Hydrogen-like atom atomic orbitals

It turns out that, even for hydrogen-like atoms, atomic orbits appear vastly more complex. Regrettably, it is impossible to obtain the exact solution even for these atoms. So in quantum chemistry different kinds of approximation are used, more or less successfully, to describe one system or another, and one atomic area or another. For instance, the factor is introduced as a multiplier in the exponent order of the radial wavefimctions to describe an orbit s compression-expansion (Slater multipUer). Sometimes, not one but two, or even several, multipliers are used, each of which is better for describing the electron density near the nucleus or far from it. These empirical modificatious for differeut atoms are given in quantum chemistry. 

To describe atoms with several electrons, one has to consider the interaction between the electrons, adding to the Hamiltonian a term of the form Ei< . Despite this complication it is common to use an approximate wave function which is a product of hydrogen-like atomic orbitals. This is done by taking the orbitals in order of increasing energy and assigning no more than two electrons per orbital. 

In order to obtain an approximate solution to eq. (1.9) we can take advantage of the fact that for large R and small rA, one basically deals with a hydrogen atom perturbed by a bare nucleus. This situation can be described by the hydrogen-like atomic orbital y100 located on atom A. Similarly, the case with large R and small rB can be described by y100 on atom B. Thus it is reasonable to choose a linear combination of the atomic orbitals f00 and f00 as our approximate wave function. Such a combination is called a molecular orbital (MO) and is written as... 

Consider a crude approximation to the ground state of the lithium atom in which the electron-electron repulsions are neglected. Construct the ground-state wave function in terms of the hydrogen-like atomic orbitals. 

The ground-state wave function for the unperturbed two-electron system is the product of two Is hydrogen-like atomic orbitals... 

The correct limiting radial behavior of the hydrogen-like atom orbital is as a simple exponential, as in (A.62). Orbitals based on this radial dependence are called Slater-type orbitals (STOs). Gaussian functions are rounded at the nucleus and decrease faster than desirable (Figure 2.2b). Therefore, the actual basis functions are constructed by taking fixed linear combinations of the primitive Gaussian functions in such a way as to mimic exponential behavior, that is, resemble atomic orbitals. Thus... 

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... 

For the 3pj and hydrogen-like atomic orbitals, sketch the following ... 

Another technique that is used is to simply increase the number of basis orbitals in the Slater determinant. This is easily done by including a larger number of the hydrogen-like atomic orbitals in the trial function. If the entire n = 2 set of atomic orbitals (2s, 2p , 2pj, 2p ) is included in a linear combination with the Is orbital for each atomic center, when... 

Now we will calculate one by one aU die integrals fliat appear in the Dirac matrix equation. The integral y ) = because die scalar product leads to the nuclear attraction integral with a hydrogen-like atomic orbital, and diis gives the result above (see Appendix H available at booksite.elsevier.com/978-0-444-59436-5, p. e91). The next integral can be computed as follows ... 

This means that the radial part of a STO has no nodes. Because of this, STOs of the same angular dependence, in contrast to the hydrogen-like atom orbitals, are not orthogonal. 

In the orbital view of bonding, atoms approach each other in such a way that their atomic orbitals can overlap to form a bond. For example, if two hydrogen atoms form a hydrogen molecule, their two spherical Is orbitals combine to form a new orbital that encompasses both of the atoms (see Figure 1.3). This orbital contains both valence electrons (one from each hydrogen). Like atomic orbitals, each molecular orbital can contain no more than two electrons. In the hydrogen molecule, these electrons mainly occupy the space between the two nuclei. 

It should be apparent that the most obvious basis set to use for an ab initio calculation is the set of hydrogen-like atomic orbitals Is, 2s, 2p, and so on that we are all familiar with from atomic structure and bonding theory. Unfortunately, these "actual" orbitals present computational difficulties because they have radial nodes... 

Just as the electronic configuration of an atom is built up by stepwise population - electron by electron - of hydrogen-like atomic orbitals, that of a diatomic molecule is constructed by successively filling the molecular orbitals derived from the hydrogen molecule ion, H2 [1]. 

That hydrogen-like atomic orbitals suffer a loss of degeneracy when more than one electron is present That electrons possess an amount of intrinsic energy governed by the spin quantum number The Pauli exclusion principle... 

We can use the quantum mechanical model of the atom to show how the electron arrangements in the hydrogen-like 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 hydrogen-like orbitals. This is called the aufbau principle. 

FIGURE 5.1 Illustration of the approximation of radial parts of hydrogen-like atomic orbitals (heavy curves) as sums of Gaussians (a) li and (b) 3p. Not to scale and fitted by trial and error (not optimized). 

The Relative Energies of Hydrogen-Like Atomic Orbitals... 

The factor is the same factor that occurs in hydrogen-like atomic orbitals. As in that case, we can choose either the complex functions or the real functions mx and < >my. The other factors are more complicated and we do not display the formulas representing them. These molecular orbitals are called exact Born-Oppenheimer molecular orbitals. They are not exact solutions to the complete Schrodinger equation, but they contain no approximations other than the Born-Oppenheimer approximation. 

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