Variational

adapted from L.Piela, Ideas of Quantum Chemistry.

Variational principle

Solving the exact Hamiltonian for complicated systems is generally undesirable, and involves several approximations to be made possible. One of the most widely used technique in approximating a solution is called the Variational principle. In mathematical terms, it states that $$ \varepsilon[\Phi] = \frac{\braket{\Phi|H|\Phi}}{\braket{\Phi|\Phi}} \label{eq1}\tag{1}$$ for some arbitrary $\Phi$ that is:

• the same coordinates (e.g. $x$, or $r$) of the corresponding Schrodinger equation
• normalizable

Then, we could say $\varepsilon \geq E_0$, where $E_0$ is the true ground state energy, and $\varepsilon = E_0$ if and only if $\Phi$ is the exact ground state wavefunction of $H$.

Proof:
We know that ${\Psi_i}$ of $H$ is a set of complete orthonormal basis. That means any function $\Phi$ can be written as: $$ \Phi = \sum_{i=0}^{\infty} c_i \Psi_i$$ where $$\sum_{i=0}^{\infty}|c_i|^2 = 1$$ Plugging into the expectation value of energy $\varepsilon=\braket{\Phi|H|\Phi}$, we get: $$\begin{gather*} \varepsilon = \braket{\Phi|H|\Phi} \\
= \braket{\sum_{j=0}^{\infty} c_j \Psi_j|H|\sum_{i=0}^{\infty} c_i \Psi_i} \\
\implies \varepsilon-E_0 = \braket{\sum_{j=0}^{\infty} c_j \Psi_j|H|\sum_{i=0}^{\infty} c_i \Psi_i} - E_0 \\
= \sum_{j=0}^{\infty}\sum_{i=0}^{\infty} c_j^* c_i E_i \braket{\Psi_j|\Psi_i} - E_0 \\
= \sum_{j=0}^{\infty}\sum_{i=0}^{\infty} c_j^* c_i E_i \delta_{ij} - E_0 \\
= \sum_{i=0}^{\infty}|c_i|^2E_i - E_0 \times 1 \\
= \sum_{i=0}^{\infty}|c_i|^2E_i - E_0 \times \sum_{i=0}^{\infty}|c_i|^2 \\
= \sum_{i=0}^{\infty}|c_i|^2(E_i - E_0) \\
\geq 0
\end{gather*} $$

With the theorem in mind, how do we actually apply it?

Variational Method

Suppose a trial function $\Phi$ and we know its analytical form. We introduce parameters $\underline{c} = (c_0, c_1, c_2, …, c_p)^T$, which we may modify, into $\Phi$. Then, for some $\underline{c}$, we can say $\Phi = \Phi(x; \underline{c})$. Recall the variational principle, equation $\ref{eq1}$, since on the right hand side we integrate out $x$, then the left hand side depends exclusively on the variational parameter, which is $\underline{c}$ in this case. i.e. $$ \varepsilon (c_0, c_1, c_2,…, c_p) = \varepsilon(\underline{c}) = \frac{\braket{\Phi(x; \underline {c})|H|\Phi(x; \underline{c})}}{\braket{\Phi(x; \underline{c})|\Phi(x; \underline{c})}}$$ And the problem amounts to an optimization problem of finding $\underline{c}$ such that $\varepsilon$ is minimized, approaching $E_0$ the number we want.
In general, it’s not an easy task, because $$\frac{\partial \varepsilon(c_0, c_1, …, c_p)}{\partial c_i} = 0 ;\text{for} ; i=0,1, 2,…,p$$ is a necessary but not sufficient condition to find the global minimum of $\varepsilon$. More details of this can be found in texts related to optimization.

The Linear Variational Principle (Ritz Method)

Here we will talk about a special case of variational method, where we represent the trial function $\Phi$ as a linear combination of some other functions ${\phi_i}$ that we know of, called the basis functions. The previously $\underline{c}$ parameter will be the linear coefficients of the basis functions. i.e. $$\Phi = \sum_{i=0}^p c_i\phi_i$$ This special case allows us to find the global minimum by just looking at the derivative. We can rewrite the equation for energy as $$\begin{gather*} \varepsilon = \frac{\braket{\sum_{i=0}^p c_i\phi_i|H|\sum_{j=0}^p c_j\phi_j}}{\braket{\sum_{i=0}^pc_i\phi_i|\sum_{j=0}^pc_j\phi_j}} \\
= \frac{\sum_{i=0}^p\sum_{j=0}^p c_i^*c_j H_{ij}}{\sum_{i=0}^p\sum_{j=0}^p c_i^*c_j S_{ij}} \end{gather*}$$ where we define $$H_{ij} = \braket{\phi_i|H|\phi_j}$$ $$S_{ij} = \braket{\phi_i|\phi_j}$$ These matrix elements can be obtained by explicitly integrating because we know the exact forms of $\phi$. Also notice that $\varepsilon$ is a function of $c_i$'s, so let’s look at its derivative. Let $$\begin{gather*} \varepsilon = \frac{\sum_{i=0}^p\sum_{j=0}^p c_i^*c_j H_{ij}}{\sum_{i=0}^p\sum_{j=0}^p c_i^*c_j S_{ij}} \\
\equiv \frac{A}{B} \end{gather*}$$

Assume all $c_i$'s are real for convenience, then for each $k=0,1,2…,p$, by the product rule we have: $$\frac{\frac{\partial A}{\partial c_k}B - A\frac{\partial B}{\partial c_k}}{B^2}$$ The individual derivatives for $A$ and $B$ are $$\begin{gather*} \frac{\partial A}{\partial c_k} = \frac{\partial \sum_{i=0}^p\sum_{j=0}^p c_ic_j H_{ij}}{\partial c_k} \\
= \sum_{j=0}^p c_j H_{kj} \end{gather*}$$ Similarly, $$\frac{\partial B}{\partial c_k} = \sum_{j=0}^p c_j S_{kj}$$ If you are having trouble thinking of these derivatives, try expanding out the individual sums with the appropriate indices, set $k=0$, and evaluate which terms are not relevant to $c_0$. Then apply this for general $k$. Now we can write $$\begin{gather*} \frac{\partial \varepsilon}{\partial c_k} = \frac{\frac{\partial A}{\partial c_k}B - A\frac{\partial B}{\partial c_k}}{B^2} \\
= \frac{\sum_{j=0}^pc_jH_{kj}}{B} - \frac{A}{B}\frac{\sum_{j=0}^pc_jS_{kj}}{B} \\
= \frac{\sum_{j=0}^pc_jH{kj} - \varepsilon c_j S_{kj}}{B} \\
= \frac{\sum_{j=0}^p c_j (H_{kj} - \varepsilon S_{kj})}{B} \end{gather*}$$ Setting the above to 0, we get $$\sum_{j=0}^p c_j (H_{kj} - \varepsilon S_{kj}) = 0 ; \text{for} ; k=0,1,2,…,p$$ This is also called the secular equation, and one way to solve it is by requiring the determinant of the matrix $\pmb{H} - \varepsilon \pmb{S}$ be 0, which reduces to diagonalizing the matrix $\pmb{S}^{-1}\pmb{H}$. The resulting eigenvalues are the energies, and the eigenvectors are the linear coefficients of basis functions that corresponds to the energy. The smallest eigenvalue corresponds to the approximated ground state energy, and the larger eigenvalues can be thought of in some sense excited state energies.