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