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Vibration-inversion-rotation spectra of ammonia
JOUBNALOF MOLECULAR SPECTROSCOPY 54, 339-349
(1975)
Vibration-Inversion-Rotation A Vibration-Inversion-Rotation
Spectra of Ammonia
Hamiltonian for NHzD and NDzH
V. DANIELIS, D. PAPOUBEK, V. SPIRKO, AND M. Ho,AK The J. Heyrocsk$ Institute of Physicd Chemistry and Hectrochemistry, Czechoslovak Academy of Sciences, Prague, Czechoslovakia
The model Hamiltonian developed previously for ammonia NH8 has been used to study the vibration-inversion-rotation energy levels of the isotopic species of ammonia NHrD and NDnH. In this model the inversion motion is removed from the vibrational problem by allowing the molecular reference configuration to be a function of the large amplitude motion coordinate. The ground state inversion-rotation energy levels of NHrD and ND,H have been calculated with the use of the zeroth-order inversion-rotation Hamiltonian, and the calculated transition frequencies have been compared with the experimental data. I. INTRODUCTION
In paper (1) we have developed a new vibration-inversion-rotation Hamiltonian for ammonia NH3 in which the inversion motion is removed from the vibrational problem by allowing the molecular reference configuration to be a function of the large amplitude coordinate. This Hamiltonian represents a modification of the formalism developed originally by Hougen, Bunker, and Johns (2) and Bunker and Stone (3) to study the vibration-rotation levels of triatomic molecules with a large amplitude bending motion. In the present paper, we shall use this formalism to study the isotopic molecules NHzD and ND2H. A similar approach has been adopted recently by Moule and Rao for the vibration-inversion-rotation problem of formaldehyde (4). The paper is divided into the following parts. In Section II, we define the reference configuration and the system of molecule-fixed axes of NHzD that are used in this paper, and discuss briefly the vibration-inversion-rotation Hamiltonian of NHzD and NDzH. In Section III, we discuss in detail the calculation of the inversion-rotation energy levels of NHzD and ND,H in the “rigid-bender” approximation using the numerical methods described in (5). Selection rules for the inversion-rotation transitions are discussed in Section IV, where the calculated and experimental transition frequencies in NH2D and NDzH are compared. II. MOLECULAR
The equal
reference
REFERENCE CONFIGURATION AND THE VIBRATIONINVERSION-ROTATION HAMILTONIAN
configuration
and fixed bond
lengths
of the (=
atomic
nuclei
of NHaD
rs), (ii) the angle p subtended 339
Copyright Q 1975 by Academic Press. Inc. All rights of reproduction in any form reserved.
is defined
by
(i) three
by the ND bond of the
340
FIG. 1. The numbering of atoms and location of the molecule-fixedaxis system for NHkD. The hydrogen nuclei HI and Ht should be replaced by DI and Dz, and D should be replaced by H for NDsH. reference configuration and the axis p which passes through the atomic nucleus X and the center of the equilateral triangle formed by the atomic nuclei HI, HZ, and D (Fig. 1). All the valence angles ar of the reference configuration are defined to be equal, thus (3+/2) 1sinp 1 = 1sin(or/2) I.
(1)
The molecule-fixed system of axes xyz has its origin at the center of mass of the reference configuration and the z-axis subtends an angle 6 with the p-axis (Fig. 1). We shall orient the xyz-axes system with respect to the reference configuration so that the xz plane bisects the valence angle HI-N-H2 (Fig. 1). The components of the position vectors ai of the ith atomic nucleus in this molecule-fixed axis system are given by al, = azz = [(3mn + mn)/2m]r0 aD%
=
-
[
(3~2~ + mN)/m]rO sinp cost + (mn/m)ro
aNz = - [(mH - mn)/m]ro an, = -
sinp cos + (m~/m)r~ cosp sine,
sinp co% - [(2mn
+
cosp sine, mD)/m]ro
(2) cosp
sine,
(3:/2)ro sinp,
a2y = (3?/2)r0 sinp, (3) UD,
=
0,
aNy
=
0,
INVERSION
al*
=
a2=
=
aNz =
[ (mH
[t3f@D+
-
aDz = [ (3%
+ -
HAMILTONIAN
mN)/2
m ] r0 sinp
FOR NH,D , ND,H
sine + (mN/m)ro cOSpco%,
sinp sine + (mN/m)rO
mN)/m]ru
sinp sine - [ (2mn +
mD)/m]ro
341
COSp
CoSe,
mD)/m]rC!
(4) COSp
COSt,
where mH, mN, and mn are the masses of the atomic nuclei H, N, and D, respectively, and m = 2mn + mD -f MN. Equation (3) of Ref. (2) can be used to calculate the relationship between the angle E and p; we obtain dc/dp
= u1/(u2
+
us sir?p),
(5)
where
Equation
(5) can be integrated
ul
=
mN(mD
u2
=
mN@mH
-
mH), +
mD),
(6)
to give
Ul
4P)
=
arctan[(F)ttanp]+const,
cu2b42
+
udli
(7)
where the integration constant may be arbitrarily chosen so that e = 0 for the value of the parameter p = 0 (i.e., const = 0). Note that for NH3 Eq. (7) gives the identity E = 0 for all values of p [cf. (I)]. We shall now specify the molecule-fixed axis system xyz by the same set of seven center-of-mass, Eckart and Sayvetz conditions used by Hougen, Bunker, and Johns (2) in their study of triatomic molecules with large amplitude bending motions [Eqs. (7) in (2) and Eqs. (4)-(5) in (I)]. Th e b asic aim of this paper, therefore, is to define a nonrigid reference configuration which follows closely the large amplitude internal motions of the atomic nuclei of the NH2D molecule [cf. (I+)]. The precise motion of the NH,D molecule during inversion is not known exactly. It is probable, for example, that the three bond lengths of NH,D do not remain exactly equal during the inversion motion. However, with the definition of the reference configuration according to Eqs. (2)-(4) the Hamiltonian for NHZD takes the same simple form discussed, e.g., in (l)-(4) and with the major part of the anharmonicity due to the inversion motion’s being accounted for by the large amplitude coordinate p. The vibration-inversion-rotation Hamiltonian can be written in the form
where all the symbols have the same meanings as in (1). The symmetry group of this Hamiltonian is the Ca, group of the permutations and permutation-inversions (6) of the elements E, (12), E*, and (12)*. The effects of the group operations on the Euler angles 0, 9, X and the inversion coordinate p can be determined as described in (I), and they are summarized in Table I. A set of vibrational
342
DANIELIS
ET AL.
TABLE I TRANSFORMATIONS OF 0, a, x, p AND THE STRETCH, BEND COORDINATES OP NHiD UNDER THE SYMMETRY OPERATIONS OF THE PERMUTATION-INVERSION GROUP Czu E
C2
E
(12)
Qv
(zx)
dY4
w*
symmetry coordinates (7) can be constructed rnn, rrN, ~ZNand the three bending coordinates
E*
?r--8
e
r+*
a
S-X
X-r
P
T-P
IDN f2N TlN (YD w Ly1
IDN IlN YZN CCD a1 012
from the three stretching (YD,(~1,~22(Fig. 1) :
coordinates
SI = 2-'(TIN+ rzN), sz = 3-+D+
al+
a2>,
s1 = rDN,
&
(9)
= 2-'(rlN- rZN),
Sg = 6+(2aD -O!l - (Yz), se = 24(cYl- arz).
The coordinates Sr and Sa to 5’~ are the linearized valence force coordinates; S2 is a redundant coordinate because of the definition of the inversion coordinate p, and it is removed from the vibrational problem. The symmetry species of these coordinates are given in Table II together with the symmetry species of various terms occurring in the Hamiltonian (1). The potential energy V can be expanded for each value of p as a Taylor series expansion in the symmetry coordinates Sr, S&e:
where V,(p) is the double minimum potential function, F,,(P) = (c~V/XS,,),, are linear force constants, and F,,(p) = (cJ*V/~S,,JS,), are quadratic force constants. The traditional GF matrix formalism (7) can be set up in terms of the symmetry coordinates(9) in the same way as was described in (1) for NHa. The “rigid-bender” Hamiltonian RiP can be obtained from the complete Hamiltonian by putting all the small amplitude vibrational coordinates and their conjugate momenta equal to zero.
INVERSION
HAMILTONIAN
FOR NHaD, ND&l
343
TABLE II THE SYMMETRY SPECIES OF THE OPERATORS AND WAVE FUNCICIONS OF NHnD AND NDIH IN THE C2. GROUP
Species
Quantity
Quantity
Species
sin p cos p sin 2p cos 2p +i+(P) tii-
(PI
J,-P, Jd" J.-P. JdP
The expressions (16)~(18)
in
WI
A* Bl BS Bl
for the quantities Imp0 (CY,p = x, y, z, p) which occur in Hi,0 [Eqs. are given for NHzD explicitly below.
Izzo = #rnHro2sin2p + [uZ cos2p cos2c + 2241sinp cosp sine cosr + (uZ + 248)sin2p sin2e]r02m-r, I,,O = (uZ + u3 sin2p)ro2m-l,
Wa) Wb)
I,,0 = QmHro2sin2p + [uZ cos2psin2e - 2~1 sinp cosp sine cost + (uz + 243)sin2p cos2e]r02m-‘, Izzo = [-
ah cos2p sine cost + 2241sinp cosp cos2~ + (uz + 243) X sin2p sine cose]ru2m-‘,
I,,“=
Wc)
(lld)
{u2(1 -zy+z43sin2pQ
+ 34th 2rnN
e+ + (2mD
dp
pI)
mH> cos2
ro2m-l,
(lle)
III. CALCULATION OF THE INVERSION-ROTATION ENERGY LEVELS OF NHnD AND NDzH IN THE RIGID-BENDER APPROXIMATION
We discuss in this section
the solution
of the equation
(Hilo - Ei,O)$i,O(B, 9, X, P> = 0
(12)
for the inversion-rotation energy levels Ei,O and wavefunctions #i:(B, a, X, p). The zeroth-order inversion-rotation Hamiltonian Hito was delined in Eqs. (16)-(18) in Ref. (1). Because NH,D and ND,H are asymmetric tops, the calculation differs from that
DANIELIS
344
ET .4L.
TableIII. The symmetry Species of the Functions Et. Cia + +
described in (1) for NH1 and can be divided into several stages as described in detail in (3) for the calculation of the rigid-bender Hamiltonian rotation energy levels. We have used the Wang transformations to factorize the infinite rotation submatrices by using the following set of basis functions : A,*
= 2-*S~k,~(9, +)[eikx f
e-ikx&,npfO(p),
(13)
where the symbol A* should be replaced by E* for k even, Ok for k odd and the subscript q by s for v2+, and by a for VZ-. The functions $Ok,rZ f (p) are the inversion functions obtained by a numerical integration of the Schrodinger equation involving the zerothorder inversion Hamiltonian Hi0 and that part of the zeroth-order rotation Hamiltonian which is diagonal in k [cf. Stage 2 in (3)]. The symbol r2* denotes an arbitrary quantum number which labels the inversion energy levels (e.g., symbols l+, l- label inversion states of 02 = 1 which are symmetric or antisymmetric with respect to inversion). The species of the functions A q* are given in Table III. It is obvious that the matrix of the inversion-rotation Hamiltonian Hi,.O factorizes into four submatrices (e.g., the O,states corresponding to VZ+combine with the E,fstates corresponding to v2’- inversion states, etc.). Parameters of the double-minimum potential function V,(p) approximated by a Gaussian perturbation to the harmonic oscillator potential (1) have been calculated for NHzD and ND2H from the observed 212= Of, O-, l+, l- energy levels. We have used several observed sets of 212= I+ and l- levels as indicated in Tables IV and V. There are considerable discrepancies between the data obtained by different authors (8-10) in the l+, l- levels of NH2D. Brewer and Swallen (8) have obtained their data by Stark-tuned laser spectroscopy, and they calculated the band origin from the observed transition frequencies using nonspecified rotational constants for the excited inversion state. Their data differ from the infrared data of Migeotte and Barker (10) by as much as 6 cm-‘. Kelly, Francke, and Feld (9) changed the assignment of the inversionrotation transitions observed by this technique, and their calculated l+ and l- levels are much closer to the Migeotte and Barker results. We have remeasured the medium resolution infrared spectra of NHzD and ND2H with a Perkin-Elmer 621 spectrometer using NH3 in the reference cell to exclude the overlapping absorption of NH3 from the NHzD and ND2H spectra (Figs. 2 and 3). It is obvious from Fig. 2 that especially the lower component frequency for NHzD as determined by Brewer and Swallen (8) is
INVERSION
Table
HAMILTONIAN
Calculated Parameter3
IV.
NH38
NH20
N03.
of
and
the
Double-Minimum
Parameter
-1 )
&idyn.
f
-
1.0156
Data
from
2007.9
2003.7
1976.0
1978.6
1 .a430
5.30
1.32
1.645
1.775
1.87
1.8017
112.37’
112.9c
112.03’
112.2@
112.30’
112.27’
112.12c
function
(sea(l)):
cur
N02Hf Hf
MB=
2047.5
potential x
for
1913.1
!?e
rc
Function’
34.5
2023.0
I()
a Double-minimum
Potential
NH20 KFFd
SC
AVi(Cm
NDzH
NO2H
NH;
Barrier
FOR NH&,
Vc(9)
b Ref.(l)1
I ’
:
k(g
Data
- ‘P/2)2+
from
a.exp ’
(8):
[-b(s
oat8
- ,G’/2)2]
from
(9);
, ’
Oata
from
(IO);
measurements
almost certainly incorrect. This is obvious also from the V,(p) parameters calculated from these data which differ considerably from the values of these parameters obtained from the data on NH, and ND;1 (Table IV). We have therefore used the data obtained from our infrared measurements for further calculations on NHzD and NDtH; they are in very good agreement with the data of Migeotte and Barker (10). It is obvious, however, from Table IV, that the double minimum potential function parameters are rather sensitive to the experimental data from which they are determined. A better adjustment of the parameters would be of course possible only after high resolution measurements on the ~2 and higher overtones become available. Table
V.
Energy
Lcvsls for
NH20
and
N02H
(cm-‘)
14NH 2D "2
Es'
14N0
MB=
KFFb
Hd
celc.'
2
Hd
ii C~lC.’
0+
O.wOf
0.000
0.0009
0.000
0-
0.407f
0.405
0.1719
0.171
BOB.8
808.8
818.4
818.4
1+
368.0
876.3
874.4
874.1
874.4
1-
391.6
894.6
894.0
894.1
894.4
2+
1518.8
1434.5
2-
1726.0
1569.5
3+
2182.0
1986.7
3-
2621.7
2351.3
4*
3121.0
2785.9
4-
3650.5
3246.1
a Ref.@);the eplitting d
cur
value 23.2
neseurementrr
g Ae!f.(l4),
cm-’
968.0 but
-1 cm not
for with
e Celculstsd
the
l+lavsl
the
transition
from
the
I.8
conelatent frcqucnciea
peraneters
denoted
with in in
the (8); TableIV
upper
etato
invsrai0”
b Ref,.(9)t by HI
’ f
Ref.(lO)t R*f.(ll)t
346
DANIELIS
1””
7
ET AL.
1””
n
665
6f5
895 mi’
665
FIG. 2. Q branches of the Y% inversion doublet of NHD. (1) pure NHs, (2) mixture of NH3 (130/G), 1 erence spectrum; 10 cm cell, 80 mm Hg. NH2D (370/o), ND&l (37%), and ND8 (13%), (3) d’ff
The program for the calculation of the inversion-rotation energy levels for NHzD according to the above described procedure has been checked by using it for the calculation of NH, energy levels [which were calculated by an independent method in (I)]. We have also found in this way that with the characteristic matrix truncated at vz = 4-, the calculated rotation-inversion levels are accurate to within a few hundredths of a cm-r, up to v2 = 4-. The calculated ground state inversion-rotation levels of NHsD and NDzH for low J values are given in Table VI. The calculated ground state rotational energy levels
1
I
805 FIG. 3.
NH9
815
I
825 cm-’
Q branches of the y2 inversion doublet of NDpH. (1) pure NHa, (2) mixture of NHa (13%), (37y0), ND&I (37%), and ND8 (139&), (3) difference spectrum; 10 cm cell, 80 mm Hg.
INVERSION Table VI.
HAMILTONIAN
FOR NH&
Cslculated Rotation Energy Lsvele for ND2H
up to
J D 3
NDaH NH2D
=K_llK
species
Cower
end
(Ground State, in cm-l)
ND2H
NH2D
’
upper
species
Lower
Species
upper
L_ 0.000
Al
0.405
1 lo;1
11.411
A2
11.653
11,l
14.679
01
15.083
16.639
82
16.879
33.554
Al
33.659
21.2
35.551
B2
35.795
21,l
41.240
Sl
41.355
22,l
51.210
A2
51.447
22.0
51.831
Al
52.015
3 30.3
65.506
A2
65.575
66.470
51
66.600
77.510
A2
77.604
85.273
Al
85.367
87.891
*2
88.090
I.06.975
81
107.131
ID7.102
92
107.254
0 00.0
Il.0
.
2 20,2
31,3 31.2 32.2 32,l 33,l 33.0
0.000
Bl
5
02
9.108
Al
11.238
A2
12.819
91
26.716
0.171
%
9.279
AZ
11.330
62 Al
12.910 26 .a77
*2
27.882
92
27.975
Al
32.606
*2
32.697
E2
38.887
01
38.957
81
39.523
Al
39.582
E2
52.038
Bl
52.173
Al
52.515
B2
61.799
81
66.218
82
66.966
%
80.903
A2
91.078
*2 62 Al 61 A2 E2
52.605 61.884 66.290 66.900 81.055 81.227
Table vxI,Obsarved and Calculated Transition Frequencies of
10.1~D0.0 ll.lclO.l ~1,0*0,0 21,2+20,2 22.0~21.2 22,1*21,1 20,2+%,0 31 ,3e30.3 31,3c22.1 42,3-1,3 '3.2-2.2
l
Ref.(l2);
NH20
347
(cm-')
LOWar Component ca1.Z. sxp.
Upper Conponenet
ll.lW
cxp.
Ref. talc.
11.411
11.102
2.866
3.026
3.674
11.246 3.672
15.687
16.234
16.493
16.879
1.667
1.892
2,474
2.241
15.079
16.036
15.074
16.461
9.410
9.865
10.199
10.207
16.289
15.675
17.093
17.020
0.627
0.895
1.436
1.083
15.070
15.023
15.667
15.390
4.757
4.793
5.523
5.314
16.758
16.199
16.609
16.616
b .Qtf.(lZ);
' Ref.(ll).
SpcclsB
348 fable VIII.Observed end Calculated Transition Frequencies of
NO2H
[cm-')
UPPer Component
Lower Component exp.
ll,o~lo,l ll,l~“o.o 11.1~l0.1 ll,o+"o,o 21,1+20,2 22.1+21.2
.Zxp.
talc.
Ref.
CSlC.
3.595
3.711
3.599
3.530
a
.A.189
11.238
11.192
11.159
a
1.924
1.959
2.253
2.222
b
.2.525
12.548
12.954
12.910
a
5.931
5.890
5.935
5.820
a
J.081
11.005
11.089
10.982
a
22,0~21,1
5.904
5.971
5.911
5.855
a
20,2+l1,1
.5.455
15.478
15.457
15.547
a
0.953
1.005
1.292
1.259
b
J.553
11.548
11.883
11.700
a
21,2+20.2 22.0+21.2 22,1+21,1 20,2+11,0 31,2c30,3 32':1%2 ..+ 31
,34-30 .3
32,2+31,2 42.2t41.3 42,3c4L3 '3 ,2f42
a
82
Ref.(U):
a,c
5.111
5.190
5.435
5.351
13.593
13.805
14.031
14.058
*
9.559
9.751
9.894
9.711
a
7.080
7.057
7.055
7.015
a
0.275
0.342
0.517
0.557
b
4.223
4.334
4.549
4.491
a.c
8.940
5.553
8.945
8.810
a
2.321
2.451
2.545
2.585
c
9.559
9.710
9.895
9.575
B
b R.f.(ll)t
= Ref.(lZ).
obtained by averaging the inversion doublets follow closely the energy levels of a rigid asymmetric rotor with the rotational constants for NHzD, A = 10.932 cm-‘, B = 7.255 cm-‘, C = 5.360 cm-’ (asymmetry parameter K = - 0.3199) and for NDzH, A = 8.406 cm-l, B = 6.055 cm-‘, C = 4.275 cm-’ (asymmetry parameter K = - 0.1383). This is because for the equilibrium value of p the principal axis c subtends an angle 8”58’ with the p axis of Fig. 1 for NHaD (- 1 l”53’ for ND?H) and the inversion wavefunctions have an appreciable amplitude only in the neighborhood of the equilibrium configuration. IV. SELECTION
RULES, TRANSITION
FREQUENCIES
The species of the component of the electric dipole moment PZ along a space fixed Z axis is AZ in the Gu group, thus the overall selection rule for the allowed inversionrotation transitions can be written as I“ x r” E Az. According to this rule, transitions between the energy levels of the same inversion doublets are not allowed in NHfD and NDtH (cf. Table VI), that is, there is no pure “inversion” spectrum in these molecules. Calculated and observed frequencies of the allowed ground state inversion-rotation transitions are given in Tables VII and VIII. They
INVERSION
HAMILTONIAN
FOR NH2D, NDnH
349
could be interpreted in terms of the a- and c-type transitions for NHzD and b- and c-type transitions for NDzH as discussed by Helminger and De Lucia (13, 14). A surprisingly good agreement between the observed and calculated transition frequencies has been obtained for ND2H (Table VIII). We believe that this is partially due to a smaller contribution from centrifugal distortion effects in an isotope with heavier atoms [cf. a similar effect between NH3 and ND% in (I)]. We hope that the introduction of the centrifugal distortion operator into the calculation will considerably improve these results. V. DISCUSSION
Results of the calculation of the inversion-rotation levels of NHzD and NDzH presented in this paper and the satisfactory agreement obtained between the calculated and observed inversion-rotation transitions (Tables VII and VIII) for low J, K values in the zeroth-order approximation demonstrate the usefulness of the formalism developed in this paper. As we have already discussed in (I) for NH,, the main advantage of this approach is that it makes it possible to study, using the zeroth-order inversion-rotation Hamiltonian, certain effects which are basically higher order effects. Furthermore, many higher order effects, e.g., Coriolis interactions or centrifugal distortion effects, could be studied quite systematically through the use of the model Hamiltonian (10). These effects are of course extremely complicated for the asymmetric tops NHzD and NDzH, and a considerable amount of further theoretical, as well as experimental, work should be done before these effects can be analyzed and understood. ACKNOWLEDGMENT We wish to thank Drs. F. C. De Lucia and P. Helminger for having made it possible for us to use their data prior to publication.
RECEIVED: May 20, 1974 REFERENCES 1. 2. 3. 4. 5. 6. 7.
D. PAPOU~EK, J. M. R. STONE, AND V. SPIRKO, J. Mol. Spectrosc. 48, 17 (1973). J. T. HOUGEN, P. R. BUNKER, AND J. W. C. JOHNS, J. Mol. Speclrosc. 34, 136 (1970). P. R. BUNKER AND J. M. R. STONE, J. Mol. Spectrosc. 41, 310 (1972). D. C. MOULE AND Ch. V. S. R. RAO, J. Mol. Spectrosc. 45, 120 (1973). V. SPIRKO, J. M. R. STONE, AND D. PAPOUSEK,J. Mol. Spectrosc. 48, 38 (1973). H. C. LONGUET-HIGGINS,Mol. Phys. 6, 44.5 (1963). E. B. WILSON, JR.; J. C. DECIUS, AND P. C. CROSS, “Molecular Vibrations,” McGraw-Hill, York, 195.5.
8. R. G. BREWER AND J. D. SWALLEN, J. Chew. Phys. 52, 2774 (1970).
9. 10. 11. 12. 13. 14.
M. J. KELLY, R. E. FRANCKE, AND M. S. FELD, J. Chem.Phys. 53, 2979 (1970). M. V. MIGEOTTE AND E. F. BARKER, Phys. Rec. 50, 418 (1936). M. T. WEISS AND M. W. P. STRANDBERG,Phys. Rev. 83, 567 (1951). M. LIIXTENSTEIN, J. J. GALLAGHER,AND V. E. DEER, J. Mol. Speclrosc. 12, 87 (1964). P. HELXINGER AND F. C. DE LUCIA, Phys. Rev. A, in press. F. C. DE LUCIA AND P. HELXINGER, Phys. Rev. A, in press.
New
(1975)
Vibration-Inversion-Rotation A Vibration-Inversion-Rotation
Spectra of Ammonia
Hamiltonian for NHzD and NDzH
V. DANIELIS, D. PAPOUBEK, V. SPIRKO, AND M. Ho,AK The J. Heyrocsk$ Institute of Physicd Chemistry and Hectrochemistry, Czechoslovak Academy of Sciences, Prague, Czechoslovakia
The model Hamiltonian developed previously for ammonia NH8 has been used to study the vibration-inversion-rotation energy levels of the isotopic species of ammonia NHrD and NDnH. In this model the inversion motion is removed from the vibrational problem by allowing the molecular reference configuration to be a function of the large amplitude motion coordinate. The ground state inversion-rotation energy levels of NHrD and ND,H have been calculated with the use of the zeroth-order inversion-rotation Hamiltonian, and the calculated transition frequencies have been compared with the experimental data. I. INTRODUCTION
In paper (1) we have developed a new vibration-inversion-rotation Hamiltonian for ammonia NH3 in which the inversion motion is removed from the vibrational problem by allowing the molecular reference configuration to be a function of the large amplitude coordinate. This Hamiltonian represents a modification of the formalism developed originally by Hougen, Bunker, and Johns (2) and Bunker and Stone (3) to study the vibration-rotation levels of triatomic molecules with a large amplitude bending motion. In the present paper, we shall use this formalism to study the isotopic molecules NHzD and ND2H. A similar approach has been adopted recently by Moule and Rao for the vibration-inversion-rotation problem of formaldehyde (4). The paper is divided into the following parts. In Section II, we define the reference configuration and the system of molecule-fixed axes of NHzD that are used in this paper, and discuss briefly the vibration-inversion-rotation Hamiltonian of NHzD and NDzH. In Section III, we discuss in detail the calculation of the inversion-rotation energy levels of NHzD and ND,H in the “rigid-bender” approximation using the numerical methods described in (5). Selection rules for the inversion-rotation transitions are discussed in Section IV, where the calculated and experimental transition frequencies in NH2D and NDzH are compared. II. MOLECULAR
The equal
reference
REFERENCE CONFIGURATION AND THE VIBRATIONINVERSION-ROTATION HAMILTONIAN
configuration
and fixed bond
lengths
of the (=
atomic
nuclei
of NHaD
rs), (ii) the angle p subtended 339
Copyright Q 1975 by Academic Press. Inc. All rights of reproduction in any form reserved.
is defined
by
(i) three
by the ND bond of the
340
FIG. 1. The numbering of atoms and location of the molecule-fixedaxis system for NHkD. The hydrogen nuclei HI and Ht should be replaced by DI and Dz, and D should be replaced by H for NDsH. reference configuration and the axis p which passes through the atomic nucleus X and the center of the equilateral triangle formed by the atomic nuclei HI, HZ, and D (Fig. 1). All the valence angles ar of the reference configuration are defined to be equal, thus (3+/2) 1sinp 1 = 1sin(or/2) I.
(1)
The molecule-fixed system of axes xyz has its origin at the center of mass of the reference configuration and the z-axis subtends an angle 6 with the p-axis (Fig. 1). We shall orient the xyz-axes system with respect to the reference configuration so that the xz plane bisects the valence angle HI-N-H2 (Fig. 1). The components of the position vectors ai of the ith atomic nucleus in this molecule-fixed axis system are given by al, = azz = [(3mn + mn)/2m]r0 aD%
=
-
[
(3~2~ + mN)/m]rO sinp cost + (mn/m)ro
aNz = - [(mH - mn)/m]ro an, = -
sinp cos + (m~/m)r~ cosp sine,
sinp co% - [(2mn
+
cosp sine, mD)/m]ro
(2) cosp
sine,
(3:/2)ro sinp,
a2y = (3?/2)r0 sinp, (3) UD,
=
0,
aNy
=
0,
INVERSION
al*
=
a2=
=
aNz =
[ (mH
[t3f@D+
-
aDz = [ (3%
+ -
HAMILTONIAN
mN)/2
m ] r0 sinp
FOR NH,D , ND,H
sine + (mN/m)ro cOSpco%,
sinp sine + (mN/m)rO
mN)/m]ru
sinp sine - [ (2mn +
mD)/m]ro
341
COSp
CoSe,
mD)/m]rC!
(4) COSp
COSt,
where mH, mN, and mn are the masses of the atomic nuclei H, N, and D, respectively, and m = 2mn + mD -f MN. Equation (3) of Ref. (2) can be used to calculate the relationship between the angle E and p; we obtain dc/dp
= u1/(u2
+
us sir?p),
(5)
where
Equation
(5) can be integrated
ul
=
mN(mD
u2
=
mN@mH
-
mH), +
mD),
(6)
to give
Ul
4P)
=
arctan[(F)ttanp]+const,
cu2b42
+
udli
(7)
where the integration constant may be arbitrarily chosen so that e = 0 for the value of the parameter p = 0 (i.e., const = 0). Note that for NH3 Eq. (7) gives the identity E = 0 for all values of p [cf. (I)]. We shall now specify the molecule-fixed axis system xyz by the same set of seven center-of-mass, Eckart and Sayvetz conditions used by Hougen, Bunker, and Johns (2) in their study of triatomic molecules with large amplitude bending motions [Eqs. (7) in (2) and Eqs. (4)-(5) in (I)]. Th e b asic aim of this paper, therefore, is to define a nonrigid reference configuration which follows closely the large amplitude internal motions of the atomic nuclei of the NH2D molecule [cf. (I+)]. The precise motion of the NH,D molecule during inversion is not known exactly. It is probable, for example, that the three bond lengths of NH,D do not remain exactly equal during the inversion motion. However, with the definition of the reference configuration according to Eqs. (2)-(4) the Hamiltonian for NHZD takes the same simple form discussed, e.g., in (l)-(4) and with the major part of the anharmonicity due to the inversion motion’s being accounted for by the large amplitude coordinate p. The vibration-inversion-rotation Hamiltonian can be written in the form
where all the symbols have the same meanings as in (1). The symmetry group of this Hamiltonian is the Ca, group of the permutations and permutation-inversions (6) of the elements E, (12), E*, and (12)*. The effects of the group operations on the Euler angles 0, 9, X and the inversion coordinate p can be determined as described in (I), and they are summarized in Table I. A set of vibrational
342
DANIELIS
ET AL.
TABLE I TRANSFORMATIONS OF 0, a, x, p AND THE STRETCH, BEND COORDINATES OP NHiD UNDER THE SYMMETRY OPERATIONS OF THE PERMUTATION-INVERSION GROUP Czu E
C2
E
(12)
Qv
(zx)
dY4
w*
symmetry coordinates (7) can be constructed rnn, rrN, ~ZNand the three bending coordinates
E*
?r--8
e
r+*
a
S-X
X-r
P
T-P
IDN f2N TlN (YD w Ly1
IDN IlN YZN CCD a1 012
from the three stretching (YD,(~1,~22(Fig. 1) :
coordinates
SI = 2-'(TIN+ rzN), sz = 3-+D+
al+
a2>,
s1 = rDN,
&
(9)
= 2-'(rlN- rZN),
Sg = 6+(2aD -O!l - (Yz), se = 24(cYl- arz).
The coordinates Sr and Sa to 5’~ are the linearized valence force coordinates; S2 is a redundant coordinate because of the definition of the inversion coordinate p, and it is removed from the vibrational problem. The symmetry species of these coordinates are given in Table II together with the symmetry species of various terms occurring in the Hamiltonian (1). The potential energy V can be expanded for each value of p as a Taylor series expansion in the symmetry coordinates Sr, S&e:
where V,(p) is the double minimum potential function, F,,(P) = (c~V/XS,,),, are linear force constants, and F,,(p) = (cJ*V/~S,,JS,), are quadratic force constants. The traditional GF matrix formalism (7) can be set up in terms of the symmetry coordinates(9) in the same way as was described in (1) for NHa. The “rigid-bender” Hamiltonian RiP can be obtained from the complete Hamiltonian by putting all the small amplitude vibrational coordinates and their conjugate momenta equal to zero.
INVERSION
HAMILTONIAN
FOR NHaD, ND&l
343
TABLE II THE SYMMETRY SPECIES OF THE OPERATORS AND WAVE FUNCICIONS OF NHnD AND NDIH IN THE C2. GROUP
Species
Quantity
Quantity
Species
sin p cos p sin 2p cos 2p +i+(P) tii-
(PI
J,-P, Jd" J.-P. JdP
The expressions (16)~(18)
in
WI
A* Bl BS Bl
for the quantities Imp0 (CY,p = x, y, z, p) which occur in Hi,0 [Eqs. are given for NHzD explicitly below.
Izzo = #rnHro2sin2p + [uZ cos2p cos2c + 2241sinp cosp sine cosr + (uZ + 248)sin2p sin2e]r02m-r, I,,O = (uZ + u3 sin2p)ro2m-l,
Wa) Wb)
I,,0 = QmHro2sin2p + [uZ cos2psin2e - 2~1 sinp cosp sine cost + (uz + 243)sin2p cos2e]r02m-‘, Izzo = [-
ah cos2p sine cost + 2241sinp cosp cos2~ + (uz + 243) X sin2p sine cose]ru2m-‘,
I,,“=
Wc)
(lld)
{u2(1 -zy+z43sin2pQ
+ 34th 2rnN
e+ + (2mD
dp
pI)
mH> cos2
ro2m-l,
(lle)
III. CALCULATION OF THE INVERSION-ROTATION ENERGY LEVELS OF NHnD AND NDzH IN THE RIGID-BENDER APPROXIMATION
We discuss in this section
the solution
of the equation
(Hilo - Ei,O)$i,O(B, 9, X, P> = 0
(12)
for the inversion-rotation energy levels Ei,O and wavefunctions #i:(B, a, X, p). The zeroth-order inversion-rotation Hamiltonian Hito was delined in Eqs. (16)-(18) in Ref. (1). Because NH,D and ND,H are asymmetric tops, the calculation differs from that
DANIELIS
344
ET .4L.
TableIII. The symmetry Species of the Functions Et. Cia + +
described in (1) for NH1 and can be divided into several stages as described in detail in (3) for the calculation of the rigid-bender Hamiltonian rotation energy levels. We have used the Wang transformations to factorize the infinite rotation submatrices by using the following set of basis functions : A,*
= 2-*S~k,~(9, +)[eikx f
e-ikx&,npfO(p),
(13)
where the symbol A* should be replaced by E* for k even, Ok for k odd and the subscript q by s for v2+, and by a for VZ-. The functions $Ok,rZ f (p) are the inversion functions obtained by a numerical integration of the Schrodinger equation involving the zerothorder inversion Hamiltonian Hi0 and that part of the zeroth-order rotation Hamiltonian which is diagonal in k [cf. Stage 2 in (3)]. The symbol r2* denotes an arbitrary quantum number which labels the inversion energy levels (e.g., symbols l+, l- label inversion states of 02 = 1 which are symmetric or antisymmetric with respect to inversion). The species of the functions A q* are given in Table III. It is obvious that the matrix of the inversion-rotation Hamiltonian Hi,.O factorizes into four submatrices (e.g., the O,states corresponding to VZ+combine with the E,fstates corresponding to v2’- inversion states, etc.). Parameters of the double-minimum potential function V,(p) approximated by a Gaussian perturbation to the harmonic oscillator potential (1) have been calculated for NHzD and ND2H from the observed 212= Of, O-, l+, l- energy levels. We have used several observed sets of 212= I+ and l- levels as indicated in Tables IV and V. There are considerable discrepancies between the data obtained by different authors (8-10) in the l+, l- levels of NH2D. Brewer and Swallen (8) have obtained their data by Stark-tuned laser spectroscopy, and they calculated the band origin from the observed transition frequencies using nonspecified rotational constants for the excited inversion state. Their data differ from the infrared data of Migeotte and Barker (10) by as much as 6 cm-‘. Kelly, Francke, and Feld (9) changed the assignment of the inversionrotation transitions observed by this technique, and their calculated l+ and l- levels are much closer to the Migeotte and Barker results. We have remeasured the medium resolution infrared spectra of NHzD and ND2H with a Perkin-Elmer 621 spectrometer using NH3 in the reference cell to exclude the overlapping absorption of NH3 from the NHzD and ND2H spectra (Figs. 2 and 3). It is obvious from Fig. 2 that especially the lower component frequency for NHzD as determined by Brewer and Swallen (8) is
INVERSION
Table
HAMILTONIAN
Calculated Parameter3
IV.
NH38
NH20
N03.
of
and
the
Double-Minimum
Parameter
-1 )
&idyn.
f
-
1.0156
Data
from
2007.9
2003.7
1976.0
1978.6
1 .a430
5.30
1.32
1.645
1.775
1.87
1.8017
112.37’
112.9c
112.03’
112.2@
112.30’
112.27’
112.12c
function
(sea(l)):
cur
N02Hf Hf
MB=
2047.5
potential x
for
1913.1
!?e
rc
Function’
34.5
2023.0
I()
a Double-minimum
Potential
NH20 KFFd
SC
AVi(Cm
NDzH
NO2H
NH;
Barrier
FOR NH&,
Vc(9)
b Ref.(l)1
I ’
:
k(g
Data
- ‘P/2)2+
from
a.exp ’
(8):
[-b(s
oat8
- ,G’/2)2]
from
(9);
, ’
Oata
from
(IO);
measurements
almost certainly incorrect. This is obvious also from the V,(p) parameters calculated from these data which differ considerably from the values of these parameters obtained from the data on NH, and ND;1 (Table IV). We have therefore used the data obtained from our infrared measurements for further calculations on NHzD and NDtH; they are in very good agreement with the data of Migeotte and Barker (10). It is obvious, however, from Table IV, that the double minimum potential function parameters are rather sensitive to the experimental data from which they are determined. A better adjustment of the parameters would be of course possible only after high resolution measurements on the ~2 and higher overtones become available. Table
V.
Energy
Lcvsls for
NH20
and
N02H
(cm-‘)
14NH 2D "2
Es'
14N0
MB=
KFFb
Hd
celc.'
2
Hd
ii C~lC.’
0+
O.wOf
0.000
0.0009
0.000
0-
0.407f
0.405
0.1719
0.171
BOB.8
808.8
818.4
818.4
1+
368.0
876.3
874.4
874.1
874.4
1-
391.6
894.6
894.0
894.1
894.4
2+
1518.8
1434.5
2-
1726.0
1569.5
3+
2182.0
1986.7
3-
2621.7
2351.3
4*
3121.0
2785.9
4-
3650.5
3246.1
a Ref.@);the eplitting d
cur
value 23.2
neseurementrr
g Ae!f.(l4),
cm-’
968.0 but
-1 cm not
for with
e Celculstsd
the
l+lavsl
the
transition
from
the
I.8
conelatent frcqucnciea
peraneters
denoted
with in in
the (8); TableIV
upper
etato
invsrai0”
b Ref,.(9)t by HI
’ f
Ref.(lO)t R*f.(ll)t
346
DANIELIS
1””
7
ET AL.
1””
n
665
6f5
895 mi’
665
FIG. 2. Q branches of the Y% inversion doublet of NHD. (1) pure NHs, (2) mixture of NH3 (130/G), 1 erence spectrum; 10 cm cell, 80 mm Hg. NH2D (370/o), ND&l (37%), and ND8 (13%), (3) d’ff
The program for the calculation of the inversion-rotation energy levels for NHzD according to the above described procedure has been checked by using it for the calculation of NH, energy levels [which were calculated by an independent method in (I)]. We have also found in this way that with the characteristic matrix truncated at vz = 4-, the calculated rotation-inversion levels are accurate to within a few hundredths of a cm-r, up to v2 = 4-. The calculated ground state inversion-rotation levels of NHsD and NDzH for low J values are given in Table VI. The calculated ground state rotational energy levels
1
I
805 FIG. 3.
NH9
815
I
825 cm-’
Q branches of the y2 inversion doublet of NDpH. (1) pure NHa, (2) mixture of NHa (13%), (37y0), ND&I (37%), and ND8 (139&), (3) difference spectrum; 10 cm cell, 80 mm Hg.
INVERSION Table VI.
HAMILTONIAN
FOR NH&
Cslculated Rotation Energy Lsvele for ND2H
up to
J D 3
NDaH NH2D
=K_llK
species
Cower
end
(Ground State, in cm-l)
ND2H
NH2D
’
upper
species
Lower
Species
upper
L_ 0.000
Al
0.405
1 lo;1
11.411
A2
11.653
11,l
14.679
01
15.083
16.639
82
16.879
33.554
Al
33.659
21.2
35.551
B2
35.795
21,l
41.240
Sl
41.355
22,l
51.210
A2
51.447
22.0
51.831
Al
52.015
3 30.3
65.506
A2
65.575
66.470
51
66.600
77.510
A2
77.604
85.273
Al
85.367
87.891
*2
88.090
I.06.975
81
107.131
ID7.102
92
107.254
0 00.0
Il.0
.
2 20,2
31,3 31.2 32.2 32,l 33,l 33.0
0.000
Bl
5
02
9.108
Al
11.238
A2
12.819
91
26.716
0.171
%
9.279
AZ
11.330
62 Al
12.910 26 .a77
*2
27.882
92
27.975
Al
32.606
*2
32.697
E2
38.887
01
38.957
81
39.523
Al
39.582
E2
52.038
Bl
52.173
Al
52.515
B2
61.799
81
66.218
82
66.966
%
80.903
A2
91.078
*2 62 Al 61 A2 E2
52.605 61.884 66.290 66.900 81.055 81.227
Table vxI,Obsarved and Calculated Transition Frequencies of
10.1~D0.0 ll.lclO.l ~1,0*0,0 21,2+20,2 22.0~21.2 22,1*21,1 20,2+%,0 31 ,3e30.3 31,3c22.1 42,3-1,3 '3.2-2.2
l
Ref.(l2);
NH20
347
(cm-')
LOWar Component ca1.Z. sxp.
Upper Conponenet
ll.lW
cxp.
Ref. talc.
11.411
11.102
2.866
3.026
3.674
11.246 3.672
15.687
16.234
16.493
16.879
1.667
1.892
2,474
2.241
15.079
16.036
15.074
16.461
9.410
9.865
10.199
10.207
16.289
15.675
17.093
17.020
0.627
0.895
1.436
1.083
15.070
15.023
15.667
15.390
4.757
4.793
5.523
5.314
16.758
16.199
16.609
16.616
b .Qtf.(lZ);
' Ref.(ll).
SpcclsB
348 fable VIII.Observed end Calculated Transition Frequencies of
NO2H
[cm-')
UPPer Component
Lower Component exp.
ll,o~lo,l ll,l~“o.o 11.1~l0.1 ll,o+"o,o 21,1+20,2 22.1+21.2
.Zxp.
talc.
Ref.
CSlC.
3.595
3.711
3.599
3.530
a
.A.189
11.238
11.192
11.159
a
1.924
1.959
2.253
2.222
b
.2.525
12.548
12.954
12.910
a
5.931
5.890
5.935
5.820
a
J.081
11.005
11.089
10.982
a
22,0~21,1
5.904
5.971
5.911
5.855
a
20,2+l1,1
.5.455
15.478
15.457
15.547
a
0.953
1.005
1.292
1.259
b
J.553
11.548
11.883
11.700
a
21,2+20.2 22.0+21.2 22,1+21,1 20,2+11,0 31,2c30,3 32':1%2 ..+ 31
,34-30 .3
32,2+31,2 42.2t41.3 42,3c4L3 '3 ,2f42
a
82
Ref.(U):
a,c
5.111
5.190
5.435
5.351
13.593
13.805
14.031
14.058
*
9.559
9.751
9.894
9.711
a
7.080
7.057
7.055
7.015
a
0.275
0.342
0.517
0.557
b
4.223
4.334
4.549
4.491
a.c
8.940
5.553
8.945
8.810
a
2.321
2.451
2.545
2.585
c
9.559
9.710
9.895
9.575
B
b R.f.(ll)t
= Ref.(lZ).
obtained by averaging the inversion doublets follow closely the energy levels of a rigid asymmetric rotor with the rotational constants for NHzD, A = 10.932 cm-‘, B = 7.255 cm-‘, C = 5.360 cm-’ (asymmetry parameter K = - 0.3199) and for NDzH, A = 8.406 cm-l, B = 6.055 cm-‘, C = 4.275 cm-’ (asymmetry parameter K = - 0.1383). This is because for the equilibrium value of p the principal axis c subtends an angle 8”58’ with the p axis of Fig. 1 for NHaD (- 1 l”53’ for ND?H) and the inversion wavefunctions have an appreciable amplitude only in the neighborhood of the equilibrium configuration. IV. SELECTION
RULES, TRANSITION
FREQUENCIES
The species of the component of the electric dipole moment PZ along a space fixed Z axis is AZ in the Gu group, thus the overall selection rule for the allowed inversionrotation transitions can be written as I“ x r” E Az. According to this rule, transitions between the energy levels of the same inversion doublets are not allowed in NHfD and NDtH (cf. Table VI), that is, there is no pure “inversion” spectrum in these molecules. Calculated and observed frequencies of the allowed ground state inversion-rotation transitions are given in Tables VII and VIII. They
INVERSION
HAMILTONIAN
FOR NH2D, NDnH
349
could be interpreted in terms of the a- and c-type transitions for NHzD and b- and c-type transitions for NDzH as discussed by Helminger and De Lucia (13, 14). A surprisingly good agreement between the observed and calculated transition frequencies has been obtained for ND2H (Table VIII). We believe that this is partially due to a smaller contribution from centrifugal distortion effects in an isotope with heavier atoms [cf. a similar effect between NH3 and ND% in (I)]. We hope that the introduction of the centrifugal distortion operator into the calculation will considerably improve these results. V. DISCUSSION
Results of the calculation of the inversion-rotation levels of NHzD and NDzH presented in this paper and the satisfactory agreement obtained between the calculated and observed inversion-rotation transitions (Tables VII and VIII) for low J, K values in the zeroth-order approximation demonstrate the usefulness of the formalism developed in this paper. As we have already discussed in (I) for NH,, the main advantage of this approach is that it makes it possible to study, using the zeroth-order inversion-rotation Hamiltonian, certain effects which are basically higher order effects. Furthermore, many higher order effects, e.g., Coriolis interactions or centrifugal distortion effects, could be studied quite systematically through the use of the model Hamiltonian (10). These effects are of course extremely complicated for the asymmetric tops NHzD and NDzH, and a considerable amount of further theoretical, as well as experimental, work should be done before these effects can be analyzed and understood. ACKNOWLEDGMENT We wish to thank Drs. F. C. De Lucia and P. Helminger for having made it possible for us to use their data prior to publication.
RECEIVED: May 20, 1974 REFERENCES 1. 2. 3. 4. 5. 6. 7.
D. PAPOU~EK, J. M. R. STONE, AND V. SPIRKO, J. Mol. Spectrosc. 48, 17 (1973). J. T. HOUGEN, P. R. BUNKER, AND J. W. C. JOHNS, J. Mol. Speclrosc. 34, 136 (1970). P. R. BUNKER AND J. M. R. STONE, J. Mol. Spectrosc. 41, 310 (1972). D. C. MOULE AND Ch. V. S. R. RAO, J. Mol. Spectrosc. 45, 120 (1973). V. SPIRKO, J. M. R. STONE, AND D. PAPOUSEK,J. Mol. Spectrosc. 48, 38 (1973). H. C. LONGUET-HIGGINS,Mol. Phys. 6, 44.5 (1963). E. B. WILSON, JR.; J. C. DECIUS, AND P. C. CROSS, “Molecular Vibrations,” McGraw-Hill, York, 195.5.
8. R. G. BREWER AND J. D. SWALLEN, J. Chew. Phys. 52, 2774 (1970).
9. 10. 11. 12. 13. 14.
M. J. KELLY, R. E. FRANCKE, AND M. S. FELD, J. Chem.Phys. 53, 2979 (1970). M. V. MIGEOTTE AND E. F. BARKER, Phys. Rec. 50, 418 (1936). M. T. WEISS AND M. W. P. STRANDBERG,Phys. Rev. 83, 567 (1951). M. LIIXTENSTEIN, J. J. GALLAGHER,AND V. E. DEER, J. Mol. Speclrosc. 12, 87 (1964). P. HELXINGER AND F. C. DE LUCIA, Phys. Rev. A, in press. F. C. DE LUCIA AND P. HELXINGER, Phys. Rev. A, in press.
New