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!
Systematically improvable auxiliary field
By Ankit Mahajan
