3.26 Hartree-Fock and hybrid functionals, including periodic systems

Periodic versions of the Hartree-Fock method and of and hybrid density functionals are implemented in FHI-aims. Generally, our experiences are very good. The implementation is stable and seems to scale well towards large systems. However, we still ask you to exercise some care. If you encounter any unexpected difficulties, consult the developers.

Periodic versions of MP2, RPA and $\mathrm{G}\mathrm{W}$ are not yet part of the main FHI-aims distribution, as they are in various stages of development. Please feel free to ask at our Slack channel regarding these methods for periodic systems. They are very important to us, and we will be happy to share them as they become ready and more usable.

Complete descriptions of the material described below, as well as extensive benchmarks, are summarized in Ref. [197], as well as Ref. [151] (for the non-periodic implementation of the “LVL” approach). Forces and the stress tensor are also implemented, with a clear description of the stress tensor given in Ref. [171].

There are two specific issues from a usability point of view which should be considered:

• •

  The RI_method “LVL” described below is implemented in a linear scaling and is therefore the only reasonable pathway for large and/or periodic systems. It is, however, slightly less accurate (for formal reasons) than the non-linear-scaling version for non-periodic systems, RI-V. Please bear this in mind. For standard solids and hybrid functionals, the effects seem very small. See reference [151] for quantitative tests. In general, RI-LVL has been extremely reliable for us.

• •

  For band structure output, the exx_band_structure_version keyword allows to toggle between a faster real-space version that works only when a relatively dense $k$-space grid has been used during the regular s.c.f. cycle and a slow(!) fallback version that is calculated in reciprocal space. The underlying reason is that the real-space Born-von Karman cell of the regular s.c.f. cycle may become too small to accommodate some $k$-vectors that are not exact reciprocal lattice vectors of the Born-von Karman cell. The slow fallback version should not simply be used by default since it can easily become the computational bottleneck – both regarding time and memory.

In periodic Hartree-Fock (and hybrid functional) implementations, the key quantity that needs to be evaluated is the exact-exchange matrix,

$$K_{ij}(𝐤)=\sum _{kl,𝐪}{D}_{kl}(𝐪)\iint 𝑑𝐫𝑑{𝐫}^{\prime }\frac{{\varphi }_{i𝐤}^{\ast }(𝐫){\varphi }_{k𝐪}(𝐫){\varphi }_{l𝐪}^{\ast }({𝐫}^{\prime }){\varphi }_{j𝐤}({𝐫}^{\prime })}{|𝐫-{𝐫}^{\prime }|}$$

(3.92)

where $𝐤,𝐪$ are the Bloch vectors, ${\varphi }_{i𝐤}(𝐫)$ is the Bloch summation of the $i$-th atomic orbital ${\varphi }_i(𝐫-𝐑)$ living in the unit cell $𝐑$, and $D_{kl}(𝐪)$ is the density matrix. $K_{ij}(𝐤)$ can be obtained from its Fourier transform,

$$K_{ij}(𝐤)=\sum _{𝐑}{e}^{i\mathrm{𝐤𝐑}}{X}_{ij}(𝐑),$$

(3.93)

where

$$X_{ij}(𝐑)=\sum _{kl}\sum _{{𝐑}^{\prime }}{D}_{kl}({𝐑}^{\prime })\sum _{{𝐑}^{\prime \prime }}\iint 𝑑𝐫𝑑{𝐫}^{\prime }\frac{{\varphi }_i(𝐫){\varphi }_k(𝐫+{𝐑}^{\prime \prime }){\varphi }_j({𝐫}^{\prime }+𝐑){\varphi }_l({𝐫}^{\prime }+{𝐑}^{\prime }+{𝐑}^{\prime \prime })}{|𝐫-{𝐫}^{\prime }|}$$

(3.94)

In FHI-aims, periodic Hartree-Fock and hybrid density functionals are implemented in two different ways. One implementation is based on the “k-space” formulation, where one computes $K_{ij}(𝐤)$ directly from Eq. (3.92). An alternative, and more efficient implementation is based on the“real-space” formulation, where one first computes $X_{ij}(𝐑)$ from Eq. (3.94), and then Fourier transform it to $K_{ij}(𝐤)$. The “real-space” implementation is used in the code by default, and the “k-space” implementation is only used for crossing-check purposes.

Both implementations are based on a localized resolution-of-identity approximation, which we termed as “RI-LVL”, in analogy to “RI-SVS” and “RI-V” introduced in Sec. 3.25. Under “RI-LVL”, the products of two normal basis functions $(i,j)$ centering at atoms $A_i$ and $A_j$ are expanded only in terms of auxiliary functions centering on these two atoms. Possible contributions of auxiliary functions from a third center are excluded in this approximation, in contrast to “RI-V”. Specifically, one has

$${\varphi }_i(𝐫-A_i){\varphi }_j(𝐫-A_j)=\sum _{\mu }{C}_{i(A_i),j(A_j)}^{\mu (A_i)}{P}_{\mu }(𝐫-A_i)+\sum _{\nu }{C}_{i(A_i),j(A_j)}^{\nu (A_j)}{P}_{\nu }(𝐫-A_j),$$

(3.95)

where $\mu$ and $\nu$ enumerate the auxiliary basis functions centering on atom $A_i$ and $A_j$ respectively. This approximation has been extensively benchmarked with respect to the more accurate “RI-V” approximation for finite systems, and with respect to other independent implementations for molecular systems. The achieved accuracy is remarkable and should be sufficiently good for production calculations.

Periodic Hartree-Fock and hybrid-functional calculations can be run in the same manner as the periodic LDA and GGA cases, by setting the keyword xc to hf or desired hybrid functionals, and setting the k_grid mesh to appropriate values. As mentioned above, by default the “real-space” periodic Hartree-Fock implementation will be invoked. There are two thresholding parameters (detailed below) which control the balance between the computational load and accuracy in the calculation. One may also switch to the “k-space” implementation of periodic Hartree-Fock and hybrid functionals for testing or comparison purposes by setting the keyword use_hf_kspace to be true (see below). The thresholding parameters do not apply to the “k-space” implementation, however.