Variation Theorem
A method for obtaining energies and wave functions by refining a function that approximates the correct wave function is provided by the variation theorem.
In the preceding sections, spectral results have been used in the deduction of atomic state symbols and electron configurations. Quantum mechanics has been a guide to the characteristics, principally that of quantization of angular momentum, of a central force system. Complex atoms can also be studied by an extension of the quantum mechanical treatment of the hydrogen atom. As a first step, we consider the helium atom.
To simplify the notation we will use the symbol _{2} = ∂^{2}/∂x^{2} + ∂^{2}/∂y^{2} + ∂^{2}/∂z^{2}. This symbol known as the laplacian can also imply the corresponding operator in polar coordinates. If we assume that the potential energy function U encompasses all the potential energy terms needed to describe this two electron atom and that ε is the total energy of the system, we can write the helium atom Schrodinger equation as
[-h^{2}/2m (_{1}^{2} + _{2}^{2} ) + U] ψ = ε ψ where _{1} and _{2} are the laplacians for the electrons labeled 1 and 2.
The potential energy expression must include the three coulombic interaction terms suggested here. When these are used for the potential energy term U, we have helium atom Schrodinger equation
[-h^{2}/2m (_{1}^{2} + _{2}^{2} ) + e^{2}/4∏ε_{0} (- 2/r_{1} – 2/r_{2} + 1/r_{1}^{2})] ψ = ε ψ
The usual procedure in solving such problems is to try to find a solution function in which the variables appear in separate factors. Here it would be convenient to deal with a solution function of the form ψ(1) ψ(2),where the first factor involves only the coordinates of electron 1 and the second only the coordinates of electron 2. But the r_{1}^{2} term depends on the coordinates of the two electrons in a way that does not allow this separation. As a result, we cannot simply solve to obtain eigenfunction value of ε. This inability to find a function that solves the Schrodinger equation is typical of all but the simplest atomic problems. There is, fortunately, a way of obtaining an approximate solution. often the approximate solution. often the approximate solution can be refined so that it is close enough to the exact solution to be of value.
The energy of a quantum mechanical system is obtained from H ψ = ε ψ when ψ is an eigenfunction. Alternatively we could write this in the expectation value form
ε = ∫ψ*Hψd/∫ψ*ψd
If Hψ = ε ψ, the right side reduces to the eigenvalue ε. Suppose, however, that we cannot find a function that satisfies H ψ = ε ψ. We could still use any well behaved function Ø to carry out the operations required for the evaluation of the integrals of
∫Ø*HØd/∫Ø*Ød
It seems reasonable that the more closely a trial function Ø approximates the true but unknown solution function ψ, the more closely the result obtained from this expression will approximate the correct energy. We thus write
ε_{appx} = ∫Ø*HØd/∫Ø*Ød
But if the solution function is unknown, we do not know which trial function most closely approximates it, and we do not know which calculated energy is close to the correct energy. An important theorem, known as the variation theorem, provides the necessary guide.
The theorem, stated without proof, is that the value of the energy calculated from the expression is less negative, i.e. the system seems less stable, than the value that would be obtained from various trial functions is the lowest values produced when these functions are used.
For systems for which an exact solution cannot be found, the procedure is to guess a trial function that is expected to be like the solution function. An energy corresponding to this trial function is calculated. If this energy is lower than the previous function is found which gives the lowest possible energy. This energy is the best approximation to the exact energy, and the corresponding trial function is, in this regard, the best approximation to the true eigenfunction.
One trial function for the helium atom problem consists of describing which electron as a 1s electron. This is acceptable, in view of the Pauli Exclusion Principle, if the electron spins are opposite.
The variation method invites more flexible functions. Such functions can be “varied” to obtain the lowest, and best, energy value. Simple modifications of the function in which Z is treated as a variable and a factor involving r_{1}^{2} is introduced. Further modifications, which produce more complex trial functions, give an energy value that is in agreement with, and probably more accurate than, the experimental value.
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