Lowest-energy wavefunction

In accordance with the variation theorem we require the set of coefficients that gives the lowest-energy wavefunction, and some scheme for changing the coefficients to derive that wavefunction. For a given basis set and a given functional form of the wavefunction (i.e. a Slater determinant) the best set of coefficients is that for which the energy is a minimum, at which point... 

The calculation proceeds as illustrated in Table 2.2, which shows the variation in the coefficients of the atomic orbitals in the lowest-energy wavefunction and the energy for the first four SCF iterations. The energy is converged to six decimal places after six iterations and the charge density matrix after nine iterations. 

The lowest energy wavefunction is a singlet, but not a dosed shell singlet (e.g., biradicals). This is an RHF-to-UHF instability. 

FIGURE 1.27 The two lowest energy wavefunctions (i <, orangei for a particle in a box and the corresponding probability densities (i] 2, blue). The probability densities are also shown by the density of shading of the bands beneath each wavefunction. 

FIGURE 6.2 The three lowest energy wavefunctions for a particle in a box. 

Kinetic energy is lowest for a diffuse wavefunction. A. Lowest energy wavefunction, B. Intermediate energy, and C. Highest energy wavefunction. 

Cross-sections along the x axis showing amplitude versus distance for A. Is orbital, and for the two lowest energy wavefunctions B. i/fg and C. i// for H2 relative to the center of the bond. The x axis is internuclear distance (Bohr radii). A represents H positions. Adapted from course notes by W. A. Goddard III, "Nature of the Chemical Bond."... 

Let s mix two atomic orbitals on different atoms to form an MO, as in Eq. 14.38. This is the LC AO method of forming molecular orbitals. We let Ca and Cb represent coefficients that tell us the proper proportion of the starting orbitals (0a and 0n) that should be present in the resulting MO. The 0 s are known quantities (possibly from the solution of the hydrogen atom, or are basis functions). We do not know what the c s should be (they are the unknowns). We will solve for the c s that create the lowest energy wavefunctions, just as with HF-SCF methods. 

The four lowest-energy wavefunctions of a onedimensional system with their corresponding energy levels are shown in the accompanying figure. Sketch the potential energy curve that leads to these solutions. 

We turn now to the eigenfunctions (2-11) for this problem. These are typically displayed by superimposing them on the energy levels as shown in Fig. 2-4 for the three lowest-energy wavefunctions. (It should be recognized that the energy units of the vertical axis do not pertain to the amplitudes of the wavefunctions.)... 

Figure 12-4 Sketches of the lowest-energy wavefunction (a) before and after perturbation and (b) the difference between them.

Determine the degeneracies of all levels for a cubical box from the lowest-energy wavefunction, described by the set of quantum numbers (1, 1, 1) to the wavefunction described by the quantum number set (4,4,4). Hint You may have to use quantum numbers larger than 4 to determine proper degeneracies. See Example 10.15. 

Based on the known wavefunctions for the harmonic oscillator, the wavefunctions for the particle in a box, and the continuity and smoothness requirements for all wavefimctions, construct a sketch showing the qualitative features of the three lowest energy wavefunctions for a particle trapped in a potential, V x), that is infinite at X = 0 and that for x > 0 is kx /2, where Ir is a constant. 

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