V(R) = D e (1 e a(r R e) ) 2, (9.1)

Transcription

1 Cheistry 6 Spectroscopy Ch 6 Week #3 Vibration-Rotation Spectra of Diatoic Molecules What happens to the rotation and vibration spectra of diatoic olecules if ore realistic potentials are used to describe the interatoic interaction? So for we have just assued that R = R e, and that no change in the bond length is possible. Clearly, for a realistic interatoic potential, such as those outlined in Figure 6., considerable changes in bond length with occur as one oves up in energy fro the ground vibrational state. As a one-diensional equation, the radial nuclear otion described by Eq. (6.4) can be solved nuerically quite siply, and there are in fact coputer routines which invert experiental data to yield diatoic potentials V(R) = E el (R) with spectroscopic precision. Considerable physical insight, however, can be gained by using siple, analytical potentials for V(R), and analyzing the resulting energy level expressions. One of the siplest and ost useful potentials was developed by Morse in the early part of this century, which we ll proceed to exaine now. 9.a. The Morse Potential Clearly, an appropriate potential ust go to sall values as R, have a iniu at R = R e, and go to very large values as R 0. A siple potential which does this, the Morse potential, is given by V(R) = D e ( e a(r R e) ), (9.) where the dissociation energy D e, the equilibriu bond length r e, and the curvature of the potential near the iniu a are three adjustable paraeters. The ajor qualitative proble with the Morse potential is that is does not go to as it should as R 0. However, we know fro our exaination of the haronic oscillator that the wavefunctions tail away rapidly to zero in the classically forbidden regions, and so as long as the potential is sufficiently large at sall R this should not pose a significant proble. If the Morse potential is placed into the radial nuclear otion equation, Eq. (6.4) on page 8 of the notes, we find: d { S(R) J(J +) dr + R + µ } h [E D e D e e a(r Re) +D e e a(r Re) ] S(R) = 0, (9.) where the substitution F(R) = RS(R) has been ade. Like all such probles in quantu echanics, the challenge is in finding the right substitution of variables that allows Eq. (9.) to be recast in a for that has well-known solutions. We ll only briefly outline the procedure here. The first set of substitutions involve the equations y = e a(r R e) and A = J(J +) h µr e. (9.3) 53

2 If these are subsituted into Eq. (9.), we find d S dy + ds y dy + µ ( E De a h y + D e y ) D e AR e y R = 0 (9.4) For A 0, the last (R e/r ) ter ust be expanded in order to generate an equation containing only y. The first three ters of the Taylor expansion are given by R e R = [ (lny/ar e )] = + ( (y )+ + 3 ) ar e ar e a Re (y ) +... (9.5) Retaining these first three ters and regrouping yields d S dy + ds y dy + µ ( E De c a h y + D e c y D e c )S = 0, (9.6) in which ( c 0 = A ar e ( 4 c = A a Re ) ar e 6 a R e ( c = A + 3 ar e a Re Eq. (9.6) can be further siplified by the substitutions ) ) (9.7) S(y) = e z/ z b/ F(z) z = dy so that it becoes where d = µ a h (D e +c ) b = 8µ a h (E D e c 0 ) (9.8) d F dz +(b+ ) df y dz + v z F = 0, (9.9) v = µ a h d (D e c ) (b+) (9.0) is the vibrational quantu nuber. Eq. (9.9) should look failiar fro your discussion of the hydrogen ato in previous classes, because it is the Laguerre equation. Fro this analysis, we know that to have sooth wavefunctions which decay to zero at appropriate boundaries the quantu nuber v = 0,,, 3,... Matheatically speaking, this condition which holds is s 0 as R, which is not strictly the boundary condition we are interested in. However, the Morse potential is large enough at sall R that this is not a serious proble. 54

3 Now that we have the values of v, is is possible to invert Eqs. (9.8) and (9.0) to discover the allowed energy levels, which are: E Jv = ω e (v+ ) ω ex e (v+ ) +J(J+)B e D J J (J+) α e (v+ )J(J+), (9.) where ω e = a D e π µ D J = 4B3 e ω e x e = hω e 4D e B e = α e = 6 h 8π µr e x e B 3 e ω e 6B e ω e (9.) if ω e,α e,b e are expressed in cycles/sec, or Hz. The great utility of the Morse potential lies in the physical significance associated with each of the ters, and the way it explicitly links experiental results with iportant physical properties of the olecule such as it s equilibriu bond length and dissociation energy. The first two ters are clearly that of a vibrating haronic oscillator and the first anharonic correction, the third and fourth ters are the rotational and centrifugal distortion constants fro section 8, and the last ter allows for the change in average oent-of-inertia due to vibration and the consequent change in rotational energy. Fro the expression for the distortion constant, we see that even if only the rotational constants are known, it is possible to predict the vibrational frequency and to place constraints on the dissociation energy D e, and this is one of the facets which akes the Morse potential so popular. The α e ter, in principal, could be either positive or negative, but in realistic situations the first ter doinates and the vibrational constants in higher vibrational states are lower than that of the ground state. Finally, the siple dependence of the results on µ enables investigators to predict the location of the spectral features of various isotopoers once a single variant has been easured. Specifically, the isotopic dependence of the constants fro a Morse analysis are: ω e µ / B e µ α e µ 3/ D J µ (9.3) 9.b. Energy levels and selection rules If even larger expansions of the potential near R e are used, the vibrational energy levels of a diatoic olecule can be expressed as G(v) = ω e (v +/) ω e x e (v +/) +ω e y e (v +/) (9.4) Again, the first ter in (9.4) is the haronic oscillator ter, and the higher order ters resultfroanharonicity. Theconstantsω e x e, ω e y e,... aretheanharonicconstants, and cause the higher v levels to be closer together than the lower v levels (see Figure 9.). For exaple, for H 35 Cl, ω e x e =5.886c, ω e y e =0.44c and ω e z e = 0.0c. A 55

4 useful suary of these constants has been given by Huber and Herzberg, 977 Constants of Diatoic Molecules. The strength of a transition between two vibrational states v and v is again proportional to the square of the transition oent R v : R v =< v µ v > (9.5) where µ is the dipole oent defined in Lecture #7. The dipole oent can be expanded in a Taylor series around the equilibriu distance R e : µ = µ e + ( ) dµ (R R e )+ ( d ) µ dr R e dr (R R e ) +... (9.6) R e so that R v becoes: ( ) dµ R v = µ e < v v > + < v R R e v > +... (9.7) dr R e Since the vibrational functions Ψ v and Ψ v are eigenfunctions of the sae Hailtonian, they are orthogonal for v v : < v v >= δ v v (9.8) so that the leading ter of Equation (9.7) is, without approxiation, R v = ( ) dµ < v R R e v >. (9.9) dr R e Thus, the strength of a vibrational band in the infrared depends on the agnitude of the derivative of the dipole oent with internuclear distance. Figure 9. shows how the dipole oent µ varies with R in a typical heteronuclear diatoic olecule. Obviously, µ 0 when R 0, since the nuclei coalesce. For neutral diatoics, µ 0 when R because the olecule dissociates into neutral atos. Therefore, between R = 0 and, there ust be a axiu value of µ. In Figure 9., this axiu occurs at R < R e, giving a negative slope dµ/dr at R e. If the axiu were at R > R e, there would be a positive slope at R e. A olecule with a relatively sall dipole oent ay still have a large dipole derivative, and, conversely, a olecule with a very large dipole oent ay have a sall dipole derivative if the dipole oent is near its axiu value at R = R e, so that dµ/dr=0atr e. Forexaple, CO,whichhasaperanentoentofonly0.D,possesses a large dipole derivative, and thus one of the strongest known infrared absorptions. A hoonuclear olecule, however, for which µ=0 at all internuclear separations, has a dipole derivative that is zero everywhere, and thus no electric dipole vibrational absorption at all. Thus, hoonuclear olecules such as H, O and N have neither vibrationally nor rotationally electric dipole allowed transitions. 56

5 ( d µ / dr) Re µ (R) 0 R e R Figure 9. Variation of dipole oent with internuclear distance in a heteronuclear diatoic olecule. In the pure haronic oscillator approxiation, only v = v v = ± transitions can occur. For real, anharonic olecules, there is no selection rule on the change in vibrational quantu nuber v, although the v = ± transitions always have vastly larger probabilities than v > transitions. Transitions with v = ± are called fundaental bands, whereas those with v = ±, ±3,... are called overtone bands. 9.c. Vibration-rotation spectroscopy Associated with each vibrational level is a stack of rotational energy levels. In the rotational spectroscopy discussed earlier, we considered transitions between rotational energy levels associated with the sae vibrational level (usually v=0). In vibrationrotation spectroscopy, we consider transitions between the sets of rotational energy levels associated with two different vibrational levels. Thus, a vibrational band, that is a transition v v, is coposed of a nuber of lines v J v J. The energy levels are given by the su of the rotational ter values F v (J) and the vibrational ter values G(v) E vr = G(v)+F v (J) = ω e (v+/) ω e x e (v+/) +...+B v J(J+) D v J (J+) (9.0) Figure 9. illustrates the rotational levels associated with two vibrational levels v and v. The selection rules on rotational quantu nuber, parity etc.. derived previously still apply. Thus, in addition to the vibrational selection rules, we have J = ±. (9.) The selection rule J = ± holds strictly only for a olecule in a Σ state. Transitions with J=0 can occur when the electronic angular oentu of the olecule is non-zero. The band origin is the ythical place where the J = J = 0 transition would occur, if it were not forbidden. The vibrational band is then coposed of a nuber of branches, which in the siplest case are: R-branch: J = J J = + Q-branch: J = 0 (9.) P-branch: J = J J = 57

6 J 5 4 ν R P R(J ) P(J ) J I( ν ) R P ν ν 0 Figure 9. Scheatic vibration-rotation spectru of a Σ heteronuclear diatoic olecule. Note that the Q branch transitions do not occur for Σ states. The frequencies of each of the lines are (neglecting distortion for now) Siilarly: ν P (J) = ν 0 +B J (J +) B J (J +) = ν 0 +B (J )J B J(J +) = ν 0 (B +B )J +(B B )J. (9.3) ν R (J) = ν 0 +(B +B )(J +)+(B B )(J +). (9.4) The appearance of such a vibration-rotation band is indicated in Figure 9.. The band appears fairly syetrical about the band center ν 0, and there is approxiately equal spacing between adjacent R-branch lines, and between adjacent P-branch lines, but there is twice as large a space between the first R and P branch lines, R(0) and P(). This spacing between R(0) and P() is called the zero gap, and it is in this region where the band origin ν 0 falls. Also, the Q branch, if present, would occur in this gap. The approxiate syetry of the band is due to the fact that B B, that is, the vibration-rotation interaction constant α is sall. Then: ν P = ν 0 BJ ν R = ν 0 +B(J +) (9.5) so that the zero gap ν(r(0) P()) = 4B. A closer look at actual spectra (see Harris & Bertolucci) reveals that the bands are not quite syetrical, but show a convergence in the R branch and a divergence in the P branch, resulting fro the fact that B and B are not quite equal. The observed energy levels can then be used to deterine these two quantities separately. 58

7 Cheistry 6 Spectroscopy Ch 6 Week # 3 Group Theory in Spectroscopy Molecular syetry is the unifying thread throughout spectroscopy and olecular structure theory. It akes it possible to classify states, and, ore iportantly, to deterine selection rules without having to do any sophisticated calculations. The application of syetry arguents to atos and olecules has its origin in group theory developed by atheaticians in the 9th century, and it is for this reason that the subject is often presented in a rigorous atheatical forulation. However, it is possible to progress quite a long way in understanding olecular syetry without a detailed atheatical knowledge of the theory of groups, and only a siple introduction to the subject will be outlined below. Besides that in Atkins & Friedan and Harris & Bertolucci, other treatents of olecular syetry and group theory are given by J.I. Steinfeld, Molecules and Radiation (974), Ch. 6, F. A. Cotton, Cheical Applications of Group Theory (963), and M. Tinkha, Group Theory and Quantu Mechanics (964). The latter book contains a ore atheatical discussion for those interested in probing the details of this subject. The great utility of group theory lies in its abstractness. Provided any set of eleents A,B,C,D,... obeys the following four conditions, they for a group: () Closure. If A and B are any two ebers of the group, then their product A B ust also be a eber of the group. () Associativity. The rule of cobination ust be such that the associative law holds. That is, if A, B, and C are any three eleents of the group, then (A B) C = A (B C). (3) Identity. The group ust contain a single eleent I such that for any eleent A of the group, A I = I A = A. I is called the identity eleent. (4) Inverse. Each eleent A of the group ust have an inverse A that is also a eber of the group. By the ter inverse we ean that A A = A A = I, where I is the identity eleent. Each olecule has a nuber of so-called syetry eleents, which together coprise the point group to which the olecule belongs (Point groups are so naed because of the fact that the syetry operations in the groups leave at least one point in space unchanged. Space groups leave lines, planes, or polyhedra unchanged, and so are very useful in the crystallographic study of solids.). As Harris & Bertolucci describe, for olecules the syetry operations (in addition to the identity operation) that ust be considered include rotation about an axis, reflection about a plane, inversion through a point, or a cobination of these operations. In what follows we ll take as an exaple the H O olecule. The syetry eleents leave the olecule in an indistinguishable orientation fro that before the operation was carried out. This does not ean that the olecule has the exact sae orientation, siply that the pattern of equivalent atos is the sae. As Figure 0. shows, water has the following syetry eleents: 59

8 C z σv (yz) O y H H σv (xz) Figure 0. An illustration of the various syetry eleents belonging to the C v group, with water as an illustrative case. (i) The identity eleent I: the syetry operation I consists of doing nothing to the olecule, so that it ay see too trivial to be of iportance. However, as noted above it is a necessary eleent required by the rules of group theory. All olecules have the identity eleent of syetry. (ii) A two-fold axis of syetry C : rotation of the olecule by π/n radians, with n=, about the z-axis produces a configuration which is indistinguishable fro the initial one. (iii) a syetry plane σ v (xz) perpendicular to the plane of the olecule: that is, reflection through the plane to an equal distance on the opposite side produces a configuration indistinguishable fro the initial one. The subscript v stands for vertical and iplies that the plane is vertical with respect to the highest-fold axis, which is C in this case (Planes that are perpendicular to the highest-fold syetry axes are called horizontal, or σ h, planes.). (iv) A syetry plane σ v (yz) in the plane of the olecule. Any planar olecule has at least one plane of syetry. A olecule with these 4 particular syetry eleents is said to have C v syetry. A useful device for thinking about groups is what is called the ultiplication table, which tabulates the products of various pairs of eleents within the group. Clearly, the eleents of the ultiplication table ust also be eleents in the group. For the C v group, the ultiplication table is as follows: Î Ĉ ˆσ v (xz) ˆσ v (yz) Î Î Ĉ ˆσ v (xz) ˆσ v (yz) Ĉ Ĉ Î ˆσ v (yz) ˆσ v (xz) ˆσ v (xz) ˆσ v (xz) ˆσ v (yz) Î Ĉ ˆσ v (yz) ˆσ v (yz) ˆσ v (xz) Ĉ Î In constructing group ultiplication tables, it is helpful to reeber than in each row or colun of the table each eleent of the group appears only once. You ll notice that 60

9 certain products in the ultiplication table above block-diagonalize into like syetry eleents. These eleents are said to be conjugate, that is for eleents P and Q there is another eleent X of the group such that P = X QX. Each group of utually conjugate eleents is called a class. For the C v group each eleent is in its own class, for the C 3v group the C 3 and C 3 eleents for a class, as do the three vertical reflection planes. Fro these tables and a consideration of how the syetry eleents affect various coordinate systes, it is possible to coe up with a variety of atrices that ultiply in the sae way as the syetry eleents do. Any set of non-null square atrices that ultiply in the sae way as the eleents of a group is said to for a representation of that group, and the order of the atrices is called the diension of the representation. Not all representations are created equal, that is, soe are ore useful than others. If the atrices of a representation can be converted by the sae siilarity transforation into the sae block diagonal for, the representation is said to be reducible, otherwise the representation is said to be irreducible. In quantu theory and spectroscopy, we ll be concerned nearly exclusively with the irreducible representations of point groups. In particular, we ll just state here (for ore inforation consult the books noted above) that in fact the full atrices are often not needed their traces alone can provide sufficient inforation. If, in a certain representation, the atrix D(ˆR) corresponds to the syetry operation ˆR, then the trace of D(ˆR) is called the character of ˆR for that representation. Tabulations of the characters of the various representations for a group are called, not surprisingly, character tables, and a wide variety of the are presented in the back of Atkins & Friedan. By construction, the classes are listed on the horizontal header of the table, while the irreducible representations are listed on the vertical header to the left. For these irreducible representations and character tables, there are five iportant theores that are useful in considering the connection between group theory and spectroscopy: () The nuber of non-equivalent irreducible representations of a group is equal to the nuber of classes in that group. () The su of the squares of the diensions of all the non-equivalent irreducible representations of a group is equal to the order of the group. (3) The su of the squares of the absolute values of the characters in any irreducible representation is equal to the order of the group. (4) The characters of two non-equivalent irreducible representations i and j satisfy χ i(ˆr)χ j (ˆR) = 0 (0.) ˆR where χ i (ˆR) and χ j (ˆR) are the characters of the syetry operation ˆR in the representations i and j, and where the su runs over the h syetry operations of the group. (5) If the characters for soe particular representation i satisfy χ i (ˆR) = h (0.) ˆR 6

10 then i is irreducible. Theore (4) iplies that two non-equivalent irreducible representations cannot have the sae set of characters, while (3) and (4) can be cobined into the so-called Great Orthogonality Theore, or χ i(ˆr)χ j (ˆR) = hδ ij (0.3) ˆR which is very useful in putting together character tables. For the C v group, which is of order 4, there are four syetry eleents and so each of the four irreducible representations ust be nondegenerate. For the C 3v group, appropriate for syetric tops like aonia or acetonitrile, the nuber of syetry eleents is six (the identity, Ĉ 3, Ĉ 3, and three ˆσ v eleents). It turns out in this case that there are three irreducible representations, and so l + l + l 3 = 6 (0.4) The only set of three integers which satisfies (0.4) is,,, and so the C 3v group ust have two non-degenerate and one doubly degenerate irreducible representations. For the C v group, the character table is as follows: Table 0. Different syetry types in the character table of the group C v Î Ĉ ˆσ xz ˆσ yz Label A z A B x B y The different irreducible representations, or syetry types, are given labels: A, A, B, and B in this case. Functions which are syetric with respect to the principle syetry axis C n are always denoted with the letter A, whereas those that are antisyetric with respect to C n are denoted with the letter B. The subscripts or then follow fro the behavior under the other eleents σ v (xz) and σ v (yz). The representation A is called the totally syetric representation, ust always be present, and is always listed first in character tables. As they should be, the character sets for each of the representations are orthogonal to each other. Also listed in Table 0. are the syetry types of the (x,y,z) coordinates (or translation operators). We ll find these to be quite useful in the consideration of selection rules in just a bit. For groups with irreducible representations that are degenerate, the letter E refers to those that are two diensional, while three, four, and five diensional irreducible representations are labeled T,G,H. If a olecule has a center of syetry (is in CO or SF 6, for exaple, a subscript g (for gerade) or u (for ungerade) is added according to whether the character is syetric or antisyetric under inversion. By convention, the identity operator is always listed first in the header row of the character table, and 6

11 the nuber within the first colun of the character table itself provides you with the diensionality of the irreducible representation. Given the abstractness of group theory, why do we care? Consider, for a oent, the electronic and vibrational wave functions of a olecule. Using the Born-Oppenheier approxiation, we know that they ust satisfy the Schrödinger equation, or Ĥψ j,v = E v ψ j,v, where Ĥ, ψ j,v, and E v are either the electronic or vibrational Hailtonian, wave functions, and energies. The subscript v lavels the energy levels and the subscript j distinguishes the wave functions belonging to each level E v. Thus, if the vth level is n-fold degenerate, j =,,,...,n. If Ô R is an operator corresponding to one of the syetry operations in the point group of a olecule, it can be shown that [ÔR,Ĥ] = 0 (0.5) The proof of this is straightforward, but tedious, and so we oit it here. Clearly the potential ters do not change if the olecular fraework is altered into one which is undistinguishable fro the original, but the kinetic energy operator is also unchanged and so the coutation relation holds (for a full discussion of these points, pp. 7-3 of Molecular Syetry by D.S. Schonland (Van Norstrand, Princeton, 965) is a good place to start). Thus, we are free to choose our wavefunctions such that they are eigenfunctions not only of the total energy but also of the syetry operations within the point group. This has two critical iplications: () The wave functions of each electronic or vibrational level of a olecule transfor according to an irreducible representation of the olecular point group, and () The degree of degeneracy of an energy level, barring accidental degeneracies, is equal to the diension of the irreducible representation to which its wavefunction belongs. Thus, even without solving the Schrödinger equation, we know the possible degeneracies at the very start! Every vibrational level of a C v olecule ust be nondegenerate, E levels are doubly degenerate, T levels are triply degenerate, etc. Given the behavior we know ust exist under the individual syetry operations of the group, we also know a great deal about the quantitative behavior of the wavefunctions (nuber of nodes, syetry, and so forth). As a siple exaple, let s again consider the water olecule, and how its olecular orbitals are assebled. First we exaine the syetry properties of soe atoic orbitals attached to the nuclei, for exaple, the s, s, and p function on O. What happens to each of these functions if the syetry operations are applied? Clearly, the s and s orbitals reain unchanged, as does the p z orbital. However, under soe syetry operations, the p x and p y orbitals are transfored into theselves, but under others, they end up as theselves. These properties are suarized in Table 0.. Now consider a s orbital attached to each of the hydrogens. Under C and σ v (xz), the s orbital on ato # is transfored into the s orbital on ato # and vice versa. Thus, thesesorbitalsonhydrogendonothaveadefinitesyetrytype, sincetheydonot transfor into + or theselves. However, the linear cobinations s(h ) + s(h )and s(h ) s(h ) do have a definite character. 63

12 The total wavefunction of the H O olecule is built up as the product of a nuber of one electron orbitals, such as those listed in Table 0., each with a definite syetry type. Thus, the total wave function has a definite syetry type, and this type can be used as the label to characterize the electronic state. For exaple, the ground state wave function of H O is totally syetric, so that the state can be labelled as X A ; the first excited electronic state is the à B state (that is, for polyatoic naing conventions for electronic states we use the group theoretical syetry of the MO, since the electron orbital angular oentu projection along a bond is no longer defined, as it is for diatoic species). As we ll see over the next couple of lectures, the vibrational wavefunctions can also be characterized according to their syetry. Table 0. Function Î Ĉ ˆσ xz ˆσ yz Label s(o) A s(o) A p z (O) A p x (O) + + B p y (O) + + B s(h ) + s(h ) s(h ) +? s(h ) + s(h ) s(h ) +? s(h ) + s(h ) A s(h ) s(h ) + + B HavingclassifiedtheelectronicorvibrationalstatesofH Oasdefinitesyetrytypes within the C v point group, we can then use the very powerful theores of group theory to iediately say whether a certain transition will be, for exaple, electric dipole allowed or not, without any calculations! The physical reasoning is exactly the sae as for the case of the parity selection rule in atos: the integrand has to be a totally syetric function, because otherwise integration over the whole coordinate space will give zero. In grouptheoretical language: the product of the syetry types of the wave functions involved and that of the dipole operator or any other operator such as the agnetic dipole or Raan (polarizability) interactions ust be A. For exaple, in the case of H O, we have to figure out whether the integrand in < à B d X A >= Ψ Ã dψ Xd 3 x (0.6) is totally syetric. One can show that the electric dipole operator d=(x,y,z) also has definite syetry. Just as the p-functions, the x-coponent transfors according to B, the y-coponent according to B, and the z-coponent according to A syetry. (These are noted in the last colun of Table 0., and along with ters for rotational and Raan 64

13 interactions they are listed in the character tables presented in Hollas). Now is where the character tables becoe very useful because we can use siple ultiplication rules for the syetry types to find the overall syetry of the intergrand of (0.6): A A = A; A B = B; B B = A; = ; = ; =, (0.7) where the sybol is called the direct product in group theoretical language. To calculate direct product for two, or ore, representations, all we need do is take the individual products of the characters of the individual syetry operations and classify the result. There is a short copendiu of direct product results for iportant groups in Appendix of Atkins and Friedan, if you are interesting in digging a little deeper. Thus, for the z-coponent of the dipole operator we obtain: B A A = B A = B A, (0.8) so that the à B X A transition cannot occur by the z-coponent of the electric dipole oent operator. For the other coponents, we find x coponent : B B A = A A = A...yes! (0.9) y coponent : B B A = A A = A A. (0.0) Thus, the transition à B X A is electric dipole allowed through the x-coponent of the operator. Siilar procedures are used to exaine whether vibrational transitions are allowed or forbidden by syetry, a process we ll look at later this week. One can show that there are only a handful of possible syetry groups to which olecules can belong; C v is certainly one of the ost coon ones. For each point group, one can label the electronic and/or vibrational states according to their syetry types or representations. The character tables of the possible representations, and their behavior under the syetry operations of a certain group, given in the back of the textbooks entioned above and in Atkins & Friedan/Harris & Bertolucci, are therefore extraordinarily useful once you know how to read the. We ll explore next the noral ode approach to olecular vibrations, and in this exploration we ll ake great use of group theory in order to siplify the calculations. 65

14 Cheistry 6 Spectroscopy Ch 6 Week # 3 The Classical Treatent of Molecular Vibrations For a gas phase olecule with N atos, there will be a total of 3N degrees of freedo in free space. Three of these will be involved in translation, which ay be factored out as deonstrated previously for diatoic systes. Syetric and asyetric tops have an additional three degrees of freedo involved in rotation, while diatoic and linear olecules only have two rotational degrees of freedo (because there is no oent of inertia about the internuclear axis). Thus, the vibrational degrees of freedo are 3N 6 for syetric or asyetric tops, and 3N 5 for linear olecules. We now define the displaceent coordinates x α = a α a α,e, y α = b α b α,e, z α = c α c α,e : (x,y,z) α = displaceent coordinates of ato α (a,b,c) α = olecule (or body ) fixed coordinates (a,b,c) α,e = olecule fixed equilibriu positions. (.) Classically, the kinetic energy of vibration is thus T = [ N (dxα ) α + dt α= ( ) dyα + dt ( ) ] dzα dt (.) The first step in siplifying this expression is to use ass weighted coordinates, that is q = / x ; q = / y ;..., q 3N = / N z N, which results in T = T = α= α= ( ) dqi or dt q i q = q q, (.3) using atrix notation, where the dots over the characters denote tie derivatives, and the last expression is the dot product of the row and colun vector containing the q i. Next we exaine the potential energy of vibration, which we ll label U(q,...,q 3N ). As we have often done in this class, to ake progress it is helpful to expand the potential energy in a Taylor series about the equilibriu body-fixed positions (a,b,c) α,e, or U = U e + i= ( ) U i + q i eq i= k= ( U q i k ) e q i q k + 6 i= j= k= ( 3 U q i q j q k ) e q i q j q k +... (.4) 66

15 Now, we can always pick our reference energy such that U e = 0. Further, if we are at or close to the equilibriu positions of the atos ( U/ q i ) e 0 since at equilibriu we are at a potential iniu. If we ake the further assuption that vibrations ay be treated haronically (i.e. that Hooke s law is valid), then the ( 3 U/ q i q j q k ) e and higher ters vanish in the Taylor series expansion of U. In atrix for we can then write where q is the transpose of q and U is the atrix of all U q Uq (.5) U ik = U q i k = ( i k ) / (k i k k ) / (.6) if the haronic approxiation is valid. The equations of otion can be solved classically using Newtonian echanics, that is F = a: F x,α = U = U q j = / U α (.7) x α q j x α q j since q j = / α x α and ( q j / x α ) = / α. Thus, a = d x α dt ( ) = d qi dt / α = α / d q j dt (.8) Fro Eqs. (.7) and (.8) it is clear that d q j dt + U q j = 0 for j =,...,3N. This 3N syste of equations can be siplified because U i= k= U ik q i q k, and only one of the sus survives the partial derivative (say k), which leads to d q j dt + k= U jk q k = 0 j =,...,3N. (.9) This set of 3N equations are not trivial to solve since each equation involves all 3N coordinates! Looking ore carefully at U will ake life a little easier. Soe properties of U include: a) U is a real, syetric atrix (i.e. U ij = U ji = Uij ). Fro atrix theory we therefore know that there is another atrix L such that: 67

16 b) LL = I, where I is the identity atrix and L is the transpose of L. This stateent says that L is unitary. Fro this we can then state: c) UL = LΛ, orl UL =Λ, whereλ ij = δ ij λ i. ThisstateentsaysthatLdiagonalizes U. Thus, d) U Iλ = 0 e) (U λ I)L = 0, where L is the th colun of L. To ake a long story short, it turns out that such a atrix L is coposed of the noralized eigenvectors of U (reeber the eigenvalues are obtained via the deterinant outlined in d); while the eigenvectors for the syste of equations in (.9) are obtained fro the atrix equations outlined in e) ). Using these atheatical tools we are now in a position to define the noral odes of the olecule. The noral odes, which we will label Q, are given in ters of the ass-weighted coordinates q in ters of In atrix ters, we have Q i = k= l ki q k or q i = k= Q = L q or q = LQ l ik Q k (.0) with the atrix L as defined above. Now, since U q Uq, we can substitute Eq. (.0) and the relations above to find U = (LQ) U(LQ) = Q L ULQ = Q ΛQ. (.) Since Λ is a diagonal atrix, this eans that U = k= λ k Q k = 3N 6 k= λ k Q k (.) Eq. (.) represents a draatic siplification in that it is now a single su, rather than the double su previously obtained in the ass-weighted coordinates. The su in the second part is reduced to 3N 6 (or 3N 5 for linear olecules) since the translational and rotational roots are zero since they don t change (a,b,c) α,e in the haronic aproxiation. By convention, these roots run fro 3N 6 to 3N and can be ignored in what follows. For the kinetic energy we have T = q q = (L Q) L Q = Q L L Q = Q Q. (.3) Thus, the classical equations of otion becoe: Q k t + U Q k = 0 k =,...,3N. (.4) 68

17 Since U is single su over the Q k (Eq..), we ay write Q k t + λ k Q k = 0 k =,...,3N, (.5) or 3N easy to solve differential equations (again, 5 or 6 of the λ k roots will be zero due to translation and rotation)! Eq. (.5) is the iportant one, for it shows that just as the principle axes diagonalized the oent of inertia tensor, noral coordinates diagonalize the kinetic energy and potential energy expressions to yield a set of 3N 6 (or 3N 5) independent differential equations. The principal axes were fairly straightforward to set up in the general case, but noral coordinates are highly variable and depend sensitively on both the syetry of the olecule and the potential. It is also worth reebering that noral ode theory depends on the haronic approxiation. We ll go through soe basic group theory and exaple cases next tie, for now we ll outline soe general results and a very siple exaple... The classical solutions to the above equations are: Q k = B k sin(λ / k t + b k) k =,...,3N q i = k= Note that if all B k = 0 except B j, then l ik B k sin(λ / k t + b k) (.6) q i = l ij B j sin(λ / j t + b j ), that is, all atos ove with the sae frequency and phase, the frequency being ν j = λ / j /π (.7). A Siple Exaple Consider the one-diensional odel of two identical asses show below in which each ass is tied to an infinite wall with a spring of force constant k and to each other by a spring of force constant k. In this case, the potential in cartesian coordinates is given by # # k k k 69

18 U = k (x x,e ) + k (x x,e ) + k [(x x ) (x,e x,e )], (.8) while the ass-weighted coordinates are siply q = (x x,e ) / and q = (x x,e ) /. Thus, in ass-weighted coordinates the potential becoes U = k q / + k q / + k (q q ) /, (.9) and the potential derivatives are given by U = k q q k (q q ) U = k q q + k (q q ) U q = k + k U q U q q = k = k + k (.0) We want to calculate U Iλ = 0, which eans (k +k ) λ k k (k +k ) λ = 0. (.) Thus, or λ + (k +k ) λ(k +k ) k = 0 λ λ (k +k ) + [k +k k +k k ] = 0. The quadratric roots of this equation are λ = (k +k ) ± which is easily solved to yield the two values 4 [(k +k ) (k +k k )] λ = k / λ = (k +k )/ (.) for the two vibrational roots. Now, to get the eigenvectors we need to solve (U λ I)L = 0 by inserting the two eigenvalues, or ( k +k λ j k k k +k λ j 70 )( Lj L j ) = 0.

19 For λ = (k )/, L = L ; while for λ = (k +k )/, L = L. Fro LL = I, L = L =, and the unitary coordinate transfor atrix L is just and Q = L q = L = ( q q ) = ( ) q q q +q for the noral coordinates. The two noral ode frequencies are given by ν = k π ν = k +k π. Pictorially: Noral Mode #: k k k Stretch or copress k only Noral Mode #: k k k which we could have guessed by syetry! Stretch/copress both k and k 7