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Computational Quantum Chemistry of Chemical Kinetic Modeling

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Computational Quantum Chemistry of Chemical Kinetic Modeling ineering ng & E P l r a o c i c e m s e s Journal of h T C e f c Gargurevich, J Chem Eng Process Technol 2016, 7:2 h o l ISSN: 2157-7048 n a o n l o r g u y o J Chemical Engineering & Process Technology DOI: 10.4172/2157-7048.1000286 ReviewResearch Article Article OpenOpen Access Access Computational Quantum Chemistry of Chemical Kinetic Modeling Ivan A Gargurevich* Chemical Engineering Consultant, Combustion and Process Technologies, 32593 Cedar Spring Court, Wildomar, CA 92595, USA Abstract The earth orbiting the sun, the electron bonded to a proton in the hydrogen atom are both manifestations of particles in motion bound by an inverse-square force and both are governed by the principle of least action (of all the possible paths the particles may take between two points in space and time, they take those paths for which the time integral of the Lagrangian or the difference, kinetic energy-potential energy, is the least) and shaped by the same Hamiltonian (or total energy) structure. For both types of motion, the invariants (or properties that are conserved) are the energy and angular momentum of the relative motion, and the symmetry is that of the rotational motion. Differences arise because the electric force bounding the electron to the proton is forty two orders of magnitude stronger than the gravitational force, and the smallness of the hydrogen atom brings about “quantum effects”: the mechanics of the microscopic particles which constitute the atom is wave-like. Yet the central concepts of mechanics are preserved in integrity: least action, invariants or conservation laws, symmetries, and the Hamiltonian structure. The discussion in the sections that follow on the quantum-mechanical treatment of molecular structures is based for the most part on the books by Pople and Murrell [2,3], and is by no means comprehensive but hopefully will elucidate the most relevant concepts for performing the estimation of thermochemical and kinetic properties of elementary reactions. The aim of quantum chemistry is to provide a qualitative and quantitative description of molecular structure and the chemical properties of molecules. The principal theories considered in quantum chemistry are valence bond theory and molecular orbital theory. Valence bond theory has been proven to be more difficult to apply and is seldom used, thus this discussion will deal only with the application of molecular orbital theory to molecular structures. In molecular orbital theory, the electrons belonging to the molecule are placed in orbitals that extend all the different nuclei making-up the molecule (the simplest approximation of a molecular orbital being a simple sum of the atomic orbitals with appropiate linear weighting coefficients, Figure 1 below for carbon Monoxide as example), in contrast to the approach of valence bond theory in which the orbitals are associated with the constituent atoms. The full analytical calculation of the molecular orbitals for most systems of interest may be reduced to a purely mathematical problem, the central feature of which is the calculation and diagonalization of an effective interaction energy matrix for the system. In ab initio molecular orbital calculations, the parameters that appear in such an energy matrix are exactly evaluated from theoretical considerations, while in semi-empirical methods experimental data on atoms and prototype molecular systems are used to approximate the atomic and molecular integrals entering the expression for the elements of the energy interaction matrix. Ab initio methods can be made as accurate as experiment for many purposes Zeener [4], the principal drawback to “high level” ab initio work is the cost in terms of computer resources which restrict it to systems of ten or fewer atoms even for the most experienced users. This is what draws the chemist to semi-empirical methods that can be easily applied to complex systems consisting of hundreds of atoms. Presently, useful semi-empirical methods are limited in execution by matrix multiplication and diagonalization, both requiring computer time proportional to N3 where N is the number of atomic orbitals considered in the calculation or basis set. Keywords: Quantum chemistry; Chemistry; Chemical kinetics; Where the solutions, Ψ the so called wave functions, would Modelling describe the behavior of all the Particles, and the quantum-mechanical Hamiltonian above is or for one electron system such as the hydrogen Quantum-Chemistry Background atom, with the electron centered on the atomic nucleus: In classical mechanics, one is concerned with the trajectories of HTV� =� + � (3) particles which theoretically can be calculated from knowledge of the initial conditions and the structure of the Hamiltonian H, or the sum of a kinetic-energy contribution T and potential-energy function V. *Corresponding author: Ivan A Gargurevich, Chemical Engineering Consultant, H=T+V (1) Combustion and Process Technologies, 32593 Cedar Spring Court, Wildomar, CA 92595, USA, Tel: 9516759455; E-mail: [email protected] The existence of the atom cannot be explained classically, but Received February 10, 2016; Accepted March 06, 2016; Published March 31, rather by the wave properties of the electron bonded to the nucleus. 2016 Schrodinger suggested that the proper way to describe the wave Citation: Gargurevich IA (2016) Computational Quantum Chemistry of Chemical character of particles was to replace the classical kinetic and potential Kinetic Modeling. J Chem Eng Process Technol 7: 286. doi:10.4172/2157- 7048.1000286 energy functions of with linear operator T,V� � and a wave equation of the form. Copyright: © 2016 Gargurevich IA . This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted � (2) use, distribution, and reproduction in any medium, provided the original author and HEΨ= Ψ source are credited. J Chem Eng Process Technol ISSN: 2157-7048 JCEPT, an open access journal Volume 7 • Issue 2 • 1000286 Citation: Gargurevich IA (2016) Computational Quantum Chemistry of Chemical Kinetic Modeling. J Chem Eng Process Technol 7: 286. doi:10.4172/2157-7048.1000286 Page 2 of 8 Where, g is any geometrical variable. h2 i � =−∇2 (4) T 2 The heat of formation for the molecule can then be obtained from 8mπ the total energy of equation (10) via 2 nN Ze AA � (5) (11) V = − ∆=−HEf r∑∑ E k +∆ Hfi r k1= i1= Where m is the mass of the electron, r is the distance of the AA∆ electron from the nucleus, Z is the atomic number, and e is the unit of Where Ek and Hfi are the electron energies and the heats of electronic charge, and in equation (5) the Laplacian ▽2 is in Cartesian formation of individual atoms, respectively? Clearly, this approach coordinates. For molecules more complex than the simple hydrogen requires the accurate knowledge of the atomic heats of formation, atom (for ·which exact solutions to the Schrodinger equation can be which may or may not be available [6]. found), the Born-Oppenheimer approximation states that because the The electronic Hamiltonian (non-relativistic) is the given by the nuclei are so much more massive than the electrons, the electrons adjust following expression in atomic units (h/2π=e=m=1) essentially instantaneously to any motion of the nuclei, consequently H� =1 ∇−2Zr−− 11 + r we may consider the nuclei to be fixed at some internuclear separation ∑2 p ∑∑A AP ∑ pq (12) in order to solve the Schrodinger equation (2) for the electronic wave p AP pq< function [5], or Where A -designate the nuclei, p, q electrons, and r is the interparticles distance. (6) Ψ≈ΨN Ψ elec The solutions to the electronic Schrodinger equation (8) are infinite Where the first term in the product of equation (6) accounts for the but for stationary, bound states only the continuous, single-valued motion of the nuclei and the second term involves the electron motion. Eigen functions i that vanish at infinity need to be considered, and the Furthermore, introducing center -of- mass and relative coordinates, electronic energies are the eigenvalues E or the nuclear wavefunction reduces to i � (13) HEΨ=i ii Ψ ΨN ≈Ψ trans (C.M.) Ψrot Ψ vib (7) The Eigen functions are normalizable and mutually orthogonal Where the center-of-mass translation, and rotational and (i.e., orthonormal) or mathematically they satisfy the condition vibrational contributions to the nuclear wave function are now Ψ* Ψ τ= Ψ Ψ =δ (14) explicitly shown. Thus, the problem of determining the structure of a ∫ i jdi | j ij alli, j complex molecule reduces to solving each Schrodinger equation for the electronic motion, the translational motion of the center of mass, and In equation (14), the integration is over the volume element for the the rotational and vibrational motion of the nuclei separately. Thus, the electron, and we have introduced the matrix or Dirac notation for the electronic energy is estimated by the equation (similarly for the other integral and δ is the Kronecker delta symbol. The electronic energy of types of motion). the system Ei is the expectation value of the Hamiltonian � or Helec (1,2,...,n)Ψ=Ψ (1,2,...,n) E (1,2,...,n) (8) elec elec elec * � � for a molecule with n electrons, and for a given internuclear ΨH Ψ d τ= Ψ |H| Ψ = E (15) distance the total energy of the system is ∫ ij i j i E0≈+ E eZ 21 Zr− The complete treatment of a quantum-mechanical problem T elec ∑ A B AB (9) involving electronic structure requires the complete solution of the AB< Schrodinger equation (8). This is only possible for one-electron Where the second term is the electrostatic internuclear repulsion systems, and for many-electron systems, where the electron repulsion energy and A,B designate the nuclei.
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