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. A better—but never used—name would therefore be hydrogen-like cations.

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. The orbitals of the different hydrogen-like atoms differ from one another in one respect only: they depend on the nuclear charge eZ (which appears in their radial part).

Hydrogen-like atoms per se do not play an important role in chemistry or physics. The interest in these atoms is mainly because their Schrödinger equation can be solved analytically, in exactly the same way as the Schrödinger equation of the hydrogen atom.

Quantum numbers of hydrogen-like wave functions

The non-relativistic wave functions (orbitals) of hydrogen-like atoms are known analytically and are labeled by three exact quantum numbers, conventionally designated n, ℓ, and m. These quantum numbers play an important role in atomic physics and chemistry, as they are useful labels for quantum mechanical states of more-electron atoms, too. Although the three quantum numbers are not exact for an atom with more than one electron, they are still approximately valid, see this article. Because the (for many-electron atoms) approximate quantum numbers n, ℓ, and m are the building bricks of the Aufbau principle (building-up principle)—the construction of the electronic configuration of atoms—they are discussed at some length in this section.

Eigenfunctions of commuting operators

Hydrogen-like atomic orbitals are eigenfunctions of a Hamiltonian H (energy operator) with eigenvalues proportional to 1/n², where n is a positive integer, referred to as principal quantum number. Observe the somewhat unexpected fact that these eigenvalues do depend solely on n.

The hydrogen orbitals are usually chosen such that they are simultaneously eigenfunctions of H and l², the square of the one-electron angular momentum vector operator

$\mathbf {l} \equiv -i\hbar \,(\mathbf {r} \times {\boldsymbol {\nabla }})\equiv (l_{x},\;l_{y},\;l_{z}),$

where ℏ is Planck's constant divided by 2π, the symbol × stands for a cross product, ∇ is the gradient operator, and r is the vector pointing from the nucleus to the electron.

From quantum mechanics it is known that a necessary and sufficient condition for the existence of simultaneous eigenfunctions of H and l ² is the commutation of the operators

$l^{2}\equiv l_{x}^{2}+l_{y}^{2}+l_{z}^{2}\quad {\hbox{and}}\quad H.$

These two operators indeed commute. (This is due to the spherical symmetry of H.) The squared operator l ² has eigenvalues proportional to ℓ(ℓ+1), where ℓ is a non-negative integer (the azimuthal quantum number, also known as the angular momentum quantum number).

Further, since l ² commutes with the three angular momentum components l_x, l_y, and l_z, it is possible to require an orbital to be an eigenfunction of any of the three components. It is conventional to choose l_z, which has an eigenvalue proportional to an integer usually denoted by m (the so-called magnetic quantum number).

Count the degenerate orbitals belonging to fixed n,

$\sum _{\ell =0}^{n-1}\sum _{m=-\ell }^{\ell }1=\sum _{\ell =0}^{n-1}2\ell +1=n^{2}.$

In other words, 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), and not SO(3) as for many-electron atoms.

Summary of quantum numbers

A hydrogen-like atomic orbital ψ_nℓm   is uniquely identified by the values of the principal quantum number n, the azimuthal quantum number ℓ, and the magnetic quantum number m. These three quantum numbers are natural numbers, their definitions and ranges are:

${\begin{aligned}H\;\psi _{n\ell m}&={\frac {E_{\mathrm {h} }}{2n^{2}}}\;\psi _{n\ell m},&\qquad n&=1,\ldots ,\infty ,&\qquad \qquad &{\hbox{(principal quantum number)}}\\l^{2}\;\psi _{n\ell m}&=\hbar ^{2}\ell (\ell +1)\;\psi _{n\ell m},&\qquad \ell &=0,\ldots ,n-1,&\qquad &{\hbox{(azimuthal quantum number)}}\\l_{z}\;\psi _{n\ell m}&=\hbar m\;\psi _{n\ell m},&\qquad m&=-\ell ,\ldots ,\ell .&\qquad &{\hbox{(magnetic quantum number)}}\\\end{aligned}}$

Here E_h is the atomic unit of energy, see below. Note that n ≥ ℓ+1.

Indication of ℓ by letters

It is very common to denote the orbitals of different angular momentum by different letters, 2s-, 3p-orbital, etc. For historical reasons ℓ = 0 orbitals are designated by s (sharp), ℓ = 1 by p (principal), ℓ = 2 by d (diffuse), and ℓ = 3 by f (fundamental) . For higher ℓ orbitals the alphabet is followed, while j orbitals are omitted. Thus we get the following association between letters and ℓ quantum numbers

${\begin{matrix}s&p&d&f&g&h&i&k\\0&1&2&3&4&5&6&7\\\end{matrix}}$

For instance, hydrogenic g-orbitals (ℓ=4) start at principal quantum number n = 5, so that we can speak of 5g-, 6g-, etc. orbitals, but a hydrogen-like 4g-orbital is not defined (i.e., does not appear as a solution of the hydrogen-like Schrödinger equation).

Spin

The set of orbital quantum numbers 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. This is an important restriction in constructing atomic states by application of the Aufbau (building up) principle.

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:

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

where

• ε₀ is the permittivity of the vacuum,
• Z is the atomic number (charge of the nucleus in unit e; 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.

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

$\left[-{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}+V(r)\right]\psi (\mathbf {r} )=E\psi (\mathbf {r} ),$

where μ is the reduced mass of the system consisting of the electron and the nucleus. Because the electron mass is about 1836 times 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

$\hbar ^{2}\nabla ^{2}={\frac {\hbar ^{2}}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r-{\frac {l^{2}}{r^{2}}}.$

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

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

where Y_lm are spherical harmonics, which are eigenfunctions of l ² with eigenvalues ${\scriptstyle \hbar ^{2}l(l+1)}$. Substituting this product, letting l ² act on Y_lm, and dividing out Y_lm, we arrive at the following one-dimensional Schrödinger equation:

$-{\frac {\hbar ^{2}}{2\mu }}\left[{\frac {1}{r}}{\frac {d^{2}}{dr^{2}}}rR(r)-{\frac {l(l+1)R(r)}{r^{2}}}\right]+V(r)R(r)=ER(r).$

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:

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

where Y_lm(θ,φ) is a spherical harmonic. Below it will be derived that the radial function (normalized to unity) is,

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

Here:

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

Note that a_μ is approximately equal to a₀ (the Bohr radius). If the mass of the nucleus is infinite then μ = m_e and a_μ = a₀.

The energy eigenvalue associated with ψ_nlm is:

$E_{n}=-{\frac {\mu }{2}}\left({\frac {Ze^{2}}{4\pi \varepsilon _{0}\hbar n}}\right)^{2}$.

As we pointed out above it depends only on n, not on l or m.

Derivation of radial function

As is shown above, we must solve the one-dimensional eigenvalue equation,

$\left[-{\hbar ^{2} \over 2m_{e}r}{d^{2} \over dr^{2}}r+{\hbar ^{2}l(l+1) \over 2m_{e}r^{2}}+V(r)\right]R(r)=ER(r),$

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

$-{\hbar ^{2} \over 2m_{e}}{d^{2}u(r) \over dr^{2}}+V_{\mathrm {eff} }(r)u(r)=Eu(r)$

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

$V_{\mathrm {eff} }(r)=V(r)+{\hbar ^{2}l(l+1) \over 2m_{e}r^{2}}.$

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,

$E_{\mathrm {h} }=m_{e}\left({\frac {e^{2}}{4\pi \varepsilon _{0}\hbar }}\right)^{2}\quad {\hbox{and}}\quad a_{0}={{4\pi \varepsilon _{0}\hbar ^{2}} \over {m_{e}e^{2}}}$.

Substitute $y=Zr/a_{0}\;$   and   $\;W=E/(Z^{2}E_{\mathrm {h} })$   into the radial Schrödinger equation given above. This gives an equation in which all natural constants are hidden,

$\left[-{\frac {1}{2}}{\frac {d^{2}}{dy^{2}}}+{\frac {1}{2}}{\frac {l(l+1)}{y^{2}}}-{\frac {1}{y}}\right]u_{l}=Wu_{l}.$

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 $\;\alpha \equiv 2{\sqrt {-2W}}$ is real and positive. The scaling of y, i.e., substitution of $\;x\equiv \alpha y$ gives the Schrödinger equation:

$\left[{\frac {d^{2}}{dx^{2}}}-{\frac {l(l+1)}{x^{2}}}+{\frac {2}{\alpha x}}-{\frac {1}{4}}\right]u_{l}=0,\quad {\hbox{with}}\quad x\geq 0.$

For x → ∞, the inverse powers of x are negligible and a solution for large x is exp(−x/2). The other solution, exp(x/2), is physically non-acceptable. For x → 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

$u_{l}(x)=x^{l+1}e^{-x/2}f_{l}(x).\,$

The equation for f_l(x) becomes,

$\left[x{\frac {d^{2}}{dx^{2}}}+(2l+2-x){\frac {d}{dx}}+(\nu -l-1)\right]f_{l}(x)=0\quad {\hbox{with}}\quad \nu =(-2W)^{-{\frac {1}{2}}}.$

Provided ν−l−1 is a non-negative integer, say k, this equation has well-behaved (regular at the origin, vanishing for infinity) polynomial solutions written as

$L_{k}^{(2l+1)}(x),\qquad k=0,1,\ldots ,$

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

$W=-{\frac {1}{2n^{2}}}\quad {\hbox{with}}\quad n\equiv k+l+1.$

The principal quantum number n satisfies $n\geq l+1$, or $l\leq n-1$. Since $\alpha =2/n$, the total radial wavefunction is

$R_{nl}(r)=N_{nl}\left({\tfrac {2Zr}{na_{0}}}\right)^{l}\;e^{-{\textstyle {\frac {Zr}{na_{0}}}}}\;L_{n-l-1}^{(2l+1)}\left({\tfrac {2Zr}{na_{0}}}\right),$

with normalization constant

$N_{nl}=\left[\left({\frac {2Z}{na_{0}}}\right)^{3}\cdot {\frac {(n-l-1)!}{2n[(n+l)!]}}\right]^{1 \over 2},$

and energy

$E_{n}=-{\frac {Z^{2}}{2n^{2}}}E_{\mathrm {h} },\qquad n=1,2,\ldots .$

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

$\int _{0}^{\infty }x^{2l+2}e^{-x}\left[L_{n-l-1}^{(2l+1)}(x)\right]^{2}dx={\frac {2n(n+l)!}{(n-l-1)!}}.$

Caveat on 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.

List of radial functions

The following list of radial functions ${\scriptstyle R_{nl}(r)}$ is copied from Ref. The scaled distance is $\rho _{n}\equiv {\frac {2Zr}{a_{0}n}}.$

${\begin{aligned}R_{10}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;2\;\\R_{20}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{2{\sqrt {2}}}}\;\left(2-\rho _{n}\right)\\R_{21}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{2{\sqrt {6}}}}\;\rho _{n}\\R_{30}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{9{\sqrt {3}}}}\;\left(6-6\rho _{n}+\rho _{n}^{2}\right)\\R_{31}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{9{\sqrt {6}}}}\;\left(4-\rho _{n}\right)\rho _{n}\\R_{32}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{9{\sqrt {30}}}}\;\rho _{n}^{2}\\R_{40}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{96}}\;\left(24-36\rho _{n}+12\rho _{n}^{2}-\rho _{n}^{3}\right)\\R_{41}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{32{\sqrt {15}}}}\;\left(20-10\rho _{n}+\rho _{n}^{2}\right)\rho _{n}\\R_{42}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{96{\sqrt {5}}}}\;\left(6-\rho _{n}\right)\rho _{n}^{2}\\R_{43}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{96{\sqrt {35}}}}\;\rho _{n}^{3}\\R_{50}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{300{\sqrt {5}}}}\;\left(120-240\rho _{n}+120\rho _{n}^{2}-20\rho _{n}^{3}+\rho _{n}^{4}\right)\\R_{51}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{150{\sqrt {30}}}}\;\left(120-90\rho _{n}+18\rho _{n}^{2}-\rho _{n}^{3}\right)\rho _{n}\\R_{52}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{150{\sqrt {70}}}}\;\left(42-14\rho _{n}+\rho _{n}^{2}\right)\rho _{n}^{2}\\R_{53}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{300{\sqrt {70}}}}\;\left(8-\rho _{n}\right)\rho _{n}^{3}\\R_{54}(r)&=\left({\frac {Z}{a_{0}}}\right)^{3/2}e^{-\rho _{n}/2}\;{\frac {1}{900{\sqrt {70}}}}\;\rho _{n}^{4}\\\end{aligned}}$