A helium atom is an
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
of the chemical element
helium
Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
. Helium is composed of
two electrons bound by the
electromagnetic force
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
to a nucleus containing two protons along with either one or two neutrons, depending on the
isotope
Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers (mass numbers) ...
, held together by the
strong force
The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the n ...
. Unlike for
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...
, a
closed-form solution
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
to the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
for the helium atom has not been found. However, various approximations, such as the
Hartree–Fock method
In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.
The Hartree–Fock method often ...
, can be used to estimate the
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
energy and
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
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
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
. The
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified 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
. 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
atomic units
The Hartree atomic units are a system of natural units of measurement which is especially convenient for atomic physics and computational chemistry calculations. They are named after the physicist Douglas Hartree. By definition, the following four ...
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 (
Pauli approximation) the wave function is a second order
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
with 4 components
, where the indices
describe the spin projection of both electrons (''z''-direction up or down) in some coordinate system. It has to obey the usual normalization condition
. This general spinor can be written as 2×2 matrix
and consequently also as linear combination of any given basis of four orthogonal (in the vector-space of 2×2 matrices) constant matrices
with scalar function coefficients
as
. A convenient basis consists of one anti-symmetric matrix (with total spin
, corresponding to a
singlet state
In quantum mechanics, a singlet state usually refers to a system in which all electrons are paired. The term 'singlet' originally meant a linked set of particles whose net angular momentum is zero, that is, whose overall spin quantum number s=0. A ...
)
and three symmetric matrices (with total spin
, corresponding to a
triplet state
In quantum mechanics, a triplet is a quantum state of a system with a spin of quantum number =1, such that there are three allowed values of the spin component, = −1, 0, and +1.
Spin, in the context of quantum mechanics, is not a mechanical ...
)
It is easy to show, that the singlet state is invariant under all rotations (a scalar entity), while the triplet can be mapped to an ordinary space vector
, with the three components
Since all spin interaction terms between the four components of
in the above (scalar) Hamiltonian are neglected (e.g. an external magnetic field, or
relativistic effects
Relativistic quantum chemistry combines relativistic mechanics with quantum chemistry to calculate elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example is an explanation for the color of ...
, like
angular momentum coupling
In quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. For instance, the orbit and spin of a single particle can interact t ...
), the four
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
s can be solved independently.
The spin here only comes into play through the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
, which for fermions (like electrons) 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 (orthogonal and normalized) one-electron eigenfunctions
:
or for the special cases of
(both electrons have identical quantum numbers, parahelium only):
.
The total energy (as eigenvalue of
) is then for all cases
(independent of the symmetry).
This explains the absence of the
state (with
) for orthohelium, where consequently
(with
) is the metastable ground state.
(A state with the quantum numbers: principal quantum number
, total spin
, angular quantum number
and total angular momentum
is denoted by
.)
If the electron-electron interaction term
is included, the Schrödinger equation is non separable.
However, even if it is neglected, all states described above (even with two identical quantum numbers, like
with
) 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 (see below).
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
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
" for helium and other few electron systems can be solved quite accurately. For example, the
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than 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.
Perturbation method
The
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified 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. is just the sum of the two hydrogenic Hamiltonians:
where
, the energy eigenvalues and
, the corresponding eigenfunctions of the hydrogenic Hamiltonian will denote the normalized energy
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s and the normalized eigenfunctions. So:
where
Neglecting the electron-electron repulsion term, the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
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 (in
atomic units
The Hartree atomic units are a system of natural units of measurement which is especially convenient for atomic physics and computational chemistry calculations. They are named after the physicist Douglas Hartree. By definition, the following four ...
, hereafter a.u.):
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
In quantum mechanics, identical particles (also called indistinguishable or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, ...
. 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
In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi- fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two elect ...
.
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 quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
, 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. (≅54.4 eV). The experimental values are ''E''
0 = −2.90 a.u. (≅ −79.0 eV) and ''V''
p = 0.90 a.u. (≅ 24.6 eV).
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 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
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 is
where is a screening constant and the quantity is the effective charge. The potential is a Coulomb interaction, so the corresponding individual electron energies are given (in a.u.) by
and the corresponding wave function is given by
If was 1.70, that would make the expression above for the ground state energy agree with the experimental value ''E''
0 = −2.903 a.u. of the ground state energy of helium. Since 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 electric charge.
The variational method
To obtain a more accurate energy the
variational principle
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those func ...
can be applied to the electron-electron potential using the wave function
After 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 ''V''
ee 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, 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
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and speci ...
corrections to get full agreement with experiment to spectroscopic accuracy.
Experimental value of ionization energy
Helium's first
ionization energy
Ionization, or Ionisation is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged atom or molecule i ...
is −24.587387936(25) eV.
This value was derived by experiment.
The theoretic value of Helium atom's second ionization energy is −54.41776311(2) eV.
[ The total ground state energy of the helium atom is −79.005151042(40) eV,][ or −2.90338583(13) ]Atomic units
The Hartree atomic units are a system of natural units of measurement which is especially convenient for atomic physics and computational chemistry calculations. They are named after the physicist Douglas Hartree. By definition, the following four ...
a.u., which equals −5.80677166 (26) Ry.
See also
* Isotopes of helium
* Araki–Sucher correction
In atomic, molecular and optical physics, the Araki–Sucher correction is a leading-order correction to the energy levels of atoms and molecules due to effects of quantum electrodynamics (QED). It is named after Huzihiro Araki and Joseph Sucher, ...
* Hydrogen molecular ion
The dihydrogen cation or hydrogen molecular ion is a cation (positive ion) with formula . It consists of two hydrogen nuclei (protons) sharing a single electron. It is the simplest molecular ion.
The ion can be formed from the ionization of a n ...
* Lithium atom
A lithium atom is an atom of the chemical element lithium. Stable lithium is composed of three electrons bound by the electromagnetic force to a nucleus containing three protons along with either three or four neutrons, depending on the isotope, ...
* List of quantum-mechanical systems with analytical solutions
Much insight in quantum mechanics can be gained from understanding the closed-form solutions to the time-dependent non-relativistic Schrödinger equation. It takes the form
:
\hat \psi\left(\mathbf, t\right) =
\left - \frac \nabla^2 + V\left(\ma ...
* Quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
* Quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
* Quantum states
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
*
* "Helium atom" on Wikiversity
References
{{DEFAULTSORT:Helium Atom
Atoms
Quantum models
Helium