Energy wavefunctions

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. 

Computation of the state energy-wavefunction pairs, T, by solving Schrodinger equation ... 

Having established the coincidence of the principal axes of g and D, Eq. (18) may be solved in closed form to obtain the energies, wavefunctions, and resonance fields in the three canonical orientations. These are listed in Table 8. By combining the expressions for the A is = 1 resonance fields in each of the canonical orientations, expressions for each of the six magnetic... 

Energies, Wavefunctions, and Resonance Fields for a Triplet State with Z > 0 > Y> X... 

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. 

O Determine the energies, wavefunctions, and other properties of a particle in a one- or three-dimensional box... 

---