1 d 2 dR(r 3(*:):) 2UE b d2R(r).2 dr(r 6 1(1+ 十 十 十 2)R()=0 R()=e·f() 2R() R(r)=0 d f 21df,「b2a(+1) f=0 2 C 2021/8/21 Che mistry De partne nt of Fudan University 6
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 6 ( ) ( ) ( ) 0 1 2 1 2 2 2 2 2 = + − + + R r r l l r e E dr dR r r dr d r 2 2 2 E a = − 2 2 2 Ze b = ( ) ( ) ( ) ( ) 0 2 1 2 2 2 2 = + + + − + − R r r l l r b a dr dR r dr r d R r ( ) ( ) 0 2 2 2 − a R r = dr d R r r → 0 2 2 1 2 2 2 2 = + + − − − − f r l l r a r b dr df r a dr d f ( ) R(r) e f (r) ar = −
Phusical chemiatry Chapter II Atomic Structure and Spectrum d-f 2a 21df,「b2a1(+1) 2 f=0 r dr 2 The coefficient of each power of r must f()=∑br k=0 be zero so we can derive the recursion relation for the constants bk k+1 水-b+a (k+1)-b bk(k+1)+2(k+1)-(+)(k+)k+2)-1(+1) The power series must be terminated a(kmax +1) for some value of k=k = n-1 E unna e anze k+1)h h2 max 2021/8/21 Che mistry De partne nt of Fudan University 7
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 7 0 2 2 1 2 2 2 2 = + + − − − − f r l l r a r b dr df r a dr d f ( ) ( ) = = k 0 k k f r b r ( 1)( 2) ( 1) 1 1 2 1 1 1 + + − + + − = + + + − + − + = + k k l l a k b k k k l l ak b a b b k k ( ) ( ) ( ) ( ) The power series must be terminated for some value of max k n = − k 1 a b (k 1 0 max + − = ) ( ) 2 2 4 2 2 4 2 2 2 2 max 2 2 k 1 z e z e E h n h = − = − + The coefficient of each power of r must be zero, so we can derive the recursion relation for the constants bk
( +1Nk+ 2)-(+1b l=0,1,2,3 a(k+1)-b The coefficients before the terms k<l-1 are zero R()=e∑b=e"r∑b L.1+1..n-I k=l k=0 This is a power series of r with n-/-1 terms n=(k+1)>l max 2021/8/21 Che mistry De partne nt of Fudan University 8
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 8 ( )( ) ( ) 1 1 1 2 1 + + − + + − + k = bk a k b k k l l b ( ) The coefficients before the terms are zero. k l −1 max max k k 0 ( ) l ar k ar l k k k l k l k R r e b r e r b r − − − + = = = = n l = + (k 1 max ) This is a power series of with terms r n−l −1 l = 0, 1, 2, 3… l, l+1,….n-1
Phusical chemiatry Chapter II Atomic Structure and Spectrum The first few radial wave-functions for the hydrogen atom are listed below: 2 /2 p 2 2 p R21()= exD 2 2r T ,0 3/2 + 3 27 exp 4√2 R 9(3an)3/2 6 p 2√2 exp 133 (3a0)3/2 2021/8/21 Che mistry De partne nt of Fudan University 9
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 9
Phusical chemiatry Chapter II Atomic Structure and Spectrum Physical significance of the solution latomic orbital y(, 0, =r(O(O)U(o There are three quantum numbers for each eigenfunction of a hydrogenlike atom. y n1, 41, my,, dT=0 The orbitals with different quantum numbers are orthogonaL 2021/8/21 Che mistry De partne nt of Fudan University 10
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 10 Physical Significance of the Solution 1.atomic orbital ( , , ) R( ) ( ) ( ) n,l,m = r = r 0 2 2 2 1 1 1 = d n ,l ,m n ,l ,m * The orbitals with different quantum numbers are orthogonal. There are three quantum numbers for each eigenfunction of a hydrogenlike atom