Wave Functions and Energies for the Ground State of

All the calculations of F2 are carried out with a simple basis set of double-zeta polarization type, the standard 6-31G(d) basis set, and are performed at a fixed interatomic distance of 1.44 A, which is approximately the optimized distance for a full Cl calculation in this basis set. Only the r bond is described in a VB fashion, and the corresponding orbitals are referred to as the active orbitals , while the orbitals representing the lone pairs, so-called spectator orbitals , remain doubly occupied in all calculations. A common point to the various VB methods we use, except the VBCI method, is that at the dissociation limit, the methods converge to two F fragments at the restricted-open-shell Hartree Fock (ROHF) level. 

A Chemist s Guide to Valence Bond Theory, by Sason Shaik and Philippe C. Hiberty Copyright 2008 John Wiley Sons, Inc. 

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Ground-State Wave Function and Energy. For the ground state of the hydrogenlike atom, we have = 1, / = 0, and m = 0. The radial factor (6.100) is... 

The simplest approach, of course, is to maintain the minimum-determinantal description and reoptimize all of the orbitals. In practice, however, such an approach is practical only in instances where die ground-state and the excited-state wave functions belong to different incduciblc representations of die molecular point group (cf. Section 6.3.3). Otherwise, the variational soludon for die excited-state wave function is simply to collapse back to the ground-state wave function And, even if the two states do differ in symmetry, the desired excited state may not be the lowest energy such state widiiii its irrep, to which variational optimization will nearly always lead. 

Insofar as the latter process docs not involve any orbital reoptimization for any particular state, it provides a wave function that is roughly equivalent in quality only to an HF wave function for the ground state. Of course, this may still be useful for a number of purposes. CIS results for six excited states of benzene are included in Table 14.2, as are results from other levels of theory that will be discussed later. The CIS results are qualitatively useful, insofar as the states are correctly ordered, and the error is fairly systematic - all states are predicted to be too high in energy by an average of 0.7 eV. The worst prediction is for the lowest excited state, which is known to have significant dynamical electron correlation, and is therefore challenging for the CIS method. 

So far we have discussed the electron density for the ground state of the H atom. When the atom absorbs energy, it exists in an excited state and the region of space occupied by the electron is described by a different atomic orbital (wave function). As you ll see, each atomic orbital has a distinctive radial probability distribution and 90% probability contour. 

Fig. 5.6. In the delta well (a) the potential energy falls abruptly to a large negative value of V over a tiny distance / and V is allowed to become infinite and / to vanish in such a way that the product VI has the finite value of —r. The well affords one bound state for a particle, whose wave function (b) has the shape of a cross-section of the wave function for the ground state of an electron in an atom Increasing the value of the single parameter r that characterizes the well is equivalent to increasing the positive charge of the nucleus of an atom.

Waals bond coordinate) and r ox q (the diatom bond coordinate). That is, the X coordinate in the Ar—HCCH potential of figure 10.6 is ignored. Figure 10.7 is a schematic illustration of the two potentials, UJjq) and VJfi) taken from Le Roy et al. (1991). Shown in this figure are also the various vibrational wave functions which enter into the problem. On the left, are the wave functions for the unperturbed diatomic in the v" = 0 and v = 1 states, while on the right are two van der Waals potential energy curves, the lower one for the ground state of the diatom (v" = 0), and the upper curve... 

Now, we assume that the functions, t(Hj, 7 = 1,. .., fV are such that these uncoupled equations are gauge invariant, so that the various % values, if calculated within the same boundary conditions, are all identical. Again, in order to determine the boundary conditions of the % function so as to solve Eq. (53), we need to impose boundary conditions on the functions. We assume that at the given (initial) asymptote all / values are zero except for the ground-state function li , and for a low enough energy process, we introduce the approximation that the upper electronic states are closed, hence all final wave functions / are zero except the ground-state function t]/ . 

Apply the particle-in-a-box basis functions to the radial equation for the hydrogen atom for the / = 0 states. Recall that in Section 6.9, we expressed the radial factor in the H-atom wave function as R r) = r F r), where F(r) = 0 at r = 0. The variation function in this problem will have the form = r F r)Yf 6, ) take the dimensionless function F,(r,) to be a linear combination of 28 pib basis functions, where the box goes from r, = 0 to 27, where r, is defined in Section 6.9. Work out the proper forms for the integrals //y and 5. Find the estimates for the lowest three / = 0 energies. For the ground-state variation function, how many pib basis functions appear with a coefficient greater than 0.1 ... 

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