Chapter 8 (Lecture 11)
Atomic Orbitals
The energy depends on the principal quantum number alone while wave function depend on the quantum numbers 𝑛, 𝑙 𝑎𝑛𝑑 𝑚.
Thus, the energy level 𝐸𝑛 has a degenaracy of 𝑛2 .
s-Orbitals
The 𝑠 orbital spherically symmetric and and they have 𝑛 radial nodes. Table shows real Hydrogenic orbitals in atomic units.
p and d Orbitals
The lowest energy solution deviating from spherical symmmetry is 2p orbitals. Consider 𝑙 = 1 functions:
𝜓210 =
1
𝑟
4√2𝜋
𝑟𝑒 −2 𝑐𝑜𝑠𝜃
And
𝜓21±1 = ∓
1
𝑟
4√2𝜋
𝑟𝑒 −2 𝑠𝑖𝑛𝜃𝑒 ±𝑖𝜙
The function 𝜓210 is real and contains 𝑟𝑐𝑜𝑠𝜃 = 𝑧 in chemical application this is designated as 2𝑝𝑧 orbital
𝜓2𝑝𝑧 =
1
4√2𝜋
𝑟
𝑧𝑒 −2
The function 𝜓21±1 are complex. Making use of equle Formula
𝑒 ±𝑖𝜙 = 𝑐𝑜𝑠𝜙 ± 𝑖𝑠𝑖𝑛𝜙
We obtain
𝜓2𝑝𝑥 =
𝜓2𝑝𝑦 =
1
√2
1
√2
(𝜓21−1 − 𝜓211 ) =
(𝜓21−1 + 𝜓211 ) =
1
𝑟
4√2𝜋
1
4√2𝜋
𝑥𝑒 −2
𝑟
𝑦𝑒 −2
The orbital 𝜓320 is like𝜓210 are real function it is known in chemistry 𝑑𝑧 2 orbital and can be expressed as
𝜓3𝑑
Periodic Table
𝑧2
=
1
81√6𝜋
𝑟
(3𝑧 2 − 𝑟 2 )𝑒 −3
A block of the periodic table of elements is a set of adjacent groups. The term appears to have been first used (in French) by
Charles Janet. The respective highest-energy electrons in each element in a block belong to the same atomic orbital type. Each
block is named after its characteristic orbital; thus, the blocks are:
s-block
p-block
d-block
f-block
g-block (hypotetical)
The block names (s, p, d, f and g) are derived from the quality of the spectroscopic lines of the associated atomic orbitals: sharp,
principal, diffuse and fundamental
The following is the order for filling the "subshell" orbitals, according
to the Aufbau principle, which also gives the linear order of the
"blocks" (as atomic number increases) in the periodic table:
1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p,
...
The "periodic" nature of the filling of orbitals, as well as emergence
of the s, p, d and f "blocks" is more obvious, if this order of filling is
given in matrix form, with increasing principal quantum numbers
starting the new rows ("periods") in the matrix. Then, each subshell
(composed of the first two quantum numbers) is repeated as many
times as required for each pair of electrons it may contain. The result
is a compressed periodic table, with each entry representing two
successive
elements:
(http://en.wikipedia.org/wiki/Block_%28periodic_table%29)
1s
2s
2p
2p
2p
3s
3p
3p
3p
4s
3d
3d
3d
3d
3d
4p
4p
4p
5s
4d
4d
4d
4d
4d
5p
5p
5p
6s
4f
4f
4f
4f
4f
4f
4f
5d
5d
5d
5d
5d
6p
6p
6p
7s
5f
5f
5f
5f
5f
5f
5f
6d
6d
6d
6d
6d
7p
7p
7p
The closest shell to the nucleus is called the "1 shell" (also called "K shell"), followed by the "2 shell" (or "L shell"), then the "3
shell" (or "M shell"), and so on farther and farther from the nucleus. The shells correspond with the principal quantum numbers (1,
2, 3, 4 ...) or are labeled alphabetically with letters used in the X-ray notation (K, L, M, …).
Each shell can contain only a fixed number of electrons: The 1st shell can hold up to two electrons, the 2nd shell can hold up to
eight (2 + 6) electrons, the 3rd shell can hold up to 18 (2 + 6 + 10), and the 4th shell can hold up to 32 (2 + 6 + 10 + 14) and so on.
Since electrons are electrically attracted to the nucleus, an atom's electrons will generally occupy outer shells only if the more
inner shells have already been completely filled by other electrons. However, this is not a strict requirement: Atoms may have two
or even three incomplete outer shells. (See Madelung rule for more details.)
The electrons in the outermost occupied shell (or shells) determine the chemical properties of the atom; it is called the valence
shell.
Each shell consists of one or more subshells, and each subshell consists of one or more atomic orbitals.
Subshells
Each shell is composed of one or more subshells, which are themselves composed of atomic orbitals. For example, the first (K)
shell has one subshell, called "1s"; the second (L) shell has two subshells, called "2s" and "2p"; the third shell has "3s", "3p", and
"3d"; the fourth shell has "4s", "4p", "4d" and "4f"; the fifth shell has "5s", "5p", "5d", and "5f" and can theoretically hold more
but the "5f" subshell, although occupied in actinides, is not filled in any element occurring naturally. The various possible
subshells are shown in the following table:
Subshell label l
Max electrons
2(2l+1)
Shells containing it
Historical name
s
0 2
Every shell
sharp
p
1 6
2nd shell and higher
principal
d
2 10
3rd shell and higher
diffuse
f
3 14
4th shell and higher
fundamental
g
4 18
5th shell and higher (theoretically) (next in alphabet after f)[3]
An atom's electron shells are filled according to the following theoretical constraints:
Subshell Shell
Shell Subshell
max
max
name name
electrons electrons
K
1s
2
2s
2
L
M
N
2
2+6=8
2p
6
3s
2
3p
6
3d
10
4s
2
4p
6
4d
10
2 + 6 + 10
= 18
2+6+
+ 10 + 14
= 32
4f
14
Magnetic Effects on Atomic Spectra—The Normal Zeeman Effect
The Dutch physicist Pieter Zeeman showed the spectral lines emitted by atoms in a
magnetic field split into multiple energy levels. It is called the Zeeman effect.
Consider the atom to behave like a small magnet. Think of an electron as an orbiting
circular current loop of I = dq / dt around the nucleus. The current loop has a magnetic
𝑒
moment μ = IA and the period T = 2πr / v. 𝜇⃗ = − 𝐿⃗⃗
where L = mvr is the
2𝑚
magnitude of the orbital angular momentum. The dipole has a potential energy:
⃗⃗
𝑉𝐵 = −𝜇⃗. 𝐵
The angular momentum
⃗⃗ causes a precession of
𝐵
Where 𝜇𝐵 =
𝑒ℏ
2𝑚
is
aligned
with
the
magnetic
moment,
𝜇𝑧 =
𝑒ℏ
𝑚 = −𝜇𝐵 𝑚𝑙
2𝑚 𝑙
and
the
torque
between
𝜇⃗
and
is called a Bohr magneton.
The potential energy is quantized due to the magnetic quantum number mℓ.
𝑉𝐵 = −𝜇𝑧 𝐵 = +𝜇𝐵 𝑚𝑙 𝐵
When a magnetic field is applied, the 2p level of atomic hydrogen is split into three different energy states with energy difference
of ΔE = μBB Δmℓ.
mℓ
Energy
1
E + μBB
0
E
−1
E− μBB
A transition from 2p to 1s