Helium electron repulsions

VV e now wish to establish the general functional form of possible wavefunctions for the two electrons in this pseudo helium atom. We will do so by considering first the spatial part of the u a efunction. We will show how to derive functional forms for the wavefunction in which the i change of electrons is independent of the electron labels and does not affect the electron density. The simplest approach is to assume that each wavefunction for the helium atom is the product of the individual one-electron solutions. As we have just seen, this implies that the total energy is equal to the sum of the one-electron orbital energies, which is not correct as ii ignores electron-electron repulsion. Nevertheless, it is a useful illustrative model. The wavefunction of the lowest energy state then has each of the two electrons in a Is orbital ... 

Although we are solving for one-electron orbitals, r /i and r /2, we do not want to fall into the trap of the last calculation. We shall include an extra potential energy term Vi to account for the repulsion between the negative charge on the first electron we consider, electron I, exerted by the other electron in helium, electron 2. We don t know where electron 2 is, so we must integrate over all possible locations of electron 2... 

Extracting an electron from helium takes less energy than expected because of electron-electron repulsion. The helium nucleus actually does pull twice as a hard as a hydrogen nucleus does, but the two electrons in helium are also repelling one another. The net effect is to make an electron in a multielectron atom easier to remove than one would expect if the other electrons were not present. 

The existence of the first HK theorem is quite surprising since electron-electron repulsion is a two-electron phenomenon and the electron density depends only on one set of electronic coordinates. Unfortunately, the universal functional is unknown and a plethora of different forms have been suggested that have been inspired by model systems such as the uniform or weakly inhomogeneous electron gas, the helium atom, or simply in an ad hoc way. A recent review describes the major classes of presently used density functionals [10]. 

Unfortunately, the Schrodinger equation for multi-electron atoms and, for that matter, all molecules cannot be solved exactly and does not lead to an analogous expression to Equation 4.5 for the quantised energy levels. Even for simple atoms such as sodium the number of interactions between the particles increases rapidly. Sodium contains 11 electrons and so the correct quantum mechanical description of the atom has to include 11 nucleus-electron interactions, 55 electron-electron repulsion interactions and the correct description of the kinetic energy of the nucleus and the electrons - a further 12 terms in the Hamiltonian. The analysis of many-electron atomic spectra is complicated and beyond the scope of this book, but it was one such analysis performed by Sir Norman Lockyer that led to the discovery of helium on the Sun before it was discovered on the Earth. 

Even in atoms in molecules which have no permanent dipole, instantaneous dipoles will arise as a result of momentary imbalances in electron distribution. Consider the helium atom, for example. It is extremely improbable that the two electrons in the Is orbital of helium will be diametrically opposite each other at all times. Hence there will be instantaneous dipoles capable of inducing dipoles in adjacent atoms or molecules. AnothCT way of looking at this phenomenon is to consider the electrons in two or more "nonpolar" molecules as synchronizing their movements (at least partially) to minimize electron-electron repulsion and maximize electron-nucleus attraction. Such attractions are extremely short ranged and weak, as are dipole-induced dipole forces. The energy of such interactions may be expressed as... 

The simplest kind of ab initio calculation is a Hartree-Fock (HF) calculation. Modem molecular HF calculations grew out of calculations first performed on atoms by Hartree1 in 1928 [3]. The problem that Hartree addressed arises from the fact that for any atom (or molecule) with more than one electron an exact analytic solution of the Schrodinger equation (Section 4.3.2) is not possible, because of the electron-electron repulsion term(s). Thus for the helium atom the Schrodinger equation (cf. Section 4.3.4, Eqs. 4.36 and 4.37) is, in SI units... 

The helium atom has two electrons and they can both fit into the Is orbital providing they have opposite spins. The other change to the diagram is that, with two electrons and electron repulsion a factor, the 2s orbital is now lower in energy than the three 2p orbitals, though these three are still degenerate. 

Problem 5.3 The calculated total radial distribution A-trr Tp for the helium atom, obtained by two methods the SCF method and ignoring electron repulsion ref. 13) is shown. Interpret. 

The possibilities of the exchange of the hydrogen electron witli either of the helium electrons is restricted in the first place by the Pauli principle. The exchange of electrons 1 and 3 results in the appearance of two electrons with parallel spins existing in one orbital which is not permitted by the exclusion principle. There remains in consequence only the possibility of the exchange of electrons 2 and 3, which because they have parallel spins, will, as has been shown in the last chapter, lead not to bond formation but to mutual repulsion. Thus no molecule HeH is formed. 

Despite its great success in accounting for the spectral lines of the H atom, the Bohr model failed to predict the spectrum of any other atom, even that of helium, the next simplest element. In essence, the Bohr model predicts spectral lines for the H atom and other one-electron species, such as He" (Z = 2), Li (Z = 3), and Be (Z = 4). But, it fails for atoms with more than one electron because in these systems, electron-electron repulsions and additional nucleus-electron attractions are present as well. Nevertheless, we still use the terms ground state and excited state and retain one of Bohr s central ideas in our current model the energy of an atom occurs in discrete levels. 

The generally accepted basis for the widely used Hund s rules for predicting the ordering of electronic states has been challenged in recent years, yet the rules appear to be valid. A reformulation of the rules in a strict SCF approximation in which many of the elements of the traditional theory are retained has been proposed, to surmount this difficulty.10 Electron repulsion in the singlet and triplet states of the helium atom, natural orbitals of several excited states of this... 

The orbital is composed largely of the s orbital on the helium nucleus in the absence of any electron-electron repulsion the electrons tend to congregate near the nucleus with the larger charge. The density matrix corresponding to this initial wavefxmction is ... 

Although the helium atom can be readily described in terms of the quantum mechanical model, the Schrodinger equation that results cannot be solved exactly. The difficulty arises in dealing with the repulsions between the electrons. Since the electron pathways are unknown, the electron repulsions cannot be calculated exactly. This is called the electron correlation problem. 

For many-electron atoms, the Schrodinger equation can be solved approximately, but very accurately nowadays, by transforming the difficult electron-electron repulsion terms into a spherically averaged form. Then, each electron can be considered to exhibit independent motion in the potential field due to the nucleus and the averaged repulsive interaction with the other electrons in the atom. This simplification leads to the Independent Particle Approximation preserving the orbital concept and first proposed by Hartree (1) for the simplest many-electron atom, helium, as the product... 

The Hartree product wave function, equation 5.2, for helium does not comply with the anti-symmetry requirement of the Pauli Principle (42,47) that electronic wave functions must change sign on exchange of the coordinates for a pair of electrons. Fock identified this defect in the overestimation of the electron-electron repulsion term, which occurs for Hartree product wave functions, while Slater showed how to overcome this problem by writing the product wave function in the form of a determinant (6,7,42,47,64). 

Figure 5.6a The familiar part of the general spreadsheet, the worksheet oneel , for the calculation of the energy of the helium atom using the sto-3g) basis set of Table 1.6 and the optimized choice for the Slater exponent, with the electron-electron repulsive potential as the extra term in cell I 7 worked out on the worksheet twoel , detail from which is shown in Figure 5.6b. Note, the A I5 master formula entry allows for the He atomic number in the calculation of the nucleus-electron potential term.

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