Tetraatomic Systems Research Articles

Nucleophilic or electrophilic interactions of C₂ with HX systems (X = F, Cl, Br).

Highly correlated ab initio wave functions within the MRCI approach are used in a comparative study of the interactions between C2 and the three hydrogen halides HX (X = F, Cl, Br). Test calculations are also presented using the UCCSD(T)-F12 approach. The asymptotic regions are investigated for different relative orientations of the two moieties. It is shown that the three systems C2 + HX are bound, for intermolecular distances close to 3 Å, through nucleophilic interactions between C2 and HX for approaches perpendicular to the C-C axis, with decreasing interaction energies from HF to HBr. For HX approaching C2 along its axis, the interactions, governed by the electrophilic character of C2 are decreasing from HBr to HF. Even though the reactions toward the molecular systems HCCX or CCHX are exothermic, activation barriers (0.58 eV and more) are calculated at short distances, preventing the direct reactions toward the corresponding tetra-atomic systems.

Reactive Scattering with Row-Orthonormal Hyperspherical Coordinates. 3. Hamiltonian and Transformation Properties for Pentaatomic Systems

The Hamiltonian for triatomic and tetraatomic systems in row-orthonormal hyperspherical coordinates has been derived previously. However, for pentaatomic systems this derivation requires nontrivial generalizations. These are presented in this paper, together with the corresponding Hamiltonian. Each of the twelve operators that contribute to this Hamiltonian is kinematic-rotation invariant. As for the triatomic and tetraatomic cases, these pentaatomic demcocratic coordinates are particularly well suited for calculations of reactive scattering in five atom systems.

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.

Incorporating the Geometric Phase Effect in Triatomic and Tetraatomic Hyperspherical Harmonics

Hyperspherical harmonics in the democratic row-orthonormal hyperspherical coordinates are very appropriate basis sets for performing reactive scattering calculations for triatomic and tetraatomic systems. The mathematical conditions for incorporating the geometric phase effect in these harmonics are given. These conditions are implemented for triatomic systems, and their explicit analytical expressions in terms of Jacobi polynomials, in both the absence and presence of the geometric phase effect, are given.

Hyperspherical Harmonics for Tetraatomic Systems. 2. The Weak Interaction Region

Hyperspherical harmonics as functions of hyperspherical coordinates appropriate for describing the weak interaction region of nuclear configuration space of tetraatomic systems are derived. They constitute an efficient basis set for calculating the corresponding local hyperspherical surface functions that avoid overcompleteness problems. Together with the hyperspherical harmonics for the strong interaction region of that space obtained previously, they should permit efficient benchmark-quality calculations of state-to-state differential and integral cross sections for those systems.

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.

Hyperspherical harmonics for tetraatomic systems

A recursion procedure for the analytical generation of hyperspherical harmonics for tetraatomic systems, in terms of row-orthonormal hyperspherical coordinates, is presented. Using this approach and an algebraic Mathematica program, these harmonics were obtained for values of the hyperangular momentum quantum number up to 30 (about 43.8 million of them). Their properties are presented and discussed. Since they are regular at the poles of the tetraatomic kinetic energy operator, are complete, and are not highly oscillatory, they constitute an excellent basis set for performing a partial wave expansion of the wave function of the corresponding Schrödinger equation in the strong interaction region of nuclear configuration space. This basis set is, in addition, numerically very efficient and should permit benchmark-quality calculations of state-to-state differential and integral cross sections for those systems.

Global fit of ab initio potential energy surfaces: II.1. Tetraatomic systems [formula omitted

Global potential energy surfaces (GPES) for general tetraatomic molecular systems which fit ab initio data can be obtained preserving the accuracy of the ab initio points. The global fitting technique is based in a procedure for tetraatomic systems including the functional form previously proposed by the authors. The global fit obtained fulfills the stringent criteria needed for molecular dynamical calculations. The program writes out as output file a Fortran-77 program in a form such that the potential and its derivatives with respect to coordinates can be evaluated readily and accurately at arbitrary geometries.

Ab initio SOFA quantum dynamics for chemical reaction

This Letter presents a sequential one-dimensional fitting approach (SOFA) to generate potential energies from a given set of ab initio data for tetraatomic systems. The SOFA approach allows one to carry out quantum dynamics calculation directly from pre-calculated ab initio energy points. Thus one could bypass the tedious and often intractable process of fitting global multi-dimensional potential energy surfaces (PES) which is a major bottleneck in computational quantum dynamics. The method of SOFA dynamics and its numerical application to the benchmark H 2+OH reaction in five mathematical dimensions are presented in this Letter.

Global nine-dimensional potential energy surface for the H5 system. II. Fit to an analytical expression

A global nine-dimensional potential energy surface (GPES) for the adiabatic ground state of the H5 system which fits the ab initio data, reported in the previous paper of this series, has been obtained (root-mean-square error 2.2 kcal/mol). The global fitting procedure is an extension of the corresponding procedure for triatomic and tetra-atomic systems including the functional form previously proposed by the authors. The H5 GPES obtained here is totally symmetric with respect to permutations of the hydrogen atoms and satisfies the criteria needed to be used in scattering calculations.

Rotation–vibration interactions in (HF)2. I. Using parallel supercomputers to calculate rotation–vibration energy levels

An algorithm for calculating rotation–vibrational energy levels and wave functions for AB–CD tetra-atomic systems is presented. By transforming the wave equation into a large sparse eigenvalue problem, we can take advantage of the implicitly restarted Lanczos method developed by Sorensen and co-workers. The algorithm has been applied to calculations of the lowest 40 bound states of (HF)2, (DF)2 and HF⋅DF with even and odd parities. The lowest 40 energies and corresponding wave functions for (HF)2 with J=0 and even parity can be calculated in 10.5 minutes on 126 processors of a CRAY T3E. The resulting energy levels are found to be in excellent agreement with the previously reported values of Zhang, et al. [J. Chem. Phys. 102, 2315 (1995)].

Reactive Scattering with Row-Orthonormal Hyperspherical Coordinates. 2. Transformation Properties and Hamiltonian for Tetraatomic Systems

The formalism for tetraatomic reactive scattering using row-orthonormal hyperspherical coordinates is presented. The transformation properties of these coordinates under kinematic rotations and symmetry operations are derived, as are the corresponding Hamiltonian and volume element. Each of the nine operators which contribute to this Hamiltonian is kinematic-rotation invariant. Continuity conditions appropriate for the absence and presence of the geometric phase associated with conical intersections are described. It is concluded that the row-orthonormal hyperspherical coordinates are particularly well suited for calculations of reactive scattering in tetraatomic systems.

Reactive Scattering with Row-Orthonormal Hyperspherical Coordinates. 1. Transformation Properties and Hamiltonian for Triatomic Systems

The formalism for triatomic reactive scattering using row-orthonormal hyperspherical coordinates is presented. The transformation properties of these coordinates under kinematic rotations and symmetry operations are derived, as is the Hamiltonian of the system. This formalism is compared with the one using symmetrized principal moment of inertia hyperspherical coordinates. Continuity conditions appropriate for the absence and presence of the geometric phase effect associated with conical intersections are considered. The potential application of these two sets of coordinates to tetraatomic systems is discussed. It is shown that the row-orthonormal coordinates are well suited for the treatment of systems of four or more atoms.