Variational Monte Carlo

This methodology uses the Variational Method, parameterised by $\theta$, in situations where the Energy , calculated as $E_{\theta }:=\frac{⟨{\psi }_{\theta }|H|{\psi }_{\theta }⟩}{⟨{\psi }_{\theta }|{\psi }_{\theta }⟩}=\frac{\int \psi (x)H{\psi }^{\ast }(x)dx}{\int \psi (x){\psi }^{\ast }(x)dx}$ , has integrals that cannot be solved analytically, and as such requires Monte Carlo to be used to calculate the integral.

Solution

The Energy of a Quantum state is described by the parameterised

$$\begin{array}{rl}{E}_{\theta }& :=⟨{\psi }_{\theta }|H|{\psi }_{\theta }⟩/⟨{\psi }_{\theta }|{\psi }_{\theta }⟩\\ & =\int \psi (x)H{\psi }^{\ast }(x)dx/\int \psi (x){\psi }^{\ast }(x)dx\end{array}$$

We will use Markov Chain Monte Carlo to calculate this integral. More specifically, we will use the Metropolis–Hastings algorithm. We start with the relevant equations

$$\{\begin{array}{rll}H& =-\frac{{\partial }^2}{2\partial x^2}+V(x)& \text{Hamiltonian}\\ \psi (R)& =\exp (-p_0(x-p_1{)}^2-p_2(x-p_3{)}^2)& \text{Wave function}\\ E_L(R)& =\frac{H\psi (R)}{\psi (R)}=V(x)-\frac{1}{2}\frac{{\partial }^2}{\partial x^2}\psi (x)& \text{"Local" energy}\\ V(x)& =\frac{a_2}{2}{x}^2+a_4{x}^4& \text{Potential energy}\\ G(R,R^{\prime },\beta )& =⟨R|e^{-\beta H}|R^{\prime }⟩& \text{???}\\ \Pi (R)& \propto |\psi (R){|}^2& \text{Sampling probability???}\end{array}$$

The goal is to calculate the integral $\int |\psi (r)|E_L(R)dR/\int |\psi (r){|}^{2}dR$. We set

$$\begin{array}{rll}{\psi }_{\beta }(r)& =⟨R|e^{-\beta H}|{\psi }_{\tau }⟩& \text{Using the R1,R2 basis}\\ & ={\int }_{\mathbb{R}}{\int }_{\mathbb{R}}⟨R|e^{-\beta H}|R_1⟩⟨R_1|e^{-\tau H}|R_2⟩{\psi }_{\tau }(R)d{R}_{1}d{R}_2& \\ & ={\int }_{\mathbb{R}}{\int }_{\mathbb{R}}⟨R|e^{-\beta H}|R_1⟩⟨R_1|e^{-\tau H}|R_2⟩{\psi }_{\tau }(R)d{R}_{1}d{R}_2& \end{array}$$

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