Atomic and Molecular Structure
COMPUTER ASSIGNMENT 1
Your task is to construct a potential energy curve for a diatomic molecule
and calculate a few properties for that diatomic.
Your calculation will be based on density functional theory. You will use
the B3LYP exchange-correlation functionals and a basis set labeled 6-31+G(d,p).
In short, you will be carrying out a number of B3LYP/6-31+G(d,p) calculations on
a diatomic (and its constituent atoms).
(1) Find the equilibrium bond length and vibrational frequency for your diatomic.
Use ‘STEP-BY-STEP’ instructions (provided separately) to create an input
file (CO.com), which looks similar to what follows below (leave out my
explanatory comments):
%chk=CO.chk
name of checkpoint file; should match input file name
%mem=200MW
memory allocation
%nproc=2
#processors
# opt freq B3LYP/6-31+g(d,p) geom=connectivity
<blank>
command line (keywords)
terminator for commands
Carbon Monoxide; R(C-O) = 1.15 A; Homework for 521
title line
<blank>
terminator for title line
01
net charge, blank space, spin multiplicity
O
0.35269710
0.06224066
C
-0.90570290
0.00000000
0.06224066
first atom; x,y,z
0.00000000 second atom;x,y,z
<blank>
terminator for molecule specification
1 2 2.0
connectivity info (uninteresting here)
2
connectivity info (uninteresting here)
<blank>
The input file must terminate with (at least) one blank line.
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This job will optimize the geometry of CO (i.e. locate the equilibrium bond
length) and compute the vibrational frequency for CO at the optimized
(equilibrium) bond length. Some additional CO properties such as the dipole
moment; partial atomic charges; rotational moments of inertia; molar enthalpy,
entropy and free energy will also be calculated. The geometry optimization is
based on energy minimization - make the molecular electronic energy as low as
possible walking around on the potential energy surface in a coordinated
("steepest descent") fashion.
The contents in the input file are generally not (upper, lower) case sensitive; the
"%xyz" expressions in the beginning of the file are exceptions, however; these
%-directives must be in lower case.
To submit the file to odin, open a terminal window; ssh over to odin0
$ssh odin0 <enter>
and submit the job for execution via
$submitg09 CO.com <enter>
$showq <enter> will show you where your job is.
All useful information will be in the CO.log file, which can be printed ($lp
CO.log) or examined in the editor. Or, open CO.log in GV and structural variables
can be read off, if the 'inquiry' button is activated, by ‘clicking’ on the appropriate
atoms; vibrational frequencies and total internal energy ('U') will be available in
‘Results’ (drop-down menu); etc.
(2) Construct the ground state potential energy curve (potential energy surface,
PES) for your diatomic by calculating the molecular electronic energy (the
potential energy, the purely electronic plus nuclear energy, U) at a number of
bond lengths at the B3LYP/6-31+G(d,p) level. Make a plot of total energy vs.
internuclear distance. Convince yourself that the total energy vs. bond length
curve looks like it should (!), except perhaps at very large bond lengths.
The PES could be constructed on a 'point-by-point' basis but that is
obviously a rather tedious procedure. It is possible to do a potential energy
surface "scan" and compute several (or many) points in one run. For example,
the input file below will successively increase the bond length from 1.20 Å in (10)
incremental steps of 0.02 Å each, for a total of 11 geometry points. The total
energies for the various points will be neatly listed toward the end of the CO.log
file
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%chk=CO.chk
%mem=200MW
%nproc=2
#B3LYP 6-31+g(d,p) scan geom=connectivity
<blank>
Carbon Monoxide; More Homework for 521
<blank>
01
C
O 1 B1
<blank>
B1=1.20 10 0.02
<blank>
1 2 3.0
2
<blank>
Scanning in the opposite direction (diminishing the bond length) is accomplished
by changing the sign of the step (e.g. to -0.03).
Note that the details of the scan (the number of steps and the step size) must be
entered manually by editing the .com file.
GaussView has a button to enter the “scan” command. After that, there are
several ways of entering the remaining information, which I will try to explain…..
Each single point (each geometry) calculation takes very little time for these very
small molecules, certainly much less than one minute per point. How many
points do you need? Enough so that you can make a decent plot (see Part (3)
below)!
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For your plot, you need corresponding bond length-total energy values. If you
open the .log file containing a ‘scan’ in GV, then the ‘Scan…’ entry in the
‘Results’ column should be highlighted. If you click on it, then a scan plot should
be generated automatically in a separate window. Right click on the mouse and
various options for manipulating the plot become available.
In the ‘File’ column, there is a ‘Save Image File…’ entry which may be used to
generate a JPEG of the scan plot; print this via $lp CO-scan.jpg.
You can also save corresponding bond length—total energy points as a text file
(.txt) and print it or mail it to yourself. Or you can read the values off the screen;
or invoke an editor (inside GV or outside) and copy the values down. Or dump
the .log file on a USB memory-stick and massage the data elsewhere. Or….
The Unix command
$ more CO.log <enter>
is useful for scrolling through a file (using the space bar).
$ tail -100 CO.log <enter>
will list the last 100 lines of CO.log on the screen.
(3) Find the equilibrium bond length (in Å) and the harmonic vibrational frequency
(in cm-1) numerically from your plot. Make a “best” parabola through three or
more points near the bottom of the well using Excel, Igor, CurveFit, three
equations in three unknowns, some other magical curve fitting program, a fancy
calculator, graph paper…use whatever tools you have at your disposal to extract
the force constant and Req from your data.
You have the 'exact' answers (Req, ) already from Part (1) above. Your results,
using parabolic interpolation (curve fitting...whatever) and proper conversion
factors(!), should be very, very close to the 'exact' answers, if you do a decent fit.
You need a fine grid for the points near the bottom of the potential energy well.
(4) Find the dissociation energy (in kcal/mol) for your diatomic. What we call
“dissociation energy” or “bond strength” is really an enthalpy, so this entails
finding the enthalpy for your diatomic and the constituent atoms (and a
subtraction of numbers). The molar enthalpy, free energy and entropy (standard
state: T = 298 K, P = 1 atm) are computed from statistical mechanics formulas as
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a by-product of a vibrational frequency calculation. They can be found in the .log
file a little below the vibrational frequencies.
The command
$grep
"xyz abc"
file1 <enter>
will search for the string 'xyz abc' (verbatim) in file1 and list the contents of the
lines in which the string occurs.
For example,
$ grep "SCF Done" CO.log <enter>
will lead you to the output line, which contains the total energy (in Hartrees).
The command
$grep “Enthalpies” CO.log <enter>
will list the line containing the enthalpy.
$grep ‘Free Energies’ CO.log <enter>
will list the line containing Gibbs free energy.
It looks weird computing the vibrational frequencies of an atom (there
aren’t any, of course) to get to the enthalpy, but G09 handles the situation and
makes the proper additions of RT to the energy to get the (atomic) enthalpy.
(5) Briefly compare the computed equilibrium bond length, vibrational frequency
and dissociation energy from steps (1) - (4) to the experimental values.
Experimental data may be found at the NIST site:
http://webbook.nist.gov/chemistry/, textbooks, or perhaps from Wikipedia.
(6) We can use rms(x) = root mean square displacement in x = <x2>1/2 as a
measure of the vibrational amplitude for a molecule. It is “easily” shown that
<x2> v = (v+0.5)ħ/(k)1/2
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where v = vibrational quantum number, k = force constant,  = reduced mass,
and ħ = "h bar" = h/2.
(a) Derive this relation for v = 0 (in other words, show that
<x2>v = 0 = (0.5)ħ/(k)1/2
by setting up the appropriate integral involving the vibrational ground-state wave
function – and integrate. Integral A.9 in Levine will come in handy, as will the use
of integrand parity (even vs. odd) to get the integral expression for the average
value into the proper form for use of A.9.
(b) Calculate <x2>1/2 (in Angstrom) for v = 0, 1, and 2 for your diatomic using the
formula above. Find the percentage increase/reduction in the bond length for v =
0, 1, and 2.
Obviously, <x2>1/2 differs from molecule to molecule. For your
information, <x2>1/2 should typically equal a few percent of the bond length for v
= 0.
(c) Stretch and compress your diatomic by <x2>1/2 for v = 0 and compute the
total energy at both points. Compare these energies to the equilibrium value
and/or to your “best” parabola. Is the computed potential energy curve in fact
harmonic w.r.t a rms distortion for v = 0? Or, asked differently, how good is the
harmonic oscillator approximation for v = 0? How about v = 3?
Hand in a brief report on your findings, including a graph with the potential
energy curve, after class on Thursday, November 3. (Typed, hand written,
mixed media, whatever…as long as I can read it)
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