12.1 Born-Oppenheimer approximation

The Born-Oppenheimer approximation is a key concept in quantum mechanics of molecules. It simplifies the complex problem of molecular structure by separating electronic and nuclear motions, allowing us to solve for electronic states while treating nuclei as fixed points.

This approximation is based on the huge mass difference between electrons and nuclei. It assumes electrons move much faster than nuclei, adapting quickly to nuclear positions. This separation enables us to tackle molecular quantum mechanics more effectively, despite some limitations in certain cases.

Born-Oppenheimer Approximation: Basis and Assumptions

Physical Basis of the Born-Oppenheimer Approximation

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• The Born-Oppenheimer approximation is based on the significant difference in mass between electrons and nuclei in molecules, with nuclei being much heavier than electrons (protons and neutrons are ~1800 times heavier than electrons)
• Due to their lighter mass, electrons move much faster than nuclei and can rapidly adjust their positions in response to nuclear motion (electrons move at velocities ~1000 times faster than nuclei)
• The timescale of electronic motion is much shorter than that of nuclear motion, allowing the electrons to quickly adapt to changes in nuclear positions

Assumptions of the Born-Oppenheimer Approximation

• The approximation assumes that the electronic motion and nuclear motion can be decoupled, allowing the total molecular wavefunction to be separated into electronic and nuclear components
• The electronic wavefunction depends parametrically on the nuclear coordinates, while the nuclear wavefunction is influenced by the average field generated by the electrons
• The approximation assumes that the nuclei are stationary on the timescale of electronic motion, allowing the electronic problem to be solved for fixed nuclear configurations
• The coupling between electronic and nuclear motions is assumed to be weak, allowing for the independent treatment of electronic structure and nuclear dynamics

Separating Electronic and Nuclear Motions

Expressing the Total Molecular Wavefunction

• The total molecular wavefunction is expressed as a product of the electronic wavefunction and the nuclear wavefunction: $\Psi(r, R) = \psi(r; R) \times \chi(R)$, where $r$ represents electronic coordinates and $R$ represents nuclear coordinates
• The electronic wavefunction $\psi(r; R)$ describes the spatial distribution of electrons for a given nuclear configuration
• The nuclear wavefunction $\chi(R)$ describes the motion of the nuclei, such as vibrations and rotations, on the potential energy surface generated by the electrons

Solving the Electronic Schrödinger Equation

• The electronic wavefunction $\psi(r; R)$ is obtained by solving the electronic Schrödinger equation for fixed nuclear configurations, with the nuclear coordinates treated as parameters
• The electronic Hamiltonian includes the kinetic energy of the electrons and the potential energy terms describing electron-nucleus attractions and electron-electron repulsions
• The solution of the electronic Schrödinger equation yields the electronic energy levels and the corresponding electronic wavefunctions for each nuclear configuration

Determining the Nuclear Wavefunction

• The nuclear wavefunction $\chi(R)$ is determined by solving the nuclear Schrödinger equation, which includes the potential energy surface obtained from the electronic problem
• The potential energy surface represents the energy of the system as a function of nuclear coordinates and is used to describe nuclear motion, such as vibrations and rotations
• The separation of electronic and nuclear motions allows for the independent treatment of electronic structure and nuclear dynamics in molecules

Born-Oppenheimer Approximation: Validity and Limitations

Validity of the Born-Oppenheimer Approximation

• The Born-Oppenheimer approximation is generally valid for ground-state molecules and low-lying excited states where the energy gap between electronic states is large compared to the nuclear kinetic energy
• The approximation works well for molecules in their equilibrium geometries or undergoing small-amplitude vibrations
• The Born-Oppenheimer approximation is widely used in quantum chemistry and has been successful in describing the electronic structure and properties of many molecular systems

Limitations and Breakdown of the Born-Oppenheimer Approximation

• The approximation breaks down when there is a strong coupling between electronic and nuclear motions, such as in cases of vibronic coupling or conical intersections
• Vibronic coupling occurs when the electronic and vibrational motions are coupled, leading to a breakdown of the Born-Oppenheimer approximation and the need for a more sophisticated treatment (Jahn-Teller effect)
• Conical intersections arise when two or more electronic states become degenerate at certain nuclear configurations, resulting in a breakdown of the adiabatic approximation and requiring a non-adiabatic treatment (photochemical reactions)
• The approximation may also fail in systems with very light nuclei, such as hydrogen-containing molecules (H2, NH3), where the nuclear motion is fast enough to couple with the electronic motion
• In cases where the Born-Oppenheimer approximation is not valid, alternative methods such as the adiabatic or diabatic representation, or non-adiabatic dynamics simulations, may be employed

Solving the Electronic Schrödinger Equation

Setting Up the Electronic Schrödinger Equation

• The electronic Schrödinger equation is set up for a fixed nuclear configuration, with the nuclear coordinates treated as parameters
• The electronic Hamiltonian includes the kinetic energy of the electrons and the potential energy terms describing electron-nucleus attractions and electron-electron repulsions
• The electronic wavefunction is expanded in terms of a basis set, such as atomic orbitals or molecular orbitals, to represent the spatial distribution of electrons (Slater determinants, Gaussian basis sets)

Solving the Electronic Schrödinger Equation

• The variational principle is applied to minimize the energy of the electronic wavefunction, leading to the Hartree-Fock equations or the Kohn-Sham equations in density functional theory
• The electronic energy and wavefunction are obtained by solving the resulting eigenvalue problem, yielding the electronic energy levels and the corresponding electronic wavefunctions
• The electronic energy, along with the nuclear repulsion term, provides the potential energy surface that governs nuclear motion
• The accuracy of the electronic structure calculation can be improved by incorporating electron correlation effects through post-Hartree-Fock methods (configuration interaction, coupled cluster) or advanced density functional approximations (hybrid functionals, dispersion corrections)

Interpreting the Results

• The electronic energy levels and wavefunctions provide information about the electronic structure of the molecule, including orbital energies, electron densities, and bonding properties
• The potential energy surface obtained from the electronic structure calculation can be used to study nuclear motion, such as vibrational frequencies, rotational constants, and reaction pathways
• The electronic structure results can be used to calculate various molecular properties, such as dipole moments, polarizabilities, and spectroscopic transitions (UV-Vis absorption, infrared spectra)