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in a 1988 review in the _ handbook on the physics and chemistry of rare earths _ c. k. jrgensen asked the rhetorical question is quantum chemistry feasible ?  and with special regard to these elements he answered sorry , not today ; perhaps next century @xcite . presently , in the new century , the structure and reactivity of @xmath0 elements is a flourishing domain of theoretical chemistry [ insert reviews .. ] the main features of the chemistry of the lanthanides and the actinides can be deduced from simple atomic calculations . in figs . [
fig : lne]-[fig : anr ] we present orbital properties extracted from numerical 4-component relativistic hartree - fock calculations@xcite averaging over the valence configuration @xmath1 for the neutral lanthanide and actinide atoms .
we have chosen this configuration , which is not the ground state configuration for all the @xmath0 elements , since it gives access to information about the @xmath2 , @xmath3 and @xmath4 orbitals . from the orbital energies in fig .
[ fig : lne ] we observe a distinct energetical separation of the @xmath5 orbitals from the @xmath6 and @xmath7 orbitals of the lanthanides which explains their chemistry dominated by the @xmath8 oxidation state .
we note that the @xmath6 levels cross the @xmath7 level towards the end of the series , a feature which may induce convergence problems in atomic calculations not exploiting the full atomic symmetry .
[ fig : ane ] shows a somewhat different situation for the actinides in that the @xmath9 levels are energetically close to the @xmath10 and @xmath11 levels at the beginning of the series , but are then strongly stabilized towards the end of the series .
these features translate into a rich oxidation chemistry for the early actinides and a restriction to the @xmath8 oxidation state for the late actinides .
this in turn explains the challenge of separating the minor actinides americium and curium from the lanthanides in the treatment of nuclear waste [ ref ] .
looking at mean radii @xmath12 , we observe in fig .
[ fig : lnr ] a distinct spatial separation of the @xmath5 , @xmath6 and @xmath7 orbitals in the lanthanides , whereas the @xmath3 and @xmath4 orbitals come quite a bit closer in the actinides .
[ discuss lanthanide contraction ] it is also interesting to observe that whereas the spin - orbit splitting in the @xmath2 shell is considerably stronger than for the @xmath3 shell , any difference in spatial extent is hardly visible for the spin - orbit components of the @xmath2 shell .
this can be understood from the fact that the @xmath2 orbitals are in general quite contracted and so any deformation of the orbitals is energetically very much more expensive than for the @xmath3 shell .
with these observations in mind it was all the more surprising when clavagura _
@xcite reported a clear example of @xmath5 participation in bonding in luf@xmath13 since lutetium is at the very end of lanthanide series where one would expect the @xmath5 orbitals to be the most inert .
the conclusion was seriously questioned by roos _
@xcite as well as ramakrishnan _
_ @xcite .
the purpose of the present paper is to investigate the possible @xmath5 participation in luf@xmath13 by an independent approach .
we employ a trick that may be useful for other purposes as well .
we extend our electronic structure analysis to all trihalides ( x = f , cl , br , i ) of lutetium as well as of lawrencium .
we thereby provide a comparison of covalency between these two elements , of relevance for the delicate problem of separation of the late actinides from the lanthanides .
the paper is organized as follows : in section [ sec : theory ] we describe our methodological approach .
computational details are given in section [ sec : comp ] . in section [ sec : results ] we present and discuss the results of our geometry optimizations and electronic structure analysis , before concluding in section [ sec : conclusion ] .
the question about the participation of @xmath2 orbitals in bonding in the lanthanides and actinides is very much a leading question :    1 .
it assumes that one can identify these atomic orbitals in the electronic structure of the molecule .
it assumes that one can unambiguously distinguish bonding from non - bonding contributions to the electronic structure of the molecule .    in order to tackle the first point we perform 4-component relativistic hartree - fock ( hf ) and kohn - sham ( ks ) calculations of the trihalides of lutetium and lawrencium ( lux@xmath13 and lrx@xmath13 , x = f , cl , br and i ) and investigate their electronic structure by projection analysis@xcite , that is we expand the molecular orbitals ( mos ) in pre - calculated orbitals ( index @xmath14 ) of the constituent atoms ( index @xmath15 ) .
@xmath16 the fragment orbitals are usually restricted to the occupied orbitals of the selected configurations of the constituent atoms of the molecule .
the expansion is completed by the polarization contribution @xmath17 which by construction is orthogonal to the fragment orbitals . once the expansion coefficients @xmath18 have been obtained a population analysis may be carried out completely analogous to , but without the basis set sensitivity of a mulliken population analysis .
the selection of fragment orbitals should be adjusted if the gross population of the polarization contribution is significant .
the next point is somewhat more difficult due to the invariance of the electronic energy under rotations of the occupied orbitals of these closed - shell molecules .
this rotational freedom can be exploited to transform from canonical hartree - fock or kohn - sham orbitals to localized orbitals , or , in the terminology of mulliken@xcite , from spectroscopic to chemical mos .
however , there is an abundance of possible localization criteria and thus perhaps not a clear answer from such an approach .
the problem would have been simpler for the trihalide of any other lanthanide than lutetium since one would then expect inert @xmath5 orbitals to form an open shell .
there would accordingly be a _ variance _ of the electronic energy with respect to rotations between the open- and closed - shell orbitals .
however , this immediately suggests the strategy that we will pursue in this paper : we will place @xmath5 orbitals from an atomic calculation in an artificial open - shell with fractional occupation close to , but not equal to 14 .
if the @xmath2 atomic orbitals now localize to the open shell , we have unambiguous evidence that they are chemically inert .
all calculations reported in this paper are based on the 4-component dirac - coulomb hamiltonian with a gaussian charge distribution as the nuclear model using the recommended values of ref .
the numerical atomic calculations were performed using the grasp code @xcite .
molecular relativistic 4-component hartree - fock and density functional ( pbe @xcite and b3lyp @xcite ) calculations were carried out using the dirac08 package @xcite .
we employed the cc - pvtz gaussian basis sets of dunning and co - workers @xcite for f , cl and br and equivalent sets of dyall and co - workers @xcite for i , lu and lr . the small component basis set for the 4-component relativistic calculations has been generated using restricted kinetic balance imposed in the canonical orthonormalization step@xcite .
all basis sets are used in uncontracted form .
hf and ks geometry optimizations were carried out using analytic and numerical gradients , respectively . for pyramidal ( @xmath19 ) and planar ( @xmath20 ) structures we employed the lower c@xmath21 and c@xmath22 symmetries , respectively .
test calculations with cc - pvdz basis sets indicate that the resported structures can be considered converged with respect to the chosen basis sets . for the projection analysis fragment orbitals
were generated by average - of - configuration hf calculations and ks calculations with fractional occupation , corresponding to ground state electronic configurations of the atoms .
in table [ tab : lux3 ] , [ tab : lrx3 ] we present hf , b3lyp and pbe calculation for lutetium and lawrencium trihalides lux@xmath13 , lrx@xmath13 ( x = f , cl , br , i ) respectively . for the former trihalides , the lux@xmath13 ,
there exist a large number of theoretical works whereas very limited works concerned the lawrencium trihalides . since we will discuss the two table separately . in table [ tab :
lux3 ] we summarise our result together with theoretical and experimental values from the literature , given also in the table the method ( or the hamiltonian ) and the basis set used to obtain these values .
firs of all looking to the bond lengths in table [ tab : lux3 ] on see that our values using b3lyp have the best agreement with experimental ones and second there is a good agreement with the literature for the respective method ( or hamiltonian ) , especially with the hf values of ref . @xcite,@xcite and the pbe of ref .
+ concerning the geometrical shape and the bond angle we see that hf give a planer geometry for all the trihalides whereas b3lyp and pbe gives a pyramidal geometry for the luf@xmath13 only and a planar geometry for the other trihlides .
increasing of the bond angle towards the heavier trihlides can be understood having in mind that the heaver halides have weaker ability to polarize the metal atom which means less distribution of the electronic density around the metal atom favoring a planar geometry with the highest bond angel of 120 grad and longer bond length towards the heavier halides as seen in table [ tab : lux3 ] .
+      we have studied the possible @xmath5 orbital participation in bonding in luf@xmath13 by the approach presented in sections [ sec : theory ] and [ sec : comp ] .
the results are summarized in table [ tab : luf3 ] .
we first calculated the neutral lutetium atom in @xmath23 with linear supersymmetry .
the resulting coefficients were exported to @xmath24 symmetry .
the lu @xmath5 orbitals were next imported into a molecular calculation on luf@xmath13 in the optimized pyramidal geometry and kept frozen in an initial calculation on the neutral molecule .
starting from the resulting molecular coefficients a series of ks calculations with fractional occupation , that is 14-@xmath25 electrons ( @xmath26 $ ] ) in 14 spinors , were carried out and the resulting molecular orbitals studied by projection analysis . in each calculation vector for each occupied orbital
was selected based on overlap with the starting molecular coefficients .
however , it is seen from table [ tab : luf3 ] that the electronic energy goes smoothly into the energy of the fully relaxed neutral molecule as the hole @xmath25 tends towards zero , indicated that we indeed obtained the ground state of the molecule for each value of @xmath25 .
+   + * conclusions*[sec : conclusion ] the presented four component relativistic result for the trihalides of lutetium and lawrencium , lux3 , lrx3 ( x= f , cl , br , i ) respectively using density functional theory ( dft ) shows that the trend of bonding is from lighter to the heavier halide atoms and between 4f/5f atoms lu and lr .    ( + 400,+400 )   energies ( in a.u . ) of the @xmath5 , @xmath6 and @xmath7 orbitals for ce - lu from 4-component relativistic hartree - fock calculations averaging over the @xmath27 ( x=1 , 14 ) valence configuration.,title="fig:",height=642 ]    ( + 300,+400 )   energies ( in a.u . ) of the @xmath9 , @xmath10 and @xmath11 orbitals for th - lr from 4-component relativistic hartree - fock calculations averaging over the @xmath28 ( x=1 , 14 ) valence configuration.,title="fig:",height=642 ]    ( + 300,+400 )   mean radius @xmath12 ( in bohr ) of the @xmath5 , @xmath6 and @xmath7 orbitals for ce - lu from 4-component relativistic hartree - fock calculations averaging over the @xmath27 ( x=1 , 14)valence configuration.,title="fig:",height=642 ]    ( + 300,+400 )   mean radius @xmath12 ( in bohr ) of the @xmath9 , @xmath10 and @xmath11 orbitals for th - lr from 4-component relativistic hartree - fock calculations averaging over the @xmath28 , ( x=1 , 14 ) valence configuration.,title="fig:",height=642 ]           36ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] 
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in this work we present a four component relativistic theoretical investigation of the trihalides of lutetium and lawrencium , lux3 , lrx3 ( x= f , cl , br , i ) respectively using density functional theory ( dft ) with different density functional and a geometrical optimisation procedure as implemented in dirac - package . 
 the results show the trend of bonding from lighter to the heavier halide atoms and between 4f/5f atoms lu and lr .