Systematically Improvable Auxiliary Field Quantum Monte Carlo

Ankit Mahajan

Reichman Group

Columbia University

Battery center meeting

Two parts

Selected CI trial states in AFQMC:

what are they good for and how to use them

with Sandeep Sharma, Joonho Lee, Jo Kurian

Response formalism for properties in AFQMC:

ground state properties using algorithmic differentiation

with Sandeep Sharma

Ab initio AFQMC

\hat{H} = t_{pr} \hat{a}_{p}^{\dagger}\hat{a}_r + \frac{1}{2} L^{\gamma}_{pr}L^{\gamma}_{qs}\hat{a}_p^{\dagger}\hat{a}_r\hat{a}_q^{\dagger}\hat{a}_s

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

single determinant trial most commonly used 

Shee et al. JCTC '19 

Lee, Malone, Morales JCTC '19

,\ \ |\psi\rangle \propto w_i\dfrac{|\phi_i\rangle}{\langle\psi_T|\phi_i\rangle}
\dfrac{\langle\psi_l|\hat{H}e^{-\tau\hat{H}}|\psi_r\rangle}{\langle\psi_l|e^{-\tau\hat{H}}|\psi_r\rangle}=

Malone, Benali et al. PRB 2020

Diamond correlation energy

Single determinant AFQMC is (surprisingly) accurate in many cases

It does struggle sometimes

Williams et al. PRX '20

AM, Lee, Sharma JCP '22

Transition metal oxide molecules (~20e, 76o)

Possible remedies

  • Improve single determinant trial state self-consistently

Qin, Shi, Zhang PRB '16

  • Correlated trial states: Jastrow, CCSD, MPS, ...
|\psi_{\text{CCSD}}\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

Hubbard-Stratonovich

Chang, Rubenstein, Morales PRB '16

AM, Sharma JCTC '21

Multideterminant states: selected configuration interaction

  • Put the most important configurations in the state using particle-hole excitations and optimize
|\psi_T\rangle = c_i \hat{E_i} |\psi_0\rangle

Selected configuration interaction 

Heat bath CI can handle fairly large spaces, e.g. (28e, 198o) calculation for \(\text{Cr}_2\) reported in Li et al. PRR '20

  • 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 \(O(N_dN^4)\)?

Selected CI local energy algorithm

E_L[\phi]=\dfrac{\langle\psi_T|\hat{H}|\phi\rangle}{\langle\psi_T|\phi\rangle},\ \text{two-body part: }\ L^{\gamma}_{ij}L^{\gamma}_{kl}\dfrac{\langle\psi_T|\hat{a}_i^{\dagger}\hat{a}_k^{\dagger}\hat{a}_l\hat{a}_j|\phi\rangle}{\langle\psi_T|\phi\rangle}

Generalized Wick's theorem:  consider \(|\psi_T\rangle = c_{ptqu}\hat{a}_t^{\dagger}\hat{a}_p\hat{a}_u^{\dagger}\hat{a}_q|\psi_0\rangle\)

AM, Sharma JCTC '21

AM, Lee, Sharma JCP '22

\(O(N_d+N^5)\)

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

or

Benzene (30e, 102o)

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

AM, Lee, Sharma JCP '22

Choosing orbital spaces

Butadiene

States like open-shell singlet excited states cannot be expressed as single determinants

Choosing orbital spaces

 \(^1A_g \rightarrow  ^1B_u\)

 (22e, 142o)

Going to the basis set limit: aTZ (22e, 318o)

AFQMC: 6.25(2) eV, Corrected experiment: 6.1 eV

\([\text{Cu}_2\text{O}_2]^{2+}\) isomerization

\(\Delta E = E(\text{bis}) - E(\text{peroxo})\)

Method
DFT (UBLYP) 36.0
DFT (UB3LYP) 52.9
DFT (UMPW1K) 74.0
CCSD(T) 30.6
CR-CCSD(TQ) 33.8
DMRG-CT 27.1
ph-AFQMC (NOCI) 32.1
fp-AFQMC 24.1(6)

kcal/mol

AM, Sharma JCTC '21

Malone, AM, Spencer, Lee '22

Two parts

Selected CI trial states in AFQMC:

what are they good for and how to use them

with Sandeep Sharma, Joonho Lee, Jo Kurian

Response properties in AFQMC:

ground state properties using algorithmic differentiation

with Sandeep Sharma

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

Adjoint algorithmic differentitation

Inputs: integrals, random numbers,...

Output: AFQMC energy

Adjoint or reverse mode: cost scaling same as that of energy \(\rightarrow\) RDM's at the cost of energy!

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

calculate gradients in the reverse sweep

Implementation details

  • The JAX library implements derivatives of many linear algebra operations
  • Memory bottleneck: checkpointing to trade memory for computational time
  • Derivatives of trial states: for mean field trials, issues due to degenerate orbitals
A=UDU^T
\bar{A}=U\left[\bar{D}+F\circ(U^T\bar{U}-\bar{U}^TU)/2)\right]U^T
F_{ij}=\dfrac{1}{\epsilon_i - \epsilon_j}

for \(i\neq j\)

and \(\epsilon_i\neq\epsilon_j\)

Molecular dipole moments (TZ basis)

https://github.com/sanshar/Dice

https://github.com/linusjoonho/ipie

https://github.com/QMCPACK/qmcpack

Future directions

  • Applications to multiple-center transition metal compounds
  • Condensed phase ab initio calculations
  • Forces, phonon couplings from adjoint AD
  • Use in embedding theories like DMET

Thank you!