Hydrogenlike Atomic Orbitals

Figure 6-3 Schematic diagram of the relative energies of the hydrogenlike atomic orbitals...

For the integrations in ab initio calculations we need the actual mathematical form of the spatial functions, and the hydrogenlike expressions are Slater functions [1]. For atomic and some molecular calculations Slater functions have been used [3]. These vary with distance from where they are centered as exp(-constant.r), where r is the radius vector of the location of the electron, but for molecular calculations certain integrals with Slater functions are very time-consuming to evaluate, and so Gaussian functions, which vary as exp(-constant.r2) are almost always used a basis set is almost always a set of (usually linear combinations of) Gaussian functions [4]. Very importantly, we are under no theoretical restraints about their precise form (other than that in the exponent the electron coordinate occurs as exp(-constant.r2)). Neither are we limited to how many basis functions we can place on an atom for example, conventionally carbon has one 1 s atomic orbital, one 2s, and three 2p. But we can place on a carbon atom an inner and outer Is basis function, an inner and outer 2s etc., and we can also add d functions, and even f (and g ) functions. This freedom allows us to devise basis sets solely with a view to getting... 

Concerning molecules, the wave function (molecular orbital) for a hydrogenlike molecule, for instance, is expanded in terms of hydrogen-like atomic orbitals Xaj(f) belonging to hydrogen-like atoms / = 1,2, respectively, as... 

R.D. ALLENDOERFER, Teaching the shapes of the hydrogenlike and hybrid atomic orbitals. J. Chem. Educ., 67, 37 (1990). 

We can use the quantum mechanical model of the atom to show how the electron arrangements in the hydrogenlike atomic orbitals of the various atoms account for the organization of the periodic table. Our main assumption here is that all atoms have the same type of orbitals as have been described for the hydrogen atom. As protons are added one by one to the nucleus to build up the elements, electrons are similarly added to these hydrogenlike orbitals. This is called the aufbau principle. 

We consider the electron in a hydrogen atom or hydrogenlike ion (He, Li +, ) orbiting around a nucleus of atomic number Z. The attractive Coulomb potential in atomic units e /ATt Q = 1) can be written as... 

In dealing with molecules, the real hydrogenlike orbitals are more useful than the complex ones. For example, we shall see in Section 15.6 that the real atomic orbitals 2p, 2py, and 2p of the oxygen atom have the proper symmetry to be used in constructing a wave function for the H2O molecule, whereas the complex 2p orbitals do not. 

Accurate representation of a many-electron atomic orbital (AO) requires a linear combination of several Slater-type orbitals. For rough calculations, it is convenient to have simple approximations for AO s- We might use hydrogenlike orbitals with effective nuclear charges, but Slater suggested an even simpler method to approximate an AO by a single function of the form (11-14) with the orbital exponent C taken as... 

For a many-electron atom, the self-consistent-field (SCF) method is used to construct an approximate wave function as a Slater determinant of (one-electron) spin-orbitals. The one-electron spatial part of a spin-orbital is an atomic orbital (AO). We took each AO as a product of a spherical harmonic and a radial factor. As an initial approximation to the radial factors, we can use hydrogenlike radial functions with effective nuclear charges. 

An important point that must not be forgotten is that fi -block metal atoms are, of course, many-electron species, and when we discuss, for example, radial distribution functions of the nd atomic orbitals, we refer to hydrogenlike atoms and, therefore, the discussion is extremely approximate. 

MO-LCAO), accepting that these atomic orbitals are actually pseudo atomic orbitals obtained by fitting one or more of the special functions to one hydrogenlike function. 

As a result of the spherical symmetry of the potential energy, the energy of the atomic orbitals for hydrogen and hydrogenlike ions depends only upon the value of the principal quantum number (n), and is given by... 

Table 1.4 summarizes the relationship between quantum numbers and hydrogenlike atomic orbitals. When Z = 0, (2/ + 1) = 1 and there is only one value of m/, so we have an s orbital. When / = 1, (2/ + 1) = 3, so there are three values of m , giving rise to three p orbitals, labeled p, Py, and p. When / = 2, (2Z + 1) = 5, so there are five values of mi, and the corresponding five d orbitals are labeled with more elaborate subscripts. In the following sections we discuss the s, p, and d orbitals separately. 

The wavefunctions and radial distribution functions for the li, 2s, and hs hydrogenlike atomic orbitals are shown in Figure 1.29 as functions of r. Note that the wavefunctions for the li, 2s, and hs orbitals have 0, 1, and 2 nodes (points where the wavefunction is zero), respectively. For a general s orbital with principal quantum number n, the number of nodes is 1. This increase in the number of nodes as... 

To get approximations to higher MOs, we can use the linear-variation-function method. We saw that it was natural to take variation functions for Hj as linear combinations of hydrogenlike atomic-orbital functions, giving LCAO-MOs. To get approximate MOs for higher states, we add in more AOs to the linear combination. Thus, to get approximate wave functions for the six lowest linear combination of the three lowest m = 0 hydrogenlike functions on each atom ... 

The formulas for the hydrogenlike ion solutions (in atomic units) of most interest in quantum chemistry are listed in Table 4-2. The tabulated functions are all in real, rather than complex, form. Problems involving atomic orbitals are generally far easier to solve in atomic units. 

Notice that each individual one-electron hamiltonian (5-4) is just the hamiltonian for a hydrogenlike ion, so it has as eigenfunctions the Is, 2s, 2p, etc., functions of Chapter 4 with Z = 2. Such one-electron functions are referred to as atomic orbitals Representing them with the symbol 0/ (e.g., 0i = Is, 02 = 2s, 0s = 2px, 04= etc.) we have, then. 

The one-electron operators in the resulting approximate hamiltonian for an atom are hydrogenlike ion hamiltonians. Their eigenfunctions are called atomic orbitals. 

Atomic orbitals are physically meaningful one-electron atom eigenfunctions for the Schrodinger equation. This gives well-known analytical expressions hydrogenlike orbitals. 

Normalized radial functions for a hydrogenlike atom are given in Table A 1.1 and plotted graphically in Fig. A 1.1 for the first ten combinations of n and /. It will be seen that the radial functions for Is, 2p, 3d, and 4f orbitals have no nodes and are everywhere of... 

The natural orbitals %2v and %3p are, in contrast to the hydrogenlike functions, localized within approximately the same region around the nucleus as the Is orbital. This means that the polarization caused by the long-range interaction is associated mainly with an angular deformation of the electronic cloud on each atom. If %2p and %3p are expanded in the standard hydrogen-like functions, an appreciable contribution will again come from the continuum. 

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