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Solving the Helium Atom Or: Why does Chemistry Exist? Matthew Reed Math 164 – Scientific Computing May 4, 2007 1. Motivation One of the first real-world calculations demonstrated in any introductory quantum mechanics class is the hydrogen atom. It is surprisingly straightforward to find the ground state. A helium atom is an atom of the chemical element helium.Helium is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with either one or two neutrons, depending on the isotope, held together by the strong force. To discuss the next-simplest system: the Helium atom. In this situation, we have two. Electrons – with coordinates r and r. 2 –orbitinganucleus with charge Z =2locatedatthe point R. Now, for the hydrogen atom we were able to ignore the motion of the nucleus by trans-formingo t the center of mass.

Exercises Up:Identical Particles Previous:Two-Electron System
Consider the helium atom, which is a good example of a two-electron system. The Hamiltonian is written
where , , , and . Suppose that the final term on the right-hand side of the above expression were absent. In this case, the overall spatial wavefunction can be formed from products of hydrogen atom wavefunctions calculated with, instead of . Each of these wavefunctions is characterized by the usual triplet of quantum numbers, , , and . Now, the total spin of the system is a constant of the motion (since obviously commutes with the Hamiltonian), so the overall spin state is either the singlet or the triplet state. The corresponding spatial wavefunction is symmetric in the former case, andantisymmetric in the latter. Suppose that one electron has the quantum numbers , , whereas the other has the quantum numbers , , . The corresponding spatial wavefunction is

(1082)

where the plus and minus signs correspond to the singlet and triplet spin states, respectively. Here, is a standard hydrogen atom wavefunction (calculated with ). For the special case in which the two sets of spatial quantum numbers, , , and , are the same, the triplet spin state does not exist (because the associated spatial wavefunction is null). Hence, onlysinglet spin state is allowed, and the spatial wavefunction reduces to
In particular, the ground state (, , ) can only exist as a singlet spin state (i.e., a state of overall spin 0), and has thespatial wavefunction

(1084)

where is the Bohr radius. This follows because
The energy of this stateis

(1086)

where is the ground state energy of a hydrogen atom. In the above expression, the factor of comes from the fact thatthere are two electrons in a helium atom.

The above estimate for the ground state energy of a helium atom completely ignores the final term on the right-hand side of Equation (1081),which describes the mutual interaction between the two electrons. We can obtain a better estimate for the ground state energy by treating (1084)as the unperturbed wavefunction, and as a perturbation. According to standard first-order perturbation theory, thecorrection to the ground state energy is

This can be written

(1088)

since . Now,
where () is the larger (smaller) of and , and is the angle subtended between and . Moreover, the so-called addition theorem for spherical harmonics states that

(1090)

However,
so we obtain

(1092)

Here, and , and . Thus, our improved estimate for the ground state energy of the helium atom is

Atomic Mass Of Helium

This is much closer to the experimental value of than our previous estimate.

Consider an excited state of the helium atom in which one electron is in the ground state, while the other is in a state characterized by the quantumnumbers , , . We can write the energy of this state as

(1094)

where is the energy of a hydrogen atom electron whose quantum numbers are , , . According to first-order perturbation theory, is the expectation value of . It follows from(1082) (with and ) that
where

(1096)

(1097)

Here, the plus sign in (1095) corresponds to the spin singlet state, whereas the minus sign corresponds to the spin triplet state. Theintegral --which is known as the direct integral--is obviously positive. The integral --which is known as the

Helium Atom Picture

exchange integral--canbe shown to also be positive. Hence, we conclude that in excited states of helium the spin singlet state has a higher energythan the spin triplet state. Incidentally, helium in the spin singlet state is known as para-helium, whereas helium in the triplet state iscalled ortho-helium. As we have seen, for the ground state, only para-helium is possible.

The fact that para-helium energy levels lie slightly above corresponding ortho-helium levels is interesting because our original Hamiltonian does notdepend on spin. Nevertheless, there is a spin dependent effect--i.e., a helium atom has a lower energy when its electrons possess parallel spins--as a consequence ofFermi-Dirac statistics. To be more exact, the energy is lower in the spin triplet state because the corresponding spatial wavefunction is antisymmetric,causing the electrons to tend to avoid one another (thereby reducing their electrostatic repulsion).

Helium Atomic Structure

Helium Atomic No

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Helium Atomic Number And Mass

Identical Particles Previous:Two-Electron SystemRichard Fitzpatrick2013-04-08