# Quantum Mechanics of Chemistry

Quantum mechanics is the study of the wavelike properties of electrons. An atomic orbital represents a region of space occupied by electrons.

Quantum mechanics is the branch of science that deals with subatomic particles, their behaviors, and their interactions. Quantum mechanics describes the behavior of an atom's electrons around its nucleus, specifically how likely it is to find electrons in particular locations around the atom's nucleus and how much energy, or the quantum of energy, it takes for an electron to move from one energy level to another.

Electrons are located in spherical and dumbbell-shaped atomic orbitals around the atomic nucleus. An orbital (or atomic orbital) is a region in which an electron has a high probability of being located. Orbitals are described by the quantum numbers s, p, d, and f, which differ from one another by their shapes. The orbitals are differentiated by a principal quantum number, which reflects how close to the nucleus the electrons in that orbital are. The 1s orbital is closest to the nucleus, and its electrons are more tightly held to the nucleus. Moving out from the nucleus, the 2s orbital is next, then three dumbbell-shaped p-orbitals, 2px, 2py, and 2pz, whose axes are at right angles to each other.

### Determining Electron Configuration

The Aufbau principle, Pauli's exclusion principle, and Hund's rule determine the electron configuration of an atom.

In addition to their negative charge, electrons also have a spin, which can have a value of $+1/2$ or $-1/2$. Pauli's exclusion principle states that paired electrons may never have the same spin value, which means they cannot have the same four quantum numbers. According to Pauli's exclusion principle, two electrons may only be in the same orbital if they have opposite spins, and therefore, only two electrons may occupy any single orbital.

An element's location in the periodic table corresponds to the principal quantum number of its highest occupied orbital. For example, the first-row elements, hydrogen and helium, contain one and two electrons, respectively, in the 1s orbital. Moving down to the second period, or row, on the table, lithium contains two electrons in the 1s orbital and one electron in the 2px orbital.

The Aufbau principle states that electrons fill orbitals in order of increasing energy. According to the Aufbau principle, electrons must fill the orbitals of the lowest available energy level before populating the higher energy levels. In other words, the 1s orbital is filled before the 2s orbital, which is filled before the 2p orbitals.

Hund's rule states that when filling degenerate orbitals (for example, the 3p orbital), electrons must first singly occupy all the empty orbitals in the subshell before pairing within the same orbital. According to Hund's rule, a single electron must populate each orbital within the same principal quantum number before those orbitals can be doubly occupied. For example, nitrogen (with seven total electrons) has two electrons in the 1s orbital, two electrons in the 2s orbital, and one electron in each of the 2p orbitals. Note that under Hund's rule, the three electrons destined for the 2p orbitals spread out individually among the three orbitals. So, rather than fully populating one orbital, the result leaves another orbital with only one electron and one empty orbital.

### Electron Configurations

Element Atomic Number (z) Number of Electrons in Orbital
1s 1s 1s 2s 2s 2s 2px 2px 2px 2py2py 2py 2px 2px 2px 3s 3s 3s
Hydrogen 1 $\uparrow$$\uparrow$$\uparrow$
Helium 2 $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$
Lithium 3 $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow$$\uparrow$$\uparrow$
Beryllium 4 $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$
Boron 5 $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow$$\uparrow$$\uparrow$
Carbon 6 $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow$$\uparrow$$\uparrow$ $\uparrow$$\uparrow$$\uparrow$
Nitrogen 7 $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow$$\uparrow$$\uparrow$ $\uparrow$$\uparrow$$\uparrow$ $\uparrow$$\uparrow$$\uparrow$
Oxygen 8 $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow$$\uparrow$$\uparrow$ $\uparrow$$\uparrow$$\uparrow$
Fluorine 9 $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow$$\uparrow$$\uparrow$
Neon 10 $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$
Sodium 11 $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow$$\uparrow$$\uparrow$
Magnesium 12 $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$ $\uparrow\downarrow$$\uparrow\downarrow$$\uparrow\downarrow$

The electron configurations of the first 12 elements follow the Aufbau principle.

### Orbitals and Bonding

The valence bond theory and molecular orbital theory help explain how atomic orbitals overlap to form covalent bonds.
Two theories of bonding, the valence bond theory and the molecular orbital theory, explain how atomic orbitals overlap to form covalent bonds. A molecular orbital is a mathematical function that gives the probability of locating an electron in a localized volume of space. The valence bond theory states that the electrons in a covalent bond remain centralized around their original nuclei and that their atomic orbitals overlap. The result is that the atoms move together until they reach the optimum distance apart, where the attractive forces between the electrons and the nuclei balance the nucleus-nucleus and electron-electron repulsion. At the optimum distance, the molecule is stable and the orbitals overlap, creating a single orbital, encompassing both atoms and their electrons.

#### Two Hydrogen Atoms Bond According to the Valence Bond Model

The molecular orbital theory is the theory that atomic electron orbitals in covalent bonds are replaced by electron orbitals that belong to the entire molecule. When two atoms meet, the theory proposes, the orbitals combine to create two new orbitals: a bonding orbital ($\sigma$) and a nonbonding orbital ($\sigma^*$). The resulting $\sigma$ orbital has a lower energy than the atomic orbital, and the $\sigma^*$ orbital has a higher energy.

#### Molecular Orbital Model

A key difference between the two theories is that the valence bond theory looks at a molecule as a collection of individual atoms bonded together, each with its own electrons. The molecular orbital theory proposes that the same electron may be shared by many or all of the atoms in the molecule.