Download Hydrogen Spectra - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides and more Slides Chemistry in PDF only on Docsity! 3.1-d: Hydrogen Spectra • Later, we will learn about selection rules for transition. For the principal quantum number, n, ∆n can be any positive (absorption) or negative (for emission) integer, resulting in the very rich form of the H atom spectrum. The Rydberg formula ( ) 21 1 32 42 H2 2 2 1 H and cm 7371094 2R here w11R nn, ch e nn o e <== −⋅= − πε µπω Paschen Balmer Lyman series En (eV) 0.00 1 − 0.85 − 1.51 − 3.40 −13.6 2 3 4 ∞ n•Emission lines fall in characteristic spectral regions The Lyman series (n1 = 1) in the ultraviolet The Balmer series (n1 = 2) in the visible The Paschen series (n1 = 3) in the near IR The Balmer series • Absorption frequencies coincide with those for emission • Transitions having n1 ≠ 1 are observed only after preparation of the excited state by some means, such as electrical discharge. • The ionization limit in absorption corresponds to a final quantum number n2 =∞. From the ground state, Io = RH, or 13.6 eV docsity.com 3.1-e: The H-Atom Orbitals ( ) ( ) ( )ϕϕ θ,YrRθ,r,ψ mnmn ll lll ⋅= Imaginary functions ( ) ϕ ϕ ϕ θ π ψ θθ π ψ θ π ψ θ π ψ θ π ψ π ψ iar oo iar oo ar oo iar ooo ar ooo ar ooo esine a r a ecossine a r a cose a r a esine a r a r a cose a r a r a e a r a r a o o o o o o 223 2 2 232 3 2 2 132 23 2 2 320 23 2 2 131 3 2 2 310 3 2 2 300 2 3 2 3 2 3 2 3 2 3 2 1 2 3 1 162 1 1 81 1 131 681 1 61 81 1 612 81 1 218271 381 1 ±− ± ±− ± − ±− ± − − ⋅⋅⋅⋅ = ⋅⋅⋅⋅⋅ = −⋅⋅⋅ = ⋅⋅⋅ −⋅ = ⋅⋅ −⋅ = ⋅ +−⋅ = ϕθ π ψ θ π ψ π ψ π ψ iar oo ar oo ar oo ar o esine a r a cose a r a e a r a e a o o o o ±− ± − − − ⋅⋅⋅⋅ = ⋅⋅⋅ = ⋅ −⋅ = ⋅ = 2 121 2 210 2 200 100 2 3 2 3 2 3 2 3 1 28 1 1 24 1 21 24 1 11 docsity.com ( ) ( ) ( ) ⋅⋅= β α θ,YrRδ,θ,r,ψ mnmn ϕϕ ll lll sm ( ) ( ) ϕϕ l ll ll im m, eθcos ⋅Θ∝θ,Y m, The ϕ-dependence is important in giving rise to selection rules on changes in the quantum number lm ( )rRn,lwhere = the radial wave function ( )ϕθ,Y m, lland = the angular wave function •The wave functions docsity.com Behaviour of the orbitals near the nucleus Electrons are progressively excluded from the neighborhood of the nucleus as l increases. Note that the s orbital as a finite, non-zero value at the nucleus. Close to the nucleus, p orbitals are proportional to r d orbitals are proportional to r2 f orbitals are proportional to r3 docsity.com
3.1-f:
Xxlay
(b)
Copa 08 aro eaten, se pb sapere gn
204 6303
w4
x of
4
24
been yeenprerrperrep rrp ee
2-10 9 10 m0
Copyright © 2006 Pearson Education, Ine., publishing as Benjamin Cummings
Contour Plots of the
* 26,
04
ao
“0
2
Seceeepereeereecneenie®
“0-10 0 0
hoesreeeprereperreperentae
D0 0 1
x
; docsit
‘Copyright @ 2006 Pearson Education, Inc., publishing as Benjamin Cumntin
.com
188
Radial distribution function
is
2p
2s
3d
3p
3s
docsity.com
Old Convention: Shells vs. Sub-shells 3d 2p 3p K n=1 L n=2 SH E LL S 1s 2s 3s M n=3 SUB-SHELLS SHELL n = 1 2 3 4 K L M N docsity.com 4.1 Angular Momentum a) General Properties: apply to all angular momenta. Any property M is assigned as an angular momentum if it obeys the following relationships: [ ] [ ] [ ] [ ] [ ] [ ] 0ˆ,ˆˆ,ˆˆ,ˆ) ˆˆ,ˆ;ˆˆ,ˆ;ˆˆ,ˆ) ˆˆˆˆ) 222 2222 === === ++= zyx yxzxzyzyx zyx MMMMMMiii MiMMMiMMMiMMii MMMMi hhh yx yx MiMM MiMM ˆˆˆ ˆˆˆ −= += − +M has Raising operator Lowering operator Two quantum numbers are needed because M has two properties: ( ) jjjjmmM jjjM jmjjmjz mjmj jj jj ,1,....1,ˆ ,...2, 2 3,1, 2 1,01ˆ ,, , 2 , 2 −+−−== =+= ψψ ψψ h h docsity.com
