Inelastic collisions of cold polar molecules in nonparallel electric
and magnetic fields
E. Abrahamsson, T. V. Tscherbul,
a

and R. V. Krems
Department of Chemistry, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada

Received 30 March 2007; accepted 21 May 2007; published online 23 July 2007
The authors present a detailed study of low-temperature collisions between CaD molecules and He
atoms in superimposed electric and magnetic fields with arbitrary orientations. Electric fields do not
interact with the electron spin of the molecules directly but modify their rotational structure and,
consequently, the spin-rotation interactions. The authors examine molecular Stark and Zeeman
energy levels as functions of the angle between the fields and show that rotating fields may induce
and shift avoided crossings between the Zeeman levels of the rotationally ground and rotationally
excited states of the molecule. The dynamics of molecular collisions are extremely sensitive to
external fields near these avoided crossings and it is shown that molecular collisions may be
controlled by varying both the strength and the relative orientation of the fields. The effects observed
in this study are due to interactions of the isolated molecules with external fields so the conclusions
should be relevant for collisions of molecules with other atoms or collisions of molecules with each
other. This study demonstrates that electric fields may be used to enhance or suppress spin-rotation
interactions in molecules. The spin-rotation interactions induce nonadiabatic couplings between
states of different total spins in systems of two open-shell species and it is suggested that electric
fields might be used for controlling nonadiabatic spin transitions and spin-forbidden chemical
reactions of cold molecules in a magnetic trap. © 2007 American Institute of Physics.
DOI: 10.1063/1.2748770
I. INTRODUCTION
The dynamics of molecules in external electromagnetic
fields has recently been a subject of many experimental and
theoretical studies.
1–9
Electric fields may be used to orient
and align molecules in the space-fixed coordinate frame.
7
Orienting and aligning molecules allow for direct measure-
ments of the anisotropy of intermolecular interactions.
10–12
Interactions of molecules with dc and laser fields have been
exploited in the experiments on selective bond breaking and
rearrangement,
13
molecular tomography,
14,15
and the design
of a molecular synchrotron.
16,17
Precise spectroscopic mea-
surements of molecular energy levels in superimposed elec-
tric and magnetic fields may provide sensitive tests of fun-
damental symmetries of nature.
18,19
A major thrust of recent
research in molecular physics has been to produce dense en-
sembles of cold and ultracold molecules.
20
Electric and mag-
netic traps have been designed to confine and isolate cold
molecules prepared in particular Zeeman or Stark energy
levels.
21–24
Experiments with cold molecules may yield de-
tailed information on intermolecular interactions and allow
for unprecedented control of inelastic collisions and chemi-
cal reactions.
18,19
For example, it was demonstrated by Mc-
Carthy et al.
3
that even extremely weak molecule-field inter-
actions such as the interaction of the nuclear spin with
magnetic fields can be used to distort molecular trajectories
in a slow molecular beam. Staanum et al.
25
and Zahzam et
al.
26
have recently reported the first experiments on ultracold
chemical reactions of alkali metal atoms with alkali metal
diatomic molecules in an optical trap. Jung et al.
27
proposed
to tune the threshold fragmentation of cold SO
2
molecules
with electric fields. Gilijamse et al.
28
carried out a crossed-
beam collision experiment with slow OH molecules pro-
duced in a Stark decelerator. All these studies are generating
an increasing demand for the development of rigorous theo-
ries for accurate simulations of molecular collisions in the
presence of external fields.
Electromagnetic fields modify molecular energy levels
and may induce inelastic transitions in collisions of
molecules.
19
In particular, spin-flipping transitions between
molecular Zeeman levels —spin relaxation— may occur in a
magnetic field.
19,29–32
The spin relaxation of cold
2

mol-
ecules was first observed in 1998 by Weinstein et al.
33
who
cooled CaH

2

 molecules in a cryogenic cell filled with
3
He buffer gas and loaded them into a magnetic trap.
Collision-induced Zeeman transitions have later been studied
for a variety of molecules in several experiments.
34,35
Spin
relaxation produces molecules in high-field-seeking Zeeman
states, which leads to trap loss and heating.
23
It is therefore
important to find mechanisms for suppressing spin relaxation
in collisions of cold molecules in order to increase the num-
ber of trapped molecules in buffer-gas loading
23,33
and
evaporative cooling experiments.
36
An adequate theoretical description of molecular align-
ment and cold collisions should be based on quantum me-
chanical calculations of dynamics in the presence of external
electric and magnetic fields. The methodology for quantum
scattering calculations in magnetic fields was developed by
a
Electronic mail: timur@chem.ubc.ca
THE JOURNAL OF CHEMICAL PHYSICS 127, 044302
2007
0021-9606/2007/1274
/044302/10/$23.00 © 2007 American Institute of Physics127, 044302-1

Volpi and Bohn
29
and by Krems and Dalgarno.
32
Bohn and
co-workers calculated cross sections for collisions of two
diatomic molecules in an electric field
37,38
and suggested the
possibility of creating novel field-linked, long-range molecu-
lar states.
39
González-Martínez and Hutson
40
and Lara et
al.
41,42
reported extensive calculations of nonreactive atom-
molecule collisions in a magnetic field. We have recently
examined the influence of combined electric and magnetic
fields on spin relaxation
43
and the rotationally inelastic
scattering
44
of polar molecules at low temperatures. In par-
ticular, we demonstrated that dc electric fields can be used to
suppress the spin relaxation of cold molecules and that the
dynamics of molecules may be extremely sensitive to exter-
nal fields when the Zeeman levels of rotationally excited and
rotationally ground manifolds intersect.
43
Here, we extend the work in Ref. 43 to present in detail
the theory of atom-molecule collisions in nonparallel electric
and magnetic fields. We examine the Zeeman and Stark en-
ergy levels of
2

molecules as functions of the angle be-
tween the fields and show that the positions and number of
the avoided crossings between rotationally ground and rota-
tionally excited states depend on the relative orientation of
the fields. The cross sections for spin relaxation near the
crossings are therefore very sensitive to the angle between
the fields. We calculate the rates for both cold and ultracold
collisions and discuss the possibility of external field control
of cold molecules. Finally, we suggest that spin-forbidden
chemical reactions of open-shell atoms in the
2
S state with
2

molecules in a magnetic trap may be stimulated or sup-
pressed by electric fields.
II. THEORY
The Hamiltonian for a
2

polar molecule in superim-
posed electric and magnetic fields can be written as
H
mol
=−
1
2
m
r
d
2
dr
2
r +
N
2

r
ˆ

2
m
r
2
+ V
r + S · N − E · d
+2
B
B · S ,
1
where 
m
is the reduced mass and V
r is the potential en-
ergy function of the diatomic molecule. The coupling be-
tween the rotational
N and spin
S angular momenta is
determined by the spin-rotation interaction constant . The
hat over the symbol denotes the unit vector. The interaction
with electric fields can be written as −E ·d=−Ed cos ,
where  is the angle between the electric field direction E
ˆ
and the molecular axis rˆ, E is the electric field strength, and
d is the electric dipole moment of the molecule. Using the
spherical harmonic addition theorem,
46
this term can be re-
written as a sum over products of spherical harmonics,
− Ed cos  =−Ed
4
3

q
Y
1q
*

r
ˆ
Y
1q

E
ˆ
 .
2
The interaction of the electron spin with the magnetic field B
is given by 2
B
B ·S, where 
B
is the Bohr magneton. We
orient the space fixed quantization axis Z along the magnetic
field direction so that only the Z component of the vector B
is nonzero and the last term in Eq.
1 reduces to 2
B
BS
Z
.
The spin-rotation interaction constant  for CaD is
0.021 cm
−1
,
47
and the dipole moment d is 2.94 D.
48
The Hamiltonian for the atom-molecule complex has the
following form
29,32
H =−
1
2R
d
2
dR
2
R +


2

R
ˆ

2R
2
+ V
R,r, + H
mol
,
3
where  is the reduced mass of the CaD–He system, R is the
distance between the center of mass of the diatomic molecule
and the atom, and
is the orbital angular momentum for the
collision. The interaction potential V
R ,r , vanishes as
R→. We used a recent ab initio potential energy surface of
Balakrishnan et al.
49
and fixed the interatomic distance r at
its equilibrium distance of 2.008 Å, which is a good approxi-
mation for collisions at low energies.
50
Following the work of Krems and Dalgarno,
32
we ex-
pand the total wave function of the collision system in a set
of uncoupled space-fixed basis functions,
NM
N
SM
S
M

 ,
4
where M
N
, M
S
, and M

denote the projections of N, S, and

on the magnetic field axis.
32
When the electric and magnetic
fields are parallel or antiparallel, the projection of the total
angular momentum M =M
N
+M
S
+M

is conserved, and the
scattering calculations can be carried out in a cycle over
M.
32,44
If the electric and magnetic fields are rotated, the
electric field couples states with different M
N
, and the pro-
jection of the total angular momentum M is no longer a good
quantum number. The R-dependent expansion coefficients
F
NM
N
SM
S
M


R of the total wave function in basis set
4 are
obtained by solving a set of close-coupled equations,
32,44

d
2
dR
2
+2E
tot
−

 +1
R
2

F
NM
N
SM
S
M


R
=2

N


,M
N

,M
S

,


,M




NM
N
SM
S
M

V
R,r,
+ H
mol
N


M
N

SM
S




M


F
N


M
N


SM
S





M




R ,
5
where E
tot
is the total energy of the system. The expressions
for the matrix elements of the operators V
R ,r , and 2BS
Z
can be found in Ref. 32. The matrix of the interaction with
electric fields
2

NM
N
− Ed cos N


M
N


=−Ed
4
3

q
Y
1q

E
ˆ

NM
N
Y
1q
*

r
ˆ
N


M
N


6
is diagonal in S, M
S
, , and M

quantum numbers. The
evaluation of the integrals in Eq.
6 provides a general ex-
pression for the matrix elements of the interaction with elec-
tric fields of arbitrary orientation,
044302-2 Abrahamsson, Tscherbul, and Krems J. Chem. Phys. 127, 044302 2007