For hydrogen-like species, the energy levels depend on the principal quantum number $n$ and are inversely proportional to the square of $n$, and they can be characterized by their nuclear charge $Z$. The energy for a hydrogen-like ion can be expressed as:

$E_n = -\dfrac{Z^2R_H}{n^2}$

where $R_H$ is the Rydberg constant for hydrogen, $Z$ is the atomic number (for Li²⁺, $Z=3$), and $n$ is the principal quantum number associated with the energy level.

Given that S₂ has energy equal to the ground state energy of the hydrogen atom ($E_n$ for hydrogen when $n=1$ and $Z=1$ is $E_1 = -R_H$), and given that S₂ has one radial node. The radial node information tells us about the principal quantum number $n$, as the number of radial nodes is given by $n-l-1$, where $l$ is the azimuthal quantum number.

The ground state of hydrogen corresponds to $n=1$. For the given species Li²⁺ to have the same energy as the ground state of the hydrogen atom but for state S₂, we can use the energy relation. Since the energy is specified to be the same as hydrogen's ground state, let's set up the equality according to the equation given and solve for $n$ specific to the condition (Li²⁺ is considered here, but the condition is about energy equivalence).

Given that S₂ has one radial node, it cannot be the ground state energy level for Li²⁺ (which would directly correspond to $n=1$), it implies a different $n$. For one radial node, the condition $n-l-1 = 1$ must be met.

The options presented are:

• 1s ($n=1$, $l=0$) - No radial nodes
• 2s ($n=2$, $l=0$) - One radial node
• 2p ($n=2$, $l=1$) - No radial nodes, due to the different $l$
• 3s ($n=3$, $l=0$) - Two radial nodes

S₁ is described as having one radial node. Based on the rule for the number of radial nodes ($n-l-1$), the only states that fits this condition directly from the options provided are $2s$ (since for a $2s$ orbital, $n=2$, $l=0$, yielding $2-0-1=1$ radial node).

Hence, the correct option for S₁ is:

Option B: 2s