The validity range of the Born-Oppenheimer (BO) approximation is studied with respect to the variation of the mass (m) of negatively charged particle by substituting an electron (e) with muon (μ) and antiproton (p¯) in hydrogen molecule cation. With the use of semiclassical quantization applied to these (ppe), (ppμ), and (ppp¯) under a constrained geometry, we estimate the energy difference of the non-BO vibronic ground state from the BO counterpart. It is found that the error in the BO approximation scales to the power of 32 to the mass of negative particles, that is, m1.5. The origin of this clear-cut relation is analyzed based on the original perturbation theory due to Born and Oppenheimer, with which we show that the fifth order term proportional to m54 is zero and thereby the first correction to the BO approximation should arise from the sixth order term that is proportional to m64. Therefore, the validity range of the Born-Oppenheimer approximation is wider than that often mistakenly claimed to be proportional to m14.

1.
M.
Born
and
R.
Oppenheimer
,
Ann. Phys.
84
,
457
(
1927
).
2.
M. S.
Child
,
Molecular Collison Theory
(
Academic
,
New York
,
1974
);
M. S.
Child
,
Semiclassical Mechanics with Molecular Approximations
(
Claredon
,
Oxford
,
1991
).
3.
W. H.
Miller
,
Adv. Chem. Phys.
25
,
69
(
1974
).
4.
J. B.
Delos
,
Rev. Mod. Phys.
53
,
287
(
1981
).
5.
H.
Nakamura
,
Nonadiabatic Transition
(
World Scientific
,
Singapore
,
2002
).
6.
A. W.
Jasper
,
B. K.
Kendrick
,
C. A.
Mead
, and
D. G.
Truhlar
, in
Modern Trends in Chemical Reaction Dynamics
, edited by
X.
Yang
and
K.
Liu
(
World Scientific
,
Singapore
,
2004
), p.
329
.
7.
Y.
Arasaki
,
K.
Takatsuka
,
K.
Wang
, and
V.
McKoy
,
Phys. Rev. Lett.
90
,
248303
(
2003
);
[PubMed]
Y.
Arasaki
,
K.
Takatsuka
,
K.
Wang
, and
V.
McKoy
,
J. Chem. Phys.
119
,
7913
(
2003
).
8.
M.
Hentschel
,
R.
Kienberger
,
C.
Spielmann
,
G. A.
Reider
,
N.
Milosevic
,
T.
Brabec
,
C.
Corkum
,
U.
Heinzmann
,
M.
Drescher
, and
F.
Krausz
,
Nature (London)
414
,
509
(
2001
).
9.
H.
Niikura
,
F.
Légaré
,
R.
Hasbani
,
A. D.
Bandrauk
,
M. Y.
Ivanov
,
D. M.
Villeneuve
, and
P. B.
Corkum
,
Nature (London)
417
,
917
(
2002
).
10.
A.
Baltuška
,
Th.
Udem
,
M.
Uiberacker
 et al,
Nature (London)
421
,
611
(
2003
).
11.
M.
Amano
and
K.
Takatsuka
,
J. Chem. Phys.
122
,
084113
(
2005
).
13.
K.
Yagi
and
K.
Takatsuka
,
J. Chem. Phys.
123
,
224103
(
2005
).
14.
K.
Sakimoto
,
Phys. Rev. A
71
,
062704
(
2005
).
15.
K.
Sakimoto
,
Phys. Rev. A
69
,
042710
(
2004
).
17.
S.
Kais
and
Q.
Shi
,
Phys. Rev. A
62
,
060502
(
2000
).
18.
O. I.
Kartavtsev
,
A. V.
Malykh
, and
V. P.
Permyakov
,
Phys. Rev. A
70
,
022504
(
2004
).
19.
S.
Kilic
,
J.-P.
Karr
, and
L.
Hilico
,
Phys. Rev. A
70
,
042506
(
2004
).
20.
E.
Lindroth
,
J.
Wallenius
, and
S.
Jonsell
,
Phys. Rev. A
68
,
032502
(
2003
).
21.
M. R.
Eskandari
and
R.
Gheisari
,
Phys. Rev. A
70
,
032505
(
2004
).
22.
D.
Bakalov
,
K.
Bakalova
,
V.
Korobov
,
H. J.
Monkhorst
, and
I.
Shimamura
,
Phys. Rev. A
57
,
3370
(
1998
).
23.
M. R.
Harston
,
S.
Hara
,
Y.
Kino
,
I.
Shimamura
,
H.
Sato
, and
M.
Kamimura
,
Phys. Rev. A
56
,
2685
(
1997
).
24.
Y.
Kino
,
M. R.
Harston
,
I.
Shimamura
,
E. A. F.
Armour
, and
M.
Kamimura
,
Phys. Rev. A
52
,
870
(
1995
).
25.
I.
Shimamura
,
Phys. Rev. A
40
,
4863
(
1989
).
26.
M. C.
Fujiwara
,
A.
Adamczak
,
J. M.
Bailey
 et al,
Phys. Rev. Lett.
85
,
1642
(
2000
).
27.
T. A.
Porcelli
,
A.
Adamczak
,
J. M.
Bailey
 et al,
Phys. Rev. Lett.
86
,
3763
(
2001
).
28.
N.
Kawamura
,
K.
Nagamine
,
T.
Matsuzaki
 et al,
Phys. Rev. Lett.
90
,
043401
(
2003
).
29.
A.
Toyoda
,
K.
Ishida
,
K.
Shimomura
,
S. N.
Nakamura
,
Y.
Matsuda
,
W.
Higemoto
,
T.
Matsuzaki
, and
K.
Nagamine
,
Phys. Rev. Lett.
90
,
243401
(
2003
).
30.
N.
Kuroda
,
H. A.
Torii
,
K.
Yoshiki Franzen
 et al,
Phys. Rev. Lett.
94
,
023401
(
2005
).
31.
A. K.
Bhatia
,
Phys. Rev. A
58
,
2787
(
1998
).
32.
V. I.
Korobov
,
Phys. Rev. A
61
,
064503
(
2000
).
33.
D. M.
Bishop
,
A. M.
Frolov
, and
V. H.
Smith
, Jr.
,
Phys. Rev. A
51
,
3636
(
1995
).
34.
A. M.
Frolov
,
Phys. Rev. A
57
,
2436
(
1998
).
35.
A. M.
Frolov
,
Phys. Rev. A
59
,
4270
(
1999
).
36.
A. M.
Frolov
,
Phys. Rev. A
67
,
064501
(
2003
).
37.
A. M.
Frolov
,
Phys. Rev. E
64
,
036704
(
2001
).
38.
K.
Sohlberg
,
R. E.
Tuzun
,
B. G.
Sumpter
, and
D. W.
Noid
,
Phys. Rev. A
57
,
906
(
1998
).
39.
Y.
Duan
,
C.
Browne
, and
J.-M.
Yuan
,
Phys. Rev. A
59
,
238
(
1999
).
40.
G.
Tanner
,
K.
Richter
, and
J.-M.
Rost
,
Rev. Mod. Phys.
72
,
497
(
2000
).
41.
M. P.
Strand
and
W. P.
Reinhardt
,
J. Chem. Phys.
70
,
3812
(
1979
).
42.
J.
Müller
,
J.
Burgdörfer
, and
D. W.
Noid
,
J. Chem. Phys.
103
,
4985
(
1995
).
43.
M.
Tabor
,
Chaos and Integrability in Nonlinear Dynamics
(
Wiley
,
New York
,
1989
);
M. C.
Gutzwiller
,
Chaos in Classical and Quantum Mechanics
(
Springer
,
New York
,
1990
);
L. E.
Reichl
,
The Transition to Chaos
(
Springer
,
New York
,
1992
);
K.
Nakamura
,
Quantum Chaos
(
Cambridge University Press
,
Cambridge
,
1993
).
44.
K.
Hotta
and
K.
Takatsuka
,
J. Phys. A
36
,
4785
(
2003
).
45.
S.
Takahashi
and
K.
Takatsuka
,
Phys. Rev. A
70
,
052103
(
2004
).
46.
P.
Kustaanheimo
and
E.
Stiefel
,
J. Reine Angew. Math.
218
,
204
(
1965
).
47.
S. J.
Aarseth
and
K.
Zare
,
Celest. Mech.
10
,
185
(
1974
).
48.
L. D.
Landau
and
E. M.
Lifshitz
,
Mechanics
(
Addison-Wesley
,
New York
,
1960
).
49.
H.
Ushiyama
and
K.
Takatsuka
,
J. Chem. Phys.
122
,
224112
(
2005
).
50.
A.
Inoue-Ushiyama
and
K.
Takatsuka
,
Phys. Rev. Lett.
78
,
1404
(
1997
);
A.
Inoue-Ushiyama
and
K.
Takatsuka
,
Phys. Rev. A
59
,
3256
(
1999
).
51.
W. H.
Miller
,
J. Chem. Phys.
53
,
3578
(
1970
).
52.
W. H.
Miller
,
J. Phys. Chem. A
105
,
2942
(
2001
).
53.
Y.
Zhao
and
W. H.
Miller
,
J. Chem. Phys.
117
,
9605
(
2002
).
54.
E. J.
Heller
,
J. Chem. Phys.
94
,
2723
(
1991
).
55.
M. A.
Sepúlveda
and
E. J.
Heller
,
J. Chem. Phys.
101
,
8004
(
1994
);
M. A.
Sepúlveda
and
F.
Grossmann
,
Adv. Chem. Phys.
96
,
191
(
1996
).
56.
G.
Campolieti
and
P.
Brumer
,
Phys. Rev. A
50
,
997
(
1994
);
[PubMed]
D.
Provost
and
P.
Brumer
,
Phys. Rev. Lett.
74
,
250
(
1995
).
[PubMed]
57.
K. G.
Kay
,
J. Chem. Phys.
100
,
4377
(
1994
);
K. G.
Kay
,
J. Chem. Phys.
100
,
4432
(
1994
).
58.
H.
Goldstein
,
Classical Mechanics
(
Addison-Wesley
,
New York
,
1980
).
59.
K.
Takatsuka
,
Phys. Rev. E
64
,
016224
(
2001
).
60.
S.
Takahashi
and
K.
Takatsuka
,
Phys. Rev. A
69
,
022110
(
2004
).
61.
H.
Ushiyama
,
Y.
Arasaki
, and
K.
Takatsuka
,
Chem. Phys. Lett.
346
,
169
(
2001
).
62.
T.
Yanao
and
K.
Takatsuka
,
J. Chem. Phys.
120
,
8924
(
2004
).
63.
M.
Klein
,
A.
Martinez
,
R.
Seiler
, and
X. P.
Wang
,
Commun. Math. Phys.
143
,
607
(
1992
).
64.
G. A.
Hagedorn
,
Commun. Math. Phys.
116
,
23
(
1988
).
You do not currently have access to this content.