This 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 thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity.
Almost all calculations of molecular wavefunctions are based on the separation of the Coulomb Hamiltonian first devised by Born and Oppenheimer.
The stationary nuclei enter the problem only as generators of an electric potential in which the electrons move in a quantum mechanical way.
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 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.
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.
Formulated with respect to this body-fixed frame the Hamiltonian accounts for rotation, translation and vibration of the nuclei.
The algebraic form of many observables—i.e., Hermitian operators representing observable quantities—is obtained by the following quantization rules: Classically the electrons and nuclei in a molecule have kinetic energy of the form p2/(2 m) and interact via Coulomb interactions, which are inversely proportional to the distance rij between particle i and j.
Since the kinetic energy operator is an inner product, it is invariant under rotation of the Cartesian frame with respect to which xi, yi, and zi are expressed.
These spectroscopic observations led to the introduction of a new degree of freedom for electrons and nuclei, namely spin.
This empirical concept was given a theoretical basis by Paul Dirac when he introduced a relativistically correct (Lorentz covariant) form of the one-particle Schrödinger equation.
The Dirac equation predicts that spin and spatial motion of a particle interact via spin–orbit coupling.
Further terms without a classical counterpart are the Fermi-contact term (interaction of electronic density on a finite size nucleus with the nucleus), and nuclear quadrupole coupling (interaction of a nuclear quadrupole with the gradient of an electric field due to the electrons).
Although it is an extremely small interaction, it has attracted a fair amount of attention in the scientific literature because it gives different energies for the enantiomers in chiral molecules.
The remaining part of this article will ignore spin terms and consider the solution of the eigenvalue (time-independent Schrödinger) equation of the Coulomb Hamiltonian.
The Coulomb Hamiltonian has a continuous spectrum due to the center of mass (COM) motion of the molecule in homogeneous space.
In the great majority of computations of molecular wavefunctions the electronic problem is solved with the clamped nucleus Hamiltonian arising in the first step of the Born–Oppenheimer approximation.
Also it is discussed in this paper whether one can arrive a priori at the concept of a molecule (as a stable system of electrons and nuclei with a well-defined geometry) from the properties of the Coulomb Hamiltonian alone.
The origin of the frame is arbitrary, it is usually positioned on a central nucleus or in the nuclear center of mass.
In the second step of the BO approximation the nuclear kinetic energy Tn is reintroduced and the Schrödinger equation with Hamiltonian
It can be shown from the invariance of V under rotation and translation that six of the eigenvectors of F (last six rows of Q) have eigenvalue zero (are zero-frequency modes).
The main effort in this approximate solution of the nuclear motion Schrödinger equation is the computation of the Hessian F of V and its diagonalization.
This approximation to the nuclear motion problem, described in 3N mass-weighted Cartesian coordinates, became standard in quantum chemistry, since the days (1980s-1990s) that algorithms for accurate computations of the Hessian F became available.
In general, the classical kinetic energy T defines the metric tensor g = (gij) associated with the curvilinear coordinates s = (si) through
It is common to follow Podolsky[6] by writing down the Laplace–Beltrami operator in the same (generalized, curvilinear) coordinates s as used for the classical form.
When we apply this recipe to Cartesian coordinates, which have unit metric, the same kinetic energy is obtained as by application of the quantization rules.
The nuclear motion Hamiltonian was obtained by Wilson and Howard in 1936,[7] who followed this procedure, and further refined by Darling and Dennison in 1940.
[8] It remained the standard until 1968, when Watson[9] was able to simplify it drastically by commuting through the derivatives the determinant of the metric tensor.
Before we do this we must mention that a derivation of this Hamiltonian is also possible by starting from the Laplace operator in Cartesian form, application of coordinate transformations, and use of the chain rule.
is the α component of the body-fixed rigid rotor angular momentum operator, see this article for its expression in terms of Euler angles.