D orbitals, wave functions

The functions of Cartesian coordinates normally included in character tables are those associated with the d orbital wave functions, namely z x2 + y2, x2 - y2, xy, xz, and yz. Since we already know how each of the individual coordinates behaves, it is simple to work out the results for these combinations. To illustrate let us take x2 - y2 ... 

The canonical real d-orbital wave functions are linear combinations of the complex eigen functions (solutions) of the hydrogenic Schrodinger equation and the orbital angular momentum operator. Thus, instead of a set of five degenerate orbitals that may be indexed by values of orbital... 

Transition metals owe their location in the periodic table to the filling of the d subshells, as you Saw in Figure 6.31. Many of the chemical and physical properties of transition metals result from the unique characteristics of the d orbitals. For a given transition-metal atom, the valence (n — l)d orbitals are smaller than the corresponding valence ns and np orbitals. In quantum mechanical terms, the (n — l)d orbital wave functions drop off more rapidly as we move away from the nucleus than do the ns and np orbital wave functions. This characteristic feature of the d orbitals limits their interaction with orbitals on neighboring atoms, but not so much that they are insensitive to... 

When M is an atom the total change in angular momentum for the process M + /zv M+ + e must obey the electric dipole selection mle Af = 1 (see Equation 7.21), but the photoelectron can take away any amount of momentum. If, for example, the electron removed is from a d orbital ( = 2) of M it carries away one or three quanta of angular momentum depending on whether Af = — 1 or +1, respectively. The wave function of a free electron can be described, in general, as a mixture of x, p, d,f,... wave functions but, in this case, the ejected electron has just p and/ character. 

In general, we do not need to know the whole energy spectrum, but we are interested in several low lying states, say, where a = 1,2,...,d. Let us further assume that the most important contributions to d exact wave functions iP, come from d configurations represented by Slater determinants in the spin-orbital formalism, where ft = 1,2,..., d. Given dominant configurations span the so-called model or reference space. To simplify... 

We first note that an isolated atom with an odd number of electrons will necessarily have a magnetic moment. In this book we discuss mainly moments on impurity centres (donors) in semiconductors, which carry one electron, and also the d-shells of transitional-metal ions in compounds, which often carry several In the latter case coupling by Hund s rule will line up all the spins parallel to one another, unless prevented from doing so by crystal-field splitting. Hund s-rule coupling arises because, if a pair of electrons in different orbital states have an antisymmetrical orbital wave function, this wave function vanishes where r12=0 and so the positive contribution to the energy from the term e2/r12 is less than for the symmetrical state. The antisymmetrical orbital state implies a symmetrical spin state, and thus parallel spins and a spin triplet. The two-electron orbital functions of electrons in states with one-electron wave functions a(x) and b(x) are, to first order,... 

Consider each boron atom to be ip3 hybridized.123 The two terminal B—H bonds on each boron atom presumably are simple a bonds involving a pair of electrons each. This accounts for eight of the total of twelve electrons available for bonding. Each of the bridging B—H—B linkages then involves a delocalized or three-center bond as follows. The appropriate combinations of the three orbital wave functions. B. d>D, (approximately spi hybrids), and (an s orbital) result in three molecular orbitals ... 

As an example, suppose for an A-electron system that energy E is approximated by an orbital functional [ , ], which depends on one-electron orbital wave functions , and on occupation numbers n, through a variational A-electron trial wave function A momentum displacement is generated by U = cxp(- 7r D), where D = r - In (he momentum representation of the orbital wave functions,... 

One feature that should be mentioned is the appearance of i (= / ) in the p and d orbital wave equations in Table 2-3. Because it is much more convenient to work with real functions than complex functions, we usually take advantage of another property of the wave equation. For differential equations of this type, any linear combination of solutions (sums or differences of the functions, with each multiplied by any coefficient) to the equation is also a solution to the equation. The combinations usually chosen for thep orbitals are the sum and difference of thep orbitals having mi = +l and... 

Calculations for best fit, using this orbitally degenerate model, employed the parameters D, J ID, and y, the last being close to 0.5 and originating in the orbital wave function, e.g., (j> ( 2) = 2 (1... 

Following Mulliken, we shall occasionally refer to one-electron orbital wave functions such as the hydrogenlike wave functions of this chapter as orbitals. In accordance with spectroscopic practice, we shall also use the symbols s, p, d, /, g, to refer to states characterized by the values 0, 1, 2, 3, 4, , respectively, of the azimuthal quantum number l, speaking, for example, of an s orbital to mean an orbital with 1 = 0. 

The function fa composed of Is hydrogenlike orbital wave functions with effective nuclear charge 2e leads to a minimum in the energy curve at r, = 1.01 A and the value 2.9 v.e. for the energy of dissociation D, into He + He+. A more accurate treatment1 can be made by minimizing the energy for each value... 

Referring back to Fig. 9-2, let us form linear combinations of the d, s, and p valence orbitals that direct large lobes at the six ligands. We first construct the orbitals that are directed toward ligands 0 and 0. We shall call these orbitals and V e, respectively. The metal orbitals that can a bond with 0 and 0 are 4s, and 4pi. Choosing the coefficients of the id, 4s, and 4ps orbitals so that 1 8 and have the desired d, r, and p character, we obtain the following hybrid-orbital wave functions ... 

C the overlap of the atomic orbital wave functions of two negative ions D the overlap of the atomic orbital wave functions of two positive ions... 

The wave function T i oo ( = 11 / = 0, w = 0) corresponds to a spherical electronic distribution around the nucleus and is an example of an s orbital. Solutions of other wave functions may be described in terms of p and d orbitals, atomic radii Half the closest distance of approach of atoms in the structure of the elements. This is easily defined for regular structures, e.g. close-packed metals, but is less easy to define in elements with irregular structures, e.g. As. The values may differ between allo-tropes (e.g. C-C 1 -54 A in diamond and 1 -42 A in planes of graphite). Atomic radii are very different from ionic and covalent radii. 

HyperClicm cati plot orbital wave fuuctious resulting fmni serni-cmpirical and ah i/iitw quan tii m m ecli an ica I calculations. It is ill tercstiu g to view both tli c u tidal properties an d th e relative sizes of the wave functions. Orbital wave functiou s can provide dietni-cal in sigh is. 

Polarization functions are functions of a higher angular momentum than the occupied orbitals, such as adding d orbitals to carbon or / orbitals to iron. These orbitals help the wave function better span the function space. This results in little additional energy, but more accurate geometries and vibrational frequencies. 

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