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Physical Chemistry II(Chapter 9‐1)
Many‐Electron Atom
Tae Kyu Kim
Department of Chemistry
Rm. 301 ([email protected])
http://cafe.naver.com/monero76
1
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SUMMARY (CHAPTER 8)• Spin → Intrinsic property of electron
1 1
2 2ss m
• Pauli’s Principle → Antisymmetic and Slater determinant
• Orbit‐Spin Angular Moment – Zeeman Effect
• Many Electron Atoms – HF‐SCF Method & Term Symbol
3
He (The Smallest Many Electron Atom)
→ Orbital Approxima on
→ Ignoring the electron correla on
0
3/2
/1 1
0
1 r ar ea
→ Spherically symmetric effec ve poten al
→ Electron spin & Indistinguishability of electrons
Due to the e‐‐e‐ repulsion, the potential energy operator no longer has the form of V(r). Whereas electron correlation ensures the minimum repulsions among electrons, here we assume that electron moves independently of one another.
Spatially averaged charge distributions
, , ,nlm nl lmr R r Y Solution is approximate due to the orbital approximation and neglect of electron correlations
4
Hartree‐Fock Self Consistent Field Method→ Antisymmetric wavefunction
modified H atom orbital
→ Separated Schrödinger equa ons
Because of the neglect of electron correlation, the effective potential is spherically symmetrical and therefore the angular part of the wave‐functions is identical to the solutions for hydrogen atom (s, p, d, … nomenclature remains intact for the one‐electron orbitals for all atoms
→ Variation method using basis functions 1
m
j i ii
r c f
→ Using an itera ve approach: this procedure is repeated for all electrons un l the solutions for the energies and orbitals are self‐consistent, meaning that they do not change significantly in a further iteration
5
Basis Functions in Variational Theory→ Basis func ons in HF‐SCF method
1
m
j i ii
r c f
→ Examples of basis func ons
0( ) exp /i i if r N r r a 2
0( ) exp /i i if r N r a
Good basis‐set is that the number of m should be kept as small as possible and the HF calculations should be done rapidly.
6
The Accuracy of HF‐SCF Method→ 1s orbital energy of He
→ Orbital energy: Effect of electron correla ons (εtotal – 2ε1s < 0)
1 01
exp /m
s i i ii
r c N r a
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HF Radial Functions→ HF radial func on for Ar → Radial probability func on (P(r))
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HF Orbital Energies→ One‐electron orbital energies from HF calculations
→ εns < εnp < εnd < …
→ εi for many electrons atom depends on the electron configuration and on the atomic charge because εi is determined in part by average distribution of all other electrons. (‐67.4 eV for Li and ‐76.0 eV for Li+)
Orbital energies depend on both n and l.
Kr – radial distributions of n = 3
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Effective Nuclear Charge→ The effective nuclear charge consider that an electron further from the nucleus
experiences a smaller nuclear charge than that experienced by an inner electrons.
→ This effect is particularly important for valence electrons and we can say that they are shielded from the full nuclear charge by core electrons closer to the nucleus.
→ The effective nuclear charge is nearly equal to the nuclear charge for the 1s orbital but falls off quite rapidly for the outermost electrons as the n increases. Whereas electrons of smaller n value are quite effective in shielding electrons with greater n values from the full nuclear charge, those in the same shell are much less effective.
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Electron Shielding of the Nuclear Charge• The energy of the orbital for specific n: s < p < d < f …
→ This difference caused by the coulombic repulsion that exists between electrons
Nucleusr
Electron under consideration
Smeared‐out charge distribution due to other electrons
→ Effec ve number of posi ve charge that electron sees at the nucleus is reduced from Z to (Z – σ): σ Screening constant
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Koopmans’ Theorem→ To a reasonable approxima on, ‐ εi for the highest occupied orbital is the first ionization energy
→ In the “frozen core” limit, it is assumed that the electron distribution in the atom is not affected by the removal of an electron in the ionization event
→ By analogy, ‐ εi for the lowest unoccupied orbital should give the electron affinity for a particular atom (less accurate)
12
Electron Configuration
→ Aufbau Principle: the relative order of orbital energies explains the electron configurations of the atoms in the periodic table
13
J. Chem. Edu. 71 (1994) 469‐471
2 1 14 3 4 3n ns d s d
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Correlation Energy
→ Hartree‐Fock calculations neglect electron correlation
Correlation Energy = True Energy – (HF total energy)
Physical Chemistry II(Chapter 9‐2)
Many‐Electron Atom
Tae Kyu Kim
Department of Chemistry
Rm. 301 ([email protected])
http://cafe.naver.com/monero
16
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SUMMARY (CHAPTER 9‐1)HF‐SCF Method1) Writing the N‐electron atomic wavefunction in terms of N
single‐electron wavefunctions or orbitals2) The procedure for determining a Hartree‐Fock wavefunction
and a Hartree‐Fock energy involves solving a set of coupled effective Schrödinger equations by an iterative method called the self‐consistent field method.
3) Hartree‐Fock atomic energies are within a percent or two of exact energies for atoms with atomic number less than 40 or so, and many other properties are in satisfactory agreement with experiment (cf. ionization energies and so on.).
This Class → Electron configura ons in terms of term symbol
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Electron Configuration: Term Symbol→ Electron configura on: C [1s22s22p2]→ The energies of different state differ and so we require more detailed designation of the electronic states of atoms: TERM SYMBOL→ The scheme is based on the idea of determining the total orbital angular
momentum L and the total spin angular momentum S and then adding L and
S together to obtain the total angular momentum J.
2S+1LJatomic term symbol L = 0 1 2 3
S P D F
2S+1 = 1 2 3
→ J (total angular momentum): Russell‐Saunders coupling vs j‐j coupling
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RS coupling (Z < 40)
→ Total (orbital and spin) angular momentum
L = ∑li S = ∑si→ z‐components of L and S
Lz = ∑lzi = ∑mi = M
Sz = ∑szi = ∑msi = Ms
→ 1s2 electron configuration
m1 m1s m2 m2s M Ms
0 +1/2 0 ‐1/2 0 0
→ M = 0 implies that L = 0 and Ms = 0 implies that S = 0
→ J = L + S Jz = Lz + Sz = (M + Ms) = MJ
→ MJ = M + Ms = 0 implies that J = 0
1S0
→ C [1s22s22p6]: ignore 1s22s2 orbital
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ns1n’s1 & np2
→ 1s12s1 electron configuration
m1 m1s m2 m2s M Ms MJ
0 +1/2 0 +1/2 0 1 1
0 +1/2 0 ‐1/2 0 0 0
0 ‐1/2 0 +1/2 0 0 0
0 ‐1/2 0 ‐1/2 0 ‐1 ‐1
→ L = 0, S = 1M = 0, Ms = 1, 0, ‐1MJ = M + Ms = 1, 0, ‐1
3S1 1S0
L = 0, S = 0M = 0, Ms = 0MJ = M + Ms = 0
→ 1s22s22p2 electron configuration: assign two electrons to six spin‐orbitals
→ The number of dis nct ways to assign N electrons to G spin‐orbitals
!
#! !
G
N G N
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np2
→ m1 m1s m2 m2s M Ms MJ
1 +1 +1/2 +1 ‐1/2 +2 0 +2
2 +1 +1/2 0 +1/2 +1 +1 +2
3 +1 +1/2 0 ‐1/2 +1 0 +1
4 +1 +1/2 ‐1 +1/2 0 +1 +1
5 +1 +1/2 ‐1 ‐1/2 0 0 0
6 +1 ‐1/2 0 +1/2 +1 0 +1
7 +1 ‐1/2 0 ‐1/2 +1 ‐1 0
8 +1 ‐1/2 ‐1 +1/2 0 0 0
9 +1 ‐1/2 ‐1 ‐1/2 0 ‐1 ‐1
10 0 +1/2 0 ‐1/2 0 0 0
11 0 +1/2 ‐1 +1/2 ‐1 +1 0
12 0 +1/2 ‐1 ‐1/2 ‐1 0 ‐1
13 0 ‐1/2 ‐1 +1/2 ‐1 0 ‐1
14 0 ‐1/2 ‐1 ‐1/2 ‐1 ‐1 ‐2
15 ‐1 +1/2 ‐1 ‐1/2 ‐2 0 ‐2
• L = 2 and S = 0 (1D) [1,3,5,12,15]
22
np2
→ m1 m1s m2 m2s M Ms MJ
1 +1 +1/2 +1 ‐1/2 +2 0 +2
2 +1 +1/2 0 +1/2 +1 +1 +2
3 +1 +1/2 0 ‐1/2 +1 0 +1
4 +1 +1/2 ‐1 +1/2 0 +1 +1
5 +1 +1/2 ‐1 ‐1/2 0 0 0
6 +1 ‐1/2 0 +1/2 +1 0 +1
7 +1 ‐1/2 0 ‐1/2 +1 ‐1 0
8 +1 ‐1/2 ‐1 +1/2 0 0 0
9 +1 ‐1/2 ‐1 ‐1/2 0 ‐1 ‐1
10 0 +1/2 0 ‐1/2 0 0 0
11 0 +1/2 ‐1 +1/2 ‐1 +1 0
12 0 +1/2 ‐1 ‐1/2 ‐1 0 ‐1
13 0 ‐1/2 ‐1 +1/2 ‐1 0 ‐1
14 0 ‐1/2 ‐1 ‐1/2 ‐1 ‐1 ‐2
15 ‐1 +1/2 ‐1 ‐1/2 ‐2 0 ‐2
• M = 1 → L = 1 with M = 0, ±1• Each of M occurs with a value of Ms = 0, ±1
• L = 1 and S = 1 (3P) [9 entries]
• M = 0 and S = 0 (L = 0 and S = 0)• 1S
23
np2
m1 m1s m2 m2s M Ms MJ
1 +1 +1/2 +1 ‐1/2 +2 0 +2
2 +1 +1/2 0 +1/2 +1 +1 +2
3 +1 +1/2 0 ‐1/2 +1 0 +1
4 +1 +1/2 ‐1 +1/2 0 +1 +1
5 +1 +1/2 ‐1 ‐1/2 0 0 0
6 +1 ‐1/2 0 +1/2 +1 0 +1
7 +1 ‐1/2 0 ‐1/2 +1 ‐1 0
8 +1 ‐1/2 ‐1 +1/2 0 0 0
9 +1 ‐1/2 ‐1 ‐1/2 0 ‐1 ‐1
10 0 +1/2 0 ‐1/2 0 0 0
11 0 +1/2 ‐1 +1/2 ‐1 +1 0
12 0 +1/2 ‐1 ‐1/2 ‐1 0 ‐1
13 0 ‐1/2 ‐1 +1/2 ‐1 0 ‐1
14 0 ‐1/2 ‐1 ‐1/2 ‐1 ‐1 ‐2
15 ‐1 +1/2 ‐1 ‐1/2 ‐2 0 ‐2
• 1D [1, 3, 5, 12, 15]J = +2, +1, 0, ‐1, ‐2
• 3P [2, 4, 6, 7, 8, 9, 11, 13, 14]J = +2, +1, +1, 0, 0, ‐1, 0, ‐1, ‐2
• 1S [10]J = 0
1D2
3P2 3P1
3P0
1S0
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J Values→ The value of J for each term symbol can be determined in terms of the values of L and S
J L S
→ J = L + S, L + S – 1, L + S – 2, …, |L – S|
→ 3P term symbol: J = (1 + 1), (1 + 1) – 1, (1 – 1) = 2, 1, 0
→ 3D term symbol: