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Perturbations between the X2Π and a4Σ− states of the NH+ ion
JOURNAL
OF MOLECULAR
SPECTROSCOPY
(1989)
I%,387401
Perturbations between the X 211 and a 421- States of the NH + Ion R. COLIN Laboratoire de Chimie Physique Molkulaire.
Universite’ Libre de Bruxelles. Brussels, Belgium
On the basis of previously published optical data (Cohn and Douglas, Canad. J. Phvs. 46,6 I 73 ( 1968)) for three isotopes ( 14NH+, “NH+ and 14ND+) of the NH+ ion, a study of the perturbations occurring between the X2rI and a?- states has been carried out using matrix diagonalization techniques. This makes it possible to derive for both states equilibrium rotational and vibrational constants, from which RKR potential energy functions, vibrational overlap integrals. electronic interaction parameters, and a pure electronic term value for the a42 - state are deduced. 6 1989 Academic Press. Inc. 1. INTRODUCTION It was shown several years ago by Cohn and Douglas ( I ) on the basis of the optical emission spectra of three isotopes ( 14NH +, I5NH’, and 14ND+) of the NH+ ion that the ground electronic state of this ion is a ‘II state and that a very low lying a4t: state (T, - 550 cm-‘) causes strong perturbations in the 2, = 0 and u = 1 levels of the ground state. Displacements as large as 46 cm-’ were observed for some rotational levels. Rotational and vibrational constants were derived by these authors for both of these states and for the more highly excited A?-, B2A, and C2Z+ states. For the X’II state, this was done in an approximate manner by fitting the “less perturbed” rotational levels of the ‘II state to the usual Hill-Van Vleck equations. By calculation of the shifts of the strongly perturbed levels it was then possible to estimate the “original” position of the perturbing 42- levels. In turn these few data were fitted to a simplified rotational energy level equation to yield approximate rotational constants for the a4Z state. No attempt was made at that time to fully analyze the perturbations by matrix techniques. In a recent paper, Kawaguchi and Amano (2) reported the observation by tunable difference frequency laser techniques of the 1-O vibrational-rotational transitions of 14NH+ in the X*II and u4Z- states, as well as between these two electronic states. A merged least-square fit of the energy levels was carried out by these authors using their infrared data, the previous optical term values of Colin and Douglas ( 1) , and two far infrared pure rotational transitions measured by Verhoeve et al. (3). All these data were fitted with suitable weight factors to a 4 X 4 effective Hamiltonian matrix and accurate rotational constants, interaction parameters, and vibronic term values were derived for the ~1= 0 and v = 1 levels of the X’II and a4Z- states of the 14NH+ isotope. Due to the possible astrophysical importance of the NH+ ion, which, however, to our knowledge, has not yet been observed in any extraterrestrial object, it was thought to be useful to extend the treatment of Kawaguchi and Amano (2) to the other two 387
0022-2852189 $3.00 Copyright 0
1989 by Academic Press. Inc.
All righU of reproduction in any form resewed.
388
R. COLIN
isotopes 14ND+ and lsNH+ by using the optical data of Colin and Douglas (I). Of course, since no rotational-vibrational or pure rotational data exist for these isotopes, the accuracy of the molecular constants derived will be less than in the case of 14NH+. However, the results presented here will allow one, by using isotopic relations to calculate vibrational constants for the X211 and a4Z- states, to derive the pure electronic term value of the a4Z- state, and to calculate therefrom limited RKR potentials and vibrational overlap integrals. These data could be of some help in the detection of NH+ in astronomical sources. Since a summary of the most important theoretical and experimental papers on NH+ published between 1967 and 1988 has been given by Kawaguchi and Amano (2), it will not be repeated here. II. ENERGY
LEVELS
The optical spectra observed by Colin and Douglas ( 1) include for all three isotopes the O-O, l-0, and 0- 1 bands of the A ‘Z--X211 transition, the O-O band of B’A-X2& and the O-O band of C2Z+--X211. The data were all obtained from high resolution spectra taken with a 10-m spectrograph giving an error in relative line measurement of kO.05 cm-’ and an absolute uncertainty of&O. 10 cm-‘. The measurements for all the observed bands in the case of r4NH+ were given in the original paper (Tables I, II and III of ( I )). For the other two isotopes, the measurements have been placed in the Depository of Unpublished Data of the Canadian Journal of Physics. The branches were identified, rotationally numbered, and led, as far as the 211 ground state is concerned, to the expected four-fold set ( Tr,, T,d, Tzc, T& of rotational energy levels for 2) = 0 and for 2) = 1 referred to the TI, (J = 0.5 ) level of ~1= 0. The energies of the levels for 14NH+ were published (Table IV of ( 1)) and the data for “NH+ and 14ND+ were placed in the Depository. In all sets of levels with the exception of the 14ND+ o = 0 set, extra levels due to perturbations with the a4Z- state are present. In the original work ( I ), several extra lines with low J values observed in the O-l bands of the A2Z --X211 transition of 14NH+ and “NH+ could not be identified and were listed without assignments. In 1983, Fame11 and Ogilvie (4) performed ab initio molecular orbital calculations to compute the potential energy curves of the X*II and a4Z- states which allowed derivation of rotational-vibrational energy levels including the interaction between both states. When adjusted to the experimental data, these calculations allowed the identification of the previously unassigned lines. It should also be mentioned that the fitting of Kawaguchi and Amano (2) removed the uncertainty in the original assignments of the Q1I=(J > 4.5 ) and Q2zc (J > 3.5 ) lines in the 0- 1 A 22 --X211 band of 14NH+. It was pointed out by Colin and Douglas (I) that these branches could well be interchanged. The fitting of Kawaguchi and Amano (2) showed that this exchange should be made. Krishnamurthy and Saraswathy (5) reached the same conclusion from their analysis of the l- 1 and 2- 1 bands of the C2Z+-X*II system of 14NH+. Since Kawaguchi and Amano (2) have listed extensively in their paper the fitted energy levels for u = 0 and v = 1 of the X211 and a4Z- states of 14NH+, they will not be given here. For the “NH+ and r4ND+ molecules, the experimental rotational energy levels for the u = 0 and tt = 1 levels of the ‘II state are presented in Tables I and II, respectively. Here again for “NH+ (J > 4.5) the Qtlc and Q22c(J > 3.5) branches of
THE X’II AND a%-
389
STATES OF NH+
TABLE I Term Values of the Rotational Levels of the X’II State of “NH+ Referred to F,, (0.5) of u = 0 (cm -‘) J
TIC(J)
K(J)
T,d(J)
0.5 1.5 2.5 3.5 4.5 5.5
0.00 33.70 95.98 187.91 309.76 461.37
(-3)’ (-0) (+4) (+4) (+8) (+lO)
0.43 34.37 96.78 188.80 310.76 462.62
(-6) (-2) (-0) (-3) (-3) (-5)
6.5 7.5 8.5
642.44 852.41 1089.83
(+Il) (+16) (+14)
644.26 855.47 1096.01
(-6) (-9) (-11)
9.5
1351.13 1415.06 1629.33 1693.63 2012.05 2363.62 2745.14 3154.59
(+2) (-4)
1365.68
(-4)
(-21)
1663.93
(-8)
1990.60 2345.17 2727.15 3136.15
(-1) (+4) (t5) (t12)
10.5 11.5 12.5 13.5 14.5
G(J)
105.26 195.36 316.45 467.87 649.26
(+0) (-1) (t3) (t2) (+4)
860.09 1099.33 1363.81 1409.33 1642.74 1687.41 2008.95
(+7) (f5) (+l) (+6) (+6) (-2) (f13)
2362.72 2745.24 3155.41
(+Q) (-1) (-10)
2882.19 3009.71 2959.52 3090.91 3036.79 3205.95 3148.36 3349.95 3288.07 3524.21 3455.04 3727.36
(to) (-1
(+5) (tl6) (+5) (-6) (f3)
3958.94
C-5)
(+9) (tll) (-13) (-7) (-11)
0.5 1.5
2939.60 (+4) 2876.09 (-16)
2.5 3.5 4.5 5.5
2908.61 (+5) 3088.34 (-16) 2966.91 (+30) 3204.50 (-3) 3349.82
(-5)
6.5
3524.23
(+0)
7.5 8.5
3727.36 3958.85
(+5) (t3)
+observed-calculated
2939.85 2897.21 3006.40 2952.89 3096.97 3037.02 3214.35 3149.08 3359.54 3288.71 3532.89 3455.50
C-10) C-26)
C-1) C-10) (+l) (f6) (+-a) (i-18)
(+31 (+20)
(-6)
f-31) (t8) (-4) C-15) (i-13)
105.04 194.77 315.37 466.19 642.49
(+I) C-1) C-2)
653.66 859.06 1099.05 1368.40
(-5) (-3) (-4) (-8) (-12) (-5)
1666.41
C-10)
1992.90
(-4)
2347.31 2729.20 3138.04
(+O) (+5) (+7)
2935.49
(+5Q)
3026.54 3065.64 3111.77 3174.68 3225.96
(+ll)
3369.02 3540.87
(-37)
(fQ) C-38) (+8) (to) (-14)
(-3)
value x10’ cm-‘.
the 0- 1 A-X band have been exchanged with respect to the original assignment ( I). As for the O-l A221---X*II band of 14ND+, where a similar situation occurs, no exchange should be made, as was shown by the fit of the corresponding levels to be presented below. III. MATRIX ELEMENTS
The matrix elements used in this analysis to represent the levels of interacting 211 and 42- states are the same as those used by Kawaguchi and Amano (2). However, since the present analysis is based exclusively on the optical data, it was expected and
R. COLIN TABLE II Tern
Values of the Rotational Levels of the X *rI State of 14ND+ Referred to F,, (0.5) of u = 0 (cm -’ )
v=o 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5
0.00 21.47 57.91 109.75 177.33 260.83 360.32 475.85 607.34 754.77 918.03 1097.02 1291.55 1501.44 1726.38 1965.99
(+1)+ C-1)
;‘y; C-1) (+‘a (+1) (+a ;:i;
0.21 21.80 58.36 110.28 177.91 261.42 360.92 476.42 607.92 755.37 918.71 1097.80 1292.59 1502.90 1728.65 1969.81
;I:; C-1) (-3) (-5) (+3)
(+3) (-3) (-2) (-2) (-1) (+0) (+l) (+0) (to) (to) (+2) (-1) (-1) (-4) (-5) (+9)
89.98 135.89 199.30 279.81 377.11 490.98 621.25 767.75 930.31 1108.83 1303.06 1512.88 1738.00 1978.19
C-2) (-2) t-11
89.96 135.79 199.06 279.41 376.49 490.12 620.15 766.36 928.56 1106.44 1303.89 1511.57 1736.53 1977.07
I:;; (+2) ;I:; (+1) (+2) (-3) (-4) (-6) (+7)
C-1) t-11 C-1)
I;:;
I::; I’:; (-4) (+0) (+O) (-4) (t2)
v=l 0.5 1.5 2.5 3.5 4.5
2147.78 2168.13 2202.83 2252.24 2316.59
(+3) (+3) (+5) (t3) (+3)
2148.31 2169.19 2204.48 2254.70 2320.13
(+2) (+0) (-1) (+l) (-1)
2230.83 2275.14 2336.06 2412.95
(+3) (-1) (-8 ) (-17)
5.5 6.5
2395.75 2489.14
(-1) (-24)
2400.98 2497.34
(-2) (-2)
2505.07 2612.15
7.5
2596.37 2656.64 2771.48 2905.70 3058.97 3229.79 3417.11
(-11) (-21) (-13) (-7) (+5) (+lO) (+13)
2609.22
(-1)
2736.58 2879.35 3037.42 3210.10
(t3) (+8) (t17) (-24)
8.5 9.5 10.5 11.5 12.5
+ observed-calculated
(-24) (t68)
2230.78 2274.93 2335.22 2407.35 2429.40 2514.08 2622.45
(+3) (to) (tl) (-10) (+17) (-2) (-3)
2661.48 2775.95
(t9) (+8)
2747.78
(-2)
2910.41 3063.96 3234.96 3422.35
(+4) (-3) (-10) (-11)
2889.06 3045.93
(+l) (+lO)
value x10’ cm-!
proven to be useless to take into account the higher order terms which were necessary in the work of Kawaguchi and Amano on 14NH+ . Thus the following parameters were set equal to zero: the centrifugal distortion spin-rotation (71,) parameter for the 211 state, the centrifugal spin-spin ( X,), the spin-spin ( X), and the spin-rotation (r ) parameters for the perturbing 42- state, and the centrifugal distortion spin-orbit coupling parameter (ED) between the 211 and 42 - states. When the optical data of the two heavier isotopes were fitted, based on weaker spectra than for 14NH+, it also proved impossible to derive the centrifugal distortion A-doubling parameters (pD and qD). As in the case of the analysis presented by Kawaguchi and Amano (2) it was also necessary to allow for two independent 211-4Z- coupling parameters EI12 and [3,2,
THE X*II AND a48-
391
STATES OF NH+
TABLE III Effective Hamiltonian Matrix Elements for Interacting ‘iI and ?Z - States L
%*
-
1 3R
T’ + B’ (X2-l) - 0’ (X’+X’-2)
(3X’-3Y2 (B’ -2D’(?zX
+l)l
T1 1. t 112
+Bt(XZ?2X+3) -dI(xfi3X’+6) t(4X’+12X)I
0
T” + A12 +(6-D”, (X*-21 -D” (X2-21’ ,H”[(X’-2)‘+3(?-lf -IX’-111
2n3,2
(0.5/3”7 5,,,
( x*-l)"'[ 8” -l/2 Y - 20” (X2-l, +H”(3(X’-lf +X*)1 z (q/2)X(X’-1)“’
_LT”- Al2 +(B”-D”)X’ - D”X‘ + H”[x6+3(xz-11z
‘J-h,2
+(X2-ljlt(P/2)X
X.J+112
uopcr signfor f Ievels lowersignfore Levels
one for each pair of D substates. The matrix used for the fitting of the rotational of the u = 0 and u = 1 levels of “NH+ and 14ND+ is presented in Table III.
levels
IV. RESULTS AND DISCUSSION
a. Constants for Individual Vibrational LeveIs The molecular constants obtained from the fits of the ZI = 0 and o = 1 levels of “NH+ and 14ND+ are presented in Table IV. The error limits given in parentheses after each constant refer to the last given decimal and correspond to one standard deviation. Table IV also presents the results of similar fits carried out on the optical data for 2, = 0 and u = 1 of the 14NH+ isotope. For the u = 0 level of this isotope a minimum variance value of the fit was reached by addition to the matrix of Table III of centrifugal distortion h-doubling parameters (pn and qo) similar to those used by Kawaguchi and Amano (2). The values of these constants are given in the footnote of Table IV.
R. COLIN
392
TABLE IV Molecular Constants for the u = 0 and u = 1 Levels of the X*II and a‘?of “‘NH+, “NH+, and 14NDt (cm-‘) ‘rNH+
“NH+
States
“ND+
v=o x=Il
T B l@D 10% A 1027 P 1o*q T
a’C-
B 103D c1 Wsol’C-
>
CI/l c3/2
0.0 15.3310 1.63 1.35 81.66 -8.0 0.226(“) 5.22(O)
0.0 15.2659
(3) (4) (2) (3) (1)
14.657 1.78
Ii;
324.57 14.597 1.80
87.22 81.12
(9) (6)
87.00 81.10
323.90
(14)
1.65 2.12 81.62 -9.5 0.238 5.04
(16) (2) (6)
0.0 8.2464 0.464
(2) (9)
-
(3) (5) (3)
81.34 -4.9 0.109 1.47
(3) (2) (13)
(26) (10) (10)
401.86 (7.826) (0.51)
(24) (a)
(16) (11)
82.91 77.77
(53) (30)
(8)
(1)
(5)
v=l X9
T B
103D 10% A 1017 P
1o*q
a%
T
B 103D <* lwsol’c-
>
:;‘: /
2903.19
(9)
14.7235 1.57
(30) (2)
(1.3) 81.82 -7.5 0.242 5.46
(c) (16) (1) (19) (17)
2868.01 13.945
(11) (9)
(15) 14.6591 (86) 1.56 (12) (2.1) 81.65 (k; -10.53 (66) 0.221 (41) 5.02 (57)
2897.21
2863.81 13.827
(15) (16)
2149.83 (84) 8.0132 (29) 0.48
(2)
81146
(15)
-6.9 0.116
(11) (17)
1.72
(14)
2358.17 7.626
(46) (22)
1.92
(16)
0.71
(37)
0.87
(30)
82.75 76.83
(33) (22)
84.07 76.39
(45) (30)
82.75 76.33
(41) (22)
(a) pi = (-4.0 f l.O)lO-’ ; qD = (-8.3 f O.S)lO-’ cm-‘. (6) from isotopic relation. (c) fixed to v=o value.
These results call for the following comments: The constants obtained for 14NH+ (V = 0 and u = 1) compare well with the corresponding more precise ones of Kawaguchi and Amano (2). This is to be expected since the latter are essentially based on the same data and the same matrix. The quality of the fits for “NH+ and 14ND+ can be appreciated by inspecting the observed-calculated values of each individual level. These are presented in Tables I and II in parentheses next to the experimental values. The average absolute error is of the order of kO.O.5cm -’ for v = 0 and +O. 10 for 2) = 1; only in a few cases does the error exceed k0.3 cm-‘.
THE X*rI
AND
a?-
STATES OF NH+
393
No third order rotational constant H could be obtained for the a4Z- state. For the heavier isotope 14ND+, the H constant for the X*II state was also undetermined. For the X*II u = 1 level of 14NH+ and “NH+, H was set equal to its value for 2, = 0. Attempts were made in each case to have only one 211-4Z- coupling parameter 4. The variance of the fits deteriorated each time by at least a factor of 10. The uncertainties of the original assignments ( 1) of the Q, rcand Q2zclines mentioned above for the O-l bands of the A*Z--X*II systems of “NH+ and 14ND+ were lifted by performing fits for each alternative assignment. The variance of the fits dropped by at least a factor of three when the assignments incorporated in Tables I and II were adopted. The six sets of levels can be classified into three categories according to the amount of perturbation present. The 14ND+ u = 0 set is a unique set in which very little perturbation occurs; no extra levels are observed. The 14NH+ 21= 0, “NH+ u = 0, and 14ND’ v = 1 sets are similar to each other in that approximately half a dozen extra levels are observed. Finally, the 14NH+ u = 1 and “NH+ v = 1 sets present a very large number of strong interactions between the *II and 42- states, especially at low J values. leading to the presence of complete extra branches in the bands which involve these levels. Figures l-3 present graphs of the rotational energies of one of each of these cases; continuous lines link together observed levels specified in the legend, dotted lines link together calculated levels. For the 14ND+ 21= 0 set shown in Fig. 1, it is clear that the interaction between the X’II and a4S- states is weak. It is therefore difficult in this case to derive good molecular constants for the perturbing 42- u = 0 level , i .e -3 Bt and Tz are strongly correlated. It was therefore chosen to fix the Bg and Di constants to values calculated by applying isotopic relations to the corresponding values for 14NH+. This made it possible to obtain a better vibronic term value Tt for the a4Z- state in order to derive the best possible vibrational constants ( see below ) . b. Equilibrium Rotational Constants For both states and all three isotopes, B,, D,, and cy, values were calculated and are presented in Table V along with the derived re values. For the calculation of the Be values in the case of the X*II state, corrections due to higher-order vibrational-rotational constants and second-order rotational interactions have been made in exactly the same manner as suggested by Kawaguchi and Amano (2). The isotopically averaged re values are r,(X*II) = 1.0691 +- 0.0001 A and r,( a42-) = 1.095 + 0.004 A, which agree with the values derived by Kawaguchi and Amano (2): r,(X211) = 1.0692 +_0.0002 A and r,( a42-) = 1.0924 -t 0.0001 A. c. Vibrational Constants Since only one vibrational quantum is measured for each state, vibrational constants can be derived only by using isotopic relations. With three isotopic values of AGfj2 it should be possible in principle to derive w,, w,x,, and weye. However, this turns out to be impossible due to the very small isotope shift between the 14Nand 15Nhydrides, the uncertainties on the vibronic term values, and the limited validity of the isotopic relations. Therefore, using the 14NH+ and i4NDf values of AGr12, one derived the
394
R. COLIN Elcnr’)
00 -
___-__
calculated
5
I
I
10
15
J
FIG. I. Rotational energies E (in cm -’ ) asa function of J for the Q components of the u = 0 level of the X*Il and a4Z- states of 14ND+.
a4Z- vibrational constants W, and w,xe given in Table V, from which the “NH+ values were calculated. These are the first precise determination of the vibrational constants of the ground and first excited states of the NH+ ion. For the ground state they can be compared to the low resolution values obtained by Kusunoki and Ottinger (6) from chemiluminescence studies on the N+ + Hz and N+ + D2 reactions (w, = 3038, W,X, = 58 cm-’ for the hydride and w, = 2218, w,x, = 3 1 cm-’ for the deuteride).
X211 and
d. “Z- Electronic Term Value From the vibrational constants it is possible to derive, separately for each isotope, the zero point energy EO for each state by using the expression
THE X%
AND
a4Z-
STATES OF NH+
395
Elcn LO{
0
5
10
15 J
FIG. 2. Rotational energies E (in cm -’ ) as a function of J for the R components of the u = 0 level of the XzIt and a4Z _ states of 14NH+. Same legend as in Fig. 1.
396
R. COLIN
Ekm,‘I
‘I, 600(l-
5001)-
3,
“NH+ v=l
400(3-
FIG. 3. Rotational energies E (in cm-‘) as a function of J for the R components of the v = 1 level of the A’% and a’?- states of 14NH+. Same legend as in Fig. 1.
where Yio are the vibrational coefficients of the Dunham expansion of the potential energy curve. As is customary, the Dunham coefficients are assimilated to the usual
THE X2JI AND a?-
397
STATES OF NH’
TABLE V Equilibrium Molecular Constants for the X211 and a4Z _ States of 14NH+, “NH+, and 14ND’ (cm -‘J 14NH+
XTI
T" WC
acr
0.0
3047.58
We&
72.19
B, a, b(J-1
15.6911 0.6075 1.0692
T= WC w.2, B.
509.41 2672.57 64.23 15.013 0.712 1.093
"ND+
IsNIl+
0.0
0.0
3040.77 71.87
2226.93 38.55
15.6253 0.6068 1.0691
509.78 2666.60 63.94
8.3781 0.2332 1.0692
537.85 1952.90 34.30
14.982 0.770 1.092
7.926 0.200 1.099
vibrational and rotational constants ( Y10= o,, Y2,,= w,x~, Y,,, = B,, - * - ) and the Yooterm is approximated by the expression
Applied to each isotope, this leads to the following electronic term values for the a4Z- state with respect to the X211 state: 14NH+: T" - T" = 505.69 cm-’ “NH+: T' - T" = 506.50 cm-’ 14ND+: T" - T" = 539.04 cm-‘. Since these values differ for each isotope, they indicate the presence of an electronic isotope shift ( 7). This is due to the term B,[L(L + 1) - A2] which was included in the constants T' and T" used in the diagonal matrix element. One may write
T,"- T,"= T=- T" - (B,[L(L+ l)-
A2])’ + {B,[L(L+ l)-
A2]}“.
where Tz - Ta is the pure, isotopically independent, electronic term value of the a4Z- state. The derivation of a pure electronic term value of a state with respect to another from the observation of transitions between these states requires a knowledge of the expectation values of (i’) for both states involved. These are accessible by ab initio calculation, but for the X211 and a42- states of NH+ they are not available in spite of the numerous calculations performed on this ion. One can estimate these L( L + 1) quantities in favourable cases by considering the correlation of the state involved with the united and separated atoms models. In the case of NH+, the a4Z- state correlates with the 0’(4S) state of the united atom and
398
R. COLIN
dissociates to N( 4S) + H+( -). It is therefore very likely that L = 0 for the molecule in its a4Z- state. For the X’II state, the situation is more uncertain since, on the one hand, this state correlates with the O+( *D) state for which L = 2 and, on the other hand, it dissociates to N+( 3P) + H(*S) which would give L = 1. One expects thus, for the X*II state, a value of L lying between 1 and 2. Since T,” - T,” should be identical for each isotope (if one neglects small corrections due to the breakdown of the Born-Oppenheimer approximation detailed by Bunker ( 7)) and since three independent isotopic values of T” - T” are available from this study, it should be possible in principle to derive T,” - T:, { L( L + 1) } u, and { L( L + 1) ) ‘. However, the accumulated uncertainties on the T” - T” values are estimated to be at least equal to the difference (0.8 cm-‘) between the T” - T” values derived for the two hydrides 14NH+ and “NH+. It was therefore decided to fix {L( L + 1)} x to zero and use the data on i4NH+ and 14ND+ to derive T,” - T: and { L( L + 1) } n. This procedure leads to T,” - T,” = 577.2 cm-’ and L” = 1.91 which, as expected from the united and separate atoms models, lies between 1 and 2, although very close to 2. It would be interesting to compare this value to an ab initio calculated value. e. Potential Energy Curves and Vibrational Overlap Integrals The rotationless potential energy curves of the X *II and a42 - states were computed on the basis of the above derived equilibrium constants using a standard RKR program. The true RKR potential functions for both states are of course limited to energies lying below the energy of the u = 1 vibrational level of the 14NH+ isotope. Above this limit, a Morse potential was assumed for the X*II state which dissociates to N+( 3P)
TABLE VI RKR Potential
Energy Functions
Radius4 A
States of NH’
of the X*II and a%-
Energy XzII (cm-‘)
0.900 0.925 0.950 0.976 1.000
5369.3 3701.9 2387.5 1408.7 718.9
1.025 1.050 1xl75 1.100 1.125 1.150 1.175 1.200 1.225 1.250 1.275 1.300 1.325 1.360
277.6 49.6 4.3 114.9 358.8 715.3 1166.8 1697.6 2294.2 2944.6 3638.4 4366.3 5122.9 5902.6
Energy a% (m-l) 5885.8 4236.4 2892.9 1840.3 1071.9 539.1 203.4 33.9 4.6 94.1 285.4 561.6 909.9 1318.6 1777.7 2278.4 2813.0 3374.9 3958.4
THE X*l-l AND &-
STATES OF NH +
3000 -
2000 -
1000 -
O0.90
1.00
1%
1.20
1.30
r(A)
FIG. 4. RKR potential energy functions of the X*11 and a%- states of NH’.
1 1.40
+ H(2S) with a dissociation energy of 36 400 cm-’ and for the a4C- state which dissociates to N(4S) + H+(-) with a dissociation energy of 28 300 cm-‘. These dissociation energies were based on the ab initio calculated value (8) of Dg(X211) = 3.39 eV for 14NH+ + N + H+. The RKR potential energy functions are given in Table VI and drawn in Fig. 4. where the observed vibrational levels for 14NHf and 14ND+ are also indicated. Such potential energy curves are of limited use, but they do make it possible to calculate the vibrational overlap integrals: ( 41;;/ ) ‘II,,,) for u’ and u” = 0 and 1. The squares of these values, the Franck-Condon factors, are given in Table VII and could be used in connection with the dipole moment functions calculated by Farnell and Ogilvie (4) to predict or compare with experimental results the relative intensities in the rotational-vibrational spectrum which involves 211-42 - transitions. Unfortunately, Kawaguchi and Amano (2) did not publish intensity measurements.
1: The 42--211 Coupling Constant The fact, already pointed out by Kawaguchi and Amano (2) for 14NH+, that two independent coupling constants between the X211 and a4Z- states (F1,2 for Q = 1
R. COLIN
400
TABLE VII Franck-Condon
Factors between the u = 0 and u = I Levels of the X*II and a4Z- States of 14NHi, lsNH+, and 14ND+
o-o
0.9708
0.9709
0.9520
o-1
0.2847
0.2833
0.4655
1-o
0.2881
0.2864
0.4691
l-l
0.9015
0.8981
0.8598
substates and [3,2 for Q = 2 substates) were necessary to fit properly the data, is intriguing. The present results show that the same situation holds for the two heavier isotopes and that similar values of .&,2 and ,& are obtained. This is not surprising since the t constant is a spin-orbit parameter which can be expressed for a given pair of interacting vibrational levels by
=
(211v” 14Z;t)(2111 2
Uili’
Sil"X)
i
= 6ucul
-
a,
where 6,v,f is the vibrational overlap integral and a is the pure electronic isotope independent spin-orbit matrix element. Using the overlap integrals derived above it is possible to calculate ~i,~ and a3/2 for each isotope and for v = 0 and 1 levels. The values are, respectively, for 14NH+, “NH+, and 14ND+, aI/2 (v = 0) = 88.5, 88.3, (85.0); alI2 (V = 1) = 87.1, 88.7, 89.2; a 312 (O = 0) = 82.3, 82.3, (79.7); ~312 (U = 1) = 80.9, 80.6, 82.3 cm-‘. If one neglects the somewhat lower values in parentheses, which were both obtained from the only very slightly perturbed 14ND+ 2) = 0 set of levels, one gets the average values all2 = 88.4 and ~312 = 81.7 cm-‘. The difference of 6.7 cm-’ is very likely due to the spin-orbit interaction of the X211ij2 substate with the excited C2ZT12 and the A2Z1/2 states and of the a421/2 substate with the C2Z:,2 state. Due to symmetry reasons the A22 iI state does not interact with the a42 1/2 substate. These types of interactions do not exist for the X2113,2 and 423/2 substates, but one should also consider a possible B2A3,2 interaction with the X2r13/2. These unequal mixings of the substates should account for the different spin-orbit interactions existing between the substates of a4Z- and X211, although it remains difficult to estimate their importance.
THE X*Il AND a%-
STATES OF NH+
401
ACKNOWLEDGMENTS Thanks are due to Dr. J. K. G. Watson for reading and improving the manuscript. I also thank Mr. M. Steinhauer for his help in the initial stages of this work. Acknowledgments are due for the financial support given by the Fonds National de la Recherche Scientifique (Belgium) under Contract FRFC 2.4567.86. RECEIVED:
March 13, 1989 REFERENCES
1. R. COLIN AND A. E. DOUGLAS, Canad. J. Phys. 46,61-73 (1968). 2. K. KAWAGUCHIAND T. AMANO, J. Chem. Phys. 88,4584-4591 (1988). 3. P. VERHOEVE,J. J. TERMEULEN,W. L. MEERTS,AND A. DYMANUS. Chem. Phys. Lett. 132,2 13-2 17 (1986). 4. L. FARNELLAND J. F. OGILVIE,.I. Mol. Specfrosc. 101, 104-132 ( 1983). 5. G. KRISHNAMURTHYAND M. SARASWATHY,Pramana 6,235-243 ( 1976). 6. I. KUSUNOKI AND C. OTTINGER,J. Chem. Phys. 80, 1872-188 I ( 1984). 7. P. R. BUNKER, J. Mol. Spectrosc. 28,422-443 ( 1968). 8. H. P. D. LIU AND G. VERHAEGEN,J. Chem. Phys. 53,735-745 ( 1970).
OF MOLECULAR
SPECTROSCOPY
(1989)
I%,387401
Perturbations between the X 211 and a 421- States of the NH + Ion R. COLIN Laboratoire de Chimie Physique Molkulaire.
Universite’ Libre de Bruxelles. Brussels, Belgium
On the basis of previously published optical data (Cohn and Douglas, Canad. J. Phvs. 46,6 I 73 ( 1968)) for three isotopes ( 14NH+, “NH+ and 14ND+) of the NH+ ion, a study of the perturbations occurring between the X2rI and a?- states has been carried out using matrix diagonalization techniques. This makes it possible to derive for both states equilibrium rotational and vibrational constants, from which RKR potential energy functions, vibrational overlap integrals. electronic interaction parameters, and a pure electronic term value for the a42 - state are deduced. 6 1989 Academic Press. Inc. 1. INTRODUCTION It was shown several years ago by Cohn and Douglas ( I ) on the basis of the optical emission spectra of three isotopes ( 14NH +, I5NH’, and 14ND+) of the NH+ ion that the ground electronic state of this ion is a ‘II state and that a very low lying a4t: state (T, - 550 cm-‘) causes strong perturbations in the 2, = 0 and u = 1 levels of the ground state. Displacements as large as 46 cm-’ were observed for some rotational levels. Rotational and vibrational constants were derived by these authors for both of these states and for the more highly excited A?-, B2A, and C2Z+ states. For the X’II state, this was done in an approximate manner by fitting the “less perturbed” rotational levels of the ‘II state to the usual Hill-Van Vleck equations. By calculation of the shifts of the strongly perturbed levels it was then possible to estimate the “original” position of the perturbing 42- levels. In turn these few data were fitted to a simplified rotational energy level equation to yield approximate rotational constants for the a4Z state. No attempt was made at that time to fully analyze the perturbations by matrix techniques. In a recent paper, Kawaguchi and Amano (2) reported the observation by tunable difference frequency laser techniques of the 1-O vibrational-rotational transitions of 14NH+ in the X*II and u4Z- states, as well as between these two electronic states. A merged least-square fit of the energy levels was carried out by these authors using their infrared data, the previous optical term values of Colin and Douglas ( 1) , and two far infrared pure rotational transitions measured by Verhoeve et al. (3). All these data were fitted with suitable weight factors to a 4 X 4 effective Hamiltonian matrix and accurate rotational constants, interaction parameters, and vibronic term values were derived for the ~1= 0 and v = 1 levels of the X’II and a4Z- states of the 14NH+ isotope. Due to the possible astrophysical importance of the NH+ ion, which, however, to our knowledge, has not yet been observed in any extraterrestrial object, it was thought to be useful to extend the treatment of Kawaguchi and Amano (2) to the other two 387
0022-2852189 $3.00 Copyright 0
1989 by Academic Press. Inc.
All righU of reproduction in any form resewed.
388
R. COLIN
isotopes 14ND+ and lsNH+ by using the optical data of Colin and Douglas (I). Of course, since no rotational-vibrational or pure rotational data exist for these isotopes, the accuracy of the molecular constants derived will be less than in the case of 14NH+. However, the results presented here will allow one, by using isotopic relations to calculate vibrational constants for the X211 and a4Z- states, to derive the pure electronic term value of the a4Z- state, and to calculate therefrom limited RKR potentials and vibrational overlap integrals. These data could be of some help in the detection of NH+ in astronomical sources. Since a summary of the most important theoretical and experimental papers on NH+ published between 1967 and 1988 has been given by Kawaguchi and Amano (2), it will not be repeated here. II. ENERGY
LEVELS
The optical spectra observed by Colin and Douglas ( 1) include for all three isotopes the O-O, l-0, and 0- 1 bands of the A ‘Z--X211 transition, the O-O band of B’A-X2& and the O-O band of C2Z+--X211. The data were all obtained from high resolution spectra taken with a 10-m spectrograph giving an error in relative line measurement of kO.05 cm-’ and an absolute uncertainty of&O. 10 cm-‘. The measurements for all the observed bands in the case of r4NH+ were given in the original paper (Tables I, II and III of ( I )). For the other two isotopes, the measurements have been placed in the Depository of Unpublished Data of the Canadian Journal of Physics. The branches were identified, rotationally numbered, and led, as far as the 211 ground state is concerned, to the expected four-fold set ( Tr,, T,d, Tzc, T& of rotational energy levels for 2) = 0 and for 2) = 1 referred to the TI, (J = 0.5 ) level of ~1= 0. The energies of the levels for 14NH+ were published (Table IV of ( 1)) and the data for “NH+ and 14ND+ were placed in the Depository. In all sets of levels with the exception of the 14ND+ o = 0 set, extra levels due to perturbations with the a4Z- state are present. In the original work ( I ), several extra lines with low J values observed in the O-l bands of the A2Z --X211 transition of 14NH+ and “NH+ could not be identified and were listed without assignments. In 1983, Fame11 and Ogilvie (4) performed ab initio molecular orbital calculations to compute the potential energy curves of the X*II and a4Z- states which allowed derivation of rotational-vibrational energy levels including the interaction between both states. When adjusted to the experimental data, these calculations allowed the identification of the previously unassigned lines. It should also be mentioned that the fitting of Kawaguchi and Amano (2) removed the uncertainty in the original assignments of the Q1I=(J > 4.5 ) and Q2zc (J > 3.5 ) lines in the 0- 1 A 22 --X211 band of 14NH+. It was pointed out by Colin and Douglas (I) that these branches could well be interchanged. The fitting of Kawaguchi and Amano (2) showed that this exchange should be made. Krishnamurthy and Saraswathy (5) reached the same conclusion from their analysis of the l- 1 and 2- 1 bands of the C2Z+-X*II system of 14NH+. Since Kawaguchi and Amano (2) have listed extensively in their paper the fitted energy levels for u = 0 and v = 1 of the X211 and a4Z- states of 14NH+, they will not be given here. For the “NH+ and r4ND+ molecules, the experimental rotational energy levels for the u = 0 and tt = 1 levels of the ‘II state are presented in Tables I and II, respectively. Here again for “NH+ (J > 4.5) the Qtlc and Q22c(J > 3.5) branches of
THE X’II AND a%-
389
STATES OF NH+
TABLE I Term Values of the Rotational Levels of the X’II State of “NH+ Referred to F,, (0.5) of u = 0 (cm -‘) J
TIC(J)
K(J)
T,d(J)
0.5 1.5 2.5 3.5 4.5 5.5
0.00 33.70 95.98 187.91 309.76 461.37
(-3)’ (-0) (+4) (+4) (+8) (+lO)
0.43 34.37 96.78 188.80 310.76 462.62
(-6) (-2) (-0) (-3) (-3) (-5)
6.5 7.5 8.5
642.44 852.41 1089.83
(+Il) (+16) (+14)
644.26 855.47 1096.01
(-6) (-9) (-11)
9.5
1351.13 1415.06 1629.33 1693.63 2012.05 2363.62 2745.14 3154.59
(+2) (-4)
1365.68
(-4)
(-21)
1663.93
(-8)
1990.60 2345.17 2727.15 3136.15
(-1) (+4) (t5) (t12)
10.5 11.5 12.5 13.5 14.5
G(J)
105.26 195.36 316.45 467.87 649.26
(+0) (-1) (t3) (t2) (+4)
860.09 1099.33 1363.81 1409.33 1642.74 1687.41 2008.95
(+7) (f5) (+l) (+6) (+6) (-2) (f13)
2362.72 2745.24 3155.41
(+Q) (-1) (-10)
2882.19 3009.71 2959.52 3090.91 3036.79 3205.95 3148.36 3349.95 3288.07 3524.21 3455.04 3727.36
(to) (-1
(+5) (tl6) (+5) (-6) (f3)
3958.94
C-5)
(+9) (tll) (-13) (-7) (-11)
0.5 1.5
2939.60 (+4) 2876.09 (-16)
2.5 3.5 4.5 5.5
2908.61 (+5) 3088.34 (-16) 2966.91 (+30) 3204.50 (-3) 3349.82
(-5)
6.5
3524.23
(+0)
7.5 8.5
3727.36 3958.85
(+5) (t3)
+observed-calculated
2939.85 2897.21 3006.40 2952.89 3096.97 3037.02 3214.35 3149.08 3359.54 3288.71 3532.89 3455.50
C-10) C-26)
C-1) C-10) (+l) (f6) (+-a) (i-18)
(+31 (+20)
(-6)
f-31) (t8) (-4) C-15) (i-13)
105.04 194.77 315.37 466.19 642.49
(+I) C-1) C-2)
653.66 859.06 1099.05 1368.40
(-5) (-3) (-4) (-8) (-12) (-5)
1666.41
C-10)
1992.90
(-4)
2347.31 2729.20 3138.04
(+O) (+5) (+7)
2935.49
(+5Q)
3026.54 3065.64 3111.77 3174.68 3225.96
(+ll)
3369.02 3540.87
(-37)
(fQ) C-38) (+8) (to) (-14)
(-3)
value x10’ cm-‘.
the 0- 1 A-X band have been exchanged with respect to the original assignment ( I). As for the O-l A221---X*II band of 14ND+, where a similar situation occurs, no exchange should be made, as was shown by the fit of the corresponding levels to be presented below. III. MATRIX ELEMENTS
The matrix elements used in this analysis to represent the levels of interacting 211 and 42- states are the same as those used by Kawaguchi and Amano (2). However, since the present analysis is based exclusively on the optical data, it was expected and
R. COLIN TABLE II Tern
Values of the Rotational Levels of the X *rI State of 14ND+ Referred to F,, (0.5) of u = 0 (cm -’ )
v=o 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 14.5 15.5
0.00 21.47 57.91 109.75 177.33 260.83 360.32 475.85 607.34 754.77 918.03 1097.02 1291.55 1501.44 1726.38 1965.99
(+1)+ C-1)
;‘y; C-1) (+‘a (+1) (+a ;:i;
0.21 21.80 58.36 110.28 177.91 261.42 360.92 476.42 607.92 755.37 918.71 1097.80 1292.59 1502.90 1728.65 1969.81
;I:; C-1) (-3) (-5) (+3)
(+3) (-3) (-2) (-2) (-1) (+0) (+l) (+0) (to) (to) (+2) (-1) (-1) (-4) (-5) (+9)
89.98 135.89 199.30 279.81 377.11 490.98 621.25 767.75 930.31 1108.83 1303.06 1512.88 1738.00 1978.19
C-2) (-2) t-11
89.96 135.79 199.06 279.41 376.49 490.12 620.15 766.36 928.56 1106.44 1303.89 1511.57 1736.53 1977.07
I:;; (+2) ;I:; (+1) (+2) (-3) (-4) (-6) (+7)
C-1) t-11 C-1)
I;:;
I::; I’:; (-4) (+0) (+O) (-4) (t2)
v=l 0.5 1.5 2.5 3.5 4.5
2147.78 2168.13 2202.83 2252.24 2316.59
(+3) (+3) (+5) (t3) (+3)
2148.31 2169.19 2204.48 2254.70 2320.13
(+2) (+0) (-1) (+l) (-1)
2230.83 2275.14 2336.06 2412.95
(+3) (-1) (-8 ) (-17)
5.5 6.5
2395.75 2489.14
(-1) (-24)
2400.98 2497.34
(-2) (-2)
2505.07 2612.15
7.5
2596.37 2656.64 2771.48 2905.70 3058.97 3229.79 3417.11
(-11) (-21) (-13) (-7) (+5) (+lO) (+13)
2609.22
(-1)
2736.58 2879.35 3037.42 3210.10
(t3) (+8) (t17) (-24)
8.5 9.5 10.5 11.5 12.5
+ observed-calculated
(-24) (t68)
2230.78 2274.93 2335.22 2407.35 2429.40 2514.08 2622.45
(+3) (to) (tl) (-10) (+17) (-2) (-3)
2661.48 2775.95
(t9) (+8)
2747.78
(-2)
2910.41 3063.96 3234.96 3422.35
(+4) (-3) (-10) (-11)
2889.06 3045.93
(+l) (+lO)
value x10’ cm-!
proven to be useless to take into account the higher order terms which were necessary in the work of Kawaguchi and Amano on 14NH+ . Thus the following parameters were set equal to zero: the centrifugal distortion spin-rotation (71,) parameter for the 211 state, the centrifugal spin-spin ( X,), the spin-spin ( X), and the spin-rotation (r ) parameters for the perturbing 42- state, and the centrifugal distortion spin-orbit coupling parameter (ED) between the 211 and 42 - states. When the optical data of the two heavier isotopes were fitted, based on weaker spectra than for 14NH+, it also proved impossible to derive the centrifugal distortion A-doubling parameters (pD and qD). As in the case of the analysis presented by Kawaguchi and Amano (2) it was also necessary to allow for two independent 211-4Z- coupling parameters EI12 and [3,2,
THE X*II AND a48-
391
STATES OF NH+
TABLE III Effective Hamiltonian Matrix Elements for Interacting ‘iI and ?Z - States L
%*
-
1 3R
T’ + B’ (X2-l) - 0’ (X’+X’-2)
(3X’-3Y2 (B’ -2D’(?zX
+l)l
T1 1. t 112
+Bt(XZ?2X+3) -dI(xfi3X’+6) t(4X’+12X)I
0
T” + A12 +(6-D”, (X*-21 -D” (X2-21’ ,H”[(X’-2)‘+3(?-lf -IX’-111
2n3,2
(0.5/3”7 5,,,
( x*-l)"'[ 8” -l/2 Y - 20” (X2-l, +H”(3(X’-lf +X*)1 z (q/2)X(X’-1)“’
_LT”- Al2 +(B”-D”)X’ - D”X‘ + H”[x6+3(xz-11z
‘J-h,2
+(X2-ljlt(P/2)X
X.J+112
uopcr signfor f Ievels lowersignfore Levels
one for each pair of D substates. The matrix used for the fitting of the rotational of the u = 0 and u = 1 levels of “NH+ and 14ND+ is presented in Table III.
levels
IV. RESULTS AND DISCUSSION
a. Constants for Individual Vibrational LeveIs The molecular constants obtained from the fits of the ZI = 0 and o = 1 levels of “NH+ and 14ND+ are presented in Table IV. The error limits given in parentheses after each constant refer to the last given decimal and correspond to one standard deviation. Table IV also presents the results of similar fits carried out on the optical data for 2, = 0 and u = 1 of the 14NH+ isotope. For the u = 0 level of this isotope a minimum variance value of the fit was reached by addition to the matrix of Table III of centrifugal distortion h-doubling parameters (pn and qo) similar to those used by Kawaguchi and Amano (2). The values of these constants are given in the footnote of Table IV.
R. COLIN
392
TABLE IV Molecular Constants for the u = 0 and u = 1 Levels of the X*II and a‘?of “‘NH+, “NH+, and 14NDt (cm-‘) ‘rNH+
“NH+
States
“ND+
v=o x=Il
T B l@D 10% A 1027 P 1o*q T
a’C-
B 103D c1 Wsol’C-
>
CI/l c3/2
0.0 15.3310 1.63 1.35 81.66 -8.0 0.226(“) 5.22(O)
0.0 15.2659
(3) (4) (2) (3) (1)
14.657 1.78
Ii;
324.57 14.597 1.80
87.22 81.12
(9) (6)
87.00 81.10
323.90
(14)
1.65 2.12 81.62 -9.5 0.238 5.04
(16) (2) (6)
0.0 8.2464 0.464
(2) (9)
-
(3) (5) (3)
81.34 -4.9 0.109 1.47
(3) (2) (13)
(26) (10) (10)
401.86 (7.826) (0.51)
(24) (a)
(16) (11)
82.91 77.77
(53) (30)
(8)
(1)
(5)
v=l X9
T B
103D 10% A 1017 P
1o*q
a%
T
B 103D <* lwsol’c-
>
:;‘: /
2903.19
(9)
14.7235 1.57
(30) (2)
(1.3) 81.82 -7.5 0.242 5.46
(c) (16) (1) (19) (17)
2868.01 13.945
(11) (9)
(15) 14.6591 (86) 1.56 (12) (2.1) 81.65 (k; -10.53 (66) 0.221 (41) 5.02 (57)
2897.21
2863.81 13.827
(15) (16)
2149.83 (84) 8.0132 (29) 0.48
(2)
81146
(15)
-6.9 0.116
(11) (17)
1.72
(14)
2358.17 7.626
(46) (22)
1.92
(16)
0.71
(37)
0.87
(30)
82.75 76.83
(33) (22)
84.07 76.39
(45) (30)
82.75 76.33
(41) (22)
(a) pi = (-4.0 f l.O)lO-’ ; qD = (-8.3 f O.S)lO-’ cm-‘. (6) from isotopic relation. (c) fixed to v=o value.
These results call for the following comments: The constants obtained for 14NH+ (V = 0 and u = 1) compare well with the corresponding more precise ones of Kawaguchi and Amano (2). This is to be expected since the latter are essentially based on the same data and the same matrix. The quality of the fits for “NH+ and 14ND+ can be appreciated by inspecting the observed-calculated values of each individual level. These are presented in Tables I and II in parentheses next to the experimental values. The average absolute error is of the order of kO.O.5cm -’ for v = 0 and +O. 10 for 2) = 1; only in a few cases does the error exceed k0.3 cm-‘.
THE X*rI
AND
a?-
STATES OF NH+
393
No third order rotational constant H could be obtained for the a4Z- state. For the heavier isotope 14ND+, the H constant for the X*II state was also undetermined. For the X*II u = 1 level of 14NH+ and “NH+, H was set equal to its value for 2, = 0. Attempts were made in each case to have only one 211-4Z- coupling parameter 4. The variance of the fits deteriorated each time by at least a factor of 10. The uncertainties of the original assignments ( 1) of the Q, rcand Q2zclines mentioned above for the O-l bands of the A*Z--X*II systems of “NH+ and 14ND+ were lifted by performing fits for each alternative assignment. The variance of the fits dropped by at least a factor of three when the assignments incorporated in Tables I and II were adopted. The six sets of levels can be classified into three categories according to the amount of perturbation present. The 14ND+ u = 0 set is a unique set in which very little perturbation occurs; no extra levels are observed. The 14NH+ 21= 0, “NH+ u = 0, and 14ND’ v = 1 sets are similar to each other in that approximately half a dozen extra levels are observed. Finally, the 14NH+ u = 1 and “NH+ v = 1 sets present a very large number of strong interactions between the *II and 42- states, especially at low J values. leading to the presence of complete extra branches in the bands which involve these levels. Figures l-3 present graphs of the rotational energies of one of each of these cases; continuous lines link together observed levels specified in the legend, dotted lines link together calculated levels. For the 14ND+ 21= 0 set shown in Fig. 1, it is clear that the interaction between the X’II and a4S- states is weak. It is therefore difficult in this case to derive good molecular constants for the perturbing 42- u = 0 level , i .e -3 Bt and Tz are strongly correlated. It was therefore chosen to fix the Bg and Di constants to values calculated by applying isotopic relations to the corresponding values for 14NH+. This made it possible to obtain a better vibronic term value Tt for the a4Z- state in order to derive the best possible vibrational constants ( see below ) . b. Equilibrium Rotational Constants For both states and all three isotopes, B,, D,, and cy, values were calculated and are presented in Table V along with the derived re values. For the calculation of the Be values in the case of the X*II state, corrections due to higher-order vibrational-rotational constants and second-order rotational interactions have been made in exactly the same manner as suggested by Kawaguchi and Amano (2). The isotopically averaged re values are r,(X*II) = 1.0691 +- 0.0001 A and r,( a42-) = 1.095 + 0.004 A, which agree with the values derived by Kawaguchi and Amano (2): r,(X211) = 1.0692 +_0.0002 A and r,( a42-) = 1.0924 -t 0.0001 A. c. Vibrational Constants Since only one vibrational quantum is measured for each state, vibrational constants can be derived only by using isotopic relations. With three isotopic values of AGfj2 it should be possible in principle to derive w,, w,x,, and weye. However, this turns out to be impossible due to the very small isotope shift between the 14Nand 15Nhydrides, the uncertainties on the vibronic term values, and the limited validity of the isotopic relations. Therefore, using the 14NH+ and i4NDf values of AGr12, one derived the
394
R. COLIN Elcnr’)
00 -
___-__
calculated
5
I
I
10
15
J
FIG. I. Rotational energies E (in cm -’ ) asa function of J for the Q components of the u = 0 level of the X*Il and a4Z- states of 14ND+.
a4Z- vibrational constants W, and w,xe given in Table V, from which the “NH+ values were calculated. These are the first precise determination of the vibrational constants of the ground and first excited states of the NH+ ion. For the ground state they can be compared to the low resolution values obtained by Kusunoki and Ottinger (6) from chemiluminescence studies on the N+ + Hz and N+ + D2 reactions (w, = 3038, W,X, = 58 cm-’ for the hydride and w, = 2218, w,x, = 3 1 cm-’ for the deuteride).
X211 and
d. “Z- Electronic Term Value From the vibrational constants it is possible to derive, separately for each isotope, the zero point energy EO for each state by using the expression
THE X%
AND
a4Z-
STATES OF NH+
395
Elcn LO{
0
5
10
15 J
FIG. 2. Rotational energies E (in cm -’ ) as a function of J for the R components of the u = 0 level of the XzIt and a4Z _ states of 14NH+. Same legend as in Fig. 1.
396
R. COLIN
Ekm,‘I
‘I, 600(l-
5001)-
3,
“NH+ v=l
400(3-
FIG. 3. Rotational energies E (in cm-‘) as a function of J for the R components of the v = 1 level of the A’% and a’?- states of 14NH+. Same legend as in Fig. 1.
where Yio are the vibrational coefficients of the Dunham expansion of the potential energy curve. As is customary, the Dunham coefficients are assimilated to the usual
THE X2JI AND a?-
397
STATES OF NH’
TABLE V Equilibrium Molecular Constants for the X211 and a4Z _ States of 14NH+, “NH+, and 14ND’ (cm -‘J 14NH+
XTI
T" WC
acr
0.0
3047.58
We&
72.19
B, a, b(J-1
15.6911 0.6075 1.0692
T= WC w.2, B.
509.41 2672.57 64.23 15.013 0.712 1.093
"ND+
IsNIl+
0.0
0.0
3040.77 71.87
2226.93 38.55
15.6253 0.6068 1.0691
509.78 2666.60 63.94
8.3781 0.2332 1.0692
537.85 1952.90 34.30
14.982 0.770 1.092
7.926 0.200 1.099
vibrational and rotational constants ( Y10= o,, Y2,,= w,x~, Y,,, = B,, - * - ) and the Yooterm is approximated by the expression
Applied to each isotope, this leads to the following electronic term values for the a4Z- state with respect to the X211 state: 14NH+: T" - T" = 505.69 cm-’ “NH+: T' - T" = 506.50 cm-’ 14ND+: T" - T" = 539.04 cm-‘. Since these values differ for each isotope, they indicate the presence of an electronic isotope shift ( 7). This is due to the term B,[L(L + 1) - A2] which was included in the constants T' and T" used in the diagonal matrix element. One may write
T,"- T,"= T=- T" - (B,[L(L+ l)-
A2])’ + {B,[L(L+ l)-
A2]}“.
where Tz - Ta is the pure, isotopically independent, electronic term value of the a4Z- state. The derivation of a pure electronic term value of a state with respect to another from the observation of transitions between these states requires a knowledge of the expectation values of (i’) for both states involved. These are accessible by ab initio calculation, but for the X211 and a42- states of NH+ they are not available in spite of the numerous calculations performed on this ion. One can estimate these L( L + 1) quantities in favourable cases by considering the correlation of the state involved with the united and separated atoms models. In the case of NH+, the a4Z- state correlates with the 0’(4S) state of the united atom and
398
R. COLIN
dissociates to N( 4S) + H+( -). It is therefore very likely that L = 0 for the molecule in its a4Z- state. For the X’II state, the situation is more uncertain since, on the one hand, this state correlates with the O+( *D) state for which L = 2 and, on the other hand, it dissociates to N+( 3P) + H(*S) which would give L = 1. One expects thus, for the X*II state, a value of L lying between 1 and 2. Since T,” - T,” should be identical for each isotope (if one neglects small corrections due to the breakdown of the Born-Oppenheimer approximation detailed by Bunker ( 7)) and since three independent isotopic values of T” - T” are available from this study, it should be possible in principle to derive T,” - T:, { L( L + 1) } u, and { L( L + 1) ) ‘. However, the accumulated uncertainties on the T” - T” values are estimated to be at least equal to the difference (0.8 cm-‘) between the T” - T” values derived for the two hydrides 14NH+ and “NH+. It was therefore decided to fix {L( L + 1)} x to zero and use the data on i4NH+ and 14ND+ to derive T,” - T: and { L( L + 1) } n. This procedure leads to T,” - T,” = 577.2 cm-’ and L” = 1.91 which, as expected from the united and separate atoms models, lies between 1 and 2, although very close to 2. It would be interesting to compare this value to an ab initio calculated value. e. Potential Energy Curves and Vibrational Overlap Integrals The rotationless potential energy curves of the X *II and a42 - states were computed on the basis of the above derived equilibrium constants using a standard RKR program. The true RKR potential functions for both states are of course limited to energies lying below the energy of the u = 1 vibrational level of the 14NH+ isotope. Above this limit, a Morse potential was assumed for the X*II state which dissociates to N+( 3P)
TABLE VI RKR Potential
Energy Functions
Radius4 A
States of NH’
of the X*II and a%-
Energy XzII (cm-‘)
0.900 0.925 0.950 0.976 1.000
5369.3 3701.9 2387.5 1408.7 718.9
1.025 1.050 1xl75 1.100 1.125 1.150 1.175 1.200 1.225 1.250 1.275 1.300 1.325 1.360
277.6 49.6 4.3 114.9 358.8 715.3 1166.8 1697.6 2294.2 2944.6 3638.4 4366.3 5122.9 5902.6
Energy a% (m-l) 5885.8 4236.4 2892.9 1840.3 1071.9 539.1 203.4 33.9 4.6 94.1 285.4 561.6 909.9 1318.6 1777.7 2278.4 2813.0 3374.9 3958.4
THE X*l-l AND &-
STATES OF NH +
3000 -
2000 -
1000 -
O0.90
1.00
1%
1.20
1.30
r(A)
FIG. 4. RKR potential energy functions of the X*11 and a%- states of NH’.
1 1.40
+ H(2S) with a dissociation energy of 36 400 cm-’ and for the a4C- state which dissociates to N(4S) + H+(-) with a dissociation energy of 28 300 cm-‘. These dissociation energies were based on the ab initio calculated value (8) of Dg(X211) = 3.39 eV for 14NH+ + N + H+. The RKR potential energy functions are given in Table VI and drawn in Fig. 4. where the observed vibrational levels for 14NHf and 14ND+ are also indicated. Such potential energy curves are of limited use, but they do make it possible to calculate the vibrational overlap integrals: ( 41;;/ ) ‘II,,,) for u’ and u” = 0 and 1. The squares of these values, the Franck-Condon factors, are given in Table VII and could be used in connection with the dipole moment functions calculated by Farnell and Ogilvie (4) to predict or compare with experimental results the relative intensities in the rotational-vibrational spectrum which involves 211-42 - transitions. Unfortunately, Kawaguchi and Amano (2) did not publish intensity measurements.
1: The 42--211 Coupling Constant The fact, already pointed out by Kawaguchi and Amano (2) for 14NH+, that two independent coupling constants between the X211 and a4Z- states (F1,2 for Q = 1
R. COLIN
400
TABLE VII Franck-Condon
Factors between the u = 0 and u = I Levels of the X*II and a4Z- States of 14NHi, lsNH+, and 14ND+
o-o
0.9708
0.9709
0.9520
o-1
0.2847
0.2833
0.4655
1-o
0.2881
0.2864
0.4691
l-l
0.9015
0.8981
0.8598
substates and [3,2 for Q = 2 substates) were necessary to fit properly the data, is intriguing. The present results show that the same situation holds for the two heavier isotopes and that similar values of .&,2 and ,& are obtained. This is not surprising since the t constant is a spin-orbit parameter which can be expressed for a given pair of interacting vibrational levels by
=
(211v” 14Z;t)(2111 2
Uili’
Sil"X)
i
= 6ucul
-
a,
where 6,v,f is the vibrational overlap integral and a is the pure electronic isotope independent spin-orbit matrix element. Using the overlap integrals derived above it is possible to calculate ~i,~ and a3/2 for each isotope and for v = 0 and 1 levels. The values are, respectively, for 14NH+, “NH+, and 14ND+, aI/2 (v = 0) = 88.5, 88.3, (85.0); alI2 (V = 1) = 87.1, 88.7, 89.2; a 312 (O = 0) = 82.3, 82.3, (79.7); ~312 (U = 1) = 80.9, 80.6, 82.3 cm-‘. If one neglects the somewhat lower values in parentheses, which were both obtained from the only very slightly perturbed 14ND+ 2) = 0 set of levels, one gets the average values all2 = 88.4 and ~312 = 81.7 cm-‘. The difference of 6.7 cm-’ is very likely due to the spin-orbit interaction of the X211ij2 substate with the excited C2ZT12 and the A2Z1/2 states and of the a421/2 substate with the C2Z:,2 state. Due to symmetry reasons the A22 iI state does not interact with the a42 1/2 substate. These types of interactions do not exist for the X2113,2 and 423/2 substates, but one should also consider a possible B2A3,2 interaction with the X2r13/2. These unequal mixings of the substates should account for the different spin-orbit interactions existing between the substates of a4Z- and X211, although it remains difficult to estimate their importance.
THE X*Il AND a%-
STATES OF NH+
401
ACKNOWLEDGMENTS Thanks are due to Dr. J. K. G. Watson for reading and improving the manuscript. I also thank Mr. M. Steinhauer for his help in the initial stages of this work. Acknowledgments are due for the financial support given by the Fonds National de la Recherche Scientifique (Belgium) under Contract FRFC 2.4567.86. RECEIVED:
March 13, 1989 REFERENCES
1. R. COLIN AND A. E. DOUGLAS, Canad. J. Phys. 46,61-73 (1968). 2. K. KAWAGUCHIAND T. AMANO, J. Chem. Phys. 88,4584-4591 (1988). 3. P. VERHOEVE,J. J. TERMEULEN,W. L. MEERTS,AND A. DYMANUS. Chem. Phys. Lett. 132,2 13-2 17 (1986). 4. L. FARNELLAND J. F. OGILVIE,.I. Mol. Specfrosc. 101, 104-132 ( 1983). 5. G. KRISHNAMURTHYAND M. SARASWATHY,Pramana 6,235-243 ( 1976). 6. I. KUSUNOKI AND C. OTTINGER,J. Chem. Phys. 80, 1872-188 I ( 1984). 7. P. R. BUNKER, J. Mol. Spectrosc. 28,422-443 ( 1968). 8. H. P. D. LIU AND G. VERHAEGEN,J. Chem. Phys. 53,735-745 ( 1970).