Triatomics-in-molecules method applied to helium cluster cations

Transcription

1 - 1 - Triatomics-i-molecules method applied to helium cluster catios Reé KALUS Departmet of hysics, Uiversity of Ostrava, 30. duba 22, Ostrava 1 Ree.Kalus@osu.cz ABSTRACT The triatomics-i-molecules (TRIM) method for calculatig potetial eergy surfaces (ES) is discussed. The method is a modificatio of Elliso's diatomics-i-molecules (DIM) approach (F. O. ELLISON, F.O. J. Am. Chem. Soc. 1963, vol. 85, p. 3540) ad cosists i expadig the electroic hamiltoia of a polyatomic system ito terms correspodig to the three- ad mooatomic fragmets. Applicatio of the geeral theory to a special case of the e clusters is preseted. KEY WORDS diatomics-i-molecule; triatomics-i-molecules; helium; cluster catios

2 INTRODUCTION Sice publicatio of the origial paper by Elliso [1], the diatomics-i-molecules (DIM) method has bee successfully used i a series of calculatios of electroic ad geometrical structure of molecules ad clusters. I particular, it has proved very useful if applied to the sigly positively charged rare gas clusters, Rg [2]. Electroic structures, equilibrium geometries, photoabsorptio spectra, ad collisio dyamics of rare-gas catios have bee studied extesively usig the semiempirical DIM surfaces. The oly exceptios are the e clusters, for which the DIM approach fails completely [3]. This is maily due to o-egligible three-body effects, which are ot take ito accout withi the miimal DIM approach [4]. Eve for the heavier rare gases (eo - xeo), for which the miimal DIM approach is fairly accurate, the may-body cotributios to the iteractio eergy are to be cosidered i accurate calculatios. At least the leadig, iduced dipole - iduced dipole three-body iteractio term is to be take ito accout [5]. I this work a modificatio of the stadard DIM method [1] is proposed which explicitly icludes all the three-body iteractio terms ito molecular electroic hamiltoia. First, the geeral theory is developed i Sectio 2, followig by part a prior work by Wu [6], the the theory is applied to a special case of the e catios i Sectio TEORY Similarly to the Elliso's DIM approach [1] ad followig the more geeral theory by Wu [6], the preset method cosists i assigig each electro to a particular atom ad partitioig the electroic hamiltoia of a polyatomic system ito three- ad mooatomic terms. This is doe i the followig way. The total amiltoia operator Ĥ ca be writte as a sum of the mooatomic hamiltoias ad the remaiig two-atomic iteractio terms Q, (1) Q ˆ ˆ Vˆ

3 - 3 - where each atomic hamiltoia, ˆ, cosists of the total kietic eergy of the electros assiged to atom ad of the terms correspodig to the iteractios of these electros with ucleus ad betwee differet pairs of the electros belogig to atom, whereas the iteractio operators, V ˆQ, iclude the terms represetig the iteractios of the electros origially assiged to atom with ucleus Q ad vice versa. I particular, the three-atomic hamiltoia correspodig to the triatom ca be writte as follows ˆ ˆ ˆ ˆ Vˆ Vˆ Vˆ. (2) Q R Q QR R Summig over the idices, Q, ad R yields i this formula ˆ 3( 1)( 2) ˆ 3( 1) ˆ N N N VQ, (3) RQ Q where N deotes the total umber of atoms i the molecule. If Eq. (3) is solved for result iserted ito Eq. (1), we get VQ ad the ˆ 1 ˆ ( 3) ˆ N 3( 1). (4) N RQ Thus, the total amiltoia operator is writte as a sum of three- ad mooatomic operators. Eq. (4) is usually called the triatomics-i-molecule (TRIM) expasio of the overall hamiltoia Ĥ. Clearly, the above procedure ca be geeralized ad the total amiltoia operator ca be expressed i terms of moo- ad K-atomic (K<N) fragmets, ˆ 1 ( N K)! ˆ ( N K) ˆ K( K 1) ( N 2)! ( K 1). (5) 1... K K... 1 Expasio (5) is ot much practical, however, because the larger K-atomic fragmets are cosidered the more cosumig prelimiary calculatios are eeded to get all the ecessary K-

4 - 4 - atomic iputs. I additio, huge techical difficulties regardig appropriate aalytical represetatio of K-particle iteractio eergies will appear if K is larger tha 3. Let ow proceed to the evaluatio of the amiltoia matrix elemets i a particular electroic wave-fuctio basis. At this stage of developmet, the procedure origially proposed by Elliso [1] ca be followed immediately. Let ad deote the atrisymmetrized ad oatisymmetrized electroic wave fuctios, respectively, Aˆ, (6) where  is the total atisymmetrizatio operator. The, sice the atisymmetrizatio operator  commutes with the total hamiltoia Ĥ, it ca be writte for the correspodig amiltoia matrix ˆ A ˆˆ A ˆ ˆ. (7) m m m m As usually, symbol deotes itegratio over the positios of all the electros, * d, the asterisk idicatig the complex cougatio. The total amiltoia matrix ca further be expressed i terms of three- ad mooatomic fragmets (cf. Eq. 4) 1 ( N 3) 3( 1) m m m N RQ (8) with A ˆ ˆ ad A ˆ ˆ. If a decompositio of the m m m m atisymmetrizatio operator, similar to that employed by Elliso [1], is used, ˆ ˆ( ) ˆ( ) A A ˆ A A, (9)

5 - 5 - with A ˆ atisymmetrizig over all the electros belogig to the triatom, ˆ ( ) A over those belogig to the remaider of the molecule, ad ˆ ( ) A cotaiig all the remaiig atisymmetrizatios, the followig holds A ˆ ˆ Aˆ ˆ Aˆ Aˆ Aˆ ˆ, (10) ( ) ( ) ( ) ( ) for both ( ) ˆ ( ) A ad ˆ ( ) A commute with the triatomic hamiltoia ˆ. I Eq. (10) deote the atisymmetrized electroic wave fuctios of the fragmet ad the remaider of the molecule, respectively, correspodig to. rovided liear combiatio of the eigevectors of the triatomic hamiltoia ˆ, ad ca be writte as a, (11), with ˆ E, uiquely iverted, E beig the correspodig eigevalues, ad if Eq. (11) ca be, (12), the the followig holds E E, (13) ˆ,,, k k k ad accordigly ˆ( ) ˆ ( ) ˆ( ) ( ) A,, k E A k. (14) k

6 - 6 - Next, if the basis wave fuctios ( ) ( ) k m are chose properly so that, for each choice of idices k ad, Aˆ is also a basis wave fuctio, tha the followig holds ˆ ˆ A,, k E, (15) p p where the first sum rus over all the idices p for which Aˆ for the particular ( ) ( ) p k choice of k ad. Cosequetly, the correspodig matrix elemets, A ˆ ˆ, read m m as follows E. (16) m,, k m p p The matrix elemets of the atomic hamiltoias, m, ca be evaluated i a similar way. The expressios are ot give here explicitly, however, for they may be eglected if the dissociatio limit is chose properly. (See also Sec. 3.) 3. ALICATION TO TE ELIUM CLUSTER CATIONS With all the three-body iteractios icluded, we believe i accordace with other authors the miimal DIM basis [4] will work for helium. Thus, we choose the basis wave fuctios i the form of ormalized Slater determiats with the positive hole localized o oe of the helium atoms: 1 aaa a N a N, 2 aaa a N a N, N aaaa an, (17)

7 - 7 - where a K deotes a 1s orbital localized o atom K occupied by a electro havig the z-compoet of spi equal +1/2 ad the stripe correspods to the z-compoet of spi equal to -1/2. Sice the spi-sesitive iteractios are ot sigificat i helium, additioal wave fuctios 1 aaa a N a N etc. eed ot be cosidered. The groud state of a eutral triatom is approximated by a ormalized Slater-type wave fuctio aaaaaa Q Q R R. (18) For a particular ioic triatom, the followig basis wave fuctios are employed, 1 2 aaaaa Q Q R R, 3 aaaaa Q R R, (19) aaaaa Q Q R ioic character of the fragmet beig idicated by small letters used i the upper idices. If all of this is take ito accout, the total amiltoia matrix ca be writte, employig the geeral procedure described i Sec. 2, as a sum of triatomic cotributios * m 1 m, (20) 3( 1) N RQ where the triatomic matrices read as follows mm E, for m,, Q R, 3 m, k k, l Ek k 1, (21) * ere, zero of the total eergy of a e + cluster correspods to the dissociated state, e + + (-1) e, ad the atomic eergies may thus be eglected.

8 - 8 - with = 1, 2, 3 for m =, Q, R, respectively, ad l = 1, 2, 3 for =, Q, R, respectively, ad m 0 (22) elsewhere. Sice overlap effects are ot importat i, are eglected for K L. K L e clusters [7], the overlap itegrals, I Eq. (21), E deotes the groud state eergy of a eutral fragmet, ad E E E are eergies correspodig to the groud state ad the first two excited states of a ioic fragmet. The expasio coefficiets k, ad kl, have the same meaig as discussed i Sec. 2. Thus, the latter coefficiets are obtaied by solvig the characteristic equatio for the ioic fragmet 3 ˆ i Ek i k, 0, (23) 1 ˆ beig the amiltoia operator for the fragmet, whereas the k, coefficiets ca be obtaied from a completeess relatio 3 k,, l kl, (24) 1 kl is the Kroecker delta, which follows from Eqs. (11) ad (12). Agai, the overlap itegrals may be eglected i Eq. (23), i i. I cotrast to the stadard DIM method, the values of both ad expasio coefficiets will deped o the particular cofiguratio of the uclei.

9 CONCLUSIONS The TRIM method for calculatig potetial eergy surfaces has bee discussed i detail ad applied to helium cluster catios. For the method to be employed successfully i calculatios, accurate iput data o the iteractios i all the triatoms, ito which the polyatomic system is fragmeted, must be available. These triatomic iputs are to be supplied from highly accurate ab iitio calculatios. I the special case of helium clusters, the ecessary triatomic iput icludes primarily three potetial eergy surfaces for the groud state ad the first two excited electroic states i helium trimer catio, e 3. At preset, extesive ab iitio calculatios are i preparatio to obtai these surfaces [8]. 5. ACKNOWLEDGEMENTS Fiacial support from the Miistry of Educatio, Sports, ad Youth, Czech Republic, (grat IZO ) ad from the Grat Agecy of the Uiversity of Ostrava (grat No. 004/2000) is ackowledged. Further, the author expresses his thaks to D. rivňák, Uiversity of Ostrava, ad I. aidarová, J. eyrovský Istitute of hysical Chemistry, rague, for helpful discussios o this work.

10 REFERENCES [1] ELLISON, F.O. J. Am. Chem. Soc. 1963, vol. 85, p [2] KUNTZ,.J., VALLDORF J. Z. hys. D 1988, vol. 8, p. 195; NAUMKIN, F.Y., KNOWLES,.J., MURRELL, J.N.: Chem. hys. 1995, vol. 193, p. 27; IKEGAMI, T., KONDOW, T., IWATA, S. J. Chem. hys. 1993, vol. 98, p. 3038; DOLTSINIS, N.L., KNOWLES,.J. Mol. hys. 1998, vol. 94, p. 981; DOLTSINIS, N.L., KNOWLES,.J. Mol. hys. 1999, vol. 96, p. 749 [3] KNOWLES,.J., MURRELL, J.., ODGE, E.J. Mol. hys. 1995, vol. 85, p. 243 [4] KUNTZ,.J., VALLDORF J. Z. hys. D 1988, vol. 8, p. 195 [5] DOLTSINIS, N.L., KNOWLES,.J. Mol. hys. 1999, vol. 96, p. 749 [6] WU, A.A. Mol. hys. 1979, vol. 38, p. 843 [7] OVCINIKOV, M. et al. J. Chem. hys. 1998, vol. 108, p [8] AIDAROVÁ, I., J. eyrovský Istitute of hysical Chemistry, rague: private commuicatio