Many-electron atoms
In constructing the hamiltonian operator for a
many electron atom, we shall assume a fixed
nucleus and ignore the minor error
introduced by using electron mass rather
than reduced mass. There will be a kinetic
energy operator for each electron and
potential terms for the various electrostatic
attractions and repulsions in the system.
Assuming n electrons and an atomic number
of Z, the hamiltonian operator is (in atomic
units):
n
n
n 1 n
1
2
ˆ
H (1,2,3,...., n)    i   (Z / ri )   1 / rij
2 i 1
i 1
i 1 j i 1
其中左邊的1,2,3,…代表spatial coordinates of
each of the n electrons. 因此1代表x1,y1,z1, or
r1, θ1, ψ1,這裡我們就不再談原子本身的動能(視為是
不動或固定能量),右邊第三項的寫法,保證 1/r12,和
1/r21不會同時出現,這樣兩電子的斥力才不會多算一
次,也不會出現 1/r22等的 case
因此上式中,對 He atom 而言,其 hamiltonian 寫法為:
1 2 1 2
ˆ
H (1,2)   1   2  (2 / r1 )  (2 / r2 )  (1 / r12 )
2
2
or ,
Hˆ (1,2)  h(1)  h(2)  1 / r
12
1 2
where, h(i )    i  (2 / ri )
2
is called one-electron operator, and 1/r12 is called
two-electron operator.如果將後面這一項省略掉的話,
那就是所謂粗略的近似法,不考慮電子間的排斥能,而
只有個別獨立電子的能量。
Hˆ app  h(1)  h(2), h(1)i (1)   ii (1)
 i is called orbital energy, i (1) called atomic orbital
1 Z2
i  
, in atomic unit (hartree)
2
2n
Where n is the principal quantum number for atomic
orbital Φi , and Z is the atomic nuclear charge in
atomic units. Φi (1) where 1 is the position coordinate
of electron 1, and the atomic orbital Φi is used for any
one-electron fN for describing the electronic
distribution about an atom.
Products of the atomic orbitals Φi’s are eigenfNs
of Happ, and the eigenvalue E is equal to the sum
of the atomic orbital energies εi’s
Hˆ appi (1) j (2)  ( i   j )i (1) j (2)  Ei (1) j (2)
Simple products and electron
exchange symmetry
For the configuration 1s12s1, the wavefN is :
 (1,2)  1s(1)2s(2) 
8

exp( 2r1 )
1

(1  r2 ) exp( r2 )
因為1和2電子無法分辨,我們必須加以修正, so that it yields an
average value for r1 and r2 that is independent of our choice
of electron labels. This means that the electron density itself
must be independent of our electron labeling scheme.
欲達到此結果,可以有兩種情形,那就是將 1s(1)2s(2)
and 1s(2)2s(1) 相加,或相減,其平方後將1和2交換才
不會變。
 (1,2) 2  [1s(1)2s(2)  1s(2)2s(1)]2 
[1s(2)2s(1)  1s(1)2s(2)]   (2,1) , or,
2
2
 (1,2) 2  [1s(1)2s(2)  1s(2)2s(1)]2 
[1s(2)2s(1)  1s(1)2s(2)]   (2,1)
2
2
 (1,2)  (1 / 2 )[1s(1)2s(2)  1s(2)2s(1)]
or , (1,2)  (1 / 2 )[1s(1)2s(2)  1s(2)2s(1)]
前式為 symmetric to the exchange of labels, 後式為
antisymmetric to the exchange of labels.
Electron spin and the exclusion
principle
Stern and Gerlach observed two bands of Ag atom in their expt.
只有兩種spin,稱αand β,為電子normalized spin fNs,
*
*

(
1
)

(
1
)
d

(
1
)

1



 (1) (1)d (1)
*
*

(
1
)

(
1
)
d

(
1
)

0



 (1) (1)d (1)
 is equivalent to summing over the possible electron
indices.
Pauli principle: wavefNs must be antisymmetric
with respect to simultaneous interchange of
space and spin coordinates of electrons, called
spin-orbitals of electrons.
Slater determinants
Slater suggested that there is a simple way to write
wavefNs guaranteeing that they will be antisymmetric for
interchange of electronic space and spin coordinates, for
example 1s12s1:
1 1s(1) (1) 1s (2) (2)
 (1,2) 
2 2s(1) (1) 2s (2) (2)
1

{1s(1) (1)2s(2) (2)  1s (2) (2)2s (1) (1)}
2
  (2,1)
try 1s(1) (1)1s(2)  (2), and 1s 2 with two  electrons
For three electrons wavefNs, 1s22s1
1s (1) (1) 1s (2) (2) 1s (3) (3)
1
 (1,2,3) 
1s (1)  (1) 1s (2)  (2) 1s (3)  (3)
3!
2 s (1) (1) 2 s (2) (2) 2 s (3) (3)
or ,
1s (1) 1s (2) 1s (3)
1
 (1,2,3) 
1s (1) 1s (2) 1s (3)
3!
2 s (1) 2 s (2) 2 s (3)
寫法為先以行的方式將各電子的 spin-orbitals寫上去,然後再
填入電子的 indices,第一行填1,第二行填2,第三行填3。
展開後,任意交換兩電子,一定會變號, antisymmetric
to the exchange of any two electron’s indices
Singlet and triplet states
有兩個電子填入同一個 orbital 時，必須是 paired
(↑↓) ，其total spin， S為0，則 2S+1=1
，其中「 2S+1=1」 就稱為 spin-multiplicity 。
其值若為1 叫 singlet，若為2，叫「doublet」，若
為3，叫「triplet」，4叫「quartet」…當然，若兩
個電子填入不同orbitals時，就可能為singlet 或是
triplet 。例如,excited state of He, 1s12s1,符合Pauli
principle wavefNs:
singlet
1
1
[ (1)  (2)   (2)  (1)]
[1s (1)2s (2)  1s (2)2 s (1)]
 s ,a (1,2) 
2
2
triplet
 (1) (2)



 1
1
[1s (1)2s (2)  1s (2)2s (1)] [ (1)  (2)   (2)  (1)]
 s ,a (1,2) 
2

 2


 (1)  (2)
如果用行列式的方式來表示時,發現無法用
單一行列式來含蓋 triplet state。
1 1s(1) (1) 1s(2) (2)
 a , s (1,2) 
2 2s(1) (1) 2s(2) (2)
equivalent to (1)
1  1 1s(1) 1s(2)
1 1s (1) 1s (2) 
 a ,s (1,2) 



s ,a
2  2 2s (1) 2s (2)
2 2s(1) 2s(2) 
展開式 equivalent to (2) and (4)
The lesson to be learned from this is that a single
Slater determinant does not always display all of the
symmetry possessed by the correct wavefN.
Paired spin, s=0,
Ms=0, 但xy平面上
的分量仍然不斷變化
This is called
Singlet.
Two electrons with parallel spins,
have a nonzero total spin angular
momentum. 有三種方式，其中
the angle between the vectors
is the same in all three cases:
the resultant of the two vectors
have the same length in each
case, but points in different
directions.
This is called triplet
與前面paired spin 比較：
two paired spin are precisely
antiparallel, however, two ‘parallel’
spins are not strictly parallel.
Next we will investigate the energies of the
states as they are described by these
wavefNs
We have known that they are eigenfNs of Happ, but not
eigenfNs of real hamiltonian, therefore, we calculate the
average values of the energy for the singlet and triplet
state wavefNs:
E
* ˆ

 Hd
   d
*
, where d includes the space and spin
since spin - orbitals are normalized , so  *d  1,
1 2 1 2 2 2 1
 E   [ 1   2    ]d
2
2
r1 r2 r12
*
We notice that the hamiltonian operator has no
interaction term on spin part, this means that the
average energy will be entirely determined by the
space parts. Therefore, the triplet state will have
the same energy, but that of the singlet state may
have a different energy. Which of these two state
energies should be higher?
*
1
1 2
*
*
*
E1    [1s (1)2s (2)  1s (2)2s (1)][  1
2
2
3
1 2 2 2 1
  2    ][1s(1)2s(2)  1s(2)2s(1)]dv(1)dv(2)
2
r1 r2 r12
展開,分為動能,位能及排斥能三部分,各別探討:
動能部分:
1
1 2
*
{ 1s (1)[  1 ]1s (1)dv(1)  2 s * (2)2 s (2)dv(2)
2
2
1
  2 s * (2)[   22 ]2 s (2)dv(2)  1s * (1)1s (1)dv(1)
2
1 2
*
  2 s (1)[  1 ]2 s (1)dv(1)  1s * (2)1s (2)dv(2)
2
1 2
*
  1s (2)[   2 ]1s (2)dv(2)  2 s * (1)2 s (1)dv(1)
2
1 2
*
  1s (1)[  1 ]2 s (1)dv(1)  2 s * (2)1s (2)dv(2)
2
1 2
*
  2 s (2)[   2 ]1s (2)dv(2)  1s * (1)2 s (1)dv(1)
2
1
  2 s * (1)[  12 ]1s (1)dv(1)  1s * (2)2 s (2)dv(2)
2
1 2
*
  1s (2)[   2 ]2 s (2)dv(2)  2 s * (1)1s (1)dv(1)
2
• The orthogonality of the 1s and 2s orbitals
caused the terms preceded by ± to vanish.
Furthermore, integrals that differ only in
the variable label ( such as those in the
2nd and 3rd terms )are equal.
動能部分:
1
1 2
*
{ 1s (1)[  1 ]1s (1)dv(1)  2 s * (2)2 s (2)dv(2)
2
2
1
  2 s * (2)[   22 ]2 s (2)dv(2)  1s * (1)1s (1)dv(1)
2
1 2
*
  2 s (1)[  1 ]2 s (1)dv(1)  1s * (2)1s (2)dv(2)
2
1 2
*
  1s (2)[   2 ]1s (2)dv(2)  2 s * (1)2 s (1)dv(1)
2
1 2
*
  1s (1)[  1 ]2 s (1)dv(1)  2 s * (2)1s (2)dv(2)
2
1 2
*
  2 s (2)[   2 ]1s (2)dv(2)  1s * (1)2 s (1)dv(1)
2
1
  2 s * (1)[  12 ]1s (1)dv(1)  1s * (2)2 s (2)dv(2)
2
1 2
*
  1s (2)[   2 ]2 s (2)dv(2)  2 s * (1)1s (1)dv(1)
2
so that this expansion becomes
1 2
1 2
*
 1s (1)[  2 1 ]1s(1)dv(1)   2s (1)[  2 1 ]2s(1)dv(1)
*
位能的部分,expansion over (-2/r1, -2/r2),類似情形得
2
2
*
 1s (1)( r1 )1s(1)dv(1)   2s (1)( r1 )2s(1)dv(1),
*
which is equal to
2
2
*
 1s (2)( r2 )1s(2)dv(2)   2s (2)( r2 )2s(2)dv(2)
*
排斥能部分,1/r12, occurs in four two-electron
integrals:
1
1
*
*
{  1s (1)2 s (2)( )1s (1)2s (2)dv(1)dv(2) 
2
r12
1
  2s (1)1s (2)( r12 )2s(1)1s(2)dv(1)dv(2)
*
*
1
   1s (1)2s (2)( )2 s (1)1s (2)dv(1)dv(2)
r12
*
*
1
   2 s (1)1s (2)( )1s (1)2 s (2)dv(1)dv(2)}
r12
*
*
1
   1s (1)2 s (2)( )1s (1)2s (2)dv(1)dv(2)
r12
*
*
1
   1s (1)2s (2)( )2 s (1)1s (2)dv(1)dv(2)
r12
*
*
Thus, the average energy value is:
1 2
2
*
E1   1s (1)[  1 ]1s (1)dv(1)   1s (1)[  ]1s (1)dv(1)
2
r1
3
*
1 2
2
*
  2s (1)[  1 ]2s (1)dv(1)   2s (1)[  ]2s (1)dv(1)
2
r1
*
1
   1s (1)2s (2)( )1s (1)2s (2)dv(1)dv(2)
r12
*
*
1
   1s (1)2s (2)( )2s (1)1s (2)dv(1)dv(2)
r12
*
*
The first two terms gives the average energy of He+ in its
1s state, and the second pair gives the energy of He+ in 2s
state, thus the final becomes,
E1  E1s  E2 s  J  K
3
Where J and K represent the last two integrals.
The integral J denotes electrons 1 and 2 as
being in ‘charge clouds’ described by 1s*1s
and 2s*2s, respectively. The operator 1/r12
gives the electrostatic repulsion energy
between these two charge clouds.因為這些是
電子雲的斥力,所以J值是正的,稱 coulomb
integral.
K is called an ‘exchange integral’ because the
two product fNs in the integrand differ by an
exchange of electrons.
K值代表the interaction between an electron ‘distribution’
described by 1s*2s, and another electron in the same
distribution. (這些分部只是數學函數,並非實質可畫出的分部
情形)。
當r1 and r2 are both smaller or
Both larger, then the fN 1s(1)2s(1)1s(2)2s(2)
will be positive. But when one r value is
smaller than R and the other is larger
than R, 此情形代表這兩電子
on opposite sides of the nodal
surface, then 1s(1)2s(1)1s(2)2s(2)
is negative.
These positive or negative contributions
to K are weighted by the fN of 1/r12
綜觀之,K值若大時,應為正值,若為負值應會是很小,可忽略。
• Since the integral K is positive, we can see
that from the derived equation that the
triply degenerate energy level lies below
the singly nondegenerate one, the
separation between them being 2K.
What is the meaning of “Fermi hole”
In triplet state the space part of the wavefN:
(1 / 2 )[1s(1)2s(2)  1s (2)2s(1)]
What would happen if these two electrons are collide ?
Which means that the coordinate of ‘1’ electron is equal to
‘2’ electron, that is, 1s(1)=1s(2), and 2s(1)=2s(2), so that,
the above equation should be vanished. That means, this
situation should never happen. This situation is called
“Fermi hole”, and it is built into any wavefN that is properly
antisymmeterized.
如果是 singlet state (symmetric space fN)
1
[1s(1)2s(2)  1s(2)2s(1)]
2
當兩電子的 coordinate 相同時, 1s(1)=1s(2),
wavefN 是否亦應該 vanish, (應與spin 無關才對),
這就是所謂的 coulomb hole. However, 在這個
wavefN下卻沒看到 vanish. Why? It is due to our
independent-electron approximations (that is, the
electrons were attracted by the nucleus but
somehow did not repel each other).
然而在 triplet state wavefN確實可由 Fermi hole 的存在,而感受
到兩電子間的距離的確較長,可是實際上的計算結果,發現其與
basis fN的設定是有很大的關係的,當basis fN越好時,其r12值卻
越小,(參照Table 5-1,列出不同 wavefN的情況下,所計算出的結
果,其 1/r12的平均值在 singlet and triplet states下有不同的趨勢。
所以wavefN的選擇有很重要的決定性)這說明了一個必須注意到
的現象,那就是:
Warning: Usable approximations to eigenfNs are
very useful in understanding, predicting, and
calculating observable phenomena. But one
must always be aware of the possibility of
significant differences existing between the real
system and the mathematical model for that
system.
suppose we take ordinary independent-electron
wavefN as our initial approximation for the
helium atom:
1s(1)1s (2)  8 /  exp( 2r1 ) 8 /  exp( 2r2 )
They are correct only if electrons 1 and 2 do not ‘see’
each other via a repulsive interaction. However, this is
not the true case. How are we going to correct it?
The Self-Consistent Field, Slater-Type
Orbitals, and the Aufbau Principle
一般作法是we can approximate this
repulsion by saying that electron 1 ‘sees’
electron 2 as a smeared out, timeaveraged charge cloud rather than the
rapidly moving point charge which is
actually present. The initial description for
this charge cloud is just the absolute
square of the initial atomic orbital occupied
by electron 2: [1s(2)] 2.
Our approximation now has electron 1 moving in the field of a
positive nucleus embedded in a spherical cloud of negative
charge by electron 2. Thus, for electron 1, the positive charge
is ‘shielded’ or ‘screened’ by electron 2. Hence electron 1
should occupy an orbital that is less contracted about the
nucleus. Let us write this new orbital in the form:
1s ' (1)   3 /  exp( r1 )...........(1)
Where ξ is related to the screened nuclear charge seen by
electron 1. Next we turn to electron 2, which we now take
to be moving in the field of the nucleus shielded by the
charge cloud due to electron 1, now in its expanded
orbital. Just as before, we find a new orbital of form (1)
for electron 2. Now, however, ξwill be different because
the shielding of the nucleus by electron 1 is different from
what was in our previous step.
• We now have a new distribution for electron 2,
but this means that we must recalculated the
orbital for electron 1 since this orbital was
appropriate for the screening due to electron 2 in
its old orbital. After revising the orbital for
electron 1, we must revise the orbital for electron
2. This procedure is continued back and forth
between electrons 1 and 2 until the value of
ξconverges to an unchanging value (under the
constraint that electrons 1 and 2 ultimately
occupy orbitals having the same value of ξ).
Then the orbital for each electron is consistent
with the potential due to the nucleus and the
charge cloud for the other electron: the electrons
move in a “self-consistent field” (SCF).
The result of such a calculation is a wavefN in much closer
accord with the actual charge density distributions.
However, because each electron senses only the timeaveraged charge cloud of the other in this approximation, it
is still an independent-electron treatment.
• The hallmark(主要特徵) of independent electron
treatment is a wavefN containing only a product
of one-electron fNs. There are no fNs of, say,
r12, which would make wavefN depend on the
instantaneous distance between electrons 1 and
2.
• Atomic orbitals that are eigenfNs for the oneelectron hydrogenlike ion are called
hydrogenlike orbitals. Since these orbitals has
radial nodes which increased the complexity in
solving integrals in quantum chemical
calculations.
Much more convenient are a class of modified
orbitals called Slater-type orbitals (STOs).
These differ from their hydrogenlike
counterparts in that they have no radial nodes.
Angular terms are identical in the two types of
orbital. The unnormalized radial term for an
STO is
R(n, Z , s )  r ( n 1) exp[ ( Z  s )r / n]
n is the principal quantum number
Z is the nuclear charge in atomic units.
s is a ' screening constant' , which has the fN of reducing
the nuclear charg Z ' seen' by an electron.
Slater constructed rules for determining the values of s that
would match the orbitals obtained from SCF calculation.
These rules, appropriate for electrons up to the 3d level, are:
(1) The electrons are divided into 1s 2s,2 p 3s,3 p 3d groups.
(2)The shielding constant s for an orbital associated with any
of the above groups is the sum of the following
contributions:
(a)比該電子更外層的電子不具遮蔽效應。
(b) 來自同層的每個電子遮蔽貢獻為0.35 (except 0.30 in the 1s
group).
(c) 來自內面一層的 s or p orbital,每個電子的貢獻為0.85, d
orbital 電子的貢獻為 1.00, 來自內面更深一層(內面第二層)
以上的電子遮蔽貢獻,不管s, p, or d orbitals,每個電子的貢獻
皆為 1.00.
For example, N atom with ground state configuration
1s22s22p3, the 2s and 2p orbital would have the same radial
part of STOs.
(n  2, Z  7, s  4  0.35  2  0.85  3.1)
 R2 s , 2 p (2,7,3.1)  r ( 21) exp[ (7  3.1)r / 2]
 r exp( 1.95r )
while for the 1s orbital, n  1, Z  7, s  0.30
R1s (1,7,0.30)  exp( 6.7r )
Slater-type orbitals are very frequently used in quantum
chemistry because they provide us with very good
approximaiton to SCF atomic orbitals with almost no effort.
The STO have no radial nodes, so it loses some
orthogonality, although the angular terms still give
orthogonality between orbitals having different l or m
quantum numbers. Therefore, STOs differing only in their n
quantum number are nonorthogonal, such as 1s, 2s,
3s,….are nonorthogonal, 2pz,3pz,… or, 3dxy, 4dxy,… are
nonorthogonal. Therefore, problem would arises if one
forgets about its nonorthognality when making certain
calculations.
Aufbau principle (building up principle): the orbital ordering:
1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 5d 4f 6p 7s 6d 5f ….
However, there is no fixing rule, it depends on the Z value of
the atoms.
Explain briefly the observation that the energy
difference between the 1s22s1 (2S1/2) state and
1s22p1 (2p1/2) state for Li is 14,904 cm-1, whereas
for Li2+ the 2s1 (2S1/2) and 2p1(2p1/2) state are
essentially degenerate. (They differ only by 2.4
cm-1).
(hint: consider the hydrogen-like orbitals but not
the Slator orbitals for the Li atom, the penetration
of 2s is larger than the 2p, so the orbital energy
of 2s is ? than 2p)
Combined spin-orbital angular
momentum for one-electron ions
j  l  s, l  s  1, , l  s . (spin - orbit coupling)
for each j value, there are (2 j  1 ) different space
orientatio ns (degenerat e in energy) which correspond
to ( 2 j  1 ) components on z axis (m j  j , j  1,, j ).
The magnitude of this coupling angular momentum is
j 
j ( j  1), or,
j ( j  1) a.u.
The symbol for the state,
2 s 1
Lj
( L  0 , 1, 2 , 3,..., refer to S, P, D, F )
such as the p1 electron configurat ion, the state symbol
1
3 1
L  1  P , s  , j  l  s, l  s  1,..l  s  ,
2
2 2
So, the state symbols are,
2
P3 / 2 , and, 2P1/ 2 , and the total angular momentum for the
former is j 
3 3
15
(  1) 
,
2 2
2
3
which has 2( )  1  4 space orientatio ns (degenerac ies)
2
3 1
1
3
and the z - component is m j  , , , 
2 2
2
2
same calculatio n for 2P1/ 2 state?
Russell – Saunders Coupling Scheme (For nonequivalent electrons)(適用於同量子數電子不在同一軌域)
用於多電子的spin-orbit coupling is weak，因此把多個電子
的 orbital momenta 一起合起來，再與多個電子合起來的
spin momenta 相互作用
 L   1   2   , S  s1  s2  
J  L  S , L  S  1 ,, L  S
Clebsch – Gordan series :描述兩個angular momentum向量加
成的可能值，如：
 1 與  2 其向量加成的可能值為 L
L  1   2 , 1   2  1 ,
 1   2  2 ,,  1   2
同理 J  L  S , L  S  1 , L  S
如：電子組態 H e  2 p 1 3 p 1 其 3D term 的 J 值有哪些？
L  2 , S  1 J  3 , 2 , 1 ;

但若 2S term 的 J 值  L  0 , S 
3
D3 , 3D2 , 3D1
1
1
,  J  ,  2S 1
2
2
2
一般而言，若L > S, 則其 J 值個數與 term symbol 的
multiplicity 相同，如上例的 3 D term case .
2
S term
但若 L < S，則就不依此法則，如上例
但 for equivalent electrons, 如 p2, (c: 1s22s22p2),就沒有
那麼單純,須考慮各種 micro states,合適保留,不合適刪除,
參考 Lowe, p156, equivalent electrons.
一般Russell – Saunders Coupling
Scheme 適用於較輕的原子 (其中
電子本身spin-orbit coupling較小)，
對於較重原子，就不適用，改為
考慮每個原子間的 j-j coupling 。
在R –S Coupling中，如果有2個電子以上，則先算2
個電子的 L，S，再和第三個電子重新做加成得到
最終的L與S，再以L+S方式求出另J值；若適合 j-j
scheme 再以另方式求出J值 ◦
1
1
1
例如 電子組態為 2 p 3 p 4 p term symbols ?
則先求出2個電子的
 1  1,  2  1 ,  3  1
s1 
L   1   2 ,  1   2  1, ...........  2 , 1, 0 S   1 , 0
1
1
1
, s 2  , s3 
2
2
2
現 couple 第三個 e -，  3 與 L   2 give L  3, 2, 1 ; with L   1
give L  2, 1, 0; , with L'  0 give L  1, therefor e, L  3, 2, 2, 1, 1, 1, 0
得 F, D, D, P, P, P,S
3 1
,
2 2
第三個電子 spin 的 coupling S  1
2 與 S   0， give 1
2
3 1 1
 S  , , ; 2 S  1  4，2，2
2 2 2
 可能的 term symbols : 4F , 2F , 4D , 2D , 4P , 2P , 4S , 2S
與 S   1， give
由 R-S coupling 求J
4
F,
2
F,
4
2
D,
D,
4
P,
2
P,
L  3. S 
3
2
L  3. S 
L  2. S 
J 
9 7 5 3
,
, ,
2 2 2 2
1
2
J 
3
2
J 
1
2
3
L 1, S 
2
1
L 1, S 
2
L2, S 
 4F9 , 4F7 , 4F5 , 4F3
7
5
,
.
2
2
7 5 3 1
, , ,
2 2 2 2
5
,
2
5
J 
,
2
3
J 
,
2
J
2

2
2
F7 , 2F5
2
2
2
2
 4D7 , 4D5 , 4D 3 , 4D 1
2
2
2
3
 2D 5 , 2D 3
2
2
2
3 1
,
 4P5 , 4P3 , 4P1
2
2
2
2 2
1
 2P3 , 2P1
2
2
2
2
4
2
S,
S,
3
L0, S 
2
J
1
L0, S 
2
3
2
1
J 
2


2
4
S1
S3
2
2
如果是heavy atom時，R-S coupling不適用，必須用 j-j
coupling。每一個電子只考慮total angular momentum (spin,
orbit加成) j，每個電子再與每個電子的相互作用，此時的
L   1   2  ..... , S  s1  s2  ... 就相對不太重要了。如
P 組態  1  1 ,  2  1 ;
2
1
1
s1  , s 2 
2
2
3
1
其每個電子的可能total angular momentum為 or
2
2
，彼此互相coupling的情況為：
pure
j-j
R-S
pure
1
S0
E
1
D2
3
3
P2
P1
3
P0
Si
Ge Sn
Pb
如圖所示，原子越大，
其能階正確值越趨近於
由 j – j 所計算的結果。
3
2
3
j1 
2
1
j1 
2
1
j1 
2
j1 
3
2
1
j2 
2
3
j2 
2
1
j2 
2
j2 

 J  3 , 2 , 1, 0 


 J  2, 1



 J  2, 1



 J  1, 0

基本上，重原子
以 J 為較可信的
quantum number.
R-S 在heavy atom 雖然不適，但其推演
出來的 term symbol 仍然有效，因為其
能階的順序仍然沒變，雖然能階差有
明顯變化。
重原子的 spin – orbit coupling
為什麼重要?
電子繞核運動，若對電子而言(站在
電子上)如同核繞電子而轉，核電荷
繞轉會在電子上產生一磁通量，與
該電子的spin magnetic moment 相互
作用，而磁通量的大小與該核電荷
成正比，因此 heavy atom，其 spin –
orbit coupling是相當顯著的◦
核
-
e
Zeeman effect
電子繞軌域運轉所產生的magnetic moment 在外加
磁場的作用下, 產生不同作用能使原本的單一能階
分裂成多條，其譜線由一條變成多條。
作用能 E   z B   B m B
e
( Bohr magneton )
其中  B   e   
2 me
加磁場
+1
P
0
-1
所以 p orbital 在沒有磁場作用
下譜線不會split,但在磁場作用下
卻split 成3條。
S
 line 為 circularly  polarized ,  line 為 plane  polarized
0
£m £k £m
• Terms where in J contains contributions from
both L and S have Zeeman splittings other than
one or two times the normal value, depending
on the details of the way L or S are combined.
The extent to which a term member’s energy is
shifted by a magnetic field of strength B is
E  g e M j B
 e : Bohr magneton (  B ), g : Lande g factor,
J ( J  1)  S ( S  1)  L( L  1)
2 J ( J  1)
g  1, when S  0 and J  L, and g  2, when L  0 and J  S
g  1
For the 3P2 term, S  1, L  1, J  2, so, g  1.5
Indicating that half of the z-component of angular momentum
is due to the orbital motion, and half is due to spin (which is
double weighted in its effect on magnetic moment).
事實上 electron spin magnetic moment 亦會和外
加磁場作用產生Zeeman splitting, 例如ESR
（Electron Spin Resonance) 就是利用此原理 。
其中 E    z B  g e  B ms B
for electron
g e  2.0023, called g factor
若是有多個電子，則以 M S  mS1  mS 2  mS3   代之
Nuclear spin magnetic moment亦會和外加磁場作用產生
Zeeman splitting, 例如NMR（Nuclear Magnetic Resonance)
就是利用此原理 。 但其
 e
B 
2M p
因為是質子，重量很大，所以作用能很小，所需要外加磁場
很大。
Angular momentum for manyelectron atoms (Equivalent electrons)
p 2 electron configurat ion
p 4 electron configurat ion
3
P2,1,0
1
D2
1
S0
, which is the ground state?
the energy order of these states?
Molecular term symbols
• 依據分子的對稱性所屬的point group, 在此
point group 的 character table下的
representation,其符號,若為小寫,即代表該
分子軌域的名稱,若為大寫,則可以代表該分
子在某一電子組態下的 term symbol.例如
H2O分子在基態下的電子組態及其分子軌域
名稱及term symbol表示方法。