Hydrogen computation

M. Holzmann, C. Pierleoni, and D. M. Ceperley (2005) Coupled Electron Ion Monte Carlo Calculations of Atomic Hydrogen. Comput. Physics Commun. 169, p. 421... 

Each row in A represents a component in the BTX system in the following order (i) benzene, (ii) ethylene, (iii) toluene, (iv) xylene, (v) diphenyl, and (vi) hydrogen. Computing rank(A) results in an answer of three, which validates that all three reactions participating in the BTX system are independent. The stoichiometric subspace is hence a three-dimensional subspace residing in re . Matrix N may be computed from null(A ) (the null space to the stoichiometric subspace). We expect the size of N to be 6 X 3, since the total number of species is 6 and rank(A) = 3. Performing the null space computation, we find that the following three vectors... 

Fig. 1. The phase diagram of condensed hydrogen computed by the quantum Monte Carlo methods. Here r is the average proton separation. The calculations treated both electrons and protons quantum mechanically except that the protons are required to have a average fee structure in the metaiiic phase and the centers of the molecules are restrained in the molecular phase. The molecular curve agrees well with experiment to 0.5 Mbar and the transition is predicted to occur around 5 Mbar. (Figure courtesy of D. M. Ceperley and B. J. Alder). 

Treat nitrogen as if it were a carbon. However, as nitrogen is trivalent, not tetravalent, subtract one hydrogen per nitrogen from the total number of hydrogens computed for the saturated hydrocarbon. A formula that can be used is... 

This interpretation is supported by model calculations for the [6,6] closed isomer of homopyracyclene, which afford large positive NICS values in the five- and six-membered rings and an even larger deshielding of the methylene hydrogens (computed 5 5 7. DFT level). 

Dougherty E P and Rabitz H 1980 Computational kinetics and sensitivity analysis of hydrogen-oxygen combustion J. Chem. Phys. 72 6571... 

Using Jacobi coordinates and reduced masses, the Hydrogen-Chlorine interaction is modeled quantum mechanically whereas the Ar-HCl interaction classically. The potentials used, initial data and additional computational parameters are listed in detail in [16]. 

A second idea to save computational time addresses the fact that hydrogen atoms, when involved in a chemical bond, show the fastest motions in a molecule. If they have to be reproduced by the simulation, the necessary integration time step At has to be at least 1 fs or even less. This is a problem especially for calculations including explicit solvent molecules, because in the case of water they do not only increase the number of non-bonded interactions, they also increase the number of fast-moving hydrogen atoms. This particular situation is taken into account... 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

DFT calculations offer a good compromise between speed and accuracy. They are well suited for problem molecules such as transition metal complexes. This feature has revolutionized computational inorganic chemistry. DFT often underestimates activation energies and many functionals reproduce hydrogen bonds poorly. Weak van der Waals interactions (dispersion) are not reproduced by DFT a weakness that is shared with current semi-empirical MO techniques. 

So called Ilydrogenic atomic orbitals (exact solutions for the hydrogen atom) h ave radial nodes (values of th e distance r where the orbital s value goes to zero) that make them somewhat inconvenient for computation. Results are n ot sensitive to these nodes and most simple calculation s use Slater atom ic orbitals ofthe form... 

There are a number of different ways that the molecular graph can be conununicated between the computer and the end-user. One common representation is the connection table, of which there are various flavours, but most provide information about the atoms present in the molecule and their connectivity. The most basic connection tables simply indicate the atomic number of each atom and which atoms form each bond others may include information about the atom hybridisation state and the bond order. Hydrogens may be included or they may be imphed. In addition, information about the atomic coordinates (for the standard two-dimensional chemical drawing or for the three-dimensional conformation) can be included. The connection table for acetic acid in one of the most popular formats, the Molecular Design mol format [Dalby et al. 1992], is shown in Figure 12.3. 

Computer Project 5-2, The result for hydrogenation of cyclopentene differs from that foi ethene by an amount that is well outside the expected error foi MM calculations. Suggest a reason for this. 

The orbitals used for methane, for example, are four Is Slater orbitals of hydrogen and one 2s and three 2p Slater orbitals of carbon, leading to an 8 x 8 secular matrix. Slater orbitals are systematic approximations to atomic orbitals that are widely used in computer applications. We will investigate Slater orbitals in more detail in later chapters. 

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