Approximation methods quantum-mechanical

Both molecular and quantum mechanics methods rely on the Born-Oppenheimer approximation. In quantum mechanics, the Schrodinger equation (1) gives the wave functions and energies of a molecule. 

The perturbation method (abr. p.m.) is one of the most important methods of approximation in quantum mechanics as well as in some fields of classical mechanics. It is usually presented in the following form. Let H0 be the operator representing some physical quantity of the unperturbed system and let i c0 be the perturbation, where k is a parameter assumed to be small. Then p.m. consists in solving problems concerning the perturbed operator H% = H0 + by expanding the results into power series of k, assuming that they are already solved for the unperturbed operator H0. 

Ligand field theory may be taken to be the subject which attempts to rationalize and account for the physical properties of transition metal complexes in fairly simple-minded ways. It ranges from the simplest approach, crystal field theory, where ligands are represented by point charges, through to elementary forms of molecular orbital theory, where at least some attempt at a quantum mechanical treatment is involved. The aims of ligand field theory can be treated as essentially empirical in nature ab initio and even approximate proper quantum mechanical treatments are not considered to be part of the subject, although the simpler empirical methods may be. 

However, in most cases a combination of kinetic and spectroscopic methods can resolve such uncertainties to a large extent. The third method is based on the study of model compounds. Model compounds are fully characterized metal complexes that are assumed to approximate the actual catalytic intermediates. Studies on the reactions of such compounds can yield valuable information about the real intermediates and the catalytic cycle. With the advent of computational speed and methods, quantum-mechanical and other theoretical calculations are also increasingly used to check whether theoretical predictions match with experimental data. 

Recent advances in computational chemistry have made it possible to calculate enthalpies of formation from quantum mechanical first principles for rather large unsaturated molecules, some of which are outside the practical range of combustion thermochemistry. Quantum mechanical calculations of molecular thermochemical properties are, of necessity, approximate. Composite quantum mechanical procedures may employ approximations at each of several computational steps and may have an empirical factor to correct for the cumulative error. Approximate methods are useful only insofar as the error due to the various approximations is known within narrow limits. Error due to approximation is determined by comparison with a known value, but the question of the accuracy of the known value immediately arises because the uncertainty of the comparison is determined by the combined uncertainty of the approximate quantum mechanical result and the standard to which it is compared. 

In a series of papers, Stock and coworkers have combined the quasiclassi-cal techniques used in the description of gas phase reactions with biomolecular force fields used in molecular dynamics (MD) simulations. This leads to nonequilibrium MD simulations, which mimic the laser excitation of the molecules by nonequilibrium phase-space initial condition for the solute and the solvent atoms. This approach is based on the following assumptions. Firstly, it is assumed that an empirical force field at least provides a qualitative modeling of the process. This is because the initial relaxation appears to be an ultrafast and generic process and because it can be expected that the strong interaction with the polar solvent smoothes out many details of the intramolecular force field. Secondly, quantum-mechanical effects are only included via the nonequilibrium initial conditions of the classical simulations. This means that the method represents a short-time approximation of quantum mechanics. 

Although the methods discussed above to incorporate electronic structure theory to model the PES and NQEs to determine the rate process are based on very different theories, a common strategy is to estimate approximately the quantum mechanical rate constant by introducing a quantum correction factor to bridge with the classical transition state theory ... 

Of the available methods, quantum mechanics (QM) attacks the problem at its deepest level. Moore (1972), in one edition of his physical chemistry text, says that, in principle, aU of chemistry could be calculated from the Schrbdinger equation. Then in a footnote, he adds Tn principle from the French, n prin-cipe, ouV, which means, "Non " Since that date, however, computers and programs have become more powerful, and much effort is being made to carry out quantum-mechanical calculations of the energetics of solvation of molecules and ions in various solvents. QM calculations are implemented in ab initio form at various levels of approximation, semiempirically also at various levels, and through density-functional theory. 

Discrete Variable Approximation in Quantum Mechanics, (b) Z. Bacicand J. C. Light, A . Rev. Phys. Chem., 40, 469 (1989). Theoretical Methods for Rovibrational States of Floppy Molecules. 

The continuum approximations applies quantum-mechanical approach only to the solute and approximates the solvent as dielectric continuum characterized by the static dielectric permittivity e. The advantage of these classes of methods is that they require only a relatively easy modification of the standard computer codes. In addition computer demands for such calculations are moderate and comparable to those of the gas phase predictions. 

The possibility of consideration of atoms as elementary subunits of the molecular systems is a consequence of Born-Oppenheimer or adiabatic approximation ( separation of electron and nuclear movements) aU quantum chemistry approaches start from this assumption. Additivity (or linear combination) is a common approach to construction of complex functions for physical description of the systems of various levels of complexity (cf orbital approximation, MO LCAO approximation, basis sets of wave functions, and some other approximations in quantum mechanics). The final justification of the method is good correlation of the results of its applications with the available experimental data and the potential to predict the characteristics of molecular systems before experimental data become available. It can be achieved after careful parameter adjustment and proper use of the force field in the area of its validity. The contributions not considered explicitly in the force field formulae are included implicitly into parameter values of the energy terms considered. 

The purpose of this chapter is to provide an introduction to tlie basic framework of quantum mechanics, with an emphasis on aspects that are most relevant for the study of atoms and molecules. After siumnarizing the basic principles of the subject that represent required knowledge for all students of physical chemistry, the independent-particle approximation so important in molecular quantum mechanics is introduced. A significant effort is made to describe this approach in detail and to coimnunicate how it is used as a foundation for qualitative understanding and as a basis for more accurate treatments. Following this, the basic teclmiques used in accurate calculations that go beyond the independent-particle picture (variational method and perturbation theory) are described, with some attention given to how they are actually used in practical calculations. 

For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantuin-seiniclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of seiniclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21]. 

The preferable theoretical tools for the description of dynamical processes in systems of a few atoms are certainly quantum mechanical calculations. There is a large arsenal of powerful, well established methods for quantum mechanical computations of processes such as photoexcitation, photodissociation, inelastic scattering and reactive collisions for systems having, in the present state-of-the-art, up to three or four atoms, typically. " Both time-dependent and time-independent numerically exact algorithms are available for many of the processes, so in cases where potential surfaces of good accuracy are available, excellent quantitative agreement with experiment is generally obtained. In addition to the full quantum-mechanical methods, sophisticated semiclassical approximations have been developed that for many cases are essentially of near-quantitative accuracy and certainly at a level sufficient for the interpretation of most experiments.These methods also are com-... 

In view of this, early quantum mechanical approximations still merit interest, as they can provide quantitative data that can be correlated with observations on chemical reactivity. One of the most successful methods for explaining the course of chemical reactions is frontier molecular orbital (FMO) theory [5]. The course of a chemical reaction is rationali2ed on the basis of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), the frontier orbitals. Both the energy and the orbital coefficients of the HOMO and LUMO of the reactants are taken into account. 

The importance of FMO theory hes in the fact that good results may be obtained even if the frontier molecular orbitals are calculated by rather simple, approximate quantum mechanical methods such as perturbation theory. Even simple additivity schemes have been developed for estimating the energies and the orbital coefficients of frontier molecular orbitals [6]. 

The first illustrative problem comes from quantum mechanics. An equation in radiation density can be set up but not solved by conventional means. We shall guess a solution, substitute it into the equation, and apply a test to see whether the guess was right. Of course it isn t on the first try, but a second guess can be made and tested to see whether it is closer to the solution than the first. An iterative routine can be set up to cany out very many guesses in a methodical way until the test indicates that the solution has been approximated within some narrow limit. 

Among the few systems that can be solved exactly are the particle in a onedimensional box, the hydrogen atom, and the hydrogen molecule ion Hj. Although of limited interest chemically, these systems are part of the foundation of the quantum mechanics we wish to apply to atomic and molecular theory. They also serve as benchmarks for the approximate methods we will use to treat larger systems. 

We shall examine the simplest possible molecular orbital problem, calculation of the bond energy and bond length of the hydrogen molecule ion Hj. Although of no practical significance, is of theoretical importance because the complete quantum mechanical calculation of its bond energy can be canied out by both exact and approximate methods. This pemiits comparison of the exact quantum mechanical solution with the solution obtained by various approximate techniques so that a judgment can be made as to the efficacy of the approximate methods. Exact quantum mechanical calculations cannot be carried out on more complicated molecular systems, hence the importance of the one exact molecular solution we do have. We wish to have a three-way comparison i) exact theoretical, ii) experimental, and iii) approximate theoretical. 

In applying quantum mechanics to real chemical problems, one is usually faced with a Schrodinger differential equation for which, to date, no one has found an analytical solution. This is equally true for electronic and nuclear-motion problems. It has therefore proven essential to develop and efficiently implement mathematical methods which can provide approximate solutions to such eigenvalue equations. Two methods are widely used in this context- the variational method and perturbation theory. These tools, whose use permeates virtually all areas of theoretical chemistry, are briefly outlined here, and the details of perturbation theory are amplified in Appendix D. 

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