9.2: Crystal Field Stabilization Energy
- Page ID
- 326239
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Crystal Field Stabilization Energy
A consequence of crystal field theory is that the distribution of electrons in the d orbitals may lead to net stabilization (decrease in energy) of some complexes depending on the specific ligand geometry and metal d-electron configurations. It is a simple matter to calculate this stabilization since all that is needed is the electron configuration and knowledge of the splitting patterns.
Definition: Crystal Field Stabilization Energy (CFSE)
The crystal field stabilization energy is defined as the energy of the electron configuration in the crystal field minus the energy of the electronic configuration in the spherical field.
\[CFSE=\Delta{E}=E_{\text{crystal field}} - E_{\text{spherical field}} \label{1}\]
The CSFE will depend on multiple factors including:
- Geometry (crystal field splitting pattern)
- Number of d-electrons
- Spin Pairing Energy (the contribution of spin pairing energy is often negligible in comparison to other contributions and is omitted at times)
For an octahedral complex, each of the more stable \(t_{2g}\) orbitals are stabilized by \(-0.4\Delta_o\) relative to spherical field whereas the higher energy \(e_g\) orbitals are destabilized by \(+0.6\Delta_o\). For a complex the stabilization or destabilization of each electron is summed. Any electrons that are paired within a single orbital, also contribute spin pairing energy (+P). The overall CFSE for a complex is expressed as a multiple of the crystal field splitting parameter \(\Delta_o\).
Example \(\PageIndex{1}\): CFSE for a high Spin \(d^7\) complex
What is the CFSE for a high spin \(d^7\) octahedral complex?
Solution
The splitting pattern and electron configuration for both spherical and octahedral ligand fields are compared below.
The energy of the spherical field \((E_{\text{spherical field}}\)) is
\[ E_{\text{spherical field}}= 7 \times 0 + 2P = 2P \nonumber \]
The energy of the octahedral crystal field \(E_{\text{oct field}}\) is
\[E_{\text{oct field}} = (5 \times -0.4 \Delta_o ) + (2 \times 0.6 \Delta_o) + 2P = -0.8 \Delta_o + 2P \nonumber \]
So via Equation \ref{1}, the CFSE is
\[\begin{align} CFSE&=E_{\text{oct field}} - E_{\text{spherical field}} \nonumber \\[4pt] &=( -0.8\Delta_o + 2P ) - 2P \nonumber \\[4pt] &=-0.8 \Delta_o \nonumber \end{align} \nonumber \]
Notice that the spin pairing energy cancels out in this case (and will when calculating the CFSE of high spin complexes) since the number of paired electrons in the octahedral field is the same as that in spherical field of the free metal ion.
Total d-electrons | Spherical Field | Octahedral Field | Crystal Field Stabilization Energy | ||||
---|---|---|---|---|---|---|---|
High Spin | Low Spin | ||||||
\(E_{\text{spherical field}}\) | Configuration | \(E_{\text{crystal field}}\) | Configuration | \(E_{\text{crystal field}}\) | High Spin | Low Spin | |
d0 | 0 | \(t_{2g}^0\; e_g^0\) | 0 | \(t_{2g}\)0\(e_g\)0 | 0 | 0 | 0 |
d1 | 0 | \(t_{2g}^1\; e_g^0\) | -2/5 \(\Delta_o\) | \(t_{2g}\)1\(e_g\)0 | -2/5 \(\Delta_o\) | -2/5 \(\Delta_o\) | -2/5 \(\Delta_o\) |
d2 | 0 | \(t_{2g}^2\; e_g^0\) | -4/5 \(\Delta_o\) | \(t_{2g}\)2\(e_g\)0 | -4/5 \(\Delta_o\) | -4/5 \(\Delta_o\) | -4/5 \(\Delta_o\) |
d3 | 0 | \(t_{2g}^3\; e_g^0\) | -6/5 \(\Delta_o\) | \(t_{2g}\)3\(e_g\)0 | -6/5 \(\Delta_o\) | -6/5 \(\Delta_o\) | -6/5 \(\Delta_o\) |
d4 | 0 | \(t_{2g}^3\; e_g^1\) | -3/5 \(\Delta_o\) | \(t_{2g}\)4\(e_g\)0 | -8/5 \(\Delta_o\) + P | -3/5 \(\Delta_o\) | -8/5 \(\Delta_o\) + P |
d5 | 0 | \(t_{2g}^3\; e_g^2\) | 0 \(\Delta_o\) | \(t_{2g}\)5\(e_g\)0 | -10/5 \(\Delta_o\) + 2P | 0 \(\Delta_o\) | -10/5 \(\Delta_o\) + 2P |
d6 | P | \(t_{2g}^4\; e_g^2\) | -2/5 \(\Delta_o\) + P | \(t_{2g}\)6\(e_g\)0 | -12/5 \(\Delta_o\) + 3P | -2/5 \(\Delta_o\) | -12/5 \(\Delta_o\) + P |
d7 | 2P | \(t_{2g}^5\; e_g^2\) | -4/5 \(\Delta_o\) + 2P | \(t_{2g}\)6\(e_g\)1 | -9/5 \(\Delta_o\) + 3P | -4/5 \(\Delta_o\) | -9/5 \(\Delta_o\) + P |
d8 | 3P | \(t_{2g}^6\; e_g^2\) | -6/5 \(\Delta_o\) + 3P | \(t_{2g}\)6\(e_g\)2 | -6/5 \(\Delta_o\) + 3P | -6/5 \(\Delta_o\) | -6/5 \(\Delta_o\) |
d9 | 4P | \(t_{2g}^6\; e_g^3\) | -3/5 \(\Delta_o\) + 4P | \(t_{2g}\)6\(e_g\)3 | -3/5 \(\Delta_o\) + 4P | -3/5 \(\Delta_o\) | -3/5 \(\Delta_o\) |
d10 | 5P | \(t_{2g}^6\; e_g^4\) | 0 \(\Delta_o\) + 5P | \(t_{2g}\)6\(e_g\)4 | 0 \(\Delta_o\) + 5P | 0 | 0 |
\(P\) is the spin pairing energy and represents the energy required to pair up electrons within the same orbital. For a given metal ion P (pairing energy) is constant, but it does not vary with ligand and oxidation state of the metal ion).
Octahedral Preference
Similar CFSE values can be constructed for non-octahedral ligand field geometries once the knowledge of the d-orbital splitting is known and the electron configuration within those orbitals known, e.g., the tetrahedral complexes in Table \(\PageIndex{2}\). These energies geoemtries can then be contrasted to the octahedral CFSE to calculate a thermodynamic preference for a metal-ligand combination to favor the octahedral geometry. This is quantified via an octahedral site preference energy defined below.
Definition: Octahedral Site Preference Energies (OSPE)
The octahedral site preference energy (OSPE) is defined as the difference of CFSE energies for a non-octahedral complex and the octahedral complex. For comparing the preference of forming an octahedral ligand field vs. a tetrahedral ligand field, the OSPE is thus:
\[OSPE = CFSE_{(oct)} - CFSE_{(tet)} \label{2}\]
The OSPE quantifies the preference of a complex to exhibit an octahedral geometry vs. another geometry. The OSPE for octahedral vs tetrahedral geometries is illustrated below.
Note: the conversion between \(\Delta_o\) and \(\Delta_t\) used for these calculations is:
\[\Delta_t \approx \dfrac{4}{9} \Delta_o \label{3}\]
which is applicable for comparing octahedral and tetrahedral complexes that involve same ligands only.
Total d-electrons | CFSE(Octahedral) | CFSE(Tetrahedral) | OSPE (for high spin complexes)** | ||
---|---|---|---|---|---|
High Spin | Low Spin | Configuration | Always High Spin* | ||
d0 | 0 \(\Delta_o\) | 0 \(\Delta_o\) | e0 | 0 \(\Delta_t\) | 0 \(\Delta_o\) |
d1 | -2/5 \(\Delta_o\) | -2/5 \(\Delta_o\) | e1 | -3/5 \(\Delta_t\) | -6/45 \(\Delta_o\) |
d2 | -4/5 \(\Delta_o\) | -4/5 \(\Delta_o\) | e2 | -6/5 \(\Delta_t\) | -12/45 \(\Delta_o\) |
d3 | -6/5 \(\Delta_o\) | -6/5 \(\Delta_o\) | e2t21 | -4/5 \(\Delta_t\) | -38/45 \(\Delta_o\) |
d4 | -3/5 \(\Delta_o\) | -8/5 \(\Delta_o\) + P | e2t22 | -2/5 \(\Delta_t\) | -19/45 \(\Delta_o\) |
d5 | 0 \(\Delta_o\) | -10/5 \(\Delta_o\) + 2P | e2t23 | 0 \(\Delta_t\) | 0 \(\Delta_o\) |
d6 | -2/5 \(\Delta_o\) | -12/5 \(\Delta_o\) + P | e3t23 | -3/5 \(\Delta_t\) | -6/45 \(\Delta_o\) |
d7 | -4/5 \(\Delta_o\) | -9/5 \(\Delta_o\) + P | e4t23 | -6/5 \(\Delta_t\) | -12/45 \(\Delta_o\) |
d8 | -6/5 \(\Delta_o\) | -6/5 \(\Delta_o\) | e4t24 | -4/5 \(\Delta_t\) | -38/45 \(\Delta_o\) |
d9 | -3/5 \(\Delta_o\) | -3/5 \(\Delta_o\) | e4t25 | -2/5 \(\Delta_t\) | -19/45 \(\Delta_o\) |
d10 | 0 | 0 | e4t26 | 0 \(\Delta_t\) | 0 \(\Delta_o\) |
\(P\) is the spin pairing energy and represents the energy required to pair up electrons within the same orbital.
Note
Tetrahedral complexes are always high spin since the splitting is appreciably smaller than \(P\) (Equation \ref{3}).
After conversion with Equation \ref{3}. The data in Tables \(\PageIndex{1}\) and \(\PageIndex{2}\) are represented graphically by the curves in Figure \(\PageIndex{1}\) below for the high spin complexes only. The low spin complexes require knowledge of \(P\) to graph, which will vary depending on the identiy of the metal ion.

From a simple inspection of Figure \(\PageIndex{1}\), the following observations can be made:
- The OSPE is small in \(d^1\), \(d^2\), \(d^5\), \(d^6\), \(d^7\) high spin complexes and other factors influence the stability of the complexes including steric factors
- The OSPE is large in \(d^3\) and \(d^8\) high spin complexes which strongly favor octahedral geometries
Applications
The "double-humped" curve in Figure \(\PageIndex{1}\) is found for various properties of the first-row transition metals, including hydration and lattice energies of the M(II) ions, ionic radii as well as the stability of M(II) complexes. This suggests that these properties are somehow related to crystal field effects.
In the case of hydration energies describing the complexation of water ligands to a bare metal ion:
\[M^{2+} (g) + H_2O \rightarrow [M(OH_2)_6]^{2+} (aq)\]
Table \(\PageIndex{3}\) and Figure \(\PageIndex{2}\) shows this type of curve. Note that in any series of this type not all the data are available since a number of ions are not very stable in the M(II) state in water.
M | ΔH°/kJmol-1 | M | ΔH°/kJmol-1 |
---|---|---|---|
Ca | -2469 | Fe | -2840 |
Sc | no stable 2+ ion | Co | -2910 |
Ti | -2729 | Ni | -2993 |
V | -2777 | Cu | -2996 |
Cr | -2792 | Zn | -2928 |
Mn | -2733 |
Graphically the data in Table 2 can be represented by:

Contributors and Attributions
Prof. Robert J. Lancashire (The Department of Chemistry, University of the West Indies)