The internal coordinate path Hamiltonian; application to methanol and malonaldehyde

The internal coordinate path Hamiltonian is introduced for the study of the vibrations of molecules which have one large amplitude motion. The Hamiltonian is represented in terms of a one path coordinate and 3N—7 normal coordinates. The variational method is used to solve the Schrödinger equation. The molecules studied are methanol and malonaldehyde. For methanol the internal coordinate is a dihedral angle, for malonaldehyde it is the difference in the distances between the migrating hydrogen and the neighbouring oxygen atoms. For methanol there is little coupling between the path and the normal coordinates and so no complications were encountered in the calculations which used harmonic surfaces generated by density functional and M?ller—Plesset theory. Fundamental frequencies were predicted. Malonaldehyde is a different story. There is substantial coupling between the path coordinate and several of the normal coordinates. This introduces many complications: an anharmonic surface is essential and large variational configuration interaction calculations are essential for convergence. Furthermore, because the Coriolis terms require the evaluation of derivatives of both the nuclear coordinates and the normal coordinate eigenvectors along the path, great care must be taken with these numerical procedures. B3LYP predicts too low a transition state which overemphasizes the large Coriolis terms near the transition state. This may be one of the reasons why our fundamental vibrations are in poor agreement with observation. It is most encouraging that the tunnelling splitting is 58 cm^?1 (obs. 21.56 cm^?1), obtained with our quartic density functional surface.     

The Kepler problem in the Snyder space

In this paper the Kepler problem in the non-commutative Snyder scenario was studied. The deformations were characterized in the Poisson bracket algebra under a mimic procedure from quantum standard formulations by taking into account a general recipe to build the non-commutative phase space coordinates (in the sense of Poisson brackets). An expression for the deformed potential was obtained, and then the consequences in the precession of the orbit of Mercury were calculated. The result could be used for finding an estimated value for the non-commutative deformation parameter.     

FURTHER DISCUSSION OF NUMERICAL ERRORS IN CFD

The methods of estimating numerical errors given in an earlier paper are extended in directions that make them useful in actual CFD applications. In particular, the method of estimating convergence error (the error due to insufficient iteration) is extended to allow the possibility of complex eigenvalues; an ad hoc method that can be applied to any case is also given. For the discretization error, which arises from the numerical approximation of the differential equation(s), methods that can be used on non-uniform drids are presented; they can be extended to unstructured grids as well. The utility of these methods is demonstrated for linear problems as well as solutions of the Navier-Stokes equations. The examples show that the estimation of errors is neither difficult nor expensive.     

New vibrational self-consistent field program for large molecules

A recently developed, general computer program that performs vibrational self-consistent field (VSCF) calculations for large molecules is described. The program, which we refer to as VSCF―95, requires as its only input a force field in mass-scaled normal coordinates. Currently, it is limited to a maximum of 200 normal modes, and the force field is limited to coupling terms involving a maximum of six normal modes, with a maximum order of six in any normal mode. As output the program returns VSCF energies for specified quantum states. We illustrate the code with two new applications. The first is to HCO, for which we use a full sixth-order force field. The second is to a model of the fullerene, C₆₀, for which we have calculated a 75,731-term force field, which includes all anharmonic terms up to fifth order, and all two-mode coupling terms up to fourth order. © 1996 by John Wiley & Sons, Inc.     

The heats of formation of the haloacetylenes XCCY [X,Y = H,F, Cl]: basis set limit ab initio results and thermochemical analysis

The heats of formation of haloacetylenes are evaluated using the recent W1 and W2 ab initio computational thermochemistry methods. These calculations involve CCSD and CCSD(T) coupled cluster methods, basis sets of up to spdfgh quality, extrapolations to the one-particle basis set limit, and contributions of inner-shell correlation, scalar relativistic effects. and (where relevant) first-order spin-orbit coupling. The heats of formation determined using W2 theory are: δH₁ ²⁹⁸(HCCH) = 54.48 kcal mol^?1, δH_f ²⁹⁸(HCCH) = 25.15 kcal mol, δH_f ²⁹⁸(FCCF) = 1.38 kcal mol^?1, δH_f ²⁹⁸(HCCC1) = 54.83 kcal mol^?1, δH_f ²⁹⁸(CICCC1) = 56.21 kcal mol^?1, and δH_f ²⁹⁸(FCCC1) = 28.47 kcal mo1^?1. Enthalpies of hydrogenation and destabilization energies relative to acetylene were obtained at the WI level of theory. So doing we find the following destabilization order for acetylenes: FCCF > ClCCF > HCCF > ClCCCl > HCCCI > HCCH. By a combination of WI theory and isodesmic reactions. we show that the generally accepted heat of formation of 1,2-dichloroethane should be revised to ?31.8 ± 0.6 kcal mol^?1, in excellent agreement with a very recent critically evaluated review. The performance of compound thermochemistry schemes, such as G2, G3, G3X and CBS-QB3 theories, has been analysed.     

The energetics of naphthalene derivatives. II. The isomeric 1— and 2—naphthyl acetates

The isomeric 1- and 2-naphthyl acetates (acetoxynaphthalenes) are at the confluence of diverse concepts, techniques and classes of organic compounds. Summing the results of literature measurements of the enthalpy of formation of their solids and of our new sublimation enthalpies reported herein, we derive gas phase enthalpies of formation of ?209.9 ± 1.4 and ?213.3 ± 1.3 kJ mol^?1 respectively. This corresponds to 2-naphthyl acetate being more stable than its 1-isomer by 3.4 ± 1.9 kJ mol^?1. We also performed MP2(full)/6-31G(d) calculations on these species, resulting in enthalpies of formation of ?212.9 ± 3.9 and ?212.2 ± 3.9 kJ mol^?1 for 1- and 2-naphthyl acetate and a difference of ?0.7 kJ mol^?1 respectively in satisfactory agreement with the above experimental results.     

Existence, Uniqueness, and Construction of the Solution of a System of Ordinary Functional Differential Equations, with Application to the Design of Perfectly Focusing Symmetric Lenses

In the design of perfectly focusing symmetric lenses, one isled, in a natural way, to a set offunctional differential equations;that is, differential equations involving composites of unknownfunctions, with initial conditions prescribed on the lens axis.This paper concentrateson those features of the equations whichmake them uniquely solvable. They are: (i) a contractivenessproperty of the equations near the axis; (ii) a uniform retardationin the arguments of thecomposite functions away from the axis.The second and third sections of this paper generalize and formalizethese properties and provide proofs of existence, uniqueness,and continuous dependence on the data for solutions of suchgeneralized systems of functional differential equations. Becauseof the lens context which motivates our study, the problem inwhich the contractiveness property (i) above holds is calledthe ‘local’ problem, and the problem in which thearguments of composite functions are uniformly retarded is calledthe ‘global’ problem. In the final section of thepaper we apply the general results of the preceding sectionsto prove existence and uniqueness of perfectly focusing symmetriclenses up to distances from the lens axis at which various typesof breakdown, discussed in the text, may occur.