Many-Electron Atoms
元素週期表是依照原子序 (Z，原子核內的質子數目，也是核外電子的數目) 排
列的：從左往右原子序逐一遞增；具有相同化學性質的元素排在同一行 (同一
族，family)。因此，氟 (fluorine，with nine electrons) 與並排的氧 (oxygen，with
eight electrons)、氖 (neon, with ten electrons)，電子數目相差 1，但化學性質差異
很大；而氟和排在下方的氯 (chlorine，with 17 electrons)，電子數目相差 8，但化
學性質卻相近。
The periodicity in chemical properties controlled by Z or the number of electrons in
neutral atoms leads to a shell structure for electrons. Each time when a shell becomes
full of electrons, the remaining electrons enter the next shell; accordingly, in the
periodic table we move from one row to the next. The electrons occupying the last
shell of an atom are called the valence electrons and those in earlier shells core
electrons. Elements from the same column, belonging to the same family, of the
periodic table have the same number of valence electrons. The similarity in properties
of the elements from the same column leads us to think that the electrons responsible
for determining chemical characteristics are those located in the outermost shell,
namely the valence electrons.
§3.1 The orbital approximation and the electron configuration
The Schrodinger equation for a many-electron atom is highly complicated because all
the electrons interact with one another. One very important consequence of these
interactions is that orbitals of the same value of n but different values of l are no
longer degenerate in a many-electron atom. Scientists adopt a simple approach based
on the hydrogenic atomic orbitals to get the electronic structure of a many-electron
atom－the orbital approximation. Each electron is supposed to occupy its own orbital,
and the wave function of an n-electron atom is written as
r1 , r2 ,  , rn    (r1 ) (r2 )    (rn )
where the individual orbitals,  (ri ) , are the hydrogenic orbitals with the nuclear
charge modified by the presence of all other electrons. These hydrogenic orbitals,
 (ri ) , are single-electron wave functions, usually called atomic orbitals of the atom.
The orbital approximation allows us to express the electronic structure of an atom by
reporting its electron configuration. The electron configuration of an atom describes
all the atomic orbitals that are populated with electrons. Several rules are used to
decide how the atomic orbitals are populated (filled) with electrons.
1. The aufbau principle or the building up principle (an algorithm for predicting the
ground-state electron configuration of an atom) states that one fills the lower energy
orbitals with electrons first. For the ground state, electrons fill the sub-shells in the
order given by Klechkowsky’s rule:
(i) The order of filling orbitals proceeds from the lowest available value for the sum n
+ l;
(ii) When two combinations have the same n + l value, the one with the smaller value
of n is filled first.
2. The Pauli exclusion principle: No more than two electrons may occupy any
given orbital, and when two electrons are in the same orbital they must be spin paired.
3. Hund’s rule of maximum multiplicity tells us how to handle the population of
degenerate orbitals－the most stable configuration for several electrons occupying
degenerate AOs is the one containing the largest number of parallel electron spins.
(1) When two electrons occupy the same orbital (same region of space) in an atom,
they repel each other, with a Coulombic energy of repulsion, ΠC, per pair of electrons.
(2) In addition, there is an exchange energy, ΠE, arising from purely QM
considerations. This energy depends on the number of possible exchanges between
two electrons with the same energy and the same spin. The energy involved in such an
exchange of parallel electrons stabilizes (lower the energy of) an electronic state,
favoring states with more parallel spins (Hund’s rule).
In summary, an atom in its ground state adopts a configuration with the greatest
number of unpaired electrons. Electrons with parallel spins behave as if they have a
tendency to stay well apart, and hence repel each other less. In essence, the effect of
spin correlation is to allow the atom to shrink slightly, so the electron-nucleus
interaction is improved when the spins are parallel.
例1. Two 2p electrons of a carbon atom, with the electron configuration 1s22s22p2,
can be placed in the p orbitals in three ways:
(i) ↑↓
(ii) ↑
↓
(iii) ↑
↑
Each of these distributions corresponds to a state of a particular energy. State (i)
involves Coulombic energy of repulsion, and its energy is higher than that of the other
two states by ΠC.
In state (iii), there are two possible ways to arrange the electrons; in other words,
there is one exchange of parallel electrons. State (iii) is lower in energy than state (ii)
by ΠE.
例2. Analyze the three 2p electrons of a nitrogen atom (with the electron
configuration 1s22s22p3) distributed differently in the p orbitals.
例3. Analyze the four 2p electrons of an oxygen atom (with the electron configuration
1s22s22p4) distributed differently in the p orbitals.
When the orbitals are degenerate, both Coulombic and exchange energies favor
unpaired distributions over paired distributions. However, if there is a difference in
energy between the orbitals involved, this difference, together with the Coulombic
and exchange energies, determines the final configuration, with the configuration of
lowest energy expected as the ground state; energy minimization is the driving force.
The approach that one subshell (e.g. s, p, d) is filled before another has any electrons
is inappropriate in some transition elements, because the 4s and 3d (or the higher
corresponding levels) are so close in energy that the sum of the Coulombic and
exchange terms is nearly the same as the energy difference between the 4s and 3d
orbitals. The lowest energy configuration for chromium (Cr) is 4s13d5 and not 4s23d4,
because the exchange energy of the half-filled 3d sub-shell with electrons of parallel
spins is sufficient to compensate for the 4s3d promotion energy. A similar effect is
responsible for the completion of the filled 3d sub-shell of copper, leading to the
4s13d10 configuration lying lowest in energy. This kind of inversion is more often
observer in the second transition metal series, e.g., Nb(5s14d4), Mo(5s14d5),
Ru(5s14d7), Rh(5s14d8), Pd(5s04d10), and Ag(5s14d10).
* Diamagnetism and paramagnetism
Consider the magnesium atom (Z = 12) with the ground state configuration
1s22s22p63s2. Since all of the electrons are paired, the spin magnetic moment of one
electron in such a pair cancels the other so that overall the magnesium atom does not
have an intrinsic spin magnetic moment. Systems with such a property are called
diamagnetic. The ground state configuration of phosphorous (Z = 15) is ~3s23p3. The
three 3p electrons each occupy a different orbital with identical spins to satisfy
Hund’s rule. Their spin magnetic moments do not cancel and the total magnetic
moment for phosphorus is non-zero. This atom is thus paramagnetic.
§3.2 Shielding and the effective nuclear charge
In many-electron atoms, each electron acts as a shield for electrons farther from the
nucleus, reducing the attraction between the nucleus and the more distant electrons.
Slater proposed a empirical method to calculate the shielding constant S (the mean
effect exercised by the inner electrons) for a specific electron.
(1) The atom’s electronic configuration is rearranged in order of increasing quantum
n and l, and grouped as follows,
(1s) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d) (4f) (5s, 5p) (5d) and so on.
(2) Electrons in groups to the right in this list do not shield electrons to their left.
(3) The shielding constant S for electrons in these groups can be determined.
(i) For ns and np valence electrons:
a. Each electron in the same group contributes 0.35 to the value of S for each other
electron in the group. (A 1s electron contributes 0.30 to S for another 1s electron.)
b. Each electron in n-1 groups contributes 0.85 to S.
c. Each electron in n – 2 or lower groups contributes 1.00 to S.
(ii) For nd and nf valence electrons:
a. Each electron in the same group contributes 0.35 to the value of S for each other
electron in the group.
b. Each electron in groups to the left contributes 1.00 to S.
(4) The effective nuclear charge on the selected electron is Zeff = Z* = Z – S.
Justification for Slater’s rules comes from the electron probability curves for the
orbitals. The s and p orbitals have higher probabilities near the nucleus than do d
orbitals of the same n recall that the wave function of a d orbital varies as r2 close to
the nucleus, where a p orbital varies as r). Therefore, the shielding of 3d electrons by
(3s, 3p) electrons is calculated as 100% effective; shielding of 3s or 3p electrons by
(2s, 2p) electrons is estimated as 85% effective because the 3s and 3p orbitals have
regions of significant probability close to the nucleus and not completely shielded by
(2s, 2p) electrons.
例1. Use Slater’s rules to calculate the shielding constant and effective charge of a 2p
electron in an oxygen atom.
例2. Use Slater’s rules to calculate the shielding constant and effective charge of a 3d
and 4s electron in a nickel atom.
例3. Use Slater’s rules to calculate the shielding constant and effective charge of a 5s,
5p, and 4d electron in a tin atom.
例4. Use Slater’s rules to calculate the shielding constant and effective charge of a 7s,
5f, and 6d electron in a uranium atom
* It is noted that the effective nuclear charge felt by the valence electrons is much
smaller than that felt by the core electrons.
An electron in an s orbital, for example, a 3s orbital, is more likely to be found close
to the nucleus than an electron in a p orbital of the same shell. Hence an s electron has
a greater penetration through inner shells than a p electron, and should experience less
shielding and be more tightly bound than a p electron.
element
Z
orbital
Zeff*
He
C
2
6
1s
1s
2s
2p
1.6875
5.6727
3.2166
3.1358
* The data are calculated from the
wave functions obtained by numerical
solution of the Schrodinger equation
for the atom.
(Atkins’ Physical Chemistry, 9e, Table
9.2)
(Atkins’ Physical Chemistry, 9e, Fig9.20)
Although the quantum number n is most important in determining the energy,
quantum number l must also be included in calculating the energy in atoms having
more than one electron. As a result of shielding and other subtle interactions between
electrons, exclusive reliance on n to rank orbital energies, which works for
one-electron species, holds only for orbitals with lowest values of n in many-electron
atoms. The following diagram provides unambiguous electron configurations for
elements hydrogen (Z = 1) to vanadium (Z = 23):
… 5p < 6s ~ 5d ~ 4f < 6p < 7s ~ 6d ~ 5f < 7p …
* For orbitals designated as comparable in energy, e.g., 4s ~ 3d, the actual order
depends on which other orbitals are occupied. The sequence of orbitals pictured above
increases in the approximate order of n 
1
l   l ,3  , where δ= 1 for an f-orbital.
2
cf. http://www.colorado.edu/physics/2000/applets/as.html
§3.3 Term symbols
普化課本中常用一個電子組態來描述位於基態 (ground state) 的多電子原子，譬
如 C 是 1s22s22p2，然而這些多電子原子的光譜線卻比這個電子組態所預測的
多。換句話說，一個電子組態包含不只一個能量狀態；科學家用“term symbol”
來描述這些產生不同光譜線的能量狀態。
A term symbol gives three pieces of information:
(i) A upper-case letter (for example, S, P, or D) indicates the total orbital angular
momentum quantum number, L.
* The convention of using lower-case letters to label orbitals and upper-case letters to
label overall states applies through spectroscopy, not just to atoms.
(ii) The left superscript in the term symbol gives the multiplicity of the term.
(iii) The right subscript in the term symbol is the value of the total angular momentum
quantum number, J.
例 1. 位於基態的 C 原子電子組態是 1s22s22p2，不去管已經填滿的 subshell，
二個電子分佈在三個 p 軌域上可以有下列 15 種方式。從 3.1 節介紹的
Coulombic energy 和 exchange energy，不難看出這 15 種分佈方式的能量有高
有低：3 種方式是二個電子在相同的 p 軌域能量最高；6 種方式是二個電子在不
同的 p 軌域，自旋相反，能量其次；6 種方式是二個電子在不同的 p 軌域，自
旋相同，能量最低。
ml
1
0
-1
↑↓
↑↓
↑↓
ML
MS
(= Σ ml)
(= Σ ms)
2
0
A
0
0
A
-2
0
A
↑
↑
1
1
↑
↓
1
0
↓
↑
1
0
B
↓
↓
1
-1
B
B
↑
B
A
↑
↑
-1
1
↑
↓
-1
0
↓
↑
-1
0
B
-1
-1
B
0
1
B
↑
A
↑
↓
0
0
↓
↑
0
0
↓
↓
0
-1
B
B
決定這個電子組態包含那些 term symbol 的步驟如下：
(A1) 找出 ML 的最大值  2，那麼 term symbol 的 upper-case letter 是 D；
(A2) 找出 ML 最大值時，MS 的最大值  0，那麼 term symbol 的 multiplicity
是 1 (= 2 × 0 + 1)；
(A3) term symbol 的 right subscript，J 是 2。
*(A4) 1D 事實上包含 5 種分佈。2 是 ML 的最大值，包含 2、1、0、-1、-2；0
是 MS 的最大值，只包含 0；因此 5 × 1 = 5。從表格中除取 5 個符合這個條件的
分佈。
(B) 在剩下的 10 個分佈中重複上述的步驟  3P2, 1, 0。
ML 的最大值是 1，MS 的最大值也是 1，所以 3P 沒有問題。這兩個最大值相加
等於 2，相減等於 0，J 的數值包含和與差以及其間差距 1 的數字。
1 是 ML 的最大值，包含 1、0、-1；1 也是 MS 的最大值，只包含 1、0、-1；因
此 3P 事實上包含 9 種分佈。從表格中除取 9 個符合這個條件的分佈。
(C) 最後剩下的分佈用上述方法找出的 term symbol 是 1S0。
The total orbital angular momentum, L
When several electrons are present, it is necessary to judge how their individual
orbital angular momenta add together. 當一個電子位於某一個軌域時，軌域的角動
量量子數只告訴我們角動量的大小；二個電子軌域角動量的和與個別軌域角動量
之間的關係是 (the Clebsch-Gordan series)
L = l1 + l2, l1 + l2 -1,
…. , ｜l1 – l2｜
化學家用大寫的英文字母表示不同的 L：
L
0
1
2
3
4
5
6
…
S
P
D
F
G
H
I
…
以 C 原子 (~2p2) 為例，L = 2，1，0；對應到 D、P、S。
The terms differ in energy on account of the different spatial distribution of the
electrons and the consequent differences in repulsion between them.
例題中我們利用角動量量子數 (角動量的大小) 與磁量子數 (角動量在外加磁場
方向的分量) 的關係，從個別電子磁量子數的和 (ML) 找出 L。
The multiplicity, the total spin angular momentum, S
多個電子的 total spin angular momentum 與個別的 spin momentum 之間同樣滿
足 the Clebsch-Gordan series
S = s1 + s2, s1 + s2 -1,
…. , ｜s1 – s2｜
例題中我們也是利用各個電子自旋角動量在外加磁場方向的分量 (ms)，加總
(MS) 再找出 S。
Term symbol 的左上角標示的是 multiplicity － 2S + 1；multiplicity 等於 1 稱
為 singlet state，表示所有電子都是兩兩配對，配對的電子擠在同一個軌域上。
multiplicity 等於 2 稱為 doublet state；multiplicity 等於 3 稱為 triplet state，以
C 原子 (~2p2) 為例，表示這二個電子自旋方向相同，分布在不同的軌域上
(Pauli’s exclusion principle)。
The energy differences between different multiplicity states result from the different
effects of spin correlation.
The total angular momentum, J
Term symbols with different J lie at different energies on account of the magnetic
spin-orbital interaction.
For atoms of low atomic number, the spin-orbital coupling is weak, then the total
angular momentum is calculated following the Russell-Saundes coupling scheme: all
the orbital angular momenta of the electrons couple to give a total L (all orbital
angular momenta are operating cooperatively) and all the spins are similarly coupled
to give a total S, then the two kinds of momenta couple to give a total J.
In heavy atoms, those with high Z, where the spin-orbital coupling is large, the
Russell-Saunders coupling scheme fails. In this case, the individual spin and orbital
momenta of the electrons are coupled into individual j values; then these momenta are
combined into a total angular momentum, J. This scheme is called jj-coupling
scheme.
A summary of the types of interaction that
are responsible for the various kinds of
splitting of energy levels in atoms. For light
atoms, magnetic interactions are small, but
in heavy atoms they may dominate the
electrostatic (charge-charge) interactions.
(Atkins’ Physical Chemistry, 9e, Fig9.32)
例 2. 如果位於激發態的 C 原子電子組態是 1s22s22p13s1，寫出包含的 term
symbol。
 3P2, 1, 0, 1P1
2p, ml
3s, ml
ML
MS
0
(= Σ ml)
(= Σ ms)
↑
↑
1
1
A
↑
↓
1
0
A
↓
↑
1
0
↓
↓
1
-1
A
↑
↑
0
1
A
↑
↓
0
0
A
↓
↑
0
0
↓
↓
0
-1
A
↑
-1
1
A
1
0
-1
↑
B
B
↑
↓
-1
0
↓
↑
-1
0
↓
↓
-1
-1
A
B
A
當我們使用 the Russell-Saunders coupling scheme 得到 term symbols 之後，判斷
這些 term symbols 的能量高低有三個規則：
(i)
The lowest energy term is that which has the greatest spin multiplicity.
(ii) For terms with the same spin multiplicity, the term which has the greatest orbital
angular momentum lies lowest in energy.
(iii) For terms with the same S and L. the order in energy is given by the following:
If the unfilled subshell is exactly or more than half full, the term with the highest J
value has the lowest energy.
If the unfilled subshell is less than half full, the term with the lowest J value has the
lowest energy.
Exercises
ex1. Derive the term symbols for a configuration of (a) d2 (b) p3 (c) f1d1 (d) d3 (e)
2s12p1 (f) 2p13d1.
ex2. 將前一題找到的 term symbols 依能量高低排序。
§3.4 Periodic properties of atoms
More than a century ago when Mendeleev proposed a tabular form of classification of
the chemical elements, there was no information available concerning the structure of
the atom (nucleus and electrons), neither a theoretical model capable of describing
their properties. This tabular form collected 63 elements only, and arranged the
elements having similar chemical properties in the same column, a chemical family. A
valuable aspect of the arrangement of atoms on the basis of similar electronic
configurations within the periodic table is that an atom’s position provides
information about its properties.
Ionization energies
The minimum energy necessary to remove an electron from a many-electron atom in
the gas phase is the first ionization energy, I1, of the element. The variation of the first
ionization energy through the periodic table is shown below:
(Atkins’ Physical Chemistry, 9e, Fig9.22)
The general trend across a period is an increase in ionization energy as the nuclear
charge increases. The transition metals have less dramatic differences in ionization
energies, with the effects of shielding and increasing nuclear charge more nearly in
balance. Overall the trends are toward higher ionization energy from left to right in
the periodic table (the major change), and toward lower ionization energy from top to
bottom (a minor change).
例. 說明 Li ~ Ne 的第一游離能。由低而高，但在 Be、B 之間與 N、O 之間卻
高低顛倒
Hint: Because boron is the first atom to have an electron in a higher energy 2p orbital
that is shielded somewhat by the 2s electrons, boron’s 2p electron is more easily lost
than the 2s electrons of beryllium. The fourth electron of oxygen shares an orbital
with one of the three previous 2p electrons, and the repulsion between the paired
electrons (Πc) reduced the energy necessary to remove an electron from oxygen.
Electron affinities
The electron affinity, Eea, is the energy released when an electron attached to a
gas-phase atom.
(Miessler, et al., “Inorganic Chemistry”, 5e, Fig2.13)
* Systems with eight valence electrons
Among the different families of the periodic table, the inert gases (also called the rare,
or noble gases) have very little chemical reactivity. The origin of this chemical
inertness is the saturation of the valence shell. Such chemical inertness is found too in
the monatomic ions which have a saturated valence shell, for example Cl + e  Cl- or
Na  Na+ + e. These ions have a considerably reduced chemical reactivity when
compared to the atoms from which they are derived. (Ksp ?!)
Electronegativity
The halogens can attain a position of chemical stability (i.e., saturated valence shell)
by the addition of a single electron to their valence shell. This property, seen for the
isolated atom, also manifests itself in molecules, so a halogen atom will always have a
tendency to attract electrons from a neighboring atom in order to complete its valence
shell. Electronegativity is the tendency of an atom in a molecule to attract bonding
electrons from neighboring atoms.
(Miessler, et al., “Inorganic Chemistry”, 5e, Fig2.14)
The peaks and valleys match for all three graphs because the electron configurations
match.
Atomic size
As the nuclear charge increases, the electrons are pulled in toward the center of the
atom, and the size of any particular orbital decreases. On the other hand, as the
nuclear charge increases, more electrons are added to the atom, and their mutual
repulsion keeps the outer orbitals large. The interaction of these two effects,
increasing nuclear charge and increasing number of electrons, results in a gradual
decrease in atomic size across each period.
For hydrogenic atoms the radius of an orbital, defined as the point where a maximum
is found for the radial distribution function, was given approximately by n2a0/Z. An
approximate expression for the radial part of the orbital wave function of
many-electron atoms was proposed by Slater.
Rn,l r   N r / a 
n 1
With

e
 Z *r
na
, where N is a normalization constant.

d 2
2
r  Rn ,l r   0 , the radius of an AO in a many-electron atom is given by
dr
n2a0/Z*. Since as n increases Z* decreases, the inner orbitals are expected to be much
more contracted than the outer orbitals. So for phosphorus
r1s = 3.6 pm,
r2s = r2p = 19.6 pm,
r3s = r3p = 99.5 pm.
The valence electrons are the most weakly bound electrons, and the electrostatic
energy between them and the nucleus is roughly proportional to –Z*/rvalence. Therefore
the valence electrons are very sensitive to outside perturbations, such as the approach
of another atom. Atomic radius gives information about the atomic polarizability, a
measure of the deformation experienced by the electron cloud under the influence of
an external electric field. Simple arguments suggest that the polarizability increases as
rvalence3.
* The values of nonpolar covalent radius are based on bond distances in nonpolar
molecules. (Miessler, et al., “Inorganic Chemistry”, 5e, Table2.8)
【The orbital approximation－atomic orbitals】
In the orbital approximation the wave function of a many-electron atom is represented
by a product of single-electron functions, r1 , r2 ,  , rn    (r1 ) (r2 )    (rn ) . These
single-electron wave functions,  (ri ) , are the hydrogenic orbitals and can be written
as a product of radial and angular parts
 n,l ,m (ri )  Rn,l ri   Yl ,m  i , i 
The angular part is identical to that found for the hydrogen atom and is determined by
the two quantum numbers l and m. The radial part, set by the values of the two
quantum numbers n and l, is determined by both the nuclear charge and the presence
of all other electrons. Its functional form is thus expected to be rather different from
the radial part of hydrogenic atoms.
In many-electron atoms, the energy of an atomic orbiral depends on two quantum
numbers, n and l. AOs with the same n and l remain degenerate and such a group of
orbitals is called a sub-shell. (The term “shell” is reserved for all orbitals with the
same value of n.) There are two general rules which allow, at least in part, the
energetic ordering of AOs:
(i) For the same value of l the orbital energy increases with increasing n.
(ii) For the same value of n the orbital energy increases with increasing l.
These two rules are not sufficient to completely fix the energetic ordering. However,
for all atoms the lowest five orbitals are found in the following order:
ε1s < ε2s < ε2p < ε3s < ε3p
Beyond this group the situation gets complicated, and the actual state of affairs
depends on the atom being considered, and sometimes its oxidation state.
It is convenient to divide the electrons of a many-electron atom into two groups－core
electrons and valence electrons. Most often the valence electrons arte regarded as all
those in orbitals associated with the largest value of the principal quantum number.
The remainders make up the core. In the case of partially filled d sub-shell all of the
electrons occupying the nd and (n+1)s orbitals are considered to be valence electrons,
since these two orbitals are close in energy. It is the valence electrons which
determine the chemical properties of the elements.