Prof. Dr Gerd ; Meyer
email: gerd.meyer@uni-koeln.de

Thomas Gloger

Jan Beekhuizen

Halides of Titanium in Lower Oxidation States

Keywords: Titanium, Dihalides, Trihalides, Complex halides, Cluster compounds Titanium, Dihalides, Trihalides, Complex halides, Clusters

New investigations on the di-and trihalides of titanium, TiX 2 and TiX 3 (X = Cl, Br, I), with their 3d 2 and 3d 1 electronic configurations, confirm the early observations and conclusions of Klemm. At sufficiently low temperatures, Ti─Ti single bonds are formed in the one-dimensional trihalides, i.e., Ti─Ti dimers are observed. Equally, in the two-dimensional dihalides, {Ti 3 } triangles occur with three single bonds. Phase transitions were detected from single-crystal or powder X-ray diffraction data, from magnetic measurements and thermal analysis. Except for the binary halides a number of ternary halides ATiX 3 (extended chains of face-sharing octahedra), K 4 Ti 3 Br 12 (triples of face-sharing octahedra), Na 2 Ti 3 Cl 8 (triangular trimers), A 3 Ti 2 X 9 (dimers of face-sharing octahedra), and A 3 TiX 6 (isolated octahedra) as well as the mixed-valent halides CsTi 2 I 7 (tetrahedra and octahedra) and Na 5 Ti 3 Cl 12 (chains of octahedra) have been observed. Except for the triangles in titanium(II) halides, cluster compounds are rare but include K 4 [{OTi 4 }I 12 }] and {CTi 6 }Cl 14 .

Introduction

In accord with its position in the periodic table as a group 4 element and, therefore, its electronic configuration [Ar]4s 2 3d 2 , titanium appears in simple halides in the oxidation states +4, +3, and +2. Of these, the oxidation states +3 and +2 are the most interesting because there titanium has one (3d 1 ) and two (3d 2 ) unpaired electrons, respectively. This gives, at sufficiently high temperatures, rise to paramagnetism. At low temperatures, collective magnetic effects may appear, or, as in the case of the titanium(II) and titanium(III) halides, chemical bonding may emerge.

First work on lower-valent titanium halides dates back to the early years of the twentieth century [1]. But only in the forties, thorough preparative work combined with magnetic measurements shed considerable light on the nature of these halides. It was Wilhelm Klemm who found that the magnetic moments dropped to low values at low temperatures, depending upon the anion X -= Cl -, Br -, I -, in both dihalides and trihalides, TiX 2 [2] and TiX 3 [3]. The drop of the magnetic moments is accompanied by a reduction of the molar volumes. Klemm has interpreted this observation as the formation of covalent bonds (or, alternatively, as antiferromagnetic behaviour).

2 Binary halides, TiX 3 and TiX 2

Titanium(III) halides: d 1 systems

The structural topology of the trihalides of titanium, TiX 3 with X = Cl, Br, I, can be deduced from a hexagonal closest packing of spheres, i.e., of halide ions. One third of the octahedral voids are filled with Ti 3+ cations, in a way that the filled [TiX 6 ] octahedra share common faces, in accord with the Niggli formula TiX 6/2 . The chains thus formed run parallel to the hexagonal c axis in the high-temperature modification named, in the following, TiX 3 -II (hexagonal, P6 3 /mcm), see Fig. 1.

For TiI 3 , structural investigations were carried out by Holze [4] and v. Schnering [5]. This latter investigation, carried out by single-crystal X-ray crystallography, exhibited for TiI 3 the hexagonal space group P6 3 /mcm. Thus, Ti─Ti bonds cannot be present as these would lead to a symmetry reduction to Pmmn (orthorhombic), see for example [6]. This low-temperature modification, MX 3 -I, was observed so far for ZrI 3 [7], MoX 3 (X = Br, I) [8], RuCl 3 [6], RuBr 3 [9], and OsI 3 [10]. Chains of [MX 6 ] octahedra run along the orthorhombic b axis and the M─M distances are not equidistant anymore as in MX 3 -II.

The crystal structures of TiBr 3 -I and TiI 3 -I were determined from single-crystal data at ambient temperature [11]. Ti─Ti distances are 292.3 (7) and 315.7 (7) pm for TiBr 3 -I and 309.6(2) and 339.6(2) pm for TiI 3 -I, respectively. The respective distances for the higher homologue ZrI 3 are 317.2(2) and 350.7(2) pm, subject to the larger zirconium atom. Luckily, it was also possible to collect data sets for TiI 3 in the temperature range of 183 to 448 K [11]. Fig. 2 shows the Ti─Ti distances derived therefrom versus temperature. An analysis of these data deduces a second-order phase transition at 317 K with an (equidistant) Ti─Ti distance of 326 pm at that temperature. These results are confirmed by a recent study of the structures and phase transition of TiI 3 [START_REF] Angelkort | 16th International Conference on Solid Compounds of Transition Elements[END_REF]. This phase transition (II → I) can be interpreted as a Peierls distortion arising from degenerate partially filled d bands at sufficiently high temperatures. At lower temperatures, the band structure of TiI 3 -I shows a strong splitting of these d bands (see, e.g., at the Г (G) point, Fig. 3) which is due to attractive interactions (covalent bonds; antiferromagnetic coupling) between the titanium atoms right below the Fermi level. From a molecular orbital point of view, σ bonds are present between every second pair of titanium atoms (Fig. 3).

Titanium(II) halides: d 2 systems

The titanium dihalides TiX 2 with X = Cl, Br, I crystallize at sufficiently high temperatures with the CdI 2 type of structure. Thus the crystal structures of TiX 3 and TiX 2 are closely related to each other. They both exhibit a hexagonal closest packing of halide ions with one third of the octahedral interstices occupied between every adjacent layer as in TiX 3 and all octahedral interstices occupied between every second layer as in TiX 2 . Thereby all [TiX 6 ] octahedra share common edges to form a two-dimensional layer according to the Niggli formula TiX 6/3 . A 3 6 net of titanium atoms is thus formed with all titanium atoms equidistant at sufficiently high temperatures, modification TiX 2 -II (Fig. 4, left). Upon cooling the 3 6 net of Ti 2+ distorts such that titanium triangles occur. These triangles are surrounded by thirteen halide ions and are further connected to the two-dimensional X-Ti-X slab as is typical for the CdI 2 type of structure. These [Ti 3 X 13 ] units (Fig. 4) are topologically also seen in Nb 3 X 8 [START_REF] Von Schnering | [END_REF] and in Na 2 Ti 3 Cl 8 [14], see below.

The phase transition TiX 2 -II → TiX 2 -I can thus be understood as a two-dimensional Peierls distortion where a symmetry reduction from P-3m1 (TiX 2 -II, CdI 2 type) to P3 (TiX 2 -I) takes place with the hexagonal a axis enlarged by √3. While the crystal structure of TiI 2 -II could be determined by powder neutron diffraction [a = 412.72(2), c = 684.74(2) pm, P-3m1, Z = 1, T = 127 K, z(I) = 0.2409(1), R p = 0.045, R wp = 0.062, χ 2 = 0.081], the crystal structure of TiBr 2 -I could only be refined to an R 1 of 0.226 by "single crystal" X-ray diffraction [11]. However, Ti─Ti distances in the [Ti 3 ] triangles of TiBr 2 -I are 300.1 (20) pm and thereby somewhat longer than the 292.3 (7) pm as observed in TiBr 3 -I. This could be interpreted in a way that in TiBr 3 -I two adjacent [TiBr 6 ] octahedra are connected through a common face while in TiBr 2 -I three adjacent [TiBr 6 ] octahedra are connected through three common edges. Therefore, it can be concluded that in both TiBr 2 -I and TiBr 3 -I Ti─Ti single bonds are formed at sufficiently low temperatures.

Powder X-ray diffraction data for TiCl 2 (Fig. 5) and TiBr 2 show the I → II phase transition (upon heating) at 640 K and 450 K and the II → I transition (upon cooling) at 530 K and 320 K, respectively. Hence, there are rather large hystereses of 110 and 130 K observed for the chloride and bromide, respectively. Magnetic measurements show the I → II phase transition for TiI 2 (upon heating) starting at 240 K and finished at 280 K (judging from the effective magnetic moment µ eff ) and the II → I transition (upon cooling) beginning at 180 K and finished at 140 K (Fig. 6). Hence, the width of the hystereses is about 100 K. Note that Klemms values of 1942 [2] fit very well the values obtained by us (Fig. 6, right). The transition temperatures increase in the sequence I → Br → Cl (with increasing "hardness" of the halide lattice) and the widths of the hystereses appear to be almost independent of the halide lattice. If we consider those temperatures alone where the phase transition is completed upon heating and cooling, we observe the following picture:

TiCl 2 430 K / 590 K, ∆ = 160 K TiBr 2 290 K / 440 K, ∆ = 150 K TiI 2
140 K / 280 K, ∆ = 140 K Difference Thermal Analysis (DTA) curves were also recorded for TiCl 2 and TiBr 2 . Transition temperatures (endo-thermic) are 525 and 350 K upon heating for TiCl 2 and TiBr 2 , respectively, and 600 and 445 K upon cooling (exothermic effect).

Differential scanning calorimetry (DSC) exhibits an endothermic effect for TiI 2 at 270 K and an exothermic effect at 172 K upon cooling. The caloric effects are rather small, enthalpies derived therefrom are 0.26 kJ/mol upon heating and 0.22 kJ/mol upon cooling (Fig. 7). 3 Ternary halides, A x Ti x X z

Titanium(III) in isolated octahedra

In all of the ternary titanium(III) halides with alkali-metal cations, Ti 3+ cations occupy octahedral interstices. There is a variety of structures known for A 3 [MX 6 ] type halides which differ in the orientation of the [MX 6 ] 3-octahedra relative to each other. Thereby the proper coordination sphere for the A + cation with respect to its size is achieved, see Table 1. 353.4 [8]; 348.5 [8]; 377.8 [11] Na 3 [TiCl 6 ] crystallizes with the cryolite-type of structure (Na 3 [AlF 6 ]) [15], Fig. 8. Its undistorted aristotype structure would be that of K 2 Na[AlF 6 ] (elpasolite). In the ambient condition structure, two Na + cations with coordination numbers of eight resemble K + in the elpasolite structure (CN = 12), the third one is in an octahedral hole, see Table 1. K 3 [TiCl 6 ] crystallizes with the K 3 MoCl 6 type of structure where the relative orientation of the [TiCl 6 ] 3- octahedra allows for coordination numbers of 8 (K1, K2) and 10 (K3), see Table 1 and Fig. 9. Rb 3 [TiCl 6 ] and Cs 3 [TiCl 6 ] both crystallize with the Cs 3 BiCl 6 type of structure [16] with coordination numbers of 8 (Cs1, Cs2) and 11 (Cs3), see Fig. 10 and Table 1.

All four compounds show at least one phase transition according to DSC measurements, at 153°C for Na 3 [TiCl 6 ], 246°C (K 3 [TiCl 6 ]), 311°C (Rb 3 [TiCl 6 ]), and 344°C (Cs 3 [TiCl 6 ]).

The crystals of all these hexachlorotitanates(III) have a yellow colour. Their absorption spectra show broad double bands ( [16], Fig. 11) as expected [17] because the e g niveaus are not entartet. Magnetic susceptibility data show a very similar behaviour for the four A 3 [TiCl 6 ] type compounds, see Fig. 11 for the example of Na 3 [TiCl 6 ]. The magnetic moment µ corresponds to the expected value of 1.73 µ B at room temperature but it is strongly dependent upon temperature.

Titanium(III) in dimers of octahedra and tetrahedra

Halides of the general composition A 3 Ti 2 X 9 contain dimers [Ti 2 X 9 ] 3-which consist of two octahedra sharing one common face, see Fig. 12. Many trivalent cations throughout the periodic table form these dimers and there is a small number of structure types of which the Cs 3 Cr 2 Cl 9 and the Cs 3 Tl 2 Cl 9 types of structure are the most abundant, see for example [18]. The A 3 Ti 2 X 9 type halides of titanium(III) belong to the Cs 3 Cr 2 Cl 9 type of structure [19], K 3 Ti 2 Cl 9 to a lower symmetry variant known as the K 3 W 2 Cl 9 type [20]. They may be derived from a ABACBC stacking of closest-packed CsCl 3 layers between which two thirds of the available octahedral interstices are occupied such that confacial double octahedra occur which are isolated from each other. The alkali-metal cations are in cuboctahedral and anti-cuboctahedral surrounding, see Fig. 13. Important structural parameters, distances and angles, within the [Ti 2 X 9 ] 3-dimers are summarized in Table 2. These show, of course, that the Ti-Ti distances grow with increasing sizes of A + and X -. The distances to the terminal halide ions (X t ) are always shorter than to the bridging ligands (X b ), d 1 /d 2 > 1. The angles α" and β (Fig. 12) lie within a narrow range for the titanium(III) dimers with α" always larger than β with small decreases and increases, respectively, with increasing A + size and almost no differences with increasing X -size. The question whether there are bonding interactions between the Ti 3+ -Ti 3+ (d 1 -d 1 ) centres, with distances ranging from 310 (K 3 Ti 2 Cl 9 ) to 364 pm (Cs 3 Ti 2 I 9 ) is not easy to answer from crystallographic data alone. However, there is a dramatic decrease in the M 3+ -M 3+ distance for Cs 3 M 2 Cl 9 from M = Sc (357 pm) to M = Ti (322 pm), 9.8%, hence from a 3d 0 to a 3d 1 electronic configuration. This is certainly significant although not quite as dramatic as the 22.7% decrease for the Cs 3 M 2 I 9 couple (M = Y → Zr) where, of course, 4d electrons are involved in metal-to-metal bonding [24], see Table 2. [Ti 2 X 9 ] 3-dimers are also present in CsTi 2 Cl 7 -II [27] which are, however, further connected via edges to chains, see Fig. 14. The structure can be understood in terms of ABAC stacking of, alternately, CsCl 3 and Cl 4 layers with Ti III occupying one quarter of the octahedral holes (Fig. 14,left). There is a second modification which was discovered first, CsTi 2 Cl 7 -I [28], where [TiCl 6 ] octahedra share common edges and vertices. The Ti-Ti distance through the common face in CsTi 2 Cl 7 -II is 322.6 pm, much the same as in Cs 3 Ti 2 Cl 9 (321.5 pm), see above. Distances through the common edges are 347.2 and 357.0 pm in II and I, respectively. Keeping the Ti-Ti distance differences in mind, it is not surprising that the 3d 1 -3d 1 interactions are much more pronounced through the common face in II than through the common edges, about 16 times as much as the interpretation of magnetic susceptibility measurements exhibited, see Fig. 15. In contrast to CsTi 2 Cl 7 where Ti III resides in octaehedral interstices, the rather rare tetrahedral coordination is observed for Ti III in CsTi 2 I 7 [29]. Two tetrahedra share one common vertex just as in Cl 2 O 7 , Fig. 16. Ti 3+ -I -distances are around 251 pm for the terminal iodide ions and 262 pm for the bridging ones. As compared to the octahedral coordination in the [Ti 2 I 9 ] 3-double octahedra, there is a roughly 10% distance reduction for the tetrahedral coordination in [Ti 2 I 7 ] -.

Table 2. M 3+ -M 3+ distances, distance ratios d 1 /d 2 , d 1 = d(M-X b ), d 2 = d(M-X t ) (b = bridging, t = terminal) and angles α" (X b -M-X b ) and β (M-X b -M),
Figure 16. Projection of the crystal structure of CsTi 2 I 7 exhibting TiI 4 tetrahedra sharing a common vertex to two symmetrically independent [Ti 2 I 7 ] -dimers (right).

Mixed-valent titanium(II, III) halides

Open trimers of octahedra sharing two common faces are present in K 4 Ti 3 Br 12 [30] and In 4 Ti 3 Br 12 [31]. The structures are very similar and show Ti-Ti distances through the shared faces of 320 and 309 pm, respectively. These bromides must be mixed-valent according to Ti 3+ -Ti 2+ -Ti 3+ . Hence, there are four electrons for this open trimer which may lead to bonding interactions. Indeed, strong antiferro-magnetic coupling is seen for In 4 Ti 3 Br 12 [31].

Edge-sharing [TiCl 6 ] 3-octahedra are present in a second mixed-valent Ti(II, III) compound, Na 5 Ti 3 Cl 12 [32]. It contains chains of [TiCl 6 ] octahedra sharing trans edges according to [TiCl 2/1 Cl 4/2 ] -, Fig. 17. Na 5 Ti 3 Cl 12 is isostructural with a number of dysprosium and thulium bromides and iodides, A 5 M 3 Br 12 and K 5 M 3 I 12 with A = K, Rb and M = Dy, Tm [33]. For an isolated trimer according to the formula [Ti 3 Cl 12 ] 5-the titanium atoms would have the charges +2, +3 and +2, the opposite as in K 4 Ti 3 Br 12 . However, it is not possible to distinguish between the three titanium atoms as they are incorporated in infinite chains and are crystallographically all equal. Therefore, as for the corresponding dysprosium and thulium bromides and iodides, there are 1.67 electrons per titanium atom in Na 5 Ti 3 Cl 12 and magnetic interactions within the chains are expected as well as three-dimensional ordering at low temperatures. An alternative to the infinite chains as observed in Na 5 Ti 3 Cl 12 are oligomers of which only the {Ti 3 } variant has been observed so far in KTi 4 Cl 11 [34] (Fig. 18), but see for the square planar [{OTi 4 }I 12 ] 4-below. KTi 4 Cl 11 is mixed-valent according to K(Ti 4+ ){(Ti 2+ ) 3 }(Cl -

Titanium(II) in trimers and in edge-and face-sharing chains

Isolated [TiX 6 ] 4-octahedra of divalent titanium appear not to be known. In Na 2 Ti 3 Cl 8 [14], the same trimers occur as in TiCl 2 at low temperatures, in Ti 7 Cl 16 and in KTi 4 Cl 11 . There is a phase transition which is driven by much the same forces as for the phase transition in TiX 2 . At ambient temperature, the Ti 2+ -Ti 2+ distances are all equal, at 372 pm. Triangular clusters are formed at decreasing temperature, the corresponding distances become unequal, 300 and 400 pm, respectively, see Fig. 19. There is a considerable volume decrease associated with the phase transition, at 183 K it is by 9% smaller than at ambient temperature. The magnetic moment per Ti II in Na 2 Ti 3 Cl 8 drops considerably around the phase transition and exhibits a pronounced hysteresis, see Fig. 21. At sufficiently high temperatures, the magnetic moment of Na 2 Ti 3 Cl 8 -II corresponds very well with that of Na 2 TiCl 4 . Hence, we then assume [TiCl 6 ] octahedra with Ti II centres which do not interact magnetically. In Na 2 TiCl 4 , [TiCl 6 ] octahedra share common edges to chains (Fig. 22), with Ti 2+ -Ti 2+ distances of 385.6 pm [36]. Below 13 K, magnetic superstructure reflections in a neutron diffraction experiment account for the onset of a three-dimensional antiferromagnetic ordering [37]. Infinite chains of [TiX 6 ] octahedra are also present in ATiX 3 type halides (A = K, Rb, Cs; X = Cl, Br, I) [38][39][40][41]. However, the chains are face-sharing according to A[TiX 6/2 ]. They crystallize essentially with the CsNiCl 3 type of structure, so-called hexagonal perovskite. Thus, AX 3 closest-packed layers are stacked in the AB fashion. One third of all octahedral interstices are occupied, all along the c direction. The structure field map of Fig. 23 shows that when the size ratio of A + to X -becomes unfavourable, there is a symmetry reduction from P6 3 /mmc to P6 3 (or P6 3 cm as reported for RbTiI 3 [38]. Fig. 24 shows that in these latter halides the [TiCl 6/2 ] chains are rotated and shifted against each other. The reason for this is to accomodate monovalent cations which are too small to fit into an anti-cuboctahedral surrounding which is, on the other hand, typical for the CsNiCl 3 -type of structure. Ti 2+ -Ti 2+ distances are little above 300 pm for the respective chlorides ATiCl 3 , around 317 in bromides and little below 340 in iodides, see for an overview [41].

Titanium halides with clusters other than dimers and trimers

With 3d elements clusters are rather rare although by no means impossible. For titanium in halides, the trimeric six-electron cluster {Ti 3 } appears to be largest cluster when these are considered without endohedral, electron-contributing atoms. An isolated octahedral cluster {Ti 6 } would afford 16 electrons for eight 3c-2e bonds. This would only be possible for a composition, for example, {Ti 6 }Cl 8 . Such a sub-chloride is not known. If an analogous structure as for {Nb 6 }Cl 14 [42], a 16-electron cluster, would be considered, an interstitial contributing four electrons is needed. Indeed, {CTi 6 }Cl 14 is the first and so far only titanium cluster of that type [43]. It is isostructural with the homologous {CZr 6 }Cl 14 [44]. In A 4 [{OTi 4 }I 12 ] (A = K, Rb) there are six electrons available for intra-cluster bonding. These six electrons occupy three low-lying molecular orbitals, see Fig. 27. Band structure calculations and the Crystal Orbital Hamiltonian Populations (COHP) derived therefrom show (Fig. 28) that there is considerable Ti-Ti bonding below the Fermi level as well as (ionic) Ti-O and Ti-I bonding. There is, however, some Ti-I bonding antibonding which must be (and is) overcompensated by Ti-Ti bonding in the same energy region. Titanium as the second transition metal has four valence electrons. In binary and ternary (quaternary) halides, the oxidation states +4 (3d 0 ), +3 (3d 1 ), and +2 (3d 2 ) are known. Of these the d 1 and d 2 "systems" are the most interesting as they offer spin, charge, and orbital ordering in low-dimensional systems. The tri-and dihalides, TiX 3 and TiX 2 (X = Cl, Br, I), are the simplest of such d 1 and d 2 systems and they are paramagnetic at sufficiently high temperatures, and undergo one-and two-dimensional Peierls distortions upon cooling. In the language of the chemist, they form σ-bonded dimers {Ti 2 } in TiX 3 and trimers {Ti 3 } in TiX 2 . The formation of {CTi 6 }Cl 14 might be regarded as a three-dimensional Peierls distortion although a single carbon atom is needed to secure a 14-electron octahedral cluster. One-and two-dimensional systems are also seen in ternary halides such as ATiX 3 (chains of facesharing [TiX 6 ] octahedra) and Na 2 TiCl 4 (chains of edge-sharing [TiX 6 ] octahedra) and in A 3 Ti 2 X 9 (dimers of face-sharing [TiX 6 ] octahedra) although magnetic interactions occur only in a way which one would not consider as a chemical bond. These are, however, observed in Na 2 Ti 3 Cl 8 and KTi 4 Cl 11 which both exhibit the same trimers as seen in TiX 2 at low temperatures. Clusters are rare in titanium halide chemistry. Except for the dimers and trimers, all the others known need an additional main-group element atom to ensure the right electron count, see {CTi 6 }Cl 14

Figure 1 .

 1 Figure 1. Crystal structure of the high-temperature modification of the titanium trihalides, TiX 3 -II (X = Cl, Br, I); right: part of one chain of [TiX 6 ] octahedra sharing common faces.

Figure 2 .

 2 Figure 2. Ti─Ti distances versus temperature for TiI 3 from single-crystal X-ray data.

Figure 3 .

 3 Figure 3. Band structure of TiI 3 -I from a DFT-LMTO ASA calculation (left); molecular orbitals from Extended Hückel MO calculations (right).

Figure 4 .

 4 Figure 4. Crystal structure of the high-temperature modification of titanium dibromide, TiBr 2 -II, and one [Ti 3 Br 13 ] unit formed within the Br-Ti-Br slabs (right) of TiBr 2 -I upon cooling.

Figure 5 .

 5 Figure 5. Powder X-ray diffraction data for TiCl 2 dependent upon temperature; left: heating, right: cooling modes.

Figure 6 .

 6 Figure 6. Effective magnetic moments µ eff versus temperature for TiCl 2 (left) and TiI 2 (right). O: Klemms data [2].

Fig. 7 .

 7 Fig. 7. Differential scanning calorimetry diagram for TiI 2 (heating and cooling rates were 8 K/min).

Figure 8 .

 8 Figure 8. Projection of the crystal structure of Na 3 [TiCl 6 ] showing the relative orientation of the [Na1Cl 6 ] and [TiCl 6 ] octahedra (left) and the coordination polyhedra around Na2 and Na1 (right, top and bottom).

Figure 9 .

 9 Figure 9. Projection of the crystal structure of K 3 [TiCl 6 ] down [010] showing the relative orientation of the [TiCl 6 ] octahedra (middle) and the coordination polyhedra around K1, K2 and K3.

Figure 10 .

 10 Figure 10. Projection of the crystal structure of Cs 3 [TiCl 6 ] (Rb 3 [TiCl 6 ] is isostructural) showing the relative orientation of the [TiCl 6 ] octahedra and the coordination polyhedra around Cs1, Cs2 and Cs3.

Figure 11 .

 11 Figure 11. Absorption spectrum above) and reciprocal magnetic susceptibility (below) for Na 3 [TiCl 6 ].

Figure 12 .

 12 Figure 12. The [Ti 2 Cl 9 ] 3-double octahedron in the crystal structure of A 3 Ti 2 X 9 type halides.

Figure 13 .

 13 Figure 13. Crystal structure of the Cs 3 Cr 2 Cl 9 type A 3 [Ti 2 X 9 ] halides; projection of the crystal structure (left) and the cuboctahedral and anti-cuboctahedral surrounding of the two symmetrically independent A + cations (right).

Figure 14 .

 14 Figure 14. Projections of the crystal structures of CsTi 2 Cl 7 -II (left) and -I (right).

Figure 15 .

 15 Figure 15. Temperature dependence of the molar magnetic susceptibility χ of CsTi 2 Cl 7 -II (insert: 1/χ; squares and circles represent different fields, 5 and 10 kOe, respectively; solid and dashed lines two model calculations, see[27]).

Figure 17 .

 17 Figure 17. Projection of the crystal structure of Na 5 Ti 3 Cl 12 exhibiting [TiCl 2/1 Cl 4/2 ] -chains which are isolated from each other.

) 11 .

 11 Six electrons are available for three σ bonds within the {Ti 3 } triangular clusters. The Ti-Ti distances are, on the average, 295.5 pm which is very similar to what is observed in Ti 7 Cl 16 = (Ti 4+ ){(Ti 2+ ) 3 } 2 (Cl -) 16[35], d = 295 pm, and to low-temperature TiBr 2 , d = 292.3(7) pm, see above.

Figure 18 .

 18 Figure 18. Projection of the crystal structure of KTi 4 Cl 11 = KTi{Ti 3 }Cl 11 exhibiting triangular six-electron titanium clusters.

Figure 19 .

 19 Figure 19. Projection of the crystal structure of Na 2 Ti 3 Cl 8 -I at 183 K with the corresponding {Ti 3 }Cl 13 units.

Figure 20 .

 20 Figure 20. Density of states (DOS) as calculated for Na 2 Ti 3 Cl 8 -I and the three energetically almost equal bonding orbitals.

Figure 21 .

 21 Figure 21. Magnetic moment µ eff versus temperature for Na 2 Ti 3 Cl 8 and for Na 2 TiCl 4 . The heating curve for Na 2 Ti 3 Cl 8 corresponds to (b), cooling to (a).

Figure 22 .

 22 Figure 22. Projection of the crystal structure of Na 2 TiCl 4 (left) and part of the [TiCl 2/1 Cl 4/2 ] 2- chains of edge-connected octahedra (right).

Figure 23 .

 23 Figure 23. Structure field map for ATiX 3 type halides exhibiting the CsNiCl 3 -type highsymmetry halides (squares, P6 3 /mmc) as well as those of lower symmetry as reported for RbTiI 3 (P6 3 cm) and for the KNiCl 3 type halides (P6 3 , open circles).

Figure 24 .

 24 Figure 24. Two projections each of the crystal structures of CsTiCl 3 (above) and KTiCl 3 (below).

  (in the Schäfer-Schnering notation[45]).

Figure 25 .

 25 Figure 25. The {CTi 6 }Cl i 12 Cl a 6 cluster unit in the crystal structure of {CTi 6 }Cl 14 .

Figure 26 .

 26 Figure 26. Projection of the crystal structure of K 4 [{OTi 4 }I 12 ].

Figure 27 .

 27 Figure 27. Molecular orbital scheme (left) for the frontier orbitals in the [{OTi 4 }I 12 ] 4-cluster unit in K 4 [{OTi 4 }I 12 ] and graphic representations of these orbitals of which the three lowest in energy are occupied by six electrons.

Figure 28 .

 28 Figure 28. Crystal Orbital Hamiltonian Population (COHP) analysis for K 4 [{OTi 4 }I 12 ].

  and K 4 [{OTi 4 }I 12 ].

Figure 1 Page

 1 Figure 1

  Figure 4

Table 1 .

 1 Mean distances <d> and coordination numbers [CN] of the cations in A 3 [TiCl 6 ] type

	chlorides
	<d>/pm	<d>/pm [CN]
	(Ti 3+ -Cl -	(A + -Cl -)
	)
	Na 3 [TiCl 6 ] 243.1	307.1 [8], 2x; 277.5 [6]
	K 3 [TiCl 6 ] 242.8	321.1 [8]; 325.7 [8]; 348.2
		[10]
	Rb 3 [TiCl 6 ] 243.8	334.0 [8]; 337.9 [8]; 365.5
		[11]
	Cs 3 [TiCl 6 ] 243.7

  see also Fig.12. Distances in pm, angles in degrees. Grey shaded: K 3 W 2 Cl 9 type, space group P6 3 /m; all the others: Cs 3 Cr 2 Cl 9 type, P6 3 /mmc.

		d(Ti-Ti) d 1 /d 2 α	β	Ref.
	K 3 Ti 2 Cl 9 310.5	1.066	85.4	76.9	11
		309.9	1.067	85.7	76.5	21
	Rb 3 Ti 2 Cl 9 314.6	1.073 84.9 77.6 16
	Cs 3 Ti 2 Cl 9 321.5	1.075 83.8 79.1 22
	K 3 Ti 2 Br 9 328.2	1.067 85.7 76.5 21
	Rb 3 Ti 2 Br 9 335.4	1.069	84.9	77.6	21
		336.6	1.070	84.7	77.8	16
	Cs 3 Ti 2 Br 9 343.4	1.074	83.8	79.0	20
		345.1	1.074	83.7	79.5	16
	Cs 3 Sc 2 I 9					23
	Cs 3 Ti 2 I 9	363.6	1.060 84.4 77.7 21
	Cs 3 Y 2 I 9	405.2	1.078 82.5 93.9 24
	Cs 3 Zr 2 I 9	312.9	1.016 93.9 64.9 24
	Cs 3 Sc 2 Cl 9 356.8	1.047 79.4 84.8 25
	Cs 3 Ti 2 Cl 9 321.5	1.075 83.8 79.1 22
	Cs 3 V 2 Cl 9 327.7	1.052 93.7 80.0 16
	Cs 3 Cr 2 Cl 9 316.7	1.070 82.7 90.5 21

Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft, Bonn (Sonderforschungsbereich 608 "Komplexe Übergangsmetall-verbindungen mit Spin-und Ladungsfreiheitsgraden und Unordnung") and by the State of Nordrhein-Westfalen as well as the Universität zu Köln. Chapter 2 has appeared in a similar version (G. Meyer, Th. Gloger, Lower Valent Titanium Halides, in: 20 Jahre Wilhelm-Klemm-Stiftung, Shaker Verlag, Aachen, 2006, p. 105-112.) to honour Professor Wilhelm Klemm and the foundation which bears his name. GM is especially grateful to his co-authors Dr. Thomas Gloger and Dr. Jan Beekhuizen) from whose theses otherwise unpublished results are reported here. We are also grateful to Dr. Dirk Hinz, Dr. Liesbet Jongen and Prof. Dr. Angela Möller who have contributed to the here reported subject over the years as is evident from the references. We are especially grateful to Prof. Dr. Anja-Verena Mudring (Bochum) who has provided much theoretical insight and to Dr. Ingo Pantenburg for continuous help in many ways. This report was first designed during a stay at Iowa State University in March of 2008 and finished during the late and early days of 2008/2009 with much support by Professor John D. Corbett. Thank you, John!


Christophe Thiery

Bruno Scherrer

Performance bound for Approximate Optimistic Policy Iteration

. This theorem applies to Least-Squares λ Policy Iteration and more generally approximate, optimistic Policy Iteration algorithms.

Theorem 1 (Performance bound for Approximate Optimistic Policy Iteration) Let (λ n ) n≥1 be a sequence of positive weights such that n≥1 λ n = 1. Let Q 0 be an arbitrary initialization. We consider an iterative algorithm that generates the sequence

(π k , Q k ) k≥1 with π k+1 ← greedy(Q k ), Q k+1 ← n≥1 λ n (B π k+1 ) n Q k + ǫ k+1 .
ǫ k+1 is the approximation error made when estimating the next value function.

Let ǫ be a uniform majoration of that error, i.e. for all k, ǫ k ∞ ≤ ǫ. Then

lim sup k→∞ Q * -Q π k ∞ ≤ 2γ (1 -γ) 2 ǫ.

Proof

Notations and main idea of the proof We will use the following notations:

• b k = Q k -B π k+1 Q k is the Bellman error, • d k = Q * -(Q k -ǫ k )
is the difference between the optimal value function and the Q k iterate (before error),

• s k = Q k -ǫ k -Q π k is the difference between the Q k iterate (before error)
and the (true) value of the policy π k ,

• β = n≥1 λ n γ n (note that 0 ≤ β ≤ γ).
The distance between the value of the optimal policy and the value of the current policy can be formulated as

Q * -Q π k ∞ = max(Q * -Q π k ) = max(Q * -Q k + ǫ k + Q k -ǫ k -Q π k ) = max(d k + s k ) ≤ max d k + max s k (1)
The idea of the proof is to compute upper bounds on d k and s k . As we will see, the bounds we will obtain will both depend on an upper on the Bellman error b k , that we derive first.

An upper bound on the Bellman error b k : As π k+1 is the greedy policy with respect to Q k , we have

B π k Q k ≤ B π k+1 Q k , which allows us to write b k = Q k -B π k+1 Q k = Q k -B π k Q k + B π k Q k -B π k+1 Q k ≤ Q k -B π k Q k = (Q k -ǫ k + ǫ k ) -B π k (Q k -ǫ k + ǫ k ) = (Q k -ǫ k ) -B π k (Q k -ǫ k ) + ǫ k -γP π k ǫ k = n≥1 λ n [(B π k ) n Q k-1 ] - n≥1 λ n (B π k ) n+1 Q k-1 + (I -γP π k )ǫ k = n≥1 λ n (B π k ) n Q k-1 ] -(B π k ) n+1 Q k-1 + (I -γP π k )ǫ k = n≥1 λ n (γP π k ) n (Q k-1 -B π k Q k-1 ) + (I -γP π k )ǫ k = n≥1 λ n (γP π k ) n b k-1 + (I -γP π k )ǫ k .
By using the fact that P π k is a stochastic matrix, we have

max b k ≤ n≥1 λ n γ n max b k-1 + (1 + γ)ǫ = β max b k-1 + (1 + γ)ǫ.
We then deduce by induction that

max b k ≤ k-1 j=0 β j (1 + γ)ǫ + β k max b 0 = 1 + γ 1 -β ǫ + O(γ k ). ( 2
)
An upper bound on d k : Let us now consider the d k term and its evolution.

d k+1 = Q * -(Q k+1 -ǫ k+1 ) = Q * - n≥1 λ n (B π k+1 ) n Q k = n≥1 λ n Q * -(B π k+1 ) n Q k . (3)
Since π k+1 is the greedy policy with respect to Q k , we have

B π * Q k ≤ B π k+1 Q k . Therefore Q * -(B π k+1 ) n Q k = B π * Q * -B π * Q k + B π * Q k -B π k+1 Q k + B π k+1 Q k - -(B π k+1 ) 2 Q k + (B π k+1 ) 2 Q k -. . . + (B π k+1 ) n-1 Q k -(B π k+1 ) n Q k ≤ B π * Q * -B π * Q k + γP π k+1 (Q k -B π k+1 Q k ) + +(γP π k+1 ) 2 (Q k -B π k+1 Q k ) + . . . + (γP π k+1 ) n-1 (Q k -B π k+1 Q k ) = γP π * (Q * -Q k ) + + γP π k+1 + (γP π k+1 ) 2 + . . . + (γP π k+1 ) n-1 (Q k -B π k+1 Q k ) = γP π * (Q * -(Q k -ǫ k )) -γP π * ǫ k + + γP π k+1 + (γP π k+1 ) 2 + . . . + (γP π k+1 ) n-1 (Q k -B π k+1 Q k ) = γP π * d k -γP π * ǫ k + γP π k+1 + (γP π k+1 ) 2 + . . . + (γP π k+1 ) n-1 b k .
As P π * and P π k+1 are stochastic matrices, we deduce

max[Q * -(B π k+1 ) n Q k ] ≤ γ max d k + γǫ + (γ + γ 2 + . . . + γ n-1 ) max b k = γ max d k + γǫ + γ -γ n 1 -γ max b k .
By using Equation 3, we obtain the following induction on max d k :

max d k+1 ≤ γ max d k + γǫ + n≥1 λ n γ -γ n 1 -γ max b k .
With the help of the Bellman error upper bound obtained earlier (Equation 2) we obtain

max d k+1 ≤ γ max d k + γǫ + n≥1 λ n γ -γ n (1 -γ)(1 -β) (1 + γ)ǫ + O(γ k ) = γ max d k + γǫ + γ -β (1 -γ)(1 -β) (1 + γ)ǫ + O(γ k )
which gives, by taking the limit superior, lim sup

k→∞ max d k ≤ γ 1 -γ ǫ + γ -β (1 -γ) 2 (1 -β) (1 + γ)ǫ. (4)
An upper bound on s k : Let us now consider the s k term from Equation 1:

s k+1 = Q k+1 -ǫ k+1 -Q π k+1 = n≥1 λ n (B π k+1 ) n Q k -(B π k+1 ) ∞ Q k = n≥1 λ n (B π k+1 ) n Q k -(B π k+1 ) ∞ Q k . (5)
It can be seen that

(B π k+1 ) n Q k -(B π k+1 ) ∞ Q k = (B π k+1 ) n Q k -(B π k+1 ) n+1 Q k + (B π k+1 ) n+1 Q k -(B π k+1 ) n+2 Q k + . . . = (γP π k+1 ) n (Q k -B π k+1 Q k ) + (γP π k+1 ) n+1 (Q k -B π k+1 Q k ) + . . . = (γP π k+1 ) n [I + γP π k+1 + (γP π k+1 ) 2 + . . .]b k .
As above, by using the stochasticity of P π k+1 , we obtain

max[(B π k+1 ) n Q k -(B π k+1 ) ∞ Q k ] ≤ γ n (1 + γ + γ 2 + . . .) max b k = γ n 1 -γ max b k .
By using Equation 5, we obtain an upper bound on max s k+1 :

max s k+1 ≤ 1 1 -γ   n≥1 λ n γ n max b k   .
With the help of the Bellman error upper bound (Equation 2) and by taking the limit superior, we have lim sup

k→∞ max s k ≤ 1 1 -γ   m≥1 λ n γ n 1 + γ 1 -β ǫ   = β (1 -γ)(1 -β) (1 + γ)ǫ. (6)
Conclusion of the proof Finally, let us get back to Equation 1 and use the upper bounds we just derived for d k (Equation 4) and s k (Equation 6):

lim sup k→∞ Q * -Q π k ∞ ≤ lim sup k→∞ max d k + lim sup k→∞ max s k = γ 1 -γ ǫ + γ -β (1 -γ) 2 (1 -β) + β (1 -γ)(1 -β) (1 + γ)ǫ. = γ 1 -γ ǫ + γ -β + (1 -γ)β (1 -γ) 2 (1 -β) (1 + γ)ǫ. = γ 1 -γ ǫ + γ (1 -γ) 2 (1 + γ)ǫ. = γ(1 -γ) + γ(1 + γ) (1 -γ) 2 ǫ = 2γ (1 -γ) 2 ǫ.


P Van Isacker

I Talmi

Effective three-body interactions in nuclei

Keywords: 21, 30, -x -Nuclear forces 21, 60, Cs -Shell model 21, 45, Ff -Three-nucleon forces

Effective three-body interactions in nuclei

P. van Isacker, I.

According to the shell model, the ground state of 40 Ca with Z = 20 protons and N = 20 neutrons has completely filled 1s, 1p, 1d and 2s proton and neutron orbits. Valence neutrons from N = 21 on occupy the 1f 7/2 orbit which is closed at the magic number N = 28. Similarly, protons added to the magic nucleus 48 Ca occupy the proton 1f 7/2 orbit. A natural description in the shell model of nuclei in this mass region is therefore obtained by restricting neutrons and protons to the 1f 7/2 orbit, an approach which was extensively used in the past [1][2][3][4]. From the energy levels of nuclei where these neutron and proton orbits are filled, it is evident that there must be strong perturbations from higher configurations. The features predicted for relative positions of levels by using two-body effective interactions between the protons and between the neutrons, are only roughly obeyed. This is not surprising due to the proximity of higher orbits. In 41 Ca, the 3/2 - level, associated with the 2p 3/2 orbit, lies only about 2 MeV above the 7/2 -ground state. Effects of configuration mixing have been considered many years ago for the calcium isotopes [5,6] and for the N = 28 isotones [7,8] with results that turned out to be in rather good agreement with experiment.

Effective interactions are determined by the model spaces for which they are intended. If pure (1f 7/2 ) n configurations are adopted, two-body interactions do not yield good agreement with experiment. Hence, three-body effective interactions have been introduced [9,10]. This approach may well reproduce level energies in a better way. The pure 1f 7/2 nuclear states, however, will be only an approximation to the correct shell-model states. Even if effective operators, other than the Hamiltonian may pro-vide some help, they cannot provide information on more complicated processes like beta decay. Recently, a detailed discussion of the effects of three-body interactions on various spectroscopic observables was presented by Zelevinsky [START_REF] Zelevisnky | Yadernaya Fizika[END_REF].

Recently, three-body interactions were introduced, in addition to two-body ones, for dealing with the proton and neutron 1f 7/2 shells [12]. The author considers his results as a "manifestation of three-body forces in 1f 7/2shell nuclei". The good agreement he obtains could be evidence of such forces if the 1f 7/2 shell was a pure one. Even within the shell model, whose wave functions are not the real ones, the 1f 7/2 shells are strongly perturbed. Such three-body interactions play a role only if the space used is a pure 1f 7/2 shell. It may happen that introducing four-body and higher interactions will improve the fit to the data [10] but this will imply a severe loss of predictive power. It seems to us that it is more reasonable to consider possible interactions with higher configurations. Attempts in this direction were made in the past and good agreement with experiment has been obtained by using only twobody forces. In fact, we aim to show that the effect of mixing with nearby configurations leads to a three-body effective interaction rather similar to those of ref. [12].

The effect of perturbations of the pure 1f 7/2 shell, due to two-body effective interactions, may assume the form of additional two-body interactions as well as three-body ones. Such effects were considered in the past in atomic spectroscopy [13,14]. The aim of this note is to derive, by adopting a simple approximation, these additional twobody and three-body interactions arising from secondorder perturbations. In pure (7/2) n configurations, all eigenstates have definite seniority for any two-body interaction. There is experimental evidence that the two lowest J = 4 levels in 52 Cr are mixtures of states with seniorities v = 2 and v = 4. This seniority mixing was obtained in ref. [12] by using a three-body interaction. This mixing, however, was obtained in refs. [7,8] due to mixing of these states with those obtained by raising one 1f 7/2 proton into the 2p 3/2 orbit.

Consider first the j n configuration. The two-body interaction V is taken to be a perturbation on the singlenucleon Hamiltonian. The first-order contribution of V is its expectation value in the states considered. In second order, the contribution to the energy of the state aJ of the j n configuration is given by

c aJ| V |cJ cJ| V |aJ E(aJ) -E(cJ) , (1)
where the summation is over all states cJ which have nonvanishing matrix elements with the state aJ. Such states are in configurations which differ from that of aJ in singlenucleon states of at most two nucleons. These are the j n-1 j ′ and j n-2 j ′ j ′′ configurations. It is well known that contributions of the interaction with the latter configurations yield, in second-order perturbation, effective twobody interactions. In the following, the contributions of configurations cJ which differ from the aJ configuration by the state of one nucleon will be considered.

The energies of all states of the j n-1 j ′ configuration cJ may differ by small amounts in the difference E(aJ) -E(cJ). In such cases it may be a good approximation to replace this difference by a constant term, independent of the quantum numbers which characterize the state with the given J in the excited configuration c. It is then possible to carry out the summation over all states cJ. In the following we make this approximation and we also assume that the energy differences in (1) are the same for all states in the aJ configuration for any number n of (identical) nucleons. This is in accordance with the strict rules of perturbation theory where these differences are determined only by the single-nucleon Hamiltonian. It is a very good approximation for electron states in atoms; in nuclei, it is not expected to be a very good one.

Using second quantized operators, explicit expressions may be obtained for the resulting two-body and threebody operators. The part of the Hamiltonian which connects the j n and j n-1 j ′ configurations is given by [START_REF] Talmi | Simple Models of Complex Nuclei[END_REF] JM

j 2 J| V |jj ′ J A † (jj ′ JM )A(jjJM ), (2)
where

A(jj ′ JM ) = A † (jj ′ JM ) † and A † (jj ′ JM ) = 1 1 + δ jj ′ mm ′ (jmj ′ m ′ |jj ′ JM )a † jm a † j ′ m ′ . (3)
From the formula (1), applied to the intermediate configurations cJ of the type jj ′ J and assuming equal denominators, the following sums are obtained:

M1M2 j 2 J 1 | V |jj ′ J 1 A † (jjJ 1 M 1 )A(jj ′ J 1 M 1 ) × j 2 J 2 | V |jj ′ J 2 A † (jj ′ J 2 M 2 )A(jjJ 2 M 2 ). ( 4
)
With the help of the anti-commutation relations between creation and annihilation operators as well as some tensor algebra identities, this expression can be evaluated. The result is the sum of a two-body interaction

JM | j 2 J| V |jj ′ J | 2 A † (jjJM )A(jjJM ), (5)
and a three-body one

J1J2JM j 2 J 1 | V |jj ′ J 1 j 2 J 2 | V |jj ′ J 2 j J 1 j ′ j J 2 J ×[A † (jjJ 1 ) × a † j ] (JM) [A(jjJ 2 ) × a j ] (JM) . (6)
Instead of using these expressions, it is simpler to reduce the matrix elements in the j n configuration to those in the j 2 and j 3 configurations, where the calculation of matrix elements is straightforward. These matrix elements define uniquely the operators. To calculate matrix elements between states aJ and bJ in the j n configuration one may use the expansion [START_REF] Talmi | Simple Models of Complex Nuclei[END_REF] Ψ(j n aJ) = a0J0 [j n-1 (a 0 J 0 )jJ|}j n aJ]Ψ(j n-1 (a 0 J 0 )j n J), (7) where [j n-1 (a 0 J 0 )jJ|}j n aJ] are coefficients of fractional parentage (c.f.p.) which express the anti-symmetric nparticle wave function Ψ(j n aJ) in terms of an antisymmetric (n-1)-particle wave function Ψ(j n-1 a 0 J 0 ) coupled with that of the n th particle j n to total angular momentum J. Since this wave function, as well as the one of the bJ state, is fully antisymmetric, it is possible to replace the matrix element of any two-body interaction V = ik Vik by V12 multiplied by n(n -1)/2, the number of such terms. Thus, we obtain

j n aJ| V |j n bJ = n(n -1) 2 j n aJ| V12 |j n bJ = n(n -1) 2 a0b0J0 j n-1 a 0 J 0 | V12 |j n-1 b 0 J 0 ×[j n-1 (a 0 J 0 )jJ|}j n aJ][j n-1 (b 0 J 0 )jJ|}j n bJ] = n n -2 a0b0J0 j n-1 a 0 J 0 | V |j n-1 b 0 J 0 ×[j n-1 (a 0 J 0 )jJ|}j n aJ][j n-1 (b 0 J 0 )jJ|}j n bJ].(8)
This procedure may be further applied to the matrix elements in the j n-1 configuration until the j 2 configuration is reached.

Similarly, it is possible to apply this procedure to any three-body interaction V = ijk Vijk and to replace it by n(n -1)(n -2) V123 /6. Using c.f.p., it is possible to calculate matrix elements in the j n configuration in terms of matrix elements in the j n-1 configuration. Thus, we obtain

j n aJ| V |j n bJ = n(n -1)(n -2) 6 j n aJ| V123 |j n bJ = n(n -1)(n -2) 6 a0b0J0 j n-1 a 0 J 0 | V123 |j n-1 b 0 J 0 ×[j n-1 (a 0 J 0 )jJ|}j n aJ][j n-1 (b 0 J 0 )jJ|}j n bJ] = n n -3 a0b0J0 j n-1 a 0 J 0 | V |j n-1 b 0 J 0 ×[j n-1 (a 0 J 0 )jJ|}j n aJ][j n-1 (b 0 J 0 )jJ|}j n bJ]. (9)
This procedure may be continued until the j 3 configuration is reached.

Let us now turn to the calculation of the effective interaction in the j 2 and j 3 configurations, due to additional shells j ′ that are not explicitly taken into account. The two-body operators may be obtained from perturbation theory in the two-nucleon configuration. The contribution in second-order perturbation to the j 2 configuration in a state with spin J 0 can be simply calculated from (1) and is given by

j ′ | j 2 J 0 | V |jj ′ J 0 | 2 E(j 2 J 0 ) -E(jj ′ J 0 ) , (10)
where the matrix elements of the interaction V are between anti-symmetric and normalized states j 2 J 0 | and |jj ′ J 0 .

To obtain the matrix elements of the three-nucleon contribution, the j 3 configuration should be considered. Results of perturbations of states in the j 3 configuration may be calculated as follows. Anti-symmetric and normalized cJ states may be expressed as

Ψ(j 2 (J 1 )j ′ J) = 1 √ 3 Ψ(j 2 (J 1 )j ′ 3 J) -Ψ(j 2 (J 1 )j ′ 1 J) -Ψ(j 2 (J 1 )j ′ 2 J) . (11)
The matrix element of V = V12 + V13 + V23 between the fully anti-symmetric states |aJ ≡ |j 3 aJ and |cJ ≡ |j 2 (J 1 )j ′ J is equal to the matrix element of 3 V12 and hence is given by

- √ 3 J0 [j 2 (J 0 )jJ|}j 3 aJ] j 2 (J 0 )j 3 J| V12 |j 2 (J 1 )j ′ 1 J - √ 3 J0 [j 2 (J 0 )jJ|}j 3 aJ] j 2 (J 0 )j 3 J| V12 |j 2 (J 1 )j ′ 2 J . (12)
To evaluate the first sum it is convenient to carry out a change of coupling transformation on the state

|j 3 j 2 (J 1 )j ′ 1 J = -|j 2 j 3 (J 1 )j ′ 1 J , |j 2 j 3 (J 1 )j ′ 1 J = J2 (-1) j+j ′ +J1+J2 Ĵ1 Ĵ2 × j J 2 J j ′ J 1 j |j 2 j ′ 1 (J 2 )j 3 J , (13)
where Ĵi ≡ √ 2J i + 1. The integration over j 3 can be carried out and yields the non-vanishing terms in which the equality J 2 = J 0 must hold, as follows

√ 3(-1) j+j ′ J0 Ĵ0 Ĵ1 [j 2 (J 0 )jJ|}j 3 aJ] × j J 0 J j ′ J 1 j j 2 J 0 | V |j 2 j ′ 1 J 0 . ( 14
)
Similarly, the second summation in ( 12) yields the following result:

- √ 3(-1) j+j ′ J0 Ĵ0 Ĵ1 [j 2 (J 0 )jJ|}j 3 aJ] × j J 0 J j ′ J 1 j j 2 J 0 | V |j 1 j ′ 2 J 0 . (15)
The two expressions may be combined yielding the result

- √ 6(-1) j+j ′ J0 Ĵ0 Ĵ1 [j 2 (J 0 )jJ|}j 3 aJ] × j J 0 J j ′ J 1 j j 2 J 0 | V |jj ′ J 0 , (16)
where again the states j 2 J 0 | and |jj ′ J 0 are antisymmetric and normalized. To obtain the second-order contribution (1) of these perturbations, this expression should be multiplied by a similar one where the summation is over all values of J ′ 0 . The second-order perturbation contribution in the j 3 configuration is therefore 6

j ′ J1 (2J 1 + 1) J0J ′ 0 Ĵ0 Ĵ′ 0 ×[j 2 (J 0 )jJ|}j 3 aJ] j J 0 J j ′ J 1 j ×[j 2 (J ′ 0 )jJ|}j 3 aJ] j J ′ 0 J j ′ J 1 j × j 2 J 0 | V |jj ′ J 0 jj ′ J ′ 0 | V |j 2 J ′ 0 E(j 3 aJ) -E(j 2 (J 1 )j ′ J) . ( 17
)
As explained above, we make the approximation that the energy denominators in (1) are the same for all states of the j n and j n-1 j ′ configurations. This means that for the j 3 configurations we assume that the energies E(j 3 aJ) ≡ E(j 3 ) and E(j 2 (J 1 )j ′ J) ≡ E(j 2 j ′ ), are independent of a, J 1 and J, and given be 3ǫ j and 2ǫ j + ǫ j ′ , respectively, where ǫ j and ǫ j ′ are single-particle energies.

Table 1: The T = 3/2 three-body interaction matrix elements in the 1f 7/2 shell obtained by Volya and the effective three-body interaction matrix elements derived with the formula (21) for various p3f7 interactions and for the gxpf1a interaction. Energies are in keV.

Z = 20

N = 28 J Volya [12] p3f7 [5] p3f7 [6] Volya [12] p3f7 [7] In this case, the summation over J 1 can be carried out directly due to the identities of Racah coefficients. The sum over even values of J 1 can be expressed as a linear combination of sums in which the summation is over all values of J 1 , yielding 2 J1 even

(2J 1 + 1) j J 0 J j ′ J 1 j j J ′ 0 J j ′ J 1 j = J1 (2J 1 + 1) j J 0 J j ′ J 1 j j J ′ 0 J j ′ J 1 j + J1 (-1) J1 (2J 1 + 1) j J 0 J j ′ J 1 j j J ′ 0 J j ′ J 1 j = δ J0J ′ 0 2J 0 + 1 + j J J 0 j j ′ J ′ 0 . ( 18
)
Substituting this result into the expression above we find

3 j ′ J0J ′ 0 Ĵ0 Ĵ′ 0 [j 2 (J 0 )jJ|}j 3 aJ][j 2 (J ′ 0 )jJ|}j 3 aJ] × δ J0J ′ 0 2J 0 + 1 + j J J 0 j j ′ J ′ 0 × j 2 J 0 | V |jj ′ J 0 jj ′ J ′ 0 | V |j 2 J ′ 0 E(j 3 ) -E(j 2 j ′ ) . ( 19
)
To obtain the matrix elements of the genuine (but effective) three-body interaction, the contribution of the twobody interaction should be subtracted. The latter is given by

3[j 2 (J 0 )jJ|}j 3 aJ] 2 j ′ | j 2 J 0 | V12 |jj ′ J 0 | 2 E(j 2 J 0 ) -E(jj ′ J 0 ) . ( 20
)
Again we assume that the energies E(j 2 J 0 ) ≡ E(j 2 ) = 2ǫ j and E(jj ′ J 0 ) ≡ E(jj ′ ) = ǫ j + ǫ j ′ , are independent of J 0 .

In that case, subtracting this contribution from (19), we find the genuine three-body matrix element to be equal to

3 j ′ J0J ′ 0 Ĵ0 Ĵ′ 0 [j 2 (J 0 )jJ|}j 3 aJ][j 2 (J ′ 0 )jJ|}j 3 aJ] × j J J 0 j j ′ J ′ 0 j 2 J 0 | V |jj ′ J 0 jj ′ J ′ 0 | V |j 2 J ′ 0 E(j 3 ) -E(j 2 j ′ ) . (21)
It should be pointed out that the second-order perturbation contribution is contained in both two-body and three-body interactions. The contribution of second order in perturbation theory is always attractive (negative). This need not be the case for the three-body interaction which may lead to some strange results. For example, if j = 7/2 and j ′ = 1/2, there is no contribution of this perturbation to the j 3 state with J = 15/2. The highest value of J 1 is J 1 = 6 which cannot couple with j ′ = 1/2 to yield J = 15/2. Still, there is a non-vanishing contribution to the matrix element with J = 15/2 and J 0 = 6 of the two-body interaction. This unphysical contribution is exactly cancelled by the corresponding contribution from the three-body interaction. Such cancellation must take place in all 7/2 n configurations.

In the calculation of matrix elements in the j 3 configuration, the origin of the unphysical terms is clear. The sum of products of two 6j-symbols in (18) is always 1/(2J 0 + 1) even if all 6j-symbols with J 1 even vanish. In such a case, the result is due to symbols with odd J 1 values. These symbols contribute to a non-vanishing symbol in which J 1 no longer appears. If in (18) J 0 = J ′ 0 , there is no contribution of terms with odd J 1 values to the three-body interaction. If, however, J 0 = J ′ 0 , the odd J 1 terms contribute with opposite signs the same amount to the twobody interactions and to the three-body ones. Thus, the two-body (20) and three-body (21) expressions calculated above, which are genuine two-body and three-body interactions, may contain unphysical terms. The contributions of two-body interactions are usually absorbed into the effective two-body interaction. If the interest is in threebody interactions whose contributions cannot be mimicked by two-body terms, the expressions obtained above should be used.

The above formalism can be applied to the 1f 7/2 shell. Several interactions are available that include effects from the 2p 3/2 shell. Two of them were derived on the basis of spectroscopic properties of the calcium isotopes [5,6] and two more from those of the N = 28 isotones [7,8]. In particular, the authors of these references give numerical values for the matrix elements involving the 2p 3/2 shell which enter the expression (21) and for the difference in single-particle energies, ǫ 2p 3/2 -ǫ 1f 7/2 . We refer to these in-teractions as p3f7, followed by the relevant reference. For comparison, we also include results obtained with a more recent interaction which considers the 2p1f shell and results from an empirical fit (starting from a microscopically derived set of matrix elements) to reproduce a large body of energy data in the 2p1f -shell nuclei. This is the gxpf1a interaction [START_REF] Honma | [END_REF] which adopts the single-particles energies ǫ 1f 7/2 = -8.6240, ǫ 2p 3/2 = -5.6793, ǫ 2p 1/2 = -4.1370 and ǫ 1f 5/2 = -1.3829 MeV.

We have used these interactions, without any modification, to calculate the effective three-body interaction which is induced when the model space is restricted to 1f 7/2 . The results are shown in Table 1. For j = 7/2 a three-nucleon state is completely specified by its total angular momentum J and there is no need for the additional label a. There are six three-nucleon states with J = 3/2, 5/2, 7/2, 9/2, 11/2 and 15/2, and each of these states defines a component of the three-body interaction. According to the expression (21), the total effective threebody interaction results from additive contributions of the different shells j ′ which can be 2p 1/2 , 2p 3/2 or 1f 5/2 . The three-body matrix elements of Volya [12] result from separate fits to the calcium isotopes and to the N = 28 isotones, and they should thus be compared with corresponding matrix elements obtained with the p3f7 interactions from refs. [5,6] and [7,8], respectively. The numbers in parentheses in Table 1 are also taken from Volya and correspond to the variances of the parameters in the fits.

There are substantial variations in the calculated effective three-body matrix elements. In particular, those derived from the Z = 20 p3f7 interactions are generally larger than those obtained with the N = 28 p3f7 interactions. This results from a combination of a larger difference ǫ 2p 3/2 -ǫ 1f 7/2 and smaller two-body matrix elements (1f 7/2 ) 2 J 0 | V |1f 7/2 2p 3/2 J 0 in the latter interactions. Note also that the effective three-body matrix elements derived from gxpf1a are closer to those obtained with the N = 28 p3f7 interactions.

There are certainly differences between these calculations and the results of Volya. However, in spite of these differences and uncertainties, it is clear that the matrix elements are correlated: with one exception, attractive (repulsive) matrix elements in the analysis of Volya turn out to be attractive (repulsive) in our analysis. The exception concerns the J = 15/2 interaction which is attractive in Volya's analysis while it is repulsive for the p3f7 interactions and essentially zero in gxpf1a.

It is believed that there are important ab initio threebody interactions and also three-body interactions due to short-range correlations between nucleons. The latter arise from admixtures of highly excited configurations. So far, no evidence was found for the effects of three-body interactions on states of valence nucleons. In cases where rather pure shell-model configurations were observed, states and energies were well determined by effective two-body interactions. Clearly, the 1f 7/2 shell for protons and for neutrons is not in this category since there are low-lying configurations whose states may well mix with the (1f 7/2 ) n states. Insisting on using pure 1f 7/2 configurations, better agreement with experiment is obtained by incorporating effective three-body interactions. We do not question the existence of such three-body interactions but we surmise that most likely they arise from renormalization effects due to admixtures of rather low-lying configurations. Our conclusion is that claims of evidence for the existence of genuine three-nucleon interaction should be treated with some skepticism if they are based on calculations in a shell-model space which seems to be too restricted.


Ming Siaw

Wong

Jean-Yves Lafaye

Patrice Boursier

DEVELOPING PMIS FOR BUSINESS PROJECTS BASED ON SOCIAL SCIENCE RESEARCH FINDINGS & ONTOLOGY MODELLING

Keywords: Project Management, Modelling, Ontology, Unified Modelling Language

The purpose of the research is to obtain a better understanding of the essential components of business project management and capture the new gained knowledge in a machine readable format to facilitate software engineering. This paper presents how this interdisciplinary link between social science and computer science can be established through ontology modelling; and demonstrates how the resulting model can be used by the requirements analysis or conceptual modelling phase in the development of a Project Management Information System (PMIS) for business projects.

INTRODUCTION

Companies today are entrusting their business projects to certified project managers, in acknowledgement of the positive causal relationship between project management effectiveness on project performance [START_REF] Bryde | Modelling project management performance[END_REF]. Nonetheless business project success rates remain low where one in three projects fails to deliver on time or within budget; and 60% fails to deliver the benefits as set out in business case (PIPC Global Project Management Survey, 2005).

The development of the project management research can be divided into 3 strands (1) hard system model which emphasizes the planning and control dimensions; (2) organization design which focuses on the organization structure as a means to achieve integration and task accomplishment; (3) broader view of projects which emphasizes context, front-end work, learning and managing exogenous factors (Winter et al, 2006). What is still lacking is approaching the problem with project performance from the holistic approach by assessing the collective influences of the various factors and demonstrating the relevance of project management in business [START_REF] Morris | Research trends in the 1990s -the need now to focus on the business benefits of project management[END_REF]. The closest work found was the investigation into project management effectiveness by considering organizational structure, technical competency, leadership ability and the characteristics of an effective project manager [START_REF] Hyväri | Project management effectiveness in project-oriented business organizations[END_REF]. The selection of these variables however, was motivated by the discrete gaps identified by previous researches with emphasis on the factors influencing the ability of the project managers in their project delivery. The path of examining project performance from an organizational perspective remains largely unexplored.

The objective of the social science research therefore, is to (1) identify the essential components of business project management based on a comprehensive understanding of collective influences of organizational variables on project success; (2) captures the findings into the model as a reference to guide the business project management practice in order to increase chances of project success, while its representation in a machine readable format expedites the injection of this new domain knowledge into a software solution. This paper will touch on all aspects of the research with focus on point (2), by outlining how the social science research findings can be transformed into a machine readable ontology model using Unified Modelling Language (UML); and provides an example of how the resulting model can be used as the foundation to support the development of an integrated Project Management Information System (PMIS) for business projects.

ONTOLOGY MODELLING

The first task of the modelling exercise is to decide on the format which could meet the stated objectives [of representing knowledge in a model that can be fed directly into a software development cycle] and ontology is chosen as the starting point because of its origin as the centre piece in the Knowledge Representation (KR) paradigm. Defined as "an explicit specification of conceptualization" [START_REF] Gruber | What is an Ontology? viewed 16 March 2007[END_REF], an ontology is virtually always the manifestation of a shared understanding of a domain that is agreed between a number of parties which in turn, facilitates inter-operability, reuse and sharing [START_REF] Uschold | Building ontologies : towards a unified methodology[END_REF].

Since ontology can take a variety of forms, it is important to understand the key dimensions of which they may differ. One of such dimensions is the degree of formality or explicitness in the vocabulary of terms and the specifications of their meanings that comprise the ontology (Ushold & Gruninger, 1996). While knowledge specified in the form of informal ontology may be contradicting the definition of ontology as an explicit specification and cannot be leveraged for automated reasoning, formal ontology can be difficult to maintain especially if the process involves non-technical users. A balance between the two extremes is therefore desired and one resolution is the use of conceptual model. This idea is not new as highlighted by Gómez-Pérez et al (2006), formal ontologies should be reverse-engineered for their underlying conceptual models so that a better understanding of the existing ontologies could be obtained before they are updated with new learning or merged/aligned with other ontologies to form the new ones. In addition, the use of conceptual model during the conceptualization phase of an ontology engineering process helps to address issue caused by the limited capacity of the formal language in capturing the problem domain (Nagypal, 2007). Last but not least, conceptual models are expressed in a form comprehensible by both the domain experts and the ontology engineers.

In the context of software engineering, ontology expresses semantics and provides reasoning, querying and navigating facilities; whilst conceptual model represents aspects of a domain to be incorporated into an information system. In which case, ontology (its semantics in particular) can be used to facilitate conceptual modelling but due to their differences in formats, the desired reusability between ontologies and conceptual models can only be achieved through indirect means, e.g. transformation through meta-model mapping (Vasilescas et al, 2006) and enriching the conceptual models with ontology-based semantic annotation [START_REF] Zouggar | Semantic Enrichment of Enterprise Modelling[END_REF]. The ideal alternative would be to develop a foundation model based on which both ontologies and conceptual models can be built upon.

The proposal for this research therefore, is to develop the Business Project Management (BProjM) ontology model using Unified Modelling Language (UML). The choice of UML specification was driven by both the need to adopt a pseudo technical standard as well as to leverage on the Model Driven Architecture (MDA) approach where the desired ontologies and information systems can be developed based on transformation of the resulting ontology model in stages. Figure 1 below indicates the positioning of the ontology model in this research as well as in relations to the software engineering and ontology engineering paths. For the software development path on the left, the developed ontology model could be fed directly into the conceptual modelling / requirement specification phase to support the development of an integrated Project Management Information System (PMIS) for business projects. For the ontology engineering path on the right, the UML Class Diagram can be translated to OWL ontology to allow further inference and exploitation of the ontology in semantic web applications. Based on OMG's Ontology Definition Metamodel (ODM) specification [http://www.omg.org/spec/ODM/1.0/], a canonical ontology can be built automatically and then appropriately enriched via annotation and model weaving techniques [START_REF] Faucher | Génération d'ontologie à partir d'un modèle métier UML annoté[END_REF].

RESEARCH METHODOLOGY

The research adopts the deductive research strategy and applies the systems school of management theory in support of the holism theme where "Organization can be viewed as open systems which take input from the environment (outputs from other systems) and through a series of activities transform or convert these inputs into output (input to other systems) to achieve some objectives" (Mullin, 2007). By examining the relevancies of the sub-systems which must work together to carry out the "series of activities" in the context of business projects, a theoretical framework with 4 constructs was developed and 3 hypotheses were derived namely (1) Project Management Competency has a direct positive impact on business project success;

(2) Comprehensiveness of the PMIS support places a positive impact on the way project management competency contribute towards business project success;

(3) The level of integration between the project organization and the parent organization places a positive impact on the way project management competency contribute towards business project success. The case study approach is adopted to validate these hypotheses as it offers the opportunity for a holistic view of a process (Patton & Applelbaum, 2003), thus permitting the attainment of a richer understanding on a well researched topic from a fresh angle.

The key component bridging the two disciplines is a consolidated form of the social science research findings which lists the essential concepts and their inter-relationships. This can be achieved by updating the original theoretical framework to reflect only the key attributes of each construct which contributes to a successful business project based on the case study findings. Given this edited theoretical framework, the following steps of developing the ontology model can then be carried out systematically:

• Define the scope and translating the research findings into requirements • Identify related research work in the past and examine whether they can be reused • Build the model based on the stated requirements by leveraging on existing work as appropriate •

Test the completeness and integrity of the model using the case study data.

CASE STUDY FINDINGS

Three companies have participated in the case study. The anchor case study is an international airline which had successfully completed a business turnaround programme; the second case study is an operator of satellite pay television services in the region whose corporate wide business transformation programme is currently on a recovery path; and the third case study is a large local banking group which has just initiated their transformation programme due to a recent investment from a foreign bank.

In summary, the case study findings had ascertained the hypotheses, in addition, it was found that (1) Business project success should also be measured at the organizational level. This findings is in agreement with [START_REF] Turner | Choosing Appropriate Project Managers: Matching Their Leadership Style to the Type of Project[END_REF] that project is a multidimensional strategy concept and Faisal (2006)'s view that project success should be evaluated in terms of production of the expected deliverables and whether these deliverables achieve the intended organization goals; (2) The project management competencies required of the business project managers are less comprehensive and technical in nature and these "Core Business Project Management Competencies" are integration management, scope management, time management and communication management. In other words, business project managers are not expected to be the master of the trade but one who facilitates the work of the domain masters; (3) An integrated programme management function is essential in maintaining on-going alignment between the project output and the organizational goal. Although the role of programme office /programme management office in marshalling projects and resources to achieve the desired strategic and/or synergy benefits is not new (Pellegrinelli, 2002), the emphasis has been placed on its configurations rather than its effectiveness especially in terms of performing on-going close-loop monitoring; (4) The use of IT in support of project work needs to be elevated to the organization level. Project learning for example, would add more value to future corporate planning if each project implementation is captured as an episode in the organization's chronology with causes and effects.

Based on the above findings, an underlying business project management theory i.e. "Business projects must be managed as an integral part of the organization" is developed. Such integration clearly demands the enablement of seamless information exchange between the project and the permanent organization, which is a system support feature less relevant to the traditional construction or engineering projects. By updating the original theoretical framework with these research findings (see figure 2), the modelling exercise is now ready to commence.

DEFINING SCOPE & REQUIREMENTS

The scope of the ontology model is provided by the edited theoretical framework where all its stated components must be represented in the ontology model. But in order for the model to be a meaningful foundation for a specialised PMIS, each of these components must be expanded with secondary level details. This is achieved by adopting relevant standard project management terms in the Project Management Body of Knowledge (PMBOK, 2004) published by the Project Management Institute (PMI) which is chosen due to its large member base and wide acceptance by the international communities. To illustrate how this is done, "Communication Management" in the "Core Business Project Management Competency" for example, can be expanded with supporting terms "communication management plan", "project records" and "project reports". Finally, based on the developed theory that the business project is an integral part of the enterprise, the ontology model must also show how these business project management components interact with their counterparts in the parent organization. As such, the consolidated requirements of the ontology model are to capture (a) essential business project management terms and their inter-relationships; (b) their interfaces with the other components in the parent organization.

REUSING EXISTING WORK

Given the intent to leverage on MDA approach to transform the resulting model into an implementable equivalent, the Business Project Management (BProjM) ontology model is built based on OMG's Business Motivation Model (BMM), Business Process Definition MetaModel (BPDM) and Organization Structure MetaModel (OSM) which collectively provide a comprehensive set of meta-classes that define the fundamentals of an enterprise.

As with the related work in the project management domain, no existing project management model in a reusable form is found. Existing project and project management ontologies on the other hand, are designed to serve very specific purposes [START_REF] Garcia | Building a Project Ontology with Extreme Collaboration and Virtual Design & Construction[END_REF]Gómez-Pérez et al, 2006;[START_REF] Dumbill | Keep project information up-to-date with the DOAP vocabulary[END_REF]) and thus, only PROMONT (PROject Management ONTology) developed by [START_REF] Abels | PROMONT -A Project Management Ontology as a Reference for Virtual Project Organizations[END_REF] can be adopted. Although it is originally developed to support product development projects in distributed virtual project environment, PROMONT summaries all major project management standards including PMBOK. It is therefore still generic enough to serve as a starting point for this research.

DEVELOPING THE ONTOLOGY MODEL USING UML

In compliance with UML standards, the research uses Class Diagram for structural specifications and Communications Diagram in conjunctions with State Machine Diagram to specify the behavioural aspect.

Structural specifications

The structural specification starts with defining Packages to group the terms by their origins namely BMM (a separate Package was defined for BMM Place Holder in accordance with OMG's BMM specification), BPDM, OSM and the to-be developed BProjM. Each essential business project management terms [in the BProjM Package] are then transformed into Classes and further grouped into sub-Packages based on their nature and purpose. Core Package for example, contains basic components of a project/programme initiative such as Phase, Task, Activity by leveraging on PROMONT's terms; whilst Project Results Package contains terms related to project performance and the measures for its success. The next important step will be to fulfil the second requirement by establishing linkages between each Class in the BProjM Package with the meta-classes in BMM, BPDM and OSM. Subject to the nature of the terms, this can be achieved in two ways by defining either (a) Inheritances or (b) Associations. Riding on the same example and highlighted by the circles in figure 3, Initiative which is a generalization of Programme and Project, is now tied to Tactical in BMM Package through an inheritance link, reflecting its nature as a special type of tactical Course Of Action. Programme Objective and Project Objective are also linked to Objective in the same manner. An example of linkage by Association on the other hand, is demonstrated by the relationship where the "Input" Class in the BProjM Package "IsAnAllocationOf" the "Asset" Class in the BMM PlaceHolder Package.

The final step of structural specification is to substantiate the Classes and Associations with more explicitness by defining Attributes, Operations and Constraints. The "Input" Class for example, is defined with (1) People, Equipment and Materials as its Attributes to reflect the type of assets which may be allocated to the project; (2) AllocateInput() and UpdateConsumption() as its Operations in support of the Association "AssignedTo" and "Consumes" respectively; (3) a Constraint indicating that during AllocateInput(), People / Equipment / Materials assigned to the project must first be available in Assets.

Behavioural specifications

To represent the dynamic aspect of the model, the exercise begins by using the State Machine Diagram to describe the behaviour of the "ruling" Class, i.e. a Class of which its evolution will trigger different instantiations of other classes at different times. In the case of business project management, its dynamism is induced by the transition of states in the business project life cycle. The "ruling" Class for BProJM ontology model is therefore Initiative which is the parent Class of Programme and Project. By using State Machine Diagram, the advancement of Initiative from one state to another as well as the conditions that govern the transition can be clearly spelt out. In a similar context but a more refined coverage, State Machine Diagram is also used to elaborate on the more sophisticated Classes, e.g. Issue and Change Request to reflect their "ruling" status in Change Management during the executing state.

Finally, for each state specified in the State Machine Diagram, a Communication Diagram is created to demonstrate the interactions among the Classes as well as the sequence of their occurrences, through invoking the predefined Operations. In addition, dependencies among Classes can also be indicated. The best example would be the communications among Classes triggered by the update of activity status during a project's execution state as illustrated by figure 4.

Testing the developed ontology model

The model is tested for completeness by instantiating the Classes using Object Diagram based on the details gathered during the case study. It is then refined based on test results and the cycle repeats itself until every aspect of the business project management practice of the participating company can be reflected in the model. One of the discrepancies uncovered in an earlier version of this model was that programme may be an Aggregation of projects as well as sub-programmes. The model has since been revised and Programme is now added with another Association that points to itself.

USING THE ONTOLOGY MODEL

Leveraging on the predefined Classes, Operations etc. in the model, the user requirements of a PMIS catering specifically for the need of business project management can now be easily captured using (a) Use-Case Diagram to describe the requirements of the different stakeholders and to provide an overview of the required system functionalities in fulfilment of those requirements; (b) Sequence Diagram to denote the series of actions that must be carried out in order to deliver the required system functionality, where existing Communication Diagrams with predefined interactions among Classes can be used as a basis and build upon with additional features and processing logics as necessary.

The Use-Case Diagram in figure 5 presents a scenario where the PMIS captures the team members' update of activity status; based on which derives the status for the task, deliverables, phase and milestones automatically; and finally formats the information according to the expectations of different stakeholders. In addition to the reduced effort in detailing user requirements, the ontology model provides the much needed context for the software engineers to better comprehend the different stakeholders' needs, thus facilitating the augmentation from conceptual modelling (i.e. "what the system should do") to requirement engineering (i.e. "why is the system like this") as proposed by [START_REF] Rolland | From Conceptual Modelling to Requirements Engineering[END_REF]. With a more in-depth understanding on the subject matter, completeness of the resulting conceptual model for the desired PMIS is assured which in turn, leads to the delivery of a quality system.

CONCLUSION : CONTRIBUTION, ISSUES & FUTURE RESEARCH

The proposed methodology can be applied to other management subject of interest as long as the findings can be expressed in the form of theoretical framework which can be expanded based on recognized standards. In which case, consistent application of this methodology over time would contribute to the development of a multi-facet enterprise ontology model. The availability of this semantic foundation should in turn, place a considerable impact on the existing landscape of software engineering where systems can now be built based on predefined knowledge specifications of the domain, rather than user input from various sources. In other words, comprehensiveness in fulfilling the domain-imposed requirements is no longer a differentiator and the enterprise application systems in the future, would compete primarily in terms of superiority in design and technicality.

The realization of such an enterprise ontology model however, would need support from a central agency to host the convergence of all related management research efforts. OMG would be in the best position to play that role since the ontology model is built upon its standards which also require on-going refinements. This modelling exercise have already identified some Classes such as those defined in the Documentation Package which should technically be applicable to the enterprise at large but there is no suitable meta-classes in BMM, BPDM nor OSM today where they could be built upon. The central agency will also play an important role in providing the forum to resolve semantic conflicts between new and existing Classes, which are expected to occur with the expansion of the ontology model.

The other concern which may potentially hamper the acceptance of the methodology is the lack of UML knowledge and understanding of the object oriented concepts among the social scientists. Thus, until unification of modelling standards across disciplines is achieved, close collaborations in the form of joint researches must be encouraged and promoted.

As an immediate next step, the comprehensiveness of the resulting business project management ontology model can be refined and further generalized by incorporating findings of more case studies. On the technical front, a potential research area would be to supplement the current ODM specifications with a solution that translates State Machine Diagram into OWL ontology.

Figure 2 .

 2 Figure 2. Social science research findings expressed in the form of theoretical framework

Figure 3 .

 3 Figure 3. BProjM ontology model -sample class diagram

Figure 4 .

 4 Figure 4. BProjM ontology model -sample communications diagram

  Figure 5. BProjM ontology model -sample use case diagramUsing "Update Work Status" as an example (see figure6), the Sequence Diagram can be drawn by first stating the user requirements in a Note and associating it to an interface definition between the actor and the system. The software engineer can then detail the series of required actions by referring to the related Communication Diagram already defined in the ontology model and sequence the standard interactions accordingly. As the Sequence Diagram is being developed, he/she can define new Classes to represent the system components, as well as adding new Operations into the existing Classes as required.In addition to the reduced effort in detailing user requirements, the ontology model provides the much needed context for the software engineers to better comprehend the different stakeholders' needs, thus facilitating the augmentation from conceptual modelling (i.e. "what the system should do") to requirement engineering (i.e. "why is the system like this") as proposed by[START_REF] Rolland | From Conceptual Modelling to Requirements Engineering[END_REF]. With a more in-depth

  user to specify the range of activity of interest -for user friendliness, default to the project which the user is currently assigned to; default the range of activity date to the current calendar week, 2) User update the status of the selected activity of interest and the actual consumption of resources allocated to that activities 3) The system should then update the status of the related tasks, phase and deliverables 4) The system automatically creates an event if a project milestone is met as a result of the activity status update description Defining the manmachine interface and the user requirements Leveraging on Communication Diagram for status update & monitoring that spells out the communications among Classes

Figure 6 .

 6 Figure 6. BProjM ontology model -sample sequence diagram