Accurate trial wave functions in AFQMC
Motivation
Projection QMC methods:
Better ∣ψ⟩ approximates ∣ψ0⟩, faster the convergence with τ
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
Exponentiating H^: [K^,V^]=0
- Exponentiating K^: orbital transformation
where ∣ϕ⟩ and ∣ϕ′⟩ are nonorthogonal determinants.
- Exponentiating V^=21∑γ(Lprγa^p†a^r)2:
xγ: auxiliary field
Sample Gaussian auxiliary fields X, propagate, and measure
CCSD as ∣ψr⟩: sampling Slater determinants from CCSD
commuting ph excitations → no Trotter error


The sign problem

Contour shift:
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)


Efficient local energy algorithm
If ∣ψl⟩ is a Slater determinant: ∣ψl⟩=∣ϕ0⟩
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!
afqmc_trials
By Ankit Mahajan
afqmc_trials
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