Various difficulties of classical physics, including inadequate description of atoms and molecules, led to new ways of visualizing physical realities, ways which are embodied in the methods of quantum mechanics. Quantum mechanics is based on the description of particle motion by a wave function, satisfying the Schrodinger equation, which in its “time-independent” form is: ((−h2/8mπ2)⛛2+V)Ψ=E Ψ or, for short: HΨ = EΨ In this equation, H, the Hamiltonian operator, is defined by H = − ((h2/8mπ2)⛛2+V, where h is Planck’s constant (6.6 10−34 Joules), m is the particle’s mass, ⛛2 is the sum of the partial second derivative with x,y, and z, and V is the potential energy of the system. As such, the Hamiltonian operator is the sum of the kinetic energy operator and the potential energy operator. (Recall that an operator is a mathematical expression which manipulates the function that follows it in a certain way. For example, the operator d/dx placed before a function differentiates that function with respect to x.) E represents the total energy of the system and is a number, not an operator. It contains all the information within the limits of the Heisenberg uncertainty principle, which states that the exact position and velocity of a microscopic particle cannot be determined simultaneously. Therefore, the information provided by Ψ(nit) is in terms of probability: Ψ2(x,t) is the probability of finding the particle between x and x + dx, at time t. The Schrödinger equation applied to atoms will thus describe the motion of each electron in the electrostatic field created by the positive nucleus and by the other electrons. When the equation is applied to molecules, due to the much larger mass of nuclei, their relative motion is considered negligible as compared to that of the electrons (Born-Oppenheimer approximation). Accordingly, the electronic distribution in a molecule depends on the position of the nuclei and not on their motion. The kinetic energy operator for the nuclei is considered to be zero. For a many-electron molecule, the Hamiltonian operator can thus be written as the sum of the electrons’ kinetic energy term, which in turn is the sum of individual electrons’ kinetic energies and the electronic and nuclear potential energy terms.
Variational finite-difference representation of the kinetic energy operator
