BSE Theory

The Bethe-Salpeter Equation (BSE) describes optical excitations including electron-hole interactions (excitons). This page provides a theoretical overview of the BSE methods implemented in QuasiX.

Introduction

While GW calculates charged excitations (adding/removing electrons), the BSE describes neutral optical excitations where an electron-hole pair is created.

Key concepts:

  • Optical excitation: Promotion of an electron from occupied to unoccupied state

  • Exciton: Bound electron-hole pair

  • Exciton binding energy: Difference between QP gap and optical gap

The BSE naturally captures:

  • Electron-hole attraction (screened Coulomb)

  • Exchange interactions (for singlet/triplet splitting)

  • Oscillator strengths (optical selection rules)

Theoretical Framework

Two-Particle Green’s Function

The BSE is the equation of motion for the two-particle correlation function:

\[L(12; 1'2') = G(1,2') G(2,1') + \int d3456 \, G(1,3) G(4,1') \Xi(34;56) L(56;2'2)\]

where \(\Xi\) is the BSE kernel containing electron-hole interactions.

BSE Hamiltonian

In matrix form, the BSE becomes a generalized eigenvalue problem:

\[\begin{split}\begin{pmatrix} A & B \\ -B^* & -A^* \end{pmatrix} \begin{pmatrix} X \\ Y \end{pmatrix} = \Omega \begin{pmatrix} X \\ Y \end{pmatrix}\end{split}\]

where:

  • \(\Omega\) are the excitation energies

  • \(X, Y\) are the electron-hole amplitudes

  • \(A\) is the resonant (particle-hole) block

  • \(B\) is the coupling (particle-hole, hole-particle) block

Matrix Elements

Resonant block (A):

\[A_{ia,jb} = (\varepsilon_a^{\text{QP}} - \varepsilon_i^{\text{QP}})\delta_{ij}\delta_{ab} + \alpha K^x_{ia,jb} - K^d_{ia,jb}\]

where:

  • \(\varepsilon_a^{\text{QP}}, \varepsilon_i^{\text{QP}}\) are QP energies from GW

  • \(K^x\) is the exchange (electron-hole exchange) kernel

  • \(K^d\) is the direct (screened Coulomb attraction) kernel

  • \(\alpha = 2\) for singlets, \(\alpha = 0\) for triplets

Exchange kernel (bare Coulomb, electron-hole exchange):

\[K^x_{ia,jb} = 2 \int d\mathbf{r} d\mathbf{r}' \, \psi_i^*(\mathbf{r})\psi_a(\mathbf{r}) \frac{1}{|\mathbf{r}-\mathbf{r}'|} \psi_j(\mathbf{r}')\psi_b^*(\mathbf{r}')\]
\[K^x_{ia,jb} = 2(ia|jb)\]

Direct kernel (screened Coulomb, electron-hole attraction):

\[K^d_{ia,jb} = \int d\mathbf{r} d\mathbf{r}' \, \psi_i^*(\mathbf{r})\psi_j(\mathbf{r}) W(\mathbf{r},\mathbf{r}') \psi_a(\mathbf{r}')\psi_b^*(\mathbf{r}')\]
\[K^d_{ia,jb} = (ij|W|ab)\]

Coupling block (B):

\[B_{ia,jb} = \alpha K^x_{ia,bj} - K^d_{ia,bj}\]

Spin Channels

Singlet Excitations

For singlet states (\(\alpha = 2\)), both exchange and direct kernels contribute:

\[A^{\text{singlet}}_{ia,jb} = \Delta_{ia}\delta_{ij}\delta_{ab} + 2(ia|jb) - (ij|W|ab)\]

Singlet excitations are optically bright (have finite oscillator strength).

Triplet Excitations

For triplet states (\(\alpha = 0\)), only the direct kernel contributes:

\[A^{\text{triplet}}_{ia,jb} = \Delta_{ia}\delta_{ij}\delta_{ab} - (ij|W|ab)\]

Triplet excitations are optically dark (zero oscillator strength due to spin selection rules).

Note

The singlet-triplet splitting arises from the exchange kernel \(K^x\). Typically, triplets lie 0.1-1 eV below singlets.

Tamm-Dancoff Approximation (TDA)

The Tamm-Dancoff approximation sets \(B = 0\), reducing the BSE to:

\[A X = \Omega X\]

This simplifies the problem to a Hermitian eigenvalue problem with several advantages:

  • Computational: Smaller matrix, standard diagonalization

  • Numerical: Always real eigenvalues (no instabilities)

  • Accuracy: Often sufficient for localized excitations

QuasiX implements BSE-TDA as the default.

When TDA is accurate:

  • Localized excitations (molecules)

  • Large HOMO-LUMO gaps

  • Single-reference ground states

When full BSE is needed:

  • Extended systems with small gaps

  • Charge-transfer excitations

  • Near-degenerate ground states

Optical Properties

Transition Dipole Moments

The transition dipole moment from the ground state \(|0\rangle\) to excited state \(|I\rangle\):

\[\boldsymbol{\mu}_{0I} = \langle 0 | \hat{\boldsymbol{\mu}} | I \rangle = \sum_{ia} X_{ia}^I \langle i | \hat{\mathbf{r}} | a \rangle\]

where \(X_{ia}^I\) are the BSE eigenvector components for state \(I\).

Oscillator Strengths

The oscillator strength measures the intensity of optical transitions:

\[f_I = \frac{2}{3} \Omega_I \sum_{\alpha=x,y,z} |\mu_{0I}^\alpha|^2\]

Selection rules:

  • \(f > 0\): Optically allowed (bright) transition

  • \(f = 0\): Optically forbidden (dark) transition

Sum rule:

\[\sum_I f_I = N_e \quad \text{(Thomas-Reiche-Kuhn sum rule)}\]

Absorption Spectrum

The absorption spectrum is constructed from discrete transitions using broadening:

\[\sigma(\omega) = \sum_I f_I \, L(\omega - \Omega_I; \gamma)\]

where \(L\) is a Lorentzian lineshape function:

\[L(\omega; \gamma) = \frac{1}{\pi} \frac{\gamma}{(\omega)^2 + \gamma^2}\]
# Generate absorption spectrum
energies, spectrum = bse.get_spectrum(
    energy_range=(4.0, 10.0),  # eV
    n_points=1000,
    broadening=0.1  # eV
)

Exciton Analysis

Exciton Binding Energy

The exciton binding energy is the difference between the quasiparticle gap (fundamental gap) and the optical gap:

\[E_{\text{bind}} = E_{\text{gap}}^{\text{QP}} - \Omega_1\]

where \(\Omega_1\) is the lowest singlet excitation energy.

Typical values:

  • Small molecules (H2O): 0.3-0.5 eV

  • Conjugated molecules (benzene): 0.8-1.2 eV

  • Extended systems: 0.01-0.1 eV

Exciton Wavefunction

The exciton wavefunction in real space:

\[\Psi^I(\mathbf{r}_e, \mathbf{r}_h) = \sum_{ia} X_{ia}^I \psi_a(\mathbf{r}_e) \psi_i^*(\mathbf{r}_h)\]

This describes the probability amplitude for finding the electron at \(\mathbf{r}_e\) and the hole at \(\mathbf{r}_h\).

Natural Transition Orbitals

The BSE eigenvector can be analyzed using natural transition orbitals (NTOs), obtained from SVD of the transition density matrix:

\[T_{ai} = X_{ia}^I \rightarrow T = U \Sigma V^T\]

The dominant NTO pairs characterize the excitation:

  • Hole NTO: \(\phi_h = \sum_i V_{i1} \psi_i\)

  • Particle NTO: \(\phi_p = \sum_a U_{a1} \psi_a\)

Implementation Details

QuasiX Implementation

  1. Static Screening: Uses \(W(\omega=0)\) from the preceding GW calculation

  2. RI Approximation: All integrals computed using density fitting

    \[(ij|W|ab) \approx \sum_{PQ} B^P_{ij} W_{PQ} B^Q_{ab}\]

  3. Davidson Solver: Iterative diagonalization for lowest eigenstates

  4. Active Space: Option to restrict to subset of occupied/virtual orbitals

Computational Workflow

  1. Run GW calculation to obtain:

    • Quasiparticle energies \(\varepsilon_n^{\text{QP}}\)

    • Screened interaction \(W\)

  2. Construct BSE matrix elements:

    • Exchange kernel from bare Coulomb integrals

    • Direct kernel from screened Coulomb integrals

  3. Solve BSE eigenvalue problem (TDA or full BSE)

  4. Compute transition dipoles and oscillator strengths

  5. Generate absorption spectrum

from quasix import G0W0Driver, BSEDriver
from pyscf import gto, scf

mol = gto.M(atom='...', basis='def2-svp')
mf = scf.RHF(mol).run()

# Step 1: GW calculation
gw = G0W0Driver(mf)
gw_result = gw.kernel()

# Step 2: BSE calculation
bse = BSEDriver(mf, gw_result=gw_result, n_states=10, spin='singlet')
bse_result = bse.kernel()

# Step 3: Analyze results
print(f"Optical gap: {bse_result.excitation_energies[0]:.3f} eV")
print(f"Exciton binding: {gw_result.gap_qp - bse_result.excitation_energies[0]:.3f} eV")

References

  1. Salpeter, E. E. & Bethe, H. A. (1951). Phys. Rev. 84, 1232. “A Relativistic Equation for Bound-State Problems”

  2. Rohlfing, M. & Louie, S. G. (2000). Phys. Rev. B 62, 4927. “Electron-hole excitations and optical spectra from first principles”

  3. Onida, G., Reining, L. & Rubio, A. (2002). Rev. Mod. Phys. 74, 601. “Electronic excitations: density-functional versus many-body Green’s-function approaches”

  4. Blase, X., Duchemin, I. & Jacquemin, D. (2018). Chem. Soc. Rev. 47, 1022. “The Bethe-Salpeter equation in chemistry: relations with TD-DFT, applications and challenges”

  5. Blase, X., Duchemin, I., Jacquemin, D. & Loos, P.-F. (2020). J. Phys. Chem. Lett. 11, 7371. “The Bethe-Salpeter Equation Formalism: From Physics to Chemistry”