Hydrogen-like atom wave functions

The requirement that the basis functions should describe as closely as possible the correct distribution of electrons in the vicinity of nuclei is easily satisfied by choosing hydrogen-like atom wave functions, t], the solutions to the Schrodinger equation for one-electron atoms for which exact solutions are available ... 

The direct variational method has been used to solve Schrodinger equation (4.2) with respect to Eqs. (4.3), (4.4), and (4.5). Hydrogen-like atom wave functions 2p and 3p have been taken as trial functions for ground and exited states respectively. Two-fermion wave functions were written in conventional form as a product of coordinate part and symmetric or asymmetric spin part for triplet (spin S = 1) or singlet (spin E = 0) state. The energy level positions 23 (S = 0), 33 (S = 0),... 

Table 1.1 Some wave functions for hydrogen and hydrogen-like atoms...

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... 

The radial factors of the hydrogen-like atom total wave functions ip r, 0, tp) are related to the functions Sni(p) by equation (6.23). Thus, we have... 

The wave functions nlm) for the hydrogen-like atom are often called atomic orbitals. It is customary to indicate the values 0, 1, 2, 3, 4, 5, 6, 7,. .. of the azimuthal quantum number / by the letters s, p, d, f, g, h, i, k,. .., respectively. Thus, the ground-state wave function 100) is called the Is atomic orbital, 200) is called the 2s orbital, 210), 211), and 21 —1) are called 2p orbitals, and so forth. The first four letters, standing for sharp, principal, diffuse, and... 

Table 6.2. Real wave functions for the hydrogen-like atom. The parameter a j,...

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... 

As the wave function is not known analytically for systems larger than a hydrogen-like atom, suitable approximate wave functions have to be found and the accuracy of Eq. (1) depends of course on the level of approximation. A survey of the various quantum chemical methods to generate approximated wave functions can be found in Refs. (22,23). Here, we shall only present the foundations of Hartree-Fock and density functional theory (DFT) needed in later sections. 

Analytic, exact solutions cannot be obtained except for the simplest systems, i.e. hydrogen-like atoms with just one electron and one nucleus. Good approximate solutions can be found by means of the self-consistent field (SCF) method, the details of which need not concern us. If all the electrons have been explicitly considered in the Hamiltonian, the wave functions V, will be many-electron functions V, will contain the coordinates of all the electrons, and a complete electron density map can be obtained by plotting Vf. The associated energies E, are the energy states of the molecule (see Section 2.6) the lowest will be the ground state , and the calculated energy differences En — El should match the spectroscopic transitions in the electronic spectrum. 

The electron which responds to both quantum and classical potential fields exhibits this dual nature in its behaviour. Like a photon, an electron spreads over the entire region of space-time permitted by the boundary conditions, in this case stipulated by the classical potential. At the same time it also responds to the quantum field and reaches a steady, so-called stationary, state when the quantum and classical forces acting on the electron, are in balance. The best known example occurs in the hydrogen atom, which is traditionally described to be in the product state tpH = ipP ipe, hence with broken holistic symmetry. In many-electron atoms the atomic wave function is further fragmented into individual quantum states for pairs of electrons with paired spins. 

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... 

Figure 1.14 Expectation value of the electric field strength for the lowest-lying states of a hydrogen-like atom in the range Z = 1-92. Electron wave functions for extended nuclear-charge distributions are employed.

---