Accurate trial wave functions in AFQMC
Motivation
Projection QMC methods:
e^{-\tau (\hat{H}-E_0)}|\psi\rangle = c_0|\psi_0\rangle + c_1 e^{-\Delta E_1\tau}|\psi_1\rangle+\dots
|\psi\rangle = c_0|\psi_0\rangle + c_1|\psi_1\rangle +\dots
Better \(|\psi\rangle\) approximates \(|\psi_0\rangle\), faster the convergence with \(\tau\)
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}
Energy estimator:
Outline
- Sampling in AFQMC
- The sign problem in AFQMC and contour shift
- Reducing noise using selected CI wave functions and efficient local energy evaluation
- Benchmark results
review paper: Motta and Zhang, arXiv:1711.02242
Sampling in AFQMC
\hat{H} = \hat{K} + \hat{V} = t_{pr} \hat{a}_{p}^{\dagger}\hat{a}_r + \frac{1}{2}v_{prqs}\hat{a}_{p}^{\dagger}\hat{a}_r\hat{a}_{q}^{\dagger}\hat{a}_{s}
Exponentiating \(\hat{H}\): \([\hat{K}, \hat{V}] \neq 0\)
e^{-\tau\hat{H}}\approx\left(e^{-\frac{\tau}{N}\frac{\hat{K}}{2}}e^{-\frac{\tau}{N}\hat{V}}e^{-\frac{\tau}{N}\frac{\hat{K}}{2}}\right)^N
- Exponentiating \(\hat{K}\): orbital transformation
e^{t_{pr}\hat{a}_p^{\dagger}\hat{a}_r}|\phi\rangle=|\phi'\rangle
where \(|\phi\rangle\) and \(|\phi'\rangle\) are nonorthogonal determinants.
- Exponentiating \(\hat{V} = \frac{1}{2}\sum_{\gamma} \left(L^{\gamma}_{pr}\hat{a}_p^{\dagger}\hat{a}_r\right)^2\):
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
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}
Sample Gaussian auxiliary fields \(X\), propagate, and measure
E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle}
CCSD as \(|\psi_r\rangle\): sampling Slater determinants from CCSD
|\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
commuting ph excitations \(\rightarrow\) no Trotter error
The sign problem
\text{Var}\left(\dfrac{\overline{N}}{\overline{D}}\right) \approx \dfrac{\text{Var}(\overline{N})}{\overline{D}^2} + \dfrac{\overline{N}^2\text{Var}(\overline{D})}{\overline{D}^4} - 2\dfrac{\overline{N}\text{Cov}(\overline{N},\overline{D})}{\overline{D}^3}
E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle} = \dfrac{\overline{N}}{\overline{D}}
Contour shift:
e^{-\frac{y^2}{2}} = \int \frac{dx}{\sqrt{2\pi}}\ e^{\frac{-x^2}{2}+ixy}
x\rightarrow x+iy
x_{\gamma} \rightarrow x_{\gamma} + i \sqrt{\tau}\langle\hat{L}_{\gamma}\rangle
In AFQMC:
Baer, Head-Gordon, Neuhauser (1998)
Selected CI trial state as \(|\psi_l\rangle\)
Zero variance principle: If \(|\psi_l\rangle\) is the exact ground state, then \(N\) and \(D\) are perfectly correlated, \(\langle\psi_0|\hat{H}|\phi_i\rangle = E_0 \langle\psi_0|\phi_i\rangle\), and the energy estimator has zero variance.
More accurate \(|\psi_l\rangle\ \rightarrow\ \) higher \(\text{Cov}(N, D)\)
E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle} = \dfrac{\overline{N}}{\overline{D}}
\text{Var}\left(\dfrac{\overline{N}}{\overline{D}}\right) \approx \dfrac{\text{Var}(\overline{N})}{\overline{D}^2} + \dfrac{\overline{N}^2\text{Var}(\overline{D})}{\overline{D}^4} - 2\dfrac{\overline{N}\text{Cov}(\overline{N},\overline{D})}{\overline{D}^3}
Efficient 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}
If \(|\psi_l\rangle\) is a Slater determinant: \(|\psi_l\rangle = |\phi_0\rangle\)
O(N^4)
If \(|\psi_l\rangle\) is a selected CI wave function: \(|\psi_l\rangle = \sum_i^{N_d} c_i |\phi_i\rangle\)
Naive way: calculating local energy of each Slater determinant as above costs \(O(N_dN^4)\)
One of the terms:
Determinants in the CI expansion are related by ph excitations \(\ \rightarrow\ \) some repeated work
Consider doubly excited determinants: \(c_{jkil} \hat{a}_j^{\dagger} \hat{a}_k \hat{a}_i^{\dagger} \hat{a}_l |\phi_0\rangle\)
One of the terms:
store intermediate
Overall scaling: \(O(N^4 + N_dN)\)
Filippi, Assaraf, Moroni (2016)
Organic molecules
Benzene: ground state energy in Hartree
| Method | DZ (30e, 108o) | TZ (30e, 258o) |
|---|---|---|
| CCSD(T) | -231.5813 | -231.8058 |
| DMRG | -231.5846(7) | - |
| SHCI | -231.586(2) | - |
| AS-FCIQMC | -231.5855(3) | - |
| ph-AFQMC (RHF) | -231.5879(4) | -231.8122(4) |
| fp-AFQMC | -231.5851(7) | -231.809(1) |
Cyclobutadiene automerization barrier (kcal/mol)
| Method | DZ (20e, 72o) | TZ (20e, 172o) |
|---|---|---|
| CCSD(T) | 15.8 | 18.2 |
| CCSDT | 7.6 | 10.6 |
| TCCSD (12,12) | - | 9.2 |
| MRCI+Q | - | 9.2 |
| fp-AFQMC | 8.4(4) | 10.2(4) |
\([\text{Cu}_2\text{O}_2]^{2+}\) isomerization
kcal/mol
Future directions
- Properties and excited states
- Importance sampling and constraints
- Embedding approaches
- Relativistic Hamiltonians
- Variational CCSD using similarity transformed Hamiltonian, other wave functions like MPS
Thank you!
afqmc_trials
By Ankit Mahajan
