ELSEVIER

4 April 1997

Chemical Physics Letters 268 (1997) 61-68

CHEMICAL PHYSICS LETTERS

Ionic versus covalent character in lanthanide complexes. A hybrid density functional study

Carlo Adamo 1, Pascale Maldivi

D~partement de Recherche Fondamentale sur la Mati~re Condens~e, Seroice de Chimie Inorganique et Biologique, Laboratoire de Reconnaissance lonique, CEA-Grenoble, 17 rue des Martyrs, F-38054 Grenoble, Cedex 9, France

Received 23 December 1996; in final form 31 December 1996

Abstract

The geometric structures and harmonic vibrational frequencies of La, Gd and Lu trihalides have been investigated by a hybrid density functional/Hartree-Fock approach coupled with a relativistic effective core potential. The adequacy of this electronic protocol is confirmed by the fairly good agreement with experimental data and with those obtained by more expensive post-HF computations. A detailed analysis of the electronic density has been performed using the natural bond localization procedure and its recent extension to natural electronic resonance theory to understand the role of charge transfer in the bonding of ianthanide complexes. This approach underlines the role of electrostatic interactions in the lanthanide- halogen bond, even if a charge transfer mechanism plays a role, especially in the more weakly bonded bromide and iodide derivatives.

I. Introduction

There has been growing interest in the investiga- tions of the physico-chemical properties and reactivi- ties of lanthanide and actinide compounds, due to their importance in various fields, such as photolumi- nescence or medical applications, as well as chemi- cal processes involved in the nuclear industry [1]. As often happens, the experimental interest has deter- mined a parallel growth of theoretical work, mainly devoted to the understanding of the nature of rare earth-ligand interactions. Nevertheless, some ques- tions remain open. For instance, the problem regard-

Permanent address: Dipartimento di Chimica, Universit~ della Basilicata, via N. Sauro 85, 1-85100 Potenza, Italy.

ing the relative weights of the covalent and ionic contributions in describing the lanthanide-ligand bonding is still a matter of controversy [2]. Although the general idea of a purely ionic Ln(III)-ligand interaction is widespread, several theoretical studies, carried out on small model systems, suggest that covalent interactions may play a non-negligible role [3-8]. These results are of particular importance for classical molecular mechanics simulations, in which, usually, rare-earth cations are modeled as triply charged hard spheres, taking into account only elec- trostatic interactions with the coordinated ligands [9]. Such theoretical analyses also provide a good start- ing point for a further comparison between lan- thanide-ligand and actinide-ligand interactions, in order to understand the different chemical behaviors observed [ 1 ].

0009-2614//97/'$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S0009-261 4(97)00177-2

62 C. Adamo, P. Maldivi / Chemical Physics Letters 268 (1997) 61-68

We have thought it interesting to carry out a systematic investigation of the nature of the chemical bonding in small model systems, namely the tri- halides of ianthanides (LnX 3, X = F to I, Ln = La, Gd and Lu). These systems have been chosen since, beyond their intrinsic interest, numerous experimen- tal [10-13] and theoretical data are available [2- 5,7,8]. Moreover, the electronic configuration of the trivalent lanthanide atoms (f0 for La, f7 for Gd and f14 for Lu) leads to an L = 0 state and therefore prevents first-order spin-orbit coupling. Further- more, in the case of La(III) and Lu(III), J = 0 and no second-order spin orbit effects can occur, whereas for Gd(III) second-order couplings are expected to be weak [14]. This discussion is based on the limit- ing model of free-ion states, which is considered to be a good approximation, even in a real molecular system. Our main goal is to gain further insight into the nature of weak rare earth-ligand interactions, paying particular attention to the role played by the charge transfer mechanism.

Unfortunately, the theoretical description and un- derstanding of lanthanide compounds face several challenging problems. First of all, the open 4f5d6s valence shell results in a multitude of quasi-degener- ate electronic states, which are often characterized by similar chemical properties. Furthermore, the elec- tron correlation in high angular momentum shells and concurrent relativistic effects on the electron dynamics play a relevant role, which are sometimes troublesome to handle in the framework of standard ab initio approaches [15]. Finally, in addition to the non-relativistic LS coupling, relativistic jj (spin- orbit) coupling is no longer a small perturbation and may significantly influence the molecular properties [16].

From a technical point of view, it has been found that the methods rooted in density functional (DF) theory are advantageous, since they provide an accu- rate description of metal-ligand interactions without losing the simple chemical interpretation arising from a single-determinant scheme [17]. In particular, sev- eral studies have recently shown that a great im- provement over standard DF results can be achieved by a partial inclusion of Hartree-Fock (HF) ex- change, leading to the so-called self-consistent hy- brid (SCH) approaches [18]. These methods are able to provide, at the same time, reliable geometric,

thermodynamic and spectroscopic parameters for a wide class of metal-ligand interactions, going from a covalent bond to weak non-covalent interactions [19-21].

Regarding relativistic effects, one of the most widely used methods relies on the use of effective core potentials (ECPs). In these ECPs, the inner shell electrons are replaced by a model potential in which relativistic effects (i.e. mass-velocity and Darwin terms) are taken into account since they originate from Dirac-Hartree-Fock calculations [22].

Within this theoretical frame, 'state of the art' investigations of the LnX 3 system become of partic- ular importance also for testing computational proto- cols. Therefore, one of the most successful SCH approach, namely B3LYP, coupled with relativistic ECP will constitute our computational tool in de- scribing the chemical features of lanthanide-halide bonding.

2. Computational details

All the computations are based on the Kohn-Sham (KS) approach to DF theory, as implemented in the Gaussian 94 package [23]. On the basis of some previous experience of transition metal complexes [20], we have selected the so called B3LYP ap- proach, which relies on the combination of HF and Becke exchange [24] with the Lee-Yang-Pan (LYP) correlation potential [25]. The ratios of the combina- tion are those determined by Becke [18] for a closely related functional.

Recently, Hay and co-workers have shown that effective core potentials (ECPs) generated from HF atomic computations can be safely used in DF com- putations as well [26]. Following this line, the rela- tivistic ECPs (RECPs) of Stevens and co-workers [22,27] have been utilized for lanthanide atoms. These RECPs explicitly consider the 5s and 5p as well as the 4f electrons in the valence space. The valence basis set has the con t r ac t i on s c h e m e (3111 /3111 /21 /52) for Gd and Lu, and in the case of La the contraction scheme is (5211/5211/311) and a single f function (exp.=0.261) has been added as the polarization function. The correspond- ing ECPs have been used for the halogen atoms

C. Adamo, P. MaMivi / Chemical Physics Letters 268 (1997) 61-68 63

[27,28], in order to avoid unphysical spurious effects arising from unbalanced basis set. The valence space, including the ns and np electrons, is described by a double-~" basis set with one d polarization function. This basis set has already been tested in the frame- work of post-HF computations, giving molecular geometries which are close to those obtained with a polarized double-~" all-electron basis set [7].

All the geometries have been fully optimized both in the planar (D3h) and in the pyramidal (C3v) structures using a non-gradient energy minimizer, and the resulting geometries characterized by com- puting an energy Hessian. All relative energies were corrected for zero-point energy (ZPE) effects.

Lanthanide-halide bond features have been ana- lyzed in terms of the natural bond orbital (NBO) scheme [29]. This method calculates an orthogonal set of localized bond orbitals (the NBOs) given the density and overlap matrices in an atomic orbital basis set. Since the localization is based on a maxi- mum occupation criterion [30], the NBO analysis provides a rigorous description of the molecular wavefunction close to the chemical concept of the Lewis structures. This scheme has been successfully applied to the study of metal-ligand interactions [ 19,30,31 ]. Furthermore, a more detailed description of the chemical bond can be obtained by the natural resonance theory (NRT) approach [32], which pro- vides an analysis of the molecular electron density in terms of resonance electronic structure and their relative weights. The NRT allows also for an evalua- tion of bond order and valency which are closely related to classical resonance theory concepts. Briefly, the NRT algorithm is based on the represen- tation of the one-electron reduced density operator, F , as an optimized 'resonance hybrid' of density operators, { F,~} [32]:

r= Ew ro, ( l )

where each F~ is the reduced density operator corre- sponding to an idealized resonance structure wave- function q*~, arising from a Lewis structure given by the NBO localization procedure. Bond orders are evaluated from the weights {w~} as:

b~ = Ew.b; ,~, (2) ot

where b~u is the integer number of bonds connecting

centers A and B in the resonance structure corre- sponding to the ~,~ wavefunction. These bond orders can be split into covalent and ionic contributions:

biA°B"=bABiAB; b~°aV=bAB(1--iAB), (3)

where the ionic part, ig8, of each A-B bond is defined as:

W "a ~ ' a a lAB gAB , (4) bAB

i~B is the ionicity of the A -B bond, as determined by the NBO coefficients c~ and c~' relative to the ~ wave function:

2 - - a ) 2

i(A~ )= (C~) 2 + ( c ~ ) 2 (5)

In the following we express the ionic character of the bond in terms of:

ionic character - (6)

Before beginning the description of our results, we feel it is necessary to add some comments concern- ing the chemical meaning of KS orbitals. It is known that the true KS orbitals (i.e. those that would corre- spond to the exact exchange-correlation potential) are an approximation of the HF orbitals. Salahub and co-workers [33] have shown that such a relationship is relevant, even if further approximations are in- volved in the derivation of the KS orbitals, as they are obtained from approximate functionals. More- over, Goerling has pointed out the physical meaning of the KS orbitals [34], showing that they are related to a well-defined approximation of the excitation energies. We believe that the relationship between HF and KS orbitals is even more stringent in the framework of a hybrid D F / H F scheme, in which the two approaches are, in some sense, mixed together. So we will refer, in the following, to the orbitais coming from a SCH approach only as orbitals.

3. Results and discussion

3.1. Geometrical features

The oPtimized geometrical parameters of the whole series of LnX 3 complexes are summarized in

64 C. Adamo, P. Maldioi / Chemical Physics Letters 268 (1997) 61-68

Table 1, together with post-HF [6-8] and experimen- tal data [11,13]. Let us start the discussion from the long-debated question of the planarity of such molecules. As underlined in Ref. [13], these halides (as all the other metal-halides) have low puckering mode frequencies and undergo large-amplitude mo- tions. For these reasons, they are often referred to as 'floppy' molecules. Most of the geometrical informa- tion on such molecules is obtained by electron diffraction. This technique provides an average pic- ture of the molecules being distributed on the puck- ering potential surface, so even a planar molecule may be detected in a pyramidal conformation [13]. As a consequence, some doubts on the quality of the experimentally observed valence angles may arise. Due to these considerations, we compare our results with published computations rather than with experi-

mental valence angles, allowing readers to draw their own conclusions from the data in Table 1.

Our computations indicate that LaX 3 complexes prefer a pyramidal arrangement with respect to a planar one. The energy gap between both conforma- tions is, as expected, small and regularly decreases in going from F (+0.41 kcal/mol) to I (+0.03 kcal/mol). The same trends are observed for Gd, but the energy difference is smaller (+0.08 kcal/mol for F), and the planar structure becomes an energy minimum for Br and I. For Lu, only the complex with F shows a pyramidal structure, the planar one lying only 0.02 kcal/mol higher in energy. All the planar structures are first-order saddle points charac- terized by one imaginary frequency. A possible ex- planation for the preference of the pyramidal struc- ture over the planar one, observed in our results, may

Table 1 Computed bond length (R, halides

,~), X - M - X bond angle (0, degrees) and relative energies (AE, kcal/mol) for all the considered lanthanide

b symm. B3LYP/RECP MC-SCF/RECP exp.

R 0 AE a R 0 AE R 0

L a F 3 D3h 2.171 0.41 C3v 2.161 115.1 0.00 2.176 c 118.6 2.139

LaCI 3 D3h 2.634 0.07 2.642 c C 3v 2.633 118.6 0.00 2.587

LaBr 3 D3h 2.796 0.07 C 3~ 2.779 118.7 0.00 2.741

Lal 3 D3h 2.977 0.03 2.938 C3v 2.973 117.9 0.00

GdF 3 D3h 2.053 0.08 2.051 c 0.70 C3~ 2.056 117.7 0.00 2.047 c 119.9 0.00 2.053

GdCI 3 D3h 2.519 0.03 2.528 c C3v 2.518 119.0 0.00 2.489

GdBr 3 D3h 2.665 2.68 d C3v 2.640

Gdl 3 D3h 2.884 2.91 d C3~ 2.841

LuF 3 D3h 1.992 0.02 1.979 c 0.40 1.968 C3~ 1.991 118.9 0.00 1.973 c 119.8 0.00

LuCI 3 D3h 2.447 2.450 c C3v 2.417 111.5

LuBr 3 D3h 2.590 2.60 d C3v 2.506

Lul 3 D3h 2.809 2.83 d C 3v 2.771

e

112.5

115.5 120.0

108.4

113.0

113.7

108.0 120.0

114.5

114.5

a Including ZPE corrections. b Ref. [13].c Ref. [6] and [8]. d Ref. [7]. e Structure C3v, but angle not reported in Ref. [13].

C. Adamo, P. Maldivi / Chemical Physics Letters 268 (1997) 61-68 65

be obtained by an NBO analysis (see below). In contrast, some previous MC-SCF computations sug- gest a pyramidal structure for LaF 3 but indicate a planar conformation for the chlorine compound [6].

In Table 1 are also reported the L n - X bond lengths estimated from gas-phase electron diffraction experiments [13]. Without going into detail a few general trends deserve some comments. Regarding the numerical quality of our computations, we note that the root-mean-square (rms) computed with re- sopect to the experimental L n - X bond lengths is 0.04 A for the 12 LnX 3 complexes. The differences be- tween the calculated and experimental values range between 0.02 ,~ (LuF 3) and 0.08 A (LuBr3). It must be pointed out that the rms are comparable to those obtained at the post-HF level [7], but, in our case, B3LYP deviation is almost constant throughout the halide series, the rms being 0.02 ,~ for LnF 3, 0.04 ,~ for LnC13, 0.05 ,~ for LnBr 3 and 0.04 A for LnI 3.

From a more chemical point of view, two main trends arise from our computations. The first one concerns the significant increase in bond lengths observed in going from fluoride to bromide com- plexes. These lengthenings (+0 .46 ~, from F to CI, + 0.15 ,~ from C! to Br and + 0.22 ,~ from Br to I) are constant and do not depend upon the nature of the metal atom, whereas the experimental values are

o

+0.45, +0.15 and +0.21 A, respectively. The par- ticularly important lengthening observed in going from F to C1 is due to a four electrons-two orbitals repulsive interaction, which is swi tched on from

fluorine to chlorine (see below). The second trend is the bond length shortening in going from LaX 3 to GdX 3, and to LuX 3, which is related to the so-called 'rare-earth contraction'. These variations are inde- pendent of the halogen, being -0 .11 A in going from La to Gd and - 0.07 A from Gd to Lu, whereas

o

the experimental values are - 0 . 1 0 and - 0 . 0 8 A, respectively. The same trends, but with greater nu- merical differences, have been found either at QCISD [5] or MC-SCF [7] levels.

In Table 2 are collected the B3LYP/ECP har- monic wavenumbers and zero point energies for the minimum energy structures, together with the experi- mental values [11]. Unfortunately, many of the ex- perimental values are, in fact, estimated by an ex- trapolation, obtained by linear functions of the atomic number, of the few available experimental infrared spectra, namely those of fluorine and chlorine com- pounds. As a consequence, the overall agreement between the theoretical and experimental values is not excellent, the average rms of the B3LYP compu- tations being 27 cm-~. This error is dramatically reduced to 9 cm- J, if only true experimental values are taken into account. The corresponding MC-SCF value is 21 cm- i [6,8] for this latter set of data.

In summary, we believe that chemically signifi- cant trends are well reproduced by our computations, with an accuracy which is constant throughout the lanthanide series. From a more technical point of view, it is sound that two different methods, namely B3LYP and MC-SCF, give results which are close to

Table 2 Computed harmonic wavenumbers (cm -] ) and zero-point energies (ZPE, kcal/mol) for all the considered lanthanide complexes. In parentheses are reported experimental values taken from Ref. [11]

symm. v] v 2 1"3 /J4 ZPE

LaF 3 C3v 527 (540) 83 (82) 512 (511) 131 (125) 2.7 LaCI 3 C 3v 317 (333) 43 (51) 316 (316) 76 (79) 1.6 LaBr 3 C 3v 258 (265) 44 (41) 245 (252) 80 (63) 1.1 Lal 3 C3v 190 (186) 32 (28) 135 (177) 18 (44) 0.9 GdF 3 C3v 569 (573) 119 (97) 553 (552) 138 (136) 2.9 GdCI 3 C3v 333 (338) 43 (56) 331 (326) 77 (84) 1.7 GdBr 3 D3h 238 (268) 47 (44) 238 (192) 64 (66) 1.1 Gdl 3 D3h 194(188) 31 (20) 138 (181) 35 (46) 0.9 LuF 3 C 3v 583 (598) 97 (106) 569 (585) 150 (147) 2.9 LuCI 3 D3h 341 (342) 75 (60) 318 (331) 78 (88) 1.7 LuBr 3 D3h 242 (242) 32 (48) 198 (263) 51 (70) 1.0 Lul 3 D3h 195 (191) 29 (34) 140 (185) 37 (49) 0.9

6 6 C. Adamo, P. Maldioi / Chemical Physics Letters 268 (1997) 61-68

each other using the same RECP. Since MC-SCF computations are more time-consuming, our results support the B3LYP/RECP protocol as a viable alter- native for the study of rare-earth complexes.

3.2. Bonding features

Up to now, all semi-empirical [3,4] and ab initio [5-7] computations on LnX 3 compounds suggest that lanthanides can be involved in a strong covalent bond. These conclusions are usually carried out on the basis of a standard electronic population analysis, such as the popular Mulliken one. Without going into further detail, the theoretical limitations of such an analysis are well known (see for instance Ref. [35]), especially concerning metal-ligand interac- tions. In contrast, the NBO analysis provides elec- tronic populations which are, in some sense, more reliable and stable to computational parameters (e.g. basis sets) [30].

In Fig. 1 are plotted the Mulliken and NBO charges of the lanthanide atoms, obtained at the B3LYP level, for the fluorine and iodine complexes, in their minimum energy conformation. From this plot, it is quite clear that each method gives a dramatically different description of the L n - X bond. In fact, the Mulliken charges indicate the presence of a bond with a partial covalent character for the fluorine complex (q = 1.701e-I), and a totally cova- lent bond for the iodine complex (q < 1.01e-I). Fur- thermore the charge born by the metal is almost constant throughout the whole lanthanide series. On

3.0-

2.5:

2.0-

~ ' 1.5-

1.0 I ~ n e

0.5 La

- - t ~ - - Mul l iken

- - , ~ - - N B O

Fluorine

o t~

& ~u

Fig. 1. Mulliken and NBO atomic charges for the LnF 3 and Lnl 3 (Ln = La, Gd, Lu) complexes.

1.0-

0.9

0.8-

.o 0.7-

_~ 0.6

- - H - - L a

- - O - - Gd

- - A - - Lu

0.5

F 6 ~ir i Fig. 2. Variation in ionic character of the Ln-X bond in the trihalide complexes of La, Gd and Lu.

the contrary, a NBO analysis indicates a strong ionic bond (q > 2.51e-1), which significantly decreases in going from Gd to Lu (q -- 2te-I for LuI3).

The role played by electrostatic and covalent con- tributions to the bonding is sketched in Fig. 2 in terms of the variation of the ionic character (as defined in Eq. 9) for all the considered lanthanide complexes. As expected, the ionic character of the L n - X bond decreases as a function of the elec- tronegativity of the ligand, reaching a minimum ( = 0.75) for iodine. The ionic contribution is almost the same for all complexes of Gd and Lu, while it is somewhat greater for the La derivatives. This dis- crepancy increases in going from F to CI and this can be related to the preference for the planar struc- ture over the pyramidal one throughout the halide series. Let us discuss this point in some detail. The enhanced stability of the C3v structure over the D3h

is due to a more favorable interaction between the occupied prr orbital of the halide iigand with the empty lanthanum dz~ orbital [3]. As a consequence, the positive atomic charge on the metal atom slightly increases in going from pyramidal to planar struc- ture. This effect is more evident in LaF 3, where the NBO charge on the metal is 2.581e-I for the C3v structure and 2.601e-t for the D3h. Meanwhile, the d population decreases from 0.35 to 0.331e-I.

For all the other lanthanides, characterized by a planar or quasi-planar rearrangement, an increase in the electronic population of the 6s orbital is also found. For instance, the population of the 6s orbital

C. Adamo, P. Maldivi / Chemical Physics Letters 268 (1997) 61-68 67

of La is 0.08te-] and 0.10le-[ in the chloride and bromide complexes respectively, but increases up to 0.25 and 0.301e-I for the corresponding Lu com- plexes. This effect is related to the decrease in the energy gap between the 6s orbital of the metal and the p orbitals of the halide, in going from La to Lu. For instance, the energy difference between the halide p and metal 6s orbital is about 0.49 hartree in LaCI 3 and 0.41 hartree in LuCI 3. Thus a significant charge transfer from the halide atoms to the metal may occur, thus accounting for the variation in the ionic/covalent ratio in going from La to Lu (Fig. 2).

In summary, two competing effects determine the molecular geometry. On one hand, a pyramidal struc- ture is favoured by the overlapping of a d.: lan- thanide orbital with the prr halide orbital, which is forbidden in the planar conformation. On the other hand, a planar structure arises when the lanthanide 6s orbital becomes closer in energy to the in-plane p orbital of the halogen ligands. This is achieved either by a decrease in the energy of the 6s orbital, due to the shielding of 4f electrons in going from La to Lu, or to the energy increase of np of the halide from F to I.

The residual covalent bonding interaction present in the Ln complexes may be described as an interac- tion of the sd hybrid of the metal atom and the two p,, orbitals of the halide. The character of this hybrid changes in going from La (s 16.7%, d83.3%) to Lu (s 28.4%, d 71.6%).

As mentioned in the previous section, the increase in bond length in going from F to CI is due to the four-electrons/two-orbitals repulsion arising from the repulsive interaction of the occupied 5p orbital of the metal atom with the occupied ns orbital of the halide moiety. In fact, the 5p orbital of lanthanide atoms (e.g. -0 .892 hartree in La) is closer to the 3s orbital of chlorine ( - 0 . 7 8 8 hartree) than to the 2s orbital of F ( - 0.377 hartree). When going from La to Lu the 5p energy decreases ( - 1 . 173 hartree in Lu), resulting in a weaker repulsion, thus giving the 'rare-earth contraction' observed for bond lengths [I].

At the end of this analysis, it must be added that the NBO procedure indicates a small variation of the f electronic population (0.01-0.021e-I) in all the complexes, thus confirming, once again, the negligi- ble role of these orbitals [10].

4. Conclusion

We have carried out a detailed analysis of the LnX s (Ln = La, Gd, Lu and X = F, C1, Br, I) com- plexes, using a hybrid density functional/Hartree- Fock method together with relativistic effective core potentials. Two main conclusions may be drawn from our computations. First, from a technical point of view we have found that this protocol combines the advantage of a computational scheme (here B3LYP method) which explicitly considers the cor- relation effects with the efficiency arising from the core potentials, without lacking in chemical accu- racy. Secondly, a natural bond orbital analysis has pointed out the relative weights of electrostatic and charge transfer interactions in these complexes. In particular we have found that charge transfer plays a significant role for relatively weak ligands, such as bromide and iodide. This latter result should give some hints for a more accurate description of metal atoms within classical molecular mechanics simula- tions.

Acknowledgements

The authors thank Professor Vincenzo Barone and Professor Robert Subra for helpful discussions and suggestions, and Professor Weinhoid for providing a copy of the latest version of the NBO program together with the Corresponding technical reports of his Institute.

References

[1] S. Cotton, Lanthanides and Actinides, MacMillan, London, 1991.

[2] J. Molnar, M. Hargittai, J. Phys. Chem. 99 (1995) 10780. [3] C.E. Myers, L.J. Norman, L.M. Loew, lnorg. Chem. 17

(1978) 1581. [4] L.L. Lohr, Y.Q. Jia, Inorg. Chim. Acta 119 (1986) 99. [5] M. Dolg, H. Stoll, H. Preuss, J. Mol. Struct. (THEOCHEM)

235 (1991) 67. [6] S. Di Bella, G. Lanza, I.L. Fragalh, Chem. Phys. Lett. 214

(1993) 598. [7] T.R. Cundari, S.O. Sommerer, L.A. Strohecker, L. Tippett, J.

Chem. Phys. 103 (1995) 7058. [8] G. Lanza, I.L. Fragalb., Chem. Phys. Lett. 255 (1996) 341. [9] B.P. Hay, Coord. Chem. Rev. 126 (1993) 177.

68 C. Adamo, P. Maldioi / Chemical Physics Letters 268 (1997) 61-68

[10] R.D. Wesley, C.W. DeKock, J. Chem. Phys. 55 (1971) 3866. [11] C.E. Myers, D.T. Graves, J. Chem. Eng. Data 22 (1977) 436. [12] B. Ruscic, G.L. Goodman, J. Berkowitz, J. Chem. Phys. 78

(1983) 5443. [13] M. Hargittai, Coord. Chem. Rev. 91 (1988) 35. [14] O. Kahn, Molecular Magnetism, VCH, New York, 1993, ch.

3. [15] G.L. Malli (Ed.), Relativistic and Electron Correlation Ef-

fects in Molecule and Solids, NATO-ASI Series B 318, Plenum Press, New York, 1992.

[16] E. van Lenthe, E.J. Baerends, J.G. Snijders, J. Chem. Phys. 101 (1994) 9783.

[17] T. Ziegler, Chem. Rev. 91 (1991) 651. [18] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [19] A. Ricca, C.W. Bauschlicher, J. Phys. Chem. 98 (1994)

12899. [20] V. Barone, C. Adamo, J. Phys. Chem. 100 (1996) 335. [21] M.C. Halthausen, C. Heinemann, H.H. Cornhl, W. Koch, H.

Schwarz, J. Chem. Phys. 102 (1995) 4931. [22] T.R. Cundari, W.J. Stevens, J. Chem. Phys. 98 (1993) 5555. [23] M.J. Frisch, G.W. Trucks, H.B. Schlegel, P.M.W. Gill, B.G.

Johnson, M.A. Robb, J.R. Cheeseman, T.A. Keith, G.A. Petersson, J.A. Montgomery, K. Raghavachari, M.A. AI- Laham, V.G. Zakrewski, J.V. Ortiz, J.B. Foresman, B. Cioslowski, B. Stefanov, A. Nanayakkara, M. Challacombe, C.Y. Peng, P.Y. Ayala, W. Chen, M.W. Wong, J.L. Andres,

E.S. Replogle, R. Gomperts, R.L. Martin, D.J. Fox, J.S. Binkley, D.J. DeFrees, J. Baker, J.P. Stewart, M. Head- Gordon, C. Gonzalez, J.A. Pople, Gaussian 94, Gaussian, Pittsburgh, PA, 1995.

[24] A.D. Becke, Phys. Rev. B 38 (1988) 3098. [25] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [26] T.V. Russo, R.L. Martin, P.J. Hay, J. Phys. Chem. 99 (1995)

17085. [27] W.J. Stevens, M. Krauss, H. Basch, P.G. Jasien, Can. J.

Chem. 70 (1992) 288. [28] W.J. Stevens, H. Basch, M. Krauss, J. Chem. Phys. 81

(1984) 6026. [29] E.D. Glendening, J.K. Badenhoop, A.E. Reed, J.E. Carpenter

and F. Weinhold, NBO 4.0, Theoretical Chemistry Institute, University of Wisconsin, Madison, WI, 1996.

[30] A.E. Reed, L.A. Curtiss, F. Weinhold, Chem. Rev. 88 (1988) 899.

[31] E.D. Glendening, J. Am. Chem. Soc. 118 (1996) 2473. [32] E.D. Glendening and F. Weinhold, University of Winscon-

sin, Theoretical Chemistry Institute Technical Report WIS- TCI-803 (1994)

[33] P. Duffy, D.P. Chong, M.E. Casida, D.R. Salahub, Phys. Rev. B 50 (1994) 4707.

[34] A. Goerling, Phys. Rev. A 54 (1996) 3912. [35] F. De Profit, J.M.L. Martin, P. Geerling, Chem. Phys. Lett.

250 (1996) 393.