Concept of Atomic Orbitals as One-Electron Wave Functions

The wave-like nature of subatomic particles was a great debate over the years, and it was put to an end by the physicist Erwin Schrodinger. He gave a mathematical equation to identify the wave nature of subatomic particles and introduced a new subject of wave mechanics where the energy of atomic orbitals could be studied.

The equation provides the relationship between the motion of the particle and its wave-like functions and its allowed energy. The theory developed by Schrodinger gives an account of the energy and spatial distributions of electrons in atoms and molecules.

Quantum mechanics uses sophisticated mathematical details which are not needed to be completely understood. We can concentrate solely on the features of wave functions deduced from the Schrodinger equation. They differ from describing those of the real waves.

On the advancement of quantum mechanics, it was deduced that an electron shows both wave-like properties and particle-like properties, which is called wave-particle duality.

Wave Function

The wave function is a mathematical function that relates the location of an electron at a given point in the space to its energy. Generally, wave functions depend on both time and position.

Wave functions for each atom have some specific properties. The significant property of a quantum number is that these are countable integers and not continuous variables such as the point on the line.

Each electron has four quantum numbers which determine its wave function. The four quantum numbers determine an electron’s wave function, but we use the average values and probabilities for the properties that cannot be determined using the quantum numbers.

A few properties of the wave function that are derived from quantum mechanics are as below:

• The wave function, independent of time, requires three variables to describe the position of an electron. The three variables are coordinated as x, y, and z.
• Wave functions have both real and imaginary properties, such as √ -1 has no physical significance.
• The probability of finding an electron at a certain point is given by the product of wave function Ѱ and its conjugate ѱ* in which all the terms containing i are replaced by -i.
• The total probability of finding an electron must be a hundred percent.
• The bonding of different atoms is determined by the relative phase of the wave function of an electron.
• Every wave function has its own unique set of quantum numbers that specifies its wave function property. The two main properties are the electron energy and spatial arrangement in an atom.
• Each wave function is related to a specific energy.

There are three main quantum numbers required for the Schrodinger theory to be applied in knowing the wave function of electrons:

• The principal quantum number denoted as “n” is the number of orbitals.
• The azimuthal quantum number denoted as “l” gives the 3D shape of the orbital.
• The magnetic quantum number denoted as “ml” provides spatial orientation of the orbitals.
• The spin quantum number represents the spin on the electron +½ or -½.

Structure of Orbitals

Orbitals can be defined as a bounded spatial region where the electrons are arranged. Each of the orbitals is represented with a number and a letter. The number represents the energy level; therefore, it starts from 1, depicting that it is closest to the nucleus – the distance between the nucleus and the orbitals increases as the numbers increase.

The letters refer to the shape of the orbital; it starts from s,p,d,f, h, i, j, k, etc. There is no particular reason for this naming.

S orbital

S-orbital is spherically bounded around the nucleus like a hollow ball. The energy level increases proportionally as moved away from the nucleus, the orbital size increases too. 1s<2s<3s…..

The probability of finding the electron decreases as the orbital size increases.

The term nodal point can be defined as the point where there is zero possibility of finding the electron. There are two types of nodal points:

• The radial nodal point determines the distance from the nucleus.
• Angular nodal points determine the direction.

The total number of nodes in s – orbital = n – 1. Therefore, the number of nodes in 2s is one and 3s is 2, and so on.

The 1s and 2s orbitals are spherical, while the s orbitals are spherical symmetric in general. This means that the chances of locating an electron at a certain distance are the same in all directions.

P orbital

Not all the electrons occupy s – orbital as we know that each has a specific energy. The second level has 2s and 2p orbitals. P orbitals are dumbbell-shaped. The nodes of p – orbital are in the center. Orbitals can accommodate a maximum of 6 electrons as they have three orbitals.

The arrangement of all the three orbits is at a right angle to each other. Each P orbital has the sections called lobes, and they are present on either side of the plane, which passes through the nucleus. The nodal points are the intersection points of the two lobes.

The lobes of P orbitals are oriented along the plane of x,y, z-axis and denoted as 2px, 2py, and 2pz as the three orbitals are identical in size, shape, and energy. They are termed degenerate orbitals.

The total number of nodes in P orbital = n – 2.

D orbital

The third energy level includes 3s, 3p, and 3d. d-orbital is double dumbbell-shaped. For d – orbital, the n value is 3 and the l = 2.

For l = 2, the ml values are –2, –1, 0, +1, and +2, resulting in five d orbitals, namely dxy, dyz, dxz, dx2–y2, and dz2. All the orbitals have the same orbital energy, but the first four’s shape is identical, and the last one, dz2, differs in shape.

Number of nodes

dxy =  2

dxz = 2

dyz = 2

dx2-y2 = 2

dz2 = 0

F orbital

At the 4th and higher level, we will find the f orbitals, which are seven in number. They have diffused shapes.

The value of l is 3; therefore, the highest n value is 4.

The values of ml corresponding to f orbital are (-3,–2, –1, 0, +1, +2, +3), therefore there are 7 f orbitals: fx(x2-y2), fy(x2-y2), fxyz, fz3, fyz, fxz2, fz(x2-y2).

Conclusion

To understand the energy levels of orbitals and their spatial arrangement in an atom, we need to understand the equation that gives a mathematical conclusion of how the energy of the atomic orbitals is calculated and how they are arranged in space.

We have understood that an electron is a particle that has both particle function and a wave function, and this is defined as the wave-particle duality. This property of electrons has been debated for years until they could conclude that electrons show wave-like properties.

After Schrodinger gave the wave function equation, it was much easier to understand the property of the electron and obtain the energies and spatial arrangement of orbitals.