Mon 22 July 2019
| tags: blog Hartree-Fock
In the following we will look at Hartree-Fock Theory, which is the starting point for
many quantum chemical methods like MP2 or CCSD.
Starting from the wave function Ansatz we will derive the Hartree-Fock equations
using the variational principle, introduce the Fock operator and talk about the
form of the Hartee-Fock equations.
In Hartree Fock theory we use a single Slater determinant
\begin{equation}
\begin{split}| \Psi^{\rm{SD}} \rangle =
\frac{1}{\sqrt{N!}} \left | \begin{array}{cccc}
\psi_1 (\vec x_1) & \psi_2(\vec x_1) & \ldots & \psi_N (\vec x_1) \\
\psi_1 (\vec x_2) & \psi_2(\vec x_2) & \ldots & \psi_N (\vec x_2) \\
\vdots & \vdots & \ddots & \vdots \\
\psi_1 (\vec x_N) & \psi_2(\vec x_N) & \ldots & \psi_N (\vec x_N) \\
\end{array}\right |\end{split}
\end{equation}
as our wave function Ansatz and optimize it variationally to solve the electronic Schrödinger equation.
Thus, we define the Hamilton operator as
\begin{equation}
\hat{H} = \sum_i \hat{h}_i + \sum_{i<j} \hat{g}_{ij} + \hat{V}_{NN} \quad,
\end{equation}
with
\begin{equation}
\hat{V}_{NN} = \sum_{A<B} \frac{Z_{A}Z_{B}}{|\mathbf{r}_{A}-\mathbf{r}_{B}|} = \sum_{A<B} \frac{Z_{A}Z_{B}}{r_{AB}}
\end{equation}
\begin{equation}
\hat{h}_i = -\frac{1}{2} \nabla^2_i - \sum_{A} \frac{Z_{A}}{r_{iA}}
\end{equation}
\begin{equation}
\hat{g}_{ij} = \frac{1}{r_{ij}}
\end{equation}
Calculating the expectation value
\begin{equation}
E[\Psi^{\rm{SD}}] = \frac{\langle{\Psi^{\rm{SD}}}| \hat{H} |{\Psi^{\rm{SD}}}\rangle }{ \langle { \Psi^{\rm{SD}} | \Psi^{\rm{SD}} } \rangle }
\label{eqn:EXPEC}
\end{equation}
under the condition that the orbitals form an orthonormal basis \(\langle{\psi_i|\psi_j}\rangle = \delta_{ij}\) we obtain
\begin{equation}
E[\Psi^{\rm{SD}}] = \sum_i h_i + \frac{1}{2} \sum_{ij} \langle {\psi_i\psi_j }||{\psi_i\psi_j}\rangle + \hat{V}_{NN}
\end{equation}
with
\begin{equation}
h_i = \langle {\psi_j}| \hat{h}_i |{\psi_i}\rangle
\end{equation}
and
\begin{equation}
\langle {\psi_i\psi_j }||{\psi_i\psi_j}\rangle = \langle {\psi_i\psi_j | \psi_i\psi_j} \rangle - \langle {\psi_i\psi_j | \psi_j\psi_i} \rangle
\end{equation}
where
\begin{equation}
\langle {\psi_i\psi_j | \psi_k\psi_l} \rangle = \int\int \psi^*_i(1) \psi^*_j(2) \hat{g}_{12} \psi_k(1) \psi_l(2)
\end{equation}
This is the so-called physicist notation. Often also the charge cloud notation
\begin{equation}
\langle {\psi_i\psi_j | \psi_k\psi_l} \rangle = \left( \psi_i\psi_k | \psi_j \psi_l \right)
\end{equation}
can be found in the quantum chemical literature. Note that the ortohonormality of the basis does not
change the generality of our Ansatz, but simplifies the equations we have to deal with. To optimize
the wave function variationally we minimize the expectation value defined above. However,
we do this under the condition of an orthonormal basis. Thus, we introduce the following Lagrangian
\begin{equation}
L = E[\Psi^{\rm{SD}}] - \sum_{ij} \varepsilon_{ij} \left(\langle {\psi_i|\psi_j} \rangle - \delta_{ij} \right)
\end{equation}
which contains this condition as Lagrange multipliers \(\varepsilon_{ij}\) . We now require that the Lagrangian
is stationary w.r.t. variations:
\begin{equation}
\begin{split}
\delta L & = 0 \\
& = \big[ \left( \sum_i \langle {\delta\psi_i}|\hat{h}|{\psi_i}\rangle + \sum_k \langle {(\delta\psi_i)\psi_j }||{\psi_i\psi_j}\rangle \right) - \sum_{ij} \varepsilon_{ij} \langle {\delta\psi_i|\psi_j} \rangle \big] + c.c. \\
& = \big[ \sum_i \langle {\delta\psi_i}|\hat{F}|{\psi_i}\rangle - \sum_{ij} \varepsilon_{ij} \langle {\delta\psi_i|\psi_j} \rangle \big] + c.c.
\end{split}
\end{equation}
where we introduced the Fock operator defined as
\begin{equation}
\hat{f}(i) = \hat{h}(i) + \hat{j}(i) - \hat{k}(i) \quad\text{and}\quad \hat{F} = \sum_i \hat{f}(i) \quad,
\end{equation}
where we also introduce the Coulomb operator
\begin{equation}
\hat{j}(i) = \sum_j \int \psi_j^{*}(2) \frac{1}{r_{12}} \psi_j(2) d\tau_2
\end{equation}
and the Exchange operator
\begin{equation}
\hat{k}(1)\psi(1) = \sum_j \psi_j(1) \int \psi_j^{*}(2) \frac{1}{r_{12}} \psi(2)d\tau_2 \quad .
\end{equation}
The Fock operator is a hermitian one particle operator. The minimization is thus equivalent to the Hartree Fock equations
\begin{equation}
\hat{F}|{\psi_i}\rangle = \sum_j \varepsilon_{ij}|{\psi_j}\rangle \quad\quad i,j\in\{occ\}
\end{equation}
\begin{equation}
\hat{F}|{\Psi^{\rm{SD}}}\rangle = \sum_i \varepsilon_{ij}|{\Psi^{\rm{SD}}}\rangle
\end{equation}
The occupied molecular orbitals are only determined up to an unitary transformation among each other.
However, they are usually completely determined by requiring that the Matrix \(\varepsilon_{ij}\)
is diagonal, which leads to canonical MOs and the canonical Hartree Fock equation
\begin{equation}
\hat{F}|{\Psi^{\rm{SD}}}\rangle = \varepsilon_{i}|{\Psi^{\rm{SD}}}\rangle
\end{equation}
Thus, the orbitals are determined as eigenfunctions of the Fock operator. The eigenvalues \(\varepsilon_i\)
are interpredted as orbital energies.
