Unfortunately, real atoms (including hydrogen atoms) are not very well described by the simple theory. There are more things contribute to the actual atomic structure:

  1. Higher order effects
    • Spin-orbit coupling
    • Relativistic
    • Nuclear spin
    • Zeman
  2. Identity particles, that leads to the Pauli exclusion principle.
  3. Multiple electrons, the electromagnetic fields associated with the electrons can interact.


Spin-Orbit Coupling

Electrons have spin, thus angular momentum, as electrons orbiting the nuclei. Those two motions generate magnetic fields that can interact. So there is some total angular momentum:

J→ = L→ + S→

The total orbital angular momentum J→ is actually quantized, that introduces a new quantum number j. One can show that J = (j (j + 1))1/2 ħ. Like angular momentum, the z component is quantized Jz = mj ħ, so that introduces yet another quantum number mj = -j, -j+1, ..., j-1, j. mj can not be bigger than j since it is just a component of j.

If angular momentum quantum number l = 0, then all we have is the electron spin contribution and so j = 1/2. And then if l > 0, then j = l ± 1/2. Thus ml and ms cease to be “good” quantum numbers and are replaced by j and mj. Now the good quantum numbers become n, l, j, mj.

So the spin-orbit effect, the coupling between “orbital angular momentum” and  “an electron’s spin momentum” causes quantum states that would otherwise have the same energy to split, that can change the degeneracy. The  splitting is significant. You will have quantum states that have the same principle quantum number and very, very different energies. The spin-orbit effect can be quite substantial and it cannot be neglected.

The Pauli Exclusion Principle

Consider the wave equation for two identical particles in a box,

(-ħ2 / 2m) [∂2Ψ/∂x12 + ∂2Ψ/∂x22] V Ψ = 0

where ψ = ψ(x1, x2) is the stationary wave function and V = V(x1, x2) is the potential energy function where the time portion is dropout. If the particles are interchanged, then the probability density of the system should be unchanged, p(x1, x2) = p(x2, x1). The particles are identical. We shouldn’t be able to tell whether they’ve been interchanged.

For this to be the case, it turns out that the wave function ψ has to be either symmetric or anti-symmetric.

ψ(x1, x2) = ψ(x2, x1)    <--- symmetric, or
ψ(x1, x2) = -ψ(x2, x1)   <--- anti-symmetric

since we have the postulate

p(r→, t) dV = Ψ*(r→, t) Ψ(r→, t) dV = |Ψ(r→, t)|2 dV

We can classify particles into Fermions and Bosons:

Spin quantum numberWave functionParticles
FermionsHalf-integerAnti-symmetric1. Electrons,
2. Protons,
3. Neutrons, or
4. Atoms or molecules with odd numbers of electrons, protons, neutrons
BosonsIntegerSymmetric1. Photons
2. Atoms or molecules with even numbers of electrons, protons, neutrons

Let’s assume that we can write the potential energy function as the sum of the potential energy acting on particle one and particle two: V = V(x1) + V(x2), then we can write the wave function for the two particles as equal to the product of two individual wave functions, where α and β denote quantum states of each particle.

ψα,β(x1, x2) = ψα(x1) ψβ(x2)

We can now construct a wave function that satisfy the symmetry requirement, where s and a stand for symmetric and anti-symmetric:

ψs,a(x1, x2) = 1/√2 [ ψs(x1) ψa(x2) ± ψs(x2) ψa(x1) ]

This clearly satisfies the symmetry requirement ψ(x1, x2) = ±ψ(x2, x1), when we switch the particles. However, if the two particles are in the same quantum state and the wave function is anti-symmetric, then:

ψ(x1, x2) = ±ψ(x2, x1)
⟹ ψ must equal to 0

This means fermions can not occupy the same quantum state at the same time. When it comes to multi-electron atoms, there is no two electrons of a multi-electron atom can assume the same quantum state and the structure of all multi-electron atoms are derived from this fact.



Multi-Electron Atoms

For multi-electron atoms, exact solutions of the wave equation are not possible, we depend on numerical solutions or other approximation solutions.

First of all, a single potential V(r) may be associated with each electron. Thus we may write wave function for a particular electron as below. Each electron may be described in terms of the same set of quantum numbers used for the single electron hydrogen atom (n, l, j, mj if spin-orbit coupling is important). The restrictions on the quantum numbers are the same.

ψn,l,ml,ms = Rn,l(r) Θl,ml(θ) Φml(φ) σms

If the Rn,l were computed and plotted versus r, we would find that the curves corresponding to a common value of n would exhibit a maximum at nearly the same value of r. We thus refer to electrons with the same n as occupying the same shell.

Shell: K L M N O P ...
    n: 1 2 3 4 5 6 ...

Spin-orbit coupling can not be ignored in multi-electron atoms and the energy is a function of both n and l. We refer to each l associated with a given n as a subshell.

Subshell: s p d f g h ...
       l: 0 1 2 3 4 5 ...
s: sharp
p: principle
d: diffuse
f: fundamental

To simultaneously specify shell and subshell give number and letter, for example: 3p, 4s, etc.

Ground level is the atom in its minimum energy configuration. You build up an atom by noting that electrons must occupy the lowest energy available subshells. The maximum number of electrons per subshell is determined by using the exclusion principle. The energy ordering of subshells has been determined historically experimentally.

When atoms have their outermost shell is filled, that is all the electrons that can fit in subshells of that outermost shell are filled. We call those atoms the noble gases: helium, neon, argon, krypton, xenon. They have high ionization potentials.

On the other hand if there’s only one electron in the outermost shell (lithium, sodium, potassium, rubidium, cesium). These have very low ionization potentials and are easier to remove an electron from an atom, so that atom is more likely to react.

Spectroscopic Term Classification

This is a notation that allows us to see right away what the structure of the atom is. We refer to the electronic energy levels of an atom as term, and denote it using a term symbol of the form:

2S+1 L J=L+S

L stands for angular momentum, S = 0, P = 1, D = 2, F = 3, G = 4, H = 5, … The subscript stands for the total angular momentum J = L + S. The leading superscript is called the multiplicity, which is the number of different values of J which are possible for given values of L and S, smaller of 2S + 1 or 2L + 1. For a given configuration and values of L and S, the different J’s may have slightly different energies. For this reason we call such states multiplets.

Multiplicity should not be confused with degeneracy, which here arises only from quantization of the total angular momentum along some axis. Recall Jz = mj ħ, and mj = -j, -j+1, ..., j-1, j. In the absence of an external field, this has no effect on energies.

gJ = 2 J + 1
gmultiplet = ∑i gJi


Real Behavior of Diatomic Molecules

A diatomic molecule can assume a number of different electrons states, just like an atom, however there are difference:

AtomsThe electron wavefunctions are approximately spherically symmetric and
L (orbital angular momentum) is quantized.
Diatomic moleculeNot spherically symmetric.
Only the component of L (orbital angular momentum) and S (spin angular momentum) along the internuclear axis is quantized.

The component of L along the inter nuclear axis is LAB = ±Λ ħ, where Λ is another quantum number, we use Greek capital letter to designate it.

Symbol: Σ Π Δ Φ Γ ...
     Λ: 0 1 2 3 4 ...

Like orbital angular momentum, spin angular momentum is quantized along the internuclear axis: SAB = ±Σ ħ, where Σ = -S, -S + 1, …, S – 1, S.

As with atomic systems, the multiplet structure is a consequence of the spin-orbit coupling. If T0 is the term without coupling T0 = 104 ~ 105 cm-1, then electron term energy would be approximately Te = T0 + CΛΣ, where C is coupling constant C = 1 ~ 103 cm-1.

For multiplet components, if Λ = 0, the number of components will be 1; if Λ > 0, the number of components will be 2S + 1.

Spectroscopic Term Classification

Like atoms, we classify the electronic energy levels of a diatomic molecule by a term symbol of the form

2S+1 Λ Λ+Σ

The main character is Λ, it will be Σ = 0, Π = 1, Δ = 2, Φ = 3, Γ = 4, … The leading superscript 2S + 1 is the multiplicity, although there is no splitting if Λ = 0.

Symmetry Properties

Symmetry properties are important in molecules. If the diatomic molecule has identical nuclei, say H2, N2, O2, etc. If you were to reflect the electron coordinates, the electronic amplitude

remains unchangedThe subscript g is used, which means even (gerade in German).
Σg, Πg, …
sign changedThe subscript u is used, which means odd (ungerade in German).
Σu, Πu, …

Depending on whether the electronic amplitude changes sign with reflection in a plane through the internuclear axis, Σ states (i.e. Λ = 0) are designated as Σ+ (no change) or Σ (change).

Total Energy

The total energy of a molecule as a sum of the electronic energy T0, plus the vibrational energy G(v), plus the rotational energy F(J).

Ttot = T0 + G(v) + F(J)

For the electronic state:

  • if Λ = 0, no orbital angular momentum, then the electronic degeneracy is ge = 2S + 1
  • if Λ > 0, with spin-orbit splitting, then ge = 2
  • if Λ > 0, without spin-orbit splitting, then ge = 2 (2S + 1)
  • vibrational degeneracy is unity gv = 1
  • rotational degeneracy is gJ = 2J + 1


Polyatomic Molecules

For a body made up of n atoms, there are 3n degrees of freedom of motion.

TranslationTakes up 3 degrees of freedom.
Rotation1. If all the atoms in the molecules are in a straight line (linear), then there are 2 degrees of rotational freedom just like the diatomic molecule.
2. For a general non-linear molecule there are 3 degrees of rotational freedom.
VibrationTakes 3n – 5 or 3n – 6 degrees of motion. For example:
1. Linear triatomic (CO2): 3 * 3 – 5 = 4 modes
2. Nonlinear triatomic (H2O): 3 * 3 – 6 = 3 modes

The electronic structure of the molecule is determined by quantum mechanics just like pure atoms. The electrons from the atoms form orbitals and these combine when making a molecule into structures that are somewhat related to the atomic orbitals. These play an important role in how molecules form and how they react, chemically.

Types of Bonds

IonicThe charge on the atoms is mostly localized on the atom.
So that you have a negative portion and a positive portion of the molecule.
They are just held together by the electrostatic force between those.
CovalentElectron orbits overlap.
1. Single bond – two electrons, one from each atom.
2. Double bond – four electrons, two from each atom.
3. Triple bond – six electrons, three from each atom.

Organic Molecules

The octet rule says that atoms tend to gain, lose, or share electrons so as to have eight electrons in their outer electron shell. The number of electrons determine how molecules are likely to react.

AlkanesSingle bondsEthane
C – C
AlkenesDouble bondsEthene
C = C
Alkynes
(Acetylene)
Triple bondsEhyne
C ≡ C
Arynes
(Aromatic compounds)
RingsBenzene
⌬

Functional groups can replace hydrogens, an unlimited range of compounds can be formed.

Quantum Numerical Methods

Quantum methods involve solving the Schrodinger wave equation H ψ = W ψ, which is an eigen equation, and there are three common solutions methods.:

  1. Ab Initio – most accurate, slow
  2. Density Functional – less accurate, faster
  3. Semi-empirical – least accurate, fastest


My Certificate

For more on Real Atoms and Molecules, please refer to the wonderful course here https://www.coursera.org/learn/quantum-mechanics


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