Shielding Constant Calculation using Slater's Rules

The shielding constant, often denoted by the Greek letter $\sigma$ (sigma), quantifies the reduction in the effective nuclear charge experienced by an electron due to the presence of other electrons in an atom. Slater's rules offer a widely used method for approximating this shielding effect.

Shielding a 2p Electron with Slater's Rules

Calculating the shielding constant for a 2p electron involves assessing the contributions from other electrons based on their location relative to this electron. Slater's rules group electrons into specific sets for this calculation:

• Electrons within the same principal quantum shell (n).
• Electrons in the immediately inner shell (n-1).
• Electrons in shells further inside (n-2, n-3, and so on).

The rules assign specific values for the shielding contribution (σ) per electron:

• Each electron in the same group (same n, same l, e.g., other 2s and 2p electrons for a 2p electron) contributes 0.35 to σ.
• Each electron in the next inner shell (n-1) contributes 0.85 to σ.
• Each electron in shells two or more steps inner (n-2, n-3, etc.) contributes 1.00 to σ.

2p Electron Shielding Calculation Example

The question asks for the shielding constant of a 2p electron. The value 3.45 is obtained when applying Slater's rules to a 2p electron in an atom like Oxygen (atomic number Z=8), which has the electron configuration $1s^2 2s^2 2p^4$. We will demonstrate the calculation using Oxygen as our example.

Step 1: Identify the Electron Configuration

Oxygen (O), with Z=8, has the electron configuration: $1s^2 2s^2 2p^4$.

Step 2: Identify Contributing Electrons

We focus on calculating the shielding experienced by one of the 2p electrons. The electrons contributing to this shielding fall into two main categories according to Slater's rules:

• Electrons in the same shell (n=2): This includes the 2s electrons and the other 2p electrons.
• Electrons in the inner shell (n=1): These are the 1s electrons.

Step 3: Apply Slater's Rules and Calculate Contributions

We calculate the shielding contributions separately:

Contribution from electrons in the same shell (n=2):

• There are 2 electrons in the 2s subshell.
• There are 4 electrons in the 2p subshell. For the specific 2p electron we are considering, there are 3 *other* 2p electrons.
• Thus, the total number of electrons in the same shell (n=2) is $2 (\text{from } 2s) + 3 (\text{other } 2p) = 5$ electrons.
• Each of these electrons contributes 0.35 according to Slater's rules.
• Total contribution from n=2 electrons = $5 \times 0.35 = 1.75$.

Contribution from electrons in the next inner shell (n=1):

• There are 2 electrons in the 1s subshell.
• Each electron in the n-1 shell contributes 0.85.
• Total contribution from n=1 electrons = $2 \times 0.85 = 1.70$.

Step 4: Sum the Contributions

The total shielding constant (σ) is the sum of these contributions:

σ = (Contribution from n=2 electrons) + (Contribution from n=1 electrons)

σ = $1.75 + 1.70$

σ = $3.45$

The calculated shielding constant for a 2p electron in Oxygen using Slater's rules is 3.45.