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Mutual Capture of Dipolar Molecules at Low and Very Low Energies. I. Approximate Analytical Treatment† E. E. Nikitin‡,§ and J. Troe*,§,⊥ Schulich Faculty of Chemistry, Technion s Israel Institute of Technology, Haifa 32000, Israel, Max-Planck-Institut fu¨r Biophysikalische Chemie, Am Fassberg 11, Go¨ttingen D-37077, Germany, and Institut fu¨r Physikalische Chemie, UniVersita¨t Go¨ttingen, Tammannstrasse 6, Go¨ttingen D-37077, Germany ReceiVed: March 8, 2010; ReVised Manuscript ReceiVed: May 12, 2010
Approximate analytical expressions are derived for the low-energy rate coefficients of capture of two identical dipolar polarizable rigid rotors in their lowest nonresonant (j1 ) 0 and j2 ) 0) and resonant (j1 ) 0,1 and j2 ) 1,0) states. The considered range extends from the quantum, ultralow energy regime, characterized by s-wave capture, to the classical regime described within fly wheel and adiabatic channel approaches, respectively. This is illustrated by the table of contents graphic (available on the Web) that shows the scaled rate coefficients for the mutual capture of rotors in the resonant state versus the reduced wave vector between the Bethe zero-energy (left arrows) and classical high-energy (right arrow) limits for different ratios δ of the dipole-dipole to dispersion interaction. 1. Introduction The dynamics of the formation of complexes in collisions of molecules at low energies is of general interest in the context of the physics of cold molecules (see, e.g., ref 1). The most general approach in the calculation of the cross section for complex formation consists of the solution of a half-collision problem, that is, in the determination of the appropriate scattering matrix calculated with standard boundary conditions at large separation between the partners and with absorbing boundary conditions at the surface of the complex (the statistical close-coupled method2-5). Within this description, the capture is accompanied by a host of inelastic processes occurring on the way to the complex boundary. This complicated problem would considerably simplify for collisions at very low energies if the relative motion were adiabatic with respect to the internal modes (electronic, vibrational, and rotational) of the partners. Then, one would first calculate adiabatic states of the colliding pair and afterward consider the uncoupled motion of the partners across the adiabatic potentials. This is the standard method of the adiabatic channel (AC) approach6 or its variants.7-11 One may question several of the assumptions underlying the AC approach, such as the neglect of nonadiabatic interactions between those states which are degenerate (or quasi-degenerate) in the free partners. The states in question are, for instance, those components that arise from the lifting of the degeneracy of the manifold of certain rotational states of the partners. Within the AC approach, the AC states, differing in the projection of the intrinsic angular momentum onto the collision axis, are assumed to be uncoupled. Therefore, the scattering matrix can be characterized, besides the exact quantum numbers J and I (total angular momentum and total parity, respectively) and asymptotic quantum numbers n,V, and j (electronic, vibrational, and rotational quantum numbers, respectively), also by a good quantum number m (the projection of the intrinsic angular †
Part of the “Reinhard Schinke Festschrift”. * To whom correspondence should be addressed. E-mail:
[email protected]. ‡ Technion s Israel Institute of Technology. § Max-Planck-Institut fu¨r Biophysikalische Chemie. ⊥ Universita¨t Go¨ttingen.
momentum onto the collision axis). The effect of the violation of this conservation of m on the probability of complex formation was discussed in a series of articles on ion-molecule capture.12-14 While the influence on the m-averaged cross section is only minor at energies where the relative motion approaches classical behavior,12,13 it substantially changes the cross sections in the quantum regime.14 This is related to the fact that, though at very low energies the radial motion conforms with the adiabatic hypothesis, the rotational motion for rotationally excited partners is influenced by Coriolis interaction. Because of the nonzero values of the angular momenta, this induces nonvanishing Coriolis coupling even in the limit of zero collision energy. This problem was discussed in ref 14, where it was shown that the nonadiabatic rotational coupling can well be accounted for by changing the AC representation into an axially nonadiabatic channel (ANC) representation with subsequent neglect of the radial coupling between ANC states. The ANC basis is obtained as the result of a transformation of the AC basis, which explicitly takes into account the Coriolis interaction. Then, the good quantum number m looses its significance, and a new quantum number arises, which can be associated with the angular momentum l of the relative motion. The change from the basis |AC〉 to |ANC〉 becomes very simple in the socalled perturbed rotor limit where the asymptotic quantum number j becomes a good quantum number. We also note that asymptotically, at very large interfragment separations, the ANC potentials approach those calculated within the perturbation approach for the so-called fly wheel (FW) basis.12,13 In the present series of two articles, we calculate rate coefficients for the capture of two identical dipolar molecules at energies where the relative motion bears quantum character. In doing this, we use the results of refs 15 and 16 for the AC potentials of two dipolar diatomic molecules. For collisions characterized by AB(j1) + AB(j2), we concentrate on the lowest nonresonant (j1 ) j2 ) 0) and resonant (j1,j2 ) 0,1 and 1,0) channels, which show quite different capture dynamics under the action of different types of interaction. For higher channels, the calculations become quite involved, but they do not lead to new physical effects, except those related to the breakdown of
10.1021/jp102098j 2010 American Chemical Society Published on Web 05/28/2010
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J. Phys. Chem. A, Vol. 114, No. 36, 2010 9763
the adiabatic approximation at the numerous avoided crossings of the AC potentials, such as illustrated in refs 15 and 16. In part I of this series, we describe an approximate analytical method which allows one to qualitatively bridge the gap between the classical rate coefficients and their low-energy quantum limit, which we call the Bethe limit. Our approach is based on the possibility of deriving an analytical expression for the classical rate coefficient within the AC approximation, on a known analytical expression for the Bethe rate within the FW approximation, and on our earlier observation that the partial s-wave capture rate coefficient provides a reliable interpolation between the classical and the Bethe rate coefficients. In part II of this series, we support these results by numerical calculations and applications to specific molecular systems. Our article is organized as follows. Section 2 describes the long-range interactions considered in this paper. In section 3, the expressions for the energy- dependent rate coefficients within the AC approach are derived. Section 4 treats the FW interaction, and the Bethe limit for the capture probability is calculated. In section 5, we provide an analytical approximation for the capture rate coefficient as a function of the reduced wave vector, and we discuss the transition from classical to quantum capture dynamics. Section 6 concludes the paper. 2. Long-Range Interaction between Two Polarizable Dipoles The long-range part of the interaction between two polarizable dipoles consists of a dispersion contribution Vdisp and a dipole-dipole component Vdip-dip. Normally, the isotropic part of the dispersion interaction is much larger than the anisotropic part, such that the latter will be neglected, and Vdisp is written as disp
V
(R) ) -C6 /R
6
(1)
where R is the distance between the centers of mass of the dipoles. In turn, Vdip-dip is totally anisotropic and reads
Vdip-dip(R, d1, d2) )
ˆ )(d2 · R ˆ) d1 · d2 - 3(d1 · R R3
(2)
where d1 and d2 are the dipole moment vectors of rotors 1 and ˆ is the unit vector directed along the collision axis R. 2 and R In our work, we express the interaction of eqs 1 and 2 in first and second order in two different basis functions that correspond to j,m,J,M and j,l,J,M representations, where j is the quantum number of the overall intrinsic angular momentum of the dipoles and J is that of the total (intrinsic plus orbital) angular momentum. Since in the following we only consider channels belonging to the 00 and 01 manifolds, j can be regarded as the quantum number of a single rotor, being either j ) 0 or 1. The AC approach uses the j,m,J,M representation with R as the quantization axis for the intrinsic angular momentum, while the FW approach uses the j,l,J,M representation with a space-fixed quantization axis. Since in both cases the Hamiltonian matrix does not depend on M (i.e., the projection of the total angular momentum J onto a space-fixed quantization axis), this quantum number will be be omitted. An additional quantum number p appears here, which specifies the symmetry with respect to permutation of dipole moments between two rotors. We consider the rotors as distinguishable such that the nuclear statistics of the rotors’ constituents is disregarded. If necessary,
it can be taken into account later, once the state-specific capture probabilities are known. 3. Classical Rate Coefficients for Capture in the AC Approximation The AC interaction potentials for nonresonant (00) and resonant (01 and 10) channels, which take into account the dispersion as well as the first-order and second-order dipoledipole interactions, read15,16
V00(R) ) -
01 Vp,m (R) ) -
C6 6
R
-
C6 R6
-
d4 6BR6
(3)
[
]
2d4 d2 d4 + (3m2 - 2) p 3 + 6 27BR 3R 270BR6 (4)
where B is the rotational constant of the rotor (in energy units). The terms proportional to d4 in eqs 3 and 4 are second-order dipole-dipole corrections with respect to free rotor states that correspond to the coupling j ) 0 f j′ ) 1 and j ) 1 f j′ ) 0,2, respectively. The term proportional to d2 is the first-order dipole-dipole correction. In eq 3, m is the projection of the intrinsic angular momentum onto the collision axis (m ) 0, (1), and p is the permutation symmetry index for dipoles located on two different centers (p ) (1). For simplicity, we neglect the small correction, proportional to d4, in the square brackets in eq 4, and we rewrite the above equations as
V00(R) ) -
01 Vp,m (R) ) -
C01 6 6
R
C00 6
(5)
R6
+ (3m2 - 2)p
d2 3R3
(6)
We pass now to the reduced variables introduced in ref 17 and introduce the effective AC potentials which contain the interaction and the centrifugal energy. For the (00) manifold of channels, we have
υ00,J(F) )
J2 1 - 6 2F2 2F
(7)
where F ) R/R600, with R600 ) (2µC600/p2)1/4; µ is the reduced mass, and J is the classical counterpart of the quantum number of the total angular momentum. In addition, υ ) V/E600, with 00 2 2 E00 6 ) p /µ(R6 ) . For a given J, there is only one channel, being either open or closed for capture. The classical scaled energydependent rate coefficient for effective potential eq 7, such as that defined in ref 17, is given by
χ (κ) )
Cl 00
(J00(κ))2 2κ
(8)
where κ is the reduced wave vector κ ) 2E/E600 and J00(κ) is found from the capture equations
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Nikitin and Troe
υ00,J(F) J)J00,F)F00 ) κ2 /2
(
dυ00,J(F) dF
)
J)J00,F)F00
)0
(9)
We note in passing that the scaled rate coefficient as defined by eq 8 is the half of the rate coefficient in reduced units. Nonetheless, in what follows, we will use the definition as given by eq 8 since it simplifies the comparison with our earlier work.17 From eqs 8 and 9, we get the explicit expressions for the scaled rate coefficient Clχ00(κ) and the rate coefficient in conventional units Clk00(E)
3 χ (κ) ) (2κ)1/3 4 2πpR00 6 Cl 00 k (E) ) µ
Cl 00
(
)
(10) Cl 00
χ (κ) κ)κ(E)
2 2 1/2 with κ(E) ) [2Eµ(R00 6 ) /p ] . The condition of the applicability of the classical approximation, J00(κ) . 1, can be written now as [2κClχ00(κ)]1/2 . 1. For the 01 manifold of channels, the effective AC potentials are
01,J υp,m (F;δ)
J2 1 p(3m2 - 2)δ ) 2 - 6 + 2F 2F F3
χ (κ, δ) )
1 6
01 (κ, δ) ∑ Clχp,m m,p
01 (Jp,m (κ, δ))2 Cl 01 χp,m(κ, δ) ) 2κ
01,J υp,m (F, δ) J)Jp,m 01 ,F
(
01,J dυp,m (F, δ) dF
)
01 ) Fp,m
)
01 ,F ) F01 J)Jp,m p,m
κ2 2
(12)
k (E, δ) )
(
2πpR01 6 µ
)
3 χ˜ (κ, δ˜ ) ) Φ1/3(κ, δ˜ )[Φ(κ, δ˜ ) + δ˜ ]Θ[κ + √3δ˜ ] 2κ
(16) with
Φ(κ, δ˜ ) )
The step function Θ on the rhs of eq 16, which appears only for negative δ, closes the capture channels when the collision energy is smaller than the height of the potential barrier for the asympotically repulsive R-3 interaction. Comparing the auxiliary ˜ potential υJ(F,δ˜ ) from eq 15 with υ˜ 01,J p,m (F;δ) from eq 11, we obtain
1 χ (κ, δ) ) (2χ˜ (κ, δ) + χ˜ (κ, 2δ) + 2χ˜ (κ, -δ) + 6 χ˜ (κ, -2δ)) (17)
(13)
)0
The asymptotic expressions of Clχ01(κ,δ) for large and small κ are determined by the capture in the field of the attractive R-6 potential and in the field of the asymptotically attractive R-3 potentials. Explicitly, we have Cl 01
χ (κ, δ) κ)κ(E)
(14) χ (κ, δ) )
Cl 01
with κ(E) ) [2Eµ(R601)2/p2]1/2. 01 (κ,δ) can be derived by The explicit expression for Clχp,m considering the capture in an auxiliary potential
J2 1 δ˜ υ˜ (F;δ˜ ) ) 2 - 6 - 3 2F 2F F J
δ˜ 2 κ2 δ˜ + 16 2 4
Cl 01
and
Cl 01
calculated from equations similar to eq 13 that determine J˜(κ,δ˜ ). After some algebraic transformations, one obtains
(11)
where F ) R/R601, with R601 ) (2µC601/p2)1/4, E601 ) p2/µ(R601)2, and δ ) 2µd2/3p2R601. For a given J, there are six channels that are either open or closed for the capture. The counterparts of eqs 8, 9, and 10 then are Cl 01
Figure 1. Classical capture rate coefficient Clχ01(κ,δ) versus κ for δ ) 4 and its large and small κ asymptotics (dotted lines).
(15)
where δ˜ that replaces the coefficient in the last term of the rhs of eq 11 can be positive or negative and assumes values of δ˜ ) -2δ, -δ, δ, and 2δ. The capture rate coefficient χ˜ (κ,δ˜ ) is
{
3 (2κ)1/3 4
for κ . δ
1 2/3 -1/3 δ κ (2 + 22/3) for κ , δ 4
(18)
The relation of Clχ01(κ,δ) to its asymptotic expression, such as that given by eq 18 for δ ) 4, is illustrated in Figure 1. Two visible kinks on the graph of Clχ01(κ,δ) are due to the opening of the capture channels for the asymptotically repulsive R-3 potentials. Note also that the increasing asymptotes in this figure describe the rate coefficient Clχ00(κ) across the whole range of κ, though in the different reduced variable.
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J. Phys. Chem. A, Vol. 114, No. 36, 2010 9765
R(δ) ) 4{0.479)n + (2δ/ √3)n}1/n
4. Low-Energy Limit for the FW Case
FW-app
Capture in the zero-energy limit only occurs in those channels for which the effective potential energy has no barrier. The probability of capture then is determined by the transmission and reflection of the incoming wave passing above the drop of the potential. Since this event occurs at large interfragment distances, the coupling of the intrinsic angular momenta to the collision axis is weak, and the AC approximation breaks down. A suitable approach for the calculation of the interaction in the capture channel, which is alternative to the method described in section 3, then is provided by a perturbation method in the FW basis. In this case, the rotational functions are quantized onto a space-fixed axis, thus properly accounting for the prevailing role of the Coriolis interaction. For the channels of the 00 manifold, the anisotropic dipole-dipole interaction vanishes, and the FW basis produces the same result as the AC basis. For the channels in the 01 manifold, on the other hand, the Coriolis interaction substantially affects the anisotropic dipole-dipole interaction which appears in second order with respect to the states of the free relative rotation. Explicitly, the FW interaction in the l ) 0 channel reads
FW 01 Vp,l)0(R)
)-
C01 6 6
(19)
The second term on the rhs of eq 19 indicates how the R-3 dipole-dipole interaction transforms into an (R-3)2/(R-2) ) R-4 interaction. Standard calculations following ref 18 show that the dependence of the interaction FWV01 p,l)0(R) on p vanishes, such that, in reduced variables, one has
≡
FW 01 υp,l)0
FW 01 υ0 (F)
1 2δ2 - 4 6 2F 3F
)-
(20)
For small κ (i.e., the Bethe limit), the explicit expression form of the capture probability in the l ) 0 channel (the Bethe limit), FW P0(κ,δ)|κ,1, then is
P0(κ, δ) κ,1 )
FW
R(δ)κ
FW
where n is a fitting parameter. The performance of the approximation of eq 24 with a value of n ) 6 illustrated is in Figure 2. Equation 21 determines the zero-energy limit of the rate coefficient FW-Bχ01(δ) given by FW
χ (κ, δ) κf0 )
FW-B 01
P0(κ, δ) κ,1 ) 2k
FW
R(δ)/2
(25) We conclude this section by emphasizing that the secondorder FW potential may represent, for large enough δ, a first term of an asymptotic (nonconverging) series. This implies that eq 25 provides only an approximation to the exact Bethe limit, but from the physical grounds, one expects that the FW approximation performs rather well. This question is answered unambiguously by the numerical calculations presented in Part II of this series. 5. Bridging the Gap between the Quantum and Classical Limits
-
R 〈j, l, J, p|Vdip-dip ||j, l', J, p〉2 (j,l,l',J))(1,0,2,1) (p2l'(l' + 1)/2µ)
(24)
In this section, bridging the gap between the quantum and classical limits of the rate coefficients will be performed with the help of an approximate expression for the partial rate coefficient of s-wave capture. Fir this case, χ01 0 (κ,δ) is expressed through the capture probability P001(κ,δ) as
χ01 0 (κ, δ) )
P01 0 (κ, δ) 2k
(26)
Adopting the FW approximation, the s-wave capture probability FW P0(κ,δ) should be found from the solution of the wave equation with the potential of eq 20 and the appropriate boundary conditions. Instead of following this, we adopt a simpler procedure which leads to an approximate but analytic result. Explicitly, FWP0(κ,δ) is recovered from its small κ limit
(21)
where the length parameter FWR(δ) can be found from the solution of the wave equation with κ ) 0. The coefficient FWR(δ) for the potential of eq 20 is known analytically19 and has the form FW
R(δ) ) 8|F(δ)| sin(π/4 - arg F(δ))
F(δ) )
Γ(3/4 - iδ2 /3) Γ(1/4 - iδ2 /3)
(22)
(23)
When δ goes from zero to large values (δ . 1), the function FW R(δ) changes from 1.91 to 8δ/31/2, that is, the two known limits correspond to a single term (the first or the second) of the rhs of eq 20. The function FWR(δ) is shown in Figure 2. It can be approximated by a simple interpolation formula
Figure 2. Plots of the function FWR(δ) (full line, eqs 20 and 21), its approximation (dotted line, eq 22), and its inverse (multiplied by 10, dash-dotted line).
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Figure 3. Classical (full lines) and s-wave quantum (dashed lines) rate coefficients for δ ) 0, 1, and 4 (from bottom to top).
of eq 21 by resorting to the Troe-Quack-Klots (TQK) extrapolation formula,20 which reads FW 01 P0 (κ, δ)
) 1 - exp(-(FWR(δ))κ)
(27)
The final expression for the interpolating rate coefficient then assumes the simple form FW 01 χ0 (κ, δ)
)
1 (1 - exp(-(FWR(δ))κ)) 2κ
(28)
For several values of δ, Figure 3 shows graphs of FWχ001(κ,δ) (dotted lines) superimposed on the graphs of Clχ001(κ,δ) (full lines). One observes crossings and avoided crossings, which, according to our interpretation, imply transitions, with decreasing κ, from the classical to the quantum regime. This interpretation is based on our earlier study of the capture in the field of an attractive R-6 potential,17 which is reproduced here by the two curves for δ ) 0. The composite curve constructed from the upper of these two curves (the dotted one to the left of the crossing point and the full one to the right) corresponds to the curve χ6app(κ) in Figure 3 of ref 17, which, in turn, provides a quite accurate approximation of the quantum capture rate coefficient. (We note in passing that this result applies also to the 00 capture manifolds when κ is properly defined; see eqs 8 and 10.) The fast convergence of the quantum rate to its classical counterpart is the consequence of large quantum effects in the barrier crossing. If the latter are ignored, the capture rates exhibit sawtooth oscillations of large amplitude. On the basis of the above, we suggest that the composite functions χ01,app(κ,δ) represent a reasonable approximation to the quantum capture rate coefficients χ01(κ,δ). The function χ01,app(κ,δ) is defined here as
χ01,app(κ, δ) )
{
FW 01 χ0 (κ, δ) Cl 01
κ < κc(δ)
χ (κ, δ) κ > κc(δ)
(29)
where κc is the point of the crossing or avoided crossing of two curves. The crucial point for the definition of χ01,app(κ,δ) by eq 29 of course is the choice of κc. Figure 3 suggests that κc is
Figure 4. Qualitative behavior of the capture rate coefficients (symbols) for δ ) 0, 1, and 4 (from bottom to top) in the intermediate quantum classical regime. Full and dashed lines correspond to classical and s-wave quantum rate coefficients. Vertical arrows mark values of κc(δ) ) 0.52, 0.24, and 0.054, and horizontal arrows mark the respective Bethe limits 0.96, 2.09, and 9.23 for the above values of δ.
quite well-defined through our approximate characterization of the energy dependence of the capture rate coefficients. This allows one to construct continuous reliable curves for χ01,app(κ,δ). Such curves are shown in Figure 4 for δ ) 0, 1, and 4. Approximately, κc(δ) can be estimated from the condition that the exponent in eq 28 for the transmission probability is about unity. This yields
κc(δ) ≈
FW
1 R(δ)
(30)
The plot of 1/FWR(δ) is shown in Figure 2. For δ ) 0, 1, and 4, the values of 1/FWR(δ) are 0.52, 0.24, and 0.054. The respective values of κc(δ), as marked by the arrows on the abscissa axis in Figure 4, indeed match the points where FW 01 χ0 (κ,δ) passes into Clχ01(κ,δ). We finally note that the shift of κc ) κc(δ) toward smaller values of κ with increasing δ simply stems from the fact that for larger δ, noticeable reflection of the wave occurs at progressively lower energies. 6. Conclusion Our study contributes to the understanding of the quantum dynamics of dipolar molecules (see, e.g., the reviews in refs 21 and 22). It leads to analytical expressions for the capture rate coefficients of two rotating dipoles with intrinsic angular momenta j1 ) 0 and j2 ) 1 and at low collision energies (i.e., below the energy of excited rotational levels), which provide a reasonable interpolation between the classical and quantum limits. The former makes use of perturbed rotor adiabatic channel potentials (with vanishing Coriolis coupling and an identification of the collision axis as the quantization axis), while the latter is based on the potential calculated within the perturbation approach in the fly wheel basis (with a prevailing role of the Coriolis coupling and a space-fixed axis as the quantization axis). This approach avoids the use of axially nonadiabatic channel potentials and considers only their adiabatic channel and fly wheel asymptotes. The present analytical approach has provided guidelines for the accurate numerical
Mutual Capture of Dipolar Molecules calculations to be presented in Part II.23 Part II will also contain explicit results for specific molecular systems. Acknowledgment. We acknowledge most stimulating discussions with Reinhard Schinke over a long period of time. We are also indebted to E. Dashevskaya and I. Litvin for their constructive comments on the present article. References and Notes (1) Cold Molecules; Krems, R. V., Stwalley, C., Friedrich, B., Eds.; CRS Press: New York, 2009. (2) Rackham, E. J.; Huarte-Larranaga, F.; Manolopoulos, D. E. Chem. Phys. Lett. 2001, 343, 356. (3) Rackham, E. J.; Gonzalez-Lezana, T.; Manolopouos, D. E. J. Chem. Phys. 2003, 119, 12895. (4) Alexander, M. H.; Rackham, E. J.; Manolopouos, D. E. J. Chem. Phys. 2004, 121, 5221. (5) Atahan, S.; Alexander, M. H.; Rackham, E. J. J. Chem. Phys. 2005, 123, 204306. (6) Quack, M.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1974, 78, 240– 252. (7) Sakimoto, K.; Takayanagi, K. J. Phys. Soc. Jpn. 1980, 48, 2076. (8) Bates, D. R. Proc. R. Soc. London, Ser. A 1982, 384, 289. (9) Clary, D. C. Mol. Phys. 1985, 54, 605.
J. Phys. Chem. A, Vol. 114, No. 36, 2010 9767 (10) Troe, J. AdV. Chem. Phys. 1992, 82, 485. (11) Ramillon, M.; McCarroll, R. J. Chem. Phys. 1994, 101, 8697. (12) Dashevskaya, E. I.; Litvin, I.; Nikitin, E. E. J. Phys. Chem. A 2006, 110, 2786. (13) Dashevskaya, E. I.; Litvin, I.; Nikitin, E. E.; Troe, J. Mol. Phys. 2010, 108, 873. (14) E Dashevskaya, E. I.; Litvin, I.; Nikitin, E. E.; Troe, J. J. Chem. Phys. 2004, 120, 9989. (15) Maergoiz, A.; Nikitin, E. E.; Troe, J. J. Chem. Phys. 1991, 95, 5117. (16) Maergoiz, A.; Nikitin, E. E.; Troe, J. Z. Phys. Chem. 1991, 172, 129. (17) Dashevskaya, E. I.; Litvin, I.; Maergoiz, A.; Nikitin, E. E.; Troe, J. J. Chem. Phys. 2003, 118, 7313. (18) Zare, R. N. Angular Momentum; Wiley: New York, 1988. (19) Eltschka, C.; Moritz, M.; Friedrich, H. J. Phys. B 2000, 33, 4033. (20) Dashevskaya, E. I.; Litvin, I.; Nikitin, E. E.; Troe, J. Phys. Chem. Chem. Phys. 2009, 11, 9364. (21) Bohn, J. L. Electric Dipoles at Ultralow Temperatures. In Cold Molecules; Krems, R. V., Stwalley, C., Friedrich, B., Eds.; CRS Press: New York, 2009; p 39. (22) Pupillo, G.; Micheli, A.; Buechler, H.-P.; Zoller, P. Condensed Matter Physics with Cold Polar Molecules. In Cold Molecules; Krems, R. V., Stwalley, C., Friedrich, B., Eds.; CRS Press: New York, 2009; p 421. (23) Dashevskaya, E. I.; Litvin, I.; Nikitin, E. E.; Troe, J. In preparation.
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