Energy of He Excited States

1
Energy of He Excited States Energy Difference
for 2s and 2p orbitals for Multi-electron
Atoms Determining the Energy of Orbitals in
Multi-electron Atoms
F. Grieman
2
He Atom Electronic Energies of Atomic States
Use Perturbation Theory to show that E(2p) gt
E(2s) Ground State (X1So) 1s2 from before ?
1s(1)1s(2) H h(1) h(2) (1/r12)
(ignore spin-orbit coupling, space
only) Perturbation treatment of He atom ground
state E E1s E1s J1s,1s Ji,j
is called Coulomb integral
(e-/e- repulsion
between
e-s in orbitals i j)
J1s,1s
lt1s(1)1s(2) 1/r12 1s(1)1s(2)gt

(5/8)Z E -2.75 H -74.8 eV
(2½)/2 ?(1)?(2)-?(1)?(2)
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Excited States 1s2s Configuration Need space
part only ?(3S1) ?- (2½)/2 1s(1)2s(2)
2s(1)1s(2) antisymmetric space ?(1S0) ?
(2½)/2 1s(1)2s(2) 2s(1)1s(2) symmetric
space H? h(1) h(2) (2½)/2
1s(1)2s(2) 2s(1)1s(2)

both at the same time
(2½)/2 E1s E2s 1s(1)2s(2) E2s
E1s 2s(1)1s(2) E1s E2s
(2½)/2 1s(1)2s(2) 2s(1)1s(2)
E E1s E2s -68.0 eV (Zero order
energy is the
same for
both.) Now First Order Correction
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H' 1/r12 electron repulsion in ?
(1So) versus ?- (3So) Anticipate results
lim 1s(1)1s(2) ?- (2½)/2
1s(1)2s(1) 2s(1)1s(1) 0 !!! r1?r2
2s(1)2s(2) ? (2½)/2 1s(1)2s(1)
2s(1)1s(1) gt 0 !!!
Electron repulsion greater for ?
(1So) So E
(1So) should be gt E- (3So) Lets Do It!
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E(1) ½lt 1s(1)2s(2) 2s(1)1s(2) 1/r12
1s(1)2s(2) 2s(1)1s(2) gt ½
??1s2(1)2s2(2) (1/r12) d?1d?2 ½ ??
1s(1)2s(2)2s(1)1s(2) (1/r12) d?1d?2
½ ?? 2s2(1)1s2(2) (1/r12) d?1d?2 ½ ??
2s(1)1s(2)1s(1)2s(2) (1/r12) d?1d?2
(2/2) J1s,2s
(2/2) K1s,2s Coulomb
Integral Exchange
Integral Coulomb repulsion btwn. charge
clouds q.m. due to indistinguishability

If ? 1s(1)2s(2), K1s,2s would not exist!!!
J1s,2s (17/81) Z K1s,2s (16/729) Z
11.4 eV 1.2 eV
for Z 2 E (1So) E E(1) E1s
E2s J1s,2s K1s,2s - 68.0 11.4
1.2 -55.4 eV E (3So)
E1s E2s J1s,2s - K1s,2s - 68.0
11.4 - 1.2 -57.8 eV Same configuration,
different terms, different energies Triplet
spatial wave function gives less e-/e- repulsion,
? lower energy
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Excited states 1s2p configuration L 1, S
1 0 3P2,1,0 1P1 Energies Same analysis
except ? (2½)/2
1s(1)2p(2) 2p(1)1s(2) E (1P1) E
E(1) E1s E2p J1s,2p K1s,2p E
(3P2,1,0) E1s E2p
J1s,2p - K1s,2p E2s E2p (Remember 1 e-
atom energies depend only on n q. ) But!!! for
He J1s,2p 13.2 eV gt J1s,2s 11.4 eV
K1s,2p 0.9 eV lt
K1s,2s 1.2 eV E (1P1) - 68.0
13.2 0.9 -53.9 eV E (3P2,1,0) -
68.0 13.2 - 0.9 -55.7 eV So, E(P
States) gt E(S States) therefore, E(2p) gt
E(2s) !!! (multi-e-



atoms) due to
Coulombic repulsion according to Perturbation
Theory
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Electronic State Energy Level Diagram
E (eV) ?
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Orbital Energy Diagram
Penetration
Higher order calculations show that penetration
causing greater nuclear attraction as
important factor in E(2s) lt E(2p)
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• Lessons Learned about Multi-electron Atoms
• E f(n, l) (penetration, screening, less e-/e-
  repulsion)
• Same configuration can give states (terms) of
  different energy
• (sym./antisym. spatial wave functions have
  different e-/e- repulsion)
• 3) States have good angular momentum quantum
  numbers
• (Term symbol used to indicate states)

Other atoms, N electrons H -½ ?iN ?2i - ?iN
Z/ri ?iN lt ?jN 1/rij ? antisymetric,
indistinguishable electrons approximation
product w.f. ? ? iN ?i (one electron
orbitals) Use Slater determinant to get Pauli
allowed wave function.
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Li Atom example, N 3
1s? 1s? 2s?
?
e 1
1s?(1) 1s?(1) 2s?(1)
2 1s?(2) 1s?(2) 2s?(2)
3 1s?(3) 1s?(3)
2s?(3)
1s?(1) 1s?(1)
2s?(1) ? (1/3!)½ 1s?(2) 1s?(2)
2s?(2) 1s?(3)
1s?(3) 2s?(3)
Determinant properties 1) identical row or
column, determinant 0 (must use 2s
orbital) 2) interchange rows ? -
determinant (P(i,j) ? antisymmetric)
?
Use diagonal abbreviation, Li 1s?(1) 1s?(2)
2s?(3)
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Energy Analysis the same but more integrals Li
example ground state (1s22s) 2S½ (E -203.5
eV) Perturbation Theory E0 E(1)
E ? E(0) E(1) -7.0566 H -192.01 eV
(5.6 error) Variational Theory Z'1s 2.694,
Z'2s 1.534 E -201.7 eV (lt 1 error)
2E1s E2s 2(-9/2) 9/8 -10.1250 H
J1s,1s 2J1s,2s - K1s,2s 15/8 2(17/27)
16/243 3.0684 H
General Results E f(n, l) E ? as n ? ltrgt
? lt1/rgt ?, less nuclear/electron
attraction E ? as l ? screening effect
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Orbital Energy Level Diagram
4
4f
4d
3
4p
3d
4s
3p
3s
2
E
2p
2s
1
1s
Shell
n
Z