4.6 Molecular Quantum Mechanics

Quantum mechanics unlocks the secrets of molecules, revealing how atoms bond and interact. It's like peering into a microscopic world where electrons dance around nuclei, creating the building blocks of matter.

The Schrödinger equation is our guide, describing molecular energy and behavior. By applying quantum principles, we can predict chemical properties, understand reactions, and explore the hidden dance of electrons in molecules.

Quantum Mechanics for Molecules

Applying Quantum Mechanics to Molecular Systems

Top images from around the web for Applying Quantum Mechanics to Molecular Systems

• 

  8.4 Molecular Orbital Theory – Chemistry View original

  Is this image relevant?

• 

  quantum mechanics - How to "read" Schrodinger's equation? - Physics Stack Exchange View original

  Is this image relevant?

• 

  Schrödinger equation - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  8.4 Molecular Orbital Theory – Chemistry View original

  Is this image relevant?

• 

  quantum mechanics - How to "read" Schrodinger's equation? - Physics Stack Exchange View original

  Is this image relevant?

1 of 3

Top images from around the web for Applying Quantum Mechanics to Molecular Systems

• 

  8.4 Molecular Orbital Theory – Chemistry View original

  Is this image relevant?

• 

  quantum mechanics - How to "read" Schrodinger's equation? - Physics Stack Exchange View original

  Is this image relevant?

• 

  Schrödinger equation - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  8.4 Molecular Orbital Theory – Chemistry View original

  Is this image relevant?

• 

  quantum mechanics - How to "read" Schrodinger's equation? - Physics Stack Exchange View original

  Is this image relevant?

1 of 3

• Molecules are composed of atoms bonded together, and the principles of quantum mechanics can be applied to understand their behavior and properties
• The Schrödinger equation can be used to describe the quantum state and energy of a molecule, taking into account the interactions between the nuclei and electrons
  • The molecular Hamiltonian operator includes terms for the kinetic energy of the nuclei and electrons, as well as the potential energy due to electrostatic interactions between particles (Coulomb interactions)
  • Solving the Schrödinger equation for molecules is more complex than for atoms due to the increased number of particles and degrees of freedom (translational, rotational, and vibrational)
• The electronic structure of a molecule determines its chemical and physical properties, including bonding (covalent, ionic), reactivity (redox, acid-base), and spectroscopic transitions (UV-vis, IR)
• The potential energy surface of a molecule describes how its energy varies with changes in nuclear coordinates, and it can be used to understand molecular geometry (bond lengths, angles), vibrational modes (stretching, bending), and reaction pathways (transition states, intermediates)

Molecular Schrödinger Equation and Hamiltonian

• The molecular Schrödinger equation is an extension of the atomic Schrödinger equation, incorporating the interactions between multiple nuclei and electrons
• The molecular Hamiltonian operator includes the following terms:
  • Kinetic energy of the nuclei (depends on their masses and velocities)
  • Kinetic energy of the electrons (depends on their masses and velocities)
  • Potential energy due to nucleus-nucleus repulsion (Coulomb interaction between positively charged nuclei)
  • Potential energy due to electron-nucleus attraction (Coulomb interaction between negatively charged electrons and positively charged nuclei)
  • Potential energy due to electron-electron repulsion (Coulomb interaction between negatively charged electrons)
• The molecular wavefunction depends on the coordinates of all the nuclei and electrons, making the Schrödinger equation a high-dimensional partial differential equation
• Approximations and numerical methods are often necessary to solve the molecular Schrödinger equation, such as the Born-Oppenheimer approximation and variational methods (Hartree-Fock, density functional theory)

Born-Oppenheimer Approximation

Separating Nuclear and Electronic Motion

• The Born-Oppenheimer approximation simplifies the molecular Schrödinger equation by separating the motion of the nuclei and electrons
  • It assumes that the nuclei are much heavier and slower-moving than the electrons, allowing the electronic and nuclear motions to be treated independently (adiabatic approximation)
  • The electronic Schrödinger equation is solved for a fixed set of nuclear coordinates, yielding the electronic energy and wavefunction (potential energy surface)
• The Born-Oppenheimer approximation allows the construction of potential energy surfaces by calculating the electronic energy at different nuclear configurations (geometries)
• The approximation is valid for most ground-state molecules and low-lying excited states, where the electronic and nuclear motions are effectively decoupled

Limitations and Beyond the Born-Oppenheimer Approximation

• The Born-Oppenheimer approximation breaks down when there is significant coupling between electronic and nuclear motions, such as in certain excited states or near conical intersections (degeneracies)
• Nonadiabatic effects, where the electronic state changes during nuclear motion, can be important in some molecular systems and require going beyond the Born-Oppenheimer approximation
  • Examples include photochemical reactions (photoisomerization), charge transfer processes (electron transfer), and energy transfer (excitation energy transfer)
  • Methods for treating nonadiabatic effects include surface hopping, multiconfigurational time-dependent Hartree (MCTDH), and ab initio multiple spawning (AIMS)
• Other corrections to the Born-Oppenheimer approximation include relativistic effects (spin-orbit coupling) and nuclear quantum effects (tunneling, zero-point energy)

Molecular Orbitals from Atomic Orbitals

Linear Combination of Atomic Orbitals (LCAO) Method

• Molecular orbitals (MOs) are constructed by combining atomic orbitals (AOs) from the constituent atoms in a molecule
• The LCAO method expresses each MO as a linear combination of AOs, with coefficients determining the contribution of each AO to the MO
  • The coefficients are determined by solving the secular equations, which involve the overlap and Hamiltonian matrix elements between the AOs (Roothaan equations)
  • The number of MOs formed is equal to the number of AOs combined, and they can be classified as bonding (lower energy, constructive interference), antibonding (higher energy, destructive interference), or nonbonding (similar energy to AOs, localized on atoms) based on their energy and nodal properties
• Symmetry considerations can simplify the construction of MOs by identifying which AOs can combine based on their symmetry properties (irreducible representations)

Basis Sets and Accuracy

• The basis set used in the LCAO method determines the accuracy and computational cost of the calculation, with larger basis sets providing more flexibility but requiring more resources
  • Minimal basis sets (STO-3G) use one basis function per atomic orbital, while split-valence basis sets (6-31G) use multiple basis functions to describe valence orbitals
  • Polarization functions (d, f) can be added to describe the distortion of atomic orbitals in the molecular environment, improving the description of bonding and molecular geometry
  • Diffuse functions (+ or ++) can be added to describe the tail of the electronic distribution, important for anions, excited states, and weak interactions (hydrogen bonding)
• The choice of basis set depends on the desired accuracy and the computational resources available, with a trade-off between accuracy and efficiency
• Convergence tests can be performed to assess the adequacy of the basis set for a given molecular system and property of interest

Electronic Structure and Bonding in Molecules

Molecular Orbital Theory and Electronic Configuration

• Molecular orbital theory provides a framework for understanding the electronic structure and bonding in molecules based on the spatial distribution and energies of the MOs
• The occupation of MOs by electrons determines the electronic configuration of the molecule, which can be represented using an MO diagram
  • The Aufbau principle states that electrons fill MOs in order of increasing energy, with lower-energy MOs filled before higher-energy ones
  • Hund's rule states that electrons occupy degenerate MOs singly with parallel spins before pairing up, maximizing the total spin and minimizing electron repulsion
  • The Pauli exclusion principle states that no two electrons in a molecule can have the same set of quantum numbers, limiting the occupation of each MO to two electrons with opposite spins
• The highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are particularly important for chemical reactivity (frontier orbitals) and spectroscopic transitions (HOMO-LUMO gap)

Visualization and Bonding Analysis

• The shape and symmetry of the MOs can be visualized using contour plots or isosurfaces, providing insight into the bonding and antibonding character of the orbitals
  • Bonding MOs have constructive interference between AOs, leading to increased electron density between the nuclei and lower energy (stabilization)
  • Antibonding MOs have destructive interference between AOs, leading to decreased electron density between the nuclei and higher energy (destabilization)
• The bond order of a molecule can be determined by the difference between the number of bonding and antibonding electrons, with higher bond orders indicating stronger bonding
  • Bond order = (number of bonding electrons - number of antibonding electrons) / 2
  • Single, double, and triple bonds have bond orders of 1, 2, and 3, respectively
• MO theory can explain the electronic spectra of molecules, with transitions between MOs giving rise to absorption or emission of light at characteristic frequencies (UV-vis, photoelectron spectroscopy)
• The MO description can be extended to more complex molecules, including those with multiple atoms and delocalized bonding, such as conjugated systems (butadiene) and aromatic compounds (benzene)