Everything about Molecular Hamiltonian totally explained
In
atomic, molecular, and optical physics as well as in
quantum chemistry,
molecular Hamiltonian is the name given to the
Hamiltonian representing the
energy of the
electrons and
nuclei in a
molecule. This
Hermitian operator and the associated
Schrödinger equation play a central role in
computational chemistry and
physics for computing properties of molecules and aggregates of molecules, such as
conductivity,
optical, and
magnetic properties, and
reactivity.
The bricks of the molecule are the nuclei, characterized by their
atomic number Z, while the electrons, which have negative
elementary charge -
q, are its mortar. The charge of a nucleus is
Zq. Electrons and nuclei are, to a very good approximation,
point charges and point masses. The molecular Hamiltonian is a sum of several terms: its major terms are the
kinetic energies of the electrons and the
Coulomb (electrostatic) interactions 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
spin.
Although it's 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 scarce. The main reason being 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
Born and Oppenheimer. 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
eigenvalue (usually the lowest) can be seen as a
function of the nuclear coordinates, which leads to a
potential energy surface. In practical calculations the surface is usually
fitted in terms of some analytic functions. In the second step of the
Born-Oppenheimer approximation 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
Born-Oppenheimer approximation—which occurs when energies of different electronic states are close—the neighboring potential energy surfaces are needed, see this
article for more details on this.
The nuclear motion Schrödinger equation can be solved in a space-fixed (laboratory)
frame, but then the
translational and
rotational (external) energies are not accounted for. Only the (internal) atomic
vibrations enter the problem. Further, for molecules larger than triatomic ones, it's quite common to introduce the
harmonic approximation, which approximates the potential energy surface as a
quadratic function 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
harmonic oscillator 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
Eckart frame) that rotates and translates with the molecule. Formulated with respect to this body-fixed frame the Hamiltonian accounts for
rotation,
translation and
vibration of the nuclei. Since Watson introduced in 1968 an important simplification to this Hamiltonian, it's often referred to as
Watson's nuclear motion Hamiltonian, but it's also known as the
Eckart Hamiltonian,
Coulomb Hamiltonian
The algebraic form of many observables—for example, Hermitian operators representing observable quantities—is obtained by the following
quantization rules:
