Stochastic electronic structure theory

Ankit Mahajan

PySCF meeting

Variational Monte Carlo

Stochastic multireference perturbation theory

Projection QMC

e^{-\tau (\hat{H}-E_0)}|\psi_r\rangle = c_0|\Psi_0\rangle + c_1 e^{-\Delta E_1\tau}|\Psi_1\rangle+\dots

|\psi_r\rangle = c_0|\Psi_0\rangle + c_1|\Psi_1\rangle +\dots

Free projection: try to manage sign problem e.g. by using accurate \(|\psi_l\rangle\) and \(|\psi_r\rangle\), numerically exact and exponentially scaling

Noise in QMC sampling worsens exponentially with \(\tau\) and system size (sign problem)

Imaginary time propagation:

Two flavors:

Constrained: use trial state to constrain projection, trial dependent bias but polynomially scaling

E(\tau)=\dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle}

Sampling in Auxiliary Field QMC

\hat{H} = \hat{K} + \hat{V} = t_{pr} \hat{a}_{p}^{\dagger}\hat{a}_r + \frac{1}{2}\sum_{\gamma} \left(L^{\gamma}_{pr}\hat{a}_p^{\dagger}\hat{a}_r\right)^2

Exponentiating \(\hat{K}\):

e^{t_{pr}\hat{a}_p^{\dagger}\hat{a}_r}|\phi\rangle=|\phi'\rangle

Exponentiating \(\hat{V}\): coupling to a scalar field

e^{-\frac{\hat{L}_{\gamma}^2}{2}} = \int \frac{dx_{\gamma}}{\sqrt{2\pi}}\ e^{\frac{-x_{\gamma}^2}{2}}e^{ix_{\gamma}\hat{L}_{\gamma}}

\(x_{\gamma}\): auxiliary field

(Thouless, 1960)

(Stratonovich, 1957)

E(\tau) = \dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle} \approx \dfrac{\int\ dX p(X)\langle\psi_l|\hat{H}\hat{\mathcal{B}}(X)|\psi_r\rangle}{\int\ dX p(X)\langle\psi_l|\hat{\mathcal{B}}(X)|\psi_r\rangle}

Free projection:

Zhang, Krakauer, Reichman, Rubenstein, ...

Phaseless:

w_{\phi(\mathbf{x})} = \Bigg\vert\dfrac{\langle\psi_T|\phi(\mathbf{x})\rangle}{\langle\psi_T|\phi\rangle}e^{\mathbf{x}.\bar{\mathbf{x}}-\bar{\mathbf{x}}^2/2}\Bigg\vert\max(0, \cos(\Delta\theta))

CCSD as \(|\psi_r\rangle\) in free projection:

|\psi_r\rangle = \exp\left(t_{ikjl}\hat{a}_i^{\dagger}\hat{a}_k\hat{a}_j^{\dagger}\hat{a}_l\right)\exp\left(t_{ik}\hat{a}_i^{\dagger}\hat{a}_k\right)|\phi_0\rangle

Benzene (30e, 102o)

E(\tau)=\dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle}

Zero variance principle

If \(|\psi_l\rangle\) is the exact ground state, \(\langle\psi_0|\hat{H}|\phi_i\rangle = E_0 \langle\psi_0|\phi_i\rangle\), and the energy estimator has zero variance. More accurate \(|\psi_l\rangle\ \rightarrow\ \) smaller variance.

E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle}

Selected configuration interaction: put the most important configurations in the state using particle-hole excitations and optimize

|\psi_l\rangle = \sum_i^{N_c} c_i |\psi_i\rangle

Benzene (30e, 102o)

E(\tau)=\dfrac{\langle\psi_l|\hat{H}e^{-\tau \hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau \hat{H}}|\psi_r\rangle}

Selected CI local energy algorithm

E_L[\phi]=\dfrac{\langle\psi_l|\hat{H}|\phi\rangle}{\langle\psi_l|\phi\rangle},\ \text{two-body part: }\ L^{\gamma}_{pr}L^{\gamma}_{qs}\dfrac{\langle\psi_l|\hat{a}_p^{\dagger}\hat{a}_q^{\dagger}\hat{a}_s\hat{a}_r|\phi\rangle}{\langle\psi_l|\phi\rangle}