Quantum mechanics Schrodinger equation

Calculations as the ones underlying the results of this section are done by solving the time-dependent Schrodinger equation. Quantum-mechanical tunneling phenomena, to be discussed in the next section, are responsible for dissociation at... 

Much of quantum chemistry attempts to make more quantitative these aspects of chemists view of the periodic table and of atomic valence and structure. By starting from first principles and treating atomic and molecular states as solutions of a so-called Schrodinger equation, quantum chemistry seeks to determine what underlies the empirical quantum numbers, orbitals, the aufbau principle and the concept of valence used by spectroscopists and chemists, in some cases, even prior to the advent of quantum mechanics. 

Schrodinger s quantum mechanical model of atomic structure is framed in the form of a wave equation, a mathematical equation similar in form to that used to describe the motion of ordinary waves in fluids. The solutions (there are many) to the wave equation are called wave functions, or orbitals, and are represented by... 

Although the Schrodinger wave equation is difficult to solve for increasingly complicated atoms and molecules, we could, if we had a large enough computer, deduce the properties of all known chemicals from this equation. Quantum mechanics, however, is even more powerful than an ordinary cookbook because it also allows us to calculate the properties of chemicals that we have yet to see in nature. 

A theory of atomic and molecular structure was advanced independently and almost simultaneously by three people in 1926 Erwin Schrodinger, Werner Heisenberg, and Paul Dirac. This theory, called wave mechanics by Schrodinger and quantum mechanics by Heisenberg, has become the basis from which we derive our modern understanding of bonding in molecules. At the heart of quantum mechanics are equations called wave functions (denoted by the Greek letter psi, if/). 

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

The central equation of (non-relativistic) quantum mechanics, governing an isolated atom or molecule, is the time-dependent Schrodinger equation (TDSE) ... 

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... 

The quantum mechanical treatment of a hamionic oscillator is well known. Real vibrations are not hamionic, but the lowest few vibrational levels are often very well approximated as being hamionic, so that is a good place to start. The following description is similar to that found in many textbooks, such as McQuarrie (1983) [2]. The one-dimensional Schrodinger equation is... 

Such a fundamental theory does exist for chemistry quantum mechanics. The dependence of the property of a compound on its three-dimensional structure is given by the Schrodinger equation. Great progress has been made both in the de-... 

The. itarting point for any discussion of quantum mechanics is, of course, the Schrodinger t-qualion. The full, time-dependent form of this equation is... 

Quantum mechanics is cast in a language that is not familiar to most students of chemistry who are examining the subject for the first time. Its mathematical content and how it relates to experimental measurements both require a great deal of effort to master. With these thoughts in mind, the authors have organized this introductory section in a manner that first provides the student with a brief introduction to the two primary constructs of quantum mechanics, operators and wavefunctions that obey a Schrodinger equation, then demonstrates the application of these constructs to several chemically relevant model problems, and finally returns to examine in more detail the conceptual structure of quantum mechanics. 

By learning the solutions of the Schrodinger equation for a few model systems, the student can better appreciate the treatment of the fundamental postulates of quantum mechanics as well as their relation to experimental measurement because the wavefunctions of the known model problems can be used to illustrate. 

Before moving deeper into understanding what quantum mechanics means, it is useful to learn how the wavefunctions E are found by applying the basic equation of quantum mechanics, the Schrodinger equation, to a few exactly soluble model problems. Knowing the solutions to these easy yet chemically very relevant models will then facilitate learning more of the details about the structure of quantum mechanics because these model cases can be used as concrete examples. ... 

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. 

Statistical mechanics is the mathematical means to calculate the thermodynamic properties of bulk materials from a molecular description of the materials. Much of statistical mechanics is still at the paper-and-pencil stage of theory. Since quantum mechanicians cannot exactly solve the Schrodinger equation yet, statistical mechanicians do not really have even a starting point for a truly rigorous treatment. In spite of this limitation, some very useful results for bulk materials can be obtained. 

The fact that an electron has an intrinsic spin comes out of a relativistic formulation of quantum mechanics. Even though the Schrodinger equation does not predict it, wave functions that are antisymmetric and have two electrons per orbital are used for nonreiativistic calculations. This is necessary in order to obtain results that are in any way reasonable. 

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 quantum mechanics methods in HyperChem differ in how they approximate the Schrodinger equation and how they compute potential energy. The ab initio method expands molecular orbitals into a linear combination of atomic orbitals (LCAO) and does not introduce any further approximation. 

Molecular quantum mechanics finds the solution to a Schrodinger equation for an electronic Hamiltonian, Hgjg., that gives a total energy, Egjg(-(R) + V (R,R). Repeated solutions at different nuclear configurations, R, lead to some approximate potential energy sur-... 

For small molecules, the accuracy of solutions to the Schrodinger equation competes with the accuracy of experimental results. However, these accurate ab initio calculations require enormous computation and are only suitable for the molecular systems with small or medium size. Ab initio calculations for very large molecules are beyond the realm of current computers, so HyperChem also supports semi-empirical quantum mechanics methods. Semi-empirical approximate solutions are appropriate and allow extensive chemical exploration. The inaccuracy of the approximations made in semi-empirical methods is offset to a degree by recourse to experimental data in defining the parameters of the method. Indeed, semi-empirical methods can sometimes be more accurate than some poorer ab initio methods, which require much longer computation times. 

It is not the intention that this book should be a primary reference on quantum mechanics such references are given in the bibliography at the end of this chapter. Nevertheless, it is necessary at this stage to take a brief tour through the development of the Schrodinger equation and some of its solutions that are vital to the interpretation of atomic and molecular spectra. 

Schrodinger postulated that the form of the hamiltonian in quantum mechanics is obtained by replacing the kinetic energy in Equation (1.20), giving... 

Electronic structure methods use the laws of quantum mechanics rather than classical physics as the basis for their computations. Quantum mechanics states that the energy and other related properties of a molecule may be obtained by solving the Schrodinger equation ... 

Quantum mechanics explains how entities like electrons have both particle-like and wave-like characteristics. The SchrOdinger equation describes the wavefunction of a particle ... 

One of the great difficulties in molecular quantum mechanics is that of actually finding solutions to the Schrodinger time-independent equation. So whilst we might want to solve... 

As I mentioned above, it is conventional in many engineering applications to seek to rewrite basic equations in dimensionless form. This also applies in quantum-mechanical applications. For example, consider the time-independent electronic Schrodinger equation for a hydrogen atom... 

We now need to investigate the quantum-mechanical treatment of vibrational motion. Consider then a diatomic molecule with reduced mass /c- His time-independent Schrodinger equation is... 

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