Alkali metal anions and trapped electrons formed by evaporating metal—ammonia solutions which contain cryptands

Volume

66, number

ALKALI

METAL ANIONS

METAL-AMMONIA

IS September

CHESIICAL PIIYSICS LETTERS

1

AND TRAPPED

SOLUTIONS

ELECTRONS

WHICH CONTAIN

1979

FORMED BY EVAPORATING

CRYPTANDS

Michael G. DaGUE, J. Steven LANDERS, Harlan L. LEWIS and James L. DYE Departmenr of Chemistr); .Wchipa,z Srate C’rriters+-.

Ensr Lansing. Jlichiqz~I -XSS2+f, (IS,1

Received 1 I June 1979

Opricel transmission spectra of thin sohent-free fims formed by evaporating ammonia from alkali metal-cryptand solutions in liquid ammonia shoxred absorption bands assigned to elkeli metal anions end trapped electrons_ Some ‘%lectride” films may be metallic since they have plasma absorptions nearly identical to those of concentrated metal-ammonia solutions.

We report here the observation by optical transspectroscopy of Na- and other alkali metal anions in thin solid films obtained by evaporation of metal-ammonia solutions which contain a cation completing agent (cryptand). By adjusting the metalcryptand ratio it is also possible to obtain films whose spectra indicate the presence of varying relative amounts of trapped eiectrons. Some solvent-free films which contain trapped electrons but not alkali metal anions are apparently metallic since they show the same type of plasma absorption as concentrated metal-ammonia solutions_ By condensing ammonia onto solvent-free films of metal only, it was also possible to obtain the first transmission spectra of concentrated metal-ammonia solutions (MAS). The spectroscopic results reported here are consistent with the formation of the cryptated cation and three types of anionic species as well as mistures of these types: (1) Alkali metal anions which show characteristic metal-dependent optical bands_ (2) Locally-trapped electrons which yield a broad infrared absorption band with a pronounced and rapid decrease in absorbance on the low-energy side. (3) Conduction electrons which exhibit a characteristic plasma absorption. This is the first report of fdms belonging to the third class and is particularly significant since it suggests the existence of “expanded metals” in which mission

the charge on the cryptated cation is balanced by conduction electrons_ The size of the cryptated cation is such that the cation-cation distance is probably greater than 8.5 A. Transmission spectra of non-metallic AlAS (
clo-(S.S.S)-

1,4,7,10,13.16-he~ro\acycloocta-

169

CIIEYICAL

PlfYSlCS

free (“dry”) films from metal-methylaminccryptand soiutions usuaIIy show bands of M- and/or trapped electrons (e;) when the corresponding species are present in solution. In the present study we used published techniques [171 to prepare dry films from ammonia solutions_ We had previously shown sxxdyt-’ iuIIy [ 171 that sampIes prepared from methylamine solutions

by sokent

cvaporzttion into a side-;lrm trap

liquid nitrogen temper;ltures were free of solvent_ Because of the higher vohtility of ammcnia and the fact that ammonia can be completely removed by evaporation from met&-ammonia solutions 2nd even from metal-amine compounds such as Li(NHj)4 and Ba(NH3)6 [I 1. the ‘-dry*- films described in this paper are undoubtediq- free of sokent_ SoIventcontaining films were prepared by esposing a dry film to ammonia vapor front a solvent reservoir \\hich was maintained at J lower temperature than the film_ The specat

tra of such tXns could be studied ;IS the films progressed from dry to liquid. the limit being reached when the liquid fiim drain& From the cel1 waIIs_ AlternativeIy. by maintaining the ditTe:ence in temperature between the solvent reservoir and the ceil at an appropriate vahte, intermediate t&e-independent spectra coufd be studied_ An ammonh sotution contrtining a sodium to sryptand mole ratio. R. of 2 gives, upon evaporation. tilnl spectrum

A in fig_ I _ Three features nearly iden-

tic31 to tbuse found

in Xa+CX!Z - Xa- Iilms from

15 September

LIXTERS

1979

methylamine

[ 171 are observed: a large peak at 15?00 cm-t (the Na- band), a high energy shoulder ok the h’a- peak, and a smdl distinct peak at 25000 cm-* _ The additional small peak at 7800 cm-t is simiIar to bands assigned to e; which were observed in films from methylamine [ 171 (but never when sodium was used)_ Because cotzcczzfz-ared b&S contain primarily soivated cations and conduction electrons, rather than solvated electrons or alkali metal anions, we were surprised to note the nearly complete formation of Xa- in films formed from ammonia soiutions. Evidentiy the excess free cations formed in the films when solvent is removed can act as efficient electron traps to form Xa- _ An ammonia-sodium-cryptand solution with R = I gives dry film spectrum B (fig_ I)_ The absorption of e, is now greater than that of Na-. The presence of some Sa’ even at this stoichiometry suggests that the cryptand complesation constant for Na+ in ammonia is small enough to permit some uncomplexed Nat and CT?31 to coexist in solution with Na*c22’, and e&_ The i\;a- and et bands are broadened relative to dry tihn spectra which contain only Na- or et bands. The general features of all the the dry-film spectra were obtained reproducibly by repeated film formation_ When dry fibs are converted to wet films by condensing ammonia, spectra such as spectrum C of fig_ 1 result for R = 1 cr 2_ The presence of enough ammonia in the film causes the Na- band to be replaced by that of e&_ Spectra of dry fihns prepared from metal-ammonia-cryptand solutions with R = 2 are shown in fig. 3for M, Rb, and Cs_ In ail cases. the lowest energy neariR band is assigned to et _ The K‘ bands are seen at I 1400,10200, spectively-

, Rb- , and Csand 9600 cm-t,

Tkese bands are broadened

re-

by the presence

of e; and, in the case of cesium, two additional absorption peaks occur similar to the shoulders observed previously for a Rb+C322 - Rb- film from methyIamine [ 171_ When these films are wet with ammonia the behavior is similar to that observed with sodium (spectrum C of fig_ I). The bl- band diminishes in intensity and the ec band grows as the film is made wet. In most cases the spectra of wet Glms appear to invohe the superposition of the absorption band of e&,, with J plasma absorption_ Sodium-ammonia solutions containing the bi-

15 Seprembrr 1979

CIILMIC_XL PHYSICSLETTCRS

Volume 66. number 1

plasma absorption

expected

for very concentrated of such an absorption by transmission spectroscopy. From the temperature difference between the film and the bulk solution, we estimate that the concentration of this film is IO-12 mole percent metal. The nature of “electride” films hI'C222 - e; _ is of parricular interest since such systems might exist as “expanded metals* [ 1 S,I9] _ Indeed, spectrum B in fig- j of a dry film prepared from an atnmoniapotassium-cryptand (2,2,2) solution with R = 1 is very similar to the metal-ammonia plasma absorption shown .IS spectrum A in the same figure and indicates possible metdlic character for this film. This view is strengthened by ESR and microwave conductivity measurements of pondered samples. These samples were prepared by evaporating all of the ammonia from potassium-ammonia-cryptand (3,Z.Z) solutions with R = I_ The X-band microwave measurements clearly showed that the samples were metallic. ESR studies of similar samples at 77 K and above did not show the broad ESR line of potassmm metal [20] (width = 13 G. g = 1.99) but rather a single narrow line (width =Z0.2 G) at the free electron g-value

hIA. and is the thirstobservation

Fkm-9

IOe3

fig_ 1. Tralumission spectra of thin solrcnt-free films from ammonia solutions xxhichcont&t cryptand (1,2,2) xtirh R = 2. A(---) potassium; B (-4

rrsium.

rubidium; C (-)

cyclic diamine DABCO (1 ,kiiazabicyclo (2,Z) OCtane, N(CH&H-,);N) to serve as a matrix give inhornogeneousdry films with no absorption bands. When only sodium is present, the dry “films“ consist of many small flecks of sodium which yield no absorption bands- When such a %lm” is wet with ammonia, spectrum Aof fig. 3 results. This spectrum sho\xs the

,

,

5

IO

(2.0023).

We estimate that the presence of 2=10% of

tile potassium as free metal could easrly have been detected with the instrument settings used. Since MAS alone or with DABCO present to provide a deposition matris give no absorption peaks when the ammonia is removed, it is unlikely that the observed properties of “eiectride” films are simply due to precipitated metal_ All of these observations susest that the residue which remains when a solution of KV222. e$, in ammonia is evaporated to dryness at low temprratures, is metallic and contains rryptated potassium cations and conduction electrons. The dry film spectrum obtained by evaporating ammonia from a solution which contained lithium and cryptand (X,1 .l) in a 2 : 1 mole ratio is shown as spectrum C in fig. 3_ The close resemblance to the spectrum of a tnetallic MAS (spectrum A) indicares that this film also has metallic chxacter. Not all “electride” films show the plasma absorbance- For example, the spectrum of a dry film from an ammonia-sodium-crytand (2,&Z) solution wirh

J$--+& 15 iT

20

2;

(cm-‘). lOA

rig. 3. Transmission spectra of thin fdms xihich bhow plasma absorptions. AI1 films wre prepared from ammonia solutions_ X (---I wet film of Na in NE13with no cryptnnd present; B i-1 dry tiIm. K;, R = I \\ithcryptand (1.2,2); C (-_) dr) film, Li, R = 2 1xit.hcryptand (2.1.1).

R = 1 (spectrum

B, fig_ 1) appears to result from locally trapped electrons_ In this case, the absorption on the low-energy side of the band of e; drops rapid171

V01ume 66. number 1

CLiEblICXL PfiYSICS LETTERS

ly to zero. indicating the absence of significant plasma absorption_ The abihty to prepare “alhalide” and “electride” salts from liquid ammonia may prove to be important in the synthesis of these compounds_ The present report is preliminary in nature and the composition and structure of these fihns need to be verified by other measurements. However, the technique of thin-film transmission spectroscopy ccrtainIy prowdes strong evidence for the existence of these new classes of materials_ We are grateful for support of this work by the US. bkttionsl Science Foundation under Grant No_ DA1R-77-22975 and for heIpful discussions with RI-R Yemen and R_ van Eck

References [ II J-C_ Thompson. UxCrons in Iiquid ~inmonia (Chrendon. Oxford. 1976) p_ 97_ [ 21 T-A_ Ucl-kun and KS_ Pitax. J_ Phyr Chem_ 65 (196 I) I527_ 131 J-C_ Thompson and K-T_ Croncn\wtt. .\d%xn Pfiyx 16 <1967) -%39_ 111 Xl-: .\Iudk, AppL Opt S ( 2969) 2fJS3_ [51 K-B_ Somox~o and J-C_ Thompson, Phys_ Rev_ AI t 1970) 376.

IS September 1979

161 W_E MueIler and 3X. Thompson, in: Metal-ammonia solutions. cd?; J..I_Lago_o\rski and NJ_ Sienko (Butterworths. London. 1970) p_ 293_ 171 J..% Vanderhoof, E-N’_ Lcmaster, W.H. M&night, J.C. Thompson and P_R Antoniewicq Phys. Rev_ A4 (1971) 4x_ [81 W.H. McKnisht and J.C. Thompson. J_ Phyn Chcm. 79 ( 1975) 29s-L (91 S_ &talon, S_ Golden and Xl_ Ottolmghhi.J_ Phys- Chem 73 (1969) 309% [IO] L Hurley, T-R Tuttle and S. Golden, inr Mxd-ammonia soIutions, cds. J-J. Lagowki and BIJ. Sienko (Butterworths. London. 1970) p_ 503 [ 111 J-L_ Dye, in: Electrons in fluids, eds. J_ Jortner and XR. Kestncr (Sprbger, Berlin. 1973) p_ 77. L121 X-T_ Lok, F-J_ Tehan and J-L. D> e. J. Phyr Chem. 76 ( 1971) 2975. [ I31 K-T_ LOB. Ph_D_dissertation, .\fichiganState University (1973) pp_ 165. I97_ [ l-l1 EJ. Tehan, B.L. Barnett and J.L. Dye, J. Am. them_ SW_ 96 (1974) 7203. [ I5 j J-L_ Dye. CA_ Andrew and SE_ Matthcrcs, J_ Phyr Chcm_ 79 (197.5) 306% [ 161 R.&. Harris and J_J_tiso\xski. J, Phyr Chem 82 (1978) 779. [ 171 J-L_ Dye. M-R. Yemen, XC_ DaGue and J.-M_ Lchn, K_ Chem Ph) s 68 ( 1978) 1665 I iSI N_ Mammzno. in: Mrtai-ammonia solutions, eds. J.J_ Lasox\rki and 3t.J. Sic&o (Buttenrorths, London. I9701 p- 369. [ 191 J-L_ Dye. in: Progress in mxrocyctic chemistry, VoI_ 1. rdr RX tzatt and J_J.Christensen (Wdey, Xcw Yorli. 1979) p. 75.

Volume 66, number I

EVIDENCE

FOR CLUSTER

IN FLUORITE

Recehed

CHEMICAL

STRUCTURE

22 February

CONTROL CRYSTALS

PHYSICS

LETTERS

OF THE DEFECT

i979

EQUILIBRIA

=

1979; in final form 1-I June 1979

The distribution of defect sites in CaFz : Er3+ has been measured b? site selection spectroscopy of annealing temperature. The results confirm a new model for the defect behwior of fluorites_

CaF, has provided a relatively simple lattice for a variety of studies on the behavior of defects and their equilibria in solids. Despite the excellent agreement that has e?risted between experiments and simple models for the defect equilibria [I-S], there has been recent evidence suggesting the equilibria are more complex than previously believed and that the equilibria are determined by defect clustering [6-l?]. In particular, it has been suggested that covalent interactions between the fluoride interstitial ions (Fi) present in large clusters ca=usesscavenging of free Ff [7,17] _ In this letter, we report the first direct evidence for the existence of this scavenging mechanism and we suggest that this mechanism is responsible for defect behavior observed by many previous workers. The traditional model for CaF2 doped Lvith trivalent cations assumes solid state equilibria exist between the intrinsic anti-Frenkel defects of interstitird fluoride (Ff) and fluoride vacancy (Vk) and between trivalent cation-Ff pairs and dissociated pairs [l--3] _ There have been a number of microscopic site distribution measurements which are not even qualitatively in agreement with the model [6,7 ,18--34]_ Several EPR measurements have shown that the ratio of associated pair concentration to cubic site (or dissociated pair) concentration decreases with increasing dopant

l

15 SepIember

_Ackno\~lcd~ement is made to the National Science roundation under grant number DMR77-07765 for support of this research.

and EPR as a function

and increases with increasing quenching temperature [(5,3-O-301 _ Both of these observations are in direct conflict with the predictions of the simple pairing model (6,20,21,~5,26,30]. To explain the behavior, Yaney et al. 191 proposed a “gettering” model where dimerized ion sites gettered Fi ions, forcing dissociation of the associated pairs to compensate for the depletion of unassociated Ff. Franklin proposed a similar model in which the gettering was accomplished by HF molecules incorporated into the lattice by the annealing process [6]_ His model does not explain the equivalent behavior of crystals annealed using a PbFz oxygen getter. Cheetham et al. [33-,331, using Bragg neutron scattering, found evidence for clustering of fluoride defects. The basic chtster was denoted the 2 I 2 : 3, cluster since it contained two sets of two Ff with different distortions and two Vk defects_ Evidence for trivalent cations being in loose association with the clusters was provided by the diffuse neutron scattering measurements of Steele et al_ 1341. Theoretical calculations showed that trivalent cations should be in close association with the clusters to stabilize them I?]_ Two problems arose from these data. First. if the cations are not in close association with the clusters, the clusters would possess net negative charges favoring dissociation. Second, the two Fi ions are closer to each other in the 2 : 2 : 2 model (2-O -4) than is allowable theoretically [X2,33] _Jn order to alleviate these problems Catlow [7j has postulated an additional coconcentration

173

V’ulumc 66. number I

CHEWCAL

talent attraction between the two F’, ions which produced an L_l ion whose negative charge is deloczrlized over the clusterDirect evidence for clustering of trivalent cations has been provided by optical spectroscopy_ Fenn et a:_ [SS]showed that the intensity of spectral lines that exhibit a non-linear intensity dependence on dopant concentration could be reduced or eliminated by quenching the crystal from 1200 K_ Tallant and \Yright (35,36j developed a technique to selectively excite spccitic sites using rr tunrrbie dye I;lser that permitted a thorough analysis of the CaF2 : Er5+ spectm_ All transitions were classifiiblc into tetmgonal and trigonal symmetries for the associated (ErCa- Fi)’ pair (A and B sites) and at least 16 sites associated with Ers+ clustering (denoted C, C’, D(I) and D(2))_ Evidence for this assigmnent was provided by several observatiors_ First. the cluster sites’ intensity rose more rapidly with concentration than did the single pair sites* intensity_ !3xond, upconversiozt processes were found in the cluster sites but not for the single pairs_ Third, comples energy transfer processes were observed for the clusters but not for the A and B sites [S71_ Fourth. substitution of other trivalen: lanthanidcs caused shifts in the transitions of clusters but not the A and B sites_ They also showed that the cluster lines that Fenn et al. [3Sl observed to disappear upon quenching corresponded to the D(l) and D(1) sites_ Further evidence for aggregation of cations has been provided by dielectric loss measurements [IO121, ionic thermocurrent experiments [ 13,14! and JlGsbauer studies of CaFz doped samples [ 15)_ Mustah et al_ [lb] made direct observation of aggregates using *9F hMR and proposed a dimerired pair identical to the 2 I 2 : 2 cluster in the unrel.txed cont~guration_ \Ye have studied the site distribution in CaF, : 0.1 mot % Er3* rrra series of tsmperntures using the optical techniques of Tallant et ai. [I?]_ In addition, the cubic site concentration was monitored relative to the tetr.rgonal site concentmtion in the srmie crystals using EPR_ The opticrd results are presented in the form of ;I series of absorption spectra for the 4115p + 4S5fz transitions [fis_ I)_ The spectral lines are de-

noted by the Tallmt-Wright convention [351 where A (tetragon.rl) and B (trigonal) are singfe pair sites, and C. c’, D(l) and D(1) are cluster sites_ All of the 174

15 September 1979

PHYSICS LETTERS

spectra are normalized to the same arbitrary absorption scale_ The tetragonal to cubic EPR peak area ratio as a function of tempenture is presented in fig. 2. The peak area ratio is proportional to the concentration ratio [39]_ The crystals were prepared by encapsulating them under vaccrrm and with PbF, present to getter oxygen. They were annealed for several days until equilibrium was reached and were then quenched quickly by directly immersing them in water_ The spectroscopic measurements were then made at 10 K_ In their pzper, Tallsnt et al. [ 1i’] presented a quasi-equilibrium model to explain their observations as well as the previously outlined behavior of the sites. The model postulated that the final site populations are determined at two different temperatures because of different mobilities of E& and c ions. The relative numbers of single pairs, dimers (C3) and trimen (C,) are established at a high temperature according to the foilowing equilibria:

3(Erca * Fir

ii;h C, _

As the temperature is Iowered the Er& mobility is frozen while the [Fi,l~ [V,l, [Er&J and [(Erca- Fir] change individually as F’r move about to establish the following low temperature equilibria: KIL

(Ert-., _ Fi)*

i- F; H “’

h;L + Fi -(3ErC-

(3Er~~ - 5Fi)”

- oft)’ _

The scavenging of interstitial fluoride by the clusten is represented by equilibria involving KzL, KsL and IiiL_ This model predicts that as the equilibration temperature of the crystal is raised above that temperature where Ffi mobility becomes appreciable but betow the temperature for appreciable Er& mobility_ the defects that have scavenged additional Ff (Le., defects like (2ErCs - 3Fi)‘, (3ErCa - 4Fi)‘, or (3Erca - 5Fi)“) will dissociate forming ciusters with

Volume 66. number 1

CHEMICAL

15 September

PHYSICS LETTERS

803 (K)

(KI

933

1

988

(K1

1083

(K)

1979

lI3800

f

:A

L 537

539 (nm 1

538 Wavelengfh

540 Wavelength

km I

I’is. I. 41*s]2-+ 4S3/2 absorption spectra of CaF2 : 0. I moI% Er quenched from indicated remprmrures. ized to the same srbitraty absorprion sxIe_ Sire xisignments of rhc lines are also indicated.

900

,

Temperature

I

1000

1

f

1100

tK1

Fig 2. The tetm~onti site to cubic site EPR pezk area ratio as a function of equilibration temperature_ The lme is for visual aid only.

At1 spectra \iere normal-

no net charge relative to the lattice and free F;. The free Ff can associate with free Er& causing the associated pair concentration to increase and the cubic site concentration to decrease. This change in dissociated to associated pairs should be directly correlated with the change in cluster sites with scavenged Fi to those without. _4t higher temperatures, the cluster species themselves will dissociate as the Er& ions become mobile. The predictions of this model are confirmed by the results presented in this letter_ The spectra of fig. 1 show that at the lowest temperature studied the C site has the Iargest absorption and the rest of the cluster sites are also present in Iqe amounts. Initially, as the temperature is raised the relative amounts of the total clusters and the tota single pairs appear to remain constant, but the reiative intensities of the individuai clusters undergo dramatic changes, the most obvious of which is the large decrease in C (and C’) site accompanied by a large increase in D(2) site. As the temperature continues to increase above 950 K all the cluster sites begin to dissociate and the 175

Volume 66. comber

L

CHEMICAL

PHYSICS LElTERS

intensities of the tetr3gonal and trigon sites increaseThis trend continues until the concentration of single pair sites xtches a mzximum ne;lr i I50 K where all cluster species (except ior some C site) are nearly eliminated_ This behavior is exactly that predicted by our modei_ Below 950 K the equilibria involving only < mobility domimte_ Since the C site (and c’ site) concentr3tion decrerrses white the D(2) site concentmtions increase 3s the tempemture is initially raised, they 3re related by the scavenging mecilanism_ The C and C’ sites 3re :hen the defects that form by scavenging an addition31 < and the D(2) sites 3re their precursor clusters_ This interpretation is aiso consistent with the EPR data_ _4t the lowest temperature the tetragonal to cubic mtio is small_ As the temperature is raised the r3tio increases, at first npidiy through the temper3turc reggon where the C site decreases and D sites incre;ise 3nd then more slowly_ This behavior indicates th3t free Fi, reksed by the C 3nd (C’) site dissochtion, associate with free Er& decnxsing the cubic site concentration while mising the single pair concentration- Above 950 I( the high tcmper3ture or ation equilibria become important and the clusters species begin to dissochte into single pair sitesThese resuhs shed new iight on the defect chemistry ofC3F2_The defect equi1ibri.t cannot be descrrbed by 3 simple model since the effectsof clustering processes must be considered_Theseeffectsrvili influence any me3surements m3dc in C3F2 by other workers_ It is anticiprrted th3t SrFz and BaF, will exhibit sin&x behavior_ Unfortunately, this means that C3Fz is not 3 simp!e model system for defect equilibria- On the other h3nd. these results suggest nxuty interesting studies of the clustering processes themselves_

References [ t 1 F_i;_t-sang_ Progr_ Solid State Chem, 3 ( 1966) t 35 [ 21 R-N’_ Lice.J_ Chem. PIQ 5_26 (1957) 1363. 131J_ Short and R Roy_ J_ Phyr Chem 67 (t963) t860_ [4j R.H. iteirc rrnd E_K_ I-‘ong_Ph>s_ Rev_ BI (1970) 1970. 151 V_V_ Osiko. Soviet P&s_ Solid State 7 (1965) liH7. [6l _A_D_Franklin. J. Chem Phys_6-t (1976) 1509. [7j C-R_& C3flow. J. Ph>s_ C9 (1976) 1859_ [Xl C_R_L C;lclow. J_ Phys_ C6 (I9731 26-l. 191 P-P_ YXILZ. D_Jl_Schaeffer snd J-I__ Wolf. Phyr Rev_ BI!
176

1.5 September

1979

[ 101 J. F~n:anella and C_ Andeen, J. Phys. C9 (1976) 1055. [ 111 A_ Edgar and H-z;- W&h, J_ Phyr CS (1975) L336. [ 121 J-H_ Chen and MS. Donough, Phyz. Rev_ 185 (1969) 453. [ 131 J-H_ Crawford Jr_ and GE_ Matthews, Semicond Insul 2 (1977) 213_ [ I4 1 R CapaIIetti, E_ Okuno, GE_ Matthews and J.H. Cm&ford Jr_, Phyr Stat Sol 47a (1978) 612 [ 151 C_ Barely. F_ Goozakz-Jimcnez, P_ Imbert and F_ Vmxet, J. Phys. Cbrm. Solids 36 (1975) 683. f 161 XR_ Blustaf;l, W_E_Jones. B-R. ,\IcGsrvey. .\I_ Greenblatt and EL Banks. J_ Chem Phys 62 (1975) 2700_ [ 171 D-R TaBant. DS_ Moore and J_C_ \Vrisht. J. Chem_ Phys. 67 (1977) 2897_ [ 18 J J_ Makovskycv,Phys_ Rev_ Letters I.5 (1965) 953_ 1191 J. Bfakovsky, Phys_ Letters 19 (1966) 647_

[ 201 E Seeemsky and W. Low, J. Chem. Phys. 64 (1976) 4240_ [?I 1 J.&l_ O’Hare. T-P_ Graham and G_T_ JohnstoR J. Chen Phys_ 64 (1976) 4249. [221 F_K_ Fang, J_ Chem Phys_ 64 (1976) 4243. 1’731 A_D_ Fmnldin. U_SN_T_LS_ AD Rept 748259 (1971). [2-l] J_M_ O’Hare. J_ Chem Ph,s_ 57 (1971) 3838_ [Xj G.K_ Mner.T_P. Graham and G.T. Johnston, J. Chem. Phys. 57 (1977) 1261 [ 26 j XD_ Franklin and S. 3la-zulIo. Proc_ Brit Ceram Sot. 19 (1971) 135. (371 XR Brolrn, IiG. Roots, J-M_ \\‘dILms. \!‘.A_ Shzmd. C_ Grater and H_F_ Kay. J_ Chem Phyr 50 (1969) 891_ 1181 Yu_E_ Voron’ko. V.V_ Osiko and LA. Shcherbakov, Soviet Phys_ JETP 29 ( 1969) 86_ t391 EZ. Cd’fanov. L.D_ Livanova, M_S_Orlov and AL_ Stolov. Soviet Phys. Solid State 11 (1970) I779_ 1301 J-11. O’IIxe, T.P. G&am and G.T. Johnston, J. Chem. Phyr 61 (1974) 1602 I311 A_D_ l’krddin, J_ Nonmetals I (1971) 17_ [Xl AX. Cheethsm, B.E5_ Eender, D. Steele. R-1. Taylor snd B_T.BL WiBis, Solid State Commun. S (1970) 171. 1331 AK. Cheecham, B_E_I_ Fender and M-J. Cooper. J. Phyr C4 (1971) 3107. 1341 D_ Steele, P-E. Childs and B-E-I=_Fender, J. Phys. C5 (1972) 2677. 1351 D.R. Tallant and J_C_ Wright, J. Chem_ Phyr 63 ( 1975) 2074. 1361 D-R. Ttiant and J-C_ Wright. Proceedings of the I1 th Rare Earth Raearch Conference (197-f)_ [371 D_R TaBant, M.P_ Miller and J.C. Wright, J. Chem. Phvs_ 65 (1976) 510_ 1381 I_&_ Fenn. J.C. Wright and F.K. Fang, J. Chem. Phys. 59 (1973) 5591, 1391 S-A_ Alt’shutrr and B-M-Kozynev, Electron pxamagnecic resonance, tramI_ C_P_ Poole (_4cadcmic Press* Ncrc York, 1964) pp- 14 ff-

CHEMICAL

Volume 66, number 1

15 September

PHYSICS LID-l-ERS

1979

ON THE USE OF PERTURBATION THEORY IN THE ASYMPTOTIC REGION OF CLASSICAL TRAJECTORIES

Reccivcd 8 .\la_r 1979; in tin.d form ?I June 1979 It is oftrcn nccc$sary to proPag.ne dnszical rrajcctoriex 10 Iniq disranccs. In this prrpcr 1%: mvcsriqnrc ZImethod in x\hicb e%wt intcgation I> cxrwd out in the close coupling region nhertxs tint order perturb&on tbcor> is used in tbr s+mptotic rc+n_ The lfI‘+ IIF system which is verb maotropic WASchosen for mvcstkatiox

I_ Introduction

Ho = (1 &t) (P_$ + P;. + P;)

in collisions between polar molecules, the long range multipole forces transfer energy even at large intermoiecular separations_ The integration of Hamilton’s equations of motion must, therefore, be carried out to distances. which are typically about 15-Z _k The integration is almost as timeconsuming in the asymptotic region as in the close coupling region when vibrating rotors are considered_ The reason is, that the vlbrationai motion has to be integrated accurately even at large separations. and the integration around the turning points of the vibrational motion requires small time steps. We shall in this paper investigate a method in which the asymptotic region is treated by classical perturbation theory. Thus the equations of motion are integrated esactly in the close coupling region for intermolecular separations less than R. and the solution is propagated to “infinity” by using perturbation theory. Test caIcuIations are carried out on one of the most anisotropic systems namely the I-IF + HF system.

The hamiltonian for the collision of two diatomic molecules is in Cartesian coordinates: -I-V(X, Y,Z,+yi,zi),

atom-atom

distance

in the two &oiecules

are denoted

by ri and the intramolecular potenriai is approximated by a Morse function: uxr(r,) = Di {I - esp [-p,(rj - F&l )‘,

(3)

where i’, is the equihbrium value of rj_ The Hamilton equations of motion

2_ Theory

H=Ho

Here p is the reduced mass for the relative motion of the two moiecules, 112~ (i = I,?) are the reduced masses of the oscillators. The relative motion is described in a coordinate system having its origin in the center of mass of molecule I_ The center of msss coordinates of molecule 3. are then
i= I,?,

where V is the intermolecular potential and:

(1)

b = (l/#&.

Ki = (l/??zj)phi.

(4a)

p

IjKi = -aHI&+

(4b)

= -aH/aQ,

are solved numerically in the “close coupling” region for R G R. , where R = (X2 f Yl + Z2)*i2 is the center of mass distance. For distances R > R. the equations of motion are solved using classical perturbation theory 177

CHEMICAL

Volume 66. number 1

from tile intermoI2cuIar potential fr is included to first order. In order to do this it is necessary to use actionangIe vzuiabks which change more siotdy in time tfian the cartesian coordinates and momenta- We use the action-angI2 variables for ttvo rigid rotor/Morse oscillators correspond@, to the hamiItonisn implies that the

which

HE = (1[2&@$

Rotation:

p+rturbation

dji/dt = 0,

Le.

ii(z) =I;-(+,) = jf,

(I2a)

fi@ = 0,

i.e.

mii = m?_, P

(I2b1

4_ii = ZWg jZ/;- = jifrnl<, 0t

+ P$>

+P;,

15 September 1979

PHYSICS LETTERS

(5) qiifr)

t F

[(I$ i- +dj

- U&(lI,

= q$ f i

+ g)’ +j;i 2q;1,

dr’

ji@*)jmr.?

_

(13e)

ro

Vibration:

where&is the rotational angulrtr momentum

and PQ the cbrssica1 action (in units of fi) for the vibrational motion of the Morse oscillator_ We furthermore have that fIlr Di = fiW&$

and

fli = (Du~~s”~~/fi)Ij~.

The a~gte corresponding to we have the follovkrg rekrtion ri=FEtf3;*

In&“[I

-(I

-+‘”

(6)

fi is denoted (see

e.g.

by Qi md ref. [2]):

cost&

0)

where $=

I --~2&;?-+‘)/U~

Iii = 0,

i-e_

It&r) = I#,

(133)

Gi = an,ofarri = cGi - 3_WzXJn~ f -“z)= +,

or c+(r) = qf f ;3irf - to)_

(I 3b)

The internal vibration-rotation coupling is now taken approximately into account in the rotational motion by replacing the equilibrium length Ff in eq. (12~) with the time-dependent oscihator length ri, i.e. we use

(8) 04)

~~ = 2Qn~,D~>‘~jfz$~_

(9)

These rest&s can be obtained using the geneurtiug function F-&r) where 0 = aF#rr and pr = iiF+

First order perturbation

theory then gives:

Pj-

(15ri)

The action-angIe

variabks for the rotationaI mcstion arei;, ?J$ (the projection off;- [4,5]) and the corresponding angks qij and &h _ In terms of these variables we obtain the foliowing equations

of motion (under_@)_

ti = n; - J

Tmnshtionr

dt a(v -I-j;/2nz~Jf)~a~i.

(I 5b)

t0

2 = W~~~~’

kX =O,

(IO)

and corresponding equations for the Y zmd 2 compo- . nents. Thus w2 g2t 0

w

(1 lb)

014

The last term in eq_ (ISb) accounts approsimatefy for the effect of the vibmtion-rotation coupling on the viimtional motion through the dependence of i-i on &. This approach is somewhat sirnpier but of course less accurate than the zpproximde solution in the BCtion-angle variables of the rotating Morse oscillator given by Porter et sl_ [6] _ An explicit comparison of the tivo methods has not been attempted in the present p3per_ Eqs_ (I 5a) and (15b) are written for the outgoing

15 September 1979

CHEUICAL PHYSICS LIXTERS

Volume 66, number 1

branch of the trajectory_ For the incoming branch the integration limits should be replaced by 0 and ti, where R(ti) = R,The intermolecular potential is normally expressed in a rotated coordinate system having its Z-asis slorig the intermolecular R-axis. Thus we have

using eqs_ (20) and that from eqs. (19): a(x/r)/arli =A,

cos g(t) + B, sin g(t),

(2h)

= A,. cos a,(t)+ By sin a,(f)_

@lb)

a(x/r)lagi = A= cos a,(r) f B= sti a,(f j.

(2lcj

a@/r)/aqj

_r[ = Ti sin -yi cos ai_

(16a)

where

-‘;- = I-’ sin yi cos rsl_,

(16b)

A, = -cos 59sin p,,,0 cos ql? - cos (&? sin qI?_ (2%) I J

zi = Ti cos r:_,

(16c)

B, = cos $! sin &I~ sin r$? - cos &? cos cl? _ J I

where Cyi_ 0:) are the polar angles of ti in the rotated coordinate system_ The derivatives in eqs. (15) are then given as a vlaqii

= (a v,,aq acijaqjf + (a v/asci> asqa+

+ (a vjasq

assijarli,

(17)

and

a v/h, = (a v/arj) ai-+a+

08)

where Ci = cos yi_SCi = sin ri cos 0; and SSi = Sinyi sin 6)_ In order to express the derivatives aCi/acli, etc. in terms of the quantities given by eqs. (1 I)-( 13) we use that [4.5] I i(l)

=x/I. = -cos

$2)

5 sin qj sin fl,, j + cos ffj cos fi,>,j,

(1W

= .l’/r = cos $ sin qi cos cl,, j + cos
F(3) = z/r = -sin

$ sin qj,

(22d)

A_ = -sin

(32e)

L)z = sin i/psin r~!!

up)

-I- [ YZ/R(X’

-I- Y’)]

sin y sin 0 = (X_v -

=y/r

Y-x)/(X’

The derivatives in eq. (l?)

-

[(X’

+ Y’)“‘r_

= q,(t) - $ _

(22:)

Eqs. (7). (I?)-(15) and (17)-122) determine n,(w) and iic-) from the intermolecular potential r/(R, ri_ Ci, SC’,. SSJ provided the values of the action-angle variables are known at t = ti .o_ Assuming that the integration for ti < t < to has been carried out in cartesian coordinates we need to express all the action-angle variables in terms of the Cartesian coordinates and momenta in order to “close” the equations. We shall again omit the index i from the equations_ Following refs. [4_5] we have

(IPc)

pL(2)= cos $ cosqj co~&,~~ - sin qj sin&,i_

+ Y’)‘p-]~/r

(Bf)

and

$. cos qj sin &zi - sin qj cos flI,zi_

- cosq sin 5_

P1@=

(33a) (3Sb)

(2%)

and j=

sin y cos 0’ =.[XZ/R(X”

$! cos qT_

JI~( 1) = -cos

(2Oa)

(23)

B_,.= -cos 5’ cos L?,,~? sin q; - sin fl,,,,,? cos qI!_ I J

(19b)

where cos $ = mn,!jand where the index i has been omitted. We furthermore have [5] cos y = (Xx + YIP t Zz)/Rr,

= cos 5' coS~,lIycosqJ9 -sin fi,,,? sin qIc_ I I

3

(22b)

G_z

+
+-jz)-)'!',

where t Y2)1/‘,:$; (2Oc)

can now be calculated

j,

= yp_ - zpp,= -j sin 5 sin flIni,

(24a)

jy =zpx -_rp,=jsin I;.co~&,~.

(24b)

i_ =xp

(24c)

Y

- ."p., =jcos $_=n;-_

179

Volume

66. number

CHEMIC_4L

1

PHYSICS

We note that i-p1 = 0 and that P_v =&r,;(I)

+jPL(I)IC

(%r)

P>_=P, $3

*iFpk

(2Sb)

[7,8] end the potential used in the present classical mechanical calculations Is that given in ref. 161. In order to amdyze the trajectories we define the rotational energy of an I-IF moIecuIe by [9] I E$

(25c)

F= = Pf ff3) +I@#).!~-

I5 September 1979

LElTERS

= B&ii

-I- 1) - C”&

-I- I)‘, ,

(28)

where

The initk-d angks at f = rjso can now be obtained as:

sinjP=*[I I

(26~~)

from (24c),

cos io = j_Jj? I

-(j[j)Z]@ 2

(Xb)

,

sin f$ = -z/r sin iP , ,

(26c)

from (Igc),

<29) cos r$o = (zpr - rp=)fj sin sCv

from (24b).

cos f.Izj- = j,-/i sin
= -jJj

sin $?

from (24a)_

(3-W

-v = ‘[=i,

sin Q,{f)

-

&

COS Q&f)]

~

(27a)

_I-= ‘@,

Sin Q#f)

-

B,.

COS Q,-(r)]

~

(27b)

-’ = ‘[-‘f,

SiE Q,&f,

P,( I ) = -4,

f?=

COS Q,.I)]

L?)S Qj(f)

+

B,

P.+(2) = A, COSQ#t)

+

B_,.sin

P+(S) = .+

+

Bz

COS Q,.f)

Sin

~

(27c)

Q,(l),

(27d)

ui
(27e)

Sin Q&t)_

+ ~&J

(26f)

Since the rmgIes in eqs_ (26c)-(269 are expressed in terms ofsin 59 a sign change in eq_ (26b) will not alter the results_ This cnn be seen from eqs_ (22) where the angks ahvqrs appear as products of two (the sign of cos $) is uniquely determined by eq_ (26a)). FinttlIy we shouId give the equations for x,_u and z at any time t~>r>to_Usingeqs_(19)and(22)wegetr

-

GO)

(26d)

_from (23~) i- Q~c)_

O_ Especirdly the eschsnge of rotational energy can take p&e at huge separations- This is shown in tabble1, where the rotational energy defined by eq. (23) is shown es a function ofR_ If we use bos-quantimion and stop the trajectory at 8 A this trajectory wouId be counted as belonging to the (jt j2 -T&) = (00 + 31) transition whereas integration to I7 _A gives contribution to the 00 d 22 transition_ Note also that the rotational ener-

Table 1 The rotz&naI

energy of txr-0eollidiq

HF moIecuIes s P func-

tion of centrr of mass distsncc The initial state was nt = 4. 112= OJ, =;z = O_The initid kinetic energy 1000 cm-* R (A)

Exact integation

Perturbation theory from 8-17 A

(270

resu1t.s

AS mentioned in section 1 the system chosen for investigrrtion WAS the HF f HF system_ We have previousIV CXried Out SzmiCI~cai Cdcuhtions on this system

180

(31)

This scheme cm be soIved iteratively starting with ui =

got

3. Xumericd

- q&-

tevv)

<‘,t

C=v)

E:,,

(eJ0

f&

feW

8.13

0.0750-1

0.00938

0.07503

10-11 1212

0.02295 0_01952

0.01199 0.01492

0.01279

0.01184

O-01951

0.01471

15.08

0.01758

0.01755

0.01764

0.01735

17.00

0.01775

0.017I7

0.01765

0.01712

0.00938

Volume

66. number

I

CHEMICAL

PHYSICS

gy of the two molecules is very well predicted using tile perturbation method from 8 to 17 A. A classical calculation of the rate constants for the detailed vib-rot transitions 1ztit1z~j2 * ~~j’t~~~~~has been published [IO] _ In this study [lo] the initial relative separation was 8 a (obtained from fig_ 8 of ref_ [IO] )_ Our results in table I show that this separation could be too small since we get significant energy transfer also at larger distances_ The maximum impact parameter in ref_ [lo] was reported to be 25 A, which may also be somewhat smali especially for rotational transitions_ For rigid rotors we have found large contributions from impact parameters between 2 and 6 a especially at low energies_ We use CIdifferent short range potential but the same long range potential as ref. [ lOI_ We have in tables 2 and 3 compared the results obtained for energy transfer AE=E

rotlrib(final - Erotllib(initial)

llsing exact integration and exact integation f perturbation theory at two different kinetic energies. The vibrational ener,v transfer is divided into two groups(+) where E,,&inal) >&,(initial) and (-)-where E,ib(final) < E,.n,(initial)_ These quantities AEtTh(i) are determined to within lo--30% and the change in rotational energy AE$ to within 5%. using perturbation theory in the asymptotic region_ The reason for this difference in acTable 1 Rotational and vibratiorzxl energy tratsfer for two colliding HF moIecules_ The initial state 1% 3s 11t = 4. )zz = O,it =& = 0. Initial kinetic enc~~- 1000 cm-’ _ The same 10 trajectories were intemted exact11 md by usiw perturbation theory. The impact pztmmcter wzs randomly chosen between 0 and 6 X. Encruies n are in

eV

_-Esxt

Perturbation S-17X

P”tibft)

0.0047 -0.01-11

&-$-)

0.0044

JJCO,

0.0183

Sk,(+)

-0.OOG9

&&-)

0.0014 -0.0157 0.0035 0.0155 -0.0067

theory from 10-17X o.oo-%s -0.0157 0 0042 O.OlSl -0

0057

-IE:ot lEw eq. (32)

0.0046

o_oo-l5

0.0047

0.6016

6_6022

0.0613

1E,,

a& (33)

0.0010

0.0016

0 0032

AE,

(average)

0.007-0

0.0019

0.0018

LETTERS

15 September

1979

Table 3 The initial states 3s given in table 2. The kinetic energy is 1000 cm-’

and 7-O trajcctorics randomly

cters betaecn

selected v.ith impact par_-

0 and 6 A xierc integyataL C\act

Encr~ies are in cV

Perturbation s-17-x

theory from 10-17x

--

=&,(+) Ah-&&-)

6 619

0.615

--0.016

-0.016 0.009s

0.0097

JE:ot lgib(+) .a-$+-)

0.011

0.024

- -0-019

-0.016 0.0052

0 0056

cot

0.0016

-0.0005

0.019 -0.015

_

0.0097 0.01-l -0.017 0.0054 0.0016

lf&

cq_ (31)

Al&

cq_ (33)

0.00-?.5

0.0034

0.0045

A&,,,

Cwcragr)

0.0046

0.0015

0.003 1

curacy is that the potential energy of the vibrating molecule (especially in an excited vibrational state) is very sensitive to the accuracy with which I-~is determined from eqs. (7) and (13b). Thus if we are interested in quantities which arise as a sum of positive and negative quantities we have to be more careful. As an example we ha\ e calculated the quantity LUY~~ which is defined as an average over those trajectories where E ,=,,(tinai) ’ Ezb(initial). Le. the vibrational excited rnoIecule (1) is deexited and molecule (2) klbrational excited. Thus AE_, =

A$,

(a)

-I-

AE\&(--+),

(32)

where the argument indicates the traject xies which should be summed over (as mentioned at ove). i.e. ~~ib(--+) should not be confused with the AI&,(--) Or glib qUantitieS given in tables 2 and 3. The quantity Mvy can also be calculated using energy conservation from

uv, = -qot

- AEfot -

AEkin_

(33)

where 4ELin is the change in kinetic energ>‘_ By evaluating this quantity in these two ways we can get an estimate of the accuracy with which it is determined_

4. Final remarks (1) The quantities c&ulated

in tables 3 and 3 are of 181

Volume 66. number 1

CIIEMICAL

PHYSICS LETTERS

course not converged_ Here we have only been interested in evaluating the use of perturbation theory - this is best done by integrating a small number of trajectories with and without use of perturbation theory in order to avoid fortuitous cancellation of errors_-i which arise as -i and Akmt (2) The quantrttes J..Evib “first” differences between Iarge numbers (the energies themselves) are determined better than 20% by using perturbation theory as.1mptotically(3 j Quantities as M, y which arise as “second” differences between Iarge numbers show larger errors(4) It appears to be wfe to use perturbation theory from IQ to 17 A (or kzger) even for the very anisotropic HF i HF system_ In this way the amount of CPU time/ trajectory can be reduced with a factor of two(5) The numbers presented here were determined using perturbation theory on the outgoing branch but not on the incoming branch of the cIassica1 trajectories. As mentioned above, the phases and momenta are slightly different when the perturbatioml treatment is used. It has previously been noticed [2,S,I 11 that, in some systems, a slight change in the initial conditions causes a large change in the outcome of the colhsion. i-e_ a reactive trajectory becomes nonreactive or the fmat vibrational transition probability changes significantly_ Such features are found especially in systems, where collision complexes may be formed_ Due to the strong anisotropy and the attractive well in the HF + HF system colhsion complexes are formed occasionally in low energy trajectories- It is therefore not possible to estimate the accuracy of the perturbational treatment by comparing individual trajectories if these are cornpIes and if the perturbational method is used on the incoming branch_ The individual trajectories may simpIy be very different_ Thus the perturbation theory can be used on the in-

15 September

1979

coming branch (1) if all trajectories are direct and (2) if multipie collisions are possible only if the phase space from which the trajectories are sampled is represented adequately by the perturbed trajectories_ The latter can be tested by comparing converged average quantities obtained using trajectories with and without perturbation theory_

Acknowledgement The Danish Natura! Science Council is acknowledged for granting computer time for this research.

References [ 11 C. tlerzberg. Spectra of dhtumic molecules(Van Nostrand. Princeton. 1950). 111CC. Rankin and \V.fI_ MiIIcr, J. Chcm_ Ph>s. 55 (1971) 3150. 131 IL Gofdstein, Ck.sicxI 3lechanics IAddison-Wesky. New York_ 1950)_ [-I] A-0. Cohen and E.A_ Marcus. J. Chem. Phyr 49 (1966) 45@9:52 (1970) 3140. (51 C-D. Billing. Chem. Phys_ 5 (1974) 241; 20 (1977) 35. 161 R-N_ Porter. LX. Raff and W_Il. !UdIer, J. Chem_ Phys. 63 (1975) 2214. 171 L-L- Pouiscn. C-D_ Bdling and J-1. Strmkld. J. Chem. Ph>r 68 (197% 5121_ [Sl C-D. Billinz and L-L_ Poulscn, J_ Chem. Phys. 65 (1978) 5128_ [91 J-T. JIuckertn;ln. J. Chem. Phys. 54 (1971) 1155; RN. Porter and IAl_ Rdf, mr Dynamics of molecular collisions. part 8. cd. WA. Mller (Pknum Press. Nen York. 1976) p_ I_ I IO) R-L- Wlhins, I- Chem. Rays_ 67 (1977) 5838. 1I 11 I’- Brumer and al- KnrpIus. Faraday Discussions $5 (1973) so.