Electronic Structure of Atoms: Quantum Mechanics and Atomic Orbitals

Quantum Mechanics and Atomic Structure

Introduction to Quantum Mechanics

Quantum mechanics provides the theoretical framework for understanding the electronic structure of atoms. Erwin Schrödinger developed a mathematical treatment that incorporates both the wave and particle nature of matter, allowing for a more complete description of atomic behavior.

• Wave Function (ψ): The wave function, designated by the lowercase Greek letter psi (ψ), describes the quantum state of an electron in an atom.

• Probability Density: The square of the wave function, $\psi^2$, gives a probability density map indicating the likelihood of finding an electron at a particular location at any instant in time.

• Example: High dot density in a probability map corresponds to a high probability of finding an electron in that region.

Quantum Numbers and Atomic Orbitals

Quantum Numbers

Solving the Schrödinger equation for atoms yields a set of wave functions called orbitals, each with a specific energy and spatial distribution of electron density. Each orbital is described by a set of quantum numbers:

• Principal Quantum Number (n): Indicates the main energy level or shell.

• Angular Momentum Quantum Number (l): Determines the shape of the orbital.

• Magnetic Quantum Number (ml): Specifies the orientation of the orbital in space.

s Orbitals

s orbitals are characterized by the following properties:

• The value of $l$ for s orbitals is 0.

• They are spherical in shape.

• The radius of the sphere increases with the value of $n$.

• For a given value of $n$, there is one s orbital.

• Example: 1s, 2s, and 3s orbitals increase in size as $n$ increases.

p Orbitals

p orbitals have distinct features:

• The value of $l$ for p orbitals is 1.

• They have two lobes with one angular node between them.

• For a given value of $n$, there are three p orbitals (px, py, pz).

• Example: The three p orbitals are oriented along the x, y, and z axes.

d Orbitals

d orbitals are more complex:

• The value of $l$ for d orbitals is 2.

• They have two angular nodes.

• For a given value of $n$, there are five d orbitals.

• Example: dxy, dyz, dzx, dx^2-y^2, dz^2.

Radial Nodes in s Orbitals

When examining the probability of finding an electron at various distances from the nucleus, s orbitals exhibit radial nodes:

• Radial Node: A region where the probability of finding an electron is zero.

• The number of radial nodes increases with the principal quantum number $n$.

• Example: The 2s orbital has one radial node, while the 3s orbital has two.

Energy Levels and Electron Configurations

Energy of Orbitals in Hydrogen

In the hydrogen atom, all orbitals with the same principal quantum number $n$ are degenerate, meaning they have the same energy.

• Degeneracy: Orbitals with the same $n$ value have equal energy in hydrogen.

Energy of Orbitals in Multi-Electron Atoms

In atoms with more than one electron, electron-electron repulsion and shielding effects cause the energies of orbitals to differ:

• Shielding: Inner shell electrons shield outer electrons from the full charge of the nucleus.

• Screening: s electrons are more effective at screening than d electrons.

• As a result, the energy order of orbitals changes compared to hydrogen.

Spin Quantum Number

In the 1920s, it was discovered that electrons possess an intrinsic property called spin, which affects their magnetic behavior:

• Spin Quantum Number (ms): Has only two allowed values: $+\frac{1}{2}$ and $-\frac{1}{2}$.

• Each electron in an atom is uniquely described by four quantum numbers: $n$, $l$, $m_l$, and $m_s$.

Pauli Exclusion Principle

The Pauli Exclusion Principle states:

• No two electrons in the same atom can have identical sets of quantum numbers.

• This principle explains the arrangement of electrons in atomic orbitals.

Electron Configurations

Electron configuration describes the distribution of electrons among the orbitals of an atom:

• Each component consists of:

  • A number denoting the energy level ($n$)

  • A letter denoting the type of orbital ($s$, $p$, $d$, $f$)

  • A superscript denoting the number of electrons in those orbitals

• The periodic table can be used as a guide to write electron configurations for each element.

• Example: The electron configuration of carbon is $1s^2 2s^2 2p^2$.

Periodic Table and Electron Configuration

The arrangement of elements in the periodic table reflects the order in which atomic orbitals are filled:

• Groups correspond to similar valence electron configurations.

• Periods correspond to increasing principal quantum number $n$.

Summary Table: Quantum Numbers and Orbitals

Quantum Number	Symbol	Possible Values	Physical Meaning
Principal	n	1, 2, 3, ...	Energy level, size of orbital
Angular Momentum	l	0 to n-1	Shape of orbital (s, p, d, f)
Magnetic	ml	-l to +l	Orientation of orbital
Spin	ms	+1/2, -1/2	Spin direction of electron

Additional info: The notes also reference reading assignments and quiz schedules, which are not included in the study guide but indicate coverage of Chapters 6 and 7 (Electronic Structure and Periodic Properties).