QUANTUM MECHANICAL MODEL OF ATOM

Classical mechanics, based on Newton’s laws of motion, successfully describes the motion of all macroscopic objects such as a falling stone, orbiting planets, etc., however, it fails when applied to microscopic objects like electrons, atoms, molecules, etc. This is mainly because of the fact that classical mechanics ignores the concept of dual behaviour of matter especially for sub-atomic particles and the uncertainty principle. The branch of science that takes into account this dual behaviour of matter is called quantum mechanics.

Quantum mechanics is a theoretical science that deals with the study of the motions of the microscopic objects that have both observable wave like and particle like properties. It specifies the laws of motion that these objects obey. 

Quantum mechanics was developed independently in 1926 by Werner Heisenberg and Erwin Schrödinger. Here, however, we shall be discussing the quantum mechanics which is based on the ideas of wave motion. The fundamental equation of quantum mechanics was developed by Schrödinger and it won him the Nobel Prize in Physics in 1933. This equation which incorporates wave-particle duality of matter as proposed by de Broglie is quite complex and knowledge of higher mathematics is needed to solve it. 

For a system (such as an atom or a molecule whose energy does not change with time) the Schrödinger equation is written as

\(\hat{H} \Psi=E \Psi\) where \(\hat{H}\) is a mathematical operator called Hamiltonian.

Schrödinger gave a recipe of constructing this operator from the expression for the total energy of the system. The total energy of the system takes into account the kinetic energies of all the subatomic particles (electrons, nuclei), attractive potential between the electrons and nuclei, and repulsive potential among the electrons and nuclei individually. Solution of this equation gives \(E\) and \(\psi\).

Hydrogen Atom and the Schrödinger Equation

When Schrödinger equation is solved for hydrogen atom, the solution gives the possible energy levels the electron can occupy and the corresponding wave function(s) \((\psi)\) of the electron associated with each energy level. These quantized energy states and corresponding wave functions which are characterized by a set of three quantum numbers (principal quantum number \(n\), azimuthal quantum number \(l\) and magnetic quantum number \(m_{l}\) ) arise as a natural consequence in the solution of the Schrödinger equation.

When an electron is in any energy state, the wave function corresponding to that energy state contains all information about the electron. The wave function is a mathematical function whose value depends upon the coordinates of the electron in the atom and does not carry any physical meaning. Such wave functions of hydrogen or hydrogen like specles with one electron are called atomic orbitals. The probability of finding an electron at a point within an atom is proportional to the \(|\psi|^{2}\) at that point. The quantum mechanical results of the hydrogen atom successfully predict all aspects of the hydrogen atom spectrum including some phenomena that could not be explained by the Bohr model.

Important Features of the Quantum Mechanical Model of Atom

Quantum mechanical model of atom is the picture of the structure of the atom, which emerges from the application of the Schrödinger equation to atoms. The following are the important features of the quantum-mechanical model of atom:

1. The energy of electrons in atoms is quantized (i.e., can only have certain specific values), for example when electrons are bound to the nucleus in atoms.

2. The existence of quantized electronic energy levels is a direct result of the wave like properties of electrons and are allowed solutions of Schrödinger wave equation.

3. Both the exact position and exact velocity of an electron in an atom cannot be determined simultaneously (Heisenberg uncertainty principle). The path of an electron in an atom therefore, can never be determined or known accurately.

4. An atomic orbital is the wave function \(\psi\) for an electron in an atom. The “one electron orbital wave functions” or orbitals form the basis of the electronic structure of atoms. In each orbital, the electron has a definite energy. An orbital cannot contain more than two electrons. In a multi-electron atom, the electrons are filled in various orbitals in the order of increasing energy. For each electron of a multi-electron atom, there shall, therefore, be an orbital wave function characteristic of the orbital it occupies. All the information about the electron in an atom is stored in its orbital wave function \(\psi\) and quantum mechanics makes it possible to extract this information out of \(\psi\).

5. The probability of finding an electron at a point within an atom is proportional to the square of the orbital wave function i.e., \(|\psi|^{2}\) at that point. \(|\psi|^{2}\) is known as probability density and is always positive. From the value of \(|\psi|^{2}\) at different points within an atom, it is possible to predict the region around the nucleus where electron will most probably be found.

Orbitals and Quantum Numbers

A large number of orbitals are possible in an atom. Qualitatively these orbitals can be distinguished by their size, shape, and orientation. An orbital of smaller size means there is more chance of finding the electron near the nucleus. Similarly shape and orientation mean that there is more probability of finding the electron along certain directions than along others. Atomic orbitals are precisely distinguished by what are known as quantum numbers. Each orbital is designated by four quantum numbers labelled as \(n, l\), \(m_{l}\) and \(m_{s}\) 

Definition of Orbital

It is three-dimensional space, where probability of finding an electron in maximum. The wave function suggests that the probability of finding an electron in three-dimensional space around the nucleus involves two aspects, radial probability, and angular probability.

Orbitals are of four types and are described as:

• \(s\) orbital: Spherical in shape, non-directional. It has only 1 orbital, therefore, can accommodate only 2 electrons.
• \(p\)-orbital: dumb-bell shaped and directional. It has 3 orbital \(\left(p_{x}, p_{y}, p_{z}\right)\). It can accommodate maximum of 6 electrons.
• \(d\)-orbital: It has double dumbbell, directional. It has 5 orbital \(\left(d_{x y}, d_{y z}, d_{z x}, d_{\mathrm{x}^{2}-\mathrm{y}^{2}}, d_{z^{2}}\right)\).It can accommodate maximum of 10 electrons.
• \(f\)-orbital: It has diffused shape. It has 7 orbital therefore, can accommodate maximum of 14 electrons.

Shells and Subshells of Orbitals

Orbitals that have the same value of the principal quantum number form a shell. Orbitals within a shell are divided into subshells that have the same value of the angular quantum number. Chemists describe the shell and subshell in which an orbital belongs with a two-character code such as 2p or 4f. The first character indicates the shell (n = 2 or n = 4). The second character identifies the subshell. By convention, the following lowercase letters are used to indicate different subshells.

\(s: l=0\)
\(p: l=1\)
\(d: l=2\)
\(f: l=3\)

Although there is no pattern in the first four letters (s, p, d, f), the letters progress alphabetically from that point (g, h, and so on). Some of the allowed combinations of the n and \(l\) quantum numbers are shown in the figure below.

The third rule limiting allowed combinations of the n, \(l\), and m quantum numbers has an important consequence. It forces the number of subshells in a shell to be equal to the principal quantum number for the shell. The n = 3 shell, for example, contains three subshells: the 3s, 3p, and 3d orbitals.

What are Quantum Numbers?

The set of numbers used to describe the position and energy of the electron in an atom are called quantum numbers. There are four quantum numbers, namely, principal, azimuthal, magnetic, and spin quantum numbers. Four quantum numbers can be used to completely describe all the attributes of a given electron belonging to an atom, these are:

• Principal quantum number, denoted by \(n\).
• Orbital angular momentum quantum number (or azimuthal quantum number), denoted by \(l\).
• Magnetic quantum number, denoted by \(m_{l}\).
• The electron spin quantum number, denoted by \(m_{s}\).

The table below shows the four quantum numbers with their symbol and possible values that describe an electron in an atom.

\(
\begin{array}{|l||c||c||}
\hline {\text { Number }} & \text { Symbol } & \text { Possible Values } \\
\hline \hline \text { Principal Quantum Number } & n & 1,2,3,4, \ldots \\
\hline \hline \text { Angular Momentum Quantum Number } & \ell & 0,1,2,3, \ldots,(n-1) \\
\hline \text { Magnetic Quantum Number } & m_{1} & -\ell, \ldots,-1,0,1, \ldots, \ell \\
\hline \hline \text { Spin Quantum Number } & m_{\mathrm{s}} & +1 / 2,-1 / 2 \\
\hline \hline
\end{array}
\)

The principal quantum number ‘\(n\)’

• The principal quantum number ‘ \(n\) ‘ is a positive integer with value of \(n=1,2,3 \ldots \ldots\). The principal quantum number determines the size and to large extent the energy of the orbital. For hydrogen atom and hydrogen like species \(\left(\mathrm{He}^{+}, \mathrm{Li}^{2+}, \ldots\right.\) etc.) energy and size of the orbital depends only on ‘ \(n\) ‘.
• The principal quantum number also identifies the shell. With the increase in the value of ‘ \(n\) ‘, the number of allowed orbital increases and are given by ‘ \(n\) ‘ All the orbitals of a given value of ‘ \(n\) ‘ constitute a single shell of atom and are represented by the following letters

\(\begin{array}{rlllll}
n & = & 1 & 2 & 3 & 4 \\
\text { Shell } & = & \mathrm{K} & \mathrm{L} & \mathrm{M} & \mathrm{N}
\end{array}\)

• Size of an orbital increases with increase of principal quantum number ‘ \(n\) ‘. In other words the electron will be located away from the nucleus. Since energy is required in shifting away the negatively charged electron from the positively charged nucleus, the energy of the orbital will increase with increase of \(n\).

The letter \(n\) designates the principal quantum number. This number can be a positive integer like 1,2,3 (shown in the figure below) etc. and indicates the energy of the orbital. Higher energy means the electron is further from the nucleus.

Azimuthal quantum number (Angular Momentum Quantum Number) ‘ \(l\) ‘

• Azimuthal quantum number. ‘ \(l\) ‘ is also known as orbital angular momentum 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 \(n-1\), that is, for a given value of \(n\), the possible value of \(l\) are \(: l=0,1,2, \ldots \ldots \ldots\) \((n-1)\)
• For example, when \(n=1\), value of \(l\) is only 0 . For \(n=2\), the possible value of \(l\) can be 0 and 1 . For \(n=3\), the possible \(l\) values are 0,1 and \(2 .\)
• Each shell consists of one or more subshells or sub-levels. The number of sub-shells in a principal shell is equal to the value of \(n\).
• For example in the first shell \((n=1)\), there is only one sub-shell which corresponds to \(l=0\). There are two sub-shells \((l=0,1)\) in the second shell \((n=2)\), three \((l=0,1,2)\) in 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.

\(\begin{array}{llllllll}\text { Value for } l: & 0 & 1 & 2 & 3 & 4 & 5 & \ldots \ldots \ldots \ldots . \\ \text { notation for sub-shell} & s & p & d & f & g & h & \ldots \ldots \ldots \ldots .\end{array}\)

Table \(2.4\) shows the permissible values of ‘ \(l\) ‘ for a given principal quantum number and the corresponding sub-shell notation.

The letter \(l\) indicates the shape of the orbital. This number is called the angular momentum quantum number and can have values from 0 to \(\mathrm{n}-1\). So in the first energy level, where \(\mathrm{n}=1\), there is 1 shape available corresponding to \(l=0\). We also call this an ” \(\mathrm{s}\) ” shape. In the 2 nd energy level, where \(\mathrm{n}=2\), there are 2 shapes available corresponding to \(l=0\), and \(l=1\). We also call this an ” \(\mathrm{s}\) ” and a ” \(\mathrm{p}\) ” shape (shown in the series of figure below).

Magnetic orbital quantum number ‘ \(m_{l}\) ‘

• Magnetic orbital quantum number gives information about the spatial orientation of the orbital with respect to standard set of co-ordinate axis. For any sub-shell (defined by ‘ \(l\) ‘ value) \(2 l+1\) values of \(m_{l}\) are possible and these values are given by :
  \(m_{l}=-l,-(l-1),-(l-2) \ldots 0,1 \ldots(l-2),(l-1), l\)
  Thus for \(l=0\), the only permitted value of \(m_{l}=0,[2(0)+1=1\), one \(s\) orbital \(]\). For \(l=1, m_{l}\) can be \(-1,0\) and \(+1[2(1)+1=3\), three \(p\) orbitals]. For \(l=2, m_{l}=-2,-1,0,+1\) and \(+2\), \([2(2)+1=5\), five \(d\) orbitals]. It should be noted that the values of \(m_{l}\) are derived from \(l\) and that the value of \(l\) are derived from \(n\).
• Each orbital in an atom, therefore, is defined by a set of values for \(n, l\) and \(m_{l}\). 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 subshell and the number of orbitals associated with it.

\(
\begin{array}{|l|l|l|l|l|l|l|}
\hline \text { Value of } l & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline \text { Subshell notation } & s & p & d & f & g & h \\
\hline \text { number of orbitals } & 1 & 3 & 5 & 7 & 9 & 11 \\
\hline
\end{array}
\)

The letter \(m_{l}\) designates the orientation of the orbital in space. It can have values between \(-l\) and \(l\). When \(l=0\) (” \(\mathrm{s}\) ” shape) only 1 orientation is possible. When \(l=0\) (” \(\mathrm{s}\) ” shape) \(m_{l}=0\). When \(l=1\) (“p” shape) 3 orientations are possible. When \(l=1\) (“p” shape) \(m_{l}=-1, m_{l}=0\), or \(m_{l}=1\).
Different \(m_{l}\) values mean different orientations in space. For an orbital with \(l=0\) only 1 orientation is possible.

Electron spin ‘ \(s\) ‘:

The three quantum numbers labelling an atomic orbital can be used equally well to define its energy, shape, and orientation. But all these quantum numbers are not enough to explain the line spectra observed in the case of multi-electron atoms, that is, some of the lines actually occur in doublets (two lines closely spaced), triplets (three lines, closely spaced), etc. 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 \(\left(\boldsymbol{m}_{s}\right)\). An electron spins around its own axis, much in a similar way as earth spins around its own axis while revolving around the sun. In other words, an electron has, besides charge and mass, intrinsic spin angular quantum number. 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 numbers \(m_{\mathrm{s}}\) which can take the values of \(+1 / 2\) or \(-1 / 2\). These are called the two spin states of the electron and are normally represented by two arrows, \(\uparrow\) (spin up) and \(\downarrow\) (spin down). Two electrons that have different \(m_{\mathrm{s}}\) values (one \(+1 / 2\) and the other \(-1 / 2\) ) are said to have opposite spins. An orbital cannot hold more than two electrons and these two electrons should have opposite spins.

To sum up, the four quantum numbers provide the following information:

• \(\boldsymbol{n}\) defines the shell, determines the size of the orbital and also to a large extent the energy of the orbital.
• There are \(n\) subshells in the \(n^{\text {th }}\) shell. \(l\) identifies the subshell and determines the shape of the orbital (see section 2.6.2). There are \((2 l+1)\) orbitals of each type in a subshell, that is, one \(s\) orbital \((l=0)\), three \(p\) orbitals \((l=1)\) and five \(d\) orbitals \((l=2)\) per subshell. To some extent \(l\) also determines the energy of the orbital in a multi-electron atom.
• \(\boldsymbol{m}_{l}\) designates the orientation of the orbital. For a given value of \(l, m_{l}\) has \((2 l+1)\) values,
  the same as the number of orbitals per subshell. It means that the number of orbitals is equal to the number of ways in which they are oriented.
• \(\boldsymbol{m}_{\mathrm{s}}\) refers to orientation of the spin of the electron.

Problem 2.17
What is the total number of orbitals associated with the principal quantum number \(n=3\)?

Solution:
For \(n=3\), the possible values of \(l\) are 0,1 and 2. Thus there is one \(3 s\) orbital \(\left(n=3, l=0\right.\) and \(m_{l}=0\) ); there are three \(3 p\) orbitals \(\left(n=3, l=1\right.\) and \(\left.m_{l}=-1,0,+1\right)\); there are five \(3 d\) orbitals \((n=3, l=2\) and \(\left.m_{l}=-2,-1,0,+1+,+2\right)\).

Therefore, the total number of orbitals is \(1+3+5=9\)

The same value can also be obtained by using the relation; number of orbitals \(=n^{2}\), i.e. \(3^{2}=9\)

Problem 2.18
Using \(s, p, d\), f notations, describe the orbital with the following quantum numbers
(a) \(n=2, l=1\), (b) \(n=4, l=0\), (c) \(n=5\), \(l=3\), (d) \(n=3, l=2\)

Solution:
\(\begin{array}{cccc} & n & l & \text { orbital } \\ \text { a) } & 2 & 1 & 2 p \\ \text { b) } & 4 & 0 & 4 s \\ \text { c) } & 5 & 3 & 5 f \\ \text { d) } & 3 & 2 & 3 d\end{array}\)

What are Orbitals?

Orbitals in Physics and Chemistry is a mathematical function depicting the wave nature of an electron or a pair of electrons present in an atom. The probability of finding an electron around the nucleus can be calculated using this function. Atomic orbital can be described as the physical bounded region or space where the electrons are present. Generally, an atom consists of electrons that are fixed inside the electronic orbitals.

Types of Orbitals

There are four common types of orbitals and let’s discuss about orbital shapes.

\(s\) Orbital: \(s\) orbital is spherically symmetrical orbital around the atomic nucleus. The energy level increases as we move away from the nucleus, therefore the orbitals get bigger. The order of size is \(1 s<2 s<3 s<4 s\).

The probability of finding an electron is maximum in \(1 \mathrm{~s}\) and decreases rapidly as we move away from it. In the case of \(2 s\) orbital, the probability density decreases sharply to zero and again starts increasing. On reaching a small maximum it decreases again and finally reaches to zero if the value of \(r\) increases further.

The nodal point is a point at which there is a zero probability of finding the electron. There are two types of nodes:

Radial nodes which calculate the distance from the nucleus and angular nodes that determines direction.

No. of radial nodes \(=n-l-1\)

No. of angular nodes \(=l\)

Total number of nodes \(=n-1\)

Nodal planes are defined as the planes of zero probability region to find the electron. The number of planes is equal to \(l\).

Shapes of Atomic Orbitals

The orbital wave function or \(\psi\) for an electron in an atom is simply a mathematical function of the coordinates of the electron. However, for different orbitals the plots of corresponding wave functions as a function of \(r\) (the distance from the nucleus) are different. Fig. 2.12(a), gives such plots for \(1 s(n=1, l=0)\) and \(2 s(n=\) \(2, l=0\) ) orbitals.

According to the German physicist, Max Born, the square of the wave function (i.e., \(\psi^{2}\) ) at a point gives the probability density of the electron at that point. The variation of \(\psi^{2}\) as a function of \(r\) for \(1 s\) and \(2 s\) orbitals is given in Fig. 2.12(b). Here again, you may note that the curves for \(1 s\) and \(2 s\) orbitals are different.

It may be noted that for \(1 s\) orbital the probability density is maximum at the nucleus and it decreases sharply as we move away from it. On the other hand, for \(2 s\) orbital the probability density first decreases sharply to zero and again starts increasing. After reaching a small maxima it decreases again and approaches zero as the value of \(r\) increases further. The region where this probability density function reduces to zero is called nodal surfaces or simply nodes. In general, it has been found that \(n s\)-orbital has \((n-1)\) nodes, that is, number of nodes increases with increase of principal quantum number \(n\). In other words, number of nodes for \(2 s\) orbital is one, two for \(3 s\), and so on.

Boundary surface diagrams

These probability density variation can be visualised in terms of charge cloud diagrams [Fig. 2.13(a)]. In these diagrams, the density of the dots in a region represents electron probability density in that region.

Boundary surface diagrams of constant probability density for different orbitals give a fairly good representation of the shapes of the orbitals. In this representation, a boundary surface or contour surface is drawn in space for an orbital on which the value of probability density \(|\psi|^{2}\) is constant. In principle many such boundary surfaces may be possible. However, for a given orbital, only that boundary surface diagram of constant probability density* is taken to be good representation of the shape of the orbital which encloses a region or volume in which the say, \(90 \%\). The boundary surface diagram for \(1 s\) and \(2 s\) orbitals are given in Fig. 2.13(b).  Boundary surface diagram for a \(s\) orbital is actually a sphere centred on the nucleus. In two dimensions, this sphere looks like a circle. It encloses a region in which probability of finding the electron is about \(90 \%\). Thus, we see that \(1 s\) and \(2 s\) orbitals are spherical in shape. In reality all the \(s\)-orbitals are spherically symmetric, that is, the probability of finding the electron at a given distance is equal in all the directions. It is also observed that the size of the s orbital increases with increase in \(n\), that is, \(4 s>3 s>2 s>1 s\), and the electron is located further away from the nucleus as the principal quantum number increases.

Note: * If probability density \(|\psi|^{2}\) is constant on a given surface, \(|\psi|\) is also constant over the surface. The boundary surface for \(|\psi|^{2}\) and \(|\psi|\) are identical.

\(p\) Orbital

The \(p\) orbitals are dumbbell-shaped. The node in the \(p\) orbital occurs at the centre of the nucleus. The \(p\) orbital can occupy a maximum of six electrons due to the presence of three orbitals. The three p orbitals are oriented right angles to each other. 

Boundary surface diagrams for three \(2 p\) orbitals \((l=1)\) are shown in Fig. 2. 14. In these diagrams, the nucleus is at the origin. Instead each \(p\) orbital consists of two sections called lobes that are on either side of the plane that passes through the nucleus. The probability density function is zero on the plane where the two lobes touch each other. The size, shape and energy of the three orbitals are identical. They differ however, in the way the lobes are oriented. Since the lobes may be considered to lie along the \(\mathrm{x}, \mathrm{y}\) or \(\mathrm{z}\) axis, they are given the designations \(2 p_{x}, 2 p_{y}\), and \(2 p_{z}\). It should be understood, however, that there is no simple relation between the values of \(m_{l}(-1,0\) and +1) and the \(x, y\) and \(z\) directions. For our purpose, it is sufficient to remember that, because there are three possible values of \(m_{l}\), there are, therefore, three \(p\) orbitals whose axes are mutually perpendicular. Like \(s\) orbitals, \(p\) orbitals increase in size and energy with increase in the principal quantum number and hence the order of the energy and size of various \(p\) orbitals is \(4 p>3 p>2 p\). Further, like s orbitals, the probability density functions for \(p\)-orbital also pass through value zero, besides at zero and infinite distance, as the distance from the nucleus increases. The number of nodes are given by the \(n-2\), that is number of radial node is 1 for \(3 p\) orbital, two for \(4 p\) orbital and so on.

\(d\)-orbital

The d orbital is cloverleaf or two dumbbells in a plane. For \(l=2\), the orbital is known as \(d\)-orbital and the minimum value of principal quantum number \((n)\) has to be 3 . as the value of \(l\) cannot be greater than \(n-1\). There are five \(m_{l}\) values \((-\) \(2,-1,0,+1\) and \(+2\) ) for \(l=2\) and thus there are five \(d\) orbitals. The boundary surface diagram of \(d\) orbitals are shown in Fig. 2.15.

The five \(d\)-orbitals are designated as \(d_{\mathrm{xy}}, d_{\mathrm{y} z}\), \(d_{\mathrm{xz}}, d_{\mathrm{x}^{2}-\mathrm{y}^{2}}\) and \(d_{\mathrm{z}_{2}}\). The shapes of the first four \(d-\) orbitals are similar to each other, where as that of the fifth one, \(d_{2}\), is different from others, but all five \(3 d\) orbitals are equivalent in energy. The \(d\) orbitals for which \(n\) is greater than \(3(4 d\), \(5 d . . .)\) also have shapes similar to \(3 d\) orbital, but differ in energy and size.

Besides the radial nodes (i.e., probability density function is zero), the probability density functions for the \(\mathrm{n} p\) and nd orbitals are zero at the plane (s), passing through the nucleus (origin). For example, in case of \(p_{z}\) orbital, xy-plane is a nodal plane, in case of \(d_{x y}\) orbital, there are two nodal planes passing through the origin and bisecting the \(x y\) plane containing \(\mathrm{z}\)-axis. These are called angular nodes and number of angular nodes are given by ‘ \(l\), i.e., one angular node for \(p\) orbitals, two total number of nodes are given by \((n-1)\), i.e., sum of \(l\) angular nodes and \((n-l-1)\) radial nodes.

\(f\) Orbitals

\(f\) orbital has diffused shape. For \(f\) orbital the value of \(\mathrm{l}=3\) thus the minimum value of principal quantum number \(\mathrm{n}\) is 4. The values of \(m_{l}\) corresponding to \(f\) orbital are \((-3,-2,-1,0,+1,+2,+3)\). For \(\mathrm{l}=3\) therefore, there are seven \(f\) orbitals.
The seven orbitals are \(f_{x}\left(x^{2}-y{ }^{2}\right), f_{y}\left(x^{2}-y^{2}\right), f_{x y z}, f_{z}^{3}, f_{y z}, f_{x z}^{2}, f_{z}\left(x^{2}-y^{2}\right)\).

Summary of Orbital progression in picture

Energies of Orbitals

The energy of an electron in a hydrogen atom is determined solely by the principal quantum number. number. Thus the energy of the orbitals in hydrogen atom increases as follows:
\(1 s<2 s=2 p<3 s=3 p=3 d<4 s=4 p=4 d\) \(=4 f<\quad\) (2.23) and is depicted in Fig. 2.16. Although the shapes of \(2 s\) and \(2 p\) orbitals are different, an electron has the same energy when it is in the \(2 s\) orbital as when it is present in \(2 p\) orbital. The orbitals having the same energy are called degenerate. The \(1 s\) orbital in a hydrogen atom, as said earlier, corresponds to the most stable condition and is called the ground state and an electron residing in this orbital is most strongly held by the nucleus. An electron in is in excited state.

The energy of an electron in a multielectron atom, unlike that of the hydrogen atom, depends not only on its principal quantum number (shell) but also on its azimuthal quantum number (subshell). That is, for a given principal quantum number, \(s\), \(p, d, f \ldots\) all have different energies. Within a given principal quantum number, the energy of orbitals increases in the order \(s<p<d<f\). For higher energy levels, these differences are sufficiently pronounced and straggering of orbital energy may result, e.g., \(4 s<3 d\) and \(6 s<5 d ; \quad 4 f<6 p\). In multielectron atoms, besides the presence of attraction between the electron and nucleus, there are repulsion terms between every electron and other electrons present in the atom. Thus the stability of an electron in a multi-electron atom is because total attractive interactions are more than the repulsive interactions. On the other hand, the attractive interactions of an electron increases with increase of positive charge \((Z e)\) on the nucleus. Due to the presence of electrons in the inner shells, the electron in the outer shell will not experience the full positive charge of the nucleus \((Z e)\). The effect will be lowered due to the partial screening of positive charge on the nucleus by the inner shell electrons. This is known as the shielding of the outer shell electrons from the nucleus by the inner shell electrons, and the net positive charge experienced by the outer electrons is known as effective nuclear charge \(\left(Z_{\text {eff }} e\right)\). Despite the shielding of the outer electrons from the nucleus by the inner shell electrons, the attractive force experienced by the outer shell electrons increases with increase of nuclear charge. In other words, the energy of interaction between, the nucleus and electron (that is orbital energy) decreases (that is more negative) with the increase of atomic number \((\boldsymbol{Z})\)

For a given shell (principal quantum number), the \(Z_{\text {eff }}\) experienced by the electron decreases with increase of azimuthal quantum number \((l)\), that is, the \(s\) orbital electron will be more tightly bound to the nucleus than \(p\) orbital electron which in turn will be better tightly bound than the d orbital electron. The energy of electrons in \(s\) orbital will be lower (more negative) than that of \(p\) orbital electron which will have less on. Since the extent of shielding from the nucleus is different for electrons in different orbitals, it leads to the splitting of energy levels within the same shell (or same principal quantum number), that is, energy of electron in an orbital, as mentioned earlier, depends upon the values of \(n\) and \(l\). Mathematically, the dependence of energies of the orbitals on \(n\) and \(l\) are quite complicated but one simple rule is that, the lower the value of \((n+l)\) for an orbital, the lower is its energy. If two orbitals have the same value of \((n+l)\), the orbital with lower value of \(n\) will have the lower energy. The Table \(2.5\) illustrates the ( \(n\) of multi-electrons atoms. 

Lastly it may be mentioned here that energies of the orbitals in the same subshell decrease with increase in the atomic number \(\left(\boldsymbol{Z}_{\text {eff }}\right)\). For example, energy of \(2 s\) orbital of hydrogen atom is greater than that of \(2 s\) orbital of lithium and that of lithium is greater than that of sodium and so on, that is, \(E_{2 s}(\mathrm{H})>E_{2 s}(\mathrm{Li})>E_{2 s}(\mathrm{Na})>E_{2 s}(\mathrm{~K})\).

Filling of Orbitals in Atom

The filling of electrons into the orbitals of different atoms takes place according to the aufbau principle which is based on the Pauli’s exclusion principle, the Hund’s rule of maximum multiplicity and the relative energies of the orbitals.

Aufbau Principle

The word ‘aufbau’ in German means ‘building up’. The building up of orbitals means the filling up of orbitals with electrons. The principle states: In the ground state of the atoms, the orbitals are filled in order of their increasing energies. In other words, electrons first occupy the lowest energy orbital available to them and enter into higher energy orbitals only after the lower energy orbitals are filled. As you have learnt above, energy of a given orbital depends upon effective nuclear charge and different type of orbitals are affected to different extent. Thus, there is no single ordering of energies of orbitals which will be universally correct for all atoms.

However, the following order of energies of the orbitals is extremely useful:
\(1 s, 2 s, 2 p, 3 s, 3 p, 4 s, 3 d, 4 p, 5 s, 4 d, 5 p, 4 f\), \(5 d, 6 p, 7 s \ldots\)

The order may be remembered by using the method given in Fig. 2.17. Starting from the top, the direction of the arrows gives the order of filling of orbitals, that is starting from right top to bottom left. With respect to placement of outermost valence electrons, it is remarkably accurate for all atoms. For example, valence electron in potassium must choose between 3d and 4s orbitals and as predicted by this sequence, it is found in 4s orbital.

Pauli Exclusion Principle

The number of electrons to be filled in various orbitals is restricted by the exclusion principle, given by the Austrian scientist Wolfgang Pauli (1926). According to this principle: No two electrons in an atom can have the same set of four quantum numbers. Pauli exclusion principle can also be stated as: “Only two electrons may exist in the same orbital and these electrons must have opposite spin.” This means that the two electrons can have the same value of three quantum numbers \(n, l\) and \(m_{l}\), but must have the opposite spin quantum number. The restriction imposed by Pauli’s exclusion principle on the number of electrons in an orbital helps in calculating the capacity of electrons to be present in any subshell. For example, subshell \(1 s\) comprises one orbital and thus the maximum number of \(p\) and \(d\) subshells, the maximum number of electrons can be 6 and 10 and so on. This can be summed up as : the maximum number of electrons in the shell with principal quantum number \(n\) is equal to \(2 n^{2}\).

Hund’s Rule of Maximum Multiplicity

This rule deals with the filling of electrons into the orbitals belonging to the same subshell (that is, orbitals of equal energy, called degenerate orbitals). It states: pairing of electrons in the orbitals belonging to the same subshell \((p, d\) or \(f)\) does not take place until each orbital belonging to that subshell has got one electron each i.e., it is singly occupied.

Since there are three \(p\), five \(d\) and seven \(f\) orbitals, therefore, the pairing of electrons will start in the \(p, d\) and \(f\) orbitals with the entry of 4 th, 6 th and 8th electron, respectively. It has been observed that half filled and fully filled degenerate set of orbitals acquire extra stability due to their symmetry.

Electronic Configuration of Atoms

The distribution of electrons into orbitals of an atom is called its electronic configuration. If one keeps in mind the basic rules which govern the filling of different atomic orbitals, the electronic configurations of different atoms can be written very easily. The electronic configuration of different atoms can be represented in two ways. For example:

(i) \(s^{a} p^{b} d^{c} \dots …………\) notation

(ii) Orbital diagram

In the first notation (i), the subshell is represented by the respective letter symbol, and the number of electrons present in the subshell is depicted, as the super script, like a, b, c, … etc. The similar subshell represented for different shells is differentiated by writing the principal quantum number before the respective subshell.

In the second notation (ii) each orbital of the subshell is represented by a box and the electron is represented by an arrow ( \(\uparrow\) ) a positive spin or an arrow \((\downarrow)\) a negative spin. The advantage of second notation over the first is that it represents all the four quantum numbers.

For example, hydrogen atom has only one electron which goes in the orbital with the lowest energy, namely \(1 \mathrm{~s}\). The electronic configuration of the hydrogen atom is \(1 \mathrm{~s}^{1}\) meaning that it has one electron in the 1s orbital. The second electron in helium ( \(\mathrm{He}\) ) can also occupy the 1 s orbital. Its configuration is, therefore, \(1 \mathrm{~s}^{2}\). As mentioned above, the two electrons differ from each other with opposite spin, as can be seen from the orbital diagram.

Writing Electron Configurations

Shells

The maximum number of electrons that can be accommodated in a shell is based on the principal quantum number \((n)\). It is represented by the formula \(2 n^{2}\), where ‘ \(n\) ‘ is the shell number. The shells, values of \(n\), and the total number of electrons that can be accommodated are tabulated below.

\(
\begin{array}{|l|l|}
\hline \text { Shell and ‘ } n \text { ‘ value } & \text { Maximum electrons present in the shell } \\
\hline \text { K shell, } n=1 & 2 * 1^{2}=2 \\
\hline \text { L shell, } n=2 & 2 * 2^{2}=8 \\
\hline \text { M shell, } n=3 & 2 * 3^{2}=18 \\
\hline \text { N shell, } n=4 & 2^{*} 4^{2}=32 \\
\hline
\end{array}
\)

 

Subshells

• The subshells into which electrons are distributed are based on the azimuthal quantum number (denoted by \({ }^{\prime} \mid\) ‘).
• This quantum number is dependent on the value of the principal quantum number, \(\mathrm{n}\). Therefore, when \(\mathrm{n}\) has a value of 4 , four different subshells are possible.
• When \(n=4\). The subshells correspond to \(I=0, I=1, I=2\), and \(l=3\) and are named the \(s, p, d\), and \(f\) subshells, respectively.
• The maximum number of electrons that can be accommodated by a subshell is given by the formula \(2^{\star}(21\) \(+1)\).
• Therefore, the s, p, d, and f subshells can accommodate a maximum of \(2,6,10\), and 14 electrons, respectively.

All the possible subshells for values of \(n\) up to 4 are tabulated below.

Thus, it can be understood that the \(1 \mathrm{p}, 2 \mathrm{~d}\), and \(3 \mathrm{f}\) orbitals do not exist because the value of the azimuthal quantum number is always less than that of the principal quantum number.

Notation

• The electron configuration of an atom is written with the help of subshell labels.
• These labels contain the shell number (given by the principal quantum number), the subshell name (given by the azimuthal quantum number) and the total number of electrons in the subshell in superscript.
• For example, if two electrons are filled in the ‘ \(s\) ‘ subshell of the first shell, the resulting notation is ‘ \(1 \mathrm{~s}^{2}\) ‘.
• With the help of these subshell labels, the electron configuration of magnesium (atomic number 12) can be written as \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2}\).

The third electron of lithium (Li) is not allowed in the \(1 s\) orbital because of Pauli exclusion principle. It, therefore, takes the next available choice, namely the \(2 s\) orbital. The electronic configuration of \(\mathrm{Li}\) is \(1 s^{2} 2 s^{1}\). The \(2 s\) orbital can accommodate one more electron. The configuration of beryllium \((\mathrm{Be})\) atom is, therefore, \(1 s^{2} 2 s^{2}\) (see Table \(2.6\)).

In the next six elements-boron \(\left(\mathrm{B}, 1 s^{2} 2 s^{2} 2 p^{1}\right)\), carbon \(\left(\mathrm{C}, 1 s^{2} 2 s^{2} 2 p^{2}\right)\), nitrogen ( \(\mathrm{N}, 1 s^{2} 2 s^{2} 2 p^{3}\) ), oxygen \(\left(\mathrm{O}, 1 s^{2} 2 s^{2} 2 p^{4}\right.\) ), fluorine \(\left(\mathrm{F}, 1 s^{2} 2 s^{2} 2 p^{5}\right.\) ) and neon \(\left(\mathrm{Ne}, 1 s^{2} 2 s^{2} 2 p^{6}\right.\) ), the \(2 p\) orbitals get progressively filled. This process is completed with the neon atom. The orbital picture of these elements can be represented as follows:

The electronic configuration of the elements sodium \(\left(\mathrm{Na}, 1 s^{2} 2 s^{2} 2 p^{6} 3 s^{1}\right)\) to argon \(\left(\mathrm{Ar}, 1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6}\right)\), follow exactly the same pattern as the elements from lithium to neon with the difference that the \(3 s\) and \(3 p\) orbitals are getting filled now. This process can be simplified if we represent the total number of electrons in the first two shells by the name of element neon (Ne). The electronic configuration of the elements from sodium to argon can be written as \(\left(\mathrm{Na},[\mathrm{Ne}] 3 s^{1}\right)\) to \(\left(\mathrm{Ar},[\mathrm{Ne}] 3 s^{2} 3 p^{6}\right)\). The electrons in the completely filled shells are known as core electrons and the electrons that are added to the electronic shell with the highest principal quantum number are called valence electrons. For example, the electrons in \(\mathrm{Ne}\) are the core electrons and the electrons from \(\mathrm{Na}\) to Ar are the valence electrons. In potassium (K) and calcium (Ca), the \(4 s\) orbital, being lower in energy than the \(3 d\) orbitals, is occupied by one and two electrons respectively.

Stability of Completely Filled and Half Filled Subshells

The ground state electronic configuration of the atom of an element always corresponds to the state of the lowest total electronic energy. The electronic configurations of most of the atoms follow the basic rules disussed in the above section. However, in certain elements such as \(\mathrm{Cu}\), or \(\mathrm{Cr}\), where the two subshells \((4 s\) and \(3 d\) ) differ slightly in their energies, an electron shifts from a subshell of lower energy \((4 s)\) to a subshell of higher energy \((3 d)\), provided such a shift results in all orbitals of the subshell of higher energy getting either completely filled or half filled. The valence electronic configurations of \(\mathrm{Cr}\) and \(\mathrm{Cu}\), therefore, are \(3 d^{5} 4 s^{1}\) and \(3 d^{10} 4 s^{1}\) respectively and not \(3 d^{4} 4 s^{2}\) and \(3 d^{9} 4 s^{2}\). It has been found that there is extra stability associated with these electronic configurations.

The completely filled and completely half-filled subshells are stable due to the following reasons:

• Symmetrical distribution of electrons: It is well known that symmetry leads to stability. The completely filled or half filled subshells have symmetrical distribution of electrons in them and are therefore more stable. Electrons in the same subshell (here 3d) have equal energy but different spatial distribution. Consequently, their shielding of one-another is relatively small and the electrons are more strongly attracted by the nucleus.
• Exchange Energy: The stabilizing effect arises whenever two or more electrons with the same spin are present in the degenerate orbitals of a subshell. These electrons tend to exchange their positions and the energy released due to this exchange is called exchange energy. The number of exchanges that can take place is maximum when the subshell is either half filled or completely filled (Fig. 2.18). As a result the exchange energy is maximum and so is the stability. You may note that the exchange energy is at the basis of Hund’s rule that electrons which enter orbitals of equal energy have parallel spins as far as possible. In other words, the extra stability of half-filled and completely filled subshell is due to: (i) relatively small shielding, (ii) smaller coulombic repulsion energy, and (iii) larger exchange energy. Details about the exchange energy will be dealt with in higher classes.