An Introduction to
Computational Chemistry
1
, 1. THEORETICAL BACKGROUND
Classical mechanics or Newtonian mechanics failed to explain the behavior of subatomic
particles like electrons. The crisis prevailed until Einstein explained these with the help of
dual nature of radiation and particles that led to Schrödinger's proposal of wave mechanics,
better known as quantum mechanics in 1926. The concept of wave function is a
fundamental postulate where the wave function can be taken as the most complete
description of a physical system. The solutions to Schrödinger’s equation describe not only
molecular, atomic or subatomic systems, but also macroscopic systems17.
1.1 Schrödinger Equation18
Obtaining solution to the Schrödinger wave equation is the major task associated with
quantum mechanics. To apply the equation, the Hamiltonian operator which accounts for the
kinetic and potential energy of particles in a system is set up and inserted into Schrödinger
wave equation. The resulting partial differential equation is solved for the wave function,
which contains information about the system. There exist two forms of Schrödinger wave
equation. These are,
1. Time dependent Schrödinger equation
(1)
It is the most general form and it gives the description of a system evolving with time. is
the Hamiltonian operator that characterizes the total energy (sum of kinetic and potential
energies) of any given wave function and takes different forms depending on the situation.
The time dependent equation contains the first derivative of the wave function with respect
to time and enables the calculation of the future wave function at any time, provided, wave
function at time t0 is known.
2. Time independent Schrödinger equation
(2)
The equation states that when the Hamiltonian operator acts on a wave function , then
is a stationary state which does not depend on time, and the proportionality constant, E, is
the energy of the state . The equation gives light to the concept of probability density,
which doesn’t change with time.
2
Computational Chemistry
1
, 1. THEORETICAL BACKGROUND
Classical mechanics or Newtonian mechanics failed to explain the behavior of subatomic
particles like electrons. The crisis prevailed until Einstein explained these with the help of
dual nature of radiation and particles that led to Schrödinger's proposal of wave mechanics,
better known as quantum mechanics in 1926. The concept of wave function is a
fundamental postulate where the wave function can be taken as the most complete
description of a physical system. The solutions to Schrödinger’s equation describe not only
molecular, atomic or subatomic systems, but also macroscopic systems17.
1.1 Schrödinger Equation18
Obtaining solution to the Schrödinger wave equation is the major task associated with
quantum mechanics. To apply the equation, the Hamiltonian operator which accounts for the
kinetic and potential energy of particles in a system is set up and inserted into Schrödinger
wave equation. The resulting partial differential equation is solved for the wave function,
which contains information about the system. There exist two forms of Schrödinger wave
equation. These are,
1. Time dependent Schrödinger equation
(1)
It is the most general form and it gives the description of a system evolving with time. is
the Hamiltonian operator that characterizes the total energy (sum of kinetic and potential
energies) of any given wave function and takes different forms depending on the situation.
The time dependent equation contains the first derivative of the wave function with respect
to time and enables the calculation of the future wave function at any time, provided, wave
function at time t0 is known.
2. Time independent Schrödinger equation
(2)
The equation states that when the Hamiltonian operator acts on a wave function , then
is a stationary state which does not depend on time, and the proportionality constant, E, is
the energy of the state . The equation gives light to the concept of probability density,
which doesn’t change with time.
2
