Atomic Structure and Quantum Chemistry

Detailed Notes for CSS Chemistry

1. Introduction

Atomic structure and quantum chemistry provide the theoretical foundation for understanding the properties and behaviour of matter. Atomic structure concerns the composition of atoms, the distribution of electrons, and the relationship between electronic arrangement and chemical properties. Quantum chemistry applies the principles of quantum mechanics to explain atomic stability, chemical bonding, molecular structure, and the interaction of matter with electromagnetic radiation.

Classical physics successfully describes the motion of ordinary objects, but it cannot adequately explain the behaviour of electrons and other microscopic particles. According to classical electromagnetic theory, an electron moving around a nucleus should continuously radiate energy and eventually collapse into the nucleus. Actual atoms, however, remain stable. Similarly, excited atoms emit radiation at specific wavelengths rather than across a continuous range.

These observations required a fundamental change in scientific understanding. Quantum theory introduced discrete energy levels, wave–particle duality, and a probabilistic description of electron behaviour. Consequently, the classical picture of electrons travelling along definite paths gave way to the modern concept of atomic orbitals.

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2. Fundamental Structure of the Atom

2.1 Subatomic Particles and Nuclear Organisation

An atom consists of a small, dense nucleus surrounded by an electronic distribution. The nucleus contains positively charged protons and electrically neutral neutrons, while electrons carry a negative charge.

The proton and neutron have approximately equal masses, whereas the electron is much lighter. A proton has roughly 1,836 times the mass of an electron. Therefore, nearly all the mass of an atom is concentrated in its nucleus, although the electronic distribution determines most of its physical size and chemical behaviour.

The typical atomic radius is of the order of 10−10 m10^{-10}\,\text{m}, whereas nuclear dimensions are generally of the order of 10−1510^{-15} to 10−14 m10^{-14}\,\text{m}. This difference explains why most of the volume of an atom lies outside the nucleus.

2.2 Atomic Number, Mass Number, and Isotopes

The atomic number, ZZ, represents the number of protons in the nucleus and determines the identity of an element. A neutral atom contains an equal number of protons and electrons.

The mass number, AA, is the total number of protons and neutrons:

A=Z+NA=Z+N

where NN is the number of neutrons.

Isotopes are atoms of the same element that possess different numbers of neutrons. For example, hydrogen has three familiar isotopes: protium, deuterium, and tritium. Their similar electronic structures give them broadly similar chemical properties, although differences in mass produce measurable isotope effects.

Atomic structure must therefore distinguish between nuclear composition, which determines isotopic identity, and electronic structure, which primarily governs bonding and chemical reactivity.

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3. Development of Atomic Models

3.1 Thomson’s Atomic Model

Following the discovery of the electron, J. J. Thomson proposed that an atom consisted of a diffuse sphere of positive charge containing embedded electrons. The model explained electrical neutrality by balancing positive and negative charges.

However, it could not account for the concentration of positive charge in a small nucleus or explain the discrete spectral lines of atoms. Its importance lies mainly in establishing that atoms contain smaller charged particles.

3.2 Rutherford’s Nuclear Model

Rutherford’s interpretation of alpha-particle scattering showed that most alpha particles passed through thin metal foil with little deflection, while a small fraction underwent large deflections.

These observations indicated that most atomic volume was relatively empty and that positive charge and most of the mass were concentrated within a very small nucleus.

Rutherford’s model established the nuclear atom, but it did not explain atomic stability. A classical electron in an orbit is an accelerating charge and should radiate energy continuously. The resulting loss of energy should cause it to spiral into the nucleus.

The model also failed to explain why atomic spectra consist of individual lines. These limitations prepared the ground for Bohr’s model and, subsequently, quantum mechanics.

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4. Electromagnetic Radiation and the Electromagnetic Spectrum

4.1 Nature of Electromagnetic Radiation

Electromagnetic radiation consists of oscillating electric and magnetic fields. In a plane electromagnetic wave, these fields are perpendicular to each other and to the direction of propagation.

The main quantities used to describe radiation are wavelength, frequency, and amplitude. Wavelength, λ\lambda, is the distance between successive equivalent points on a wave. Frequency, ν\nu, is the number of oscillations passing a point per second.

In a vacuum:

c=λνc=\lambda\nu

where:

c≈3.00×108 m s−1c\approx 3.00\times10^8\,\text{m s}^{-1}

Wavelength and frequency are inversely related. Radiation with a shorter wavelength has a higher frequency.

4.2 Regions of the Electromagnetic Spectrum

The electromagnetic spectrum extends from radio waves to gamma rays. In order of increasing frequency and photon energy, the major regions are:

Radio→Microwave→Infrared→Visible→Ultraviolet→X-rays→Gamma rays\text{Radio}\rightarrow\text{Microwave}\rightarrow \text{Infrared}\rightarrow\text{Visible}\rightarrow \text{Ultraviolet}\rightarrow\text{X-rays}\rightarrow\text{Gamma rays}

Different regions interact with matter in different ways. Microwave radiation commonly produces molecular rotational transitions, infrared radiation produces vibrational transitions, and visible or ultraviolet radiation can produce electronic transitions.

These associations are useful generalisations rather than absolute boundaries. The actual transition energy depends on the atom or molecule under investigation.

4.3 Spectroscopic Significance

Spectroscopy studies how matter absorbs, emits, or scatters radiation. Because atoms and molecules possess characteristic energy levels, their spectra provide information about composition, structure, and bonding.

An electronic transition occurs when the energy difference between two states matches the energy of an absorbed or emitted photon:

ΔE=hν=hcλ\Delta E=h\nu=\frac{hc}{\lambda}

Thus, spectroscopy provides experimental access to the otherwise invisible energy structure of atoms and molecules.

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5. Planck’s Quantum Theory

5.1 The Problem of Blackbody Radiation

A blackbody is an ideal object that absorbs all incident electromagnetic radiation. Its emitted spectrum depends on temperature.

Classical physics failed to describe the observed distribution of blackbody radiation at short wavelengths. In particular, the Rayleigh–Jeans treatment predicted an unlimited increase in emitted energy at high frequencies, a failure known as the ultraviolet catastrophe.

5.2 Quantisation of Energy

Max Planck resolved this problem by proposing that the material oscillators responsible for radiation exchange energy in discrete amounts.

The energy quantum is:

E=hνE=h\nu

where hh, Planck’s constant, is:

h=6.62607015×10−34 J sh=6.62607015\times10^{-34}\,\text{J s}

In Planck’s original treatment, oscillator energies occurred in integral multiples:

En=nhνE_n=nh\nu

The central implication is that energy exchange at the microscopic level is not always continuous. Certain systems can absorb or release only specific amounts of energy.

5.3 Importance for Chemistry

Quantisation explains why atoms possess discrete energy states and why they absorb or emit radiation at particular frequencies.

It also establishes an important distinction between photon energy and light intensity. The energy of an individual photon depends on frequency. At a fixed frequency, increasing intensity increases the number of photons incident per unit area per unit time.

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6. The Photoelectric Effect

6.1 Experimental Observations

The photoelectric effect is the emission of electrons from a material when electromagnetic radiation of sufficiently high frequency strikes its surface.

Experiments established that electron emission requires a minimum frequency, called the threshold frequency. Below this frequency, ordinary photoelectric emission does not occur even when the incident intensity increases.

Above the threshold, greater intensity generally increases the number of emitted electrons, provided the other conditions remain unchanged. However, the maximum kinetic energy of the electrons depends on frequency rather than intensity.

6.2 Einstein’s Explanation

Einstein proposed that radiation transfers energy in discrete packets called photons. Each photon has energy hνh\nu.

An electron must acquire sufficient energy to overcome the work function, ϕ\phi, of the material. Any remaining energy appears as kinetic energy:

hν=ϕ+Kmax⁡h\nu=\phi+K_{\max}

Therefore:

Kmax⁡=hν−ϕK_{\max}=h\nu-\phi

At the threshold frequency, ν0\nu_0:

ϕ=hν0\phi=h\nu_0

The maximum kinetic energy can also be measured through the stopping potential, VsV_s:

Kmax⁡=eVsK_{\max}=eV_s

Hence:

eVs=hν−ϕeV_s=h\nu-\phi

A graph of stopping potential against frequency is linear, with slope h/eh/e.

6.3 Scientific Significance

The photoelectric effect demonstrates the particle-like character of light. Classical wave theory alone could not explain the threshold frequency or the dependence of electron kinetic energy on radiation frequency.

The phenomenon does not invalidate the wave description of light. Instead, it shows that light exhibits different measurable properties under different experimental conditions.

6.4 Worked Example

Suppose radiation of wavelength 300 nm300\,\text{nm} strikes a metal with a work function of 2.00 eV2.00\,\text{eV}.

Using:

E(eV)≈1240λ(nm)E(\text{eV})\approx\frac{1240}{\lambda(\text{nm})}

the incident photon energy is:

E=1240300=4.13 eVE=\frac{1240}{300}=4.13\,\text{eV}

Therefore:

Kmax⁡=4.13−2.00=2.13 eVK_{\max}=4.13-2.00=2.13\,\text{eV}

The corresponding stopping potential is approximately:

Vs=2.13 VV_s=2.13\,\text{V}

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7. Bohr’s Atomic Model

7.1 Principal Postulates

Bohr proposed that an electron in a hydrogen-like atom can occupy only certain allowed stationary orbits. While occupying one of these states, the electron does not continuously emit radiation.

Radiation is absorbed or emitted when the electron changes from one allowed state to another:

hν=∣Ef−Ei∣h\nu=|E_f-E_i|

Bohr also introduced quantisation of orbital angular momentum:

mevr=nℏm_evr=n\hbar

where:

ℏ=h2π\hbar=\frac{h}{2\pi}

and n=1,2,3,…n=1,2,3,\ldots.

These assumptions combined classical orbital motion with a new quantum restriction.

7.2 Derivation of the Allowed Radius

For a one-electron species with nuclear charge +Ze+Ze, electrostatic attraction provides the centripetal force:

mev2r=Ze24πε0r2\frac{m_ev^2}{r} = \frac{Ze^2}{4\pi\varepsilon_0r^2}

From angular-momentum quantisation:

v=nℏmerv=\frac{n\hbar}{m_er}

Substituting and rearranging gives:

rn=4πε0ℏ2mee2n2Zr_n=\frac{4\pi\varepsilon_0\hbar^2}{m_ee^2}\frac{n^2}{Z}

Thus:

rn=a0n2Z\boxed{r_n=a_0\frac{n^2}{Z}}

where:

a0=5.29×10−11 ma_0=5.29\times10^{-11}\,\text{m}

is the Bohr radius.

These expressions use the approximation of a stationary nucleus. More accurate calculations replace the electron mass with the electron–nucleus reduced mass.

7.3 Derivation of the Energy Levels

The kinetic energy of the electron is:

T=12mev2=Ze28πε0rT=\frac12m_ev^2 =\frac{Ze^2}{8\pi\varepsilon_0r}

Its electrostatic potential energy is:

V=−Ze24πε0rV=-\frac{Ze^2}{4\pi\varepsilon_0r}

Therefore, the total energy is:

E=T+V=−Ze28πε0rE=T+V=-\frac{Ze^2}{8\pi\varepsilon_0r}

Substitution of the allowed radius gives:

En=−13.6Z2n2 eV\boxed{E_n=-13.6\frac{Z^2}{n^2}\,\text{eV}}

The negative sign indicates that the electron is bound to the nucleus. The energy reference E=0E=0 corresponds to a free electron infinitely far from the nucleus.

For hydrogen:

E1=−13.6 eV,E2=−3.40 eV,E3=−1.51 eVE_1=-13.6\,\text{eV},\qquad E_2=-3.40\,\text{eV},\qquad E_3=-1.51\,\text{eV}

The levels become progressively closer together as nn increases.

7.4 Successes and Limitations

Bohr’s model explains the principal spectral lines and ionisation energies of hydrogen and hydrogen-like ions such as He+\mathrm{He^+} and Li2+\mathrm{Li^{2+}}.

However, it does not adequately explain many-electron atoms, spectral intensities, or the full details of line splitting. Its assumption of definite electron trajectories is also incompatible with the modern quantum description.

Bohr’s model is therefore historically and mathematically important, but its orbits should not be confused with quantum-mechanical orbitals.

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8. The Hydrogen Spectrum

8.1 Origin of Spectral Lines

An excited hydrogen atom emits a photon when its electron moves from a higher energy level to a lower one.

Combining Bohr’s energy expression with the photon-energy equation gives the Rydberg formula:

1λ=RH(1nf2−1ni2)\boxed{ \frac{1}{\lambda} = R_H\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right) }

where ni>nfn_i>n_f for emission and:

RH≈1.097×107 m−1R_H\approx1.097\times10^7\,\text{m}^{-1}

For a hydrogen-like ion, the corresponding expression includes a factor of Z2Z^2, with a small reduced-mass correction to the Rydberg constant.

8.2 Spectral Series

The series is determined by the final energy level.

The lines within a series converge as the initial quantum number increases. At the series limit, nin_i approaches infinity.

8.3 Worked Example: The First Balmer Line

For the transition ni=3n_i=3 to nf=2n_f=2:

1λ=1.097×107(14−19)\frac1\lambda = 1.097\times10^7 \left(\frac14-\frac19\right) 1λ=1.097×107(536)\frac1\lambda = 1.097\times10^7\left(\frac5{36}\right)

Hence:

λ≈6.56×10−7 m=656 nm\lambda\approx6.56\times10^{-7}\,\text{m} =656\,\text{nm}

This is the red hydrogen-alpha line.

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9. Wave–Particle Duality and de Broglie’s Hypothesis

9.1 Wave and Particle Properties

Light exhibits interference and diffraction, which are wave phenomena, while the photoelectric effect demonstrates discrete energy transfer.

Louis de Broglie extended this duality to matter. He proposed that a particle with momentum pp has an associated wavelength:

λ=hp\boxed{\lambda=\frac{h}{p}}

For a nonrelativistic particle:

λ=hmv\lambda=\frac{h}{mv}

The wavelength is appreciable for microscopic particles such as electrons but extraordinarily small for ordinary macroscopic objects.

9.2 Electron Diffraction

Electron-diffraction experiments, including the Davisson–Germer experiment, confirmed the wave properties of electrons. Electrons scattered by a crystal produced diffraction patterns consistent with de Broglie’s relation.

A crystal acts as a diffraction structure because the separation of its atomic planes is comparable to the wavelength of suitably accelerated electrons.

This evidence established that matter waves are experimentally significant rather than merely mathematical constructions.

9.3 Electrons Accelerated Through a Potential Difference

If an electron initially at rest is accelerated through a potential difference VV, then, in the nonrelativistic approximation:

eV=12mev2eV=\frac12m_ev^2

Therefore:

p=2meeVp=\sqrt{2m_eeV}

and:

λ=h2meeV\boxed{\lambda=\frac{h}{\sqrt{2m_eeV}}}

A convenient numerical form is:

λ(nm)≈1.227V(volts)\lambda(\text{nm})\approx\frac{1.227}{\sqrt{V(\text{volts})}}

For an accelerating potential of 150 V150\,\text{V}:

λ≈0.100 nm\lambda\approx0.100\,\text{nm}

This wavelength is comparable to atomic spacings in crystals.

9.4 Connection with Bohr’s Quantisation

A standing-wave interpretation of a circular Bohr orbit requires an integral number of wavelengths around the circumference:

2πr=nλ2\pi r=n\lambda

Using λ=h/(mv)\lambda=h/(mv):

mvr=nh2πmvr=\frac{nh}{2\pi}

This reproduces Bohr’s angular-momentum condition. However, it remains a historical bridge to quantum mechanics rather than the modern description of an electron’s motion.

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10. Heisenberg’s Uncertainty Principle

10.1 Mathematical Statement

Heisenberg’s uncertainty principle states that a quantum state cannot possess arbitrarily small spreads in both position and the corresponding component of momentum:

Δx Δpx≥ℏ2\boxed{\Delta x\,\Delta p_x\geq\frac{\hbar}{2}}

Here, Δx\Delta x and Δpx\Delta p_x are standard deviations of measurement outcomes for identically prepared systems.

For nonrelativistic motion:

Δx Δvx≥ℏ2m\Delta x\,\Delta v_x\geq\frac{\hbar}{2m}

10.2 Physical Interpretation

The uncertainty principle is not simply a statement about defective instruments. It reflects the mathematical and physical character of quantum states.

A wave with a precisely defined wavelength has a well-defined momentum but is spread over space. To localise a particle, waves of different wavelengths must be combined. This necessarily introduces a spread in momentum.

Therefore, precise localisation and precise momentum cannot be achieved simultaneously.

10.3 Consequences for Atomic Structure

A classical orbit requires a definite position and momentum at every instant. Quantum mechanics does not generally permit such a description for an electron in an atom.

Instead, it predicts probability distributions. The concept of an orbital therefore replaces the concept of a definite trajectory.

The uncertainty principle also helps explain why an electron cannot simply collapse into an arbitrarily small region near the nucleus. Extreme confinement would imply a large momentum spread and a correspondingly large kinetic-energy contribution.

10.4 Worked Example

If an electron is localised with:

Δx=1.0×10−10 m\Delta x=1.0\times10^{-10}\,\text{m}

then:

Δvx≥1.055×10−342(9.11×10−31)(1.0×10−10)\Delta v_x\geq \frac{1.055\times10^{-34}} {2(9.11\times10^{-31})(1.0\times10^{-10})}

Thus:

Δvx≥5.79×105 m s−1\Delta v_x\geq5.79\times10^5\,\text{m s}^{-1}

The substantial uncertainty illustrates why classical trajectories are unsuitable for atomic electrons.

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11. Wavefunctions and the Quantum-Mechanical Description

11.1 Meaning of the Wavefunction

A quantum state is represented by a wavefunction, usually denoted by ψ\psi. The wavefunction contains the information needed to predict the probabilities of measurement outcomes.

The wavefunction itself is not an ordinary material wave or a directly observable electron-density distribution. Its physical significance emerges through the Born interpretation:

∣ψ∣2=ψ∗ψ|\psi|^2=\psi^*\psi

For a single particle, this quantity represents probability density.

The probability of finding the particle within a small volume element dτd\tau is:

dP=∣ψ∣2dτdP=|\psi|^2d\tau

11.2 Normalisation

Because the particle must be somewhere in space, the total probability must equal one:

∫all space∣ψ∣2dτ=1\boxed{\int_{\text{all space}}|\psi|^2d\tau=1}

A wavefunction satisfying this condition is normalised.

For a physically acceptable bound state, the wavefunction must be square-integrable, single-valued, and consistent with the boundary conditions. In regions of finite, nonsingular potential, it and its first derivative are normally continuous.

11.3 Superposition

Quantum states can be combined through linear superposition:

ψ=c1ψ1+c2ψ2\psi=c_1\psi_1+c_2\psi_2

If ψ1\psi_1 and ψ2\psi_2 are orthonormal energy eigenstates, then ∣c1∣2|c_1|^2 and ∣c2∣2|c_2|^2 give the probabilities of obtaining their respective energies on measurement.

Superposition underlies interference and the construction of molecular orbitals from atomic basis functions.

11.4 Orbit and Orbital

An orbit is a definite path, as used in Bohr’s model. An orbital is a one-electron wavefunction.

The familiar drawings of orbitals usually show surfaces enclosing a chosen fraction of the probability distribution. These surfaces are visual representations; they are not rigid boundaries beyond which an electron cannot occur.

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12. Operators, Eigenvalues, and Expectation Values

12.1 Operators in Quantum Mechanics

Measurable physical quantities are represented mathematically by operators.

For motion along the xx-axis, the position operator is multiplication by xx:

x^=x\hat{x}=x

The momentum operator is:

p^x=−iℏ∂∂x\hat{p}_x=-i\hbar\frac{\partial}{\partial x}

The kinetic-energy operator is:

T^=−ℏ22m∇2\hat{T}=-\frac{\hbar^2}{2m}\nabla^2

The total-energy operator is the Hamiltonian:

H^=T^+V^\hat{H}=\hat{T}+\hat{V}

12.2 Eigenfunctions and Eigenvalues

An eigenvalue equation has the form:

A^ψ=aψ\hat{A}\psi=a\psi

Here, ψ\psi is an eigenfunction of A^\hat{A}, and aa is the corresponding eigenvalue.

If a system is in an eigenstate of an observable, measuring that observable yields the corresponding eigenvalue with certainty in the ideal theory.

For energy:

H^ψ=Eψ\hat{H}\psi=E\psi

12.3 Expectation Values

For a normalised wavefunction, the expectation value of an observable is:

⟨A⟩=∫ψ∗A^ψ dτ\langle A\rangle=\int\psi^*\hat{A}\psi\,d\tau

The expectation value represents the average of many measurements on identically prepared systems. It need not equal the result of any individual measurement.

For example, a particle may have an average position at a point where the probability density is actually zero.

12.4 Commutators and Uncertainty

The commutator of two operators is:

[A^,B^]=A^B^−B^A^[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}

For position and momentum:

[x^,p^x]=iℏ[\hat{x},\hat{p}_x]=i\hbar

This nonzero commutator is the mathematical basis of the position–momentum uncertainty relation.

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13. Schrödinger’s Wave Equation

13.1 Time-Dependent Equation

The time-dependent Schrödinger equation describes the evolution of a nonrelativistic quantum state:

iℏ∂Ψ∂t=H^Ψ\boxed{ i\hbar\frac{\partial\Psi}{\partial t} = \hat{H}\Psi }

For a single particle in a potential VV:

iℏ∂Ψ∂t=[−ℏ22m∇2+V]Ψi\hbar\frac{\partial\Psi}{\partial t} = \left[-\frac{\hbar^2}{2m}\nabla^2+V\right]\Psi

13.2 Time-Independent Equation

When the Hamiltonian has no explicit time dependence, stationary states can be obtained from:

−ℏ22m∇2ψ+Vψ=Eψ\boxed{ -\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi }

In one dimension:

−ℏ22md2ψdx2+V(x)ψ=Eψ-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+V(x)\psi=E\psi

The equation relates the spatial form of a wavefunction to its energy and the potential in which the particle moves.

13.3 Origin of Quantisation

Energy quantisation arises when only particular solutions satisfy the physical boundary conditions and normalisation requirements.

For a bound system, these restrictions often permit only discrete energies. This is more fundamental than simply assuming that allowed energies exist.

Not every quantum system has exclusively discrete energies. A free particle, for example, has a continuous range of allowed energies.

13.4 Stationary States

The full wavefunction of an energy eigenstate is:

Ψ(r,t)=ψ(r)e−iEt/ℏ\Psi(\mathbf r,t)=\psi(\mathbf r)e^{-iEt/\hbar}

Although its phase changes with time:

∣Ψ(r,t)∣2=∣ψ(r)∣2|\Psi(\mathbf r,t)|^2=|\psi(\mathbf r)|^2

Therefore, its probability density remains stationary. This provides a quantum description of atomic stability without requiring an electron to travel around a classical orbit.

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14. Particle in a One-Dimensional Box

14.1 Model and Assumptions

The particle-in-a-box model considers a particle confined between two impenetrable walls separated by a distance LL.

The potential is:

V(x)=0for 0<x<LV(x)=0 \quad\text{for }0<x<L

and infinite outside the box.

Inside the box, the Schrödinger equation becomes:

−ℏ22md2ψdx2=Eψ-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}=E\psi

Its general solution is:

ψ(x)=Asin⁡kx+Bcos⁡kx\psi(x)=A\sin kx+B\cos kx

where:

k2=2mEℏ2k^2=\frac{2mE}{\hbar^2}

14.2 Boundary Conditions and Allowed Energies

The wavefunction must vanish at both walls:

ψ(0)=0,ψ(L)=0\psi(0)=0,\qquad\psi(L)=0

The first condition gives B=0B=0. The second requires:

sin⁡(kL)=0\sin(kL)=0

Therefore:

kL=nπkL=n\pi

where n=1,2,3,…n=1,2,3,\ldots.

The resulting energies are:

En=n2h28mL2\boxed{E_n=\frac{n^2h^2}{8mL^2}}

The normalised wavefunctions are:

ψn(x)=2Lsin⁡(nπxL)\boxed{ \psi_n(x)=\sqrt{\frac2L}\sin\left(\frac{n\pi x}{L}\right) }

14.3 Physical Interpretation

The lowest possible energy is:

E1=h28mL2E_1=\frac{h^2}{8mL^2}

The value n=0n=0 is excluded because it gives a wavefunction that is zero everywhere and cannot be normalised.

The nonzero ground-state energy is a form of zero-point energy. It reflects the impossibility of confining a particle while simultaneously giving it precisely zero momentum.

The energy increases as n2n^2, decreases as particle mass increases, and decreases as the square of the box length increases.

14.4 Nodes

A node is a position where the wavefunction, and therefore the probability density, is zero.

The nnth box eigenfunction has n−1n-1 internal nodes. Higher-energy states have more nodes and greater spatial variation.

14.5 Chemical Application

The model provides a qualitative description of delocalised electrons in conjugated molecules. Increasing the length over which electrons are delocalised generally reduces the spacing between relevant electronic energy levels.

Consequently, extended conjugation commonly shifts absorption towards longer wavelengths. The model is approximate because real molecules do not contain perfectly rigid walls or a uniform internal potential.

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15. Quantum-Mechanical Treatment of the Hydrogen Atom

15.1 Coulomb Potential

For a hydrogen-like atom:

V(r)=−Ze24πε0rV(r)=-\frac{Ze^2}{4\pi\varepsilon_0r}

Because the potential depends only on distance from the nucleus, spherical polar coordinates are convenient.

The wavefunction separates into radial and angular parts:

ψnlml(r,θ,ϕ)=Rnl(r)Ylml(θ,ϕ)\boxed{ \psi_{nlm_l}(r,\theta,\phi) = R_{nl}(r)Y_l^{m_l}(\theta,\phi) }

The radial function describes variation with distance, while the spherical harmonic describes angular dependence.

15.2 Energy Levels and Degeneracy

The nonrelativistic Coulomb solution gives approximately:

En=−13.6Z2n2 eVE_n=-13.6\frac{Z^2}{n^2}\,\text{eV}

The principal energy levels agree with Bohr’s result, but the quantum treatment does not assign definite circular trajectories.

Within this approximation, the energy depends only on nn. Thus, the 2s2s and 2p2p orbitals have the same energy. Orbitals with equal energies are described as degenerate.

Relativistic effects, spin-dependent interactions, and quantum-electrodynamic corrections introduce smaller splittings beyond this basic model.

15.3 The Hydrogen 1s1s Wavefunction

In the fixed-nucleus approximation, the normalised hydrogen ground-state wavefunction is:

ψ1s=1πa03e−r/a0\psi_{1s} = \frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0}

It is spherically symmetric and decreases exponentially with distance.

The probability density is:

∣ψ1s∣2=1πa03e−2r/a0|\psi_{1s}|^2 = \frac{1}{\pi a_0^3}e^{-2r/a_0}

The electron distribution therefore has no sharply defined outer boundary.

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16. Quantum Numbers

16.1 Principal Quantum Number, nn

The principal quantum number takes positive integral values:

n=1,2,3,…n=1,2,3,\ldots

It identifies the main shell and strongly influences orbital size. For hydrogen-like atoms, it also determines the energy in the basic nonrelativistic treatment.

Larger values of nn generally correspond to more spatially extended electronic distributions.

16.2 Orbital Angular-Momentum Quantum Number, ll

For a given nn:

l=0,1,2,…,n−1l=0,1,2,\ldots,n-1

The conventional labels are:

l=0→s,l=1→p,l=2→d,l=3→fl=0\rightarrow s,\quad l=1\rightarrow p,\quad l=2\rightarrow d,\quad l=3\rightarrow f

The magnitude of orbital angular momentum is:

L=l(l+1) ℏ\boxed{L=\sqrt{l(l+1)}\,\hbar}

An ss orbital has l=0l=0 and therefore zero orbital angular momentum. This illustrates a major difference from Bohr’s picture of a circulating electron.

16.3 Magnetic Quantum Number, mlm_l

The allowed values are:

ml=−l,−l+1,…,0,…,+lm_l=-l,-l+1,\ldots,0,\ldots,+l

The component of angular momentum along a chosen axis is:

Lz=mlℏL_z=m_l\hbar

There are 2l+12l+1 orbitals within a subshell. Thus, an ss subshell contains one orbital, a pp subshell three, and a dd subshell five.

16.4 Electron Spin and msm_s

The electron has intrinsic spin angular momentum with:

s=12s=\frac12

Its allowed spin projections are:

ms=+12or−12m_s=+\frac12\quad\text{or}\quad-\frac12

Spin should not be interpreted literally as a small charged sphere rotating about its own axis. It is an intrinsic quantum property.

16.5 Shell and Subshell Capacities

Three spatial quantum numbers specify a hydrogenic orbital; adding a spin projection specifies a spin-orbital. A shell contains n2n^2 spatial orbitals and can accommodate up to 2n22n^2 electrons. These allowed combinations underlie the organisation of electronic configurations. Purdue University: Quantum Numbers and Electron Configurations

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17. Shapes of Atomic Orbitals and Nodal Structure

17.1 Shapes of ss, pp, and dd Orbitals

All ss orbitals are spherically symmetric. Higher ss orbitals have additional radial structure and radial nodes.

The familiar real pp orbitals have two lobes separated by a nodal plane. They are labelled pxp_x, pyp_y, and pzp_z according to orientation.

Most familiar real dd orbitals have four lobes. The dz2d_{z^2} orbital has two main lobes along the zz-axis and a toroidal region around the centre.

Different colours or signs on orbital diagrams indicate the phase of the wavefunction, not positive and negative electrical charges.

17.2 Radial and Angular Nodes

For hydrogenic orbitals:

Radial nodes=n−l−1\text{Radial nodes}=n-l-1 Angular nodes=l\text{Angular nodes}=l Total nodes=n−1\text{Total nodes}=n-1

A 2s2s orbital has one radial node and no angular node. A 2p2p orbital has no radial node and one angular node.

A 3p3p orbital has one radial node and one angular node, while a 3d3d orbital has no radial node and two angular nodes.

17.3 Probability Density versus Radial Probability

Probability density at a point and probability within a spherical shell are different quantities.

For a spherically symmetric state:

P(r)=4πr2∣ψ(r)∣2P(r)=4\pi r^2|\psi(r)|^2

More generally, when the angular function is normalised:

P(r)=r2∣Rnl(r)∣2P(r)=r^2|R_{nl}(r)|^2

For hydrogen 1s1s, the probability density is greatest at the nucleus. However, the radial probability is zero at r=0r=0 because the spherical-shell volume vanishes there.

The radial probability reaches its maximum at:

r=a0r=a_0

Thus, the most probable electron–nucleus distance differs from the position of maximum probability density.

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18. Many-Electron Atoms: Shielding and Penetration

18.1 Electron–Electron Repulsion

In a many-electron atom, each electron experiences attraction to the nucleus and repulsion from other electrons.

The electron–electron repulsion terms couple the motions of the electrons, preventing the simple separation that makes the hydrogen atom analytically solvable. Approximation methods are therefore necessary.

18.2 Shielding and Effective Nuclear Charge

Other electrons partially screen the nuclear attraction experienced by a particular electron. A useful approximate expression is:

Zeff=Z−σZ_{\text{eff}}=Z-\sigma

where σ\sigma represents a shielding constant.

Effective nuclear charge is a model-dependent measure rather than a single exact charge experienced uniformly throughout an orbital.

18.3 Penetration

Penetration describes the extent to which an electron’s probability distribution reaches regions close to the nucleus.

Within the same principal shell, the usual penetration order is:

s>p>d>fs>p>d>f

Greater penetration generally allows an electron to experience stronger nuclear attraction and less shielding. Consequently, in many-electron atoms, subshells with the same nn generally follow:

E(ns)<E(np)<E(nd)<E(nf)E(ns)<E(np)<E(nd)<E(nf)

This splitting distinguishes many-electron atoms from the ideal hydrogenic case.

18.4 Periodic Trends

Across a period, increasing effective nuclear attraction generally contracts atomic size and raises ionisation energy. Down a group, additional shells generally increase atomic size and place valence electrons farther from the nucleus.

These broad patterns contain exceptions because subshell energies, electron pairing, and electronic configurations also influence stability.

For example, oxygen has a lower first ionisation energy than nitrogen partly because removing a paired 2p2p electron from oxygen relieves electron–electron repulsion.

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19. Electronic Configuration

19.1 Aufbau Principle

The Aufbau principle provides a practical procedure for constructing approximate ground-state configurations by occupying available low-energy orbitals.

A commonly used sequence is:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s,…1s,\ 2s,\ 2p,\ 3s,\ 3p,\ 4s,\ 3d,\ 4p,\ 5s,\ldots

The n+ln+l rule helps predict this order. Orbitals with lower n+ln+l usually fill first; where values are equal, the orbital with lower nn generally fills first.

However, this is an empirical guide rather than an exact law. Orbital energies change with nuclear charge, electron occupancy, and ionisation.

19.2 Pauli Exclusion Principle

The Pauli exclusion principle states that no two electrons can occupy the same spin-orbital.

In the usual atomic orbital description, this means that no two electrons can have the same set of four quantum numbers. A spatial orbital can therefore contain at most two electrons with opposite spin projections.

At a deeper level, the total electronic wavefunction must change sign when the coordinates, including spin, of two electrons are exchanged.

19.3 Hund’s Rule

For a set of degenerate orbitals, the lowest-energy arrangement generally maximises total spin before electron pairing occurs.

Thus, the three 2p2p electrons of nitrogen occupy separate pp orbitals with parallel spins:

N:1s22s22p3\mathrm{N}:1s^22s^22p^3

The underlying explanation involves exchange effects and the resulting electron distribution. It should not be described simply as a classical attraction between parallel spins.

19.4 Exceptions: Chromium and Copper

Chromium has the ground-state configuration:

Cr:[Ar] 3d54s1\mathrm{Cr}:[Ar]\,3d^54s^1

Copper has:

Cu:[Ar] 3d104s1\mathrm{Cu}:[Ar]\,3d^{10}4s^1

These differ from the simplest filling prediction because the relevant configurations are close in energy. Their relative stability reflects the balance of orbital energies, repulsion, exchange, and correlation.

The familiar statement that half-filled and filled subshells are especially stable is useful, but it is not a complete quantitative explanation.

19.5 Formation of Transition-Metal Ions

When transition metals form cations, the outer nsns electrons are generally removed before the (n−1)d(n-1)d electrons.

For example:

Fe:[Ar] 3d64s2\mathrm{Fe}:[Ar]\,3d^64s^2 Fe2+:[Ar] 3d6\mathrm{Fe^{2+}}:[Ar]\,3d^6 Fe3+:[Ar] 3d5\mathrm{Fe^{3+}}:[Ar]\,3d^5

The fact that 4s4s appears before 3d3d in a simple filling sequence does not imply that it always remains lower in energy.

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20. Atomic Spectra, Selection Rules, and Term Symbols

20.1 Allowed and Forbidden Transitions

An energy difference is necessary for a spectral transition, but it does not alone determine the transition probability. The interaction between the radiation field and the states also matters.

For hydrogenic electric-dipole transitions, important selection rules are:

Δl=±1\Delta l=\pm1 Δml=0,±1\Delta m_l=0,\pm1

For instance, 2p→1s2p\rightarrow1s is electric-dipole allowed, whereas 2s→1s2s\rightarrow1s is electric-dipole forbidden.

“Forbidden” does not mean absolutely impossible. Such transitions may occur through weaker mechanisms, including two-photon emission or higher-multipole processes.

20.2 Orbital and Spin Coupling

For many-electron atoms, individual orbital angular momenta combine to give total orbital angular momentum LL, while individual spins combine to give total spin SS.

Under the LS-coupling approximation, these combine to produce total angular momentum JJ:

J=∣L−S∣, ∣L−S∣+1,…,L+SJ=|L-S|,\ |L-S|+1,\ldots,L+S

20.3 Term Symbols

Atomic terms and levels are represented using:

2S+1LJ{}^{2S+1}L_J

The quantity 2S+12S+1 is the spin multiplicity. Capital letters S,P,D,F,…S,P,D,F,\ldots represent L=0,1,2,3,…L=0,1,2,3,\ldots.

For example, the hydrogen ground level is:

2S1/2{}^2S_{1/2}

Term symbols allow a compact description of angular momentum and help explain fine structure and spectral transitions.

20.4 Zeeman and Stark Effects

The splitting or shifting of atomic energy levels in an external magnetic field is the Zeeman effect. The corresponding effect of an electric field is the Stark effect.

These phenomena demonstrate that atomic energy levels respond to external fields and provide evidence about angular momentum and the structure of quantum states.

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21. Approximation Methods in Quantum Chemistry

21.1 Why Approximation Is Necessary

Exact analytic solutions are available for only a limited number of idealised systems. Real atoms and molecules usually contain several interacting particles.

The objective of an approximation method is to preserve the essential physics while making the mathematical problem tractable.

21.2 Variational Method

For a normalised trial wavefunction, the calculated energy satisfies:

Etrial=⟨ψtrial∣H^∣ψtrial⟩≥E0\boxed{ E_{\text{trial}} = \langle\psi_{\text{trial}}|\hat H|\psi_{\text{trial}}\rangle \geq E_0 }

where E0E_0 is the exact ground-state energy of the same Hamiltonian.

Parameters in the trial wavefunction are adjusted to minimise the calculated energy. Within the chosen trial space, this produces the best variational approximation.

The principle underlies many computational methods, including Hartree–Fock theory.

21.3 Perturbation Theory

Perturbation theory begins with a solvable reference Hamiltonian and adds a correction:

H^=H^0+λH^′\hat H=\hat H_0+\lambda\hat H'

For a nondegenerate reference state, the first-order energy correction is:

En(1)=⟨ψn(0)∣H^′∣ψn(0)⟩E_n^{(1)} = \langle\psi_n^{(0)}|\hat H'|\psi_n^{(0)}\rangle

The method is useful when the additional interaction is sufficiently small relative to the reference problem. Degenerate states require a modified treatment.

21.4 Hartree–Fock Theory and Correlation

Hartree–Fock theory approximates the electronic wavefunction using a single Slater determinant. Each electron moves in an effective field determined by the others, and the equations are solved self-consistently.

It includes exchange through antisymmetry but does not fully describe correlated electron motion. More advanced methods improve the treatment of this electron correlation.

21.5 Born–Oppenheimer Approximation

Because nuclei are much heavier than electrons, electronic motion can often be treated with the nuclei held at fixed positions.

The electronic energy is calculated for different nuclear arrangements, producing a potential-energy surface that helps describe molecular geometry, vibrations, and reactions.

The approximation becomes less reliable when electronic states approach closely and electronic and nuclear motions become strongly coupled.

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22. Quantum-Mechanical Description of Chemical Bonding

22.1 Origin of a Chemical Bond

A chemical bond forms when an arrangement of interacting atoms has a lower total energy than an appropriate separated-atom reference.

Bond formation involves a balance between electron–nucleus attraction, electron–electron repulsion, nucleus–nucleus repulsion, and electronic kinetic energy.

A convincing explanation of bonding must therefore consider the entire energy balance rather than attributing bonding solely to electrostatic attraction.

22.2 Valence Bond Theory

Valence bond theory commonly describes a covalent bond through overlapping atomic orbitals and a spin-coupled electron pair.

For hydrogen, the interaction between two 1s1s atomic orbitals produces a lower-energy singlet bonding state over a suitable range of internuclear distances. At excessively short distances, strong repulsive contributions raise the energy.

Valence bond theory is particularly useful for explaining localised bonds, resonance structures, and directional bonding.

22.3 Hybridisation

Hybridisation constructs directional orbitals through linear combinations of atomic orbitals on the same atom.

In the standard localised description, spsp hybridisation gives two directions separated by 180∘180^\circ, sp2sp^2 gives three coplanar directions separated by 120∘120^\circ, and sp3sp^3 gives four tetrahedral directions separated by approximately 109.5∘109.5^\circ.

These models help describe molecules such as acetylene, ethene, and methane.

Hybridisation is a representational model. It should not be treated as a directly observed preliminary event that atoms must undergo before bonding.

22.4 Sigma and Pi Bonds

A sigma bond has bonding electron density with cylindrical symmetry about the internuclear axis.

A pi bond results from an interaction such as the sideways overlap of parallel pp orbitals and has a nodal plane containing the internuclear axis.

In a common localised description, a double bond contains one sigma and one pi bond, while a triple bond contains one sigma and two pi bonds. The directional character of pi bonding helps explain restricted rotation around double bonds.

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23. Molecular Orbital Theory

23.1 Linear Combination of Atomic Orbitals

Molecular orbital theory describes electrons using orbitals that extend over a molecule.

For two suitable atomic basis functions:

ψbonding=N+(ϕA+ϕB)\psi_{\text{bonding}}=N_+(\phi_A+\phi_B) ψantibonding=N−(ϕA−ϕB)\psi_{\text{antibonding}}=N_-(\phi_A-\phi_B)

Constructive combination generally increases electron density between the nuclei and produces a bonding orbital. Destructive combination produces an internuclear node and an antibonding orbital.

Effective interaction requires appropriate symmetry, sufficient overlap, and reasonably compatible orbital energies.

23.2 Bond Order

A simple molecular-orbital bond order is:

Bond order=Nb−Na2\boxed{\text{Bond order}=\frac{N_b-N_a}{2}}

where NbN_b and NaN_a are the numbers of electrons in bonding and antibonding orbitals.

Within comparable species, a higher bond order generally corresponds to a shorter and stronger bond.

23.3 Hydrogen and Helium

For hydrogen:

H2:(σ1s)2\mathrm{H_2}:(\sigma_{1s})^2

Therefore:

Bond order=2−02=1\text{Bond order}=\frac{2-0}{2}=1

For the simplest description of helium dimer:

He2:(σ1s)2(σ1s∗)2\mathrm{He_2}:(\sigma_{1s})^2(\sigma_{1s}^{*})^2

The bond order is zero. This predicts the absence of an ordinary covalent bond, although extremely weak helium dimers can exist through dispersion interactions.

23.4 Oxygen and Paramagnetism

A central success of molecular orbital theory is its explanation of oxygen’s paramagnetism.

In the ground state of O2\mathrm{O_2}, two electrons occupy separate degenerate π∗\pi^* orbitals with parallel spins. These unpaired electrons account for attraction to a magnetic field.

The bond order of oxygen is two. Adding an electron to form superoxide places it in an antibonding orbital and lowers the bond order to 1.51.5. Adding another electron to form peroxide lowers it to one.

This comparison demonstrates how electronic occupancy influences both bonding and magnetism.

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24. Quantum Tunnelling and Zero-Point Motion

24.1 Quantum Tunnelling

Classically, a particle cannot cross a potential barrier if its energy is below the barrier height. In quantum mechanics, a finite barrier can permit a nonzero transmission probability.

Within the classically forbidden region, the wavefunction typically decays rather than becoming immediately zero.

Tunnelling becomes more significant for lighter particles and narrower barriers. It contributes to electron transfer, proton-transfer processes, and some reaction-rate isotope effects.

24.2 Harmonic Oscillator and Molecular Vibrations

Near an equilibrium bond length, a molecular potential can often be approximated by a harmonic potential:

V(x)=12kx2V(x)=\frac12kx^2

The allowed vibrational energies are:

Ev=(v+12)hν\boxed{E_v=\left(v+\frac12\right)h\nu}

where:

v=0,1,2,…v=0,1,2,\ldots

and:

ν=12πkμ\nu=\frac{1}{2\pi}\sqrt{\frac{k}{\mu}}

Here, kk is the force constant and μ\mu is the reduced mass.

The ground-state vibrational energy is:

E0=12hνE_0=\frac12h\nu

Thus, even the lowest vibrational state possesses zero-point motion. Isotopic substitution changes the reduced mass and therefore shifts vibrational frequencies.

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25. Important Conceptual Distinctions

25.1 Quantisation Does Not Mean Every Energy Is Discrete

Bound states frequently have discrete energies because of confinement and boundary conditions. Unbound states can form an energy continuum.

Therefore, the statement that “energy is always quantised into discrete levels” requires qualification.

25.2 Probability Is Not a Definite Hidden Orbit

An orbital does not merely show uncertainty about an otherwise ordinary classical path. It is part of a fundamentally quantum description that predicts distributions of measurement outcomes.

25.3 Orbital Phase Is Not Electrical Charge

Positive and negative signs in a wavefunction indicate phase. All electron density carries negative charge, regardless of the sign of the orbital amplitude.

Phase matters because wavefunctions interfere constructively or destructively.

25.4 Hydrogenic and Many-Electron Energies Differ

In the ideal hydrogenic Coulomb problem, orbital energies depend only on nn. In many-electron atoms, shielding, penetration, and electron interactions remove much of this degeneracy.

Consequently, hydrogenic energy formulas should not be applied directly to neutral many-electron atoms without a justified approximation.

25.5 Electron Spin Is Intrinsic

Spin is not literal rotation of the electron about its own axis. The classical analogy can introduce contradictions and should be used cautiously.

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26. How to Develop a Strong CSS Answer

26.1 Combine Explanation with Mathematics

A strong answer should state the physical problem, explain the relevant theory, develop the necessary equations, and interpret the result.

For example, an answer on the particle in a box should explain the potential, solve the Schrödinger equation, apply the boundary conditions, derive the energy expression, and discuss zero-point energy and nodes.

An equation without explanation demonstrates less understanding than an equation connected to its assumptions and chemical significance.

26.2 State the Limits of Each Model

The assumptions of a model define where it works. Bohr’s model applies effectively to one-electron species but does not provide a general theory of many-electron atoms. The particle-in-a-box model illustrates confinement but only approximates delocalised molecular electrons.

Similarly, hybridisation describes bonding geometry within a chosen model, while molecular orbital theory is particularly useful for delocalisation and magnetic behaviour.

26.3 Prepare the Central Derivations

The principal derivations to practise are Bohr’s radius and energy, the Rydberg relation, the de Broglie wavelength of an accelerated electron, and the energy levels of a particle in a one-dimensional box.

For each derivation, define the symbols, identify the assumptions, preserve consistent units, and explain the final dependence. For instance, the relationship En∝L−2E_n\propto L^{-2} immediately shows why stronger confinement raises energy.

26.4 Use Diagrams Purposefully

Useful examination diagrams include hydrogen energy levels, orbital shapes with nodes, radial probability curves, particle-in-a-box wavefunctions, and molecular-orbital energy diagrams.

Each diagram should support an argument. A molecular-orbital diagram of oxygen, for example, should explicitly connect the occupation of the two π∗\pi^* orbitals with paramagnetism.

For additional university-level study, MIT’s physical chemistry lecture notes provide a structured progression through quantum principles, model systems, atomic structure, and molecular bonding. MIT OpenCourseWare: Physical Chemistry Lecture Notes

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27. Conclusion

Atomic structure and quantum chemistry explain how microscopic behaviour produces observable chemical properties. The failure of classical physics to account for atomic stability and line spectra led to energy quantisation, wave–particle duality, and the probabilistic framework of quantum mechanics.

Schrödinger’s equation provides the central mathematical description, while quantum numbers, orbitals, and electronic configurations connect that description to atomic properties. Approximation methods extend the theory to many-electron systems, and quantum theories of bonding explain molecular stability, geometry, spectroscopy, and magnetism.