**Anharmonically coupled local-mode model: An application**

**to silane vibrational overtones (*)**

XI-WENHOU(1)(**), MIXIE(2) and ZHONG-QIMA(3)

(1*) Institute of High Energy Physics - P. O. Box 918(4), Beijing 100039, PRC*
*Department of Physics, University of Three Gorges - Yichang 443000, PRC*
(2*) Graduate School, Chinese Academy of Sciences - Beijing 100039, PRC*
(3*) Institute of High Energy Physics - P. O. Box 918(4), Beijing 100039, PRC*
(ricevuto il 18 Settembre 1996; approvato il 27 Gennaio 1997)

**Summary. — An anharmonically coupled local mode model for a tetrahedral**
molecule is constructed with new higher-order coupling terms, and applied to the
vibrational spectrum of silane by properly chosen parameters. Our results are better
than those of other models.

PACS 33.20 – Molecular spectra. PACS 33.20.Tp – Vibrational analysis.

**1. – Introduction**

Since Child and Lawton [1] proposed a harmonically coupled local mode model in 1981, the local mode model has in recent years been developed into an efficient theory on the description of the excited vibrational states in some symmetrical molecules and

of the rotational energy level structures near the local mode limit in the XH2, XH3, and

XH4 type symmetrical hydrides. The early successful applications were on the local

mode nature of stretching vibrational states in some symmetrical molecules [2], later the relations between the local-mode and the normal-mode stretching vibrations in

*symmetrical systems, called “the x, K relations” [3, 4] or “beyond the x, K*

relations” [5, 6], were established. The local-mode model has even been extended to

include the bending vibrations in XH2[5], XH3[7, 8], and XH4[9]. Furthermore,

rotational energy levels of the stretching vibrational states of the same systems in the local-mode limit were widely investigated, and the local-mode relations between the vibration-rotation parameters were set up, which showed a good agreement with experiments [10]. The most pleasing aspect in the local-mode theory is that it offers a clear physical picture with the least parameters.

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) E-mail address: HOUXWHBEPC3.IHEP.AC.CN

Recently, the study of highly vibrational overtone spectra has become active. Using a three-parameter local-mode model, Holonen and co-workers successfully explained the observed stretching vibrational spectra of arsine and stibine [11]. Algebraic approaches based on different algebra chains [12-16] were presented to study vibrations of polyatomic molecules. An algebraic approach to describe the complete vibrational spectra of methane was proposed by Lemus and Frank [15], where they had to eliminate the spurious vibrational degrees of freedom from both the basis and the Hamiltonian. Law and Duncan in their recent paper [6] obtained the extended, anharmonically coupled local-mode Hamiltonians for the X-H stretching vibrations of

XH4 and XH3 systems from the corresponding full, unconstained normal-mode

Hamiltonians, and set up the relationships between the extended local mode and the
*normal-mode parameters: “beyond the x, K relations”, by which they calculated the*
local-mode parameters of the methyl halides from the normal-mode parameter sets. In
this paper, we will first construct the Hamiltonian of the tetrahedral molecules by
techniques of group theory, and then apply it to the overtones of silane with the
properly chosen four parameters, by which the normal-mode constants will be
*calculated with Law and Duncan’s “beyond the x, K relations” [6]. The conclusion is*
given in the final section.

**2. – Hamiltonian**

We shall limit our attention to the stretching vibrations of tetrahedral molecules. How to include the bending vibrations was discussed in our previous paper [9], where harmonically coupled Morse oscillators were used. Using the techniques of group theory, we first set up the Hamiltonian to describe the stretching motions of a molecule

XH4.

*In order to achieve that, we introduce four sets of shift operators a*†

*j* *(aj) ( 1 GjG4)*

to describe the motions of the four equivalent X-H bonds:

.
/
´
*a*†
*j* *Nnjb 4 (nj*1 1 )1 /2*Nnj*1 1 b ,
*ajNnjb 4nj*1 /2*Nnj*2 1 b ,
(1)

*where nj* *is the number of phonons in bond j. The total number of phonons, denoted by*

*n, is preserved. The usual boson commutation rules for these operators follow*
automatically from their definitions in terms of eq. (1):

.
/
´
*[ai, aj*†*] 4aiaj*†*2 aj*†*ai4 dij*,
*[ai, aj] 40 , [ai*†*, aj*†] 40 ,
(2)

*where d is the Kronecker delta.*

From the point of view of group theory, these operators can be combined into the

*irreducible tensor operators belonging to the representations A15F2of Td*. Neglecting

the mixture of the states with different total number of phonons, we can express the Hamiltonian as a graded and symmetrized power series in operators. The character table, the representation matrices of generators, and the Clebsch-Gordan coefficients

*of the group Td* were given in ref. [15]. Since the symmetrical combination of

*(A15F2*)*7 (A15F2) is equivalent to 2 A15E52F2, there are seven independent Td*

interaction up to fourth order now can be explicitly expressed in terms of the shift
operators as follows:
(3) *H*s4

### !

*j*

*a*†

*j*

*aj[vm1 xm(aj*†

*aj1 1 ) ] 1 l*1

### !

*i c j*

*a*†

*i*

*aj1 l*2

### !

*i Ej*

*(a*†

*i*

*ai)(aj*†

*aj*) 1

*1l*3

### !

*i c j*

*a*†

*i(ai*†

*ai1 aj*†

*aj) aj1 l*4

### !

*i c j*

*(a*†

*i*)2

*aj*2

*1 l*5

### !

*i c j c k c i*

*(a*†

*i*

*ai) aj*†

*ak*1

*1l*6

### !

*i c j Ekci*

*(ai*†

*ai*†

*ajak1 aj*†

*ak*†

*aiai) 1l*7

### !

*i EjckEl*

*j c l c i c k*

*ai*†

*aj*†

*akal*,

*where vm* *and xm* *are the constants in the Morse potential, l*1 is the harmonically

coupling constant, and the others are the anharmonical parameters. The summation of
*the last term in eq. (3) runs over all different i Ej, and kEl.*

*The harmonically coupled local mode model can be obtained when one sets li*4 0

*(i 42, 7) in eq. (3). It is a natural way that the Morse potential exists when the*
interactions are taken up to fourth order. In addition, the Hamiltonian (3) is in
agreement with Law and Duncan’s one. However, Law and Duncan obtained it by
transforming the unconstrained normal-mode Hamiltonian. Thanks to the “beyond the
*x, K relations” [6], that offer the more appropriate relationship between the *

normal-mode and the local-normal-mode descriptions of the stretching vibrations in XH4systems. We

can calculate the normal-mode parameters in the next section in terms of those relations.

**3. – Application to silane**

The descriptions of the vibrations in silane already exist in the literature. Halonen
*and Child [17] analyzed its stretching vibrations for n G2 with a three-parameter*
harmonically coupled local-mode model. Leroy and Michelot [16] proposed an algebraic
*model with four parameters based on a U( 5 ) dynamical group to describe its stretching*
*modes for n G4, without discussing the relations between the four parameters and*
those in other models. In order to understand better the anharmonicity in silane, we
will apply the extended, anharmonically coupled local-mode Hamiltonian (3) to its
stretching vibrations.

Before the application we have to deal with the problem how to choose the
parameters in the Hamiltonian (3). The number of the observed vibrational levels from
*n 41 to 4 would not be large enough to determine all coupling constants in eq. (3). On*
the other hand, although all the fourth-order terms in eq. (3) are equivalent from the
point of view of group theory, the coupling parameters are quite different in

*magnitudes, for example, xm* is usually greater than the other anharmonic ones. The

*term we are mostly interested in is the coupling term with l2*in eq. (3). It is such a term

that plays an important role in different algebraic approaches [12-16] to vibrational
spectra of polyatomic molecules. This term has the largest effect on states with quanta
equally distributed between two bonds. In order to keep the extended local-mode
model with the least parameters, it is natural to choose such four coupling constants as
*vm, xm, l*1*, and l*2, which means setting l*i4 0 (i 4 3, 7) in eq. (3). Calculation shows*

TABLE* _{I. – Observed and calculated vibrational levels of silane for n G4 (units: cm}*21).
Level

*E(o)*

*E( o ) 2E( c )a*

*E( o ) 2E( c )b*
*E( o ) 2E( c )c*
*E( o ) 2E( c )d*
*1000 A*1 2186.870 [18] 0.07 0.318 20.313 0.229
*1000 F*2 2189.194 [18] 0.194 0.061 20.213 20.035
*2000 A*1 4308.867 [19] 0.667 20.253 20.733 20.375
*2000 F*2 4309.348 [19] 20.052 0.181 20.291 0.054
*1100 F*2 4378.40 [20] 20.1 0.530 0.680 0.460
*3000 F*2 6362.0 [21] 0.1 20.083 20.396 20.183
*2100 F*2 6497.48 [21] 22.12 20.265 0.413 20.245
*4000 F*2 8347.4 [21] 0.4 0.178 0.284 0.169
*s* 1.01 0.38 0.57 0.36

describing the stretching vibrations of silane is now assumed to be
*H*s4

### !

*j*

*a*†

*j*

*aj[v 1xm(aj*†

*aj1 1 ) ] 1 l*1

### !

*i c j*

*a*†

*i*

*aj1 l*2

### !

*i c j*

*a*†

*i*

*aiaj*†

*aj*. (4)

We can now calculate the Hamiltonian matrix elements in the appropriate symmetrized bases, then fit the experimental data by a least-squares optimization to determine the four local-mode parameters. The observed and calculated stretching vibrational levels are presented in table I where our results are also compared with those of recent studies.

The root-mean-square deviations are calculated unweightedly [16]:

*s*2

4 1

*m 2p*

## !

*[E( c ) 2E( o ) ]*

2_{,}

*where m is the total number of experimental data, and p the number of parameters.*

*The experimental energies E(o) in cm*21 _{with their refs. [18-21] are listed in the}

*second column of the table. The energies E( c )a* _{are calculated by Halonen and}

*Child [17] with m 44, nG2, p43, and s41.01 cm*21_{, and the energies E( c )}b_{by Leroy}

*and Michelot [16] with m 48, nG4, p44, and s40.38 cm*21_{. The calculated results of}

*the present paper are listed in the last two columns as the energies E( c )c* _{calculated}

*under the condition l*2*4 0 with m 4 8, n G 4, p 4 3, and s 4 0.57 cm*21, and the energies

*E( c )d*

*with m 48, nG4, p44, and s40.36 cm*21_{. The E( o ) 2E( c ) are the differences}

between the observed data and the calculated results.

The four parameters to fit the 8 experimental data are given as follows in unit
cm21_{:}
.
/
´
*vm*4 2256.427 ,
*l*14 2 0.647,
*xm*4 2 33.923 ,
*l*24 0.751 .
(5)

normal mode constants in unit cm21_{:}

### .

### /

### ´

*v*14 2253.360,

*x*134 2 33.548,

*T*

_{334 1.715,}

*v*24 2255.948 ,

*x*334 2 15.059,

*K*11334 2 34.299,

*x*

_{114 2 8.200 ,}

*G*334 5.145 ,

*K*13334 2 137.194 . (6)

To our knowledge, only the normal mode parameters of CH4and isotopomers were

*calculated by ab initio method [22]. We are looking forward to the normal mode*
parameter set for silane calculated by that method for comparison with those given
above.

**4. – Conclusion**

The extended, anharmonically coupled local-mode Hamiltonian (3) for the X-H

stretching vibrations of a molecule XH4has been obtained in terms of the techniques of

group theory. Choosing parameters properly, we have successfully explained the stretching vibrations of silane with the smaller root-mean-square deviations, which shows that the anharmonicity in silane should be described with at least two parameters. In addition, we have given the local-mode and the normal-mode parameters by eq. (5) and (6). The last term in eq. (4) describes the interaction between the phonons in different bonds, which can be regarded as a modification to Morse

*potential. Although the Morse constant xm* is greater than the anharmonic parameter

*l*2 *from eq. (5), the anharmonic constant l2* is in the same magnitude as the harmonic

*one l*1, therefore we have too enough reasons to neglect such interaction when the

highly excited vibrations are taken into account. We believe that such interaction may, to some extent, play a role in the predictions for the higher states. The predicted values for the energies of the local mode bands (5000) and (6000) are 10 264.44 and

12113.80 cm21_{, respectively. Other values can be obtained from us upon request.}

Our scheme can be extended to include other motions and interactions, and applied to other symmetric molecules. The investigations at that direction are in progress.

* * *

This work was supported by the National Natural Science Foundation of China and Grant No. LWTZ-1298 of the Chinese Academy of Sciences.

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