Ab initio SCF and CEPA investigations of stable lithium clusters

L’olunte 67, number I

CHCNlC_.\L I’IIYSICS LLTTERS

I Norsnber

1979

V.uious approaches of two Liz molecules are investigated in a search tor st.tbIe Liq ciusrer configurations and favourabk pathxtrs formin: them. The Ioriest energy minimum is found for .I si&ct state in .t rhombic structure with its short d&onJl cloe to the equdzbrium inrernuckn dbtoncc of free Liz. The stability of this arrangement is traced to the bindmg potenti. of the p-x funcrions of its ccntra1 Liz substructure_ This sugyzsts J fl&t squ~rc bipyramidal configyxarion for Lig ahi& is indeed found w rcprescnt .I IocaI minimum- Binding cnergcs for lLiz - Liq .md Liz + Liq - Lig arc caIcul.ttcd as 15 kcaI/moIe and 14 kd/moIe, respcctwelg. Formation of both clusttrs IS possible \~ithout surpxsing energy barriers. ilarmonic %ibrationJI frequcncie\ xe gnen for LL and ionizxtion potentmls are cakulated for Lie and Lis.

l_ Introduction hlolecular beam and mzttri?c isoI3tion techniques have made possible the sepamtion ofsm311 metal &tom rtggregstes. For example. the ionistltion potenthIs of XI and K cIusters have been measured using the first technique [I-S] , whererts the eIectronic spectrum of some nobel metal clusters (Ag and Au cIusters) have been obtrtined using the latter [4] _ The study of the bonding properties in small metal clusters should contribute towards 3 better understanding of problems related to heterogeneous catalysis as well as to nucIe3tion phenomena- Therefore, numerous theoretical papers employing vttrious semiempirical and ab initio techniques h3ve been devoted to this subject [S- 191. In most of them. geometry optimizrttion has been carried out only fcr highIy symmetric nuclear configurations. A very c3refuI investigation of the Li3 ground state potential energy surface (including electron correlation) has been performed by Gerber and Schumacher [ 15]_ For Li4, different structures h3ve been considered in SCF caIcuIations by Janoschek [6,1 l] ) but his basis set seems

to be insufficient to describe binding in this system sppropriatelyIn this work we made an attempt to find energetically Dvourdble geometric31 configurarions of Li4 which result from different 3pproaches of Li7 mole-

cuIes, allowing for geometrical relaxation of kdividual Liz subsystems_ The electronic structure of the mosf st3bIe nuclear configuration of Li3 suggests a bipyramidal configuration for Li6, the geometrical prtr3meters of which have been optimized. Finally, we calculated the ionization energies for Li,, Si, and Jli6 in their stable geometries-

2. Methods and basis sets Most of the calculstions have been carried out within the HF SCF approximation, making partial use of analytically caIcuIated energy gradients [20] _ Electron correlstion effects, investigated at selected geometries, have been taken into account by means of the coupIed electron pair approximation CEPA [21] _

119

The basis sets ofg;lussian type orbit& are described in table I _ where also rcsutrs i-or Li, are given. Somcwh;lr IkrtuitousIy. the smaltest hasis X yields an equilibrium bond Iengih quite close to the ! I I’knit vaIue I?? j _ Ester&on of the basis scc to two v;lfcncc she11 s t-unctions (basis 5) increases the bond length T, to 5.32 bohr (Jr, = 0.06 bohr). but augmsr2irrg zk p sex rc3Paw I’uncri0ns 3g3in gives gffod

agreement (within O-02 bohr or 0.4%) with the IIFlimit v&e, Correhtion cffkts (valence sheli phts intenhell correiation is included in all cidculations) taken into account by CEP=X shortens the bond length by O-16 bohr (basis B) and 020 bohr (basis D), respectively_ For the most flexible basis set D, the error in re ;tmounts to OA%. As is we11known. RHF SCF xcotmfs only for a small fraction of the dissociation energy of Liz (I 6% at the HF iimit)_ Our CEPA calcuhtions account for 72 (basis B) and 83% (basis D) of the correlation contribution to DC_ Several 3pproaches of two Liz molecules were cdc&ted with the SCF method using basis sets A and c_ Differences between the bond length snd energies of optimal geometries obtained utilizing these two different basis sets are even smaIIer than in the c&e of the I& moIecuIe_ Consequentiy, basis set A is employed in the SCF cakufations for extensive investigation of the energy hypersurface of Li4_ The CEPA geometry optimiwtioon has been carried out with basis set B. and minima so obtained were investigated using the larger basis set D_

I20

3_ SCF results for I i., and Li6 clusters For ;I first inspection of the Lit ground state cncr=T hypersurtke, diffkrent approaches of two Liz nioiecuics with fixed intramoIecuhr distances (rt = r, = r = 52 bohr) h;tve been considered_ The corrcaponding energies -3sfunctions of the internloleculx dikxmcc R obtained with the SCf- method and basis set A are illustrzrted in Gg_ I_ Energy minima have been obtsined for pzraiiei-in-phne-end-on approach (A) and for perpendic&r-in-pkme-symmetrkd (0) apprortch (“T-form”)_ The long-range attraction in approscb (a) isstronger than in approach (A) due to the more favourable orientation of the quadrupole moments. The coliinear-end-on (0) approach as well as the perpendicular-out-of-plane-end-on (A) approach exhibit very flat repulsive curves_ Considerably steeper repulsive tunes have been obtained for paralIeI-symmetrical-out-of-plane (m) and in-plane (0) approacJles_ The qualitative features of the curves remain unckm~ed when the Iarger basis set C is employed_ Moreover, relaxation of the intramoieculrtr distnnce r in the Liz subsystems does not infkence the shape of these enerv curves considerably_ Gradient c&xktions have been performed in the neighborhood of the two energy minima in R of fig,_ l _ Two cxtrema, one corresponding to 3 “‘rhombic”, the other to a “T-shaped” configuration, have been found. In the former, the equilibrium lengths of the diagonals were caIcuIated to 5.07 bohr and IO-78 bohr (basis A), whiIe in the latter both intramolecular distances are

Volume 67. number 1

CtfEVIC4L

PHYSIC-S LLTCRS

1 November

1979

AE(kcallmole) &/.q

,;---f

-6 7

6

1

,

48

a0 distance

of 2Li,/

1

11 2

1

IU.

R

5

ciu

I‘& 2. In-pl.mc interconrcrsian of Lie (T-form) into Li4 ?FzO&?&;s.I)z=s?ssez ‘%_S;;.@>: :;xe: WeTg? =s1 f_.lrt$s rf g uitft ~11other yeomctricxl p~mmeters optimized. Dnsh-dotted line: enct~~ ,ts 3 function of 9 u ith all other geometrical panmcters kept constant (nt T-form ralues).

obtained at 5.35 bohr, the intermolecuhtr distance R &nounting to 6-Z bo!tr_ Bllsis ZLyields dissociation energies (into two Lil molecules) of 13.3 and 6.2 ksaI/moIe for the “rI&bus” and the “T-form”, respectveily. While the “rhombus” is clearly an energy minimum, tbe “T-form” may be interconverted into the more stabIe “rhombus” without activation energy_ Taking -+ = s+t - 9, as the reaction coordinate and ntininti&ng the energy with respect to the other coordinates, the energy profile shown in fig. 2 resultsHarmonic vibrational frequencies have been calcuIated for the “rhombus” from SCF/gradient calculations employing basis B (table 2)_ The Iargest frequency at 334 cm-l, which corresponds to one of the Al0 fundamentals, is practicahy indistinguishable from tge cakuhtted vaiue for Liz _ Inspection of the L-matrix shows that this vibrational mode is clearly dominated by the motion of the two Li atoms along the small diagonal. The other two Rarnan-active fun-

d.mtent& are cakul~ted at 221 cm-* (Ale; predominantly stretching of the long diagonaI) and 163 cm-1 (B1,). Infrared-active modes are calculated at 7-95 (B&l, 169 (BSu) and 109 cm-l (B,,; out-of-plane motion). The reason for the stability of the rhombic form may be understood by inspecting the structure of the occupied molecular orbitals. The lowest valence orbital is of aI,, symmetry and would favour 3 square geometry ov& other planar structures. The nodal plane of the second orbital may either intersect two opposite Li, “bonds” or go through two diagonal Li atoms. In the first case, Jahn-Teller distortion would Iead to rectangular structures which. due to the antibonding force of the second orbital, end-up as two isolated Liz molecules as shows the energy dependence of approach (0) in fig. 1_ In the second case, the antibonding effect Is first relaxed by lengthening of the diagonal distance perpendicular to the nodal 121

C) R &ms

rhwnbic form.

pkme but is t?~ulIy b&ncxd by the bonding cffn-t wl;icll is main& due to the admhture ofp--r funcrtions of the Liz subsystem in the nodal phnc of the secvnd orbital_ As 1 rc’sponse of the Iengthening of one dizf:onal. tfrc al-_ orbit31 contracts ground the shorter diagonai tn~~xuses a substructure close to the free Li, n~oIccuIe_ But also the bonding inhuence of the a tr orbital contributes appreciabIy to the stability of thr?rhomb as is demonstrated by cakulations without p-z functions of tlte Lil subsystems_ When oniy the pohrization functions in the directions of the diag0naIs are taken into account as a Iocal minimum the rhontbic form with approximately the sa.me geome:ry as with p-rr functions is obtained with the energy sIightIy higher than tlte energy of two isolated Liz molecufes- The energy lowering and disappearance of the barrier between the “r!lombus” and the two isohted Liz molecules is due to the inchrsion of aii p functions in the LiS plane_ Plots of the two valence orbit& of the rhombus are given in fig_ 3_ A hluIIiien population an&&s shows that the central Liz has 3 charge defkct of 0.50 electron charges and that its p-z functions carry about O-4 electron charges. The binding effect of the highest valence orbited is aIso demonstrated by the Iengthening of both diagonal distances in tlte mtion by about 03 bohr (rabble4, footnote c)_ The Ieast-motion approach of forming the Li* rhombus from two Liz molecules is of course energetiu.Uy unfzvourabIe as may seen from fig. 4, it

1-i 3_ PIots of the f=st (4 and second (b) timce of the rhomb-

orbitais

shows however the origin of the bonding in Lid in a transparent way_ The approach (I) from fig_ 1 has been modified so that for each intermolecular dis-

1 November

*

I

1

1

3

5

I

7 d.stance

1

1

I

9

11

13

of

L

15 R

2~8~ f au

ture in the short diagonal .md the binding cJpdbiIity of its p-r; functions, one is mmlediately led to the concIusion thar it most f3vourJble configuration for the Li6 cluster should be 3 flat square bipyramidal one with a build-in centrJ1 Li2 substructure v.ith short interatomic distmce. Its two x-type orbitals support the binding of four equivalent “peripher& Li atoms. This configuration has indeed been found to represent a 10-I minimum of the energy surface N ith equilibrium v3lues for short and long dkgonal dist.mces (A.94 snd 10.02 bohr, respectively) very similrtr to that of the rhombus (5.07 and 10.78 bohr. respectively). Basis A yields 3 dissociation energy (into three Li-, molecules) of23_4 kc3I/moIe_ It should be noted that both clusters, the rhombic LiA .ts %veIlJS the bipyrdmidal Lib. c3n be forn;ed without activation energy. The corresponding pathwtys require the cent& Liz structure to be build up from 3rorns of different fr3gments (see fig. I, approach “)_ For Li, + Li,, one of the Liz shown represents the centml subsystem with the remaining ntoms placed perpendicu1.u to the pI3ne of approach.

4_ hestisation method

tance R one intramolecular Liz distance ‘1 wds optimized while the other one (at ) has been kept equal to 52 bohr (comprtre hg_ 4 where zllso the optimized intr3moIecuI~r distance r? is given 3s d function of the intermo1ecuI.u dist3nce R). For R = 0 the “rhombic” form is obtained. For intermolecular dist,mce R > 45 bohr the repulsion-like energy dependence is very similsr to the energy curve for the 3pproJch (I) from fig. I _ The chnrges dt all Li atoms in this R intervrt1 are the same_ For R < 4.0 bohr the energy is dmstic3lly decre3sing with decreasing li and charge separation omong Li atoms occurs_ A crossing of the MO tunes t&es place in the region between R = 4-O .md R = 1.5 bohr. The HOMO for larger distances Ilas 3 nod31 plane between the two Liz subsystems tind for sm3IIer distances after the orbital crossing the I fob10 h3s the nodal plane going through the Liz subsystem with fixed intmmolecular distance ‘t _ Since the stability of the rhombic structure appears to stem from the formation of 3 Liz substruc-

i979

of

Li4 and

Lig

clusters with

CEPA

The influence of the correIation effects on the reof the ground state energy hypersurface which in the SCF appro&i indicate stable Lil clusters, has been studied with the CEPA method employing basis set B_ In the case of the “rhombic” configuration, the diagonal r-, between the more distztnt Li atoms is substantially-shortened in comparison with the optimized 1-7 distance obtained from SCF calculations for the same basis set B. In contrast, the distance rl between closer lying Li 3toms remains almost unchsnged in both SCF and CEPA approaches. The CEPA method ako yields an mtermolecular distrtnce R of the “T-form” cluster drastically shorter than the SCF method. The two intrrtmolecular distances rl and r, for LiT subsystems of the “T-form”, 3re of sinlila> length-and nearly equal to the bond length of Li2 molecule. The optimized geometries for Liz, two configumtions of Li4 and the bipyr3midaI Li6 cIuster obtained by the CEPA method employing basis set B are given in table 3, where also tke corresponding CEP_4 gions

123

TabIt 3 TotA cmrgirs zd

bitr.!rng rncrgia of smdi I_izn cIuswrs a) ___ ._____--_-___.--,-_^.-__ - --

________

. Mrrlloll

cxsrrr

--_

___

Liz

r=

III

____2Li - - ----’

_____________

-

- ___-__I_

11a.G 1)

Ihsb _--_.-_

----

I _Sovembcr 1979

Clil UK’AL PII’LSICs LI-l-r1 Its

SoIumc 67. numbu I

_ )--rtE< Li2 I _ _---_--

A(ii,,z -- -. --

5.16

SC1I’S0 a Cl I’:\

- 1-1.678981 - I-?_7U3637 -I-l.7O391-l

u

-_--_-

. _

E( Li,,r) -

L(Lr,,rWIt(Li,,,)

-11.664a31 - I -I.89298 1 -I-l.S93175

rq

R b)

rI = 5.10 rz = 10.23

SCII’S8 CI Cl- I?A

-29.372776 -29.42SIJ-f -79 _ _433603

-0.014812 -0_02U-?66 -0.V25775

-29.71564 1 -29_80121 I -29.SlOS91

-0.016779 -O.OI~2-x3 -0.02?54

14

-I-C)

rr ~5-16 R=5_59 ‘2 s 5.1s

SC-F I’S0 CI CFI’A

-‘9- _363369 -29.4 I.?292 -29_?lS796

-O_Ci35106 -0.006618 -0.01090

-29,73472s -19.791522 X9.793674

-0.005S66 -0.00555s -0.011324

Lk

Bd)

rr =5.12 rz = 9.60

Xl’ PSO C-2 CLPA

---t4_071100 -4-l_156180 -a_1 72523

-0.035 I5 -0_05-I47 -0.0607S

-4.Z.634554 -Al.?22676 --13.7-l 1779

-0_041261 -0.043733 -0.062254

energka, and for comparison

SCF and PXD CI [Z 11

Idted (CEPA,

basis D) at 15-4 kcal/nloIe and 23-7

energies, for these ciusters are I&cd_ The czdcuL~tion.s

kcal/mole,

with the Iarger basis set D have been performed

_gy of Li3 is only 09 kczd/mole Iargcr thzn that of

the

mOSt

stabk

‘Mornbic”

Liz. Li,

arrangement

ized by correlation

and Li6 conformations. r?ppars

*

Lij

Xi,,

Tile

to be further stabil-

CEPA

The binding ener-

Note that the zero point ener-

to table 2. energies per atom, crtlculated with

method employing

basis set D, are 10.3 for

Liz, l-l.1 for Lia “rhombus”

and Lia f- Liz + Li6 zre calcu-

Ionizxion potuntblsof small Li,,

according

The atomization

effects and is now by S3 kcal

lower than the “T’* configuration. gies for lLi7

for

respectively.

Li6 “bipyr;lmidal”

dustrrs in au

and 16.S kczd/mole for

form_ Sicce caictdated

-

CIusicr

IP

Ektsis

hoopmans

sex-

CEI’A

Liz

VI 1.2)

B D

0_1so-l O_ISO9

0.1599 O-I613

O-1842 0.1892

=TR

vi a)

I% D B B D

0_15.53 0.1567 0.2234 0.2220

O_f-t14 0.1szs 0-I 395 0.1978 0.1975

0.1650 0-l 708 0.1641 0.2163 Il.2207

B D B D

0.1595 0_1594 O-2619 02604

0_14is O-1420 0.2318 0.2311

0.16-1-l O-1690 0.2300 0.2325

a1 5) VI?bl Lib B

v! a) v2 b)

x) VI mans f& vertical ionization po!ential from the h&hest occupied MO_ b) v2 muns second vertical ionization potential from the orbital bcIow_ =)a1 means fmt adltitiul ionization potential_ Distsnces~t minimum energy. cation: rt = 5AO;fz = 10.6 au_ d, JExperimenti due of VI is O.L893 ou [31_ 124

ionization

k-olume 67. number I

CFII \IIC.\L I’IIYSICS

enCr@es dre closeI_v related to espcrimenrally obtainztbIe photoionization threshold energies [X3], it seems to be useful to probide corresponding thcoreticaI data. The ionization potentials for Li-, _ “rhombic” form of Lia and “bipyramidal” form of fi6 clusters as obtained by SCF and CEPr\ methods with both bais sets I.3and D dre compared with vahles obtained from Koopmans theorem in table 4. The CEPk\ method with basis set D yields very good agreement with the eupcrimentaIIy known rd!ue of the ioriiration potentirti for Liz (cf. table -?)_ In sutumary, aheady simple IIF cttIcuIJtions describe sttlbIe contigurJtions of even Li-clusters well, and bond lengths and the norn1.d frequencies can be calculated qualitatively at this level of approGnation. ft is necessary to take correlation effects into account for determination of dissociation energies, ionirations potentials and for quantitative detrrmination of the bond lengths in Li clusters.

5_ Conclusion It 1t.1~been found both with the SCF and CEP,L\ methods that the energetically frtvourttble form of J_iGh.~sa “rhombic” contiguration. This form represents the absolute minimum of the corresponding ground state energy hypersurt%ce_ The Li atoms in this configuration are dit-ferently chJrgcd_ The nodal pIane in the highest occupied molecular orb&d which goes through the Li atoms lying on the shorter diagonal of the “rhombus” expkins the occurrence of positive charges on these atoms. It is expected that .m analogous situation wiI1 exist in 311wses of alkali four-atom cluster. in agreement with the resuits of &cuktions using pseudo-potential techniques on N+ clusters [17]_ A “rhombic” geometry w;1sJISO found for the Liz02 molecule (3%-301, where the substantial difference of eIectronegstivity between the lithium and oxygen atoms pIays of course a very

important role. Nevertheless, it is interesting to note that the Li 3-p functions are stronger occupied than the 2s functions and that they considerably stttbdize one of the antibonding Q functions of 0, (see ref. [30 J , table 2). .A second extremunl found 0x1 the hypersurface of Li4 arrangenlents (“T-form”) with &her energy than the “rhombic” configuration is 3 saddle point.

Ll.TTCRS

1 Norember

1979

It is worth mentioning tI13L the Iinear chain, the tetra1ledr.d and square arrangements xe not energeticdIy tkvourable. The “bipyr.mlid.d” con@uratIon of Li6 is found as .I stable cluster. with four Li &oms having negative chxges ~11d the remdining two being positiveIy charged_ TJG “bipyrsmidal” arrmgement can be considercd JS s&htly deformed section of the bee lattice. By trtking the atomil&ion energies per atom as ~1 meclsurc of cluster stability. our results show an incredse in stability with cluster size. Still, the Li6 CIUSter sIlows ~1 atomization energy which is only about one 11JIf Of the experiIiwntal Vahle for JtOini/dtiOn of the bee lattice (0.055 13 bohr [3 1J )_

Xckno\\Iedgement We thank the Informatik-Rechenbetrieb, Technische Universitzt Berlin, and ZEDAT, Freie L*niversitHt Berlin. for providing us with 3 generous aIIocation of computer time. Computation time provided by the Regionales Hochschulrechenzentrum Kaiserslautern is also gratefully acknowledged. One of the authors (H.O.B.) is grateful to the Fond der Deutschen

Chemischen Industrie for granting We Jppreciate valuable discussions

him

3

scholarship_

with Dr. V.

BonGC-Kouteck?.

References [I] E.J. Robbins, R-E. Leclenb, and P_ Wdhs. AdvJn. Ph>s_ 16 (1967) 739.

[ 21 PJ. Foster. R-E. Leckenby and E.J. Robbins, J. Phys. [;I [-I] [Sj [61 171 [8] 191 [IO] [Ii 1

B’ (1969) 178s A. Harmam, S. Leutwyler, E. Schumacher and L. \V&te, IIelv. Cbim. Acta 61 (1978) 153. W_ Schulze, H-U- Brcker and H. Abe’, Chem. Phys. 3.5 (1978) I77_ A.L. Companion. D-L_ Seibel Jr_ and AJ. Stxshnk, J. Chem. Phks. 48 (1968) 3637. R. Janoschek. Acta Phys. Acad. Sci. Hung. 27 (1969) 373_ B-T_ Pickup, Proc- Roy_ Sot. A 333 (1973) 69_ .‘I. St011 and H. Preuss, Phys Stat. Sol. 53 (1972) 519_ R-F. MarsbaLI, RJ. BIint snd A.B. Runz, Solid State Commun. 18 (1976) ?31_ J-G. Fripht. K-T_ Chow. M Boudart, J-B. Diamond and K.11. Johnson, J. bfoI_ Cat. 1 (1975) 59. R. Janoschek, J. bloI_ Struct. 6 (1970) 283.

125

126