The Vibron Model for Methane: Stretch–Bend Interactions

JOURNAL OF MOLECULAR SPECTROSCOPY ARTICLE NO.

184, 277–287 (1997)

MS977335

The Vibron Model for Methane: Stretch–Bend Interactions Laurent Wiesenfeld Laboratoire de Spectrome´trie Physique, Universite´ Joseph-Fourier-Grenoble, BP 87, F-38402 Saint-Martin d’He`res Ce´dex, France Received January 14, 1997; in revised form May 6, 1997

The full vibrational spectrum of methane is calculated, with up to four quanta of excitation. An anharmonic local mode model, the vibron model, is used. It includes all stretch and bend modes as well as Fermi 1:2 resonances between them. All energy levels below 6000 cm01 are calculated and fit against ‘‘experimental’’ lines. With the help of nine independent parameters, an overall precision of 8.8 cm01 is obtained. Vibrational intensities of the infrared active lines are given for overtones below 9000 cm01 . q 1997 Academic Press 1. INTRODUCTION

There have been many attempts at various levels of sophistication to try to calculate vibrational spectra of very symmetric molecules, benzene (C6H6 ) and methane (CH4 ) being the most prominent of them. These molecules are very important in many aspects of physical chemistry, yet their experimental vibrational spectra are very difficult to analyze. For methane, many infrared lines are known but their assignment is not straightforward, since it cumulates the difficulties: (i) CH is one of the most anharmonic bonds, (ii) H being the lightest element, the associated rotational constant is the largest so that various vibrational bands overlap, and (iii) methane being of tetrahedral (Td ) symmetry, eigenvalues are degenerate 1, 2, or 3 times, complicating further the analysis. Let us recall that the usual analysis of rotation– vibration bands fails for the F2 three times degenerate levels that are infrared active for tetrahedral molecules. So, for methane, it is impossible to extract straightforwardly the vibrational terms from an experimental spectrum, without resorting to models incorporating a rovibrational analysis. These models have been used in the past especially by the Dijon school. Champion and Hilico (1, 2) performed a complete analysis of the experimental lines from 0 to 6000 cm01 , by use of multi-parameter rovibrational Hamiltonian. However this type of model is in no way easy to handle nor easily extensible to higher overtones. Therefore, a simpler and less precise model would be of great usefulness in order to facilitate the analysis and somehow predict the position of higher overtones. In this paper, I wish to present a vibron analysis for methane, with up to four quanta of vibrational excitation. The simulation includes stretches and bends on the same footing and is especially focused on their interactions through the ubiquitous Fermi 1:2 resonance. The simple one-dimensional (or u(2)) vibron model is especially useful for that purpose (3, 4). In this model, each internal coordinate (for methane, local stretches and local bends) is associated with

a local anharmonic oscillator, represented in second quantization by its spectrum-generating algebra u(2). By splitting in the very beginning the different types of internal coordinates, the model includes at once the different scales of energies for stretches (one quantum of a stretching mode is about 3000 cm01 ) and bends (about 1500 cm01 ). Since each vibration is anharmonic, the relevant CH anharmonicity of about 2% is included at zero-order in the model, allowing the various interactions to properly scale as one goes up in the overtones. The construction of the full vibrational Hamiltonian is nearly straightforward, by careful inspection of the point-group symmetry and of the various equivalent stretches or bends that characterize methane. Once the Hamiltonian is written up, the various matrix elements are easily found through purely algebraic formulas, derived from Wigner–Eckhardt-type of equations. This way of thinking has been very successful for other molecules that present the same type of difficulties as methane:benzene, its substituted species (6), and the stretches of octahedral XF6 molecules (7). The vibron model has many advantages over simple Dunham expansions or harmonic Hamiltonian of comparable complexity. On the one hand, one builds a full Hamiltonian matrix so that all resonant interactions are properly treated and wavefunctions are obtained, for further analysis and possibly for infrared/Raman intensities calculations. On the other hand, the Hamiltonian being anharmonic (anharmonic operators acting on an anharmonic base), it is unnecessary to add various extra terms in order to properly take into account high order resonances in high overtones, since they are built in the Hamiltonian at once. As another consequence of this built-in anharmonicity, it is not necessary to include either the various anharmonicities corresponding to each type of local motion (or symmetry motion) or the cross anharmonicities corresponding to interaction between modes. Also, working in a second quantization scheme, matrix elements are easily computed for any number of quanta of excitation without ever resorting to numerical/analytical

277 0022-2852/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.

AID

JMS 7335

/

6t1e$$$201

08-20-97 13:36:49

mspal

278

LAURENT WIESENFELD

integration. As a further advantage, it has been shown (9) that the diagonalization of a Hamiltonian correctly written in the framework of the vibron model automatically provides eigenvalues and eigenvectors that carry irreducible representations of the point-group symmetry of the molecule at hand. This feature allows one not to bother with the complicated problem of determining the actual symmetry coordinates for the molecule. There have also been attempts to construct the vibrational Hamiltonian out of very large algebras (like U(4)stretch # U(5)bend ) (10). The very complexity of these methods and the lack of transparent physical meaning of the various terms arising in the Hamiltonian seem to reserve these algebraic methods for extremely demanding fits. It must be underlined that very few papers correctly included the Fermi stretch–bend interactions in methane. Even if some papers dealt either with pure stretches ( 8) or neglected the Fermi resonance altogether (11, 12), one cannot consider these fits as being representative of methane dynamics. Very recently, Halonen (14) published a study correctly including all degrees of freedom of methane, as well as their couplings, in a local symmetry mode framework, very comparable to our scheme, but in position/momentum space. By use of many parameters, he obtained a fit of very good quality that compares favorably with the ‘‘experimental’’ data of the Dijon school. After a description of the Hamiltonian (Section 2), we shall fit and extrapolate the methane spectrum up to 4 quanta of excitation (1000 levels, degeneracy included) (Section 3). A conclusion will follow. 2. THE VIBRATIONAL HAMILTONIAN FOR METHANE

2.1. Construction of a General Hamiltonian Before constructing the full Hamiltonian, let us recall how one u(2) spectrum-generating algebra is a model of an anharmonic oscillator. As has been stated in the preceding section, each of the local modes, whether stretch or bend, is represented by an anharmonic oscillator, whose eigenvalues are En Å E0 / v( £ 0 x£ 2 )

[1]

x£ õ 1. This type of spectrum is usually associated with the eigenvalues of a one-dimensional Morse oscillator (Eq. [2]), for stretch degrees of freedom and a modified Po¨schl–Teller potential for bending degrees of freedom (Eq. [3], see (13) for a full derivation):

F

VMorse Å V0 1 0 exp

r 0 re a

G

2

[2]

VPoF schl – Teller Å V0

JMS 7335

/

6t1e$$$202

08-20-97 13:36:49

[3]

Both potentials are isospectral if proper renumbering of eigenvalues are made. It has been shown at several instances that both spectra may be represented in a second-quantization formalism, by the spectrum-generating algebra chain u(2) . o(2). The Hamiltonian is built by use of the Casimir (invariant) operators of the o(2) algebra, especially the quadratic Casimir operator, C. Eigenstates are labeled by two quantum numbers, (i) N that characterizes the size of the representation of u(2), hence the total number of bound levels of the oscillator, and (ii) 0 £ £ õ N, the usual vibrational quantum number. The Hamiltonian is written as H Å aC, where C has the following matrix elements: » N£ÉCÉN£*… Å 04(N£ 0 £ 2 ) d££ = .

[4]

A proper rescaling C Å C /( 04(N 0 1)) yields the usual » N£ÉCÉN£*… Å ( £ 0 x£ 2 ) d££ =

[5]

with N á 1. N01 For H Å E0 / vC at the harmonic limit, N r ` , one recovers the usual transition energy E1 0 E0 Å v. This treatment is valid for both stretch and bend local modes. Now, the local oscillators of the molecule have to be coupled in order to construct a valid vibrational hamiltonian. This coupling must allow for both descriptions of local modes or normal (symmetry) modes type of vibrational spectrum. If we particularize to oscillators i and j, two chains of subalgebra correspond to these two descriptions: chain 1: ui (2) ! u j (2) . uij (2) . oij (2)

[6]

chain 2: ui (2) ! u j (2) . oi (2) ! oj (2) . oij (2).

[7]

Chain 1 corresponds to the symmetry modes description and chain 2, to the local mode description. Let us make the convention that we consider as diagonal the local basis, chain 2. The invariant operators of the oi (2) are labeled by Ci , recalling the local mode to which they belong. Then, two types of interactions between modes i and j are considered: first, a diagonal part, called the two-body Casimir operator, Cij , belonging to both chains 1 and 2 and second, a diagonal and off-diagonal operator Mij , called the Majorana operator. The Majorana operators allow for exchange of vibron quanta between bonds. At the harmonic limit, these operators are

Copyright q 1997 by Academic Press

AID

b2 . cosh 2br

mspal

279

STRETCHES AND BENDS IN METHANE

TABLE 1 Comparison between Anharmonic and Harmonic Operators

2.2. Methane Vibrational Hamiltonian With help of the preceding operators, we are now able to set up the vibrational Hamiltonian for methane. For each category of motion (stretch, bend, stretch–bend couplings), the structure of the Hamiltonian must be an image of the internal symmetries of methane, so that the Hamiltonian operator remains invariant under the various point-group symmetry operations. If the Hamiltonian is constructed in that way, all the eigenstates of the Hamiltonian will be characterized by a symmetry species that corresponds to the Td group: A1 , A2 , E, F1 , F2 . 2.2.1. Stretches. The internal coordinates are the CH bond lengths. All bonds are equivalent, so that the local Hamiltonian comes into a Casimir part that is completely symmetric in the bonds:

responsible for the local mode to normal mode transition. In the algebraic representation they are the invariant operators of the uij (2) algebras, in chain 1. Matrix elements are the following, for the relevant reduced matrix operators,

1 £ 2i £ 2j ( £i / £j ) 2 / / Ni / Nj Ni Nj

Nj Ni 1 / £j 0 2£i £j Ni / Nj Ni / Nj Ni / Nj

The total stretch Hamiltonian is thus

[9]

4

iÅ1

iõjÅ1

iõjÅ1

[10] TABLE 2 Numbering of Bond Angles

and Mij is symmetric. It is also possible to a build ladder-like operator, from which Fermi operators may be constructed. For the Fermi 1:2 resonance between s and b modes, Fsb , we have ( £s , stretching mode; £b , bending mode): » Ns£s 0 1Nb£b / 2É FsbÉNs£s Nb£b … q 1 £s (Nb 0 £s / 1) Ns / Nb

[11]

q

1 ( £b / 1)( £b / 2)(Nb 0 £b )(Nb 0 £b / 1)

and the matrix Fsb is symmetric. To show the reader the easy analogy between these operators and usual creation/ annihilation operators in the harmonic basis (a, a † operators), Table 1 shows the correspondence. Copyright q 1997 by Academic Press

6t1e$$$202

4

2.2.2. Bends and the Spurious Mode. The internal coors dinates that describe the bends are the six HCH angles. They are numbered as indicated in Table 2. We have six diagonal

q

/

4

Hstretch Å cs ∑ Ci / c *s ∑ Cij / ms ∑ Mij . [13] It has three independant parameters, Cs, C*s , and Ms.

1 (Ni 0 £i )(Nj 0 £j / 1)

JMS 7335

[12]

iõjÅ1

[8]

q 01 Å £j ( £i / 1) (Ni / Nj )

AID

iõjÅ1

4

» Ni £i / 1Nj £j 0 1É MijÉNi £i Nj £j …

Å

i Å1

S Å ∑ Mij .

» Ni £i Nj £jÉ MijÉNi £i Nj £j … Å £i

4

The Majorana also are very simple, since all bonds hold the same relation to all the other ones:

» Ni £i Nj £jÉCijÉNi £i Nj £j … Å

4

Hdiag,stretch Å H1 Å cs ∑ Ci / c *s ∑ Cij .

08-20-97 13:36:49

mspal

280

LAURENT WIESENFELD

Casimir operators, all equal, C5 , . . . , C10 . The Majorana operators are written as M5,6 , M5,7 , . . . , M9,10 . The diagonal part of the Hamiltonian is readily constructed: 10

10

iÅ5

iõjÅ5

Hdiag,bend Å H2 Å cb ∑ Ci / c *b ∑ Cij .

[14]

H Å cb ∑ Ci / c *b iÅ5,11

∑ Cij / mA1 P(A1 ) iõjÅ5,11

[19]

/ mE P(E) / mF2 P(F2 ),

with the following relations lI Å mA1 0 mE

There are two types of bend–bend interactions: either the two bends have one bond in common, or none. We thus define, with the above numbering conventions,

lII Å mA1 / 2mE 0 3mF2 mE Å 0 mF2 .

10

SI Å ∑ Mij

The final bending Hamiltonian is thus written as

iõjÅ5 i/j x15

10

10

iÅ5

iõjÅ5

Hbend Å cb ∑ Ci / c *b ∑ Cij and 10

SII Å ∑ Mij . iõjÅ5 i/jÅ15

The Hamiltonian now is 10

10

iÅ5

iõjÅ5

H Å cb ∑ Ci / c *b ∑ Cij / lI SI / lII SII .

[15]

It has four independent parameters, namely cs , c s* , lI , and lII . In methane, there are 5 1 3 0 6 Å 9 internal coordinates, 4 stretches, i.e., 5 bends. However, we defined 4 / 6 Å 10 internal coordinates; there is thus one symmetry mode in the bending part that does not correspond to actual internal motion of the molecule. This mode is called the spurious mode (or Goldstone mode). By inspection of the Td character table, the only spurious mode in the internal bending coordinates is of A1 species. It corresponds to a scalar conservation rule: the sum of all spherical angles around the C atom is equal to 4p. In what follows, we shall have to identify this mode and properly decouple it from all other modes of the molecule, whether bend or stretch. To remove the spurious mode, let us project the Hamiltonian operator [15] on the three symmetry species that appear for one quantum of excitation, A1 , E, F2 . The three projectors are P(F2 ) Å 03SII

[16]

P(E) Å 0 SI / 2SII

[17]

P(A1 ) Å 06Nsnq I / SI / SII .

[18]

Ns is the vibron number of the A1 mode, nq the number of quanta of excitation, and I the identity matrix. The Hamiltonian [15] may be rewritten now as the sum of five terms, two diagonal and three projectors,

There are two ways to treat the spurious A1 mode now: either put its frequency at zero or reject it at /` . The former way has the distinctive disadvantage that all mixed overtones, of the type A1 (spurious) / E or F2 , will appear at a frequency in between actual lines in the spectrum. It is thus absolutely necessary to reject those lines as far as possible from the meaningful spectrum. Also, the Hamiltonian [20] decouples the spurious mode perfectly for one quantum of excitation, but only to the order of O(N 2 ) for higher overtones, as due to Majorana couplings. To overcome those difficulties, it is advisable to diminish as much as possible the anharmonicity of the spurious mode oscillator since it is completely decoupled at the harmonic limit. To keep numerical stability and to reject sufficiently far the spurious mode, a value of Nspurious Å 10 6 has been used, together with a characteristic frequency of 10 5 cm01 . 2.2.3. Stretch–Bend Interactions: Fermi Couplings. Two main types of exchange between stretches and bends are in principle possible in methane, namely 1:1 and 1:2 resonances. While we have constructed the symmetryadapted 1:1 stretch–bend exchange operators, with help of the relevant Majorana operators, they proved to be of no practical importance in the fit that follows. This is a wellknown fact in this type of molecules because of the large difference of energy attached to the stretching and bending modes. We shall thus no longer consider those interactions. However, Fermi 1:2 resonance is known to play a key role in the spectrum of methane. This resonance is so important that it is impossible to determine without it the harmonic frequency of the main stretching symmetry mode, the so-called n3 mode. It was a prerequisite of any anharmonic model that Fermi resonances were possible to be included in the total Hamiltonian. With help of the Fermi matrix, Eq. [11], let us build the various symmetry variants of Fermi operators. A Fermi operator, by a slight generalization of

Copyright q 1997 by Academic Press

AID

JMS 7335

/

6t1e$$$203

[20]

/ mEF2 [P(E) 0 P(F2 )] / mA1 P(A1 ).

08-20-97 13:36:49

mspal

281

STRETCHES AND BENDS IN METHANE

TABLE 3 The Five Nonequivalent Fermi Operators from Which the Symmetry Operators Are Built through the Relevant Exchange Operators from the Group Td

Eq. [11], may occur with three indices, one stretch and two bends: Fs;b1,b2 . Several symmetry-adapted Fermi operators are thus possible. The series of the five symmetry-unrelated operators is given in Table 3; all members of each series may be found by making the appropriate exchanges of stretches and bends. A full Hamiltonian has to be constructed with help of all these operators, FI , . . . , FV . However, two of those are not relevant in what follows. By inspection of the eigenvectors/ eigenvalues of the symmetry-adapted FIV, it is seen that this operator couples directly the overtones of the spurious mode to the stretch modes. It is thus not interesting to include this operator. The last operator FV has been added in a first time to the full Hamiltonian, but its numerical importance proved to be negligible. The full Fermi Hamiltonian is thus HF Å fI FI / fII FII / fIII FIII / fV FV

[21]

but for all practical purposes, only the first three terms of Eq. [21] are retained. 3. RESULTS

better than 2 10 03 cm01 . Taking these vibrational levels as ‘‘experimental,’’ at the level of precision that we reach here, is thus totally reliable, in this spectral region. Further up, up to 6200 cm01 (Table 6), few experimental lines are known with high resolution. We thus compare our extrapolated levels with two calculations that should be of good quality, Halonen (14) and Hilico (2). Those two calculations agree well one with another even if they are based on completely different assumptions. The former is identical to the Hamiltonian employed in the lower spectral region, while the latter is very similar to our approach. The Hamiltonian is expanded on local Morse oscillators, expressed in a symmetrized basis. It includes various relevant resonances. Since they agree one with another with a precision of about 1 cm01 , we take this number as their absolute precision against ‘‘experimental’’ bandheads. All levels in this range have at most 4 quanta of excitation. In the next ranges, up to 9000 cm01 , levels with 5 and 6 quanta are coupled to levels with 4 quanta. Since these states are not included in our scheme now, we cannot consider those eigenvalues as fully calculated levels but as simple indications of their actual positions. We shall limit ourselves to quote the infrared bright levels, with an indication of the intensity of the corresponding line. 3.1. Energy Levels Up to 6200 cm01

As stated in the Introduction, a straightforward analysis of the experimental lines is impossible for methane. There is thus no possibility, unlike in many other cases, to compare vibrational energy levels to bandheads found by a simple extrapolation of rovibrational experimental lines. The vibrational levels have to be found using full rovibrational analysis. Such an analysis has been performed for all levels in the spectral range 0–5000 cm01 (Table 5), which involve states with at most 3 quanta of excitation, by Hilico et al. (1). They employed a very complete Hamiltonian, written in an harmonic base but including up to sixth order of interaction and projected onto the various symmetry species appearing in the eigenfunctions. The inclusion of infrared, Raman, and microwave results, with J values up to 25, allow those authors to have an overall precision

We have thus undertaken a simulation of the vibrational levels of methane up to 4 quanta of excitation (1000 levels, including the degeneracies). Let us recall first the Hamiltonian we shall use, as well as the number of independent parameters. The full, practical Hamiltonian is 4

iÅ1

/ cb

JMS 7335

/

6t1e$$$204

08-20-97 13:36:49

4

iõjÅ1 10

10

iÅ5

iõjÅ5

/ fI FI / fII FII / fIII FIII ,

mspal

iõjÅ1

∑ Ci / c *b ∑ Cij

/ mEF2 [P(E) 0 P(F2 )]

Copyright q 1997 by Academic Press

AID

4

Hvibration Å cs ∑ Ci / c *s ∑ Cij / ms ∑ Mij

[22]

282

LAURENT WIESENFELD

TABLE 4 Results of the Fit

extrapolated lines are in general of the same quality as those fitted, indicating that the anharmonic approach we chose is indeed stable. For ovetones higher than 6000 cm01 , Fermi couplings with levels with more than 4 quanta should occur since the 5n4 band should be seen around 6400 cm01 . We thus restrain ourselves to the bright F2 levels, as explained in the next section. 3.2. Vibrational Transition Moments Since we have wavefunctions at our disposal, it is most interesting to compute vibrational transition moments, for lines originating from the ground state at least. For the simple model at hand, it is enough to resort to a model of dipole transition that is as simple as possible. One local transition operator is modeled as »£i ÉtO i É0 … Å e 0 b£i .

[23]

Now, the vibrational intensities are originating from both bending and stretching transitions. Even if actual transitions are coherent superpositions of both types of transitions, we shall suppose that an incoherent superposition is of sufficient precision for our purposes. If Tˆ is the overall dipole operator, we have É» cÉTO É0 …É2 Å É» cÉTO stretchÉ0 …É2 / É» cÉTO bendÉ0 …É2 .

[24]

Each of the Tˆ operators is a representation of the F2x ,y ,z symmetry coordinates, either in bend or in stretch. Namely, we have the following expressions of the operators, where the numbering scheme is the same as before: 1 F2x ,stretch TO Å (tO 1 0 tO 2 / tO 3 0 tO 4 ) 2

Note. Parameters of the Hamiltonian (22). All digits shown are significant. Vibron numbers, Nbend Å 76; Nstretch Å 38. Overall precision, s Å 8.8 cm01 .

where the Fermi operators F are taken from Table 3. We have nine parameters to adjust, as well as the two vibron numbers Nstretch and Nbend . The first part concerns the fit of the nine free parameters, together with the vibron numbers to the experimental lines below 5000 cm01 . We have fitted 35 lines with nine free parameters, since the vibron number is not actually included in the fitting procedure but set at a given value that produces a good fit afterward, with the supplementary constrain 2Nstretch Å Nbend . Results are shown in Table 4 for the parameters and in Table 5 for the energy levels. From 5000 to 6200 cm01 , very few experimental lines are known with precision, even if broad band (15) and solidstate spectra (16) have been recorded (see Table 6). The calculated lines are now merely extrapolated from the lower part of the spectrum. They are compared to the lines calculated in the large fits of Halonen (14) and Hilico (2). These

F2y ,stretch TO Å

1 (tO 1 0 tO 2 0 tO 3 / tO 4 ) 2

F2z ,stretch

TO Å

1 (tO 1 / tO 2 0 tO 3 0 tO 4 ) 2

F2x , bend

TO Å

q

F2y , bend

TO Å

q

F2z , bend

TO Å

q

JMS 7335

/

6t1e$$$204

08-20-97 13:36:49

1 (tO 8 0 tO 7 ) 2 1 (tO 10 0 tO 5 ). 2

There remains to set an absolute intensity and the b parameter, Eq. [23]. Absolute intensities are fitted against the experimental integrated band intensities of the two most intense experimental bands, the F2 transitions at 1310 and 3019 cm01 (resp., n4 and n3 bands). b is set at a value of 1.3, roughly

Copyright q 1997 by Academic Press

AID

1 (tO 9 0 tO 6 ) 2

mspal

TABLE 5 Lines under 5000 cm01

Note. Total standard deviation for these 33 lines, s Å 9 cm01 . Remarks: (1) Energy level from full rovibrational analysis; (2) Calculated level only; (3) Adapted from the solid-state low resolution spectra (16).

AID

JMS 7335

/

6t1e$$7335

08-20-97 13:36:49

mspal

TABLE 6 Methane Vibrational Spectrum Calculation from 5000 to 6200 cm01

Note. Experimental data in this range of energy are scarce. We shall thus compare our calculations with Halonen (14) and Hilico (2). The attribution provided at the beginning of each series is provided as a guide to the various lines. Remarks: (1) Energy level from full rovibrational analysis; (2) Calculated level only; (3) Adapted from the solid-state low resolution spectra ( 16).

AID

JMS 7335

/

6t1e$$7335

08-20-97 13:36:49

mspal

STRETCHES AND BENDS IN METHANE

285

TABLE 6 —Continued

fitting the decrease in intensities from the HITRANS basis (19). Let us note that the n2 , E line, readily observed in infrared, is forbidden in principle and occurs probably because of rotation–vibration interactions. Table 7 lists all transition moments to F2 levels with a value above 10 024 cm01 /mol cm02 , covering thus a dynamical range of seven orders of magnitude. We included also levels above 6000 cm01 , since sizeable theoretical transition moments are indications of a possible experimental line, indicated in the table where relevant. All bands listed originate from the ground state. 4. CONCLUSION

Calculation of the vibrational energy levels and infra-red transition moments have been successfully obtained by use

of the vibron model. With a minimal number of nine free parameters, including stretches, bends, and Fermi resonances, an overall precision of 8.8 cm01 is obtained. Indeed, this calculation shows the crucial importance of the Fermi couplings. As a particular example, it would have been impossible otherwise to find relatively correct positions of the successive 2n4 , n2 / n4 , n1 , n3 , and 2n2 levels. Their overtones (3n4 , rrr) fall at a place that would have been incompatible with the position of the main n4 , n2 lines. Also, all three Fermi operators are needed in order to correctly move the various symmetry species. The stability of the present calculation, from adjusted levels (E £ 5000 cm01 ) to extrapolated levels (5000 cm01 õ E £ 6200 cm01 ) is very encouraging. It shows that even a simple vibron Hamiltonian like that used here contains all important features, at least up to 4 quanta of excitation.

Copyright q 1997 by Academic Press

AID

JMS 7335

/

6t1e$$$204

08-20-97 13:36:49

mspal

286

LAURENT WIESENFELD

TABLE 7 Largest Vibrational Transition Moments from the Ground State

Note. Only moments larger than 1 1 10 024 cm01 /mol cm02 are displayed. Lines marked with (*) have their intensities normalized against experimental integrated band intensity (19). Remarks: (1) Energy level from full rovibrational analysis; (2) Calculated level only; (3) Adapted from the solid-state low resolution spectra (16).

AID

JMS 7335

/

6t1e$$7335

08-20-97 13:36:49

mspal

287

STRETCHES AND BENDS IN METHANE

It must be underlined that the Hamiltonian used in this simulation is built in a minimal way. We have used only as many parameters as the symmetry of the problem requires, including all local modes and their 1:1 and 1:2 resonances. The anharmonicity is built in the model at zero order and appears only through one physical quantity (Nbend Å 2Nstretch ). The various cross-anharmonicities do not appear as explicit, adjustable quantities but as geometrical means between the anharmonicities of the local oscillators, through the matrix element expressions (Eqs. [5] and [9] – [11]). Similarly, high order harmonic resonances appear in an expansion of the various anharmonic resonances on an harmonic base. In this way, it may be understood that the simple vibron Hamiltonian gathers at the beginning of its expansion all terms essential in a qualitative understanding of the molecule, terms that are spread over several orders in a harmonic expansion and all the more difficult to obtain as more quanta of excitation are relevant. The expansion into anharmonic local modes, but in a position/momentum space, is very similar to the expansion presented here, even though their couplings between modes is different from ours (separation of momentum and coordinate couplings vs one single coupling). However, because of the need of explicitly symmetrized coordinates, the operators corresponding to the various resonances must be calculated individually and in the various representations of Td . Since this major difficulty is circumvented here, while retaining most of the features of the position/momentum model, the vibron model should be very effective for many classes of highly symmetric molecules. Improvement of the present calculation could be of two different kinds. First, the anharmonicity is only treated in second order, and quartic terms should be of great help, as has been shown, e.g., in the large scale simulation of CS2 (20) or other similar three atomic molecules (4). Also, the two F2 , 3n4 levels show very clearly that terms representing the Darling–Dennison type of resonances are of importance in this type of molecule. An expansion with terms containing products and squares of operators should vastly improve precision. The chromophore bearing molecule CHD3 has been the subject of many experimental studies, at high resolution. A logical next step of the present work will thus be an extension

of the methane vibrational Hamiltonian to its various isotopomers. ACKNOWLEDGMENTS I thank F. Iachello and S. Oss for their warm hospitality during several stays in Yale or Trento. I thank J. M. Champion, D. Permogorov (Grenoble), M. S. Child (Oxford), and J. P. Champion (Dijon) for numerous discussions and J. P. Hilico for communicating data prior to publication. Part of this work has been supported by NATO Grant CRG 940136. Laboratoire de Spectrome´trie Physique is affiliated to C.N.R.S., under the name UMR5580.

REFERENCES 1. J. C. Hilico, J. P. Champion, S. Toumi, Vl. G. Tyuterev, and S. A. Tashkun, J. Mol. Spectrosc. 168, 455 (1994). 2. J. C. Hilico and J. P. Champion, private communication. 3. S. Oss, Adv. Chem. Phys. XCIII, 455 (1996). 4. F. Iachello and R. D. Levine, ‘‘Algebraic Theory of Molecules.’’ Oxford Univ. Press, Oxford, 1994. 5. F. Iachello and S. Oss, J. Chem. Phys. 99, 7337 (1993). 6. F. Iachello and S. Oss, J. Mol. Spectrosc. 153, 225 (1992). 7. F. Iachello and S. Oss, Phys. Rev. Lett. 66, 2976 (1991). 8. L. Halonen and M. S. Child, Mol. Phys. 46, 239 (1982). 9. A. Frank and R. Lemus, Phys. Rev. Lett. 68, 413 (1992). 10. V. Boujut, The`se de Doctorat, Universite´ de Bourgogne, Dijon, 1996.; C. Leroy, F. Collin and M. Loete, J. Mol. Spectrosc. 175, 289 (1996). 11. A. Frank and R. Lemus, J. Chem. Phys. 101, 8321 (1994). 12. Z. Q. Ma, X. W. Hou, and M. Xie, Phys. Rev. A 53, 2173 (1996). 13. S. Flu¨gge, ‘‘Practical Quantum Mechanics,’’ Sect. 39. Springer-Verlag, New York, 1974. 14. L. Halonen, J. Chem. Phys. 106, 831 (1997). 15. A. R. W. McKellar, Can. J. Phys. 69, 1027 (1989). 16. P. Calvani, S. Cunsolo, S. Lupi, and A. Nucara, J. Chem. Phys. 96, 7372 (1992). 17. J. C. Hilico, J. Physique 31, 289 (1970); W. L. Barnes, J. Susskind, R. H. Hunt, and E. K. Plyler, J. Chem. Phys. 56, 5160 (1972); B. Bobin, J. Phys. 33, 345 (1972); D. L. Gray and A. G. Robiette, Mol. Phys. 32, 1609 (1976); B. Bobin and G. Guelachvili, J. Phys. 39 (1978); J. P. Champion, G. Pierre, H. Berge, and J. Cadot, J. Mol. Spectrosc. 79, 281 (1980); G. Pierre, J. C. Hilico, C. de Bergh, and J. P. Maillard, J. Mol. Spectrosc. 82, 379 (1980); A. de Martino, R. Frey, and F. Pradere, Chem. Phys. Lett. 100, 329 (1983); Mol. Phys. 55, 731 (1985). 18. A. de Martino, R. Frey, and F. Pradere, Chem. Phys. Lett. 95, 200 (1983). 19. L. S. Rothman et al., J. Quant. Spectrosc. Radiat. Transfer 48, 469 (1992). 20. J. M. Champion, The`se de Doctorat, Universite´ Joseph-Fourier-Grenoble, 1997.

Copyright q 1997 by Academic Press

AID

JMS 7335

/

6t1e$$$205

08-20-97 13:36:49

mspal