Helium atom
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. Unlike for hydrogen, a closed-form solution to the Schrödinger equation for the helium atom has not been found. However, various approximations, such as the Hartree–Fock method, can be used to estimate the ground state energy and wavefunction of the atom.
Introduction
The quantum mechanical description of the helium atom is of special interest, because it is the simplest multi-electron system and can be used to understand the concept of quantum entanglement. The Hamiltonian of helium, considered as a three-body system of two electrons and a nucleus and after separating out the centre-of-mass motion, can be written aswhere is the reduced mass of an electron with respect to the nucleus, and are the electron-nucleus distance vectors and. The nuclear charge, is 2 for helium. In the approximation of an infinitely heavy nucleus, we have and the mass polarization term disappears. In atomic units the Hamiltonian simplifies to
It is important to note, that it operates not in normal space, but in a 6-dimensional configuration space . In this approximation the wave function is a second order spinor with 4 components, where the indices describe the spin projection of both electrons in some coordinate system. It has to obey the usual normalization condition. This general spinor can be written as 2x2 matrix and consequently also as linear combination of any given basis of four orthogonal constant matrices with scalar function coefficients
as. A convenient basis consists of one anti-symmetric matrix
and three symmetric matrices
It is easy to show, that the singlet state is invariant under all rotations, while the triplet can be mapped to an ordinary space vector,
with the three components
, and.Since all spin interaction terms between the four components of in the above Hamiltonian are neglected, the four Schrödinger equations can be solved independently.
The spin here only comes into play through the Pauli exclusion principle, which for fermions requires antisymmetry under simultaneous exchange of spin and coordinates
Parahelium is then the singlet state with a symmetric function
and orthohelium is the triplet state with an antisymmetric function.
If the electron-electron interaction term is ignored, both spatial functions can be written as linear combination of two arbitrary
one-electron eigenfunctions :
or for the special cases
of :.
The total energy is then for all cases .
This explains the absence of the state for orthohelium, where
consequently is the metastable ground state.
If the electron-electron interaction term is included, the Schrödinger equation is non separable.
However, also if is neglected, all states described above
cannot be written as a product of one-electron wave functions:
— the wave function is entangled.
One cannot say, particle 1 is in state 1 and the other in state 2, and measurements cannot be made on one particle without affecting the other.
Nevertheless, quite good theoretical descriptions of helium can be obtained within the Hartree–Fock and Thomas–Fermi approximations.
Hartree–Fock method
The Hartree–Fock method is used for a variety of atomic systems. However it is just an approximation, and there are more accurate and efficient methods used today to solve atomic systems. The "many-body problem" for helium and other few electron systems can be solved quite accurately. For example, the ground state of helium is known to fifteen digits. In Hartree–Fock theory, the electrons are assumed to move in a potential created by the nucleus and the other electrons. The Hamiltonian for helium with two electrons can be written as a sum of the Hamiltonians for each electron:where the zero-order unperturbed Hamiltonian is
while the perturbation term:
is the electron-electron interaction. H0 is just the sum of the two hydrogenic Hamiltonians:
where
Eni, the energy eigenvalues and, the corresponding eigenfunctions of the hydrogenic Hamiltonian will denote the normalized energy eigenvalues and the normalized eigenfunctions. So:
where
Neglecting the electron-electron repulsion term, the Schrödinger equation for the spatial part of the two-electron wave function will reduce to the 'zero-order' equation
This equation is separable and the eigenfunctions can be written in the form of single products of hydrogenic wave functions:
The corresponding energies are :
Note that the wave function
An exchange of electron labels corresponds to the same energy. This particular case of degeneracy with respect to exchange of electron labels is called exchange degeneracy. The exact spatial wave functions of two-electron atoms must either be symmetric or antisymmetric with respect to the interchange of the coordinates and of the two electrons. The proper wave function then must be composed of the symmetric and antisymmetric linear combinations:
This comes from Slater determinants.
The factor normalizes. In order to get this wave function into a single product of one-particle wave functions, we use the fact that this is in the ground state. So. So the will vanish, in agreement with the original formulation of the Pauli exclusion principle, in which two electrons cannot be in the same state. Therefore, the wave function for helium can be written as
Where and use the wave functions for the hydrogen Hamiltonian. For helium, Z = 2 from
where E = −4 a.u. which is approximately −108.8 eV, which corresponds to an ionization potential V = 2 a.u.. The experimental values are E = −2.90 a.u. and V = 0.90 a.u..
The energy that we obtained is too low because the repulsion term between the electrons was ignored, whose effect is to raise the energy levels. As Z gets bigger, our approach should yield better results, since the electron-electron repulsion term will get smaller.
So far a very crude independent-particle approximation has been used, in which the electron-electron repulsion term is completely omitted. Splitting the Hamiltonian showed below will improve the results:
where
and
V is a central potential which is chosen so that the effect of the perturbation is small. The net effect of each electron on the motion of the other one is to screen somewhat the charge of the nucleus, so a simple guess for V is
where S is a screening constant and the quantity Ze is the effective charge. The potential is a Coulomb interaction, so the corresponding individual electron energies are given by
and the corresponding wave function is given by
If Ze was 1.70, that would make the expression above for the ground state energy agree with the experimental value E0 = −2.903 a.u. of the ground state energy of helium. Since Z = 2 in this case, the screening constant is S = 0.30. For the ground state of helium, for the average shielding approximation, the screening effect of each electron on the other one is equivalent to about of the electronic charge.
Thomas–Fermi method
Not long after Schrödinger developed the wave equation, the Thomas–Fermi model was developed. Density functional theory is used to describe the particle density, and the ground state energy E, where N is the number of electrons in the atom. If there are a large number of electrons, the Schrödinger equation runs into problems, because it gets very difficult to solve, even in the atom's ground states. This is where density functional theory comes in. Thomas–Fermi theory gives very good intuition of what is happening in the ground states of atoms and molecules with N electrons.The energy functional for an atom with N electrons is given by:
Where
The electron density needs to be greater than or equal to 0,, and is convex.
In the energy functional, each term holds a certain meaning. The first term describes the minimum quantum-mechanical kinetic energy required to create the electron density for an N number of electrons. The next term is the attractive interaction of the electrons with the nuclei through the Coulomb potential. The final term is the electron-electron repulsion potential energy.
So the Hamiltonian for a system of many electrons can be written:
For helium, N = 2, so the Hamiltonian is given by:
Where
yielding
From the Hartree–Fock method, it is known that ignoring the electron-electron repulsion term, the energy is 8E1 = −109 eV.
The variational method
To obtain a more accurate energy the variational principle can be applied to the electron-electron potential Vee using the wave functionAfter integrating this, the result is:
This is closer to the experimental value, but if a better trial wave function is used, an even more accurate answer could be obtained. An ideal wave function would be one that doesn't ignore the influence of the other electron. In other words, each electron represents a cloud of negative charge which somewhat shields the nucleus so that the other electron actually sees an effective nuclear charge Z that is less than 2. A wave function of this type is given by:
Treating Z as a variational parameter to minimize H. The Hamiltonian using the wave function above is given by:
After calculating the expectation value of and Vee the expectation value of the Hamiltonian becomes:
The minimum value of Z needs to be calculated, so taking a derivative with respect to Z and setting the equation to 0 will give the minimum value of Z:
This shows that the other electron somewhat shields the nucleus reducing the effective charge from 2 to 1.69. So we obtain the most accurate result yet:
Where again, E1 represents the ionization energy of hydrogen.
By using more complicated/accurate wave functions, the ground state energy of helium has been calculated closer and closer to the experimental value −78.95 eV. The variational approach has been refined to very high accuracy for a comprehensive regime of quantum states by G.W.F. Drake and co-workers as well as J.D. Morgan III, Jonathan Baker and Robert Hill using Hylleraas or Frankowski-Pekeris basis functions. One needs to include relativistic and quantum electrodynamic corrections to get full agreement with experiment to spectroscopic accuracy.