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Unit VII Multielectron Atoms
A. Electron Spin
- Introduction a. The idea was introduced initially when it was observed that some lines in Na spectrum that are predicted to be a singlet actually appear as a doublet. b. The electron behaves like a spinning top with a z component of ± ħ /2. c. This classical picture is not accurate; the spin is strictly a nonclassical (quantum mechanical) concept. d. Spin is an intrinsic (built-in) angular momentum possessed by elementary particles.
- Spin quantum numbers a. New spin-angular-momentum quantum numbers are introduced, s and m (^) s , that are analogues to the orbital-angular-momentum quantum numbers l and m (or more specific ml ). b. The spin quantum number m (^) s determines the z component of the electron spin angular momentum just as m (^) l determines the z component of the electron orbital angular momentum in hydrogen atom. □ The idea of spin is though more general than the case of electrons in atoms. □ Similar to ml = – l ,…,+ l , m (^) s can take 2 s + 1 values: m (^) s = – s ,…,+ s. □ The analogy is not complete because a given species of elementary particle can have only one value for s. c. The value of s may be half-integral ( 21 , 23 ,…) as well as integral (0,1,…).
□ Examples: ○ electrons, protons, neutrons have s = 21 ⇒ ms =+ 21 ,− 21 ○ 35 Cl nuclei have s = 23 ⇒ ms =+ 23 , + 21 ,− 21 ,− 23 ○ 12 C nuclei have s = 0 ⇒ ms = 0 ○ photons have s = 1 ⇒ ms =− 1 , 0 ,+ 1 d. The particles with half integer spin are called fermions, and the particles with integer spin are called bosons. e. Because s = 21 only and cannot have big values, the spin cannot be classical (cannot have classical behavior).
- Spin eigenfunctions a. There are two operators associated with the spin (similar to the angular momentum): S ˆ^2 and S ˆ^ z. b. For the electron ( s = 21 ; ms =± 21 ), there are two eigenfunctions
associated with the spin (similar to the spherical harmonics) α and β
with the properties: ( )
β ( ) β β
2 2 2
2 2 2
h h
h h
⎟ ⎠
= + = ⎛^ +
= + = ⎛^ +
S s
S s s
○ similar to L ˆ^2 Ylm (θ ,φ)= l ( l + 1 )h^2 Ylm (θ, φ)
h h
h h
ˆ^1
ˆ^1
z s
z s
S m
S m
○ similar to L ˆ z Ylm (θ ,φ)= m h Ylm (θ, φ)
□ Spin eigenfunction α looks like Y 1^1 // 22 , and β looks like Y 1 −/^12 /^2.
d. The one-electron wavefunctions are called spin orbitals. □ Spin orbitals are normalized. e. The case of He atom:
2
1 2
ψ ( 1 , 2 )= 1 s α( 1 ) 1 s β( 2 )=ψ 1001 ψ 100 −
□ But the electrons are indistinguishable (our label cannot make a distinction between the electrons) so the wavefunction
ψ ( 2 , 1 )= 1 s α( 2 ) 1 s β( 1 ) is equivalent.
□ Linear combinations of the two wavefunctions are also possible:
Ψ 1 = ψ ( 1 , 2 )+ ψ( 2 , 1 )= 1 s α ( 1 ) 1 s β( 2 )+ 1 s α( 2 ) 1 s β( 1 )
Ψ 2 = ψ ( 1 , 2 )− ψ( 2 , 1 )= 1 s α ( 1 ) 1 s β( 2 )− 1 s α( 2 ) 1 s β( 1 )
□ Out of the two possible linear combinations, only Ψ 2 is antisymmetric:
f. Slater determinants □ Antisymmetric wavefunctions are represented by Slater determinants:
1 ( 2 ) 1 ( 2 )
( 1 , 2 )^1
s s
s s
○ This is called determinantal wavefunction. □ More general (the case of N electrons):
( 1 , 2 ,.... )^1
1 2
1 2
1 2
u N u N u N
u u u
u u u
N
N
N
N
N Ψ =
where u are orthonormal spin orbitals. □ Properties of the wavefunction written is determinantal form: ○ Ψ = 0 if two columns are the same. ○ Ψ changes sign if columns are interchanged.
B. Hartree-Fock Method
- The case of helium atom a. Determining ground-state energy using variational method and a trial
function of the form ψ ( r 1 , r 2 )= φ( r 1 ) φ( r 2 ) the minimum-energy limit
that one can obtain will be E = − 2. 8617 E h. □ The result is too high compared to the experimental value of E = − 2. 9033 E h. b. This is the best value of the energy that can be obtained using a trial function of the form of a product of one-electron wavefunction. □ This is called the Hartree-Fock limit. c. The idea of one-electron orbitals (wavefunctions) is preserved in the Hartree-Fock approximation and is abandoned in more accurate methods. d. To improve the variational results one should include explicitly the interelectronic distance: □ Example: ψ ( r 1 (^) , r 2 , r 12 )= e − Zr^1^^ e − Zr^2 [ 1 + cr 12 ]. e. Variational method with a trial function with 1078 parameters gives very good results.
- Description of the Hartree-Fock method a. Hartree-Fock equations are solved using self-consistent field (SCF) method known also as HF. b. Example for He: □ Write the two-electron wavefunction as a product of one-electron wavefunctions (orbitals) and assume the same wavefunction for both electrons:
ψ ( r 1 , r 2 )= φ( r 1 ) φ( r 2 )
e. The final wavefunctions φ i are self-consistent orbitals and are called
Hartree-Fock orbitals.
f. In practice, the trial function φ is (or can be) a sum of Slater orbitals
and what are optimized are the linear coefficients (and maybe the
exponent ζ in the e–^ ζ r^ term).
g. The energies ε i are called orbital energies.
□ Koopman’s theorem given an interpretation for ε i :
○ It is the ionization energy of the electron from the i th^ orbital. □ Example: Argon configuration: 1 s 2 2 s^22 p^63 s^23 p^6 Ionization Process Koopman’stheorem calculationHF experimental 1 s^2 2 s^22 p^63 s^23 p^6 → 1 s 2 s^2 2 p^63 s^23 p^6 + e − 311.35 308.25 309. 1 s^2 2 s^22 p^63 s^23 p^6 → 1 s^2 2 s 2 p^63 s^23 p^6 + e − 32.35 31. 1 s^2 2 s^22 p^63 s^23 p^6 → 1 s^2 2 s^22 p^53 s^23 p^6 + e − 25.12 24.01 23. 1 s^2 2 s^22 p^63 s^23 p^6 → 1 s^2 2 s^22 p^63 s 3 p^6 + e − 3.36 3.20 2. 1 s^2 2 s^22 p^63 s^23 p^6 → 1 s^2 2 s^22 p^63 s^23 p^5 + e − 1.65 1.43 1. Experimental values are determined using photoelectron spectroscopy. The values are in MJ/mol
- Correlation energy a. When the wavefunction for a two-electron system is written as a
product of two one-electron wavefunctions ψ ( r 1 , r 2 )= φ( r 1 ) φ( r 2 ), the 2
electrons are considered to be independent of each other (or just to have an average potential created by the other one) and the electrons are said to be uncorrelated. b. Define the correlation energy: CE = E (^) exact− E HF □ Example: CE = − 110 kJ/molfor the He atom. □ The Hartree-Fock energy ( E (^) HF) is about 99% exact but is missing the correlation energy.
C. Electron Configurations and Atomic Term Symbols
- Electron configurations a. The electron configuration gives the subshells (or the orbitals) that are occupied and how many electrons are in which orbital. □ Example: C atom 1 s^22 s^2 2 p^2 b. Relative energies of the atomic orbitals determine their occupation, and the order in which they are occupied is from bottom to top (i.e., from the most stable to the least stable). □ The order is: 1 s , 2 s , 2 p , 3 s , 3 p , 4 s , 3 d , 4 p , 5 s , 4 d , 5 p , … c. The electron configuration doesn’t say anything about the four quantum numbers of each electron. d. For each configuration there are a number of such possibilities. e. The energies are different for each of these possibilities so there is a need for a more detailed description of the electronic states of atoms than the one provided by electronic configuration.
- Russell-Sounders coupling a. Russell-Sounders coupling (called RS coupling) is a method of providing more detailed information on the electronic state in an atom in a form of an atomic term symbol. □ An atomic term symbol is also called a spectroscopic term symbol. b. The general idea is: □ Determine the total orbital angular momentum L □ Determine the total spin angular momentum S □ Vectorially add L and S to get total angular momentum J c. Define atomic term symbols as:
J
2 S + 1 L
□ J is the total angular momentum quantum number
k. Define a microstate as a set of m (^) l and m (^) s values for each electron in the atom. l. The degeneracy (i.e., the number of microstates) of a term: □ For a term written as 2 S^ +^1 L : ( 2 S + 1 )( 2 L + 1 ) □ For a term written as 2^ S^ +^1 LJ : 2 J + 1
m. Recall that the number of distinct ways to assign N electrons into G spin orbitals belonging to the same subshell (equivalent orbitals) is given by:
!( )!
N G N
G
n. The term symbols are read as spin multiplicity (as a word describing a number) followed by L (as a letter) followed by J (as a number). □ 2 D (^5) / 2 is read as “doublet D five halves” (⇒ S = 21 , L = 2 , J = 25 ) □ 1 P 1 is read as “singlet P one” (⇒ S = 0 , L = 1 , J = 1 ) □ 3 P 0 is read as “triplet P zero” (⇒ S = 1 , L = 1 , J = 0 ) □ 4 S 3 / 2 is read as “quartet S three halves” (⇒ S = 23 , L = 0 , J = 23 )
o. The term symbol of the ground state electronic state is given by Hund’s rules: □ The state with the largest value of S is the most stable (has the lowest energy), and stability decreases with decreasing S. □ For states with the same value of S , the state with the largest value of L is the most stable. □ If few states have the same value of L and S. ○ for a subshell that is less than half filled, the state with the smallest value of J is the most stable. ○ for a subshell that is more than half filled, the state with the largest value of J is the most stable.
p. Examples of possible terms and the term symbol of the ground state for a certain configuration: □ The ns^2 configuration:
2
1
=
L
l
l
M
m
m
2
1
=
S
s
s
M
m
m L = 0, S = 0 ⇒ J = 0
○ The only possible term symbol and therefore the terms symbol of the ground state is 1 S 0. □ Same result is obtained for the np^6 configurations and in every situation when a subshell is completely filled. □ The 1 s^12 s^1 configuration: ○ The total number of microstates (possibilities of putting two electrons in two spin orbitals): 2 × 2 = 4 ○ A microstate is represented by a number representing m (^) l and a superscript + or – representing the sign of m (^) s. ○ For the case of only 2 electrons, one possibility to determine the possible terms is to make a table to show all possible microstates:
M (^) L 1
M S
0 0 + 0 +^0 + 0 –^ ; 0– 0 +^0 – 0 –
○ The two 0 + 0 –^ microstates are not equivalent because the two orbitals are not equivalent. L = 0, S = 1 ⇒ 3 S 1 term 3 microstates L = 0, S = 0 ⇒ 1 S 0 term 1 microstate Total 4 microstates ○ The ground state is 3 S 1 because of its highest spin multiplicity.
D. Atomic Spectra
- The use of atomic term symbols to describe atomic spectra a. The energy of the hydrogen atom depends only on principal quantum number n. b. It was experimentally observed that various levels show small splitting. □ The reason for this fine splitting is the spin-orbit coupling. c. Hamiltonian should include a small term (i.e., a perturbation) that define the spin-orbit interaction which represents the interaction of the magnetic moment associated with the spin of the e –^ with the magnetic field generated by the electric current produced by the electron’s own orbital motion. d. The increased spectral complexity caused by the spin-orbit coupling is called fine structure. e. Any state of the atom, and any spectral transition, can be specified using term symbols. □ By convention, in writing the expression for the transition, the upper term precedes the lower term. f. Transitions in atomic spectra are restricted by some selection rules: □ Δ S = 0 (no change of the overall spin ⇒ the light does not affect the spin) □ Δ L = 0 , ± 1 □ Δ l =± 1 □ Δ J = 0 , ± 1 (except J = 0 → J = 0 transition) g. The selection rules show that a transition is accompanied by a change in the angular momentum of an individual electron.
- The spectrum of the hydrogen atom a. The n = 2 → n = 1 in Bohr model appear at 82258.19 cm –^. Configuration Term Symbol Energy / cm– 1 s 1 s^2 S 1 / 2 0. 2 p 2 p^2 P 1 / 2 82258. 2 s 2 s^2 S 1 / 2 82258. 2 p 2 p^2 P 3 / 2 82259. □ Allowed transitions: ○ 2 p^2 P 3 / 2 → 1 s^2 S 1 / 2 ○ 2 p^2 P 1 / 2 → 1 s^2 S 1 / 2 □ The two allowed transitions make the first line in Lyman series to appear as a doublet. b. The 3 d → 2 p ( n = 3 → n = 2) transition shows a fine structure:
□ The line also appear as a doublet. c. The term symbols and their energies for various atoms are tabulated.
Hartree-Fock method
Hartree-Fock approximation
self-consistent field method
effective one-electron Hamiltonian operator
Hartree-Fock equation
Fock operator
Hartree-Fock orbitals
Koopman’s theorem
correlation energy
electron configuration
Russell-Sounders coupling
total angular momentum quantum number
total orbital angular momentum quantum number
total spin angular momentum quantum number
atomic term symbol
spin multiplicity
Hund’s rules
microstate
fine splitting
spin-orbit coupling
fine structure
