Hydrogenic Atomic Orbital

Introduction

A mathematical function representing the location and wave-like activity of an electron in an atom is called an atomic orbital in atomic theory and quantum theory.

Each orbital in an atom is defined by a set of values for the three quantum numbers 𝑛, 𝓁, 𝓂_𝑙, which correspond to the energy, angular momentum, and angular momentum vector component of the electron, respectively (the magnetic quantum number).

The atomic orbital model, a new framework for viewing the submicroscopic activity of electrons in matter, is built around atomic orbitals. In this approach, a multi-electron atom’s electron cloud can be thought of as being built up (roughly) in an electron configuration that is a result of smaller hydrogen-like atomic orbitals.

“Hydrogenic orbitals” are the orbitals derived by solving the Schrödinger equation for the hydrogen atom, and they serve as the foundation for the atomic orbitals of all elements, not only hydrogen.

Table of Contents

• Atomic Orbital Concept
  • Types of orbitals
  • Names of Atomic Orbitals and the Relationship Between the Different Quantum Numbers
• Hydrogenic Atomic Orbital or Hydrogen-like Orbitals
• The Schrödinger equation
• Frequently Asked Questions

Atomic Orbital Concept

Atomic orbitals are mathematical functions that shed light on the wave nature of electrons (or pairs of electrons) that circle atom nuclei. These mathematical functions are frequently used in quantum mechanics and atomic theory to determine the likelihood of finding an electron (belonging to an atom) in a certain location around the nucleus of the atom.

It’s vital to remember that the properties of each atomic orbital are determined by the values of the quantum numbers below:

• The principal quantum number (referred as ‘n’)
• The azimuthal quantum number (abbreviated as ‘l’) is also known as the orbital angular momentum quantum number
• The magnetic quantum number (abbreviated as’m_l‘)

Types of orbitals

The hydrogen-like “orbitals,” which are exact solutions to the Schrödinger equation for a hydrogen-like “atom,” can be atomic orbitals (i.e., an atom with one electron). For the radial functions R(r), there are commonly three mathematical forms that can be used as a starting point for calculating the properties of atoms and molecules with many electrons:

1. The exact solutions of the Schrödinger Equation for one electron and a nucleus for a hydrogen-like atom yield the hydrogen-like atomic orbitals. The part of the function that is dependent on the nucleus distance r has nodes (radial nodes) and decays as e^{-(constant x distance)}.
2. The Slater-type orbital (STO) lacks radial nodes but decays from the nucleus in the same way as the hydrogen-like orbital.
3. There are no radial nodes or decays in the Gaussian type orbital (Gaussians).

Names of Atomic Orbitals and the Relationship Between the Different Quantum Numbers

The primary quantum number (n) and the azimuthal quantum number (l) are commonly combined to form the name of an atomic orbital. The atomic orbitals’ simple names and the azimuthal quantum number’s associated value are listed below.

• The s orbital, with the azimuthal quantum number equal to 0.
• The p orbital, with the azimuthal quantum number equal to 1.
• The d orbital, with the azimuthal quantum number equal to 2.
• The f orbital, with the azimuthal quantum number equal to 3.
• The g orbital, with the azimuthal quantum number equal to 4.
• The h orbital, with the azimuthal quantum number equal to 5.

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What are Atomic Orbitals?

Hydrogenic Atomic Orbital or Hydrogen-like Orbitals

The atomic orbitals that are estimated for systems with a single electron, such as the hydrogen atom, are the simplest. An atom of any other element ionised down to a single electron has orbitals that are quite similar to hydrogen’s. The atomic orbitals are the eigenstates of the Hamiltonian operator for the energy in the Schrödinger equation for this system of one negative and one positive particle.

The orbital electron does not traverse a simple planetary orbit in the contemporary idea of a hydrogen atom. Rather, we’re talking about an atomic orbital, in which the likelihood of finding an electron in a specific volume at a given distance and direction from the nucleus is only a probability.

Unique values of three quantum numbers: 𝑛, 𝓁, 𝓂_𝑙, are used to identify a specific (hydrogen-like) atomic orbital. A hydrogen atom’s electron has access to a number of discrete atomic orbitals. The energy, size, and form of these orbitals vary, and detailed mathematical explanations for each are conceivable.

A hydrogen atom’s most stable or ground state is called 1s¹. In the 1s state, the electron is closest to the nucleus on average (i.e., it is the state with the smallest atomic orbital). The spherically symmetrical 1s orbital. This means that the chance of finding an electron at a certain r distance from the nucleus is unaffected by the direction of travel. The 1s orbital will be represented as a sphere centred on the nucleus with a radius large enough to ensure that the electron will be found within the boundary surface (0.80 to 0.95).

The 2s orbital is essentially similar to the 1s orbital, although it is larger and thus more diffuse, as well as having greater energy. There are three equal-energy orbitals for the primary quantum number 2 termed 2p orbitals, which have a different shape than the s orbitals. These are depicted in the Figure, where we can see that the axes travelling through the tangent spheres of the three p orbitals are at right angles to one another. Also because p orbitals are not spherically symmetrical.

The 3s and 3p states have comparable properties to the 2s and 2p states, however, they have higher energy. The 3d, 4d, 4f, orbitals have even higher energies and geometries, but they aren’t crucial for bonding in most organic molecules, at least not in carbon compounds containing hydrogen and elements in the first main row (Li – Ne) of the periodic table.

The Schrödinger equation

The Schrödinger wave equation is a mathematical expression that describes the energy and position of an electron in space and time while accounting for the electron’s matter wave nature within an atom. It is based on three basic factors. They are:

• Classical plane wave equation,
• Broglie’s Hypothesis of matter-wave, and
• Conservation of Energy

The Schrödinger equation describes the shape of the wave functions or probability waves that influence the motion of some smaller particles in great detail. Furthermore, we may determine the quantum numbers as well as the orientations and shapes of orbitals where electrons are found in a molecule or an atom by solving the Schrödinger equation.

The time-dependent Schrödinger equation and the time-independent Schrödinger equation are the two equations.

The time-dependent Schrödinger equation is written as follows:

OR

In a position basis, the time-dependent Schrödinger equation is as follows:

where, i = imaginary unit, Ψ = time-dependent wavefunction, h² is h-bar, V(x) = potential and = Hamiltonian operator.

Read More:

Schrodinger Wave Equation