Quantum Mechanical Model of Atom

Full Notes

Hydrogen Atom and the Schrödinger Equation

In 1926, Erwin Schrödinger introduced a mathematical model that treated electrons as wave-like entities. His equation doesn’t give a simple path for an electron to follow. Instead, it gives us a wave function (Ψ) – a mathematical expression that holds all the information about an electron’s behavior.

• Ψ is the wave function (not physically observable, but essential),
• H is the Hamiltonian operator (total energy of the system),
• E is the energy of the electron.

The real insight comes from Ψ², which gives the probability density – essentially, where we’re most likely to find the electron in a given region of space. This region is what we call an orbital.

In the quantum mechanical model, electrons are no longer viewed as "orbiting" the nucleus but are instead "spread out" in a region of space, with non-perfect boundaries.

2.6.1 Orbitals and Quantum Numbers

Just like every house has an address, every electron in an atom is uniquely described by a set of four quantum numbers. These numbers tell us the shell, shape, orientation, and spin of the orbital an electron occupies.

We’ll explore them here one by one:

1. Principal Quantum Number (n)

This number tells us the main energy level or shell of the electron.

• Values: n = 1, 2, 3, …
• Larger n → higher energy level → farther from the nucleus
• Number of orbitals in a shell = n²
• Maximum electrons in a shell = 2n²

2. Azimuthal Quantum Number (l) (orbital angular momentum)

This defines the three-dimensional shape of the orbital and the sub-shell the electron is in.

• For a given value of n, l can be any integer value from 0 to n−1.
• For example - when n = 1, then l = 0
  when n = 2, then l can be 0 or 1
  when n = 3, then l can be 0, 1 or 2

Number of orbitals in a subshell = 2l + 1

3. Magnetic Quantum Number (m or m_l)

This tells us the orientation of the orbital in space.

• Values: m = −l to +l
• For Example: For p-orbitals (l = 1), m = −1, 0, +1
  three possible values for m means the electron orbitals can ‘point’ in three directions within the p sub-shell. These are described as p_x, p_y, p_z

4. Spin Quantum Number (s or m_s)

Electrons have a property called spin – commonly described as being described as the ‘spin’ or rotation of the electron. There are only two possible values for an electrons spin in a given orbital, given values of +½ and −½.

• Values: +½ or −½ (often represented with up or down arrows).
• No two electrons in the same orbital can have the same spin.
• Each orbital can hold maximum two electrons with opposite spins.

Summary of Quantum Numbers and Their Meaning

• Principal Quantum Number (n):
  Defines the shell of the electron. It determines the size of the orbital and mainly its energy.
• Azimuthal Quantum Number (l):
  Defines the subshell and the shape of the orbital. For each value of n, there are n subshells.
  Each subshell contains:
  • 1 s-orbital (l = 0)
  • 3 p-orbitals (l = 1)
  • 5 d-orbitals (l = 2)
• Magnetic Quantum Number (m_l):
  Describes the orientation of the orbital in space. Each subshell has (2l + 1) orbitals corresponding to different values of m_l.
• Spin Quantum Number (m_s):
  Refers to the spin orientation of an electron: either +½ or −½.

2.6.2 Shapes of Atomic Orbitals

What Is an Orbital?
An orbital is a region in space around the nucleus where there is a high probability of finding an electron. But it's important to know the orbital wave function (ψ) given by the Schroedinger equation itself has no physical meaning — it’s just a mathematical tool.

However, the square of the wave function – ψ² – does have meaning.

It tells us the probability density: how likely we are to find an electron at a particular point in space and at a particular distance (r) from the atoms nucleus.

• ψ² is high ⇒ high chance of finding the electron there.
• ψ² is low or zero ⇒ low or no chance of finding the electron.

Radial plots show this graphically.

Understanding radial Plots for 1s and 2s Orbitals

Part (a): Orbital Wave Function ψ(r) vs. Distance r (Left Graph)
This graphs shows how the wave function ψ(r) changes as you move away from the nucleus (i.e., increasing distance r, in nanometers).

• (1s Orbital – n=1, l=0):
  The wave function ψ(r) starts at a very high value at r = 0 (the nucleus). It decreases rapidly as distance increases. This indicates that the electron is most likely to be found very close to the nucleus.
• (2s Orbital – n=2, l=0):
  ψ(r) starts high at the nucleus but quickly drops to zero (this is a node - where there is no chance of finding the electron). Then it rises again to a smaller peak before slowly tapering off.

Interpretation: The shape of ψ(r) is different for 1s and 2s, which reflects their different energy levels and spatial electron distributions.

Part (b): Probability Density ψ²(r) vs. Distance r (Right Graph)
This graph is far more meaningful – it shows ψ²(r), the probability density of the electron, i.e., how likely it is to find the electron at different distances from the nucleus.

• (1s Orbital):
  Maximum probability is very close to the nucleus - as r increases, ψ² drops sharply, meaning the electron is highly concentrated near the nucleus.
• (2s Orbital):
  Starts high near the nucleus, drops to zero at the node (no electron found there), Then rises again to a peak – this is where the electron is likely found in the outer shell. Beyond that, the probability density decreases with distance.

Interpretation:
The 2s orbital has one node (as expected for an orbital where n – 1 = 1). The 1s orbital has zero nodes.

Key Takeaways

• Wave function (ψ) tells you nothing directly physical — but ψ² (the square) tells you the actual probability density.
• 1s orbitals are most dense near the nucleus, with no node.
• 2s orbitals have a node, meaning a region where the electron will never be found.
• These plots visualise how electrons are distributed in atoms – a foundation for understanding atomic structure.

Comparing 1s and 2s Orbitals
The 1s orbital has maximum probability density close to the nucleus, and it decreases smoothly as you move away. The 2s orbital behaves differently:

• Starts high near the nucleus
• Drops to zero (a node)
• Then increases again to a smaller peak
• Finally decreases to near zero at far distances

Key point:
A node is a point or surface where the probability of finding the electron is zero. The number of nodes = n – 1

Example:
2s has 1 node
3s has 2 nodes
4s has 3 nodes

How Do We Visualise Orbitals?

There are two helpful ways we can visualise the orbital shapes predicted by probability densities: charge cloud diagrams and boundary surface diagrams.

• Charge Cloud Diagrams
  These show a cloud of dots. Denser dots = higher probability of finding the electron.
• Boundary Surface Diagrams
  These show a clear boundary enclosing 90% probability of where the electron is likely to be.

For s-orbitals, the shape is always a sphere – same in all directions (spherically symmetric).

Important Note:
We don’t draw a boundary for 100% probability because technically, the electron has some chance (however tiny) of being found at any distance from the nucleus.

Boundary surface diagrams for orbital shapes

s-Orbitals

• Shape: Spherical
• Found in every energy level.
• Larger n → bigger size.
• Symmetric in all directions.

p-Orbitals

• Shape: Dumbbell
• Starts from n = 2.
• Three orientations: p_x, p_y, p_z — each along a different axis.

d-Orbitals

• Shape: Four-lobed (cloverleaf) or donut-shaped
• Found from n = 3 onward.
• Five orientations.

f-Orbitals

Complex shapes. Found from n = 4 onward. Seven orientations.

All these above shapes help explain bonding patterns, molecular geometries, and chemical reactivity.

2.6.3 Energies of Orbitals

Not all orbitals at the same principal level have equal energy – in multi-electron atoms, energy depends on both n and l.

(n + l) Rule

• Orbitals with lower (n + l) are filled first.
• If (n + l) is same, lower n fills first.
• Order of filling (increasing energy):
  1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s …

Filling of Orbitals in Atom

Nature always seeks the most stable, lowest energy arrangement. This guides the order in which orbitals are filled:

Aufbau Principle

• Electrons fill orbitals starting from the lowest energy level, moving upward step by step.
• Think of it as pouring water into a stack of bowls — you fill the lowest one first before moving to the next.

Pauli Exclusion Principle

• No two electrons in an atom can have the same set of four quantum numbers.
• This means an orbital can hold:
  • At most two electrons
  • And they must have opposite spins

Hund’s Rule of Maximum Multiplicity

• When multiple orbitals of the same energy are available (like the three p orbitals), electrons prefer to occupy them singly first, with parallel spins.
• This minimizes electron-electron repulsion and makes the atom more stable.

Electronic Configuration of Atoms

We now have all the tools to write electronic configurations — a shorthand for showing where electrons live inside an atom.

Example:

• Hydrogen: 1s¹
• Carbon: 1s² 2s² 2p²
• Oxygen: 1s² 2s² 2p⁴

Use superscripts to show the number of electrons in each orbital.

Configurations often use noble gas shorthand:

Example: Na = [Ne] 3s¹

Stability of Completely Filled and Half-Filled Subshells

Some configurations are unexpectedly stable:

For Example:

• Chromium (Z = 24): [Ar] 4s¹ 3d⁵
• Copper (Z = 29): [Ar] 4s¹ 3d¹⁰

Why? Because half-filled and completely filled subshells offer:

1. Symmetrical Distribution
   Electrons are spread evenly → reduced repulsion → greater stability
2. Exchange Energy
   Unpaired electrons with the same spin can exchange places — this creates a more stable arrangement due to quantum mechanical effects.

So atoms sometimes "borrow" an electron from one orbital to stabilize another.

Summary

• Schrödinger’s equation describes electrons as waves and leads to orbitals.
• Ψ² gives probability density for where electrons are likely found.
• Four quantum numbers define shells, subshells, orientation and spin.
• Orbital energies in multi-electron atoms follow the (n + l) rule.
• Aufbau, Pauli and Hund’s rules determine electron configurations.