The Schrodinger equation and some of its solutions

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. 

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THE SCHRODINGER EQUATION AND SOME OF ITS SOLUTIONS Table 1.3 Some values of the nuclear spin quantum number / 19... 

In this chapter we introduce the SchrSdinger equation this equation is fundamental to all applications of quantum mechanics to chemical problems. For molecules of chemical interest it is an equation which is exceedingly difficult to solve and any possible simplifications due to the symmetry of the system concerned are very welcome. We are able to introduce symmetry, and thereby the results of the previous chapters, by proving one single but immensely valuable fact the transformation operators Om commute with the Hamiltonian operator, Jf. It is by this subtle thread that we can then deduce some of the properties of the solutions of the Schrodinger equation without even solving it. 

Since its eigenvalues correspond to the allowed energy states of a quantum-mechanical system, the time-independent Schrodinger equation plays an important role in the theoretical foundation of atomic and molecular spectroscopy. For cases of chemical interest, the equation is always easy to write down but impossible to solve exactly. Approximation techniques are needed for the application of quantum mechanics to atoms and molecules. The purpose of this subsection is to outline two distinct procedures—the variational principle and perturbation theory— that form the theoretical basis for most methods used to approximate solutions to the Schrodinger equation. Although some tangible connections are made with ideas of quantum chemistry and the independent-particle approximation, the presentation in the next two sections (and example problem) is intended to be entirely general so that the scope of applicability of these approaches is not underestimated by the reader. 

When it comes to the course and dispersion of more complicated bands, this is easily illustrated by two other one-dimensional examples. Note that the above Bloch formula for the construction of tp k) at some k value did not depend on the orbital involved the plus/minus sign changes only resulted from the exponential pre-factor. Since Bloch s theorem just depends on some solution of the Schrodinger equation, and this may be another atomic orbital or, equally well, a molecular orbital, let us first assume, in Scheme 2.2, a onedimensional chain of, say, nitrogen atoms where each N carries a set of one 2s... 

The term "ab initio means "from first principles" it does not mean "exact" or "true". In ab initio molecular orbital theory, we develop a series of well-defined approximations that allow an approximate solution to the Schrodinger equation. We calculate a total wavefunc-tion and individual molecular orbitals and their respective energies, without any empirical parameters. Below, we outline the necessary approximations and some of the elements and principles of quantum mechanics that we must use in our calculations, and then provide a summary of the entire process. Along with defining an important computational protocol, this approach will allow us to develop certain concepts that will be useful in later chapters, such as spin and the Born-Oppenheimer approximation. 

The solutions to the Schrodinger equation for the particle in a box and the free particle exhibit some of the most important features of quantum mechanics. We have seen one case of quantized energy (the particle in a box) and one case of energy that is not quantized (the free particle). We have seen that the Schrodinger equation and its boundary conditions together dictate the nature of the wave functions and energy eigenvalues that can occur. We have also seen that the principle of superposition applies A valid wave function can be a linear combination of simpler wave functions. 

In addition to initial conditions, solutions to the Schrodinger equation must obey eertain other eonstraints in form. They must be eontinuous funetions of all of their spatial eoordinates and must be single valued these properties allow T T to be interpreted as a probability density (i.e., the probability of finding a partiele at some position ean not be multivalued nor ean it be jerky or diseontinuous). The derivative of the wavefunetion must also be eontinuous exeept at points where the potential funetion undergoes an infinite jump (e.g., at the wall of an infinitely high and steep potential barrier). This eondition relates to the faet that the momentum must be eontinuous exeept at infinitely steep potential barriers where the momentum undergoes a sudden reversal. 

In the previous chapter we considered a rather simple solvent model, treating each solvent molecule as a Langevin-type dipole. Although this model represents the key solvent effects, it is important to examine more realistic models that include explicitly all the solvent atoms. In principle, we should adopt a model where both the solvent and the solute atoms are treated quantum mechanically. Such a model, however, is entirely impractical for studying large molecules in solution. Furthermore, we are interested here in the effect of the solvent on the solute potential surface and not in quantum mechanical effects of the pure solvent. Fortunately, the contributions to the Born-Oppenheimer potential surface that describe the solvent-solvent and solute-solvent interactions can be approximated by some type of analytical potential functions (rather than by the actual solution of the Schrodinger equation for the entire solute-solvent system). For example, the simplest way to describe the potential surface of a collection of water molecules is to represent it as a sum of two-body interactions (the interac-... 

For all but the simplest systems the Schrodinger equation must be solved approximately. It is assumed that the true wavefunction, which is too complicated to be found directly, can be approximated by a simpler function. For some types of function it is then possible to solve the electronic Schrodinger equation numerically. Provided the assumption made regarding the form of the function is not too drastic, a good approximation will be obtained to the correct solution. Electronic structure theory consists of designing sensible approximations to the wavefunction, with an inevitable trade-off between accuracy and computational cost. 

A numerical solution of the Schrodinger equation in Eq. [1] often starts with the discretization of the wave function. Discretization is necessary because it converts the differential equation to a matrix form, which can then be readily handled by a digital computer. This process is typically done using a set of basis functions in a chosen coordinate system. As discussed extensively in the literature,5,9-11 the proper choice of the coordinate system and the basis functions is vital in minimizing the size of the problem and in providing a physically relevant interpretation of the solution. However, this important topic is out of the scope of this review and we will only discuss some related issues in the context of recursive diagonalization. Interested readers are referred to other excellent reviews on this topic.5,9,10... 

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