The general analytic expression for S4-symmetry-invariant potential functions of tetra-atomic homonuclear molecules

The potential energy of a homonuclear X₄ molecule is an invariant of the atom permutation group S₄ acting on the internal coordinates of X₄. It is shown by means of invariant theory that six algebraically independent invariants and five additional invariants are required to express the general invariant function for this group. Explicit expressions for these 11 invariants are given.     

Upper and lower bounds to partition functions for molecules with double minimum potentials

Upper and lower bounds to the vibrational partition function q for molecules with double minimum potentials are derived. Both the exact lower (Gibbs-Bogoliubov) and upper (Golden-Thompson) bound to q are evaluated analytically for the harmonic oscillator perturbed symmetrically or asymmetrically by a gaussian barrier. For the quadratic-quartic oscillator only the lower bound can be evaluated analytically, whereas the upper bound leads to a strongly divergent series resisting conventional surnmability techniques. In this case the classical partition function is used as an upper bound.     

Calculation of rotational partition functions by an efficient Monte Carlo importance sampling technique

The evaluation of the classical rotational partition function represented by a configuration integral over all external and internal rotational degrees of freedom of nonrigid chain polyatomic molecules is described. The method of Pitzer and Gwinn is used to correct the classical partition function for quantum mechanical effects at low temperatures. The internal rotor hindrance and all coupling arising from the external and internal rotational degrees of freedom are explicitly taken into account. Importance sampling Monte Carlo based on the adaptive VEGAS algorithm to perform multidimensional integration is implemented within the TINKER program package. A multidimensional potential energy hypersurface is calculated with the MM3(2000) molecular mechanics force field. Numerical tests are performed on a number of small n-alkanes (from ethane to octane), for which the absolute entropies calculated at three different temperatures are compared both with the experimental values and with the previous theoretical results. The application of a more efficient importance sampling technique developed here results in a substantial reduction of statistical errors in the evaluation of the configuration integral for a given number of Monte Carlo steps. Error estimates for the calculated entropies are given, and possible sources of systematic errors, and their importance for a reliable prediction of the absolute entropy, are discussed.     

Iterative solutions with energy selected bases for highly excited vibrations of tetra-atomic molecules

The use of energy selected bases (ESB) with iterative diagonalization of the Hamiltonian matrix is described for vibrations of tetra-atomic systems. The performance of the method is tested by computing vibrational states of HOOH below 10,000 cm(-1) (1296 A+ symmetry states) and H(2)CO below 13,500 cm(-1) (729 A(1) symmetry states). For iterative solutions, we tested both the implicitly restarted Lanczos method (IRLM) and the standard (nonreorthogonalizing) Lanczos approach. Comparison with other contracted basis approach as well as direct product grid representation shows superior performance of the ESB/IRLM approach. Of the two systems, H(2)CO is found to be more challenging than HOOH since it has much stronger couplings among vibrational modes, which leads to a drastically larger primitive basis set. For H(2)CO we also discuss some interesting behavior of the molecule in the high internal energy regime.     

Calculation of some thermodynamic properties of air plasmas: Internal partition functions,plasma composition,and thermodynamic functions

The thermodynamic properties of air plasmas are calculated in the framework of the C.L.T.E. hypothesis, for pressures varying from 1 to 200 atm, in a temperature range from 1000 to 30,000 K. In these calculations, the neutral mon--, di-, and triatomic species, as well as their positive and negative ions, are taken into account. From the internal partition function values, the plasma compositions and the thermodynamic functions (specific enthalpy, Gibbs and Helmholtz free energies and entropy) are calculated.     

Calculation of single-ion partition coefficients

Single-ion partition coefficients have been calculated for NBu ₄ ⁺ and I⁻ in the system water—chloroform. The standard free energy of transfer of each ion was separated into an electrostatic and a neutral term. The electrostatic term was calculated with the aid of the Born equation. In the case of I⁻ the neutral term was calculated with the aid of the xenongas assumption, whereas in the case of NBu ₄ ⁺ the tetrabutylmethane assumption was used. The results of the calculation were compared with the experimentally found partition coefficient of the completely dissociated ion-pair NBu₄I.     

Calculating energy levels of isomerizing tetra-atomic molecules. II. The vibrational states of acetylene and vinylidene

A general, full-dimensional computational method for the accurate calculation of rotationally and vibrationally excited states of tetra-atomic molecules is further developed. The resulting computer program may be run in serial and parallel modes and is particularly appropriate for molecules executing wide-amplitude motions and isomerizations. An application to the isomerizing acetylene/vinylidene system is presented. Large-scale calculations using a coordinate system based on orthogonal satellite vectors have been performed in six dimensions and vibrational term values and wave functions for acetylene and vinylidene states up to approximately 23 000 cm(-1) above the potential minimum have been determined. This has permitted the characterization of acetylene and vinylidene states at and above the isomerization barrier. These calculations employ more extensive vibrational basis sets and hence consider a much higher density of states than in any variational calculations reported hitherto for this system. Comparison of the calculated density of states with that determined empirically suggests that our calculations are the most realistic achieved for this system to date. Indeed more states have been converged than in any previous study of this system. Calculations on lower lying excited states of acetylene based on HC-CH diatom-diatom coordinates give nearly identical results to those based on orthogonal satellite vectors. Comparisons are also made with calculations based on HH-CC diatom-diatom coordinates.     

The rotational partition function for linear molecules

An important function in the statistical treatment of a gas of linear molecules is This sum is convenient to use mainly when α is large and alternate expressions, generally asymptotic expansions, are often used when α is small. In this paper, the sum is evaluated to yield a single expression that is valid for large and small values of α. The expression is composed of three terms, each of which involves the theta functions of Jacobi. One term is in the form of an integral, but is small relative to the other two and easily evaluated by numerical means. The expression is readily differentiated and can be used for the general evaluation of the rotational partition function for gases of linear molecules at all temperatures. This revised version was published online in July 2006 with corrections to the Cover Date.     

Calculation of conformations of organic molecules

The most stable conformation of the molecule has a minimum of potential energy that arises due to the competition between the tendency of the valency angles to take ideal values and that of non-bonded atoms to be situated at an equilibrium distance. The equilibrium distance is equal to the sum of intermolecular radii determined by measuring the distances between atoms of adjacent molecules in the crystal. This idea is suggested as underlying a method of computation that enables to solve two problems. Firstly, the structure of the molecule being known, it becomes possible to ascertain the points of the interaction curve of the non-bonded atoms and secondly, with the interaction curve known, one can calculate the optimal configuration of the molecule.

To illustrate the suggested theory the conformation of molecules of some cyclic hydrocarbons has been calculated. It is to be stressed that the investigation of the conformation of strained molecules affords the main means of studying the interaction of non-bonded atoms.     

A derivation of local composition expressions from partition functions

This paper presents a method for developing local composition expressions from partition functions and the two-liquid theory. The assumptions required for the derivation are discussed. It is shown that the local composition expressions are theoretically founded only in the vicinity of complete randomness.     

Potential energy functions of polyatomic molecules

A direct method is proposed for determining polyatomic potential energy functions, expressed in terms of normal coordinates, which yield a given set of vibrational excitation energies. The method is a modification of the semiclassical technique for computing vibrational energy levels of Percival and Pomphrey. The technique is used to derive potential functions for the NO₂, SO₂ and ClO₂ molecules. With these potentials twenty two higher vibrational excitations energies have been predicted for these molecules and these results differ from the experimental values by at most 3 cm^?1. The computed potential functions are not unique despite the apparent accuracy of the vibrational energy levels. Comparison with the RKR method indicates that the present method must be extended to include rotational perturbations.     

Numerical generation of hyperspherical harmonics for tetra-atomic systems

A numerical generation method of hyperspherical harmonics for tetra-atomic systems, in terms of row-orthonormal hyperspherical coordinates-a hyper-radius and eight angles-is presented. The nine-dimensional coordinate space is split into three three-dimensional spaces, the physical rotation, kinematic rotation, and kinematic invariant spaces. The eight-angle principal-axes-of-inertia hyperspherical harmonics are expanded in Wigner rotation matrices for the physical and kinematic rotation angles. The remaining two-angle harmonics defined in kinematic invariant space are expanded in a basis of trigonometric functions, and the diagonalization of the kinetic energy operator in this basis provides highly accurate harmonics. This trigonometric basis is chosen to provide a mathematically exact and finite expansion for the harmonics. Individually, each basis function does not satisfy appropriate boundary conditions at the poles of the kinetic energy operator; however, the numerically generated linear combination of these functions which constitutes the harmonic does. The size of this basis is minimized using the symmetries of the system, in particular, internal symmetries, involving different sets of coordinates in nine-dimensional space corresponding to the same physical configuration.     

On the best quasiclassical adiabatic approximation for partition functions

An explicit proof is given that no sequence of quasi-classical adiabatic approximations in which intrinsic symmetry restrictions are relaxed in any way can give an appropriately approximated partition function smaller than the one directly obtainable in the absence of such a sequence.     

Rapid calculation of partition functions and free energies of fluids

The partition function (Q) is a central quantity in statistical mechanics. All the thermodynamic properties can be derived from it. Here we show how the partition function of fluids can be calculated directly from simulations; this allows us to obtain the Helmholtz free energy (F) via F = -k(B)T ln Q. In our approach, we divide the density of states, assigning half of the configurations found in a simulation to a high-energy partition and half to a low-energy partition. By recursively dividing the low-energy partition into halves, we map out the complete density of states for a continuous system. The result allows free energy to be calculated directly as a function of temperature. We illustrate our method in the context of the free energy of water.     

Potential functions for alkali halide molecules

Repulsion and dispersion parameters for alkali–metal halide diatomic molecules were computed by ionic Rittner and truncated Rittner models with radial dependent repulsion terms. Experimental data on the bond energies, the equilibrium interionic distances, and the spectroscopic frequencies were employed for the purpose. The polarizabilities used were also computed from the experimental dipole moments of alkali–metal halides. The potential parameters obtained were compared with parameters from other sources and checked for consistency. The computed potential parameters of alkali–metal halide monomer molecules were used to predict the energetics and geometries for alkali–metal halide dimer molecules. The predicted values are in good agreement with experiment and other calculations indicating the consistency and reliability of the potential employed. Although the magnitude of repulsive and dispersive energy terms varies with potential functions employed, the difference between the two for a molecule is constant. The repulsive term is more sensitive than the attractive term. The uncertainty in the exponential repulsion results in an inaccurate representation of the attractive contribution. Introduction of the radial-dependent repulsion term changes the relative magnitudes of repulsive and dispersive parameters and hence the relative contribution to the total potential with monomers. But this has no significant effect on the energetics and geometries of the dimers.     

Calculation of vibrational constants of diatomic molecules

We apply Wigner's effective-harmonic-oscillator (EHO) function with the Morse potential to the calculation of vibrational constants of diatomic molecules. A one-particle system is described by the operator representation of Wigner's function. We find the EHO parameters both for a single molecule and for an equilibrium ensemble of molecules. We obtain numerical values of the constants for the iodine molecule.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 24, No. 4, pp. 485–487, July–August, 1988.     

Analysis of wave functions for open-shell molecules

During the past decade we have looked at several ways to track the distribution of unpaired electrons during chemical reactions and in different spin states. These methods were inspired by our previous work on singlet di-radicals where the spin density is zero yet there are clearly singly occupied orbitals. More recently we have been concerned with analysis of wave functions for single molecule magnets. This review discusses the mathematical framework by which open-shell systems can be described, in addition to methods that extract the effectively unpaired electron density, the spin state of atoms in a molecule, and other useful properties from a molecular wave function. Some of the difficulties associated with using broken spin Slater determinants to evaluate the exchange coupling parameters in the Heisenberg Hamiltonian are also mentioned.