Hydrogen-like atom


A hydrogen-like atom is any atom or ion with a single electron. Examples of hydrogen-like atoms are H, He+, Li2+, Be3+ and so on, as well as any of their isotopes. These ions are isoelectronic with hydrogen and are sometimes called hydrogen-like ions. The non-relativistic Schrödinger equation and relativistic Dirac equation for the hydrogen atom and hydrogen-like atoms can be solved analytically, owing to the simplicity of the two-particle physical system. The one-electron wave function solutions are referred to as hydrogen-like atomic orbitals. Hydrogen-like atoms are of importance because their corresponding orbitals bear similarity to the hydrogen atomic orbitals.
The definition of hydrogen-like atoms can be extended to also include any system with only one valence electron. Examples such atoms include, but are not limited to, all alkali metals such as Rb and Cs and singly ionized alkaline earth metals such as Ca+ and Sr+. In such a case, the hydrogen-like atom includes a positively charged core consisting of the atomic nucleus and any core electrons, as well as a single valence electron. Because helium is common in the universe, the spectroscopy of singly ionized helium is important in EUV astronomy, for example, of DO white dwarf stars.
Other systems may also be referred to as "hydrogen-like atoms", such as muonium, positronium, certain exotic atoms, or Rydberg atoms.

Rydberg atoms

Highly excited states of neutral atoms are well described in terms of one electron around a nucleus of a single positive charge resembling a hydrogen atom. These states are called Rydberg atoms. They have important applications in astrophysics, including in the dynamics of the primordial gas of the Big Bang.
The hydrogen-like approximation becomes poor for Rydberg atoms, even if they have only one valence electron, because the wavefunction of the valence electron is non-negligible in the ion core where the core electrons can no longer effectively screen the outer electrons.
As discussed below, the 1/r potential in the hydrogen atom leads to an electron binding energy given by
where is the Rydberg constant, is the Planck constant, is the speed of light and is the principal quantum number. For alkali atoms with small orbital angular momentum, the spectrum is still described well by the Rydberg formula with an angular momentum dependent quantum defect, :
The largest shifts occur when the orbital angular momentum is zero and these are shown in the table for the alkali metals:
ElementConfiguration
Li2s1.590.41
Na3s1.631.37
K4s1.772.23
Rb5s1.813.19
Cs6s1.874.13

Schrödinger solution

In the solution to the Schrödinger equation, which is non-relativistic, hydrogen-like atomic orbitals are eigenfunctions of the one-electron angular momentum operator and its z-component. A hydrogen-like atomic orbital is uniquely identified by the values of the principal quantum number, the angular momentum quantum number, and the magnetic quantum number. The energy eigenvalues do not depend on or, but solely on. To these must be added the two-valued spin quantum number, setting the stage for the Aufbau principle. This principle restricts the allowed values of the four quantum numbers in electron configurations of more-electron atoms. In hydrogen-like atoms all degenerate orbitals of fixed and, and varying between certain values form an atomic shell.
The Schrödinger equation of atoms or ions with more than one electron has not been solved analytically, because of the computational difficulty imposed by the Coulomb interaction between the electrons. Numerical methods must be applied in order to obtain wavefunctions or other properties from quantum mechanical calculations. Due to the spherical symmetry, the total angular momentum of an atom is a conserved quantity. Many numerical procedures start from products of atomic orbitals that are eigenfunctions of the one-electron operators and. The radial parts of these atomic orbitals are sometimes numerical tables or are sometimes Slater orbitals. By angular momentum coupling many-electron eigenfunctions of are constructed.
In quantum chemical calculations hydrogen-like atomic orbitals cannot serve as an expansion basis, because they are not complete. The non-square-integrable continuum states must be included to obtain a complete set, i.e., to span all of one-electron Hilbert space. This was observed as early as 1928 by E. A. Hylleraas, and later by Harrison Shull and Per-Olov Löwdin.
In the simplest model, the atomic orbitals of hydrogen-like atoms/ions are solutions to the Schrödinger equation in a spherically symmetric potential. In this case, the potential term is the potential given by Coulomb's law:
where
After writing the wave function as a product of functions:
, where are spherical harmonics, we arrive at the following Schrödinger equation:
where is, approximately, the mass of the electron, and is the reduced Planck constant.
Different values of give solutions with different angular momentum, where is the quantum number of the orbital angular momentum. The magnetic quantum number is the projection of the orbital angular momentum on the z-axis. See for the steps leading to the solution of this equation.

Non-relativistic wavefunction and energy

In addition to and, a third integer, emerges from the boundary conditions placed on. The functions and that solve the equations above depend on the values of these integers, called quantum numbers. It is customary to subscript the wave functions with the values of the quantum numbers they depend on. The final expression for the normalized wave function is:
where:
parity due to angular wave function is.

Quantum numbers

The quantum numbers, and are integers and can have the following values:
For a group-theoretical interpretation of these quantum numbers, see . Among other things, this article gives group-theoretical reasons why and.

Angular momentum

Each atomic orbital is associated with an angular momentum. It is a vector operator, and the eigenvalues of its square are given by:
The projection of this vector onto an arbitrary direction is quantized. If the arbitrary direction is labelled 'z', the quantization is given by:
where is restricted as described above. Note that and commute and have a common eigenstate, which is in accordance with Heisenberg's uncertainty principle. Since and do not commute with, it is not possible to find a state that is an eigenstate of all three components simultaneously. Hence the values of the x- and y-components are not sharp, but are given by a probability function of finite width. The fact that the x- and y-components are not well-determined, implies that the direction of the angular momentum vector is not well determined either, although its component along the z-axis is sharp.
These relations do not give the total angular momentum of the electron. For that, electron spin must be included.
This quantization of angular momentum closely parallels that proposed by Niels Bohr in 1913, with no knowledge of wavefunctions.

Including spin–orbit interaction

In a real atom, the spin of a moving electron can interact with the electric field of the nucleus through relativistic effects, a phenomenon known as spin–orbit interaction. When one takes this coupling into account, the spin and the orbital angular momentum are no longer conserved, which can be pictured by the electron precessing. Therefore, one has to replace the quantum numbers, and the projection of the spin by quantum numbers that represent the total angular momentum, and, as well as the quantum number of parity.
See the next section on the Dirac equation for a solution that includes the coupling.

Solution to Dirac equation

In 1928 in England Paul Dirac found an equation that was fully compatible with special relativity. The equation was solved for hydrogen-like atoms the same year by the German Walter Gordon. Instead of a single function as in the Schrödinger equation, one must find four complex functions that make up a bispinor. The first and second functions correspond to spin "up" and spin "down" states, as do the third and fourth components.
The terms "spin up" and "spin down" are relative to a chosen direction, conventionally the z-direction. An electron may be in a superposition of spin up and spin down, which corresponds to the spin axis pointing in some other direction. The spin state may depend on location.
An electron in the vicinity of a nucleus necessarily has non-zero amplitudes for the third and fourth components. Far from the nucleus these may be small, but near the nucleus they become large.

Quantum numbers

The eigenfunctions of the Hamiltonian, which means functions with a definite energy, have energies characterized not by the quantum number only, but by and a quantum number, the total angular momentum quantum number. The quantum number determines the sum of the squares of the three angular momenta to be . These angular momenta include both orbital angular momentum and spin angular momentum. The splitting of the energies of states of the same principal quantum number due to differences in is called fine structure. The total angular momentum quantum number ranges from to.
The orbitals for a given state can be written using two radial functions and two angle functions. The radial functions depend on both the principal quantum number and an integer, defined as:
where is the azimuthal quantum number that ranges from to. The angle functions depend on and on a quantum number, which ranges from to by steps of. The states are labeled using the letters S, P, D, F etc. to stand for states with equal to,,,, etc., with a subscript giving. For instance, the states for are given in the following table :
, F5/2F5/2F5/2F5/2F5/2F5/2
, D3/2D3/2D3/2D3/2
, P1/2P1/2
, S1/2S1/2
k = −2, ℓ = 1P3/2P3/2P3/2P3/2
k = −3, ℓ = 2D5/2D5/2D5/2D5/2D5/2D5/2
k = −4, ℓ = 3F7/2F7/2F7/2F7/2F7/2F7/2F7/2F7/2

These can be additionally labeled with a subscript giving. There are states with principal quantum number, of them with any allowed except the highest for which there are only. Since the orbitals having given values of and have the same energy according to the Dirac equation, they form a basis for the space of functions having that energy.

Energy

The energy, as a function of and, is:
Note that if were able to be more than then we would have a negative value inside the square root for the S1/2 and P1/2 orbitals, which means they would not exist. The Schrödinger solution corresponds to replacing the inner bracket in the second expression by. The accuracy of the energy difference between the lowest two hydrogen states calculated from the Schrödinger solution is about 9 ppm, whereas the accuracy of the Dirac equation for the same energy difference is about 3 ppm. The Schrödinger solution always puts the states at slightly higher energies than the more accurate Dirac equation. The Dirac equation gives some levels of hydrogen quite accurately, others less so. The modifications of the energy due to using the Dirac equation rather than the Schrödinger solution is of the order of, and for this reason is called the fine-structure constant.

General solution of Dirac equation

In the general case, the solution to the Dirac equation for quantum numbers,, and, is:
where the Ωs are columns of the two spherical harmonics functions shown to the right. signifies a spherical harmonic function:
in which is an associated Legendre polynomial.
Here is the behavior of some of these angular functions. The normalization factor is left out to simplify the expressions.
From these we see that in the S1/2 orbital, the top two components of have zero orbital angular momentum like Schrödinger S orbitals, but the bottom two components are orbitals like the Schrödinger P orbitals. In the P1/2 solution, the situation is reversed. In both cases, the spin of each component compensates for its orbital angular momentum around the z-axis to give the right value for the total angular momentum around the z-axis.
The two spinors obey the relationship:
To write the functions and ) let us define a scaled radius :
with
where is the energy given above. We also define as:
and are based on two generalized Laguerre polynomials of order and :
where is a normalization constant involving the gamma function:
is small compared to because when is positive the first terms dominate, and is big compared to, whereas when is negative the second terms dominate and is small compared to. Note that the dominant term is quite similar to corresponding the Schrödinger solution – the upper index on the Laguerre polynomial is slightly less, as is the power of . The exponential decay is slightly faster than in the Schrödinger solution.
The normalization factor makes the integral over all space of the square of the absolute value equal to.

Special case when ''k'' = −''n''

When, then and are:
With now reduced to:
Notice that because of the factor, is small compared to. Also notice that in this case, the energy is given by
and the radial decay constant by

1S orbital

Here is the 1S1/2 orbital, spin up, without normalization:
Note that is a little less than, so the top function is similar to an exponentially decreasing function of except that at very small it theoretically goes to infinity. But the value of the surpasses 10 only at a value of smaller than, which is very small unless is very large.
The 1S1/2 orbital, spin down, without normalization, comes out as:
We can mix these in order to obtain orbitals with the spin oriented in some other direction, such as:
which corresponds to the spin and angular momentum axis pointing in the x-direction. Adding times the "down" spin to the "up" spin gives an orbital oriented in the y-direction.

2P1/2 and 2S1/2 orbitals

To give another example, the 2P1/2 orbital, spin up, is proportional to:
Notice that when is small compared to the "S" type orbital dominates.
For the 2S1/2 spin up orbital, we have:
Now the first component is S-like and there is a radius near where it goes to zero, whereas the bottom two-component part is P-like.

Negative-energy solutions

In addition to bound states, in which the energy is less than that of an electron infinitely separated from the nucleus, there are solutions to the Dirac equation at higher energy, corresponding to an unbound electron interacting with the nucleus. These solutions are not normalizable, but solutions can be found which tend toward zero as goes to infinity. There are similar solutions with. These negative-energy solutions are just like positive-energy solutions having the opposite energy but for a case in which the nucleus repels the electron instead of attracting it, except that the solutions for the top two components switch places with those for the bottom two.
Negative-energy solutions to Dirac's equation exist even in the absence of a Coulomb force exerted by a nucleus. Dirac hypothesized that we can consider almost all of these states to be already filled. If one of these negative-energy states is not filled, this manifests itself as though there is an electron which is repelled by a positively-charged nucleus. This prompted Dirac to hypothesize the existence of positively-charged electrons, and his prediction was confirmed with the discovery of the positron.

Beyond Gordon's solution to the Dirac equation

The Dirac equation with a simple Coulomb potential generated by a point-like non-magnetic nucleus was not the last word, and its predictions differ from experimental results as mentioned earlier. More accurate results include the Lamb shift and hyperfine structure.