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Internuclear potential and equilibrium structure of the nitrous oxide molecule from rovibrational data
JOURNAL
OF MOLECULAR
SPECTROSCOPY
135,389-409 (1989)
Internuclear Potential and Equilibrium Structure of the Nitrous Oxide Molecule from Rovibrational Data J.-L. TEFFO Laboratoire de Physique Molkulaire et Atmosph&ique. Tour 13, Universitk Pierre et Marie Curie et CNRS, 4, Place Jussieu, 7S252 Paris Cedex OS, France
AND
A. CH~DIN Laboratoire de MMoroiogie Dynamique, CNRS, Ecole Polytechnique, Route Dkpartementale 36, 91128 Palaiseau Cedex, France
The internuclear potential up to sextic terms and the equilibrium structure of the nitrous oxide molecule are reinvestigated using the algebraized contact transformation method previously applied to carbon dioxide. Infrared spectroscopic data for a large number of vibrational levels belonging to six isotopic species of NzO am included in the refinement. A good agreement between calculated and experimental values and a significant improvement of the determination of the NN and NO bond lengths are obtained. 0 1989Academic press, hc.
I. INTRODUCTION
Since the first determination of the quartic force field of N20 made by Suzuki using standard perturbation theory ( I ), several authors have attempted to improve the modeling of the internuclear potential of this molecule. In 1976, Chkdin et al. (2) and Amiot (3) reported the first calculation of the sextic force field using a new algebraic approach for solving the vibrational eigenvalue problem of linear triatomic molecules (4). However, this determination, like that of Suzuki, relied upon vibrational data only. The problem was then reinvestigated by Lacy and Whiffen (5)) and more recently by Kobayashi and Suzuki (6), by means of a direct numerical diagonalization method. These authors obtained a sextic force field of N20 in terms of curvilinear internal coordinates from a set of vibrational band centers and of rotational band constants. Successfully extended to the rovibrational eigenvalue problem in the case of the carbon dioxide molecule ( 7, 8), the algebraized contact transformation method is here applied to N20 in order to reinvestigate the internuclear potential of this molecule. Moreover, we address the problem of the equilibrium structure of the N20 molecule, which cannot be separated from that of a meaningful determination of the internuclear potential. The computational method is summarized in Section II. The experimental data set is described in Section III, and the results are reported and discussed in Section IV.
389
0022-2852189 $3.00 Copyright 8
1989 by Academic Press, Inc.
All rights of reproduction in any form reserved.
390
TEFFO AND CHCDIN
FIG. 1. Instantaneousconfigurationof N20.
II. SUMMARY
OF THE METHOD
The expansion of the potential energy function I’ is carried out with respect to the three dimensionless internal coordinates c;, =
-:NN
)
t2
=
Aa,
[3
=
r231No-N0
,
where r12, r23, and ACYare defined in Fig. 1, r NN and rNo being the two equilibrium bond lengths. The expansion of the potential function being truncated at the fourth order of approximation in the perturbation theory, the N20 potential is a polynomial containing 47 coefficients up to the sixth power in the &‘s. From an initial determination, the first part of the computational program sets up the rotation-vibration energy matrix of the molecule. The Hamiltonian His expanded in a basis made of symmetrized products obtained from the dimensionless normal coordinates qi, their conjugate momenta pi, and the components of the angular momentum J. At the order of magnitude 71the basis is of degree n + 2 with respect to these elementary operators. The perturbation treatment of H is then performed by means of two successive contact transformations previously encoded ( 7, 9, 10). The calculation of the transformed Hamiltonian HT reduces to a matrix product H= = TH,
(2)
where T is the transformation matrix. The nonvanishing matrix elements of HT in the basis of the harmonic oscillator wavefunctions are listed in Table I. Each one is a product of a square root part ( 7) by a polynomial in the quantum numbers vI, v2, 12, v3, and J, the coefficients of which are called the spectroscopic constants denoted C in the following. It is worth noting that two Fermi resonances are included in the diagonalization scheme, while the Coriolis couplings (which would correspond to the missing codes 6 and 7 in Table I) are removed by the contact transformation; this is reasonable since few levels are affected by this interaction. Moreover, keeping it would lead to infinite Hamiltonian matrices. However, this means that levels coupled by strong Coriolis resonances cannot be included in the data set. The second part of the program solves the inverse eigenvalue problem using a leastsquares fit procedure. If we denote by E the set of theoretical rotation-vibration energies and by F the set of potential coefficients, the Jacobian matrix dE/dF is calculated from the product
INTERNUCLEAR
391
POTENTIAL OF N20
TABLE I Nonvanishing Matrix Elements (u, , u2, 12, q 1HT 111, + Au,, u2+ Au2, I2+ Al,, uj + Au3) 3v I
dv2
?lV 3
Code
ie2
0
0
0
0
1
-1
2
0
0
2
-2
0
1
0
3
-1
-2
1
0
4
0
0
0
2
5
-1
2
0
2
a
-1
2
0
-2
9
0
-4
1
0
10
-2
-4
0
0
11
-4
0
2
0
12
-3
2
1
0
13
aE
aE
%=dCXaHT
ac
-X-
aHT aF
’
where -dE/dC is the matrix of the partial derivatives of the energies with respect to the spectroscopic constants C defined previously, and its elements are the coefficients of the linearized expressions of the energies in terms of the spectroscopic constants (11). -aC/aHT is the matrix of the coefficients occurring in the expression of the spectroscopic constants in terms of the parameters of the transformed Hamiltonian. These coefficients are computed in the first part of the program. - dHT/aF is the matrix of the partial derivatives of the latter parameters with respect to the potential coefficients. It is written as
The matrix a V/a F is the matrix used to transform the V (4) expansion of the potential into the V(q) expansion. It is also computed at the beginning of the calculation ( 12). Once the dE/dF matrix is known, it is possible to start the iterative process for the adjustment of the potential. It is important to note that the values of the internuclear distances enter as variable parameters in this calculation, contrary to the previous calculation (2) where these distances appeared as fixed parameters derived from rotational analysis only ( 13). This was not quite a satisfactory approach since the equilibrium structure of the molecule is linked to its vibrational potential energy. Therefore, instead Of considering the tW0 rNN and rNo distances as Constant parameters, we have treated them as two additional variables that are adjusted together with the potential coefficients throughout the fitting process. For this purpose, we have introduced the partial derivatives dHT/& in the Jacobian. This was done by means of the approximation
392
TEFFO
AND CHBDIN
dHT
aH;
ar=ar-ar,
aH,
(5)
where Ho = ; C hcwi(p; 1
+ qf) + ; B,(J:- + J;)
(6)
is the zeroth-order Hamiltonian. Since the matrix elements of p’ + qf and Ji + Jf. are not r-dependent, the evaluation of aHo/& reduces to the calculation of the partial derivatives of the WI’sand of Be. Practically, this approximation means that the partial derivatives, with respect to rNN and YNo, of all the other spectroscopic constants are neglected. The validity of this approximation was numerically tested in the CO2 case, which is more simple (14), and was found satisfactory. The eight quantities dw;/dr and aBe/& are given in the Appendix. III. EXPERIMENTAL
DETAILS
As in the CO2 case ( 7, 8) the potential was not fitted directly to the experimental rotation-vibration energies E,(J) but to the usual band constants defined as E,(J)
= G, + B,J(J+
1) - D,J’(J+
l)*.
(7)
This means that the data concerning the vibrational levels for which Eq. (7) is poorly significant cannot be included in the data set. Most of the experimental data used in this work were taken from the extensive N20 studies of Amiot and Guelachvili (15, 16) and Amiot ( 17, 28). The source of the other data is mentioned in Table VIII. Six isotopic species have been included in the least-squares refinement. The highest levels are the u3 = 4 levels which lie about 9000 cm-’ above the fundamental. The data weights are inversely proportional to the squares of the experimental uncertainties, the greatest weights being assigned to the rotational constants of the ground states for which microwave results are available ( 19, 20). IV. RESULTS
AND DISCUSSION
As in Ref. (2), we used in this fitting process the expansion of the potential with respect to the 5‘coordinates of Eq. ( 1)) the 47 coefficients of this expansion being labeled with an f symbol. Actually, starting from the quartic potential of Suzuki ( 1)) a preliminary adjustment was performed with an expansion of the potential with respect to “quasi-normal” coordinates (4)) the corresponding coefficients being labeled with an F symbol. The main interest of this preliminary calculation is that the F’s have the same order of magnitude as the coefficient of the expansion of the potential with respect to the q’s for any isotopic species, which is a convenient situation to estimate the convergence behavior of the potential. The quasi-normal coordinates are linear combinations of the [ coordinates having the same symmetry. Since their choice is rather arbitrary, we kept the numerical values of the coefficients used by Amiot (3), leading to the following definition of the quasi-normal coordinates: R, = 16.14& + 22.47E3;
R; = 55.5[:;
R3 = 19.02& - 9.22&.
(8)
INTERNUCLEAR
393
POTENTIAL OF N20
Moreover, instead of rNN and rNo, the two variables
Dl =
D2 = r NO-TNN
INN + rN0,
(9)
were introduced in the fit because previous studies had shown a good convergence of D, values.
The converged F coefficients obtained at the end of the preliminary adjustment were then transformed into f coefficients according to Eq. (8), and a final step of refinement of thef’s was implemented, the rNN and rNo distances being frozen to the best values obtained from D1 and D2 as detailed below. 1. Equilibrium
Structure
The values of D, , D2, rNN, and rNo are reported and compared with those of other authors in Table II. As expected, the total length of the molecule is much more accurately determined than the two bond lengths are. Since we have used an approxi-
TABLE II Equilibrium Structure of Nitrous Oxide (A) References
r VW
r >0
r N.+rNO
NO-fN N
CU)
1.1266(S)
1.1856(5)
2.31230(3)
0.0590(10)
($6)
l-1284(3)
1.1841(3)
2.31244
0.0557
CB)
1.1282(l)
1.1843(l)
2.312535(S)
0.0561(2)
(22)
1.1281(8)
l-1842(8)
2.3123
0.0561
(38)
1.1277(12)
1.1846(11)
2.3123
0.0569
(51
1.12598(3)
1.18624(3)
2.31222
0.06026
1.1850896
2.312381(41
0.05779(7)
1.127292
this work
b
Atomic Masses (a.m.u.)
’ The
r
Conversion Factor’
I’N
14.003074002
I’N
15.00010897
Be (cm-‘) xIB (a.m.u.Aa)
“0
15.99491463
16.85763143
“0
16.9991312
L80
17.9991603
uncertainties
are
standard
deviations
from
the
various
least
square
determinations. b See text. ’ From
the 1986 adjustment of physical
change with the factor
used in ref.(s)
constants is -1.15
(35). ‘c 10“.
for example the relative
394
TEFFO AND CHkDIN
mation in the evaluation of the Jacobian with respect to TNNand rNo, we further tested the D, and D2 values in the following way: given the sum rNN + rNo = D,, several refinements were performed in which r m and rNo were constrained to different values and the final values reported in the table correspond to the minimum of residuals achieved. The masses and the conversion factor hNA/8?r2c used are given at the bottom of Table II. Our results are on the average in good agreement with the previous determinations, the main improvement being the simultaneous fit of the sextic force field as well as the internuclear distances to rovibrational data. For example, this allows one to take into account fourth-order contributions such as the yli’s which are generally neglected in the derivation of the equilibrium B, rotational constants from the ground state B0 rotational constants. Similar fourth-order corrections were carried out by Lacy and Whiffen (5)) but in their final calculation, only r NNand rNo were fitted to semiempirical B, values obtained from B, - B. estimates from the force field. Table III lists for the 12 N20 isotopic species the calculated values of B, and BO together with the differences with the experimental BO’s. The very good agreement between observation and calculation shows the quality of the results. 2. Internuclear Potential Table IV gives the expansion of the potential with respect to the quasi-normal coordinates. The F coefficients are given together with their standard deviation and TABLE III Equilibrium and Ground States Rotational Constants of N20 (cm -‘) B
e
B
0
B. (exp.)-B
0.4211207
0.4190098
1.2
0.4210675
0.4189803
0.6
0.4068672
0.4048552
1.9
0.4068474
0.4048584
l.Y
0.4087021
0.4066708
1.1
0.4086114
0.4066033
4.7
0.3947137
0.3927782
2.8
0.3946690
0.3927561
0.9
0.3975365
1.2
0.3974036
0.3954661
1.6
0.3837843
0.3819172
2.2
0.3837088
0.3818637
2.0
0
'
INTERNUCLEAR
POTENTIAL OF NzO
TABLE IV Expansion of the Potential with Respect to Quasi-normal Coordinates (cm-‘)
’ One standard deviation. b Mean absolute value of the F coefficients.
395
396
TEFFO
AND
CHfiDIN
their mean absolute value at each order of the expansion showing on the average a good convergence. Note that the coefficients nonsignificant in the fit were constrained to zero. The expansion of the potential with respect to dimensionless internal coordinates is reported in Table V and compared with previous determinations. Once again nonsignificant coefficients have been constrained to zero. It is seen that our solution is rather close to that of Kobayashi and Suzuki (6) up to the second order of approximation. The discrepancies which occur for higher order coefficients may be explained first by the fact that the number of significant coefficients is different and second because these authors assumed a Morse-type potential for the two stretching modes; hence the corresponding quartic and sextic coefficients were constrained according to &iii
= f 5/4f
$5
Jiiiii
=
31fii/360f:;
i= 1,3.
(10)
One can observe that the values derived in this study for the sextic coefficients are much larger than those of Ref. (6). This may be related to the question of convergence of the perturbation series at higher order since the first-order coefficient FlJ3 is found to be as large as that for CO2 (40). In Table VI are gathered the potential expansions with respect to the dimensionless normal coordinates for the 12 isotopic species of N20. 3. Spectroscopic Constants In Table VII we give for the 12 isotopic species of NzO the spectroscopic constants, i.e., the coefficients occurring in the polynomial expansion of the matrix elements of the Hamiltonian with respect to the quantum numbers vl, 212,la, ~3, J. These constants are identified in the table by the corresponding quantum numbers, the number following each constant being the coupling code defined in Table I. In the first column are listed successively the usual U”S, then the second-order x0’s and the fourth-order y’s. The second column gives the rotational constants Bo, (Y’, Do, y, 8, and H (JJ stands for J( J + 1) - 1:). The last four constants which are denoted in Ref. ( 7), all, a ,I[, a311,and aJl[, are contributions to the constants xii, ylu, y311,and TN, respectively. The last two columns of the table give the vibrational and rotational off-diagonal constants. 4. Vibrational Energy Levels and Eflective Molecular Band Constants From the spectroscopic constants (Table VII) it is possible to calculate, in the range of vibrational energies considered here (under 10 000 cm-‘), any rotation vibration energy level and the effective molecular constants G,, B,, and 0, for any given vibrational level not affected by a strong Coriolis resonance. Such constants are available on request to the authors. In the case of strong Coriolis interactions, one can always calculate from the potential coefficients the spectroscopic constants associated with the Coriolis coupling, and then one can compute rotation vibration energies including this coupling. Practically, on
INTERNUCLEAR
POTENTIAL OF N20
TABLE V Expansion of the Potential with Respect to Dimensionless Internal Coordinates (cm-‘)
Note. All values are to be multiplied by I X 106. ’ One standard deviation. b Transformed coefficients from original results.
391
398
TEF’FO AND CHBDIN TABLE VI Expansion of the Potential with Respect to Dimensionless Normal Coordinates for 12 Isotopic Species of NzO (cm -’ )
INTERNUCLEAR
POTENTIAL
OF NzO
400
TEFFO AND CHEDIN TABLE VII Spectroscopic Constants for 12 Isotopic Species of NzO (cm -’ )
( 1) Identification of the constant. (2) Coupling number.
INTERNUCLEAR POTENTIAL OF N20 TABLE VII-Continued
v
”
JJ
1
“*“*JJ1 Y2Y3JJ 1 ““J:JJ1
c v I
“iJJJJt “JJJJJ ’1 JJJJJJ
JJ
1
“JJ
1
“;JJ
1
-5.2832737.lE-04
“JJ
1
3.23333368F03
J:JJ
1
” ”
JJ
4.04858399E-01 1.80519828E-03
I .634m763E-07
1
-1.336175481-05
“‘“‘JJ
1
-9.35341768E-06
“‘“*JJ
1
-2.071627631-05
L’L3JJ
1
“*v*JJ
’
“*“oJJ ” ” JJ
’ 1
4.477235*9E-06 -9.6922944%-06 2.93379766E-05 -4.21866729E-06
“:;JJ
1
“*JJJJ V3JJJJ
’ 1
1.34737356E-09
JJJJJJ
1
-t.3644391*8-,4
LK
I
-2.55354082E-0.
” L K
1
-3.61W.O866E-05
“i,
1
-4.76602072E-04
2.39151952E-OS
K
I. I( JJ
1
SPECTROSCOPIC
-4,41358171E-10
1.529706278-07
CONSTANTS
OF THE MOLECULE
N
N
0
,cu-I,
t
”
1
1.*33006*5E+03
JJ
1
3.95576144E-01
c
2
“’
1
5.840905151’02
“JJ
1
I .8ooo6947E-03
”
2
2.34427515E-01
v*
1
*.*3191307E+o3
“:JJ
1
-5.43841314E-04
Y’
2
3.02931056E-01
V3Y
1
-3.*8*98535E+m
“JJ
1
3.22453745F03
Y2
*
1.640448*9E-o1
Y’ “*
“‘V’ 1 a
1
-5.12355997ElOO
J:JJ
I
1.581634541-07
J:
2
1.20642531E-04
J:
-1.8117506*E+O‘
v
402
TEFFO
AND
CHEDIN
c
c
”
Y’ Y2 .A c v
Y’ Y2 2 c c c c
c ”
“’ v2 J: c
INTERNUCLEAR
POTENTIAL
OF N20
TABLE VII-Continued
c
JJ
1
“JJ
1
v
“;JJ
1
“’
“JJ
1
ZJJ
1
J?
““JJ
1
c
+;JJ
1
”
“,“oJJ l. L JJ
’ ,
““JJ
1
v* Y’ v* J:
V2V2JJ 1 V2Y3JJ , ““J:JJ1
c ” “’
1
v*
“‘JJJJ
1
“*JJJJ
1
J:
JfJJJJ
1
c
LK
1
c
“LK
1
c c
“‘L
K
1
L3K
JJ
1
15
SPECTROSCOPIC
OF TN? MJOLECULE
17 N
cl
(CM-1,
“JJ
1
”
2
2.4,7683928-01
2.2r25*23,E+03
“‘JJ
1
-5.45041448E-04
“’
2
3.0,9,674,E-01
-3.47350152ElOO
“*JJ
1
3.235wc69E-03
v*
2
2.04542,53E-0,
-5.
J”JJJ
1
,.545,58KE-07
J”J
2
,.,9,58535E-04
1
-9.79353283E-06
t
3
6.350,9943E-0,
1
-8,588757,,E-06
”
3
2.49522777E-02
” ”
-5.77083586E-0,
“‘Y’JJ 1 2
JJ
1.737402496-03
t
14
5.0275915eE+02
-2.7449014aE+o,
3.92778245E-0,
N
JJ
,4313237840
1
CONSTANTS
1.236887471+03
2
,
-,.8587cJ48*E*0,
TEF’FO AND
CHEDIN
TABLE VII-Continued
”
“’ .I: c v
Y’ Y2 ,: c c c c
c
c ” 1
“;“*JJ
1
“1”3JJ1’
L L JJ
“*“*JJ’
INTERNUCLEAR
POTENTIAL
OF NzO
405
TABLE VII-Continued
c ”
“’ v2 JZ c ”
Y’ v2 .I: c ”
Y’ Y2
2 c
c c c
the few observed cases, a good approximation is obtained by considering a diagonalization scheme similar to that used for CO2 ( 7). The results of our computation are summarized and compared with experimental values in Table VIII. It is seen that the agreement is quite good. We have also reported in this table the results of Refs. (5, 6)) and one can observe that in the present case the agreement between experiment and theory is much better when using the contact transformation method. Finally, it should be emphasized that all the present results were obtained within the Born-Oppenheimer framework.
406
TEFFO
AND
CHkDIN
TABLE VIII Summary
of the Fit to Experimental
Data
This work DATA
c
Y
BYP
REF @I EXP. REFS
D”_
G ”
REF@)
I
Byb-B
Ebb-Be CY
19 21 22 23
0
-1-r-r-1
10
90
g,a.26 -r-l 21 _~_~_r_~ 28 29 30 31
93
zz.3,
_
_._
‘SN
‘80 2
ALL EXP-CALC R.M.S
35
39
49
60
63
20
21
20
18
L1,34
267
319
333
0.047
(cm-‘)
’ The constants
30
of
2.16x10-’
+--22 _
1.03
5.64~10‘~
the e and f levels
of
a II state
(and also
of
a A state
6.0x10-”
III the
I
2.65
case
of
3.04x10-’
Dy) are
different. * The constants
Bv of
APPENDIX:
a n state
PARTIAL
is
the
average
of
the B(e)
and B(f)
components.
DERIVATIVES OF THE w,‘S AND THE TO THE INTERNUCLEAR DISTANCES
B,‘S WITH
RESPECT
Transforming the quadratic potential expanded in terms of the dimensionless & coordinates defined in Eq. ( 1)
(A-1) one gets
(A-2) with hl = (2?rC01)~ = -(c,
+ &)/2
- [(c, + c~)~/4 - o]“2
(A-3)
INTERNUCLEAR POTENTIAL OF N20
x2 = (27rcw~)* =
ML
.I-22
m1m2m&~
x3 = (27XX.+)* = -(c,
(A-4)
+ c2)/2 + [(c, + c2)*/4 - D]“*,
(A-5)
where MI, = m,rng:
+ m2m3r: + m1m3(rl + r2)*
(A-6) (A-7)
fi3
c,=___
(A-8)
m2rlr2
fu
D=-.-_-
filf33
-f?3
(A-9)
r:rt
wm2m3
M=ml+m2+m3.
(A-10)
In the above equations, rl and r2 stand for rNN and rNo, respectively. Note that the introduction of the dimensionless & coordinates instead of the curvilinear ones makes w, and 03 explicitly r-dependent. Equations (A-3) to (A-5) are homogeneous rl and r2 functions of degree -2, so one gets according to the Euler theorem dXi
rl
ar, +
dXi r2
G
=
-2Xi9
dwi dwi rI + r2 = -ai, drl ar2
i=
1,2,3
(A-11)
i=
1,2,3.
(A-12)
Therefore, for each Xi, one needs to calculate only one of the partial derivatives. From (A-4) one finds -aA2c 8a2c2u2
ah
$f?= mvjf$r2 tml(rl I
I
+ r2) +
m2r21
+
m2r21.
-2h2m3r2 =
Mzerl
[ml(rl
r2)
+
(A-13)
The following properties -$C,fC*)=-25 n
?z= 1,2
(A-14)
408
TEFFO AND CHBDIN
and Eqs. (A-3) and (A-5) finally lead to dXj
=
&ZcZwi
$)l
_
n
arrl
2(Cnhi rn(b
+ -
D,
;
(i,j)
=
(1,
3), (3, 1).
(A-15)
xj)
From (A-6 ) , one gets
aBe
drl-
-
9
[m3(r,
+ r*) +
my,].
(A-16)
e
It is then easily verified that 2B, I, au2 aB,_ _---.
ah
RECEIVED:
w2 rn dr, ’
(n,ml = (1,2), (2, 1).
(A-17)
January 30, 1989
REFERENCES 1. I. SUZUKI, J. Mol. Spectrosc. 32, 54-13 ( 1969). 2. A. CHBDIN,C. A~IIOT, AND Z. CIHLA, J. Mol. Spectrosc. 63,348-369 ( 1976). 3. C. &dIOT, Thesis,CNRS No. A012433 (1976). 4. Z. CIHLA AND A. CHkDIN, J. Mol. Spectrosc. 40, 337-355 ( 1971). 5. M. LACYAND D. H. WHIFFEN,Mol. Phys. 45,241-252 (1982). 6. M. KOBAYASHIAND I. SUZUKI,J. Mol. Spectrosc. 125,24-42 ( 1987). 7. A. CHBDIN, J. Mol. Spectrosc. 76,430-491 (1979). 8. A. CHBDIN AND J.-L. TEFFO,J. Mol. Spectrosc. 107,333-342 ( 1984). 9. A, CHEDINAND Z. CIHLA, J. Mol. Spectrosc. 45,475-488 ( 1973). 10. A. CHBDINAND Z. CIHLA, J. Mol. Spectrosc. 49,289-309 ( 1974). 11. A. CHBDINAND Z. CIHLA, Cab. Phys. 199-200, 129-141 (1967). 12. A. CHBDINAND Z. ~IHL.A,J. Mol. Spectrosc. 47,542-553 (1973). Z3. J. L. GRIGGS,K. NARAHARI RAO, L. H. JONES,AND R. M. POTTER,J. Mol. Spectrosc. 25, 34-61 (1968). 14. J.-L. TEFF~ AND J. F. OCILVIE,to be published. 15. C. AMIOT AND G. GUELACHVILI,J. Mol. Spectrosc. 51,475-491 ( 1974). 16. C. AIUIOTAND G. GUEJXHVILI, J. Mol. Spectrosc. 59, 171-190 ( 1976). 17. C. AMIOT, J. Mol. Spectrosc. 59, 191-208 (1976). 18. C. AMIOT, J. Mol. Spectrosc. 59, 380-395 (1976). 19. B. A. ANDREEV, A. V. BURENIN, E. N. KARYAKIN, A. F. KRUPNOV, AND S. M. SHAPIN, J. Mol. Spectrosc. 62, 125-148 (1976). 20. A. BAUER,J.-L. TEFFO,A. VALENTIN,AND T. K. M~C~BBIN, J. Mol. Spectrosc. 120,449-454 ( 1986). 21. W. B. OLSON, A. G. MAKI,AND W. J. LAFFERTY,J. Phys. Chem. Rej: Data 10,1065-1084 ( 1981). 22. K. JOLMA,J. KAUPPINEN,AND V. M. HORNEMAN,J. Mol. Spectrosc. 101,278-284 ( 1983). 23. G. GUELACHVILI,Canad. J. Phys. 60, 1334-1347 (1982). 24. L. R. BROWN AND R. A. TOTH, J. Opt. Sot. Amer. B: Opt. Phys. 2, 842-856 ( 1985). 25. J. S. WELLS, D. A. JENNINGS,A. HINZ, AND J. S. MURRAY, J. Opt. Sot. Amer. B: Opt. Phys. 2,857861 (1985). 26. J. S. WELLS,A. HINZ, AND A. MAKI, J. Mol. Spectrosc. 114,84-96 (1985).
INTERNUCLEAR
POTENTIAL
OF NzO
409
27. C. R. POLLOCK,F. R. PETERSEN, D. A. JENNINGS, J. S. WELLS, AND A. G. MAKI, J. Mol. Spectrosc. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.
107,62-7 1 ( 1984). B. G. WHITFORD,K. J. SIEMSEN, H. D. R~ccrus, ANDG. R. HANES,Opt. Commun. 14,70-74 ( 1975). M. J. REISFELD AND H. FLICKER, Appl. Opt. 18, 1136-l 138 (1979). J. M. KRELLAND R. L. SAMS,J. Mol. Spectrosc. 51,492-507 ( 1974). R. FARRENCQ AND J. DUPRB-MAQUAIRE, J. Mol. Spectrosc. 49,268-279 (1974). J. DUPRB-MAQUAIRE AND P. PINSON,J. Mol. Spectrosc. 58,239-249 ( 1975). J. PERRIZO,L. A. PUGH,AND K. NARAHARIRAO, J. Mol. Specfrosc. 57, 397-401 ( 1975). A. W. MANTZ, K. NARAHARIRAO, L. H. JONES,AND R. M. POI-IER,J. Mol. Spectrosc. 30,513-530 (1969). E. R. COHENAND B. N. TAYLOR,CODATA Bull. 63, ( 1986). K. NARAHARIRAO, Amer. N.Y. Acad. Sci. 200, 15-54 ( 1973). J. K. G. WATSON,J. Mol. Spectrosc. 48,479-502 (1973). M. D. HARMONYAND W. H. TAYLOR,J. Mol. Spectrosc. 118, 163-173 (1986). C. H. Tow AND A. L. S~HALOW,“Microwave Spectroscopy,”McGraw-Hill, New York, 1955. G. AMAT, Chem. Phys. 71,363-375 ( 1982).
OF MOLECULAR
SPECTROSCOPY
135,389-409 (1989)
Internuclear Potential and Equilibrium Structure of the Nitrous Oxide Molecule from Rovibrational Data J.-L. TEFFO Laboratoire de Physique Molkulaire et Atmosph&ique. Tour 13, Universitk Pierre et Marie Curie et CNRS, 4, Place Jussieu, 7S252 Paris Cedex OS, France
AND
A. CH~DIN Laboratoire de MMoroiogie Dynamique, CNRS, Ecole Polytechnique, Route Dkpartementale 36, 91128 Palaiseau Cedex, France
The internuclear potential up to sextic terms and the equilibrium structure of the nitrous oxide molecule are reinvestigated using the algebraized contact transformation method previously applied to carbon dioxide. Infrared spectroscopic data for a large number of vibrational levels belonging to six isotopic species of NzO am included in the refinement. A good agreement between calculated and experimental values and a significant improvement of the determination of the NN and NO bond lengths are obtained. 0 1989Academic press, hc.
I. INTRODUCTION
Since the first determination of the quartic force field of N20 made by Suzuki using standard perturbation theory ( I ), several authors have attempted to improve the modeling of the internuclear potential of this molecule. In 1976, Chkdin et al. (2) and Amiot (3) reported the first calculation of the sextic force field using a new algebraic approach for solving the vibrational eigenvalue problem of linear triatomic molecules (4). However, this determination, like that of Suzuki, relied upon vibrational data only. The problem was then reinvestigated by Lacy and Whiffen (5)) and more recently by Kobayashi and Suzuki (6), by means of a direct numerical diagonalization method. These authors obtained a sextic force field of N20 in terms of curvilinear internal coordinates from a set of vibrational band centers and of rotational band constants. Successfully extended to the rovibrational eigenvalue problem in the case of the carbon dioxide molecule ( 7, 8), the algebraized contact transformation method is here applied to N20 in order to reinvestigate the internuclear potential of this molecule. Moreover, we address the problem of the equilibrium structure of the N20 molecule, which cannot be separated from that of a meaningful determination of the internuclear potential. The computational method is summarized in Section II. The experimental data set is described in Section III, and the results are reported and discussed in Section IV.
389
0022-2852189 $3.00 Copyright 8
1989 by Academic Press, Inc.
All rights of reproduction in any form reserved.
390
TEFFO AND CHCDIN
FIG. 1. Instantaneousconfigurationof N20.
II. SUMMARY
OF THE METHOD
The expansion of the potential energy function I’ is carried out with respect to the three dimensionless internal coordinates c;, =
-:NN
)
t2
=
Aa,
[3
=
r231No-N0
,
where r12, r23, and ACYare defined in Fig. 1, r NN and rNo being the two equilibrium bond lengths. The expansion of the potential function being truncated at the fourth order of approximation in the perturbation theory, the N20 potential is a polynomial containing 47 coefficients up to the sixth power in the &‘s. From an initial determination, the first part of the computational program sets up the rotation-vibration energy matrix of the molecule. The Hamiltonian His expanded in a basis made of symmetrized products obtained from the dimensionless normal coordinates qi, their conjugate momenta pi, and the components of the angular momentum J. At the order of magnitude 71the basis is of degree n + 2 with respect to these elementary operators. The perturbation treatment of H is then performed by means of two successive contact transformations previously encoded ( 7, 9, 10). The calculation of the transformed Hamiltonian HT reduces to a matrix product H= = TH,
(2)
where T is the transformation matrix. The nonvanishing matrix elements of HT in the basis of the harmonic oscillator wavefunctions are listed in Table I. Each one is a product of a square root part ( 7) by a polynomial in the quantum numbers vI, v2, 12, v3, and J, the coefficients of which are called the spectroscopic constants denoted C in the following. It is worth noting that two Fermi resonances are included in the diagonalization scheme, while the Coriolis couplings (which would correspond to the missing codes 6 and 7 in Table I) are removed by the contact transformation; this is reasonable since few levels are affected by this interaction. Moreover, keeping it would lead to infinite Hamiltonian matrices. However, this means that levels coupled by strong Coriolis resonances cannot be included in the data set. The second part of the program solves the inverse eigenvalue problem using a leastsquares fit procedure. If we denote by E the set of theoretical rotation-vibration energies and by F the set of potential coefficients, the Jacobian matrix dE/dF is calculated from the product
INTERNUCLEAR
391
POTENTIAL OF N20
TABLE I Nonvanishing Matrix Elements (u, , u2, 12, q 1HT 111, + Au,, u2+ Au2, I2+ Al,, uj + Au3) 3v I
dv2
?lV 3
Code
ie2
0
0
0
0
1
-1
2
0
0
2
-2
0
1
0
3
-1
-2
1
0
4
0
0
0
2
5
-1
2
0
2
a
-1
2
0
-2
9
0
-4
1
0
10
-2
-4
0
0
11
-4
0
2
0
12
-3
2
1
0
13
aE
aE
%=dCXaHT
ac
-X-
aHT aF
’
where -dE/dC is the matrix of the partial derivatives of the energies with respect to the spectroscopic constants C defined previously, and its elements are the coefficients of the linearized expressions of the energies in terms of the spectroscopic constants (11). -aC/aHT is the matrix of the coefficients occurring in the expression of the spectroscopic constants in terms of the parameters of the transformed Hamiltonian. These coefficients are computed in the first part of the program. - dHT/aF is the matrix of the partial derivatives of the latter parameters with respect to the potential coefficients. It is written as
The matrix a V/a F is the matrix used to transform the V (4) expansion of the potential into the V(q) expansion. It is also computed at the beginning of the calculation ( 12). Once the dE/dF matrix is known, it is possible to start the iterative process for the adjustment of the potential. It is important to note that the values of the internuclear distances enter as variable parameters in this calculation, contrary to the previous calculation (2) where these distances appeared as fixed parameters derived from rotational analysis only ( 13). This was not quite a satisfactory approach since the equilibrium structure of the molecule is linked to its vibrational potential energy. Therefore, instead Of considering the tW0 rNN and rNo distances as Constant parameters, we have treated them as two additional variables that are adjusted together with the potential coefficients throughout the fitting process. For this purpose, we have introduced the partial derivatives dHT/& in the Jacobian. This was done by means of the approximation
392
TEFFO
AND CHBDIN
dHT
aH;
ar=ar-ar,
aH,
(5)
where Ho = ; C hcwi(p; 1
+ qf) + ; B,(J:- + J;)
(6)
is the zeroth-order Hamiltonian. Since the matrix elements of p’ + qf and Ji + Jf. are not r-dependent, the evaluation of aHo/& reduces to the calculation of the partial derivatives of the WI’sand of Be. Practically, this approximation means that the partial derivatives, with respect to rNN and YNo, of all the other spectroscopic constants are neglected. The validity of this approximation was numerically tested in the CO2 case, which is more simple (14), and was found satisfactory. The eight quantities dw;/dr and aBe/& are given in the Appendix. III. EXPERIMENTAL
DETAILS
As in the CO2 case ( 7, 8) the potential was not fitted directly to the experimental rotation-vibration energies E,(J) but to the usual band constants defined as E,(J)
= G, + B,J(J+
1) - D,J’(J+
l)*.
(7)
This means that the data concerning the vibrational levels for which Eq. (7) is poorly significant cannot be included in the data set. Most of the experimental data used in this work were taken from the extensive N20 studies of Amiot and Guelachvili (15, 16) and Amiot ( 17, 28). The source of the other data is mentioned in Table VIII. Six isotopic species have been included in the least-squares refinement. The highest levels are the u3 = 4 levels which lie about 9000 cm-’ above the fundamental. The data weights are inversely proportional to the squares of the experimental uncertainties, the greatest weights being assigned to the rotational constants of the ground states for which microwave results are available ( 19, 20). IV. RESULTS
AND DISCUSSION
As in Ref. (2), we used in this fitting process the expansion of the potential with respect to the 5‘coordinates of Eq. ( 1)) the 47 coefficients of this expansion being labeled with an f symbol. Actually, starting from the quartic potential of Suzuki ( 1)) a preliminary adjustment was performed with an expansion of the potential with respect to “quasi-normal” coordinates (4)) the corresponding coefficients being labeled with an F symbol. The main interest of this preliminary calculation is that the F’s have the same order of magnitude as the coefficient of the expansion of the potential with respect to the q’s for any isotopic species, which is a convenient situation to estimate the convergence behavior of the potential. The quasi-normal coordinates are linear combinations of the [ coordinates having the same symmetry. Since their choice is rather arbitrary, we kept the numerical values of the coefficients used by Amiot (3), leading to the following definition of the quasi-normal coordinates: R, = 16.14& + 22.47E3;
R; = 55.5[:;
R3 = 19.02& - 9.22&.
(8)
INTERNUCLEAR
393
POTENTIAL OF N20
Moreover, instead of rNN and rNo, the two variables
Dl =
D2 = r NO-TNN
INN + rN0,
(9)
were introduced in the fit because previous studies had shown a good convergence of D, values.
The converged F coefficients obtained at the end of the preliminary adjustment were then transformed into f coefficients according to Eq. (8), and a final step of refinement of thef’s was implemented, the rNN and rNo distances being frozen to the best values obtained from D1 and D2 as detailed below. 1. Equilibrium
Structure
The values of D, , D2, rNN, and rNo are reported and compared with those of other authors in Table II. As expected, the total length of the molecule is much more accurately determined than the two bond lengths are. Since we have used an approxi-
TABLE II Equilibrium Structure of Nitrous Oxide (A) References
r VW
r >0
r N.+rNO
NO-fN N
CU)
1.1266(S)
1.1856(5)
2.31230(3)
0.0590(10)
($6)
l-1284(3)
1.1841(3)
2.31244
0.0557
CB)
1.1282(l)
1.1843(l)
2.312535(S)
0.0561(2)
(22)
1.1281(8)
l-1842(8)
2.3123
0.0561
(38)
1.1277(12)
1.1846(11)
2.3123
0.0569
(51
1.12598(3)
1.18624(3)
2.31222
0.06026
1.1850896
2.312381(41
0.05779(7)
1.127292
this work
b
Atomic Masses (a.m.u.)
’ The
r
Conversion Factor’
I’N
14.003074002
I’N
15.00010897
Be (cm-‘) xIB (a.m.u.Aa)
“0
15.99491463
16.85763143
“0
16.9991312
L80
17.9991603
uncertainties
are
standard
deviations
from
the
various
least
square
determinations. b See text. ’ From
the 1986 adjustment of physical
change with the factor
used in ref.(s)
constants is -1.15
(35). ‘c 10“.
for example the relative
394
TEFFO AND CHkDIN
mation in the evaluation of the Jacobian with respect to TNNand rNo, we further tested the D, and D2 values in the following way: given the sum rNN + rNo = D,, several refinements were performed in which r m and rNo were constrained to different values and the final values reported in the table correspond to the minimum of residuals achieved. The masses and the conversion factor hNA/8?r2c used are given at the bottom of Table II. Our results are on the average in good agreement with the previous determinations, the main improvement being the simultaneous fit of the sextic force field as well as the internuclear distances to rovibrational data. For example, this allows one to take into account fourth-order contributions such as the yli’s which are generally neglected in the derivation of the equilibrium B, rotational constants from the ground state B0 rotational constants. Similar fourth-order corrections were carried out by Lacy and Whiffen (5)) but in their final calculation, only r NNand rNo were fitted to semiempirical B, values obtained from B, - B. estimates from the force field. Table III lists for the 12 N20 isotopic species the calculated values of B, and BO together with the differences with the experimental BO’s. The very good agreement between observation and calculation shows the quality of the results. 2. Internuclear Potential Table IV gives the expansion of the potential with respect to the quasi-normal coordinates. The F coefficients are given together with their standard deviation and TABLE III Equilibrium and Ground States Rotational Constants of N20 (cm -‘) B
e
B
0
B. (exp.)-B
0.4211207
0.4190098
1.2
0.4210675
0.4189803
0.6
0.4068672
0.4048552
1.9
0.4068474
0.4048584
l.Y
0.4087021
0.4066708
1.1
0.4086114
0.4066033
4.7
0.3947137
0.3927782
2.8
0.3946690
0.3927561
0.9
0.3975365
1.2
0.3974036
0.3954661
1.6
0.3837843
0.3819172
2.2
0.3837088
0.3818637
2.0
0
'
INTERNUCLEAR
POTENTIAL OF NzO
TABLE IV Expansion of the Potential with Respect to Quasi-normal Coordinates (cm-‘)
’ One standard deviation. b Mean absolute value of the F coefficients.
395
396
TEFFO
AND
CHfiDIN
their mean absolute value at each order of the expansion showing on the average a good convergence. Note that the coefficients nonsignificant in the fit were constrained to zero. The expansion of the potential with respect to dimensionless internal coordinates is reported in Table V and compared with previous determinations. Once again nonsignificant coefficients have been constrained to zero. It is seen that our solution is rather close to that of Kobayashi and Suzuki (6) up to the second order of approximation. The discrepancies which occur for higher order coefficients may be explained first by the fact that the number of significant coefficients is different and second because these authors assumed a Morse-type potential for the two stretching modes; hence the corresponding quartic and sextic coefficients were constrained according to &iii
= f 5/4f
$5
Jiiiii
=
31fii/360f:;
i= 1,3.
(10)
One can observe that the values derived in this study for the sextic coefficients are much larger than those of Ref. (6). This may be related to the question of convergence of the perturbation series at higher order since the first-order coefficient FlJ3 is found to be as large as that for CO2 (40). In Table VI are gathered the potential expansions with respect to the dimensionless normal coordinates for the 12 isotopic species of N20. 3. Spectroscopic Constants In Table VII we give for the 12 isotopic species of NzO the spectroscopic constants, i.e., the coefficients occurring in the polynomial expansion of the matrix elements of the Hamiltonian with respect to the quantum numbers vl, 212,la, ~3, J. These constants are identified in the table by the corresponding quantum numbers, the number following each constant being the coupling code defined in Table I. In the first column are listed successively the usual U”S, then the second-order x0’s and the fourth-order y’s. The second column gives the rotational constants Bo, (Y’, Do, y, 8, and H (JJ stands for J( J + 1) - 1:). The last four constants which are denoted in Ref. ( 7), all, a ,I[, a311,and aJl[, are contributions to the constants xii, ylu, y311,and TN, respectively. The last two columns of the table give the vibrational and rotational off-diagonal constants. 4. Vibrational Energy Levels and Eflective Molecular Band Constants From the spectroscopic constants (Table VII) it is possible to calculate, in the range of vibrational energies considered here (under 10 000 cm-‘), any rotation vibration energy level and the effective molecular constants G,, B,, and 0, for any given vibrational level not affected by a strong Coriolis resonance. Such constants are available on request to the authors. In the case of strong Coriolis interactions, one can always calculate from the potential coefficients the spectroscopic constants associated with the Coriolis coupling, and then one can compute rotation vibration energies including this coupling. Practically, on
INTERNUCLEAR
POTENTIAL OF N20
TABLE V Expansion of the Potential with Respect to Dimensionless Internal Coordinates (cm-‘)
Note. All values are to be multiplied by I X 106. ’ One standard deviation. b Transformed coefficients from original results.
391
398
TEF’FO AND CHBDIN TABLE VI Expansion of the Potential with Respect to Dimensionless Normal Coordinates for 12 Isotopic Species of NzO (cm -’ )
INTERNUCLEAR
POTENTIAL
OF NzO
400
TEFFO AND CHEDIN TABLE VII Spectroscopic Constants for 12 Isotopic Species of NzO (cm -’ )
( 1) Identification of the constant. (2) Coupling number.
INTERNUCLEAR POTENTIAL OF N20 TABLE VII-Continued
v
”
JJ
1
“*“*JJ1 Y2Y3JJ 1 ““J:JJ1
c v I
“iJJJJt “JJJJJ ’1 JJJJJJ
JJ
1
“JJ
1
“;JJ
1
-5.2832737.lE-04
“JJ
1
3.23333368F03
J:JJ
1
” ”
JJ
4.04858399E-01 1.80519828E-03
I .634m763E-07
1
-1.336175481-05
“‘“‘JJ
1
-9.35341768E-06
“‘“*JJ
1
-2.071627631-05
L’L3JJ
1
“*v*JJ
’
“*“oJJ ” ” JJ
’ 1
4.477235*9E-06 -9.6922944%-06 2.93379766E-05 -4.21866729E-06
“:;JJ
1
“*JJJJ V3JJJJ
’ 1
1.34737356E-09
JJJJJJ
1
-t.3644391*8-,4
LK
I
-2.55354082E-0.
” L K
1
-3.61W.O866E-05
“i,
1
-4.76602072E-04
2.39151952E-OS
K
I. I( JJ
1
SPECTROSCOPIC
-4,41358171E-10
1.529706278-07
CONSTANTS
OF THE MOLECULE
N
N
0
,cu-I,
t
”
1
1.*33006*5E+03
JJ
1
3.95576144E-01
c
2
“’
1
5.840905151’02
“JJ
1
I .8ooo6947E-03
”
2
2.34427515E-01
v*
1
*.*3191307E+o3
“:JJ
1
-5.43841314E-04
Y’
2
3.02931056E-01
V3Y
1
-3.*8*98535E+m
“JJ
1
3.22453745F03
Y2
*
1.640448*9E-o1
Y’ “*
“‘V’ 1 a
1
-5.12355997ElOO
J:JJ
I
1.581634541-07
J:
2
1.20642531E-04
J:
-1.8117506*E+O‘
v
402
TEFFO
AND
CHEDIN
c
c
”
Y’ Y2 .A c v
Y’ Y2 2 c c c c
c ”
“’ v2 J: c
INTERNUCLEAR
POTENTIAL
OF N20
TABLE VII-Continued
c
JJ
1
“JJ
1
v
“;JJ
1
“’
“JJ
1
ZJJ
1
J?
““JJ
1
c
+;JJ
1
”
“,“oJJ l. L JJ
’ ,
““JJ
1
v* Y’ v* J:
V2V2JJ 1 V2Y3JJ , ““J:JJ1
c ” “’
1
v*
“‘JJJJ
1
“*JJJJ
1
J:
JfJJJJ
1
c
LK
1
c
“LK
1
c c
“‘L
K
1
L3K
JJ
1
15
SPECTROSCOPIC
OF TN? MJOLECULE
17 N
cl
(CM-1,
“JJ
1
”
2
2.4,7683928-01
2.2r25*23,E+03
“‘JJ
1
-5.45041448E-04
“’
2
3.0,9,674,E-01
-3.47350152ElOO
“*JJ
1
3.235wc69E-03
v*
2
2.04542,53E-0,
-5.
J”JJJ
1
,.545,58KE-07
J”J
2
,.,9,58535E-04
1
-9.79353283E-06
t
3
6.350,9943E-0,
1
-8,588757,,E-06
”
3
2.49522777E-02
” ”
-5.77083586E-0,
“‘Y’JJ 1 2
JJ
1.737402496-03
t
14
5.0275915eE+02
-2.7449014aE+o,
3.92778245E-0,
N
JJ
,4313237840
1
CONSTANTS
1.236887471+03
2
,
-,.8587cJ48*E*0,
TEF’FO AND
CHEDIN
TABLE VII-Continued
”
“’ .I: c v
Y’ Y2 ,: c c c c
c
c ” 1
“;“*JJ
1
“1”3JJ1’
L L JJ
“*“*JJ’
INTERNUCLEAR
POTENTIAL
OF NzO
405
TABLE VII-Continued
c ”
“’ v2 JZ c ”
Y’ v2 .I: c ”
Y’ Y2
2 c
c c c
the few observed cases, a good approximation is obtained by considering a diagonalization scheme similar to that used for CO2 ( 7). The results of our computation are summarized and compared with experimental values in Table VIII. It is seen that the agreement is quite good. We have also reported in this table the results of Refs. (5, 6)) and one can observe that in the present case the agreement between experiment and theory is much better when using the contact transformation method. Finally, it should be emphasized that all the present results were obtained within the Born-Oppenheimer framework.
406
TEFFO
AND
CHkDIN
TABLE VIII Summary
of the Fit to Experimental
Data
This work DATA
c
Y
BYP
REF @I EXP. REFS
D”_
G ”
REF@)
I
Byb-B
Ebb-Be CY
19 21 22 23
0
-1-r-r-1
10
90
g,a.26 -r-l 21 _~_~_r_~ 28 29 30 31
93
zz.3,
_
_._
‘SN
‘80 2
ALL EXP-CALC R.M.S
35
39
49
60
63
20
21
20
18
L1,34
267
319
333
0.047
(cm-‘)
’ The constants
30
of
2.16x10-’
+--22 _
1.03
5.64~10‘~
the e and f levels
of
a II state
(and also
of
a A state
6.0x10-”
III the
I
2.65
case
of
3.04x10-’
Dy) are
different. * The constants
Bv of
APPENDIX:
a n state
PARTIAL
is
the
average
of
the B(e)
and B(f)
components.
DERIVATIVES OF THE w,‘S AND THE TO THE INTERNUCLEAR DISTANCES
B,‘S WITH
RESPECT
Transforming the quadratic potential expanded in terms of the dimensionless & coordinates defined in Eq. ( 1)
(A-1) one gets
(A-2) with hl = (2?rC01)~ = -(c,
+ &)/2
- [(c, + c~)~/4 - o]“2
(A-3)
INTERNUCLEAR POTENTIAL OF N20
x2 = (27rcw~)* =
ML
.I-22
m1m2m&~
x3 = (27XX.+)* = -(c,
(A-4)
+ c2)/2 + [(c, + c2)*/4 - D]“*,
(A-5)
where MI, = m,rng:
+ m2m3r: + m1m3(rl + r2)*
(A-6) (A-7)
fi3
c,=___
(A-8)
m2rlr2
fu
D=-.-_-
filf33
-f?3
(A-9)
r:rt
wm2m3
M=ml+m2+m3.
(A-10)
In the above equations, rl and r2 stand for rNN and rNo, respectively. Note that the introduction of the dimensionless & coordinates instead of the curvilinear ones makes w, and 03 explicitly r-dependent. Equations (A-3) to (A-5) are homogeneous rl and r2 functions of degree -2, so one gets according to the Euler theorem dXi
rl
ar, +
dXi r2
G
=
-2Xi9
dwi dwi rI + r2 = -ai, drl ar2
i=
1,2,3
(A-11)
i=
1,2,3.
(A-12)
Therefore, for each Xi, one needs to calculate only one of the partial derivatives. From (A-4) one finds -aA2c 8a2c2u2
ah
$f?= mvjf$r2 tml(rl I
I
+ r2) +
m2r21
+
m2r21.
-2h2m3r2 =
Mzerl
[ml(rl
r2)
+
(A-13)
The following properties -$C,fC*)=-25 n
?z= 1,2
(A-14)
408
TEFFO AND CHBDIN
and Eqs. (A-3) and (A-5) finally lead to dXj
=
&ZcZwi
$)l
_
n
arrl
2(Cnhi rn(b
+ -
D,
;
(i,j)
=
(1,
3), (3, 1).
(A-15)
xj)
From (A-6 ) , one gets
aBe
drl-
-
9
[m3(r,
+ r*) +
my,].
(A-16)
e
It is then easily verified that 2B, I, au2 aB,_ _---.
ah
RECEIVED:
w2 rn dr, ’
(n,ml = (1,2), (2, 1).
(A-17)
January 30, 1989
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INTERNUCLEAR
POTENTIAL
OF NzO
409
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