Self-Consistent NEGF-DFT

This section details the self-consistent procedure combining NEGF and DFT calculations, including density matrix construction, convergence strategies, and practical considerations.

Theory Overview

Basic Procedure

The self-consistent NEGF-DFT cycle:

Initial Guess
- Start with DFT density
- Set up contact parameters
- Define integration grid
NEGF Step
- Calculate Green’s functions
- Construct density matrix
- Update chemical potentials
DFT Step
- Generate Fock matrix
- Update electronic structure
- Check convergence
Iterate until convergence

Implementation Classes

The gauNEGF package provides two main classes for NEGF-DFT calculations:

NEGF Class (scf.py)
- Energy-independent self-energies [Damle2002]
- Constant broadening
- Simple contact models
- Faster calculations
- Suitable for quick tests and initial setup
NEGFE Class (scfE.py)
- Energy-dependent self-energies
- Temperature effects
- Advanced contact models (Bethe lattice [Jacob2011], 1D chain)
- Not approximate
- Longer calculations (10-100x compute)

Mathematical Details

Density Matrix

The non-equilibrium density matrix has two components:

\[P = P_{eq} + P_{neq}\]

which are given by the definite integrals (assuming T=0):

\[ \begin{align}\begin{aligned}P_{eq} = -\frac{1}{\pi} \Im \int_{-\infty}^{E_F} G^r(E) dE\\P_{neq} = -\frac{1}{2\pi} \int_{E_F}^{E_F+V/2} G^r(E)\Gamma(E)G^a(E) dE + -\frac{1}{2\pi} \int_{E_F}^{E_F-V/2} G^r(E)\Gamma(E)G^a(E) dE\end{aligned}\end{align} \]

For the energy-independent case (NEGF), Γ(E) is constant. For the energy-dependent case (NEGFE), both G(E) and Γ(E) vary with energy.

Implementation

Integration Methods

Energy-Independent Case (NEGF):

from gauNEGF.scf import NEGF

# Initialize with constant self-energies
negf = NEGF('molecule', basis='lanl2dz')
negf.setSigma(lContact=[1], rContact=[6], sig=-0.1j)  # Simple constant self-energy

Energy-Dependent Case (NEGFE):

from gauNEGF.scfE import NEGFE

# Initialize with energy-dependent self-energies
negf = NEGFE('molecule', basis='lanl2dz')
negf.setContactBethe([1,2,3], [4,5,6], latFile='Au', T=300)  # Bethe lattice with temperature

Convergence Acceleration

Density mixing strategies (applicable to both NEGF and NEGFE):

The Pulay mixing method [Pulay1980] is a powerful convergence acceleration technique that uses information from previous iterations to predict the optimal density matrix. This method is particularly effective for systems with challenging convergence behavior.

# Simple mixing
negf.SCF(damping=0.02, pulay=False)

# Pulay mixing (DIIS)
negf.SCF(damping=0.02, pulay=True, nPulay=4)  # Use 4 previous iterations

Fermi Energy Search

The Fermi energy is found by a root-finding search on the contact electron count (NEGFE only). The search algorithm is selected via the fermiMethod keyword on gauNEGF.scfE.NEGFE.setVoltage():

# Default search (Muller's method, falls back to bisection)
negf.setVoltage(qV)

# Explicit method selection
negf.setVoltage(qV, fermiMethod='poly')    # 3rd-order polynomial fit
negf.setVoltage(qV, fermiMethod='secant')  # for strongly-coupled contacts
negf.setVoltage(qV, fermiMethod='bisect')  # always-safe fallback

For 1D auto-extract contacts, do not pass an explicit Fermi energy to setVoltage; let the search run. See Configuration and Tuning for the full method comparison and selection guidance.

Practical Considerations

Choosing Between NEGF and NEGFE

Guidelines for selecting the appropriate class:

Use NEGF when:
- Quick initial tests are needed
- System is well-described by constant self-energies
- Temperature effects are negligible
- Performance is critical
Use NEGFE when:
- Accurate transport properties are needed
- Temperature effects are important
- Realistic contact models are required
- Energy-dependent effects are significant

Convergence Issues

Common problems and solutions:

Charge Oscillations
- Reduce mixing parameter
- Increase Pulay vectors
- Check contact parameters
Slow convergence
- Add broadening (eta) to surfG
- Change fermi solver
- Reduce system/basis size

Example Workflows

Basic NEGF Calculation

Quick test with energy-independent self-energies:

from gauNEGF.scf import NEGF

# Initialize system
negf = NEGF('molContact', basis='lanl2dz')
negf.setSigma(lContact=[1], rContact=[2], sig=-0.05j)
negf.setVoltage(0.0)

# Run SCF
negf.SCF(conv=1e-4, damping=0.02)

Production NEGFE Calculation

Accurate calculation with temperature effects:

from gauNEGF.scfE import NEGFE

# Initialize system
negf = NEGFE('molecule', basis='lanl2dz')
negf.setContactBethe([[1,2,3], [4,5,6]], latFile='Au', T=300)

# Use defaults for fermiMethod -- see /guides/config_tuning for tuning
negf.setVoltage(0.0)
negf.SCF(conv=1e-4, damping=0.02)

See Bethe Lattice Contacts and Configuration and Tuning for production workflows and Fermi-search tuning.

Next Steps

Continue to Transport Calculations for details on calculating transport properties.

[Damle2002]

Damle, P., Ghosh, A. W., & Datta, S. (2002). First-principles analysis of molecular conduction using quantum chemistry software. Chemical Physics, 281(2-3), 171-187. DOI: 10.1016/S0301-0104(02)00496-2

[Pulay1980]

Pulay, P. (1980). Convergence acceleration of iterative sequences. The case of SCF iteration. Chemical Physics Letters, 73(2), 393-398. DOI: 10.1016/0009-2614(80)80396-4

[Jacob2011]

Jacob, D., & Palacios, J. J. (2011). Critical comparison of electrode models in density functional theory based quantum transport calculations. The Journal of Chemical Physics, 134(4), 044118. DOI: 10.1063/1.3526044