The Geometry of Formaldehyde

JOURNAL OF MOLECULAR SPECTROSCOPY 179, 65–72 (1996)ARTICLE NO. 0184

The Geometry of Formaldehyde

Stuart Carter* and Nicholas C. Handy†

*Department of Chemistry, University of Reading, Whiteknights, Reading RG6 2AD, United Kingdom; and †Department of Chemistry,University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

Received February 16, 1996; in revised form April 22, 1996

A new six-dimensional variational code is presented for the determination of the rovibrational energy levels of fouratom molecules which are centrally connected. Internal (displacement) coordinates are used. Using formaldehyde asthe first example we have attempted to refine a quartic potential surface and simultaneously to optimize the geometry.The parameters have been adjusted to obtain as good agreement as possible with the lowest 65 J Å 0 observed vibrationallines of H2CO and the fundamentals of D2CO, and the J Å 1 observed rotational lines of the fundamentals of bothH2CO and D2CO. This is a highly nonlinear problem, and it is not possible to refine separately the geometry and forcefield. We have simultaneously optimized the geometry and quadratic force field, and separately the cubic and quarticforce field. Fermi and Coriolis resonances have been reproduced. The predicted geometry, with uncertainties, is CH Å1.1003 { 0.0005 A, CO Å 1.2031 { 0.0005 A, and HCO Å 121.62 { 0.057. q 1996 Academic Press, Inc.

1. INTRODUCTION To do this we have had to extend our J Å 0 variationalvibrational tetra-atomic code to a J x 0 variational rovibra-tional tetra-atomic code. Bramley and Handy (BH) haveRecently, we presented a force field (1) for formaldehydealready presented such calculations on linearly connectedwhich was derived by refining its force constants such thattetra-atomics (e.g., HCCH) (5) using a preliminary version,good agreement was obtained between the observed 65 low-which did not take advantage of the fact that Ka is almost aest JÅ 0 vibrational levels (with respect to the ground state)good quantum number for reasonably low values of J . Weand their calculated values. For such studies we use thepresent, in Section 2, the details of our new J x 0 code forvariational method. In (1) we described our new variationalcentrally connected four-atom molecules. Section 3 reportsvibrational code for these six-dimensional calculations, forthe results of our optimizations for formaldehyde, using Jwhich we use internal coordinates. We described our expan-Å 0 and J Å 1 observed data for H2CO and D2CO.sion set and also the method by which we updated the force

constants in successive iterations. We commenced with thequartic ab initio force field of Martin et al. (2) and we also 2. THEORYused their corresponding equilibrium geometry. Our final

Handy (6) has already presented the J Å 0 vibrationalmean absolute error was 1.1 cm01 , which we viewed askinetic energy operator for formaldehyde, obtained using thesatisfactory because the observed dispersed fluorescencechain rule and a computer algebra program. The internaldata (which we used) of Bouwens et al. (3) had an errorcoordinates were the three bond lengths CH1(r1) , CH2(r2) ,bar of 1 cm01 . In that paper we compared the potential withand CO(r3) , the two angles H1CO and H2CO (u1 , u2) , andthe one of Burleigh et al. (4) .the book angle f. We believe that this coordinate system isWe recognized that such a force field cannot be com-optimum in two ways: ( i) it allows a systematic contractionpletely accurate because the equilibrium geometry had notscheme, commencing from one-dimensional contractedbeen refined. There is considerable uncertainty with the ge-functions, and (ii) because these coordinates mimic normalometry of formaldehyde. In Table 1 may be found a varietycoordinates, this allows for accurate assignments of the spec-of predicted geometries. Note that suggested bond lengthstrum without examining the wavefunction in detail. The ex-vary by up to 0.01 A, and even the ab initioists appear totra terms needed to form the rovibrational kinetic energybe uncertain by 0.003 A, in particular because of the effectoperator are given in the Appendix. They involve Jx , Jy ,of core electrons which are often not included in such calcu-

lations. Thus it is of particular interest to see if we can refine Jz , J 2x , J 2

y , J 2z , [Jx , Jy]/ , [Jy , Jz]/ , [Jz , Jx]/ , where CO is

along the y-axis, and the bisector of the book angle is alongthe geometry to a high degree of accuracy. In our previousstudies no observed data had been included for rotational the y-axis. In the generation of such operators and the evalua-

tion of their matrix elements, considerable care has to belevels. Furthermore no attempt had been made to verify theforce field for another isotope. The purpose of the present taken with signs. Indeed we cannot be absolutely certain

that our new code is correct, but the fact that our results forpaper is to include rotational level data in the optimizations.

650022-2852/96 $18.00

Copyright q 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

AID JMS 7055 / 6t0d$$$141 07-24-96 16:02:48 mspa AP: Mol Spec

CARTER AND HANDY66

TABLE 1S(S/

u ){dl1u10 (u1)dl2

u20 (u2) / dl2u20 (u1)dl1

u10 (u2)}cosSv2 fDPredictions for the Equilibrium Geometry of Formaldehyde

1 D 101(0) / S(S/

u ) i{dl1u10 (u1)dl2u20 (u2)

0 dl2u20 (u1)dl1

u10 (u2)}sinSv2 fDD 100(/) . [4]

In this notation S(S/g , S/

u ) are stretching expansion func-tions involving the three bond lengths; dl

u0 are Jacobi func-tions; DJ

0K are rotational functions; and l, u1 , u2 , v, and Kare related integers. For K Å 0, u1 Å u2 Å 0; for K Å 1, u1

Å 1, u2 Å 0. These functions guarantee that all singularitieson the acetylene surface are exactly canceled. However,

formaldehyde are entirely plausible is one check. Further- whereas in acetylene the singular point u Å p occurs atmore we have verified that our linearly connected energies the equilibrium geometry, in formaldehyde the equilibriumof acetylene for J Å 0 and J Å 1 agree with the original geometry is approximately equidistant between the two sin-values in (7) . gular points u Å 0, p. It is therefore possible to relax the

In our previous paper we gave details of our primitive relations between these integers, and therefore we decidedsix-dimensional expansion set for the J Å 0 calculations. to use the above expansion functions with u1 , u2 Å 0 every-The same stretch basis is used here. For the angular basis where. In this case the Jacobi functions become Legendrewe observe that the rovibrational kinetic energy operators functions.for acetylene and formaldehyde are very similar. Indeed all In order to optimize the angular expansion set we firstthe terms of the acetylene-like operator are present in the performed a one-dimensional contraction in u, reducingformaldehyde-like operator. There are some extra terms in the number of functions from 2 ∗ NBF4,5 Legendre func-the formaldehyde operator arising because of the central tions to NBF4,5 contracted functions. Three similar one-connection. This discussion means that the previous analysis dimensional contractions were performed for theof BH (5) on the selection of angular basis functions for cos (vf ) , cos ( (v / 2 )f ) , and sin ( (v / 2 )f ) functions,HCCH can be carried over to formaldehyde. That analysis yielding NV6 contracted functions from NBF6 primi-concentrated on the fact that the sin02u singularities in the tives. Then the lowest energy f contraction of each typekinetic energy operator had to be eliminated when the opera- was used in the two-dimensional u1 , u2 contraction, yield-tor acts on the basis functions. The result of the analysis ing NV45 two-dimensional contracted functions fromprovided the following expansion functions with S0

g symme- NBF4∗ NBF5 starting functions. This gave both A1 andtry: B2 symmetry combinations. These NV45 u functions, of

both symmetries in the K Å 1 case, were then combinedwith the appropriate NV6 f functions and the appropriate

K Å 0 rotational functions, and three-dimensional u1 , u2 , f con-tractions were performed, yielding N3D2 angular expan-S(S/

g ){dl1u10 (u1)dl2u20 (u2)sion functions from 2 ∗NV45 ∗NV6 starting functions( for K Å 0, the size is NV45 ∗NV6) . These are called K-/ dl2

u20 (u1)dl1u10 (u2)}cos(vf)D 1

00 [1]diagonal contractions, the elements of different symmetrybeing connected by the Jz parts of the kinetic energyS(S/

u ){dl1u10 (u1)dl2u20 (u2)operator.

Next, we use these optimized three-dimensional bend-0 dl2u20 (u1)dl1u10 (u2)}cos(vf)D 100 [2]

ing functions with the optimized three-dimensionalstretch functions (5 ) in order to perform a K-diagonalK Å 1‘‘vibrational’’ analysis for both K Å 0 and K Å 1. Itremains to obtain the final rovibrational energies by con-

S(S/g ){dl1

u10 (u1)dl2u20 (u2) 0 dl2

u20 (u1)dl1u10 (u2)}cosSv2 fD tracting the K Å 0 and K Å 1 functions using the DK {

1 parts of the kinetic energy operator (note that in the1 D 1

01(0) / S(S/g ) i{dl1u10 (u1)dl2u20 (u2) kinetic energy operator given in the Appendix, there is

no direct term involving DK { 2 ) .The first step in the rovibrational analysis is to collect/ dl2u20 (u2)dl1u10 (u2)}sinSv2 fDD 1

00(/) [3]subsets of the K Å 0 and K Å 1 contracted functions. In

Copyright q 1996 by Academic Press, Inc.


THE GEOMETRY OF FORMALDEHYDE 67

both cases, the K-diagonal matrices are formed from bands have the same symmetries on several occasions, albeitfor different values of Ka in some instances. Because ofN3D1 stretching functions and N3D2 bending functions,

noting that ( for example ) there will be (/ // ) and (0 /0 ) this, the specific ‘‘observed’’ rotational levels were extractedfrom the diagonalization of a 9∗9 matrix which was con-contributions to the same overall K-diagonal symmetric

block (and also (/ /0 ) and (0 // ) contributions to the structed by us from the data in (10) . In particular, n4 andn6 have the same symmetries for both sets of Ka Å É1É. Thissame overall K-diagonal asymmetric block ) ; see (5 ) .

This means that the total size of the K-diagonal matrix causes the classic Coriolis resonance, as the correspondingJ Å 0 levels are separated by only 82 cm01 . This turnedis 2 ∗N3D1 ∗N3D2. A total of N6D eigenfunctions and

corresponding eigenvalues of this matrix are stored, com- out to be one of a number of problems in our subsequentoptimization of the force field, because this interaction mustmencing with the function of lowest energy. In doing this,

we minimize waste, such that the top functions for both be reproduced. The J Å 0 and J Å 1 levels for the groundvibrational state and the six fundamentals are given in TableK Å 0 and K Å 1 will correspond to approximately equal

energies. The next step is to insert the eigenvalues of 2. In that table the corresponding data are also given forD2CO, derived from Refs. (13–15) .these functions along the diagonal elements of K Å 0

and K Å 1 block matrices ( total size N6D) , and use the We commenced with a study of H2CO with the force fieldpresented in our previous paper (1) , and with the MLTcorresponding eigenvectors, together with the appropriate

DK { 1 matrix element given in ( 8 ) in order to construct geometry (given in Table 1) which was used by us to obtainthat force field. An initial J Å 1 variational calculation gavethe off-diagonal block connecting the K Å 0 and K Å 1

functions. This procedure is related to the two-step proce- rotational levels which were in error by up to {0.1 cm01 ,implying that the rotational constants (and hence the geome-dure of Tennyson and Sutcliffe (9 ) . Note that in (8 ) , the

matrix elements correspond to rotational functions which try) needed considerable adjustment. We started by observ-ing that, relative to J Å 0, the Ka level for J Å 1 is givenobey standard commutation, whereas the operator in the

Appendix was derived for functions appropriate to a mol- by B / C . Both B and C are dominated by the CO bondlength. The CO bond length was therefore adjusted fromecule-fixed axis, which obey nonstandard commutation.

To convert from one to the other merely involves chang- 1.2096 to 1.2036 A; as a result the error in these levelswas reduced from Ç0.1 to Ç0.003 cm01 . Those key forceing the signs of all the J-operators in TVR .

For both K-diagonal and DK { 1 matrices, we use a constants involving the CO bond were then rerefined againstJ Å 0 data, and as a result the original accuracy in (1) wasGIVENS diagonalizer, and only calculate eigenvalues and

eigenvectors required in subsequent stages. To be safe, we retrieved. For the two Ka Å 1 rotational levels those for Kc

Å 1 are given by A / C , whereas those for Ka Å 0 are givencalculate N6D/2 / 20 K-diagonal functions from a total of2∗N3D1∗N3D2, and 10 final rovibrational functions (for J by A / B . B and C have been improved by changing the

CO bond length; the A rotational constant does not involveÅ 1) are sufficient for comparison with all of the availableexperimental data. the CO bond, but is determined by the geometry of the CH2

group. Therefore the CH bond length was next adjusted,from 1.1033 to 1.1011 A, which partially improved some of3. THE GEOMETRY AND FORCE FIELD OFthe Ka Å 1 levels. The force field was again rerefined. WeFORMALDEHYDEkept the cubic force constants in our list of key force con-stants ‘‘frozen’’ during these initial refinements.Except for the six fundamentals, the lowest 65 J Å 0

vibrational wavenumbers were taken from data of Bouwens At this stage problems began to arise. We noticed that theKa Å 1 rotational levels for n4 and n6 were in substantialet al. (3) . These data have an estimated maximum uncer-

tainty of 1 cm01 . Much more accurate data for the vibrational error, Ç{0.08 cm01 , indicating the importance of this Cori-olis resonance. It was possible to remove much of this errorground state and six fundamentals were taken from the pa-

pers which describe the individual spectroscopic analyses of by a simultaneous change of the CH bond length and theHCO angle, without destroying the ground state rotationalthese bands (10–12) . These data consist of the J Å 0 vibra-

tional wavenumber, effective rotational constants, centrifu- constants. Although such a change again removed the accu-racy of the J Å 0 levels, a refinement of the force field wasgal distortion constants, and Coriolis coupling constants,

and, with the exception of n3 , n4 , and n6 , the three asymmet- possible in such a way that all the H2CO data were wellreproduced. We were unsure whether the removal of theric-top J Å 1 rotational levels for formaldehyde were ob-

tained by diagonalizing 3∗3 matrices corresponding to Ka Coriolis resonance by geometry alteration alone was correctbecause another argument (due to I. M. Mills) suggests thatÅ 01, 0, /1. There are extra coupling terms involved for

n3 , n4 , and n6 . Although these levels are A1 , B1 , and B2 , it can be removed by adjustment of some quadratic forceconstants. The Coriolis zeta constant depends upon the Lrespectively, for J Å 0, for J Å 1 the Ka Å 0 levels are B1 ,

A1 , and A2 , but the Ka Å 1 levels are A2 or A1 , B2 or B1 , vectors, in this case those specific to the Q4 and Q6 modes(these are the out-of-plane and wag motions) . The natureand B1 or B2 (for Kc Å 1 or 0), respectively. Hence, these



CARTER AND HANDY68

TABLE 2J Å 0 and J Å 1 Rovibrational Levels in cm01 of the Ground Vibrational State and

the Fundamentals of H2CO and D2CO (See Text) (J Å 1 Are Relative to J Å 0)

of Q4 is fixed by symmetry, but the wag is linked by symme- appropriate to use atomic masses (i.e., mass of carbon Å 12unified atomic mass units (Å12 1 1.6605402 1 10027 kg)) .try to S5 and S6 , which are the B2 symmetry coordinates for

asymmetric stretch and bend. This argument suggests that Because of this mass question, we do not believe that it ispossible for the model to reproduce all the rotational (J Åthe Coriolis constant will depend upon F55 , F56 , and F66 ,

the force constants in symmetry coordinates. Furthermore 1) levels to better than 0.001 cm01 . There has been muchin the literature on this problem; see for example Meyer,we must be uncertain whether the data for H2CO alone are

sufficient to determine the force field and geometry. Botschwina, and Burton (16) .On performing the rovibrational calculations on D2COIt was therefore appropriate to study another isotope. We

chose D2CO because it has C2£ symmetry. Because we are using the above geometry, it was found that both the funda-mentals and the rotational energies were generally in goodhoping to obtain rotational energy levels to an accuracy an

the order of 0.001 cm01 , it was important to reconsider agreement with observed values, the principal exception be-ing n1 (CD symmetric stretch, 2061 cm01) and n2 (COthe atomic masses and indeed the whole problem we are

attempting. We are trying to find a surface (the same for all stretch, 1702 cm01) fundamentals, which were in error by/8.4 and 06.3 cm01 , respectively. We found that this dis-isotopes) which reproduces the rovibrational levels. This is

not quite the Born–Oppenheimer problem, but more like the crepancy was eliminated by a small change in the quadraticCO–CH force constant. Because the fundamentals are muchclassical particle rolling on a surface. To go beyond the B–

O approximation, one introduces the diagonal Born–Oppen- further apart in H2CO, they were relatively unaffected.The above discussion and other less apparent discrepanc-heimer correction, a mass-dependent surface correction,

which is not like our problem. In our case the model can be ies between our calculated values and observed values con-vinced us that a much more systematic approach was needed.improved by including the mass of the associated electrons

into the nuclear mass. Thus for our model problem it is We therefore decided to determine the geometry (r , u, R)




TABLE 3Vibrational Band Origins (cm01 ) of H2CO, Calculated by the Variational Method, Compared with Experiment

and the quadratic force field (10 unique constants) through D2CO), and 0.028 cm01 (n1 , Ka Å 1, J Å 1, D2CO). Thusalthough the mean errors are acceptably small, there area rigorous least-squares adjustment analysis. Therefore each

of these 13 adjustables was slightly changed and new errors some individual errors which we have been unable to elimi-nate, presumably because of coupling between the geometry(for H2CO and D2CO) for the fundamentals and ground state

(J Å 0 and J Å 1) were determined, that is, for the 54 data and cubic/quartic force constants. We suspect that such anelimination will move the geometry a little (e.g., 0.0005 A,given in Table 2. The result of this least-squares calculation

(using weights of unity for vibrational data and 100 for 0.057 pessimistically) .The final force field is given in Table 4. It is encouragingrotational data) showed that the Coriolis resonance is re-

moved by adjustment of the cited force constants and that to note how close it is to the starting ab initio force field ofMartin et al. (2) .as a result the above geometry was in error. This demon-

strates that an incomplete optimization using the variationalmethod can lead to both incorrect potentials and incorrect 4. SUMMARYgeometries.

Having refined the geometry and quadratic force field, we In this paper we have described how we have used our JÅ 0 and J Å 1 variational code to refine the quartic forcethen rerefined some cubic and quartic constants against the

remaining 59 J Å 0 data of H2CO. We then returned to the field and the geometry of formaldehyde using 113 observeddata. It is a very nonlinear problem and we entered manyabove quadratic force field and geometry and refined these

again. At this stage there was little movement and so we false minima (as described in the previous section) on ourroute to a successful refinement. We are reasonably confi-stopped the refinement process. The resulting levels for the

65 J Å 0 vibrational levels are given in Table 3, where it dent of our predicted geometry; Botschwina (17) informedus that he was confident of his bond lengths (determined abis seen that the mean absolute error is 0.8 cm01 ( i.e., less

than the experimental error of 1 cm01 and a substantial im- initio) , and we are very close to his values given in Table1. If anything he considered his CO bond distance to beprovement on our earlier value of 1.1 cm01 (1)) , and that

the mean absolute error in the 42 J Å 1 levels (see Table slightly overestimated and his CH bond distance to beslightly underestimated ({0.0005 A approx). This makes2) is 0.0051 cm01 . The error in the fundamentals of D2CO

is 0.74 cm01 . We observe that maximum errors were 4.2 our prediction more certain. Our bond angle seems to be alittle smaller than most others in Table 1, but we have nocm01 (2n3n4n6 , J Å 0, 5353 cm01) , 1.9 cm01 (n3 , J Å 0,



CARTER AND HANDY70

TABLE 4reason to suspect our value. We do observe that throughoutThe Complete Force Field for Formaldehyde Derived inour refinement ÉCHÉsin u stayed remarkably constant at

This Work (the Notation Gives Powers of Dr1 /r1 , Dr2 /r2 ,0.93701 A. Our best estimate for the equilibrium geometryDr3 /r3 , Du1 , Du2 , Df ) with Units of Hartrees for the Threeof formaldehyde is CH Å 1.10034 A, CO Å 1.20312 A, andBond Stretches, Hartrees/radn for the Three AnglesHCO Å 121.627.

We have gained considerable knowledge from this optimi-zation problem. We are encouraged that we have been ableto solve it, because that means that provided there are enoughobserved data (including more than one isotope), it is possi-ble to refine to a high accuracy the force field and geometryof a four-atom molecule. In hindsight we did not proceedby the optimum route; it would have been much more effi-cient to have included three extra linear terms in Dr , DR ,Du in the adjustable potential, and then use the standardHellmann–Feynman approach in the least-squares procedure(i.e., evaluate »fiÉVjÉfi …) . We now have this in place andit is working well on our next candidate molecule, H2CS.

5. APPENDIX: ROTATIONAL TERMS IN THE KINETICENERGY OPERATOR FOR FORMALDEHYDE

This Appendix, when taken in conjunction with AppendixA of (6) , gives the derived kinetic energy operator TVR fora four-atom system such as formaldehyde and with a sym-metrically embedded molecule-fixed frame. The volume ele-ment for integration is sin u1sin u2sin bdr1dr2dr3du1du2d-fadbdg.

Nuclear masses are M1 , M2 , M3 , and M4 . The reducedmasses m1 , m2 , and m3 are then defined as

1m1

Å 1M1

/ 1M4

;1m2

Å 1M2

/ 1M4

;1m3

Å 1M3

/ 1M4

.

Terms in TVR are obtained by multiplying each operator Note. The coordinates are defined by Martin, Lee, and Taylor. Only thoseconstants which are not symmetry-related are given.by 01

2 and the corresponding prefactor; \ is taken to be 1.

ÌÌr1

iJO x : 0 2 cos(f /2)sin u1

M4r3

ÌÌr1

iJO y :2 sin(f /2)sin u1

M4r3

ÌÌr1

iJO z :sin f sin u1

M4Scot u2

r3

0 cosec u2

r2D

ÌÌr2

iJO x : 0 2 cos(f /2)sin u2

M4r3

ÌÌr2

iJO y : 0 2 sin(f /2)sin u2

M4r3




ÌÌr2

iJO z : 0sin f sin u2

M4Scot u1

r3

0 cosec u1

r1D

ÌÌr3

iJO x : 0

ÌÌr3

iJO y : 0

ÌÌr3

iJO z : 0

ÌÌu1

iJO x : 2 cos(f /2)S0 cos u1

M4r1r3

/ 1m3r

23D

ÌÌu1

iJO y : 2 sin(f /2)S cosu1

M4r1r3

0 1m3r

23D

ÌÌu1

iJO z : sin fScos u1cot u2

M4r1r3

0 cos u1cosec u2

M4r1r2

0 cot u2

m3r23

/ cosec u2

M4r2r3D

ÌÌu2

iJO x : 2 cos(f /2)S0 cos u2

M4r2r3

/ 1m3r

23D

ÌÌu2

iJO y : 2 sin(f /2)S0 cos u2

M4r2r3

/ 1m3r

23D

ÌÌu2

iJO z : sin fS0 cot u1cos u2

M4r2r3

/ cosec u1cos u2

M4r1r2

/ cot u1

m3r23

0 cosec u1

M4r1r3D

ÌÌf

iJO x : 02 sin(f /2)Scot u1

m3r23

/ cot u2

m3r23

0 cosec u1

M4r1r3

0 cosec u2

M4r2r3D

ÌÌf

iJO y : 2 cos(f /2)S0 cot u1

m3r23

/ cot u2

m3r23

/ cosec u1

M4r1r3

0 cosec u2

M4r2r3D

ÌÌf

iJO z :2 cot u1cosec u1

M4r1r3

0 2 cot u2cosec u2

M4r2r3

0 cosec2u1S 1m1r

21

/ 1m3r

23D / cosec2u2S 1

m2r22

/ 1m3r

23D

iJO x :cos(f /2)

2 Scot u1

m3r23

/ cot u2

m3r23

/ 4 sin u1

M4r1r3

/ 4 sin u2

M4r2r3

0 cosec u1

M4r1r3

0 cosec u2

M4r2r3D

iJO y :sin(f /2)

2 S0 cot u1

m3r23

/ cot u2

m3r23

0 4 sin u1

M4r1r3

/ 4 sin u2

M4r2r3

/ cosec u1

M4r1r3

0 cosec u2

M4r2r3D

iJO z :sin f

M4Scot u1sin u2

r2r3

0 sin u1cot u2

r1r3

/ sin u1cosec u2

r1r2

0 cosec u1sin u2

r1r2D

JO 2x : 0 1

m3r23



CARTER AND HANDY72

JO 2y : 0 1

m3r23

JO 2z : 0 cos f cot u1cot u2

2m3r23

/ cos f cot u1cosec u2

2M4r2r3

/ cos f cosec u1cot u2

2M4r1r3

0 cos f cosec u1cosec u2

2M4r1r2

/ cot u1cosec u1

2M4r1r3

/ cot u2cosec u2

2M4r2r3

/ 12m3r

23

0 cosec2u2

4 S 1m2r

22

/ 1m3r

23D 0 cosec2u1

4 S 1m1r

21

/ 1m3r

23D

[JO x , JO y]/ : 0

[JO x , JO z]/ :sin(f /2)

2 S0 cot u1

m3r23

/ cot u2

m3r23

/ cosec u1

M4r1r3

0 cosec u2

M4r2r3D

[JO y , JO z]/ :cos(f /2)

2 S0 cot u1

m3r23

0 cot u2

m3r23

/ cosec u1

M4r1r3

0 cosec u2

M4r2r3D

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