Can the first ionization energy for a specific element be calculated?

I'm taking a first-year Inorganic Chemistry course in college, and stumbled upon this problem:



> 
> For $\ce{Mn}$ and $\ce{P}$, calculate the first ionization energy and explain the difference between the obtained values.
> 
> 
> 


Now, my course materials (written by our professor) apparently say that the first ionization energy can be computed using this formula. I say apparently because the "course materials" are poorly written PowerPoint slides with loads of unexplained terminology and abbreviations, so I may be misunderstanding.


$$
\text{IE} = \frac{Z\_\mathrm{eff}^2}{n\_\mathrm{eff}^2} \cdot 13.6~\mathrm{eV}
$$


where $Z\_\mathrm{eff}$ is the atomic number minus the differentiating electron's shielding constant (calculated using Slater's rules) and $n\_\mathrm{eff}$ is pulled out of a table of values (see [here](https://en.wikipedia.org/wiki/Slater%27s_rules#Motivation)).


Unfortunately, it only seems to produce (somewhat) numerically correct results for $\ce{H}$ and $\ce{Li}$, and stops being useful for comparing them at $Z=5$. I wrote [an implementation](https://gist.github.com/xndc/51154373025bb513b612) of the algorithm in Mathematica and compared the results it produces to the first ionization energies returned by `ElementData` (suitably converted from molar to regular), and it's a very rough approximation.


Much too rough, in fact, to be usable for the problem above: it returns $47.38$ instead of $7.43$ for $\ce{Mn}$ and $45.71$ instead of $10.48$ for $\ce{P}$. It sorts them the wrong way around, in other words.


My question is, what am I missing? As far as I can tell from Wikipedia, there's no way to simply *calculate* ionization energies - they have to be measured. Is there some obvious detail I haven't spotted? Another way to interpret the problem text? Some elementary knowledge I don't have?




I tried to find out empirically whether your formula makes sense. The data I used was effective nuclear charges from [this Wikipedia page](https://en.wikipedia.org/wiki/Effective_nuclear_charge) (always the minimal value, Pt was discarded), and ionization energies already gathered for the answer to [this question](https://chemistry.stackexchange.com/questions/9340/is-there-a-relation-between-the-ionization-energy-and-the-electron-affinity) on this site. The principal quantum number was also manually added to the data set.


I explored the data in many ways and tried to find a simple expression which could give an estimate for where the ionization energy should lie.


First, I looked at what your formula would give:


[![Proposed formula from question.](https://i.stack.imgur.com/HfGSO.png)](https://i.stack.imgur.com/HfGSO.png)


It is quite obvious that your formula describes two atoms perfectly and is absolutely terrible for the overwhelming majority of elements in the periodic table.


I present to you the best guess so far, based on simple manual adjustments and some exploration of the data:


$$ E\_\text{ion} / \mathrm{eV} = 6 \left(\frac{Z\_\text{eff}}{n^{1.5}} \right)^2 + 4$$


Below is the graphical representation:


[![Plot demonstrating the accuracy of the formula presented above.](https://i.stack.imgur.com/2hqV0.png)](https://i.stack.imgur.com/2hqV0.png)


The error will still be large for many atoms, I estimate the median error to be about 2 eV. But if you want to use a simple formula to calculate the first ionization barrier, this would be the one I would use.


The data exploration was done in Python and is recorded in [this](https://gist.github.com/tschoppi/027242ddd56c7e12e8de) iPython notebook. Feel free to have a look at it and even come up with a better model.


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