**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.

**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 1

*s*orbital is closest to the nucleus, and its electrons are more tightly held to the nucleus. Moving out from the nucleus, the 2

*s*orbital is next, then three dumbbell-shaped

*p*-orbitals, 2

*p*

_{x}, 2

*p*

_{y}, and 2

*p*

_{z}, whose axes are at right angles to each other.

### Determining Electron Configuration

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 1*s* orbital. Moving down to the second period, or row, on the table, lithium contains two electrons in the 1*s* orbital and one electron in the 2*p _{x}* 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 1*s* orbital is filled before the 2*s* orbital, which is filled before the 2*p* orbitals.

**Hund's rule** states that when filling degenerate orbitals (for example, the 3*p* 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 1*s* orbital, two electrons in the 2*s* orbital, and one electron in each of the 2*p* orbitals. Note that under Hund's rule, the three electrons destined for the 2*p* 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 |
2p 2_{x}p 2_{x}p_{x} |
2p2_{y}p 2_{y}p_{y} |
2p 2_{x}p 2_{x}p_{x} |
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$ |

### Orbitals and Bonding

**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

**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.