CROATICA CHEMICA ACTA CCACAA 73 (4) 1087¿1098 (2000) ISSN-0011-1643 CCA-2704 Original Scientific Paper Variational Study of Fermionic Helium Dimer and Trimer in Two Dimensions # Leandra Vranje{ and Sre}ko Kili}* Faculty of Natural Science, University of Split, 21000 Split, Croatia Received March 15, 2000; revised June 17, 2000; accepted July 28, 2000 In variational calculation, we have obtained the binding energy of helium 3 dimer in two dimensions. The existence of one bound sta- te, with the binding energy –0.014 mK, has been definitively found. Also, the existence of a binding state of helium 3 trimer having spin –1/2 with the energy below –0.0057 mK is indicated. This re- opens the question of the existence of the gas phase of many he- lium 3 atoms on the surface of superfluid helium 4. Key words: helium dimer, helium trimer, binding energy of helium 3 dimer. INTRODUCTION About thirty years ago, it was demonstrated1 that, in dilute bulk 3He– 4He solution, atoms of 3He prefer to float on the surface of the 4He rather than to be dissolved in the bulk. All atoms in the solution are pulled down by gravity. A 3He atom is less massive than a 4He atom and therefore its zero point motion energy is greater than that of 4He (approximately by a factor 1.3). Due to this motion, it tends to have no 4He nearby. This ten- dency leads it to sit on the surface of the 4He, where it has an empty space above. Thus, a 3He atom at low temperatures (below 0.1 K), on the surface of bulk liquid 4He behaves as a spin –1/2 Fermi particle in two dimensions. # Dedicated to Academician Krunoslav Ljolje on the occasion of his 70th birthday. * Author to whom correspondence should be addressed. 1088 L. VRANJE[ AND S. KILI] In our recent papers,2–4 we have considered binding of helium diatomic molecules in confined and unconfined geometries. It has been shown that in infinite space helium fermionic dimer exists only in two dimensions. In con- fined geometry, two helium atoms were studied in 2 and 3 dimensions. Mo- tion of atoms has been confined by spherically external holding potentials.2 Using a similar procedure, diatomic helium molecules have been studied in the external holding potential that depends on one coordinate as well.4 All the considered systems might be thought as models for the interactions be- tween helium atoms in a specific real physical environment. For example, in solid matrices, where helium dimers form the condensation seed for helium clusters, in nanotubes, with a diameter between 10 and 100 Å, and in »con- densation« on a solid or liquid substrate. We are not convinced that the atoms of 3He form a gas on the surface. This doubt is based on the fact that there is one bound state of two 3He at- oms in 2 D space with the binding energy of about –0.02 mK.2 This result was achieved after numerical solving of the Schrödinger equation. Of cour- se, a variational calculation is desired as well. No successful variational cal- culation showing the binding of helium 3 dimer in 2 D has been done so far. The first goal of this paper is to derive a trial radial wave function and perform variational calculation in finding the binding energy of fermionic helium 3 dimer in 2 D and the mean value of the internuclear distance. The second goal is to examine the possibility of the existence of helium 3 trimer with spin –1/2. DERIVATION OF THE TRIAL WAVE FUNCTION Very good trial wave functions describing the ground state of helium 4 dimer and the molecule consisting of one atom of helium 4 and one atom of helium 3 were obtained and used in Ref. 2. They describe a short-range cor- relation between two atoms, like in the Jastrow wave function for liquid he- lium state. Long-range correlations are described by decreasing the expo- nential function. Comparing our results with the numerical solution of the Schröedinger eq., we found that the best form was a product of the func- tions, which describe short and long range correlations divided by the squa- re root of the distance: 1 é æ aög ù y= expê –ç ÷ – srú , (1) r ëê è r ø ûú where a, g and s are variational parameters. STUDY OF FERMIONIC HELIUM DIMER AND TRIMER 1089 Our experience showed that this function, although very good for helium 4 dimer and 4He–3He molecule, was not good enough to give the bound state of the fermionic helium dimer. This dimer is very large (the largest molecule we know) and the behaviour of the wave function in-between short and long range is very important. Using Gnuplot graphics and data from the numeri- cal solution of the Schrödinger equation, we were able to construct the fol- lowing trial wave function. 6 ==11 y ()r fr ()å fri (), (2) r r i=1 where é æ a öa3 ù = ê ç 2 ÷ ú Î[] fr11() a exp– , rrr12, ëê è r ø ûú = b3 Î[] fr21() b (ln(– r b 2 )) , rrr23, é c3 ù æ rr– ö fr()= c exp–ê ç 4 ÷ ú , rrrÎ[], 31ê ç ÷ ú 34 ë è c2 ø û é d 3 ù æ rr– ö fr()= d exp–ê ç 4 ÷ ú , rrrÎ[], 41ê ç ÷ ú 45 ë è d2 ø û é e3 ù æ rr– ö fr()= e exp–ê ç 4 ÷ ú , rrrÎ[], 51ê ç ÷ ú 56 ë è e2 ø û = Î¥[] fg61exp (– sr ) , rr6, and r1 =1Å,r2 = 2.97 Å, r3 = 34.57 Å, r4 = 165.1 Å, r5 = 228.5 Å, r6 = 2000 Å. It has 17 parameters a1, a2, a3, b1, b2, b3, c1, c2, c3, d1, d2, d3, e1, e2, e3, g1, s and 8 of them are independent. Namely, using the continuity of the wave function and the first derivative in points r2,tor6, one finds the following nine equations among the parameters; there would be ten, but our wave function has its maximum at point r = r4 and the equation which demands continuity of the first derivative disappears (with the constraint that d2 > 1): 1090 L. VRANJE[ AND S. KILI] é a3 ù æ a ö a expê –ç 2 ÷ ú = brb (ln( – ))b3 , (3) 1 ê ç ÷ ú 122 ë è r2 ø û æ öb3 ç a2 ÷ a2 b3 1 ç ÷ = , (4) è r2 ø r2 rb22–ln(–) rb 22 é c3 ù æ rr– ö brb(ln( – ))b3 = c expê –ç 43÷ ú , (5) 132 1ê ç ÷ ú ë è c2 ø û æ öc3 –1 b3 1 c3 ç rr43– ÷ = ç ÷ , (6) rb22–ln(–) rb 32c2 è c2 ø = cd11, (7) é d 3 ù é e3 ù æ rr– ö æ rr– ö d expê –ç 54÷ ú = e expê –ç 54÷ ú , (8) 1 ê ç ÷ ú 1 ê ç ÷ ú ë è d2 ø û ë è e2 ø û æ öde33––1 æ ö 1 d3 ç rr54––÷ e3 ç rr54÷ ç ÷ = ç ÷ , (9) d2 è d2 ø e2 è e2 ø é e3 ù æ rr– ö e expê –ç 64÷ ú = gsrexp(– ) , (10) 1 ê ç ÷ ú 16 ë è e2 ø û æ öe3 –1 e3 ç rr64– ÷ ç ÷ = s .(11) e2 è e2 ø We chose coefficients a2, a3, b2, c1, c3, d2, d3 and s as variational parame- ters, and for the others, using relations (3)–(11), we obtained the following expressions: = dc11, (12) STUDY OF FERMIONIC HELIUM DIMER AND TRIMER 1091 a æ a ö 3 a =ç 2 ÷ 3 b3 ç ÷ (–rb22 )ln(– rb 22 ), (13) è r2 ø r2 1/c é c ù 3 = 3 c3 –1 crbrb23332ê(– )ln(– ) (–)rr43 ú , (14) ë b3 û é æ öc3 ù ç rr43– ÷ expê –ç ÷ ú ê è c ø ú = ë 2 û bc11 , (15) b3 (ln(rb32 – )) (ln(rb – ))b3 = 22 ab11 , (16) é a3 ù æ a2 ö expê –ç ÷ ú ê ç ÷ ú ë è r2 ø û d æ d ö 3 []ç 2 ÷ 1 lnsr (64 – r ) ç ÷ è rr– ø d e = 54 3, (17) 3 æ ö ç rr64– ÷ lnç ÷ è rr54– ø æ e ö1/e3 e =ç 3 ÷ (–)rr11–/e3 , (18) 2 è s ø 64 é ed33ù æ rr– ö æ rr– ö ed= expêç 54÷ – ç 54÷ ú , (19) 11 êç ÷ ç ÷ ú ëè e2 ø è d2 ø û é e3 ù æ rr– ö gee= sr6 expê –ç 64÷ ú . (20) 11 ê ç ÷ ú ë è e2 ø û The coefficients are given in the order in which they are calculated. 1092 L. VRANJE[ AND S. KILI] VARIATIONAL CALCULATION OF THE DIMER Having derived the trial wave function we performed the variational cal- culation ^ òyy* H rrd E £ , (21) òyy* rrd where 2 ^^h æ ®®ö HVrr=+––D ç ||÷ , (22) 2m è 12ø ®® m 3 ´ –27 || = m/2 is the reduced mass of He, m = 5.00649231 10 kg and r= rr12– . Then, the expression for the energy can be written in the form ¥ æ h 2 ö ç 2 2 ÷ ò rrVrd ç ()y () r+Ñ (y ()) r ÷ 0 2m £ è ø E ¥ . (23) ò rrd y 2 () r 0 For the interatomic potential, we used ab initio the SAPT potential by Korona et al.6 After adding the retardation effects (SAPT1 and SAPT2 ver- sions), Janzen and Aziz13 showed that SAPT potential recovers the known bulk and scattering data for helium more accurately than all the other ex- isting potentials.