Hydrogen-like atom

In physics and chemistry, a hydrogen-like atom (or hydrogenic atom) is an atom with one electron. Except for the hydrogen atom itself (which is neutral) these atoms carry positive charge e(Z-1), where Z is the atomic number of the atom and e is the elementary charge. Because hydrogen-like atoms 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. These orbitals differ from one another in one respect only: the nuclear charge eZ appears in the radial part of the wave function.

Hydrogen-like atoms per se do not play an important role in chemistry or physics. The interest in these atoms is caused mainly by the quantum mechanical fact that their Schrödinger equation can be solved as easily as the equation for the hydrogen atom, because Z enters the problem in a trivial way.

Quantum numbers

Hydrogen-like atomic orbitals are eigenfunctions of a Hamiltonian (energy operator) with eigenvalues proportional to 1/n², where n is a positive integer. Further the orbitals are usually chosen such that they are also eigenfunctions of the square of the one-electron angular momentum vector operator

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{l} \equiv -i\hbar\, (\mathbf{r}\times \boldsymbol{\nabla}) \equiv (l_x,\; l_y,\; l_z), }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar} is Planck's constant divided by 2π. Since l ² ≡ l_x² + l_y² + l_z² commutes with the three angular momentum components, it is possible to require an orbital to be an eigenfunction of any of the three. It is convention to choose l_z, which has an eigenvalue proportional to an integer usually denoted by m. The square l ² has an eigenvalue proportional to l(l+1), where l is a non-negative integer.

It is important to note that the energies of the hydrogen-like orbitals do not depend on the angular momentum quantum numbers l and m, but solely on the principal quantum number n. The degeneracy (maximum number of linearly independent eigenfunctions of same energy) of energy level n is equal to n². This is the dimension of the irreducible representations of the symmetry group of hydrogen-like atoms, which is SO(4).

A hydrogen-like atomic orbital is uniquely identified by the values of the principal quantum number n, the angular momentum quantum number l, and the magnetic quantum number m. These quantum numbers are integers and we summarize their ranges:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} n &=1,2,3,4, \ldots,\\ l &=0,1,2,\ldots,n-1, \\ m &=-l,-l+1,\ldots,l-1,l. \end{align} }

This set must be augmented by the two-valued spin quantum number ms = ±½ in application of the exclusion principle. This principle restricts the allowed values of the four quantum numbers in electron configurations of more-electron atoms: it is forbidden that two electrons have the same four quantum numbers.

Schrödinger equation

The atomic orbitals of hydrogen-like atoms are solutions of the time-independent Schrödinger equation in a potential given by Coulomb's law:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 (charge of the nucleus in unit e),
• e is the elementary charge (charge of an electron),
• r is the distance of the electron from the nucleus.

The Schrödinger equation is the following eigenvalue equation of the Hamiltonian (the quantity in large square brackets):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ -\frac{\hbar^2}{2 \mu} \nabla^2 + V(r) \right] \psi(\mathbf{r}) = E \psi(\mathbf{r}), }

where μ is the reduced mass of the system consisting of the electron and the nucleus. Because the electron mass is about 1836 smaller than the mass of the lightest nucleus (the proton), the value of μ is very close to the mass of the electron m_e for all hydrogenic atoms. In the derivation below we will make the approximation μ = m_e. Since m_e will appear explicitly in the formulas it will be easy to correct for this approximation if necessary.

In this article (in which l ² is defined without Planck's constant and imaginary unit i) it is shown that the operator ∇² expressed in spherical polar coordinates, can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar^2 \nabla^2 = \frac{\hbar^2}{r}\frac{\partial^2}{\partial r^2} r - \frac{l^2}{r^2}. }

The wave function is written as a product of functions in the spirit of the method of separation of variables:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(r, \theta, \phi) = R(r)\,Y_{lm}(\theta,\phi)\,}

where Y_lm are spherical harmonics, which are eigenfunctions of l ² with eigenvalues Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar^2 l(l+1)} . Substituting this product, we arrive at the following one-dimensional Schrödinger equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \frac{\hbar^2}{2\mu}\left[ \frac{1}{r} \frac{d^2}{d r^2} r R(r) - \frac{l(l+1)R(r)}{r^2}\right] + V(r)R(r) = E R(r), }

Wave function and energy

In addition to l and m, there arises a third integer n > 0 from the boundary conditions imposed on R(r). The expression for the normalized wave function is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{nlm} = R_{nl}(r)\, Y_{lm}(\theta,\phi).}

Below it will be derived that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{nl} (r) = \sqrt {{\left ( \frac{2 Z}{n a_{\mu}} \right ) }^3\frac{(n-l-1)!}{2n[(n+l)!]} } e^{- Z r / {n a_{\mu}}} \left ( \tfrac{2 Z r}{n a_{\mu}} \right )^{l} L_{n-l-1}^{2l+1} \left ( \tfrac{2 Z r}{n a_{\mu}} \right ) }

where:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{n-l-1}^{2l+1}} are the generalized Laguerre polynomials in the definition given here.
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{\mu} = {{4\pi\varepsilon_0\hbar^2}\over{\mu e^2}}}

Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{\mu}\,} is approximately equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0\,} (the Bohr radius). If the mass of the nucleus is infinite then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = m_e\,} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_\mu = a_0\,} .

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{n} = -\frac{\mu}{2} \left( \frac{e^2}{4 \pi \varepsilon_0 \hbar n}\right)^2 } . (Energy eigenvalues. As we pointed out above they depend only on n, not on l or m).

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{lm} (\theta,\phi)\,} function is a spherical harmonic.

Derivation of radial function

As we just saw, we must solve the one-dimensional eigenvalue equation,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left \{ - {\hbar^2 \over 2m_e r} {d^2\over dr^2}r +{\hbar^2 l(l+1)\over 2m_e r^2}+V(r) \right \} R(r)=ER(r), }

where we approximated μ by m _e. If the substitution u(r) = rR(r) is made, the radial equation becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -{\hbar^2 \over 2m_e} {d^2 u(r) \over dr^2} + V_{\mathrm{eff}}(r) u(r) = E u(r)}

which is a Schrödinger equation for the function u(r) with an effective potential given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{\mathrm{eff}}(r) = V(r) + {\hbar^2l(l+1) \over 2m_e r^2}.}

The correction to the potential V(r) is called the centrifugal barrier term.

In order to simplify the Schrödinger equation, we introduce the following constants that define the atomic unit of energy and length, respectively,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_\textrm{h} = m_e \left( \frac{e^2}{4 \pi \varepsilon_0 \hbar}\right)^2 \quad\hbox{and}\quad a_{0} = {{4\pi\varepsilon_0\hbar^2}\over{m_e e^2}}} .

Substitute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = Zr/a_0\,} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = E/(Z^2 E_\textrm{h})\,} into the radial Schrödinger equation given above. This gives an equation in which all natural constants are hidden,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ -\frac{1}{2} \frac{d^2}{dy^2} + \frac{1}{2} \frac{l(l+1)}{y^2} - \frac{1}{y}\right] u_l = W u_l . }

Two classes of solutions of this equation exist: (i) W is negative, the corresponding eigenfunctions are square integrable and the values of W are quantized (discrete spectrum). (ii) W is non-negative. Every real non-negative value of W is physically allowed (continuous spectrum), the corresponding eigenfunctions are non-square integrable. In the remaining part of this article only class (i) solutions will be considered. The wavefunctions are known as bound states, in contrast to the class (ii) solutions that are known as scattering states.

For negative W the quantity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \equiv 2\sqrt{-2W}} is real and positive. The scaling of y, i.e., substitution of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \equiv \alpha y } gives the Schrödinger equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ \frac{d^2}{dx^2} -\frac{l(l+1)}{x^2}+\frac{2}{\alpha x} - \frac{1}{4} \right] u_l = 0, \quad \hbox{with}\quad x \ge 0. }

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \rightarrow \infty} the inverse powers of x are negligible and a solution for large x is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp[-x/2]} . The other solution, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp[x/2]} , is physically non-acceptable. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \rightarrow 0} the inverse square power dominates and a solution for small x is x^l+1. The other solution, x^-l, is physically non-acceptable. Hence, to obtain a full range solution we substitute

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_l(x) = x^{l+1} e^{-x/2}f_l(x).\, }

The equation for f_l(x) becomes,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ x\frac{d^2}{dx^2} + (2l+2-x) \frac{d}{dx} +(\nu -l-1)\right] f_l(x) = 0 \quad\hbox{with}\quad \nu = (-2W)^{-\frac{1}{2}}. }

Provided Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu-l-1} is a non-negative integer, say k, this equation has polynomial solutions written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^{(2l+1)}_{k}(x),\qquad k=0,1,\ldots , }

which are generalized Laguerre polynomials of order k. We will take the convention for generalized Laguerre polynomials of Abramowitz and Stegun. Note that the Laguerre polynomials given in many quantum mechanical textbooks, for instance the book of Messiah, are those of Abramowitz and Stegun multiplied by a factor (2l+1+k)! The definition given in this article coincides with the one of Abramowitz and Stegun.

The energy becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = -\frac{1}{2n^2}\quad \hbox{with}\quad n \equiv k+l+1 . }

The principal quantum number n satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \ge l+1} , or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l \le n-1} . Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = 2/n} , the total radial wavefunction is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{nl}(r) = N_{nl} \left(\tfrac{2Zr}{na_0}\right)^{l}\; e^{-{\textstyle \frac{Zr}{na_0}}}\; L^{(2l+1)}_{n-l-1}\left(\tfrac{2Zr}{na_0}\right), }

with normalization constant

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_{nl} = \left[\left(\frac{2Z}{na_0}\right)^3 \cdot \frac{(n-l-1)!}{2n[(n+l)!]}\right]^{1 \over 2}}

which belongs to the energy

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_n = - \frac{Z^2}{2n^2}E_\textrm{h},\qquad n=1,2,\ldots . }

In the computation of the normalization constant use was made of the integral

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^\infty x^{2l+2} e^{-x} \left[ L^{(2l+1)}_{n-l-1}(x)\right]^2 dx = \frac{2n (n+l)!}{(n-l-1)!} . }

Completeness of hydrogen-like orbitals

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.