Variational Methods for Quantum Systems

Abstract

The variational principle states, that the energy expectation value of any trial function is bounded from below by the exact ground state energy. Therefore, the ground state can be approximated by minimizing the energy of a trial function, which involves certain parameters that have to be optimized. In this chapter we study two different kinds of quantum systems. First we apply the variational principle to one- and two-electron systems and calculate the ground state energy of the Helium atom and the Hydrogen molecule. If the trial function treats electron correlation explicitly, the calculation of the energy involves inseparable multidimensional integrals which can be efficiently evaluated with the variational quantum Monte Carlo method. In a second series of computer experiments we study models with a large number of variational parameters. We simulate excitons in a molecular aggregate which are coupled to internal vibrations. The number of parameters increases with the system size up to several hundreds and the optimization requires efficient strategies. We use several kinds of trial functions to study the transition from a delocalized to a self trapped state.

Problems

In the first three computer experiments, we use the variational quantum Monte Carlo method to calculate the groundstate energy. The Metropolis algorithm with $N_{w}$ walkers is used to evaluate the integral

$$ E(\kappa , R)=\frac{<\psi _{\kappa }H\psi _{\kappa }>}{<\psi _{\kappa }\psi _{\kappa }>}=\int d^{3}r\quad \frac{|\psi _{\kappa }(\mathbf {r})|^{2}}{\int |\psi _{\kappa }(\mathbf {r}')|^{2}d^{3}r'}E_{loc}(\mathbf {r}). $$

Adjust the maximum trial step to obtain an acceptance ration of about 1 and study the influence of the number of walkers on the statistical error.

Problem 24.1

Optimize the effective nuclear charge $\kappa$ for the hydrogen molecular ion $H_{2}^{+}$ as a function of R and determine the equilibrium bond length. The trial function has the form

$$ {\psi _{trial}=\sqrt{\frac{\kappa ^{3}}{\pi }}\mathrm {e}^{-\kappa r_{a}}+\sqrt{\frac{\kappa ^{3}}{\pi }}\mathrm {e}^{-\kappa r_{b}}} . $$

Problem 24.2

For the Helium atom we use a trial wavefunction of the Slater-Jastrow type

$$ \psi _{trial}=\mathrm {e}^{-\kappa r_{1}}\mathrm {e}^{-\kappa r_{2}}\mathrm {e}^{\alpha r_{12}/(1+\beta r_{12})}\frac{1}{\sqrt{2}}\left( \uparrow (1)\downarrow (2)-\uparrow (2)\downarrow (1)\right) $$

to find the optimum parameters $\alpha ,\beta ,\kappa$.

Problem 24.3

In this computer experiment we study the hydrogen molecule $H_{2}$. The trial function has the form

$$\begin{aligned} \psi _{trial}= & {} \left\{ C\left[ \mathrm {e}^{-\kappa r_{1a}-\kappa r_{2b}}+\mathrm {e}^{-\kappa r_{1b}-\kappa r_{2a}}\right] +(1-C)\left[ \mathrm {e}^{-\kappa r_{1a}-\kappa r_{2a}}+\mathrm {e}^{-\kappa r_{1b}-\kappa r_{2b}}\right] \right\} \\\times & {} \exp \left\{ \frac{\alpha r_{12}}{1+\beta r_{12}}\right\} . \end{aligned}$$

Optimize the parameters $\kappa ,\beta , C$ as a function of R and determine the equilibrium bond length.

Problem 24.4

In this computer experiment we simulate excitons in a molecular dimer coupled to molecular vibrations. The energy of the lowest exciton state is calculated with the dressed exciton trial function including a frequency change of the vibration

$$ \psi _{trial}=\frac{1}{\sqrt{2}}|1>\left( \frac{2\kappa }{\pi }\right) ^{1/4}\mathrm {e}^{-\kappa (q_{-}+\alpha )^{2}}+\frac{1}{\sqrt{2}}|2>\left( \frac{2\kappa }{\pi }\right) ^{1/4}\mathrm {e}^{-\kappa (q_{-}-\alpha )^{2}}. $$

The parameters $\kappa ,\alpha$ are optimized with the Newton-Raphson method. Vary the exciton coupling V and the reorganization energy $\lambda ^{2}/2$ and compare with the numerically exact values.

Problem 24.5

In this computer experiment we simulate excitons in a molecular aggregate coupled to molecular vibrations. The energy of the lowest exciton state is calculated with different kinds of trial functions

• the dressed exciton

  $$ \varPsi _{MF}=\frac{1}{\sqrt{N}}\sum _{n}|n>G^{n}\prod _{n=1}^{N}\pi ^{-1/4}\mathrm {e}^{-(q_{n}+\alpha _{n})^{2}/2} $$

• the soliton

  $$ \varPsi _{sol}=\sum _{n}\varphi _{n}|n>\prod _{n=1}^{N}\pi ^{-1/4}\mathrm {e}^{-(q_{n}+\alpha _{n})^{2}/2} $$

• the delocalized soliton

  $$ \varPsi _{delsol}=\frac{1}{\sqrt{N}}\sum _{m}\sum _{n}\varphi _{n}|n+m>G^{m}\prod _{n=1}^{N}\pi ^{-1/4}\mathrm {e}^{-(q_{n}+\alpha _{n})^{2}/2}. $$

The system size can be varied from a dimer (N=2) up to chains of 100 molecules. The N equilibrium shifts $\alpha _{n}$ and the N excitonic amplitudes $\varphi _{n}$ are optimized with the methods of steepest descent or conjugate gradients. The optimized parameters are shown graphically. Vary the exciton coupling V and the reorganization energy $\lambda ^{2}/2$ and study the transition from a delocalized to a localized state. Compare the different trial functions.