《结构化学 Structural Chemistry》课程教学资源（课件讲稿）Chapter 2 Atomic Structure

2.1.2 The solution --- separation of variablesSubstitute (r, , Φ)=R(r)@(0)@(Φ)intotheequationaayZe11Q18元msin0=0Eh200arsin?sineaA4元8dadV/[R(r)O(0)Φ())and multiply witha0aR(0(0)sinaOrR(r) arae@(0)sin 0Now both sides1d8元1are variable-=0Og?h?Φ(Φ)sin4元8independent!221a8元aR(r)mZeEh2OrR(r)Or4元%a11asin12a0Φ(Φ)sin?adDsin060H

2.1.2 The solution - separation of variables Substitute (r, q, f)  R(r)(q)F(f) into the equation 2 2 2 2 2 2 2 2 2 2 0 1 1 1 8 ( ) (sin ) ( ) 0 sin sin 4 me Ze r E r r r r r h r     q  q q q q f                 and multiply with /[ ( ) ( ) ( )] 2 r R r  q F f ) 0 4 ( ( ) 8 ( )sin 1 ) ( ) (sin ( )sin 1 ) ( ) ( ( ) 1 2 0 2 2 2 2 2 2 2      F F            r r ze E h m r R r r R r r e    f f f q q q q q q q ] ( ) ( )sin 1 ) ( ) (sin ( )sin 1 [ ) 4 ( 8 ) ( ) ( ( ) 1 2 2 2 2 0 2 2 2 2 f f q f q q q q q q      F F               r r ze E h m r R r r R r r e Now both sides are variable￾independent!

Radial part1a8元aR(rzemletR eq.Eh?R(r)Orr4元Yetunsolvable!a1a@(d)10HAngular partsin-B0d?a0Φ(Φ)sin (0)sin 0 a0Now multiply with sin?andLet1asin 04CBsinsin2map?一a00(0) 00Φ(d)asino0(0+βsin0=mHeq(sinda00(0) 00Unsolvable yet!1ad(Φ)solvable now!-mΦ0p2eq12Φ(d)

           2 0 2 2 2 2 4 1 8 r r ze E h m r R r r R r r let e ) ( ) ( ) ( ( )  Angular part  Radial part R eq.  f f q f q q q q q q     F F       2 2 2 ( ) ( )sin 1 ) ( ) (sin ( )sin 1 2 2 2 ( ) ( ) 1 ) sin ( ) (sin ( ) sin f f f  q q q q q q q   F F         2 2   m       q q q q q q q ) sin ( ) (sin ( ) sin 2 2 2 ( ) ( ) 1  m   F F f f f   eq.  F eq. Yet unsolvable! Unsolvable yet! solvable now! 2  m Let Now multiply with sin2q and

d(d)equationd()a.+m2@(Φ) = 0dp?Itssolutionincomplexform@=Aeimd→Φ=AemLet m=ml,NownormalizeΦ2T" A?e-im§ eimg d = 2元A?Φ*Φ =-→A=(V2元)-1= V1/2元eime = V1/2元(cosm@+isinmΦDm?:Φ(Φ)=Φ(Φ+2元（Boundarycondition!)im2元im2元Φ=eim(+2元）=eimbimm=0,±1,±2:Thevaluesofmmustbem(m):magnetic quantumnumber12

a. F(f) equation ( ) 0 ( ) 2 2 2  F  F f f f m d d 2 1 2 2 0 2 2 0 F F F       d A e e d A i m i m f   f f  * F (f )  F (f  2 ) The values of m must be m  0, 1,  2,  m (ml ): magnetic quantum number 1 2   A  (  )   (cos f sin f ) f e m i m i m F  1/2  1/2  i mf i m f 2 i mf i m2 e  e  e  e (  ) 1 2   im  e Its solution in complex form:  F = Aei|m|f ; Now normalize F: m? （Boundary condition!） Let m = |m|, F = Aeimf

m = AeimpD(m=0,±1,±2,.:Complexfunctionm Ae+img = A(cosmg±isinmp)In case m0DtmHow to transform these complex functions into real ones forpractical use?UsethesuperpositionprincipleLeti =Φm +Φ-m= 2Acosm= BcosmdP2 = Φm -Φ-m = 2iAsinm = B'sinmdNow normalize P, and P2P1 = /1/元 cosmg"(Bcos md)dgPiP,ddIP2 = /1/元 sinmgB = /1/ 元B3元=112

 f f  f f iA m B m A m B m m m m m sin 'sin cos cos  F  F    F  F     2 Let 2 2 1 How to transform these complex functions into real ones for practical use? Complex function F  Ae (m 0,1,2, ) i m m f (cos f sin f ) f Ae A m i m i m F m          f f f   1 1/ 2 2 0 2 2 0 1 1        B B d ( Bcos m ) d *   f   f m m sin cos 1/ 1/ 2 1   In case m0, Now normalize 1 and 2 : Use the superposition principle

The solutionsof @(@)equationcomplexformrealformm11072元2元10cos1cosd±1L元V元1Dsin1sind±12元√元120cos2Dcos20±22元V元11Dsin2sin2g±2122元√元

The solutions of F(f) equation 1 2 1 1 1 2 1 1 2 1 2 1 0 1 1 1 1 0 0 f   f     f f sin cos sin cos  F  F  F  F  F  F      i i e e m complex form real form f   f   f f 2 1 2 1 2 2 1 2 1 2 2 2 2 2 2 2 sin cos sin cos  F  F  F  F      i i e e