A complex with maximum spin angular momentum

The correct option is [FeF₆]^3–.

Concept and Formula:
The spin angular momentum ($S$) of a transition metal complex depends directly on its total spin quantum number ($S$), which is determined by the number of unpaired electrons ($n$) in the d-orbitals of the central metal ion.
The spin angular momentum is given by the formula:
$S=\sqrt{s(s+1)}\hslash =\sqrt{\frac{n}{2}(\frac{n}{2}+1)}\frac{h}{2\pi }$
Here, $s$ is the total spin ($s=\frac{n}{2}$), $n$ is the number of unpaired electrons, and $\hslash =\frac{h}{2\pi }$ is the reduced Planck's constant. From this relation, we can see that a higher number of unpaired electrons ($n$) directly results in a greater spin angular momentum.

Step-by-Step Analysis of the Complexes:

1. [FeF₆]^3–:
• Central metal ion: Fe³⁺ (Iron has atomic number 26, configuration: [Ar] 3d⁶ 4s²).
• Oxidation state of Fe: $x+6(-1)=-3\Rightarrow x=+3$.
• Electronic configuration of Fe³⁺: [Ar] 3d⁵.
• Fluoride ion (F^–) is a weak-field ligand (according to the spectrochemical series) and cannot pair up the d-electrons.
• As shown in the attached orbital diagram for Fe⁺³ ($3d^5$), all five d-orbitals are singly occupied with one electron each (represented as '1' in the five boxes of the 3d orbital).
• Number of unpaired electrons ($n$) = 5.

2. [Fe(CN)₆]^3–:
• Central metal ion: Fe³⁺ ([Ar] 3d⁵).
• Cyanide ion (CN^–) is a strong-field ligand.
• It causes pairing of the 3d electrons, forcing them to occupy the lower-energy t_2g orbitals.
• Pairing configuration: (t_2g)⁵ (e_g)⁰, which results in two pairs of electrons and only one unpaired electron.
• Number of unpaired electrons ($n$) = 1.

3. [Fe(H₂O)₆]²⁺:
• Central metal ion: Fe²⁺ ([Ar] 3d⁶).
• Water (H₂O) is a weak-field ligand and does not force pairing.
• The six 3d electrons are arranged such that one orbital contains paired electrons and the remaining four orbitals contain one electron each.
• Number of unpaired electrons ($n$) = 4.

4. [V(H₂O)₆]²⁺:
• Central metal ion: V²⁺ (Vanadium has atomic number 23, configuration: [Ar] 3d³ 4s²).
• Electronic configuration of V²⁺: [Ar] 3d³.
• In an octahedral field, these three electrons occupy the three t_2g orbitals singly.
• Number of unpaired electrons ($n$) = 3.

Conclusion:
Comparing the number of unpaired electrons ($n$) for each complex:
• [FeF₆]^3–: $n=5$
• [Fe(CN)₆]^3–: $n=1$
• [Fe(H₂O)₆]²⁺: $n=4$
• [V(H₂O)₆]²⁺: $n=3$
Because [FeF₆]^3– has the highest number of unpaired electrons ($n=5$), it exhibits the maximum spin angular momentum.