Ab-initio Methods

Quantum theory is based on the Schrödinger equation:

Ψ describes the state of the system as a function of coordinates. This function, called the state function or wave function, contains all the information that can be determined about the system. is the Hamiltonian (i.e., energy) operator of the system; and E is the energy of that particular state. Schrödinger equation for molecular systems can only be solved approximately. Methods which approximate maybe classified into two categories namely ab-intio and semiempirical. While Semi empirical methods uses parameters that reduces the computational time especially the time consuming mathematical terms in Schrödinger equation. Ab-intio methods uses all the mathematical term and thus are time consuming.
            The term ab-initio implies a rigorous, non-parameterized molecular orbital treatment derived from first principles. However, this is not completely true. There are a number of simplifying assumptions in ab-initio theory, but the calculations are more complete, and therefore more expensive, than those of the semi-empirical methods. It is possible to obtain chemical accuracy via ab-initio calculations, and this approach is especially favored in situations in which little or no experimental information is available.

1.1 Shrodinger Wave Equation

The time dependent Schrodinger  equation (Leach,2001).for a single particle of mass m  is given as

(1.1)

Here the particle is under the influence of the external field V ( usually the electrostatic potential due to the nuclei of the molecule) and the position r (given by r = xi + yj + zk) at any time t. is Planck's constant (h) divided by   and i is the square root of  -1. Ψ is the wavefunction which describes the particle's motion. This is the vital quantity from which a number of properties for e.g energy states etc can be derived.  When the external potential  V  is independent of time then the  wave function can be written as the product of a spatial part and a time part:

(1.2)

We shall only consider situations where the potential is independent of time, which enables the time-dependent Schrodinger equation to be written in the more familiar, time-independent form:

(1.3)

Here, E is the energy of the particle and we have used the abbreviation  ∇² (pronounced 'del-
squared').

(1.4)

It is usual to abbreviate the left-hand side of  equation (1) to HΨ, where H  is the  Hamiltonian operator.

(1.5)

This reduces the shrodinger equation to HΨ = EΨ.The Schrodinger equation can be solved exactly for only a few problems, such as the particle  in a box, the harmonic oscillator, the particle on a ring, the particle on a sphere and the  hydrogen atom.A common feature  of these problems is that it is necessary to impose certain requirements (often called boundary conditions) on possible solutions to the equation. Thus, for a particle in a box with infinitely  high walls, the wavefunction is required to go to zero at the boundaries. For a particle on a  ring the wavefunction must have a periodicity of  2π  because it must repeat every traversal of  the ring. An additional requirement on solutions to the Schrodinger equation is that the  wavefunction at a point r, when multiplied by its complex conjugate, is the probability of  finding the particle at the point (this is the Born interpretation of the wavefunction). The  square of an electronic wavefunction thus gives the electron density at any given point. If  we integrate the probability of finding the particle over all space, then the result must be  1 as the particle must be somewhere. Thus

(1.6)

Here  Ψ * is the complex conjugate of the wavefunction Ψ.