The absorption spectrum of HNF in the region 3800–5000 Å

JOURNAL

OF MOLECULAR

SPECTROSCOPY

The Absorption

33, 311-344 (1970)

Spectrum

of HNF

3800-5000 C. Division

of Pure

Physics,

National

M.

in the Region

A

WOODMAN*

Research

Council

of Canada, Ottawa, Canada

An absorption spectrum occurring in the region 3800-5000 8, obtained in the flash photolysis of HNF, , WELS investigated at, high resolution. The main conclusions of earlier workers are confirmed, namely that the absorption is due to a type-C transition of the HNF molecule, which is bent in both ground and excited states. The bond angles for the two states are 105 and 125”, respectively. Seven members of a progression in the upper-state bending frequency are observed. Axis-switching effects accompanying the large change in bond angle generate prominent “Q- and ‘Q-branches. Most of the rotational lines are resolved into spin-doublets, and the splitting pattern provides an excellent example of the effects of spin-rotation interaction in a nearly-symmetric top. The spin-rotation coupling constants in the two states indicate that they correlate with the same % state of the linear molecule. The previously suggested possibility, that the off-diagonal elements of the spin-rotation coupling tensor might not be symmetrical, is considered, and matrix elements for the skewsymmetric part of the tensor are given. The OlOfXKl band shows evidence of rotational coupling between the vibrations pp and Y~which is diagonal in N and K and proportional to K2. The theory of this type of coupling is developed. A. INTRODUCTION

The absorption spectrum discussed below was first observed at low resolution by Goodfriend and Woods (1) in the flash photolysis of difluoroamine, HNF;. A transient band system was discovered which, extended from 4000 to 5000 A, and was most intense at zero time delay between the peaks of the photolysis and source flashes. It appeared to consist of a progression of five regularly spaced bands, each containing discrete features which were tentatively identified as K-type subbands associated with a perpendicular transition of a prolate nearlysymmetric top. Similar experiments with DNFZ produced a transient spectrum in the same region, but with much smaller spacing between the bands. On the basis of the correlation diagram for HAB molecules of Walsh (2), the band systern was attributed to a zA’-2AM (type-C) transition of the HNF molecule, the * Present England.

address:

Chemistry

Department, 311

University

of Reading,

Reading

RG6 2AD,

WOODMAN

812

individual bands forming a progression in the upper state bending frequency II 7J2. The present paper describes the detailed analysis of the spectrum of HNIC obtained at high resolution. This analysis shows that all the tentative conclusions of Goodfriend and Woods are correct. Accurate rotational constants for the two electronic states have been obtained, and from them the molecular geomet’ries in the upper and lower states have been determined. In addition, the p-type doubling resulting from interaction between electron spin and overall rotation has been analyzed in detail. This analysis yields parameters which show that’ tllc ground and excited electronic states both correlate with the same ‘11 state in the linear configuration. Analysis of perturbations occurring in the 010-000 band indicates that they are due to internction of V: Lvith the vi (N-k’ stretck ing) vibration, and enables the frequency of t,his vibration to be estimnt~ed.

B.

EXPEBIMENT.4L

MF:THOl)S

The HNFy was prepared from tetrafluorohydrazine and thiophenol b3. the method of Freeman et al. (5). The violently explosive nature of HNP, at low t.empcratures in the presence of air R-M confirmed. However, gaseous HSF, could be kept at room temperature without appreciable decomposition for at. least a week. No attempt was made to prepare DNE‘, . The flash photolysis apparatus was described previously by ltamsay (4). The photolysis flash lasted less than 10 psec, and the source flash was timed to occur just aft.er the photolysis flash had reached its peak. The photolysis produc.ts, which presumably contained HIT, att.acked the silver mirrors of the multiple-reflection absorption cell (despite a protective layer of ,llgI<‘r ), and this limited the number of flashes possible before the mirrors had to be replaced. The best spectra mere obtained using HSl’, at tl pressure of S mm Hg without additional inert gas, changing the gas every four flashes. The path length \V:W lci m (eight traversals of a 3-m cell ). The spectra were recorded photographically with a 3%ft concave grating spectrograph; the grating had 15 000 lines/in. and was used in the second and third orders. A total of 12-20 flashes gave a satisfactory exposure on Kodak Ila-0 plates, prebaked for 24 hr at GO”. The plates Ivere measured on a photoelectric comparator of the Tomlks-Fred type. The root mean square error in the measurement of sharp lines is estimated to be 0.04 eniF’, though the stronger Jines were measured rather more accurately. C. APPEARANCE

OF THE

SPECTRUM

The overall appearance of the spectrum of HNP’ is similar to tl+t of HCF (5) HNO (6, 7). Seven bands are observed between 5000 and 3SO0 A, forming 1We shall follow the established prartice of denoting the bending frequency by Y? , wcn thollgh it will be shown in Sect. I;: that the N-F stretchiug frequency is probably higher.

ABSORPTION

SPECTRUM

FIG. 1. The 010-000 baud (a) in t,he region perturbation discussed in Sect. E.

FIG. 2. The K’ = 5 +-K”

of the balld origin,

= 4 subband

313

OF HNF

(b) ill the region

uf the

of the 000-000 band.

progression, with the vibrational interval decreasing monotonicahy from 1074 to 962 cm-‘. The 010-000 band in the region of the band origin is shown in Fig. la. In each band the K-structure is strongly degraded to the violet, indicating that the rotational constant 4-B is much larger in the excited than in the ground state. Thus there is a widening of the bond angle in the excited state and, as expected, a progression in the bending frequency v2 is observed. A a single

WOODMAN 8-7

-v---.

u-L‘_ I *0

5

IO

--c--L 15

.

v

._

.

20

FIG. 3. Calculated and observed doublet line split,tings in the ‘R branches of the 000-000 ba.ud. The continuous lines indicate the splittings calculated using Eqs. (7), modified for centrifugal distort.ionbyreplacing B by B - ??A NK and E.~by lao(1 - 2K2AK/A). Parameter Set 1 was used (see Tabie III). Large deviations iI1 the 4-3 and 3-2 subbands are attributed to perturbations (see Table II).

rapid increase in A---B with the vibrational quantum number ~2’ confirms that it is the bending frequency which is involved in the progression. Of the seven bands observed, the second and third in order of decreasing wavelength are the strongest. The longest wavelength band at 4964 8 is moderately strong, however, and the complete absence of any further bands at higher wavelengths indicates that this is the 000-000 band. Additional evidence for this vibrational assignment comes from an analysis of perturbations in the 010-000 band (Sect. E). Additional heads which do not fit into the AZ< = ~1 ,series of subbands are assigned as AK = 0, =t2 subbands, made allowed by the :wi+ sxvitching mechanism which has been shown by Hougen and Watson (8) to bc important when a large change in bond angle occurs. These extra subbands arc more prominent in the higher members of the progression. The HNF molecule has an odd number of electrons, and the doublet splitting of many of the rotational lines can be observed (see Figs. 2 and 3). The separation of the doublets is greatest at high K and low N, and is too small to be resolved in subbands involving the lowest values of K. This type of dependence of the spin splitting on the rotational quantum numbers is well accounted for bJ the theory developed by Van Vleck (9) and Raynes (10) and has previously been

ABSORPTION

SPECTRUM

OF HNF

315

Table I Observed

sub-band heads in the 2A'-2A" transition

of

in order of'wavelength

Aa&

v

-1

vat (cm

1

Assignment

4974.2

20097.9

(000-000)

4973.5

20100.8

(000-000)

4968.2

20112.6

(000-000)

4967.6

20124.9

(000-000)

4963.3

20142.4

(000-000)

4962.9

20143.8

(000-000)

4961.2

20150.8

(000-000)

4957.0

20167.7

(000-000)

4953.7

20181.4

(000-000)

4951.7

20189.5

(000-000)

4941.9

20229.5

(000-000)

4926.6

20292.3

(000-000)

4922.1

20310.9

(000-000)

4907.7

20370.4

(000-000)

4898.6

20408.2

(000-000)

4886.1

20460.4

(000-000)

4878.9

20490.5

(000-000)

4872.2

20519.0

(000-000)

4844.8

20634.8

(000-000)

4843.3

20641.4

(000-000)

4812.5

20773.4

(000-000)

4808.7

20790.0

(000-000)

4780.4

20912.7

(000-000)

HNF,

WOODMAN Table I (cont'd)

h

air

V

Assignment

vat

4770.8

20955.0

sQ5

(000-000)

4747.6

21057.5

rQ8

(000-000)

4746.8

21061.0

'Q&o)

(glO-!lOC)

4745.3

21067.4

(@lo-Olin)

4720.3

21179.3

(OlO-300)

4719.4

21183.0

(OlO-r)OO)

4715.9

21199.1

(010-000)

4713.6

21209.1

(OlO-rJO0)

4712.1

21215.8

(OlO-CIOC)

4708.9

21230.4

(010-00~)

4707.0

21239.1

(010-000)

4701.1

21265.8

(OlU-000)

4690.3

21314.6

(010-000)

4688.0

21325.1

(010-000)

4671.6

21400.0

‘&$(b)

(010-000)

4669.7

21408.7

‘Q2 (b)

(010-000)

4666.5

21423.3

‘Q4(a)

(010-000)

4663.1

21439.0

rQ2(a)

(010-000)

4647.6

21510.3

'Qs(a)

(010-000)

4646.4

21515.9

?J3 (b)

(010-000)

4641.6

21538.4

'&j(a) (010-000)

4628.5

21599.0

'Q,(b) (010-000)

4624.4

21618.5

qQ6

(010-000)

4623.9

21620.6

'Q2(a)

(010-000)

4621.3

21633.0

rQ,(b)

(010-000)

ABSORPTION

SPECTRUM

OF HNF

317

Table I (cont'd)

x

air

V

vat

Assignment

4615.7

21659.0

rQ+(a) (010-000)

4591.1

21775.1

'93(a) (010-000)

4586.2

21798.4

(010-000)

4555.6

21945.2

(010-000)

4555.2

21946.7

(010-000)

4522.5

22105.3

(010-000)

4515.7

22138.8

(020-000)

4513.7

22148.5

(o2o-ooo)

4502.5

22203.5

(020-000)

4495.4

22238.7

(020-000)

4492.7

22252.8

(020-000)

4489.0

22270.5

(020-000)

4485.3

22288.8

(020-000)

4483.8

22296.3

(020-000)

4482.1

22304.6

(020-000)

4475.7

22336.7

(020-000)

4469.7

22366.7

(020-000)

4465.9

22385.4

(020-000)

4460.5

22412.7

(020-000)

4444.0

22495.9

(020-000)

4440.0

22516.4

(020-000)

4417.4

22631.4

(020-030)

4401.3

22714.2

(020-000)

4391.9

22762.8

(020-000)

318

WOODMAN Table I (cont'd)

h ^<_

;_-__

4387.8

22784.3

rQ4

(020-000)

4365.5

22900.4

SQ3

(020-000)

4357.1

22944.4

rQ5

(020-000:

4329.2

23092.4

sQ4

(020-000)

4325.3

23113.1

rQ6

(020-000~

4293.5

23284.6

pQ1

(030-000)

4290.5

23300.9

qQO

(030-000)

4289.5

23306.4

nQ3

(030-000)

4286.7

23321.4

qQ1

(030-000)

4282.9

23341.9

4277.4

Assignment

P

Qo

(030-000)

23372.3

PQ+

(030-000)

4273.6

23393.0

qQ2

(030-000)

4265.0

23440.1

rQ1

(030-000)

4241

23573

4213.6

23726.3

rQ3

(030-000)

4183.2

23898.5

pQ4

(030-000)

4151.0

24083.7

rQ5

(030-000)

4124.7

24237.3

‘Q2

(040-000)

4116.2

24287.6

pQ1

(040-000)

4113.5

24303.4

‘Qo

(040-000)

4105.6

24350.4

rQO

(040-000)

4084.2

24478.0

rQ,

(040-000)

4053.8

24661.5

rQ2

(040-000)

3957.7

25259.9

pQ1

(050-000)

r

Q2

(030-oooj*

ABSORPTION

SPECTRUM

319

OF HNF

Table I (cont'd)

A

air

Assignment

'vat

3955.3

25275.3

qQ0

(050-000)

3815.5

262o1.g

pQ1

(060-000)

3813.1

26217.8

qQO

(060-000)

*Not measured Goodfriend

in present work, wavelength

and Woods

taken from

(1).

detected in NO2 by Huber (11) and in several dihydrides. It throws considerable light on the electronic structure of the states involved in the transition (Sect. G). Minor small-scale perturbations, affecting two or three lines at a time and producing displacements or additional splittings which are never greater than 2-3 cm-l, occur in all the bands. The 010-000 band however shows large perturbations (up to 12 cm-‘) affecting whole subbands, which are interpreted in Sect. E as a second-order rotational resonance interaction between Ye’ and ~31. The third and subsequent members of the progression exhibit complicated and overlapping rotational structure, which probably results from more complicated interactions of the type observed in the 010-000 band. No progressions in vl and v3 are observed, and no sequences (hot bands) have been detected. The wave numbers of the subband heads in the seven bands observed are listed in Table I, with their assignments. D. ROTATIONAL

ANALYSIS

OF THE

oOO-000 BAND

The HNF molecule is a nearly symmetric top, and therefore its spectrum may be analyzed in terms of quantum numbers N (total angular momentum excluding spin) and K (component of N along the top axis). The selection rules are AN = f 1 or 0 and AK = f 1, though subbands with AK = 0 or &2 are weakly allowed by the axis-switching mechanism. The assignment of the K-numbering to the AK = f 1 subbands which account for the strongest features of the OO(H300 band was straightforward. The 2-l and l-2 subbands show strong asymmetry splitting and the 1-O and O-l subbands show anomalous degradation, due to the first-order contribution of the asymmetry to the K = 1 term-values. Once these four subbands had been assigned, the assignment of K’ and KN to the remaining AK = fl subbands fol-

WOODMAN

320

lowed logically. It was then possible, using the rather sharp “Q- and ‘Q-heads which characterize the spectrum, to obtain a set of approximate term-values for the K-dependent contribution to the energy; these term-values were then used to check the assignments of the “Q-, ‘Q-, and “Q-branches generated by the axisswitching mechanism. To interpret the fine structure (involving J), we must consider the rotational Hamiltonian in detail. Following Van Vleck (9) and Raynes (IO), but with a slight correction which is discussed in the “Appendix,” we write the effective (i.e., contact-transformed) rotational Hamiltonian for an asymmetric top molecule in a doublet state in cm-’ units as .K, = ;? (AN,2 + BN,,” + CN,2)

E’or a molecule which is planar but not orthorhombic, only four of the ~~6~6are independent, and Only fk? of the cap (ha , Ebb, E,, , eat,, and cba) are nonzero. In this analysis, we used the set of constants A,, AN=, AR, aN, 8X due to Watson (2.2) to represent the effects of centrifugal distortion. Of these five, only four are linearly independent for a planar molecule (so that, for example, 6R may be written as a function of the other four) and neither 8# nor 6K, which give matrix elements linking states differing by 2 in K, was determined with any significant degree of accuracy by the data available. Therefore these two parameters were set equal to zero. In addition to the terms of Eq. (1) it was necessary to include two kinds of higher-order terms. Firstly, higher-order distortion constants in K were necessary to fit the subband origins satisfactorily. Secondly, it has been pointed out by L)ixon and Duxbury (13) that fourth-order spin-rotation coupling terms can be important when centrifugal distortion effects are large. In particular, if T,,,, is large, a term ~aan,,NUaS,must be added, where the contribution of centrifugal distortion to qaauais 7aaaaEaa c.d. 1) aaaa = “A From Eq. (51) of Watson

(12 ), we have

~ao,a = -4CA, Anticipating the results, appreciably to qaaaa, and N,“S, will be replaced by be written as the sum of

(2)

+ ANK + A,).

(3 >

we note that only AK is large enough to contribute the contributions of ANR and A, will be ignored. Also, Na2J,Sa (see “Appendix”). The Hamiltonian may then two terms: 32 = x, + x,,

(4)

ABSORPTION

SPECTRUM

OF HNF

321

where X, = ;

(AN,2 + BN:

+ CN,2) (5)

- ; (AEN:

+ A,&N,2

+ Ad4)

-

;

H, Nafi- ; Lx N:,

and 1 -

Z&Z N," _ A@

JaX, -I- mJbSb l

(6)

Evaluation of the matrix elements of the Hamiltonian in the ( JNK) representation is now straightforward. The elements of X, , involving only N and its components, are given by the usual expressions for a nonrigid asymmetric top. Elements of JaSp have been tabulated by Van Vleck (9) and Raynes (IO), although some modifications to their expressions are given in the “Appendix” to the present paper. Elements of the fourth-order term in Na2J,Sa are obtained by multiplying the corresponding expressions for J,S, by K2fi2 (since J,S, is itself diagonal in K). The effects of the diagonal elements Ebbo f the spin-rotation interaction tensor on the spectrum of a nearly-symmetric top have been described by Raynes (10). The component E.~causes spin-doubling which is greatest at high K and low N, and this is obviously the most important effect in HNF. The components eJ,and ccc make a contribution to the p-type doubling which is proportional to N + $$ and independent of K. For all states except those with K = 1, this doubling depends only on ~,b+ E,, ; for states with K = 1, the doubling is proportional to tbb+ ecef $5 (~,a-- eec). Examination of the spectrum of HNF showed no evidence of p-type doubling in the 1-O and O-l subbands, except possibly at the very lowest values of N, where err,,might contribute. Therefore @, and ccc are zero in both states to within experimental accuracy. Upper limits for these parameters are considered later. The remaining elements, Eaband Eba, link states differing in K by fl and in the limit of high N are proportional to K and N, respectively (see “Appendix”). It was easily shown that &b and ebo would have to be moderately large to produce an observable effect in the spectrum, and that the most conspicuous result would be a p-type doubling in the O-l and 14 subbands. Since no splitting was observed in these subbands, 6aband EZ,~ were also taken as zero. The 000-000 band of HNF may therefore be regarded as a type-C transition of a nearly-symmetric top, with a p-type doubling caused by coupling of the spin angular momentum to rotation about the prolate axis. Assignment of the quantum numbers NN and N’ to individual rotational lines in the R branches was made by counting from the first members of the branch (which have N” = K” ),

WOODMAN

322

Table II

i('-K" N"

Wave-numbers

of assigned

lines in the (000-000) band

8-7

7-6

6-5

r'R7

rR6

rRS

5-4 pRII

FP1,

20 525.60 529.59 20 649.68 654.40" 652.09 655.90

528.53 531.37 530.68 533.20

785.54 790.04x

654.40* 657.70

:;:::5 *

926.28 931.48

787.91 791.67

656.83 659.65

535.08* 537.06

928.64 933.25*

790.04x 793.81

658.93 661.53

537.18 539.02

930.97 935.14

792.43 795.65

661.17 663.43"

$?:E

933.25* 937.00 ;;::;z

794.74 797.59 796.88 799.55

663.43" 665.53Y 665.53' 667.49

937.64 940.81

799.04 801.55

;z;::;

801.20 803.49X

667.63 669.53 669.83 671.54

;;t:;;

803.49W 805.53"

277::;:

944.23 946.87

805.53~ 807.62"

946.27

20 783.05 788.30 20 923.85 929.73

20 498.70" 496.85

498.70*

:Z::Z

496.76

::;:;:

493.14 494.71

545.84 547.13

492.61

489.33 ::;::; 550.10 551.23

487.64 488.65

552.25 553.30

486.69

807.62* Bog.54

674.11 675.58 676.42 677.72

::;:z:

g50.84*

K::;:

678.41 679.79

556.52 557.48

950.84" 952.89

812.08 813.82

680.63 681.87

494.75 481.83 482.78 479.98 480.97

814.20 815.78

683.98

::z1. 3 560.84 561.66

816.41 817.91

684.99 686.08

563.02 563.82

476.15 477.11

818.55 P20.06

687.09 688.19

565.19 565.93

474.38 475.17

Y20.68

689.24 690.28

567.33 568.06

691.39

569.48 570.23 571.59

479.00

ABSORPTION

SPECTRUM

OF HNF

323

Table II (cont'd) Wave-numbers

lines in the (000-000) band 3-2

4-3

K 1 _K”

N”

of assigned

rR3

rp,

r%c

?R2d

rp,

20 315.35 317.42

2 3

20 413.49 416.34

4

415.94 418.14*

320.01 321.19

5

418.14* 420.32*

322.16 323.06

6

420.32* 421.74t

324.24t 324.71t

7

422.85 424.43

327.04 327.51

8

425.24 426.28

328.83 329.44

293.99t

9

427.17 428.24

;;Si;

292.75

332.85 333.66

290.42

10

20 388.53 390.10

429.22

430.26

g;:

11

431.36 432.32

g:;

12

433.37 434.34

382.88 383.88

20 337.13 337.60

13

435.61 436.40

380.86 381.90

;;;:i:

14

437.66 438.61

379.06 379.88

341.26 341.60X

15

439.92 440.61

;;;:;z

343.33 343.68"

16

441.98 442.67

375.16 376.09

345.32 345.78"

17 18

446.28 446.87

19

448.36 448.96

20

450.47 450.94

21 22 23 24

452.64 453.23 454.87 455.38 457.02 457.41 457.22 459.55

20 295.93” 296.82 294.37t 293.06

290.10 288.51 289.07

20 337.26 337.74

341.60" 341.90

345.7a* 346.21

;;:::;

348.04 348.35

372.10

350.18 350.54

286.58 287.12

WOODMAN Table II(cont'd) Wave-numbers

K

t

-K”

I<"

of assigned

lines in the (000-000) band 1-O

2-l rRIC

rRld

r‘RO

rpO

20 173.99 3

176.10

4

178.29 180.52

6

182.66 20 2k7.478

184.91

249.96

187.04t

249.59

252.32

189.53

150.76

10

251.37

254.66"

191.91

149.18'"

11

252.92

194.32

147.33*

12

259.61

196.77

145.60

13

256.74

261.90*

199.22

14

258.26

264.60

201.74

142.42

15

260.04

267.09

204.31

140.94

16

261.90*

206.92

139.46

17

263.47

209.55

138.13

18

265.14

212.27

136.58

15

266.75

215.07

135.30

20

268.48

217.88

8

21

20 247.47s

220.81

20 152.51

Table II (cont'd) Wave-numbers

of assigned

lines in the (000-000) band

O-l pR1

pQ1

pP,

l-1

o-o

'Qld

qQ,

20 118.34

3 4

20 134.48

116.25

5

136.45

114.13

6

138.30

111.95

7

140.22

20 125.65

109 70

a

142.07

125.98

107.58

9

143.83

126.34

105.35

10

145.60

126.72

103.07

11

147.33*

127.17

100.83

12

149.18"

127.63

098.55

13

150.95

128.13

096.27

14

152.50

128.69

093.91

15

154.20

129.25

091.55

20 153.25

16

155.91

129.86

089.24

153.49

17

157.49

130.52

086.79

153.83

18

159.17

131.25

084.43

19

160.83

131.89

082.01

154.58

20

162.41

132.67

079.60

154.98

21

163.93

133.57

077.13

155.42

22

134.48

074.61

23

135.30

071.93

20 143.20

144.19 144.53

156.27

24

168.41

156.80

25

169.89

157.35

26

171.35

144.97

146.03 146.49

158.53

27 28

147.79

29

148.45

30

148.77

Lines assigned are marked

mere thanonceare

marked

t. 325

*.

Strongly

perturbed

lines

WOODMAN

326

and then in the P branches by the method of combination differences. Some assignments were not certain until the analysis was substantially complete. The wave numbers of the assigned lines are listed in Table II. The main complication in the analysis arose from the fact that it is not sufficient to regard the midpoints of the spin-doublets as the position which a single line would occupy in the absence of spin-doubling. To understand this, we may consider Eqs. (13) of Raynes (10) which, in the absence of centrifugal distortion, neglecting asymmetry effects, and with @, = tee = 0, become &(N,

K)

= BN(N

+

1) +

(A -

B)K” [$&ah”

+ B(N -

+

1)

~a,sK” + B2 (N + 1)“11” (7)

Fz(N,K) = BN(N + 1) + (A - B)K2 - BN +

(f/4&K2 -

E,,~?K' + B2Na)"2.

(F1 ,Fzindicate states with J = N + $5,N - $2,respectively.) These equations can only be simplified by assuming that eaaK' <
(8) F,(N,K) 2 BN(N + 1) + (A - B)K* - gcaa Even to this level of approximation, the mean position of the doublet is dependent on E,, , and can deviate as far as Wean from the position it would occupy if eaa were zero. If these expressions were accurate, it would be a relatively simple matter to calculate corrections. However, the approximations made in expanding the square root make this an unreliable procedure.2 The analysis was therefore carried out in two stages. First, a computer program was written which fitted, by the method of least squares, the wave numbers of the observed lines to the term values of a doublet symmetric top, using the accurate expression for the spin splitting [Eq. (7), with centrifugal distortion terms added]. For this fit, only the branches with AK = +l, K" = 2, 3,4, 5, 6, 7 were used. (For these branches, assignment of N-numbering was straightforward, and asymmetry splittings were very smaI1.) It was at this stage that the necessity for including a fourth-order spin-rotation constant vaaaawas noticed. With only the second-order constants caa, very poor agreement between computed and observed line positions was obtained, while fits on each subband separately gave much better agreement, but with significantly different vaIues of & and & for 2Equation (8) in fact defines a more accurate value of Herzberg’s K (G. Herzberg, 1966, “Electronic Spectra of Polyatomic Molecules,” p. 88), i.e., ~~~(1 - ~,./4&, rather than eaa . However, this expression differs from that of Herzberg only by the term in &B, and if this constant. becomes large, the equation is no longer applicable.

ABSORPTION

SPECTRUM

OF HNF

327

each subband. Inclusion of fourth-order constants with values given by Eq. (2) gave good agreement for all six subbands simultaneously. However, two different parameter sets were found to give equally satisfactory fits, corresponding to the two possible ways of assigning J-numbering to the components of the spin doublets. In both cases, E:~and & were of opposite sign, but in one case & was positive, and in the other E: was positive. Using these values for E:~and & it was possible to compute corrections to the mean positions of the p-type doublets which could then be used to determine the rotational constants accurately. The two sets of corrections were of opposite sign. The rotational constants were determined using the computer program of Birss and Ramsay (S9), which can make least-squares fits on upper and lower state parameters simultaneously, using centrifugal distortion constants of the symmetric top type up to the sixth order. It would have been more satisfactory if it had been possible to conduct fits on the upper and lower states separately, using the method of combination differences. However, this was impossible because in several subbands, only one branch was sufficiently strong and well resolved to be used for analysis. The branches included in the fit were 'R, ,rRg,‘Rg , 'R4,'P4, 'Ra,'Ps,*Rz,rP~,'Ro,%, 'RI,'&I,and 'PI.Nearly all the subbands with AK = - 1 overlapped each other too strongly to enable correct assignments to be made. In the AK = +1 subbranches with K" > 4, the P branches were too weak to be observed; this was because of their intrinsic weakness due to low H&l-London factors, combined with the fact that the HNF was rotationally cold. Had the P branches been more clearly visible, measurement of combination differences would have made it possible to determine unambiguously the signs of & and e:, . It was also necessary to compensate for the absence of P-type subbands by including in the fit the positions of some of the Q-heads of the Q- and X-type subbands with AK = 0, +2. The Q-branches were double due to spinrotation coupling, and in each case one branch (the low-wavelength component) formed a very sharp head (see Fig. 2). Using the spin-rotation constants already determined, it was possible to assign values of N to the heads, and to correct their positions for the effects of spin-rotation interaction. The subbands “Qo and "&I,which are only allowed by the axis-switching mechanism, could not be included in the fit because the computer program could not take account of the selection rules peculiar to this type of transition (8).3 A certain amount of trial and error was required to determine which rotational parameters were well determined by the available data. It was found that the fit was significantly worse unless both H RN and H,' were included, but that the constants AN were very poorly determined and could be set equal to zero. The final parameter sets consisted of A, B, C, Ax, AN=, and H, for both states. These constants exhibited one feature which appeared physically unlikely: while AK’ 3The higher “Q- and “Q-heads were included as transitions to the incorrect asymmetrydoubling component in the excited state; these transitions follow Type A selection rules.

32s

ABSORPTION

SPECTRUM TABLE

OF HNF

329

IV

MEAN GEOMETRIES FOR THICLOWEST VIBRATIONAL LIWICLS OF THF: Two

ELECTRONIC

STATES OF HNF

r(N-H) T(N-F) L HNF

(assumed)

A2A’

PA”

1.03 A 1.34 A 125”

1.06 A 1.37 ;i 105”

was four times greater than AKN,reflecting the fact that the upper state is closer to linearity, H,’ and H,” were similar in magnitude. It was therefore thought that a more reasonable set of constants might be obtained if H,” was constrained to zero, while an eighth-order constant L xt was introduced for the upper state (which is shown in subsequent sections to have a low barrier to linearity). This was done by fitting the residuals to an expansion in KNz and K”, truncated at K”’ and K@, with the coefficient of K"" constrained to the value -H,“. This change in the parameter set caused large changes in AKN (which decreased from S.10e3 to 3.10e3 cm-‘) and H,' (which increased from 0.14 X 10e3 to 0.19 X lop3 cm-l), but negligible changes in the remaining parameters. The standard deviation was slightly improved. The final parameters are given in Table III, and are averages of the results obtained from two independent measurements of the band. The errors given for each parameter are standard deviations; differences between the fits obtained with the two different sets of measurements indicate that the true errors are somewhat greater (because of small systematic errors in measurement, and perturbations in the upper state). This applies particuIarly to the inertial defects, which are probably only accurate to h.007 amu 8” (i.e., 10 % ). The principal difference between the two parameter sets (apart from the signs of E:~and & ) is in the values of A’NKand A{K . This reflects the fact that at low N and high K, the midpoint of each p-type doublet is displaced in a direction which depends on the sign of cae ; hence Buff = B - K2a,, has different values in the two cases. The rotational constants of a planar triatomic molecule do not determine the geometry unless data for isotopically substituted molecules are available. Consequently, since no isotopic spectra were obtained, in order to obtain a value for the bond angle it is necessary to estimate one of the bond lengths. By comparing X-H internuclear distances in other small molecules we estimate r,“-, to be 1.06 _& in the ground state, and 1.03 A in the excited state, yielding the bond angles and internuclear distances given in Table IV. The N-E’ bond lengths fall within 0.05 8 of the value of 1.32 8 for NF (X32-) (14) and 1.37 8 for NFs (15, 16).

The inertial defects given in Table III are in moderate agreement with values which can be predicted using the theory of Herschbach and Ilaurie (17) for the

WOODMAN

330

inertial defects of unsymmetrical planar triatomic molecules. According to these authors, the inertial defect in the 000 vibrational state is a simple function of the vibrational frequencies and the Coriolis coupling constants for vibrational angular momentum about the c-axis. For the electronically excited state, we take V? = 1070 cm-l, v3 = 1120 cm-’ (see Sect. E), and estimate VI = 3800 cm-l. Noting that the Coriolis constants for HAB molecules are dominated by the motion of the H atom, and that the motion of thisatom in v2 is roughly parallel to its direction in vs , we take {& = 0. Then {fz and [I) , of which {& is probably the larger (though this is unimportant, since in this case v2‘v v3), are connected by the sum rule &” + <;s2 + c&z = 1. Applying Herschbach and Laurie’s Eq. (49a), but with the “dominant” term being & rather than & , we find Aiarc = 0.061 amu 8”. For the ground state, the frequency v2 is not known, but, comparing with HCF (which has similar bond angles in the two states), we may estimate vzn = 1300 cm’ and thence A$, = 0.068. Disagreement between which is satisfactory in view the observed and calculated values is then -lo%, of the limited accuracy of the measurements and the approximations in the calculations. To place upper limits on the Ebband ccc, we note that in all the branches involving K”, K’ 5 1, no splitting is observed up to N = 20. Detailed treatment of these observations, using a rather conservative limit of 0.1 cm-’ for the minimum detectable splitting, shows that all di$erences between & , trc , & and & are less tlran 0.04 cm-l. The observations are, however, compatible with a value of $ (tlb + EFc+ ELb+ EWE) of up to 0.1 cm-‘, although it would be rather surprising if a11four parameters were as Iarge as 0.1 cm-’ but within 0.04 cm-’ of each other. The off -diagonal spin-rotation coupling constants, &b and @a , link states differing in K by f 1. These parameters will therefore only affect the spectrum in the second order, and so could conceivably be quite large (-0.2 cm-‘) without being detected. b:. PARTIAL

ROTATIONAL

ANALYSIS

OF THE

OlSOOO BAND

The second member of the progression, the 010-000 band with origin at 4712 A, is generally similar to the OOCMOOband, but subbands involving the K’ = $4, 5 levels in the upper state each consist of two components with separations of 20-30 cm-‘. A plot of approximate K-dependent contributions to the energy for the upper state obtained by taking K-dependent combination differences she%-s a large perturbation (of roughly f10 cm-‘) at K’ = 4. By “smoothing over” the perturbation, it can be deduced that the effective value of A’ for this vibrational state is about 2.5 to 3 cm-’ greater than in the 21’ = 0 state: hence ,a’ is large and negative. The region of the spectrum affected by the perturbation is sho\vn in Fig. lb. Rotational numbering cannot be established except for a few lines because P-branches are weak, and because numerous local perturbations interrupt the

ABSORPTION

SPECTRUM

OF HNF

331

regular spacing. However, certain facts about the main perturbation are obvious. It is approximately independent of N, since the Q-heads remain relatively sharp. There is no drastic effect on the spin-doubling, which indicates that interaction with high vibrational levels of the ground state is not involved (see Sect. G). Intensities of “Q- relative to ‘Q-branches are not greatly affected and so the selection rule AK = 0 is established for the perturbation. (In any case, A-axis Coriolis coupling is impossible in a nonlinear triatomic molecule). Consideration of these facts indicates that the perturbation involves two sets of K-levels which, in the absence of the perturbation, would coincide at K’ = 4, the separation at this value of K’ being about 20 cm-‘. Using the fact that the intensity is higher in the low-energy subband for K’ = 3, about equal for the two components with K’ = 4, and higher in the high-energy component with K’ = 5, it is deduced that the effective value of (A - B) for the perturbing state is lower than for the 010 state, while the origin is higher in energy. The most likely state to occur in this position is the vi = 1 (i.e., 001) vibrational level of the same electronic state. The YSvibration is, to first order, the N-F stretching vibration, and is expected to occur between 1100 and 1300 cm-‘. The effective A value of this state (determined by a;“) would not be much different from that for the 000 state, since the moment of inertia about the A-axis would not be much affected by displacements in the N-F bond length. The lowest term in the expansion of the Hamiltonian which could cause these two sets of K-levels to interact, homogeneously in N, is the term in N,qz~a (where N, is the total angular momentum about the z, or A, axis exclusive of spin and qt and and q3 are dimensionless normal coordinates). The possibility that a term of this kind might generate perturbations has been mentioned by Nielsen (18), who has called the effect “second-order rotational resonance interaction”. Contributions to the Hamiltonian (expressed in cm-’ units) involving the operator products Nz2q,q,l may be written

The constants (II,“,are then equivalent to the well-known constants (ysAwhich describe the linear variation of the rotational constant A with the vibrational quantum number us . Terms involving the a$ are not normally included in the vibration-rotation Hamiltonian, but will be important if the frequencies vs and usIlie close together. Approximating ws = wsr (w denoting a normal frequency in cm-’ units), c&r is given by A

Qbar

= -2A

2

WOODMAN

3332

This may be compared with the equation for 01;’ given by Oka et al. (19).4 The t,erm involving the third-order force constants is generated by contact transformation of the basic vibration-rotation Hamiltonian, as is part of the term in &.CV . The significant matrix change of one vibrational

elements quantum

involving between

(v,va,K (SC’ ( v,&llu,~=~lK)

afS, link states v, aud v,l :

differing

= -c~:gK~(&&)~‘~,

by the ex-

(11)

where ~~ represents the larger of II, and aus& 1. To consider the likely effect of a perturbation of this type on the excited state energy levels, we note that the asymmetry splittings are negligibly small at K’ 2 4, and we also neglect the effects of centrifugal distortion and variation of the rotational constants B, C (defining B) with the vibrational quantum numbers. Then the Hamiltonian matrix for the 010 and 001 vibrational levels factorises into a number of 2 X 2 submatrices of the form ~3+ BN(N

+ 1) + (A,,,, - L? - azA)K2

C

-c&i K2

This submatrix

may be diagonalized

1 (is)

-c&K’ ~3+ BN(N

+ 1) + (Aor, - l? - a3”)K2

’

to give the eigenvalues

,‘,i (~2 + ~a) + &V (N + 1) + [Am -

B -

>i (azA + c&K*

The behavior predicted by this formula can be described as follows. At very low values of K, terms in K2 under the square root can be neglected and the subband origins occur at positions close to those they would occupy if CY&were zero. At some larger value of K, given by Kz,, = (~2 - va)/(aLA - asA), and provided ‘4 that LYZ - asA has the same sign as ~‘2- v3 , a resonance occurs, and the basis functions mix in ratios close to 1: 1. In this region, subbands from the two bands which, in the absence of the interaction due to C& , would occur very close to one another, repel each other and acquire approximately equal intensity. The separation at resonance is 2c&Kf,, . The point of closest approach, however, occurs at a smaller value K given by

KLL = At high values

of K, the term

(V?- vB)(ci2A - ai4)

(cYpA-

values

(14)

@A)2 + 4&jtend

to new limiting

values

given by

ABSORPTION

f

SPECTRUM

OF HNF

$5 [(agA- (ysAy + 4a&y}K2

333

+ RN(N

+ 1).

Thus at high speeds of rotation, the modes of vibration which occur are linear combinations of vz and va, the ratios of the contributions tending towards values determined by the relative magnitudes of azA - CQ~and a(& . In fact the limiting modes of vibration as K + cc are those which diagonalize the aA matrix, rather than (as in the nonrotating molecule) those which diagonalize the GF matrix. When plotted against K” (Fig. 4), the calculated term values fall on the two parts

I$’

200 -

F[(OlO,OOl~,K] km-‘)

-

CALCULATED (hyperbola)

-------

.,

I 5

ASYMPTOTES -

UNPERTURBED

I 15

I IO

TERM VALUES

TERM

VALUES

I

I

20

25

K*-

FIG. 4. Effect of perturbation by & : the calculated term values with P$ - ~3 = 50 cm-r, aZA - a3A = 3 cm-‘, & = f1.25 cm-i, and >i(A olo + A& = 6 cm-l. (This artificially low value of M(Aolo + Aoal) does not affect the extent of the interaction, but condenses the graph.) A is the center of the hyperbola formed by the term values, and marks the point of closest approach of the two sets of K-levels. B is the point where the two sets of levels would intersect if my?“, were sero. Note that although the perturbation is homogeneous (linking states of the same symmetry) the matrix element responsible is proportional to K2. Therefore the diagram resembles that for a heterogeneous perturbation (increasing with J) in a diatomic molecule.

WOODMAN

X3-2

t PROPORTION OF

v1

06

INEACH

EIGENFUNCTION

K-

FIG. 5. Mixing ratios of ~2 and ~3 as a function Fig. 4, (b) with Y? - vg = -50 cm-l.

of R, (a) with the same parameters

as for

of a hyperbola. Equation (15) then gives the asymptotes of this hyperbola. The intersection of the asymptotes (point A) defines the point of closest approach of the two curves, and is not the same as the point B where the two bands would cross if the interaction were not present. Figure 5 shows the mixing ratios of v2 and v3as a function of K. One pair of curves depicts the mixing which accompanies an avoided crossing (using the same parameters as Fig. 4). The other pair of curves represents the situation when vz - vaand atA - CYST’ have opposite sign so that the K-levels do not cross for a positive value of K’; however, even in this case, mixing of the vibrations induced by c& can be significant. On the other hand, in the absence of the conspicuous effects in the spectrum which occur when the unperturbed energy levels actually cross, the perturbation might well go undetected, and its effect on the spectrum be attributed to the higher order parameters pse4(coefficients of (us + ?/i)K” in the energy expansion). It may now be understood why it is that perturbations of this type (“secondorder rotational resonance”) are rare, and have not hitherto been detected, even though predicted theoretically by Nielsen (18). They will only occur when two fundamental bands have the same symmetry, lie rather close together in frequency, and have widely differing a-values, so that a difference in the vibrational energy can be canceled by differences in the rotational energy for attainable values of the quantum number K. Only very rarely will these conditions be simultaneously satisfied, and, among triatomic molecules, the phenomenon is probably restricted to triatomic molecules of the HAB type. Using the observed separations between the Q-branches for the subbands with K’ = 3, 4, 5, it is possible to determinecrzA - (Y~“,vp - vaand j a& 1. (Thesign ot

ABSORPTION

SPECTRUM

OF HNF

33.5

0.8 -

T PROPORTION INTENSITY EACH

OF IN

COMPONENT

-o-o--o_o o-o-o-o-~-.J10

=

(a) lb1

K-

FIG. 6. Calculated intensities of the subbands, using the parameters which fit the subband frequencies best (YZ- ~2 = 47.1 cm-l, CQA - ~~a-4= 2.07 cm-‘, (Y& = 0.53 cm-l), (a) using the model which gives zero intensity in Y%at K = 0,(b) using the model which gives zero intensity in the low-frequency component as K -+m .

& determines the relative phases of the combining vibrations. In the absence of other interactions between these two vibrations, and without knowledge of the relative signs of the Franck-Condon factors and reliable intensity measureA ments, it cannot be determined.) We find that ~2 - L+= -47 cm-‘, a2’ -a3 = -2.07 cm-’ and & = ~tO.53 cm-‘. These values must be regarded as very approximate since the positions of the subband heads are only approximations to the subband origins, and centrifugal distortion has been neglected. Using these values, it is possible to estimate the contributions of u2and v3to the upperstate wave functions for each value of K. If we assume that the 001-000 transition has zero intensity in the absence of mixing (i.e., the Franck-Condon factor is zero) then these contributions are proportional to the relative intensities (see Fig. 6). However, these intensities are clearly in poor agreement with experiment (e.g., they suggest that for the 3-2 subband, one component is 30 times as intense as the other). If, on the other hand, we take the intensities at high K to be in the ratio 1: 0, the intensities of subbands near the resonance are considerably altered, and are in satisfactory agreement with observation. This model for the intensities is quite plausible : it is possible, for example, that the vibrational modes prevailing at high K (i.e., those which diagonalize the OI~matrix) are pure bending and stretching motions, and that only the bending motion has a nonzero Franck-

WOODMAN

336

Condon factor. This model would also predict that the subbands involving the v3 = 1, K = 0 states should have $& of the intensity of the corresponding transitions to uq = 1, K = 0. There are some very weak heads in the predicted region, but they are too indistinct for their assignment to be certain. Using the values of the parameters obtained from the separations between the perturbed components, it is possible to de-perturb the subband origins and obtain rough values for A,,,, and AK,,,10 of 30.3 and 0.6 cm-‘, respectively. The vibrational frequencies (strictly AG,,, values) are ~2’ = 1074 cm-’ and YS’ = 1121 f 5 cm-l. The latter frequency lies close to the values of A&,2 reported by Douglas and Jones (14) for ?\TFin its two known states, 1124 cm-’ in X32- and llS0 cm-’ in 6’9 F. HIGHER

BANDS

OF THE

PROGRESSION

The progression in ~2’ can be followed up to vz’ = 6. The wave numbers of the subband heads are given in Table I. By regarding the subband heads as approximate subband origins (which will not involve errors greater than 2 cm-‘), and using the ground state parameters obtained from the 000-000 band, one can arrive at the K-dependent contributions to the energy F, (K) and the vibrational contributions G (0~~0) listed in Table V. As the upper-state vibrational quantum number increases, the bands become weaker, and it also appears that the intensities of the subbands with high values of K’ decrease relative to those with lower values of K’. For the highest two members of the progression, the 050-000 and 060-000 bands, the only features observed are the “&I and ‘&o-heads, separated by (approximately) the groundstate rotational constant (AN--B”). The vibrational interval decreases steadily, while simultaneously the upper-state rotational constant (A’-B’) increases rapidly, reaching a value of 48 cm-’ at VZ’ = 4, indicating that the mean bond angle in the excited state is increasing. This is accompanied by an increase in the axis-switching angle, which causes a greater proportion of the total intensity to occur in the AK = 0, +2 subbands. These changes are characteristic of the behavior of bent molecules with a small energy separation between linear and bent configurations, discussed by Johns @O).” It is shown in the next section that for HNF the upper state correlates in the linear configuration with one component of a %I state, while the ground state correlates with the other component. For bent molecules in general, it is possible in favorable cases to use the term-vaiues for the vibrational and K-dependent contributions to the energy to determine the shape of the potential-energy curve and to estimate the height of the barrier. In the upper ‘A” state of HCN, for example, the potential function is quite well determined by the term-values with 5 This is often called “quasi-linear” behavior. However, all bent molecules will show these effects if sufficiently vibrationally excited, and it, is probably better to keep the term “quasi-linear” for electronic states where the jirst vibrational level lies above the hump.

ABSORPTION

SPECTRUM TABLE

OF HNF

337

V

TERM VALUES G(O,U~,O),VIBRATIONAL INTERVALS AG21,+1/~,~~~ K-TYPE ROTATIONAL TERM VALUES F,(K) (cm-l units) Ground state 2)2:

G(0, ~2 , AG,,+w :

0) :

0 0

Excited state, TOOD = 20141.3 0 0 1074.1

1 1074.1 1053.1

2 2127.2 1032.9

3 3160.1 1003

0 29 116

0 35 134

4 4163 972

5 5135 942

6 6077

0 40 156

0 48 190

0

0

423

1 2

0 16.6 66.3

0 26.5 105.6

3

149.1

236.1

293

338

4

264.8

415.8

512

574

5

413.0

642.5

780

861

6 7 8

593.4 805.7

913.5 1226.5 1578.6

1088 1440

1197

K=O

996 1323 1696

vz 5 8, K 5 3 (20). However, when the electronic wave function in the linear configuration is orbitally degenerate, the problem is less simple, because only the 0 states are determined by one of the potential energy functions. For K= states with K f 0,the energies depend on both potential functions. Thus for the upper ‘A” state of HCCl, which, as was shown by Merer and Travis (21), probably correlates with a linear ‘A-state, it was necessary to use only the data relating to the K = 0 levels to determine an approximate potential function. Only in the case of NH2 , studied by Dixon (22), has it been possible to use a22the v,K-states to treat both Renner components simultaneously. It is doubtful whether, in the case of HNF, meaningful results would be obtained using only the K = 0 levels, while to conduct a complete fit on both electronic states would be difficult in the complete absence of information on the vibrational levels in the ground state. For HNF, there is also the problem that ~2and ~3are close together in the excited state, so that ~2is probably rather an “impure” bending vibration. Moreover, ~2and v3 are known to interact (see Sect. E), and this may cause additional complications. However, using the fact first noted by Dixon (25) that for bent molecules in general, the barrier height is marked by a minimum in the plot of AGYZf1,2against ys[G (VZ,K) + G(v2+ 1,K)] it is clear that the barrier is at least 6000 cm-’ above the lowest vibrational level; and considering

WOODMAN

338

the trend in the rotational constant A (obtained from the spacing between the K = 1 and K = 0 levels) it would seem that it probably lies below 10 000 cm-‘. It is not immediately obvious why the vibrational interaction which produces such obvious effects in the 010400 band does not have an equally conspicuous effect on the higher bands. However, there are several reasons why such effects would be rather unpredictable. Firstly, there will be more than two interacting vibrational levels: for example, the 24’ Ievel (involved in the 02CkOOO band ) can interact with Q’ + Q’ and 2~3’. Secondly, the parameters involved in the interaction, in particular v2 - v3 and aTA - aSA, must be replaced by the more general forms G(0, up , 0) - G(0, v2 - 1, 1) and -Ao,,o + AO,v,--l,~; and since neither G nor A varies linearly with v2 , these quantities will be different in each case. In fact, since AG,i+l/z falls steadily as v2 increases, v2 and vz may be expected to move out of resonance. Finally, it is very likely that as this happens, anharmanic terms in jzPz3and f2333will become important which will mix, for example, the functions 1020) and loll> in such a way that although vz is not a pure bending vibration, the level 020, in which 1020) is combined with a small amount of loll), corresponds much more closely to a pure bending state. If the interpretat,ion of & suggested in Sect. E is correct,, i.e., that it arises because A is affected mainly by bending displacements, but that v2 and v3 correspond to linear combinations of bending and stretching motions, then it may be expected that the transformation generated by the anharmonic force constants will bring terms in a$ on to the diagonal of the Hamiltonian matrix, and so no K-dependent perturbation will be observed.6 C:. CONCLUSION:

ELECTRONIC

STRUCTURE

OF THE

HNF

MOLEGULF:

The rotational analysis of the 000-000 band and the parameters obtained, together with the isotopic evidence of Goodfriend and Woods (1 ), show that the absorption is due to a type-C transition of a molecule of the HAB type. The only molecule of this type that can be obtained as a primary product of the photolysis of HEFT is HNF. The only other possible species would be HNN and HFF, and both of these molecules would be far too unstable to occur as secondary products. There can therefore be little doubt that the spectrum is due to HNI’. Following Goodfriend and Woods (1) we apply the diagram of WaIsh (9) for HAB molecules, and deduce that the ground state is

(a’s,)2(u’)~(u’)2(a”)2(u’)2(6’SJ(a”)

?=1”9

6 It has been drawn to my attention by Dr. J. Ii. G. Watson that an alternative explanation of the interaction between t.he 010 and 001 vibrational states can be given in terms of a Fermi resonance involving t,he 4th order force c0nstant.s. This explanation would require thatfrzps +fzazx = =t 27 cm-‘. Such a perturbation, unlike second-order rotational resonance, worlld involve an off -diagonal matrix element independent of K, and in principle would give a different energy level pattern. With the doubling visible in only three subbands, however, it is not possible to distinguish between these t.wo possibilities, and it is likely that the observed interaction between ~2’ and ~3’ results from significant contributions of bot,h t.ypes.

ABSORPTION

SPECTRUM

OF HNF

339

while the first excited state is probably (cz’sj3)2(a’)2 (a’)2 (a”)2 (a’)2 (cs’Sd) (a”)”

2A’.

Both these states correlate with the linear configuration 2 2 2 U~UU7r7r

4-3 2rII;.

This situation would be very similar to that for NH2, where the two known states also correlate with a df3 “II, configuration. There is however an alternative assignment for the upper state, which is equally compatible with the symmetry and the higher bond angle: (u’s,)2 (u’)2(a’)2 (a” )” (u’)2 (&a)2 (6’)

3A,

which correlates with 2 2 2 (TBUUTTU

4_2_

2 + z .

To decide which of the two possible assignments for the upper state is correct, we consider the spin-rotation coupling constants E,,~.The detailed interpretation of these parameters for XH2 molecules has been discussed by Dixon ($5). The spin-rotation constants ~~8arise mainly from cross-terms between the operators N,L, and ZT,L;S~~(where the latter term represents the spin-orbit coupling for one-electron orbitals in the state considered) (9). Applying second-order perturbation theory, Dixon deduced that each element of the eUbtensor can be regarded as a sum of contributions from various possible one-electron promotions. In the case of NH2, relatively accurate SCFMO calculations were available, and it was arises from the one-electron calculated that by far the greatest contribution to E(I(I! excitation which correlates with the 7~.+-- a promotion for the linear molecule, and is responsible for the absorption bands of NH2. Two approximate models were investigated, the “directive hybridization” model, which yielded r-values smaller than those observed, and the “free precession” model, which yielded high values, but was easier to apply. Also needed for the calculation were a one-electron spinorbit coupling constant for a closely related molecule, which for NH2 was the 311state of NH, and the excitation energy for the ? +- r electronic transition. For HNF, the corresponding one-electron spin-orbit coupling constant would be that for NF. No states of NF are known which could yield a value, either directly or indirectly, of this coupling. However, considering the similar molecules NO and CF, a value of 100 cm-’ appears to be reasonable. Then we may apply Eq. (7) of Dixon et al. (2’4) coo = and obtain the spin-rotation

4& * AEvertieal’

(16)

coupling constant in the case of pure precession.

340

WOODMAN

In this equation, A is the rotational constant, { is the spin-orbit coupling constant, and AEvertia1is a mean excitation energy, which may be taken to be the wave number separation between the interacting potential curves at the equilibrium angle of the state considered. For a a3 configuration, the - sign applies to the lower state, and the + sign to the upper state (95). For the lower state, AEvelticalis best estimated by adding to the excited state energy To,, the vibrational energy corresponding to the Franck-Condon maximum, -u1.5 v~‘. Hence AEveltioalfor the lower state is about 21 700 cm-’ and similarly for the upper state AEvertical1: 18 000 cm-‘. With these values, we calculate f~C%lC

&a

= -0.33

‘Cdl2 GLa

=

cm -l

+0.61 cm-’

While the precise calculated values depend on the value chosen for {, their signs and their ratio do not. Clearly, there is good agreement with the paramet.er set 1 (Table III) and total disagreement with para.meter set 2. Also, if the excited state were to correlate wit.h the linear “2’ configuration, this would not be in the relationship of pure precession to any other state, and therefore & would be small, like &, and &. Therefore, this alternative assignment for the upper state can be eliminated. Strictly speaking, these values of Ese relate to components along the N-E internuclear axis. For the lower state, this axis makes an angle 0 of S” with the principal axis, while for the upper state the angle is smaller. It is easily shown that, though this causes a negligible change in caa, it generates an off-diagonal coupling el::ment tllbequal to ?+a3 sin 20 = 0.05 cm-‘. Such a low value of 66 would, honever, by undetectable. This explanation of the spin-rotation constants taa as resulting from a mutual interaction between ground and excited electronic states also indicates that where an accidental resonance occurs between ground and excited states, of the type predicted by Dixon (.22), this should be accompanied by highly anomalous behavior of the spin-splitting. The two electronic states have opposite values of E,~= , so that when the wave functions mix in the ratio 1: 1, the p-type doubling in the two components should be determined by the sum e:, + & and difference I Since an effect of this kind does not occur in the perturbation in the %a ~2’ band discussed in Sect. E (the Q-branches, for example, remaining fairly line-like), this alternative explanation of the perturbation can be excluded. On the other hand, weak interactions of this type are almost certainly responsible for the numerous small-scale perturbations observed in all the bands. The spectrum shows no detectable diffuseness, but this does not necessarily imply that the dissociation limit lies above 26 000 cm-‘. The situation concerning predissociation is complicated, since the products could be H + NE’, NH + F, or HI; + S, though the first of these possibilities might be thought most likely. The

eta

ABSORPTION

SPECTRUM

341

OF HNF

excited state of HNF, with symmetry 2A’, does. not correlate with the ground states of NF (“Z) and H (“X). The ground state 2A”, on the other hand, does have the required symmetry, though the electronic configurations for HNF and its dissociation products are very different. This is similar to the situation for HCO, studied by Johns et al. (26). The fact that even in the region of the small heterogeneous perturbations marking resonances between the excited state and vibrationally excited levels in the ground state, no predissociation is observed, is evidence that the higher levels of the ground state are not strongly predissociated. APPENDIX:

THE

SPIN-ROTATION

COUPLING

HAMILTONIAN

According to Van Vleck (9) the effective Hamiltonian X,, for spin-rotation interaction for a molecule in a doublet state can be written

x,,

= ;

(ao (N, 4%+ Ns E$, + N, S,) + a(2Ns 8, - N, & - NT 8,) + b(Nz 8, -

Ng A%;,> + c(Nz

+ d(N,&

8, +

Nt,

8,)

(17)

+ N, 8,) + e(N, A%+ N, &,)I.

Here sZ, &, , & are components of the reversed spin angular momentum, and the coefficients a0 , a, * - * e are all real. The general derivation gives corresponding terms for spin-spin coupling in 8,s~ : for a molecule in a doublet state, all these terms vanish except the symmetrical one in &” + &,:,”+ s?, which is a constant equal to 9@“, and may be taken into the electronic term whenever it occurs. (In general, for a spin-g state, && is equal to x’n (6S,, -ie,,&), where 6,,~ = 1 if cy = ~3and is zero otherwise, and ears7= + 1 if (Y,/3, y are x, y, x in cyclic order, -1 if they are in anticyclic order, and zero otherwise; summation over y is implied. This identity may be demonstrated by multiplying together the Pauli spin matrices. > Raynes (10) noted that the coefficient of N,& in Eq. (17) need not be the same as that of N$&, and has rewritten the Hamiltonian with nine independent terms, involving constants cl and & (coefficients of N,&, and N,S, , respectively), etc. Clearly the equation is then completely equivalent to

(18) However, the operators N and s do not commute, because N is defined as J + s, where J and 9 commute (9). Equation (18) therefore gives rise to terms which are not Hermitian. These can easily be eliminated however, by rewriting x,, in one of the two forms (19):

The final term, as usual, can be incorporated into the electronic energy.

WOODMAN

342

Similar corrections are necessary for higher-order general expansion of X,, might be written

x,, = 2 2 x @=Iv=l ul=a.b,c

...

c

Bv=a,b,c

spin-rotation

terms. A

a,,,,...a,s,e,...s,JIIIJ,,

(30)

(Since J and 8 commute, terms in which the J=‘s and I!$‘s are interspersed can be omitted.) Following the general approach of Watson (Id), we restrict the expansion to Hermitian terms, and apply the commutation rules to bring the expansion into a “standard form”: X,,

=

5

apprijk ’ (J,PJbqJc’S~iS~S,” j- Z~S~&iJCrJgqJap)

(21)

P,q.r,i,i,k=o

where p + q + r + i + j + lc must be even to satisfy time-reversal symmetry (i.e., X,, must be invariant to the simultaneous reversal of all angular momenta), and all the aaqr;ikare real. A similar expansion may be written in powers of the components of N and S: X8,

=

p QV$k_I 91 *‘,‘,

baq,iik(N,PNbqN,rSailSbjlSl,k +

&‘%bj&iN,‘NbqN,P).

(22)

The b&jk are then linearly related to the ai,,iik by the commutation rules and the fact that N = J + s. For a molecule in a doublet state, the operator s,&$ may be written j”jfi (56,~ - ie~&?,.)), and therefore the expansions (21) and (22) may be restricted to terms which are first-order in 8. The second-order terms in Eq. (22) then reduce to Eq. (19), while the fourth-order terms may be written

where CX,8, y are axes which are written in alphabetical order (e.g., %&a occurs in the expansion, but vbaaa does not) ; Summation of CC,P, y, 6 over a, b, c is implied. However, the matrix elements of the Hamiltonian in this form are difficult to calculate in the IJNK) representation. Writing N, as J, + 8, and simplifying the resulting terms in N,N,$,& , we obtain a form which, though less elegant, is easier to handle:

ABSORPTION

SPECTRUM

343

OF HNF

The operator N,NoJ,& is then regarded as the product of N,No , which occurs in the rigid-rotor Hamiltonian, and J,& , which occurs in the second-order spinrotation Hamiltonian whose matrix elements in the IJNK) representation have been tabulated by Raynes (10). The second pair of terms in the braces generates contributions to the rotational constants A, B, C, which are small and probably always negligible. In the upper *A’ state of HNF, qaacro= 0.001s cm-‘, so that the contribution to A is 0.0009 cm-’ , which is well within the standard deviation for this parameter. The remaining terms in (24) make infinitesimal contributions to the Q. Although Raynes pointed out that the coefficients cl and CP , dl and dz , and er and e2 need not be equal, he did not determine the matrix elements of these terms separately. Following Van Vleck (9) who took linear combinations of the operators N,8, which transform as spherical harmonics, we divide the offdiagonal part of the c tensor into a symmetric and an antisymmetric (or skewsymmetric) part. This contribution to the Hamiltonian may then be written off-d&z

X8,

=

c(Ji?&, + J,i%) + d (J&

+ ct (Jz&, - J$L)

+ J,&) + e (J$$ + Jz&,)

+ dt (J,& - J$‘,) + et (J,I% - J&,).

(25)

The matrix elements in (Jai?@+ J&‘m), involving the coefficients in c, d, and e, are just those given by Van Vleck and Raynes. The operator combinations (Jai& - Jp%) transform as axial vectors (T-vectors), and the corresponding matrix elements may be evaluated by the usual methods (27). Using the notation of Raynes (10) and his choice of axes, these elements are (NKI :X$=w-aymIN-1K) (NK( X::w-sym IN-1

K&l)

= -#N

(N2 -

= xNg(N,

K2)1’2~+4(N),

&K)(d+ =Fie+)$(N).

(26)

The coefficients c, ct, d, dt, e, et are then related to the en6parameters of Lin (28) by the following equations: hb

= -d+d+,

Qa = -d

- d+,

be = -e+e+,

Qc = --c - C+,

&a = -e

ecb

- et,

=

-c+iG.

(27)

ACKNOWLEDGMENTS I wish to thank Mr. J. Shoosmith for much practical assistance in obtaining the spectra; Dr. J. W. C. Johns for making his least-squares program available to me; Drs. F. W. Birss and D. A. Ramsay for the asymmetric rotor program; Drs. P. R. Bunker, T. Oka, and J. Pliva for theoretical discussions on several aspects of this work; Dr. A. E. Douglas for bringing the earlier work on HNF to my attention; and Dr. G. Herzberg for much encouragement and helpful criticism of the completed manuscript. RECEIVED: August 7, 1969

344

WOODMAN REFERENCES

1. 2. 3. Q. 5.

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