Particle-particle ladder based basis-set corrections applied to atoms and molecules using coupled-cluster theory

TL;DR

This paper introduces a basis-set correction method for coupled cluster calculations that reduces computational cost while maintaining accuracy for atoms and molecules, especially in energy calculations.

Contribution

The study applies a finite basis-set correction based on the particle-particle ladder term to coupled cluster theory, enabling reduced basis sets without loss of accuracy.

Findings

Basis set reductions by two cardinal numbers are feasible for atomization energies, ionization potentials, and electron affinities.

Reaction energies can be accurately computed with basis set reductions by one cardinal number.

The correction technique is adaptable to other many-electron theories without complex integrals.

Abstract

We investigate the basis-set convergence of electronic correlation energies calculated using coupled cluster theory and a recently proposed finite basis-set correction technique. The correction is applied to atomic and molecular systems and is based on a diagrammatically decomposed coupled cluster singles and doubles correlation energy. Only the second-order energy and the particle-particle ladder term are corrected for their basis-set incompleteness error. We present absolute correlation energies and results for a large benchmark set. Our findings indicate that basis set reductions by two cardinal numbers are possible for atomization energies, ionization potentials and electron affinities without compromising accuracy when compared to conventional CCSD calculations. In the case of reaction energies we find that reductions by one cardinal number are possible compared to conventional CCSD calculations. The employed technique can readily be applied to other many-electron theories without the need for three- or four-electron integrals.

Figures

Tables

Table 1. Table 1: Index notation for different orbital subspaces of the complete one-electron basis. The orbitals refer to the Hartree-Fock spatial orbital basis set over which the wavefunction amplitudes are defined.

	Occ. orbitals	Virt. orbitals	Complement
$i,j,k,l,m,n$	Yes	No	No
$a,b,c,d$	No	Yes	No
$C$	No	Yes	Yes
$P,Q$	Yes	Yes	Yes

Table 2. Table 2: Valence electron correlation energies for H 2 O and F 2 in mH. The CCSD CBS limit values are -297.9 mH and -601.17 mH, respectively  Knizia  et al. ( 2009 ) .

H2O
	AVDZ	AVTZ	AVQZ
CCSD	-227.11	-273.05	-288.21
CCSD-PPL	-291.44	-298.22	-299.50
CCSD-F12a Knizia et al. (2009)	-293.22	-298.50	-299.37
CCSD-F12 Shiozaki et al. (2008)	-289.86	-295.40	-297.23
F2
	AVDZ	AVTZ	AVQZ
CCSD	-435.39	-538.91	-575.10
CCSD-PPL	-580.92	-598.32	-602.67
CCSD-F12a Knizia et al. (2009)	-590.81	-599.65	-602.89
CCSD-F12 Shiozaki et al. (2008)	-584.83	-594.98	-599.01

Table 3. Table 3: Statistical analysis of basis set incompleteness errors of correlation energy contributions to the reaction energies. Mean absolute deviations (MAD), root mean squared (RMS) and maximum deviations (MAX) have been computed for all reactions listed in Table  4 . The CBS limit reference results have been approximated using CCSD/[45]. F12a/b values were extracted from the supplemental information from Ref.  Knizia  et al. ( 2009 ) . Given F12a/b values are total valence electron energies including RHF contribution with CABS singles corrections.

	MAD	RMS	MAX
CCSD/AVDZ	10.906	13.837	43.053
CCSD/AVTZ	4.451	6.485	18.707
CCSD/AVQZ	1.674	2.545	8.838
CCSD/AV5Z	0.863	1.305	4.532
CCSD-PPL/AVDZ	5.908	9.231	35.917
CCSD-PPL/AVTZ	1.918	2.493	7.005
CCSD-PPL/AVQZ	1.070	1.310	2.852
CCSD-F12a/AVTZKnizia et al. (2009)	1.047	1.392	4.625
CCSD-F12a/AVQZKnizia et al. (2009)	0.395	0.507	1.145
CCSD-F12b/AVTZKnizia et al. (2009)	0.891	1.392	5.313
CCSD-F12b/AVQZKnizia et al. (2009)	0.457	0.585	1.304

Table 4. Table 4: List of closed shell reactions.

NR	Reaction	NR	Reaction	NR	Reaction
1	CO + H2 $\to$ HCHO	2	CO + H2O $\to$ CO2 + H2	3	CH3OH + HCl $\to$ CH3Cl + H2O
4	H2O + CO $\to$ HCOOH	5	CH3OH + H2S $\to$ CH3SH + H2O	6	CS2 + 2 H2O $\to$ CO2 + 2 H2S
7	C2H6 + H2 $\to$ 2 CH4	8	HNCO + H2O $\to$ CO2 + NH3	9	CH4 + Cl2 $\to$ CH3Cl + HCl
10	Cl2 + F2 $\to$ 2 ClF	11	CO + Cl2 $\to$ COCl2	12	CO2 + 3 H2 $\to$ CH3OH + H2O
13	HCHO + H2 $\to$ CH3OH	14	CO + 2 H2 $\to$ CH3OH	15	C2H4 + H2 $\to$ C2H6
16	SO3 + CO $\to$ SO2 + CO2	17	H2 + Cl2 $\to$ 2 HCl	18	C2H2 + H2 $\to$ C2H4
19	SO2 + H2O2 $\to$ SO3 + H2O	20	CO + 3 H2 $\to$ CH4 + H2O	21	HCN + 3 H2 $\to$ CH4 + NH3
22	H2O2 + H2 $\to$ 2 H2O	23	CO + H2O2 $\to$ CO2 + H2O	24	2 NH3 + 3 Cl2 $\to$ N2 + 6 HCl
25	3 N2H4 $\to$ 4 NH3 + N2	26	H2 + F2 $\to$ 2 HF	27	CH4 + 4 H2O2 $\to$ CO2 + 6 H2O
28	2 NH3 + 3 F2 $\to$ N2 + 6 HF

Table 5. Table 5: Statistical analysis of basis set incompleteness errors of computed atomization energies (AEs), ionization potentials (IPs) and electron affinities (EAs) as depicted in Fig.  3 . The mean absolute deviations (MAD), root mean squared (RMS) and maximum deviations (MAX) have been computed for basis set incompleteness errors of correlation energy contributions to the AEs, IPs and EAs only. The CBS limit reference results have been approximated using CCSD/[45]. F12 results have been taken from Ref.  Knizia  et al. ( 2009 ) .

	AEs (kJ/mol)			IPs (meV)			EAs (meV)
	MAD	RMS	MAX	MAD	RMS	MAX	MAD	RMS	MAX
CCSD/AVDZ	47.729	54.503	161.095	261.93	281.35	491.35	196.07	252.97	892.96
CCSD/AVTZ	19.972	22.046	41.375	93.85	105.82	189.12	65.20	76.31	153.93
CCSD/AVQZ	7.110	8.594	17.384	37.75	42.12	76.82	24.62	28.52	52.90
CCSD/AV5Z	3.648	4.401	8.903	19.33	21.57	39.32	12.61	14.60	27.07
CCSD-PPL/AVDZ	6.500	7.934	22.225	35.37	47.74	126.31	30.06	37.07	72.05
CCSD-PPL/AVTZ	1.852	2.309	6.196	13.27	16.66	40.71	12.42	14.56	30.88
CCSD-PPL/AVQZ	0.559	0.730	2.106	6.07	7.19	17.63	8.79	9.71	21.74
CCSD-F12a/AVDZKnizia et al. (2009)	5.972	7.031	18.202	39.02	51.96	143.45	26.79	34.93	98.90
CCSD-F12a/AVTZKnizia et al. (2009)	1.534	1.859	4.102	7.70	9.23	23.97	10.28	11.76	18.14
CCSD-F12a/AVQZKnizia et al. (2009)	1.910	2.167	4.654	8.73	9.10	14.35	10.37	11.38	18.08
CCSD-F12b/AVDZKnizia et al. (2009)	8.390	10.230	27.434	60.47	70.84	176.81	39.24	50.22	132.47
CCSD-F12b/AVTZKnizia et al. (2009)	1.717	2.144	6.138	21.80	44.69	44.69	11.92	14.83	30.35
CCSD-F12b/AVQZKnizia et al. (2009)	0.554	0.701	1.714	4.24	5.20	12.07	3.15	3.76	8.13

Table 6. Table 6: RMS deviations of basis set incompleteness error for REs, AEs, IPs, and EAs. Shown are the MP2 term, the ppl contributions, and the energy contributions from the terms denoted as rest. Reference energy of the rest energy is AV5Z, whereas the other two references are obtained from [45]-extrapolation, respectively. CCSD results from Tables  3 and 5 are presented once more for the sake of readability.

	REs (kJ/mol)	AEs (kJ/mol)	IPs (meV)	EAs (meV)
CCSD/AVDZ	13.837	54.503	281.35	252.97
CCSD/AVTZ	6.485	22.046	105.82	76.31
CCSD/AVQZ	2.545	8.594	42.12	28.52
CCSD/AV5Z	1.305	4.401	21.57	14.60
$\Delta E^{\mathrm{rest}}$/AVDZ	3.230	15.199	92.60	63.87
$\Delta E^{\mathrm{rest}}$/AVTZ	1.726	4.834	30.54	25.07
$\Delta E^{\mathrm{rest}}$/AVQZ	0.360	0.789	3.35	2.63
$\Delta E^{\mathrm{driver}}$/AVDZ	14.396	71.041	298.31	349.61
$\Delta E^{\mathrm{driver}}$/AVTZ	6.457	26.294	126.15	106.33
$\Delta E^{\mathrm{driver}}$/AVQZ	2.719	11.370	59.38	49.51
$\Delta E^{\mathrm{driver}}$/AV5Z	1.391	5.823	30.40	25.35
$\Delta E^{\mathrm{ppl}}$/AVDZ	2.201	24.111	111.64	135.07
$\Delta E^{\mathrm{ppl}}$/AVTZ	1.616	9.864	54.45	54.14
$\Delta E^{\mathrm{ppl}}$/AVQZ	0.906	4.397	26.16	25.30
$\Delta E^{\mathrm{ppl}}$/AV5Z	0.464	2.252	13.40	12.95

Equations

$$E_{\mathrm{c}}^{\mathrm{CCSD}}=T_{ij}^{ab}\left(2⟨ij|ab⟩-⟨ji|ab⟩\right),$$

$$E_{\mathrm{c}}^{\mathrm{CCSD}}=E^{\mathrm{driver}}+E^{\mathrm{ppl}}+\underset{=E^{\mathrm{rest}}}{\underbrace{E^{\mathrm{phl}}+E^{\mathrm{hhl}}+E^{\mathrm{phr}}+...}},$$

$$E^{\mathrm{driver}}=W_{ij}^{ab}⟨ab|ij⟩$$

$$E^{\mathrm{ppl}}=W_{ij}^{ab}⟨ab|cd⟩T_{ij}^{cd}.$$

$$W_{ij}^{ab}=\frac{2⟨ij|ab⟩-⟨ji|ab⟩}{{ϵ}_i+{ϵ}_j-{ϵ}_a-{ϵ}_b}.$$

$${t^{(1)}}_{ij}^{ab}\approx ⟨ab\left|(-1)\frac{t_{ij}}{\gamma }{e}^{-\gamma r_{12}}\right|ij⟩.$$

$$E^{\mathrm{ppl}}\approx W_{ij}^{ab}⟨ab|cd⟩⟨cd\left|(-1)\frac{t_{ij}}{\gamma }{e}^{-\gamma r_{12}}\right|ij⟩$$

$$E^{\mathrm{ppl}}\approx W_{ij}^{ab}(-1)\frac{t_{ij}}{\gamma }\left[⟨ab\left|\frac{e^{-\gamma r_{12}}}{r_{12}}\right|ij⟩\underset{=-⟨ab|Cl⟩⟨Cl|e^{-\gamma r_{12}}|ij⟩}{\underbrace{+⟨ab|kl⟩⟨kl|e^{-\gamma r_{12}}|ij⟩-⟨ab|Pl⟩⟨Pl|e^{-\gamma r_{12}}|ij⟩}}-⟨ab|kQ⟩⟨kQ|e^{-\gamma r_{12}}|ij⟩\right]$$

$$\Delta E(X)=E(\mathrm{CBS})-\mathrm{E}(\mathrm{X}).$$

$$E_{\mathrm{c}}^{\mathrm{CCSD}-\mathrm{PPL}}=E_{\mathrm{c}}^{\mathrm{CCSD}}(X)+\Delta E^{\mathrm{driver}}(X)+\Delta E^{\mathrm{ppl}}(X),$$

$$\Delta E^{\mathrm{ppl}}(X)=\frac{E^{\mathrm{driver}}(\mathrm{CBS})}{E^{\mathrm{driver}}(X)}{E}^{\mathrm{ppl}}(X)-E^{\mathrm{ppl}}(X)$$

Full text

I Introduction

Quantum chemical many-electron theories that employ anti-symmetrized one-particle functions (Slater determinants) as a basis for the many-electron wavefunction exhibit a frustratingly slow convergence of central quantities like the ground state energy to the complete basis-set limit. A large fraction of the computational cost involved in many-electron theory calculations of atoms, molecules and solids originates from the need to include large numbers of one-electron basis functions necessary to achieve the desired level of precision. Many techniques have been developed to accelerate the convergence to the complete basis-set limit including explicitly correlated methods, transcorrelated methods or simple yet less efficient basis-set extrapolation techniques Hättig et al. (2012); Grüneis et al. (2017); Knizia et al. (2009); Boys et al. (1969a, b); Ten-no (2000); Feller (2013); Shepherd et al. (2012). More recently a density-functional theory based approach has also been employed to correct for basis set incompleteness errors Giner et al. (2018).

It has been known since the early days of electronic structure theory that the slow convergence of the wavefunction expansion in Slater determinants is due to short-ranged interelectronic correlation. As the electrons coalesce, a derivative discontinuity or ‘cusp’ must arise, so that a divergence in the kinetic energy operator cancels an opposite one in the potential Kato (1957); Pack and Byers Brown (1966); Carlsson and Ashcroft (1982); Pan and Sahni (2003). Explicitly correlated methods account for the cusp condition in an a priori manner and are commonly referred to as F12 theories, where F12 stands for a two-electron correlation factor that enables a compact expansion of the wavefunction at short interelectronic distances Hättig et al. (2012); Werner et al. (2007); Kutzelnigg and Klopper (1991). F12 theories introduce, however, the need for additional many-electron integrals such as three- and sometimes even four-electron integrals. Due to their large computational overhead, explicitly correlated methods are generally only beneficial for more complex parent methods. Furthermore they cause substantial additional effort in their computer code implementation. Nonetheless, thanks to their improved basis-set convergence and reliability, they have become an indispensable tool for quantum chemical calculations of large systems Knizia et al. (2009); Hättig et al. (2010, 2012); Kong et al. (2012); Tew et al. (2010); Zhang and Valeev (2012); Werner et al. (2010); Ten-no and Noga (2012); Ten-no (2012); Shiozaki and Hirata (2010).

In this work we apply a recently proposed basis-set extrapolation technique for coupled cluster singles and doubles (CCSD) theory to atoms and molecules. The employed technique is based on a diagrammatic decomposition of the coupled cluster correlation energy and was previously employed to study the uniform electron gas Irmler et al. (2019). An analysis of the decomposed correlation energy for large basis sets in the uniform electron gas shows that the slow convergence of the CCSD energy originates from the second-order correlation energy and the so-called particle-particle ladder term. Both contributions converge at the same rate but with opposite sign to the complete basis-set (CBS) limit. As a consequence MP2 theory exhibits a larger basis-set incompleteness error than CCSD theory. This observation is in agreement with prior findings for molecules and atoms where it has been referred to as interference effect Petersson et al. (1988); Nyden and Petersson (1981); Petersson and Nyden (1981). Furthermore related findings have been discussed in the context of explicitly correlated methods Kutzelnigg and Klopper (1991); Noga and Kutzelnigg (1994); Vogiatzis et al. (2011). We stress that the asymptotic convergence of the ppl contribution has already been studied by Klopper and Kutzelnigg using a partial wave expansion in third-order perturbation theory in Ref. Kutzelnigg and Klopper (1991).

Based on the observation of only two dominant channels in the basis-set limit, we have devised a basis-set correction that is based on re-scaling the partice-particle ladder (ppl) correlation energy contribution by a factor that is estimated from the corresponding basis-set incompleteness error of the MP2 correlation energy, which can be obtained in a computationally significantly cheaper manner albeit requiring a non-negligible effort Liakos et al. (2013). Previously this scheme has successfully been applied to estimate the CCSD correlation energy of semiconducting and insulating solids Irmler et al. (2019). In this article we investigate the performance of the technique for a large set of molecules. We study total energies, reaction energies, atomization energies, ionization potentials, and electron affinities of a large benchmark set taken from Ref. Knizia et al. (2009). We stress that the presented scheme is related to the CCSD-F12a approximation, which corrects for the basis-set incompleteness errors in the same terms, albeit using a more sophisticated explicitly correlated ansatz Adler et al. (2007). Furthermore we note in passing that a similar rescaling approach of the perturbative triples electronic correlation energy contribution has also been applied in Ref. Knizia et al. (2009).

We emphasize that in addition to explicitly correlated methods, there exist a wide range of basis set extrapolation techniques that can also be used to rapidly converge correlation energies to the complete basis set limit Feller (2013); Ranasinghe and Petersson (2013). However, in this work we compare to explicitly correlated methods that are widely-used for the prediction of reliable CBS limit reference values. For the sake of consistency we employ the same molecular geometries as employed for CCSD-F12a and CCSD-F12b published in Ref. Knizia et al. (2009).

II Theory

Here, we employ a recently proposed method to correct for the basis-set incompleteness error of CCSD correlation energies. The central premise of this method is that the CCSD correlation energy,

[TABLE]

can be decomposed into different diagrammatic contributions such that Irmler et al. (2019)

[TABLE]

where $E^{\rm driver}$ corresponds to the MP2 correlation energy

[TABLE]

and the particle-particle ladder term is defined as

[TABLE]

$T_{ij}^{cd}$ is computed from the CCSD singles ($t_{i}^{a}$) and doubles ($t_{ij}^{ab}$) amplitudes such that $T_{ij}^{ab}=t_{ij}^{ab}+t_{i}^{a}t_{j}^{b}$. $t_{i}^{a}$ and $t_{ij}^{ab}$ are obtained by solving the corresponding amplitude equationsBartlett and Musiał (2007); Cizek and Paldus (1971). $W_{ij}^{ab}$ is given by

[TABLE]

The employed indices are explained and summarized in Table 1. Einstein summation convention applies to repeated indices throughout this work.

We note that the employed decomposition of the CCSD correlation energy is achieved by replacing the doubles amplitudes in Eq. (1) with the corresponding right hand side of the amplitude equations using converged CCSD amplitudes. For the sake of brevity, we define $E^{\rm rest}$ such that it contains all remaining contributions to the CCSD correlation energy including terms such as the particle-hole ladder, hole-hole ladder and the particle-hole ring. We stress that the labelling of the contributions to the correlation energy on the right hand side of Eq. (2) is inspired by the corresponding terms in the amplitude equations and does not imply that the ppl term includes particle-particle ladder contributions to the correlation energy only. The latter holds only in the case of $t_{ij}^{ab}={t^{(1)}}_{ij}^{ab}$ where ${t^{(1)}}_{ij}^{ab}=\frac{\left\langle ab|ij\right\rangle}{\epsilon_{i}+\epsilon_{j}-\epsilon_{a}-\epsilon_{b}}$. We note that a similar labeling of the terms in the amplitude equations is used by Shepherd et al. in Ref. Shepherd et al. (2014). We refer the reader to Ref. Irmler et al. (2019) for a more detailed analysis of the different diagrammatic contributions to the CCSD correlation energy in periodic systems.

The MP2 correlation energy, $E^{\rm driver}$, converges as $1/L^{3}$ using a correlation consistent atom-centered Gaussian basis set, where $L$ refers to the cardinal number of the employed basis set. We will argue in the following that the first two terms on the right hand side of Eq. (2) exhibit the same convergence rate when approaching the complete basis set limit, constituting the dominant source of the basis set incompleteness error of the CCSD correlation energy. To better understand the convergence behaviour of $E^{\rm ppl}$, we approximate the first-order amplitudes using the following expression that follows from F12 theory

[TABLE]

$t_{ij}$ are geminal amplitudes determined by the Kato cusp conditions Kato (1957). The expression above implies that the first-order doubles amplitudes can be approximated using an expression that is similar to the corresponding electron repulsion integral, where the Coulomb kernel has been replaced by the Slater-type correlation factor that depends on a parameter $\gamma$. Approximating the CCSD doubles amplitudes $t_{ij}^{ab}$ on the right hand side of Eq. (4) by Eq. (6) and disregarding the contribution of single amplitudes, we can approximate the particle-particle ladder contribution to the correlation energy by

[TABLE]

We now replace the summation over the virtual orbital indices $c$ and $d$ in Eq. (7) by the closure relation $|c\rangle\langle c|=|P\rangle\langle P|-|k\rangle\langle k|$, yielding the following approximate expression for $E^{\rm ppl}$

[TABLE]

We stress that the particle-particle ladder contribution to the correlation energy is positive (for $t_{ij}^{ab}\approx{t^{(1)}}_{ij}^{ab}$), whereas the last three terms in the brackets on the right hand side of Eq. (8) yield a negative contribution. According to this, the leading contribution to the correlation energy must be the first term in the brackets on the right hand side of the above expression. This term converges at the same rate to the complete basis set limit as $E^{\rm driver}$. In particular the $\frac{e^{-\gamma r_{12}}}{r_{12}}$ kernel exhibits a singularity at $r_{12}=0$ that causes an asymptotic basis set convergence rate which is identical to MP2 theory ($1/L^{3}$). In short, we have shown above that $E^{\rm ppl}$ defined by Eq. (4) will converge to the CBS limit at the same rate as $E^{\rm driver}$ if the following approximations hold ${t}_{ij}^{ab}\approx{t^{(1)}}_{ij}^{ab}\approx\left\langle ab\left|(-1)\frac{t_{ij}}{\gamma}e^{-\gamma r_{12}}\right|ij\right\rangle$.

We now put the analysis carried out above to a numerical test. Figure 1 depicts the convergence of the absolute basis set incompleteness error in the decomposed correlation energy contributions of He, Ne, H2O and F2. The errors are retrieved as a function of the cardinal number of the employed aug-cc-pV$X$Z basis. Basis set incompleteness errors are defined by

[TABLE]

We refer to channel decomposed correlation energies obtained in a given basis set $X$ by $E^{\rm driver}(X)$, $E^{\rm ppl}(X)$ and $E^{\rm rest}(X)$. For the results shown in Figure 1, we estimate the CBS values using an aug-cc-pV6Z basis for He and H2O, and an aug-cc-pV7Z basis for Ne and F2. We note that $\Delta E^{\rm driver}(X)$ and $\Delta E^{\rm ppl}(X)$ are parallel on a logarithmic scale, whereas $\Delta E^{\rm rest}(X)$ exhibits a faster convergence rate compared to the other terms. From these results and the analysis carried out above as well as for the uniform electron gas Irmler et al. (2019), we conclude that $E^{\rm driver}$ and $E^{\rm ppl}$ are the only two dominant channels in the basis-set limit of coupled-cluster singles and doubles theory. Furthermore we do not identify a fundamental difference in the discussed behaviour between any of these systems.

Based on the analysis above and in agreement with Ref. Irmler et al. (2019), it is reasonable to propose a finite basis set error correction to the CCSD correlation energy that requires the CBS limit estimate of the second-order correlation energy $E^{\rm driver}({\rm CBS})$ only. We define the approximation to the CBS limit estimate of the CCSD correlation energy, CCSD-PPL, as

[TABLE]

where we employ the following approximation to the particle-particle ladder correlation energy contribution

[TABLE]

III Computational Details

We have modified the coupled cluster code in the open-source quantum chemistry package PSI4 Parrish et al. (2017) such that the $E^{\rm ppl}(X)$ contribution to the CCSD correlation energy is computed separately at the end of each CCSD calculation using fully converged CCSD amplitudes. The required modifications to the exisiting CCSD code are minor and we expect that other computer code implementations of coupled cluster theory can also be modified in such a simple manner. Throughout this work, correlation energies always correspond to valence electron correlation energies only.

We employ aug-cc-pVXZ basis sets for first-row elements and aug-cc-pV(X+d)Z basis sets for second-row elements. These basis sets will be denoted as AVXZ throughout the article. For reaction energies (REs), atomization energies (AEs), ionization potentials (IPs), and electron affinities (EAs) CBS limit estimates are obtained from AVQZ and AV5Z CCSD correlation energies using the $E_{n}=E_{\textrm{CBS}}+a/n^{3}$ extrapolation formula and will be denoted as CCSD/[45]. Likewise the MP2 CBS limit estimates, needed for CCSD-PPL calculations, are also obtained using a [45] extrapolation. The employed equilibrium geometries have been taken from Ref. Knizia et al. (2009).

IV Results

IV.1 Total energies

We first seek to assess the efficiency of CCSD-PPL for total valence electron correlation energies of the closed shell molecules H2O and F2. Table 2 summarizes the calculated valence electron correlation energies for AVXZ basis sets with $X=\mathrm{D,T,Q}$ and results obtained using F12 approaches taken from Refs. Knizia et al. (2009); Shiozaki et al. (2008) The valence electron correlation energies shown in Table 2 demonstrate that CCSD-PPL converges to the CBS limit with respect to the employed basis set at a similar rate as F12 theories. Compared to CCSD theory, CCSD-PPL allows for a reduction by approximately two cardinal numbers in the employed basis set size while capturing a similar fraction of the correlation energy.

A careful analysis of the energy contributions from the different channels shows for H2O that $|\Delta E^{\rm rest}|$ is 12.8 mH and 3.5 mH for AVDZ and AVTZ, respectively. These numbers are even larger than the total deviations of the CCSD-PPL results compared to the CBS limit (see Table 2). For F2 we observe a similar behaviour. Therefore we conclude that the good performance of CCSD-PPL for total correlation energies using small basis sets, such as AVDZ, is partly due to error cancelation between $\Delta E^{\rm rest}$ and $\Delta E^{\rm ppl}$.

IV.2 Reaction energies

To further assess the basis set convergence of CCSD-PPL, we calculate reaction energies using a benchmark set developed by Knizia et al. Knizia et al. (2009). However, instead of the full set containing 54 closed-shell molecules, and 50 open-shell molecules, we restrict our analysis to a subset of 28 reactions, containing small closed-shell molecules only. The investigated reactions are listed in Table 4.

Figure 2 depicts basis set incompleteness errors of the computed correlation energy contributions to the reaction energies for CCSD/AVTZ, CCSD/AV5Z and CCSD-PPL/AVTZ. We note that the errors of CCSD/AVTZ results are significantly larger than corresponding CCSD-PPL/AVTZ results, whereas CCSD/AV5Z exihibits smaller errors than CCSD-PPL/AVTZ. This indicates that in contrast to total correlation energies, CCSD-PPL does not allow for a reduction by two cardinal numbers in the basis compared to CCSD, while capturing the same fraction of the CBS limit correlation energy contribution to the reaction energies. This observation is also reflected in the statistical analysis of the computed results summarized in Table 3. CCSD/AVXZ is closer to the CBS limit than CCSD-PPL/AV(X-2)Z for RMS, MAD and MAX. However, CCSD-PPL/AV(X-1)Z and CCSD/AVXZ exhibit similar basis set incompleteness errors, indicating the a reduction by one cardinal number can be achieved using CCSD-PPL for reaction energies. By comparing CCSD/AVXZ to CCSD-PPL/AVXZ, we find that the RMS, MAD and MAX of basis set incompleteness errors are reduced by approximately a factor of two, except for AVDZ, where this factor is smaller. However, this could be related to non-negligible basis set incompleteness errors of the underlying HF reference. In contrast to CCSD-PPL, both F12a and F12b approximations allow for a reduction by two cardinal numbers while achieving a similar level of accuracy as corresponding CCSD results. In particular, F12a and F12b achieves CCSD/AV5Z quality using AVTZ basis sets only. For a comparable accuracy the PPL scheme requires AVQZ basis sets.

IV.3 Atomization energies, ionization potentials, and electron affinities

As a further test, we have calculated atomization energies (AEs), ionization potentials (IPs), and electron affinities (EAs) using CCSD and CCSD-PPL. Figure 3 depicts the CBS error in the valence electron correlation energy contributions.

For the atomization energies shown in Fig. 3 (top panel), we observe a monotonically decreasing error for increasing basis set size. CCSD-PPL/AVDZ achieves on average the same level of precision as CCSD/AVQZ. The same conclusion can be drawn from the statistical analysis given in Table 5. It is shown that not only mean absolute and root mean squared deviations are reduced, but also the maximum deviation. A comparison of CCSD-PPL to CCSD-F12a reveals that on average CCSD-F12a achieves a slightly better agreement with CBS limit values for AVDZ and AVTZ. For the AVQZ basis set, CCSD-PPL is closer to the CBS limit estimates than CCSD-F12a and shows comparable deviations as CCSD-F12b.

In the case of the basis set errors of the ionization potentials shown in Fig. 3 (middle panel), and summarized in Table 5, a slightly different situation arises. Although both the MAD and RMS deviations of CCSD/AVQZ and CCSD-PPL/AVDZ are comparable, CCSD-PPL shows a few outliers which are significantly larger in magnitude. However, this behavior is no more present for CCSD-PPL/AVTZ and CCSD/AV5Z, where not only mean absolute deviations but also the maximum deviations are comparable. The comparison of CCSD-PPL to F12 methods shows superior results of F12a for AVDZ and AVTZ basis sets, and better values for F12b for AVQZ basis sets.

Similar conclusions can be drawn for the basis set errors of electron affinities depicted in Fig. 3 (bottom panel) and listed in Table 5. Here CCSD-PPL/AVDZ shows larger maximum and mean deviations than CCSD/AVQZ. However, CCSD-PPL/AVTZ and CCSD/AV5Z yield very similar results, indicating that the PPL scheme can reduce the required basis set size by two cardinal numbers compared to CCSD theory. The comparison with F12 methods leads to the same conclusions as for the ionization potentials.

IV.4 Further analysis

There are significant differences in the CCSD-PPL basis-set convergence between the reaction energies on the one hand and atomization energies, ionization potentials, and electron affinites on the other hand. As mentioned before, the two presented F12 variants achieve a reduction of the basis set incompleteness error by two cardinal numbers for all explored quantities. A similar reduction is achieved using CCSD-PPL for AEs, IPs and EAs. However, in the case of REs, CCSD-PPL achieves a reduction by one cardinal number only. In order to investigate this disparity further, we perform an analysis of the channel decomposed basis set error, summarized in Table 6. We first investigate the error for $\Delta E^{\mathrm{r}est}$, which corresponds to the basis set error of the hypothetically perfect CCSD-PPL scheme assuming driver and ppl contributions in the CBS limit. We find that the basis set convergence of $\Delta E^{\mathrm{r}est}$ is much better balanced for all investigated properties, achieving a reduction by almost two cardinal numbers. In particular $\Delta E^{\mathrm{r}est}/AVTZ$ is very close to $CCSD/AV5Z$ for AEs, REs and IPs. We note that the convergence for EAs is slightly slower, which agrees with findings from Ref. Knizia et al. (2009) that have been obtained using explicitly correlated methods and show that EAs converge more rapidly if additional diffuse basis functions are employed.

The above discussion shows that the employed approximation to $\Delta E^{\rm{ppl}}$ as defined by Eq. (10) is less reliable for REs and achieves results for AEs, IPs and EAs that are partly fortuitously close to the CBS limit. To better understand the disparity between REs and the other properties, Table 6 also lists the RMS basis set errors of the particle-particle ladder and driver terms for all employed basis sets. We find that the ratio between driver and ppl RMS basis set errors is larger for REs compared to the AEs, IPs and EAs especially when using small basis sets such as AVDZ and AVTZ. Furthermore Table 6 shows that the basis set error of the ppl contributions to the REs are smaller than for AEs. Consequently, the simple rescaling procedure defined by Eq. (10) is likely to introduce non-negligible errors for the reaction energies.

In passing we also note that the basis set errors of the combined rest and ppl contributions partly cancel (not shown explicitly in Table 6), which is in agreement with findings by Ranasinghe and Petersson in their CCSD studies Ranasinghe and Petersson (2013).

V Summary and Conclusions

We have applied a recently introduced finite basis-set correction method for coupled cluster singles and doubles theory to atoms and molecules. The method is based on a decomposed correlation energy and corrects for the basis-set incompleteness error in the second-order and the particle-particle ladder term of the CCSD energy only. Compared to CCSD, CCSD-PPL allows for a reduction of atom centered correlation consistent Gaussian basis sets by approximately two cardinal numbers, while capturing a similar fraction of the CBS limit correlation energy contributions to total energies, atomization energies, ionization potentials and electron affinities. We have also studied reaction energies using CCSD-PPL and CCSD, finding that the PPL scheme allows for reductions in the basis set size by one cardinal number only. In passing we note that F12 theories achieve a reduction by two cardinal numbers for all properties studied in this work including reaction energies. We have shown that the discrepancy in the CCSD-PPL basis set convergence between reaction energies and the other properties can be attributed to the CBS limit approximations to the particle-particle ladder term. Nonetheless, we conclude that CCSD-PPL exhibits a good trade-off between computational cost and basis set reduction for CCSD calculations in atoms, molecules and solids. Future work will focus on more accurate and equally efficient approximations to this term.

We stress that it is possible to transfer the outlined method to other widely-used many-electron theories that allow for a decomposition of the electronic correlation energy such as full configuration interaction and related theories. Finally, we note that the employed decomposition of the electronic correlation energy will potentially also be useful for the further development and improvement of correlation factors used in F12 theories.

VI Acknowledgements

Support and funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (Grant Agreement No 715594) is gratefully acknowledged. A. G. gratefully acknowledges many helpful discussions on F12 theory with Seiichiro Ten-no and David Tew.