Reactions of half-sandwich rhodium(III) and iridium(III) compounds with pyridinethiolate ligands: Mono-, di-, and tri-nuclear complexes

Neutral trinuclear metallomacrocycles, [Cp^*RhCl(μ-4-PyS)]₃ (3) and [Cp^*IrCl(μ-4-PyS)]₃ (4) [Cp^* = pentamethylcyclopentadienyl, 4-PyS = 4-pyridinethiolate], have been synthesized by self-assembly reactions of [Cp^*RhCl₂]₂ (1) and [Cp^*IrCl₂]₂ (2) with lithium 4-pyridinethiolate, respectively. In situ reaction of complex 3 with three equivalent of lithium 4-pyridinethiolate resulted in [Cp^*Rh(μ-4-PyS)(4-PyS)]₃ (5) containing both skeleton and pendent 4-PyS groups. Chelating coordination of 2-pyridinethiolate broke down the triangular skeleton to give mononuclear metalloligands Cp^*Rh(2-PyS)(4-PyS) (6) and Cp^*Ir(2-PyS)(4-PyS) (7) [2-PyS = 2-pyridinethiolate], which could also be synthesized from Cp^*RhCl(2-PyS) (10) and Cp^*IrCl(2-PyS) (11) with lithium 4-pyridinethiolate. The coordination reactions of 6 with complexes 1 and 2 gave dinuclear complexes [Cp^*Rh(2-PyS)(μ-4-PyS)][Cp^*RhCl₂] (8) and [Cp^*Rh(2-PyS)(μ-4-PyS)][Cp^*IrCl₂] (9), respectively. Molecular structures of 3, 4, 6 and 11 were determined by X-ray crystallographic analysis. All the complexes have been well characterized by elemental analysis, NMR and IR spectra.     

Uranium-sulfilimine chemistry. The preparation of Cp2*UCl2(HNSPh2) and its hydrolysis with HNSPh2 · H2O

The reaction of Cp₂*UCl₂ with HNSPh₂ produces Cp₂*UCl₂(HNSPh₂), which is the first structurally characterized complex of a sulfilimine. The hydrolysis of Cp₂*UCl₂(HNSPh₂) with HNSPh₂ · H₂O yields a tetrauranium cluster whose heavy atom structure has been determined by x-ray diffraction and which is formulated as a U^IV/U^V complex: [Cp*(Cl)(HNSPh₂)U(μ₃-O)(μ₂-O)₂U(Cl)(HNSPh₂)₂]₂.     

Coordination‐driven self‐assembly of Cp*Rh‐Based Rectangles,Cages and Their Host−Guest Binding Study

Supramolecular coordination‐driven self‐assembly rectangle 1a have be formation of distorted shapes, with a strong π···π interactions between the L1 and anthracene in 1b ( 1a  ? anthracene) that changed the size to desired and supported by single‐crystal Xray diffraction data. The formation of 1c( 1a  ? DMF) has been further corroborated by the single‐crystal and 1b also can reaction with decaborane to form ortho‐carborane complex 1d . A self‐assembled triangular prism cage 2a consisting of binuclear half‐sandwich metal precursors [Cp^*₂Rh₂(μ‐BiBzIm)]Cl₂ (BiBzIm = 2,2′‐bisbenzimidazole) ligands and L2 were found to capsulated a triphenylene in cage. Another two kinds of prismatic cages ( 3a and 3b ) were obtained from the reactions of the bis‐chelating‐coordinated [Cp^*₂Rh₂(μ‐CA)]Cl₂(CA = chloranilate) with L2 and L3 in the presence of AgOTf (OTf = CF₃SO₃) in CH₃OH, and cage 3b have a perfect sized cavity to self‐assembled with anthracene in it.     

π-Coordination de doubles liaisons azoteazote et carboneazote sur les fragments Cp2NbH et Cp2TaH

The trihydrides Cp₂MH₃ (M = Nb, Ta) react with diphenyl diazomethane giving Cp₂M(diazo)H complexes in which the diazo molecule is η²-N,N-bonded to the metal. When Cp₂TaH₃ is treated with the isonitrile CNCMe₂CH₂C(Me)₃, the complex Cp₂Ta[CH(CN)NR]H (R = CMe₂CH₂CMe₃) is obtained which contains an η²-coordinated CN bond.     

Kinetics and Mechanism of the Gold-Catalyzed Hydroamination of 1,1-Dimethylallene with N-Methylaniline

The mechanism of the intermolecular hydroamination of 3-methylbuta-1,2-diene ( 1 ) with N-methylaniline ( 2 ) catalyzed by (IPr)AuOTf has been studied by employing a combination of kinetic analysis, deuterium labelling studies, and in situ spectral analysis of catalytically active mixtures. The results of these and additional experiments are consistent with a mechanism for hydroamination involving reversible, endergonic displacement of N-methylaniline from [(IPr)Au(NHMePh)]⁺ ( 4 ) by allene to form the cationic gold π-C1,C2-allene complex [(IPr)Au(η²-H₂C=C=CMe₂)]⁺ ( I ), which is in rapid, endergonic equilibrium with the regioisomeric π-C2,C3-allene complex [(IPr)Au(η²-Me₂C=C=CH₂)]⁺ ( I′ ). Rapid and reversible outer-sphere addition of 2 to the terminal allene carbon atom of I′ to form gold vinyl complex (IPr)Au[C(=CH₂)CMe₂NMePh] ( II ) is superimposed on the slower addition of 2 to the terminal allene carbon atom of I to form gold vinyl complex (IPr)Au[C(=CMe₂)CH₂NMePh] ( III ). Selective protodeauration of III releases N-methyl-N-(3-methylbut-2-en-1-yl)aniline ( 3 a ) with regeneration of 4 . At high conversion, gold vinyl complex II is competitively trapped by an (IPr)Au⁺ fragment to form the cationic bis(gold) vinyl complex {[(IPr)Au]₂[C(=CH₂)CMe₂NMePh]}⁺ ( 6 ).     

Stacking reactions of the borole complex Cp*Rh(η5-C4H4BPh) with the dicationic fragments [Cp*M]2+ (M = Rh or Ir)

The reaction of the (borole)rhodium iodide complex [(η-C₄H₄BPh)RhI]₄ with Cp^*Li afforded the sandwich compound Cp^*Rh(η-C₄H₄BPh) (4). The reactions of compound 4 with the solvated complexes [Cp^*M(MeNO₂)₃]²⁺(BF ₄ ⁻ )₂ gave triple-decker cationic complexes with the central borole ligand [Cp^*Rh(η-η⁵:η⁵-C₄H₄BPh)MCp^*]²⁺(BF ₄ ⁻ )₂ (M = Rh (5) or Ir (7)). The structure of complex 4 was established by X-ray diffraction. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1525–1527, September, 2006.     

1.2-Bis(pentamethylcyclopentadienyl)tetrachlorodisilane and its reduction to decamethylsilicocene

Reaction of pentamethylcyclopentadienyl(pentachloro)disilane (2), prepared from hexachlorodisilane and potassium pentamethylcyclopentadienide (Cp^*K), with a further equivalent of Cp^*K leads selectively to the title compound Cp^* ₂ Si ₂ Cl ₄ (3) which was characterized by NMR and X-ray structural data. Dehalogenation of 3 with four equivalents of sodium naphthalenide offers an alternative route for the synthesis of decamethylsilicocene (1). Dedicated to Professor Mitsuo Kira on the occasion of being honoured with the Wacker Silicon Award 2005.     

The structure of copolymers of vinyl cyclohexane with styrene

Copolymerization of vinyl cyclohexane (monomer-1) with styrene was investigated in the presence of the stereospecific complex catalyst TiCl₃ + Al(iso-C₄H₉)₃. Monomer reactivity ratios were r₁ = 0·177 ± 0·051 and r₂ = 2·117 ± 0·370. The monomer unit distributions in the copolymers were estimated by comparison of the i.r.-spectra of copolymers and the isotactic homopolymers using absorption bands at 565 and 1084 cm^?1 which correspond to the vibrations of styrene blocks containing ? 5 styrene units and the band at 985 cm^?1 characterizing polystyrene crystallinity. The data indicate the tendency towards alternation in the copolymerization. Analysis of the experimental and literature data led to the conclusion that distribution of the units in copolymers of vinyl cyclohexane with α-olefins is determined by the nature of the α-olefin. The following activity series is proposed for α-olefins in their copolymerization with vinyl cyclohexane in the presence of catalytic systems based on titanium salts and organo-aluminium compounds: propylene >; 4-methylpentene-1 >; styrene >; 3-methylbutene-1 ～ vinyl cyclohexane.     

Triazamethylenemethane complexes of zirconium and tantalum

Addition of [Li₂(THF)₄][C(NPh)₃] (2) to a THF solution of Cp^*ZrCl₃ (Cp^*=C₅Me₅) yields, after recrystallization in Et₂O, the zwitterionic species Cp^*[C(NPh)₃]ZrCl₂Li(Et₂O)(THF) (3). Treating 3 with excess methylaluminoxane (MAO) affords a homogeneous Ziegler–Natta catalyst for ethylene polymerization. Addition of LiNPh₂ to 3 allows for Cl substitution to give the new product Cp^*[C(NPh)₃]Zr(NPh₂)ClLi(THF)₂ (4). A single crystal diffraction study of 4 reveals that the [C(NPh)₃] ligand is η²-bound. The group 5 complex Cp^*[C(NPh)₃]TaMe₂ (5) was prepared by addition of 2 to Cp^*TaMe₂Cl(OSO₃CF₃). The X-ray diffraction structure of 5 shows that the [C(NPh)₃] ligand is η²-bound to tantalum and that, when compared to 4, there is less electron delocalization across the inner core of [C(NPh)₃].     

Pyridylmethylamines as Ligands in Iron Halide Complexes – Coordination Behaviour Depending on the Halide,the Denticity of the Amino Ligand and the Oxidation State of Iron

2‐Pyridylmethylamine (amp) and 8‐aminochinoline (ach) readily form the following complexes with iron halides in methanol: [(amp)₂FeCl₂] ( 1a ), [(amp)₂FeBr₂] ( 1b ), [(ach)₂Fe(MeOH)₂]Br₂ ( 1c ), and [(amp)FeCl₂(μ‐OMe)]₂ ( 2 ). Methanol was chosen as a solvent because these reactions are rather complex in ether. For example, FeCl₃ forms the ionic complex pair [(dme)₂FeCl₂] [FeCl₄] ( 3 ) with 1,2‐dimethoxyethane (dme). The reaction of FeBr₂ with tridentate di(2‐pyridylmethyl)amine (dpa) and tetradentate 1,2‐dipyridyl‐1,2‐diaminoethane (dpdae) yields the complexes [(dpa)₂Fe]Br₂·2 MeOH ( 4 ) and [(dpdae)₂Fe] [FeBr₄] ( 5 ), respectively. Crystallographic and magnetochemical investigations show the high‐spin configuration for the complexes 1 and 2 , whereas the short Fe‐N distances of 4 clearly indicate a low‐spin state. Compound 2 exhibits an antiferromagnetic exchange interaction with a coupling constant J = ?29.4 cm^?1 (H;af = ?J S;af_A·S;af_B).     

Synthesis and Spectroscopic Characterisation of a New Class of Heterobimetallic Zirconocene(IV) Compounds

Reactions of Cp₂ZrCl₂ with homometallic complexes of aluminium containing one residual hydroxy group Al(OGO)(OGOH) and Al(L)(OGOH) [where G=G¹=CMe₂CMe₂ (1a); G=G²=CMe₂CH₂CHMe (1b); G= G³=CMe₂CH₂CH₂CMe₂ (1c) and L=L¹=OC₆H₄CH=NCH₂CH₂O, G=G¹ (2a); L=L¹, G=G² (2b); L=L¹, G=G³ (2c); L=L²=OC₁₀H₆CH=NCH₂CH₂O, G=G¹ (2d); L=L², G=G² (2e); L=L², G=G³ (2f)] in THF using Et₃N as HCl acceptor affords novel heterobimetallic compounds of the types Al(OGO)₂Zr(Cl)Cp₂ and Al(L)(OGO)Zr(Cl)Cp₂, respectively. All of these derivatives have been characterised by elemental analyses, molecular weight measurements, and spectroscopic [IR, NMR (¹H and ²⁷Al)] studies.     

Zur Bildung der Cyclotetraphosphane cis- und trans-P4(SiMe3)2(CMe3)2 bei der Umsetzung von (Me3C)PCl2 mit LiP(SiMe3)2 · 2 THF

Formation of the Cyclotetraphosphanes cis- und trans-P₄(SiMe₃)₂(CMe₃)₂ in the Reaction of (Me₃C)PCl₂ with LiP(SiMe₃)₂ · 2 THF The mechanism of the reaction of (Me₃C)PCl₂ 1 with LiP(SiMe₃)₂ · 2 THF 2 was investigated. With a mole ration of 1:1 at ?60°C quantitatively (Me₃C)(Cl)P? P(SiMe₃)₂ 3 is formed. This compound eliminates Me₃SiCl on warming to 20°C, yielding Me₃Si? P?P? CMe₃ 4 (can be trapped using 2,3-dimethyl-1,3-butadiene in a 4+2 cycloaddition), which dimerizes to produce the cyclotetraphosphanes cis-P₄(SiMe₃)₂(CMe₃)₂ 5 and trans-P₄(SiMe₃)₂(CMe₃)₂ 6 . Also with a mole ratio of 1:2 initially 3 is formed which remarkably slower reacts on to give [(Me₃Si)₂P]P₂P? CMe₃ 8 . Remaining LiP(SiMe₃)₂ cleaves one Si? P bond of 8 producing (Me₃)₂P? P(CMe₃)? P(SiMe₃)₂Li. Via a condensation to the pentaphosphide 10 and an elimination of LiP(SiMe₃)₂ from this intermediate, eventually trans-P₄(SiMe₃)₂(CMe₃)₂ 6 is obtained as the exclusive cyclotetra-phosphane product.     

Reaktionen von Nitrosylkomplexen. XIII. Synthese neuer zwei- und dreikerniger heterobimetallischer Komplexe mit NO-Brückenliganden

Reactions of Nitrosyl Complexes. XIII. Synthesis of Novel Di- and Trinuclear Heterobimetallic Complexes with Bridging NO Ligands By reaction of [{Cp′Fe(μ-NO)}₂] with [Cp′Mn(CO)₂ · (THF)] (Cp′ = μ⁵-C₅H₄Me) in THF [Cp₃′Fe₂Mn(μ-CO)₂(μ-NO) · (μ₃-NO)] 1 is formed in high yield. The reaction of [{Cp′Fe(μ-NO)}₂]Na with [Cp′Mn(CO)₂NO]BF₄ in DME/acetone yields besides known [{Cp′Mn(CO)(NO)}₂] 2 the novel complex [Cp₂′FeMn(μ-NO)₂NO] 3 . By interaction between [Cp′Mn(CO)₂(THF)] and 3 , [Cp₃′FeMn₂(μ-CO)(μ-NO)₂ · (μ₃-NO)] 4 is formed. The complex 4 represents the hitherto unknown missing link in the series of the isoelectronic clusters [Cp₃′Mn₃(μ-NO)₃(μ₃-NO)], 1 , and [Cp₃′Fe₃(μ-CO)₃(μ₃-NO)]. Attempts to synthesize the unknown complex [(Cp′FeNO)₂ · Cr(CO)₅] by addition of carbene analogous Cr(CO)₅ fragments to the Fe=Fe bond in [{Cp′Fe(μ-NO)}₂] only led to very low yields of [Cp₂′FeCr(CO)₅] 5 . The new complexes were characterized by mass, NMR and IR spectra.     

Synthesis, characterization and reactivity of the molybdenum(VI) complex [MoCl(NAr)2(CH2CMe2Ph)] (Ar=2,6-Pr2C6H3)

The compounds [MoCl(NAr)₂R] (R=CH₂CMe₂Ph (1) or CH₂CMe₃(2); Ar=2,6-Prⁱ₂C₆H₃) have been prepared from [MoCl₂(NAr)₂(dme)] (dme=1,2-dimethoxyethane) and one equivalent of the respective Grignard reagent RMgCl in diethyl ether. Similarly, the mixed-imido complex [MoCl₂(NAr)(NBu^t)(dme)] affords [MoCl(NAr)(NBu^t)(CH₂CMe₂Ph)] (3). Chloride substitution reactions of 1 with the appropriate lithium reagents afford the compounds [MoCp(NAr)₂(CH₂CMe₂Ph)] (4) (Cp=cyclopentadienyl), [MoInd(NAr)₂(CH₂CMe₂Ph)] (5) (Ind=Indenyl), [Mo(OBu^t)(NAr)₂(CH₂CMe ₂Ph)] (6), [MoMe(NAr)₂(CH₂CMe₂Ph)] (7), [MoMe(PMe₃)(NAr)₂(CH₂CMe ₂Ph)] (8) (formed in the presence of PMe₃) and [Mo(NHAr)(NAr)₂(CH₂CMe₂P h)](9). In the latter case, a by-product {[Mo(NAr)₂(CH₂CMe₂Ph) ]₂(μ-O)}(10) has also been isolated. The crystal structures of 1, 4, 5 and 10 have been determined. All possess distorted tetrahedral metal centres with cis near-linear arylimido ligands; in each case (except 5, for which the evidence is unclear) there are α-agostic interactions present.     

Synthesis and properties of bis(biphenyl)chromium(<Emphasis Type="SmallCaps">i</Emphasis>) 1,4-di(2-cyanoisopropyl)-1,4-dihydrofulleride and 1-(2-cyanoisopropyl)-1,2-dihydrofullerene

Radical-ion salts bis(biphenyl)chromium(i) 1,4-di(2-cyanoisopropyl)-1,4-dihydrofulleride [(Ph₂)₂Cr]^+·[1,4-(CMe₂CN)₂C₆₀]^−· and bis(biphenyl)chromium(i) 1-(2-cyanoisopropyl)-1,2-dihydrofulleride [(Ph₂)₂Cr]^+·[1,2-(CMe₂CN)(H)C₆₀]^−·, the salt bis(biphenyl)chromium(i) (2-cyanoisopropyl)fulleride [(Ph₂)₂Cr]^+·[(CMe₂CN)C₆₀]⁻, and neutral 1-(2-cyanoisopropyl)-1,2-dihydrofullerene 1,2-(CMe₂CN)(H)C₆₀ have been synthesized for the first time. The compounds [(Ph₂)₂Cr]^+·[1,4-(CMe₂CN)₂C₆₀]^−· and [(Ph₂)₂Cr]^+·[1,2-(CMe₂CN)(H)C₆₀]^−· decompose in THF to form [(Ph₂)₂Cr]^+·[(CMe₂CN)C₆₀]⁻, whose protonation affords 1,2-(CMe₂CN)(H)C₆₀. 1,4-Di(2-cyanoisopropyl)-1,4-dihydrofullerene 1,4-(CMe₂CN)₂C₆₀ and 1,2-(CMe₂CN)(H)C₆₀ are stable in vacuo up to 513 K. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1935–1939, September, 2008.     

Reaktion von Cp2Ti(Me3SiC?CSiMe3) mit O-Trimethylsilyl-benzaldoxim unter Bildung eines Nitril- und eines Titana-Diazacyclopenten-Komplexes

Reaction of Cp₂Ti(Me₃SiC?CSiMe₃) with O-Trimethylsilyl-benzaldoxime under Formation of a Nitrile and a Titana-Diazacyclopentene Complex Cp₂Ti(Me₃SiC?CSiMe₃) ( 1 ) reacts with O-trimethylsilyl-benzaldoxime ( 4 ) to the titanocene-(η²-benzonitrile) complex 5 and 2,5-diaza-1-titana-3-cyclopentene 7 . The structure of 7 was obtained by X-ray crystal structure analysis ( 7 : orthorhombic, space group P2₁2₁2₁, Z = 4, a = 7.955(2), b = 12.630(3), c = 19.426(4) Å).     

Synthese und Molekülstruktur von Barium-bis[N,N′-bis(trimethylsilyl)benzamidinat] · DME · THF

Synthesis and Molecular Structure of Barium Bis[N,N′-bis(trimethylsilyl)benzamidinate] ° DME ° THF Barium bis[N,N′-bis(trimethylsilyl)benzamidinate] · thf · dme crystallizes in the monoclinic space group P2₁/n with a = 1 122.0(2), b = 2 190.7(4), c = 1 840.2(3) pm, β = 98.04(1)° and Z = 4 containing a metal center in a distorted monocapped trigonal prismatic surrounding. The barium dibenzamidinate moiety is sent with an angle of 120°, although this leads to different Ba? N distances of 273 and 282 pm originating from the interligand repulsion of the trimethylsilyl groups and the dme substituent. The 1,3-diazaallyl fragment with C? N bond lengths of 132 pm shows a delocalisation of the anionic charge.     

Synthesis and Characterization of Chlorobis(cyclopentadienyl) Titanium(IV) and Zirconium(IV) O,O’-Dialkyl and Alkylene Dithiophosphates

Abstract

Reactions of O,O′-dialkyl and alkylene dithiophosphoric acids with bis (cyclopentadienyl) titanium(IV) and zirconium (IV) dichloride in a 1:1 molar ratio in refluxing benzene proceeds with elimination of HCl and formation of the substituted derivatives, Cp₂MCl[S₂P(OR)₂] (where R = Et, Pr-n, Pr-i, Bu-i and Ph), Cp₂MCl[S₂POGO] (where G = ?CH₂CMe₂CH₂?, ?CH₂CEt₂CH₂? and ?CMe₂CMe₂?), (M = Ti and Zr). The complexes are dark red and yellow solids, soluble in common organic solvents and monomeric in nature. These have been characterized on the basis of elemental analyses, molecular weight determinations, IR, and NMR (¹H, ¹³C, and ³¹P).

GRAPHICAL ABSTRACT      

Bildung und struktur der Cyclophosphane P4(CMe3)2[P(CMe3)2]2 und P4(SiMe3)2[P(CMe3)2]2

Formation and Structure of the Cyclophosphanes P₄(CMe₃)₂[P(CMe₃)₂]₂ and P₄(SiMe₃)₂[P(CMe₃)₂]₂ n-Triphosphanes showing a SiMe₃ and a Cl substituent at the atoms P¹ and P², like (Me₃C)₂P? P(SiMe₃)? P(CMe₃)Cl 3 or (Me₃C)₂P? P(Cl)? P(SiMe₃)₂ 4 are stable only at temperatures below ?30°C. Above this temperature these compounds lose Me₃SiCl, thus forming cyclotetraphosphanes, P₄(CMe₃)₂[P(CMe₃)₂]₂ 1 out of 3 , P₄(SiMe₃)₂[P(SiMe₃)₂]₂ 2a (cis) and 2b (trans) out of 4 . The formation of 1 proceeds via (Me₃C)₂P? P?PCMe₃ 5 as intermediate compound, which after addition to cyclopentadiene to give the Diels-Alder-adduct 6 (exo and endo isomers) was isolated. 6 generates 5 , which then forms the dimer compound 1 . Likewise (Me₃C)₂P? P?P-SiMe₃ 8 (as proven by the adduct 7 ) is formed out of 4 , leading to 2a (cis) and 2b (trans). Compound 1 is also formed out of the iso-tetraphosphane P[P(CMe₃)₂]₂[P(CMe₃)Cl] 9 , which loses P(CMe₃)₂Cl when warmed to a temperature of 20°C. 1 crystallizes monoclinically in the space group P2₁/a (no. 14); a = 1762.0(15) pm; b = 1687.2(18) pm; c = 1170.5(9) pm; β = 109.18(5)° and Z = 4 formula units in the elementary cell. The molecule possesses E conformation. The central four-membered ring is puckered (approx. symmetry 4 2m; dihedral angle 47.4°), thus bringing the substituents into a quasi equatorial position and the nonbonding electron pairs into a quasi axial position. The bond lengths in the four-membered ring of 1 (d (P? P) = 222.9 pm) are only slightly longer than the exocyclic bonds (221.8 pm). The endocyclic bond angles \documentclass{article}\pagestyle{empty}\begin{document}$ \bar \beta $\end{document}(P/P/P) are 85.0°, the torsion angles are ±33° and d (P? C) = 189.7 pm.     

Production of electronically excited atoms. H + Br2 → HBr + Hb(*), H + HBr → H2 + Br(*)

By measurement of infrared chemiluminescence we have obtained for the branching ratio of the room temperature reaction H + Br₂ (1), k*₁/k₁ = 0.015 ± 0.004 and for H + HBr (2), k*₂/k₂ ? 0.013. For H + Br₂ → HBr(υ^· ? 6) + Br (1), the detailed rate constant k(υ^* = 6) = 0.014 ± 0.003 relative to k(υ^· = 4) = 100.