Spectral Symbol Determination

The question asks for the spectral symbol for the ground state of an atom whose outermost shell configuration is $3d^3$. The ground state spectral symbol is represented in the form $^{2S+1}L_J$. To determine this symbol, we need to find the total spin angular momentum ($S$), the total orbital angular momentum ($L$), and the total angular momentum ($J$).

Electron Configuration and Hund's Rules

We have 3 electrons in the 3d subshell. A d subshell has $l=2$, and it contains 5 orbitals with magnetic quantum numbers ($m_l$) ranging from $-l$ to $+l$, i.e., $-2, -1, 0, +1, +2$. According to Hund's rules, for the ground state:

• Electrons occupy different orbitals within a subshell to maximize spin multiplicity.
• Electrons in different orbitals will have parallel spins if possible.
• Orbitals are filled starting with the highest possible $m_l$ value (usually positive) when determining L.

For the $3d^3$ configuration, the three electrons will occupy three different d orbitals with parallel spins to maximize $S$. Let's assign the electrons to the orbitals with the highest $m_l$ values first with parallel spins (e.g., spin up, $m_s = +\frac{1}{2}$).

Calculating S (Total Spin Angular Momentum)

The total spin angular momentum $S$ is the sum of the individual spin magnetic quantum numbers ($m_s$). For three electrons with parallel spin +1/2:

\(S = \sum m_s = +\frac{1}{2} + (+\frac{1}{2}) + (+\frac{1}{2}) = \frac{3}{2}\)

The spin multiplicity is $2S+1$.

\(2S+1 = 2(\frac{3}{2}) + 1 = 3 + 1 = 4\)

This gives the superscript in the spectral symbol.

Calculating L (Total Orbital Angular Momentum)

The total orbital angular momentum $L$ is the sum of the individual orbital magnetic quantum numbers ($m_l$) for the electrons in their assigned orbitals:

\(L = \sum m_l = (+2) + (+1) + (0) = 3\)

The value of $L$ corresponds to a letter symbol for the spectral term:

• L=0: S
• L=1: P
• L=2: D
• L=3: F
• L=4: G

Since $L=3$, the term symbol is F. This gives the letter in the spectral symbol.

Calculating J (Total Angular Momentum)

The total angular momentum $J$ is determined by the values of $L$ and $S$. For a subshell that is less than half-filled, the ground state value of $J$ is $|L-S|$. A d subshell can hold 10 electrons (2(2*2+1) = 10). The $3d^3$ configuration has 3 electrons, which is less than half-filled (5 electrons).

So, for the ground state of a less than half-filled subshell:

\(J = |L - S| = |3 - \frac{3}{2}| = |\frac{6}{2} - \frac{3}{2}| = |\frac{3}{2}| = \frac{3}{2}\)

For subshells that are more than half-filled, $J = L+S$. For half-filled subshells, $J=S$.

Assembling the Spectral Symbol

Combining the values calculated:

• Spin multiplicity ($2S+1$) = 4
• Term symbol (L) = F (for L=3)
• Total angular momentum (J) = 3/2

The ground state spectral symbol is $^{2S+1}L_J = ^4F_{3/2}$.