Lecture 30.
Multielectron Atoms Revisited.
Spin-Orbit Interaction in Multielectron Atoms.
Lande’s Interval Rule.
Electrons in atom are distributed among single-particle states. Each of those states is
characterized by the principal quantum number n, by the orbital quantum number l, by the
magnetic quantum number m l , and by the spin quantum number ms . A set of states with the
same n is called a shell. A set of states with the same (n, l ) is called a subshell. A set of states
(
)
with the same n, l , m l is called an orbital.
To first approximation (ignoring electron mutual repulsion altogether), all single-particle
states of the nth shell have the same energy
2
1  1  m Z 2 e4

,
ε n = − 2 
2n  4πε 0 
h2
(30.1)
where Z e is a nucleus charge. The ground state of the atom has the lowest energy. Therefore, if
electrons were bosons, the ground state energy of any atom would be
2
1  1  m Z 2 e4

E 1 = − 
Z,
2  4πε 0 
h2
(30.2)
where Z is the number of electrons in the atom. But electrons are identical fermions subject to
(
)
the Pauli exclusion principle, so only two can occupy any given orbital, n, l , m l . We
remember that there are n 2 orbitals for the nth shell. Therefore, the nth shell can accommodate
2n 2 electrons. As a result, the formula (30.2) applies to the ground states of hydrogen and
helium only. For the ground state of lithium (Z = 3) we have
2
2
1  1  m 32 e 4
1 1  1  m 32 e 4



2
1 = −275. 4 eV.
−
E 1 = − 
2  4πε 0 
2 2 2  4πε 0 
h2
h2
Hartree method takes into account mutual electron repulsion, see Lecture 20. The
outcome of the method is that all single-particle states of the same subshell (n, l ) have the same
1
energy; within a given shell n, the state with the lowest energy has l = 0 and the energy
increases with increasing l. There are l + 1 orbitals for the (n, l ) subshell. If we now augment
Hartree method by Pauli exclusion principle, we obtain that the (n, l ) subshell can accommodate
2 (2 l + 1) electrons. Keep in mind that single-particle states with l = 0 are called s-states, single-
particle states with l = 1 are called p-states, single-particle states with l = 2 are called d-states,
single-particle states with l = 3 are called f-states. For example, the ground state electron
configuration of carbon
(1 s ) 2 (2 s ) 2 (2 p ) 2
(30.3)
tells us that there are two electrons in the subshell (1, 0 ) , two electrons in the subshell (2, 0 ) ,
and two electrons in the subshell (2, 1) .
The total orbital momentum and the total spin of the filled shell are zero. As a result, the
operators of the total orbital momentum L̂ and the total spin Ŝ of an atom are formed from the
operators l̂ i and ŝ i of the electrons of the unfilled shell.
Lˆ =
∑ ˆl ,
i
Sˆ =
i
∑ sˆ .
(30.4)
i
i
The operators L̂ and Ŝ commute with the Hamiltonian of an atom. Therefore, there is a
common set of eigentstates
ψ E , L, M L , S , M S = E L M L S M S .
(30.5)
If we apply Hartree method, for example, to carbon, we obtain that all multiplets
E 0 ML 0 MS
(one state), E 0 M L 1 M S
(three states),
E 1 ML 0 MS
(three states),
E 1 M L 1 M S (nine states), E 2 M L 0 M S (five states), E 2 M L 1 M S (fifteen states) have
the same energy. Therefore, there are, in fact, 36 possible states of carbon atom within the
ground state electron configuration (1 s ) 2 (2 s ) 2 (2 p ) 2 ! All those states have the same energy
according to Hartree method.
More sophisticated methods (like Hartree-Fock method or some other methods) showed
that the atom energy levels depend on S and L as well, i.e.
E = E ( L, S ) .
(30.6)
2
Therefore, the set of (L, S ) defines the atom term (the multiplet with the same energy) which
2 S +1
can be described as
carbon are
3
P
(
1
S
L . The degeneracy of the term is (2 L + 1)(2 S + 1) . Possible terms for
( E 0 M L 0 M S ),
E 1 ML1 MS
),
1
D
(
3
S
(
E 2 ML 0 MS
E 0 ML1 MS
),
3
D
(
),
1
P
E 2 ML1 MS
(
).
E 1 ML 0 MS
),
The question is
which term is the ground one (has the lowest energy). In order to answer this question, we need
to know the relation (30.6) in the explicit form, which is a very complicated problem.
Fortunately, there is also empirical rule – a so called Hund’s first rule saying that the term
with the highest total spin will have the lowest energy; for a given total spin, the term with
the highest total angular momentum, consistent with overall antisymmetrization, will have
the lowest energy. As a result, the ground term for carbon is
3
P . There are still nine states
(2 ⋅1 + 1)(2 ⋅1 + 1) of the same energy within that term.
The total angular momentum of the atom (the total angular momentum of the unfilled
subshell) is
Jˆ = Lˆ + Sˆ .
(30.7)
As usual, in addition to a set of commuting operators Ĥ , L̂2 , Ŝ 2 , L̂ z , Ŝ z with common
eigenstates (30.5), there is a set of commuting operators Ĥ , L̂2 , Ŝ 2 , Ĵ 2 , J z with common
eigenstates
E J MJ LS =
∑ LS M M
L
S
J MJ
E L S ML MS ,
(30.8)
ML, MS
M L +M S =M J
which describe (2 L + 1)(2 S + 1) degenerate states of the term
varies from L − S
2 S +1
L as well. Remember that J
to L + S .
Let me now include the spin-orbit interaction
Uˆ s. o. = A Lˆ Sˆ
(30.9)
in the Hamiltonian of the atom. The consequence is that operators L̂ and Ŝ (in particular, L̂ z and
Ŝ z ) don’t commute with Ĥ anymore. In general, each term
structure of the term will be formed).
3
2 S +1
L will be split (the fine
Let me investigate this splitting in details. I consider the operator Uˆ s.o. to be a
perturbation. Since Jˆ 2 = Lˆ 2 + 2Lˆ Sˆ + S 2 , it takes the form
Jˆ 2 − Lˆ 2 − Sˆ 2
Uˆ s.o. = A
.
2
(30.10)
Now we can see that Uˆ s. o. has a diagonal form in the basis E J M J L S . Therefore, despite the
fact that this basis is a set of degenerate states, each particular state E J M J L S
is shifted in
accordance with the first-order perturbation theory for the nondegenerate case:
J ( J + 1) − L(L + 1) − S (S + 1)
.
∆ E J = J M J Uˆ s.o. J M J = A
2
(30.11)
Since the values of L and S are the same for all the components of a term, and we are now
interested only in their relative position, we can write the energy of the term splitting in the form
∆ E J , relative = A
J ( J + 1)
.
2
(30.12)
The intervals between adjacent components (with numbers J and J − 1 ) are consequently
∆ E J , J −1 = A J .
(30.13)
This formula gives what is called Lande’s interval rule.
The constant A can be either positive or negative. For A > 0 the lowest component of the
term
2 S +1
L is one with the smallest possible J, i.e. J = L − S ; such multiplets are said to be
normal. For A < 0 the lowest component of the term
2 S +1
L is one with the largest possible J, i.e.
J = L + S ; such multiplets are said to be inverted. In fact, Hund’s second rule defines the sign
of A: if the electron configuration is such that the unfilled subshell is no more than half
filled then the term with J = L − S has the lowest energy; if the unfilled subshell is more
than half filled than the term with J = L + S has the lowest energy.
Let me go back to carbon. Due to spin-orbit interaction, the carbon ground term 3 P is
split into the terms
(2 J + 1) states.
3
P0 ,
3
P1 ,
3
P2 . Each of those terms is still degenerate and contains
In accordance with (30.11) and Hund’s second rule, the ground term of carbon
is 3 P0 . Well, it happens that the ground term of carbon is nondegenerate.
4