Variational calculations for the rovibrational states of Si212C and Si213C

SPECTROCHIMICA ACTA PART A

ELSEVIER

Spectrochimica Acta Part A 52 (1996) 1581 1592

Variational calculations for the rovibrational states of Si~2C and Si13C Feng Wang ~'b'*, Ellak I. von Nagy-Felsobuki b ~Guelph-Waterloo Center Jbr Graduate Work in Chemistry, and Department of Chemistry, University of Waterloo, Waterloo, Ont., N2L 3GI, Canada bDepartment of Chemistry, The University of Notcastle, Callaghan, N.S.W. 2308, Australia Received 21 October 1995: revised 16 March 1996

Abstract

Rovibrational states of the ground electronic state of Si~2C and Si~3C isotopomers have been calculated variationally. The potential energy surface used in the calculations was obtained from an MP2/TZ2Pfab initio surface of Barone et al. (V. Barone, P. Jensen and C. Minichino, J. Mol. Spectrosc., 154 (1992) 252) by applying the restrictions of 100 ° < ~ < 150 °. The ab initio surface was refitted to a fourth-order polynomial with an Ogilvie-Tipping variable using a multi-dimensional least-squares procedure. The force field was then embedded in an EckartWatson vibration-rotation Hamiltonian, from which low-lying vibrational states and rovibrational states of Si2~2C and Si~3C were obtained. The calculated vibrational states (100) and (001) of Si~2C and the ~3C isotopic shifts agree well with a recent experiment (J.D. Presilla-Marquez and W.R.M. Graham, J. Chem. Phys., 95 (1991) 5612). Also, the calculations support the vibrational transition at 658.2 cm-~ found by Kafafi et al. (Z.H. Kafafi, R.H. Hauge, L. Fredin and J.L. Margrave, J. Chem. Phys., 87 (1983) 797). The rotational energies of these isotopomers for the lowest six vibrational states are given as are the rotational constants for Si~2C and Si~3C. Keywords: Energy levels; Isotopic shift; Rovibrational; Silicon carbide; Vibrational calculations; Vibrational states

I. Introduction

Silicon carbide species are of astrophysical interest and play an important role in material science, for example, in semiconductors, crystals and diamonds. In spectrometric studies, SiC2 (about 9%) and Si2C (about 8%) [1,2] have been recognized as major molecular species in vapors * Corresponding author. Current address: Department of Chemistry, The University of Melbourne, Parkville, Victoria 3052, Australia. Fax: + 61 3 9347 6883.

above silicon carbide. Though not identified, Si2 C [3] was speculated to be responsible for the blue/ green absorption bands first observed in 1926 by Memill [4] and Sanford [5] in the spectra of certain carbon-rich stars and cometary atmospheres. Unlike SIC2, SiRC is more difficult to observe spectroscopically [6] because high temperatures (about 2000°C) are required to vaporize it [7]. The failure to detect the fluorescence spectrum of Si2C is attributed to its low quantum yield [8]. As a result, experimental data for SiRC are limited [9].

0584-8539/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S0584-8539(96)01714-X

1582

F. Wang, E.1. yon Nagy-Felsobuki / Spectrochimica Acta Part A 52 (1996) 1581 I592

Table 1 Comparison of the electronic structural calculations of the

Si2C

Model

Rsic//~

/_ SiCSi /deg

E/E h

Ogl/cm

SCF/6-31G(2d) SCF/DZ+2P SCF/TZ + 2P + f SCF/EXT+2P CISD/TZ+2P

1.670 1.671 1.668 1.669 1.685 1.682 1.703 1.706

124.4 133.3 136.4 128.2 119.7 117.6 119.5 117.1

-615.59355 -615.60355 - 615.64462 -615.65611 -615.92606 -615.95315 -616.23328 -615.98125

876 823 803 848 860 883 808 830a 839.5a

CCSD/TZ+2P+f

MP2/6- 31 lG(2d) MP2/TZ+ 2P+f Expt.

I

~2/Cm- I

¢o3/cm-i

Ref.

134 94 66 117 145 151 131 144a

1393 1429 1441 1401 1298 1300 1223 1220a 1188.4~

[9] [6] [13] [13] [13] [13] [12] [14] [11]

(172) a'b

"Values of v rather than ~o between which there is an anharmonicity. b Data in parentheses are from results of force constants adjustment calculations. See Ref. [11]. Of the data available, Weltner and McLeod [2] have tentatively assigned a few bands at 4893, 5015 and 5303/k in the Ne matrix spectrum, but a band at 4893 /k was shown by Verma and Nagaraj [10] to belong to SIC2. A decade later, Kafafi et al. [8] studied the infrared (IR) spectra of matrix isolated Sil2C and Si~3C and observed two vibratonal frequencies at 1188.4 and 658.2 cm ~ respectively. Their analysis was later supported by an ab initio calculation of Grev and Schaefer [6]. However, a more recent experiment by Presilla-Marquez and G r a h a m [11] using a Fourier transform infrared (FTIR) spectrometer showed that the vibrational frequency of Si2C at 1188.4 cm 1 corresponded to a v3 asymmetric stretching mode, whereas the 658.2 c m - t absorption given by Kafafi et al. [8] was "misassigned" [11]. Instead, Presilla-Marquez and G r a h a m [11] identified a v~ symmetric stretching mode at 839.5 c m - ~ which was also supported by the ab initio calculations of Rittby [12] and Bolton et al. [13]. Moreover, Rittby [12] predicted that the intensity of the bending mode would be very low, which was confirmed by its absorption band at 1354.8 c m - ~ as a v2 + v3 combination band, which suggested that the bending mode v2 was at about 166.4 cm A number of theoretical calculations have been performed on Si2C over the past decade [6,7,12 14]. However, predictions for its molecular geometry from the calculations are not in good agreement, and the results depend largely on the models (level of theory and basis set) used [13].

Table 1 summarizes the previous theoretical calculations on Si2C. Nevertheless, both experimental data [8,11] and theoretical calculations [6,7,12-14] confirmed that the energetically most favored stationary point on the Si2C (tA0 potential energy surface is a closed shell molecule with C2v symmetry [6]. Although there have been some studies on Si2C, most theoretical calculations focused on its equilibrium geometry in the electronic ground state [6,7,9,12]. Vibrational fundamental frequencies (v) of the molecule (see Table 1) were approximately calculated as harmonic frequencies (co) rather than obtained from quantum mechanical solutions of the nuclear Schr6dinger equation. Experiment provides the v values but not harmonic frequencies (co) values, between which there is a difference of the order of 1-3%, given the usual difference due to anharmonicity between v and co [13]. In this regard, Barone et al. [14] performed the first comprehensive rovibrational analysis of the Si2C using an M P 2 / T Z 2 P f ab initio potential energy surface. Their calculations confirmed the vibrational frequencies v~ and v2 observed by the F T I R experiment [ll]. However, the calculated vibrational states seem method-dependent in their study. For example, the (011) state was calculated at 1308.0 cm ~ and 1342.4 cm 1 using a perturbative treatment and the MORBID method [14], respectively, giving a difference of 34.4 cm 1. Moreover, the vibrational transition at 658.2 c m - 1 observed by Kafafi et al. [8] did not appear in

F. Wang, E.1. yon Nagy-Felsobuki / Spectrochimica Acta Part A 52 (1996) 1581-1592

the calculations. Instead, such an observation was presumably caused by some other molecule [14]. As a result, further work is still needed to provide detailed understanding for the vibration-rotation states of this molecule. We would like to present calculations of vibrational and rovibrational states of the ~2C and ~3C isotopomers of the Si2C using an Eckart-Watson Hamiltonian embedded in rectilinear displacement coordinates. The vibrational band origins of the Si~2C and Si~3C were calculated up to approximately 1200.00 cm-~. The rovibational states of these isotopomers (using their lowest six vibrational states) were calculated up to J~< 5. This approach has been successful in analyzing the rovibrational states of H2O÷ and D20 + [15] and a series of alkali metal cations such as N a f [16], Li2K+ [17], K2Li + [18], KNa2+ [19], Li2Na +, LiNa2+, KLiNa + [20] and Li + [21].

2. Potential energy function The potential energy surface of Si2C used in the calculations was based on the ab initio calculations performed by Barone et al. [14]. The surface of the ground electronic state (1A1) of Si2C was generated using an MP2/TZ2Pf model, freezing all core electrons in a correlation treatment. The equilibrium geometry was predicted to be a closed shell molecule with CRv symmetry (i.e. Si-C bond length of 1.706 ~ and /_SiCSi angle of 117.1° [14]). This structure gave a minimum energy at -615.981246 Eh, 9.01 kJ mol-1 below the linear local minimum at an Si-C separation of 1.700/k. In our calculation, a 71 point surface (PES-2) was obtained from the original 90 point surface (PES1) [14] by applying the restrictions 100°~<~< 150°, thereby ensuring small amplitude of vibrations and according with the vibrational energies reported in Table III of Ref. [14]. Because the masses of the Si atom are sufficient to confine the significant amplitude of the lowlying vibrational wavefunctions in the vicinity of the energy minimum, the immediate neighborhood of the linear structure gives a negligible contribution [14].

1583

It is important to obtain a force field that accurately interpolates a discrete potential energy surface. In this regard, a multi-dimensional leastsquares procedure was employed to fit the discrete surface to a fourth-order polynomial using an Ogilvie-Tipping variable [22] and zeroing als-G2o singular values. Table 2 gives the force field of the PES-2. The (2'2)12 value of the fit is 4.539 x 10 5

Eh. 3. Computational procedure A Czv Eckart-Watson vibration-rotation Hamiltonian [23,24] in the vibrational t-coordinates was used in this study. The Hamiltonian takes into account a full description of both mechanical anharmonicity and Coriolis vibrationrotation coupling. The full rovibrational eigenvalue problem was given by Table 2 Analytical representation of the Si2C PES-2 surface Variable a

Coefficient

1

p3p2 + p3p~ P~P3+ P~P3 P~P~+ P2P~ P ~P2 p~p]+p2~p3 P 2p2P3+ PIP ~P3 plp2p2~

-615.981246 0.00000 0.00000 1.76120 0.24641 -0.20191 -0.04450 -3.35184 -0.59560 --0.07959 0.42330 0.02748 0.20803 2.57410 --0.83609 1.22233 -0.35704 2.29017 1.38499 --0.69747 -- 2.45292 -- 1.41920

(Z2) 1'2

4.539 x 10 -5 E h

Pl + P2 P3

p~+p] p2

PiP2 PIP3 p~+p~ p~

P~P2+ P~P, P ~P3 + P ~P3

p~ p2 + p2 p~ P lP2P3 p4+p4 p~

2 ( R , - RT) . P i - - - , t = 1, 2 and 3; where R e is the equilibrium

Ri + R7

bond length of Si2C.

1584

F. Wang, E.1. yon Nagy-Felsobuki /Spectrochimica Acta Part A 52 (1996) 1581-1592

-615.0

V(h ,0,0) V(O,t2,0 )

•

! .........

:: ............... V(O,O,t3 ) -615.2

L' t

-615.4 l- ',

',

U.!

"

-615.6

LU

-

•

',

~t

"

:

~

."

"

1

-615.8

.

-

"

~

~

•

-616.0 -3.0

-2.0

-1.0

J

-

-

:

/

0.0

'

1.0

2.0

t i (i=1,2,3) / a o Fig. 1. One-dimensional potential curves along the vibrational t-coordinates. ~ v ~'/rv = (/'tvib +/-Irot q-/~c )~rv "n- Erv klJrv

(1)

where the rovibrational Hamiltonian /-trv consists of a "pure" vibrational Hamiltonian (/4vib), a rotational Hamiltonian (/-Iro,) and a Coriolis coupling operator (g~). The trial rovibrational wavefunctions were constructed from a configurational product of vibrational basis functions using plus and minus combinations of symmetric-top rotor basis functions [24,25]. ,e~,~, = y, c.Kj % Rh~M n,K

(2)

The "pure" vibrational Hamiltonian was given by [24]

/~vib ~.~

1 ~-~ ~ 2 - 2--~ ,=, ~t~

1_ 8E ~

1 ( C~ 2rzz(t, ) t3 ~ -

/ ~ + 1~(4' t2, t3)

t2

(~)2

(3)

Here Mr is the reduced mass of the molecule and the third term is a Watson operator which is a sum of the diagonal elements of the reciprocal effective moment of the inertial tensor operator I'~a, where (~, fl = x, y and z).

1585

F. Wang, E.I. yon Nagy-Felsobuki I Spectrochimica Acta Part A 52 (1996) 1581-1592

A description of the variational solution of the vibrational eigenvalue problem for the SizC isotopomers is as follows. A three dimensional (3D) vibrational wavefunction was constructed using three one dimensional (1D) eigenfunctions which are finite-element method (FEM) solutions of the corresponding 1D eigenvalue problems and was pruned using a nodal cut-off criterion: all configurations containing no more than 13 nodes were included. At each dimension of the t-coordinates, 1000 finite-elements were employed within the following integration domains: tiE[--2.5, 2.5], t2E [ --2.0, 1.5] and / 3 E [ - 2.5, 2.5] all in a0, which were so determined to ensure that the 1D wavefunctions decayed to 10 ~0 a0~12 in the classical forbidden regions. Position and kinetic energy integrals were evaluated using a 16 point Gaussian quadrature scheme, whereas the potential energy integrals were evaluated using the discrete variable representation (i.e. DVR or HEG [26] scheme) with 20 × 20 × 20 quadrature points. The 3D vibrational eigenvalue problem was then solved variationally. R~'M R j2M Rj1M

JOM

i

t

+ R JIM + R j2M

R%M

i 0 0 0 0 0 0 0

-.. -.. 0 0 0 0 0 ...

0 0 i 0 0 0 1 0

0 0 0 0 0 0 i 0 0 ~ 1 0 0 0 0 0

0 0 0 -i 0 1 0 0

0 0 -i 0 0 0 1 0

1

'"

0

0

0

0

0

In the vibrational representation, V,, the rovibrational Hamiltonian of Carney et al. [24] was rearranged as [27]

~'~ -- (v,, IBr~ Iv,,,> 1

= Ev + 5 ( B , x + Bvy l f l

where Ev is the vibrational eigenvalue matrix and is diagonal in the vibrational representation, J~ is the a-component of the total angular momentum operator, B,/~ are the rotational constants (~ = fl) and centrifugal distortion constant (~ ¢ fl, where ~, fl = x, y, z), and F is the Coriolis coupling matrix. Here B~/~ and F were calculated in the vibrational representation [24] 1 ..... = 7

(F> .... :

(5)


--

t2

IV,-)

(6)

where #~/~ was the reciprocal of the instantaneous effective inertial tensor operator. Matrix elements of the rovibrational Hamiltonian super-matrix may be complex if the ordinary symmetric-top functions ~ J K M a r e used as rotational basis functions in Eq. (2). Therefore, a transformation was employed in order to ensure that the rovibrational super-matrix contains real matrix elements [24,27,28] ....0 0 0 0 0 ... ...

__i "~ 0 0 0 0 0

(~ J( -- K') M ~i J'''M (I)j( _ 2 ) M (I)JI -- I ) M

dPJOM

(7)

f~JJ I M

0 0

(~J...M

1 J

(~)J K' M

f~ J2.44

Here i = x / - 1 and K ' = IKI. As a result, an nvib X ( 2 J + 1)-order super-matrix (where nvib is the number of vibrational basis functions) can be constructed for any given combinations of nvib and J. Diagonalizations of the resulting rovibrational super-matrices give the rovibrational eigenstates.

1

+ [8== - 5 (B,.,. + Sl,.,. )]~ 1 + 5 (~-<-" - sl,.,, )O]x -

4. Results and discussion

~)

+ B.,.,, (),. ,~,,+ a~,.J,. ) + iFa~

(4)

Fig. 1 gives the 1D cuts of the potential energy surface along the individual vibrational coordinates, whereas Fig. 2 gives three two-dimensional

1586

F. Wang, E.I. yon Nagy-Felsobuki / Spectrochimica Acta Part A 52 (1996) 1581-1592

(2D) contour plots of the potential function in the vibrational t-coordinates. In Fig. 1, the cut along the asymmetric stretching vibrational coordinate V(0,0,t3) is generally a harmonic curve in the vicinity of the equilibrium geometry; the cut along the symmetric stretching coordinate, V(tl, O, 0), is a Morse-like curve and the cut along the bending coordinate V(0, t2, O) is also a Morse-like curve but is more anharmonic and the potential well is shallower. Therefore, it can be expected from these 1D potential energy curves that the low-lying vibrational energy states will contain a number of bending vibrational states, some symmetric stretching states and a few asymmetric stretching states. The lowest 10 vibrational band origins for the Si{2C and Si~3C are given in Table 3. The assignment of vibrational band origins (v,, v2, v3) was rather difficult because of the mixed nature of the wavefunctions. Table 3 also lists percentage of the "dominant" configurational basis functions in the wavefunctions. Vibrational transitions (100) and (001) for Si~2C were assigned as 811.5 cm i and 1200.9 cm -1, respectively, which agree with 839.5 cm ~ and 1188.4 cm of the FTIR experiment [11]. The (100) and (001) bands of Sil3C were predicted by this calculation to be at 795.2 cm i and 1165.3 cm -~, respectively, and agree quite well with the experimental data [11] which are 821.6 cm -1 and 1153.4 cm i respectively. Moreover, it is interesting to note that a vibrational transition at 658.2 cm 1, which was observed by Kafafi et al. [8] but claimed by Presilla-Marquez and Graham [11] to be "misassigned", was also supported by our calculations of 692.6 cm 1 for Si~2C and 674.2 cm i for Si~3C, respectively. However, instead of being a fundamental frequency [8], our calculations assign this band as a mixture of the (110) and (020) bands. Table 3 also compares the vibrational band origins of Si{2C with the calculations of Barone et al. [14]. In the present calculations, zero-point energy (ZPE) was obtained at 1091.2 cm -1, which is in good agreement with Barone et al. [14] who obtained 1087.1 cm i and 1085.6 cm i using the perturbative treatment and the MORBID method, respectively. The low-lying vi-

brational transitions calculated in this work and in Ref. [14] agree well with the available experimental data [11]. However, there are discrepancies between the two calculations with respect to the other vibrational states and their assignments. Some features obtained in our calculations are not seen in the calculations of Ref. [14]. For example, the stretching and bending vibrational states were heavily mixed in our calculations which enabled these states to be mixed states rather than overtone bands alone as they were in Ref. [14]. Because the Eckart-Watson Hamiltonian and rectilinear vibrational coordinates which are linearly related to normal coordinates are used in the current calculations, the vibrational energies and their assignment are a direct result of the calculation. However, in Ref. [14], the vibrational energies were generated indirectly in the following manner: the force field was obtained from fitting the potential energy surface PES-1 in the region 100°~< c~< 150 ° (i.e. it is essentially the PES-2) and an analytical transformation to normal coordinates followed by a standard second-order perturbative treatment. The vibrational band origins of Sil3C were shifted from those of Si~2C due to the different masses of the C atom. ZPE values were calculated as 1091.2 cm 1 for Sit2C and 1063.5 cm ~ for Si~3C, s h i f t e d - 2 7 . 7 cm -1. The 13C isotopic shifts of (100), (010) and (001) vibrational transitions are summarized in Table 4, together with the MORBID calculations of Barone et al. [14] and some recent MP2/6-311G(2d) calculations of Rittby [12]. Table 4 also gives the experimental results of Presilla-Marquez and Graham [11]. Excellent agreement was obtained between the theoretical calculations and experimental data. The rotational constants (Bx,., B~,y and Bzz ), centrifugal distortion constant (Bxv) and the Coriolis coupling (F) matrices spanned by the lowest six vibrational states are given in Table 5 for both Si~2C and Si~3C. The diagonal elements are the dominant elements in the rotational constant matrices. The elements B,-x were the leading rotational contributions to the rovibrational Hamiltonian super-matrix, whereas the elements

F. Wang, E.I. yon Nagy-FeL~obuki

•

Spectrochimica Acta Part A 52 (1996) 1581-1592

1.0

-1.0 0~ "2" -4 0

-3.0

-2.0

-1.0

O0

1.0

2.0

tl / a.u. 3.0

.

.

.

.

i

.

.

.

.

i

~

.

.

.

.

i

.

.

.

.

i

.

.

.

.

:3 1.0

-2.5

-I,5

-O.S

0.5

1,5

2.5

3.5

t~ / a.u.

4.0

2.0

.~

0.0

-2.0

-4.0 -4.0

*2.0

0.0

2.0

t 2 / a.u.

Fig. 2. Two-dimensional potential energy contour plots of Si2C.

4.0

1587

F. Wang, E.I. yon Nagy-Felsobuki / Spectrochimica Acta Part A 52 (1996) 1581-1592

1588

Table 3 Low-lying vibrational band origins of Si2C and the assignment (cm -~) (v I v2v~ )

This work

Expt. [11]

Wt.%

Band origin

77.5 27.6 12.4 13.3, 9.7, 36.9 8.6 20.0, 13.9, 75.0

1091.22 a 139.27 290.27 459.94 692.55 811.52 930.08 946.45 1102.91 1200.86

Barone et al. [14] (v t v2v3 )

Harmonic

Perturb.

MORBID

000 010 020 030 040 100 110

1098.6 144.5 289.0 433.5 578.0 830.3 974.8

1087.1 122.0 235.9 341.2 807.2 916.2

1085.6 133.4 258.7 376.2 485.0 811.1 928.3

001 Ol 1

1220.3 1364.8

1200.3 1308.0

1195.3 1342.4

si52c 000 010 000, 010, 110, 100 110 020, 200, 001

220 100 020

200 030

8.4 9.2

13.1 9.5

(172) b

839.5

1188.4

Si~-~C 000 010 000, 010, I10, 100 110 020, 200, 001

78.0 28.8 13.2 13.2, 9.5, 38.7 8.9 20.9, 13.3, 75.6

220 100 020

200 030

7.5 8.9

13.2 10.5

1063.49 a 137.46 284.98 449.65 674.20 795.15 903.51 928.02 1080.27 1165.32

821.6

1153.4

Zero-point energy (ZPE). b Data in parentheses are from results of force constants adjustment calculations. See Ref. [ll].

BI:,, and B= were rather small. Centrifugal distortion elements Bx,, however, were expected to be small for eigenstates near the potential energy minimum so that the diagonal elements were not necessarily larger in value than the offdiagonal elements. Finally, diagonal terms of the Coriolis coupling matrix were not appreciable, Table 4 Isotopic I~C shifts of the fundamental vibrational frequencies (cm ') State

(100) (010) (001)

This work

-16.4 - 1.8 -35.5

Barone et al.

Rittby

Expt.

[14]

[12]

[ll]

-16.3 - 1.3 -35.5

-16.7 - 1.5 -37.2

(-2.0) a

" See footnote b of Table 3.

-17.8 -35.0

because for an Eckart-Watson Hamiltonian, the Coriolis coupling necessarily gives small numerical diagonal elements. These features are reflected in Table 5. Table 6 gives the rovibrational energies up to J ~< 5 of the lowest six vibrational states of Si2t2C and Si~3C. The rotational states are labeled J, ca.Kc. As expected, for a given vibrational state, the rotational energies increase as the total angular momentum quantum number J increases. The first and fifth excited vibrational states of both isotopomers were in excellent agreement with the MORBID calculations [14], whereas the agreement of the second, third and fourth excited vibrational states between two calculations were not so good. For instance, the 55o rotational energies of the Si~2C of the second, third and fourth vibrational excited states

F. Wang, E.L yon Nagy-Felsobuki / Spectrochimica Acta Part A 52 (1996) 1581 1592

1589

Table 5 R o t a t i o n a l , centrifugal d i s t o r t i o n c o n s t a n t s a n d Coriolis c o u p l i n g m a t r i x elements of the Si2t2 C a n d Si2~3 C ( c m - ~) vi

v/

B.'~,.

B,~,.

B~.:

B ~,.,.

Fb

1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5 6

2.2334 --0.3434 2.4333 0.0444 -0.5170 2.5221 -0.0104 -0.0956 -0.5800 2.5020 -0.0086 0.0556 -0.1468 0.6174 2.4824 -0.1931 0.0764 -0.0154 -0.0092 0.0144 2.3850

0.1411 0.0072 0.1396 0.0011 0.0097 0.1397 0.0002 0.0013 0.0106 0.1410 -0.0001 - 0.0005 -0.0011 -0.0114 0.1414 -0.0021 0.0005 -0.0001 0.0002 - 0.0001 0.1380

0.1324 0.0051 0.1313 0.0008 0.0069 0.1315 0.0001 0.0010 0.0076 0.1323 -0.0001 - 0.0004 -0.0009 -0.0081 0.1326 -0.0025 0.0006 -0.0001 0.0002 -0.0002 0.1301

0.8952 -0.4891 0.8084 - 0.7918 -0.8667 0.9214 -0.2426 0.9169 -0.4703 0.8905 -0.7685 0.1622 -0.4312 0.1845 0.8058 0.9323 0.5583 0.7816 -0.3972 -0.5654 0.8627

x x x x x x x x x x x x x x x x x x x x x

10-12 10-13 10-L, 10-1~ 10-13 10- 12 10- 13 10-14 10-13 10 -12 10-13 10-13 10-13 10-13 10- 12 10- 13 10-14 10- 14 10- 13 10-13 10 -12

- 0 . 2 5 4 7 x 10-11 --0.4734 X 10 t2 0.5438 x 10 II 0 . 2 3 2 5 x 10 17 - 0 . 5 8 8 7 x 10 L~ 0.3147 x 10 II 0 . 7 6 6 9 x 10 I-' - 0 . 1 2 2 4 x 10 ii 0.4717 x 10- i_~ 0 . 5 6 6 8 x 10 17

1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 1 2 3 4 5 6

2.0908 0.3202 2.2813 -0.0400 -0.4858 2.2720 --0.0089 -0.0877 0.5499 2.3568 0.0072 0.0501 -0.1373 -0.5867 2,3366 0,1744 0.0664 -0,0126 0.0080 0.0137 2.2230

0.1411 -0.0071 0,1320 -0.0011 0.0096 0.1396 0.0002 --0.0013 --0.106 0.1408 0.0001 -0.0005 -0.0011 0.0114 0.1413 0.0022 0.0005 -0.0001 -0.0002 -0.0003 0.1382

0.1319 -0.0050 0.1308 -0.0008 0.0067 0.1308 0.0001 --0.0010 -0.0074 0.1317 0.0001 -0.0003 -0.0009 0.0079 0.1320 0.0026 0.0006 -0.0001 -0.0001 -0.0002 0.1297

0.8042 x 0.5316 x 0.8502x -0.2923 x -0.3471 x 0.9450x --0.5000 x 0.6089 x 0.5839 × 0.8628 x -0.1221 x -0.2053 x 0.8729 x -0.9890 x 0.9429 x -0.4531 x O. 1075 x 0.3369 x 0.7801 x -0.3319× 0.9083 x

10- 12 10- 13 10 12 10 13 10-13 10 12 10-13 10 -13 10 13 10 ~2 10- 13 10 13 10 t4 10 -14 10 -12 10 12 1O- 13 10 13 10 13 10 13 10 12

-0.7791 x 10-]9 0 . 2 1 1 9 x 1 0 II - 0 . 5 0 0 1 x 10 18 - 0 . 9 9 5 9 x 10 i~ - 0 . 1 1 5 5 x 10 - I I - 0 . 5 0 4 4 x 10 18 - 0 . 6 3 0 4 x 10 12 0 . 2 9 5 0 x 10 tl 0.9025 x 10 - I I - 0 . 1 5 6 5 x 10 Is 0.1095 x 10 -I~' 0 . 7 5 3 3 x 10 1~ - 0 . 1 3 4 0 x 10 - I I - 0 . 4 4 9 8 x 10 Ii - 0 . 2 7 0 2 x 10 t9 0 . 4 5 0 9 x 10 tl 0.1388 x 10 - m 0 . 8 3 2 4 x 10 I_, - 0 . 5 8 1 1 × 10 i_, 0.9173 x 10 12 - 0 . 6 5 0 4 x 10 18

si~:c 1 2 2 3 3 3 4 4 4 4 5 5 5 5 5 6 6 6 6 6 6

0.1194× 10 - 1 7 - 0 . 3 6 8 3 x 10 - I I 0.4324 x 1 0 - t 7 - 0 . 2 5 2 3 x 10 13 - 0 . 3 6 1 6 x 10 ii 0 . 4 6 9 9 x 1 0 17 - 0 . 2 0 2 6 x 10 t2 0 . 7 2 1 7 x 10 12 - 0 . 4 9 5 3 x 10 it 0.2803 x 10 - w -0.7063×10 12

SiI:3C 1 2 2 3 3 3 4 4 4 4 5 5 5 5 5 6 6 6 6 6 6

a S y m m e t r i c matrices. b A n t i - s y m m e t r i c matrix.

were 63.50 cm-1, 63.42 cm-I and 62.96 cm-~, respectively, in this calculation, which were not in agreement with the rotational states of

71.35 cm-1, 84.43 cm-I respectively given by the [14].

and 103.87 c m calculations

MORBID

F. Wang, E.L yon Nagy-Felsobuki / Spectrochimica Acta Part A 52 (1996) 1581 1592

1590

Table 6 Rotational energies on the low-lying vibrational states of the Si~3C and Si2~3C (cm-t)

JKa.KcEvib

0.000

139.267

290.271

459.936

692.548

811.521

lol 1i i Im

0.274 2.365 2. 374

0.271 2.564 2. 572

0.271 2.563 2.662

0.273 2.635 2.643

0.274 2.615 2.624

0.268 2.514 2.522

2o2 2t2 21j 221 220

0.820 2.903 2.929 9.193 9.193

0.813 3.098 3.122 9.988 9.988

0.815 3.187 3.212 10.354 10.354

0.820 3.173 3.198 10.286 10.286

0.822 3.155 3.181 10.209 10.209

0.804 3.054 3.066 9.972 9.972

3o3 313 3t 2 3_,, 321 331 330

1.641 3.711 3.763 10.014 10.014 20.440 20.440

1.626 3.899 3.948 10.802 10.802 22.225 22.225

1.627 3.988 4.038 11.168 11.168 23.078 23.078

1.640 3.979 4.031 11.106 11.106 22.953 22.953

1.644 3.964 4.016 11.031 11.031 22.782 22.782

1.609 3.836 3.883 10.597 10.597 21.783 21.783

4o4 414 413

2.734 4.787 4.874 11.109 11.109 20.258 20.258 36.060 36.060

2.709 4.967 5.048 11.887 11.887 21.949 21.949 39.219 39.219

2.712 5.057 5.140 12.253 12.253 22.823 22.823 40.804 40.804

2.733 5.055 5.142 12.199 12.199 22.735 22.735 40.655 40.655

2.740 5.042 5.129 12.127 12.127 22.657 22.657 40.358 40.358

2.681 4.893 4.971 11.671 11.671 21.484 21.484 38.432 38.432

4.101 6.133 6.263 12.477 12.478 22.906 22.906 37.432 37.432 55.986 55.986

4.064 6.301 6.424 13.243 13.244 24.669 24.669 40.578 40.578 60.888 60.888

4.067 6.392 6.516 13.609 13.610 25.519 25.519 42.161 42.161 63.498 63.498

4.099 6.400 6.530 13.565 13.566 25.411 25.411 42.020 42.020 63.416 63.416

4.110 6.390 6.521 13.497 13.498 25.247 25.247 41.726 41.726 62.964 62.964

4.021 6.214 6.331 13.012 13.013 24.202 24.202 39.779 39.779 59.658 59.658

Si52C

423 422 432 431 441 440 5o5 515 514 524 523 533

532 542 541 551 55o

Both Si212C and Sigl3C are asymmetric-tops with Ray's asymmetric parameters [28] o f - 0 . 9 9 1 7 a n d - 0.9906 respectively. The spectroscopic constants can be obtained from fitting the diagonal elements of the corresponding rotational constant elements given by Table 5, to the expression [29]

3 ( ,v,+l ) l)(1)

x=xo- Z

i=!

+

v,+ 5

vj+g

+...

(8)

where X represents the rotational constants,

Bxx (A), Bw (B) and B= (C) respectively. Eq.

1591

F. Wang, E.I. yon Nagy-Felsobuki / Spectrochimica Acta Part A 52 (1996) 1581 1592 Table 6 (continued) Rotational energy energies on the low-lying vibrational states of the

JKa.KcEvib

Si~2Cand

Si~3C (cm-L)

0.000

137.459

284.980

449.646

674.197

795.151

1m 1II I io

0.273 2.222 2.231

0.270 2.411 2.420

0.270 2.504 2.513

0.273 2.489 2.498

0.273 2.470 2.479

0.268 2.353 2.361

2o2 21, 211 22t 22o

0.819 2.759 2.786 8.624 8.624

0.811 2.943 2.970 9.382 9.382

0.811 3.035 3.062 9.759 9.759

0.818 3.025 3.052 9.705 9.705

0.820 3.007 3.052 9.645 9.645

0.804 2.880 2.905 9.160 9.160

3o3 313 312 322 32L 331 330

1.638 3.564 3.619 9.443 9.443 19.164 19.164

1.623 3.742 3.794 10.193 10.193 20.863 20.863

1.622 3.834 3.886 10.569 10.569 21.736 21.736

1.635 3.829 3.883 10.522 10.522 21.642 21.642

1.640 3.812 3.868 10.445 10.445 21.467 21.467

1.607 3.671 3.721 9.950 9.950 20.334 20.334

404 4t4 413 423 42z 432 431 441 44o

2.729 4.637 4.730 10.536 10.536 20.258 20.258 33.803 33. 803

2.704 4.807 4.893 11.275 11.276 21.946 21.946 36.806 36.806

2.704 4.898 4.986 11.651 11.651 22.818 22.818 38.420 38.420

2.725 4.900 4.992 11.611 11.612 22.732 22.732 38.322 38. 322

2.733 4.887 4.980 11.538 11.538 22.559 22.559 38.019 38.019

2.679 4.726 4.810 11.023 11.023 21.409 21.409 35.871 35.871

5o5 515 514 524 523 533 53, 542 541 551 550

4.093 5.979 6.117 I 1.902 11.903 21.625 21.625 35.172 35.172 52.482 52.482

4.056 6.137 6.267 12.628 12.629 23.301 23.301 38.163 38.163 57.138 57.138

4.056 6.228 6.359 13.003 13.004 24.171 24.171 39.773 39.773 59.772 59.772

4.087 6.240 6.377 12.974 12.975 24.094 24.094 39.683 39.683 59.761 59.761

4.099 6.230 6.369 12.904 12.905 23.925 23.925 39.382 39.382 59.301 59.301

4.019 6.045 6.171 12.364 12.365 22.75 t 22.751 37.216 37.216 55.683 55.683

Si ~-~C

(8) was truncated to the first order in the fitting procedure. Table 7 compares the fitted rotational constants of Si~2C with those from the second order perturbative treatment [14]. 5. Conclusions Low-lying vibrational band origins, rovibrational states and spectroscopic constants of Si~2C and Si~3C

have been calculated using the MP2/TZ2Pfab initio potential energy surface of Barone et al. [14] in the region 100°~< 7 < 150°. The discrete surface was then refitted to a fourth-order polynomial with an Ogilvie-Tipping variable using a least-squares procedure. The potential function was embedd in the an Eckart-Watson Hamiltonian spanned by the t-coordinates, which was solved directly using numerical techniques such as the FEM and the HEG

1592

F. Wang, E.1. yon Nagy-Felsobuki / Spectrochimica Acta Part A 52 (1996) 1581-1592

grid technique. The calculated vibrational frequencies of (100) and (001) were in good agreement with the experimental data [11] and with the calculations of Barone et al. [14] using the MOR~ID method. The vibrational transition at 658.2 cm- i observed by Kafafi et al. [8] was also supported by our calculation. More detailed calculations of the rovibrational states of Si2C require an extensive ab initio potential energy surface and more detailed experiments.

Acknowledgements Calculations were performed using an IBM RS6000 UNIX work station and a VAX/VMS 6620s. We would like to thank the support of the Computing Center at the University of Newcastle. F. Wang wishes to acknowledge an Overseas Postgraduate Research Award, a University of Newcastle Postgraduate Research Award (Australia) and an award of NSERC international research fellowship (Canada). Finally, we wish to thank the referee for important comments on the manuscript.

Table 7 Comparison of the rotational constants of the ground vibrational state of Si~2C (cm ~) Rotational constant

This work

Barone et al. [14]~

Ao A~ &l Be Co C~ ~A ~ ~A ~ :~ :~3B ~ ~ ~

2.5558 2.2334 0.1381 0.1411 0.1303 0.1324 0.0238 0.0246 --0.3575 0.0000 0.0001 0.0036 0.0000 0.0001 0.0023

2.3094 2.1313 0.1392 0.1428 0.1311 0.1338 --0.2009 --0.1178 --0.0373 0.0022 0.0030 0.0020 0.0017 0.0023 0.0015

" From the second order perturbative treatment.

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