Subject: Computing Vibrations

From: [email-masked] (Dave Young)
Date: Sun, 13 Aug 1995 14:19:50 -0400

Hello all,

	I have written the following short introduction for an introductory 
graduate class in computational chemistry in order to give the students
a handle on what is going on before delving into equations and computations.  
I am posting it here for your enjoyment and comments.  Please, let me
know of I am over looking any important techniques.

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		Computational Modeling of Molecular Vibrations

   				   David Young

				Chemistry Department
		  	     Michigan State University
				E. Lansing, MI 48824

	The vibrational motion of a molecule is observed experimentally
via infrared and Raman spectroscopy.  These techniques can be used to
determine a molecule's structure and environment since those factors affect
the vibrational frequencies.  In order to gain such useful information,
it is necessary to determine what vibrational motion corresponds to each
peak in the spectra.  This assignment can be quite difficult due to the
large number of closely spaced peaks possible even in fairly simple 
molecules.  In order to aid in this assignment, many workers use computer
simulations to calculate the vibrational frequencies of molecules.  The
following is a brief description of the various computational techniques
available.

	The simplest description of a vibration is a harmonic oscillator
which describes springs exactly and pendulums with small amplitudes 
fairly well.  A harmonic oscillator is defined by the potential energy
being proportional to the square of the distance displaced from an 
equilibrium position.  In a classical treatment of a vibrating object, the 
motion is fastest at the equilibrium position and comes to a complete stop 
for an instant at the turning point, where all of the energy is potential 
energy.  The probability of finding the object is highest at the turning 
point and lowest at the equilibrium point.

	A quantum mechanical description of a harmonic oscillator uses
the same potential energy function, but gives radically different results.
In a quantum description, there are no turning points.  There is some
probability of finding the object at any displacement, but that
probability becomes very small (decreasing exponentially) at large distances.
The energy is quantized, with a quantum number describing each possible
energy state and only certain energies possible. Very small objects, such
as atomic particles behave according to the quantum description with 
low quantum numbers (<100).  Macroscopic objects under a quantum description 
will have very large quantum numbers with energy spacings that are too close
together to measure and a probability distribution that becomes identical to
the classical result in the limit of infinite quantum numbers.  The fact that
classical mechanics is a special case of quantum mechanics for large masses is
called the "correspondence principle".

	The vibration of molecules is best described using a quantum 
mechanical approach.  However, molecules do not behave according to a 
harmonic oscillator description.  Bond stretching is better described
by a Morse potential and conformational changes have a sine wave type
behavior.  However, the harmonic oscillator description is used as an
approximate treatment for low vibrational quantum numbers.

	The quantum mechanics equation (called the Schrodinger equation)
has never been solved exactly for any chemical system other than the
hydrogen atom.  However, many ways are known to approximate the solution.
Approximation methods known as ab initio methods use mathematical 
approximations only.  Frequencies computed with ab initio methods and
a quantum harmonic oscillator approximation tend to be 10% too high (due to the
difference between a harmonic potential and the true potential)
except for the very low frequencies (below about 200 wave numbers)
which are often quite far from the experimental values.  Many studies are
done using ab initio methods and multiplying the resulting frequencies
by 0.9 to get a good estimate of the experimental results.

	Semiempirical methods are another means of approximating the 
Schrodinger equation.  In a semiempirical treatment, the computation is done 
much faster by neglecting part of the computation and using experimentally 
determined values to correct for the resulting errors.  Vibrational 
frequencies from semiempirical calculations tend to be qualitative in 
that bond stretches have high frequencies, bond angle bending lower 
frequencies, torsions even lower, etc.  However, the actual values are
erratic.  Some values will be close while others are too low or too high.
The density-functional theory methods give frequencies with this same
erratic behavior, but a somewhat smaller deviation from the experimental
results.

	Some computer programs will output a set of frequencies containing six
values near zero for the three degrees of translation and three degrees
of rotation of the molecule.  Other programs will use a more sophisticated
technique to avoid computing these extra values, thus reducing the computation
time.  Before frequencies can be computed, the program must compute the
geometry of the molecule since the normal vibrational modes are centered
at the equilibrium geometry.  When a negative frequency is computed, it 
indicates that the geometry of the molecule corresponds to a maximum of 
potential energy with respect to the positions of the nuclei.  The transition 
state of a reaction is characterized by having one negative frequency.
	
	It is possible to compute vibrational frequencies from ab initio
methods without using the harmonic oscillator approximation.  For a diatomic
molecule, the quantum harmonic oscillator energies can be obtained by 
knowing the second derivative of energy with respect to the bond length
at the equilibrium geometry.  For a non-harmonic oscillator energy,
the entire bond dissociation curve must be computed, which requires far more
computer time.  Like wise, computing anharmonic frequencies for any
molecule requires computing at least a sampling of all possible nuclear
motions.  Due to the enormous amount of time necessary to compute all
of these energies, this sort of calculation is very seldom done.

	Another method for computationally describing molecules is
called molecular mechanics.  It is a non-quantum method in which the
forces acting on the atoms are modeled as simple algebraic equations
such as harmonic oscillators, Morse potentials, etc.  All of the constants
for these equations are usually obtained from experimental results.
A set of equations and their constants is called a force field.  A force
field can be designed to describe the geometry of the molecule only or 
specifically created to describe the motions of the atoms.  Calculation of 
the vibrational frequencies by determining the geometry then using a 
harmonic oscillator approximation can yield usable results if the force 
field was designed to reproduce the vibrational frequencies.  NOTE: Many 
of the force fields in use today were not designed to reproduce vibrational 
frequencies in this manner.  When using this method, there is not necessarily 
a 10% error between the results and the experiments, since the parameters 
may have been created by determining what harmonic parameters would reproduce 
the experimental results, thus building in the correction.  As a general
rule of thumb, mechanics methods do well if the compound being examined
is similar to those used to create the parameters.  Molecular mechanics 
does not do so well if the structure is significantly different from the 
compounds in the parameterization set.

	Another technique built around molecular mechanics is 
a dynamics simulation.  In a dynamics simulation, the atoms move around
for a period of time following Newton's equations of motion.  This
motion is a super-position of all of the normal modes of vibration
so frequencies can not be determined directly from this simulation.
However, the spectra can be determined by doing a Fourier transform
on these motions.  The motion corresponding to a peak in this spectrum 
is determined by taking just that peak and doing the inverse Fourier 
transform to see the motion.  This technique can be used to calculate 
anharmonic modes, very low frequencies and frequencies corresponding to 
conformational transitions.  However, a fairly large amount of computer time 
may be necessary to get enough data from the dynamics simulation to get a 
good spectra.

	Another related issue is the computation of the intensities 
of the peaks in the spectra.  Peak intensities depend upon the probability 
that a particular wavelength photon will be absorbed or Raman scattered.  
These probabilities, can be computed from the wave function by first computing 
the transition dipole moment.  Some types of transitions turn out to have a 
zero probability due to the molecules symmetry or the spin of the electrons.  
This is where spectroscopic selection rules come from.

	In conclusion it is possible to use computational techniques
to gain insight into the vibrational motion of molecules.  There are a 
number of computational methods available which are have varying degrees 
of accuracy and difficulty.  These methods can be powerful tools if
the user is aware of their strengths and weaknesses.

Further Information

The seminal text on molecular vibrations is
E. B. Wilson Jr., J. C. Decius, P. C. Cross "Molecular Vibrations : The
Theory of Infrared and Raman Vibrational Spectra" Dover (1980)

For an introductory level overview of computational chemistry see
G. H. Grant, W. G. Richards "Computational Chemistry" Oxford (1995)

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				Dave Young
				[email-masked]

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Q:  How do you avoid making the errors of the HF central field approximation
    in the first place in stead of spending all of your time trying to
    correct for them.

A:  Quantum Monte Carlo
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