# Homework 7B: ab initio Calculations

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

from Professor Delmar Larsen, UC Davis

This homework will use the ab initio site by Perri at Sonoma State U. Follow instructions on Homework 6B for an introduction and how to create an account.

To start the calculations

## Binding (total) Electronic Energy

Binding energy $$E_Tot$$ is the total electronic energy for the system and it does not include nuclear energies (e.g., vibration and rotational). The "chemical" reaction associated with this energy for the MM atom with nn electrons is

$M \rightarrow ne^- + M^{n+} \label{Eq1}$

The expression for Binding Energy would be calculated from the energies EE of each species:

$E_{tot} = \underbrace{nE(e^-) + E(M^{n+})}_{products} - \underbrace{E(M)}_{reactant} \label{Eq2}$

Since kinetic energy of any species is not included in this discussion, only the energy of the $$M$$ species needs to be explicitly calculated.

## First Ionization Energy

The First Ionization energy is the energy necessary to remove the weakest bound electron. The "chemical" reaction associated with this energy for the MM atom with nn electrons is

$M \rightarrow e^- + M^{+} \label{Eq3}$

The expression for First Ionization Energy would be calculated from the energies $$E$$ of each species:

$E_{I_1} = \underbrace{E(e^-) + E(M^{+})}_{products} - \underbrace{E(M)}_{reactant} \label{Eq4}$

Since kinetic energy is not included in this discussion, only the energy of the $$M$$ and $$M^+$$ species needs to be explicitly calculated.

## Second Ionization Energy

The second ionization energy is the energy required to remove the next weakest bound electron. The "chemical" reaction associated with this energy for the $$M$$ atom with $$n$$ electrons is

$M^+ \rightarrow e^- + M^{2+} \label{Eq5}$

The expression for Second Ionization Energy would be calculated from the energies $$E$$ of each species:

$E_{I_2} = \underbrace{E(e^-) + E(M^{2+})}_{products} - \underbrace{E(M^+)}_{reactant} \label{Eq6}$

Since kinetic energy is not included in this discussion, only the energy of the $$M^+$$ and $$M^{2+}$$ species needs to be explicitly calculated.

Note: The binding energies and Ionization Energy are related. For an atom ( $$M$$ ) with $$n$$  atoms, the binding energy is determined by

$E_{tot} = \sum_i^n - E_{I_n} = -E_{I_1} - E_{I_2} - ... - E_{I_{n-1}} - E_{I_n} \label{Eq7}$

## Electron Affinity

The electron affinity is the change of energy when a neutral atom acquires an electron to generate an anion. The "chemical" reaction associated with this energy for the $$M$$ atom with $$n$$ electrons is

$M + e^- \rightarrow M^{-} \label{Eq8}$

The expression for electron affinity would be calculated from the energies $$E$$ of each species:

$E_{EA} = \underbrace{E(M^{-})}_{products} - \underbrace{E(M) + E(e^-)}_{reactant} \label{Eq9}$

Since kinetic energy is not included in this discussion, only the energy of the $$M^−$$ and $$M$$ species needs to be explicitly calculated.

Calculating electron affinities is harder to do than with ionization energies, especially for closed shell systems (all electrons paired). Approximations can be made to simplify this problem.

## Building the first Two rows of the Periodic Table

Run the single-point calculations on the first 18 atoms of the periodic table (Hydrogen to Argon) and fill in the below table. You only need to extract the total energy (binding energy) of each system for the table, but you need to address the spin of the system explicitly (see  Dry Lab for details on multiplicity in quantum calculations).

• If the spin multiplicity of the atom is not 1, then use the UHR (Unrestricted Hartree-Fock) method,
• If the spin multiplicity of the atom is 1, you can use either the RHF (Restricted Hartree-Fock) or UHR method (should give the same results).

Use the aug-cc-pvdz basis set for both calculations. If you select the RHF method and have a non-unity spin multiplicity, the program will give you an error.

##### Note

If the program crashes:

You'll may have to use HCORE to calculate the orbitals which provides an initial starting point by just diagonalizing the matrix of orbitals instead.

1. Click Do more calculations
2. Click Create Input File
3. Change: GUESS=HUCKEL
4. To: GUESS=HCORE
5. Click Submit Job

When you recalculate a new atom you will have to change the multiplicity for the molecule (from the table you are making below) and then you "create" the input file and swap to the next atom in the input (see arrow above) by changing these two aspects:

1. the name of the atom to the standard chemical abbreviation and
2. the number after that is the atomic number $$Z$$.

Atom Nuclear Charge ($$Z$$ ) Electronic Configuration Spin Multiplicity $$2S+1$$  $$E_{Tot}$$ $$E_{I_1}$$ $$E_{EA}$$
Hydrogen 1 $$1s^1$$ 2 -313 313
Helium 2
Lithium 3
Beryllium 4
Boron 5
Carbon 6
Nitrogen 7
Oxygen 8
Fluorine 9
Neon 10
Sodium 11
Magnesium 12
Aluminum 13
Silicon 14
Phosphorus 15
Sulfur 16
Chorine 17
Argon 18

## Submit for Credit:

Submit the above table filled out for the total binding electronic energy of each atom (do not copy from others)

• Plot up this binding energy, first ionization energy and electron affinity vs. Z on your favorite software (e.g., Desmos, Excel, Matlab, or Plot2 for mac users)
• You should notice several periodic trends in each of the three plots. Explain the origin of the trends of each plot and be as specific as possible within the context of atomic theory.
• The general argument for octets is based on the special "stability" of full shell systems. Does that argument make sense within the context of the definitions above and the simulations you ran and tabulated above? For example is $$E_{tot}$$ for  Ne more stable (lower) than $$E_{tot}$$ for Na? Why or why not does this justify the octet rule?
• Compare the calculated Electron Affinities to the "correct" ones. What conclusion can be drawn from this comparison?

Homework 7B: ab initio Calculations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.