Accurate trial wave functions in AFQMC
Motivation
Projection QMC methods:
e−τ(H^−E0)∣ψ⟩=c0∣ψ0⟩+c1e−ΔE1τ∣ψ1⟩+…
e^{-\tau (\hat{H}-E_0)}|\psi\rangle = c_0|\psi_0\rangle + c_1 e^{-\Delta E_1\tau}|\psi_1\rangle+\dots
∣ψ⟩=c0∣ψ0⟩+c1∣ψ1⟩+…
|\psi\rangle = c_0|\psi_0\rangle + c_1|\psi_1\rangle +\dots
Better ∣ψ⟩ approximates ∣ψ0⟩, faster the convergence with τ
E(τ)=⟨ψl∣e−τH^∣ψr⟩⟨ψl∣H^e−τH^∣ψr⟩
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
H^=K^+V^=tpra^p†a^r+21vprqsa^p†a^ra^q†a^s
\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 H^: [K^,V^]=0
e−τH^≈(e−Nτ2K^e−NτV^e−Nτ2K^)N
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 K^: orbital transformation
etpra^p†a^r∣ϕ⟩=∣ϕ′⟩
e^{t_{pr}\hat{a}_p^{\dagger}\hat{a}_r}|\phi\rangle=|\phi'\rangle
where ∣ϕ⟩ and ∣ϕ′⟩ are nonorthogonal determinants.
- Exponentiating V^=21∑γ(Lprγa^p†a^r)2:
e−2L^γ2=∫2πdxγ e2−xγ2eixγL^γ
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}}
xγ: auxiliary field
E(τ)=⟨ψl∣e−τH^∣ψr⟩⟨ψl∣H^e−τH^∣ψr⟩≈∫ dXp(X)⟨ψl∣B^(X)∣ψr⟩∫ dXp(X)⟨ψl∣H^B^(X)∣ψr⟩
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(τ)≈∑i⟨ψl∣ϕi⟩∑i⟨ψl∣H^∣ϕi⟩
E(\tau)\approx\dfrac{\sum_i\langle\psi_l|\hat{H}|\phi_i\rangle}{\sum_i\langle\psi_l|\phi_i\rangle}
CCSD as ∣ψr⟩: sampling Slater determinants from CCSD
∣ψr⟩=exp(tikjla^i†a^ka^j†a^l)exp(tika^i†a^k)∣ϕ0⟩
|\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 → no Trotter error
The sign problem
Var(DN)≈D2Var(N)+D4N2Var(D)−2D3NCov(N,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}
E(τ)≈∑i⟨ψl∣ϕi⟩∑i⟨ψl∣H^∣ϕi⟩=DN
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−2y2=∫2πdx e2−x2+ixy
e^{-\frac{y^2}{2}} = \int \frac{dx}{\sqrt{2\pi}}\ e^{\frac{-x^2}{2}+ixy}
x→x+iy
x\rightarrow x+iy
xγ→xγ+iτ⟨L^γ⟩
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 ∣ψl⟩
Zero variance principle: If ∣ψl⟩ is the exact ground state, then N and D are perfectly correlated, ⟨ψ0∣H^∣ϕi⟩=E0⟨ψ0∣ϕi⟩, and the energy estimator has zero variance.
More accurate ∣ψl⟩ → higher Cov(N,D)
E(τ)≈∑i⟨ψl∣ϕi⟩∑i⟨ψl∣H^∣ϕi⟩=DN
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}}
Var(DN)≈D2Var(N)+D4N2Var(D)−2D3NCov(N,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
EL[ϕ]=⟨ψl∣ϕ⟩⟨ψl∣H^∣ϕ⟩, two-body part: LprγLqsγ⟨ψl∣ϕ⟩⟨ψl∣a^p†a^q†a^sa^r∣ϕ⟩
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 ∣ψl⟩ is a Slater determinant: ∣ψl⟩=∣ϕ0⟩
O(N4)
O(N^4)
If ∣ψl⟩ is a selected CI wave function: ∣ψl⟩=∑iNdci∣ϕi⟩
Naive way: calculating local energy of each Slater determinant as above costs O(NdN4)
One of the terms:
Determinants in the CI expansion are related by ph excitations → some repeated work
Consider doubly excited determinants: cjkila^j†a^ka^i†a^l∣ϕ0⟩
One of the terms:
store intermediate
Overall scaling: O(N4+NdN)
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) |
[Cu2O2]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!
Accurate trial wave functions in AFQMC
afqmc_trials
By Ankit Mahajan
afqmc_trials
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