Hydrogen-like atom

A hydrogen-like ion is any atomic nucleus with one electron and thus is isoelectronic with hydrogen. Except for the hydrogen atom itself (which is neutral), these ions carry the positive charge e(Z-1), where Z is the atomic number of the atom. Examples of hydrogen-like ions are He⁺, Li²⁺, Be³⁺ and B⁴⁺. Because hydrogen-like ions are two-particle systems with an interaction depending only on the distance between the two particles, their (non-relativistic) Schrödinger equation can be solved in analytic form. The solutions are one-electron functions and are referred to as hydrogen-like atomic orbitals.

Hydrogen-like atomic orbitals are eigenfunctions of the one-electron angular momentum operator L and its z component L_z. The energy eigenvalues do not depend on the corresponding quantum numbers, but solely on the principal quantum number n. Hence, a hydrogen-like atomic orbital is uniquely identified by the values of: principal quantum number n, angular momentum quantum number l, and magnetic quantum number m. To this must be added the two-valued spin quantum number m_s = ±½ in application of 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 n and l, m and s varying between certain values (see below) form an atomic shell.

The Schrödinger equation of atoms or atomic 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 (approximate) wavefunctions or other properties from quantum mechanical calculations. Due to the spherical symmetry (of the Hamiltonian), the total angular momentum J of an atom is a conserved quantity. Many numerical procedures start from products of atomic orbitals that are eigenfunctions of the one-electron operators L and L_{z. The radial parts of these atomic orbitals are sometimes numerical tables or are sometimes Slater orbitals. By angular momentum coupling many-electron eigenfunctions of J² (and possibly S²) 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 (E > 0) states must be included to obtain a complete set, i.e., to span all of one-electron Hilbert space.

Mathematical characterization

The atomic orbitals of hydrogen-like 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:

$V(r)=-{\frac {1}{4\pi \epsilon _{0}}}{\frac {Ze^{2}}{r}}$

where

• ε₀ is the permittivity of the vacuum,
• Z is the atomic number (number of protons in the nucleus),
• e is the elementary charge (charge of an electron),
• r is the distance of the electron from the nucleus.

After writing the wave function as a product of functions:

$\psi (r,\theta ,\phi )=R(r)Y_{lm}(\theta ,\phi )\,$

(in spherical coordinates), where $Y_{lm}$ are spherical harmonics, we arrive at the following Schrödinger equation:

$\left[-{\frac {\hbar ^{2}}{2\mu }}\left({1 \over r^{2}}{\partial \over \partial r}\left(r^{2}{\partial R(r) \over \partial r}\right)-{l(l+1)R(r) \over r^{2}}\right)+V(r)R(r)\right]=ER(r),$

where $\mu$ is, approximately, the mass of the electron. More accurately, it is the reduced mass of the system consisting of the electron and the nucleus.

Different values of l give solutions with different angular momentum, where l (a non-negative integer) is the quantum number of the orbital angular momentum. The magnetic quantum number m (satisfying $-l\leq m\leq l$) is the (quantized) projection of the orbital angular momentum on the z-axis. See here for the steps leading to the solution of this equation.

Non-relativistic wavefunction and energy

In addition to l and m, a third integer n > 0, emerges from the boundary conditions placed on R. The functions R and Y 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:

$\psi _{nlm}=R_{nl}(r)\,Y_{lm}(\theta ,\phi )$

$R_{nl}(r)={\sqrt {{\left({\frac {2Z}{na_{\mu }}}\right)}^{3}{\frac {(n-l-1)!}{2n(n+l)!}}}}e^{-Zr/{na_{\mu }}}\left({\frac {2Zr}{na_{\mu }}}\right)^{l}L_{n-l-1}^{2l+1}\left({\frac {2Zr}{na_{\mu }}}\right)$

where:

• $L_{n-l-1}^{2l+1}$ are the generalized Laguerre polynomials in the definition given here.
• $a_{\mu }={{4\pi \varepsilon _{0}\hbar ^{2}} \over {\mu e^{2}}}={{m_{\mathrm {e} }} \over {\mu }}a_{0}$

Here, $\mu$ is the reduced mass of the nucleus-electron system, that is, $\mu ={{m_{\mathrm {N} }m_{\mathrm {e} }} \over {m_{\mathrm {N} }+m_{\mathrm {e} }}}$ where $m_{\mathrm {N} }$ is the mass of the nucleus. Typically, the nucleus is much more massive than the electron, so $\mu \approx m_{\mathrm {e} }$.

• $E_{n}=-\left({\frac {Z^{2}\mu e^{4}}{32\pi ^{2}\epsilon _{0}^{2}\hbar ^{2}}}\right){\frac {1}{n^{2}}}=-\left({\frac {Z^{2}\hbar ^{2}}{2\mu a_{\mu }^{2}}}\right){\frac {1}{n^{2}}}$.
• $Y_{lm}(\theta ,\phi )\,$ function is a spherical harmonic.

parity due to angular wave function is -1 whole to the power l.

Quantum numbers

The quantum numbers n, l and m are integers and can have the following values:

$n=1,2,3,4,\dots \,$

$l=0,1,2,\dots ,n-1\,$

$m=-l,-l+1,\ldots ,0,\ldots ,l-1,l\,$

See for a group theoretical interpretation of these quantum numbers this article. Among other things, this article gives group theoretical reasons why $l<n\,$ and $-l\leq m\leq \,l$.

Angular momentum

Each atomic orbital is associated with an angular momentum L. It is a vector operator, and the eigenvalues of its square L² ≡ L_x² + L_y² + L_z² are given by:

$L^{2}Y_{lm}=\hbar ^{2}l(l+1)Y_{lm}$

The projection of this vector onto an arbitrary direction is quantized. If the arbitrary direction is called z, the quantization is given by:

$L_{\mathrm {z} }Y_{lm}=\hbar mY_{lm},$

where m is restricted as described above. Note that L² and L_z commute and have a common eigenstate, which is in accordance with Heisenberg's uncertainty principle. Since L_x and L_y do not commute with L_z, it is not possible to find a state which 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 (see Bohr model) in 1913, with no knowledge of wavefunctions.

Including spin-orbit interaction

Further information: Azimuthal quantum number § Addition of quantized angular momenta

In a real atom the spin interacts with the magnetic field created by the electron movement around the nucleus, a phenomenon known as spin-orbit interaction. When one takes this into account, the spin and angular momentum are no longer conserved, which can be pictured by the electron precessing. Therefore one has to replace the quantum numbers l, m and the projection of the spin m_s by quantum numbers which represent the total angular momentum (including spin), j and m_j, as well as the quantum number of parity.

See also

• Rydberg atom
• Positronium
• Exotic atom
• Two-electron atom

References

• Gerald Teschl (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. American Mathematical Society. ISBN 978-0-8218-4660-5.
• Tipler, Paul & Ralph Llewellyn (2003). Modern Physics (4th ed.). New York: W. H. Freeman and Company. ISBN 0-7167-4345-0