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Spectroscopy Cheat Sheet, Study notes of Chemistry

Spectroscopy Cheat Sheet. Atomic Spectroscopy. • Energy of hydrogen atom ... Vibrational Spectroscopy. • Anharmonic (Morse) Oscillator energies.

Typology: Study notes

2022/2023

Uploaded on 05/11/2023

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Spectroscopy Cheat Sheet
Atomic Spectroscopy
Energy of hydrogen atom
En=ÃZ2ħ2
2µa2
µ!1
n2(1)
aµ=4π²0ħ2
µe2=me
µ
a0(2)
Mean radii of hydrogen atom
¿1
rÀZ
n2(3)
r=aµn2(1 +(1 l(l+1)/n2)/2) (4)
Spin-orbit coupling arises due to electron spin magnetic moment in-
teracting with magnetic moment generated by nuclear charge orbiting
electron in electron rest frame. This effect is proportional to the mag-
netic interaction 1/r3®and the nuclear charge Z, so overall you
would expect SO coupling to be proportional to Z4. In fact
¿1
r3À=2
a3
µn3l(l+1)(2l+1) (5)
Selection rules. For optically allowed transitions
n=anything (6)
l= ±1 from conservation of angular momentum and parity (7)
ml=0,±1 from conservation of angular momentum (8)
s=0 from conservation of spin (9)
ms=0 (10)
j=0,±1 (no j=00) conservation of angular momentum (11)
mj=0,±1 (12)
These are the same for many electron atoms with l,ml,j,mj,s,msre-
placed by L,ML,J,MJ,S,MS.
Zeeman effect splits levels with angular momentum
EZeeman =µBgJBM J(13)
gJ=1+J(J+1) L(L+1) +S(S+1)
2J(J+1) (14)
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Spectroscopy Cheat Sheet

Atomic Spectroscopy

  • Energy of hydrogen atom

En =

Z^2 ħ^2 2 μ a^2 μ

n^2

a μ =

4 π≤ 0 ħ^2 μ e^2

me μ

a 0 (2)

  • Mean radii of hydrogen atom 〈 1 r

Z

n^2

〈r〉 = a μ n^2 (1 + (1 − l(l + 1)/n^2 )/2) (4)

  • Spin-orbit coupling arises due to electron spin magnetic moment in- teracting with magnetic moment generated by nuclear charge orbiting electron in electron rest frame. This effect is proportional to the mag- netic interaction ∝

1/r^3

and the nuclear charge Z, so overall you would expect SO coupling to be proportional to Z^4. In fact 〈 1 r^3

a^3 μ n^3 l(l + 1)(2l + 1)

  • Selection rules. For optically allowed transitions ∆n = anything (6) ∆l = ±1 from conservation of angular momentum and parity (7) ∆ml = 0 , ±1 from conservation of angular momentum (8) ∆s = 0 from conservation of spin (9) ∆ms = 0 (10) ∆ j = 0 , ±1 (no j = 0 → 0) conservation of angular momentum (11) ∆m (^) j = 0 , ± 1 (12) These are the same for many electron atoms with l, ml , j, m (^) j, s, ms re- placed by L, ML, J, MJ , S, MS.
  • Zeeman effect splits levels with angular momentum EZeeman = μ B gJ BMJ (13)

gJ = 1 +

J(J + 1) − L(L + 1) + S(S + 1)

2 J(J + 1)

Vibrational Spectroscopy

  • Anharmonic (Morse) Oscillator energies

En = ħ ω e(n + 1/2) − ħ ω e xe(n + 1/2)^2 (15)

  • Dissociation energy.

xe =

ħ ω e 4 De

  • Selection rules from dipole moment operator. Mechanical an electronic anharmonicity leads to overtones.
  • P-branch ∆J = −1, R-branch ∆J = +1. Q-branch ∆J = 0 appears when angular momentum can be conserved in other ways e.g. in vibrational or electronic degrees of freedom (e.g. via Lambda-doubling in NO or OH).