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Effective rotational Hamiltonian of the local mode vibrational states
osl4-8539/92 s5.oo+o.oo @ 1992 Pergamon Press plc
Spccrochimica Acta, Vol. 48A. No. 2, pp. 193-198,19!%? Printed in Great Britain
Effective rotational Hamiltonian
of the local mode vibrational states
QING-SHI
ZHU
Dalian Institute of Chemical Physics, Chinese Academy of sciences, Dalian 116012, People’s Republic of China (Received 31 May 1991; in final form 18 July 1991; accepted 16 August 1991) Abstract-The effective rotational Hamiltonians of the local mode states are derived for XY,, XY, and XY4 molecules by averaging the vibration-rotation Hamiltonians using the corresponding local mode vibrational wavefunctions. The results indicate that in a local mode vibrational state the molecular symmetry is reduced by the coupling of Hz term. A tetrahedral molecule reduces to a Cs, symmetric top, a symmetric top becomes an asymmetric top, while the degree of asymmetry increases for an asymmetric top. The vibrational dependences of rotational constants and vibration-rotation coupling parameters predicted by present theory are in good agreement with those obtained experimentally from SiH, local mode vibrational bands.
IN THE local mode model, a molecule containing n X-Y bonds is considered as a set of n Morse oscillators, coupled to one another by cross-terms in both kinetic and potential
energies [l]. This model is preferable for molecules with small inter-bond couplings. In the limiting case, the inter-bond coupling is negligible, and the molecule can be viewed as a set of independent single bond oscillators. Then the vibrational energy, once excited into one bond, will stay there for a relatively long time. Such states, called local mode vibrational states, are thought to be ideal for bond-selective chemistry [2]. The rotational energy level structure of local mode vibrational states is one of the major topics of molecular overtone spectroscopy. HALONEN and ROBIETTE [3] have carried out a systematic study on the rotational energy level structure in the local mode limit for XY2, XY, and XY,, molecules. They found that, under some assumptions they made about the local mode limit (“the strict local mode limit”), the stretching vibrational Coriolis coefficients disappear and a simple relationship between the coeefficients of the vibrationally diagonal and off-diagonal Hz terms is obtained. Constraining the coefficients to this simple relationship, they calculated numerically the rotational energy level structure of the v11v3 pair of XY3 and XY4 molecules, and showed that the rotational energy level structure of the local mode state of a tetrahedral molecule is similar to that of a C,, symmetric top. This CfUpattern has been found by ZHU and co-workers in the high-resolution Fourier transform spectra of the GeI-I., (3000) band [4] and SiH,, (nOOO), n = 3,4,5 bands [5]. The rotational constants and coupling parameters obtained from these local mode states show striking vibrational dependences [5,6], which require a more general theory to explain. Besides, the physical origin of the symmetric top rotational structure for these tetrahedral molecules is by no means apparent from the present numerical calculations [3-51. The more serious problem confronting us is that some rather drastic assumptions were introduced in Ref. [3] in order to obtain results in simple form. It is of great interest to derive the general rotational level structures of the local mode states, using only the basic condition about the local mode limit: the interbond coupling parameter I = 0 [ 11. This work is devoted to these problems. The effective rotational Hamiltonian is derived for XY2, XY3 and XY4 molecules by averaging the vibration-rotation Hamiltonian over all the vibrational coordinates using the local mode vibrational wavefunctions. The results indicate that in a local mode vibrational state, the molecular symmetry is generally reduced owing to the coupling of Hz term. A tetrahedral molecule reduces to a C3, symmetric top, a symmetric top becomes an asymmetric top, while the degree of asymmetry increases for an asymmetric top. The vibrational dependence of rotational constants and vibration-rotation coupling parameters predicted by present 193
QING-SHI ZHU
194
theory are in good agreement with those obtained experimentally vibrational bands.
from SiH, local mode
THELOCALMODESTATE
The vibrational wavefunctions of a XY, molecule with negligible inter-bond coupling (including the coupling with bending as the indirect inter-bond coupling), 1= 0, can be expressed as the products of bond oscillator eigenfunctions: UlU2.
. . &I> = bJb2).
* * Id
(1)
where u, denotes the vibrational quanta in the nth bond. In particular, In 0 . . . 0) denotes a local mode state with II vibrational quanta in bond 1 and null everywhere else. Considering the bond oscillators as Morse oscillators, the matrix elements of bond displacement coordinate r and its conjugate momentum p are known [l]. Their diagonal elements are:
where k = do,, o is the Morse frequency and w, is the anharmonicity. Since k is usually quite large, for example, k = 46 and 65 for Hz0 and D20, (ulr]u) may be omitted in many cases. In fact, any bond oscillator model for which the matrix elements (ulplu) and (u]r]u) are negligible is applicable here. The vibration-rotation Hamiltonian has been thoroughly discussed [7]. Here we only consider its first few terms. Among them, the Hi2 term can be ignored since it has no non-zero vibrationally diagonal matrix element. The H2, term can also be omitted since all Y atoms move along the molecular bonds when inter-bond coupling is negligible, so that the stretch vibrational Coriolis coefficients vanish. Therefore, only the Z& terms will be considered in addition to the pure rotational term Z&. Furthermore, since the coupling between stretch and bend is negligible, we can only consider the stretch part of Hamiltonian.
XY2 MOLECULE
For an XY,-type molecule, we have the stretch vibration-rotation
Hamiltonian:
where 4 : = i_(q: + p :), q1q3 = i( q1q3 +plp3). Replacing the normal mode coordinates the bond displacement coordinates: q1=$$r.+d
PI=&(P.+P*).
&=&r/b),
then averaging H,, by the local mode wavefunction rotational Hamiltonian is: (nO]H,,lnO)= B;J:+
p3= &
(PC!-Pb),
by
(4)
InO), we find that the effective
By,J;+ B:JO+3d,,()nIP21n)-(OIi210)(J~Jx+JIJI),
whereB$= Bf -~((nIPZln)+ (Oli;‘lO))(a~+a~), 5=x, y, z. This can be transformed
(3 into
Effective rotational Hamiltonians
195
(nop”,lno) = B&J:, + B&J;, + B&J:,
(6)
the diagonal form:
where
B$*= +(B”,+ BE) + 3((B”,- By + [d&lPln)
- (0lP”l0))]‘)‘”
B$ = +(B”,+ B:) - 3((B”,- B:)* + [d13((nJf21n) - (olP’lo))]y* BE;,= By,.
(7)
It is obvious that an XY2 molecule remains an asymmetric top. The degree of asymmetry increases for an asymmetric top in the local mode state. But the larger the coupling parameter dr3 and the higher the vibrational overtone state, the higher is the degree of asymmetry. In the strict local mode limit, applying the relationship [3]:
(8) Eqn (7) reduces to: B;;, = BE + +[d&li*ln)
- (OlP’lO))]
B:;, = B; - +[d,,(- )] BzR= By,.
(9)
XY,MOLECULE
For an XY3 molecule the stretch vibration-rotation H,,=BxJ~+ByJ;+
Hamiltonian
is:
BzJ~-$t’,[(q2,-~~y)(J~-J~+2&q,y(J~y+J,J,)]
+2r3[(4~-q~y)(JxJr+JIJx)+241r43y(JyJ=+JrJy)l +3af;[4~4~(J~J,+J,J,)+41ci~,(JyJ1+J,Jy)1,
w-3
where f3, is the Z-type Coriolis resonance term parameter conventionally denoted as q3. Replacing the normal coordinates by the bond displacement coordinates:
and averaging H,, by local mode wavefunction
I&O), we get:
(nOO~H&lO)=(B”+E)J~+(BY-E)J;+BzJ:+G(J,Jz+JrJx),
(12)
where
03) SA(A) 4812-E
’
QING-SHI ZHU
1%
Solving the secular equation: B”+ E+A
0 BY-E-A
0 G the effective rotational Hamiltonian
G 0
0
=o,
(14)
B’-A
can also be transformed
to the diagonal form:
(nOO1H&OO) = B”‘J:, + By’J;, + B”Jf,,
(15)
where B”‘=:{(B”+B”+E)+[(B’-B”-E)2+4G2]1’2} B”=+{(B’+B”+
E)-
[(B’- B”- E)2+4G2]1’2}
BY’=By-E.
(16)
There are three different roots of Eqn (14), indicating that a C,, symmetric top molecule generally becomes an asymmetric top in the local mode vibrational state. In the strict local mode limit, applying the relationship [3]: B”=BZ=BY=B,
a?=af=a;=a5=-2e3=-4~r3=-a;;=~a~
(17)
Eqn (16) reduces to: B”‘=BY’=B-+a;
,
B”=B+*af,
(18)
which indicates that in the “strict” local mode state, an XY3 molecule remains a symmetric top. This result agrees with the numerical calculation in Ref. [3].
XY,MOLECULES
The vibration-rotation Hamiltonian written in the usual way as [7,8]:
of tetrahedral
molecules
considered
H,, = BJ* + Hb + am T(220) + amT(224),
here is
(19)
where Hh is the interaction term between y1 and v3 as discussed by SUSSKIND [9] and Bmss [lo]. T(220) and T(224) are the tensor Coriolis terms [8]. Including only the stretch vibration (vl and v3), they are: Ho,={B-al~~-
b3+(~~+hd14~J2+
.x (am+ 1%d4iaJ~ a
+ P13&43z+(a2~ -8an4)Ci~43~l(J,J~+J~J,) + PI34 & + (a220-8aus)Q3y4311(JyJr+JrJy) +[d134143y+(a2u,-8a,)43=ql,l(J,J,+J,J,). In a short communication local mode limit:
(20)
[ll], applying the simple ratio derived in Ref. [3] for the
(21) we obtained the effective rotational Hamiltonian for XY, molecules. In this work a more general effective rotational Hamiltonian is derived.
EffectiverotationalHamiltonians
For tetrahedral mode coordinates
197
molecules with negligible bond-angle force constants, the normal can be replaced by the symmetrized bond displacement coordinates: %=!dra+rb+rc+rd),
Pl
q&=+(re-rb+rc-
=&a
+Pb
rd)9
Plr=+(Po-pb+Pc-pd)
rd),
P3y=3(P.-pb-pc+pd)
+Pc
+pd)
(22) q3Y
=
+kO
-
rb
-
rc +
q3r=+(ra+rb-rc-rd)p
pL=+(po+pb-pc-pd).
Averaging H,, by the local mode vibrational wavefunction (nOOO(H”,~nOOO) = B”J2 +
$(d,3
+
auo
-
f&MJJ,
+
JyJx
+
~nOOO),we obtained: J,J,
+J,J,
+
JyJz
+JJ,),
(23)
where B, = B - (al + 3a3)((nIP21n) + 3(Olf*jO)), (43
+
a220
-
h24h
=
(43
+
a220
-
(W214- (Olf210)),
Sa224)
which are the effective rotational constant and coupling parameters states. Then rotating axes to let z’ point along r,,:
we
l/fi
- lIti
l/ti
lIti
lIti
l/d3
(24)
of the local mode
0
w
have:
(nOOOlH,,~nOOO) = B, J2 + :(d13 + am - 8am),(3J~. -J’), which gives rise to the symmetric-top-type
rotational
(26)
energy levels with the effective
rotational constant
& = B, - iW13 + au0
-
&A,
(& - Bed= $(43+ aa - &da.
(27)
When the ratio (18) is applied, Eqn (24) becomes identical with that given in Ref. [ll]: &
=
B,
-
3(43)u9
(&-
Be,)
=
%43L,
(28)
which agrees very well with the experimental results [5,11]. It is interesting to note that this general effective rotational Hamiltonian is obtained by using only the assumption )3= 0, and the observed symmetric top rotational structure of tetrahedral molecules is one of its natural consequences. Therefore the rather drastic assumptions made in Ref. [3] about the local mode limit are not necessary for general local mode states.
DISCUSSION
There is a problem deserving further discussion. For molecules with negligible interbond couplings, the local mode states InO. . . 0) are the stretching vibrational eigenstates. However, for tetrahedral molecules they are not dipole-active from the ground state owing to symmetry restriction on the transitions to the Al state. It is shown in this work that, at the local mode limit, the Al and F2 states can be coupled by the vibrationrotation term Z& to give rise to a dipole-active rovibrational eigenstate at the local mode limit. The striking spectra of GeH4 [4] and SiI-I, [5] can be interpreted as arising from transitions to such states.
198
QING-SHIZHU
Acknowledgements-1 am very grateful to Dr M. S. Child for helpful discussions, and would also like to thank the Natural Science Foundation of China for supporting this work.
REFERENCES [l] [2] [3] [4]
M. S. Child and L. Halonen, Adu. Chem. Phys. 57, 1 (1984). F. F. Crim, Science 249, 1387 (1990). L. Halonen and A. G. Robiette, J. Chem. Phys. 84, 6861 (1986). Q.-S. Zhu, B. A. Thrush and A. G. Robiette, Chem. Phys. Lett. 150, 181 (1988); Q.-S. Zhu and B. A. Thrush, J. Gem. Phys. 92, 2619 (1990). (51 Q.-S. Zhu, B.-S. Zhang, Y.-R. Ma and H.-B. Qian, Chem. Phys. Len. 164,5% (1989); Q.-S. Zhu, B.-S. Zhang, Y.-R. Ma and H.-B. Qian, Spectrochim. Acta 46A, 1217 and 1323 (1990). [6] Q.-S. Zhu, H.-B. Qian, H. Ma and L. Holonen, Chem. Phys. Letr. 177,261 (1991). [7] D. Papousek and M. R. Ahev, Molecular VibrutionabRotational Specrra. Elsevier, Amsterdam (1982). [8] A. G. Robiette, D. L. Gray and F. W. Birss, Molec. Phys. 32, 1591 (1976). [9] J. Susskind, J. Chem. Phys. 56, 5152 (1972). [lo] F. W. Birss, Mofec. Phys. 31,491 (1976). [ll] M. S. Child and Q.-S. Zhu, Chem. Phys. Lett., 184, 41 (1991).
Spccrochimica Acta, Vol. 48A. No. 2, pp. 193-198,19!%? Printed in Great Britain
Effective rotational Hamiltonian
of the local mode vibrational states
QING-SHI
ZHU
Dalian Institute of Chemical Physics, Chinese Academy of sciences, Dalian 116012, People’s Republic of China (Received 31 May 1991; in final form 18 July 1991; accepted 16 August 1991) Abstract-The effective rotational Hamiltonians of the local mode states are derived for XY,, XY, and XY4 molecules by averaging the vibration-rotation Hamiltonians using the corresponding local mode vibrational wavefunctions. The results indicate that in a local mode vibrational state the molecular symmetry is reduced by the coupling of Hz term. A tetrahedral molecule reduces to a Cs, symmetric top, a symmetric top becomes an asymmetric top, while the degree of asymmetry increases for an asymmetric top. The vibrational dependences of rotational constants and vibration-rotation coupling parameters predicted by present theory are in good agreement with those obtained experimentally from SiH, local mode vibrational bands.
IN THE local mode model, a molecule containing n X-Y bonds is considered as a set of n Morse oscillators, coupled to one another by cross-terms in both kinetic and potential
energies [l]. This model is preferable for molecules with small inter-bond couplings. In the limiting case, the inter-bond coupling is negligible, and the molecule can be viewed as a set of independent single bond oscillators. Then the vibrational energy, once excited into one bond, will stay there for a relatively long time. Such states, called local mode vibrational states, are thought to be ideal for bond-selective chemistry [2]. The rotational energy level structure of local mode vibrational states is one of the major topics of molecular overtone spectroscopy. HALONEN and ROBIETTE [3] have carried out a systematic study on the rotational energy level structure in the local mode limit for XY2, XY, and XY,, molecules. They found that, under some assumptions they made about the local mode limit (“the strict local mode limit”), the stretching vibrational Coriolis coefficients disappear and a simple relationship between the coeefficients of the vibrationally diagonal and off-diagonal Hz terms is obtained. Constraining the coefficients to this simple relationship, they calculated numerically the rotational energy level structure of the v11v3 pair of XY3 and XY4 molecules, and showed that the rotational energy level structure of the local mode state of a tetrahedral molecule is similar to that of a C,, symmetric top. This CfUpattern has been found by ZHU and co-workers in the high-resolution Fourier transform spectra of the GeI-I., (3000) band [4] and SiH,, (nOOO), n = 3,4,5 bands [5]. The rotational constants and coupling parameters obtained from these local mode states show striking vibrational dependences [5,6], which require a more general theory to explain. Besides, the physical origin of the symmetric top rotational structure for these tetrahedral molecules is by no means apparent from the present numerical calculations [3-51. The more serious problem confronting us is that some rather drastic assumptions were introduced in Ref. [3] in order to obtain results in simple form. It is of great interest to derive the general rotational level structures of the local mode states, using only the basic condition about the local mode limit: the interbond coupling parameter I = 0 [ 11. This work is devoted to these problems. The effective rotational Hamiltonian is derived for XY2, XY3 and XY4 molecules by averaging the vibration-rotation Hamiltonian over all the vibrational coordinates using the local mode vibrational wavefunctions. The results indicate that in a local mode vibrational state, the molecular symmetry is generally reduced owing to the coupling of Hz term. A tetrahedral molecule reduces to a C3, symmetric top, a symmetric top becomes an asymmetric top, while the degree of asymmetry increases for an asymmetric top. The vibrational dependence of rotational constants and vibration-rotation coupling parameters predicted by present 193
QING-SHI ZHU
194
theory are in good agreement with those obtained experimentally vibrational bands.
from SiH, local mode
THELOCALMODESTATE
The vibrational wavefunctions of a XY, molecule with negligible inter-bond coupling (including the coupling with bending as the indirect inter-bond coupling), 1= 0, can be expressed as the products of bond oscillator eigenfunctions: UlU2.
. . &I> = bJb2).
* * Id
(1)
where u, denotes the vibrational quanta in the nth bond. In particular, In 0 . . . 0) denotes a local mode state with II vibrational quanta in bond 1 and null everywhere else. Considering the bond oscillators as Morse oscillators, the matrix elements of bond displacement coordinate r and its conjugate momentum p are known [l]. Their diagonal elements are:
where k = do,, o is the Morse frequency and w, is the anharmonicity. Since k is usually quite large, for example, k = 46 and 65 for Hz0 and D20, (ulr]u) may be omitted in many cases. In fact, any bond oscillator model for which the matrix elements (ulplu) and (u]r]u) are negligible is applicable here. The vibration-rotation Hamiltonian has been thoroughly discussed [7]. Here we only consider its first few terms. Among them, the Hi2 term can be ignored since it has no non-zero vibrationally diagonal matrix element. The H2, term can also be omitted since all Y atoms move along the molecular bonds when inter-bond coupling is negligible, so that the stretch vibrational Coriolis coefficients vanish. Therefore, only the Z& terms will be considered in addition to the pure rotational term Z&. Furthermore, since the coupling between stretch and bend is negligible, we can only consider the stretch part of Hamiltonian.
XY2 MOLECULE
For an XY,-type molecule, we have the stretch vibration-rotation
Hamiltonian:
where 4 : = i_(q: + p :), q1q3 = i( q1q3 +plp3). Replacing the normal mode coordinates the bond displacement coordinates: q1=$$r.+d
PI=&(P.+P*).
&=&r/b),
then averaging H,, by the local mode wavefunction rotational Hamiltonian is: (nO]H,,lnO)= B;J:+
p3= &
(PC!-Pb),
by
(4)
InO), we find that the effective
By,J;+ B:JO+3d,,()nIP21n)-(OIi210)(J~Jx+JIJI),
whereB$= Bf -~((nIPZln)+ (Oli;‘lO))(a~+a~), 5=x, y, z. This can be transformed
(3 into
Effective rotational Hamiltonians
195
(nop”,lno) = B&J:, + B&J;, + B&J:,
(6)
the diagonal form:
where
B$*= +(B”,+ BE) + 3((B”,- By + [d&lPln)
- (0lP”l0))]‘)‘”
B$ = +(B”,+ B:) - 3((B”,- B:)* + [d13((nJf21n) - (olP’lo))]y* BE;,= By,.
(7)
It is obvious that an XY2 molecule remains an asymmetric top. The degree of asymmetry increases for an asymmetric top in the local mode state. But the larger the coupling parameter dr3 and the higher the vibrational overtone state, the higher is the degree of asymmetry. In the strict local mode limit, applying the relationship [3]:
(8) Eqn (7) reduces to: B;;, = BE + +[d&li*ln)
- (OlP’lO))]
B:;, = B; - +[d,,(
(9)
XY,MOLECULE
For an XY3 molecule the stretch vibration-rotation H,,=BxJ~+ByJ;+
Hamiltonian
is:
BzJ~-$t’,[(q2,-~~y)(J~-J~+2&q,y(J~y+J,J,)]
+2r3[(4~-q~y)(JxJr+JIJx)+241r43y(JyJ=+JrJy)l +3af;[4~4~(J~J,+J,J,)+41ci~,(JyJ1+J,Jy)1,
w-3
where f3, is the Z-type Coriolis resonance term parameter conventionally denoted as q3. Replacing the normal coordinates by the bond displacement coordinates:
and averaging H,, by local mode wavefunction
I&O), we get:
(nOO~H&lO)=(B”+E)J~+(BY-E)J;+BzJ:+G(J,Jz+JrJx),
(12)
where
03) SA(A) 4812-E
’
QING-SHI ZHU
1%
Solving the secular equation: B”+ E+A
0 BY-E-A
0 G the effective rotational Hamiltonian
G 0
0
=o,
(14)
B’-A
can also be transformed
to the diagonal form:
(nOO1H&OO) = B”‘J:, + By’J;, + B”Jf,,
(15)
where B”‘=:{(B”+B”+E)+[(B’-B”-E)2+4G2]1’2} B”=+{(B’+B”+
E)-
[(B’- B”- E)2+4G2]1’2}
BY’=By-E.
(16)
There are three different roots of Eqn (14), indicating that a C,, symmetric top molecule generally becomes an asymmetric top in the local mode vibrational state. In the strict local mode limit, applying the relationship [3]: B”=BZ=BY=B,
a?=af=a;=a5=-2e3=-4~r3=-a;;=~a~
(17)
Eqn (16) reduces to: B”‘=BY’=B-+a;
,
B”=B+*af,
(18)
which indicates that in the “strict” local mode state, an XY3 molecule remains a symmetric top. This result agrees with the numerical calculation in Ref. [3].
XY,MOLECULES
The vibration-rotation Hamiltonian written in the usual way as [7,8]:
of tetrahedral
molecules
considered
H,, = BJ* + Hb + am T(220) + amT(224),
here is
(19)
where Hh is the interaction term between y1 and v3 as discussed by SUSSKIND [9] and Bmss [lo]. T(220) and T(224) are the tensor Coriolis terms [8]. Including only the stretch vibration (vl and v3), they are: Ho,={B-al~~-
b3+(~~+hd14~J2+
.x (am+ 1%d4iaJ~ a
+ P13&43z+(a2~ -8an4)Ci~43~l(J,J~+J~J,) + PI34 & + (a220-8aus)Q3y4311(JyJr+JrJy) +[d134143y+(a2u,-8a,)43=ql,l(J,J,+J,J,). In a short communication local mode limit:
(20)
[ll], applying the simple ratio derived in Ref. [3] for the
(21) we obtained the effective rotational Hamiltonian for XY, molecules. In this work a more general effective rotational Hamiltonian is derived.
EffectiverotationalHamiltonians
For tetrahedral mode coordinates
197
molecules with negligible bond-angle force constants, the normal can be replaced by the symmetrized bond displacement coordinates: %=!dra+rb+rc+rd),
Pl
q&=+(re-rb+rc-
=&a
+Pb
rd)9
Plr=+(Po-pb+Pc-pd)
rd),
P3y=3(P.-pb-pc+pd)
+Pc
+pd)
(22) q3Y
=
+kO
-
rb
-
rc +
q3r=+(ra+rb-rc-rd)p
pL=+(po+pb-pc-pd).
Averaging H,, by the local mode vibrational wavefunction (nOOO(H”,~nOOO) = B”J2 +
$(d,3
+
auo
-
f&MJJ,
+
JyJx
+
~nOOO),we obtained: J,J,
+J,J,
+
JyJz
+JJ,),
(23)
where B, = B - (al + 3a3)((nIP21n) + 3(Olf*jO)), (43
+
a220
-
h24h
=
(43
+
a220
-
(W214- (Olf210)),
Sa224)
which are the effective rotational constant and coupling parameters states. Then rotating axes to let z’ point along r,,:
we
l/fi
- lIti
l/ti
lIti
lIti
l/d3
(24)
of the local mode
0
w
have:
(nOOOlH,,~nOOO) = B, J2 + :(d13 + am - 8am),(3J~. -J’), which gives rise to the symmetric-top-type
rotational
(26)
energy levels with the effective
rotational constant
& = B, - iW13 + au0
-
&A,
(& - Bed= $(43+ aa - &da.
(27)
When the ratio (18) is applied, Eqn (24) becomes identical with that given in Ref. [ll]: &
=
B,
-
3(43)u9
(&-
Be,)
=
%43L,
(28)
which agrees very well with the experimental results [5,11]. It is interesting to note that this general effective rotational Hamiltonian is obtained by using only the assumption )3= 0, and the observed symmetric top rotational structure of tetrahedral molecules is one of its natural consequences. Therefore the rather drastic assumptions made in Ref. [3] about the local mode limit are not necessary for general local mode states.
DISCUSSION
There is a problem deserving further discussion. For molecules with negligible interbond couplings, the local mode states InO. . . 0) are the stretching vibrational eigenstates. However, for tetrahedral molecules they are not dipole-active from the ground state owing to symmetry restriction on the transitions to the Al state. It is shown in this work that, at the local mode limit, the Al and F2 states can be coupled by the vibrationrotation term Z& to give rise to a dipole-active rovibrational eigenstate at the local mode limit. The striking spectra of GeH4 [4] and SiI-I, [5] can be interpreted as arising from transitions to such states.
198
QING-SHIZHU
Acknowledgements-1 am very grateful to Dr M. S. Child for helpful discussions, and would also like to thank the Natural Science Foundation of China for supporting this work.
REFERENCES [l] [2] [3] [4]
M. S. Child and L. Halonen, Adu. Chem. Phys. 57, 1 (1984). F. F. Crim, Science 249, 1387 (1990). L. Halonen and A. G. Robiette, J. Chem. Phys. 84, 6861 (1986). Q.-S. Zhu, B. A. Thrush and A. G. Robiette, Chem. Phys. Lett. 150, 181 (1988); Q.-S. Zhu and B. A. Thrush, J. Gem. Phys. 92, 2619 (1990). (51 Q.-S. Zhu, B.-S. Zhang, Y.-R. Ma and H.-B. Qian, Chem. Phys. Len. 164,5% (1989); Q.-S. Zhu, B.-S. Zhang, Y.-R. Ma and H.-B. Qian, Spectrochim. Acta 46A, 1217 and 1323 (1990). [6] Q.-S. Zhu, H.-B. Qian, H. Ma and L. Holonen, Chem. Phys. Letr. 177,261 (1991). [7] D. Papousek and M. R. Ahev, Molecular VibrutionabRotational Specrra. Elsevier, Amsterdam (1982). [8] A. G. Robiette, D. L. Gray and F. W. Birss, Molec. Phys. 32, 1591 (1976). [9] J. Susskind, J. Chem. Phys. 56, 5152 (1972). [lo] F. W. Birss, Mofec. Phys. 31,491 (1976). [ll] M. S. Child and Q.-S. Zhu, Chem. Phys. Lett., 184, 41 (1991).