Physical Chemistry 2021/8/21 Che mistry De partne nt of Fudan University 1
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 1 Physical Chemistry
Phusical chemiatry Chapter II Atomic Structure and Spectrum Hydrogen-like Atom: the model consists of a proton fixed at the origin and an electron that interacts with the proton through a coulombic potential 2 2 H 2 X The three spherical coordinates are associated spherical coordinate with the three spatial quantum numbers 2u r ar Or rasin 60o sin.o 方2r1a2,O 1 十 00 r2sin20 002 jy 十 e+ 2021/8/21 Che mistry De partne nt of Fudan University 2
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 2 Hydrogen-like Atom: the model consists of a proton fixed at the origin and an electron that interacts with the proton through a Coulombic potential. r Ze H V 2 2 2 2 2 2 2 = − + = − − ˆ spherical coordinate = + + + − r Ze E r r r r r r 2 2 2 2 2 2 2 2 2 1 1 1 2 sin sin sin
Phusical chemiatry Chapter II Atomic Structure and Spectrum Note that the angular and radial terms can be separated, we suggest that we can write the wavefunction as a product of radial and angular parts. (;0,)=R(r)Y(0,) Then the angular part is separated into two parts Y(O,d)=(O(0) and three parts are substituted into Schrodinger equation, we have 1 a OR ao sin 6-+ e+ r ar ar osin 6 ae a0 snap ao 2021/8/21 Che mistry De partne nt of Fudan University 3
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 3 Y(,)= ( )() + = − + + r Ze E r r R r R r 2 2 2 2 2 2 1 2 1 1 2 sin sin sin Note that the angular and radial terms can be separated, we suggest that we can write the wavefunction as a product of radial and angular parts. : (r,,)= R(r)Y(,) Then the angular part is separated into two parts: and three parts are substituted into Schrodinger equation, we have
Now the schrodinger equation can be written as three separate equations. 102OR2/2 radial equation E k r ar a 九2 sin e d de sin 6-+ksin 0 colatitude equation o de de 1dΦ azimuthal equation op do 2021/8/21 Che mistry De partne nt of Fudan University
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 4 k r Ze E r r R r R r = + 2 2 2 1 2 2 2 2 2 1 m d d − = 2 2 k m d d d d + = sin sin sin colatitude equation azimuthal equation radial equation Now the Schrodinger equation can be written as three separate equations
Phusical chemiatry Chapter II Atomic Structure and Spectrum We have seen that the azimuthal wave functions are e 2丌 This solution imposes the constraint the m be a quantum number and have values n=0,士1,±2,士3, sin e d de sin e +ksm2=m2k=1(+1) o de When this equation is solved it is found that k must equal l(l+1) with l=0,1,2,3.. and as above n=0,±1,土2,土3,, O(0)=P(cos0)=(1-cos 2 G(cos 0) G(e)=2a,cose 2021/8/21 Che mistry De partne nt of Fudan University 5
Physical ChemistryI Chapter II Atomic Structure and Spectrum 2021/8/21 Chemistry Department of Fudan University 5 2 2 k m d d d d + = sin sin sin ( ) im e − = 2 1 ( ) P(cos ) ( cos ) G(cos) m 2 2 = = 1− − = = l m n n an G z 0 ( ) cos k = l(l +1) We have seen that the azimuthal wave functions are This solution imposes the constraint the m be a quantum number and have values m = 0, ±1, ±2, ±3, … When this equation is solved it is found that k must equal l(l+1) with l = 0, 1, 2, 3… and as above m = 0, ±1, ±2, ±3, …
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 kl 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