Orbitals wavefunctions

The additional terms in the molecular orbital wavefunction correspond to states with the two electrons in the same orbital, which endows ionic character to the bond The... 

An important conceptual, or even philosophical, difference between the orbital/wavefunction methods and the density functional methods is that, at least in principle, the density functional methods do not appeal to orbitals. In the former case the theoretical entities are completely unobservable whereas electron density invoked by density functional theories is a genuine observable. Experiments to observe electron densities have been routinely conducted since the development of X-ray and other diffraction techniques (Coppens, 2001).18... 

FIGURE 3.13 These contours indicate the amplitude of the sp hybrid orbital wavefunction in a plane that bisects it and passes through the nucleus. Each sp hvbrid orbital points toward the corner of a tetrahedron. 

In the uncorrelated limit, where the many-electron Fock operator replaces the full electronic Hamiltonian, familiar objects of HF theory are recovered as special cases. N) becomes a HF, determinantal wavefunction for N electrons and N 1) states become the frozen-orbital wavefunctions that are invoked in Koopmans s theorem. Poles equal canonical orbital energies and DOs are identical to canonical orbitals. 

P has been computed using Hartree-Fock atomic orbital wavefunctions and can be found in several published tabulations14 17 and in Appendix 1. Because of the (r 3) dependence of P, dipolar coupling of a nuclear spin with electron spin density on another atom is usually negligible. 

We first note that a localized NBO description is an equally valid way to view a delocalized molecular-orbital wavefunction. In the case of N2, for example, 0mo is expressed in terms of doubly occupied MOs 0, i = 1-7,... 

We now use a Pauling-like approach to show how hybrid orbitals for a variety of combinations of s, p, and d orbitals may be formulated.10 We assume that the radial dependences of the s, p, d orbitals are similar so that they can be neglected. The angular parts of the orbital wavefunctions are given by the following expressions (in the usual spherical coordinates 9, ) ... 

The development of localized-orbital aspects of molecular orbital theory can be regarded as a successful attempt to deal with the two kinds of comparisons from a unified theoretical standpoint. It is based on a characteristic flexibility of the molecular orbital wavefunction as regards the choice of the molecular orbitals themselves the same many-electron Slater determinant can be expressed in terms of various sets of molecular orbitals. In the classical spectroscopic approach one particular set, the canonical set, is used. On the other hand, for the same wavefunction an alternative set can be found which is especially suited for comparing corresponding states of structurally related molecules. This is the set of localized molecular orbitals. Thus, it is possible to cast one many-electron molecular-orbital wavefunction into several forms, which are adapted for use in different comparisons fora comparison of the ground state of a molecule with its excited states the canonical representation is most effective for a comparison of a particular state of a molecule with corresponding states in related molecules, the localized representation is most effective. In this way the molecular orbital theory provides a unified approach to both types of problems. 

What has been said applies to approximate as well as to ab-initio molecular orbital wavefunctions,i.e. those obtained by solving the self-consistent-field equations exactly. Hence, the localized orbital approach also offers an attractive tool for bridging the gap between rigorous quantitative calculations and qualitative chemical intuition. The experience gained so far has shown that interpretations suggested by the localized orbital picture correspond closely to intuitive chemical thinking. 

For the purpose of this presentation we limit ourselves to closed-shell systems, i.e. those systems consisting of an even number, 2N, of electrons which doubly occupy N space orbitals uy,u2,...uN> each with a and 3 spin. The appropriate molecular orbital wavefunction is then given by... 

The results discussed so far are a consequence of the determinantal form of the molecular orbital wavefunction jj. They are valid for an arbitrary choice of molecular orbitals uk. Of particular interest are, however, those space orbitals which render p optimal, i.e. that molecular orbital (MO) approximation which is closest to the true wavefunction. The optimal molecular orbitals are those which minimize 3C iii), and Fock 6) has shown that they are the solutions of the integro-differential equations... 

In view of the preceding considerations it should be emphasized that it is incorrect to talk about the self-consistent-field molecular orbitals of a molecular system in the Hartree-Fock approximation. The correct point of view is to associate the molecular orbital wavefunction of Eq. (1) with the N-dimen-sional linear Hilbert space spanned by the orbitals t/2,... uN any set of N linearly independent functions in this space can be used as molecular orbitals for forming the antisymmetrized product. 

The lower-energy bonding molecular orbitals result when atomic orbital wavefunctions enhance each other in the region of the nuclei. The atoms are held together by attraction between the nuclei and the electrons in the bonding molecular orbital, and /Ab = Va + W... 

Based on Mulliken population analysis of ab initio molecular orbital wavefunctions for inner-sphere complexes (28). 

Returning to Figure 4, we note the two similar results obtained at 5.0 8 which indicate that the determination of from SCF molecular orbital wavefunctions is about the same... 

As an example, to construct the character table for the Oh symmetry group we could apply the symmetry operations of the ABg center over a particularly suitable set of basis functions the orbital wavefunctions s, p, d,... of atom (ion) A. These orbitals are real functions (linear combinations of the imaginary atomic functions) and the electron density probability can be spatially represented. In such a way, it is easy to understand the effect of symmetry transformations over these atomic functions. 

The value of the radial part of an atomic orbital wavefunction must tend to zero at very large distances from the nucleus. 

Extraction of information from p may not be as elegant as from P. For example, the Woodward-Hoffmann rules follow fairly transparently from the symmetries of molecular orbitals (wavefunctions), but deriving them from p requires using a dual descriptor function [1]. 

Such second-order molecular properties as spin-spin coupling depend upon distortion of electron clouds by additional external perturbations that is, in the NMR experiment they depend upon the electronic motion induced by an applied magnetic field. Theories for such second-order molecular properties require a study of the change in the molecular-orbital wavefunctions, which may be found by using a perturbation method to describe the effects occurring when a magnetic field is applied.8-1065-67... 

Wavefunction Vq refers to the situation where both electrons are on nucleus a or nucleus b, i.e., ionic structures. Now it is obvious that the valence bond wavefunction covalent structure, while the molecular orbital wavefunction t has an equal mixture of covalent and ionic contributions. Similarly, expanding the wavefunction in eq. (3.2.27) yields... 

The optimal c2/ci ratio is -0.73. More importantly, the improved valence bond wavefunction p + and the improved molecular orbital wavefunction t/fj are one and the same, thus showing these two approaches can lead to identical quantitative results. 

The 7r energy levels, along with the molecular orbital wavefunctions, are pic-torially displayed in Fig. 7.1.12. Since there are six n electrons in benzene, orbitals a2u and e % are filled, giving rise to a 1 /V ig ground state with the total n energy... 

Relationship between the coefficient matrices for the hybrid and molecular orbital wavefunctions... 

Fig. 2.5. Transmission difference spectra AT/T of quaterthiophene in solution. Photobleaching (PB) is seen as a rise towards the low wavelength edge and develops outside the experimentally accessible spectral range. Stimulated emission (SE) and photoinduced absorption (PA) from Si are seen at 500 nm and 675 nm, respectively. They decay in about 600 ps and a new PA band appears, at 560 nm, assigned to triplet-triplet transitions. A clear isosbestic point occurs at 600 nm. The data are an example of intersystem crossing in photoexcited molecules induced by spin-orbit wavefunction mixing...

The matrix elements Hab.ca, Aat,ca are calculated using spin valence functions constructed from atomic orbitals. The energies E%, Ef, Ef, Ef are the exact values for the particular states of the participating atoms and may be taken from spectroscopic tables. The corresponding quantities E%, etc., with tildes are the values for these same spectroscopic states obtained from calculations using the orbital wavefunctions. In essence, equation (116) prescribes how the matrix elements Hab.ca, which are obtained from an ab initio spin valence calculation, are to be corrected in order to eliminate known atomic errors. 

The equivalence that exists between the MO and VB resonance descriptions of the three-electron bond is only exact when one AO per centre is used to accommodate the electrons. This has been discussed in some detail by Murrell and Ralston [34], and elaborated further in VB calculations for a variety of systems by Hiberty and coworkers [8(a),35]. However, when four or more AOs are used to accommodate the three electrons, the VB structure XII or XIII may still be associated with an orbital wavefunction, and an extended VB-MO equivalence may be developed [36]. We shall use H2 to demonstrate this result for the simplest case, which involves the use of four AOs to accommodate the... 

---