Atomic Orbitals

Atomic orbitals are regions around the nucleus of an atom where electrons are likely to be found. They are described by quantum mechanics and are characterized by specific quantum numbers. Each type of atomic orbital has a distinct shape, size, and orientation, which corresponds to the probability distribution of finding an electron within that orbital.

In this article, we will learn in detail about atomic orbitals, their names, definition, significance and how they are related to different quantum numbers.

What are Atomic Orbitals?

Atomic orbitals are spherical paths around the atomic center where the electrons are most likely to be occupied. According to quantum mechanics, electrons do not follow classical paths but are subject to wave functions that show the electron probability distributions in the atom. The atomic orbitals emerge from the Schrödinger equation, where energy is maintained with what an electron is doing in the atom.

Each orbital is described by a specific group of numbers, those being the quantum numbers, that define the size and shape of the orbital as well as the direction it is pointing in. The principal quantum number (n) represents the orbital’s energy and, as such, determines how far from the nucleus the orbital is located. The azimuthal quantum number (l), on the other hand, indicates the shape of the orbital, while the magnetic quantum number (m_l) reflects the orientation of the orbital in space. Moreover, apart from the orbital angular momentum that is given by the orbital quantum number (ml), electrons also have the spin angular momentum, which is the spin quantum number (m_s).

Atomic orbitals that form spherical or dumbbell shapes are represented as the s and p orbitals, respectively, while the more complex configurations for the d- and f-levels are shown for the higher energy levels. Orbitals are important in determining the fundamental properties of molecules. For instance, each orbital can only accommodate a maximum of two electrons with opposite spins because of the Pauli Exclusion Principle.

Names of Atomic Orbitals

Atomic orbitals are regions around the nucleus where electrons are likely to be found most of the time. These orbitals are arranged differently and also have characteristic energies. These atomic orbital names are classified by their main quantum numbers (principal quantum number n) azimuthal quantum numbers (l) and magnetic quantum number (m_l)

The main types of atomic orbitals, labelled by their principal quantum number (n) and azimuthal quantum number (l), are as follows:

• s-orbital
• p-orbital
• d-orbital
• f-orbital

These orbitals are discussed in detail below:

s-orbital

The s (or spherical) orbital is the first atomic orbital in quantum mechanics in terms of its energy. It is known for its rounded form, and its main features are represented by the principal quantum number (n) and the azimuthal quantum number (l). Here are some key points about the s orbital:

1. Shape: S-orbital is spherically-shaped, which means that there is constant probability of finding an electron at any distance from the nucleus in any direction.

2. Principal Quantum Number (n): A principal quantum number is an indicator of the energy level of the electron as well as the size of the orbital. If an s orbital has been filled, the principal quantum number is only a positive integer (n = 1, 2, 3,…).

3. Azimuthal Quantum Number (l): The azimuthal quantum number for the s-orbital is always 0. That means that the shape of that orbital is spherically symmetric.

4. Electron Capacity: The distribution of s orbitals can contain two electrons, and if we apply the Pauli exclusion principle, it holds true that none of the two electrons have the same four quantum numbers.

5. Energy: In multi-electron atoms, the s orbitals occupy the position of the lowest energy states in the same principal level (n) of the orbitals p, d, and f.

6. Designation: The s orbitals carry the notation in the format of their principal quantum number followed by the letter “s”. For example, with the s orbital in the first energy level receiving the symbol 1s, in the second energy level as 2s, and so on.

p-orbital

p-orbital is ranked second atomic orbital in terms of energy level in quantum mechanics. Here are the key points about the p orbital

1. Shape: In contrast to the round shape of an s orbital, the p orbital can be represented by a dumbbell or a figure of eight with two lobes disconnected by a nodal plane in which the probability localization is zero. The lobes are positioned on the three mutually perpendicular axes (x, y, and z) in space.

2. Principal Quantum Number (n): p-orbital starts from second shell and hence has the principal quantum number starting from 2 onwards (n = 2, 3, 4,…). They are written as 2p, 3p, 4p etc.

3. Azimuthal Quantum Number (l): The azimuthal quantum number for the p orbital is 1, which means the sigmoidal shape of the orbitals. Contrary to the fact that the s orbital has no directional properties, the p orbital has some directional qualities owing to its dumbbell shape.

4. Electron Capacity: Since, azimuthal quantum number of p-orbital is 1, its magnetic quantum number is given as -1 to 1 i.e. -1, 0 and 1 corresponding to three sub-orbitals. According to Pauli Exclusion Principle there can be at maximum of two electrons resulting to total six electrons in the orbital.

5. Designation: The designation of orbital specifies the particular axis along which the electrons are located. In the case of the second energy level (n = 2), we have three p orbitals that are named 2px, 2py, and 2pz.

d-orbital

The d-orbital is ranked third in terms of increasing energy level. The details about d-orbital is discussed below:

1. Shape: The d-orbitals are usually complex, with four cloverleaf shapes probing four regions of space, which are similar to doughnut-shaped ones. These lobes all have their own unique characteristics, and they all occur along their own axis in three-dimensional space.

2. Principal Quantum Number (n): The principal quantum number of d orbital starts with 3 onwards. For example, 3d, 4d, 5d etc.

3. Azimuthal Quantum Number (l): The azimuthal quantum number for d orbital is 2.

4. Electron Capacity: There are total of five d-orbitals and each contain two electrons. Hence, there are total ten electrons in d-orbitals

6. Orientation: There are five d orbitals per total (n) that are designated for the first to the fifth principal energy levels, with dxy, dxz, dyz, dx²-y², and dz² as their names. These orbitals can be considered to have certain orientations with respect to the coordinate axes and, therefore, help make the bond in the molecular orbitals more directional.

d-orbitals are important since they determine the electronic configurations of transition metals and their compounds. However, electrons enter d- orbitals along with the s and p orbitals. It is the presence of d electrons that causes transition metals due to which they had variable oxidation state and the form colored compounds and complex ions.

f-orbital

f-orbital is the highest energy orbital among the four orbitals. f-orbitals are found in lanthanides and actinides configurations. The electrons in f subshell gives distinctive properties to these elements, including their magnetic properties, coordination property, and radiation behavior.

1. Shape: In f-orbitals, different lobes and additional splashes formed like an elongated doughnut shape are involved. Therefore, their shapes are more complex. They have bell-shaped structure and oriented in different directions, they form extremely complex and elegant 3D shapes.

2. Principal Quantum Number (n): The principal quantum number for f-orbital starts with four onwards. For example, 4f, 5f, 6f etc.

3. Azimuthal Quantum Number (l): The azimuthal quantum number for f-orbital is 3.

4. Electron Capacity: Each f-orbital can take up to two electrons per orbital and there are seven atomic orbitals. Hence, f-orbital can have maximum of fourteen electrons.

5. Orientation: There are total seven f-orbitals. They are represented by: fxyz, f(x2-y2), fz³, fz(x²-y²), fxyz², fyz³, and fzx³. Those states can have different orientations relative to the centroid and can give up the detailed description of electronic configurations of heavy elements.

Electronic Configuration using Atomic Orbital

Electronic configuration is a way of representing how electrons are distributed among the various atomic orbitals in an atom. The notation follows the Aufbau principle, Pauli exclusion principle, and Hund’s rule.

Here’s how you can represent the electronic configuration of an atom using atomic orbitals:

Aufbau Principle: Electrons fill orbitals starting from the lowest energy level (closest to the nucleus) and proceed to higher energy levels in order of increasing energy.

Pauli Exclusion Principle: Each orbital can hold a maximum of two electrons, and these electrons must have opposite spins.

Hund’s Rule: When filling orbitals of equal energy (degenerate orbitals, such as p orbitals), electrons occupy empty orbitals singly with parallel spins before pairing up.

For example, let’s consider the electronic configuration of carbon (atomic number 6):

Carbon has 6 electrons.

The electronic configuration of carbon can be represented as 1s²2s²2p²

This means that the first two electrons fill the 1s orbital, the next two fill the 2s orbital, and the remaining two electrons occupy two of the three available 2p orbitals.

Relationship between Atomic Orbitals and Quantum Numbers

Quantum numbers are a set of four parameters that describe the unique quantum state of an electron in an atom. These numbers arise from the solution of the Schrödinger equation for the hydrogen atom and provide a rule for understanding the electronic structure of atoms. Each electron in a orbital has four quantum numbers. These are discussed below:

Four Quantum Numbers

1. Principal Quantum Number (n): Denoted by “n”, it determines the energy level of an electron and the size of the orbital. It can take integer values starting from 1 (1, 2, 3, …).

2. Azimuthal Quantum Number (l): It is denoted as ‘l’, and defines the shape of the orbital. It takes value from from 0 to (n − 1). The azimuthal quantum number for different atomic orbitals are s (l = 0), p (l = 1), d (l = 2), f (l = 3), and so on.

3. Magnetic Quantum Number (m_l): It is denoted as “m_l”. It refers to the direction of the orbital in space. It takes value from -l to l.

4. Spin Quantum Number (m_s): It is denoted as “m_s” which defines the spin of electrons in a room or the angular momentum of electrons. It takes the value +1/2 or -1/2, which indicates electron spin states.

Rules for Relationship between Quantum Numbers and Atomic Orbitals

1. Pauli Exclusion Principle: It states that no two electrons can have all the four quantum numbers same.

2. Quantization of Angular Momentum: The component of the angular momentum quantum number only takes integer values from 0 to (n-1) with no value in between them.

3. Magnetic Quantum Number and Angular Momentum: The values of the magnetic quantum number “m_l” can be negative, zero, or positive, being equal to -l, 0, and +l. This means there will be 2l+1 possible values of m_l

4. Spin-Orbit Coupling: Spin-orbit coupling is a quantum mechanical interaction that arises from the coupling between an electron’s intrinsic spin angular momentum and its orbital angular momentum due to the presence of an external electromagnetic field. It is is responsible for the splitting of spectral lines in atoms into multiple closely spaced lines, known as fine structure.

The quantum numbers and the relationships between them are very important for the understanding of electron behavior—electrons in atoms and molecules, mainly the inner states and configurations of electrons within the overall system.

Table for Relation between Quantum Numbers and Atomic Orbital

The relation between Quantum Numbers and Atomic Numbers are discussed in the following table

Principal Quantum Number	Azimuthal Quantum Number	Possible Atomic Orbitals
n = 1	l = 0 to n – 1 = 0 l = 0 (s orbital)	1s Atomic Orbital
n = 2	l = 0 to n – 1 = 0, 1 l = 0 (s-orbital) l = 1 (p-orbital)	2s Atomic Orbital 2p Atomic Orbital
n = 3	l = 0 to n – 1 = 0, 1, 2 l = 0 (s-orbital) l = 1 (p-orbital) l = 2 (d-orbital)	3s Atomic Orbital 3p Atomic Orbital 3d Atomic Orbital
n = 4	l = 0 to n – 1 = 0, 1, 2, 3 l = 0 (s-orbital) l = 1 (p-orbital) l = 2 (d-orbital) l = 3 (f -orbital)	4s Atomic Orbital 4p Atomic Orbital 4d Atomic Orbital 4f Atomic Orbital

Conclusion

Atomic orbitals establish the basis of quantum mechanics, as they provide an model for the description of the electrons’ behavior within the atomic structure. These orbitals specify where the electrons are to be found in space, their energy levels, and the orientation of the tiny quantum universe.

Atomic Orbitals Frequently Asked Questions

What are atomic orbitals?

Atomic orbitals are the space around atoms where an electron is probably to occupy. They may be specified by quantum numbers for different energy levels and forms.

How does the orbital momentum (l) differ from the spin quantum number (s)?

The principal quantum number (n) is the one that defines energy level of an electron while spin quantum number tells about the orientation of the atomic orbitals

What is Atomic Orbital Theory?

Atomic orbital theory is a fundamental concept in quantum mechanics that describes the behavior of electrons in atoms.

What are four quantum numbers?

The four quantum numbers are principal quantum numbers, azimuthal quantum numbers, magnetic quantum numbers and spin quantum numbers

What is Pauli Exclusion Principle?

Pauli Exclusion Principle states that no two electrons in an atom can have all the four quantum numbers same