On the Sociobiology of Hawks and Doves. A phase diagram for integration and segregation

Competing groups in a population will be integrated or segregated depending on their contest strategies. In this work a population of a fixed proportion of hawks and doves is supposed to be able to employ two different contest strategies, one more competitive than the other one. Energies are derived for populations employing these strategies and these energies depend on the availability of the resource for which hawks and doves compete. The energy for the less competitive strategy is lower than the other one when the resource is abundant. In that case hawks and doves can be in cohabitation in all proportions. If, however, the resource is scarce, the energy of the more competitive strategy is lower than the other one. In that case complete segregation of hawks and doves into colonies will result. The situation is akin to the phase pressure diagram of a binary solution with eutectic point, miscibility gap in the liquid phase and complete miscibility in the vapour phase. This revised version was published online in August 2006 with corrections to the Cover Date.     

An accurate numerical multicenter integration for molecular orbital theory

A powerful and accurate numerical three‐dimensional integration scheme was developed especially for molecular orbital calculations. A multicenter integral is decomposed into the sum of single‐center integrals using nuclear weight functions and calculated using Gaussian quadrature rules. The decomposed single‐center integrands show strong anisotropy. With a careful selection of the Gaussian quadrature rule according to the anisotropy, it is possible to obtain an accuracy of 13 digits with a small number of integration points for the overlap integrals, normalization integrals, and molecular integrals for the hydrogen molecule. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 509–523, 1999     

A method for selection and use of numerical integration points for molecules with cylindrical symmetry

Boys and Handy [1] have discussed the solution of the bivariational equations with restricted numerical integration. One of the weaknesses of the method was that in the numerical summations over points, some points arose with r _ij= 0 and non-zero weights. This makes the method quite impractical for the Schrodinger Hamiltonian (because of the singularity at r _ij= 0), and it cannot be advantageous for the transcorrelated Hamiltonian C^–1HC because there will be some discontinuous higher derivatives at r _ij=0. Here it is shown how the symmetry of cylindrically symmetric molecules can be used to eliminate such points, without losing any of the advantages of the overall method, such as the convergence of the eigensolutions. It is also shown how the primary numerical integration points (z _i, r_i) may be chosen in any calculation such that each is associated with an equal amount of one-electron density. The choice of the angular coordinates are governed by the removal of the r _ij=0 points and maintaining the natural orthogonality between orbitals of different symmetry types. The method has been programmed and found to be practical, although no new molecular calculations have yet been performed. It is to be hoped that these points will give a basis for new transcorrelated calculations on diatomic molecules.This paper was presented during the session on numerical integration methods for molecules of the 1970 Quantum Theory Conference in Nottingham. It has been revised in the light of the interesting discussion which followed.     

Binary alloys as a model for the multicultural society

The model of regular solutions, that may be applied to binary alloys (e.g. Au?Pt, Si?Ge) has been compared to binary societies: blacks—non-blacks in the US, catholics—non-catholics, foreigners—German citizen. The excellent agreement of phase diagrams and intermarriage data encourages a calculation of the multicultural society by functions of thermodynamics: Solubility corresponds to integration, miscibility gap to segregation, free enthalpy to happiness and temperature to tolerance of a society. Only a high level of tolerance will integrate ghettos and lead to a peaceful multicultural society.