Communication: A novel implementation to compute MP2 correlation energies without basis set superposition errors and complete basis set extrapolation

Abstract

By using a formulation based on the dynamical polarizability, we propose a novel implementation of second-order Møller-Plesset perturbation (MP2) theory within a plane wave (PW) basis set. Because of the intrinsic properties of PWs, this method is not affected by basis set superposition errors. Additionally, results are converged without relying on complete basis set extrapolation techniques; this is achieved by using the eigenvectors of the static polarizability as an auxiliary basis set to compactly and accurately represent the response functions involved in the MP2 equations. Summations over the large number of virtual states are avoided by using a formalism inspired by density functional perturbation theory, and the Lanczos algorithm is used to include dynamical effects. To demonstrate this method, applications to three weakly interacting dimers are presented.

I. INTRODUCTION

The ab initio methods that approximate the correlation energy on top of a Hartree-Fock (HF) ground-state are well established and widely used within the quantum chemistry community.¹ With respect to density functional theory (DFT),² these methodologies allow for a more systematic and rigorous improvement of the accuracy of the correlation energy. In the context of these post-HF methodologies, the simplest approximation is probably the Møller-Plesset perturbation theory to the second order (MP2).³ This approach is equivalent to the traditional Rayleigh-Schrödinger perturbation theory applied to a HF ground-state and truncated to the second order. While MP2 does not fully reach chemical accuracy, this approximation is often preferred over more accurate but computationally expensive methodologies based on coupled-cluster theories.⁴ Additionally, the interest in MP2 is increasing also in the DFT community due to the promising level of accuracy of double-hybrid functionals.⁵

The MP2 approximation is typically implemented in codes using Gaussian-type orbitals (GTOs) as a basis set and is widely available through several quantum chemical codes. In general, the MP2 correlation energy is characterized by a slow convergence with respect to the basis set size and complete basis set (CBS) extrapolation is typically required (this is the case also for other post-HF methods).⁶ Additionally, when studying the interaction between molecules (and/or atoms), the accuracy of atom-centered GTOs depends on the relative distance between monomers. Indeed, as monomers get closer, they can “borrow” each other’s basis set and the relative accuracy at different distances changes. This issue introduces the so-called basis set superposition error (BSSE).

By using an unconventional formulation based on response functions (polarizabilities), we discuss a new implementation to compute MP2 correlation energies within the plane-wave (PW) basis set. Since PWs are delocalized in space and not associated with specific atoms, this approach intrinsically avoids the BSSE.⁷ Additionally, the convergence with respect to the PW basis set size can be systematically tested by tuning a single parameter, the kinetic-energy cut-off. However, in numerical applications, the number of PWs is usually much larger than the number of GTOs. This is particularly evident for molecules and reduced dimensionality systems, where the use of large supercells leads to a huge number of PWs (up to several hundreds of thousands). Accordingly, a brute force MP2 implementation within PWs is problematic. In this work, this issue is overcome by using a polarizability-based expression for the MP2 correlation energy^8–10 and a compact auxiliary basis set to represent response functions; this basis set is computed from the projective dielectric eigenpotential (PDEP) technique first introduced by Wilson, Gygi, and Galli.¹¹ More generally, the MP2 implementation of this work takes advantage of a series of ideas introduced for different types of applications involving polarizabilities (or, equivalently, dielectric matrices): (1) The use of density functional perturbation theory techniques to eliminate the explicit use of virtual states;^12–16 (2) the use of a compact auxiliary basis set to represent response functions;^11,17–24 and (3) the use of the Lanczos algorithm to include dynamical effects and compute diagonal and off-diagonal matrix elements of the polarizability.^15,25,26 As shown by the results in Sec. III, these numerical techniques make MP2 calculations feasible even for very large numbers of PWs (>160 000) and avoid the need for a complete basis set extrapolation. This represents a completely new paradigm for post-HF calculations of the ground-state correlation energy.

It is important to mention that the MP2 approximation has already been implemented within the PW framework in the VASP code.²⁷ While this implementation is not affected by the BSSE, the MP2 correlation energy is computed for different values of the cut-off and then extrapolated to the CBS with an analytic expression. This approach has been applied to the Li atom, the LiH molecule, and simple periodic solids, all of which require small cells and consequently a small number of PWs. Other methods use a hybrid PW basis set and a GTO basis set but still involve the BSSE and require CBS extrapolation.^28,29

Within the GTO framework, the slow convergence of the correlation energy with respect to the basis set size can be to a large extent avoided by using explicitly correlated methods to better satisfy the cusp condition.³⁰ Notably, the MP2-F12 approach is characterized by a much faster basis-set convergence with respect to the conventional implementation of MP2.

In many respects, the present work is different from previous numerical approaches. Our methodology does not converge the total correlation energy but rather converges correlation energy differences, which are the relevant quantities for applications in physics and chemistry. This can be achieved without any extrapolation technique by taking advantage of the high accuracy of the PW basis set coupled with a smaller auxiliary basis set that allows for a compact and accurate description of the polarizability. Our approach does not change the scaling of standard MP2 implementations but allows for calculations involving hundreds of thousands of PWs. To demonstrate the method, in Sec. III, we will present a numerical study of three weakly interacting dimers. While the current code includes only the Γ point in the Brillouin zone integration, extensions to solids are in principle feasible³¹ and will be the subject of future work.

II. THEORY AND IMPLEMENTATION

In this section, we will follow the notation of Ref. 10. In general, we will use the subscript c (and $c^{\prime }$) to denote conduction (namely, virtual) states with corresponding energy ${𝜖}_c$ and space orbital ${\varphi }_c\left(\mathbf{r}\right)$; similarly, the subscript $v$ (and $v^{\prime }$) will be used for valence (namely, occupied) orbitals. The formalism will be developed in a spin-restricted way. From Eq. (15) of Ref. 10, we have the following expression for the MP2 correlation energy:

$$E_c^{\mathrm{\text{MP2}}}=-\frac{1}{4}{\int }_{-\infty }^{\infty }\frac{d\omega }{2\pi }\mathrm{\text{Tr}}\left\{{\mathbf{\chi }}^0\left(i\omega \right)\mathit{B}{\mathbf{\chi }}^0\left(i\omega \right)\mathit{K}\right\}$$ (1)

$$=-\frac{1}{2}\sum _{vc,v^{\prime }{c}^{\prime }}\frac{K_{vc,v^{\prime }{c}^{\prime }}{B}_{v^{\prime }{c}^{\prime },vc}}{{𝜖}_c+{𝜖}_{c^{\prime }}-{𝜖}_v-{𝜖}_{v^{\prime }}},$$ (2)

where $\mathrm{\text{Tr}}$ denotes the trace operator, $\omega$ the frequency, and

$${\mathbf{\chi }}^0\left(i\omega \right)=2\mathfrak{R}\frac{1}{i\omega +{𝜖}_v-{𝜖}_c}$$ (3)

is the independent electron polarizability. The matrix elements of the matrices K and B in Eqs. (1) and (2) are defined as

$$\begin{array}{rcl}{K}_{vc,v^{\prime }{c}^{\prime }}& =& 2⟨{\varphi }_v{\varphi }_c|V|{\varphi }_{v^{\prime }}{\varphi }_{c^{\prime }}⟩\\ & =& 2\int {\varphi }_v\left(\mathbf{r}\right){\varphi }_c\left(\mathbf{r}\right)V\left(\mathbf{r}-{\mathbf{r}}^{\prime }\right){\varphi }_{v^{\prime }}\left({\mathbf{r}}^{\prime }\right){\varphi }_{c^{\prime }}\left({\mathbf{r}}^{\prime }\right)d\mathbf{r}d{\mathbf{r}}^{\prime }\end{array}$$ (4)

and

$$\begin{array}{rcl}{B}_{vc,v^{\prime }{c}^{\prime }}& =& K_{vc,v^{\prime }{c}^{\prime }}-K_{vc,v^{\prime }{c}^{\prime }}^x\\ & =& 2⟨{\varphi }_v{\varphi }_c|V|{\varphi }_{v^{\prime }}{\varphi }_{c^{\prime }}⟩-⟨{\varphi }_v{\varphi }_{c^{\prime }}|V|{\varphi }_{v^{\prime }}{\varphi }_c⟩,\end{array}$$ (5)

where V is the Coulomb potential. The definition in Eq. (2) clearly corresponds to the traditional expression for the MP2 correlation energy.¹ While the formalism developed in Ref. 10 was based on an approximate exchange kernel, it is important to mention that the MP2 correlation energy in Eq. (1) is exact and strictly equivalent to Eq. (2). This can be readily verified by analytical evaluation of the integral over $\omega$. This work will use the formulation based on the independent electron polarizability ${\mathbf{\chi }}^0$ [Eq. (1)]. While already known in the literature,^8–10 this expression is not typically used in MP2 numerical implementations. Indeed, Eq. (1) requires a frequency integration that makes this formulation less appealing than the traditional expression in Eq. (2). However, on the imaginary axis, the polarizability is a smooth function, and similar numerical integrals are routinely computed to obtain random phase approximation (RPA) correlation energies.^{10,17,18,22,32–34} Up to this point, all the matrices have been expressed in a electron-hole representation based on products of valence and conduction states. Traditionally, solid state implementations rather use the PW basis set to represent response functions.^35,36 In order to make calculations faster and avoid the CBS extrapolation, here we use a more compact representation based on the projective dielectric eigenpotential (PDEP) technique.¹¹ Within this framework, the independent electron polarizability is diagonalized iteratively to obtain the eigenvalue/eigenvector decomposition ${\mathbf{\chi }}^0={\sum }_G{\mathrm{\lambda }}_G|{\mathrm{\Phi }}_G⟩⟨{\mathrm{\Phi }}_G|$. Since the eigenvalues ${\mathrm{\lambda }}_G$ decay rapidly to 0, the summation over G can be truncated to a small number of terms. This technique can be used directly to diagonalize ${\mathbf{\chi }}^0$ at finite frequencies $\omega$ and to compute the correlation energy at the random phase approximation level of theory.^17,18 The PDEP technique will be used only in the static limit $\omega =0$ to generate an auxiliary basis set that will also be used for the dynamical case.^26,37 Additionally, the use of a simple approximation for ${\mathbf{\chi }}^0$ based on the kinetic energy will significantly speed up calculations.²² In general, the number of auxiliary basis vectors (N_aux) generated by the PDEP method is much smaller than the number of PWs (N_PW). Since our implementation scales as $N_{PW}\times N_{aux}^2\times N_v^2$ (N_v being the number of occupied states), the use of a compact auxiliary basis set is crucial for good numerical performance.

By using the completeness relation $\mathit{I}={\sum }_G|{\mathrm{\Phi }}_G⟩⟨{\mathrm{\Phi }}_G|$ of the eigenvectors of the static (kinetic) ${\mathbf{\chi }}^0$, we can write Eqs. (4) and (5) as

$$K_{vc,v^{\prime }{c}^{\prime }}=2\sum _G⟨{\varphi }_v{\varphi }_c|V^{\frac{1}{2}}|{\mathrm{\Phi }}_G⟩⟨{\mathrm{\Phi }}_G|V^{\frac{1}{2}}|{\varphi }_{v^{\prime }}{\varphi }_{c^{\prime }}⟩,$$ (6)

$$K_{vc,v^{\prime }{c}^{\prime }}^x=\sum _G⟨{\varphi }_v{\varphi }_{c^{\prime }}|V^{\frac{1}{2}}|{\mathrm{\Phi }}_G⟩⟨{\mathrm{\Phi }}_G|V^{\frac{1}{2}}|{\varphi }_{v^{\prime }}{\varphi }_c⟩.$$ (7)

By introducing Eqs. (6) and (7) in the expression for the MP2 correlation energy, Eq. (1), and by using the cyclic property of the trace, we obtain

$$E_c^{\mathrm{\text{MP2}}}=-\frac{1}{2}{\int }_0^{\infty }\frac{d\omega }{2\pi }\mathrm{\text{tr}}\left\{{\mathit{C}}^2\left(i\omega \right)-\mathit{Z}\left(i\omega \right)\right\},$$ (8)

where

$$C_{QR}\left(i\omega \right)=4\sum _{vc}⟨{\mathrm{\Phi }}_Q|V^{\frac{1}{2}}|{\varphi }_v{\varphi }_c⟩\mathfrak{R}\frac{1}{i\omega +{𝜖}_v-{𝜖}_c}⟨{\varphi }_v{\varphi }_c|V^{\frac{1}{2}}|{\mathrm{\Phi }}_R⟩$$ (9)

and

$$Z_{QR}\left(i\omega \right)=2\sum _{v{v}^{\prime }}\sum _G{J}_{QG}^{v{v}^{\prime }}\left(i\omega \right)L_{GR}^{v{v}^{\prime }}\left(i\omega \right),$$ (10)

with

$$J_{QG}^{v{v}^{\prime }}\left(i\omega \right)=2\sum _c⟨{\mathrm{\Phi }}_Q|V^{\frac{1}{2}}|{\varphi }_v{\varphi }_c⟩\mathfrak{R}\frac{1}{i\omega +{𝜖}_v-{𝜖}_c}⟨{\varphi }_{v^{\prime }}{\varphi }_c|V^{\frac{1}{2}}|{\mathrm{\Phi }}_G⟩$$ (11)

and

$$\begin{array}{rcl}{L}_{GR}^{v{v}^{\prime }}\left(i\omega \right)& =& 2\sum _{c^{\prime }}⟨{\mathrm{\Phi }}_G|V^{\frac{1}{2}}|{\varphi }_v{\varphi }_{c^{\prime }}⟩\\ & & \times \mathfrak{R}\frac{1}{i\omega +{𝜖}_{v^{\prime }}-{𝜖}_{c^{\prime }}}⟨{\varphi }_{v^{\prime }}{\varphi }_{c^{\prime }}|V^{\frac{1}{2}}|{\mathrm{\Phi }}_R⟩.\end{array}$$ (12)

Equation (9) corresponds to the representation of ${\mathbf{\chi }}^0$ on the auxiliary basis set vectors $\mathrm{\Phi }$. The previous equations can be conveniently expressed by introducing the projector onto the conduction state subspace $\widehat{P}={\sum }_c|{\varphi }_c⟩⟨{\varphi }_c|$. For example, C_QR becomes

$$\begin{array}{rcl}{C}_{QR}\left(i\omega \right)& =& 4\sum _{vc}⟨{\mathrm{\Phi }}_Q|V^{\frac{1}{2}}|{\varphi }_v{\varphi }_c⟩\mathfrak{R}\frac{1}{i\omega +{𝜖}_v-{𝜖}_c}⟨{\varphi }_v{\varphi }_c|V^{\frac{1}{2}}|{\mathrm{\Phi }}_R⟩\\ & =& 4\sum _{vc}⟨{\varphi }_v|{\mathrm{\Phi }}_Q{V}^{\frac{1}{2}}|{\varphi }_c⟩⟨{\varphi }_c|\\ & & \times \mathfrak{R}\frac{1}{i\omega +{𝜖}_v-\widehat{H}}|{\varphi }_c⟩⟨{\varphi }_c|V^{\frac{1}{2}}{\mathrm{\Phi }}_R|{\varphi }_v⟩\\ & =& 4\sum _v⟨{\varphi }_v|{\mathrm{\Phi }}_Q{V}^{\frac{1}{2}}\widehat{P}\mathfrak{R}\frac{1}{i\omega +{𝜖}_v-\widehat{H}}\widehat{P}{V}^{\frac{1}{2}}{\mathrm{\Phi }}_R|{\varphi }_v⟩,\\ & & \end{array}$$ (13)

where $Ĥ$ is the Hartree-Fock Hamiltonian. Analogous equations hold for $J_{QG}^{v{v}^{\prime }}$ and $L_{GR}^{v{v}^{\prime }}$.¹⁰ Since the projector onto the conduction state subspace can be expressed in terms of occupied states only $\widehat{P}=\widehat{I}-{\sum }_v|{\varphi }_v⟩⟨{\varphi }_v|$, the calculation of the MP2 correlation energy as expressed in Eq. (8) does not require any explicit reference to virtual states.¹³ Once the auxiliary basis set elements $\mathrm{\Phi }$ are fixed, the matrix elements involved in the last line of Eq. (13) can be efficiently computed by using the Lanczos algorithm developed in previous works.^15,25,26 Equation (8) is our final working equation that allows us to efficiently implement MP2 within the PW basis set. As already mentioned, the use of the plane wave basis set avoids the basis set superposition error and, in principle, the need for the complete basis set extrapolation. Specifically, the CBS extrapolation can be avoided by using a compact auxiliary basis set that is significantly smaller than the plane wave basis set and that allows for a systematic convergence of correlation energy differences.

III. RESULTS AND DISCUSSION

In this section, we will discuss how the methodology presented in Sec. II works in practice to compute MP2 correlation energies. This approach has been implemented as a separate module in the Quantum Espresso (QE) package, which uses plane-waves and pseudopotentials.³⁸ The calculation of the MP2 correlation energy requires first a self-consistent HF calculation, which can be performed using the standard QE implementation.¹⁸ In order to simulate molecules within periodic boundary conditions, we use the supercell approach and truncate the Coulomb interaction between periodically repeated images by using a spherical Coulomb cut-off.³⁹ It is important to mention that HF pseudopotentials are not currently available within the QE package. For this reason, we used norm-conserving pseudopotentials generated for the Perdew-Burke-Ernzerhof⁴⁰ (PBE) functional from the library generated by Schlipf and Gygi.^41,42 For the three dimers considered below, the use of these pseudopotentials leads to a slight overbinding (within 0.035 kcal/mol) at the HF level with respect to all-electron values computed with the aug-cc-pV5Z basis set. This small influence of pseudopotentials is certainly due to the weak nature of the interactions involved in these systems. In applications requiring the breaking of covalent bonds, the effect of pseudopotentials is usually much stronger,¹⁰ and the use of the projector augmented-wave method might be necessary to reach better accuracy.⁴³

To validate our method, we chose three small dimers from the S22 test set:⁴⁴ the NH₃ dimer, the C₂H₄ dimer, and the C₂H₄–C₂H₂ dimer. We start by discussing in detail the NH₃ dimer. In Fig. 1, we show the convergence of the correlation energy contribution to the binding energy of the ammonia dimer with respect to N_aux. Specifically, $\mathrm{\Delta }{E}_c^{MP2}=E_c^{MP2}\left(dimer\right)-E_c^{MP2}\left(monomer1\right)-E_c^{MP2}\left(monomer2\right)$. The number of auxiliary basis set vectors N_aux has been chosen to be 20 times the number of electrons in the dimer and in each monomer. This guarantees a good level of convergence, with all the values of $\mathrm{\Delta }{E}_c^{MP2}$ between N_aux = 240 and N_aux = 320 contained in a 0.028 kcal/mol interval. It is important to highlight that the number of auxiliary basis vectors (N_aux = 320) is much smaller than the number of plane-waves, that for the largest supercell is beyond 100 thousand. Convergence with respect to other computational parameters has also been carefully tested. The different curves in Fig. 1 have indeed been computed for different values of the supercell size ($25\times 25\times 25$ and $30\times 30\times 30$ ${\mathrm{a}}_0^3$) and different values of the kinetic energy cut-off E_cut (50, 60, 70 Ry). The change of all these parameters leads to differences in $\mathrm{\Delta }{E}_c^{MP2}$ smaller than 0.01 kcal/mol at N_aux = 320. Similar convergence tests have been carried out also for the C₂H₄ dimer and the C₂H₄–C₂H₂ dimer. In this case, supercells of $30\times 30\times 30$ ${\mathrm{a}}_0^3$ and $35\times 35\times 35$ ${\mathrm{a}}_0^3$ were necessary to reach a level of convergence similar to the ammonia dimer ($N_{PW}>160000$). The final values for the MP2 binding energies are reported in Table I and compared with previous results in the literature.^44,45 Our results are in excellent agreement with previous work based on localized Gaussian basis set implementations (within 0.03 kcal/mol of the results of Ref. 44 and within 0.06 kcal/mol of the results of Ref. 45). The small residual differences might be ascribed to the use of pseudopotentials in our implementation or numerical errors associated with the CBS in GTO approaches. We emphasize here once again that our implementation does not require any CBS extrapolation or BSSE correction to reach convergence.

FIG. 1.

Convergence with respect to the auxiliary basis set size of the correlation energy contribution to the binding energy of the NH₃ dimer. The curves have been computed for different values of the kinetic energy cut-off E_cut and for different values of the supercell size.

TABLE I.

MP2 interaction energies (kcal/mol) obtained with the present plane-wave implementation compared with the results in the literature obtained with Gaussian-type orbitals (GTOs); CBS denotes the complete basis set limit; CP denotes counterpoise corrections for the basis set superposition error.⁷

System	This work Plane-waves	CBS (Ref. 44) GTO	CBS (Ref. 45) GTO	CBS (Ref. 45) with CP corrections GTO
NH3 dimer	−3.20	−3.20	−3.19	−3.15
C2H4 dimer	−1.64	−1.62	−1.60	−1.58
C2H4–C2H2 dimer	−1.72	−1.69	−1.68	−1.66

IV. CONCLUSIONS

In conclusion, we discuss a novel implementation based on the PW basis set to compute MP2 correlation energy differences. This method avoids basis set superposition errors and does not require complete basis set extrapolation techniques. This has been achieved by using a formulation of MP2 based on the dynamical polarizability and by taking advantage of a series of numerical techniques and algorithms to efficiently compute response functions. As a proof of principle, we demonstrated this method by computing the MP2 binding energy of three weakly interacting molecular dimers. Additional work will be necessary in the future to further improve the numerical efficiency of this method. Numerical limitations are related to the high computational cost of evaluating the HF exchange within the PW basis set. Indeed, the use of the Lanczos algorithm to compute the resolvent in Eq. (13) requires the repeated application of the HF Hamiltonian. Improved efficiency without loss of accuracy might be reached by decreasing the kinetic energy cut-off used to evaluate the exchange term.^46,47 Since PWs are intrinsically characterized by periodic boundary conditions, our methodology is suitable for generalization to treat periodic solids,³¹ and this will be considered in future extensions of our approach.