Properties and excited states in AFQMC

Ankit Mahajan

Reichman group, Columbia University

Stochastic electronic structure methods workshop

Two parts

Selected CI trials and excited states in AFQMC

Response formalism for properties in AFQMC

with Sandeep Sharma

with David Reichman, Shiwei Zhang, Sandeep Sharma, Joonho Lee

Ab initio AFQMC

sign problem \(\rightarrow\) phaseless constraint \(\rightarrow\) trial dependent bias

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

Shee et al. JCTC '19 

Lee et al. JCTC '19

Hao et al. PRB '20

\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
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

Lee, et al.  Journal of Chemical Theory and Computation 18.12 (2022)

For single reference molecules, AFQMC/HF is usually more accurate than CCSD

Properties in AFQMC

  • Mixed estimator:
\langle \hat{O}\rangle = \dfrac{\langle\psi_T|\hat{O}\hat{P}|\psi_I\rangle}{\langle\psi_T|\hat{P}|\psi_I\rangle}
\langle \hat{O}\rangle_{\text{corrected}} = 2\dfrac{\langle\psi_T|\hat{O}\hat{P}|\psi_I\rangle}{\langle\psi_T|\hat{P}|\psi_I\rangle} - \dfrac{\langle\psi_T|\hat{O}|\psi_T\rangle}{\langle\psi_T|\psi_T\rangle}
  • Backpropagation:
\langle \hat{O}\rangle = \dfrac{\langle\psi_T|\hat{P}'\hat{O}\hat{P}|\psi_I\rangle}{\langle\psi_T|\hat{P}'\hat{P}|\psi_I\rangle}
  • Response formulation:
\langle\hat{O}\rangle = \dfrac{d E(\hat{H}+\lambda \hat{O})}{d\lambda}\bigg\vert_{\lambda=0}

1. Analytical derivatives

2. Finite difference with correlated sampling: multiple                calculations for different observables

AM, Lee, Sharma JCP '22

CO (aQZ)

Adjoint algorithmic differentitation

Inputs: integrals, random numbers,...

Output: AFQMC energy

Adjoint mode: cost scaling same as that of energy \(\rightarrow\) 1-RDM at the cost of energy!

Sorella, Capriotti JCP '10; Song, Martinez, Neaton JCP '20; Zhang, Chan '22; ...

calculate gradients in the reverse sweep

# H Finite difference AD AFQMC w/o SR AD AFQMC w/ SR
10 -31.4545(8) -31.45513(9) -31.45675(6)
20 -81.786(1) -81.7873(2) -81.7891(1)
30 -139.6266(3) -139.6285(2)

Systematic error in ammonia dipole moment (dz basis)

\dfrac{\langle\psi_T|\hat{H}\hat{P}|\psi_T\rangle}{\langle\psi_T|\hat{P}|\psi_T\rangle}

changes in the trial due to perturbation

Dipole moments in the continuum limit

Self-consistent AFQMC

CO (aTZ)

Stochastic error scaling with system size

Hydrogen chains in minimal basis, \(d =1.6\) au

comparison of scaling of stochastic error in energy vs site occupation

Systematic error scaling with system size

1e energy of hydrogen chains

Two parts

Selected CI trials and excited states in AFQMC

with David Reichman, Shiwei Zhang, Sandeep Sharma, Joonho Lee

Response formalism for properties in AFQMC

with Sandeep Sharma

  • Include the most important configurations using particle-hole excitations
|\psi_T\rangle = c_i \hat{E_i} |\psi_0\rangle

Selected CI trials

  • In real space QMC: reduced cost scaling due to the algorithm of Filippi, Assaraf, Moroni (JCP '16)
  • In AFQMC: cost of single determinant trial local energy is \(O(N^4)\). Can we avoid \(O(N_dN^4)\)?

\(O(NN_d+N^4)\)

  • Yes, we can!

AM, Sharma JCTC '21

AM, Lee, Sharma JCP '22

\(\text{H}_{50}\) (50e, 50o)

AM, Lee, Sharma JCP '22

 \(^1A_g \rightarrow  ^1B_u\)

Butadiene (22e, 142o)

Open-shell singlet excited states

Modified AFQMC

AFQMC propagation

collapses to closed-shell

open-shell walker

Projection to prevent collapse:

orthogonalize periodically to inserted orbitals

Ma, Zhang, Krakauer '13

Minimal CAS trials

Open-shell singlet excited states

Thank you!