^{1,a)}and Garnet Kin-Lic Chan

^{1}

### Abstract

We investigate the dynamical mean-fieldtheory (DMFT) from a quantum chemical perspective. Dynamical mean-fieldtheory offers a formalism to extend quantum chemical methods for finite systems to infinite periodic problems within a local correlation approximation. In addition, quantum chemical techniques can be used to construct new *ab initio* Hamiltonians and impurity solvers for DMFT. Here, we explore some ways in which these things may be achieved. First, we present an informal overview of dynamical mean-fieldtheory to connect to quantum chemical language. Next, we describe an implementation of dynamical mean-fieldtheory where we start from an *ab initio* Hartree–Fock Hamiltonian that avoids double counting issues present in many applications of DMFT. We then explore the use of the configuration interaction hierarchy in DMFT as an approximate solver for the impurity problem. We also investigate some numerical issues of convergence within DMFT. Our studies are carried out in the context of the cubic hydrogen model, a simple but challenging test for correlation methods. Finally, we finish with some conclusions for future directions.

This work was supported by the Department of Energy (DOE), Office of Science. We acknowledge useful conversations with A. J. Millis, C. A. Marianetti, D. R. Reichman, E. Gull, and G. Kotliar.

I. INTRODUCTION

II. AN INFORMAL OVERVIEW OF DMFT

A. Summary of Green's function formalism

B. DMFT equations

C. The impurity problem and solver in the discrete bath formulation

D. Eliminating double counting in DMFT through Hartree–Fock theory

III. DMFT ALGORITHM

IV. BENCHMARK DMFT STUDIES

A. The cubic hydrogen solid model

B. A configuration interaction impurity solver

C. DMFT numerics: self-consistency

D. DMFT numerics: convergence with bath size

V. CONCLUSIONS

### Key Topics

- Green's function methods
- 63.0
- Mean field theory
- 36.0
- Density functional theory
- 20.0
- Configuration interaction
- 19.0
- Wave functions
- 18.0

## Figures

Spectral functions (density of states) from FCI, CISD, and RHF calculations for cubic hydrogen, at various lattice constants.

**(A)** *a* _{0} = 1.40 Å, 9 bath orbitals, 300 frequency points. **(B)** *a* _{0} = 2.25 Å, 9 bath orbitals, 300 frequency points.

**(C)** *a* _{0} = 2.50 Å, 9 bath orbitals, 300 frequency points. **(D)** *a* _{0} = 6.00 Å, 9 bath orbitals, 300 frequency points.

Spectral functions (density of states) from FCI, CISD, and RHF calculations for cubic hydrogen, at various lattice constants.

**(A)** *a* _{0} = 1.40 Å, 9 bath orbitals, 300 frequency points. **(B)** *a* _{0} = 2.25 Å, 9 bath orbitals, 300 frequency points.

**(C)** *a* _{0} = 2.50 Å, 9 bath orbitals, 300 frequency points. **(D)** *a* _{0} = 6.00 Å, 9 bath orbitals, 300 frequency points.

Spectral function (density of states) obtained with CISD as a solver during the iterations of the self-consistency cycle for cubic hydrogen, at various lattice constants.

**(A)** *a* _{0} = 1.40 Å, 9 bath orbitals, 300 frequency points. **(B)** *a* _{0} = 2.25 Å, 9 bath orbitals, 300 frequency points.

**(C)** *a* _{0} = 2.50 Å, 9 bath orbitals, 300 frequency points. **(D)** *a* _{0} = 6.00 Å, 9 bath orbitals, 300 frequency points.

Spectral function (density of states) obtained with CISD as a solver during the iterations of the self-consistency cycle for cubic hydrogen, at various lattice constants.

**(A)** *a* _{0} = 1.40 Å, 9 bath orbitals, 300 frequency points. **(B)** *a* _{0} = 2.25 Å, 9 bath orbitals, 300 frequency points.

**(C)** *a* _{0} = 2.50 Å, 9 bath orbitals, 300 frequency points. **(D)** *a* _{0} = 6.00 Å, 9 bath orbitals, 300 frequency points.

Fitting accuracy for the real part of the hybridization for various numbers of bath orbitals. The number of frequencies employed was 128 and β = 128.

Fitting accuracy for the real part of the hybridization for various numbers of bath orbitals. The number of frequencies employed was 128 and β = 128.

Fitting accuracy for the imaginary part of the hybridization for various numbers of bath orbitals. The number of frequencies employed was 128 and β = 128.

Fitting accuracy for the imaginary part of the hybridization for various numbers of bath orbitals. The number of frequencies employed was 128 and β = 128.

Fitting accuracy with different number of bath orbitals for the Hartree–Fock impurity spectral function of cubic hydrogen. The number of frequencies employed was 128 and β = 128.

Fitting accuracy with different number of bath orbitals for the Hartree–Fock impurity spectral function of cubic hydrogen. The number of frequencies employed was 128 and β = 128.

Spectral function (density of states) obtained with CISD solver for different number of bath orbitals for cubic hydrogen, *a* _{0} = 2.25 Å.

Spectral function (density of states) obtained with CISD solver for different number of bath orbitals for cubic hydrogen, *a* _{0} = 2.25 Å.

## Tables

General DMFT loop structure. Note that the DMFT self-consistency is carried out on the imaginary frequency axis.

General DMFT loop structure. Note that the DMFT self-consistency is carried out on the imaginary frequency axis.

DMFT self-consistency for . Note that all calculations are done on the imaginary frequency axis.

DMFT self-consistency for . Note that all calculations are done on the imaginary frequency axis.

Total weight of CI coefficients of different classes of determinants [Hartree–Fock (HF), singly-excited (S), doubly-excited (D)] in the ground-state wavefunction of the impurity model as a function of lattice constant *a* _{0}.

Total weight of CI coefficients of different classes of determinants [Hartree–Fock (HF), singly-excited (S), doubly-excited (D)] in the ground-state wavefunction of the impurity model as a function of lattice constant *a* _{0}.

Impurity model natural orbital occupancies for cubic hydrogen (nine bath orbitals) as a function of lattice constant *a* _{0}.

Impurity model natural orbital occupancies for cubic hydrogen (nine bath orbitals) as a function of lattice constant *a* _{0}.

Natural orbital occupancies obtained with CISD solver during the iterations of self-consistent cycle for cubic hydrogen, *a* _{0} = 2.5 Å, 9 bath orbitals, for exact parameters used to converge self-consistency see supplementary material.

Natural orbital occupancies obtained with CISD solver during the iterations of self-consistent cycle for cubic hydrogen, *a* _{0} = 2.5 Å, 9 bath orbitals, for exact parameters used to converge self-consistency see supplementary material.

Impurity natural orbital occupancies obtained with CISD solver for cubic hydrogen at lattice constants 2.25 Å, using 5, 9, and 19 bath orbitals.

Impurity natural orbital occupancies obtained with CISD solver for cubic hydrogen at lattice constants 2.25 Å, using 5, 9, and 19 bath orbitals.

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