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. To calculate the integrals, we used the Romberg extrapola- tion method5 and by a minimization procedure obtained the binding energy of – 0.014 mK. Values of variational parameters for this energy are: a2 = 2.873 Å, a3 = 3.698, b2 = 1.55 Å, c3 = 5.9, d2 = 573 Å, d3 = 2.0, s = 0.0009318 –1 Å . The value of parameter c1 does not affect the binding energy, but only the normalization integral; in our calculation it has the value c1 = 0.03588. We also used the boundary points r3 ,r4 ,r5 and r6 as variational parame- ters. Their final values are r3 = 19.5 Å, r4 = 199 Å, r5 = 282 Å and r6 = 1200 Å. Other parameters, when calculated from expressions (12)–(20) are: a1 = 0.01988, b1 = 0.01456, b3 = 0.547, c2 = 217.6 Å, d1 = 0.03588, e1 = 0.03634, e2 = 1262.9 Å, e3 = 3.634 and g1 = 0.05257. In the limit r ®¥, the wave function has the asymptotic form e –sr0 , whe- 2m re s is determined by relation s = (–e ). The value of s coincides with 0 0 h 2 0 the value of s, which confirms the correct asymptotic behaviour of the wave function Y. STUDY OF FERMIONIC HELIUM DIMER AND TRIMER 1093
The function f(r) and its first derivative are shown in Figure 1.
fr() 0.04 f'() r 0.035
0.03
0.025
0.02
0.015
0.01
0.005
0 1 10 100 1000 10000 r (Å)
Figure 1. The figure shows the radial wave function f(r) (solid line) and the first de- rivative f '(r) (dashed line) of helium 3 in 2 D for the parameters determined by mi- nimization of the energy.
We also calculated the mean value of internuclear distance
ò rrrry 2 ()d <>=r , (24) òy 2 ()rrd r
ò rrrr22y ()d <>=r 2 , (25) òy 2 ()rrd r and D()rr=<22 > – <> r. (26)
D 3 The obtained values of
é c3 ù æ ||rr– ö fr()= c exp–ê ç 4 ÷ ú , rrrÎ+[], d , (27) 31ê ç ÷ ú 34 ë è c2 ø û
Î+[]d and the function f4(r) is defined for rr45d, rwhere = 1.1 Å. From the d condition that the function and the first derivative are continuous in r4 + , two relations for parameters d1 and d2 are obtained,
æ ö1/d 3 d cd/ =ç 3 ÷ 3311–/e3 d2 ç ÷ c2 d , (28) è c3 ø
é dc33ù æ ddö æ ö dc= êç ÷ – ç ÷ ú . (29) 11êç ÷ ç ÷ ú ëè dc22ø è ø û
The relations for other parameters (13)–(20) are left unchanged. With the wave function defined in this way, no singularities in the second deriva- tive of function Y are expected, and therefore the variational calculation can be performed using relation (23) as well as the following relation for the en- ergy
¥ æ ö 2 h ò rrVrd ç ()y () r+ yy () rD () r÷ 0 2m £ è ø E ¥ , (30) ò rrd y 2 () r 0 where the kinetic energy is expressed through the Laplace operator, D . The minimization of energy in both cases gave the same value of – 0.014 mK, which is the same as the one obtained using the function where there is no d displacement from the maximum in r4. Thus, we can be certain of the ap- plied numerical procedures. STUDY OF FERMIONIC HELIUM DIMER AND TRIMER 1095
CALCULATION OF TRIMER WITH SPIN –1/2
7 3 3 In 1979, Cabral and Bruch considered the binding of He2 and He3. They performed a variational calculation, with the interatomic potentials available at the time, and concluded that both molecules are probably not bound in 2 D. Our results for the dimer led us to extend variational calcula- tion to trimer binding. Since 3He atoms are fermions, they form spin –1/2 trimers and spin –3/2 trimers. The results from Ref. 7 indicate that spin –1/2 trimer has a lower energy and therefore we studied only that case. The chosen form of the variational wave function, following Refs. 7 and 8 is
=+ yFasXX F sa, (31) where Xs and Xa are spin doublets, symmetric and antisymmetric, respecti- f f vely, under the exchange of particles 1 and 2 while a and s are space wave functions that are, respectively, antisymmetric and symmetric under the ex- change of particles 1 and 2. Spin +1/2 projections of the doublets are
==1 Xsaz(/)12 (ab123 a – baa 123 ), (32) 2
==1 Xssz(/)(12 2aab123 – ab 123 a – baa 123 ), (33) 6 a b where i ( i) are the usual spin up (down) eigenstates of a spin 1/2 particle and subscript i is the particle label. In the calculation for the space wave functions, we combine the following forms:
==y ==x ffyaa0120(–)yy yand, ffyaa0120(–)xx y, (34) then ==y 1 [] ffyss031320(–)(–)yy+yy y 3 and
==x 1 ffyss031320(–)(–)xx+xx y, (35) 3
fr()()()fr fr = 12 13 23 y 0 , (36) r12 r13 r23
d where f(rij) is the new dimer wave function (2), with the modification. The constructed wave function is antisymmetric under the exchange of particles 1096 L. VRANJE[ AND S. KILI]
1 and 2 and symmetric under cyclic exchange of particles 1,2 and 3. There- fore,8 it is also antisymmetric under the exchange of particles 2 and 3 as well as 1 and 3. The Hamiltonian of the system is
h 2 H =– (DD1 +++++ D )()()()Vr Vr Vr . (37) 2m 2 3 12 13 23 Again, a variational ansatz was used to calculate the binding energy (21). Using the fact that the Hamiltonian is spin independent and symmet- ric under the exchange of x and y coordinates, we managed to express en- ergy by the following relations:
II+++ I I E £ p ka kb kc , (38) In where
¥¥ 2p ®® ®® =+2 2 2 2 || ICn òòòrrddd( rr qr–(,,–) rr12 ry ) rrr1 r , (39) 3 0 110 220 1 2 0 1 2 2
¥¥2p ®® ®® =+2 2 2 || ICp 2 òòrrddd() V() r rrò qr – rr12 ry(,rr , r1 – r ), (40) 0 11 10 220 1 2 0 12 2
h 2 ¥ æ f"()r 1 ö ¥ IC=+–2 ò rrddç 1 ÷ ò rr. ka 0 11ç 2 ÷ 0 22 m è f ()r1 4r1 ø
2p ®® ® ® 2 + 2 2 || ò d(qr–(,,–) r12 r r ) y r r r1 r , (41) 0 1 2 0 12 2
h 2 ¥¥æ f'()r 1 ö æ f'()r 1 ö IC= –2 òòrrddç 1 – ÷ rrç 2 – ÷ . kb 0 11ç ÷ 0 22ç ÷ m è f ()rr112 ø è f ()rr222 ø
2p ®® ® ® 2 + 2 2 || ò d(qqrrrryrrrrcos –12 ) ( , ,1 – ) , (42) 0 1 2 0 12 2
2 h ¥¥æ f'()r ö 2p ®® = 2 ç 1 1 ÷ 2 || IC–3 ò rrdd– òòrr dqY(, r r , r12 – r )(43) kc 0 1 1 ç ÷ 0 220 0 12 m è f ()rr112 ø
®®®= ®®®= q ® ® C is a constant, r1 r12– r , r 2 rr13– and is the angle between r1 and r 2 . The fact that expressions for the energy were reduced to three-dimensional integrals enabled us to perform the calculations using the same numerical methods as in the dimer case. After time consuming numerical calculations, STUDY OF FERMIONIC HELIUM DIMER AND TRIMER 1097
3 we found that the upper bound of the binding energy of He3 trimer was – 0 .0057mK. Using the same wave function, we derived the average distance r Dr between atoms of < 1> = 4503 Å and 1 = 3633 Å.
DISCUSSION
To the best our knowledge, function (2) is the first trial form that in va- riational calculation led to the binding of helium 3 dimer. In this case, one- dimensional Romberg integration with high accuracy has been performed. The results are in good agreement with those obtained by numerical solving of the Schrödinger equation. Having an appropriate two-body function, we were able to construct a special form of the Cabral-Bruch trimer wave function describing the state with spin 1/2. We performed a very accurate Romberg integration of three- dimensional integrals and found an upper bound to the binding energy of –0.0057 mK. This shows that helium 3 trimer with spin –1/2 is bound in two dimensions. (As it was showed in paper,4 binding of diatomic helium mole- cules is significantly increased if they are close (about 3 Å) to the surface of liquid helium. It means that binding of trimers could be experimentally ob- served in the future.) This result is quite a new one. It opens the question of the phase of many helium 3 atoms on the surface of liquid helium 4. So far, it has been believed that they form a two-dimensional gas. A qualitative estimation of our result for the trimer may be done as well. r D r 3 The obtained values of < 1> = 4503 Å and 1 = 3633 Å show that He3 is a large molecule. Let us assume that there is a homogenous monolayer gas (or liquid) with the average distance between particles as in helium 3 trimer, then its concentration is 4.9 ´ 1012 m–2. This concentration is several orders lower than the one of 3%, which is the upper limit for the attractive interac- tion between two 3He atoms in helium 3–helium 4 film.14 Consequently, it may be concluded that in our case the necessary condition for the binding of three helium atoms is satisfied. Recently, new interatomic helium potentials have appeared. Van Mourik and Dunning computed a new ab initio potential energy curve9 that lies be- tween the HFD-B3-FCI1 (Ref. 12) and SAPT2 (Ref. 13) potentials, being closer to SAPT2 potential. Other authors10,11 maintain that, according to their calcu- lations, SAPT potential is insufficiently repulsive at short distances. In papers,2,3 the binding energy of helium molecules was calculated us- ing two different potentials, HFD-B3-FCI1 and SAPT. The obtained results did not differ substaintially for these two cases. Therefore, we do not expect that the calculations with new, more precise potentials would change our re- sults appreciably. 1098 L. VRANJE[ AND S. KILI]
Acknowledgements. – We are indebted to Professor E. Krotscheck for many stim- ulating discussions and R. Zillich for providing us with data concerning the numeri- cal solution of the Schrödinger equation for helium 3 dimer in 2 D.
REFERENCES
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SA@ETAK Varijacijska analiza dimera i trimera helija-3 u dvije dimenzije
L. Vranje{ i S. Kili}
Varijacijskim prora~unom dobijena je energija vezanja dimera helija-3 u dvije di- menzije. Definitivno je utvr|eno postojanje jednog vezanog stanja, s energijom veza- nja od – 0.014 mK. Tako|er je odre|ena energija vezanja odgovaraju}eg trimera spina –1/2 od – 0.0057 mK. Ovaj rezultat otvara dvojbe o tome da atomi helija-3 tvore plin- sku fazu na ravnoj povr{ini suprafluidnog helija-4.