Classical theory of rotational excitation of diatomic molecules. Rotor–rotor coupling

Physica A 258 (1998) 395–413

Classical theory of rotational excitation of diatomic molecules. Rotor–rotor coupling R.E. Kolesnick ∗ Physics Department, University Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK Received 3 November 1997

Abstract The new concept of two momenta coupling is developed to treat the Coriolis coupling and the centrifugal one for two rotors because they are important. A new method of classical canonical transformations is applied for two rotors coupling. The classical theory of two rotors scattering is studied. The obvious advantages of angle–action variables to describe angular momenta of the internal motion of Rotors have been employed. Canonical magnitudes of the dynamical variables have been generated by the F4 transformation function for a partly coupled set and for the fully coupled one. Such a generating function was obtained by extending Miller’s function. Relative motion of molecules is treated by classical trajectories. The theoretical model is worked out to study transport properties of gases with rotational degrees of freedom. The structure of the resulting expressions under consideration are shown to be dierent in principle from the expressions obtained by other approaches. The interplay between the dierent types of coupling in uencing the transport properties is feasible to determine. The eect of coupling on energy transfer can be studied in the framework of such an approach. Previously, the usage of the approximative treatment of momentum coupling was imperative. The method is designed to incorporate recent improved formulas for calculating internal coordinates and their derivatives c 1998 Elsevier Science B.V. All rights to ensure the most optimal calculation sequence. reserved. Keywords: Angular momentum; Rotor; Angle–action variables; Coupling

1. Introduction The enormous progress made in recent years in the techniques for detecting and analysing excited states of polyatomic molecules has prompted the development of theoretical quantum mechanical methods to determine and characterize these states [1]. The dynamics of molecular scattering can often be approximated by classical or ∗

Tel.: +44 191 22 7399; fax: +44 191 222 7361; e-mail: [email protected].

c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$19.00 PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 0 4 3 - 0

396

R.E. Kolesnick / Physica A 258 (1998) 395–413

semi-classical mechanics. For heavy particles, quantum eects such as tunnelling, interference, large vibrational spacing and so on are of little importance. This is especially true if quantities such as generalized cross-sections [2] averaged over internal rotational degrees of freedom are calculated. This is fortunate, since the solution of the classical equations of motion usually poses no numerical problems even for a large number of atoms. Thus, classical trajectory calculations still remain a main instrument to study molecular scattering nowadays. So, the choice of the right trajectory where the total angular momentum is conserved is still very important, in spite of previous considerations. Such models of collisions can be useful in applications of rare ed gas dynamics such as probing the rotational population in an expanding supersonic molecular beam [3], for gas phase chemical reactions modelling [4], for studying scattering [5] where angular momentum J plays an important role, and collinear approximations or calculations at J = 0 are totally inappropriate. One might want to use the coupled representation because it employs the canonical constants of motion and it facilitates the choice of a relevant kinetic scheme to describe the dynamics. The usage of the full set of dynamical variables including the total angular momentum is important for a number of applications. For example, in practice it is necessary to introduce the quantity of rotational energy per molecule to study the kinetics of the rotational levels population and so on. There are in the literature only a few articles [6,7] which are devoted to studying the angular momentum coupling of two rotors. In practice, while studying molecular dynamics, quite often one needs to apply semiclassical methods to express the Hamiltonian in angle–action variables. Such a way of setting up the Hamilton’s equations brings computational advantages [7]. The potential energy surface, (PES) very often depends on angles, which should be connected with the action–angle variables directly. A direct geometrical attack is not very elegant and is still unsuccessful, because of the very complex geometry of the problem. It is not feasible to consider a 3-D geometry coupling procedure and so an analytical approach should be implemented. Even for small molecules it is quite dicult to develop accurate classical models of collisions in action–angle variables, but for such cases it is possible to avoid approximations at all. First, the classical transformation between uncoupled and coupled angular momentum given by Miller [8] was de ned in terms of an F4 generating function, which is de ned in the space of classical actions [13]. Unfortunately, such a transformation is still dicult to implement, because, in general the PES depends on the coupled actions and the conjugate angle variables. We extend Miller’s generating function for our space of angle–action variables. Nevertheless, a complementary way [9] was proposed to transform between position coordinates and angle–action variables. It provided the possibility to cover all the angles together in a single framework. In the present article we develop the classical theory of transformations for canonical angles, by associating these angles with rotational matrices directly. The geometrical angles are included in the description conveniently via canonical actions, the rst ones are specifying the position of interatomic vectors. The multiplication of one rotation matrix, M by a vector

R.E. Kolesnick / Physica A 258 (1998) 395–413

397

is equivalent to carrying a vector from one frame of reference to another. So, the Cartesian coordinates of the rotor axis and canonical variables should be connected directly in the framework of our method. Thus, the M-matrix approach is the parallel matrix formulation to the F4 classical transformation, which has been derived as an addition of classical actions [8]. This approach provides a possibility of nding connections between uncoupled and coupled canonical angular momentum variables quite easily. It is easier to implement elaborate accurate models for problems in molecular dynamics, especially for small molecules. As applications, we consider the rotational dynamics of two coupled rotors, which is of interest for the last decade [10,11]. In the present study we intend to analyse further the behaviour of the PES [12] and its eciency in inducing rotationally inelastic collisional processes only. The rigid-rotor model will allow us to focus more speci cally on those features of the interaction which govern the inelasticity at the temperatures which are relevant to the processes mentioned above. In this paper, we derive several useful formulae for two coupled rotors whose Hamiltonian has been expressed in action–angle variables to facilitate solving the trajectory problem. Finally it leads to an easily programmed prescription for setting Hamiltonian equations of coupled rotating diatomic molecules in classical trajectory calculations. In Section 2 we discuss the new approach to the general theory of rotor–rotor scattering in angle–action variables. The derivation of expressions for two partly coupled rotors is described in Section 3. The fully coupled case is considered in Section 4 and a summary and conclusions are presented in Section 5. 2. General theory 2.1. Classical model of collisions for two rotors expressed in angle–action variables In the space- xed system of coordinates the Hamiltonian of two rigid rotors with moments of inertia I1 and I2 and masses M1 and M2 , respectively, in conventional polar coordinates is given by H (R; rˆ1 ; rˆ2 ; P; j1 ; j2 ) =

j2 P2 j2 + 1 + 2 + V (R; 1 ; 2 ; ) ; 2 2I1 2I2

(1)

where j2i = p&2i + p’2 i =sin2 &i ; i = 1; 2 are squares of the rotational angular momenta j1 ; j2 of the rotors,  is their reduced mass, and &i and ’i are the spherical polar angles of the internuclear axis i. Here R is the vector joining the centres of mass of the molecules, P the momentum of their relative motion and ri lies along the axis of rotor de ned by i. The potential V depends only on R and the three angles 1 ; 2 and ˆ · rˆi and cos = rˆ1 · rˆ2 . cos i = R Here we are going to introduce angle–action variables for solving the classical trajectory problem. In de ning the angle variables care has to be taken in the speci cation

398

R.E. Kolesnick / Physica A 258 (1998) 395–413

of their origin. The uncoupled angle–action variables are de ned as follows: m1 ; m2 and ml are the projections of j1 ; j2 and l is the orbital angular momentum of the rotors, on the OZ axis respectively, and qm1 ; qm2 and qml the corresponding conjugate angles, where qm1 ; qm2 and qml are measured from the Y axis to nˆ j1 ; nˆ j2 and nˆ l respectively, which are de ned to be along the direction of z × j1 ; z × j2 and z × l respectively. The angles q1 ; q2 and ql are conjugate to the momenta j1 ; j2 and l measured from nˆ j1 ; nˆ j2 and nˆ l to r1 ; r2 and R respectively. Here cos l = ml =l. Then the rotation (qml ; l ; ql ) carries the position vector R initially along the OY axis, with l along OZ to its nal position, where the conventions of Rose for Euler angles have been adopted. For the fully uncoupled set of dynamical variables in the space- xed frame ˆj1 and rˆ1 are speci ed by (qm1 ; 1 ; q1 ) and similarly for ˆj2 and rˆ2 . Here cos i = mi =ji . For this uncoupled case we have rˆ1 = M(qm1 ; 1 ; q1 )† (0; 1; 0)† ;

(2)

rˆ2 = M(qm2 ; 2 ; q2 )† (0; 1; 0)† ;

(3)

ˆ = M(qml ; l ; ql )† (0; 1; 0)† ; R

(4)

where † denotes the transpose and the matrix M(; ; ) is the rotation matrix of Rose [14]. Hence cos 1 can be written ˆ · rˆ1 = (0; 1; 0)M (qml ; l ; ql )M(qm1 ; 1 ; q1 )† (0; 1; 0)† cos 1 = R

(5)

and similarly for 2 and . The intermolecular potential V depends on (qm1 − qml ); (qml − qm2 ) and (qm2 − qm1 ) so (m1 + m2 + ml ), the Z component of the total angular momentum, J = j1 + j2 + l is conserved. The uncoupled set of variables leads to 14 equations of motion. 2.2. De nition of partly coupled set. Canonical transformation from uncoupled to partly coupled set of angle–action variables for two rotors The partly coupled set of variables corresponds to the case of two momenta coupled from three, which can be done in three ways. To introduce the partly coupled set of angle–action variables we shall follow the quantal coupling scheme [15]. Now we get 14 equations of motion as for the case of two uncoupled rotors. The following transformations are required: the Cartesian coordinates are transformed to the uncoupled set of angle–action variables. These uncoupled variables are converted to the set of partly coupled variables which result from coupling j1 with j2 to give the total rotational angular momentum of the system, j12 = j1 + j2 . The usual quantal partly coupled set of variables [15] uses j12 . The two alternative partly coupled schemes involve h1 = j1 + l and h2 = j2 + l. The variables PR ; l; ml ; R; ql ; qml describe the relative rotor–rotor motion and are unchanged. The projection of j12 on the space- xed Z-axis shall be designated m12 and the

R.E. Kolesnick / Physica A 258 (1998) 395–413

399

angles conjugate to j12 and m12 are designated Q12 and Qm12 , respectively. A generating function which describes the transformation from the uncoupled angle–action variables p = ( j1 ; j2 ; l; m1 ; m2 ; ml ; PR ) and q = (q1 ; q2 ; ql ; qm1 ; qm2 ; qml ; R) to the partly coupled variables P = ( j1 ; j2 ; j12 ; l; m12 ; ml ; Pr ) and Q = (q1 ; q2 ; Q12 ; ql ; Qm12 ; qml ; R) may be shown to be F2 (q; P) = F3 (p; q0 ) + F4 (p; p0 ) + F4 (p0 ; P) :

(6)

First, we need the fairly trivial transformation F3 (p; q0 ), where p0 = (j1 ; j2 ; l; m1 ; m12 ; ml ; PR ) and q0 = (q1 ; q2 ; ql ; qm1 ; qm2 ; qml ; R). Because the transformation (p; q) to (p0 ; q0 ) is a point momentum transformation and F4 (p0 ; p) is thus zero. The next transformation F4 transforms from ‘intermediate’ momenta p0 to ‘new’ momenta P. It is possible to show that according to de nition [8] for our set of variables the F4 can be constructed as a one-dimensional integral 0

F4 (p ; P) = −

Zm1

dp40 q40 (p40 ) ;

(7)

and q40 = qm1 − qm2 and q40 is given as a function of the ‘intermediate’ and ‘new’ momenta by j 2 − j 2 − j 2 − 2p0 (m12 − p40 ) q40 (p40 ) = ± arccos p12 2 1 022 2 4 : 2 ( j2 − p4 )( j1 − (m12 − p40 )2 ) The p40 is the intermediate momentum or variable of integration. The integral in Eq. (7) in our case of angle–action variables can be found as follows: Zm1

dp40 q40 (p40 ) = ± ˜ ;

(8)

where ˜ = j1 1 + j2 2 + j12 3 + m1 1 + m2 2 + 2m12 3 ; so m2 = m12 − m1 and 1 ; 2 ; 3 are angles de ned as follows:   2 ) − 2m2 j12 −m1 ( j12 + j22 − j12 ; 1 = arccos ( j1 ; j2 ; j12 )j1   2 ) 2m1 j22 + m2 ( j22 + j12 − j12 2 = arccos ; ( j1 ; j2 ; j12 )j2   2 2 − m12 ( j12 + j22 − j12 ) 2m1 j12 3 = arccos ; ( j1 ; j2 ; j12 )j12  2  2 − j22 − 2m12 m1 j1 + j12 1 = arccos ; 2j12 j1  2  2 − j12 − 2m2 m12 j + j12 2 = arccos 2 ; 2j12 j2

(9) (10) (11) (12) (13)

400

R.E. Kolesnick / Physica A 258 (1998) 395–413

 3 = arccos

2 + j12 + j22 − 2m1 m2 −j12 2j1 j2

( j1 ; j2 ; j12 ) =

q

 ;

4 − j 4 + 2j 2 j 2 + 2j 2 j 2 + 2j 2 j 2 ; −j24 − j12 1 12 2 2 1 1 12

(14) (15)

with j2i = ji2 − m2i ; i = 1; 2; 12. The angles have the following geometrical interpretation. ˆ are de ned to be along the direction of zˆ × j12 and j2 × j1 , The unit vectors nˆ j12 , and N ˆ nˆ j1 );  − (N ˆ nˆ j2 ) and respectively. The angles 1 ; 2 ; 3 can be identi ed with  − (N ˆ ˆ ˆ (Nnˆ j12 ), respectively, where (Nnˆ j1 ) is the angle measured from N to nˆ j1 in sense of ˆ nˆ j12 ) etc. The angles 1 ; 2 and 3 can be identi ed increasing q1 , and similarly for (N with qm1 − Qm12 ; qm2 − Qm12 and Qm12 , respectively [16]. Thus ˆ nˆ j1 )) + j2 ( −(N ˆ nˆ j2 )) +j12 (N ˆ nˆ j12 ) ˜ = j1 ( − (N +m1 qm1 +m2 qm2 +m12 Qm12 :

(16)

Since an identity transformation for j1 and j2 cannot be obtained with an F4 -type generator we introduce the corresponding F2 generator F2 (q; P) = q1 j1 + qm1 m1 + q2 j2 + qm2 m2 + Qm12 m12 − (˜ − ( j1 + j2 )) ;

(17)

where the transformation has been carried out in two stages and we are using the continuity of F4 and the de nition of the identity transformation F2 . From this equation, we ˆ nˆ j1 ); Q2 = @F2 =@j2 = q2 + (N ˆ nˆ j2 ) and Q12 = @F2 =@j12 = obtain Q1 = @F2 =@j1 = q1 + (N ˆ − (Nnˆ j12 ). Similarly qm1 = @F2 =@m1 ; qm2 = @F2 =@m2 ; Qm12 = @F2 =@m12 . We can perform a similar derivation for the other partly coupled cases such as h1 and h2 . All of them are equivalent if no approximations are employed. 3. Cartesian and angle–action variables for a partly coupled set 3.1. Partly coupled set of angle–action variables First, we have to de ne positions and properties of interatomic unit vectors rˆ1 ; rˆ2 ˆ which are in the direction between the center of masses. Let S1 be the reference and R frame of coordinates OX1 Y1 Z1 . Let S10 be the frame obtained from S1 by the rotation (qm1 ; #1 ; Q1 ), where angles qm1 ; qm2 ; qml ; Qm12 ; Q12 are conjugate to action variables m1 ; m2 ; ml ; m12 ; j12 and #1 satis es cos #1 = ˆj1 · ˆj12 =

2 j22 − j12 − j12 : 2j1 j12

(18)

One can make similar manipulations for #2 , by rotating the S2 frame to S20 . We make the same arrangement for l and get #l by rotating the frame Sl to Sl0 , respectively. Let the S 00 frame be obtained from the frame S10 by the rotation (Qm12 ; 12 ; Q12 ), where cos 12 = m12 =j12 . We have taken j1 ; j2 and l to be lying along the OZ10 axis, OZ20 and OZl , respectively, and rˆ1 ; rˆ2 and R to be respectively along OY1 ; OY2 , and OYl . Here cos #l = ˆl · ˆj12 .

R.E. Kolesnick / Physica A 258 (1998) 395–413

401

ˆ as a product of rotation matrices in the Then, we can get expressions for rˆ1 ; rˆ2 ; R, appropriate geometrical representation rˆ1 = M(Qm12 ; 12 ; Q12 )M(qm1 ; #1 ; Q1 )† (0; 1; 0)† ;

(19)

rˆ2 = M(Qm12 ; 12 ; Q12 )M(qm2 ; #2 ; Q2 )† (0; 1; 0)† ;

(20)

ˆ = M(Qm12 ; 12 ; Q12 )M(qml ; #l ; Ql )† (0; 1; 0)† : R

(21)

ˆ · rˆ1 ; R ˆ · rˆ2 . Through these From these we can readily get the scalar products rˆ1 · rˆ2 ; R the angle variables q1 ; q2 ; Q12 ; ql and actions j12 ; j1 ; j2 ; l appear in the Hamiltonian. The partly coupled case corresponds to the case of pairwise two momenta coupled from three given momenta which can be done in three ways. They are j12 = j1 + j2 ; h1 = j1 + l; h2 = j2 + l. None of j12 ; h1 ; h2 are constants of the motion. We couple the qm1 ; qm2 ; qml pairwise. They are conjugate to m1 ; m2 ; ml , which are projections of j1 ; j2 and l to Z-axis of coordinates. Then we have to eliminate the intermediate (qm1 − qm2 ); (qm1 − qml ) and (qm2 − qml ) variables. As a result we get 14 equations in coupled variables as in the case of uncoupled variables. The total angular momentum can easily be introduced as a parameter. Such a system of coordinates can be more suitable to perform the classical coupled states approximation. We can show that the magnitudes of j12 ; h1 and h2 are connected. Thus, any two of them are connected to the third. This follows directly from geometrical considerations. We can nd magnitudes of h1 ; h2 in terms of j12 ; Q12 and the coupled action variables. So, we can nd the magnitudes of cos( j1 ; l) and cos(j2 ; l) in terms of cos( j1 ; j2 ). The rotation of partly coupled rotors as a whole goes about an axis parallel to j12 . So, the relative position of the triangle formed by j1 ; j 2 ; j12 and the triangle formed by J; l; j12 would be de ned by the angle Q12 . The angle between the directions j1 × j12 and l × j12 in origin would be equal to Q12 . On the basis of geometrical considerations the expressions for h1 ; h2 would be q h12 = l 2 sin #l + ( j 22 − l 2 cos 2 #l ) − 2l cos #l (j 22 − l 2 cos #l ) cos Q12 ; (22) h22 = l 2 sin #l + ( j12 − l 2 cos 2 #l ) − 2l cos #l

q (j12 − l 2 cos #l ) cos( − Q12 ) : (23)

The action variables j1 ; j2 ; j12 and l and conjugate angle Q12 have been chosen to describe completely the partly coupled rotors. Thus, we need not use all the intermediate actions h1 ; h2 ; j12 . It would be enough to take only one of them for instance, j12 . 3.2. Expressing the potential in terms of the partly coupled variables. Space con guration coupling of two momenta from three j1 ; j2 and l for Now we consider the derivation of the expressions for the angles 1 ; 2 and j1 and j 2 space coupling con guration. The orbital momentum l would couple with j1

402

R.E. Kolesnick / Physica A 258 (1998) 395–413

and j 2 pairwise in that case and its position de ned by the angle Qml ; #l and Ql . The expression for cos 1 could be written from Eqs. (19) and (21) ˆ · rˆ1 = (0; 1; 0)M(qml ; #l ; Ql )M(qm1 ; #1 ; Q1 )† (0; 1; 0)† cos 1 = R

(24)

= (cos Q1 cos Ql + cos #1 cos #l sin Q1 sin Ql ) cos(qm1 − qml ) + (sin Q1 cos Ql cos #1 − cos Q1 sin Ql cos #l ) sin(qm1 − qml ) + sin Q1 sin Ql sin #1 sin #l :

(25)

The cosine of the angle between j1 and l can be written in terms of #1 and #l can be written as [17] cos( j1 ; l) = cos #1 cos #l + sin #1 sin #l cos(qm1 − qml ) =

−l 2 − j12 + h12 : 2j1 l

(26)

So, the general expression for cos 1 in action–angle variables can be derived from Eqs. (24)–(26). The intermediate variables (qml − qm1 ); (qml − qm2 ) and (qm1 − qm2 ) should be eliminated. It would be given by cos( j1 ; l) − cos #l cos #2 sin #1 sin #l + (sin Q1 cos Ql cos #1 − cos Q1 sin Ql cos #l ) s (cos( j1 ; l) − cos #l cos #1 ) 2 × 1− (sin #1 sin #l ) 2

cos 1 = (cos Q1 cos Ql + cos #1 cos #l sin Q1 sin Ql )

+ sin Q1 sin Ql sin #1 sin #l :

(27)

The expression for cos 2 could be written in the general form ˆ · rˆ2 = (0; 1; 0)M(qml ; #l ; Ql )M(qm2 ; #2 ; Q2 )† (0; 1; 0)† cos 2 = R

(28)

= (cos Q2 cos Ql + cos #l cos #2 sin Q2 sin Ql ) cos(qm2 − qml ) + (sin Q2 cos Ql cos #2 − cos Q2 sin Ql cos #l ) sin(qm2 − qml ) + sin Q2 sin Ql sin #2 sin #l :

(29)

Cosine of the angle between j 2 and l in terms of #j2 and #l can be written as in [17]: cos( j 2 ; l) = cos #2 cos #l + sin #l sin #2 cos(qm2 − qml ) =

l 2 + j22 − h22 : 2lj2

(30)

So, the general expression for cos 2 in action–angle variables can be derived from Eqs. (28)–(30) and is given by: cos( j 2 ; l) − cos #l cos #2 sin #2 sin #l + (sin Q2 cos Ql cos #2 − cos Q2 sin Ql cos #l ) s (cos( j 2 ; l) − cos #l cos #2 ) 2 × 1− + sin Q2 sin Ql sin #2 sin #l : (31) (sin #2 sin #l ) 2

cos 2 = (cos Q2 cos Ql + cos #2 cos #l sin Q2 sin Ql )

R.E. Kolesnick / Physica A 258 (1998) 395–413

The expression for cos cos

403

could be written in the general form

= rˆ1 · rˆ2 = (0; 1; 0)M(qm1 ; #1 ; Q1 )M(qm2 ; #2 ; Q2 )† (0; 1; 0)†

(32)

= (cos Q1 cos Q2 + cos #1 cos #2 sin Q1 sin Q2 ) cos(qm1 − qm2 ) + (sin Q1 cos Q2 cos #1 − cos Q1 sin Q2 cos #2 ) sin(qm1 − qm2 ) + sin Q1 sin Q2 sin #1 sin #2 :

(33)

Cosine of the angle # between j1 and j 2 in terms of #1 and #2 can be written as in [17]: cos # = cos #1 cos #2 + sin #1 sin #2 cos(qm1 − qm2 ) =

2 j12 + j22 − j12 : 2j1 j2

(34)

Hence cos

cos # − cos #1 cos #2 sin #1 sin #2 + (sin Q1 cos Q2 cos #1 − cos Q1 sin Q2 cos #2 ) s (cos # − cos #1 cos #2 ) 2 × 1− + sin Q1 sin Q2 sin #1 sin #2 : (sin #1 sin #2 ) 2

= (cos Q1 cos Q2 + cos #1 cos #2 sin Q1 sin Q2 )

(35)

So, we have obtained angle–action variables for two partly coupled rotors. There are enough conditions to obtain a closed set of variables, so that cos( j1 ; l) and cos( j 2 ; l) are obtained from the geometrical equations. The expression for J can be written as 2 + l 2 − 2lj12 cos #l : J 2 = j12

(36)

Obviously, J is a constant of motion and that condition would be important. Now, it is possible to introduce it as a parameter of the system and keep it under control when one is making simulations. It provides the possibility to reduce the dimension of the equations set. 3.3. Generating function eects uncoupled set to partly coupled. Connections for j1 ; j2 coupling case Having obtained angle–action variables for the partly coupled set we can evaluate the F4 -generator. According to de nitions (9) – (13), we express 1 ; 2 ; 3 ; 1 ; 2 and 3 angles in terms of given geometrical angles 1 ; 2 ; 12 and actions variables j1 ; j2 ; j12   −cos 1 cos # − cos 2 ; (37) 1 = arccos sin 1 sin #   cos 2 cos # + cos 1 ; (38) 2 = arccos sin 2 sin #

404

R.E. Kolesnick / Physica A 258 (1998) 395–413



 −cos 12 cos #1 + cos 1 3 = arccos ; sin # sin 12   cos #1 − cos 12 cos 1 ; 1 = arccos sin 12 sin 1   cos #2 − cos 12 cos 2 ; 2 = arccos sin 12 sin 2   cos # − cos 1 cos 2 : 3 = arccos sin 1 sin 2

(39) (40) (41) (42)

So, we thus connect uncoupled and partly coupled canonical variables. In such a case the Hamiltonian depends on Q12 ; j12 . The generating function F4 does not depend on l and Ql in the partly coupled case. 3.4. Generation of coordinates in case of plane con guration of momenta j1 and j2 coupled Let consider the case, when (qml − qm1 ); (qml − qm2 ) and (qm1 − qm2 ) are xed and equal to , or 0. Such a simpli cation means that the projections of j1 and l on the XY plane should align along the ascending line. It was implied that OZ is lying in the plane of j1 , j 2 , j12 , l. Of course, some features of the scattering would be lost in this simpli ed con guration and its validity should be studied. That simpli cation facilitates calculations, but the physical sense of it is still not clari ed. Probably, it corresponds to several coherent conditions [18] of molecules. The expressions for 1 ; 2 ; would be written in a general form for that case ˆ · rˆ1 = cos 1 = −cos Q1 cos Ql − (ˆl · ˆj1 ) sin Q1 sin Ql ; R

(43)

ˆ · rˆ2 = cos 2 = cos Q2 cos Ql + (ˆl · ˆj2 ) sin Q2 sin Ql ; R

(44)

rˆ1 · rˆ2 = cos = cos Q1 cos Q2 + (ˆj1 · ˆj2 ) sin Q1 sin Q2 :

(45)

We note that apart from the signs Eqs. (43) – (45) looks like the atom–rotor expression, when l + j1 = J , i.e. we are using l × j1 as a polar axis. The Ql has the same origin, as it is in case of h2 or h1 . The results are in a more detailed form   2 j12 + j12 − j22 cos #l sin Q1 sin Ql ; cos 1 = −cos Q1 cos Ql − j1 sin #l + 2j12 j1 (46)  2 j12 + j22 − j12 cos #l sin Q2 sin Ql ; cos 2 = −cos Q2 cos Ql + j2 sin #l + 2j12 j2 

(47) cos = cos Q1 cos Q2 +

2 j12 − j12 − j22 sin Q1 sin Q2 : 2j1 j2

(48)

R.E. Kolesnick / Physica A 258 (1998) 395–413

405

j1 = |ˆj1 × ˆj12 | = 1=2(ˆj1 ; ˆj2 ; ˆj12 ) ;

(49)

j2 = |ˆj2 × ˆj12 | = 1=2(ˆj1 ; ˆj2 ; ˆj12 )j1 =j2 ;

(50)

cos #l = |ˆl · ˆj12 | :

(51)

3.5. Generating function for the xed con guration coupling of j1 and j2 Now we consider the particular case of two rotational momenta coupling. Such a situation arises when 12 = 0 in the expressions (40) – (42). The phases i of polar angles would be unde ned. It means that there is no such classical canonical transformation which eects from the uncoupled to partly coupled set of variables with j12 aligned in a xed position along OZ. In this way we should get an non-Hamiltonian dynamical system. It might be possible to avoid this unde nement, but we shall not consider such a case. 4. Canonical transformation from partly coupled to fully coupled set of angle–action variables for two rotors 4.1. De nition of the fully coupled set of angle–action variables In this section we are going to introduce the fully coupled set of angle–action variables to yield obvious advantages of canonical transformation usage in angle–action variables. Such a scheme involves the total angular momentum J variable and its projection to Z-axis M . Since the total angular momentum of the system is conserved, now instead of 14 equations of motion we get only 11 for the case of two rotors. The following transformations are required. The partly coupled variables j1 ; j2 , etc. are transformed to the set of fully coupled variables which results from the coupling of the rotational angular momentum of the rotors, j12 with the orbital angular momentum, l to give the total angular momentum J = j12 + l. The usual quantal close coupling method employs J [15]. After that we have to get back to the Cartesian system of coordinates to determine r1 ; r2 , etc. conveniently. The variables PR ; l; ml ; R; ql ; qml , describe the relative rotor– rotor motion in the fully uncoupled set of coordinates. The projection of J on the space xed OZ axis shall be designated M and the canonical angles conjugate to J and M are denoted by Q J and QM . First, we need the fairly trivial transformation F2 (m12 ; ml ; m12 ; M ). The next transformation F4 transforms from ‘old’ momenta p = ( j12 ; l; ml ; m12 ), to ‘new’ momenta P = (m12 ; j12 ; M; J ). A generating function F4 which eects the transformation from the partly coupled angle–action variables j1 ; j2 ; j12 ; l; m12 ; ml and their conjugate angle variables to the fully coupled variables j1 ; j2 ; j12 ; l; J; M; qj1 ; qj2 ; Qj12 ; ql ; Q J ; QM may be shown to be F4 ( j1 ; j2 ; l; j12 ; m12 ; ml ; j1 ; j2 ; j12 ; l; J; M ) = F4 (j12 ; m12 ; l; ml ; j1 ; j2 ; J; M ) :

(52)

406

R.E. Kolesnick / Physica A 258 (1998) 395–413

It was shown in Section 2, that F4 can be expressed as a one-dimensional integral for the case of arbitrary oriented J ZM F4 (p; P) = −

dp04 q40 (p40 ) ;

(53)

where F4 is the classical generator of the transformation between old and new momenta, and q40 = Qm12 − Qml ; p = ( j1 ; j2 ; m12 ; j12 ) are ‘old’ momenta while P = (m12 ; j12 ; M; J ) are ‘new’ momenta and the intermediate momentum, p40 is given as a function of the ‘old’ and ‘new’ momenta by 2 J 2 − j12 − l 2 − 2p04 (M − p40 ) q40 (p40 ) = ±arccos p : 2 − (M − p0 ) 2 ) 2 (l 2 − p402 )(j12 4

But F4 in Eq. (53) transforms m12 ; j12 to M; J , not just a one-dimensional transformation. p40 is the intermediate momentum or variable of integration. The integral in Eq. (53) can be found, as follows: Zm12 dp04 q40 (p40 ) = ±˜ ;

(54)

where ˜ = j12 1 + l2 + J3 + ml 1 + m12 2 + 2M3 ; ml = M − m12 ,

 2 2 + l 2 − J 2 ) − ml (j12 + J 2 − l2) −M ( j12 ; (j12 ; l; J )j12   2 2 ) + M (l 2 + j12 − J 2) m12 (l2 + J 2 − j12 ; 2 = arccos (j12 ; l; J )l   2 2 − l 2 ) − ml (J 2 + l 2 − j12 ) m12 (J 2 + j12 ; 3 = arccos (j12 ; l; J )J   2 j12 + J 2 − l 2 − 2Mm12 ; 1 = arccos 2J j12  2  2 − 2ml M l + J 2 − j12 ; 2 = arccos 2J l   2 + l 2 − 2m12 ml −J 2 + j12 ; 3 = arccos 2j12 l q 4 + 2j 2 l 2 + 2j 2 J 2 + 2l 2 J 2 = 2|j × J| ; ( j12 ; l; J ) = −l4 − J 4 − j12 12 12 12 1 = arccos



(55) (56) (57) (58) (59) (60) (61)

with l2 = l 2 − ml2 ; J2 = J 2 − M 2 . The polar angles have the following geometriˆ de ned to be along the direction of cal interpretation. The unit vectors nˆ j12 ; nˆ l ; nˆ J ; N

R.E. Kolesnick / Physica A 258 (1998) 395–413

407

zˆ × j12 ; zˆ × l; zˆ × J and j12 × l, respectively. Thus nˆ l (nˆ j12 ) is along the direction of the ascending node from which ql (qj12 ) is measured. The angles 1 , 2 and 3 can be identi ed with  − (Nˆ nˆ j12 );  − (Nˆ nˆ l ) and (Nˆ nˆ J ) respectively, where (Nˆ nˆ j12 ) is the angle measured from Nˆ to nˆ j12 in sense of increasing qj12 , and similarly for (Nˆ nˆ J ). The angles 1 ; 2 and 3 can be identi ed with qm12 − QM ; Qml − QM and QM , respectively, where Qm12 ; qml ; QM are the angles measured from the Y axis to nˆ j12 ; nˆ l and nˆ J , respectively. Thus ˜ = j12 ( − (Nˆ nˆ j12 )) + l( − (Nˆ nˆ l )) + J(Nˆ nˆ J ) +ml ql + m12 Qm12 + MQM :

(62)

Since an identity transformation for M; j12 and l cannot be obtained with an F4 -type generator we introduce the corresponding F2 -generator F2 ( j1 ; j2 ; j12 ; l; J; M ) = Q12 j12 + Qm12 m12 + ql l + QM M + qml ml − (˜ − ( j12 + l)) ;

(63)

where the transformation has been carried out in two stages, using the continuous property of F4 and the de nition of F2 . From this equation, we obtain Qj12 = @F2 =@j12 = Q12 + (Nˆ nˆ j12 ). Similarly Ql = @F2 =@l = ql + (Nˆ nˆ l ); Q J = @F2 =@J = −(Nˆ nˆ J ). The J; M are constants of motion for fully coupled rotors.

4.2. Cartesian and angle–action variables for the fully coupled case The canonical transformation from the partly coupled to the fully coupled set of canonical variables is written as F4 ( j12 ; l; m12 ; (M − m12 ); j1 ; j2 ; l; J; M ) : For the fully coupled case, the canonical angle variables are Qj1 ; Qj2 ; Q J ; QM ; Ql and ˆ Expressions for rˆ1 ; rˆ2 ; R ˆ would be the angle cos J = M=J . Here cos j12 = ˆj12 · J. following: rˆ1 = M(QM ; J ; Q J )M(Qm12 ; j12 ; Qj12 )M(Qm1 ; j1 ; Qj1 )† (0; 1; 0)† ;

(64)

rˆ2 = M(QM ; J ; Q J )M(Qm12 ; j12 ; Qj12 )M(Qm2 ; j2 ; Qj2 )† (0; 1; 0)† ;

(65)

ˆ = M(QM ; J ; Q J )M(Qm12 ; j12 ; Qj12 )M(Qml ; l ; Ql )† (0; 1; 0)† ; R

(66)

where angle j1 ; j2 ; l we consider as given and chosen to be sin j1 = | ˆj1 × ˆj12 | = 1=2( ˆj1 ; ˆj2 ; ˆj12 ) ;

(67)

sin j2 = | ˆj2 × ˆj12 | = 1=2(ˆj1 ; ˆj2 ; ˆj12 )j2 =j1 ;

(68)

ˆ sin l = |ˆl × ˆj12 | = 1=2(ˆl; ˆj12 ; J)J=l :

(69)

and which can be expressed in terms of the canonical actions J; j1 ; j2 ; j12 ; l; M in the three various geometrical con gurations.

408

R.E. Kolesnick / Physica A 258 (1998) 395–413

4.3. Expressing the potential in terms of the fully coupled variables. Space geometry coupling of two momenta j12 and l The expression for cos 1 can be written in terms of the fully coupled variables using Eqs. (64) and (66) ˆ · rˆ1 = (0; 1; 0)M(Qml ; l ; Ql )M(Qmj ; j1 ; Qj1 )† (0; 1; 0) cos 1 = R 1

(70)

= (cos Qj1 cos Ql + cos j1 cos l sin Qj1 sin Ql ) cos(Qmj1 − Qml ) + (sin Qj1 cos Ql cos j1 − cos Qj1 sin Ql cos l ) sin(Qmj1 − Qml ) + sin Qj1 sin Ql sin j1 sin l :

(71)

The cosine of the angle between j1 and l in terms of j1 and l can be written as in [17]: cos(l ; j1 ) = cos l cos j1 + sin l sin j1 cos(Qml − Qmj1 ) :

(72)

Hence, cos 1 = (cos Qj1 cos Ql + cos l cos j1 sin Qj1 sin Ql )

cos(l ; j1 ) − cos l cos j1 sin l sin j1

+ (sin Qj1 cos Ql cos j1 − cos Qj1 sin Ql cos l ) s (cos(l ; j1 ) − cos l cos j1 ) 2 × 1− + sin Qj1 sin Ql sin l sin j1 : (sin j1 sin l ) 2

(73)

The expression for the cos 2 could be written in the general form from Eqs. (64) and (65) ˆ · rˆ2 = (0; 1; 0)M(Ql ; l ; Ql )M(Qmj ; j2 ; Qj2 )† (0; 1; 0)† cos 2 = R 2

(74)

= (cos Qj2 cos Ql + cos j2 cos l sin Qj2 sin Ql ) cos(Qmj2 − Qml ) +(sin Qj2 cos Ql cos j2 − cos Qj2 sin Ql cos l ) sin(Qmj2 − Qml ) + sin Qj2 sin Ql sin j2 sin l :

(75)

Cosine of the angle between l and j 2 in terms of j2 and l can be written as in [17]: cos( j 2 ; l) = cos j2 cos l + sin j2 sin l cos(Qmj2 − Qml ) :

(76)

So, the general expression for cos 2 in action–angles variables can be easily derived from Eqs. (74)–(76). In the general formulation it can be written as cos 2 = (cos Qj2 cos Ql + cos l cos j2 sin Qj2 sin Ql )

cos(l ; j 2 ) − cos j2 cos l sin l sin j2

+ (sin Qj2 cos Ql cos j2 − cos Qj2 sin Ql cos l )

R.E. Kolesnick / Physica A 258 (1998) 395–413

s ×

1−

409

(cos(l ; j 2 ) − cos l cos j2 ) 2 (sin l sin j2 ) 2

+ sin Qj2 sin Ql sin l sin j2 :

(77)

In the fully coupled case we can express the angles (73) and (77) in terms of given l and J; j1 ; j2 ; j12 via geometrical description. The relations between angles are listed below cos( j1 ; l) =

−h12 + l 2 + j12 ; 2lj1

(78)

cos( j 2 ; l) =

−h22 + l 2 + j22 : 2lj2

(79)

From these equations one can take cos(j1 ; l) and cos( j 2 ; l) to substitute to the Eqs. (70) and (74). So that, the description would be closed. The expression for cos may be written in the general form cos

= rˆ1 · rˆ2 = (0; 1; 0)M(Qmj1 ; j1 ; Qj1 )M(Qmj2 ; j2 ; Qj2 )† (0; 1; 0)†

(80)

= (cos Qj1 cos Qj2 + cos j1 cos j2 sin Qj1 sin Qj2 ) cos(Qmj1 − Qmj2 ) +(sin Qj1 cos Qj2 cos j1 − cos Qj1 sin Qj2 cos j2 ) sin(Qmj1 − Qmj2 ) + sin Qj1 sin Qj2 sin j1 sin j2 :

(81)

The cosine of the angle  between j1 and j 2 can be written in terms of j1 and j2 as in [17]: cos  = cos j1 cos j2 + sin j1 sin j2 cos(Qmj1 − Qmj2 ) 2 − j12 − j22 )=2j1 j2 : = ( j12

So, the general expression for the cos Eqs. (80)–(82) and is given by cos

(82) in action–angle variables can be derived from

= (cos Qj1 cos Qj2 + cos j1 cos j2 sin Qj1 sin Qj2 )

cos  − cos j1 cos j2 sin j1 sin j2

+ (sin Qj1 cos Qj2 cos j1 − cos Qj1 sin Qj2 cos j2 ) s ×

1−

(cos  − cos j1 cos j2 ) 2 + sin Qj1 sin Qj2 sin j1 sin j2 : (sin j1 sin j2 ) 2

(83)

So, we have obtained the Hamiltonian for two coupled rotors in the fully coupled set of angle–action variables representation.

410

R.E. Kolesnick / Physica A 258 (1998) 395–413

4.4. Generating function eects from partly coupled to fully coupled set of variables. Relations for j12 and l coupling The generating function F4 for that case can be evaluated as well. According to de nitions (9)–(13), we express the polar angles 1 ; 2 ; 3 ; 1 ; 2 and 3 in terms of geometrical angles   −cos j12 cos l − cos l 1 = arccos ; (84) sin j12 sin l   cos l cos l + cos j12 ; (85) 2 = arccos sin l sin l   −cos l cos j12 + cos j12 3 = arccos ; (86) sin l sin j12 ! ˆ − cos j12 cos J cos( ˆj12 ; J) ; (87) 1 = arccos sin j12 sin J ˆ − cos J cos l cos( ˆl ; J) 2 = arccos sin J sin l

!

cos( ˆl ; ˆj12 ) − cos j12 cos l 3 = arccos sin j12 sin l

;

(88)

! :

(89)

We can connect coupled and partly coupled canonical angles, to get details regarding the generating function. Since the Hamiltonian is independent of Q J and Q M ; J; M are constants of motion. 4.5. Generating function for the plane geometry coupling j12 and l It was implied in Section 3, that OZ is lying in the plane of l ; j1 ; j 2 ; J. The expressions for 1 ; 2 ; would be written as ˆ · rˆ1 = cos 1 = −cos Qj1 cos Ql − ( ˆl · ˆj1 ) sin Qj1 sin Ql ; R

(90)

ˆ · rˆ2 = cos 2 = cos Qj2 cos Ql + ( ˆl · ˆj 2 ) sin Qj2 sin Ql ; R

(91)

rˆ1 · rˆ2 = cos = cos Qj1 cos Qj2 + ( ˆj 1 · ˆj 2 ) sin Qj1 sin Qj2 :

(92)

The results would be written in more detailed form: cos 1 = −cos Qj1 cos Ql −

2 2 + j12 − j22 )(j12 − l2 − J 2) j1 l + ( j12 × sin Qj1 sin Ql ; 2 4j12 j1 J

(93)

R.E. Kolesnick / Physica A 258 (1998) 395–413

411

cos 2 = cos Qj2 cos Ql +

2 2 + j22 − j12 )(j12 − l2 + J 2) j2 l + ( j12 × sin Qj2 sin Ql ; 2 j J 4j12 2

cos = cos Qj1 cos Qj2 −

2 j12 − j12 − j22 sin Qj1 sin Qj2 ; 2j1 j2

(94) (95)

where j 1 = | ˆj 1 × ˆj12 | = 1=2( ˆj1 ; ˆj 2 ; ˆj12 ) ;

(96)

j2 = | ˆj 2 × ˆj12 | = 1=2( ˆj1 ; ˆj 2 ; ˆj12 )j2 =j1 ;

(97)

ˆ l = | ˆl × ˆj12 | = 1=2( ˆj12 ; ˆl ; J)J=l :

(98)

We note that the position of axis OZ can take up the three dierent con gurations relative to the triangle formed by j1 ; j 2 ; l; J. Here we consider only the case when OZ is in the plane formed by the j1 ; j 2 ; l; J and compare it with the case, when J is parallel to OZ. 4.6. Generating set of coordinates for the case of the xed total momenta J The obtainment of the particular cases can be useful for consistency of consideration. So, the xed total momenta alignment was of particular interest for a long time. This geometry corresponds to the case when J = 0, it means that the angles Q J = 0; Q M = 0. For such a case, as we can see, the generating function is unde ned. So, in such a simpli ed case there is no canonical transformation eecting a change from the partly coupled set of coordinates to the fully coupled one. 5. Summary and conclusions In the absence of a comprehensive model for the coupling of two rotors a new concept was developed and a new method for the coupling was formulated. Thus, we found an approach to the problem. By using an M-matrix approach the classical theory of scattering of two coupled rotors has been studied in detail. Now, it is possible to take the classical trajectory where total angular momentum of coupled rotors is conserved. This paper makes principal steps in the direction to describe properly the total angular momentum motion. Various possible methods of reduction to particular cases are considered for classical rotor–rotor collisions. Thus, some of the simpli ed geometrical con gurations of two coupled rotors were distinguished. They are: the xed con guration [6] of J and the plane con guration of it against an arbitrary con guration. The method can also be applied, if the pairs are coupled by combinations of angular momenta and in our applications the pairs combined by many Coriolis and centrifugal terms are presented. It is found that our model disagrees with the previous

412

R.E. Kolesnick / Physica A 258 (1998) 395–413

one. First, the accurate description of the coupling of two rotors to introduce angle– action variables has been given. The partly coupled set of coordinates and the fully coupled one has been considered separately. A detailed analysis of dierent geometrical types of coupling schemes has been carried out. We have classi ed and developed some theoretical models of coupling and introduced action–angle variables of coupled rotors. The theoretical model is worked out to study transport properties of gases with rotational degrees of freedom. The structure of resulting expressions under consideration show that they dier in principle from the expressions that were obtained by other approaches. It is feasible to determine the interplay between dierent types of coupling in uencing the transport properties. The eect of coupling on energy transfer can be determined in the framework of such a method. The choice of the right classical trajectory is provided. The principal result is an analytical prescription to write the Hamiltonian for two rotors in the case, when the total angular momentum is orientated arbitrarily, this facilitates solving the equations of motion of diatomic molecules in trajectory calculations. Such results can give far-reaching consequences in the development of kinetic models for rotational kinetics. Thus, for calculations of state-to-state rates and cross-sections one must use the j 1 ; j 2 ; j12 ; J set to denote the rotational state of the diatomic molecule instead of just j1 and j2 . The Classical coupled states approximation for the uncoupled case (see Section 2.2) was considered recently with encouraging results [19]. The analysis will be continued to other aspects of the problem, such as classical S-matrix development and numerical results will be reported. Previously, the usage of approximative treatment of momenta coupling was imperative. The general method will undoubtedly serve well for other molecular simulation techniques as they are developed. Acknowledgements This work was supported by a grant from the Royal Society – NATO Postdoctoral Fellowship Programme. The author thanks Prof. A.S. Dickinson for useful discussions. References [1] O.V. Prezdo, V.V. Kisil, Phys. Rev. A 56 (1997) 162. [2] F.R.W. McCourt, J.J.M. Beenakker, W.E. Kohler, I. Kuscer, Nonequilibrium Phenomena in Polyatomic Gases, vol. 1, Oxford Science Publications, Oxford, 1990, chap. 4.2. [3] P.L. James, I.R. Sims, I.W.M. Smith, Chem. Phys. Lett. 272 (1997) 412. [4] A.J.C. Varandas, W. Wang, Chem. Phys. 215 (1997) 167. [5] R.J. Cross, J. Chem. Phys. 95 (1991) 1. [6] A.F. Turfa, D.E. Fitz, R.A. Marcus, J. Chem. Phys. 67 (1977) 4463. [7] N.J. Smith, J. Chem. Phys. 85 (1987). [8] W.H. Miller, Adv. Chem. Phys. 25 (1974) 69. [9] R.E. Kolesnick, Rotational Relaxation in the Nonequilibrium Gas Fluxes, PhD thesis, Mathematics and Mechanics Department, Sakt-Petersburg State University, 1984. [10] W.M. Huo, S. Green, J. Chem. Phys. 104 (1996) 7590. [11] A.E. DePristo, M.H. Alexander, J. Chem. Phys. 66 (3) (1976) 1334.

R.E. Kolesnick / Physica A 258 (1998) 395–413 [12] [13] [14] [15] [16] [17] [18] [19]

413

D. Capilletti, F. Vecchiocattivi, F. Pirani, F.R.W. McCourt, Chem. Phys. Lett. 248 (1996) 237. H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1980. M.E. Rose, Elementary Theory of Angular Momentum, NY, 1995. D. Secrest, Theory of Rotational and Vibrational Energy Transfer in Molecules, Annu. Rev. Phys. Chem. 24 (1973) 389. A.S. Dickinson, W.-K. Liu, J. Phys. Chem. 90 (1986) 3612. R.M. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Wiley, New York, 1988. A.M. Perelomov, Generalised Coherent Conditions, Springer, Berlin, 1994. R.E. Kolesnick, A.S. Dickinson, Abstr. XX-ICPEAC, 1997, Vienna.