Some observations on the relationship between charge and 13C chemical shift in cyclohexadienylic anions and cations

Carbon-13 chemical shifts δ^± in a variety of cyclohexadienylic cations and anions have been separated semi-empirically into contributions from (a) the hypothetical neutral framework, (b) charge effects and (c) interactions between ions and between ions and solvent molecules. Carbon-13 shifts for sites in neutral model compounds, δ⁰, approximate contribution (a) the shift differences for the model carbons δ^±_3,5-δ⁰_3,5 are assumed to reflect electric field effects (b). Thus what remains, δ^±_s-δ⁰_s- (δ^±_3,5-δ⁰_3,5) comes from the charge. The summation of these terms for each of the ions comes to ca. 165 ppm. This shows that a changein charge of ±1 of the neutral model, with correction for the electric field, brings about a change in shift of 165 ppm, thus supporting the linearity of charge with shift.     

A two-parameter family of an extension of Beatty sequences

Beatty sequences ⌊nα+γ⌋ are nearly linear, also called balanced, namely, the absolute value of the difference D of the number of elements in any two subwords of the same length satisfies D?1. For an extension of Beatty sequences, depending on two parameters s,t∈Z_>0, we prove D?⌊(s-2)/(t-1)⌋+2, and D?2s+1. We show that each value that is assumed, is assumed infinitely often. Under the assumption (s-2)?(t-1)² the first result is optimal, in that the upper bound is attained. This provides information about the gap-structure of (s,t)-sequences, which, for s=1, reduce to Beatty sequences. The (s,t)-sequences were introduced in Fraenkel [Heap games, numeration systems and sequences, Ann. Combin. 2 (1998) 197-210; E. Lodi, L. Pagli, N. Santoro (Eds.), Fun with Algorithms, Proceedings in Informatics, vol. 4, Carleton Scientific, University of Waterloo, Waterloo, Ont., 1999, pp. 99-113], where they were used to give a strategy for a 2-player combinatorial game on two heaps of tokens.     

Arylallylic lithium compounds, internally versus externally coordinated: comparison of their structures and dynamic behavior via X-ray crystallography and NMR

A comparison of externally coordinated arylallyllithiums with internally coordinated arylallyllithiums using a combination of X-ray and NMR studies shows that 2-bis(2-methoxyethyl)aminomethyl-1-phenylallyllithium 7, and 2-bis(2-methoxyethyl)aminomethyl-1,3-diphenylallyllithi um 8, are indeed fully internally coordinated with all phenyls endo, and lithium is close to one terminal allyl carbon. By contrast, among the externally coordinated analogs 1-phenylallyl-lithium.(HMPT)4 5, and 1-phenylallyllithium.(THF) 6, both phenyls are exo and the proximity of lithium to anion was only detected in compound 6. Lithium-7 NMR clearly identifies the Li(HMPT)4+ complex in 5, whereas a 7Li{1H} HOESY experiment reveals 7Li to be coordinated to THF in 6 and close to the PhC terminal allyl carbon. Carbon-13 NMR shows that all of the above compounds are fully delocalized despite differences among the sites of lithium and their separations from the anions. Changes in the 13C NMR line shapes show that the diphenyl compound 8 undergoes a very fast 1,3-lithium sigmatropic shift, and all of the phenyls in the above compounds undergo fast rotation around their phenyl Cipso-Callyl bonds. Barriers to rotation, DeltaH from NMR line shapes for 5, 6, 7, and 8 respectively, are 19.8, 14.6, 10.2, and 8.9 kcal x mol-1. The decrease in barriers is clearly correlated with a decrease in separation of lithium from an allyl terminal carbon and implies that especially among the latter three, 6, 7, and 8, lithium is involved in the mechanism for phenyl rotation. This question is discussed.     

Effects of ligands on ion-pairing behavior of benzylic lithium compounds

The $\text{1}\text{1}$ adduct of t-butyllithium and α-methylstyrene (II) has been generated in cyclopentane in the presence of a variety of ether and t-amine ligands as well as unsolvated, giving stable solutions in every case. NMR spectra of the solvated species are the same for all ligands but differ from that of the unsolvated compound. The results are consistent with a salt which contains a conjugated t-benzylic anion and exists as a loose ion-pair in the presence of ligands and as a tight ion-pair in cyclopentane alone. In contrast, benzyllithium behaves like a tight ion-pair in the presence of all ligands tried. Steric hindrance to tight ion-pairing at C_α of II is concluded to be responsible for the results observed.A covalently-bonded dilithium compound, 4,4-dimethyl-2-lithio-2-(p-lithiophenyl)pentane has been generated.     

Complementing and exactly covering sequences

The following is proved (in a slightly more general setting): Let α₁, …, α_m be positive real, γ₁, …, γ_m real, and suppose that the system [nα_i + γ_i], i = 1, …, m, n = 1, 2, …, contains every positive integer exactly once (= a complementing system). Then $\text{α}{}_{\mathrm{i}}\text{α}{}_{\mathrm{j}}$ is an integer for some i ≠ j in each of the following cases: (i) m = 3 and m = 4; (ii) m = 5 if all α_i but one are integers; (iii) m ? 5, two of the α_i are integers, at least one of them prime; (iv) m ? 5 and α_n ? 2ⁿ for n = 1, 2, …, m ? 4.For proving (iv), a method of reduction is developed which, given a complementing system of m sequences, leads under certain conditions to a derived complementing system of m ? 1 sequences.     

Geodetic contraction games on graphs

LetG = (V, E) be a finite graph. The setL of labeled vertices is initially empty. Two players A and B move alternately (A first), by choosing an unlabled vertexu V\L; thenu itself and all vertices on shortest paths betweenu and any vertex ofL are adjoined toL. WhenL =V, the game is over. Innormal play, the first player unable to move loses and his opponent wins. The outcome is reversed formisère play. We resolve the game by determining its winning strategies for the following cases: trees in normal play; cycles in normal and misère play; and complete graphsK _m with rays all of lengthn, in normal play.Partial support by Canadian Research Grants 69-0695 and A-4792 during visits at the University of Calgary and Simon Fraser University is gratefully acknowledged.     

Games played by Boole and Galois

We define an infinite class of 2-pile subtraction games, where the amount that can be subtracted from both piles simultaneously is an extended Boolean function f of the size of the piles, or a function over GF(2). Wythoff's game is a special case. For each game, the second player winning positions are a pair of complementary sequences. Sample games are presented, strategy complexity questions are discussed, and possible further studies are indicated. The motivation stems from the major contributions of Professor Peter Hammer to the theory and applications of Boolean functions.