Schrödinger equation

The evolution of a quantum system is described by the Schrödinger equation

$$i\hbar \frac{\partial}{\partial \tau} \varphi ({\vec r},\tau) = \hat H \varphi ({\vec r},\tau),$$ (2.6)

Instead of considering the time-evolution we will look for the ground state properties. That can be done by introducing the imaginary time $\tau = i t/\hbar$. We rewrite the Schrödinger equation and introduce a constant energy shift $E$, whose meaning will become clearer later:

$$-\frac{\partial}{\partial \tau} \varphi ({\vec r},\tau) = (\hat H -E) \varphi ({\vec r},\tau),$$ (2.7)

The formal solution $\psi ({\vec r},\tau) = e^{-(\hat H - E)\tau} \psi ({\vec r},0)$ can be expanded in eigenstate functions of the Hamiltonian $\hat H\phi_n~=~E_n\phi_n$, where we order the eigenumbers in an increasing order $E_0 < E_1 < ...$

$$\psi ({\vec r},\tau) = \sum\limits_{n=0}^\infty c_n \phi_n({\vec r},\tau) = \sum\limits_{n=0}^\infty c_n \phi_n({\vec r},0) e^{-(E_n - E)\,\tau}$$ (2.8)

The amplitudes of the components change with time, either increasing or decreasing depending on the sign of $(E_n-E)$. At large times the term that corresponds to the projection on the ground state dominates the sum. In other words all excited states decay exponentially fast and only contribution from ground state survives

$$\psi ({\vec r},\tau) \to c_0~\phi_0({\vec r},0)~e^{-(E_0 - E)\,\tau},\qquad \mbox{when~~~} \tau \to \infty$$ (2.9)

In the long time limit the wave function remains finite only if $E$ is equal to $E_0$. The ground state energy $E_0$ will be estimated in a different way (Sec. 2.7.1), but the fact that its estimation used for the energy shift value $E$ leads to a stable normalization of $\psi({\vec r},\tau)$ proves in a different way that the estimator is correct (for the implementation see Eq. 2.34).

The Hamiltonian of a system of $N$ particles interacting via pair-wise potential $V_{int}$ and subjected to an external field $V_{ext}$ in a most general form can be written as

$$\hat H = -\frac{\hbar^2}{2m} \sum\limits_{i=1}^N \Delta_i + \sum\... ... V_{int}(\vert\vec r_i -\vec r_j\vert) + \sum\limits_{i=1}^N V_{ext}(\vec r_i),$$ (2.10)

The Schrödinger equation (2.7) reads as^2.2

$$-\frac{\partial}{\partial \tau} \psi ({\vec r},\tau) =-D \Delta_{... ...psi ({\vec r},\tau) + V({\bf R}) \psi ({\vec r},\tau) - E \psi ({\vec r},\tau),$$ (2.11)

where we introduced the notation $D = \hbar^2/2m$ and terms depending only on particle coordinates are denoted as $V({\bf R}) =\sum\limits_{i<j}^N V_{int}(\vert\vec r_i -\vec r_j\vert) + \sum\limits_{i=1}^N V_{ext}(\vec r_i)$.

The efficiency of the method can be significantly improved if additional information on the wave function is used. The idea is to approximate the true wave function $\psi({\vec r},\tau)$ by a trial one $\psi_T({\vec r},\tau)$ and let the algorithm correct the guess done. This approach is called importance sampling and consists in solving the Schrödinger equation for the modified wave function

$$f({\vec r},\tau) = \psi_T ({\vec r},\tau) \psi ({\vec r},\tau)$$ (2.12)

Another reason for using the product of wave functions as the probability distribution instead of sampling $\psi$ is that the average over the latter is ill defined $\langle A\rangle = \int A\,\psi\,{\bf dR}/\int\psi\,{\bf dR}$, on the contrary the average over the product of wave functions has the meaning of the mixed estimator $\langle A\rangle = \int \psi_T A\,\psi\,{\bf dR}/ \int\psi_T\psi\,{\bf dR}$. From (2.11) it follows that the distribution function $f$ satisfies the equation

$$-\frac{\partial}{\partial \tau} f ({\vec r},\tau) = -D\Delta_{\bf... ...bla_{\bf R}({\bf F}f({\vec r},\tau)) + (E^{loc}({\bf R}) - E) f({\vec r},\tau),$$ (2.13)

here $E^{loc}$ denotes the local energy which is the average of the Hamiltonian with respect to trial wave function^2.3

$$E^{loc}({\bf R}) = \psi^{-1}_T({\bf R}) \hat H \psi_T({\bf R})$$ (2.14)

and ${\bf F}$ is the drift force which is proportional to the gradient of the trial wave function and points in the direction of the maximal increase of $\psi _T$^2.4

$${\bf F}= \frac{2}{\psi_T({\bf R})}\nabla_{\bf R}\psi_T({\bf R})$$ (2.15)