The Helium Atom

The stability of the solar system is one of the most important unsettled questions of classical mechanics. Even a simplified version of the solar system, the three-body problem, presents a formidable challenge. An important breakthrough occurred when Poincare, with some assistance from his Swedish colleague Pragmen, proved in 1892 that, apart from some notable exceptions, the three-body problem does not possess a complete set of integrals of the motion. Thus, in modern parlance, the three-body problem is chaotic. 

The helium atom is an atomic physics example of a three-body problem. On the basis of Poincare s result we have to expect that the helium atom is classically chaotic. Richter and Wintgen (1990b) showed that this is indeed the case the helium atom exhibits a mixed phase space with intermingled regular and chaotic regions (see also Wintgen et al. (1993)). Thus, conceptually, the helium atom is a close relative of the double pendulum studied in Section 3.2. Given the classical chaoticity of the helium atom we are confronted with an important question How does chaos manifest itself in the helium atom  

The focus of this chapter is not on the accurate computation of helium energy levels, but on the development of a global understanding of the different dynamical regimes of the helium atom, and on identifying them in the helium spectrum. 

There are a total of six possible combinations of the four quantum numbers for this case. 

Finally, because spin is part of the properties of an electron, its observable values should be determined from the electrons wavefimction. That is, there should be a spin wavefunction part of the overall T. A discussion of the exact form of the spin part of a wavefunction is beyond our scope here. However, because there is only one possible observable value of the total spin (s = ) and only two possible values of the z component of the spin (m = or — ), it is typical to represent the spin part of the wavefunction by the Greek letters a and /3, depending on whether the quantum number is -l- or —respectively. Spin is unaffected by any other property or observable of the electron, and the spin component of a one-electron wavefunction is separable from the spatial part of the wavefunction. Like the three parts of the hydrogen atom s electronic wavefunction, the spin function multiplies the rest of T. So for example, the complete wavefunctions for an electron in a hydrogen atom are 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

FIGURE 12.3 Definitions of the radial coordinates for the helium atom. 

In order to properly write the complete form of the Schrodinger equation for helium, it is important to understand the sources of the kinetic and potential energy in the atom. Assuming only electronic motion with respect to a motionless nucleus, kinetic energy comes from the motion of the two electrons. It is assumed that the kinetic energy part of the Hamiltonian operator is the same for the two electrons and that the total kinetic energy is the sum of the two individual parts. To simplify the Hamiltonian, we will use the symbol V, called del-squared, to indicate the three-dimensional second derivative operator  

The Schroedinger equation involves six independent variables. Three coordinates specify each electron. The wavefunction is expressed as a function of these six coordinates. 

The coordinate r is in terms of the coordinates of each electron. In Cartesian coordinates, it is given as follows. 

The ri2 term cannot be separated into coordinates for either electron making the Hamiltonian for the two-electron system inseparable. In order to solve the Schroedinger equation, an approximation technique is needed. 

A natural choice of approximation techniques to use for this system is perturbation theory. The Hamiltonian in Equation 8-17 can be divided into the unperturbed Hamiltonian,, consisting of the two hydrogen-like Hamiltonians, and the first-order perturbing Hamiltonian,, that consists of the electron-electron repulsion term l/rij. 

In the unperturbed system, the two electrons do not interact with one another. This means that the electrons are independent, and their motions are separable. The unperturbed wavefunction consists of a product of two  

The helium atom is similar to the hydrogen atom with the critical difference that there are two electrons moving in the potential field of a nucleus with a double positive charge (Z = 2) (Eig. 8-1). 

If the Hamiltonian were to operate on an exact, normalized wave function for helium, the energy of the system would be obtained 

As a naive or zero-order approximation, we can simply ignore the V12 term and allow the simplified Hamiltonian to operate on the Is orbital of the H atom. The result is 

We can compare this result with the experimental first and second ionization potentials (IPs) for helium 

If we compare the calculated total ionization potential, IP = 4.00 hartiees, with the experimental value, IP = 2.904 hartiees, the result is quite poor. The magnitude of the disaster is even more obvious if we subtract the known second ionization potential, IP2 = 2.00, from the total IP to find t c first ionization potential, IPi. The calculated value of IP2, the second step in reaction (8-21) is IP2 = Z /2 = 2.00, which is an exact result because the second ionization is a one-election problem. For the first step in reaction (8-21), IPi (calculated) = 2.00 and IPi(experimental) = 2.904 — 2.000 =. 904 hartiees, so the calculation is more than 100% in error. Clearly, we cannot ignore interelectronic repulsion. 

In the helium atom two electrons revolve about the nucleus (nuclear charge 2e) we have therefore 6 co-ordinates to deal with instead of 3, with the result that an exact solution is no longer possible. For the purpose of obtaining a general idea of the possible states, an exact solution is, however, not at all necessary following Bohr, we can in the first place neglect the mutual interaction of the electrons, and for a first approximation treat the problem as if the two electrons moved undisturbed in the field of the nucleus. Afterwards, the interaction can be taken into account by the methods of the theory of perturbations. 

We shall therefore (as with the hydrogen atom) associate three quantum numbers with each of the two electrons  

Another limiting case of the coupling relations is that in which the spin vectors and the orbital vectors are compounded separately into resultants I and s, so that for two electrons we have the diagram of fig. 11. The vectors Zj and Zg rotate round the total orbital moment Z, 

Experimental results show that as a rule (in helium and the alkaline earths) the second case of coupling is the one which occurs, it is called the normal coupling case or, after its discoverers, the Eussell-Saunders coupling. This is the only case we shall consider here although the case first described also actually occurs, as well as intermediate stages between these two extreme cases. 

We may remind the reader of the similar circumstances in the case of the abnormal Zeeman effect, where the 

2s orbital and at 1.90ao and 7.10aQ for the 3s orbital. It is a general characteristic that ns orbitals have n — 1) nodes. It is also noted that the distance at which the maximum probability occurs increases as the value of n increases. In other words, a 3s orbital is larger than a 2s orbital. Although not exactly identical, this is in accord with the idea from the Bohr model that the sizes of allowed orbits increase with increasing n value. 

We know that the general form of the wave equation is 

In order to solve the wave equation for the hydrogen atom, it is necessary to transform the Laplacian into polar coordinates. That transformation allows the distance of the electron from the nucleus to be expressed in terms of r, 9, and (p, which in turn allows the separation of variables technique to be used. Examination of Eq. (2.40) shows that the first and third terms in the Hamiltonian are exactly like the two terms in the operator for the hydrogen atom. Likewise, the second and fourth terms are also equivalent to those for a hydrogen atom. However, the last term, e2/r12, is the troublesome part of the Hamiltonian. In fact, even after polar coordinates are employed, that term prevents the separation of variables from being accomplished. Not being able to separate the variables to obtain three simpler equations prevents an exact solution of Eq. (2.40) from being carried out. 

When an equation describes a system exactly but the equation cannot be solved, there are two general approaches that are followed. First, if the exact equation cannot be solved exacdy, it may be possible to obtain approximate solutions. Second, the equation that describes the system exactly may be modified to produce a different equation that now describes the system only approximately but which can be solved exactly. These are the approaches to solving the wave equation for the helium atom. 

If the problem is approached from the standpoint of considering the repulsion between the electrons as being a minor irregularity or perturbation in an otherwise solvable problem, the Hamiltonian can be modified to take into account this perturbation in a form that allows the problem to be solved. When this is done, the calculated value for the first ionization potential is 24.58 eV. 

We now reconsider the helium atom from the standpoint of electron spin and the Pauli principle. In the perturbation treatment of helium in Section 9.3, we found the zeroth-order wave function for the ground state to be 15(1)15(2). To take spin into account, we must multiply this spatial function by a spin eigenfunction. We therefore consider the possible spin eigenfunctions for two electrons. We shall use the notation a(l)a(2) to indicate a state where electron 1 has spin up and electron 2 has spin up a(l) stands for a(Wji). Since each electron has two possible spin states, we have at first sight the four possible spin functions  

There is nothing wrong with the first two functions, but the third and fourth functions violate the principle of indistinguishability of identical particles. For example, the third function says that electron 1 has spin up and electron 2 has spin down, which does distinguish between electrons 1 and 2. More formally, if we apply Pyi to these functions, we find that the first two functions are symmetric with respect to interchange of the two electrons, but the third and fourth functions are neither symmetric nor antisymmetric and so are unacceptable. 

What now Recall that we ran into essentially the same situation in treating the helium excited states (Section 9.7), where we started with the functions ls(l)2s(2) and 2s(l)ls(2). We found that these two functions, which distinguished between electrons 1 and 2, are not the correct zeroth-order functions and that the correct zeroth-order functions are 2 [ls(l)2s(2) 2s(l)ls(2)]. This result suggests pretty strongly that instead of a(l)j8(2) and /3(l)a(2), we use 

Therefore, the four normalized two-electron spin eigenfunctions with the correct exchange properties are 

We now include spin in the He zeroth-order ground-state wave function. The function ls(l)ls (2) is symmetric with respect to exchange. According to the Pauli principle, the overall wave function including spin must be antisymmetric with respect to interchange of the two electrons. Hence we must multiply the symmetric space function ls(l)ls(2) by an antisymmetric spin function. Tliere is only one antisymmetric 

Like the solar system, it is not possible to find a set of coordinates where the Schrodinger equation can be solved analytically for more than two particles (i.e. for many-electron atoms). Owing to the dominance of the Sun s gravitational field, a central field approximation provides a good description of the actual solar system, but this is not the case for an atomic system. The main differences between the solar system and an atom such as helium are  

The total energy calculated by this wave function is simply twice the orbital energy, -4.000 au, which is in error by 38% compared with the experimental value of -2.904 au. Alternatively, we can use the wave function given by eq. (1.41), but include the electron-electron interaction in the energy calculation, giving a value of -2.750 au. 

A better approximation can be obtained by taking the average repulsion between the electrons into account when determining the orbitals, a procedure known as the Hartree-Fock approximation. If the orbital for one of the electrons were somehow known, the orbital for the second electron could be calculated in the electric field of the nucleus and the first electron, described by its orbital. This argument could just as well be used for the second electron with respect to first electron.The goal is therefore to calculate a set of self-consistent orbitals, and this can be done by iterative methods. 

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Dalgarno A and Lynn N 1957 Properties of the helium atom Proc. Phys. Soc. London 70 802... 

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 ... 

The final wavefunction stUl contains a large proportion of the Is orbital on the helium atom, but less than was obtained without the two-electron integrals. 

Write the Hamiltonian for the helium atom, which has two electrons, one at a distance r and the other at a distance r2-... 

A Self-Consistent Field Variational Calculation of IP for the Helium Atom... 

Use Mathcad to calculate the first approximation to the SCF energy of the helium atom... 

For the helium atom ground state, which we shall later generalize to many election atoms and molecules. 

All of our orbitals have disappeared. How do we escape this terrible dilemma We insist that no two elections may have the same wave function. In the case of elections in spatially different orbitals, say. Is and 2s orbitals, there is no problem, but for the two elechons in the 1 s orbital of the helium atom, the space orbital is the same for both. Here we must recognize an extr a dimension of relativistic space-time... 

To satisfy the Pauli exelusion prineiple, the eleetronie wave funetion must be antisymmetrie. This eondition can be met in the exeited state of the helium atom by taking the produet of an antisymmetrie space part sueh as... 

If we expand Eq. (10-7) and simplify aeeording to the symmetry of the problem, (Richards and Cooper, 1983) the integral breaks up in the way it did for the helium atom excited state... 

Example The electron configuration for Be is Is lsfi but we write [He]2s where [He] is equivalent to all the electron orbitals in the helium atom. The Letters, s, p, d, and f designate the shape of the orbitals and the superscript gives the number of electrons in that orbital. 

The analogy is even closer when the situation in oxygen is compared with that in excited configurations of the helium atom summarized in Equations (7.28) and (7.29). According to the Pauli principle for electrons the total wave function must be antisymmetric to electron exchange. 

CBS extrapolation is illustrated in the following figure which depicts the extrapolation for the helium atom ... 

The first floor of the atom can hold only two electrons. When two hydrogen atoms, each with one electron, come together, each nucleus is in effect sharing two electrons, and as far as each nucleus is concerned, its shell is filled. The helium atom already contains two electrons its first floor is filled. That is why helium is unreactive. Wlien two helium atoms collide, rather than stick together, they simply bounce apart. (.See Table 1.)... 

Since the nucleus has positive charge, it attracts electrons (each with negative charge). If a nucleus attracts the number of electrons just equal to the nuclear charge, an electrically neutral atom is formed. Consider a nucleus containing two protons, a helium nucleus. When the helium atom has two electrons as well (2— charge), an electrically neutral helium atom results ... 

Figure 16-3D shows the simplified representation of the interaction of two helium atoms. This time each helium atom is crosshatched before the two atoms approach. This is to indicate there are already two electrons in the Is orbital. Our rule of orbital occupancy tells us that the Is orbital can contain only two electrons. Consequently, when the second helium atom approaches, its valence orbitals cannot overlap significantly. The helium atom valence electrons fill its valence orbitals, preventing it from approaching a second atom close enough to share electrons. The helium atom forms no chemical bonds. ... 

To make matters worse, the use of a uniform gas model for electron density does not enable one to carry out good calculations. Instead a density gradient must be introduced into the uniform electron gas distribution. The way in which this has been implemented has typically been in a semi-empirical manner by working backwards from the known results on a particular atom, usually the helium atom (Gill, 1998). It has thus been possible to obtain an approximate set of functions which often serve to give successful approximations in other atoms and molecules. As far as I know, there is no known way of yet calculating, in an ab initio manner, the required density gradient which must be introduced into the calculations. 

A. Historical Development and Basic Ideas. Hyllcraas Work on the Helium Atom.248... 

Mainly for considerations of space, it has seemed desirable to limit the framework of the present review to the standard methods for treating correlation effects, namely the method of superposition of configurations, the method with correlated wave functions containing rij and the method using different orbitals for different spins. Historically these methods were developed together as different branches of the same tree, and, as useful tools for actual applications, they can all be traced back to the pioneering work of Hylleraas carried out in 1928-30 in connection with his study of the ground state of the helium atom. 

To test the accuracy and convenience of the method of superposition of configurations, the problem of the ground state of the helium atom has recently been reexamined by several authors. According to Hylleraas (1928), the total wave function may be expressed in the form... 

TABLE VI. Ground State of the Helium Atom Obtained by Superposition of Configurations ... 

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