Schrodinger equation force field methods

The most fundamental level of modeling of any chemical system employs quantum mechanics. Quantum mechanical (QM) treatments are required to understand many important chemical and biological properties of nucleic acids. Moreover, empirical force-field methods, employed to study the conformations of polynucleotides, rely on quantum calculations to obtain crucial parameters that are difficult to measure experimentally, such as atom-centered charges for calculating electrostatic interactions. The obtain a description of a chemical system using QM one solves the time-independent Schrodinger equation with or without the use of empirical parameters. 

Schrodinger equation. When the molecule is too large and difficult for quantum mechanical calculations, or the molecule interacts with many other molecules or an external field, we turn to the methods of molecular mechanics with empirical force fields. We compute and obtain numerical values of the partition functions, instead of precise formulas. The computation of thermodynamic properties proceeds by using a number of techniques, of which the most prominent are the molecular dynamics and the Monte Carlo methods. 

In summary, this method solves the Schrodinger equation at several intervals of time for the largest possible sample that can be solved with present computational resources. It also creates a force field to compute forces with a classical molecular dynamics procedure in a system containing the largest number of particles that is practical to be used with MD methods. When the time intervals of the ab initio calculations coincide with the time intervals of the molecular dynamics, and when the electron density distribution is used to compute the forces instead of the force field, this method is equivalent to the well known Car-Parrinello method. Evidently, this latter method is limited to a... 

In contrast to force-field calculations in which electrons are not explicitly addressed, molecular orbital calculations, use the methods of quantum mechanics to generate the electronic structure of molecules. Fundamental to the quantum mechanical calculations that are to be performed is the solution of the Schrodinger equation to provide energetic and electronic information on the molecular system. The Schrodinger equation cannot, however, be exactly solved for systems with more than two particles. Since any molecule of interest will have more than one electron, approximations must be used for the solution of the Schrodinger equation. The level of approximation is of critical importance in the quality and time required for the completion of the calculations. Among the most commonly invoked simplifications in molecular orbital theory is the Bom-Oppenheimer [13] approximation, by which the motions of atomic nuclei and electrons can be considered separately, since the former are so much heavier and therefore slower moving. Another of the fundamental assumptions made in the performance of electronic structure calculations is that molecular orbitals are composed of a linear combination of atomic orbitals (LCAO). 

In summary, the two problem areas of state-of-the-art mobility calculations are the neglect of inelasticity of molecular collisions, especially with respect to rotation, and poor quality or absence of force fields for ion-molecule interactions. However, the impossibility of rigorously solving the Schrodinger equation for polyatomic molecules has stimulated rather than precluded continuous improvement and application of approximate quantum chemistry methods. 

Shortly after the development of VBT, an alternative model, known as MOT, was introduced by the American physicist Robert Mulliken (and others) around 1932. MOT is a delocalized bonding model, where the nuclei in the molecule are held in fixed positions at their equilibrium geometries and the Schrodinger equation is solved for the entire molecule to yield a set of MOs. In practice, it is possible to solve the Schrodinger equation exactly only for one-electron species, such as H2. Whenever more than one electron is involved, the wave equation can only yield approximate solutions because of the e/ectron correlation problem that results from Heisenberg s principle of indeterminacy. If one cannot know precisely the position and momentum of an electron, it is impossible to calculate the force field that this one electron exerts on every other electron in the molecule. As a result of this mathematical limitation, an approximation method must be used to calculate the energies of the MOs. 

Systems of this size prohibit the use of a computational approach based solely on quantum mechanics, a method that can only provide an exact solution for systems with limited number of electrons. To study the dynamics of proteins, an approximate solution to the Schrodinger equations, such as the one given by molecular dynamics using force fields, is required. This type of molecular dynamics is based on three approximations ... 

The quantity qt is the charge on nucleus i, and /Oe (t /) is the quantum mechanical electron probability density corresponding to the electronic state Z. Note that the right side of Eq. (14) is the force on nucleus i, as it would be calculated from classical electrostatics if /oe(t Z) was known. Thus, this equation justifies the description of (short-ranged) intra-molecular interactions via empirical force fields. We remark that Eq. (14), at least in principle, allows quantum MD calculations. For fixed positions of the nuclei, /oe(t Z) may be computed solving Schrodinger s equation numerically. Subsequently, the nuclei are displaced according to Eqs. (1) and (14). However, this procedure is prohibitively slow, and in practice other methods are used. 

Ab initio methods solve the molecular Schrodinger equation associated with the molecular Hamiltonian based on different quantum-chemical methodologies that are derived directly from theoretical principles without inclusion of any empirical or semiempirical parameters in the equations. Though rigorously defined on first principles (quantum theory), the solutions from ab initio methods are obtained within an error margin that is qualitatively known beforehand thus all the solutions are approximate to some extent. Due to the expensive computational cost, ab initio methods are rarely used directly to study the physicochemical properties of flotation systems in mineral processing, but their application in developing force fields for molecular mechanics (MM) and MD simulation has been extensively documented. (Cacelli et al. 2004 Cho et al. 2002 Kamiya et al. 

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