# 11.1: Overview of Quantum Calculations

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## Multielectron Electronic Wavefunctions

We could symbolically write an approximate two-particle wavefunction as \(\psi (r_1, r_2)\). This could be, for example, a two-electron wavefunction for helium. To exchange the two particles, we simply substitute the coordinates of particle 1 (\(r_l\)) for the coordinates of particle 2 (\(r_2\)) and vice versa, to get the new wavefunction \(\psi (r_1, r_2)\). This new wavefunction must have the property that

\[|\psi (r_1, r_2)|^2 = \psi (r_2, r_1)^*\psi (r_2, r_1) = \psi (r_1, r_2)^* \psi (r_1, r_2) \label{9-38}\]

Equation \(\ref{9-38}\) will be true only if the wavefunctions before and after permutation are related by a factor of \(e^{i\varphi}\),

\[\psi (r_1, r_2) = e^{i\varphi} \psi (r_1, r_2) \]

so that

\[ \left ( e^{-i\varphi} \psi (r_1, r_2) ^*\right ) \left ( e^{i\varphi} \psi (r_1, r_2) ^*\right ) = \psi (r_1 , r_2 ) ^* \psi (r_1 , r_2) \label{9-40}\]

If we exchange or permute two identical particles twice, we are (by definition) back to the original situation. If each permutation changes the wavefunction by \(e^{i \varphi}\), the double permutation must change the wavefunction by \(e^{i\varphi} e^{i\varphi}\). Since we then are back to the original state, the effect of the double permutation must equal 1; i.e.,

\[e^{i\varphi} e^{i\varphi} = e^{i 2\varphi} = 1 \]

which is true only if \(\varphi = 0 \) or an integer multiple of π. The requirement that a double permutation reproduce the original situation limits the acceptable values for \(e^{i\varphi}\) to either +1 (when \(\varphi = 0\)) or -1 (when \(\varphi = \pi\)). Both possibilities are found in nature, but the behavior of elections is that the wavefunction be antisymmetric with respect to permutation \((e^{i\varphi} = -1)\). A wavefunction that is antisymmetric with respect to electron interchange is one whose output changes sign when the electron coordinates are interchanged, as shown below.

\[ \psi (r_2 , r_1) = e^{i\varphi} \psi (r_1, r_2) = - \psi (r_1, r_2) \]

Blindly following the first statement of the Pauli Exclusion Principle, that each electron in a multi-electron atom **must **be described by a different spin-orbital, we try constructing a simple product wavefunction for helium using two different spin-orbitals. Both have the 1s spatial component, but one has spin function \(\alpha\) and the other has spin function \(\beta\) so the product wavefunction matches the form of the ground state electron configuration for He, \(1s^2\).

\[ \psi (\mathbf{r}_1, \mathbf{r}_2 ) = \varphi _{1s\alpha} (\mathbf{r}_1) \varphi _{1s\beta} ( \mathbf{r}_2) \label{8.6.1}\]

After permutation of the electrons, this becomes

\[ \psi ( \mathbf{r}_2,\mathbf{r}_1 ) = \varphi _{1s\alpha} ( \mathbf{r}_2) \varphi _{1s\beta} (\mathbf{r}_1) \label{8.6.2}\]

which is different from the starting function since \(\varphi _{1s\alpha}\) and \(\varphi _{1s\beta}\) are different spin-orbital functions. However, an antisymmetric function must produce the same function multiplied by (–1) after permutation, and that is not the case here. We must try something else.

To avoid getting a totally different function when we permute the electrons, we can make a linear combination of functions. A very simple way of taking a linear combination involves making a new function by simply adding or subtracting functions. The function that is created by subtracting the right-hand side of Equation \(\ref{8.6.2}\) from the right-hand side of Equation \(\ref{8.6.1}\) has the desired antisymmetric behavior. The constant on the right-hand side accounts for the fact that the total wavefunction must be normalized.

\[\psi (\mathbf{r}_1, \mathbf{r}_2) = \dfrac {1}{\sqrt {2}} [ \varphi _{1s\alpha}(\mathbf{r}_1) \varphi _{1s\beta}( \mathbf{r}_2) - \varphi _{1s\alpha}( \mathbf{r}_2) \varphi _{1s\beta}(\mathbf{r}_1)]\]A linear combination that describes an appropriately antisymmetrized multi-electron wavefunction for any desired orbital configuration is easy to construct for a two-electron system. However, interesting chemical systems usually contain more than two electrons. For these multi-electron systems a relatively simple scheme for constructing an antisymmetric wavefunction from a product of one-electron functions is to write the wavefunction in the form of a determinant. John Slater introduced this idea so the determinant is called a Slater determinant.

The Slater determinant for the two-electron wavefunction for the ground state \(H_2\) system (with the two electrons occupying the

\(\sigma_{1s}\) molecular orbital)\[ \psi (\mathbf{r}_1, \mathbf{r}_2) = \dfrac {1}{\sqrt {2}} \begin {vmatrix} \sigma_{1s} (1) \alpha (1) & \sigma _{1s} (1) \beta (1) \\ \sigma _{1s} (2) \alpha (2) & \sigma_{1s} (2) \beta (2) \end {vmatrix} \]

We can introduce a shorthand notation for the arbitrary **spin-orbital**

\[ \chi_{i\alpha}(\mathbf{r}) = \varphi_i \alpha\]

or

\[ \chi_{i\beta}(\mathbf{r}) = \varphi_i \beta\]

as determined by the \(m_s\) quantum number. A shorthand notation for the determinant in Equation 8.6.4 is then

\[ \psi (\mathbf{r}_1 , \mathbf{r}_2) = 2^{-\frac {1}{2}} Det | \chi_{1s\alpha} (\mathbf{r}_1) \alpha \chi_{1s\beta} ( \mathbf{r}_2) \beta| \]

The determinant is written so the electron coordinate changes in going from one row to the next, and the spin orbital changes in going from one column to the next. The advantage of having this recipe is clear if you try to construct an antisymmetric wavefunction that describes the orbital configuration for uranium! Note that the normalization constant is

\((N!)^{-\dfrac {1}{2}}\)

for a system of \(N\) electrons.

The generalized Slater determinant for a multe-electrom atom with N electrons is then

\[ \psi(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N)=\dfrac{1}{\sqrt{N!}} \left| \begin{matrix} \chi_1(\mathbf{r}_1) \alpha & \chi_1(\mathbf{r}_1) \beta& \cdots & \chi_{N/2}(\mathbf{r}_1) \beta\\ \chi_1(\mathbf{r}_2) \alpha & \chi_2(\mathbf{r}_2)\beta & \cdots & \chi_{N/2}(\mathbf{r}_2)\beta \\ \vdots & \vdots & \ddots & \vdots \\ \chi_1(\mathbf{r}_N) \alpha & \chi_2(\mathbf{r}_N)\beta & \cdots & \chi_{N/2}(\mathbf{r}_N) \beta \end{matrix} \right| \label{slater}\]

In a modern *ab initio *electronic structure calculation on a closed shell molecule, the electronic Hamiltonian is used with a single determinant wavefunction. This wavefunction, \(\Psi\), is constructed from molecular orbitals, \(\psi\) that are written as linear combinations of contracted Gaussian basis functions, \(\varphi\)

\[\varphi _j = \sum \limits _k c_{jk} \psi _k \label {10.69}\]

The contracted Gaussian functions are composed from primitive Gaussian functions to match Slater-type orbitals. The exponential parameters in the STOs are optimized by calculations on small molecules using the nonlinear variational method and then those values are used with other molecules. The problem is to calculate the electronic energy from

\[ E = \dfrac {\int \Psi ^* \hat {H} \Psi d \tau }{\int \Psi ^* \Psi d \tau} \label {10.70}\]

or in bra-ket notation

\[ E = \dfrac {\left \langle \Psi |\hat {H} | \Psi \right \rangle}{\left \langle \psi | \psi \right \rangle}\]

The the optimum coefficients \(c_{jk}\) for each molecular orbital in Equation \(\ref{10.69}\) by using the Self Consistent Field Method and the Linear Variational Method to minimize the energy as was described previously for atoms.

The variational principle says an approximate energy is an upper bound to the exact energy, so the lowest energy that we calculate is the most accurate. At some point, the improvements in the energy will be very slight. This limiting energy is the lowest that can be obtained with a single determinant wavefunction (e.g., Equation \(\ref{slater}\)). This limit is called the *Hartree-Fock limit*, the energy is the *Hartree-Fock energy*, the molecular orbitals producing this limit are called *Hartree-Fock orbitals*, and the determinant is the *Hartree-Fock wavefunction*.

David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules")