The stability and specrta of isotopic forms of the ammonium radical: An ab initio CI study of the dissociation barrier

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chemicalphysi~~96'<198~81-95‘-.~~':..;No&~&d.Amsterdam I : _ I . . .. _

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-STABIL,l-W

AN AB INITIO

Depanme~~ of Chemdy,

F&i& ? m._

&THE

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&MONIl_iM .--- ‘- :_-.

Queen3 lhicem2y. Kingron. Ontario.Cant@

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RADIkAIi t1.. _;:._ ., ._;y _-.- 1_

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-AhD SPECI-RA OF ISOTOPIti CI ,STU?Y OF THE DI$SOCL+~ON : J~~~M~~KASPAR+V~~~~~-H_~MITHJ~_ --TEE

-_ _ ,:.~~ -. -,= .;-_;:‘_ .Y:; _ _.._: _-1-z .r. _ 81,:

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K7L 3N6

and BIair N. McMASTER Depanmenr of P&xi=

McGiIf Uniceniy. Manrrd

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Canada H3A -12%

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Received 25 July 1984; in fmaI form 12 Dcccmber 1984

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.NH, + H) has been calculated at an ab-i&o CI In contrast to the UHF remIts. the tao-poinr viirational Icvds inside the barrier arc bound for aII isotopic spccics Ne,. ND, land NT& Foi NH,. o&e q-iund Vibrationallevel is found while both ND, 2nd NI, have two quasibound Ie&k It is suggested that the latter correspond to the ‘stable” and “dissociative” form of the radical found in ncut&izcd ion.!Scwxal emission bands arc &&cd to vibrational transitionsof the dcctronic Rydbag 3p2TL-3s’A, transition The potenthI bank

for the disociation of the -onimn

radical (NH, -

Icml and a vibrational and tmmcling-lifetime analysis was made +refrom.

1. Intro4iuction Hemberg’s discovery [l] that several emission bands observed _in electric discharges in ammonia are Rydberg transitions of the NH, .radieal has stimulated interest again in this molecule whose molecuk orbital structure was first discussed by Mull&en [2] over half a century ago. Despite considerabIe p,mgress made in both the experimental [3-111 and theoreticaI realm [12-171, the questions of the ground-state stability and possible dissociation channels of this Rydberg radical a& not yet resolved_ There are two primary reasons why this particular molecule_ presents diff$uIties for .the theorist: As this paper is go_mg to show*._the consideration of, electron correlation is of crucial importance for the stab&&on of NH, relative to its disociatioti products and for the shape of the dissociation. barrier. This shape, in turn, determines the run& ber of possible molecular states inside the barrier. and their lifetimes_ If. ekctron .correIation is taken

into account the radical is electronically stable towards dissociation into either NH, f H or-NH, + Hz_ The instability of NH, is therefore solely due to its larger number of vibrational degrees of freedom compared with those of its _dissociation products. Experimentahy, strong isotopic dependencies of the lifetimes of the radical are observed. In their’ neutralized-ion beam studies,. Williams and Porter -.: : [3] found that NH, decays in less than 1 &The molecular beam profile of ND,, in.contras~~ &hibits a “stable” state with a lifetime loriger- than 20 us and :a. “dissociative” .state with a lifetime shorterthanq+ps[4]: : .-. ..,. : Further data on the stability of-ND, come from spectroscopic studies [5,6,8,9]. The, emission _+eci : trum of the r&Cal ~is&aracterized~by. two bana systti; the C spectrum eontainiug the _S?hust~. =-- band and Sch uster-likebands, and the .high-pre& sure D spectrum ~&n%ining~the !Schlilertram5tion [US]. -In-ckrast to the-D -bands which &bit a ~. well-resolved ~otatioid ~strudure was MelI %s : do&

0301~1&/85/SO3.30 6 I%sevier Skier& Publishers B-V_ : ~.-~ (North-Holland Physics Publishing Division)

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-_’ _

blet -splitting,.the C spectrum lacks any obvious doubkt structure and its broad I&widths are indicative of a quite unstabIe Io%%rstate. To understand these features on a fundamental Ievet, we have calculated the Born-Oppenheimer ekctronic enew curve, inchxdingekctron correIation, for the dissociation of a hydrogen atom (NH, - NH, + H)_ This cume was used subsequently as an effectivz one-dimensional potentia& in which a NH, *‘particIe” and a hydrp=en atom would move Inside the dissociation barrier, two vibrations for NH, and three for ND, and NT, were found as solutions of the the corresponding two-particle Schr&iinger equation These osciIIar.ions axrespend to local rather than normaI modes and are in exceIIent n umericaI agreement with the energy difference between the SchiiIer band and the c band of the SchXer D spectrum_ From this we propose that part of the notorious difficulty tl.4-14,16,17] to expIain the emission spectrum of the radical is a consequence of the unusuaIIy large an~harmonicity of *he 3s’A, ground state vibntions that couples them into IocaI modes rather than leaking 2hcm normaL This situation is usually found or& in highIy excited states [191_ Tunneling Lifetimes were determined in the same framework of a pseudodiatomic. using the semicIassicaI approach of LeRoy and co-workers [20-221 and Connor and Smith [23)_ Since the zero-point vibration inside the barrier is bound reIative to the electronic dissociation Iimit in aI1 three isotopes NH,, ND, and ma, it would seem that the molecules in their vibrationd and elec*mnic grour.d state are stabIe tou-ards decay by tunneiins This finding is difficult to reconcile with the assumption that the decay of the ammonium radical is mainIy due to tunneling_ Furthermore. when zero-point energies are added to the electronic energies of the radical and its dissociation products, the ND, and NT, rxiicaIs arc stabIe by 0.5 and 13 kcaI/moI relative to their respective loss of a deuterium or tritium atom. whiIe NH, is sIighzIy unstabIe to the loss of a hydrogen atom by 1-l kcaI/moL in Her&erg’s terminoIogg fl] NH, would be called a Rydberg molecule while ND, and NT, are not Two choices of zero-point energies were used in the discussion_ both of which

are quite similar showing a basic consistency in the arguments used for their derivation (see sections

3.3 and 35). Another resuh at odds with experiment concerns the nature of the Schuster band which our data suggest is a 3p2Tz-3s*Ai transition of vibrationally hot ammonia radicals. This would imply

considerably larger isotope shifts than suggested by the mixed-isotope bands found by Iierzberg and Hougen [S]_ The isotope shifts of the S&i&r band [lo] on the other hand, are in good agreement with our caIcuIated values_ Experimental studies of the NT, moIecuIe would be very desirable indeed fo heip resolve some of these difficulties. Since the first vibrational excitation of the ground state (3s’A,, R= 1) has a computed lifetime of the order of milliseconds the Schuster-like bands ought to be fairIy sharp and well resolved, provided our interpretation of those bands is correct_

2, Methods Our ab initio calcuiations for the ground-state potent&J curves [configuration ‘?A,(la: 2a: ltq 3a’,)] were made using the GAUSSIAN 80 set of programs [24]_ The SCF energies for the NH, molecule were obtained at the unrestricted Hartree-Fock level (UHF); an estimate of the correlation energy was obtained from second-order and third-order unrestricted Meller-Piesset perturbation theory (UMP2 and UMP3, see ref_ [25])_ The CI caIcuIations included al1 single and double excitation (SDCI) from the reference configuration; the variational energy was then corrcctcd for size-con@ency (SDCI (s.c.c_), see ref. [26J). The basis set that was used throughout this work was 6-311G**R [13] which is the standard 6-311G” basis [27] including polarization functions on nitrogen (d-exponent 0.913) and hydrogen (p-exponent 0.75) and augmented with a diffuse sp shell to represent the Rydberg 3a, orbitaI [13]. In our previous study 1131the best choice for this exponent -was found to be O-025 for both the s and p sheIIs_ The vibrational and rotationa bound and quasibound states inside. the dissociation barrier of

NH,+ NH, .f. H for thenvarious isotopic molecules were computed with LeRoy’s program -1201 for two.particIe~systems.An effective onedimensional potential V,(x) is conStructedby adding a centrifugal potential- J(J t i)h2/2bx’ to a fit to the computed electronic potentid points, and the radial !SchrWir&er equation for this effective potential is solved. Energies of the quasibound IeveIs are determined using the Airy-function boundary coklition method [21,22] and the tunneling predissociation level width is then calculated using a uniform semicIassicalapproximation iutU]-

3. Resultsand disc&ion. ._ .~ _-_ _ --

..- .:

3.1. Ground-state geometry

. .. _-

--

:. ~~~~~-~ ~. .._...-.; -. ’ T&k e&ihbrium geometries of NH,, NH:%d NH, we&determined at-the SDCI Ieve-using-a sphne fit to the potential points. The. results-are summarized in table 1. EIectron correlation causes the e&ilibrium distance id NH, to increase by 0.021 A relative to the unrestricted Hartree-Foch distances[13]: The re- : sdtant (N-H)-bond length of 1.041 A is 0.015 A huger than the bond in the cation NH:. Experi-

Table1 Equilibzium geometries and grotmd-state energies:Distances HF NH,

NH:

NH,

I.020 - 56.70683

R,P-W E

”

C,,)

1.041 - 56.93642

I.023

&W-H)

l_flol 107.9 - 562127

I.011 106.7 =’ - 56.41429

1.010 1065 =’ - 56.42524

1.010 106.6 =’ -56.41881

1.012 1065 - 56.42876

E

- 0.49981

E

-I_1325 - 55.5808

- 1.1662 - 55.7500

I.012 1052 - 555806 1.024 =’ 103.3 =’

E

- 55.7429 ”

- 55.7469 *t

- 55.7497

- 55.7565

- 55.6974

- 55.6904 *’

-55.6940

LHNH

E R,(N-H) &NH

- 55.4182 1230 180

- 555633

- 555578 *’

- 555629 *’

E

-55.3116

- 55.4664 J’

- 55.4726 d’

value for R,
d’

1.066 1047 for NH: in halide crystal 1.032-1.041 A [28]_ = 1.012 $ LHNH = 106J0 [ZS]. Expzrimental toral energy:

Cl T&se &a are duiwd from an angle variation with the SDCI(s.c.c.) optimized equilibrium 9 Gcomeuv optimization on UHF IeveL The Cl did not include single exitsdons. =’ EqJelinl;ntal geomcuy [3oj_

’

- 1.1683

- 55.5293 0.987 o 143.6 ~0

&iNH

*

- 1.1683

R,(N-I-0

R,WW

=’ Er~crimcntal b’ Experimcati

- 56.91922

1.026 - 56.77444

LHNH

NHJZ’A,.

SDCI(s.cc)

1.035 - 56.92430

- 56.76492

4

Da)

SDCI

1.038 - 56.93199 1.023 - 56.77081

E R,(N-H) LHNH E

NH,(laBI.

UMP3

1.025 - 56.75737

1.012

E

Cz,)

LJMP2

- 5655903

1’

LHNH

NH,(l’A,.

in & angles in degret. energies in hartme

E

R,
Expetimcnrai dismncc: 1.004 ri [31]_ Expaimenlal angk 1440 [31b

.;~_::~..-_ ;::,

- 56.5601 au.

diswxe

R,(N-H)

= 1.0116 A.

mental data are not available for either moIecule. but very recently Watson [S] found from his analysis of the rotational structure of the 3p’Tz - 3s ‘A, of the bond Iength of bandinND,anincrease O-0132 A due to this electronic transition. an indication that for the 3p’T2 Rydberg state the molecular core of NH, is almost f&e the free cationThis couch&on is also supPorted by previous calculations of the quantum defect (13.la The experimental situation for ammonia is more satisfactory_ The agreement of the N-H distance and the HNH angle with the experimental data 1291 is excellent_ Again, the Hartree-Fock bond lengths are too short by O-01 A and the angles HNH too huge 3.2

: ,. -. SorQc pmpatia of NH,
Table 2

ioniz&on potentials (IP); disociation

ena~y @I’ aqd ban& height E, for KH, - NH, f H’; dissohtion cneqy D2n for NH, 4 NH= i Hz: Rydbqg transitions (using CI meqjs of NH, and NH: and Ihe UHF via%ual orbital energy for 3p)

adiabatic IP (cv) VCnical IP (ev) D;”

(cl’)

0:”

(ev)

En (ev) E, (a-‘) at R(N-H’)

(A) 3&x,-3p’rz 3s ‘_A,-M ‘T= 3s%,-M’E

HF

sDcI(rcc)

4.Q22 4.0?s -0.155 -0.177 Oxi 6035 1373 I50 CL50 265

4.405 4.426 0214 0_317 0.64 5182 I-427 1-89 2x9 3.04

To determine the dissociation limit of NH, for hydrogen funneling, the energy of the hydrogen atom wm computed with the same 311G basis used for NH, and added to the ammonia energy_ In agreement with our previous calculations at the UMP3 Ievel[l3] and the results of Cardy et al 1151 using their SCF+ Ci (CIPSI) method (both of which were made at their respective optimized UHF geometries), the NH, radical becomes eIectrot&ally stableby 4.9 kcal/moi when electron correlation is included (table 2) On the UHF level it is unstable by = 35 kcal/mol[l3]_ The difference in correIation energy between NH, and NH, is = 0%O-4 eV (table 3) Since the NH: molecule is isoelectronic to NH, its correlation energy is almost the same as in ammonia and

(eS)

for mokuks

in their gameus:

optimized

with the SDCI(s_azQ

mccbd

-E

XH, NH,

NH: NH&‘&) _~H,...H’]b’ HZ

=’

212 287 3.47

the ionization potential (IP) of NH, is increased by O-4-4-41 eV From 4-02 eV at the UHF level [13]_ The smah difference in the equilibrium distances results in small differences between the electronic vertical and adiabatic IPs. The ionization potential is still some 0.3 eV too small when compared with Porter’s experimental estimate of 4.73 + 0.06 eV [4]. We attribute this difference to changes in the zero-point vibrational energy which will be discuss& in sections 3.3 and 3.5. Another important consequence of electron corre!ation concerns the relative stability of NH, versus NH, + Hz_ Whereas on the UHF level the dtsso&tion of NH, into these products is more exothermic than inka NH, -f- H, it becomes endothermic by 0.32 eV when electron correlation en-

-l-*3

esqies

5.07

=’ Rcf_ [17k

Total energies

Cocrdation

x0*’

LIMP2

UMP2

SDCI

SDcI(rcc_)

5812 5.495 5Aos 4275 5.90 0.757

6.160 S-793 s-774 4-673 6227 0917

5950 5.618 5614 4_601 6-019 O-974

6281 5.888

-

5.872 4_7a5 6.378 O-955

-. ‘.

/SponrOofiroropicf-to/lhr

ergy is takeuinto account. (Note, however, that already on the UHF level j13], the dissociation barrier is larger than foil the NH;-, NH, -I- H channet) _ 1 3.3. Zero-point energier (ZPgj _: Since the instability of the NH, radical is a consequence of vibrational motion only, consideration of the changes in total ZPES is very important for judgement about thermodynamic stabilitiesThe sum over all zero-point energies determined by Cardy et aL [15] from their UHF calculations of the various normal modes is 137 eV and therefore =O_lS eV larger than expected from the fragmentation energy analysis of the beam experiments of GeIlene et aL [4]_ Another indication that the ground-state vibrational frequencies may be considerably lower than the computed harmonic frequencies comes from the anaIysis of the SchiiIer emission_ Watson [8] proposes an assignment of a vibrational state with P= 1964 cm-’ to the breathing A, mode in ND,_ This may be compared to our calculated harmonic vibration of 2198 cm-’ (table 4) On the other hand the single-bond oscillation frequencies caIcuIated from the radial Schrodinger equations (eq. (3) in section 3.4) agree much better with experiment. For example, the first vibrationaI excitation n=O-n= 1 of the ND, radical is computed to be 1970 cm-’ (table 8 in section 3-4).

Table4 Force axstams

(mdyn/&

and harmonic frequencies (cm-‘)

UHF

SDCI

F&f_ [16]

Rcf_ [IS]

&(A,) NH,(3sA,) ND,(3sA,). NT..(3sA,)

6.801 3385 2394 1957

57311 3iO7 2198 17%

7-604 3476 2459 2010

6.7584 3374 2386

Ever,) NH,(3sA,) ND,<3sA,) NT,(3sA,)

-

55024 3187 2351 1997

7-420 3687 2720 2310

6.6060 3492 2578 -

L -’

f I*

*)

-1 in NH,.

f, is 654 mdyn/A

556.

7.47

O-057

0.046

6.64 O-038

and f,_ is O-Cm mdYn/~

[321-

anmwnhmlmdicnl

~:

--

.-.. .- ~.-..85i:

me gooij agreement ‘with Wat&& value [S];lS& __ well as the axiharmonicity of this-vibrati_o&ugge& : that tlie vibtitio_&.‘of the “amrr&iti~rrIi@I ‘a& St local rather -&an normaLmodes_ In pa.rticuIar_~the A, breathing mode -and the three &ymmetric stretching modes should be replaced by four local N-II bond oscihations. Note that the dissociative local mode ;S basically an asymmetrid stretch 310; tion with some admixture of the breathing modeThe local mode model. is generahy appropriate only for a description of high-energy states since it is usuaIly only for large-amplitude vibrations that anharmonic terms (and Off-diagonal c&plirigsbetween normal modes) become large. In the case of. NH,, however, we propose that the instability towards dissociation may produce enough anharmonicity to localize vibrational quanta into a single N-H oscihator. We have used two estimates for the zero-pomt energies of the ammonium radical isotopes_ The first oqe is based on the computed energiesof the stretching~ modes (table 8 in section 3.4) and -the frequencies of the bending modes which were derived from the experimental emission spectrum as outlined in section 3_5_ The second. estimate is based on the experimental data of Gelltne et aL [4] who showed from fragmentation energies and known changes iri deuterium- and tritium-substituted ammonia that isotopic substitution in NH, should produce changes in zero-point energy comparable to those found in CH,. These ZPEs have been estimated assuming a simple proportionality between the ZPEs of the ammonia series NHd I ND, I NT, and the methane series CH, I CD, t CT& These two estimates of the total zero-point energies are quantitatively very similar (table 5) To determin e dissociation energies, the ZPES of the ammonia molecule have been taken into account using the experimental data from table 72 in ref. [29]_ The NH, radic&is then ~found to be. unstable ‘by 12-1.8 kcaI/moI whereas the ND, and NT, radicals ‘are stable by O-3-0.6 and .1.2 kcal/mol, respectively (table 5). The ZPEs for the ammonium canon were obtained from the caIcuIated frequencies of Yamaguchi and Schaefer [32].-One set-was derived from their DZ CI- values, another from their .DZ- CI

-rabk5

cnagics.dissodcion

Zao-poht

cnagia

and adiabatic ioniza-

areineV

Kioaporcn*AUencrgits

come smaller and so do the ionization potentials (see tabIe 5)_

ZPE NH, -’ 0.89

SiD3 -’ O-f%

NT, 056

NH* LX5 a’

O-72 -

1.18”

ND, 085 b’ 0.86 =a

;t’H;

ND,’

139 dJ 133 =I

LO1 SJl 0.9% =’

D,(NH,

-

D,@iD.

0.05

O-02 O-01

- 0-0s

It is proposed that the potential barrier for the dissociation NH, + NH, -t- H’ is due to the MOor-state Rydbergization (in Mull&en’s terminology [36D of a hydrogen Is atomic orbital interacting with the highly antibonding mokcukr orbital o& from NH,_ The change in hybridization along the dissociation coordinate is demonstrated by the geometry relaxation accompanying the dissociation. The barrier can be understood as the result of an avoided crossing between the potential curves of the 3s state of NH, dissociating to an excited 3s state of NH, (plus H’) and the valence-shell repulsive state of NH, -I- H’ which rises rapidly in energy as R(N-H’) is decreaxd- This vaience-shell state does not seem to connect to an excited stationary state of NH, and we view it as the discrete component of a time-dependent resonant state corresponding to a dissociative recombination channel of NHf . For the UHF points the geometry relaxation is shown in table 6. Two regions corresponding to

=T &s6 0_82 -

o,W?,

ND3 f Dj

NH, f H)

-

XT3 +-I-)

om

b’ =’

tabIe 5. we obtain an ionization potentiai of 4-72-4-75 eV_ In the other isotop& the ZPE corrections be-

w

IWDa)

Tabk Gcoq

6

&ax&on for the s%Aatic bsis SN 6-3ltG*-R)_ (Cs.. qxnmc~_ ncion is -tial Hanra-Fock

frequencies for the caq vaIue cm-’ for the vibration was the smaller more experimental 3343 [34] For NH: _ The difference in ZPE for NH, and NH: is OS-O-24 cV and the adiabatic ionization potential is therefore 4564.65 eV; on the other hand, if the thermochemical relation 1351

IP(NH,)

is use&

= D(H-NH,)

+ IP(H) - PA(NH,)

= D(H-NH,)

t4.8

then,

eV

from the dissociation

(1) energies in

NH,_._H’ disochion The I~-& of qximi-

R(N-H’)

R,W-H)

(A)

(4

O-90 O-95 I.02 I-10 1.20 130 1.40 150 1.60 1.80 2x0 250

1.020 1.020

1 IO-0

108_94

io9.8

1_020 1.019 I.016 I.010 1.005 1.002 1_w2 1.001 I_001 1.001

lfB471 109.1 108-6 108.4 108-7 109_2 1095 1102 I to-7 111.1

109.14 109_471 1OM4 11033 11052 11023 lO9_74 109.44 108.73 10821 107_79

- 56-6923 - 56.7024 -56_7068 - 56_iO28 - 56-6915 - 56.6814 - 56-6795 - 56.6831 - 566880 - 56.6966 - 56.7025 - 56_7095

cc

1_001

111.0

107_9

-56.7125

LHNH’

rHNH

UHFenugy (ban=:

-

..

the two

crossing states can be identified where-the geometry of the NH, fragment changes very little as R(N-H’) is increased_ Inside the barrier the HNH angles are almost tetrahedral and the-N.-H bond length is 0.02 -A longer than in NH,; when the hydrogen atom H’ is dissociated beyond the barrier maximum at 1.4 A, the N-H bonds and HNH angles shrink to their values in ammonia_ Therefore bond lengths and angIes of the NH, fragment in the interior region were left at their values optimized by SDCI(s_c_c_) for the tetrahedral ground state of NH,_ For _the reaction coordinate R(N-H’) larger than l-4 + the geometry of the point with R(N-H’) = 1.8 A was chosen to represent the NH, part_ For this point (R(N-H’)= 1.8 A) bond lengths and HNH’ angles were optimized. keeping C,, symmetry_ The NH, fragment is then found to have N-H-bond lengths 1.0135 A and angles HNH’ = lll” (which corresponds to HNH = 107_9”). For the special point (transition state) with R(N-H’) = l-4 A the R(N-H) distance was optimized. Geometries and potentird energies are Listed in table 7. The dramatic Table 7 Total energies for the diiation NH,___ H’. from SDCI@C_C-) cakuhtiuns. A fit to these poinls ax used as the potential V(x) of an effecfiv~ diaromic (NHB)-H’ with disranccs x(H)

from thecenterof massin NH,

IO H’: x(D) and x(T) arc the and (Pii,)-T. Distances

corresponding d-ktancu for (NDs)-D arcinA.cnergicsinhaaUec r(D)

d-n

SDcI(rcc)

080 0.85 O-87 O-90 O-95 1.04 l-08 110 1.30 135 1.40 150

O-8616 09116 O-9316 0.9616 1.0116 1.1016 l-1416 11616 13616 -1.4116 l&O7 15645

0.9045 0.9545 O-9745 l_OOa t -0545 1.1445 I.1845 13045 1_4045 1.4545 -15030 l&w5

0.9361

- 56.867542

O-9861 l_f3O61 1.0361 1.0861 1.1?61 12161 13361 I.4361 1.4861 15341 1.6426

- 56398687 - 56.907782 - 56-918467 - 56.929915 - 56-936416 - 56-935564 - 56.927727 - 56917327 - 56914187 -56.912951 -56913543

1.60 1.80 LOO 220 240 260

1.6645 lA645 LO645 22645 2-4645 26645

l-7095 1.9095 2x095 23095 25095 27095

1.7426 l-9426 21426 23426 25426 27426

-

00

m

00

00

-

R(N-H’)

x(H)

56.915402 56-920183 56-923356. 56.925248 56-926516 56.927375 56928566

Can perha&> best- .. appreciated by comparing the di&ciation c ._ curves from the SCF calculations (frg .I) and influe&

of electron

correlation

be

SDCI(s.c_c.)

calculations (fig. 2) ‘. correlation diminishes the _potent& barrier from 6035 to .5182~cm-*, without ~&t&a-: bly shifting its position. More important than the. decrease of the barrier height, however, is. the change in the shape of the barrier itself. : It has been suggested by various authors that :the instability of NH, is essentially due fo a.tunneling process_ WiIIiams and Porter [3] es& mated the barrier for dissociation to be between’ 2904 and 3872 cm-‘, using a rate constant

Electron

k=l/t=Aexp(-E-,-JkT). This value is considerably smaller than the cornputed barrier, which is perhaps indicative that tunneIing processes are essential To study this proposition it was assumed that the dynamics of a dissociating NH, radical is that of a two-body collision between a hydrogen atom -and an ammonia moIecuIe interacting via the one-dimeni sional potential V(x)_ Note that the effective distance x in this two-body modeI is not R(N-H’) but rather the distance between H’ and the centre of mass in the NH, fragment_ The interaction is further modified by a centrifugal potential J( J + l)li’/2px’, where p is the reduced mass _

p = mH8mNH,/(mh’H,

+ mHa)-

(2)

Isotope effects are taken into account via the reduced mass and the changes in the position of the centre-of-mass. The vibrational (n = 0, 1, -. - ) and a rotational (J = 0, 1, _ __) bound and quasibound IeveIs are then found from the radial Schrodinger equation

(d’;dx’+(Zlr/h”)[E--J(J+l)h’~~x’l)~(x)=ol

I’(x) (3)

The results of this analysis for the SDCI(s.cc.) potentiai are summ arized in tables 8-12 The zero-point vibrational level is calculated to be bound in all three isotopic species. Only one quasibound vibrational level (n = 1) with a short lifetime of 2 ps is found in NH,, whereas two

0--

0.8

1.0

L2

1X

1.6

1.8

to

2.2

24

26

i y

00

*HZ

R (NH’)IA-I Fs I_ Potential ~LVIT for the disoh&m of NH, in:0 NH,tH. from UHF caIcuhtions_ The viirationai Ievcls with rotational q-m number J= 0 are shown The s+mboIs H, and D,. tith R. the xibrational quantumnume. refer to ihe dissxiation NH,_._H and ND,___D. nspazSwzly_ Ou rhe E&r. the ~dence of zunneIing tiferimes on the energy for states with I= 0 is ZNote that tk htdica& dkocikon energy for NH, into NH= i Hz iz nzot-efawxcable

qua&ound levels (n= 1 and n =2) appear in ND, and NT, inside the barrier (fig_ 2)_ It is tempting to assume that the lower one of those

TabIe 9 Lcxl energies of bound and fw two quasibound roratioual stares in the dkaciatiwt mode Reference energy is the dksociahonLmirTbeenergyuuicsarean-l /

WH,...H KDs...H hl,___H NHJ__.D ND>___D NT&D NH,___T ND,.__T WI-&T

1440 1434 1429 1054 1046 1040 888 878 8x?

2632 2623 2617 1985 1970 1959 1679 1660 - 1647

4E(n-qJ=O)-~isthedccrronEcencrlpy~ w E(n-I;J=O)--E(n-O. J-O& =J E(n==t. i-0)--E(n-‘l. / -0%

1791 1786 1782 .X613 1599 I589

_

0 I 2 3 4 5 6 7

8 9 10 11 12 I3 14

15

NH,_..H (n=O)

ND,...D (n=O)

NT&T (n=O)

-284 -256 -200 -116 -4 136 303

-677 -664 -636 -5% -541 -473 -391 -2%

-853 -844 - 827 -800 -764 -720 -666

-107 -65 70 218

-604 -533 -453 -365 -267 - 161

:. _ - 46 77

go9

. . . .

J=12- -.

40(}0

'E o

0.'I

I0003000 2000

~ -

"

uJ'

I'°'2 p-~

"°""~"~"°"-.-..__.~ Ho -

-1000

lO-~Opsec

1-3O.sec

2 msec

Tl

0

_

: . :-

psec

_

--

. -

NHa

.

-

~

_

.

- " ~ '::__ ~-_ _

--

2-200 sec

-::_/..-

. " :.... .

,,,NH 3 +H

_ .

_

DO

-

0.8

~0

~.2

~4

~.6

~.8

2.0.

2.2

2.4

2.6

RtNH 1) [,~] Fig_ 2- Potential curve for the dissociation of N H 4 into N H a + H. from SDCI(s.c.c.) calculations. The vibrational levels with rotational quantum number J ffi 0 are shown- The symbols H . . D. and T.. with n the vibrational quantum number, refer to the dissociations " N H 3 . . . H . N D a . . . D a n d NT3...T. rcspe~ively. O n the fight, the dependea~c~ of tunneling life times o n the energy for states with. rotational quantum numbers J = 0 and J = 12 is illustrated. T h e energy of the dissociation product N H 2 + H 2 is also indicated and is !¢:~ favourabl¢. ( D 1 i n fig. 2 ) r e p r e s e n t s t h e " ' s t a b l e " f o r m s e e n i n t h e N D 4 m o l e c u l a r b e a m [4] w h e r e a s t h e u p p e r o n e ( D e i n fig_ 2 ) w o u l d c o r r e s p o n d t o t h e " d i s Table 10 Collisional time-delay results for energy E. width F and tunneling predi.s~3ciation lifctin~ a- for a f ~ " quasibound levels of N H a . . . H . Energies are in cm - z relative to the elo~tronic energy minlmu.l~ Emm ~ --56.9364178 au E (an -l)

J 0

5

1859

6 7 14 15

2026 2221 4322 " 4723

19

6558

- 29 12479 0 - -- 4071 2

9 15

~ -

~ (s) 6×10

F (cm -l ) -3

3x10 -4 4x10 -s 1 × 10 -9

4×10 -m 9 × 1 0 -a2 3×10-14 2 X 1 0 - a" 2x10 -Iz

4148

2xlO-tZ

5206 -7027

2X10 -la 3 x IO w

-

8x10 -m 2x10 -s 1 xl0 -7 5xI0 -a 1x10 -2 0.6 181 2_8 3.0 3.3 22 177

sociative" form with a very short lifetime (0.2 ps). The zero-point vibrational state should, in this interpretation, not play any decisive role in theinstability of the molecule and the fragmentation energies derived from the edge-structure inneut r * l h , e d i o n - b e a m e x p e r i m e n t s ( t a b l e 3 i n reg. [4]) could not be interpreted as dissociation energies from true gr0und-state molecules z : -. .. T h e m a j o r d L ~ c r e p a n d y in t h i s p i c t u r e is t h a t t h e l i f e t i m e o f t h e Fn-st q u a s i b o u n d N D 4 s t a t e i s o f t h e o r d e r o f 0 . 0 7 l~S w h e r e a s e ~ a e t i m c n t a l l i f e t i m e s a r e considerably longer (table 13): Before concludlng that the experimental lifetimes are not- Vmneling times but characteristiC--for, a'.different process (collision for example) it is worthwhileto consider . o t h e r p o s s ~ i l i t i e s . : I f t h e N D 4 m o l e c u l e w a s v i b r a - '. tiona]ly cold (n =13) but rotationalIy very:hot.(J = 19), the effective potential Would be sufficiently m o d i f i e d b y c e n t r i f u g a l e n e r g i e s tO a l l o w t h e z e r o point vibrational state to dissociate in a few micro-

J. Jtbsjmra aL / Spmm

90

ofiwlqpicffonnz of the ommoskm rzu&cal

Table 11 ing

prca;aociariocl tirecialeT

for a few quyibouod

ND3__.D_ Ehergia arc in cm-’ minimum && -569364178 v R

I

IO II 12

13 I4

15

& (Cm-Q 1793 I942 2103 2278 2466 2667 2s80 3106 3344 3595 3358 4134 442l 5352

refativt BP

7 (s) IO6 10’ 27 I_S 02 2x10-’ 3x1o-3 6x10-’ 1 x10-*

:

0 1 2 3 9 15 19

3016 3029 3Gs 3094 3599 4560 %Z

2x10-”

22

6223 7098 9IOl

3x10-” ~xIO-‘~ 5x10-‘=

0.3 1s 9 lo9

4su2

2x10-‘3

25

4813 as7 4E72 6cm2

2x10-” 2x IO-‘= 2x lo-= 4xlo-‘a

25 31 0 1 2 3 14

7x lo-’ 7x10-s 6x IO-* 5X10-’ 5xlo-9 2x lo-‘*

14 15 16 22 26 32 37 39 59

4x10-‘* 5x10-” 2x lo-‘3 3x10-“ 3x10-” 2x10-‘0 2x 10-9 9x 10-9 5x10-9 2x10-3

7324 1322

3x10-5

of

r(cm-‘)

16 17 18 19 20 21 22 25 30 41

6X1O-6 2x 10-e 5x10-’ 2x10-* 2xlf)-‘” 6X10-”

kvds

to the ckamnic

l-able I2 Collkionai timedday results &nergy &. width r andtunneling pruikmiatiou Efetime 7 for a few qmsiid-kwls of m--___T_Encrgies~inan-‘reIati~~t0tbed&aicalefgy minimum &&-56.9364178 a”

8x1o-T 3x10-6 1x10-5 3x10-* 3x10-’ 83 7x10-5 8x10-’ 9x 10-s 1x10-* 1 x lo-’ 3x10-’

26 27 29 148

This is. however. only possible if the molecuIa are very far indeed from thermal equipartitioning between their degrees of freedom; by assuming the usual Etolczmann distribution the average rotational level at 300 K would be J = 3. Another possibility is that the form of the potential irrcide the barrier is wrong_ due to a neglect of some of the contributions of electron

1932 2073 3093 3937 5436 6887 7516 15006

5x 10’ 6x10’ 05 1xio-3 1 x10-b 1x10-= 1 x10-9 5x10-”

6x 1O-4 3x10-3 113

2517

2x 10-3

2x10-9

1X10’” 5x1o-9 5x10-b

0 1 2 9 15 19 22 25 30 44

2525 2542 2901 3538 4128 4655 5253 6400 lo)74

2x10-3 2x10-3 9x10-s 2x10-6 9x lo-’ 1x10-’ I x1o-9 5x10-” 5xlo-‘a

2xlo-p 3x10-9 6x10-’ 3x 10-d 5x10-5 4x 10-a 4x 10-3 0.1 101

0 1 2

4105 4113 4130

7x lo-” 7x10-” 7x10-”

7x10-2 7x10-= 8X10-2

3 9 14 22 28

4’154

6x IO-”

9x10-’

4472 4956

2x10-” 5x10-=

025 1.1

6117 7269

5x10-‘= 3x 10-I’

1:

decrease the barrier width and the predissociation time of D,. it is a major effect of eiectron correlation that

seconds_

correlation_ Various resonable modifications of the potentiaI in this region did not change the lifetimes appreciably; indeed. if it is assLIm& that the Izsgcst error is made at the barrier maximum any improvement in the barrier energy .ten& to

Table 13 Lifetime acimara Ref_

(ND,)

i5l

IO-‘0

for NH,

=-2x10-5 13.41 ‘stable“disso&ati~r” <15x [61

<8x10-’

[lOI

-n - x slate”

isoropes (ia s)

x0-7

> 3x10-*=

(NH,)

Melhod

-=lo-“m

emission line broadening ncutnlizcd ion-beam prcfik photoc~cal modulalion anission tine

broadening

I

--

.

.I

_.

: ..

-. -.

.~_ ..;. _..

,

:_-

--..__

_ ____,. _:->

.L- Kizrpor the.

poten&

&me

o&de

U

0L -/ Sp&&

of isoropiicforms of&

thenbarrier becomes

I4l ,,;& .~.

off&&-+ ~&&

I. -. &[q;

__;z;L:.

‘r_-~.~-~_-;. :: .,;;-;‘;i-_

'..; __ -;::...~1,_: __ .. . .:^ :_;_y_ : tie

._ &&&-&~.~.

-_ ..-.

at-l285o-&-t in’N~~~-~.~-~‘:~~~-~~i. stronger. dependence 1 a& O-1 :&&isofi of ttietingtimes -on.- the :&ci @o&ion of -the -cm_!- [lOI coksponditig to a: Iifetime of?a.few~‘i quasibound _ livei rektive to, the dissociation : picokconds_ _rP&e values should be comp~_ed :asymptote_ To ~detet+ae the sensitivity of cakuwith our computed -values of -infiite -lifetime~‘foi lated lifetimes on the shape of thebotential in thi$ ~the~n=O‘sta~eand70000psf~rthen_~l-stat~--’-. We mention at this point recent &~e&nentsby region;. the potential ‘energies were shifted uniformly by various amount& AE for distances Gehene and Porter [ll] who produced _..NH, I : molecular be&is with the~r&IicaI solvated by am: .. R(N-H)> 1.8 A. It is found (fig 3) that lifetimes monia moIec&s.‘ It seems t+& the hfetixke of. thein the window between 20 ps (log 4.7) Andy80 -ILs. species increases considerably by solvkion; Future (log 4.1) require shifts of 680-780 cm-’ or 0.08-0.1 theoretical studies of these duster effects should-. eV_ Such shifts could occur. due to changes in help to clarify-the validity of the dynam&aI treatzero-point energy of the ND, fragment during tunneling ment of the polyatomic system as presented here and the relation between thecreticaI and experi: That the higher vibrational states of the radicai ._. . mental results. are involved in the mokcular beam data can be inferred from the fohowing arguments. From the data in table 5 and tables lo-12 it follows that the 3.5 Rydberg transitions -maximum fragmentation energies (FE) for tunneiing from the n = 1 vibration in NH, should be Two band systems are prominent in the emis= 0.38 eV_ In ND, the FE for the n = 1 tunneling sion spectrum of the ammonium radicaLThe sowould be 022 and 0.44 eV for the n = 2 tunneling; called Schuster band is a broad continuous feature in NT, 0.15 eV for n = 1 and 0.35 eV for n = 2. at 2.19 eV in NH, and at 2.13 eV- in ND,; the ThevaIuesforn=linNH,andn=2inND,are SchliIer band, observed at l-87 eV in NH, and. 1.84 eV in ND,, is considerably better resolved in reasonable agreement with the FEs derived from fragmentation continua (table 1 in ref_ [4])_ [1,5]. Other bands of similar intensity and broadness as t.he~Schuster band found by SchiiIer et al_ Neverthdess recent experimen:aI data do sug[18] have not yet been identified. In a recent paper gest that the molectde as computed is too stable_ [lo], Herzberg published a detailed compiIatiou_ of AIberti et al [9] found a Lifetime of 30 ps for the the bands between18500 and 12700 cm”- attri!khiiIer transition into the vibrational n = 0 state buted to the ND, molecule. in ND,, in agreement with results by Gellene et al. It is suggested; on the basis of our calculated value of 1970 cm-’ for the local stretching vibration in ND,, that the bands between 12859 and -8 .. 12845 cm-’ correspond to 3p ---, 3s (0 i-1) transi-7 . . tions into the A, and T,- stretching modes. Gorresponding transitions should be found in NH, at -6 .12435 cm-‘, a value close to the 12515 cm-r band -. = reported by Schiiler et al. (181; Herzberg [lo] pro -s .-2Op.X posed that the ND, “band -at 13680 cm-’ and '"PIOT \ -4 ._ :BOpacc Schiiler’s NH, bandy at-i3486 cm’:? correspond to the vz(E) bending mode, with frequencies 1134: -3 -. cm-’ in ND, and 1581-cm-’ in-NH.,_ Further-.-2 ‘more, we assign tentatively the emissio~_at 13730 cm-! in ND, and at ~13602 .&n-l -in NH,. to the -400 *__ 0 -r 400. . 8m_C& 1 vJT*) beIding~ v+-ation, iyith frequ~ti* ,lfIS3~ --Y_ cm-’ in ND, and -1465 cm-* in NH,: Fig_3. Lifetimes for ND, vcxsus shift of ourcr-tier potential rdati\-et00esDc1potcadalcurvc The vibrationsin NTi~ have values 1647 &n-r very

fl&

This~causes

a much

(for the At and T, stretching modes) and csti-m mated values of 927 cm-’ for the pI(E) 944 cm-* for the Q(T) bending vibratiow respectkIy_ From thcsc f&&ncks and the data in table 8, the tourIzeropoint energiesare 1.182 eV for NH,, 0.861 eV for NH, and O-722 eV for NT,_- These values are very close to those derived from the experimentaIdata of GeIIene et al [4] (table 5 and section 33)_ We have caLdated the vibrational transitions for the ekctronic 3p’T, + 3s ‘A, and 3d’Tz + 3s%, transitionsassuming that the initial state (or upper state) vibrations are the anharmonic modes of NH: given by Yamaguchi and Scttaefer [33]_ For the pj(l) mode the sIightIy different (and more recent) experimentaI value 3343 cm-’ [34) for NH: was chosen_ This seems a reasonable choice since the experimentallyobserved change in bond Iength IS>] together with our geometry optimization t-es&s suggest the simikity of the NHf ground state and the 3p’Tz state of NH,_ The same assumption was made fcr the 3d’Tz state_ The final (or ground) state vibrations were assumed to be the IocaI N-H modes (data in table S)_ With these assumptions the changes in zeropoint energy between 3s and 3p states are 0.148 eV in NH,, 0.115 eV in ND4 and 0.102 eV in NT,_ The ekctronic tmnsition energy has been determined from the difference in the SDCI groundstate energies of NH, and NHf, pIus the virtual orbitaI energy of the 3p MO in NH: (table 2)_ By adding the zeropoint energy chauges one arrived at 2038 eV in NH, and 2005 eV in ND,_ Since the actual O-O transitions are at 1_867 and l-836 eV, respfxtively~ thecorrect ekctronic energy difference should be at = l-72 eV_ The same correction for the 3d-3s would give 272 eV_ By using the corrected ekctronic energy values, and considering only excitations of single Tz asymmetric stretch modes one gets the rentIts summarized in table 1A The data do not support any assignment to known lines invoking 3d-3s transitiOnS_ An attempt was therefore made to explain the experimentaI bands in terms of -3p-3s transitions between viirationaIIy excited states_ An example of such a fit is shown in table 15 where we used

Table14 Rydbcrg uansitioa~ in the ammonium mdicaI_ The viirational numbers for the upper and to& slilte~ M &o~ed qm by tx,, ind R/. mpa&Iy (see &on 36)

NH,

ND,

o-o

1.87

O-I a-2 i-0 l-l 1-2 2-o 2-1 2-2

154

1.83 159 IX I214 I_90

“U-Q 3p=T,-3r2A,

228 1-96 L70 238

1.6s z44 220 198

M=TZ-3S=A, O-O

28-I

283

O-1

254

259

I-O 1-x l-2

328 236

237 3.14 290 268

2-o 2-1

310 33s

3.44 320

O-2

Z-2

liquid ND,’ The SchKtIer should therefore 1.822 eV or the various 3p + 3s (0 --, 1) transitions ought 13053 cm-’ (pt and Do)and (Pi) and cm-’ The Schuster wouId appear where all vibrations singly excited. 16600 and cm-’ involve some O-O transitions cited stretch osciIIations decay of the molecule. These~bands are therefore Schuster-like. On the the emission 14930 cm-t is O-O with l-l only betweenshould therefore appear Schukr The band in NH, corresponding wouId appear around 2.32 eV (!khuIer’s -DA)While

assignments

unique nor

186TI

18932[DH]

I7235

17784

17689[AH J

16601

The

16170

16891

15967

16252

15624

15587[B,]

15361

15359

15392[BH j

I4990

14930[a,]

147%

15061

15061[ad

14812

14812[ao]

14652

mmsirions arc 3p’T,:

+(E)_ d?Ii)

:1dW

..

dA,)’

r:(4) -’

16534[B,,]

-

3s’A,z

[I

” Ref_ [18je <’ Ref_ (IO]: in brackets the classification of ref. [lS]_

+z.(E) dT2)

-

QCG)

particularlyconvincing, since they rest on doubtful assumptions, they do show that large isotope ef-Fec~ are to be expected ln the Schuster band; ‘indeed, they have to be much larger than 556 cm-’ so that the two Schuster emissions in NH, and ND, could not correspond to the same transitionIn his recent paper, -Hetzberg [IO] shows the main Scbiiler bands of the intermediate isotopes NH@. NH,D, and NHD, appearing between the ND, and NI-I, :Schiiler- bands. Thenmedian isotope shift is 62 cm-‘. Assuming theseshifts are dominated by the change in local stretchitigmode zero-point energy~a shift of 54 cm-’ is -expected

(the average change in ZPE for *he stretching vibration in the ionization NH,+NHf is 223 cm-‘, in ND,-+ ND:. 169 cm-‘)_ In a previous study, bands of the intermediate isotopes have been found between the Schuster bands of NH, and ND, [S].. The average isotope shift on the replacement of a single H by D is; however, only - 112 cm-‘Such a small shift cannot be accommodated if the Schuster band belongs -to’ a hot .. vibrational 3p-3s transition_The observed n&ted- ~isotope bands shouid~thereforebelong to xjjfferent 1. vibrational b,ands.and add$lonal transition5should -be. found. outside~_theNH&ND, gap;- Further: -~ : more:assuming that the-Schusterband is electio;i-~_

icaIIy hot 3p A+3s. the smaII shift seems to exclude the 3s’&, sta:e as fmaI state since the very I anharmonic ground-smte frequencies, tpaether -- with the presumabIy more harmonic upper-state viirations~-result in averagedshifts of some 200 cm-= _ This would pose another problem: from our scattered wzwe-l_SD-transition state calarlations [17J we concluded that al1 transitionsin the Schuster-band regime ought to involve the 3p’T7 state (with the possibie exception of 3d-3s). Furthermore the origin of the diffuser& of the Schuster band would be uncIear_ The nature of the Schuster band remains puzzIing but the results of this study do support an assignment to an eiectronic 3p --, 3s transition between vibrationaIIyhot molecuks. Further work is in progress that should ckrify the situation in the excited states [37J_

It is proposed that the instability of the NH, radicaI. predkociation lifetimes and IocaI N-H oscilhttiousarc cIoseIy related aspects of the radical NH, and its isotopic forms_ The central feature that connects these aspects is the local character of the vibrations_ In contrast to the CH, mokcuIe it is more app_topriateto think in terms of uncoupkd N-H bond oscillations than in terms of harmonic normaI modes_

_~clmowIedgeulent The “,enerou.sassistanceof ProfessorR_J_LeRoy in the analysis of the dkociation .potentiaI is greatryappreciated_We also wish to thank Professor DR_ SaIahub for his advice and critical comments_ as weII as Professors G_ Her&erg, RF_ Porter and J_K_G_Watson for enbghtenikg discussions and preprints_ We arr indebted to Dr_ J. Mrozek for his assistance in the initial stages of this work FikmciaI support of this msearch by the Natural Sciences and Engineering Research CounciI of Canada (NSERCC) is gratefuIIy acknowiedgeh We thank NSERCC-and Queen*s University for providing the funds to purchase the

Perk+Elmer 3251 computer on which m&t _of ~necesmry caIcuIations were mad& ~This research was facihtated by arr Ontario-Quebec echangegrant-L ..-~ Lhe

References [l] G- Hazbq.

Famday Discusions Ghan

Sot 71 (1981)

(21 z lUuBiken. J- Ghan Phys 1 (1933) 492 [31 B-W- Wilkins and RF- Porter, J_ Ghan Phys_ 73 (1980) 5598. [4] G-i- GdIcng DA. Clury and RF- Pomr, J_-Chcm Phyx 77 (1982) 347l. [S] G- Hazbeq and J-T- Houges J- MoL Spcctry_ 97 (1983) (61 ‘2 Whi~akcr_ B J_ SuBivau_G-C- Bjoxkhmd.H-R Wcndt and H-E Hunzikcr. J- Chau Phyz- 80 (1984) %l. I_r] J-K-G- Watson. J. ‘MoL Spccuy 103 (1984) 125. [S] J-KG. Warson. J_ MoL Spatry_ 107 (1984) 124. [P] F- Alberti. K-P. Hubcr and J_K_G_Warson, J_ Md Spcctry_ 107 (1984) 133_ [IO] G- Henbag. J- Asmphys. Astr_ 5 0984) 131_ 1111GJ_ Gdkne and RF- Porter. J_ l’h>x Chcm_ 88 0984) 6680. [I21 E Brodawik. J- Mrozck and V-H- S~~I!I Jr_ Chcm_ Phys_ 66 (1982) 417: Erratum 83 (1984) 490. 1131B.h‘- h¶cMamr. J. Mrozck and V_K Smith Jr-. Chum Fhyz- 73 (1982) 131_ 1141S Baynor aud D-R Hcrschu J_ Ph_yxGhan 86 0982) 3592 [iSI H_ Cardy. D- Lio&rd. A Dargdos and E Poquec. C&cm Phys_ 77 (1983) 287_ 116) S Havriliak and H-F_ King. J_Am_ Ghan Sot_ 105 (1983) [17] ;: Kaspar and V-H- Smidx Jr_ Chun Fhys 90 (1984) 47_ [X8] H- Schiikr. A- Ml&cl and AE Griin. 2 Nansfd lOa (1=55)

1.

[lP] OS Moncnwa. B-R Haxy and MA- Mohammadi. J. Chau- Phjx 75 (l%l) 4800. and rcfcraxcs tllcrein @)j RJ- LeBoy. Further lmpromi Compurcr Progxam for SolvimS.theRadialSdu&BquEquationforBoundand Quasiid (orbiting Rkmamx) Levels. Chemical. Physics Research Report CKZOR Univcssity of Waterloo (1984)_ (21) RL LcRoy akd ~RB_~Banswin. J- Ghan P&s_ 54 (19ll) 5114. m] RL LdCoy + W_-R Jiu. J. Chcm Phys- 69 (1978) Mti 123) J-N-L- Connor and AD- Smith. ,MoL Phyr 43 (l981) 397[t4] J-S_ Sink& RA Whiteside. R a-R Scqcr. DJ_ Del=rccs. H-B- S&k@. S. TopioL L-R Kahn and J+ Pa@& GAUSSIAN 80. QCPE No_ 406 (1980) [251 JA_Po&JSBinkkyandRSc&cr_lntanJ_Quantum CBan 10s 0976) lp6] JA Pople. R Zic&r and R- Kris+ab. Intm~J_ Quantum ckau_11s(1977)149_~ in

Nomad.

Prinaton. 1966).