Mueller M.R. - Fundamentals of Quantum Chemistry[c] Molecular Spectroscopy and Modern Electronic Structure Computations (Kluwer, 2001)
.pdf
56 |
Chapter 4 |
Since the 
eigenfunctions are orthonormal (see Equation 2-24), the previous integral is zero when 
and one when 
The result above must be positive since
are positive. Therefore:
or
completing the proof.
Variation theory states that the energy calculated from any trial wavefunction will never be less than the true ground-state energy of the system. This means that the smaller the value of 
the closer it is to the true ground-state energy of the system and the more 
represents the true ground-state wavefunction. The trial wavefunction is set up with one or more adjustable parameters, 
making the function flexible to minimize the value of 
An n number of adjustable parameters will set up an n number of differential equations:
Increasing the number of adjustable parameters improves the result, however it also increases the complexity of the problem. The variational approach is demonstrated in the following example.
Techniques of Approximation |
57 |
Example 4-1
Problem: A particle with a mass m is confined to a 1-dimensional box. Use the following trial wavefunction:
where N is the normalization constant and p is the adjustable parameter. Note that this function is well behaved at the boundary conditions since the function is zero at 
and 
Solution: The first step is to write the Hamiltonian for the problem (see Section 2.4).
The next step is to solve for 
in terms of the adjustable parameter p (Equation 4-1).
58 |
Chapter 4 |
Now the derivative of 
with respect to p is taken and set equal to zero to solve for p.
The solutions are |
and -4.6331. These values are now |
substituted into the expression for |
|
|
for |
|
for |
Since 
results in a lower value for the energy of the ground-state this value is adopted. This optimizes the trial wavefunction to:
The energy obtained using this trial function can now be compared to the
.true ground-state energy for a Particle-in-a-Box given in Equation 2-19.
True Ground-state Energy:
Ground-state Energy from Trial Wavefunction:
Techniques of Approximation |
59 |
The optimized trial wavefunction can now be normalized (see Section 2.3).
A useful approach to obtaining a trial wavefunction is to form it from a linear combination of functions 
such that the combination coefficients, 
become the adjustable parameters.
60 |
Chapter 4 |
The functions 
are not varied in the calculation and constitute what is called the basis set. The value of 
is computed as follows:
where
To find the minimum value of 
Equation 4-5 is differentiated with respect to each coefficient and in turn set
in each case.
4.2TIME INDEPENDENT NON-DEGENERATE PERTURBATION THEORY
The idea behind perturbation theory is that the system of interest is “perturbed” or changed slightly from a system whereby the solution is known. This can occur in two different ways: a) a new problem that has similarities to another system of which the solution is known (this happens often in chemistry) or b) the molecule or atom experiences some type of external perturbation such as a magnetic field or electromagnetic radiation
Techniques of Approximation |
61 |
(this is important in the case of spectroscopy). At this point the discussion will be limited to time-independent systems with non-degenerate quantum states. A time-independent perturbation is one in which the perturbation is not a function of time.
The Hamiltonian for the system of interest is divided into parts: the part representing system with a known solution, and then into a number of additional parts that correspond to perturbations from the known system to the system of interest.
The term in the equation above with a superscript zero corresponds to the Hamiltonian for the system with a known solution (unperturbed system), 
The rest of the terms correspond to additional terms that perturb the known system. The term 
is a first-order perturbation, the term 
is a second-order perturbation, and so on. The idea is that each order of perturbation is a slight change from the previous order.
Example 4-2
Problem: Consider a Particle-in-a-Box with a sinusoidal potential inside:
The term 
is a constant. Write the different orders of the Hamiltonian for the particle.
Solution: The complete Hamiltonian is first written for the particle.
This Hamiltonian can be broken down into two parts: that of the Particle-in- a-Box Hamiltonian, 
and that of the first-order perturbation, 
62 |
Chapter 4 |
The solution for the perturbed system can now be developed. The variable 
is introduced as a scalar quantity that acts as a “tunable dial” for the perturbation in the range of 
When 
is equal to zero, there is no perturbation resulting in the unperturbed system. When the value of 
is unity, the system experiences the full perturbation. At the end of the derivation, the value of 
will be set at unity removing it from all of the expressions and the perturbation will be entirely reflected in the first and higher order perturbing Hamiltonians. The Hamiltonian for the perturbed system can be written as an expansion series in terms of 
The wavefunction for the system of interest at a quantum level n, 
can also be written as a sum of correction terms from the unperturbed wavefunctions, 
in an expansion series of 
Likewise, the energy for the perturbed system for a quantum level n can also be written as a sum of corrective terms in energy from the unperturbed system, 
in an expansion series of 
Equations 4-6 through 4-8 can now be applied to the Schroedinger equation for the problem.
Techniques of Approximation |
63 |
In Equation 4-9, 
is the variable and the equation can be expanded and grouped in terms of orders of 
Since 
can take on any value from zero to unity, each term for the power of 
in Equation 4-10 must be individually equal to zero. Instead of just one equation, the original Schroedinger equation, there are now an infinite number of equations since the expansion in terms of powers of 
is infinite. Generally perturbation computations are only taken to the second-order and so these equations are shown below.
terms (zero-order):
terms (first-order):
terms (second-order):
The reason for introducing the variable 
was to produce the separate equations 4-11a-c. Now the value of 
can be set at unity. This means that the full perturbation is reflected in the first and higher-order Hamiltonians 
The Equations 4-6 through 4-8 can now be rewritten with
64 |
Chapter 4 |
The goal now is to determine the energy and wavefunction corrections for the perturbed system using Equations 4-12a-c. The most common practice is to take the corrections to second-order. Therefore, the discussion here will be limited to the first and second-order energy correction and the first-order wavefunction correction. The zero-order wavefunction and energy given in Equation 4-1la are already known as they correspond to the unperturbed system.
To obtain the first-order energy correction for the 
quantum level of the perturbed system, 
Equation 4-11b is multiplied by the complex conjugate of the unperturbed wavefunction 
and integrated over all space for the perturbed system.
The above equation can be simplified by realizing that 
is orthonormal.
This equation can be further simplified by realizing that 
is hermitian (see Section 2.5 and Equation 2-20).
Techniques of Approximation |
65 |
This equation can be readily rearranged to solve for the first-order energy correction of the nth level of the perturbed system.
The interpretation of this result is that the first-order energy correction is a kind of average of the effect of the perturbation on the unperturbed wavefunction. The perturbation effect will be greatest at the antinodes of the wavefunction and the least at the nodes.
Example 4-3
Problem: Consider a particle in a 1-dimensional box with a potential 
in the middle 10% of the box.
Potential 
within the region of: 