Random-phase approximation and its applications in computational chemistry and materials science

Abstract

The random-phase approximation (RPA) as an approach for computing the electronic correlation energy is reviewed. After a brief account of its basic concept and historical development, the paper is devoted to the theoretical formulations of RPA, and its applications to realistic systems. With several illustrating applications, we discuss the implications of RPA for computational chemistry and materials science. The computational cost of RPA is also addressed which is critical for its widespread use in future applications. In addition, current correction schemes going beyond RPA and directions of further development will be discussed.

Appendices

Appendix 1: RI-RPA implementation in FHI-aims

In this section, we will briefly describe how RPA is implemented in the FHI-aims code [134] using the resolution-of-identity (RI) technique. More details can be found in Ref. [133]. For a different formulation of RI-RPA see Ref. [30, 40]. We start with the expression for the RPA correlation energy in Eq. (23), which can be formally expanded in a Taylor series,

$$ E^{\rm RPA}_{\rm c} = -\frac{1}{\pi} \int\limits_0^\infty d\omega \sum_{n=2}^\infty \frac{1}{2n} \hbox{Tr}\left[\left( \chi^0(i\omega)v \right)^n \right]. $$

(52)

Applying RI to RPA in this context means to represent both χ⁰(iω) and v in an appropriate auxiliary basis set. Eq. (52) can then be cast into a series of matrix operations. To achieve this we perform the following RI expansion

$$ \psi_i^\ast({\bf r})\psi_j({\bf r}) \approx \sum_{\mu=1}^{N_{\rm aux}} C_{ij}^\mu P_\mu({\bf r}) , $$

(53)

where P _μ(r) are auxiliary basis functions, C ^μ _ij are the expansion coefficients, and N _aux is the size of the auxiliary basis set. Here C serves as the transformation matrix that reduces the rank of all matrices from N _occ*N _vir to N _aux, with N _occ, N _vir and N _aux being the number of occupied single-particle orbitals, unoccupied (virtual) single-particle orbitals, and auxiliary basis functions, respectively. The determination of the C coefficients is not unique, but depends on the underlying metric. In quantum chemistry the “Coulomb metric” is the standard choice where the C coefficients are determined by minimizing the Coulomb repulsion between the residuals of the expansion in Eq. (53) (for details see Ref. [133] and references therein). In this so-called “RI-V” approximation, the C coefficients are given by

$$ C_{ij}^{\mu} = \sum_{\nu}(ij|\nu)V_{\nu\mu}^{-1} , $$

(54)

where

$$ (ij|\nu)=\iint \frac{\phi_i({\bf r})\phi_j({\bf r})P_\nu({\bf r^{\prime}})}{|{\bf r}-{\bf r^{\prime}}|} {\rm d}{\bf r} {\rm d}{\bf r^{\prime}} , $$

(55)

and

$$ V_{\mu\nu}=\int \frac{P_\mu({\bf r}) P_\nu({\bf r^{\prime}})}{|{\bf r}-{\bf r^{\prime}}|} {\rm d}{\bf r} {\rm d}{\bf r^{\prime}}. $$

(56)

In practice, sufficiently accurate auxiliary basis set can be constructed such that N _aux ≪ N _occ*N _vir, thus reducing the computational effort considerably. A practically accurate and efficient way of constructing auxiliary basis set {P _μ(r)} and their associated {C ^μ _ij } for atom-centered basis functions of general shape has been presented in Ref. [133].

Combining Eq. (20) with (53) yields

$$ \begin{aligned} \chi^0({\bf r},{\bf r^{\prime}}, i\omega)& = \sum_{\mu\nu} \sum_{ij} \frac{(f_i-f_j)C_{ij}^\mu C_{ji}^\nu} {\epsilon_i - \epsilon_j - i\omega} P_\mu({\bf r})P_\nu({\bf r^{\prime}}) \\ & = \sum_{\mu\nu} \chi^0_{\mu\nu}(i\omega) P_\mu({\bf r})P_{\nu}({\bf r^{\prime}}), \end{aligned} $$

(57)

where

$$ \chi^0_{\mu\nu}(i\omega) = \sum_{ij} \frac{(f_i-f_j)C_{ij}^\mu C_{ji}^\nu}{\epsilon_i - \epsilon_j - i\omega}. $$

(58)

Introducing the Coulomb matrix

$$ V_{\mu\nu} = \iint {\rm d}{\bf r}{\bf r^{\prime}} P_\mu({\bf r}) v({\bf r},{\bf r^{\prime}})P_{\nu}({\bf r^{\prime}}), $$

(59)

we obtain the first term in (52)

$$ \begin{aligned} E_{\rm c}^{(2)} = & -\frac{1}{4\pi} \int\limits_0^\infty d\omega \int\cdots\int {\rm d}{\bf r} {\rm d}{\bf r}_1 {\rm d}{\bf r}_2 {\rm d}{\bf r^{\prime}} \\ &\quad \times \chi^0({\bf r},{\bf r}_1) v({\bf r}_1,{\bf r}_2) \chi^0({\bf r}_2,{\bf r^{\prime}}) v({\bf r^{\prime}},{\bf r}) \\ = & -\frac{1}{4\pi} \int\limits_0^\infty d\omega \sum_{\mu\nu,\alpha\beta} \chi^0_{\mu\nu}(i\omega) V_{\nu\alpha} \chi^0_{\alpha\beta}(i\omega) V_{\beta,\mu} \\ = & -\frac{1}{4\pi} \int\limits_0^\infty {\rm d}\omega \hbox{Tr}\left[\left(\chi^0(i\omega)V\right)^2\right]. \end{aligned} $$

(60)

This term corresponds to the second-order direct correlation energy also found in MP2. Similar equations hold for the higher order terms in (52). This suggests that the trace operation in Eq. (23) can be re-interpreted as a summation over auxiliary basis function indices, namely, $\hbox{Tr}\left[AB\right] = \sum_{\mu\nu}A_{\mu\nu}B_{\nu\mu},$ provided that χ⁰(r, r′, iω) and v(r, r′) are represented in terms of a suitable set of auxiliary basis functions. Equations (23), (58), and (59) constitute a practical scheme for RI-RPA. In this implementation, the most expensive step is Eq. (58) for the construction of the independent response function, scaling as N ²_aux N _occ N _vir. We note that the same is true for standard plane-wave implementations [112] where N _aux corresponds to the number of plane waves used to expand the response function. In that sense RI-based local-basis function implementations are very similar to plane-wave-based or LAPW-based implementations, where the plane waves themselves or the mixed product basis serve as the auxiliary basis set.

Appendix 2: Error statistics for G2-I, S22, and NHBH38/NHTBH38 test sets

Tables 2, 3, and 4 present a more detailed error analysis for the G2-I, S22, and NHBH38/NHTBH38 test sets. Given are the mean error (ME), the mean absolute error (MAE), the mean absolute percentage error (MAPE), the maximum absolute percentage error (MaxAPE), and the maximum absolute error (MaxAE).

Table 2 ME (in eV), MAE (in eV), MAPE (%), MaxAPE(%) for atomization energies of the G2-I set obtained with four RPA-based approaches in addition to PBE, PBE0 and MP2. A negative ME indicates overbinding (on average) and a positive ME underbinding. The cc-pV6Z basis set was used

Table 3 ME (in meV), MAE (in meV), MAPE (%), and MaxAPE(%) for the S22 test set [144] obtained with five RPA-based approaches in addition to PBE, PBE0, and MP2 obtained with FHI-aims. The basis set “tier 4 + a5Z-d” [133] was used in all calculations

Table 4 ME, MAE, and MaxAE (in eV) for the HTBH38 [187] and NHTBH38 [188] test sets obtained with four RPA-based approaches in addition to PBE, PBE0, and MP2, as obtained using FHI-aims. The cc-pV6Z basis set was used in all calculations. Negative ME indicates an underestimation of the barrier height on average