The ** variational method** is a mathematical technique for approximating the energy state of a system, most often the ground state of a multi-electron system.

To illustrate the method, we begin with the expectation value for the ground state energy of a system, :

where is the ground state wavefunction of the system.

###### Question

Derive eq293.

###### Answer

Eq293 is obtained by multiplying the Schrodinger equation from the left by and integrating over all space.

As mentioned in an earlier article, is, by definition, the lowest energy value of the system. If we replace (the mathematical description of the ground state energy of the system) with any wave function that describes any energy state of the system,

where .

If is normalised,

Eq295 is called the ** variational principle**, i.e. the principle behind the variational method. It is evident that eq295 is a functional: a function of a function . Therefore, to estimate the ground state energy of a system, we substitute an appropriate trial wave function that depends on one or more arbitrary parameters (

**) into eq295, and minimise with respect to the parameters.**

*variational parameters*For example, we can estimate the ground state energy of a hydrogen atom using the trial wavefunction , where is the variational parameter. Substituting , in eq295, and using the identity , we have

Clearly, eq296 describes a parabola with a minimum value. Setting , solving for , and substituting the expression for back into eq296, we have

where is the estimated ground state energy of hydrogen.

In other words, we vary the variational parameter to arrive at a good approximation of .

To determine the ground state energy of a multi-electron system, the variational method is used together with the Hartree self-consistent method, which will be explained in subsequent articles.

###### Question

Provide a proof of the variational principle.

###### Answer

Since any well-behaved wavefunction can be expressed as a linear combination of a complete orthonormal set of basis wavefunctions, we substitute in eq295.

Since , we have . So,

is, by definition, the lowest energy value of the system. So, and is zero if and only if . Therefore, .