Anthony R. Lacey University of Sydney, Sydney, N.S.W. 2006. Australia Most undergraduate students receive a sound instruction in the understanding of simple harmonic motion as applied to vibrating diatomic molecules and the resulting interplay between vibrational frequency, force constant, and reduced mass of the molecule. Unfortunately, not many undergraduate . oroerams can afford the time to teach students the routine procedures of performing a normal coordinate analysis of a uolvatomic molecule; as a consequence, students do not get feeling for how primary force constants and atomic masses will affect the vibrational frequencies of a polyatomic molecule. They learn that molecules and groups within the molecule can be characterized by group frequencies observed in infrared spectra but have no fundamental feeling for why this should be so. T h e purpose of this experiment is to teach students the fundamentals underlying the normal coordinate analysis orocedure. The vibrational frequencies and the form of the 3N - 6 normal vibrations of an N atomic molecule are characteristic of two features of the molecular structure:
a
(1) the atomic masses and the geometrical distribution uf the vibrating nuclei; (2) the force field that tends to restore the molecule tu its internal equilibrium configuration during any distortion.
Generally the atomic masses and molecular geometry are known for a molecule, hut the force field or potential field is not generally known. The usual spectroscupic procedure is to determine the experimental vihrational frequencies and then to refine the force field calculation until the calculated frequencies match the observed freauencies. This is a rather difficult and sometimes ambiguous exercise and so to make the problem easier for student evaluation the reverse procedure will be used. Given the values for the force field one calculates the allowed frequencies and then examines the effect that changing the force field has on the vihrational frequencies.
Theory The mathematical representation of the force field may he developed in the following way. Suppose we define 3N - 6 internal disolacement coordinates R, (i = 1to 3N - 6). which are just suttirimt t o spwil, rhc ron~igurilti~n 01 the moirrule c .,mnlrtt,lv. " Bv " "mtrrnal" wr mean thur translation :mcl rotation of the molecule are ignored. By "displacement coordinates" we mean that thev measure displacements frum the equilihrium configuraton so that every R, = 0 a t equilibrium. The coordinates are frequently chosen to be displacements in bond lengths and angles. The total potential energy of the
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Journal of Chemical Education
molecule as a function of the coordinates V(R;) may be expanded as a Taylor series about the equilihrium configuration.
The first term, V,, is trivial since it defines the arbitrary zero of the energy scale. The coefficients in the second term all are zero since all derivatives are taken in the equilihrium configuration in which, by definition, V is a minimum with. respect to all R,. The coefficients in the third term, being second derivatives in the equilibrium configuration, are the harmonic force constants, and are usually written:
Higher order terms in eq 1 provide the anharmonic force constants, but we will neglect these in this discussion because they are very small for polyatomic molecules. Confining ourselves to the harmonic approximation we will now consider the calculation of vihrational frequencies ( w ) and normal modes from an assumed set of force constants and the (known) atomic masses and equilibrium geumetry of the molecules. One very simplifying feature is the effect of symmetry. The normal vibrations always reflect the symmetry properties of a molecule. Furthermore, the numher of independent fnrce constants is reduced, since symmetrically equivalent force constants are required to be equal a t ~ dothers zero. The first step toward solving the equations of motion of any system consists of deriving expressions for the kinetic and potential energies in terms of some convenient set of cuordinates. The most convenient will he the 3N - 6 displacement coordinates, R,, described above. 'She potential energy V is then given by eq 1 in the harmonic approximatiun. T h a t is,
For small displacements a similar expression holds for the kinetic energy involving the time derivatives of the coordinates R, = (dR,lat); that is,
The coefficients M are functions of the atomic masses and equilihrium geometry of the molecules. Although their de-
termination is straightforward in principle, it is tedious in practice. For this reason, the kinetic energy is generally characterized by an alternative equation involving the momenta P, conjugate to the coordinates R such that 2T =
z1
GjjPjPj
L
(5)
j
(eqs 8 and 10) and the transformation coefficients Ljk = (aRjIaQk), with j = 1 to n (eq 7). for each coordinate Qk. These define, respectively, the vibration frequency wk, and the relative contributions of the internal coordinates Rj, for the hth normal vihration. These frequency parameters and transformation coefficients are given by the following set of linear equations:
I t is beyond the scope of this exercise but it can be shown' that the coefficients G and M are related by the following:
where 6,k = 1 if i = h and 6,k = 0 if i Z k. The set of coefficients F,, define the force field. The set of coefficients G , define the kinetic properties of the system. These are the two features that control the frequencies and modes of the normal vibration. The numerical values of the coefficients will depend on the choice of coordinates and in general there will be nonzero cross terms in the expressions for V and T. However, the form of eqs 3, 4, and 5 is always the same in any set of coordinates. There are now several wavs of deriving eanations for the modes and frequencies of the normal {ihiations, hut the most illuminating approach is to look for a new set of displacement coordinates Q, in terms of which the expresthat is, have zero sion for v and T are both coefficients for all the cross terms. We definethe newcoordinates Q by a linear transformation from the original coordinates R.
with j = 1t o n and involving n2transformation coefficients L,h = (aR,/aQk). This transformation can always be chosen in such a way that the expressions for V and T take the form
In addition to heing free of cross terms, the coefficients of the diagonal terms in eq 9 have been chosen t o he unity: the form of eqs 8 and 9 then prove to he just sufficient fully to define the coordinates 0.In terms of these coordinates the molecular vibrations take the form of n independent simple harmonic motions, one in each coordinate Qk, the frequency wk of the hth vibration heing given by Solution of the Schrodinger equation for the system gives a wave function that is a product of harmonic oscillator wave functions in the independent coordinates, and the corresponding sum of harmonic oscillator energy terms is
The n quantum numhers u k define the state of the system, and the parameters wk define the frequencies of the fundamental transitions. The key to our problem is thus to determine the normal coordinates Q, that is, to find the frequency parameter hk
' Readers requiring a more detailed analysis of the procedures outlrned in th~ssection are referred to Wlson. E. 6.: Decus, J. C : Cross. P C. Molecular Vibrarions. McCnaw-Hill: New York. 1955.
C[F,
- M .ri ] LJ = 0
(12)
with i = 1t o n and where the Lj's are now the coefficients of the ith eigenvalue. If the coefficients G areused tocharacterize the kinetic energy then in place of eq 12 we have
with i = 1t o n . This equation has the form of n eauations in the n unknowns Lip ~ e o n d i t i o nof self consistenEy requires that the determinant of their coefficients he zero.
This "secular equation" is a polynomial of degree n in A, whose roots, hk, are the desired frequency parameters. Substituting ha for h in eq 13 will give the corresponding set of transformation coefficients L,k, j = 1ton. In fact solution of eq 13 will only give the ratio of the coefficients L, to one another so they must he normalized to satisfy eqs 8 and 9. The X's and L's are known as the characteristic roots and vectors of the secular determinant ( 1 4 ) . It is interestinpl toobserve rheeffert of the cross terms in I.' and G , in the original expressions for the potential and kinetic energies, on the form of the normal vibrations. I t is these cross terms that cause the original coordinates R, to mix together to form the normal coordinates Qk in which the vibrations occur. For, if the cross terms were zero, the potential and kinetic eneraies would alreadv be diaeonal: thus the Q's would be esseGtially identical to the R'H (sinee this is their defining nronertv). and there would he nomixinaof the coordinatesR In [he d o r m a ~vibrations. I t can therefore he said that the coordinates R mix toaether to form the normal vibrations due to two distinct causes: rl) the presenre of cros.; terms in the kinetic enerm, or "kinetic efierts" and (2) the presence of cross terms i n t h e potential energy, or effects of the force field. We shall see later that when the R's are chosen to he pure bond-stretching and angle-bending coordinates, the cross terms in the potential energy are often small compared to the diagonal terms, however, even if the force field is completely diagonal in a particular set of coordinates, there will still eenerallv be some mixine of these coordinates to furm the normal vibrations,uwing tothe kineticeffects.This fact is often not tull\~ in aualitatke diicuasims " aoureciated .. of the form of normal vibrations. It remains todiscuss the choire of the initial coordinates R in terms of which the solution for normal coordinutrs is carried out. The three considerations which influence the choice are: (1) ease of calculation of the coefficients G,>,(2) ease of solution of the secular eq 13, and (3) the physical significance of the force constants F, in terms of the chosen coordinates. Some alternative choices, with the relative merits of each, are described below, and are illustrated using the water molecule (Fig. 1) as an example. Cartestan displacement coordmates for each atom in the molecule, in X, Y , and Z directions, are chosen in some convenient way relative to the equilibrium configuration. These have the advantage that the coefficients M are extremely simple, heing the atomic masses for the diagonal elements and zero for the cross terms; in fact,
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757
where m, is the mass of the tth atom. Cartesian coordinates have two serious disadvantages. Firstly, there are six too many of them, 3N instead of (3N - 6 ) ;in fact, they include the six overall translations and rotations of the molecule, in which we are not really interested. This leads to six superfluous zero roots in the secular equation. For the H20 molecule, for example, it is necessary to-solve a 9 X 9 secular equation to obtain the three vihration frequencies. Secondly, the nhvsical significance of the force constants in terms of Carte" sian coordinates is inconvenient and hard tovisualize, and is comnlicated hv the six redundant coordinates. For these reasons, they are rarely used to set up the secular equation. Internal ualence coordinates are displacements in the hond lengths and inter-bond angles in the molecule. Because thev are internal coordinates, only (3N - 6) are required, and the redundant coordinates a;e avoided. ~ o r e & e r ,in terms of changes in hond lengths and inter-bond angles, the force constanis have the most convenient chemicalsignificance, since the diagonal terms give directly the resistance to the stretching" and hendine of the bonds. The disadvantaee of these coordinates is the difficulty of deriving expressions for the kineticcoefficientsM,,, but this has been circumvented by Wilson et al.', who have described a straightforward method of calculatine the coefficients G,,. which is alwavs applicable. The securar equation may t h i s he set up and
.
-
-
solved in the form of eq 13, and this is, in fact, how most normal coordinate calculations are now carried out. A further advantage of internal valence coordinates is that the force field is approximately diagonal for most molecules in this form, cross terms generally being rather small in magnitude. It should he noted, however, that the kinetic energy is not generally diagonal, so that there is always some mixing to form the normal vibrations. This should he remembered when using the terminology "H-stretching", "CH2-deformation", "CH2-rock", etc., as descriptive of the normal vibrations; strictly this terminology should he reserved for the internal coordinates and should only he used for a normal vihration when the corresponding internal coordinate R j predominates in the form of the vihration. For the water molecule.. (3N . - 6). = 3.. and there are three vihration frequencies and three internal coordinates. The latter are chosen to he chances in the two hond leneths and the inter-bond angle, 6rl, 6r2,and 6n (Fig. 1). The secular equation then involves a 3 X 3 determinant, whose roots would give the three vibration frequencies and whose vectors would give the relative contributions of 0-HI and 0-Hz stretching, and (HOH) bending to each normal vihration. Symmetry coordinates are simple linear combinations of the previous internal coordinates chosen to take advantage of the molecular symmetry. They are chosen to have the simplest possible transformation properties under the rotations and reflections that leave the eauilihrium confieuration unaltered; this means, for example, that the effect of a twofold rotation or a olane of reflection on anv. svmmetrv . coordinates is either to reverse its sign or leave it unaltered. (The effect of threefold, and higher, rotation axes is more complicated and will not he discussed here.) In the water molecule there are two planes of symmetry, the XZ and the YZ planes, and one twofold rotation axis, the Z axis (Fig. 1). The symmetry coordinates S might he chosen to he
-
S, = 2-'"(6r,
Figure 1. Coordinates for the water molecule. Cartesian cwrdinates: [x,y,z,. X Z Y Z Z ~x. ~ Y s . ? ~internal ]: coordinates: [6r,. br,. be]: symmetry coordinates: [ S , = J2(6rr 6r21, S3= 4216r, - brd. S2 = 601. Ail these coordinates measure displacements from the equilibrium configuration.
+
F gLre 2 Doagrammat c oilurtrat on of h e effect of symmetry coordonafes in factor 21ng tne s e w ar determmant nonzero o OCrs me Shaded. la) nternal
coordinates. (b) symmetry coordinates.
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Journal o f Chemical Education
+ br,)
(16)
We observe that S, and Szare symmetric-that is, are unaltered-by reflection in the XZ plane or by twofold rotation about the Z axis, hut SQchanges sign under these symmetry operations, since they have the effect of interchanging the two hydrogen atoms. Thus SQhas different symmetry properties from SI and S p and is said to he of a different species; the symbols A, and Bz are the conventional notations for the symmetry species in this case. In asimilar way the symmetry coordinates of any molecule can he divided into groups, or species, having different symmetry properties. The advantage of symmetry coordinates lies in the fact that, as indicated earlier, the normal vibrations and normal coordinates also have these simple properties, and may he similarlv divided into species: this means that the normal coordinates of a particuiar species are obtained by comhining symmetry coordinates of that species only. The secular equation (eq 14) factorizes in terms of symmetry coordinates by breaking into nonzero blocks along the main diagonal with zero cross terms connecting the blocks; the cross terms Fi,and Gjj,are always zero between two coordinates Si and Sj of different species. Thus the problem of solving the original (3N - 6) square secular equation reduces to several smaller problems that are correspondingly simpler; this is illustrated diagrammatically in Figure 2. For the water molecule, the 3 X 3 eouation in terms of 6rl. .,6r9. -. and 6n factorizes into a 2 X 2 equation in S1and S2and a 1 X 1 equation in SS.(The normal coordinate Q2... is thus identical to the svmmetrv coordinate S,, apart from a normali7atio1i f x t o r , since there is nu uther coonlinnre of the same ssmmt:trv with which S could mix.) Symmetry coordinates are generally used as a halfway
nates and the internal coordinates (6rl, 6r2, r&). This transformation matrix is given by L = U'L.
Figure 3. Normal coordinate displacements In water.
If we now solve the columns of the I matrix we will find that when appropriately normalized the L matrix will define the normal coordinates in terms of the symmetry coordinates. The columns of the L matrix are solved by substituting the roots XI, Xz, A3 back into eq 13 as follows:
I:$[
I[']
["""'"
0.317 = 8.651~ 0.0714 1.5953 L,,
Expanding we get 8.6546L1,+0.317 L z l = 8 . 6 5 7 8 L , , 0.0714 L,, 1.5953 L,, = 8.6578 L,,
+
Only one equation is necessary; the other is redundant. We can only get a ratio of Lll and Lpl from this, but together with the normalization requirement that
This time the columns of the matrix tell us the ratio of displacements in the three coordinates (&I, 6rz, r.64 for each normal mode. The forms of the normal coordinates implied by the ahove matrix enables us to make the sketches in Fieure 3. The H-stretching vihration Q1 is seen to Fnvolve a small amount of angle bend the angle increasing slightly as the bonds stretch; the bending vihration involves some Hstretching, the (0-H) bonds contracting slightly as the angle opens up, but the antisymmetric stretching vihration involves no change in the bond angle at all. I t is interesting to observe how small the mixing between stretching and bending vibrations is in this molecule. A similar result holds for all hydrogen stretching vibrations, owing to the very light mass of the hydrogen atom. The practice of referring to "H-stretching" and "HOHbending" is justified in this case. Exercise The students are e x ~ e c t e dto use the examde for water ahove to calculare the Lihrational frequencies i n d the form the of the normal modes of vibration of HS.'l'hey are given . force constants of HnS as
we find that
f,
[2]
][',I,, K;;]
Similarly for Xz
8.6546 0.317 0.0714 1.5953
= 1.5921
1
to give
[:I [-3 =
The normalized solution for La3is unique and equal to unity. The whole L matrix is thus
I
0.9999 -0.0448 0 0.0101 0.9990' 0 0 1
[.
One further requirement of normalization is that LL' = 8. This is a consequence of satisfying eqs 8 and 9. T o ensure this is so each column of the ahove matrix must be multiplied by the JSn, JSZ2, JS33. respectively. That is, S,, = 1.0390 8,, = 2.1405 S,, = 1.0702
JS,, = 1.0193 JSzz = 1.4630 JS,, = 1.0345
The complete L matrix is then Qz S~ 1.0192 -0.0655 s2 0.0103 1.4616
[ Q1 O
;I
result of their different symmetry properties. We can now proceed one step further and obtain the complete transformation matrix between the normal coordi760
mdynek'
4.04
fa
,dyneA-I
= 0.487
and the HSH angle as 92.0'. (These are not definitive values but are quite adequate for the purpose of the exercise.) The students are then asked t o write a short program on their electronic calculators or use the one already stored on our mainframe computer to solve the roots of the 9 3 matrix for any value of the force constants, atomic masses, or HSH angle. By doing so the students can explore the effects of changing (1) (2) (3) (4)
the primary force constants, the cross terms of the force constant matrix, the masses of the central and peripheral atoms, and the bond angle
on the molecular frequencies. An example of such calculations is shown in Table 2. The results of the calculations are used to show that the primary stretching and bending force constants are the ones that most effect the vibrational frequencies. The cross terms
Table 2.
Data for H,S
Atomic Masses 63
0 1.0345 Thus the normal coordinate QI involves simultaneous displacements of S1, Sz, S3 in the ratio of 1.0192:0.0103:0, etc. The fact that S3is not involved in Q1 or Q2 is a necessary sa
=
f , = -0.108 mdyne kL f,, = 0.120 mdyne k'
0.9999 = [0.0101]
Journal of Chemical Education
XY2 X
Y
Angle Deg.
Force Constants f,
f,
32.06 1.008 92.0 4.04 -0.108 32.06 1.008 92.0 8.00 -0.108 32.06 1.008 92.0 4.04 -0.108 32.06 1.006 92.0 4.04 -0.200 32.06 1.006 92.0 4.04 -0.108
f,
f,
Frequencies sym. asym. stretch bed
0.120 0.487 2611 0.120 0.487 3696 0.120 0.240 2610 0.120 0.467 2580
2684 3752 2664 2713 0.200 0.487 2618 2684
1290 1296 898 1290 1269
have only a minimal effect. Changing the mass of the central atom has very little effect compared to changing the peripheral atoms, showing how the concept of "group frequencies" can be sustained. I t can also be seen that the O-H group behaves like a harmonic oscillator; when D is substituted for H the frequency varies approximately as 1/J2. The angular orientation plays a minor role in the frequency determination of the vibrations. Concluslons We have had this experiment operating in our senior undergraduate laboratory for some years, and students express considerable satisfaction in working through the exercise. An addition to the exercise is to get the students to work through the symmetry procedures as outlined in Cotton2 to establish the symmetry species and spectral allowedness of the vibrations of H2S. They can record the infrared spectrum of a 10-cm path length of H2S gas between 625 and 4000 cm-I on any commercial infrared spectrometer.
The spectrum is inherently complicated3.4 but shows the main hand systems centered on 1290 cm-I (uz, al) and 2684 cm-I (u3, hl). The vl (al) band centered on 2611 cm-I does not show up under these conditions but is the only band strongly apparent in the Raman spectrum. Students are asked to discuss possible reasons why this should be so when all three hands are both infrared and Raman allowed by symmetry. The infrared spectrum also shows a strong com, is a bination hand centered a t 3790 em-' ("1 "2, a ~ )which combination of the strong system a t 1290 cm-I and the band not seen in the infrared spectrum a t 2611 cm-I.
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2Cotton. F. A. Chemical Applications of Group Theory; WileyInterscience: New York, 1971. Sprague, A. D.; Nielsen, H. H. J. Chem. Phys. 1937, 5,85. *Allen, H. C.: Plyler, E. K. J. Chem. Phys. 1958, 25, 1132.
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