Chem. Rev. 1988, 8 6 , 709730
709
Ab Initio Calculations of Vibrational Spectra and Their Use in the Identification of Unusual Molecules
B. ANDES HESS, JR.' and LAWRENCE J. SCHAAD
Department of Chemistry, Vanderbilt Universiv, Nashville, Tennessee 37235
PETR CARSKY and RUDOLF ZAHRADNiK
J. Heyrovskp Institute of Physical Chemistrv and Eiectrochemistry, Czechoslovak Academy of Sciences, Michova 7, 12 1 38 Prague 2, Czechoslovakia Received December 12, 1985 (Revised Manuscript Received March 27
Chem. Rev.
1988,
86,
709730
709
Contents
Ab
Initio Calculations
of
Vibrational Spectra and Their Use in theIdentification
of
Unusual Molecules
B.
ANDES
HESS,
JR.'
and
LAWRENCE
J.
SCHAAD
Department
of
Chemistry, Vanderbilt Universiv, Nashville, Tennessee 37235
PETR
CARSKY
and
RUDOLF
ZAHRADNiK
J.
Heyrovskp Institute
of
Physical Chemistrv and Eiectrochemistry, Czechoslovak Academy
of
Sciences, Michova
7,
12
1
38 Prague 2, CzechoslovakiaReceived December 12, 1985 (Revised Manuscript Received March 27, 1986)
I.
Introduction
I
I.
Theoretical Approaches to Vibrational Spectra
A.
Harmonic Approximation
8.
Perturbation Treatment
of
Anharmonicky
C.
Classification
of
ab Initio Calculations
A.
Force Constants and
Frequencies
B.
Intensities
C.
Use
ofEmpirical ParametersIV.
A
Test
of
the Method
V.
Examples ofAppiicatlons
A. HNO
B.
Cyciobutadiene
C.
[l.l.l]Propeilane
D.
Methylenecyciopropene
E.
Cyciopropenyiidene
F.
ThiireneMoleculesCalculations
111.
Computation
of
IR
Spectra
VI.
Some Predictions of
IR
Spectra ofUnknown
VII.
Compilation ofab Initio Vibrational
VIII.
Summary
I.
Introductlon
709
71
1
71
1
712
713
714
714
715
716
717
718718
719
722
722
723
723
723726
726
The synthesis of new compounds and the determination of their structures is one of the main goals inchemistry. In recent decades various kinds of spectroscopy have played a major role in structural determination. Apart from the direct determination ofstructure by means
of
Xray analysis, the most usefulamong these spectroscopies are electronic (ultravioletand visible), vibrational (infrared and
Raman),
nuclearmagnetic resonance, and mass spectrometry.
A
full use
of
experimental spectroscopy for the purpose of structure determination requires the availabilityof a theoretical tool
that
can
provide computed spectraldata in a completely independent way and with anaccuracy sufficient for a meaningful comparison withthe experimental data.
An
deal procedure is the following. First record the spectrum of a compound of
unknown
structure. Assign
to
it one or more plausiblestructures, and compute theoretical spectra for
all
thesestructures. Compare experimental and theoreticalspectra, and accept or reject the assumed structure onthe basis of agreement or disagreement between thetwo.Historically, quantum chemistry had an impact first
in
the field of electronic spectroscopy. In the
late
fiiies
and
early sixties the quantum chemistry of all but thesimplest systems was dominated by semiempiricaltechniques such as the PariserParrPople
(PPP)
method.' Calculations of this type were applied mostly inthe field of electronic spectroscopy to interpret theelectronic spectra of conjugated aelectron systems.
An
example,2 he main goal of which was structure determination, concerns the benzenoid hydrocarbon zethrene
(I).
A
hydrocarbon
to
which the structure of zethrenewas srcinally assigned was actually either a derivativeof acepleiadiene
(11)
or the hydrocarbon
(111).
Com
883
00
00
00
I
II
Ill
parison of theoretical spectral data for
1111
with twoavailable absorption curves showed that the assumedzethrene was actually
I11
and that a newly synthesizedhydrocarbon was really zethrene.Since the
1960s
he development of larger and fastercomputers together with elaborate ab initio programshas allowed the routine computation of theoretical
vi
brational spectra for molecules of up to, at present,about ten firstrow atoms. During the same period theavailability of lowtemperature matrix isolation hasmade it possible
to
isolate highly reactive molecules andto obtain their spectra. This technique coupled withthe advent of Fouriertransform IR spectroscopy hasprovided a powerful new tool to the experimentalist.
0
1986
American
Chemical
Society
710
Chemical
Reviews.
1966.
Vol.
86.
No.
4
B.
Andes
Hess.
Jr.. is
Professor
and
Chahman
of
the
Deprmwmt
of
Chemistry
a1
Vanderbill University.
He
was
bom
h
WMngton.
DE,
in
1940.
received
a
B.A. (Wllllams
College)
In
1982.
an
M.S.
(Yale)
h
1963.
and
a
W.D.
(Yale)
n 1966
under
Professor
Kenneth
B.
Wberg.
After
two
years of postdoctoral
studles
wiih ProfessorVkgil BoekelheMe at the University of Oregon.
he
plned
the
Department
of
Chemistry at Vanderbin University
as
an assistant
professor
in
1968.
He
spent a year
(19731974)
at
the
Heyow
Institute.
Czechoslovak
Academy of Sciences. Prague wnh Dr.
Rudolf
Zahradnik as a National Academy
of
Sclences ExchangeScientist. His research nterests include the
thmry
of aromaticity.ab initio wmputatbn of vibratonal
spectra
and
tkaetkal
shdies
of
organic reaction mechanisms.
Q
\
.
A
1
LJ.
schaad.
WhosereJearch
hieresbareh~chemlsby.
b
~ofessor
f Chemistry. varderbin
university.
ashv vine.
TN.
~e
was
ban
(1930)
h
cokmbus.
OH.
W
A.B.
deqee
is
I
arvard
h
1952.
and his W.D. was
done
at
the
Massachusetts
1
of
Technology
under
the
direction
of
C.
G.
Swain. After postdoaoral
study
wiih C.
A.
Coulson (Oxford)
and
Harrison Shuil (Indiana).
he
joined Vanderbin University in
1961.
MEBrsky
was
tan
h
1942
h
slavakla
and
he
hk
RNX
from
Charles University in Prague in
1964.
He
then
!dmdDr.
Zakadnik's
gwp
t
the
CzechoslovakAcademy of
sdenees
and
received his
CSc.
(equivalent to Ph.D.) in
1968.
He
was
a
postdoctoral fellow
at
the
University
of
Winburg
(1968)
and visningprofessor
at
Vanderbin University in Nashville
(1982)
end
PasteuUniversity in Strasboug
(1984).
His research interests Center on
chmkal
amtiom
of
molecu$r
Uway
and
computational
chemistry.
Hess
et
al.
Rudolf
Zahradnk
was
barn
1928
and graduated
from
th9
Prague
Institute
of
Technology
(1952).
In
1956
he received his C.Sc.
wee
(appoximately
equivalent
to
Ph.D.).
and
in
1968.
the
D.Sc.
At present
he
is
the
Head
of
the Group of
Theay
of
the
ChemicalReactivity in J. Heyrovskg Institute of Physical Chemistry andElectrochemistry. Czechoslovak Academy
of
Sciences. Prague.From
1965
to
1984
he
spent several periods
as
visiting professorat universities n Wuzburg. Darmstadt. Groningen. Giessen.
Basb.
Sandal.
Osaka. Erlangen. and Leipzig. In
1970
he
received a NSF(Washington) senior fellowship. In
1981
he was elected in theInternational Academy
of
Quantum Molecular Sciences.
He
actsas a member
of
ediorial boardsofJournal
of
Molecular slrucfure
(THEOCHEM).
Reactiviiy
and
Structure (Springer). and ChemicalReviews, and
is
a member
pf
the
advisoly board
of
TheoreficaChemica Acta. Dr. Zahradnik
has
authored or coauthored about
300
papers, a number
of
textbooks. and
7
books. These worksdeal mainly wiih molecular orbital theory. theory
of
chemical re
activity.
and weak intermolecular interactions in chemistry andbiology.
For example
the
highly reactive thiirene
was
preparedby the following ~cheme.~
&gee;
he then
became
Assodate
Professor
O
Charles
vntvenny.
L
Also
present in the product mixture were ethynylmercaptan and thioketene.
IR
spectroscopy was thetool by which the presence of thiirene was identified.
A
spectrum
of
a reaction mixture obtained by the decomposition of thiadiazol
is
shown in Figure
l.
It is
to
be expected that the spectra
so
obtained are of lowrmlution and
that
the assignment is uncertain
because
of side product formation.The aim of this review is
to
show the utility of theoretical vibrational spectra in such experiments. It
tums
out
that
the accuracy of
these
spectra
is
sufficient
to
give correct overall patterns, but fine details are notyet
to
be trusted. The next sections outline the theoretical background of vibrational calculations.
A
practical chemist, whose interest is seeing whethertheoretical vibrational spectra can be of use
to
him,might profitably skip these sections at
first
reading and
turn
o
the examples of sections
IV
and
V.
Such readersshould however keep in mind the distinction betweentwo ways of calculating vibrational spectra. In both
a
force constant matrix
is
diagonalized
to
give molecularvibrational frequencies. In the ab initio method of thisreview, these force constanta are derived by
as
rigorous
as
possible
a
solution of the electronic SchrBdingerequation
of
the molecule in the fixednucleus approximation.
In
the other method, which
is
older and more
Calculations
of
Vibrational SpectraChemicalReviews,
1986,
Vol.
86,
No.
4
711
systems such as liquids, clusters, or floppy molecules,for which normal mode analysis
is
inappropriate. Thesubject of the review by Schrader, Bougeard, and Niggemann6 is most closely related to our paper.
It
alsodeals with structure determination by infrared and
Raman
spectra. Their review is however more general,and the results of only a few ab initio calculations arepresented as illustrative examples. Recent reviews byFogarasi and Pulay7 summarize the progress made inthe last several years in ab initio methods and computational techniques (analytical computation of secondand higher derivatives of energy, basis set and correlations effects), and are recommended as an extensionof our sections IIC and
IIIA.
0
I
I
I1
3000
2000
1600 I200 800
c m1
Figure
1.
Infrared spectrum
of
the photolyzed thiadiazole? Thearrows mark bands assigned to thiirene.
bond
length

Figure
2.
Schematic representation of a potential curve fromthe fixed nucleus calculation (solid line) and ita harmonic approximation (dashed line).
usual among experimental spectroscopists, the forceconstants are simply guessed. This older method canbe useful if the molecule studied is similar enough toothers with known force constants, but since there areusually more force constants than vibration frequencies,there are many ways
to
choose force constants
to
produce exact agreement with experimental frequencies.
It
is
common
to
find in the literature a molecule studiedby two groups who have produced different, but equallyaccurate and equally plausible, sets of force constants.One application of ab initio vibrational calculations isin the choice between these sets of experimental forceconstants.However it
is
in the case of unusual molecules, such
as
reactive intermediates, where analogiesare poor and guessed force constants particularlydoubtful, that we suggest ab initio calculations will beof greatest use.There are several review articles
in
the literature
with
titles similar to
ours
though their scope is different. Asstated by Schutte4 he purpose of his paper is
to
examine the progress which has been made since
1926
inthe ab initio calculation of both the vibrational frequencies of molecules
as
well
as
the forces acting uponindividual atoms when the equilibrium of the moleculehas been disturbed . Molecular orbital calculations aredealt with only very briefly, and the paper concernsmostly diatomic molecules.
A
review by Fredkin,
Ko
mornicki, White, and Wilson5 focuses on treatment of
II.
Theoretlcal Approaches to VlbratlonalSpectra
A.
Harmonic Approximatlon
molecule with
M
nuclei and
N
electronsConsider the Schrodinger equation for a polyatomic
c
c
@>a
a=i
rab
j>i
i=irjj
where
ma
nd
2,
are the mass and charge (in units ofelectron charge) of nucleus
a,
m,
and
e
are the electronicmass and charge,
ria
is the distance between electron
i
and nucleus
a,
nd
rij
and
ra6
are defined analogously.The terms on the left of eq
1
give, in order, the kineticenergy of the nuclei, that of the electrons, the electronnuclear attraction, the nuclearnuclear repulsion,and the electronelectron repulsion. The wavefunctiondepends upon the
3(M
+
N)
coordinates of theelectrons and nuclei. Since the nuclei are heavier andmove more slowly than the electrons, it is usual toneglect the nuclear kinetic energy term in eq
1
byholding the nuclei stationary. The resulting equationdescribes the motion of the
N
electrons in the field ofthe fured nuclei, and the energy
E
of these electrons
will
depend upon the chosen relative nuclear positions. Inthe case of a diatomic molecule there is only a singlerelative position, the bond length
Ri2;
and the plot of
E(R12)
w.
R12
will
be
as
shown by the solid line
in
Figure
2.
For a polyatomic molecule the electronic energy inthis fixednuclear approximation will depend on allnuclear positions
so
that
E
=
E(R
...RM),
hough
E
isconstant for those combinations of nuclear displacements that give translation or rotation of the molecule
as
a whole.Born and Oppenheimer8 showed that if the nuclearkinetic energy
is
treated as a perturbation on the
fix
ednuclear Hamiltonian, the firstorder energy correction vanishes at
all
positions of nuclear equilibrium (i.e.,for stable molecules). The secondorder energy correction is given by an equation describing the nuclearmotion
712
Chemical Reviews,1980,
Vol.
80,
No.
4
Hess
et
al.
where the first term in eq
2
is the kinetic energy operator for the nuclei and
E,,
is the secondorder energycorrection and gives the energy of the nuclear motion.The potential
V(R,
..
RM)
onsists
of the quadratic termsin a power series expansion of
E(R
...RM)
about theequilibrium position,
1
3M
a2E


AXiAXj
+
...
(3)
2
ij=ldxiaxj
where
X1,
2..Xm
are the Cartesian coordinates of the
A4
nuclei. The first term
E(X10...X3Mo)
dds only aconstant
to
all allowed
E,,
and can be ignored for thepurposes of mqst vibrational calculations. Since
E
is
a minimum at the equilibrium geometry, alland the second term in eq
3
vanishes.
If
all terms oforder higher than second are neglected, the potential
is
a quadratic function of the nuclear displacements
AXi
1
3M
2
lJ'1
E(X1
..
X3M)

Fii
AXiAXj V(R1
..
M)
(4)
where the force constants
Fij
are given by
F..
=

~
dxiaxj
2E
I
(5)
Equation
4
is
known
as
the harmonic approximation.In the diatomic case this corresponds to replacing thesolid curve of Figure
2
by the parabola shown by thedashed line. As can be seen, this approximation maybe satisfactory
for
small displacements from the equilibrium position
R120,
ut it
cannot
be accurate for largedistortions of the molecule.
A
transformation to massweighted coordinates,
pi
=
(mi)*/2AXi,
ollowed by rotation of the coordinates tocoincide with the principal axes of the quadratic formin eq
4,
gives an expression for
V
containing onlysquared terms
1
3M
where the
Q,
are coordinates relative to the principalaxes and are called normal coordinates .
As
a resultof these transformations eq
2,
which depends on
3M
variables, separates into
3M
equations, each dependingupon a single
Q,.
Further, each of these
is
a harmonicoscillator equation with force constant
hi.
For a nonlinear polyatomic molecule there turn outto be six of the
A,
with value zero. These correspondto the three translational motions and three rotationsof the entire molecule. The
six
zerofrequency motions
can
be
removed by working in a coordinate system withthe srcin at the center
of
mass and rotating with themolecule. The remaining
3M

6
degrees of freedomare usually specified by internal coordinates such asbond lengths and bond angles.
As
a result, fewer forceconstants need be evaluated, and those that are evaluated have chemical interpretation. Minor difficultiesarise because the transforrqation to these coordinatesis not linear except in the limit of infinitesimal displacements, and because in particular cases there may
TABLE
.
Harmonic and Anharmonic VibrationalFrequencies (cm')
of
the HCOOH Molecule Given by abInitio SCF 431G Calculations
obsd
frequency, calcd frequency (cm')mode cm' harmonic anharmonic
OH
35703769 3629
CH
29443160 3047
c=o
17761909 1882
co
11051150 1140
LHCO
13871510 1490
LOCO
625659 653
LCOH
12231366 1347Reference 13.
be no nonredundant set of simple internal coordinates.All of this is taken care of in the commonly used
GF
matrix formulation of the vibrational problem byWil~on.~
6.
Perturbation Treatment
of
Anharmonicity
Most of the vibrational calculations discussed insections
IVVI
and
listed
in Tables
IV
and
V
have usedthe
harmonic
approximation of
eq
4,
but in recent yearsthere has been progress in the ab initio calculation ofcubic and quartic force
coestants.
These are the thirdand fourth order terms that were dropped from eq
3,
and their availability allows the possibility of includinganharmonicity
in
ab initio treatments of the vibrationalproblem. Prior to
1980
his was rarely done, but onemay anticipate it being done more commonly in thenear future. Perturbation theory yields the followingformula for the anharm0nicity:'O
1
1
xrr
=
16
4ww

16
C4,2[(8
4
3
d)/~s(4
41
(7)
Equation
7
contains the quadratic, diagonal and semidiagonal cubic, and diagonal quartic force constants.The
wk
are the harmonic frequencies. The quarticconstant
+rm
may be estimated easily1' (for stretchingvibrations) or it may be calculated numerically.Equation
7
does not yield anharmonicities of high accuracy, but it represents the only method suitable forpractical application
to
polyatomic molecules of arbitrary structure. Equation
7
is relatively simple. Theonly complication met in the calculation of
xW
is thatthe force
constants
4
and
4
are
derivatives of energywith respect to normal coordinates. They are nottherefore force
constants
obtained by the differentiationof energy with respect
to
internal coordinates
(J
nd
F,,
in the notation
of
the next sections). The conversion
F

may be performed by a rather complicatednonlinear transformation.12 Alternatively,
4w8
may beobtained from the changes of the energy gradient alongnormal coordinates. This way of obtaining
4rrs
s conceptually simple, but computationally it is not economic.We have selected two examples of the ab initio calculation of anharmonicity by means of
eq
7.
The fiist13
is
for the formic acid molecule. The level of the ab initiocalculations
was
modest,
so
only a rather approximatepotential
was
obtained. From the results presented inTable
I
it is seen that the inclusion of anharmonicityby means of eq
7
leads to better agreement betweentheory and experiment, though the error in the com
8