Energies of Orbitals

Energy of Orbitals: Hydrogen vs. Multi-Electron Atoms

The energy of an electron in an orbital is not a fixed, universal number. It depends critically on whether the atom has one electron (like hydrogen) or many electrons. This distinction is the central idea of this section.

Energy in the Hydrogen Atom: Dependence Only on $n$

In a hydrogen atom (or any one-electron system), the only electrical interaction is the electrostatic attraction between the single electron and the nucleus. There are no other electrons to cause repulsion. As a result, the energy of an orbital is determined solely by the principal quantum number $n$.

The energy of orbitals in hydrogen increases in the following order:

This means that for a given value of $n$, all subshells ($s$, $p$, $d$, $f$) have exactly the same energy. For example, the $2s$ and $2p$ orbitals have different shapes, but an electron in either one possesses the same energy. Orbitals that share the same energy are called degenerate orbitals.

The $1s$ orbital corresponds to the most stable condition, called the ground state. An electron in this orbital is most strongly held by the nucleus. An electron in any $2s$, $2p$, or higher orbital is in an excited state.

Energy in Multi-Electron Atoms: Dependence on Both $n$ and $l$

The situation changes dramatically when an atom has more than one electron. In a multi-electron atom, the energy of an electron depends on both its principal quantum number ($n$) and its azimuthal quantum number ($l$). This means that for a given $n$, the $s$, $p$, $d$, and $f$ subshells all have different energies.

Within a given principal quantum number, the energy of orbitals increases in the order:

For higher energy levels, these differences become so pronounced that the energy ordering can "stagger" — meaning an orbital from a higher shell can have lower energy than an orbital from a lower shell. Examples include:

Why Do Subshells Have Different Energies in Multi-Electron Atoms?

The root cause is electron-electron repulsion. In a hydrogen atom, the only force is attraction between the electron and the nucleus. In a multi-electron atom, there are two types of interactions:

1. Attraction between each electron and the nucleus.
2. Repulsion between each electron and every other electron.

The overall stability of an electron comes from the fact that the total attractive interactions outweigh the repulsive ones. However, the repulsive interactions — especially those between outer-shell electrons and inner-shell electrons — are very important.

Shielding and Effective Nuclear Charge ($Z_{\text{eff}}$)

Because of the inner-shell electrons, an electron in an outer shell does not experience the full positive charge of the nucleus ($Ze$). The inner electrons partially "screen" or "shield" the outer electron from the nucleus. The net positive charge that an outer electron actually feels is called the effective nuclear charge ($Z_{\text{eff}} e$).

Despite this shielding, the attractive force experienced by an outer electron increases as the nuclear charge $Z$ increases. In other words, the orbital energy (the energy of interaction between the nucleus and the electron) becomes more negative (lower) as the atomic number increases.

The Role of Orbital Shape

Both the attractive and repulsive interactions depend on the shell ($n$) and the shape ($l$) of the orbital. The key difference lies in how effectively electrons in different subshells shield outer electrons, and how close an electron in a given subshell can get to the nucleus.

• An electron in a spherical $s$ orbital spends more time close to the nucleus compared to an electron in a $p$ orbital of the same shell.
• An electron in a $p$ orbital spends more time near the nucleus than an electron in a $d$ orbital of the same shell.

This has two consequences:

1. Shielding: Electrons in $s$ orbitals shield outer electrons from the nucleus more effectively than $p$ orbital electrons, which in turn shield more effectively than $d$ orbital electrons.
2. Binding: Because an $s$ electron spends more time near the nucleus, it experiences a greater effective nuclear charge ($Z_{\text{eff}}$) than a $p$ electron in the same shell. The $p$ electron experiences a greater $Z_{\text{eff}}$ than a $d$ electron.

Since a higher $Z_{\text{eff}}$ means a stronger attraction and a lower (more negative) energy, the energy of electrons in a given shell follows the same order:

This is the reason for the splitting of energy levels within the same shell in multi-electron atoms.

The $(n + l)$ Rule: A Simple Guide to Orbital Energies

The exact mathematical dependence of orbital energy on $n$ and $l$ is complicated, but a simple rule — the $(n + l)$ rule — helps predict the relative energies of orbitals.

Table 2.5 (the book's own table) illustrates this rule up to 4p.

Continuing the same rule beyond the book's table: $5s\,(n+l=5)$ → $4d, 5p, 6s\,(=6)$ → $4f, 5d, 6p, 7s\,(=7)$ — which completes the familiar filling order used in Section 2.6.4.

Applying the rule:

• $2s$ vs $2p$: $2s$ has $(n+l) = 2+0 = 2$. $2p$ has $(n+l) = 2+1 = 3$. Since $2 < 3$, $2s$ has lower energy than $2p$.
• $3s$ vs $2p$: $3s$ has $(n+l) = 3+0 = 3$. $2p$ has $(n+l) = 2+1 = 3$. They have the same $(n+l)$ value. The rule says the orbital with the lower $n$ has the lower energy. Since $n=2$ for $2p$ and $n=3$ for $3s$, $2p$ has lower energy than $3s$. …