Hydrogen-Like Atom Energy

Q: What is a hydrogen-like atom?

It is an atom or ion with only one electron, such as He$^+$, Li$^{2+}$ or C$^{5+}$. Its nucleus has charge $+Ze$, where $e$ is the elementary charge, and the single electron feels that charge.

Q: Why does the energy scale with $Z^2$?

The electron's potential energy grows in proportion to $Z$, and each orbit shrinks in proportion to $1/Z$. Together these make the energy scale as $Z^2$.

In one sentence: An atom or ion with only one electron has energy levels like hydrogen’s, but a larger nuclear charge pulls every level deeper, so the electron is held more tightly.

$E_n = -\frac{Z^2}{n^2} \times 13.605693\ \text{eV}$

Read it out loud: “E sub n equals negative Z squared over n squared, times 13.605693 electron volts. E sub n is the energy of level n, in electron volts. Z is the nuclear charge, which is the number of protons in the nucleus. n is the level number: 1, 2, 3 and so on. The minus sign means the nucleus holds the electron. An electron volt (eV) is a small unit of energy.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Powers of ten and the electron volt (eV): An electron volt is a very small unit of energy: the energy an electron gains when it is pushed through one volt. One electron volt equals $1.602176634 \times 10^{-19}$ J. The exponent $-19$ means move the decimal point 19 places to the left, so the same energy in joules is written 0.0000000000000000001602 J, with 18 zeros after the decimal point. Electron volts keep the numbers short. To multiply powers of ten, add the exponents.
Atomic number $Z$ , protons and the nucleus: Every atom has a central core called the nucleus, made of protons and, usually, neutrons. Protons carry positive electric charge. The number of protons is the atomic number, written $Z$ . Hydrogen has one proton, so $Z = 1$ . Helium has two protons, so its nucleus has $Z = 2$ . A neutral atom has as many electrons as protons, and the electrons’ negative charge balances the protons’ positive charge.
Ions and the plus sign, like He $^+$: An ion is an atom that has lost or gained electrons, so it carries a net electric charge. The small plus sign after the element name shows how many electrons are missing. He $^+$ is helium with one electron removed, so its nucleus with two protons holds only one electron. The bigger the nuclear charge $Z$ , the stronger the pull on each electron, because the protons’ charge is larger.
Energy levels and the number $n$: An electron bound to a nucleus can only have certain energies, called energy levels. They are numbered by whole numbers $n = 1, 2, 3, \dots$ , which is called the principal quantum number. Level 1, the lowest, is the ground state. Higher numbers have more energy, and the levels get closer together as $n$ grows, like rungs on a ladder that bunch up near the top.
Squares, like $Z^2$ and $\frac{1}{n^2}$: A squared number is the number multiplied by itself. $2^2 = 2 \times 2 = 4$ , and $3^2 = 9$ . In this formula $Z^2$ means the nuclear charge times itself, so $Z = 2$ gives $Z^2 = 4$ . The fraction $\frac{1}{n^2}$ is one divided by the level number squared. For $n = 2$ it is $\frac{1}{4}$ , and for $n = 3$ it is $\frac{1}{9}$ .
Binding energy and the minus sign: The minus sign in $E_n$ means the electron is bound, held by the nucleus. We set the energy of a free electron, far away and at rest, to zero. A bound electron has less energy than that, so its energy is negative. The size of the negative number is the energy needed to pull the electron free. Pulling an electron free of an atom is called ionization.
The Rydberg energy, 13.605693 eV: The number $13.605693$ eV is the Rydberg energy. It is the energy needed to remove the electron from a hydrogen atom in its ground state, and it is about 13.6 electron volts. It is built from fixed constants of nature. In joules it is about $2.18 \times 10^{-18}$ J, which is why the electron volt is the better unit here.

What each symbol means

Symbol	Meaning
$E_n$	The energy of the electron in level $n$ , in electron volts (eV). It is negative because the electron is bound to the nucleus. Level 1 gives the most negative value, and larger $n$ gives values closer to zero, where a free electron sits.
$Z$	The nuclear charge in units of the proton’s charge. It counts the protons in the nucleus, and it has no unit. Hydrogen has $Z = 1$ , singly ionized helium He $^+$ has $Z = 2$ , and carbon with five electrons removed, C $^{5+}$ , has $Z = 6$ .
$n$	The principal quantum number, a whole number 1, 2, 3 and so on that labels the energy level. It has no unit. A larger $n$ means a higher level, with less binding, so the electron sits farther from the nucleus on average.
$13.605693\ \text{eV}$	The Rydberg energy, the energy scale of the whole hydrogen pattern, about 13.6 electron volts. It is the ionization energy of hydrogen from its ground state. It is built from fixed constants of nature, so every hydrogen-like level is a fixed fraction of it.

The formula piece by piece

$n$: Start with the level number $n$ , a whole number that labels the energy level. Level 1 is the ground state, and level 2 is the next one up. The value of $n$ decides how deep the level sits in the energy pattern.
$Z^2$: Square the nuclear charge $Z$ . This is the factor that makes heavier nuclei hold their electrons more tightly. Doubling $Z$ multiplies every energy by four, because $2^2 = 4$ .
$\frac{Z^2}{n^2}$: Divide the two together, $Z^2$ over $n^2$ . This fraction says how strongly bound level $n$ is, in units of the Rydberg energy. For hydrogen, $Z = 1$ and $n = 1$ give exactly one, so the ground state is the full Rydberg energy.
$13.605693\ \text{eV}$: Multiply by the Rydberg energy, 13.605693 eV, which sets the scale of the energies in electron volts. Put a minus sign in front, and the result is the energy of the level, which is negative because the electron is bound.
$E_n = -\frac{Z^2}{n^2} \times 13.605693\ \text{eV}$: The whole formula in one sentence: the energy of level $n$ equals minus $Z$ squared divided by $n$ squared, times 13.605693 electron volts.

What the formula is doing, and why

The hydrogen energy formula also works for any atom or ion stripped down to a single electron. Examples are He $^+$ , Li $^{2+}$ and C $^{5+}$ , which are called hydrogen-like or one-electron ions. The electron still feels one nucleus, so the same kind of energy ladder applies, with the strength of the pull set by $Z$ . What goes in is $Z$ and $n$ . What comes out is the energy of that level, in electron volts.

Why $Z^2$ ? The nucleus pulls on the electron with an electric force that grows with the number of protons. A nucleus with $Z$ protons pulls $Z$ times as hard, and the electron’s orbit also shrinks in proportion to $1/Z$ . Together, the stronger pull and the smaller orbit scale the energy with $Z^2$ . Going from hydrogen ( $Z = 1$ ) to He $^+$ ( $Z = 2$ ) multiplies every energy level by $2^2 = 4$ .

Why $1/n^2$ ? Each level is one of the allowed energies of the atom. The energies come in fractions of $1/n^2$ , so the levels crowd together as $n$ grows. For a given $Z$ , going from level 1 to level 2 needs $\frac{3}{4} Z^2 \times 13.6$ eV. The gap is largest between the lowest two levels, and it becomes smaller for each higher level.

Why the minus sign? The nucleus holds the electron. We set the energy of a free electron, far away and at rest, to zero. A bound electron has less energy than that, so its energy is negative. The most negative level, $n = 1$ , is the most tightly held. The size of $E_1$ is the energy needed to free the electron from the ground state.

Where it applies. The formula is very close to measured values for one-electron ions, such as He $^+$ . It is approximate for heavier ions, because the nucleus moves slightly and relativistic effects grow with $Z$ . It does not describe neutral helium or any atom with two or more electrons, since those electrons push on each other and change the energies.

An everyday picture

Think of the staircase from the hydrogen atom, built in a stronger building. More protons in the nucleus means a stronger pull, so every step sits deeper below the zero line. The steps keep their shape, but each one is scaled by $Z^2$ . Doubling the nuclear charge makes every step four times deeper.

Worked example: Ground state and first jump of He$^+$

He $^+$ is helium with one electron left, so its nucleus has $Z = 2$ and its single electron starts in level $n = 1$ . We want the ground-state energy $E_1$ , and the energy and wavelength of the photon given off when the electron falls from level 2 to level 1. Use 13.605693 eV for the constant and $hc = 1239.84$ eV nm, a shortcut that combines Planck’s constant and the speed of light.

Identify the numbers. Helium has 2 protons, and He $^+$ has lost one electron, so its nucleus has $Z = 2$ . The electron starts on level 2 and drops to level 1, so the two levels are $n_2 = 2$ and $n_1 = 1$ . Level 1 is the ground state, the lowest energy the electron can have.
Find the ground-state energy. Put $Z = 2$ and $n = 1$ into the formula: $E_1 = -\frac{2^2}{1^2} \times 13.605693$ eV. Since $2^2 = 4$ and $1^2 = 1$ , this is $-4 \times 13.605693 = -54.42277$ eV, which we round to $-54.4228$ eV.
Find the level 2 energy. Now $n = 2$ : $E_2 = -\frac{2^2}{2^2} \times 13.605693$ eV. The fraction $\frac{4}{4}$ is exactly 1, so $E_2 = -13.605693$ eV, which is the same as the ground state of hydrogen. Doubling $n$ from 1 to 2 raised the energy from $-54.4$ eV to $-13.6$ eV.
Find the energy of the photon. When the electron falls from level 2 to level 1, the energy it gives off is the upper level minus the lower level: $E_{\text{photon}} = E_2 - E_1 = (-13.6057) - (-54.4228)$ eV. Subtracting a negative number is the same as adding its size, so $E_{\text{photon}} = 40.8171$ eV.
Find the wavelength. Use $\lambda = \frac{hc}{E}$ with $hc = 1239.84$ eV nm. The eV in the top and the eV in the bottom cancel, which leaves nanometers. So $\lambda = \frac{1239.84\ \text{eV nm}}{40.8171\ \text{eV}} \approx 30.38$ nm.
Convert the energy to joules to see how small it is. One electron volt is $1.602176634 \times 10^{-19}$ J, so $40.8171$ eV is $40.8171 \times 1.602 \times 10^{-19} \approx 6.54 \times 10^{-18}$ J. That is a tiny amount of energy for one photon, but it is much bigger than the energy of a single photon of visible light.
Compare with hydrogen and with visible light. Hydrogen’s own jump from level 2 to level 1 gives only about 10.2 eV, which is one quarter of the He $^+$ value, and it has a wavelength of about 121.5 nm. A wavelength of about 30 nm is extreme ultraviolet, far shorter than visible light, which runs from about 400 nm to 700 nm.

Answer: The He $^+$ ground state is $-54.42$ eV. The electron falling from level 2 to level 1 gives off a photon of about 40.82 eV, with a wavelength of about 30.38 nm, in the extreme ultraviolet. Hydrogen’s same jump is only about 10.2 eV, because the energies of He $^+$ are four times larger.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Using $Z = 1$ for every ion. Count the protons in the nucleus, not the electrons. He $^+$ has $Z = 2$ because helium’s nucleus has two protons.
Forgetting the $Z^2$ . Doubling $Z$ makes every energy four times bigger, not twice as big, because the energy scales with $Z$ squared.
Dropping the minus sign. The bound levels are negative, and a photon’s energy is the higher level minus the lower one, which comes out as a positive number.
Using this formula for a neutral atom with several electrons. It needs exactly one electron, so neutral helium, which has two, does not follow it.

Where you will meet it

Astronomy, where lines from ionized helium reveal very hot gas around stars and in the Sun’s atmosphere.
Plasma physics and fusion research, where the light from highly ionized atoms helps show the temperature of the plasma.
Estimating the ionization energies of hydrogen-like heavy ions, which grow with $Z^2$ .
Introductory atomic physics, where one-electron ions are the first step beyond hydrogen.

Questions people ask

What is a hydrogen-like atom?

It is an atom or ion with only one electron, such as He $^+$ , Li $^{2+}$ or C $^{5+}$ . Its nucleus has charge $+Ze$ , where $e$ is the elementary charge, and the single electron feels that charge.

Why does the energy scale with $Z^2$?

The electron’s potential energy grows in proportion to $Z$ , and each orbit shrinks in proportion to $1/Z$ . Together these make the energy scale as $Z^2$ .

Can I use this for a molecule?

No. Molecules have several nuclei and several electrons, so the one-nucleus formula does not apply.

Hydrogen Energy Levels: An electron in a hydrogen atom can sit only on certain energy steps, and when it jumps between steps it gives off or takes in light of one exact color.
Rydberg Formula: The Rydberg formula gives the exact color of the light that a hydrogen atom gives off when its electron drops from one energy level to a lower one.
Bohr Radius: The Bohr radius is the typical size of a hydrogen atom, about 0.0529 nanometers, and it comes from combining a few fixed constants of nature.