# Why does the GW approximation give accurate quasiparticle energies? The cancellation of vertex corrections quantified

Arno Förster, Fabien Bruneval

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Arno Förster, Fabien Bruneval. Why does the GW approximation give accurate quasiparticle energies? The cancellation of vertex corrections quantified. Journal of Physical Chemistry Letters, 2024, 15 (51), pp.12526. 10.1021/acs.jpclett.4c03126. cea-04995686

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# Why does the GW approximation give accurate quasiparticle energies? The cancellation of vertex corrections quantified

Arno Förster

Theoretical Chemistry, Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands

Fabien Bruneval

Université Paris-Saclay, CEA, Service de recherche en Corrosion et Comportement des Matériaux, SRMP, 91191 Gif-sur-Yvette, France

(*fabien.bruneval@cea.fr)

(*a.t.l.foerster@vu.nl)

(Dated: October 23, 2024)

Hedin's GW approximation to the electronic self-energy has been impressively successful to calculate quasiparticle energies, such as ionization potentials, electron affinities, or electronic band structures. The success of this fairly simple approximation has been ascribed to the cancellation of the so-called vertex corrections that go beyond GW. This claim is mostly based on past calculations using vertex corrections within the crude local-density approximation. In this study, we explore a wide variety of non-local vertex corrections in the polarizability and the self-energy, using first-order approximations or infinite summations to all orders. In particular, we use vertices based on statically screened interactions like in the Bethe-Salpeter equation. We demonstrate on realistic molecular systems that indeed the two vertices in Hedin's equation essentially compensate. We further show that the consistency between the two vertices is crucial to obtain realistic electronic properties. We finally consider increasingly large clusters and extrapolate that our conclusions would hold for extended systems.

*Introduction.* Spectroscopic properties of many-electron systems are often described in terms of effective equations for single- and two-particle Green's functions first formulated by Hedin [1]. Hedin's equations start from the Dyson equation for the single-particle Green's function *G* and express the corresponding self-energy Σ in terms of a dynamically screened electron-electron interaction *W* and a vertex function Γ.

Practical calculations have to approximate the vertex function. The most drastic of these approximations is the GW approximation (GWA) [1–3], where the vertex function is reduced to delta functions. First applied to extended systems [1, 4–10] and later to small metal clusters [11–13] and molecules [14–21], it is by now widely used to describe quasiparticle (QP) levels and band structures in systems as diverse as complex molecules [22, 23], molecule-metal interfaces [24–26], dye-sensitized solar cells [27–29] or Moiré materials [30–32].

In weakly correlated systems, the GWA is relatively accurate for two reasons. First, the dynamical screening of the electron-electron interaction at large distances captures a significant source of electron correlation [33, 34]. While this seems natural in extended systems, it is remarkable that the GWA often gives highly accurate QP energies in atoms and molecules with sometimes only a few electrons [20, 35–37]. This hints towards major cancellations between higher-order terms in the self-energy as a second reason for the success of the GW approximation.

However, despite numerous studies [16, 38–82], these cancellations are still poorly understood. The partial cancellation of vertex corrections in *W* and in *Σ* has first been demonstrated for aluminum [38] and silicon [40, 41, 50]. While vertex corrections improve fully self-consistent GW (scGW) band gaps and satellites [43, 57, 58, 61, 65, 77, 83, 84], almost all GW calculations replace the interacting *G* by an effective non-interacting *G*⁽⁰⁾ that may be judiciously chosen to achieve high accuracy for molecular QP energies [82, 85, 86]. Many

authors argue that partial cancellations of vertex corrections in *W* and *Σ* in combination with the QP approximation to *G* is another reason for the success of the GWA in practice [87] but this subject is debated.

Including the very same vertex consistently in *W* and *Σ* allows one to quantify these cancellations rigorously. Following this strategy, previous work has demonstrated the Hartree-Fock (HF) vertex to improve over GW QP excitations and satellites in atoms and small molecules [63, 72, 79]. The resulting self-energy is self-screening free [88, 89] but comes with the disadvantage that its beyond-GW contribution is expanded in terms of the bare Coulomb interaction instead of the screened one. Especially in larger systems where screening effects are potentially strong, a screened TDHF vertex should be more realistic. Patterson has recently performed such calculations [81, 90], albeit within the Tamm-Dancoff approximation (TDA) in *L* and *Σ*. Within the TDA the same vertex has also been used by Cunningham *et al.* [66, 91] within quasiparticle self-consistent GW (qsGW) [87, 92–94] but without any vertex correction in *Σ*.

We here build on these works and further explore the maze of vertex corrections, which is still today mostly unmapped. Our quantitative conclusions are based on well-established molecular benchmarks where accurate wavefunction method-based results offer unambiguous references. We consider bare and screened exchange vertices which we consistently include in the expression for *W* and *Σ*. For a wide range of molecules, including one-dimensional and two-dimensional models of graphene and passivated silicon clusters, we demonstrate far-reaching cancellations of vertex corrections, which rationalizes the success of the GW approximation from small molecules to extended systems.

*Theory.* We write the self-energy in the form [95–97]
$$ \Sigma_{xc}(1,2) = iv(1^+,2)G(1,2) + iv(1^+,3)G(1,4)I(4,6,2,5)L(5,3,6,3), \quad (1) $$

where integers $n = (r_n, \sigma_n, t_n)$ collects spatial coordinates, spin and time, $v$ is the usual 2-point Coulomb interaction and $I(1,2,3,4) = i\delta\Sigma(1,3)/\delta G(4,2)$ is the 4-point irreducible kernel. Integration over repeated indices is implied. In the following, we focus on closed shells only and therefore assume spin compensation. The two-particle correlation function $L$ is obtained through the solution of a Bethe-Salpeter equation (BSE)

$$ L(1,2,3,4) = L^{(0)}(1,2,3,4) + L^{(0)}(1,5,3,6)I(6,7,5,8)L(8,2,7,4), \quad (2) $$

where we introduced the non-interacting correlation function $L^{(0)}(1,2,3,4) = -iG(1,4)G(2,3)$ and the very same kernel $I$ as in $\Sigma$ appears. Complemented with the Dyson equation for $G$, eqs. (1) and (2) yields a self-consistent scheme that is completely equivalent to Hedin's equations [98]. The present scheme has the major advantage that the 3-point vertex $\Gamma$ only appears implicitly and its explicit calculation is effectively replaced by the solution of the BSE in Eq. (2).

We follow previous work [63, 68, 69, 72, 79, 85] and exclusively work with a Hartree-Fock (HF) Green's function

$$ G^{(0)}(\mathbf{r}, \mathbf{r}', \omega) = \sum_{i}^{\text{occ}} \frac{\varphi_i(\mathbf{r}) \varphi_i(\mathbf{r}')}{\omega - \varepsilon_i - i\eta} + \sum_{a}^{\text{virt}} \frac{\varphi_a(\mathbf{r}) \varphi_a(\mathbf{r}')}{\omega - \varepsilon_a + i\eta} \quad (3) $$

expressed in terms of HF orbitals $\varphi$ and HF eigenvalues $\varepsilon$. The indices $i, j, k, \dots$ denote occupied and $a, b, c, \dots$ unoccupied (or virtual) states. $\eta$ is an infinitesimal positive real number. Since HF is diagrammatic, arbitrariness in the choice of $G^{(0)}$ is avoided. Moreover, for small molecules, HF orbitals are known to be close to true Dyson orbitals [99]. For the kernel we choose the expression

$$ I(3,5,4,6) = \delta(3,4)\delta(5,6)v(3,6) - \delta(3,6)\delta(4,5)W_0(3,4), \quad (4) $$

For $W_0 = 0$, the GWA is recovered and with $W_0 = v$, one obtains the time-dependent HF (TDHF) self-energy [79]. Another possibility is to set $W_0 = W(\omega = 0)$ where $W$ denotes the screened Coulomb interaction calculated within the random-phase approximation (RPA). $L$ then turns into the usual BSE implemented in many electronic structure codes, with the important difference that it is constructed with HF eigenvalues instead of the GW ones. By choosing a static approximation to $I$, only the electron-hole part of $L$ will contribute to $\Sigma$. Eq. (2) turns into a function of a single frequency which can be solved exactly by diagonalization in the particle-hole representation [100]. This part of $L$ is [101]

$$ L(\mathbf{r}_5, \mathbf{r}_3, \mathbf{r}_6, \mathbf{r}_3, \omega) = -i \sum_S \left[ \frac{\chi_S(\mathbf{r}_5, \mathbf{r}_6) \chi_S^*(\mathbf{r}_3, \mathbf{r}_3)}{\omega - \Omega_S + i\eta} - \frac{\chi_S(\mathbf{r}_3, \mathbf{r}_6) \chi_S^*(\mathbf{r}_6, \mathbf{r}_5)}{\omega + \Omega_S - i\eta} \right] \quad (5) $$

where $\Omega_S$ are the neutral excitation energies of the system and the amplitudes

$$ \chi_S(\mathbf{r}_1, \mathbf{r}_2) = \sum_{ia} X_{ia}^S \varphi_a(\mathbf{r}_1) \varphi_i^*(\mathbf{r}_2) + \sum_{ia} Y_{ia}^S \varphi_i(\mathbf{r}_1) \varphi_a^*(\mathbf{r}_2), \quad (6) $$

are expressed in terms of resonant and anti-resonant transition matrix elements $X$ and $Y$. We use the consistent notations for $X$ and $Y$ as those used in the usual Casida's equations solution [102]. The correlation part of the self-energy can now be written as $\Sigma = \Sigma^o + \Sigma^v$ with the contributions

$$ \Sigma_{pq}^{o}(\omega) = \sum_{S} \sum_{k} \frac{1}{\omega - \varepsilon_{k} + \Omega_{S} - i\eta} \times \left[ \sum_{ia} 2(ai|v|qk)(X_{ia}^{S} + Y_{ia}^{S}) - (ka|W_{0}|qi)X_{ia}^{S} - (ki|W_{0}|qa)Y_{ia}^{S} \right] \times \left[ \sum_{jb} (bj|v|pk)(X_{jb}^{S} + Y_{jb}^{S}) \right] \quad (7) $$

and

$$ \Sigma_{pq}^{v}(\omega) = \sum_{S} \sum_{c} \frac{1}{\omega - \varepsilon_{c} - \Omega_{S} + i\eta} \times \left[ \sum_{ia} 2(ai|v|qc)(X_{ia}^{S} + Y_{ia}^{S}) - (ci|W_{0}|qa)X_{ia}^{S} - (ca|W_{0}|qi)Y_{ia}^{S} \right] \times \left[ \sum_{jb} (bj|v|pc)(X_{jb}^{S} + Y_{jb}^{S}) \right] \quad (8) $$

The factor of 2 comes from spin-summation and is absent in the exchange terms. The four-center integrals for $v$ and $W_0$ are defined with the chemists' notation:

$$ (pq|v|rs) = \int d\mathbf{r} \int d\mathbf{r}' \varphi_p^*(\mathbf{r}) \varphi_q(\mathbf{r}) v(\mathbf{r}, \mathbf{r}') \varphi_r^*(\mathbf{r}') \varphi_s(\mathbf{r}') . \quad (9) $$

For a detailed derivation, we refer to the SI. Other authors have already used this scheme presented there with the TDHF kernel [63, 64, 68, 79] and we extend it here by using a screened exchange kernel. Based on the prior knowledge about BSE success and the TDHF mixed performance to describe neutral excitations [103], we expect this improvement to be significant.

The Dyson-like structure of the equations ensure that the kernel is consistently included to infinite-order in $L$ and hence in $\Sigma$. We go beyond approaches that include the vertex in $\Sigma$ to first order only [40, 45, 47], leading for instance to $G3W2$ vertex-corrections [57, 60, 61, 82] and approximations like its completely statically screened version [53, 76], SOSEX [36, 55, 104], or subsets of $G3W2$ [54, 58, 70]. Using different kernels in $L$ and $\Sigma$ is possible, but we show here that the kernels should be kept consistent. The different approximations are summarized in Table I.

### Results.

In the following, we discuss numerical results. We first test different vertex corrected schemes on the GW100 test set of first molecular ionization potentials [105]. We perform all calculations with MOLGW [106] and BAND [107, 108] using the def2-qzvpp basis set and use the corresponding CCSD(T) values from Ref. 109 as reference.

Figure 1 shows their error distributions of several vertex-corrected schemes compared to CCSD(T) together with mean absolute deviations (MAD) and root mean square deviations
<table><caption>Table I. Summary of the different infinite order approximations used in this work.</caption><thead><tr><th>I(6, 7, 5, 8) in L</th><th>L</th><th>I(3, 5, 4, 6) in Σ</th><th>Σ</th><th>Vernacular name</th></tr></thead><tbody><tr><td>v(6, 8)δ(6, 5)δ(7, 8)</td><td>L<sup>TDH</sup></td><td>v(3, 6)δ(3, 4)δ(5, 6)</td><td>GW @L<sup>TDH</sup></td><td>standard GW</td></tr><tr><td>v(6, 8)δ(6, 5)δ(7, 8) - v(6, 5)δ(6, 8)δ(7, 5)</td><td>L<sup>TDHF</sup></td><td>v(3, 6)δ(3, 4)δ(5, 6)</td><td>GW @L<sup>TDHF</sup></td><td>GW with TDHF screening</td></tr><tr><td>v(6, 8)δ(6, 5)δ(7, 8) - W<sub>0</sub>(6, 5)δ(6, 8)δ(7, 5)</td><td>L<sup>BSE</sup></td><td>v(3, 6)δ(3, 4)δ(5, 6)</td><td>GW @L<sup>BSE</sup></td><td>GW with BSE screening</td></tr><tr><td>v(6, 8)δ(6, 5)δ(7, 8) - v(6, 5)δ(6, 8)δ(7, 5)</td><td>L<sup>TDHF</sup></td><td>v(3, 6)δ(3, 4)δ(5, 6) - v(3, 4)δ(3, 6)δ(4, 5)</td><td>Σ<sup>TDHF</sup></td><td>TDHF self-energy</td></tr><tr><td>v(6, 8)δ(6, 5)δ(7, 8) - W<sub>0</sub>(6, 5)δ(6, 8)δ(7, 5)</td><td>L<sup>BSE</sup></td><td>v(3, 6)δ(3, 4)δ(5, 6) - W<sub>0</sub>(3, 4)δ(3, 6)δ(4, 5)</td><td>Σ<sup>BSE</sup></td><td>BSE self-energy</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>Figure 1. Errors of GW and several vertex corrected schemes with respect to CCSD(T) in eV of the HOMO of the molecules contained in the GW100 set for infinite vertex resummation (panel a) and first-order only (panel b).</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2. Vertex corrections in eV of the highest occupied molecular orbital of the molecules in the GW100 set. Besides the rare gases, the molecules are sorted by decreasing electronegativity of the element most represented in the HOMO.</figcaption></figure>

154 (RMSD). The leftmost violin in Fig. 1a) shows the errors of GW @L<sup>TDHF</sup> and GW @L<sup>BSE</sup> which only include the vertex in L. Therefore, our results temper the strong conclusions of Lewis et. al. [69], when they claim that the efforts to improve the screening part of the self-energy are worsening the results. However, GW @L<sup>TDHF</sup> and GW @L<sup>BSE</sup> (to a lesser extent) lead to major errors for some molecules. The kernel in Σ balances the sizable effect of the kernel in L, leading to

151 for GW @L<sup>TDHF</sup> and GW @L<sup>BSE</sup> which only include the vertex in L. Therefore, our results temper the strong conclusions of Lewis et. al. [69], when they claim that the efforts to improve the screening part of the self-energy are worsening the results. However, GW @L<sup>TDHF</sup> and GW @L<sup>BSE</sup> (to a lesser extent) lead to major errors for some molecules. The kernel in Σ balances the sizable effect of the kernel in L, leading to
168 consistent improvements over GW @RPA.

169 To understand this behavior, we show that the vertex corrections systematically have opposite signs in Fig. 2 and therefore 170 partially and sometimes completely cancel. The orange bars 171 show the magnitude of the vertex correction beyond TDH in 172 *L* (corresponding to the third violin in Fig. 1a)), and the blue 173 bars the magnitude of the vertex correction in Σ beyond GW 174 (corresponding to the last violin in Fig. 1a)). The red boxes 175 show the difference between GW @RPA and Σ<sup>BSE</sup> @L<sup>BSE</sup>, 176 which is the sum of the blue and orange bars. The BSE kernel 177 describes the electron-hole interaction missing in GW @RPA 178 which stabilizes the cation and therefore lowers the HOMO 179 energy. In some cases exceeding 0.6 eV, this effect is sizable for most molecules in GW100 and frequently the HOMO 180 energy is overcorrected. The vertex correction in Σ has the 181 opposite effect and reduces the HOMO energy further. The 182 effect of the vertex is generally stronger for *L* than for Σ and 183 therefore the combination of both vertex corrections lowers 184 the HOMO. Both vertices combined lead to the observed improvement of Σ<sup>BSE</sup> @L<sup>BSE</sup> over GW @RPA. 185

186 As shown in the SI (Fig. S1), a similar picture is obtained 187 for the TDHF screening and self-energy approximations. Our 188 results qualitatively agree with Ref. 79. With average values 189 of 0.41 and -0.21 eV, the effect of the individual vertex cor- 190 rections in *L* and Σ is significantly larger. However, with 0.2 191 eV on average, the combined effect of the vertex correction is 192 comparable to BSE. The BSE vertex correction accounts for 193 higher-order vertex diagrams not included in Σ<sup>TDHF</sup> @L<sup>TDHF</sup>. 194 The smaller magnitudes of the vertex corrections in *L* and Σ 195 with the BSE vertex indicate further cancellations between 196 these higher-order diagrams. 197

198 Further insight into the cancellation of vertex corrections is 199 provided in Fig. 1b) where we show the errors of the same 200 vertex-corrected schemes as in Fig. 1a), but in all cases trun- 201 cated to first order. For the polarizability, this means that RPA 202 screening is modified by including only the first-order ladder 203 diagram [44, 45, 60]. Including the same vertex diagram in 204 Σ the SOSEX self-energy is obtained with the bare vertex[82] 205 and the screened vertex leads to a second-order term similar to 206 SOSEX but with the bare *v* replaced by the statically screened 207 one. In this scheme, the vertex correction is consistent in *L* 208 and Σ since the next-to-leading order diagram is added to both 209 quantities. We refer to the SI (section S3) for detailed deriva- 210 tions. 211

212 Including the kernel to first order only has generally a much 213 smaller effect than the infinite-order resummations (0.17 eV 214 on average vs. 0.41 eV for *L* and -0.13 vs. -0.21 eV 215 for Σ) As shown in the Supporting Information Fig. S2 216 and Fig. S3, adding the same vertex correction to *W* and 217 Σ results in HOMO energies almost indistinguishable from 218 GW @RPA. The same conclusion has already been drawn 219 based on results for the band gap of silicon [40] and for a one- 220 dimensional semiconductor [42] and we confirm here its va- 221 lidity for molecules. This almost complete cancellation of the 222 next-to-leading order terms in *L* and Σ rationalizes the good 223 performance of GW @RPA for calculating QP energies.

224 Finally, in Fig. 3, we show the magnitude of the different 225 BSE vertex corrections for the HOMO and LUMO energies

<figure><img src="image_4.png" /><figcaption>Figure 3. Screened exchange vertex corrections in eV of the HOMO (top), LUMO (bottom) for linear acenes, non-linear acenes, and passivated silicon clusters of increasing size.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4. Bare exchange vertex corrections in eV of the HOMO (top) and LUMO (bottom) for linear acenes, non-linear acenes, and passivated silicon clusters of increasing size.</figcaption></figure>

of molecules of systematically increasing size: Linear acenes
227 ranging from a single benzene ring ($C_6H_6$) up to hexacene ($C_{26}H_{16}$), coronene and circumcoronene, as well as passivated silicon clusters with up to 37 silicon atoms. For the BSE vertex, we find the magnitude of the vertex correction to be almost independent of the system size and with about 0.1 eV to be rather small. The effect on the LUMO is with 0.2 eV about twice as large. This observation is consistent with Ref. 68. While initially increasing, the magnitude of the vertex correction stays approximately constant for the linear acenes and silicon clusters. We also notice that the first-order truncation of $\Sigma^{BSE}$ is always a good approximation.

Figure 4 shows the same information for the TDHF vertex. As for GW100, the magnitudes of the individual vertex corrections in $\Sigma$ and $L$ are larger than for the BSE vertex. Also, the total vertex correction is much larger than for the BSE vertex. Moreover, we observe especially for the linear acenes that the infinite-order resummation of the TDHF vertex in $\Sigma^{TDHF}$ leads to a rapidly increasing vertex correction for the HOMO. At the same time, its first-order approximation (SOSEX) goes to almost zero. The opposite can be observed for the LUMO. This inconsistency indicates the importance of screened vertices for larger systems.

Conclusions. Several vertex-corrected schemes have been investigated over the last decades to improve over the simple GWA for QP energies. Cancellations between vertex corrections have been observed early on for simple (model) systems [38, 41, 42]. Despite immense implications for practical GW calculations, these results have never been generally confirmed using realistic, non-local vertices.

With this work, we have filled this gap. To rationalize the success of the GWA for calculating QP energies, we have investigated several vertex corrections beyond the GWA. We benchmarked these methods for systems ranging from small and medium molecules in the GW100 set, over linear and non-

linear acenes, to silicon clusters. We have used the TDHF vertex as obtained from the HF self-energy, which adds infinite-order particle-hole diagrams to $L$ and $\Sigma$ as well as a BSE vertex which statically screens these diagrams. Especially for larger molecules it becomes decisive to use a screened vertex correction.

By restricting the series of ladder diagrams to first-order only, we have also performed calculations that only include the next-to-leading order correction to $L$ and $\Sigma$. Both corrections effectively cancel for HOMO QP energies, suggesting an order-by-order expansion of $L$ and $\Sigma$ beyond GW@RPA to be inefficient. Despite being of low order in perturbation theory, it accounts for the most important signatures of electron correlation for a charged excitation.

We have rationalized why schemes that add a vertex correction to either the response function or the self-energy have been unsuccessful [69, 82]. The cancellations between these vertices are far-reaching and they both must be included to obtain systematic improvements over GW. To improve over GW@RPA, infinite-order resummations of the vertex function are needed in both $L$ and $\Sigma$. Moving forward, dynamical vertex corrections could be explored. These would allow for the inclusion of the yet missing particle-particle channel to the self-energy which is important in the strongly correlated regime [75, 110].

ACKNOWLEDGMENTS

Part of this work was performed using HPC resources from GENCI–TGCC (Grant 2024-gen6018). AF acknowledges the use of supercomputer facilities at SURFsara sponsored by NWO Physical Sciences, with financial support from The Netherlands Organization for Scientific Research (NWO).

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