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 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.
Historically, the first attempt to obtain the helium spectrum from quantum mechanics was done by Albrecht Unsöld in 1927. Egil Hylleraas obtained an accurate approximation in 1929. Its success was considered to be one of the earliest signs of validity of Schrödinger's wave mechanics.

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 as
where is the reduced mass of an electron with respect to the nucleus, and are the electron-nucleus distance vectors and. It operates not in normal space, but in a 6-dimensional configuration space . The nuclear charge, is 2 for helium. In the approximation of an infinitely heavy nucleus, we have and the mass polarization term disappears, so that in operator language, the Hamiltonian simplifies to:
The wavefunction belongs to the tensor product of combined spin states and combined spatial wavefunctions, and since this Hamiltonian only acts on spatial wavefunctions, we can neglect spin states until after solving the spatial wavefunction. This is possible since, for any general vector, one has that where is a combined spatial wavefunction and is the combined spin component. The Hamiltonian operator, since it only acts on the spatial component, gives the eigenvector equation:
which implies that one should find solutions for where is a general combined spatial wavefunction. This energy, however, is not degenerate with multiplicity given by the dimension of the space of combined spin states because of a symmetrization postulate, which requires that physical solutions for identical fermions should be totally antisymmetric, imposing a restriction on the choice of based on solutions. Hence the solutions are of the form: where is the energy eigenket spatial wavefunction and is a spin wavefunction such that is antisymmetric and is merely some superposition of these states.
Since the Hamiltonian is independent of spin, it commutes with all spin operators. Since it is also rotationally invariant, the total x, y or z component of angular momentum operator also commutes with the Hamiltonian. From these commutation relations, and also commutes with the Hamiltonian which implies that energy is independent of and. Although the purely spatial form of the Hamiltonian implies that the energy is independent of, this would only be true in the absence of symmetrization postulate. Due to the symmetrization postulate, the choice of will influence the type of wavefunction required by symmetrization postulate which would in turn influence the energy of the state.
Other operators that commute with the Hamiltonian are the spatial exchange operator and the parity operator. However a good combination of mutually commuting operators are:,,, and. Hence the final solutions are given as:
where the energy is fold degenerate. For electrons, the total spin can have values of 0 or 1. A state with the quantum numbers: principal quantum number, total spin, angular quantum number and total angular momentum is denoted by.
States corresponding to, are called parahelium and are called orthohelium . Since the spin exchange operator can be expressed in terms of dot product of spin vectors, eigenkets of spin exchange operators are also eigenkets of. Hence parahelium can also be said to be the spin anti-symmetric state or orthohelium to be spin symmetric state.
The singlet state is given as:
and triplet states are given as:
as per symmetrization and total spin number requirement. It is observed that triplet states are symmetric and singlet states are antisymmetric. Since the total wavefunction is antisymmetric, a symmetric spatial wavefunction can only be paired with antisymmetric wavefunction and vice versa. Hence orthohelium has a symmetric spin wavefunction but an antisymmetric spatial wavefunction and parahelium has an antisymmetric spin wavefunction but a symmetric spatial wavefunction. Hence the type of wavefunction of each state is given above. The degeneracy solely comes from this spatial wavefunction. Note that for, there is no degeneracy in spatial wavefunction.
Alternatively, a more generalized representation of the above can be provided without considering the spatial and spin parts separately. This method is useful in situations where such manipulation is not possible, however, it can be applied wherever needed. Since the spin part is tensor product of spin Hilbert vector spaces, its basis can be represented by tensor product of each of the set, with each of the set,. Note that here but are in fact orthogonal. In the considered approximation, the wave function can be represented as a second order spinor with 4 components, where the indices describe the spin projection of both electrons in this coordinate system. The usual normalization condition,, follows from the orthogonality of all elements. This general spinor can be written as 2×2 matrix:
If the Hamiltonian had been spin dependent, we would not have been able to treat each these components independently as shown previously since the Hamiltonian need not act in the same manner for all four components.
The matrix can also be represented as a 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 are spherical vector tensor representations of an ordinary space vector, with the three components:
Since all spin interaction terms between the four components of in the above Hamiltonian are neglected, the four Schrödinger equations can be solved independently. This is identical to the previously discussed method of finding spatial wavefunction eigenstates independently of the spin states, here spatial wavefunctions of different spin states correspond to the different components of the matrix.
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 spatial function and orthohelium is the triplet state with an antisymmetric spatial function.

Approximation methods

Following from the above approximation, effectively reducing three body problem to two body problem, we have:
This Hamiltonian for helium with two electrons can be written as a sum of two terms:
where the zero-order unperturbed Hamiltonian is
while the perturbation term:
is the electron-electron interaction. is just the sum of the two hydrogenic Hamiltonians:
whereare independent Coulomb field Hamiltonian of each electron. Since the unperturbed Hamiltonian is a sum of two independent Hamiltonians, the wavefunction must be of form where and are eigenkets of and respectively. However, the spatial wavefunction of the form need not correspond to physical states of identical electrons as per the symmetrization postulate. Thus, to obtain physical solutions symmetrization of the wavefunctions and is carried out.
The proper wave function then must be composed of the symmetric and antisymmetric linear combinations:or for the special cases of :.
This explains the absence of the state for orthohelium, where consequently is the metastable ground state.
Note that all wavefunction obtained thus far cannot be separated into wavefunctions of each particle i.e. the wavefunctions are always in superposition of some kind. In other words, one cannot completely determine states of particle 1 and 2, or measurements of all details, of each electrons cannot be made on one particle without affecting the other. This follows since the wavefunction is always a superposition of different states where each electron has unique. This is in agreement with Pauli exclusion principle.
We can infer from these wavefunctions that.
The corresponding energies are:
A good theoretical descriptions of helium including the perturbation term can be obtained within the Hartree–Fock and Thomas–Fermi approximations.
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 with high numerical accuracy. For example, the ground state energy of helium has been computed to 40 digits, , but the difference between the value and experiment is not understood.

Ground state of Helium: Perturbation method

Since ground state corresponds to state, there can only be one representation of such wavefunction whose spatial wavefunction is:
We note that the ground state energy of unperturbed helium atom as:Which is 30% larger than experimental data.
We can find the first order correction in energy due to electron repulsion in Hamiltonian :
The energy for ground state of helium in first order becomes compared to its experimental value of. A better approximation for ground state energy is obtained by choosing better trial wavefunction in variational method.