💫Intro to Quantum Mechanics II Unit 13 – Atomic and Molecular Spectra

Key Concepts and Fundamentals

• Quantum mechanics provides a mathematical framework for describing the behavior of matter and energy at the atomic and subatomic scales
• The wave-particle duality of matter and energy states that particles can exhibit wave-like properties and waves can exhibit particle-like properties
  • Electrons in atoms behave as standing waves, leading to discrete energy levels
• The Bohr model of the atom introduced the concept of stationary states and energy levels, laying the foundation for understanding atomic spectra
• The Schrödinger equation is a fundamental equation in quantum mechanics that describes the wave function of a quantum-mechanical system
  • Solutions to the Schrödinger equation for a given potential energy function yield the allowed energy levels and wave functions of the system
• The Heisenberg uncertainty principle states that the product of the uncertainties in the position and momentum of a particle is always greater than or equal to $\frac{h}{4\pi}$, where $h$ is Planck's constant
• The Pauli exclusion principle states that no two identical fermions (particles with half-integer spin) can occupy the same quantum state simultaneously

Atomic Structure and Energy Levels

• Atoms consist of a positively charged nucleus surrounded by negatively charged electrons
• Electrons in an atom occupy discrete energy levels, which are determined by the solutions to the Schrödinger equation for the Coulomb potential of the nucleus
• The principal quantum number $n$ represents the main energy level and the average distance of an electron from the nucleus
• The angular momentum quantum number $l$ describes the shape of the electron's orbital and takes integer values from 0 to $n-1$
  • The magnetic quantum number $m_l$ represents the orientation of the orbital in space and takes integer values from $-l$ to $+l$
• The spin quantum number $m_s$ describes the intrinsic angular momentum of the electron and can have values of $+\frac{1}{2}$ or $-\frac{1}{2}$
• The energy levels in an atom are split into sublevels due to the electron-electron interactions and the coupling of the electron's orbital angular momentum and spin angular momentum
  • The fine structure is caused by the spin-orbit coupling, while the hyperfine structure arises from the interaction between the electron's magnetic moment and the magnetic moment of the nucleus

Spectroscopic Notation and Selection Rules

• Spectroscopic notation is used to describe the electronic states of atoms and molecules
• The term symbol for an atomic state is given by $^{2S+1}L_J$, where $S$ is the total spin angular momentum, $L$ is the total orbital angular momentum (represented by letters S, P, D, F, ...), and $J$ is the total angular momentum (orbital + spin)
• The parity of an atomic state is denoted by a superscript "o" for odd parity and "e" for even parity
• Selection rules govern the allowed transitions between energy levels in atoms and molecules
  • The electric dipole selection rules for atoms are:
    • $\Delta l = \pm 1$
    • $\Delta m_l = 0, \pm 1$
    • $\Delta S = 0$
    • Parity must change (odd $\leftrightarrow$ even)
• The selection rules arise from the conservation of angular momentum and the symmetry properties of the wave functions involved in the transitions

Types of Spectra: Emission and Absorption

• Emission spectra are produced when atoms or molecules emit photons as they transition from higher to lower energy levels
  • Emission lines appear as bright lines on a dark background
• Absorption spectra are produced when atoms or molecules absorb photons, transitioning from lower to higher energy levels
  • Absorption lines appear as dark lines on a bright background (Fraunhofer lines in the solar spectrum)
• The wavelength of the emitted or absorbed photon is related to the energy difference between the levels involved in the transition by the equation $E = \frac{hc}{\lambda}$, where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength
• The intensity of the spectral lines depends on the population of the energy levels and the transition probabilities between them
  • The population of energy levels is governed by the Boltzmann distribution at thermal equilibrium
• The natural broadening of spectral lines is caused by the finite lifetime of the excited states, as described by the Heisenberg uncertainty principle
  • Other broadening mechanisms include Doppler broadening (due to the motion of atoms or molecules) and pressure broadening (due to collisions between particles)

Molecular Bonding and Energy States

• Molecules are formed by the bonding of atoms through the sharing or transfer of electrons
• Covalent bonds involve the sharing of electrons between atoms, resulting in the formation of molecular orbitals
  • Molecular orbitals are formed by the linear combination of atomic orbitals (LCAO) and can be bonding, antibonding, or non-bonding
• Ionic bonds involve the transfer of electrons from one atom to another, resulting in the formation of ions with opposite charges
• The potential energy curve of a diatomic molecule describes the variation of the potential energy as a function of the internuclear distance
  • The equilibrium bond length corresponds to the minimum of the potential energy curve
• The Born-Oppenheimer approximation allows the separation of electronic and nuclear motions in molecules, simplifying the calculation of molecular energy levels
• Molecular energy levels are characterized by electronic, vibrational, and rotational states
  • The electronic states are determined by the configuration of electrons in the molecular orbitals
  • Vibrational states arise from the oscillations of the nuclei about their equilibrium positions
  • Rotational states are associated with the rotation of the molecule about its center of mass

Rotational and Vibrational Spectra

• Rotational spectra arise from transitions between rotational energy levels in molecules
• The rotational energy levels of a diatomic molecule are given by $E_J = \frac{\hbar^2}{2I}J(J+1)$, where $J$ is the rotational quantum number and $I$ is the moment of inertia of the molecule
• The selection rule for rotational transitions is $\Delta J = \pm 1$, resulting in a series of equally spaced lines in the microwave or far-infrared region
• Vibrational spectra arise from transitions between vibrational energy levels in molecules
• The vibrational energy levels of a diatomic molecule can be approximated by the harmonic oscillator model, with energy levels given by $E_v = \left(v + \frac{1}{2}\right)h\nu$, where $v$ is the vibrational quantum number and $\nu$ is the fundamental vibrational frequency
  • Anharmonicity in real molecules leads to deviations from the equally spaced energy levels predicted by the harmonic oscillator model
• The selection rule for vibrational transitions is $\Delta v = \pm 1$, resulting in a series of lines in the infrared region
• Vibrational-rotational spectra involve transitions between both vibrational and rotational energy levels, leading to a complex pattern of lines in the infrared region
  • The selection rules for vibrational-rotational transitions are $\Delta v = \pm 1$ and $\Delta J = \pm 1$
  • The P, Q, and R branches in vibrational-rotational spectra correspond to transitions with $\Delta J = -1, 0, and +1$, respectively

Electronic Transitions in Molecules

• Electronic transitions in molecules involve the promotion of an electron from a lower-energy molecular orbital to a higher-energy molecular orbital
• The selection rules for electronic transitions in molecules are similar to those for atoms, with additional considerations for the symmetry of the molecular orbitals involved
  • The transition dipole moment integral must be non-zero for an electronic transition to be allowed
  • The symmetry of the initial and final electronic states must be compatible with the symmetry of the transition dipole moment operator
• The Franck-Condon principle states that electronic transitions occur vertically on the potential energy curves, without changes in the nuclear coordinates
  • The intensity of vibrational bands in an electronic transition depends on the overlap of the vibrational wave functions of the initial and final states (Franck-Condon factors)
• The coupling of electronic and vibrational motions in molecules leads to the formation of vibronic states and the appearance of vibrational progressions in electronic spectra
• The Jablonski diagram is a graphical representation of the electronic states and the radiative and non-radiative transitions between them
  • Radiative transitions include absorption, fluorescence, and phosphorescence
  • Non-radiative transitions include internal conversion (between states of the same multiplicity) and intersystem crossing (between states of different multiplicities)

Applications and Experimental Techniques

• Atomic and molecular spectroscopy has a wide range of applications in various fields, including:
  • Analytical chemistry: Identification and quantification of elements and compounds
  • Astrophysics: Study of the composition and properties of stars, planets, and interstellar medium
  • Environmental monitoring: Detection of pollutants and trace gases in the atmosphere
  • Materials science: Characterization of the electronic and structural properties of materials
• Experimental techniques used in atomic and molecular spectroscopy include:
  • Absorption spectroscopy: Measures the absorption of light as a function of wavelength or frequency
    • Examples: UV-Vis spectroscopy, infrared spectroscopy, X-ray absorption spectroscopy
  • Emission spectroscopy: Measures the emission of light from excited atoms or molecules
    • Examples: Flame emission spectroscopy, inductively coupled plasma (ICP) spectroscopy, fluorescence spectroscopy
  • Raman spectroscopy: Measures the inelastic scattering of light by molecules, providing information on vibrational and rotational modes
  • Laser spectroscopy: Uses tunable lasers to achieve high resolution and sensitivity in spectroscopic measurements
    • Examples: Laser-induced fluorescence (LIF), cavity ring-down spectroscopy (CRDS), two-photon spectroscopy
• Fourier-transform spectroscopy (FT-IR, FT-Raman) improves the signal-to-noise ratio and resolution of spectroscopic measurements by using an interferometer and Fourier analysis
• Spectroscopic databases, such as HITRAN and NIST Atomic Spectra Database, provide extensive data on the spectroscopic properties of atoms and molecules, facilitating the interpretation and analysis of experimental spectra