½ * ª ~~ ~ A1 ' B1 ' = D1 ' ~~ ~ A2 ' B2 ' = D2 ' ˜ ½ I ª ~~ ~ A3 ' B3 ' = D3 ' ~~ B',C ' ° block-factored matrix( ½ * I ª . ½, ª , @ Å á eª 8 I ½ ª * a ¨ group representation8 I ª . (group multiplication gable ) (8I e + 6 W 1 1 0 0 0 0 0 2 0 0 0 0 block-factored matrix0 0 0 3 0 0 0 0 0 0 1 1 4 0 4 0 2 0 0 × 2 0 2 0 1 0 0 0 0 3 2 0 7@ 1 3 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 2 block-factored matrix8 ½ I ª 0 4 0 8 0 0 = 2 0 2 0 1 0 0 0 0 1 0 1 .) 0 0 0 0 0 0 0 0 3 0 0 0 0 13 3 10 0 10 3 8 0 2 5 9 1 7 0 0 0 0 ~~~~~ ½ ª E , A, B , C , D } 8 K * { a “ reducible representation” 8 e representation0 @ » ½ * ª 7@ . 8K ½ 2 @ E ά > * great Orthogonality ~ L » E ² 7 . @ 7 representationH » k ª * representation irreducible = (k irreducible Theorem ½ ª I . valence bond theory ∑ [ Γ ( R ) ][ Γ ( R ) ] i mn j * m 'n ' R = @@ @@ » E < . h δ ijδmm' δnn' li l j R : various operations in the group ΓI : I-th irreducible representation m,n : m,n8 e ° ( ½ ª * 8I e ª * .( complex number0 7 @@ @@ 7 ∑ Γ ( R) i mn Γ j ( R ) mn = 0 i f i ≠ j mn Γi ( R ) m 'n ' = 0 mn Γi ( R ) mn = R ∑ Γ ( R) i R ∑ Γ ( R) i R i f m ≠ m' and/or n ≠ n' h li h : the order of a group (order : the number of elements in a group) @ ) l i : the dimension of the i -the representation irre.rep.( ¨ ∑l 2 i 1) ªºî =h R The sum of the squares of the dimensions of the irr. rep. Of a group is equal to the order of the group. ∑ [ χ ( R)] 2) 2 i =h R ( χ i : character) ∑ χ ( R) χ i 3) R j ( R ) = 0 when i ≠ j : vector¸ M ½ * . 4)In a given representation (reducible or irreducible) the characters of all matrices belonging to operations in the same class are identical. 5) The number of irreducible representations of a group is equal to the number of classes in the group. < 1> C2v symmetry groupø M group element ½ { E, C2, σv, σ v’ } 4¸ M ½. ª class . 1) 5 : class 4øM K½ 2) 1: number of ir. rep. ø M ½ ª 2 2 l12 + l2 + l32 + l4 = 4 (order = symmetry operation C2v point group 4 3) Γ1è M ª ½ * 1 ir. rep.È M ª . ½ ir. rep. E C2 σ v σ v ' 111 1 vector ) l1 = l2 = l3 = l4 M l ≥1 ¸ M Γ1 4 (set of ir. rep.) (1 2ø M ª . ½ +1 +1 +1 = 4 ) È M . ½ ir. rep.è M 2( ª ½ 2 2 2 – 1è Mª ½ 2 .È M 7 ½ vector Y * ª » E C2 σ v σ v ' Γ1 1 1 Γ2 Γ3 1 1 −1 −1 1 −1 1 −1 Γ4 1 1 1 @ 1 −1 −1 ir. rep.è M ª ½ * Õ * ¬ > Î .( 3) vector 1 à < 2 +1 4 vector < " o . 2> C3v group { E, 2C3 , 3σ v } 3 class, order = 6 number of ir. rep. = 3 ( 1 -> 5) 2 l12 + l2 + l32 = 6 8 li 2 ir. rep. 1, 1, 2 . 1 1 2 group character 1 1 E 1 Γ1 2C3 1 . ir. rep.( ¤ ¸ º ª * C3v group I 3 * ª 3σ v 1 2 -> 1 + 2 × 1 + 3 × 1 = 6 3 I 3 © ˜ 1 * ª 1 1 vector ˜ — vector 3 – 1, 3 ª ir. rep. E 2C3 3σ v 1 1 1 (all elements in the same class must have representations with the same character.) 1 1 −1 Γ1 Γ2 I 3 ª * ir. rep 2 χ 3 (E) = 2 3 -> :3 . ir. rep. character. ir. rep.Ø I ª ½ * vector ½ I * . ∴ ∑ χ1 ( R ) χ 3 ( R) = 1 ⋅ 2 + 2 ⋅1⋅ χ 3 (C3 ) + 3 ⋅1 ⋅ χ 3 (σ v ) = 0 R ∑χ 2 ( R) χ 3 ( R ) = 1 ⋅ 2 + 2 ⋅1 ⋅ χ 3 (C3 ) − 3 ⋅1 ⋅ χ 3 (σ v ) = 0 R ∴ χ 3 (C3 ) = −1, χ 3 (σ v ) = 0 3 C3v Γ1 Γ2 Γ3 +1Ø I 2 ir. rep. C3 o ` ir. rep. character ½ Iª * E 2C3 3σ v 1 1 1 1 1 −1 2 −1 0 character 2 2 + 2(−1) 2 + 3(0) 2 = 6 2: ( @ @ reducible rep.ø Ì , reducible representation O direct sum ½ X < 1> C3v group C 3v Γ1 Γ2 Γ3 Γa Γb E 1 1 2 5 7 3σ v 1 −1 0 −1 −3 2C3 1 1 −1 2 1 ½ * ª Γa( O , . Γa = a1Γ1 ⊕ a2 Γ2 ⊕ a3Γ3 A a1 ,a2 ,a3 ø O ½ * ª ai = 1 ∑ χ ( R) χ i ( R) hR ∴ a1 = 1 [1⋅1⋅ 5 + 2 ⋅1⋅ 2 + 3 ⋅1⋅ −1] = 1 6 1 [1⋅1⋅ 5 + 2 ⋅1⋅ 2 + 3 ⋅ −1⋅ −1] = 2 6 1 a3 = [1⋅ 2 ⋅ 5 + 2 ⋅ −1 ⋅ 2 + 3 ⋅ 0 ⋅ −1] = 1 6 a2 = ∴ Γa = Γ1 ⊕ Γ2 ⊕ Γ2 ⊕ Γ3 O ͽ Nª Ab 1 [1⋅1⋅ 5 + 2 ⋅1⋅1 + 3 ⋅1⋅ 3] = 0 6 1 a2 = [1 ⋅1⋅ 5 + 2 ⋅1 ⋅ 2 + 3 ⋅ −1⋅ −1] = 3 6 1 a3 = [1⋅ 2 ⋅ 5 + 2 ⋅ −1 ⋅ 2 + 3 ⋅ 0 ⋅ −1] = 2 6 ∴ Γb = Γ2 ⊕ Γ2 ⊕ Γ2 ⊕ Γ3 ⊕ Γ3 a1 = ½ ir. rep. G ª ?h F ? * ª » Õ ¬ > Î . . *Character Table E 2C3 3σ v 1 1 1 1 1 −1 2 −1 0 C3v A1 A2 E z x2 + y2 , z 2 Rz ( x, y )( Rx , R y ) ( x 2 − y 2 , xy )( xz, yz ) Schoenflies symbol Mulliken symbol i)1 representation A 2 representation E 3 representation B T ii) 2π n (Cn axis) χ (C n ) = −1 ⅲ)A ⅳ C2 8 P ª ½ ‘”’ σ h ⅳ)’’’ hT ⅴ)center of inversion8 2 ) A, anti symmetric P ½ C 2 h P symmetric ½ X σv @ Õ * ¼ Î > . antisymmetric u(ungerade) x, y, z u *½ ; ª X ø rotationØ s ª » Raman spectrum selection rule ê € _ Õ ¬ > .T ½ P . Chapter 5 Atomic Spectroscopy 1.atom P ½ * (basis set : set P ª ½ basis set u p vector ? l inear combination8 T ( )spherical harmonics YLM (θ , φ ) A basis * hP ª T½ ˆ ① L operatorÐ 2ª ½ * ② ¿* © ª » ˆ ½ * Lz operatorH Pª Pª ½ ③ YLM u¼ p ½ vector ª p u . (spherical harmonics) 2 = Θ LM (θ )Φ M (φ ) ˆ ④ L2YLM ( . * g(gerade) P ½ z, ( x, y ), Rz  column column IR ( χ (Cn ) = 1 symmetric ) B. Bh P subscript 1 ½ . ③ⅳ X = h L( L + 1)YLM . (simultaneous eigenfunction) . .) * ˆ ˆ L2YLM = MhYLM ⑤L ⅳ 0,1,2,… M– ° . L, -L+1, …, L-1, L © » . ⑥Spherical harmonics is orthonormalized YLM (θ ,φ ) YL 'M ' (θ ,φ = δ LL 'δ MM ' *½ :' 8ª (conservation of angular momentum) ÿ (system)Ð E @ ² ² © » (conserved or unchanged physical quantity) @ 7 @ @ * ©* » + 6 W . E ² » ( )¨ ² »E . à ° v ª * •` ”ª º n ~ ƒ † @ @ » ² E @ 1 á J W ° : :@ @ 2 6 + ¬* Õ 4ë —º » ² ¬* Õ B + 6 ÿ ¹ ² $ ° 3ë —º d ¹E 8 . (8 ' : ª )Ø » * Ì @² » , ƒ† ~ (linear momentum)8 ½ '* ª E p ’E » ‘† 6 Z+ ` ² ¹ ˜ .X Z ª ½( {ª * . º ² E @ » + > Î ¬ > Î . 6 +p ². E . . > . è Pª U½ Schr dinger time-dependentÍO o ½ ª ih ˆ dA ˆ ∂A ˆˆ → = [ A, H ] + ih dt ∂t , ˆˆ [ A, H ] = 0 A ˆ 'ª Ab ½ “ constant of motion”(è U ½ ª * ä ª´ . explicitly ) ° ˆ ½* Au O ª . dˆ A =0A dt eigenvalue a ˆ A ° å j 8 Good Quantum Number ½ 'ª number( . @² » E @ 7 @ Central force field( . eigenfunction )˜ ( £ basis@ @ Õª * . > Î áJ Å ¸ Ì + 6 good quantum ‡ . (good quantum # )potential ( dL ) = r × F = torque dt |r |Ð ½ * ª ¿» © * central force field r × F = 0 →| r || F | sin θ *]Ø –º ª Õ Î | r || P | sin θ ¸’ L = r ×P ( á Å W 6 + r,P ½ ª * ¸ ¹ “ , XO ª ô½ * ) a ∂ ∂ ˆ Lx = −ih ( y − z ) ∂z ∂y ∂ ∂ ˆ L y = −i h ( z − x ) ∂x ∂z ∂ ∂ ˆ Lz = −ih ( x − y ) ∂y ∂x commutatorØ ½ ª P ˆˆ ˆ [ L x , L y ] = ih L z ˆˆ ˆ [ L y , Lz ] = ih Lx ˆˆ ˆ [ L z , L x ] = ih L y 1 ˆˆˆ Lx , L y , Lz → Ø → @ » ² ˆ 2 [L 2 Ø W •* ñª * . 7 @ ˆ ˆˆ ˆˆ , Lx ] = [ L2 , L y ] = [ L2 , Lz ] = 0 ÈS P ª ˆ ˆ· L2 =| L |2 ∠ ² E E ² » @ ½. * ª ~ † 6 + » E ² ¹ ÿ E E ² ¹ ¹ ` . ∂ ∂ ˆ Lx = ih (sin θ + cot θ cosθ ) ∂θ ∂φ ∂ ∂ ˆ L y = −ih (cosφ − cot θ sin φ ) ∂θ ∂φ ∂ ˆ Lz = −ih ∂φ Â, 1∂ ∂ 1 ∂2 ˆ L2 = − h 2 ( sin θ + ) sin θ ∂θ ∂θ sin 2 θ ∂φ 2 *raising and lowering operators ˆ ˆ ˆ L+ = Lx + iL y ˆ ˆ ˆ L− = Lx − iL y ˆ ˆ * L+  L− B ½ª M eigenvalue 1 ß ` . , ˆ L±YLM = h [ L( L + 1) − M ( M + 1)]1 / 2 YLM ±1 *The spherical harmonics YLM =| LM > are a convenient set of basis functions to construct a matrix representation of the angular momentum operators. ⇒ ˆ* LM basisX O L2 A Lz € »ª ½ˆ Õ ¬ Î > ˆ ˆ L2 A ˆLz ( YLM A !) matrix element ˆ ˆˆ LM | L2 | L' M ' = L( L + 1)h 2δ LL 'δ MM ' ˆ ˆˆ LM | Lz | L' M ' = Mh δ LL 'δ MM ' * YLM A basisH O ª ˆ ˆ L+  L− …7 @ @ . ˆ ˆ ˆ ˆ L + L− ˆ L − L− ˆ Lx = + , Ly = + 2 2i *L=1, M=1,0,-1X O ª basis function ˆ * ( ) 1,1 , 1,0 , 1,−1 Η ½ i * ª column vector operatorX ½ i * ª ˆ . . 1 0 0 = 1,1 , 1 = 1,0 0 0 1 ˆ2 = 2 h 2 0 L 0 10 ˆ Lz = h 0 0 00 0 ˆ = 2h 0 L+ 0 0 ˆ L− = 2h 1 0 0 ˆ=h1 Lx 2 0 0 , 0 = 1,−1 1 00 10 01 0 0 −1 10 01 00 00 00 10 10 01 10 0 −1 0 ih ˆ Ly = 1 0 −1 2 010 * ( Chemical Physics) Hamiltonian ˆ H  spin ¿ 8 Pª W½ eigenvalue(energy) ª » * ² E @ @ Atom > Î ( E ² system8 P* ½ ª @ º ² . Schr dinger½ X# o | .( V ! ˆ ½H  ½! * ª @ ! . Molecular spectroscopy atomic ? )8P W½ @ » 7 ˆ ½ H  matrix elementØ P ª* Î 1/3è a+ ¨ Õ Õ operator( Pª ½ * eigenfunction(wave equation)8 W @ ÅW J spectroscopyPª ¸ R ˆ basis set å º * ª 7 matrix representation( orbital angular momentum ! ª Ðh ½ 2 @ operator rotation v ibration8 Pª ½ * applied quantum mechanics . 1. Good Quantum Number and Not So Good Quantum Numbers; ①”Good” quantum #’s are eigenvalues of operators which commute with the Hamiltonian and thus correspond to physical properties which can be specified exactly. 2 In the presence of a perturbation that mixes states with different eigenvalues, the corresponding quantum # is no longer strictly good, but may be considered to be “approximately good” in the case of a sufficiently weak perturbation. * OZ » Ĉ O r 4 (X !ª ½ / ) Θ . Θ ¼ Õ ! ¼ ¸ Î w Î basis set P½ Qª ½ * ª , * ”perturbation” ª¸ Q * ª * basis 9 ª ¼ state ¿ © ,( ÕÎ > > Î ¬ Î . c0 . 2.1-electron atom (hydrogen-like ion) ˆ L2 LM L = L( L + 1)h 2 LM L ˆ Lz LM L = M L h LM L 3 ˆ S 2 α = S ( S + 1)h 2 α = h 2 α 4 3 ˆ S2 β = h2 β 4 1 1 α S = , S z = + A eigenstate 2 2 β 1 1 S = , S z = − A eigenstate 2 2 h ˆ Sz α = α 2 h ˆ Sz β = − β 2 Q ΞQ * ª 2 h Ze ˆ H0 = − ∇2 − 2µ r 2 ³ . * ˜ Î basis setX Q ª ½ ! ˆ ˆΘ H << H 0 Ξ ½ * ˆ H ' << H 0 Õ ¬ » ª * Z» ² º ² . . contribution( » ( ª * ”Z ¶º Ž ˆ ˆ ˆ H = H0 + H ' > Î c0ψ 0 + c1ψ 1 + s • “ @ à ô * ª Õ ¬ “ basis set” g @ O @ P 5 Ð 2 Õ¬ ! !¼ ¨ Î ˆ H  eigenfunction Ψ ˆ H 0 A eigenfunction {ψ n } ˆ ˆ H 0ψ n = E nψ n → HΨ = E n ' Ψ c1 , c2 A 10 @ @ • g ô Ψ contribution * ª · ¬Î ˆˆˆˆ ① L2 , Lz , S 2 , S z p ˆ H 0 A commute ② L, M L , S , M S  good quantum number ˆˆˆ ③ L constant of motion.( Lx , L y , Lz  H 0 A commute ) *spin-orbit coupling @ @ Å á W 6 + → → B ' = − c12 v × E ( ¨ a p magnetic fieldÈ Q ½ ª . ) → eS , spin angular momentumX Q intrinsic magnetic moment, µ s = ½ me → Spin-orbit coupling electron orbital motion ˆ magnetic momentÐ Ø . , → → H SO = − µ s ⋅ B → 2 1 ∂U (r ) r ( U (r ) = − Ze ) E=− ⋅ r e ∂r r → → v= → p me magnetic field . electron spin → → 1→ → 1p 1 ∂U r 1 1 ∂U → → B' = − 2 v × E = − 2 × (− )= ⋅ p× r e ∂r r c c me eme c 2 r ∂r → → → → p× r = − L A → eS 1 1 ∂U → =− ⋅ ( − L) me eme c 2 r ∂r H SO = = l 1 1 ∂U → → L⋅ S me2 c 2 r ∂r Z →→ L⋅ S me2 c 2 r 3 ¼ *ª →→ H SO ∝ L⋅ S > Î ¬ Õ ð × ù 1 2 ˆ →→ 2 ∧ * H0 = − Qª * ½ 2µ . . H SO I H SO = a L⋅ S A Qª *½ ˆ ∇2 − @ Ze 2 r @ . @² » 7 (spin-orbital coupling) ∧ ∧ →→ H = H 0 + a L⋅ S ,( 7 →→ ∧ ½ ª H 0 H Q* ∧ 2 good q.n. ∧ ∧ A, H = 0 ) good q.n. 2 ∧ ½ H ¸ Mª* @ ½ ª a L⋅ S ¸ M* ( . ∧ ∧ ∧ ½ L , L Z , S , S Z ÷ M H0 commute . →→ L⋅ S = Lx S x + L y S y + Lz S z good quantum number( N* ¼ ª L, M L , S, M S Qª ½* ˆ _ . Õ >Î good q.n. ˆ . ? , ∧ H0A ∧ 2 ∧ ∧ L , L⋅ S = 0 C . ∧ 2 ∧ ∧ 2 ∧ ∧ 2 ∧ ( L , L x = L , L y = L , Lz = 0 A 2 ∧ ∧ * ª » L A S° › Î ¬ Õ variable T * ½ ª orbital a.m. spin a.m.è T commute .) ½ ∧ 2 ∧ ∧ S , L⋅ S = 0 é ½ ªT ∧ ∧ ∧ ∧ L, L⋅ S = L z , L x S x + L z , L y S y A = iAL y S x − iAL x S y = iA( L y S x − L x S y ) ≠ 0 ∧ S z , L ⋅ S = iA( Lx S y − L y S x ) ≠ 0 é½ xT ª ¤ D ! ª * ) ¬ Î hamiltonian M S¸ T ª * → → → good q.n. ¨ ! ) ¸ L A conserveè ∧ L x , H SO ≠ 0 , … → ª© 2 J = L+ S Ð ½ ¿ ª * → J A spin-orbit coupling (significantly ¸ T ª ½ [ J z , L ⋅ S] = [ J x , L ⋅ S] = [J y , L ⋅ S] = 0 ) * èT è ½ ( ⅰ) ½ ª * ½. ª conserve . ( S(Ø è . → L A conserve ∧ L z , H SO ≠ 0 , * L( still good q.n. ML () spin-orbit couplingØ T ª ½ * correlation, , electron-electron repulsion ) H 0 = H1 + H 2 H1 = − 2 2m1 ∇ 2 + V (r1 ) good q.n. T* ½ ª ? ) H2 = − 2 ∇ 2 + V (r2 ) 2m 2 [ Lx1 , H 1 ] = [ L y1 , H 1 ] = [ Lz1 , H 1 ] → → L1 , H 0 = L1 , H 1 = 0 A → → L 2 , H 0 = L 2 , H1 = 0 → A L1 ( ⅱ) ⅳ È î 1 P two particle ½) ª , → ) L2 ( interact 2 ,¨ U î P ª interaction )( @ 7 • @ conserved. central force field (¨ î ½ ª H = H 1 (r1 ) + H 2 ( r2 ) + V ' (r12 ) → L1 , H = [V ' (r12 )] • → → → → î ª L1 , L 2 ¨ U *ª conserveÈ ½ . nonzeroÈ ½ î* ª → @ L = L1 + L 2 @ á J6 W + € ‘ a constant of motion’ ⅲ) ⅳ system spin-orbit coupling electron-repulsionø Õ ª î¬* P ) * LS-coupling → → ½ * J 1 , J 2 A c.o.m(constant of motion)˜ Uª @ ⅰ) light atom¨ → @ U S-O coupling ½ → ½ ª ª * ½ S È Uª* .¨ î → → → L 2 A S 2 A couple ˜î ½ ª → → J A c.o.m → S-O interaction è ª * ½ ª ½. ª → J1A → J2A J 1 A J 2 A couple → JA . → Õ ¬ > . ά ( → ½ L1 A L 2 ¨ U couple → → S A L A couple Russel-Saunders coupling( P ⅱ) Spin-orbit couplingⅳ r˜ U Zh U ª, ½ L1 A S 1 A couple ) correlationÈ î → LS-coupling → (ø ¹* ªî !) ¨ î ½ ª . ½ S 1 A S 2 ˜ U couple h í > Î j j-coupling atomÈ U electron correlation(U ½ î¨ ( . )h í heavy atom inner orbital → ½ J h Uª* → ½ L h Uª* . ˜ Uª ½ ½. ª P - .h í ½ ª couple . * ª ½ j j-coupling H @ @ Å W J 8 + ³ è 7@ ⅰ)¨ @ L A S A good q.n. → ⅱ)⑤ Vª ½ → J 1 A J 2 A good q.n. ⅰ)V ½ ⅱ)( ? → → V ½ ¨ ¨ ½. ª good q.n. ç ª ·* €! ª . €! ª º ? ? good( almost good) q.n. eigenftn basis ftn( Ù · õ * . * atomic term symbols ① Spin-orbit coupling⑤ V* ½ ª 2 2 ∧ ∧ ∧ ∧ ∧ H , L , S , L z , S z all commute with each many electron atom other so that the waveftn Ψ is a simultaneous eigenftn of all 5 operators. ⇒ E, L, S, M L , MSS are constants of motion! ⇒ These 5 quantum #’s can be used to label the waveftns Ψ = nLM L SM S ② total degeneracy H î spin-orbit coupling( aº × g = (2 L + 1)(2 S + 1) 2 S +1 term symbol * p2 configration H L ½ ª * term configuration : a way of loading electrons into spin-orbitals in a manner consistent with the Pauli exclusion principle spin-orbital : spin¨ I ½ ª * • Orbital H ? À * ª ¸ ½ * ª Õ Î orbital : 1s22s22p2 → LS-coupling ⅰ) ⅳ electron V ½ l1 = l 2 = 1 A H Vª ½ * * (2p13p1 H p orbital¨ V ½ ½ ª ) L=2, 1, 0 D, P, Sh Vª ½ . 1 A 2 S=1,0 S1 = S 2 = ∴ 1S, 3S, 1P, 3P, 1D, 3D¨ V ª ½ ⅱ) ⅳ electron V ½ term p-orbital¨ V ½ . * (2 equivalent p electrons) C2=15(spin-orbital( Q 6 ) ½ 6 Pauli( @ @. ml 1 ↑ ↑ ↑ ↑ ↑ ↓ ML 0 ↑ ↑ ↓ ↓ ↓ ↑ ↑ ↑ ↓ ↑ ↓ 0 0 -1 -1 -1 0 0 0 0 0 0 ↓ ↓ ↓ ↑ ↓ ↓ 1 1 1 0 ↓ ↓ ↑ ↓ ↓ ↓ 1 0 ↑ ↑ ↑ ↓ ↑ MS -1 ↑ ↓ ↑ ↓ ↑ ↓ ½ 2Ð ¿* ª » * 2 electron Q ½ inequivalent electron( @ p orbital @ two inequivalent p electronQ X ½ À 15 ,8 Q W ª L è ¼* * Jª À q . two . S À Õ ¬ > Î term . ① 3D : L=2, S=1 È ± M L=2, M S=1 15( . ② 3P : L=1, M L =1, 0, -1 S=1, M S=1, 0, -1 ⇒ degeneracyⅳ 3 × 3 = 9 ③ S : L=0, S=1 3 ∴ M L =0, M S=1, 0, -1( Q ½ (Q ½ * 6( è ²ª * . *! ④ 1D : L=2 → M L=2, 1, 0, -1, -2 S=0 → M S=0 ⇒ degeneracyⅳ 5 ⑤õ ° *Õ ª ⇒ 1S *( 7 Î M L =0, M S=0 . @ @ : Hund’s rule ① The term with the highest multiplicity 2S+1 lies lowest in energy. @ 7 @ @ term ② Of the terms of the same multiplicity, the term with the highest L value lies lowest in energy. ∴ ⑤X ª ½ Õ 3 P<1D<1S > Î * p5 configuration p1X U term symbol ½ 3 * D∠½ term symbol ª (2S+1)(sL+1)ˆ â .(d9 d1 S, L LS-coupling ½ spin-orbit coupling ª X U ä ª * ˆU â½ H SO = a L⋅ S A matrix element significantly large spin-orbit couplingˆ U energy( ½ ∧ →→ .) 7 @ ˜ @ H X q . degenerate term degeneracy energy split . . . →2 →→ → → → → J = J ⋅ J = L+ S ⋅ L+ S 2 2 ∧ ∧ ∧ ∧ = L + S + 2 L⋅ S  H SO a∧2 ∧2 ∧2 = J − L − S 2 ∧2 L JM J LS = ∧2 S JM J LS = ∧2 J JM J LS = • XU ä L( L + 1) JM J LS 2 S ( S + 1) JM J LS 2 J ( J + 1) JM J LS : ª basis U* ½ ª ∧ 2 LM L SM S JM J LS s →→ ½ H SO = a L⋅ S A matrix elementˆ U ª (1 E SO) = JM J LS H SO JM J LS = Xä U 12 aA [ J ( J + 1) − L( L + 1) − S ( S + 1) ] 2 ª term symbolˆ U * ½ ª L, SX U J ½ . X *º µª ? Jè X * ½ Î ¬ Õ J 1@ J Å + 6 ³ è . E J +1 − E J = aA2 ( J + 1) aÂ2 1 – 2 → 2aÂ2 J: 0 – 1 → A ( ) 2S 2P orbitalh Xª ½ * HX Œ ( ªrepulsionh * Œ 7 zero-th order HX Œ ½ ª @ @ term• v ½ ª * ª * .( h X ª Œ½ ÎÕ ¬ ( ˆ ŒX H ee x½ )HX Œ ), ª 2h X ª * ½ * spin-orbit coupling split . ※Selection rules for atomic absorption and emission ① one-electron atoms ∆ = ±1 , ∆m = 0 , ± 1 ② multielectron atoms ∆L = 0 , ± 1 (L = 0 not → L =0) ∆S = 0 ③ ∆J selection rule: ∆J = 0 , ± 1 ( J = 0 not → J =0 ) ( ∆L ④ ∆S = 0 A selection rule hX ½ * spin stateH Œ Œ H so = aL ⋅ S ξ ˜X ½ x Œ ½ ª * ∆J ∆L selection rule spin good.q.n. h X ½ ª A H so h X ª ½ ½ ª H so = aL ⋅ S ½* ª ) { ( s-o couplingH X ª ) A ˆˆˆ ½ H , L2 , S 2 , J 2 , J z } ˜ X commute H so = ∑ ξ (ri )aˆ i ⋅ si ξ X ˆ ½ ) J good q.n. L S almost good q.n. , ˆ L2 Ψ ~ L( L + 1)Â2 Ψ ˆ S 2 Ψ ~ S ( S + 1)Â2 Ψ ※selection ruleⅳ transition moment integral˜ X Œ M = ∫ Ψ f µ Ψi dτ * . Spin-orbital coupling * ª . *.( ˆ , si → i ˆ aŒ x èº y ½ * ª * atomic wave ftn ° parity atomic transition (X ¢ ½ ª * µ = − er µ odd parityè Z*ª ½ dipole moment orbitalè Z ª ½ ∆ ) Z ½ one-electron . M = −e ∫ R * n '' (r )Y *'m ' (θ , φ )r R* nY *m (θ , φ )r 2 sin θdrdθdφ r = xi + yj + zk = r (sin θ cos φ i + sin θ sin φ j + cos θ k ) g ≅ O á W J z-¥ ¸ + ³ , ª cos θ k {¢½* ½* ª ∞ π 2π 0 0 0 M z = −eN ∫ r 3 Rn '' Rndr ∫ Pm ' (θ ) cos θPm (θ ) sin θdθ ∫ e −im'φ eimφ dφ ' (€ ² ½* Pm (θ ) , φ θ >¬ Î m P (θ ) associated Legendre polynomial( èZ ¥ ° ½ eimφ ¸ Z ª .) Z` ² !) ∆m = 0 , ∆ = ±1 • M z A nonzero ° * ª ' º ² » . ∆m = ±1 , ∆ = ±1 M A x° ¸Z ¥ M x A nonzero ° . ∆n ª. * ※Ø¥ ½ Zª z¥ ¸ laser ½ ª * P ³ polarizedp ³ transition moment( @ 7 ½ ª * Õ ¬ > Î @ laserÐ 2 ¿ » ½ ©* * . ∆m = 0 , ± 1 , ∆ = ±1 selection rule ∆m = 0 , ± 1 for hydrogenlike ※single-photon, electric-dipole-allowed selection ruleⅳ ∆ = ±1 , atoms. ※й ² @ E ²¹ 0 ¾» * ª @ @ @ ¾ » * ª 1 Ì + 6 W ? . ※Atoms in an external electric field : precession of angular momentum ⅳ ¸¥ ½ * ª gyroscopeØ Z* ½ ª (@ 7 @ @ system Ø ² Iw Ø Z ª ½ L Z ª * wð ½ . Gyroscope L( . precession( 7 precession speed¸ Z ½ * ª precession¸ Z ½ * ª ² » x, y, . ÎÕ ¬ )XZ ¢ f lywheel( @ precessionè Z ½ * ª ªñ ´ *. ª Z * ª L ' X¢ (L=I ⋅ w) . θ( . 7 Ø . Ü ¥ º * Chapter6. Vibrational-Rotational spectroscopy of a diatomic molecule ¾ © ª * ¾ª © » 8 € [ (È *¯ ¿ª * Ð 2 » * ª @ ª * [ dτ ¸„ ª * approx. ¨ a ¾ ª » * Õ ¬ @ » @ @ > © » .! ! + 6 , [˜ „ * Ì . (ψ 2 ( r ) dτ  M6 W @@ @ » ² E x . B-O º ² @ » . ¹ ² E @ @ ½ coupling” ª “ @ 7 .@ ^ +p V @ E ² áJ Å @ 7 B-O app.ÐE ² 7( @ Born-Oppenheimer approximation ) Oppenheimer 7 E ÅW áJ È ½ ¯ * . . W+ 6 BornÐ E @ ²E ²¹ . !) ° p + V á Å + Hamiltonian@ @ áW Å J 1. @ @ È ¯ M W ¾ » ª * !¨ a €[ ƒ ¾» ©ª 1 ² E ÌM 6 + » .( B-O„¸ ½ [ * ¬ > !! ) H (qe , q N ) = H (qe ) + H (q N ) → Ψ ( qe , q N ) = ψ e ( qe ) + ψ N ( q N ) ψ N (q N ) Η Z ª 2. ½ . “ Divide and conquer!” !) p² » E ¹ ï ² B-O ƒ 8 ° ²¹ E E º ² H (q N ) È ½ * ?» ¸ E ² PE ¹ ² ¾ © » ² E * center of massX ¯½) Ȫ translation internal motion 3. ° @² » E ª¾ *© * ª E² @² E ° . ° . ° Ψmolecule = ψ e ⋅ψ vib ⋅ψ rot ⋅ψ trans À „ @. v ibrating and rotation diatomic moleculeÈ ½ *ª * 22 − 2 µ ∇ + V (r ) ψ (r ,θ , φ ) = Eψ (r ,θ , φ ) ∙∙∙∙∙∙∙ ①H Z ª Nuclear motion( ΨN (q N ) = ψ trans ⋅ψ int . internal motion rotation v ibration 7@@ . ? R (r ) → ψ vib YLM (θ , φ ) → ψ rot 38 ( . 1 ∂ ∂ 1 ∂2 ˆ A . L2 = − 2 sin θ +2 sin θ ∂θ ∂θ sin θ ∂φ 2 ˆ L2 operator ( , spherical harmonics ( Ym (θ , φ ) )ø : eigenfunction( Z ½ ½ ª ? ˆ ½ ª L A kinetic energy Z* Laplacian operator( ∇ 2 ) 2 ˆ 2∂ ∂ 1 L2 ∇= + 2− 2 2 r ∂r ∂r r 2 ((( Ø X ½ . ∙∙∙∙∙∙∙∙∙∙② ① Â2 2 ∂ ∂ L2 − + 2 + + V (r ) ψ = Eψ ∙∙∙∙∙∙∙∙③ 2 µ r ∂r ∂r 2 2 µr Ψ ( r ,θ , φ ) = Ym (θ , φ ) ⋅ R( r ) à Xª ½ Ψ ( r ,θ , φ ) = Ym (θ , φ ) ⋅ R( r ) ( ¼ ‚ª Z ø: * ª >¬ Î , R(r ) à X Õ ½ * ¬ >. Î . 2 ③ L ψ 2 µr 2 À X , 2 L2 ( + 1) ψ= Ym (θ , φ ) ⋅ R (r ) ∙∙∙∙∙∙∙∙∙∙④ 2 µr 2 2 µr 2 ˆ ( L2Ym (θ , φ ) ④ − ① 2 2 ∂R ∂ 2 R ( + 1) L2 Ym + 2 + Ym R + V (r )Ym R = EYm R 2µ r ∂r ∂r 2 µr 2 Ym ø: − 2 = ( + 1)Ym ) r A θ ,φ ½* ª R(r ) X 8½ X . 2 2 ∂R ∂ 2 R ( + 1) + − R + V (r ) R = ER ∙∙∙∙∙∙∙∙∙⑤ 2µ r ∂r ∂r 2 r2 ⑤ø: ½* ª ø Z : ½ 2-body problem radial Schrodinger eq. Ø : ½ * ª . * ⑤ Hydrogen atom@ V (r ) A 12 kx 2 @ − e2 r Z ½ . * =0 → s =1 → p ⅳ . X ½ @ diatomic molecule W J Å À+ ½ X [ƒ ½ 1 k (r − re ) 2 ⑤ 2 R(r ) x ] Ym (θ , φ ) ø ]* ª ½ ª ½ V (r ) = x ]½ª ptl harmonicè *] ½ ª 7 R(r ) H ( ª ½ 2H † ² È V p Î ² » @ E Õ ½ * ª à. . @ ¨ a Ψint (r ,θ , φ ) ª ½ * " @ 2@ . áJ Å ½ æ] S ≡ r − re ] S ( s ) = rR ( r ) ξ º ² » ² 6+ ª * 8 ! . . ds = dr ∂S ∂r ∂R ∂r = R(r ) + r =1) ( ∂r ∂s ∂s ∂s ∂ 2 S ∂R ∂R ∂2R 2 ∂R ∂ 2 R = + + r 2 = r( + ) ∂s 2 ∂r ∂r ∂r r ∂r ∂r 2 ξ]ª * − 1 ∂ 2 S l (l + 1) S S S − + V ( s + re ) = E 2 2 2 µ r ∂s r r ( s + re ) r 2 r − 2 ∂ 2 S l (l + 1) − 2 µ ∂s 2 ( s + re ) 2 _ ¸ æ x S + V ( s + re ) S = ES ? ½! ª ¸ i) è í ½* ª − l( ,l =0 @ ¸ _ ½ . . ∂ 2 S ( s) + V ( s + re ) S ( s ) = ES ( s ) 2 µ ∂s 2 2 V ( s + re ) ≅ @ V= @ 12 ks í 2 i i) l > 0 C ½ ª . ÅW J +¨ a harmonic oscillator” H ( “ ½ ª . @ @á W J6 ˆ R .( ?) ª ½ ,V − eff ( s + re ) = l (l + 1) 2 + V ( s + re ) η ]½ªU 2µ ( s + re ) 2 .HÈ * ª ∂ 2 S (s) + V eff ( s + re ) ⋅ S ( s ) = E ⋅ S ( s ) 2 µ ∂s 2 2 ⋅ V eff ( s + re ) Taylor expansion ∞ f ( x + a) = ∑ n =0 . f ( n ) (a ) n x n! 1 1 2 s 6 s 2 24 s 3 = 2 − 3 + 4 − 5 + ( s + re ) 2 re re re re 1 1 2(r − re ) 2 = 2− + r 2 re re3 _ Ø ª , Taylor È h * ª . H ½ r ~ re _ð . . @ 7 r = re ¨ ( r ~ re >¬ Î V eff 0 x½ È rH ½*ª . s 0H ] ª ½ * a) _ 1 A re H] È ½ r2 @ v ibrational ptl ²º _ (l (l +1) ) @ r8 _ ª ½ * @ 7 @ ( Õ Î¬ 1 ) r2 ¨ . . ˜]ª * − 2 2 ∂ 2 S ( s) l (l + 1) + + V ( s + re ) S ( s ) = E ⋅ S ( s ) ( 2 2 2 µ ∂s 2 µre harmonic oscillator rigid rotor( @@ È ˜ ptl Ex ] ª ½ l (l + 1) 2  2I 7 @@ @ » E ² $ ƒ + 6 Z † ∂ 2 S (s) + V ( s + re ) S ( s ) = E ' ( s ) 2µ ∂s 2 2 E' = E − l (l + 1) 2  2I V ( s + re ) = ks 2 , 2 harmonic ptl . 8 _ *ª 12 ks ) 2 l (l + 1) A >¬ Î . ½ ª − ¨ , V ( s + re ) = E ² » coupling(È ½ ª* ˜ ) zero. J ( J + 1) 2 1 J ( J + 1) 2  = ( n + ) + ω  2I 2 2I E = E '+ (l ² » @ 7 @J .) , 1 1 1 = 2~ 2 2 ( s + re ) r re ( ø » @E 2ˆ ¤ * ª ½ b) a ¨ 7 ÎÕ ¬ ) v ibrational E rotational Eø ` ½ ºE re à (centrifugal distortion : ø` Þ 0 @ *. _ ½ .@ á @Å ª) * @ , 1 1 1 2s = 2~ 2− 3 2 ( s + re ) r re re V ( s + re ) = ks 2 2 V eff ( s + re ) = a + bs + V ( s + re ) = a + bs + J ( J + 1) a = 2µre2 V ( eff 2 J ( J + 1) µre3 ,b = − =− a re k b2 b2 ( s + re ) = ( s + ) + a − 2 k 2k • »* • ª ptlX ` ª ½ ptl˜ _ ª ½ Õ Î¬ E = E '+ a − b2 Þ½* `ª 2k − b k energy 2 b 2k 2 1 J ( J + 1) 1 = ( n + ) + ω − 2 2 µre2 2k ø 2 ks 2 2 centrifugal distortion X Ù 2 J ( J + 1) 3 µre ½ ª * . € ½ _ . W 6 ^ @ . * ª ½ bs ï h * ª ½ re8 ] ª ½ @8 re 7 re rotation0 re8 ½ª * I@ @ @ . rotation v ibration couple @. W Å J + ¨a J ( J + 1) 2  2I rigid rotor8 ½ª 8 . c) Vibration-Rotation coupling ;˜ ˆ_ ½ ½ ª * * Rotation >¬ Î @E » ² .0 @ @ 2 J ( J + 1) 1 2s 6 s 2 2 − 3 + 4 r 2µ re e re 12 ks 2 1 VJeff = a + bs + (k + 2c ) s 2 2 € @ . VJeff ( s + re ) = V ( s + re ) + V ( s + re ) = 7 s1 8 ¿ © V Jeff . @² » @ 7 sï 2 ª * a O @@ á potential˜ `* ½ ª W J6 0 = 6 J ( J +) 1 + 24 µ µre @ µ @ v ibrational @ E @ E ² ²¹ E ï @ ² » freq.0 8 ƒ @ , 2x` g ª * Õ Î ¬ parabola @ . . ! .@ ^ * ª ¼ Ü º ² ` * ª ¼ ² » . ) @ ? @ @ @² » º ² E ² » H` Ö½ @ E ²¹ Perturbation Theory(P t r e ‘0 ’ 7 . 2 k vibration-rotation coupling effect p » ` g a k +c 2 ω= J parabola 7 p + 6 E ² . » E ² *, ƒ ~ † @ . ¸ d “ Perturbation theory expresses the solution to one problem in terms of another problem that has been solved previously” ³ è ²º 8 ƒ “ @ ” . ²º E “ »E ² , È` ª d* ½ Õ > ά À D O * H½ Ö ` y º í » ² ¬> 7 , + 7 @ ² » system exact solution @ . 0 ,0 @ A λH ' @ A H (0) R A H (0) p ²º ½ a] Practically speaking, ( À D áJ Å . + 6 AA A . , H = H ( 0 ) + λH ' ξ ½* ` ª ºE @ » Õ basis ά ½* ª . ! Î Schrodinger eq. exact solution@ @ Hamiltonian( À * ƒ† ~Z ½Õ ª O ½ a À M .” ,( @@ . @ 8 ? exact solutionh ½ `* ª H(0)˜ ` solution ½ . . A H ( 0 ) A eigen function (zeroth order eigenftn) basis ¨a M . r O . O ,è ³ ²º O . time-independent perturbation method @ @ Ð Å á6 W J ½ ª * ¨ Ü a ¨ time-dependent perturbation method spectroscopic transition rate * ª ,ˆ º E @ À ® a . . . ...
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This note was uploaded on 09/22/2011 for the course CHEM 222 taught by Professor Linda during the Spring '11 term at Edmonds Community College.

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