A new procedure is presented for building a general kinetic energy operator expressed as a polynomial series expansion of symmetry-adapted curvilinear coordinates for semirigid polyatomic molecules. As a starting point, the normal-mode Watson kinetic energy part is considered and then transformed into its curvilinear counterpart. An Eckart molecular fixed-frame is thus implicitly used. To this end, we exploit symmetry at all stages of the calculation and show how group-theoretically based methods and Γ-covariant tensors help properly invert nonlinear polynomials for the coordinate changes. Such a linearization procedure could also be useful in different contexts. Unlike the usual normal mode approach, the potential part initially expressed in curvilinear coordinates is not transformed in this work, making convergence of the Hamiltonian expansion generally faster. For dimensionality reduction, the final curvilinear kinetic and potential parts are expanded in terms of irreducible tensor operators when doubly and triply degenerate vibrations are involved. The procedure proposed here is general and can be applied to arbitrary Abelian and non-Abelian point groups. Illustrative examples will be given for the H2S (C2v), H2CO (C2v), PH3 (C3v), and SiH4 (Td) molecules.

1.
X.
Chapuisat
,
A.
Nauts
, and
G.
Durand
,
Chem. Phys.
56
,
91
(
1981
).
2.
N.
Handy
,
Mol. Phys.
61
,
207
(
1987
).
3.
R.
Islampour
and
S.
Lin
,
J. Mol. Spectrosc.
147
,
1
(
1991
).
4.
X.
Chapuisat
and
C.
Iung
,
Phys. Rev. A
45
,
6217
(
1992
).
5.
A.
Császár
and
N.
Handy
,
J. Chem. Phys.
102
,
3962
(
1995
).
6.
Y.
Justum
,
M.
Menou
,
A.
Nauts
, and
X.
Chapuisat
,
Chem. Phys.
223
,
211
(
1997
).
7.
H.
Wei
and
T.
Carrington
, Jr.
,
J. Chem. Phys.
107
,
9493
(
1997
).
8.
H.
Wei
and
T.
Carrington
, Jr.
,
Chem. Phys. Lett.
287
,
289
(
1998
).
9.
F.
Gatti
and
C.
Iung
,
Phys. Rep.
484
,
1
(
2009
).
10.
J.
Frederick
and
C.
Woywod
,
J. Chem. Phys.
111
,
7255
(
1999
).
11.
J.
Makarewicz
and
A.
Skalozub
,
Chem. Phys. Lett.
306
,
352
(
1999
).
12.
M.
Mladenovic
,
J. Chem. Phys.
112
,
1070
(
2000
).
13.
M.
Mladenovic
,
J. Chem. Phys.
112
,
1082
(
2000
).
14.
T.
Carrington
, Jr.
,
J. Chem. Phys.
113
,
7097
(
2000
).
15.
D.
Schwenke
,
J. Chem. Phys.
118
,
10431
(
2003
).
16.
S. N.
Yurchenko
,
W.
Thiel
, and
P.
Jensen
,
J. Mol. Spectrosc.
245
,
126
(
2007
).
17.
R.
Islampour
,
Mol. Phys.
101
,
2489
(
2003
).
18.
J.
Pesonen
,
J. Chem. Phys.
128
,
044319
(
2008
).
19.
K.
Sadri
,
D.
Lauvergnat
,
F.
Gatti
, and
H. D.
Meyer
,
J. Chem. Phys.
136
,
234112
(
2012
).
20.
X.-G.
Wang
and
T.
Carrington
, Jr.
,
J. Chem. Phys.
138
,
104106
(
2013
).
21.
C.
Fábri
,
E.
Mátyus
, and
A.
Császár
,
Spectrochim. Acta, Part A
119
,
84
(
2014
).
22.
A.
Protasevich
and
A.
Nikitin
,
J. Chem. Phys.
116
,
000044
(
2018
).
23.
W.
Kopp
and
K.
Leonhard
,
J. Chem. Phys.
145
,
234102
(
2016
).
24.
P. R.
Bunker
and
P.
Jensen
,
Molecular Symmetry and Spectroscopy
(
NRC-CNRC
,
Ottawa
,
1998
).
25.
B.
Podolsky
,
Phys. Rev.
32
,
812
(
1928
).
26.
C.
Eckart
,
Phys. Rev.
47
,
552
(
1935
).
27.
J. D.
Louck
and
H. W.
Galbraith
,
Rev. Mod. Phys.
48
,
69
(
1976
).
28.
L. C.
Biederharn
and
J. D.
Louck
,
Angular Momentum in Quantum Physics
(
Cambridge University Press
,
New York
,
1985
).
29.
A.
Yachmenev
and
S.
Yurchenko
,
J. Chem. Phys.
143
,
014105
(
2015
).
30.
A.
Dymarsky
and
K.
Kudin
,
J. Chem. Phys.
122
,
124103
(
2005
).
31.
S.
Krasnoshchekov
,
E.
Isayeva
, and
N.
Stepanov
,
J. Chem. Phys.
140
,
154104
(
2014
).
32.
V.
Szalay
,
J. Chem. Phys.
140
,
234107
(
2014
).
33.
T.
Szidarovszky
,
C.
Fábri
, and
A. G.
Császár
,
J. Chem. Phys.
136
,
174112
(
2012
).
34.
J. K. G.
Watson
,
Mol. Phys.
15
,
479
(
1968
).
35.
E. B.
Wilson
,
J. C.
Decius
, and
P. C.
Cross
,
Molecular Vibrations
(
Dover Publications
,
1955
).
36.
M.
Ndong
,
A.
Nauts
,
L.
Joubert-Doriol
,
H. D.
Meyer
,
F.
Gatti
, and
D.
Lauvergnat
,
J. Chem. Phys.
139
,
204107
(
2013
).
37.
D.
Lauvergnat
and
A.
Nauts
,
J. Chem. Phys.
116
,
8560
(
2002
).
38.
D.
Lauvergnat
,
J.
Luis
,
B.
Kirtman
,
H.
Reis
, and
A.
Nauts
,
J. Chem. Phys.
144
,
084116
(
2016
).
39.
J.
Nissinen
,
K.
Liu
,
R.-J.
Slager
,
K.
Wu
, and
J.
Zaanen
,
Phys. Rev. E
94
,
022701
(
2016
).
40.
V.
Szalay
,
J. Chem. Phys.
142
,
174107
(
2015
).
41.
V.
Szalay
,
J. Chem. Phys.
146
,
124107
(
2017
).
42.
P.
Jensen
,
Comput. Phys. Rep.
1
,
1
(
1983
).
43.
A. R.
Hoy
,
I. M.
Mills
, and
G.
Strey
,
Mol. Phys.
24
,
1265
(
1972
).
44.
M.
Rey
,
A. V.
Nikitin
, and
V. G.
Tyuterev
,
J. Chem. Phys.
136
,
244106
(
2012
).
45.
M.
Rey
,
A. V.
Nikitin
, and
V. G.
Tyuterev
,
J. Chem. Phys.
141
,
044316
(
2014
).
46.
M.
Rey
,
A. V.
Nikitin
, and
V. G.
Tyuterev
,
Phys. Chem. Chem. Phys.
15
,
10049
(
2013
).
47.
M.
Rey
,
A. V.
Nikitin
, and
V. G.
Tyuterev
,
J. Phys. Chem. A
119
,
4763
(
2015
).
48.
M.
Rey
,
I.
Chizhmakova
,
A.
Nikitin
, and
V.
Tyuterev
,
Phys. Chem. Chem. Phys.
20
,
21008
(
2018
).
49.
A.
Nauts
and
X.
Chapuisat
,
Mol. Phys.
55
,
1287
(
1985
).
50.
G. O.
Sørensen
,
Top. Curr. Chem.
82
,
97
(
1979
).
51.
E.
Mátyus
,
G.
Czakó
, and
A. G.
Császár
,
J. Chem. Phys.
130
,
134112
(
2009
).
52.
A.
Nauts
and
D.
Lauvergnat
,
Mol. Phys.
116
,
3701
(
2018
).
53.
A.
Nikitin
,
M.
Rey
, and
V.
Tyuterev
,
J. Chem. Phys.
142
,
094118
(
2015
).
54.
K.
Makino
and
M.
Berz
,
Nucl. Instrum. Methods Phys. Res., Sect. A
558
,
346
(
2006
).
55.
V.
Tyuterev
,
S.
Tashkun
,
M.
Rey
,
R.
Kochanov
,
A.
Nikitin
, and
T.
Delahaye
,
J. Phys. Chem. A
117
,
13779
(
2013
).
56.
M.
Rey
,
A. V.
Nikitin
, and
V. G.
Tyuterev
,
Mol. Phys.
108
,
2121
(
2010
).
57.
V.
Tapia
,
J. Phys. A: Math. Theor.
40
,
5525
(
2007
).
58.
M.
Forger
,
J. Math. Phys.
39
,
1107
(
1998
).
59.
P.
Cassam-Chenaï
,
G.
Dhont
, and
F.
Patras
,
J. Math. Chem.
53
,
58
(
2014
).
60.
G.
Dhont
,
F.
Patras
, and
B.
Zhilinski
,
J. Phys. A: Math. Theor.
48
,
035201
(
2015
).
61.
V.
Boujut
and
F.
Michelot
,
J. Mol. Spectrosc.
173
,
237
(
1995
).
62.
V.
Boujut
and
F.
Michelot
,
J. Mol. Spectrosc.
184
,
257
(
1997
).
63.
F.
Michelot
and
M.
Rey
,
Eur. Phys. J. D
30
,
181
(
2004
).
64.
F.
Michelot
and
M.
Rey
,
Eur. Phys. J. D
33
,
357
(
2005
).
65.
J. P.
Champion
,
M.
Loëte
, and
G.
Pierre
, in
Spherical Top Spectra
, edited by
K. N.
Rao
and
A.
Weber
(
Academic Press
,
San Diago
,
1992
).
66.
V.
Boudon
,
J. P.
Champion
,
T.
Gabard
,
M.
Loëte
,
F.
Michelot
,
G.
Pierre
,
M.
Rotger
,
C.
Wenger
, and
M.
Rey
,
J. Mol. Spectrosc.
228
,
620
(
2004
).
67.
A. V.
Nikitin
,
J. P.
Champion
, and
V. G.
Tyuterev
,
J. Mol. Spectrosc.
182
,
72
(
1997
).
68.
A. V.
Nikitin
,
M.
Rey
,
J. P.
Champion
, and
V.
Tyuterev
,
J. Quant. Spectrosc. Radiat. Transfer
113
,
1034
(
2012
).
69.
M.
Rey
, “
Vibrational and rotational irreductible tensor operators for Abelian and non-Abelian point groups
” (unpublished).
70.
J.-Q.
Chen
,
J.
Ping
, and
F.
Wang
,
Group Representation Theory for Physicists
, 2nd ed. (
World Scientific
,
Singapore
,
2002
).
71.
M.
Rey
,
F.
Michelot
, and
V.
Tyuterev
,
Phys. Rev. A
78
,
022511
(
2008
).
72.
R.
Glauber
,
Phys. Rev.
131
,
2766
(
1963
).
73.
H.
Yuen
,
Phys. Rev. A
13
,
2226
(
1976
).
74.
M.
Nieto
and
D.
Truax
,
Phys. Rev. Lett.
71
,
2843
(
1993
).
75.
C.
Caves
,
Phys. Rev. D
23
,
1693
(
1981
).
76.
D.
Stoler
,
Phys. Rev. D
1
,
3217
(
1970
).
77.
D.
Stoler
,
Phys. Rev. D
4
,
1925
(
1971
).
78.
M.
Rey
,
V.
Boudon
,
C.
Wenger
,
G.
Pierre
, and
B.
Sartakov
,
J. Mol. Spectrosc.
219
,
313
(
2003
).
79.
O. L.
Polyansky
,
P.
Jensen
, and
J.
Tennyson
,
J. Mol. Spectrosc.
178
,
184
(
1996
).
80.
A.
Yachmenev
,
S. N.
Yurchenko
,
P.
Jensen
, and
W.
Thiel
,
J. Chem. Phys.
134
,
244307
(
2011
).
81.
A. V.
Nikitin
,
F.
Holka
,
V. G.
Tyuterev
, and
J.
Fremont
,
J. Chem. Phys.
130
,
244312
(
2009
).
82.
A.
Owens
,
S.
Yurchenko
,
A.
Yachmenev
, and
W.
Thiel
,
J. Chem. Phys.
143
,
244317
(
2015
).
83.
A. V.
Nikitin
,
M.
Rey
, and
V. G.
Tyuterev
,
J. Mol. Spectrosc.
305
,
40
(
2014
).
84.
D.
Viglaska
,
M.
Rey
,
A.
Nikitin
, and
V.
Tyuterev
,
J. Chem. Phys.
149
,
174305
(
2018
).
85.
D.
Viglaska
,
M.
Rey
,
T.
Delahaye
, and
A.
Nikitin
,
J. Quant. Spectrosc. Radiat. Transfer
230
,
142
(
2019
).
86.
T.
Johnson
,
R.
Sams
, and
S.
Sharpe
,
Proc. SPIE
5269
,
159
(
2004
).
87.
J.
Hougen
,
P.
Bunker
, and
J.
Johns
,
J. Mol. Spectrosc.
34
,
136
(
1970
).
88.
K.
Chubb
,
P.
Jensen
, and
S.
Yurchenko
,
Symmetry
10
,
137
(
2018
).
89.
I.
Nazrul
,
Tensors and their Applications
(
New Age International Publisher
,
New Delhi
,
2006
).
90.
F. D.
Murnaghan
,
Vector Analysis and the Theory of the Relativity
(
The John Hopkins Press
,
1922
).
You do not currently have access to this content.