NOTES 5
Approximation Techniques
Most problems of quantum chemistry cannot be solved exactly, that is there is no
analytical solution. It is necessary to make some sort of approximation.
Four ways:
1) Use classical mechanics as a guide to the quantum mechanical solution and take note
of the small magnitude of Planck's constant. This is the Semiclassical Approximation.
2) Guess the form of the wavefunction of the system and use the variation principle to
help find the best values of the adjustable parameters in the wavefunction. The Variation
Principle basically says that the average energy derived from using a wavefunction that is
not the exact solution to Schrodingers equation will always be higher than the true value.
Thus finding the adjustable parameter values that minimize the energy give you the best
values and the best approximate wavefunction.
3) In the case that the exact solutions for a system that resembles the true system are
known, the differences in the Hamiltonians of the two systems can be used as a guide to
finding the best approximation to the true solution. This procedure is the main tactic of
Perturbation Theory. Perturbation Theory is most useful when one is interested in the
response of atoms and molecules to electric and magnetic fields. If the fields change with
time, then Time-Dependent Perturbation Theory is necessary.
4) The Self Consistent Field Theory is essentially a set of procedures which is an iterative
method for solving Schrodinger's Eqn. for systems of many particles that is based on the
Variation Principle.
1. Semiclassical Approximation
How might classical mechanics suggest the form of a wavefunction and the
corresponding energy?
Note it can be written:
ħ2 d2 /dx2 + px2  = 0
(A)
where px = {2m(E-V)}1 / 2
This is the classical expression for the linear momentum of a particle of mass m.
If the potential energy is uniform, the solution to equation (A) is e+ / - p x / ħ is a first
approximation to .
The WKB approximation says the solution should be of the form:
 = c++ + c-If the potential was uniform
+/- = e+ - p x / ħ
But if the potential is varying slowly then replace product px by a function S+- so
+/- = e+ - i S + - / ħ
The WKB Method then seeks to find an expression for S+-.
putting this solution into eqn. (A):
-(dS+-/dx)2 +/- iħ (d2 S+-/dx2 ) + p2 = 0
(B)
No approximation are used up to here, the WKB Approximation says on can find S+- by
expanding it in powers of ħ.
S+- = S+-( 0 ) + ħS+-( 1 ) + ħ2 S+-( 2 ) + ……..
One can think of the transition from Q.M. to C.M. as corresponding to the replacement of
ħ by zero! and in retaining low-order terms in ħ one is bridging from classical to quantum
mechanics.
Substitution of the expansion in terms of the powers of ħ, into (B) and a collection of the
terms in the same power of ħ gives:
{-(dS+-( 0 ) /dx)2 + p2 } + ħ{-2(dS+-( 0 ) /dx)(dS+-( 1 ) /dx) +/- i(d2 S+-( 0 ) /dx2 )} + … = 0
The coefficient of each power of ħ is set to zero.
1st Order WKB Approximation
Consider only the terms up to ħ, yields
ħ( 0 ) term: -(dS+-( 0 ) /dx)2 + p2 = 0 implies
S+-( 0 ) = S+-( 0 ) (0) + 0∫x p dx
(C)
ħ( 1 ) term:
implies:
And if the constants of integration are absorbed into the coefficients c+-, the 1st order
WKB wavefunction becomes
 = 1/p {
The probability density given by this expression is proportional to 1/p and is thus
inversely proportional to the speed of the particle.
This is consistent with the classical picture of particle motion: the faster a particle is
traveling, the less time it is found in a given region.
In classically allowed regions (where E>V), p is real and the solution may be written isn
some cases as:
 =
The constants C and  are related to c+ and c- using Euler's relation, e- i x = cos x + i sin x
C cos = i(c+ - c-)
C sin = (c+ + c-)
In classically forbidden regions (E<V), p is imaginary, and the solution is a linear
combination of exponentially decaying and increasing functions.
Also the approximation is not applicable to the classical turning points as p = 0 at those
points and the expression diverges. It is also not applicable when the potential energy is
very sharply varying.
Analysis shows that the semiclassical approximation is valid where:
|dV/dx| << {2m|E-V|}3 / 2 / (mħ)
Also to link the wavefunctions across a turning point where the WKB expressions are not
valid assume the potential energy function close to the turning point xtp can be written:
V = E + (x-xtp)(dV/dx)xtp (dV/dx)xtp = E + (x-xtp)F
For this type of a potential energy Schrodinger's eqn. is
ħ2 d2 /dx2 - 2m(x-xtp) F = 0
Solutions to this are the Airy function. To cross the turning point, then one needs to
match the oscillatory behavior on one side of xtp to an Airy function and extend the Airy
function to the other side and match that Airy function to the oscillating function on the
other side. This must be done at all of the turning points.
Turns out the for such confining potentials the wavefunction is well-behaved only for
This eqn. provides a semiclassical formula for the energy levels associated with the
stationary states "n" of the particle.
This is known as the Bohr-Sommerfeld quantization
As you might expect, the WKB approximation is useful in scattering problems.
2. Time Independent Perturbation Theory
In perturbation theory, we suppose that the Hamiltonian for the problem we are solving
can be expressed as:
H = H(0) +  H(1) + 2 H( 2 ) .....
where H0 is a Hamiltonian that has an exact solution with known eigenfunctions and
eigenvalues.
H( 0 ) m( 0 ) = Em( 0 ) m( 0 )
H1 is a Hamiltonian which is a slight change from the exact solution.
Suppose that the true energy is the sum:
E = E0 + E( 1 ) + 2 E( 2 ) + ...
E0 is the eigenvalue for H0n 0n = E0n 0n, which is an exact analytical solution
E(1) is the first order correction, E(2) is the second order correction, etc.
The true wavefunction can also be written as  = (0) + (1) + 2 (2) + ....
Where 0 is the eigenfunction shown above, 1 is the first order correction, etc.
Need to solve H = E.
same power of  gives
Putting the above eqns into this and collecting terms with the
Because  is an arbitrary parameter, the coefficient of each power of  must equal zero
separately giving:
Often only need lst order correction to the wavefunction which is sufficient to give
higher order corrections to the energy.
The wavefunction is written as a linear combination of the unperturbed wavefuction of
the system because they are a complete basis set of functions:
o( 1 ) = n Cnn( 0 )
The sum is over all the states if the ideal model.
And substitution into the 2nd equation above allows solution for E0( 1 )
E0( 1 ) =
The first order correction to the wavefunction can be found by:
giving ck.
So
 = 0( 0 ) + k0 {Hk0( 1 ) / (E0( 0 ) - Ek( 0 ) ) } k( 0 )
Here the ideal model is distorted by mixing into it the other states of the system. This
mixing is expressed by saying that the perturbation induces virtual transitions to these
other states of the model system. But in fact, the distorted state is beinig simulated as a
linear superposition of the unperturbed states of the system:
For a particular state k
a) makes no contribution to the superposition if Hk0( 1 ) = 0
b) makes a larger contribution (for a magnitude of the matrix element) the smaller the
energy difference |Eo( 0 ) - Ek( 0 ) |
Second Order Correction to the Energy
Use the same technique to extract the second-order correction
Write the second-order correction to the wavefunction as the linear combination
o( 2 ) =
Substitute in and multiply by left ket <0|
Left hand side is zero and
E0( 2 ) =
plugging in the coefficients cn
E0( 2 ) =
Example
Comments on Perturbation
Role of Symmetry
Degenerate States
3. Variation Theory
Based on the Variation Theorem which states that the true ground state energy is always
less than the energy that can be derived from a "trial wavefunction" which is a
wavefunction that is not the exact solution to H=E
That being the case one can try to optimize the trial function, if it has adjustable
parameters to make it have the lowest energy possible, and then this is the best possible
form of the trial wavefunction. We do this from the calculation of the average or
expectation value of the energy based on the trial wavefunction.
1st Step
Guess the form of the Trial Wavefunction, t
2nd Step
Plug it into the expression for the average energy also called the RAYLEIGH RATIO.
<E> = ∫t* H t d / ∫t* t d ]
Again typically the trial function is expressed in terms of one or more parameters that can
be varied until the Rayleigh Ratio is minimized.
Example
Rayleigh Ritz Method
Perhaps the most computer-friendly way to implement this method is to represent the trial
function as a linear combination of fixed basis functions with variable coefficients. The
coefficients then are treated as the variable sot be changed and optimized to find the
minimal average energy.
The trial wavefunction is taken to be
t = i ci i
(i are the basis functions)
<E> =
Finding the minimum value of this ratio involves
Thus solutions to the equations
j cj (Hij - ESij) = 0
Sij is the overlap integral
This is a set of simultaneous eqns. for the coefficients cj.
And the solution is that the following secular determinant should be zero.
det |Hij - ESij| = 0
The set of values of E are the roots of this eqn., and the lowest value is the best one.
Example
The variation principle leads to an upper bound for the ground state energy of the system.
One could also use it to determine an upper bound for the first excited state by
formulating a trial function that is orthogonal to the ground state function.

Hellmann-Feynman Theorem
Here the Hamiltonian depends on a parameter P (like the internuclear distance etc.) The
exact wavefunction is a solution of Schrodinger's Eqn., and so it and its energy also
depend on the parameter P.
Well then how does the energy of the system vary as the parameter is varied?
dE/dP = <H/P>
Example
4. Time-Dependent Perturbation Theory
Any sort of interaction of a molecule or atom with an electric field means that the system
is exposed to a field that changes in time, oscillates for as long as the perturbation is
imposed. Time-dependent perturbation theory is necessary then to calculate the transition
probabilities in spectroscopy and the intensities of spectral lines.
For time-dependent pertubation theory
Start off with perturbation wavefunction that depends on the time:
H = H0 + H1(t)
H( 1 ) (t) may for instance be 2H( 1 ) cost
Must use time dependent form of Schrodinger’s Eqn. (See Postulates) Hhd/dt
/
If E1( 0 ) and E2( 0 ) are the stationary energies of two of the states and their corresponding
time independent wavefunctions are 1( 0 ) and 2( 0 ) ; they are related to the timedependent unperturbed wavefunctions by
H( 0 ) n( 0 ) = iħ n( 0 ) /t where n( 0 ) (t) = n( 0 ) e- i E n t / ħ
When the perturbation H( 1 ) (t) is on, the state of the system is expressed as a linear
combination of the time-dependent basis functions:
(t) = c1(t)1( 0 ) (t) + c2(t)2( 0 ) (t)
The state may evolve with time as well
The probability that at any time t the system is in a state n is |cn(t)|2
Substituting the linear combination into Schrodingers eqn.:
Each basis function satisfies the unperturbed time-dependent equation so
Then the time variation of the coefficients can be extracted to give:
After simplification on obtains:
c1 H11( 1 ) (t) + c2 H12( 1 ) (t) e- i ω 2 1 t = i ħ dc1/dt
Often the time-dependent perturbation has no diagonals so H11( 1 ) (t) = H22( 1 ) (t) = 0
leaving:
When the perturbation is absent, the matrix elements H12( 1 ) (t) = H21( 1 ) (t) = 0
so dc1/dt = dc2/dt = 0. So coeff. don't change from intial values and the state is
(t) =
Even though (t) oscillates with time, the probability of finding the system in either of
the state is constant. In absence of the perturbation the system is frozen at the initial
composition.
Case of a constant perturbation applied at t =0
fig
H12( 1 0 (t) =
Then
dc1/dt =
dc2/dt =
Further differentiation gives
The general solutons tot eh 2nd order differential eqn. are:
Now suppose that at t=0 the system is definitely in state 1. Then c1(0) = 1 and c2(0) = 0
These are initial conditions which lead to the solutions that:
c1(t) =
c2(t) =
Exact solutions
Rabi Formula
If interested in the probability finding the system in one of the two states as a function of
time, when the system was initially in state 1 at t=0, the time the constant perturbation is
applied. These probabilities are P1(t) = |c1(t)|2 and P2(t) = |c2(t)|2
One finds
P2(t) =
P1(t) = 1 - P2(t) for a two state system
Now consider a couple of special cases:
1. degenerate pair of states
ω21 = 0.
The probability of finding the system in state 2, if it was definitely in state 1 at t =0 is:
P2(t) =
Basically the system oscillates between the two states. The frequency is governed by the
magnitude of V and strong perturbations drive the system between its two states more
rapidly than weak perturbations.
2. Energy levels are widely separated
Widely separated in comparison with the strength of the perturbation, ω212 >>4|V|2
then
P2(t) ≈
Here the populations of the two states oscillate, but P2(t) never rises above 4|V|2 /ω212
Many level system
Requires dealing with Nt h order differential eqns. for an N-level system.
Use an approximation
Base the approximation on the idea that the perturbation is so weak and applied for so
short a time that all the coefficients remain close to their initial values.
Then if the system is in the state |i> at t=0, all the coeff. other than ci are close to zero
throughout the period for which the perturbation is applied and any coeff. of the state |f>
that is zero initially is given by:
cf(t) = 1/(iħ) ∫ ci(t)Hfi(t)ei ω f i t dt
and ci(t) is ~ 1
This ignores the possibility that the perturbation can take the system from its initial state
to the final by some indirect route which sequences several transitions.
So the perturbation is assumed to act only once.
Can be represented by a Feynman diagram
What about for a slowly switched constant perturbation
Such a perturbation is
H( 1 ) (t) = 0
for t<0
= H( 1 ) (1-e- k t ) for t≥0
H( 1 ) is a time-independent operator and for slow switching, k is mall and positive
The result is that for a state that is initially unoccupied:
cf(t) =
and for very long times after the perturbation has reached its final value, t>> 1/k, then
and the denominator in the second term can be replaced by -iωfi. This leads to
Which is just the result that would be obtained from first order time-independent
perturbation theory.
What about an oscillating perturbation?
Such as electromagnetic radiation in a spectrometer
A perturbation oscillating with an angular frequency  and turned on at t=0 has the
form
H( 1 ) (t) = 2H( 1 ) cos t = H( 1 ) (ei ω t + e- i ω t )
for t≥0
one obtains
cf(t) =
Can calculate the probatility of finding the system in the discrete state |f> after time t if
initially it was in state i at time t=0 is
Pf(t)
Basically the same as the expression for a static perturbation applied to a two-level
system.
Note that the probability depends on the frequency offset.
Transition to continuum of states
Now consiter the final state is a part of a continuum of states. The molecular density of
state is M(E).
The total transition probability is given by:
And the if we express the transition frequency fi in terms of the energy E/ħ (set energy
of initial state to be zero of energy)
One finds
and the integral can be simplified to give
define the transition rate, W, as the rate of change of probatility of being in an initially
empty state:
W = dP/dt
It follows that
W=
which is called Fermi's Golden Rule.
These are related to Einsteins transition probabilities
Lifetime and Energy uncertainty