Undergraduate
  1. General chemistry  1.1. Introduction to Chemistry  1.2. Atomic Structure  1.2.1. Subatomic particles  1.2.2. Atomic Model  1.2.3. Quantum numbers  1.2.4. Electron Configuration  1.2.5. Periodic trends  1.2.6. Isotopes and their applications  1.3. Chemical bond  1.3.1. Ionic bond  1.3.2. Covalent bonds  1.3.3. Metal Bond  1.3.4. Hybridization  1.3.5. Molecular geometry and VSEPR theory  1.3.6. Bond polarity and dipole moment  1.4. Stoichiometry  1.4.1. Mole concept  1.4.2. Empirical and molecular formula  1.4.3. Limiting reactant and excess reactant  1.4.4. Percent Yield and Purity  1.4.5. Dilution calculation  1.5. States of matter  1.5.1. Gas Laws  1.5.2. Matter and Intermolecular Forces  1.5.3. Solid and Crystal Structures  1.5.4. Phase diagram  1.5.5. Plasma and supercritical fluid  1.6. Chemical reactions  1.6.1. Types of Chemical Reactions  1.6.2. Balancing Chemical Equations  1.6.3. Thermochemical reactions  1.6.4. Reaction energy profiles  1.7. Solutions and Mixtures  1.7.1. Concentration units  1.7.2. Syndrome properties  1.7.3. Solubility and Solubility Rules  1.7.4. Henry's Law and Raoult's Law  1.7.5. Partition coefficient  1.8. Chemical equilibrium  1.8.1. Law of mass action  1.8.2. Le Chatelier's principle  1.8.3. Reaction quotient  1.8.4. To use this online calculator for Equilibrium Constant, enter Equilibrium Constant (Eq.) and hit the calculate button. Here is how the Equilibrium Constant calculation can be explained with given input values -> 0.05 = 0.05/0.  1.8.5. Acid-base balance  1.9. Acids and bases  1.9.1. Arrhenius, Brønsted–Lowry, and Lewis definitions  1.9.2. pH and pOH  1.9.3. Acid-base titration  1.9.4. Buffer Solution  1.9.5. Acid-base indicator  1.9.6. Polyprotic Acids and Bases  1.10. Electrochemistry  1.10.1. redox reactions  1.10.2. Galvanic and Electrolytic Cells  1.10.3. Nernst equation  1.10.4. Electrolysis  1.10.5. Faraday's laws of electrolysis  1.11. Thermodynamics  1.11.1. Laws of Thermodynamics  1.11.2. Enthalpy, entropy and Gibbs free energy  1.11.3. Spontaneity of reactions  1.11.4. Thermodynamic cycles (Hess's law, Carnot cycle)  1.12. Kinetics  1.12.1. Rate law and reaction order  1.12.2. Activation Energy and Catalyst  1.12.3. Collision theory  1.12.4. Reaction mechanism  1.12.5. Transition state theory
  2. Organic chemistry  2.1. Structure and relationships  2.1.1. Hybridisation in Carbon Compounds  2.1.2. Resonance and aromatization  2.1.3. Molecular orbitals  2.1.4. Inductive and mesomeric effects  2.2. Hydrocarbons  2.2.1. Hydrocarbons  2.2.2. Alkene  2.2.3. Alkynes  2.2.4. Aromatic compounds  2.2.5. Cycloalkanes and Conformation  2.3. Functional Group  2.3.1. Alcohols and phenols  2.3.2. Ethers and epoxides  2.3.3. Aldehyde and ketone  2.3.4. Carboxylic acids and derivatives  2.3.5. Amin  2.3.6. Esters and amides  2.3.7. Thiols and sulfides  2.4. Stereoscopic  2.4.1. Isomerism in Stereochemistry  2.4.2. Chirality and optical activity  2.4.3. Structural analysis  2.4.4. Diastereomers and Enantiomers  2.5.1. Substitution reactions  2.5.2. Elimination reactions  2.5.3. Addition reactions  2.5.4. Radical reactions  2.5.5. Rearrangement reactions  2.5.6. Pericyclic Reactions  2.6. Spectroscopy and structural analysis  2.6.1. Infrared Spectroscopy  2.6.2. Nuclear magnetic resonance spectroscopy  2.6.3. Mass Spectrometry  2.6.4. UV-visible spectroscopy  2.6.5. X-ray crystallography  2.7. Polymer chemistry  2.7.1. Addition and Condensation Polymers  2.7.2. Polymerization mechanism  2.7.3. Biodegradable Polymer  2.7.4. Conducting and smart polymers
  3. Inorganic chemistry  3.1. Coordination chemistry  3.1.1. Ligands and coordination compounds  3.1.2. Crystal field theory  3.1.3. Molecular Orbital Theory in Coordination Compounds  3.1.4. Jahn–Taylor distortion  3.2. Main Group Chemistry  3.2.1. Alkali and alkaline earth metals  3.2.2. Halogens and Noble Gases  3.2.3. Transition metals and their complexes  3.2.4. Lanthanides and Actinides  3.3. Solid state chemistry  3.3.1. Crystal lattices and unit cells  3.3.2. Defects in solids  3.3.3. Electrical and magnetic properties  3.3.4. Superconductivity  3.4. Bio-inorganic chemistry  3.4.1. Metal ions in biology  3.4.2. Enzymatic reactions with metals  3.4.3. Metal poisoning and detoxification  3.4.4. Metalloproteins and Metalloenzymes
  4. Physical chemistry  4.1. Quantum Chemistry  4.1.1. Wave–particle duality  4.1.2. Schrödinger Equation  4.1.3. Quantum Numbers and Orbitals  4.1.4. Atomic and molecular spectroscopy  4.2. Statistical mechanics  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Partitioning functions  4.2.3. Bose–Einstein and Fermi–Dirac statistics  4.3. Chemical thermodynamics  4.3.1. Gibbs Free Energy  4.3.2. Phase transition  4.3.3. Thermodynamic efficiency  4.4. Surface chemistry  4.4.1. Absorption and Catalysis  4.4.2. Colloids and Emulsions
  5. Analytical Chemistry  5.1. Classical methods  5.1.1. Gravimetric Analysis  5.1.2. titrimeter  5.2. Instrumental methods  5.2.1. Chromatography  5.2.2. Spectroscopy  5.2.3. Electroanalytical methods  5.2.4. X-ray diffraction
  6. Industrial Chemistry  6.1. Petrochemistry  6.2. Chemical engineering principles  6.3. Green Chemistry  6.4. Pharmaceuticals and Drug Synthesis
  7. Biochemistry  7.1. Carbohydrates  7.2. Proteins and enzymes  7.3. Lipids and membranes  7.4. Nucleic acids  7.5. Metabolism and Bioenergetics  7.6. Enzyme kinetics and mechanism

Undergraduate → Physical chemistry → Quantum Chemistry ↓


Quantum Numbers and Orbitals


In the fascinating world of quantum chemistry, one of the most fundamental concepts to understand are quantum numbers and orbitals. These concepts form the backbone of how we understand the structure of atoms and the behavior of electrons in an atom. Each electron occupies a particular set of orbitals defined by quantum numbers, helping to define how atoms interact in chemical reactions.

Quantum numbers: Addresses of electrons

Quantum numbers are like an address system for the electrons in an atom, detailing where you can find each electron within the atom. There are four elementary quantum numbers: the principal quantum number (n), the azimuthal quantum number or angular momentum quantum number (l), the magnetic quantum number (m_l), and the spin quantum number (m_s).

Principal quantum number (n)

The principal quantum number, represented by n, indicates the main energy level or shell of the electron within the atom. These energy levels are like circular orbits at increasing distances from the nucleus. As n increases, the energy and size of the orbital increases, meaning that the electrons are found farther from the nucleus.

n can take positive integer values: 1, 2, 3, 4, etc. Higher principal quantum numbers correspond to higher energy levels:

n = 1 (first shell)
n = 2 (second shell)
n = 3 (third shell)

For example, the one-electron hydrogen atom has its electron in n = 1 state in its ground state, which is the lowest energy state.

Azimuthal quantum number (l)

The azimuthal quantum number, also known as the angular momentum quantum number, is represented by l and it defines the shape of the orbital. For a given principal quantum number n, l can have integer values ranging from 0 to n-1.

For n = 1, l = 0
For n = 2, l = 0, 1
For n = 3, l = 0, 1, 2

These values correspond to different sizes of orbitals:

  • l = 0: s orbital (spherical)
  • l = 1: p orbital (dumbbell shaped)
  • l = 2: d orbital (cloverleaf shaped)
  • l = 3: f orbital (complex shape)

Magnetic quantum number (m_l)

The magnetic quantum number, denoted by m_l, specifies the orientation of the orbital in space relative to the three axes (x, y, z). For a given l value, m_l can have integer values ranging from -l to +l.

For l = 0, m_l = 0
For l = 1, m_l = -1, 0, 1
For l = 2, m_l = -2, -1, 0, 1, 2

Each orbital can contain two electrons, which differ in a property defined by the fourth quantum number, which is the spin of the electron.

Spin quantum number (m_s)

The spin quantum number, m_s, describes the intrinsic spin of the electron, which is a fundamental quantum property. It has two possible values: +1/2 or -1/2. This property allows each orbital to hold two electrons with opposite spins.

According to the Pauli exclusion principle, no two electrons in the same atom can have the same set of four quantum numbers. Therefore, in any orbital, the spins of the two electrons must be opposite (+1/2 and -1/2).

Understanding orbitals

The concept of orbitals

In quantum chemistry, an orbital is a mathematical function that describes the wave-like behavior of electrons in atoms. An orbital can be viewed as a region of space where there is a high probability of finding an electron. Orbitals are defined by three quantum numbers n, l and m_l.

Unlike the orbits in the Bohr model, the orbits do not determine a fixed path for the electrons, but rather they define a cloud-like region where the electrons are likely to be found.

Looking at orbitals helps us understand how electrons are arranged in atoms and how they participate in chemical bonding.

s-orbitals

S-orbitals are spherical and exist at every energy level starting from n=1. The size of the s-orbital increases with the increase of the principal quantum number, n.

Shape: Round
Example:
1s, 2s, 3s indicate s orbitals at different energy levels.

The above circle is a representation of the s-orbital, where the electron has an equal probability of being found at any point around the nucleus.

p-orbitals

p-orbitals are dumbbell-shaped and exist at energy levels where n >= 2. For each principal quantum number n beyond 2, there are three p-orbitals (l = 1), oriented along the x, y and z axes. These are called p_x, p_y and p_z.

Shape: Dumbbell
Example:
2p, 3p, 4p indicate p orbitals at the corresponding energy levels.

The ellipses above depict the shape of a p-orbital. Notice how they extend in opposite directions, indicating a dumbbell shape.

d-orbitals

d-orbitals have more complex shapes than s and p orbitals. These orbitals exist from n=3 onwards, where l = 2. There are five possible d-orbitals: d_xy, d_yz, d_zx, d_{x^2-y^2}, and d_{z^2}.

Shape: Cloverleaf
Example:
3d, 4d indicates d orbitals at different energy levels.

The SVG shows how d-orbitals can be viewed with the lobes oriented in specific spatial arrangements. This cloverleaf shape allows d-orbitals to participate in complex bonding scenarios.

f-orbitals

The f-orbitals are even more complex, appearing at the energy level n=4 where l=3. The shapes are highly complex, and these orbitals are important in the chemistry of the lanthanide and actinide elements.

Example:
4f, 5f indicate f orbitals at different energy levels.

Observing f-orbitals is challenging because they have complex shapes and contain multiple lobes. These orbitals have a significant impact on the chemical properties and bonding behavior of heavy elements.

Role of quantum numbers in electron configuration

Quantum numbers not only help in finding the position of the electron within the atom but also play a crucial role in writing the electron configuration of an element. The electron configuration shows the distribution of electrons among different orbitals. It determines how atoms bond and interact chemically.

Example: helium atom

1s2

The configuration shows that helium's two electrons completely fill the 1s orbital. Each set of orbitals can hold a certain number of electrons, determined by the exclusion principle and the limitations of quantum numbers.

Example: oxygen atom

1s2 2s2 2p4

This configuration shows that the first two electrons are in the 1s orbital, the next two in the 2s orbital, and the remaining four in the 2p orbital, giving oxygen its unique chemical properties.

Creation of atoms: the Aufbau principle

Aufbau principle is a method used to determine the electron configuration of an element. It states that electrons occupy the lowest energy orbitals first. To understand this concept, it is necessary to know the increasing order of orbital energy levels, e.g.

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s

This theory helps predict the arrangement of electrons (filling of orbitals) in multi-electron atoms, essentially giving rise to the periodic table of elements.

Conclusion

Quantum numbers and orbitals are central concepts in quantum chemistry, defining the spatial distribution and energy states of electrons. The state of an electron in an atom is expressed through quantum numbers, which contribute significantly to the chemical properties and reactivity of the atom. Understanding these fundamental principles allows chemists to predict the behavior of atoms in various chemical contexts, leading to advanced discoveries in chemistry and related fields.


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Quantum Numbers and Orbitals

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