Orbitals and Quantum Numbers

Orbitals and Quantum Numbers

An atom contains a large number of possible orbitals. These orbitals can be distinguished qualitatively by three characteristics: size, shape, and orientation. An orbital of smaller size means there is a greater chance of finding the electron near the nucleus. Similarly, shape and orientation tell us that the probability of finding the electron is higher along certain directions than along others.

Atomic orbitals are precisely distinguished by what are known as quantum numbers. Each orbital is designated by three quantum numbers labelled as $n$, $l$, and $m_l$.

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The Principal Quantum Number ($n$)

The principal quantum number $n$ is a positive integer with values $n = 1, 2, 3, \dots$

$n$ determines the size and, to a large extent, the energy of the orbital. For the hydrogen atom and hydrogen-like species ($\text{He}^+$, $\text{Li}^{2+}$, etc.), the energy and size of the orbital depend only on $n$.

The principal quantum number also identifies the shell. All orbitals of a given value of $n$ constitute a single shell of the atom. Shells are represented by letters:

As $n$ increases, the number of allowed orbitals increases and is given by $n^2$.

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The Azimuthal Quantum Number ($l$)

The azimuthal quantum number $l$ is also known as the orbital angular momentum quantum number or subsidiary quantum number. It defines the three-dimensional shape of the orbital.

For a given value of $n$, $l$ can have $n$ values ranging from $0$ to $n-1$:

For example:

• When $n = 1$, the only possible value of $l$ is $0$.
• For $n = 2$, the possible values of $l$ are $0$ and $1$.
• For $n = 3$, the possible $l$ values are $0$, $1$, and $2$.

Each shell consists of one or more sub-shells or sub-levels. The number of sub-shells in a principal shell is equal to the value of $n$. In the first shell ($n = 1$), there is only one sub-shell corresponding to $l = 0$. There are two sub-shells ($l = 0, 1$) in the second shell ($n = 2$), three ($l = 0, 1, 2$) in the third shell ($n = 3$), and so on.

Each sub-shell is assigned an azimuthal quantum number $l$. Sub-shells corresponding to different values of $l$ are represented by the following symbols:

Table 2.4 shows the permissible values of $l$ for a given principal quantum number and the corresponding sub-shell notation:

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The Magnetic Orbital Quantum Number ($m_l$)

The magnetic orbital quantum number $m_l$ gives information about the spatial orientation of the orbital with respect to a standard set of coordinate axes.

For any sub-shell (defined by a particular $l$ value), $2l+1$ values of $m_l$ are possible. These values are given by:

Thus:

• For $l = 0$: the only permitted value of $m_l$ is $0$. $[2(0)+1 = 1$, one s orbital$]$
• For $l = 1$: $m_l$ can be $-1$, $0$, and $+1$. $[2(1)+1 = 3$, three p orbitals$]$
• For $l = 2$: $m_l = -2, -1, 0, +1, +2$. $[2(2)+1 = 5$, five d orbitals$]$

For example, an orbital described by the quantum numbers $n = 2$, $l = 1$, $m_l = 0$ is an orbital in the p sub-shell of the second shell.

The following chart gives the relation between the sub-shell and the number of orbitals associated with it:

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The Electron Spin Quantum Number ($m_s$)

The three quantum numbers $n$, $l$, and $m_l$ are not enough to explain the line spectra observed in the case of multi-electron atoms. Some of the lines actually occur in doublets (two lines closely spaced), triplets (three lines closely spaced), and so on. This suggests the presence of a few more energy levels than predicted by the three quantum numbers.

In 1925, George Uhlenbeck and Samuel Goudsmit proposed the presence of the fourth quantum number known as the electron spin quantum number ($m_s$). An electron spins around its own axis, much like the Earth spins around its own axis while revolving around the Sun. In other words, an electron has, besides charge and mass, an intrinsic spin angular momentum.

Spin angular momentum of the electron — a vector quantity — can have two orientations relative to the chosen axis. These two orientations are distinguished by the spin quantum number $m_s$, which can take the values of $+\frac{1}{2}$ or $-\frac{1}{2}$. These are called the two spin states of the electron and are normally represented by two arrows: $\uparrow$ (spin up) and $\downarrow$ (spin down).

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Summary of Information from the Four Quantum Numbers

The four quantum numbers provide the following information:

(i) $n$ defines the shell, determines the size of the orbital, and to a large extent the energy of the orbital.

(ii) There are $n$ sub-shells in the $n$th shell. $l$ identifies the sub-shell and determines the shape of the orbital. There are $(2l+1)$ orbitals of each type in a sub-shell — one s orbital ($l = 0$), three p orbitals ($l = 1$), five d orbitals ($l = 2$) per sub-shell. To some extent, $l$ also determines the energy of the orbital in a multi-electron atom.

(iii) $m_l$ designates the orientation of the orbital. For a given value of $l$, $m_l$ has $(2l+1)$ values, the same as the number of orbitals per sub-shell. This means the number of orbitals is equal to the number of ways in which they are oriented.

(iv) $m_s$ refers to the orientation of the spin of the electron.

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Orbit vs. Orbital

An orbit, as proposed by Bohr, is a circular path around the nucleus in which an electron moves. A precise description of this path of the electron is impossible according to the Heisenberg uncertainty principle. Bohr orbits, therefore, have no real meaning and their existence can never be demonstrated experimentally.

An atomic orbital, on the other hand, is a quantum mechanical concept and refers to the one-electron wave function $\psi$ in an atom. It is characterized by three quantum numbers ($n$, $l$, and $m_l$), and its value depends upon the coordinates of the electron. …