Determination of a new molecular constant from overtone vibrational spectra

JOURNAL

OF MOLECULAR

SPECTROSCOPY

114, 1-12 (1985)

Determination of a New Molecular Constant from Overtone Vibrational Spectra E. S. MEDVEDEV Instituteof Chemical Physics,Academy of Sciences of the USSR, 142432 Chernogolovka, MoscowRegion, USSR The quasiclassical Landau-Liftitz formula is applied to higher overtone vibrational tmnsitions. The transition probabilities are expressed in terms ofthe repulsive branch of the molecular potential, which can be characterized by a constant /3describing its rapid rise in the region of strong interatomic repulsion. Values of @are calculated from the higher overtone band intensities for both diatomic molecules and the local vibrations in polyatomic molecules. o 1985 ACZXI~C ~esn h. 1. INTRODUCTION

The n - m vibrational transition probability is governed by the dipole moment matrix element,

which depends on the dipole moment, d(q), and implicitly on the molecular potential, E(q), as functions of the displacement from the equilibrium position, q = I - r,. It is usually supposed that E(q) is known well enough and that only d(q) needs to

be reconstructed from the experimental band intensities. To this end various representations of d(q) have been used, e.g., the Taylor (1-8) and Fourier (9) series expansions, an expansion in vibrational wave functions (10, II), exponential functions (2, 7,12,13), Pad&approximants (14) etc. This approach does not make any distinction between the fundamental and the overtone transitions regarding the physical mechanisms involved. It is implicitly suggested that the overtone band intensities can only be utilized to refine the dipole moment functions derived from the fundamental spectra. In fact, Stine and Noid (9) have shown that the fundamental and the first two overtone matrix elements allow one to accurately reconstruct d(q) in the classical region, the higher overtones contributing insignificantly. The question then arises as to what kind of information about E(q) and d(q) is contained in the higher overtone band intensities. The purpose of the present work is to show that the overtone transition, unlike the fundamental, is a classically forbidden process. It can be described in terms of tunneling into the region where E(q) > en, cm, (2) with a probability mainly dependent on the potential alone. Hence the higher overtone band intensities should provide new information about E(q) not available from the

1

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E. S. MEDVEDEV

intensities of the classically allowed processes like the fundamental vibrational and electronic Franck-Condon transitions. The method is based on the quasiclassical approximation, which can be applied in two different ways. If the lower state energy is high enough as compared to the quantum of the harmonic frequency, that is, if em % fiw,

(3)

and 6” - t, is not too large, the matrix elements (1) can be calculated quasiclassically as the coefficients of the Fourier series expansion of d(q(t)), where q(l) is a relevant classical trajectory (15). This method has been applied to the fundamental and lower overtone transitions (9) and is not considered here. In the other case which is appropriate for the higher overtone transitions, one has %I- %I b ito,

(4)

and the matrix elements (1) are expressed by the Landau-Lifshitz formula (15). The latter has been derived assuming both the upper and the lower states to be quasiclassical, i.e., condition (3) should be fulfilled. However, we have shown (16, 17) that this condition is not necessary and the Landau-Lifshitz formula applies as well to the higher overtone transitions (4) involving even the m = 0 ground vibrational state. Essential features of the Landau-Lifshitz derivation are outlined in the next section. 2. THE TRANSITION

MATRIX

ELEMENT

The first and the most important step is a transformation of Eq. (1) to a more suitable form assuming E(q) to be an analytic function in the complex q plane. The upper state wave function is expanded in a sum, Xn(4) = xi(q) + X&),

(5)

where x;(q) are exact solutions to the Schradinger equation such that xi(q) and x;(q) decay in the upper (Imq - +a) and lower (fmq - -co) half-planes, respectively. On the real axis they are complex conjugates to one another, so Eq. (1) is recast in the form &, = 2Re

s

xmkM~)xnfW~.

(6)

While representing the same matrix element, d,,, Eqs. ( 1) and (6) are quite different regarding the behavior of their integrands. The function xn(q) decays on the real axis at q - -+co whereas x:(q) increases [and so does x;(q)], the integral (6) still being convergent. For this reason, different portions of the real q axis contribute to Eqs. (1) and (6). It is well known that Eq. (I), when integrated numerically, involves the displacements essentially in the classical region, as the product xm(q)xn(q) decays very rapidly in the tunneling region Eq. (2). On the other hand, the product xm(q)xi(q) decays much slower, so the displacements outside the classical region emerge upon integrating Eq. (6). The above Landau-Lifshitz transformation is valid, if E(q) is an analytic function. Any analytic function, when defined on a limited interval of the real axis, is thereby defined everywhere over the complex q plane. Therefore, the same quantity, d,,,, can

MOLECULAR

OVERTONE

SPECTRA

3

be expressed in terms of different parts of the potential E(9) since these are not independent of one another, being representatives of the unique analytic function. Further insight into the properties of Eqs. (1) and (6) can be gained using the quasiclassical approximation. In both the integrals the integration paths can be displaced from the real axis into the upper half-plane to avoid all turning points. Along these new paths the wave functions can be represented in the quasiclassical form as (1.5-17) (7)

(8) and (9) where a, and a, are the left turning points, i.e., a, < a, < 0. The momentum branch is chosen such that ~~(9) and xn(q) decrease on the real axis as q - -co, i.e.

(10) at q -c a,, etc. Equations (7)-(9) show that, in the quasiclassical limit of A - 0, the integrands of both Eqs. (1) and (6) oscillate, suggesting that the saddle-point method be applied. However, for Eq. (1) no isolated saddle points exist far enough from the classical region. Indeed, the product xm(q)xn(q) decreases on the real axis as g - &co whereas it increases in the upper half-plane as Imq - +oo (it also does so in the lower one as Imq - -co) (16, 17). This implies a whole “saddle region” involving the classical range of nuclear motion together with the turning points. Displacing the integration path of Eq. (1) into the complex plane thus turns out to be useless as the path will inevitably cross this saddle region where Eqs. (7)-(9) are no longer valid because of the proximity of the turning points. As a result, Eq. (I) cannot be simplified in this way and it has to be evaluated numerically. On the other hand, Eq. (6) displays a saddle point resulting from the equation (18. 19) Pn = Pm.

(11)

which is satisfied at a point q*, where E(9) tends to infinity and the momentum difference behaves like an-am a l/\IEo (12) at 4 - q* . Being interested here only in the dominant exponential factor of the transition matrix element, we can simply take the integrand value at q = q*, which immediately leads to the well-known Landau-Lifshitz formula &, a P exp(-a,,),

(13)

where ~~~=j+(~p,dy-~p,d9)

(14)

4

E. S. MEDVEDEV

and p is a slowly varying factor involving the dependence on both d(q) and E(q). Its validity is restricted only by the condition (4), which results in OmnB 1,

(15)

and implies an exponentially small transition probability. Equation (14) can be simplified further to be more suitable for applications. A typical diatomic potential is composed of a left (repulsive) branch and a right (attractive) one. The attractive potential tends to a finite limit at large interatomic separations whereas the repulsive one increases steeply at large negative displacements from the equilibrium position. It follows that q* should originate from the repulsive branch of E(q), i.e., Req, 4 a,,. Tn practice, only approximate values of E(q) within a limited range of the reaI nuclear displacements are available from experimental data, and various analytic representations of E(q) are used to fit them as closely as possible. Since the choice of the appropriate analytic function E(q) is not unique, it can always be made in such a way that q* lies on the real axis at large negative displacements where the strong interatomic repulsion is operating. A typical example is the Morse function, E(q) = D(1 - e-@)*,

D = kfw*/2a*,

(16)

which has q* =- 00. Then Eq. (14) is recast in the final form

where use was made of Eq. ( 10). Equations (13) and ( 17) express the transition matrix elements in terms of the repulsive branch of E(q). In fact, only a limited portion of the repulsive branch contribute to Eq. ( 17) because of rapid convergence of the difference between the two action integrals involved. This latter behavior means that there exists a point ql, q* < q1 < a,, such that the interval from q1 to q* contributes insignificantly. In view of Eq. ( 15), any contribution to 6,, of the order of unity can be neglected since it is safely absorbed into the preexponential factor p of Eq. (I 3). Then q1 is estimated from the condition ‘1/2M q* Au,, = (S fi s QI

- ~)dC?

= 1.

(1s)

Multiplying and djviding the integrand by the sum of the square roots and assuming E(q1) g %,

cm7

(19)

one obtains

Further calculations need an explicit form of E(q). In atomic collisions the repulsive potentials are well described by the simplest exponential function E(q) = cFzgq,

(21)

where the constaats C and 28 have been estimated for a number of atomic pairs (28).

MOLECULAR

OVERTONE

5

SPECTRA

It seems natural to use Eq. (2 1) to describe the molecular repulsive potentials far from equilibrium. Inserting Eq. (21) into Eq. (20) immediately gives 4,

=

_$In[(!$%3].

(22)

It is thus found that contributions to Eq. (17) are obtained from the displacements from a,(a3 to ql, where the potential energy varies from tn(c,) to (23) D and cybeing the corresponding Morse parameters. As will be seen later on, typically p 2 2a, so that (24) E(q,)lD = [(a - ~n)/2W~. It is seen that very large negative displacements are important in Eq. (17) since the repulsive potential E(q,) exceeds the dissociation energy for the higher overtone transitions. 3. THE OVERTONE

TRANSITION

AS A TUNNELING

PROCESS

Now we are in a position to discuss the mechanism of the overtone transition as compared to that of the fundamental. In the latter case the energy change is AC= hw and the momentum change at a given displacement is approximately equal to Ap = @p,&)A~ x hwfv,

(25)

where v is a characteristic velocity. The momentum change is evidently small, being of the order of the quantum uncertainty in the momentum, Ap = )i/6x = tt/(v/w),

(26)

resulting from the finite range, 6x, of the nuclear motion limited by the turning points. Additionally, the separations between the respective turning points are small as well. It follows that the fundamental transition can occur with a high probability within the classical region by either a short position jump from one turning point to another nearby while retaining the momentum p = 0 unchanged or by a small momentum jump at a fixed position. On the other hand, the overtone transition with a large energy change (4) needs large jumps or large, abrupt changes in the momentum. Both events are strongly forbidden for heavy classical nuclei, resulting in a low transition probability. Since this picture relates to the classical range of nuclear displacements, one could say that Eq. (1) squared gives the probability of the above position and/or momentum jumps. In the preceeding section an equivalent representation of the transition matrix element was given expressing d,,,, in terms of the repulsive branch of E(q) far enough from the classical region. This implies a different description of the mechanism involved. The nuclei move into the classically forbidden region to reach a point q, where they can change their energy while maintaining nearly constant momentum as is suggested by Eq. (11). This is nothing else but the Franck-Condon principle previously

6

E. S. MEDVEDEV

extended to the tunneling processes in atomic collisions (28) and to electronic radiationless transitions (I 9). To avoid confusion, it should be noted that a unique mechanism rather than two different ones is involved in the overtone transition because Eqs. (1) and (6) give precisely the same transition matrix element. The overtone transition is equally described in two different ways like any other tunneling process. For example, the usual tunnel penetration through a potential barrier can be regarded as a long jump from the left turning point to the right one. Here also the quantum mechanical matrix element describes the probability of the jump whereas the quasiclassical one (15) describes the tunneling probability. 4. INTENSITY

DISTRIBUTIONS

IN THE OVERTONE

SPECTRA

Equations ( 13) and ( 17) enable us to gain information about the repulsive branch of E(q). Assuming the latter to be of the form (21), we shall first derive an intensity distribution in the overtone absorption spectrum versus the upper level energy E,. From Eq. (17) one has (27) where q* = -co and the derivative with respect to the lower integration limit vanishes because a, is a turning point. Changing the integration variable to 2zdz = E’(q)dq = -F(q)dq

z=V&&,

(28)

transforms (27) to

ah -=-

aen

s

112Mm- dz h

0 m

’

(29)

where q is a function of z and E, according to Eq. (28). Inserting (2 1) into (29) and integrating the latter with respect to E, gives 6,,

=

Const. + 5 G,

(30)

where n- 2Mw ‘I2 x=P As the oscillator strength&, form

(4 fi is proportional logf,,

and

(31)

to d2,,, the final result is written in the

= Const. - a\lTISI,

(32)

where a = X/in 10. In Eqs. (30) and (32) the constants are functions of m only. Equation (32) is a normal intensity distribution law (NIDL) for overtone absorption spectra derived from the dominant exponential factor of the transition matrix element (13). It is represented by a straight line on a semilogarithmic plot of the intensity versus the square root of the reduced upper state energy. From the slope of this line

MOLECULAR

7

OVERTONE SPECTRA

the new molecular constant @introduced in Section 2 to describe the repulsive branch of the molecular potential can be calculated as (33) where p is in A-‘, it? is the reduced mass in atomic units, and w is the harmonic frequency in units of lo3 cm-‘. In deriving the NIDL, the preexponential factor p of Eq. (13) has been neglected as it is a slowly varying function of the quantum numbers and generally does not affect the overall intensity distribution in the higher overtone spectra. However, there is an important exception when p passes through zero at some value of the energy difference, en - E,. If this zero falls between two adjacent levels, n and n + 1 (at a fixed value of m), then the intensities of either one or both of the n, n + 1 t* m transitions will be anomalously low. An expression for p, based on a more accurate saddle point evaluation, was derived in Ref. (21) and the p values were calculated for a number of diatomic molecules using available data on the dipole moment functions. As a result, the 12- 0 anomalous absorption bands were predicted for HF(n = 6) HCl(n = 4, 5), HBr(n = 3) HI(n = 1, 2), and CO(n = 5). 5. COMPARISON WITH EXPERIMENTAL

DATA

Figures 1 and 2 show experimental data (22-27) on local vibrations in polyatomic molecules. The points are seen to be on straight lines in accordance with the NIDL of Eq. (32). Line slopes, a, along with standard deviations, A, of the experimental points are indicated in the figure captions. Data from Refs. (7, 28) on benzene were also processed (21,29) and an average value ofa = 4.73 + 0.06 was obtained. Inserting it along with &? = 0.930 (20) and W = 3.157 (28) into Eq. (33) gives ,f3= (3.8 1 t 0.05)

-6

CH

1

2

Jz

FIG. 1. Common logarithm of the experimental relative oscillator strengths in the overtone absorption spectra of the local C-H vibrations versus square root of the upper state energy in units of hw. l and X. Liquid phase benzene, data from Refs. (22) and (23) respectively; 0, average data (24, 25) for the C-H vibrations in aromatic and aliphatic groups in 10 liquid-phase polyatomic molecules (benzene, toluene, xylene, tetramethylsilane, hexane, etc.). The straight lines are least-squares fits. (1) line slope, a = 4.79; standard deviation, A = 0.10 (two last crosses with large deviations were not included in the calculation). (2) a = 4.79, A = 0.01.

8

E. S. MEDVEDEV

FIG. 2. Common logarithm of the experimental relative oscillator strengths in the overtone absorption spectra of the local Si-H vibrations in SiH4 (26) (open circles, n 2 k = 6) and C-H vibrations in (CF,)sCH (27) (full circles, n 3 k = 2) versus square root of the upper state energy in units of &I. The straight lines are least-squares fits. (1) a = 5.20, A = 0.08; (2) a = 5.20, A = 0.05.

A-‘. The above values for aand Wfor C-H and the values W = 2.256 (26), A? = 0.966 for Si-H were used in calculating /?Ifrom the data of Fig. 2. Data on the halomethanes CHX3 and CHzXz (X = Cl, Br) (30) were also used to calculate p for the C-H bonds in these molecules (22, 29). Figure 3 shows data on HC1(31-34). The predicted 4 - 0 and 5 - 0 anomalous bands at abscissae ~2.0 and 2.2, respectively, are clearly seen. From the slope and the standard deviation of the NIDL straight line, the p value was obtained using

-2 HCI -4 -

”

-6 .

i -8-

-10

I

*

1

1.4

5

1.8

2.2

I

Jz

FIG. 3. The experimental dipole moment matrix elements squared from the HCl overtone absorption spectrum (31-34). The abscissa is the same as in Fig. 1, the ordinate is the common logarithm of the matrix elements squared (do, in Debye). There are the anomalous bands with low intensities at n = 4 and 5. The straight line is a least-squares fit to the experimental points except for the anomalous ones. a = 6.65, A = 0.21.

MOLECULAR OVERTONE SPECTRA

9

M = 0.980 and ; = 2.991 (20). Reddy (33) has qualified as surprising the fact that the 5 - 0 and 6 - 0 band intensities are very similar in contrast to the usual intensity decrease in the higher overtone bands. Here this finds a natural explanation: the 6 0 band intensity is a normal one whereas that of the 5 - 0 band is anomalously low. Data on HBr (35) and HI (36) were also analyzed and the predicted 3 - 0 (HBr) and 1, 2 - 0 (HI) anomalous absorption bands were revealed (21, 29). The /3 values were also calculated using the normal band intensities. Figure 4 shows data on CO from Refs. (37-39). The 0 value was obtained using &? = 6.86 and 0 = 2.170 (20). Chackerian (4) predicted the value of & to be + 1.275 X 10e6 Debye whereas our calculations (21) result in a negative do, matrix element of much lower magnitude. Unfortunately, the exact anomalous band intensities could not be evaluated by the present method because near the anomaly the p factor is extremely sensitive to the dipole moment function. The NIDL for overtone emission spectra was also derived and applied (21, 40) to the HF, DF (6), and OH (41) spectra. The calculated ,6 values are collected in Table I together with the corresponding values of the Morse parameter, CY. 6. DISCUSSION

At least three overtone band intensities are needed to check the NIDL and to calculate the p values. In this sense, all the available data, which we were aware of, were used and the NIDL was proved to be valid for all the diatomic systems investigated. This suggests Eq. (2 1) to be a valuable representation of the repulsive branch of E(q) far away from the equilibrium. It follows that p can be regarded as a good molecular spectroscopic constant.

1.2

1.6

2

2.4 djii

FIG. 4. The experimental dipole moment matrix elements squared (&, in Debye) from the CO overtone absorption spectrum (37-39). The axes are the same as in Fig. 3. The straight line is a least-squares fit to the experimental points at n = 2,3,4; a = 9.37, A = 0.14. In view of the anomaly predicted, the experimental point (not yet observed) at abscissa 2.31 corresponding to the 5 - 0 band is expected to be much below the straight line.

10

E. S. MEDVEDEV TABLE I

The NIDL (Normal Intensity Distribution Law) Slopes and the Molecular Spectroscopic Parameters a and /3 (in A-‘) for Diatomic Molecules and for Local Vibrations in Polyatomic Molecules Diatomic

a

cYa

P

Remark

C-H

4.73

1.77

3.81+0.05b

UC in benzene

C-H

4.00

1.77

3.7620.07

LM in CH2Xzd

C-II

5.80

1.77

3.1OLO.05

LM in CHXgd

C-H

5.20

1.77

3.4620.04

ud in (cF313CH

Si-H

5.20

1.39

2.98+0.10

LM in SiH4

OH

5.54

2.16

3.57~0.22~

HF

6.76

2.26

3.09+0.14e

HCl

6.65

1.75

2.71~0.06

HBr

5.06

1.63

3.37LO.14

HI

4.34

1.52

3.6820.08

co

9.37

2.33

4.3320.13

_ aThe Morse parameter (Y = 7.7027 (M (3x,)"' bStatisticaluncertainty 'Local mode dX = Cl, Br eFrom the emission spectra (other data from the absorption spectra)

The present theory shows that the intensity distributions in the higher overtone spectra are governed, as a rule, by the molecular potential, being quite insensitive to the dipole moment function. This is in line with Ferguson and Parkinson’s conclusions (2). On the other hand, Meredith and Smith (3) have calculated the HF n - 0 matrix elements for both quadratic and cubic dipole moment functions and found that the former decreased monotonically with n whereas the latter exhibited an anomaly at n = 6; i.e., the 6 - 0 matrix element was very small in absolute magnitude and, at n 2 6, even the signs become reversed relative to the signs of the matrix elements of the quadratic function [see Figs. 7, 14, 17 and Appendix 2 of Ref. (3)]. We have explained this effect (21) as being due to the vanishing of the p factor [see Eq. (13)] near the n = 6 HF level. The Meredith-Smith anomalies have also been predicted and discovered (21) in the overtone absorption spectra of HCl, HBr, and HI. The CO 5 - 0 anomalous band was predicted but not yet observed. Except for such anomalies the intensity distributions in the overtone spectra are always governed by the molecular potentials. It is seen from Table I that p > (Y,which implies that the true molecular potentials

MOLECULAR

OVERTONE

SPECTRA

11

rise much more rapidly than the corresponding Morse values. This conclusion will still be valid if one takes into account that in actuality, /3 should be compared with a different parameter, &(>a), characteristic of the potential behavior within a region between the left turning point, a,, and the extreme asymptotic limit of q 4 0 and E(q) [Morse, Eq. (16)] = D exp(-2aq). As is seen from Eq. (17), just such an intermediate region rather than the above limit provides a major contribution to the transition matrix element. We have calculated [see Table I in Ref. (21)] and plotted (unpublished) log d& versus G for the Morse potential (16) with (Y= 1.77 A-’ corresponding to the C-H local vibrations. The points with n = 3 through 10 are on a perfect straight line with a = 6.30 f 0.04, resulting in (Y’= 2.86 A-’ [calculated by Eq. (33)]. Inspection of Table I shows that p > (Y’,i.e., the true molecular potentials rise within the above intermediate region more rapidly than the corresponding Morse ones. Amrein et al. (27) have given a convergence rule for the higher C-H overtone band intensities in a row of similar compounds (C,F,ClkH). It states that the intensity spread among the molecules is greatly reduced in going from the fundamental band to the high overtones, and all the intensities converge to a similar value. Burberry and Albrecht (24, 25) have noted the universality of the C-H overtone intensities in large polyatomic molecules and derived average values shown in Fig. 1. Similar results have also been obtained for halomethanes (30). The present theory provides a straightforward explanation of the above findings. In the case of the n - 0 transitions with n B 1, the repulsive branch of E(q) at small interatomic separations contributes to the transition matrix element whereas both d(q) and the attractive branch of E(q) at larger separations are of importance at n = 1. Evidently, in the latter case the C-H band intensities are affected to a greater extent by specific molecular structure. ACKNOWLEDGMENTS I am very grateful to Professor V. 1. Osherov and Dr. V. G. Ushakov for steady support and many helpful discussions. Thanks are also due to Professor M. Quack for valuable comments and for the preprint of Ref. (27).

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E. S. MEDVEDEV

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