Molecular Hamiltonian

In <a href="/facts/Atomic,_molecular,_and_optical_physics/6HeJIr7P">atomic, molecular, and optical physics</a> and <a href="/facts/Quantum_chemistry/H9AXV0lx">quantum chemistry</a>, the molecular Hamiltonian is the <a href="/facts/Hamiltonian_(quantum_mechanics)/5XwK2HmD">Hamiltonian</a> operator representing the <a href="/facts/Energy/wd8PrQM7">energy</a> of the <a href="/facts/Electron/L0KBo5cW">electrons</a> and <a href="/facts/Atomic_nucleus/orpIm5Xl">nuclei</a> in a <a href="/facts/Molecule/N9ljSfFq">molecule</a>. This operator and the associated <a href="/facts/Schr%C3%B6dinger_equation/3csgSIWF">Schrödinger equation</a> play a central role in <a href="/facts/Computational_chemistry/KyuCgRV3">computational chemistry</a> and <a href="/facts/Computational_physics/1ltJ109X">physics</a> for computing properties of molecules and aggregates of molecules, such as <a href="/facts/Thermal_conductivity/qWnSG3nQ">thermal conductivity</a>, <a href="/facts/Specific_heat/9P0U6YgI">specific heat</a>, <a href="/facts/Electrical_conductivity/akyCKvMy">electrical conductivity</a>, <a href="/facts/Optics/p0vq4S1T">optical</a>, and <a href="/facts/Magnetism/eo75vApl">magnetic properties</a>, and <a href="/facts/Reactivity_(chemistry)/PuJ2pmRT">reactivity</a>.
The elementary parts of a molecule are the nuclei, characterized by their <a href="/facts/Atomic_numbers/bgAr581S">atomic numbers</a>, Z, and the electrons, which have negative <a href="/facts/Elementary_charge/lPcj6CoQ">elementary charge</a>, −e. Their interaction gives a nuclear charge of Z + q, where q = −eN, with N equal to the number of electrons. Electrons and nuclei are, to a very good approximation, <a href="/facts/Point_charge/4MPToN8W">point charges</a> and point masses. The molecular Hamiltonian is a sum of several terms: its major terms are the <a href="/facts/Kinetic_energy/aJRxUd4p">kinetic energies</a> of the electrons and the <a href="/facts/Coulomb%27s_law/V0lR5NKv">Coulomb (electrostatic) interactions</a> between the two kinds of charged particles. The Hamiltonian that contains only the kinetic energies of electrons and nuclei, and the Coulomb interactions between them, is known as the Coulomb Hamiltonian. From it are missing a number of small terms, most of which are due to electronic and nuclear <a href="/facts/Spin_(physics)/4Jvp7e8P">spin</a>.
Although it is generally assumed that the solution of the time-independent Schrödinger equation associated with the Coulomb Hamiltonian will predict most properties of the molecule, including its shape (three-dimensional structure), calculations based on the full Coulomb Hamiltonian are very rare. The main reason is that its Schrödinger equation is very difficult to solve. Applications are restricted to small systems like the hydrogen molecule.
Almost all calculations of molecular wavefunctions are based on the separation of the Coulomb Hamiltonian first devised by <a href="/facts/Born%E2%80%93Oppenheimer_approximation/hxlUcBrX">Born and Oppenheimer</a>. The nuclear kinetic energy terms are omitted from the Coulomb Hamiltonian and one considers the remaining Hamiltonian as a Hamiltonian of electrons only. The stationary nuclei enter the problem only as generators of an electric potential in which the electrons move in a quantum mechanical way. Within this framework the molecular Hamiltonian has been simplified to the so-called clamped nucleus Hamiltonian, also called electronic Hamiltonian, that acts only on functions of the electronic coordinates.
Once the Schrödinger equation of the clamped nucleus Hamiltonian has been solved for a sufficient number of constellations of the nuclei, an appropriate <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalue</a> (usually the lowest) can be seen as a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of the nuclear coordinates, which leads to a <a href="/facts/Potential_energy_surface/YDUNcSM8">potential energy surface</a>. In practical calculations the surface is usually <a href="/facts/Least_squares/50jIwbxC">fitted</a> in terms of some analytic functions. In the second step of the <a href="/facts/Born%E2%80%93Oppenheimer_approximation/hxlUcBrX">Born–Oppenheimer approximation</a> the part of the full Coulomb Hamiltonian that depends on the electrons is replaced by the potential energy surface. This converts the total molecular Hamiltonian into another Hamiltonian that acts only on the nuclear coordinates. In the case of a breakdown of the <a href="/facts/Born%E2%80%93Oppenheimer_approximation/hxlUcBrX">Born–Oppenheimer approximation</a>—which occurs when energies of different electronic states are close—the neighboring potential energy surfaces are needed, see this <a href="/facts/Born%E2%80%93Oppenheimer_approximation/hxlUcBrX">article</a> for more details on this.
The nuclear motion Schrödinger equation can be solved in a space-fixed (laboratory) <a href="/facts/Frame_of_reference/15yYfB9M">frame</a>, but then the <a href="/facts/Translation_(physics)/KlpWmnfX">translational</a> and <a href="/facts/Rotation/yChkz49x">rotational</a> (external) energies are not accounted for. Only the (internal) atomic <a href="/facts/Vibration/sGgDqG71">vibrations</a> enter the problem. Further, for molecules larger than triatomic ones, it is quite common to introduce the harmonic approximation, which approximates the potential energy surface as a <a href="/facts/Quadratic_function/mKxHIC3U">quadratic function</a> of the atomic displacements. This gives the harmonic nuclear motion Hamiltonian. Making the harmonic approximation, we can convert the Hamiltonian into a sum of uncoupled one-dimensional <a href="/facts/Harmonic_oscillator/7IQNToGt">harmonic oscillator</a> Hamiltonians. The one-dimensional harmonic oscillator is one of the few systems that allows an exact solution of the Schrödinger equation.
Alternatively, the nuclear motion (rovibrational) Schrödinger equation can be solved in a special frame (an <a href="/facts/Eckart_conditions/r2Xhy47L">Eckart frame</a>) that rotates and translates with the molecule. Formulated with respect to this body-fixed frame the Hamiltonian accounts for <a href="/facts/Rotation/yChkz49x">rotation</a>, <a href="/facts/Translation/2foI82so">translation</a> and <a href="/facts/Vibration/sGgDqG71">vibration</a> of the nuclei. Since Watson introduced in 1968 an important simplification to this Hamiltonian, it is often referred to as Watson's nuclear motion Hamiltonian, but it is also known as the Eckart Hamiltonian.

Molecular Hamiltonian open-in-new

Molecular Hamiltonian