Stochastic electronic structure theory
Describing correlated electrons
- Lattice models:
- Ab initio descriptions:
\hat{H}_{\text{Hubbard}} = -t\sum_{\langle ij\rangle,\sigma}(\hat{a}_{i\sigma}^{\dagger}\hat{a}_{j\sigma}+\hat{a}_{j\sigma}^{\dagger}\hat{a}_{i\sigma}) + U\sum_i\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}
\hat{H}_{\textit{Ab initio}} = \sum_{pr} t_{pr} \hat{a}_{p}^{\dagger}\hat{a}_r + \sum_{prqs} v_{prqs}\hat{a}_{p}^{\dagger}\hat{a}_r\hat{a}_{q}^{\dagger}\hat{a}_{s}
The quantum many-body problem and Monte Carlo
|\psi\rangle=
Number of configurations increases exponentially with system size
Quantum Monte Carlo: sample properties without storing full wave functions
Exact simulations \(\rightarrow\) fermion sign problem
\hat{H}|\psi\rangle = E|\psi\rangle
i\dfrac{d |\psi\rangle}{d t} = \hat{H}|\psi\rangle
Variational Monte Carlo
Stochastic multireference perturbation theory
Auxiliary field QMC
Outline
- Sampling and the sign problem in free projection
- Reducing noise using selected CI wave functions
- Phaseless constraint and trial state bias
- Symmetry projected states in VMC
- Auxiliary field QMC
- Variational MC
Free projection AFQMC
e^{-\tau (\hat{H}-E_0)}|\psi_r\rangle = c_0|\Psi_0\rangle + c_1 e^{-\Delta E_1\tau}|\Psi_1\rangle+\dots
|\psi_r\rangle = c_0|\Psi_0\rangle + c_1|\Psi_1\rangle +\dots
- Better \(|\psi_l\rangle\) and \(|\psi_r\rangle\) approximate \(|\Psi_0\rangle\), 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}
Mixed energy estimator:
- Numerically exact but noise in QMC sampling worsens exponentially with \(\tau\)
Imaginary time propagation:
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 \(\hat{H}\): \([\hat{K}, \hat{V}]\neq 0\)
- 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
(Thouless, 1960)
(Stratonovich, 1957)
Zhang, Krakauer, Reichman, Rubenstein, ...
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}
Coupled cluster 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
Benzene (30e, 102o), Hilbert space dimension ~ \(10^{35}\)
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}
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_{-\infty}^{\infty} \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)
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}
Selected configuration interaction: put the most important configurations in the state using particle-hole excitations and optimize
|\psi_l\rangle = \sum_i^{N_c} c_i |\psi_i\rangle
Benzene (30e, 102o)
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}
Selected CI 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}
Generalized Wick's theorem: consider \(|\psi_l\rangle = c_{ptqu}\hat{a}_t^{\dagger}\hat{a}_p\hat{a}_u^{\dagger}\hat{a}_q|\psi_0\rangle\)
Benzene (30e, 102o)
\([\text{Cu}_2\text{O}_2]^{2+}\) 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
(32e, 108o)
Phaseless AFQMC
- Bias depends on the trial state used
- There is a trade-off between bias and variance
- Phaseless constraint elminates the sign problem at the expense of a bias
|\psi_0\rangle = \sum_{\phi} w_{\phi}\dfrac{|\phi\rangle}{\langle\psi_T|\phi\rangle}
w_{\phi(\mathbf{x})} = \Bigg\vert\dfrac{\langle\psi_T|\phi(\mathbf{x})\rangle}{\langle\psi_T|\phi\rangle}e^{\mathbf{x}.\bar{\mathbf{x}}-\bar{\mathbf{x}}^2/2}\Bigg\vert\max(0, \cos(\Delta\theta))
\(\text{H}_{50}\) (50e, 50o)
Ni
Excited states of conjugated systems
Butadiene: (22e, 142o)
Nickel porphyrin: (122e, 406o)
| Method | ||
| NEVPT2 | 6.72 | 6.74 |
| CCSD | 6.31 | 7.08 |
| AFQMC / sCI | 6.50(5) | 6.67(5) |
| Exact* | 6.2 | 6.5 |
eV
| Method | |
| CASSCF (4e, 4o) | 3.8 |
| CCSD | 2.55 |
| AFQMC / sCI (50k) | 3.0(1) |
| AFQMC / sCI (100k) | 2.8(1) |
| Experiment | 2.3-2.4 |
eV
1\ ^1B_{2u}
1\ ^1B_{u}
2\ ^1A_{g}
Other properties
| Species | Exact | CCSD | ph-AFQMC |
| 0.986 | 0.991 | 0.985(2) | |
| 0.990 | 0.992 | 0.986(2) | |
| CO | 0.090 | 0.099 | 0.086(3) |
- Can be evaluated as derivatives of energy
- Derivatives can be calculated just as efficiently as energy (automatic differentiation)
a.u.
Small molecule dipole moment calculations:
NH\(_3\)
H\(_2\)O
Outline
- Sampling and the sign problem in free projection
- Reducing noise using selected CI wave functions
- Phaseless constraint and trial state bias
- Symmetry projected states in VMC
- Auxiliary field QMC
- Variational MC
Variational Monte Carlo (VMC)
E = \dfrac{\langle \psi|H|\psi\rangle }{\langle \psi|\psi\rangle}
Strategy:
- Parametrize the wave function: \(|\psi(\mathbf{p})\rangle\), choose initial \(\mathbf{p}\)
- Calculate energy and gradient: Markov chain Monte Carlo
\ \dfrac{\langle \psi(\mathbf{p})|H|\psi(\mathbf{p})\rangle }{\langle \psi(\mathbf{p})|\psi(\mathbf{p})\rangle} = \sum_{\mathbf{n}} \rho_{\mathbf{n}} E_L[\mathbf{n}]
- Optimize: smart gradient descent to change parameters
\propto|\langle \mathbf{n}|\psi(\mathbf{p})\rangle|^2
Ground state minimizes
\text{walker}: |\mathbf{n}\rangle
McMillan (1965)
Symmetry projection in VMC
Symmetry breaking \(\rightarrow\) more variational freedom
Break the symmetry under a projector, to retain good quantum numbers
|\psi\rangle=\hat{P}\hat{\mathcal{J}}|\phi\rangle
Symmetries: spin, number, complex conjugation, ...
Example: complex conjugation in \(\text{H}_2\) near dissociation
|\text{RHF}\rangle = (a_{1\uparrow}^{\dagger}+a_{2\uparrow}^{\dagger})(a_{1\downarrow}^{\dagger}+a_{2\downarrow}^{\dagger})|0\rangle = (a_{1\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{2\downarrow}^{\dagger})|0\rangle
|\text{cRHF}\rangle = (e^{i\pi/4}a_{1\uparrow}^{\dagger}+e^{-i\pi/4}a_{2\uparrow}^{\dagger})(e^{i\pi/4}a_{1\downarrow}^{\dagger}+e^{-i\pi/4}a_{2\downarrow}^{\dagger})|0\rangle\\
\quad = (ia_{1\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}+a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}-ia_{2\uparrow}^{\dagger}a_{2\downarrow}^{\dagger})|0\rangle
\hat{K}|\text{cRHF}\rangle = (a_{1\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}+a_{2\uparrow}^{\dagger}a_{1\downarrow}^{\dagger})|0\rangle
Imada, Sorella, Neuscamman, ...
\hat{\mathcal{J}} = \exp\left(\sum_{p\sigma,q\gamma} v_{p\sigma,q\gamma}\hat{n}_{p\sigma}\hat{n}_{q\gamma}\right)
Density-density correlation function: 18 site 2D Hubbard model (\(U/t=4\))
| d (Bohr) | Exact (DMRG) | Jastrow-SzPfaffian | Jastrow-KSzPfaffian |
| 1.6 | -0.5344 | -0.5327(2) | -0.5337(2) |
| 1.8 | -0.5408 | -0.5389(2) | -0.5400(2) |
| 2.5 | -0.5187 | -0.5167(2) | -0.5180(2) |
H\( _{50} \) linear chain (50e, 50o)
Hartree/particle
Quantum spin liquid in \(\text{Ba}_4\text{Ir}_3\text{O}_{10}\)?
- Insulator with T-linear heat capacity
- Interactions ~ 500 K but orders at 0.2 K
- Not geometrically frustrated
G. Cao, et al. (2020) 1901.04125
Possible in \(\text{Ba}_4\text{Ir}_3\text{O}_{10}\): \(U(1)\) QSL with \(|\psi_0\rangle\) a metallic free fermion state
Gutzwiller projection with VMC:
Savary, Balents (2016)
Summary and outlook
- Combining ideas from QMC and quantum chemistry is a useful approach
- Scalability and parallelizability mean they can be used for studying much larger correlated systems
- QMC techniques are very flexible, so can be employed in a variety of problems
- Possible future directions:
- AFQMC: excited states, dynamics, solids,...
- VMC: neural network states, spin liquids,...
Thank you to:
- The committee
- Sandeep and the Sharma group
- Friends and family
- Funding from NSF and CU Boulder
- Teachers, mentors, collaborators
defense
By Ankit Mahajan
