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Molecular constants appearing in reduced Hamiltonian for C3v molecules
JOURNAL
OF MOLECULAR
SPECTROSCOPY
133.467-468 (1989)
NOTES Molecular Constants Appearing in Reduced Hamiltonian for Csv Molecules Recently, Lobodenko et al. (I ) showed how to avoid correlation problems in fitting the parameters of the rotation-vibration Hamiltonian to experimental data for Coriolis-interacting v,(A i), v,(E) states of C,, molecules. By applying the block diagonal contact transformation they found the determinable combinations of molecular parameters. It is the purpose of this note to complement their results by specifying in more detail some of the parameters appearing in the reduced Hamiltonian and to show the modifications which should be made if a molecule is a quasi-spherical top, such as AsHr or PH3. Let us denote by symbol H’ the Hamiltonian (e.g., Table V of Ref. (2)) after the contact transformation removed the operators nondiagonal in principal vibrational quantum numbers (except those connecting interacting Y., v, states). This Hamiltonian is not identical to H specified by Eqs. (I)-( 5) of Ref. (1) because the form of HE in Bq. (5) suggests that H is already the result of a block diagonal contact transformation of H’, H = exp(-iS,)H’exp(iS,),
(I)
where S, = Sr + S, and both Ss and S, are specified by Eqs. (50), (5 1) of Ref. (3). We need to know explicitly only the operator & which can be written as S, = -is,(J:
- J?)
(2)
with c’
” = 3(Gb - Bb) ’
d = r,/4,
J, = J,k
iJy.
If the truncated Hamiltonian H in Ref. ( 1) is considered, the relevant part of the transformation in Eq. (1) is H = H’ + i[HbZ + H& + Hb4, &] + i[H&, S,] -[[H&,
&I, &l/2,
(4)
where H’,. are the terms of degree m in q, p (or a+, a) and degree n in J.. The transformation (4) brings the following results: (i) It eliminates from the Hamiltonian quartic and sextic rotational terms with the Ak = +3 matrix elements (3, 4). (ii) It changes the values of molecular constants in the following way, H, = H; - 22:
HJK = H;K + 1st
HK=HK+ d, = d: - 6[G;{:/e’
14[
Hm = H;,
- 30[
hg = h; - [ BTK,= EC: + 3@Bmrlc’
(5)
with 5 = c”/(GO - B,,).
(6)
Symbol hs replaced the symbol As used in Ref. ( I ). The effect of transformation on quadratic and quartic constants can be found in Eq. (53) of Ref. (3). Thus the constants H,. * - B
0022-2852189 $3.00 Copyright 0
1989 by Academic Prw
Inc.
All rights of reproduction in any form mcwd.
468
NOTES s3 =
h;/(6c’)
has beeen shown (4) to eliminate the Ak = +6 matrix elements. With this choice 5 = hi in Eq. (5) and in the rotational Hamiltonian Hi of Ref. (1) the term A,(/$ + J!) is replaced by the terms (f
+
f,J2)[Jz,
J:
+
P],
+
c&,
J:
+
511,
and c, c,, eKare given by Eqs. (23)-(25) of Ref. (4). In conclusion, insofar as fitting of the observed spectra is concerned, the only suggested change as compared to Ref. ( 1) is the above given modification of H$ for quasi-spherical tops. If the determined combinations of molecular parameters are used to calculate the molecular force fields it should be kept in mind that the values of some parameters appearing in Eqs. (lo), (I 7)-( 19) of Ref. (I) are modified according to Eq. (5) with Cgiven either by Eq. (6) or equal to hi. REFERENCES 1. E.I. LOBODENKO, O.N. SULAKSHINA, V.I. FEREVALOV,AND V.G. TYUTEREV, J.Mol. Spectrosc. 126, 159-170 (1987). 2. J. K. G. WATSON, in “Vibrational Spectra and Structure” (J. During, Ed.), Vol. 6., pp. l-89, Elsevier, Amsterdam, 1977. 3. M. R. ALIEV AND J. K. G. WATSON, in “Molecular Spectroscopy: Modem Research” (K. Narahari Rao, Ed.), Vol. III, pp. l-67, Academic Press, San Diego, 1985. 4. K. SARKA, J.Mol. Spec~rosc. 133 (1989). KAMILSARKA
Departmentof PhysicalChemistry, Faculty of Pharmacy, Comenius University, Odboj&ovIO, 832 32 Bratislava,Czechoslovakia ReceivedAugust 29, 1988
OF MOLECULAR
SPECTROSCOPY
133.467-468 (1989)
NOTES Molecular Constants Appearing in Reduced Hamiltonian for Csv Molecules Recently, Lobodenko et al. (I ) showed how to avoid correlation problems in fitting the parameters of the rotation-vibration Hamiltonian to experimental data for Coriolis-interacting v,(A i), v,(E) states of C,, molecules. By applying the block diagonal contact transformation they found the determinable combinations of molecular parameters. It is the purpose of this note to complement their results by specifying in more detail some of the parameters appearing in the reduced Hamiltonian and to show the modifications which should be made if a molecule is a quasi-spherical top, such as AsHr or PH3. Let us denote by symbol H’ the Hamiltonian (e.g., Table V of Ref. (2)) after the contact transformation removed the operators nondiagonal in principal vibrational quantum numbers (except those connecting interacting Y., v, states). This Hamiltonian is not identical to H specified by Eqs. (I)-( 5) of Ref. (1) because the form of HE in Bq. (5) suggests that H is already the result of a block diagonal contact transformation of H’, H = exp(-iS,)H’exp(iS,),
(I)
where S, = Sr + S, and both Ss and S, are specified by Eqs. (50), (5 1) of Ref. (3). We need to know explicitly only the operator & which can be written as S, = -is,(J:
- J?)
(2)
with c’
” = 3(Gb - Bb) ’
d = r,/4,
J, = J,k
iJy.
If the truncated Hamiltonian H in Ref. ( 1) is considered, the relevant part of the transformation in Eq. (1) is H = H’ + i[HbZ + H& + Hb4, &] + i[H&, S,] -[[H&,
&I, &l/2,
(4)
where H’,. are the terms of degree m in q, p (or a+, a) and degree n in J.. The transformation (4) brings the following results: (i) It eliminates from the Hamiltonian quartic and sextic rotational terms with the Ak = +3 matrix elements (3, 4). (ii) It changes the values of molecular constants in the following way, H, = H; - 22:
HJK = H;K + 1st
HK=HK+ d, = d: - 6[G;{:/e’
14[
Hm = H;,
- 30[
hg = h; - [ BTK,= EC: + 3@Bmrlc’
(5)
with 5 = c”/(GO - B,,).
(6)
Symbol hs replaced the symbol As used in Ref. ( I ). The effect of transformation on quadratic and quartic constants can be found in Eq. (53) of Ref. (3). Thus the constants H,. * - B
0022-2852189 $3.00 Copyright 0
1989 by Academic Prw
Inc.
All rights of reproduction in any form mcwd.
468
NOTES s3 =
h;/(6c’)
has beeen shown (4) to eliminate the Ak = +6 matrix elements. With this choice 5 = hi in Eq. (5) and in the rotational Hamiltonian Hi of Ref. (1) the term A,(/$ + J!) is replaced by the terms (f
+
f,J2)[Jz,
J:
+
P],
+
c&,
J:
+
511,
and c, c,, eKare given by Eqs. (23)-(25) of Ref. (4). In conclusion, insofar as fitting of the observed spectra is concerned, the only suggested change as compared to Ref. ( 1) is the above given modification of H$ for quasi-spherical tops. If the determined combinations of molecular parameters are used to calculate the molecular force fields it should be kept in mind that the values of some parameters appearing in Eqs. (lo), (I 7)-( 19) of Ref. (I) are modified according to Eq. (5) with Cgiven either by Eq. (6) or equal to hi. REFERENCES 1. E.I. LOBODENKO, O.N. SULAKSHINA, V.I. FEREVALOV,AND V.G. TYUTEREV, J.Mol. Spectrosc. 126, 159-170 (1987). 2. J. K. G. WATSON, in “Vibrational Spectra and Structure” (J. During, Ed.), Vol. 6., pp. l-89, Elsevier, Amsterdam, 1977. 3. M. R. ALIEV AND J. K. G. WATSON, in “Molecular Spectroscopy: Modem Research” (K. Narahari Rao, Ed.), Vol. III, pp. l-67, Academic Press, San Diego, 1985. 4. K. SARKA, J.Mol. Spec~rosc. 133 (1989). KAMILSARKA
Departmentof PhysicalChemistry, Faculty of Pharmacy, Comenius University, Odboj&ovIO, 832 32 Bratislava,Czechoslovakia ReceivedAugust 29, 1988