In this chapter we study the energy levels of two-electron atoms, principally helium. A two-electron atom is a system of three interacting particles, a problem extremely difficult to analyze within classical mechanics. Since the mass of the nucleus is much larger than that of the electrons, the nuclear motion may be ignored and the problem reduces to two interacting particles moving in an external potential. This is still difficult to solve within classical mechanics. The old Bohr-Sommerfeld quantum mechanics also had little success with this problem. Thus it was a critical triumph for wave mechanics to provide correct calculations of the energy levels of two-electron atoms. Indeed it is remarkable that very little effort is required, as we shall see, to give a good estimate for the ground state energy. Further calculations for two-electron atoms provided six-figure agreement between theory and experiment, a striking proof of the validity of wave mechanics.
In this chapter we study the energy levels of two-electron atoms, principally helium. A two-electron atom is a system of three interacting particles, a problem extremely difficult to analyze within classical mechanics. Since the mass of the nucleus is much larger than that of the electrons, the nuclear motion may be ignored and the problem reduces to two interacting particles moving in an external potential. This is still difficult to solve within classical mechanics. The old Bohr-Sommerfeld quantum mechanics also had little success with this problem. Thus it was a critical triumph for wave mechanics to provide correct calculations of the energy levels of two-electron atoms. Indeed it is remarkable that very little effort is required, as we shall see, to give a good estimate for the ground state energy. Further calculations for two-electron atoms provided six-figure agreement between theory and experiment, a striking proof of the validity of wave mechanics.
The problem of interacting particles can usually be reduced in quantum mechanics to that of one particle, as can be done in classical mechanics. Normally, this problem is one of an electron orbiting around, or being affected by, a positive atomic core or a scattering center with the interaction governed by the Coulomb potential. In general, this is a multi-dimensional problem, and not one of the simpler one-dimensional problems with which we have been concerned in the previous chapters. Once we begin to treat multiple dimensions, then state degeneracy begins to arise more frequently, and the most common problem treated is that of the hydrogen atom vs. the helium atom. These extra dimensions provide more degrees of freedom and more complexity. In the hydrogen atom, there is only a single electron, and the problem is solvable. On the other hand, adding a second electron for the helium atom makes the problem extremely difficult to solve, as it has become a three-body problem in which the interactions are difficult to handle.
Although many of the principles of quantum mechanics can be adequately illustrated by considering systems that consist of only one particle subject to external forces, there are a number of important phenomena that are manifest only in systems containing two or more particles, and these will be the subject of the present chapter. We shall begin with some general statements and go on to consider the case of two interacting particles subject to no external forces, when we shall find that, in this case, the problem can be separated into one describing the behaviour of the centre of mass of the system and another describing the relative motion. We shall then consider the case of two noninteracting particles and show that these can often be treated independently as would be expected. In the case of indistinguishable particles, however, we shall find that a symmetry is imposed on the wave function that ensures that the behaviour of such particles is coupled even when they do not interact. We shall introduce the problem of interacting systems containing more than two identical particles, explaining what is meant by “fermions” and “bosons” and how their properties compare. These results will be illustrated by discussing the energy levels and optical spectra of the helium atom, starting with the approximation that the two electrons do not interact and extending the discussion to the realistic case using perturbation theory. The chapter closes with a consideration of the effects of particle indistinguishability on scattering theory.
Abstract A theory of the helium atom was developed by Heisenberg, following his famous principle of “resonance.” His work is based upon Schrodinger’s equation, and in order to allow for the spin of the electrons he introduces extra energy terms, expressing the classical energy of two appropriate magnets. It is the perturbation by this spin energy which produces the triplet separations. Heisenberg’s calculation of these separations is open to three criticisms: (1) He assumes that certain simple forms are correct first approximations to the wave-functions; whereas, owing to degeneracy, it is necessary to take linear combinations of these forms. (2) He estimates the mean values of the spin energy by means of a model built up of precessing vectors. (3) He neglects the radius of the inner orbit in comparison with that of the outer orbit, even when the principal quantum number of the latter is only 2; this is the least satisfactory of several necessary simplifications. Schrödinger’s equation is likely to be supplanted as the foundation of wave-mechanics by the equation recently put forward by Dirae. This fits the spin of the electron neatly into the theory of relativity, and produces the doublets of the hydrogen-like atom in a beautiful manner. Dirac’s q-number theory has been translated into wave-mechanics by Darwin. Apart altogether from the above criticisms of Heisenberg, it seemed expedient to proceed to the theory of an atom with two electrons on the basis of the new equation. Calculations which are independent of the spin, such as the approximate energylevels, and the separation between ortho- and para-terms, are the same on either theory, and are not the concern of this paper. We deal here with spin effects, such as the fine structure of the triplets and intercombinations between ortho- and para-states.
Abstract This chapter begins by defining the electronic structure problem of matter. Following the introduction of the system of atomic units, the electron-nuclear many-body problem is then formulated as a Schrödinger equation of the coupled electron-nuclear many-body system. The electronic and nuclear degrees of freedom are then decoupled using the Born-Oppenheimer approximation, which defines the electronic many-body problem. An illustration of potential-energy curves of a diatomic molecule is given, as well as an example of an interacting and a noninteracting two-electron system such as the helium atom. The chapter concludes by discussing the N-electron wave function, arguing that it becomes meaningless when the number of electrons, N, becomes large, due to the exponential increase of complexity.
In 1926, Schrödinger [365] recognized that the variational theory of elliptical differential equations with fixed boundary conditions could produce a discrete eigenvalue spectrum in agreement with the energy levels of Bohr's model of the hydrogen atom. This conceptually startling amalgam of classical ideas of particle and field turned out to be correct. Within a few years, the new wave mechanics almost completely replaced the ad hoc quantization of classical mechanics that characterized the “old” quantum theory initiated by Bohr. Although the matrix mechanics of Heisenberg was soon shown to be logically equivalent, the variational wave theory became the standard basis of methodology in the physics of electrons. The nonrelativistic Schrödinger theory is readily extended to systems of N interacting electrons. The variational theory of finite N -electron systems (atoms and molecules) is presented here. In this context, several important theorems that follow from the variational formalism are also derived.
Models for the Helium Atom.---(1) Bohr's model is unsatisfactory because it gives too great a value for the ionizing potential and is not in accord with the optical and magnetic properties of helium. Now the chemical evidence suggests that each electron in an atom has its own separate orbit, and that these orbits are closely interrelated. Accordingly two new models are considered. (2) In the double circle model the two electrons are assumed to move in two separate, parallel, circular orbits. The model, however, is unstable, for the ionizing potential, computed by applying the quantum theory, comes out negative. Another objection is that the magnetic moment is not zero. (3) In the semi-circular model each electron is assumed to oscillate back and forth along an approximately semi-circular path in accord with classical mechanics, each being brought to rest at the end of its path by the repulsion of the other. Assuming the maximum angular momentum of each equal to $\frac{h}{2\ensuremath{\pi}}$, the absolute dimensions of the model were computed. The total energy comes out less than for the Bohr model, and the ionizing potential, 25.62 volts, agrees closely with the experimental value. The magnetic moment is zero.Application of the Quantum Theory to Coupled Electrons.---The success of the semi-circular model in giving the correct value for the ionizing potential of helium suggests that in the case of coupled electrons the quantum theory should be applied not to the momentum of the individual electrons according to the relation $\ensuremath{\int}\mathrm{pdq}=\frac{h}{2\ensuremath{\pi}}$, but rather to the momentum which, by being relayed from one electron to another, passes in each direction around the nucleus.