Timothy J. McLarnan

A study of the arsenic, black phosphorus, and other structures derived from rock salt by bond‐breaking processes. I. Structural enumeration

The structures of arsenic and black phosphorus can be regarded as distortions of the simple cubic or NaCl structure. The presence of more than four electrons per site results in the breaking of three of the six linkages around each atom in the NaCl structure. A generalization of Polya’s enumeration theorem is proven which allows counting of the 36 possible ways of breaking bonds to produce structures containing trigonal pyramidal coordination at each site and having the same size unit cell as NaCl. Larger cells are considered briefly.

Observed wurtzite derivatives and related dipolar tetrahedral structures

Abstract Eight lower-symmetry ordered derivatives of the wurtzite type and an additional six related dipolar tetrahedral structure types were observed. All these structures are based on hexagonal closest packing of anions. Their mutual symmetry relationships can be summarized in a Barnighausen diagram. Dipolar tetrahedral structures are characterized by centrosymmetric space groups, thus resulting in a dipolar distribution of the coordination tetrahedra. Dipolar tetrahedral structures often exhibit disorder and random statistical occupations and several good ionic conductors are found among them. The wurtzite-type derivatives are characterized by polar space groups and all coordination tetrahedra in them are pointing in the same direction. Some simple, geometrically possible wurtzite derivatives have never been observed. Their absence can be rationalized on the basis of a poor geometrical fit between tetrahedral chains as demonstrated by computer simulations of their hypothetical structures. The ratio ϵ (quotient of the thickness of one hexagonal layer in [001] over one measure of the mean diameter of one atom within this layer) is useful for ordering the wurtzite derivatives on the basis of their chemical composition, since ϵ increases from the oxides, over the nitrides to the sulfides. The ϵ ratio also distinguishes clearly between the oxidic wurtzite types (0.79) and the tetrahedral rutile types of β-BeO derivatives (0.86), while the dipolar tetrahedral types have ϵ values around 0.82, that is, close to the value expected for hexagonal closest packing.

A study of the arsenic, black phosphorus, and other structures derived from rock salt by bond‐breaking processes. II. Band structure calculations and the importance of the gauche effect

Band structure calculations based on the extended Huckel method order energetically the 36 arsenic‐like structures having the NaCl size unit cell in a way which is compatible with the observed crystal structures. Black phosphorus is the most stable of the 36, and the arsenic layer type is number seven. Energy differences among these structures can be explained by different energetic contributions from syn‐, gauche, and anti‐arrangements, of both bonded and nonbonded pairs of atoms, and by a destabilization accompanying small bond angles. The order of stability is gauche ≳ anti ≳syn. Lone pair repulsions are important in determining the total energy, but play only a minor role in the relative energies of these structures. The energies ascribed to all these local geometries are compatible with molecular orbital calculations on small molecules and with observed molecular conformations described by the ’’gauche effect.’’

The combinatorics of cation-deficient close-packed structures

Abstract Polyas enumeration theorem is used to derive an algorithm for counting all hexagonal close-packed structures with a unit cell of given size, having composition MX 2 , where X is an hcp anion, M is an octahedrally coordinated cation, and no face sharing is permitted between octahedra. Generalizations of this algorithm to enumerate ordered derivatives of these structures, hcp structures with tetrahedral instead of octahedral cations, and similar structures having different stacking sequences among the close-packed layers are sketched.

Enumeration of wurtzite derivatives and related dipolar tetrahedral structures

Abstract Polyas theorem and related results are used to count and classify by space group all possible wurtzite derivatives with small unit cells having composition ABX2, AB2X3, AB3X4, or ABC2X4. The same arguments are applied to the dipolar tetrahedral structures, which resemble wurtzite but have a different pattern of occupancy of the tetrahedral voids in the hcp anion framework. Covalent molecular orbital and electrostatic calculations are used to study the two real and eight hypothetical structures for Li3PO4. Both predict shared edges to be destabilizing, in keeping with Paulings rules.

The structures of transition metal-transition metal alloys

Abstract A structure map using the average electron count and d orbital energy difference as indices is used to sort transition metal alloys of stoichiometry AB . The gross features of the map are mimicked by tight-binding calculations. The inclusion of s orbitals on the metal atoms appear to be important in the determination of alloy structure in some parts of the calculated map. The correct coloring of the elemental lattice as a function of electron count is reproduced by calculation (i.e., AuCd vs WC and CsCl vs CuTi). Two new stability fields for the WC and CuTi structures are predicted. The calculations fail to really distinguish bcc, fcc, and hcp derivative structures in the region of 6–8 d + s valence electrons per atom. In this part of the structure map the calculations appear to be sensitive to small geometrical changes.

The enumeration of reaction pathways using Burnside's lemma

Formulas are given for the number and statistical weights of reaction pathways from one molecular conformation to another via a given transition state. Arbitrary mechanistic assumptions can be incorporated into this enumeration. These formulas arise as straightforward generalizations of an expression for the number of double cosets in any group, which can be proven in one line using Burnsides lemma.