Advanced Chemistry - Mono Mole

January 13, 2026February 10, 2026

General selection rules for scattered radiation

Selection rules for scattered radiation describe the allowed changes in a molecule’s state during light scattering.

According to the time-dependent perturbation theory, the transition probability $\vert T\vert^2$ between the orthogonal states $\psi$ and $\psi'$ is proportional to the square of the corresponding transition matrix element $\vert\langle\psi'\vert\boldsymbol{\mathit{\hat{\mu}}}\vert\psi\rangle\vert^2$ . For example, in conventional rotational spectroscopy, this involves the electric dipole moment operator $\boldsymbol{\mathit{\hat{\mu}}}$ , and transitions occur only for molecules with a permanent dipole moment.

Transitions associated with scattering radiation, however, arise from an induced dipole moment rather than a permanent one. It follows, with reference to eq1, that the transition probability is given by:

$\vert T\vert^2=\vert\langle\psi'\vert\boldsymbol{\mathit{\hat{\mu}}}_{ind}\vert\psi\rangle\vert^2=E^2\vert\langle\psi'\vert\boldsymbol{\mathit{\alpha}}\vert\psi\rangle\vert^2$

Thus, a necessary condition for scattered-radiation transitions is that the matrix element $\boldsymbol{\mathit{E}}\vert\langle\psi'\vert\boldsymbol{\mathit{\alpha}}\vert\psi\rangle$ be non-zero. To examine this condition further, we consider how the molecular polarisability $\boldsymbol{\mathit{\alpha}}$ depends on the molecular coordinate $q$ , which may represent a rotational angle or a vibrational displacement. This dependence can be expressed through a Taylor series about the equilibrium configuration:

$\boldsymbol{\mathit{\alpha}}=\boldsymbol{\mathit{\alpha}}_0+\biggr$\frac{\partial\boldsymbol{\mathit{\alpha}}}{\partial q}\biggr$_0q+\cdots$

Substituting this series (and neglecting higher-order terms) into $T=\boldsymbol{\mathit{E}}\vert\langle\psi'\vert\boldsymbol{\mathit{\alpha}}\vert\psi\rangle$ gives:

$T=\boldsymbol{\mathit{E}}\biggr\[\boldsymbol{\mathit{\alpha}}_0\langle\psi'\vert\psi\rangle+\biggr$\frac{\partial\boldsymbol{\mathit{\alpha}}}{\partial q}\biggr$_0\langle\psi'\vert q\vert\psi\rangle\biggr\]$

Because the wavefunctions are orthogonal, the first term vanishes unless $\psi'=\psi$ , which corresponds to Rayleigh scattering. Consequently, a Raman transition can occur only if $\biggr$\frac{\partial\boldsymbol{\mathit{\alpha}}}{\partial q}\biggr$_0\neq 0$ , i.e. the polarisability of the molecule must change as it undergoes rotational or vibrational motion.

When $\psi'=\psi$ , we have $\vert T\vert^2=\vert T\vert^2_{Rayleigh}=\vert\boldsymbol{\mathit{E}}\vert^2\vert\boldsymbol{\mathit{\alpha}}_0\vert^2$ , which is the transition probability for Rayleigh scattering. When $\psi'\neq\psi$ , we have the transition probability for Raman scattering:

$\vert T\vert^2=\vert T\vert^2_{Raman}=\vert\boldsymbol{\mathit{E}}\vert^2\biggr\vert\biggr$\frac{\partial\boldsymbol{\mathit{\alpha}}}{\partial q}\biggr$_0\biggr\vert^2\vert\langle\psi'\vert q\vert\psi\rangle\vert^2$

Since $\biggr$\frac{\partial\boldsymbol{\mathit{\alpha}}}{\partial q}\biggr$_0$ represents a small perturbation of the electron cloud of the molecule, $\biggr\vert\biggr$\frac{\partial\boldsymbol{\mathit{\alpha}}}{\partial q}\biggr$_0\biggr\vert^2\ll \vert\boldsymbol{\mathit{\alpha}}_0\vert^2$ . Furthermore, the magnitude of the transition matrix element $\langle\psi'\vert q\vert\psi\rangle$ is never large: for vibrational Raman scattering it is proportional to the small zero-point vibrational amplitude, while for rotational Raman scattering it is a dimensionless angular matrix element bounded by unity. Consequently,

$\vert T\vert^2_{Raman}\ll\vert T\vert^2_{Rayleigh}$

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Raman spectroscopy (instrumentation)

Raman spectroscopy relies on a compact yet sophisticated set of instruments designed to probe molecular rovibrational states through photon-matter interactions.

The laser provides monochromatic, coherent and intense light required to induce Raman scattering. Common laser types include diode lasers, Nd:YAG lasers and argon-ion lasers, operating at wavelengths ranging from the ultraviolet to the near-infrared region. The sample compartment is designed to accommodate solids, liquids or gases with minimal preparation. Raman instruments may use backscattering (180°), right-angle (90°) or forward-scattering geometries, depending on the application. In the above diagram, scattered radiation perpendicular to the laser beam is focused onto a pinhole.

The pinhole acts as a spatial filter that controls the amount of light arriving at the detector and helps define the resolution of the final spectrum. After passing through the slit, the scattered radiation still contains a mixture of Rayleigh, Stokes and anti-Stokes frequencies and is therefore dispersed into its individual wavelength components using a reflective diffraction grating. Each groove on the grating acts as a secondary source of light, generating spherical wavefronts in accordance with Huygen’s principle. Neighbouring reflected wavefronts interfere with one another to produce constructive diffraction patterns as defined by:

$n\lambda=d(sin\theta_i+sin\theta_m)\;\;\;\;\;\;\;\;3$

where

$n$ is the order of the diffraction (with the detector typically positioned to record $n=1$ ).
$\lambda$ is the wavelength of the light.
$d$ is the spacing between the grooves.
$\theta_i$ is the angle between the incident ray and grating’s normal vector.
$\theta_m$ is the angle between the diffracted ray and grating’s normal vector.

Question

Derive eq3.

Answer

With respect to the above diagram, the difference in path length between two adjacent incident rays is $dsin\theta_i$ , and that between two adjacent diffracted rays is $dsin\theta_m$ . Since the incident ray and the diffracted ray are on opposite sides of the normal, the total path difference is $d(sin\theta_i+sin\theta_m)$ , assuming that both angles are positive. This difference must be an integer multiple of the light’s wavelength for constructive interference to occur. Therefore, $n\lambda=d(sin\theta_i+sin\theta_m)$ .

Since the groove spacing is fixed, each wavelength is diffracted at a unique angle, causing light reflected from different grooves to travel slightly different path lengths. Constructive interference occurs when these wavefronts arrive in phase, producing intensity maxima at specific angles, while destructive interference reduces or cancels the signal at other angles. This process results in the spatial separation of wavelengths at the focal plane of the instrument.

A charge-coupled device (CCD) is positioned at this focal plane to detect the dispersed light. Its location is precisely calculated to receive the first-order ( $n=1$ ) diffracted rays. The CCD consists of a two-dimensional array of light-sensitive pixels fabricated on a silicon substrate and is widely used due to its high sensitivity, low-noise quality, and ability to simultaneously record a broad spectral range. When photons strike a pixel, they are absorbed by the silicon, generating electron–hole pairs via the photoelectric effect. The number of electrons produced is proportional to the intensity of the incident light. Because each wavelength is focused onto a distinct position on the detector, individual pixels record the intensity corresponding to a single Raman frequency, yielding the final spectrum of intensity $I$ versus frequency (or versus Raman shift $\Delta\nu=\nu_{incident}-\nu_{scattered}$ ).

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January 13, 2026March 2, 2026

Raman scattering

Raman scattering is the inelastic scattering of light by molecules in which the exchanged energy reveals information about the molecules’ vibrational or rotational states.

This effect arises from the interaction between incident light and the internal energy structure of molecules, and is best understood by contrasting it with ordinary absorption. When a photon collides with a molecule and its frequency corresponds exactly to the energy difference between two stationary molecular states, the molecule can absorb the photon and undergo a real electronic, vibrational or rotational transition. In this situation, the photon’s energy matches the gap between two eigenstates of the unperturbed molecular Hamiltonian, making true absorption possible.

However, most photons do not have energies that coincide with any allowed transition. In such cases, absorption cannot occur, yet the photon may still interact with the molecule. This is the domain of Raman scattering, which is an inelastic scattering process rather than a resonant absorption phenomenon. The mechanism begins when the electric field associated with the incident photon interacts with the molecule’s charge distribution. Although the photon is detected as a particle, it propagates as an electromagnetic wave, and its oscillating electric field transiently polarises the molecule. In other words, an applied electric field $\boldsymbol{\mathit{E}}$ can distort the molecule and induce an electric dipole moment $\boldsymbol{\mathit{\mu}}_{ind}$ , with a stronger field resulting in a larger dipole moment for a given molecule:

$\boldsymbol{\mathit{\mu}}_{ind}=\boldsymbol{\mathit{\alpha}}\boldsymbol{\mathit{E}}\;\;\;\;\;\;\;\;1$

where $\boldsymbol{\mathit{\alpha}}$ is the polarisability of the molecule, which is a measure of the degree to which the electrons in the molecule can be displaced relative to the nuclei.

Since $\boldsymbol{\mathit{E}}=\boldsymbol{\mathit{i}}E_x+\boldsymbol{\mathit{j}}E_y+\boldsymbol{\mathit{k}}E_z$ and $\boldsymbol{\mathit{\mu}}_{ind}=\boldsymbol{\mathit{i}}\mu_x+\boldsymbol{\mathit{j}}\mu_y+\boldsymbol{\mathit{k}}\mu_z$ , eq1 can also be written in matrix form as:

$\begin{pmatrix}\mu_x\\\mu_y\\\mu_z\end{pmatrix}=\begin{pmatrix}\alpha_{xx}&\alpha_{xy}&\alpha_{xz}\\\alpha_{yx}&\alpha_{yy}&\alpha_{yz}\\\alpha_{zx}&\alpha_{zy}&\alpha_{zz}\end{pmatrix}\begin{pmatrix}E_x\\E_y\\E_z\end{pmatrix}\;\;\;\;\;\;\;\;2$

Here, $\alpha_{mn}$ is the $mn$ -component of $\boldsymbol{\mathit{\alpha}}$ , where $n$ denotes the direction of the applied electric field and $m$ denotes the direction of the induced dipole moment component.

Question

Why is $\boldsymbol{\mathit{\alpha}}$ expressed as a rank-two tensor, known as the polarisability tensor, in eq2?

Answer

In general, the spatial distribution of electrons in a molecule is constrained by the geometry of the nuclei and the bonding orbitals. Consequently, an electric field applied in one direction can induce a redistribution of electron density in any direction permitted by those constraints, as is the case for a non-spherically symmetric molecule such as glycerol (C₃H₈O₃). As an example, if the electric field lies along the laboratory $z$ -axis, then the distortion of the electron cloud are measured by the polarisability components $\alpha_{xz}$ , $\alpha_{yz}$ and $\alpha_{zz}$ .

For spherically symmetric particles, such as an atom or a spherical rotor, the same distortion is induced regardless of the direction of the applied field, resulting in all off-diagonal elements of the tensor being zero and the diagonal elements being equal, $\alpha_{xx}=\alpha_{yy}=\alpha_{zz}$ (see above diagram). The polarisability tensor of a diatomic molecule, whether homonuclear or heteronuclear, also has zero off-diagonal elements. However, $\alpha_{xx}=\alpha_{yy}=\alpha_{\perp}$ while $\alpha_{zz}=\alpha_{\parallel}$ , where the $z$ -axis is chosen to be the molecular axis. Typically, $\alpha_{\parallel}=\alpha_{\perp}$ .

Furthermore, $\boldsymbol{\mathit{\alpha}}$ is symmetric (e.g. $\alpha_{xy}=\alpha_{yx}$ ) because the work done to polarise a molecule depends only on the final state of the electric field, not the path taken to reach it. If a field is applied first in the $x$ -direction and then the $y$ -direction, the net change in potential energy due to the molecule’s polarisability must be the same as if the fields were applied in the reverse order.

The redistributed electrons transform the molecule from its ground state into a short-lived virtual energy state. Unlike a real excited state, this state is an eigenstate of the perturbed Hamiltonian. Because it exists only during the presence of the perturbing field, the virtual state lasts on the order of femtoseconds, with its existence permitted by the time–energy uncertainty principle. Since the virtual state is not stable, the molecule quickly relaxes. If it relaxes back to the same stationary state it occupied before the interaction, the photon re-emitted has the same energy as the incident photon. This elastic process is known as Rayleigh scattering, and it accounts for the vast majority of scattered photons (see diagram below).

But if the molecule relaxes to a different stationary vibrational or rotational state than the one it started in, energy must be conserved by adjusting the energy of the scattered photon accordingly. If the molecule ends up in a state of higher internal energy, the scattered photon necessarily loses that amount of energy and emerges with a lower frequency (Stokes scattering). Conversely, if the molecule relaxes to a lower one, the scattered photon gains the excess internal energy and is emitted with a higher frequency (anti-Stokes scattering). These two types of scattering are collectively known as the Raman effect or Raman scattering.

Thus, Raman scattering provides a window into the vibrational and rotational structure of molecules: the energy shifts — Stokes and anti-Stokes — encode the quantised energy differences between molecular states. By measuring these shifts, Raman spectroscopy reveals the “fingerprint” of molecular vibrations, enabling powerful chemical identification and structural analysis without requiring the incident light to resonate with any real molecular transition.

However, due to the instantaneous orientation of the molecule, different photons colliding with it may lead to different extents of polarisation according to the polarisation tensor. The transition probability for Raman scattering is even lower, at about one in ten million. Therefore, Raman spectroscopy requires a high-power laser source, with the complete instrumentation described in the next article.

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January 13, 2026February 10, 2026

Determining infrared and Raman activities of molecular vibrations using character tables

Character tables provide a systematic group-theoretical framework for determining whether molecular vibrational modes are infrared– or Raman-active by examining how those modes transform relative to the electric dipole moment and polarisability operators.

The first step is to work out the symmetries of the normal modes of vibration of a molecule. For example, H₂O lying in the $xz$ -plane belongs to the $C_{2v}$ point group. Two of its three normal modes (symmetric stretch and bend), as explained in an earlier article, transform according to the $A_1$ irreducible representation, while the remaining one (asymmetric stretch) transforms according to $B_1$ .

The $C_{2v}$ character table (see above table) shows that the linear functions (or Cartesian coordinates) $z$ and $x$ also transform according to $A_1$ and $B_1$ respectively. This correspondence means that the collective atomic displacements associated with each of the three H₂O normal modes transform in the same way under the symmetry operations of the group as the corresponding Cartesian vector component.

The second step is to recognise that an IR-active vibrational transition requires a change in the molecular dipole moment during the vibration. The dipole moment is a vector quantity whose non-zero components transform as the linear functions $x$ , $y$ and $z$ . If a normal mode transforms according to the same irreducible representation as one of these components, then the corresponding component of the dipole moment changes as the atoms are displaced from their equilibrium positions.

Therefore, the modes that transform according to $A_1$ and $B_1$ are IR-active.

For vibrations to be Raman-active, we refer to the quadratic functions because the components of the polarisability tensor transform in the same way as these functions. In this case, both the $A_1$ and $B_1$ modes are Raman-active.

Question

Why are IR and Raman activities mutually exclusive for vibrational modes of centrosymmetric molecules?

Answer

See this article for explanation.

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January 13, 2026May 17, 2026

Determining the normal modes of CO₂ using the descent of symmetry method

The descent of symmetry method offers an efficient approach to determining the normal modes of CO₂.

The conventional procedure for determining the normal modes of a molecule involves the following steps:

1. Forming a basis set of unit displacement vectors $B=(x_1,y_1,z_1,x_2,y_2,z_2,\cdots)$ representing the instantaneous motions of the atoms.
2. Determining geometrically the transformed vectors $B'=(x'_1,y'_1,z'_1,x'_2,y'_2,z'_2,\cdots)$ for each symmetry operation associated with the point group of the molecule.
3. Constructing the symmetry operation matrix $R$ such that $B'=RB$ .
4. Evaluating the trace of each matrix $R$ to obtain the character of the reducible representation, and decomposing each representation to determine the irreducible representations of the normal modes of the molecule.

However, CO₂ is a linear molecule belonging to the $D_{\infty h}$ point group (click this link for details), whose infinitely many symmetry elements render the above procedure impractical. The descent of symmetry method circumvents this difficulty by replacing $D_{\infty h}$ with a suitable finite subgroup, while preserving the essential symmetry properties of all $3N-9$ degrees of freedom. This is justified because

- the symmetry operations of both the parent group and its subgroup return the molecule to physically indistinguishable configurations; and
- both the parent group and its subgroup act on the same set of basis functions (e.g. the Cartesian displacement vectors), from which reducible representations of either group are generated and subsequently decomposed to obtain the irreducible representations of the respective group.

As a result, all the normal modes of a molecule that transform according to certain irreducible representations of a parent group must also transform according to irreducible representations of the subgroup. The final step is to match the irreducible representations of both groups through a process known as correlation.

To proceed, we select an appropriate subgroup of $D_{\infty h}$ for constructing the transformation matrices. While not strictly required, it is generally advantageous for the chosen subgroup to contain symmetry elements belonging to the same classes as those of the parent group, in order to minimise ambiguity during the correlation step. A common and convenient choice is the $D_{2h}$ point group, which includes the key symmetry operations of $E$ , $C_{2}$ , $\sigma$ and $i$ preserved from the parent group.

Next, we follow steps 1 to 3 by imposing the $D_{2h}$ symmetry operations on the set of nine unit displacement vectors $B=(x_1,y_1,z_1,x_2,y_2,z_2,x_3,y_3,z_3)$ . For example, the transformation matrix for inversion $i$ is:

$R_i=\begin{pmatrix}0&0&0&0&0&0&-1&0&0\\0&0&0&0&0&0&0&-1&0\\0&0&0&0&0&0&0&0&-1\\0&0&0&-1&0&0&0&0&0\\0&0&0&0&-1&0&0&0&0\\0&0&0&0&0&-1&0&0&0\\-1&0&0&0&0&0&0&0&0\\0&-1&0&0&0&0&0&0&0\\0&0&-1&0&0&0&0&0&0\end{pmatrix}$

where $B'=R_iB$ .

The eight transformation matrices form a reducible representation $\Gamma_{3N}$ of the $D_{2h}$ point group with the following traces:

Using eq27a, $\Gamma_{3N}$ is decomposed to $\Gamma_{3N}=A_g\oplus B_{2g}\oplus B_{3g}\oplus 2B_{1u}\oplus 2B_{2u}\oplus 2B_{3u}$ . Subtracting the translational modes ( $B_{1u},B_{2u},B_{3u}$ ) and the rotational modes ( $B_{2g},B_{3g}$ ) gives the vibrational representation:

$\Gamma_{vib}=A_g\oplus B_{1u}\oplus B_{2u}\oplus B_{3u}$

The correlation step involves comparing the characters of symmetry operations common to both $D_{\infty h}$ and $D_{2h}$ , using their respective character tables. In particular:

- Since rotation about the $z$ -axis (intermolecular axis) is not a physical degree of freedom for a linear molecule, characters associated with $C_{2}(z)$ in $D_{2h}$ are not used in the correlation. Instead, the characters of $\infty C'_2$ in $D_{\infty h}$ are compared with those of $C_2(x)$ and $C_{2}(y)$ in $D_{2h}$ .
- Only the reflection planes $\sigma_{xz}$ and $\sigma_{yz}$ in $D_{2h}$ correspond to $\infty\sigma_v$ in $D_{\infty h}$ ( $\sigma_{xy}$ does not).

This leads to the following correlation between the irreducible representations of the two groups:

It is evident that the vibrational mode transforming according to the totally symmetric irreducible representation $A_g$ in $D_{2h}$ , and correspondingly according to $\Sigma^+_g$ (also totally symmetric) in $D_{\infty h}$ , represents the symmetric stretching motion of CO₂.

Two of the four vibrational modes are degenerate and transform according to $\Pi_u$ when the full $D_{\infty h}$ symmetry of CO₂ is considered. These two modes describe bending motions in mutually perpendicular planes and share the same vibrational frequency. Upon descending from $D_{\infty h}$ to $D_{2h}$ , this degeneracy is lifted, and the doubly degenerate bending mode splits into two symmetry-distinct components transforming as $B_{2u}$ and $B_{3u}$ .

The remaining vibrational mode transforms according to $\Sigma^+_u$ and is antisymmetric with respect to inversion, identifying it as the antisymmetric stretching mode.

Question

How do we determine the IR and Raman activities of CO₂ vibrational modes?

Answer

See this article for details.

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January 13, 2026February 10, 2026

Exclusion rule in vibrational spectroscopy

The exclusion rule in vibrational spectroscopy states that, for molecules with a centre of symmetry, a vibrational mode that is infrared-active is Raman-inactive, and a mode that is Raman-active is infrared-inactive.

From a group‐theoretical perspective, the exclusion rule follows from the symmetry properties of the operators that govern infrared and Raman transitions. A vibrational mode is IR-active if its irreducible representation transforms like one of the linear basis functions (Cartesian coordinates) $x$ , $y$ , or $z$ , because the electric dipole moment operator has the same symmetry. It is Raman-active if its irreducible representation appears in the symmetry of the quadratic basis functions $x^2$ , $y^2$ , $z^2$ , $xy$ , $xz$ , or $yz$ , which correspond to components of the polarisability tensor.

A point group to which a centrosymmetric molecule belongs always includes the inversion symmetry operator. Each of the basis functions $x$ , $y$ , or $z$ changes sign (e.g. $x\rightarrow -x$ ) when it is inverted through the origin. On the other hand, quadratic basis functions, which are products of two linear functions, do not change sign upon inversion because both linear functions constituting the quadratic function change sign (e.g. $x^2\rightarrow(-x)^2\rightarrow x^2$ and $xy\rightarrow(-x)(-y)\rightarrow xy$ ).

This implies that the character corresponding to the inversion operation of an irreducible representation associated with a linear basis function is always different from that associated with a quadratic basis function. It follows that irreducible representations that are ungerade (u) can be IR-active but cannot be Raman-active, while irreducible representations that are gerade (g) can be Raman-active but cannot be IR-active. In other words, no vibrational mode of centrosymmetric molecules can be both IR- and Raman-active, which is the essence of the mutual exclusion rule.

This principle is clearly illustrated by the vibrational spectrum of CO₂, which has four vibrational modes belonging to the $D_{\infty h}$ point group. The symmetric stretching vibration transforms as the $\Sigma^+_g$ irreducible representation and is therefore Raman-active but IR-inactive. In contrast, the asymmetric stretching vibration transforms as $\Sigma^+_u$ , which only has the same symmetry as the $z$ function, making it IR-active but Raman-inactive. The two bending vibrational modes are doubly degenerate and transform as the $\Pi_u$ irreducible representation, which corresponds to the $x$ and $y$ functions. Consequently, the bending modes are also IR-active and Raman-inactive. Thus, no mode is simultaneously IR- and Raman-active.

Accordingly, the IR spectrum of CO₂ contains absorption bands corresponding to the asymmetric stretch and the bending vibrations, whereas the Raman spectrum shows a strong band associated with the symmetric stretch (see diagram below).

Question

Is the vibrational mode of O₂ IR-active or Raman-active?

Answer

O₂ vibrates by simply moving its two atoms closer and farther apart. Since both atoms are identical, this stretching mode is perfectly symmetric and transforms according to the $\Sigma^+_g$ irreducible representation of the $D_{\infty h}$ point group. Therefore, it is Raman-active but not IR-active.

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December 6, 2025December 7, 2025

Tight-binding model

The tight-binding model incorporates the Hückel approximation to describe the electronic structure of solids by treating electrons as localised around atoms and hopping between neighbouring sites.

Consider a linear chain of $N$ identical atoms located at position $n=1,2,\cdots,N$ , with $\vert n\rangle$ representing the s-orbital of atom $n$ . According to the Hückel approximation, the Hamiltonian $\hat{H}$ is defined by two parameters:

1) $\alpha=\langle n\vert\hat{H}\vert n\rangle$ is the energy of an electron localised on a single atom.

2) $\beta=\langle n\vert\hat{H}\vert n\pm 1\rangle$ is the energy associated with an electron hopping between adjacent atoms.

3) All other matrix elements are zero.

To find the eigenvalues $E$ and eigenstates $\vert \psi\rangle$ satisfying the Schrödinger equation $\hat{H}\vert\psi\rangle=E\vert\psi\rangle$ , we adopt the linear combination of atomic orbitals (LCAO) approach, in which

$\vert \psi\rangle=\sum_{n=1}^Nc_n\vert n\rangle\;\;\;\;\;\;\;\;1$

Multiplying the Schrödinger equation on the left by the bra $\langle n\vert$ gives:

$\langle n\vert\hat{H}\vert\psi\rangle=E\langle n\vert\psi\rangle\;\;\;\;\;\;\;\;2$

Substituting eq1 into eq2, and noting that the states are orthonormal, yields:

$\sum_{m=1}^Nc_m\langle n\vert\hat{H}\vert m\rangle=E\sum_{m=1}^Nc_m\langle n\vert m\rangle=Ec_n\;\;\;\;\;\;\;\;3$

Expanding the first summation in eq3 and applying the Hückel approximation results in:

$\beta c_{n-1}+\alpha c_n+\beta c_{n+1}=Ec_n\;\;\;\;\;\;\;\;4$

Substituting the trial solution $c_n=Asin(n\theta)$ into eq4, and imposing the boundary conditions of $c_0=c_{N+1}=0$ , gives:

$\beta Asin\[(n-1)\theta\]+\alpha Asin(n\theta)+\beta Asin\[(n+1)\theta\]=EAsin(n\theta)$

This rearranges to:

$\beta A\{sin\[(n-1)\theta\]+sin\[(n+1)\theta\]\}=(E-\alpha)Asin(n\theta)$

Using the trigonometric identity $sin(A-B)+sin(A+B)=2sinAcosB$ , with $A=n\theta$ and $B=\theta$ yields:

$2\beta sin(n\theta)cos\theta=(E-\alpha)sin(n\theta)$

which simplifies to:

$E=\alpha+2\beta cos\theta\;\;\;\;\;\;\;\;5$

Since $c_{N+1}=0$ , we have $c_{N+1}=Asin\[(N+1)\theta\]=0$ . So, $(N+1)\theta=k\pi$ , which when substituted into eq5 gives:

$E_k=\alpha+2\beta cos\frac{k\pi}{N+1}\;\;\;\;\;\;\;\;6$

where $k=1,2,\cdots,N$ .

Although $k$ is an integer due to the condition $(N+1)\theta=k\pi$ , its specific range $k=1,2,\cdots,N$ in eq6 follows the fact that each eigenvalue $E_k$ is associated with one of $N$ linearly independent eigenstates $\vert\psi\rangle$ described by eq1.

Question

Show that $E_0$ and $E_{N+1}$ are trivial solutions.

Answer

If $k=0$ , then $\theta=0$ and $c_n=Asin(n\theta)=0$ everywhere, which is a trivial solution. If $k=N+1$ , then $\theta=\pi$ and $c_n$ is again zero everywhere. Therefore, the non-trivial solutions correspond to $k=1,2,\cdots,N$ .

An electron localised on an atom resides in a bound state with energy measured relative to the vacuum level ( $E=0$ ). Therefore, the conventional value of $\alpha=\langle n\vert\hat{H}\vert n\rangle$ is negative. For s-orbitals, $\beta$ is also negative because $\beta=\langle n\vert\hat{H}\vert n\pm 1\rangle$ corresponds to an integral of the form (positive wavefunction) $\times$ (negative potential) $\times$ (positive wavefunction). It follows that the lower-energy states occurs when $k$ is small in eq6 (so that the cosine term is close to +1 for large $N$ ). In other words, $E_1$ and $E_N$ represent the lowest and highest energy states of the system respectively (see diagram below).

One important application of the tight-binding model is its role in explaining band theory, which will be explored in the next article.

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Band theory

Band theory describes how atomic orbitals in a solid combine to form extended molecular orbitals, whose closely spaced energies effectively create continuous bands.

Consider a linear chain of $N$ identical atoms located at position $n=1,2,\cdots,N$ , with $\vert n\rangle$ representing the s-orbital of atom $n$ . When two atoms overlap, molecular orbital (MO) theory states that two MOs are formed: a bonding orbital and an antibonding orbital. For three atoms, three MOs are produced — bonding, non-bonding and antibonding — while the orbital overlap of four atoms results in four MOs: two bonding and two antibonding orbitals. In general, a chain of $N$ atoms generates $N$ molecular orbitals. If $N$ is large, the energy levels of these $N$ MOs merge to form an energy band (see diagram above).

Mathematically, the energies are derived using the tight-binding model:

$E_k=\alpha+2\beta cos\frac{k\pi}{N+1}$

where $\alpha=\langle n\vert\hat{H}\vert n\rangle$ , $\beta=\langle n\vert\hat{H}\vert n\pm 1\rangle$ and $k=1,2,\cdots,N$ .

To show that the energies merge to form an energy band, we analyse the energy separation:

$E_{K+1}-E_k=2\beta\biggr\[cos\frac{(k+1)\pi}{N+1}-cos\frac{k\pi}{N+1}\biggr\]\;\;\;\;\;\;\;\;7$

Using the trigonometric identity $cos(A+B)=cosAcosB-sinAsinB$ ,

$cos\frac{(k+1)\pi}{N+1}=cos\frac{k\pi}{N+1}cos\frac{\pi}{N+1}-sin\frac{k\pi}{N+1}sin\frac{\pi}{N+1}$

As $N\rightarrow\infty$ , $cos\frac{(k+1)\pi}{N+1}\rightarrow 1$ and $E_{k+1}-E_k\rightarrow 0$ in eq7.

For example, the 1s-orbitals of Na, which has the electronic configuration 1s²2s²2p⁶3s¹, overlap to form the 1s band. Similarly, the 2s orbitals form the 2s band, and so on (see diagram above). The various Na bands do not overlap because the energy differences between different Na atomic orbitals (AOs) are much greater than the separation $E_{N}-E_1$ within each type of AO. Another characteristic of the band-MO diagram of Na is that the 3s band is only partially filled. In contrast, the 3s and 3p bands of Mg overlap to form a continuous band. Nevertheless, this merged band is still only partially filled because each 3p AO is initially unoccupied.

In metals, the Fermi energy $E_F$ — the energy of the highest occupied electronic state at absolute zero — lies within a partially filled band. However, it is more meaningful to associate frontier energies with the Fermi level $\mu$ , defined as the energy level at which the probability of finding an electron is 50% at thermodynamic equilibrium according to the Fermi-Dirac distribution, for any temperature above 0 K.

Electronic states of metals near the Fermi energy at absolute zero arise from the overlap of many AOs and are extremely closely spaced in energy. Because the density of states varies only weakly in this region and the gaps between occupied and unoccupied states are extremely small, increasing the temperature from 0 K to room temperature excites only a tiny fraction of electrons into slightly higher-energy levels. This redistribution is so small that the energy at which the occupation probability is 50% at room temperature remains essentially the same as the Fermi energy at 0 K. Consequently, electrons can easily move into the empty states just above the Fermi level when an electric field is applied, giving metals their high electrical conductivity. This conductivity decreases at higher temperatures due to increased collisions between moving electrons and the vibrating lattice atoms (phonons), which disrupt the flow of charge.

On the other hand, solid Ne is a non-conductor. Each Ne atom has the electronic configuration 1s²2s²2p⁶, with the 2p band completely filled in the solid. The next available band (derived from 3s orbitals) lies far higher in energy, creating a large band gap. With no empty states near the top of the filled band, electrons cannot be thermally promoted into any empty states, and the material remains insulating. In such insulators, the Fermi level lies within the band gap.

The main types of semiconductors are intrinsic (undoped), p-type and n-type (see diagram above). They also possess a band gap between the valence band and the conduction band. For an intrinsic semiconductor, such as GaAs, the valence band is completely filled at absolute zero, while the conduction band is completely empty. Therefore, the Fermi level lies near the middle of the band gap. Although the gap (1.42 eV) is much larger than the thermal energy at room temperature (0.026 eV), it is still small enough that a small but statistically significant number of electrons can be thermally excited across it. Once promoted into the conduction band, these electrons can be driven by an applied electric field to produce a current. Unlike metals, the conductivity of semiconductors increases with temperature because the number of thermally generated charge carriers grow exponentially, easily outweighing the increased scattering from lattice vibrations.

When dopants are introduced into intrinsic semiconductors, they create additional electronic states within the band gap. In a p-type semiconductor, acceptor impurities introduce energy levels slightly above the valence band. At 0 K, these acceptor levels are empty, while the valence band remains fully occupied. It follows that the Fermi level lies between the valence band and the acceptor levels. Electrical conductivity arises when electrons are thermally promoted from the valence band into the acceptor levels, leaving behind mobile holes in the valence band. These holes serve as the majority carriers responsible for current flow.

n-type semiconductors contain donor levels positioned just below the conduction band. At 0 K, these donor levels are fully occupied by electrons supplied by the dopant atoms, with the Fermi level lying between the donor levels and the conduction band. When electrons are promoted from the donor levels into the conduction band, they become free to move, producing a current.

In conclusion, band theory provides a molecular-orbital perspective on how electronic structure governs conductivity across different materials and forms the foundation for devices such as transistors and solar cells.

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December 6, 2025December 7, 2025

Superconductor

A superconductor is a material that, below a certain critical temperature, conducts electricity with zero electrical resistance and expels magnetic fields.

The first superconductor was discovered in 1911, when Dutch physicist Heike Kamerlingh Onnes observed that mercury’s electrical resistance suddenly vanished at about 4.2 K. In the years that followed, additional elements were identified as superconductors, with critical temperatures $T_c$ below 23 K. These superconducting elements are mostly transition metals that are bunched into certain parts of the periodic table (elements highlighted in yellow in the above diagram). In contrast, the noble metals (such as copper, silver and gold) and the alkali metals generally do not become superconducting at ambient pressure.

Metals normally resist the flow of electricity. Electrons, the carriers of current, constantly scatter off vibrating atoms (phonons), impurities and even each other. Yet, below a certain temperature, some metals suddenly lose all electrical resistance.

How can electrons stop scattering entirely? The answer lies in a profound insight developed by John Bardeen, Leon Cooper and Robert Schrieffer, known as the BCS theory.

According to this theory, electrons stop acting like isolated particles. Instead, they form highly coordinated pairs that move in perfect harmony, allowing current to flow without energy loss. At first glance, the idea that electrons can pair seems impossible. After all, they repel each other electrically. But the key insight is that electrons can interact indirectly through the crystal lattice. When an electron moves through a lattice, it slightly attracts the positively charged ions nearby, creating a tiny distortion. This distortion, in turn, can attract a second electron (see diagram below). This effective attraction is extremely weak, only acts for electrons near the Fermi surface, and operates within a narrow energy range. Such a phonon-mediated interaction is generally isotropic (the same in all directions) in momentum space, or at least does not have a strong directional dependence.

In 1956, Leon Cooper showed that if even the tiniest attraction exists between two electrons just above the Fermi level, they will inevitably form a bound pair — a Cooper pair. Essentially, the enormous density of available electronic states near the Fermi surface acts like a magnifying glass for even the tiniest attractions. Because so many states are packed closely together, two electrons with a weak attractive interaction have an overwhelming number of ways to pair up. This abundance of possibilities amplifies the effect of the weak attraction, making it sufficient to bind electrons into a Cooper pair. Even interactions that would be negligible elsewhere in the energy spectrum become decisive near the Fermi surface, ensuring that pairing always occurs when conditions are right.

A Cooper pair isn’t two electrons clinging together like magnets. It is a quantum superposition extending over thousands of lattice spacings, characterised by:

- A total spin of 0 (singlet)
- A total momentum of 0 (electrons at $\boldsymbol{\mathit{k}}$ and $\boldsymbol{\mathit{-k}}$ )
- A highly delocalised wavefunction that strongly overlaps with others

The wavefunction of the Cooper pair is given by:

$\psi=\sum_{\boldsymbol{\mathit{k}}}g(\boldsymbol{\mathit{k}})c^{\dagger}_{\boldsymbol{\mathit{k}}\uparrow}c^{\dagger}_{\boldsymbol{\mathit{k}}\downarrow}\vert 0\rangle$

where

$g(\boldsymbol{\mathit{k}})$ is a function describing the amplitude
$c^{\dagger}_{\boldsymbol{\mathit{k}}\uparrow}$ is an operator that creates an electron with momentum $\boldsymbol{\mathit{k}}$ and spin $\uparrow$
$c^{\dagger}_{\boldsymbol{\mathit{k}}\downarrow}$ is an operator that creates an electron with momentum $\boldsymbol{\mathit{-k}}$ and spin $\downarrow$
$\vert 0\rangle$ is the vacuum state where the system has zero particles

Question

Why does the Cooper pair have zero momentum?

Answer

In conventional superconductors, a Cooper pair with a total linear momentum of zero minimises kinetic energy, making this configuration energetically favourable. As the phonon-mediated interaction is approximately isotropic in momentum space, the Cooper pair’s spatial wavefunction is spherical with zero angular momentum $L=0$ .

Electrons are fermions and the total wavefunction must be antisymmetric under exchange of the two electrons. The total wavefunction of the pair is the product of a spatial part, which is symmetric ( $L=0$ ), and a spin part, which must then be antisymmetric, forming a singlet.

Because the wavefunctions overlap so extensively, all Cooper pairs in the metal lock into a single quantum state. This creates a kind of “quantum super-traffic-jam,” where every pair is aware of every other pair through a shared phase. When billions of Cooper pairs occupy the same quantum state, they form a macroscopic quantum condensate, a unified wavefunction stretching across the entire metal. This condensate is rigid and tiny disturbances cannot knock individual pairs out of step.

How then is the material able to conduct electricity without resistance? Resistance arises when electrons scatter off imperfections or vibrations. Cooper pairs, however:

- Are spread out over huge distances
- Share a common quantum phase
- Cannot scatter individually

For a Cooper pair to scatter, all pairs must do so simultaneously, which requires a prohibitive amount of energy. Small impurities or low-energy phonons simply cannot disrupt this concerted motion. Once the condensate begins moving, it keeps moving forever unless something strong enough breaks the pairs. Furthermore, breaking a Cooper pair costs energy, creating the superconducting energy gap. This gap shields the condensate from thermal fluctuations, explaining why superconductors exhibit zero electrical resistance and persistent currents at low temperatures where phonon-mediated interactions are not disrupted by thermal motion of lattice ions. In other words, electrons in a superconductor below its critical temperature move together forever, without resistance.

This remarkable ability to carry persistent currents without energy loss is directly exploited in technologies that require stable, intense magnetic fields. For example, in magnetic resonance imaging (MRI) machines, niobium-titanium coils are cooled below their critical temperature (10 K) with liquid helium to form persistent currents. These currents generate strong and extremely stable magnetic fields, which are essential for producing high-resolution images of internal tissues. Because the currents circulate indefinitely without decay, MRI magnets do not require continuous power input to maintain their fields, making them highly efficient and reliable. Beyond medical imaging, the same principle is exploited in quantum computing, where persistent, lossless currents are critical.

The second defining property of a superconductor is its ability to expel magnetic fields from its interior, such that $\boldsymbol{\mathit{B}}_{inside}=0$ . This phenomenon is called the Meissner effect, and it occurs even if an external magnetic field was already present before the material was cooled.

In a normal conductor, magnetic fields penetrate the material almost uniformly (see diagram above), limited only by weak and short-lived effects such as induced eddy currents. If the external magnetic field changes with time, Faraday’s law generates eddy currents in the conductor, but these currents quickly decay because the material has finite electrical resistance. As they die out, they no longer oppose the applied field, allowing the magnetic field to fully penetrate the bulk.

In contrast, when a superconductor cools below $T_c$ , Cooper pairs form and create a coherent superconducting condensate. In the presence of an external magnetic field, the condensate develops spatial variations that raise its kinetic energy, which in turn induces circulating screening currents along the surface. These currents generate magnetic fields that exactly oppose the applied field inside the material. Because the resistance is zero, the screening currents persist indefinitely without any power source, maintaining complete magnetic field expulsion.

Consequently, when a magnet is brought near a superconducting material, the induced screening currents produce a magnetic field that repels the magnet. This repulsive force can counteract gravity, allowing the magnet to float or levitate above the superconductor. Conversely, if the superconductor is placed above a magnet, it can also hover, held in place by the interaction between its induced currents and the magnetic field.

This principle of magnetic levitation can be harnessed to create frictionless transportation. In 1986, high-temperature superconductors ( $T_c$ of up to 138 K) were discovered. These materials, typically containing copper oxide and other metals such as barium and yttrium, are exemplified by YBa₂Cu₃O_x( $T_c\approx 90\;K$ ), where $6.5\leq x\leq 7.2$ .

$4BaCO_3+Y_2(CO_3)_3+6CuCO_3\rightarrow 2YBa_2CO_3O_x+13CO_2$

In maglev trains, YBa₂Cu₃O_x magnets are mounted on the train and cooled inside well-insulated chambers using liquid nitrogen. These magnets interact with permanent magnets or electromagnets embedded in the track, causing the train to levitate and eliminating friction with the track. By carefully controlling the magnetic fields along the track, the train can also be propelled forward. Changing the position or intensity of the track’s magnetic fields induces forces on the superconducting magnets via electromagnetic induction, pushing or pulling the train along its path. The combination of frictionless levitation and contactless propulsion allows maglev trains to achieve very high speeds with minimal energy loss, making them a highly efficient transportation technology.

Question

Why is Yba₂Cu₃O_x a superconductor when the individual elements are not superconducting?

Answer

YBa₂Cu₃O_x has a layered, perovskite-like crystal structure, with alternating layers of CuO₂ planes and other layers containing yttrium and barium (see above diagram for the unit cell of YBa₂Cu₃O₇). The CuO₂ planes are where superconductivity primarily occurs, while the other layers act as charge reservoirs and provide structural support for the lattice. This arrangement creates an energetically favourable environment in which the electronic properties necessary for superconductivity can emerge.

Within the CuO₂ planes, strong hybridisation between copper $3d_{x^2-y^2}$ orbitals and oxygen $2p$ orbitals creates an extensive two-dimensional network of electronic states. The superconductivity itself is enabled by controlling the oxygen content through a process known as doping, which introduces holes (positively charged carriers) into the planes. These holes are the microscopic charge carriers that pair up to form Cooper pairs.

Furthermore, the two charge carriers that form a Cooper pairs in the CuO₂ planes are much more tightly bound than in conventional superconductors. This results in a very short coherence length (the characteristic size of a Cooper pair) of only a few nanometres, compared with the tens to hundreds of nanometres typical in phonon-mediated superconductors such as NbTi. The short coherence length reflects a much stronger effective attractive interaction, likely arising from magnetic (spin-fluctuation) mechanisms rather than phonons. This stronger pairing interaction is one of the key reasons YBa₂Cu₃Oₓ can sustain superconductivity at temperatures far higher than those of conventional superconductors.

In essence, superconductivity in YBa₂Cu₃O_x is not due to the individual elements, but emerges from the specific arrangement of copper and oxygen atoms in the planes, combined with proper doping and crystal structure.

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December 6, 2025February 22, 2026

Fermi-Dirac distribution

The Fermi–Dirac distribution gives the probability that a quantum state of a given energy is occupied by a fermion at any temperature above absolute zero, accounting for the Pauli exclusion principle.

It is essential for describing the behaviour of electrons in solids and underpins our understanding of electrical and thermal conductivity in metals and semiconductors. Because many modern technologies, such as transistors, lasers and integrated circuits, depend on the behaviour of electrons in materials, the Fermi–Dirac distribution is a foundational tool in both solid-state chemistry and electronic engineering.

To derive the Fermi-Dirac distribution, consider a system of fixed-volume containing electrons occupying discrete single-particle energy levels $E_i$ , each with degeneracy $g_i$ . If $n_i$ electrons occupy the $g_i$ available states of energy, where $0\leq n_i\leq g_i$ , then the total number of electrons $N$ and the total energy $E$ of the system are

$N=\sum_in_i\;\;\;\;\;\;\;\;1$

$E=\sum_in_iE_i\;\;\;\;\;\;\;\;2$

where $i=0,1,2,\cdots$

Replicas of this system form a microcanonical ensemble, meaning that only configurations $\{n_0,n_1,n_2,\cdots\}$ with the same fixed $N$ and $E$ are allowed.

Because electrons are indistinguishable fermions and each single-particle state can be occupied by at most one electron (Pauli exclusion principle), the number of ways $W_i$ to place $n_i$ electrons among the $g_i$ states at energy $E_i$ is

$W_i=\frac{g_i!}{n_i!$g_i-n_i$!}\;\;\;\;\;\;\;\;3$

For example, if two electrons $n_i=2$ occupy three degenerate states ( $g_i=3$ ), the system can be found in any of the three microstates 110, 101, or 011 at different instants in time.

Question

Is eq3 a combination?

Answer

Yes, it is. It counts the number of ways to choose $n_i$ occupied states out of $g_i$ available single-particle states, where the electrons are indistinguishable (order does not matter) and the degenerate states are distinct because each state is defined by a unique set of quantum numbers.

It follows that the total number of microstates corresponding to a particular configuration is

$W_i=\prod_iW_i\;\;\;\;\;\;\;\;4$

Taking the natural logarithm of eq4 and substituting eq3 into it gives:

$lnW=lnW_1+lnW_2+\cdots=\sum_ilnW_i=\sum_iln\frac{g_i!}{n_i!$g_i-n_i$!}$

Using Stirling’s approximation. where $lnx!=xlnx-x$ , yields:

$lnW=\sum_i\[g_ilng_i-$g_i-n_i$ln$g_i-n_i$-n_ilnn_i\]\;\;\;\;\;\;\;\;5$

The possible configurations that define the microcanonical ensemble are restricted by eq1 and eq2. For example, the configurations $\{N,0,0,\cdots\}$ and $\{N-1,1,0,\cdots\}$ generally have different total energies and therefore cannot both belong to the same microcanonical ensemble. Within the allowed set of configurations (see above diagram for an illustration), all corresponding microstates are equally probable. The most probable equilibrium configuration is therefore the one with the largest number of ways of achieving it, i.e. the one whose $W$ (or $lnW$ ) is maximal.

The total differential of $lnW$ is:

$dlnW=\sum_i\biggr$\frac{\partial lnW}{\partial n_i}\biggr$_{all\;other\;n\neq n_i}dn_i\;\;\;\;\;\;\;\;6$

Hence, we want to solve for $dlnW=0$ . With reference to Step 2 of the derivation of the Boltzmann distribution using the Lagrange method of undetermined multipliers, eq6 becomes:

$\sum_i\biggr\[\biggr$\frac{\partial lnW}{\partial n_i}\biggr$-a-bE_i\biggr\]dn_i=0\;\;\;\;\;\;\;\;7$

where we have chosen minus signs for the 2^nd and 3^rd terms for convenience.

Since $dn_i$ varies independently, eq7 holds only if each coefficient is zero:

$\biggr$\frac{\partial lnW}{\partial n_i}\biggr$-a-bE_i=0\;\;\;\;\;\;\;\;8$

Substituting eq5 into eq8 gives:

$\sum_j\biggr\[\frac{\partial g_jlng_j}{\partial n_i}-\frac{\partial$g_j-n_j$ln$g_j-n_j$}{\partial n_i}-\frac{\partial n_jlnn_j}{\partial n_i}\biggr\]-a-bE_i=0\;\;\;\;\;\;\;\;9$

where we have changed the summation index from $i$ to $j$ in eq9 to discriminate the summation variable from the differentiation variable.

Since $g_j$ does not depend on $n_i$ ,

$\sum_j\biggr\[-\frac{\partial$g_j-n_j$ln$g_j-n_j$}{\partial n_i}-\frac{\partial n_jlnn_j}{\partial n_i}\biggr\]-a-bE_i=0\;\;\;\;\;\;\;\;10$

All terms in the summation of eq10 goes to zero except when $j=i$ . So,

$-\frac{\partial$g_j-n_j$ln$g_j-n_j$}{\partial n_i}-\frac{\partial n_jlnn_j}{\partial n_i}-a-bE_i=0$

Carrying out the differentiation and rearranging the result yields:

$n_i=\frac{g_i}{e^{a+bE_i}+1}\;\;\;\;\;\;\;\;11$

As mentioned above, $n_i$ is the number of electrons, which is equivalent to the number of occupied states at that level. To transform eq11 into a probability distribution, we write:

$n_i=f_ig_i\;\;\;\;\;\;\;\;12$

where $f_i$ is the probability (a fraction between 0 and 1) that a single state at energy $E_i$ is occupied. Thus, $f_ig_i$ gives the expected number of electrons occupying that energy level.

Substituting eq12 into eq11 gives:

$f_i=\frac{1}{e^{a+bE_i}+1}\;\;\;\;\;\;\;\;13$

To evaluate the parameters $a$ and $b$ , consider a system (with energy $E$ ) of the microcanonical ensemble that is partitioned by a rigid but permeable divider into two subsystems A and B, where A and B have energies $E_A$ and $E_B=E-E_A$ respectively. The total entropy $S$E,V,N$$ of the system is

$S=S_A+S_B$

Because the total system is also isolated, the second law of thermodynamics demands that $S$ be maximised at equilibrium. The total differential of $S$ with respect to $E_A$ is:

$\frac{dS}{dE_A}=\biggr$\frac{\partial S_A}{\partial E_A}\biggr$_{V_A,N_A}+\biggr$\frac{\partial S_B}{\partial E_B}\biggr$_{V_B,N_B}\frac{dE_B}{dE_A}=0$

From $E_B=E-E_A$ , we have $\frac{dE_B}{dE_A}=-1$ . Thus, the condition for maximum entropy under energy exchange between the two subsystems at equilibrium is

$\biggr$\frac{\partial S_A}{\partial E_A}\biggr$_{V_A,N_A}=\;\biggr$\frac{\partial S_B}{\partial E_B}\biggr$_{V_B,N_B}\;\;\;\;\;\;\;\;14$

Eq14 suggests that the two subsystems share a common physical property at thermal equilibrium. If we regard the electrons in each of the subsystems as a collection of non-interacting particles that move freely, much like molecules in a gas, they collectively form an electron gas. We can then use the fundamental thermodynamic equation $dE_A=TdS_A-pdV_A+\mu dN_A$ to describe system A, where $\mu$ is the chemical potential (also known as the Fermi level) of the electron gas, which has units of energy per particle instead of energy per mole. Since $dV_A=0$ and $dN_A=0$ , we have $\biggr$\frac{\partial S_A}{\partial E_A}\biggr$_{V_A,N_A}=\frac{1}{T}$ . Similarly, $\biggr$\frac{\partial S_B}{\partial E_B}\biggr$_{V_B,N_B}=\frac{1}{T}$ . This means that the change in entropy with respect to the change in energy is the same throughout the system:

$\biggr$\frac{\partial S}{\partial E}\biggr$_{V,N}=\frac{1}{T}\;\;\;\;\;\;\;\;15$

Substituting the statistical entropy $S=klnW$ into eq15 yields:

$k\biggr$\frac{\partial lnW}{\partial E}\biggr$_{V,N}=\frac{1}{T}\;\;\;\;\;\;\;\;16$

Substituting eq8 into eq6 gives:

$dlnW=a\sum_idn_i+b\sum_iE_idn_i\;\;\;\;\;\;\;\;17$

Substituting the derivatives of eq1 and eq2 into eq17 results in:

$dlnW=adN+bdE\;\;\;\;\;\;\;\;18$

Substituting eq18 back into eq16 yields

$b=\frac{1}{kT}\;\;\;\;\;\;\;\;19$

Repeating the above logic, the total differential of $S$ with respect to $N_A$ , where $N_A+N_B=N$ , is:

$\frac{dS}{dN_A}=\biggr$\frac{\partial S_A}{\partial N_A}\biggr$_{V_A,E_A}+\biggr$\frac{\partial S_B}{\partial N_B}\biggr$_{V_B,E_B}\frac{dN_B}{dN_A}=0$

Substituting $\frac{dN_B}{dN_A}=-1$ into the above equation gives $\biggr$\frac{\partial S_A}{\partial N_A}\biggr$_{V_A,E_A}=\;\biggr$\frac{\partial S_B}{\partial N_B}\biggr$_{V_B,E_B}$ , and using the fundamental thermodynamic equation results in:

$\biggr$\frac{\partial S_A}{\partial N_A}\biggr$_{V_A,E_A}=\;\biggr$\frac{\partial S_B}{\partial N_B}\biggr$_{V_B,E_B}=-\frac{\mu}{T}=\biggr$\frac{\partial S}{\partial N}\biggr$_{V,E}\;\;\;\;\;\;\;\;20$

Substituting $S=klnW$ and eq18 into eq20 yields:

$a=-\frac{\mu}{kT}\;\;\;\;\;\;\;\;21$

Substituting eq19 and eq21 back into eq13 gives:

$f_i=\frac{1}{e^{\frac{E_i-\mu}{kT}}+1}\;\;\;\;\;\;\;\;22$

which is the Fermi-Dirac distribution function.

Eq22 gives the probability that an energy state $E_i$ is occupied by a fermion at temperature $T$ . It plays a central role in solid-state chemistry. When the energy of the state equals the Fermi level ( $E_i=\mu$ ), the occupancy becomes $f_i=0.5$ at any $T> 0$ . In other words, the Fermi level $\mu$ is the energy level at which the probability of finding an electron in a material is 50% at thermodynamic equilibrium for any temperature above 0 K.

The Fermi level, together with band theory, is particularly useful for understanding the electrical conductivity in metals. In semiconductors, it determines how the occupation of states in the conduction and valence bands changes with temperature and doping.

Question

What is the definition of Fermi energy?

Answer

The Fermi energy is the highest occupied single-particle state in a system of non-interacting electrons at 0 K. In metals, the Fermi level approaches the Fermi energy at absolute zero. Although the terms Fermi energy and Fermi level are often used interchangeably in chemistry and physics, especially when discussing properties at or near absolute zero, their strict theoretical definitions distinguish them.

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