Wavefunction radial

The function R(r) is called the radial wavefunction it tells us how the wavefunction varies as we move away from the nucleus in any direction. The function Y(0,c[>) is called the angular wavefunction it tells us how the wavefunction varies as the angles 0 and c > change. For example, the wavefunction corresponding to the ground state of the hydrogen atom ( = 1) is... 

What does this equation tell us For this wavefunction, the angular wavefunction Y is a constant, 1/2tti/2 , independent of the angles, which means that the wavefunction is the same in all directions. The radial wavefunction R(r) decays exponentially toward zero as r increases, which means that the electron density is highest close to the nucleus (e° =1). The Bohr radius tells us how sharply the wavefunction falls away with distance when r = a ), t i has fallen to 1/e (37%) of its value at the nucleus. 

FIGURE 1.34 The radial wavefunctions of the first three s-orbitals of a hydrogen atom. Note that the number of radial nodes increases (as n 1), as does the average distance of the electron from the nucleus (compare with Fig. 1.32). Because the probability density is given by ip3, all s-orbitals correspond to a nonzero probability density at the nucleus. 

For the one-center valence contribution, there are essentially three factors that control its value (a) the radial wavefunction of the 3d orbitals, (b) the covalent dilution of the 3d orbitals with ligand orbitals, and (c) the occupation pattern of the 3d shell. An additional factor may be low-symmetry induced 3d/4p mixing. We will focus on the first three factors here. 

A. Radial wavefunction. With respect to the radial wavefunction, a pronounced dependence of the value on the oxidation state and configuration of the... 

Fig. 5.8 Molecular radial wavefunctions for the ferric complex FeCLt compared to the radial wavefunctions of the free ions Fe ", Fe ", and Fe " (taken from [79])...

The normalized radial wavefunctions for hydrogenlike atoms can be expressed by... 

The term in square brackets is the body-fixed radial wavefunction and has been discussed previously (see Ref. 80, Appendix B). Defining this body-fixed radial wavefunction as... 

For dissimilar pairs, the parameter ys equals zero and we have Eq. 5.36. Like pairs of zero spin are bosons and all odd-numbered partial waves are ruled out by the requirement of even wavefunctions of the pair this calls for ys = 1. In general, for like pairs, the symmetry parameter ys will be between -1 and 1, depending on the monomer spins (fermions or bosons) and the various total spin functions of the pair. A simple example is considered below (p. 288ff.). If vibrational states are excited, the radial wavefunctions xp must be obtained from the vibrationally averaged potential, Fq(R). The functions gf(R) and gM(R) are similar to the pair distribution function, namely [294]... 

The radial wavefunctions tp R ,Et)/R needed for the computation of Eq. 6.55 are solutions of the Schrodinger equation of relative motion,... 

An important property of the wavefunction is its normalization, and we have yet to normalize the radial coulomb radial functions. Following the approach of Merzbacher, we can find an approximate WKB radial wavefunction, good in the classically allowed region, given by6... 

The minus sign is for radial wavefunctions R e r) —I-r( as r — 0. On the other hand, the m = 0 states also include the low states for which the radial matrix element is large, and the extreme m = 0 states have Stark shifts of approximately 3n2E/2. 

Values of (r-4) and (r-6) for hydrogenic wavefunctions have been calculated analytically, and the expressions are given in Table 2.3. Examining the forms of Table 2.3, it is apparent that both (r-4) and (r-6) exhibit n-3 scalings, due to the normalization of the radial wavefunction. However, they exhibit different i dependences (r-4) scales as 5, and (r-6) scales as 8. As a result of the very different i scalings the contributions of the dipole and quadrupole polarizabilities to the quantum defect are easily separated by measurements of the intervals between several i series. Furthermore, for high t, (r -4> , and as a result, for high ,... 

For this wavefunction, the angular wavefunction Y is a constant, l/2ir1/2, independent of the angles, and the radial wavefunction decays exponentially toward 0 as r increases. The quantity a0 is called the Bohr radius when the values of the fundamental constants are inserted, we find a0 = 52.9 pm. The expressions for a number of other atomic orbitals are shown in Table 1.2. 

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