Exact Solutions of the Schroedinger Equation

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. 

The problem is heated in elementary physical chemishy books (e.g., Atkins, 1998) and leads to a set of eigenvalues (energies) and eigenfunctions (wave functions) as depicted in Fig. 6-1. It is solved by much the same methods as the hamionic oscillator in Chapter 4, and the solutions are sine, cosine, and exponential solutions just as those of the harmonic oscillator are. This gives the wave function in Fig. 6-1 its sinusoidal fonn. 

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 

The hydrogen atom is a three-dimensional problem in which the attractive force of the nucleus has spherical symmetr7. Therefore, it is advantageous to set up and solve the problem in spherical polar coordinates r, 0, and j). The resulting equation can be broken up into three parts, one a function of r only, one a function of 0 only, and one a function of j). These can be solved separately and exactly. Each equation leads to a quantum number 

These are three of the four quantum numbers familiar from general chemistry. The spin quantum number s arises when relativity is included in the problem, introducing a fourth dimension. 

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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... 

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. 

Equations A.5, A.9, and A.10 define the BO approximation. Note that two approximations (neglect of the second term of the left-hand side of eq. A.7 and off-diagonal terms in A.6) were made in order to obtain this result. Therefore, we see that the BO expression A.10 is not an exact solution of the total Schroedinger equation A.l. 

While this particular problem of mechanical stability can be solved in principle by means of the Schroedinger equation, exact solutions have never been obtained for any polyatomic molecules except H2 and Hf. Later on we shall consider some of the approximate treatments which have been made. 

The divergence problem in the Born expansion can be eliminated by introducing projection operators which allow the troublesome part of the problem to be treated in a more exact way. The solution to the Schroedinger equation can be written as an integral equation... 

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. 

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. 

The Schroedinger equation cannot be solved exactly except for very simple systems like the hydrogen atom. For molecules, we must be satisfied with an approximate solution of H V(r) = is F(r). In recent years, owing to the work of Pople, Gordon, and others, agreement between MO approximations and such experimental results as exist has been brought to a level that makes quantum thermochemistry competitive with experimental thermochemistry in reliability. 

In general, the development of approximation methods for the solution of the many-electron Schroedinger equation is a challenge for physicists because no exact numerical solutions can be found apart from very few cases of a small number of electrons, such as the helium atom. The main difficulty arises because of the electron-electron interaction, which is a two-particle operator. Thus, increasing the accuracy of solutions implies increasing the computer time needed for the numerical calculations, and the cost becomes prohibitive even for molecules with a few atoms. 

Thus we have formally, and exactly, converted the master equation to a Schroedinger equation. This has the substantial advantage that we can apply well-known approximations in quantum mechanics to obtain solutions to the master equation. In particular we refer to the W.K.B. approximation valid for semiclassical cases, those for which Planck s constant formally approaches zero. The equivalent limit for (3.8) is that of large volumes (large munbers of particles). Hence we seek a stationary solution of (3.8), that is the time derivative of Px X,t) is set to zero, of the form... 

This equation is formally like the two-mathematical dimensional Schroedinger equation for a collinear reaction with the addition of the constant rotational energy Ej and the adiabatic bending energy. The solutions without the constant Ej are denoted CEQB (collinear exact quantum with adiabatic bending energy). Thus, the solutions l o(r.R) can be related to the CEQB ones as follows. 

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