The Schroedinger Equation

The behavior of a single electron in an isolated atom can be exactly determined (neglecting relativity) by solving the Schroedinger equation 

The Hamiltonian for a single electron in orbit around a fixed nucleus of charge Z is 

The energies associated with these atomic orbitals are given by 

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Most problems in chemistry [all, according to Dirac (1929)] could be solved if we had a general method of obtaining exact solutions of the Schroedinger equation... 

The reason a single equation = ( can describe all real or hypothetical mechanical systems is that the Hamiltonian operator H takes a different form for each new system. There is a limitation that accompanies the generality of the Hamiltonian and the Schroedinger equation We cannot find the exact location of any election, even in simple systems like the hydrogen atom. We must be satisfied with a probability distribution for the electron s whereabouts, governed by a function (1/ called the wave function. 

We cannot solve the Schroedinger equation in closed fomi for most systems. We have exact solutions for the energy E and the wave function (1/ for only a few of the simplest systems. In the general case, we must accept approximate solutions. The picture is not bleak, however, because approximate solutions are getting systematically better under the impact of contemporary advances in computer hardware and software. We may anticipate an exciting future in this fast-paced field. 

All solutions of the Schroedinger equation lead to a set of integers called quantum numbers. In the case of the particle in a box, the quantum numbers are n= 1,2,3,. The allowed (quantized) energies are related to the quantum numbers by the equation... 

In the few two- and three-dimensional cases that pemiit exact solution of the Schroedinger equation, the complete equation is separated into one equation in each dimension and the energy of the system is obtained by solving the separated equations and summing the eigenvalues. The wave function of the system is the product of the wave functions obtained for the separated equations. 

We have said that the Schroedinger equation for molecules cannot be solved exactly. This is because the exact equation is usually not separable into uncoupled equations involving only one space variable. One strategy for circumventing the problem is to make assumptions that pemiit us to write approximate forms of the Schroedinger equation for molecules that are separable. There is then a choice as to how to solve the separated equations. The Huckel method is one possibility. The self-consistent field method (Chapter 8) is another. 

Three major approximations are made to separate the Schroedinger equation into a set of smaller equations before carrying out Huckel calculations. 

We assume that the nuclei are so slow moving relative to electrons that we may regard them as fixed masses. This amounts to separation of the Schroedinger equation into two parts, one for nuclei and one for electrons. We then drop the nuclear kinetic energy operator, but we retain the intemuclear repulsion terms, which we know from the nuclear charges and the intemuclear distances. We retain all terms that involve electrons, including the potential energy terms due to attractive forces between nuclei and electrons and those due to repulsive forces... 

Using atomic units, the Schroedinger equation for ground-state hydrogen is... 

The reason the Schroedinger equation for molecules cannot be separated appears in the last term, involving a sum of repulsive energies between electrons. To... 

With this new approximation, the ry term does not appear [it is hidden in V i)] and the Schroedinger equation becomes separable into n equations, one for each elechon... 

It has been known for more than a century that hydrocarbons containing double bonds are more reactive than their counterparts that do not contain double bonds. Alkenes are, in general, more reactive than alkanes. We call electrons in double bonds 71 electrons and those in the much less reactive C—C or CH bonds Huckel theory, we assume that the chemistry of unsaturated hydrocarbons is so dominated by the chemistry of their double bonds that we may separate the Schroedinger equation yet again, into an equation for potential energy. We now have an equation of the same fomi as Eq. (6-8), but one in which the Hamiltonian for all elections is replaced by the Hamiltonian for Ji electrons only... 

It is a property of linear, homogeneous differential equations, of which the Schroedinger equation is one. that a solution multiplied by a constant is a solution and a solution added to or subtracted from a solution is also a solution. If the solutions Pi and p2 in Eq. set (6-13) were exact molecular orbitals, id v would also be exact. Orbitals p[ and p2 are not exact molecular orbitals they are exact atomic orbitals therefore. j is not exact for the ethylene molecule. 

By systematically applying a series of corrections to approximate solutions of the Schroedinger equation the Pople group has anived at a family of computational protocols that include an early method Gl, more recent methods, G2 and G3, and their variants by which one can anive at themiochemical energies and enthalpies of formation, Af and that rival exper imental accuracy. The important thing... 

In hybrid DET-Gaussian methods, a Gaussian basis set is used to obtain the best approximation to the three classical or one-election parts of the Schroedinger equation for molecules and DET is used to calculate the election correlation. The Gaussian parts of the calculation are carried out at the restiicted Hartiee-Fock level, for example 6-31G or 6-31 lG(3d,2p), and the DFT part of the calculation is by the B3LYP approximation. Numerous other hybrid methods are currently in use. 

It is the gradient-squared term that caught Frieden s attention. Frieden noticed that it almost always appears in the Langrangians for various physical phenomena. The Langrangian for classical mechanics, for example, is dq/dt) — V. The La-grangian for the Schroedinger equation contains the term Vtp. The reader can perhaps recall at least a half-dozen other simple examples from basic physics (see Table 1.1 in [frieden98]). 

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

Since HF has a closed-shell electronic structure and no low-lying excited electronic states. HF-HF collisions may be treated quite adequately within the framework of the Born-Oppenheimer electronic adiabatic approximation. In this treatment (4) the electronic and coulombic energies for fixed nuclei provide a potential energy V for internuclear motion, and the collision dynamics is equivalent to a four-body problem. After removal of the center-of-mass coordinates, the Schroedinger equation becomes nine-dimensional. This nine-dimensional partial differential... 

Moreover, instead of describing the electrons by the Dirac equation, that is fully taking into account relativistic effects due to the high acceleration voltage, a modified form of the Schroedinger equation is used, in which electron energy and wavelength are replaced by the equivalent relativistically corrected expressions [85]. 

Just as the first term in the Schroedinger equation describes the kinetic energy of an electron system, and the second term deals with the potential energy, so are these the two parts of any simple model. Electrostatic forces provide attraction between the atoms, while kinetic energy keeps the system from collapsing by exerting a quantum mechanical Schroedinger pressure. ... 

Calculating Eigenfunctions and Eigenvalues of the Schroedinger Equation on a Grid. 

Fig. 2.26 When 4He is replaced with 3He, the secondary Rh2+ peak disappears even though 3HeRh2+ ions are still formed. At first glance, this strong isotope effect is most surprising since one would expect 3HeRh2+ to field dissociate more easily than 4HeRh2+ because of its smaller reduced mass. This peculiar isotope effect is the result of a center of mass transformation in the applied field, as can be understood from the Schroedinger equation of eq. (2.63), already explained in...

In the Schroedinger equation we do not really talk of Van der Waals forces, Coulomb forces, and so on. We simply have a Hamiltonian that includes all those forces and that one attempts to solve for as well as is feasible. Today for systems with 10 to 20 electrons good solutions can be obtained. Few are, however, present in the literature concerning biological systems, where gross oversimplification is still accepted. 

X is not necessarily an eigenvalue. But if s is sufficiently small, X can be regarded as an approximate eigenvalue and

Schroedinger equation of motion that the state

[Pg.56]

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