The Time-Independent Schrodinger Equation

How the wave function gives us information on other properties besides the position is discussed in later chapters. 

The postulates of thermodynamics (the first, second, and third laws of thermodynamics) are stated in terms of macroscopic experience and hence are fairly readily understood. The postulates of quantum mechanics are stated in terms of the microscopic world and appear quite abstract. You should not expect to fully understand the postulates of quantum mechanics at first reading. As we treat various examples, understanding of the postulates will increase. 

Quantum mechanics provides the law of motion for microscopic particles. Experimentally, macroscopic objects obey classical mechanics. Hence for quantum mechanics to be a valid theory, it should reduce to classical mechanics as we make the transition from microscopic to macroscopic particles. Quantum effects are associated with the de Broglie wavelength A = h/mv. Since h is very small, the de Broglie wavelength of macroscopic objects is essentially zero. Thus, in the limit A 0, we expect the time-dependent Schrodinger equation to reduce to Newton s second law. We can prove this to be so (see Prob. 7.59). 

A similar situation holds in the relation between special relativity and classical mechanics. In the limit v/c— 0, where c is the speed of light, special relativity reduces to classical mechanics. The form of quantum mechanics that we will develop will be nonrelativistic. A complete integration of relativity with quantum mechanics has not been achieved. 

The time-dependent Schrodinger equation (1.13) is formidable looking. Fortunately, many applications of quantum mechanics to chemistry do not use this equation. Instead, the simpler time-independent Schrodinger equation is used. We now derive the 

We begin by restricting ourselves to the special case where the potential energy V is not a function of time but depends only on x. This will be true if the system experiences no time-dependent external forces. The time-dependent Schrbdinger equation reads 

We now restrict ourselves to looking for those solutions of (1.16) that can be written as the product of a function of time and a function of x  

Capital psi is used for the time-dependent wave function and lowercase psi for the factor that depends only on the coordinate x. States corresponding to wave functions of the form (1.17) possess certain properties (to be discussed shortly) that make them of great interest. [Not all solutions of (1.16) have the form (1.17) see Problem 3.41.] Taking partial derivatives of (1.17), we have 

Integrating both sides of this equation with respect to t, we have 

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If the Hamiltonian operator does not contain the time variable explicitly, one can solve the time-independent Schrodinger equation... 

We normally take the constant of integration i7o to be zero. Solution of the time-independent Schrodinger equation can be done exactly. We don t need to concern ourselves with the details, I will just give you the results. 

Thus we wish to solve the time-independent Schrodinger equation... 

To solve the time-independent Schrodinger equation for the nuclei plus electrons, we need an expression for the Hamiltonian operator. It is... 

The total wavefunction will depend on the spatial coordinates ri and ra of the two electrons 1 and 2, and also the spatial coordinates Ra and Rb of the two nuclei A and B. I will therefore write the total wavefunction as totfRA. Rb fu fi)-The time-independent Schrodinger equation is... 

If we are interested in describing the electron distribution in detail, there is no substitute for quantum mechanics. Electrons are very light particles, and they cannot be described even qualitatively correctly by classical mechanics. We will in this and subsequent chapters concentrate on solving the time-independent Schrodinger equation, which in short-hand operator fonn is given as... 

The description of electronic distribution and molecular structure requires quantum mechanics, for which there is no substitute. Solution of the time-independent Schrodinger equation, Hip = Eip, is a prerequisite for the description of the electronic distribution within a molecule or ion. In modern computational chemistry, there are numerous approaches that lend themselves to a reasonable description of ionic liquids. An outline of these approaches is given in Scheme 4.2-1 [1] ... 

As an illustration of the application of the time-independent Schrodinger equation to a system with a specific form for F(x), we consider a particle confined to a box with infinitely high sides. The potential energy for such a particle is given by... 

Solve the time-independent Schrodinger equation for this particle to obtain the energy levels and the normalized wave functions. (Note that the boundary conditions are different from those in Section 2.5.)... 

If the potential energy of a system is an even function of the coordinates and if (q) is a solution of the time-independent Schrodinger equation, then the function is also a solution. When the eigenvalues of the Hamiltonian... 

The time-independent Schrodinger equation for the two-particle system is... 

Most semi-empirical models are based on the fundamental equations of Hartree-Fock theory. In the following section, we develop these equations for a molecular system composed of A nuclei and N electrons in the stationary state. Assuming that the atomic nuclei are fixed in space (the Born-Oppenheimer approximation), the electronic wavefunction obeys the time-independent Schrodinger equation ... 

The purpose of most quantum chemical methods is to solve the time-independent Schrodinger equation. Given that the nuclei are much more heavier than the electrons, the nuclear and electronic motions can generally be treated separately (Born-Oppenheimer approximation). Within this approximation, one has to solve the electronic Schrodinger equation. Because of the presence of electron repulsion terms, this equation cannot be solved exactly for molecules with more than one electron. 

The initial purpose of pioneer quantum mechanics was to provide the theoretical framework to account for the structure of hydrogen and the nuclear model of atoms in general. The final result, a quantum theory of atomic structure can be discussed in terms of the time-independent Schrodinger equation, in its most general form... 

Calculations of the electronic structure of molecules, crystals and surfaces are often performed in atomic units. They are defined by setting the most important constants equal to unity h — eo — me — 1, where me is the electronic mass. The Coulomb law is written in electrostatic units V(r) = q/r, so that the time-independent Schrodinger equation for the hydrogen atom takes on the simple form ... 

The basic problem is to solve the time-independent electronic Schrodinger equation. Since the mass of the electrons is so small compared to that of the nuclei, the dynamics of nuclei and electrons can normally be decoupled, and so in the Born-Oppenheimer approximation the many-electron wavefunction P and corresponding energy may be obtained by solving the time-independent Schrodinger equation in which the nuclear positions are fixed. We thus solve... 

Technically, the time-independent Schrodinger equation (2) is solved for clamped nuclei. The Hamiltonian is broken into its electronic part, He, including the nuclear Coulomb repulsion energy, and the nuclear Hamiltonian HN. At this level, mass polarization effects are usually neglected. The wave function is therefore factorized as usual (r,X)= vP(r X)g(X). Formally, the electronic wave function d lnX) and total electronic energy, E(X), are obtained after solving the equation for each value of X ... 

The Hartree-Fock approach derives from the application of a series of well defined approaches to the time independent Schrodinger equation (equation 3), which derives from the postulates of quantum mechanics [27]. The result of these approaches is the iterative resolution of equation 2, presented in the previous subsection, which in this case is solved in an exact way, without the approximations of semiempirical methods. Although this involves a significant increase in computational cost, it has the advantage of not requiring any additional parametrization, and because of this the FIF method can be directly applied to transition metal systems. The lack of electron correlation associated to this method, and its importance in transition metal systems, limits however the validity of the numerical results. 

Only a very concise description of quantum chemistry is presented in this section the reader is referred to standard texts for a more complete treatment. " The aim of quantum chemistry or ab initio electronic structure theory is the solution of the time-independent Schrodinger equation ... 

The first of these equations is called the time-independent Schrodinger equation it is a so-called eigenvalue equation in which one is asked to find functions that yield a constant multiple of themselves when acted on by the Hamiltonian operator. Such functions are called eigenfunctions of H and the corresponding constants are called eigenvalues of H. 

The quantum theory of molecular collisions in external fields described in this chapter is based on the solutions of the time-independent Schrodinger equation. The scattering formalism considered here can be used to calculate the collision properties of molecules in the presence of static electric or magnetic fields as well as in nonresonant AC fields. In the latter case, the time-dependent problem can be reduced to the time-independent one by means of the Floquet theory, discussed in the previous section. We will consider elastic or inelastic but chemically nonreac-tive collisions of molecules in an external field. The extension of the formalism to reactive scattering problems for molecules in external fields has been described in Ref. [12]. 

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