- Email: [email protected]
Density-Based Analysis of Spin-Resolved MP2 Method
CHAPTER SIXTEEN
Density-Based Analysis of Spin-Resolved MP2 Method Mateusz Witkowski, Szymon Śmiga1, Ireneusz Grabowski Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Torun, Poland 1 Corresponding author: e-mail address: [email protected]
Contents 1. Introduction 2. Computational Details 3. Methodology and Results 3.1 Proportionality of the Spin-Resolved MP2 Densities 3.2 Two-Dimensional Scan 3.3 Dipole Moments 4. Final Remarks Acknowledgments References
279 281 282 282 284 288 288 290 290
Abstract The extensive study of the spin-resolved second-order Møller–Plesset method in the context of the electron density is performed. It was found the well-defined proportionality of the same- and opposite-spin parts of the MP2 correlated electronic density. We have rationalized the value of the scaling parameter used in the foundation of the SOS-MP2 (Jung et al., 2004) method from the density point of view. Our analysis is complemented by the calculations of the dipole moments using differently parameterized spin-resolved MP2 methods.
1. INTRODUCTION Since 2003, when Grimme proposed the spin-component-scaled MP2 method,1 a rapid development of spin-resolved (SR) methods is observed in the field of quantum chemistry. Many useful variants have been proposed, analyzed, and implemented2 in different levels of approximations and different application areas.3–7 However, those based on the second-order Møller–Plesset
Advances in Quantum Chemistry, Volume 76 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2017.05.004
#
2018 Elsevier Inc. All rights reserved.
279
280
Mateusz Witkowski et al.
perturbation theory (MP2)8 energy expression are the most interesting ones from point of view of a simplicity and computational cost. Rescaling of the same-spin (SS) and opposite-spin (OS) part of the MP2 correlation energy proposed in Ref. 1 leads to a spin-resolved MP2 method (SR-MP2) EcSRMP2 ¼ cOS E MP2ðOSÞ + cSS EMP2ðSSÞ
(1)
with EcMP2ðOSÞ ¼ EcMP2ðSSÞ ¼
1 X X jðiσ aσ j jτ bτ Þj2 2 σ , τ6¼σ ijab Eiσ + Ejτ Eaσ Ebτ
1 X X jðiσ aσ jjσ bσ Þj2 2 σ ijab Eiσ + Ejσ Eaσ Ebσ
1 X X ðiσ aσ jjσ bσ Þðaσ jσ jbσ iσ Þ , 2 σ ijab Eiσ + Ejσ Eaσ Ebσ
(2)
(3)
where cOS and cSS are the OS and SS coefficients, respectively. In the above a, b, and i, j label the virtual and occupied indexes, respectively, while σ and τ label the spin indexes. Such simple modification of the MP2 energy expression results in significant improvement in the accuracy of calculated properties which, in some cases, can be compared to those obtained from a more advanced methods such as the coupled-cluster single double and perturbative triple [CCSD(T)]9 approach. At the same time the SR-MP2 method retains the computational simplicity of its original counterpart as well as other important features like size-consistency and orbital-rotational invariance. In the initial work1 the cOS ¼ 6/5 and cSS ¼ 1/3 scaling parameters have been optimized with respect to a training set of reaction energies. Defined in this manner the “spin-component scaled” MP2 method (SCS-MP2) gives rather good improvement in the quality of atomization energy, barrier heights,1,10 heats of formation,11 or molecular geometries.1,10,12 Another interesting variant of SR-MP2 method is the one, proposed by Jung et al. dubbed “scaled-opposite spin” MP2 (SOS-MP2).10 In this simplified method, the SS contribution to the correlation energy in Eq. (1) is modeled by proper selection of the cOS ¼ 1.3 parameter, while cSS ¼ 0. In general, SOS-MP2 leads to a similar quality of results as the SCS-MP2 method. However, due to neglect of the SS term in Eq. (1), computational efficiency
Density-Based Analysis of Spin-Resolved MP2 Method
281
increases significantly, leading to reduction of the numerical cost from OðN 5 Þ to OðN 4 Þ or even less.13,14 Many other useful parameterizations of Eq. (1) have been proposed, based on semiempirical7,15–18 as well as theoretical considerations.19,20 Furthermore, the spin-resolved concept was also applied in the context of the density functional theory (DFT) and ab initio DFT,21,22 where Eq. (1) was utilized as the correlation energy functional in the double-hybrid23–27 and second-order optimized effective potential (OEP) method,28–31 respectively. The latter studies28,30 are especially interesting because they have revealed that their exists almost linear proportionality between the OS and SS part of the correlation energies, Kohn–Sham correlation potentials and correlated electronic densities (although, the proportionality factors are different for different quantities). This suggests that similar proportionality of densities may also exist in the SR-MP2 method. To date, however, such an analysis was not carried out. Thus to fill this gap, in this chapter we devote special attention to the density analysis of spin-resolved methods. The work is organized as follows. In Section 3.1 we discuss the proportionality of the spin-resolved densities and its implications. Next, in Section 3.2 we extend this analysis by presenting two-dimensional scans of the SR-MP2 densities with respect to the CCSD(T). Finally, in Section 3.3 we investigate the impact of these results on the dipole moments which are especially sensitive to the quality of the electron density.
2. COMPUTATIONAL DETAILS The main quantity used in our analysis is the electron density calculated obtained from the MP2 and CCSD(T) methods. The densities are calculated from the relaxed density matrices32–34 constructed using the Lagrangian approach.35–37 In the case of SR-MP2 method the SS and OS parts of correlated densities have been obtained by a scaling of adequate parts in the Lagrangian. Calculations have been carried out in NWChem software package.38 The basis sets employed are the spherical cc-pVXZ and aug-cc-pVXZ basis sets, where X¼(D,T,Q).39–42 All electrons were included in the calculation of correlated quantities. The calculations were performed for several atomic: Ne, Ar, Kr, Zn, and molecular systems, i.e., Cl2, CS, F2, He2, N2, OCS, P2, and those listed in Table 1. For the latter ones, the experimental geometries were taken from Ref. 43. As in our previous studies16,17,28,30 the CCSD(T)9 results were used as the reference benchmark data.
282
Mateusz Witkowski et al.
3. METHODOLOGY AND RESULTS In this section we discuss several aspects related to the SR-MP2 method in the context of electron density. In particular we analyze the proportionality of the spin-resolved densities by examining its relative parts and the possibility of reconstruction of the CCSD(T) density by proper scaling of the SR-MP2 density. Due to large qualitative and quantitative resemblance of densities obtained within all basis sets, for the sake of clarity of the discussion, we report only those obtained with aug-cc-pVQZ basis set.
3.1 Proportionality of the Spin-Resolved MP2 Densities In order to demonstrate all features of second-order spin-resolved densities we utilize the concept of difference total densities (DTD) distribution, defined with respect to the total Hartree–Fock (HF) density [ρHF(r)] ðrÞ ¼ ρmethod ðrÞ ρHF ðrÞ, ρmethod c
(4)
where ρmethod(r) is the total density gained from correlated (e.g., MP2 or CCSD(T)) calculations. The DTD was successfully used as a testing tool in the development of new methods, especially in the context of DFT.28,44–48 We note that in case of atoms, the ρc(r) is closely related to another quantity, namely the difference radial-density (DRD) distributions,44 defined as ðrÞ: DRDmethod ðrÞ ¼ 4πr 2 ρmethod c
(5)
Thus, the latter one is used for atoms. For the clarity of discussion, henceforth we will refer to both quantities as correlated densities. In Fig. 1 we present the SS and OS parts of MP2 correlated densities obtained for Ne and Ar atoms and for the CO molecule. Similar results have been observed in the case of other systems (not reported here). One can immediately observe the clear point-by-point proportionality of the spinresolved parts of the MP2 correlated densities. All minima and maxima positions are almost ideally aligned with each other, even in the molecular case. This in turn suggests that SS correlation effects can be effectively modeled by properly rescaling the OS contributions only, which is more favorable from the computational point of view. We remark, however, that the relative density scaling factor γ which links SS and OS correlated densities SS ρOS c ðrÞ ¼ γρc ðrÞ
(6)
283
4pr 2rc (a.u.)
4pr 2rc (a.u.)
Density-Based Analysis of Spin-Resolved MP2 Method
0.04 0.03 0.02 0.01 0 −0.01 −0.02 −0.03 0.02
Ne, g = 2.3 OS SS γ SS
0 −0.02 −0.04
Ar, g = 2.1
−0.06 0
1
2 r (a.u.)
rc (a.u.)
0.05
3
4
0 −0.05 −0.1 −3
CO, g = 2.1 −2
−1
0 z (a.u.)
1
2
3
Fig. 1 Correlated density distributions for Ne (top), Ar (middle), and CO (bottom, plot along the molecular axis) calculated for SS and OS parts of MP2 density. On each plot, the value of γ factor rescaling SS correlated density has also been shown. See the text for details.
is not a general constant. It slightly depends on the choice of basis set as well as the molecular system itself. For example, in the case of the CO molecule for the aug-cc-pVDZ and aug-cc-pVQZ, γ changes from 2.3 to 2.1, respectively. Nevertheless, in general, for most systems and basis sets, the proportionality parameter ranges between the 1.7 and 2.5. For the systems presented in Fig. 1, the γ values are γ ¼ 2.3 for Ne and γ ¼ 2.1 for Ar and CO. We underline here that proportionality of resolved density is the general feature observed in all considered systems in our set, which is the first important finding of this study. Furthermore, above results confirm the previous findings,2,16–18 based mostly on the analysis of SS and OS parts of the SR-MP2 correlation energy expression. The density-based analysis presented here is more demanding than those based on the correlation energy, because we scale not just one number (like in the correlation energy case) but the scaling holds for the whole space where the electron density is calculated. Hence, this kind of analysis can be crucial for understanding and rationalizing the behavior of spin-resolved-based methods.
284
Mateusz Witkowski et al.
3.2 Two-Dimensional Scan In the previous section we have shown that SS and OS parts of the MP2 correlated densities display a very high degree of proportionality, and then that proper scaling of one term (e.g., the SS one) can reproduce general spatial behavior of another one (OS correlated density), with relatively good accuracy. In order to better understand and make possible use of this issue in its practical application, we perform more quantitative analysis. To this end, we conducted a full two-dimensional scan of the space spanned by the cOS and cSS parameters, analyzing, in each point of space the error indicator function (the integrated density differences—IDD30) defined as Z IDDðcOS , cSS Þ ¼ drρSRMP2 ðr; cOS ,cSS Þ ρCCSDðTÞ ðrÞ: (7) c c where ρcCCSDðTÞ is the correlated CCSD(T) density. The first term in Eq. (7) is the correlated spin-resolved density calculated for the fixed pair of scaling coefficients as ρSRMP2 ðr;cOS , cSS Þ ¼ cOS ρOSMP2 ðrÞ + cSS ρSSMP2 ðrÞ: c c c
(8)
The cOS and cSS coefficients have been varied in the range between 0.0 and 2.0 with the 0.1 interval. The integration in Eq. (7) runs over whole threedimensional space on a very accurate grid of spatial points. One can note that the indicator defined by Eq. (7) tells explicitly how accurately the SR-MP2 method can reproduce the CCSD(T) reference ρSRMP2 ), for a fixed pair of coefficients. density, i.e., we obtain (ρCCSDðTÞ c c It also shows implicitly, the global qualitative relative proportionality of OS and SS parts of MP2 correlated densities. As before the scans have been done for all investigated systems. However, in Fig. 2, we report only the scan obtained for a CO molecular system which presents general trend observed in our results. Inspection of the figure confirms the observations made in Section 3.1, i.e., the proportionality of the OS and SS parts of the MP2 correlated densities. Furthermore we see that the SR-MP2 method can reproduce the reference CCSD(T) densities with quite good accuracy. We note that similar findings have been observe in Ref. 28, however, in the context of ab initio DFT. Nonetheless, several other features may be observed (a) The IDD indicator shows the minima for a continuous set of parameter pairs, which can be obtained by a simple linear relation cOS ¼ αcSS + β, with α and β being system-dependent constants. This relation indicates the evident proportionality of the spin-resolved densities.
285
Density-Based Analysis of Spin-Resolved MP2 Method
0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05
0.5 IDD (a.u.)
0.4 0.3 0.2 0.1 0 0
0.5
1 COS
1.5
2 0
0.5
1
1.5
2
CSS
Fig. 2 The IDD scan in the function of cOS and cSS parameters for CO molecule.
(b) The performance of standard SCS-MP2 and SOS-MP2 methods, which yield almost identical IDD’s, i.e., 0.07 and 0.065, respectively, is better than in the MP2 case where IDD ¼ 0.09. The global minimum, with lower IDD ¼ 0.06, is found for cOS ¼ 1.1 and cSS ¼ 0.1 pair of coefficients. (c) The absolute minimum value of IDD is nonzero. This is due to the fact that the IDD is an integrated quantity which describes how correlation effects are reproduced in the density for all points in the space. This, in turn, means that the reference CCSD(T) density cannot be reproduced by a simple scaling of second-order spin-resolved densities. This may lead to the conclusion that wherever the MP2 method fails badly (e.g., in systems with multireference character) also the simple rescaled methods will lead to similar predictions. (d) For the case where SS part of the density is neglected (cSS ¼ 0), it is always possible to find a value of the cOS parameter such that the IDD error is minimized and its value is close to the global absolute minimum. This is a direct consequence of the proportionality found here and described in Section 3.1. In fact, the SS compensation term is proportional to the inverse of γ (present in Eq. (6)) and gives ðrÞ: ρcCCSDðTÞ ðcOS + γ 1 cSS ÞρOSMP2 c
(9)
This, in turn, justifies assumed proportionality used in the foundation of the SOS-MP2 method. According to SOS-MP2 definition,10 in order to compensate the lack of SS part of the correlation, one needs to
286
Mateusz Witkowski et al.
increment the OS scaling factor of SCS-MP21 method by about 1/9. This means that the proportionality coefficient γ for SOS-MP2 method equals 10/3 3.3 which is slightly larger than those obtained in the present work (γ ¼ 1.7 2.5). Due to the latter feature we have also performed the one-dimensional scans along the cOS axis with cSS ¼ 0. The plots obtained for Ne, Ar, and CO are presented in Fig. 3. Note that all plots have a parabolic profile with the clear minimum (indicated by blue-dotted vertical line) obtained at cmin OS ¼ 1.22 for min Ne, cmin ¼ 1.48 for Ar and c ¼ 1.14 for CO. Similar results (not reported) OS OS have been obtained for the rest of the systems with the minima spreads in the range between 1.0 1.5. This feature shows evidently that the cOS parameter for which the minimum IDD value is obtained (cmin OS ) is strictly system-dependent quantity. This, in turn, leads to the conclusion that the universal value of the parameter, which works with the same accuracy for all systems cannot be found. Thus, as in the SOS-MP2 method, it seems reasonable to consider an average optimal copt OS parameter which in our case is 0.1 0.08
Ne
0.06 0.04 0.02 0 0.06 0.05 0.04 0.03 0.02 0.01 0 0.25 0.2 0.15 0.1 0.05 0
Ar
CO
0
0.2
0.4
0.6
0.8
1 COS
1.2
1.4
1.6
1.8
2
Fig. 3 The IDD scans along the cOS axis (cSS ¼ 0) for Ne (top), Ar (middle), and CO (bottom). Blue (dotted) line indicates the minimum value of IDD error (Ne—cmin OS ¼ 1.22, Ar— min cmin OS ¼ 1.48, CO—cOS ¼ 1.14) of the IDD error. Red (dash-dotted) line indicates the average optimal value of copt OS ¼ 1.25.
287
Density-Based Analysis of Spin-Resolved MP2 Method
4pr 2rc (a.u.)
4pr 2rc (a.u.)
obtained as an overall average value of all cmin OS parameters minimizing IDD opt error, what gives cOS ¼ 1.25 (indicated in Fig. 3 by dash-dotted red line). Interestingly, this value is relatively close to the one proposed in SOSMP2 method (i.e., cOS ¼ 1.3).10 However, the correlation energies calculated with the copt OS ¼ 1.25 coefficient (optimized for the correlated densities) are slightly worse than those obtained with the cOS ¼ 1.3 one, which was optimized to reproduce the CCSD(T) correlation energies. For the systems considered in this chapter, the MAE errors of correlation energies with respect to CCSD(T) data are 0.037 for SOS-MP2 and 0.054 for the SR-MP2 with copt OS ¼ 1.25. To conclude this section, in Fig. 4 we report the correlated densities calmin culated with optimal copt OS and minimizing cOS parameter compared to the reference CCSD(T) ones. Again, we see that for the cmin OS one can reproduce the reference CCSD(T) data with a very good accuracy. The correlated densities calculated using the copt OS parameter also perform very well, being slightly min worse than in the case of cOS value. The SOS-MP2 correlated densities (not reported in Fig. 4) are in line with our optimal ones, which is not surprising due to relatively small difference in the cOS parameter value. 0.02 0 −0.02 −0.04 −0.06 −0.08
min
cos
opt cos
0.04 0.02 0 −0.02 −0.04
min
Ar, cos = 1.48 0
1
2 r (a.u.)
0.05 rc (a.u.)
min
Ne, cos = 1.22
CCSD(T)
3
4
0 −0.05 min
−0.1 −3
CO, cos = 1.14 −2
−1
0 z (a.u.)
1
2
3
opt Fig. 4 The correlated densities calculated for OS-MP2 density with cmin OS , cOS ¼ 1.25 scaling parameters for Ne (top), Ar (middle), and CO (bottom). For comparison the CCSD(T) correlated densities are also plotted.
288
Mateusz Witkowski et al.
3.3 Dipole Moments To link the results of the density analysis conducted in the previous sections, with a more tangible quantity we present in Table 1 the assessment of the dipole moments calculated using different SR-MP2 methods (SCS-MP2, SOS-MP2, and with the copt OS coefficient). The dipole moment is an important quantity characterizing the charge distribution of a molecular electronic system being directly related to the electronic density. Thus, it can be also used as a kind of measure of the quality of electron density. For comparison, several other standard methods have been considered, i.e., HF and MP2. To assess the quality of the results, we report in Table 1 the root mean square errors (RMS), the mean absolute errors (MAE) and finally the mean absolute relative errors (MARE) calculated with respect to the CCSD(T) reference data. Note that only systems with nonzero dipoles are considered and the aug-cc-pVTZ basis set was used in the calculations. At first glance one can note that all SR-MP2 methods show a reduction of all types of error compared to the standard MP2. The results obtained with optimized cOS parameter are comparable with those from the SOSMP2 method, yielding MAE and MARE of 0.26, 4.1% and 0.25, 3.6%, respectively. Slightly larger errors are observed for the SCS-MP2 method which yields MAE of 0.035 and MARE of 6.7%. Nevertheless, we note that the performance of all SR-MP2 methods is good, being roughly twice better than the standard MP2. It must be noted, that in the case of NNO molecule the MP2 method predicts a qualitatively incorrect dipole moment 0.052 D compared with +0.115 D for CCSD(T) (the experimental value is +0.167 D43). The error is reduced in all SR-MP2 methods for which the correct sign and quantitatively accurate predictions of dipole moment are obtained. The errors obtained for the different SR-MP2 methods are minor, which can be a result of the pretty flat minimum in the density scan (see Fig. 3). So, all the parameterizations used in SCS-MP2, SOS-MP2, and with our copt OS parameter seem to work similarly and give the dipole moments of almost the same quality. Certainly because of the lower computational cost of the opposite part of the MP2 method, the one parameter variant (i.e., with cSS ¼ 0) should be used in practical calculations.
4. FINAL REMARKS The density-based analysis presented in this work indicated several aspects related to SR-MP2 methods. First of all, we have shown the evident
Table 1 Dipole Moments (in Debye) Obtained for Several Spin-Resolved Methods SR-MP2 CCSD(T)
HF
MP2
SCS-MP2 SOS-MP2 This Worka
CH2CF2
1.416
1.559
1.352
1.398
1.420
1.426
ClF
0.893
1.114
0.893
0.909
0.917
0.925
CH3CN
4.517
4.918
4.533
4.513
4.503
4.519
H2S
0.973
1.091
0.999
0.988
0.982
0.986
HCOOH 1.379
1.641
1.351
1.358
1.363
1.373
HNO
1.615
1.945
1.617
1.618
1.621
1.634
HNO3
2.194
2.558
2.122
2.138
2.148
2.164
CHF3
1.466
1.635
1.473
1.480
1.483
1.489
SiO
3.048
3.713
2.955
3.014
3.042
3.068
SO
1.372
1.763
1.312
1.339
1.352
1.368
SO2
1.653
2.026
1.571
1.614
1.635
1.650
N2 O 3
2.440
3.566
2.378
2.408
2.426
2.470
SOCl2
1.493
1.943
1.411
1.457
1.492
1.509
FCN
2.191
2.333
2.226
2.195
2.180
2.186
H2CO
2.360
2.841
2.355
2.372
2.380
2.398
H2CS
1.674
2.165
1.632
1.642
1.646
1.666
H2 O
1.838
1.980
1.851
1.851
1.850
1.855
HCCF
0.713
0.895
0.683
0.727
0.748
0.754
HCl
1.089
1.209
1.120
1.107
1.101
1.105
HCN
3.007
3.293
3.030
3.029
3.028
3.038
HF
1.803
1.937
1.814
1.811
1.809
1.814
NH3
1.507
1.611
1.521
1.524
1.525
1.528
CO
0.138
0.254
0.287
0.247
0.228
0.209
NNO
0.115
0.618
0.052 0.052
0.103
0.123
0.063
0.035
0.026
0.025
0.046
0.026
0.019
0.020
6.7
4.1
3.6
Relative to RMSb CCSD(T) MAEc d
MARE [%] 12.6
SOS-MP2 method with copt OS ¼ 1.25. Root mean square error. c Mean absolute error. d Mean absolute relative error. The HF and CCSD(T) data are also included for comparison. The calculations have been performed in aug-cc-pVTZ basis set. a
b
290
Mateusz Witkowski et al.
proportionality of the SS and OS parts of the MP2 correlated densities. This physical feature is important because it provides a rationale for the wellknown SOS-MP2 method, which up to now have been based mainly on the empirical observations focusing only on the correlation energy. Moreover, we have shown that the cOS is a system-dependent parameter which optimized for particular system leads to the significant improvement of the representation of correlation effects in density. This means that all optimal-parameter-based SR-MP2 methods (e.g., SCS-MP2, SOS-MP2) benefit rather from the statistical error cancelation than the actual better representation of correlation effects. Nonetheless, despite this small drawback, the spin-resolved methods can lead to relative good increase of accuracy of the results. Furthermore, due to relatively small numerical cost (especially the SOS variant), the method can be applied to relatively large molecular systems. In conclusion, the analysis provides a valuable insight for practical and deeper understanding of the SR-MP2 methods what can be essential for the further development of SR-MP2 type approximation for different applications (e.g., noncovalent interaction as in Refs. 16 and 17). Additional work can be foreseen. For example, a similar type of analysis can be performed for other spin-resolved methods, e.g., SCS-CCSD.3
ACKNOWLEDGMENTS This work was partially supported by the National Science Center under Grant No. DEC2016/21/D/ST4/00903 and DEC-2013/11/B/ST4/00771. Conflict of interest: The authors declare no competing financial interest.
REFERENCES 1. Grimme, S. Improved Second-Order Møller-Plesset Perturbation Theory by Separate Scaling of Parallel- and Antiparallel-Spin Pair Correlation Energies. J. Chem. Phys. 2003, 118(20), 9095–9102. https://doi.org/10.1063/1.1569242. 2. Grimme, S.; Goerigk, L.; Fink, R. F. Spin-Component-Scaled Electron Correlation Methods. WIREs Comput. Mol. Sci. 2012, 2(6), 886–906. https://doi.org/10.1002/ wcms.1110. 3. Takatani, T.; Hohenstein, E. G.; Sherrill, C. D. Improvement of the Coupled-Cluster Singles and Doubles Method Via Scaling Same- and Opposite-Spin Components of the Double Excitation Correlation Energy. J. Chem. Phys. 2008, 128(12), 124111. https:// doi.org/10.1063/1.2883974. 4. Hellweg, A.; Grun, S. A.; Hattig, C. Benchmarking the Performance of SpinComponent Scaled CC2 in Ground and Electronically Excited States. Phys. Chem. Chem. Phys. 2008, 10, 4119–4127. https://doi.org/10.1039/B803727B. 5. Pitonak, M.; Rezac, J.; Hobza, P. Spin-Component Scaled Coupled-Clusters Singles and Doubles Optimized Towards Calculation of Noncovalent Interactions. Phys. Chem. Chem. Phys. 2010, 12, 9611–9614. https://doi.org/10.1039/C0CP00158A.
Density-Based Analysis of Spin-Resolved MP2 Method
291
6. Lochan, R. C.; Jung, Y.; Head-Gordon, M. Scaled Opposite Spin Second Order Møller-Plesset Theory With Improved Physical Description of Long-Range Dispersion Interactions. J. Phys. Chem. A 2005, 109(33), 7598–7605. https://doi.org/10.1021/ jp0514426. PMID: 16834130. 7. Distasio, R. A., Jr.; Head-Gordon, M. Optimized Spin-Component Scaled SecondOrder Møller-Plesset Perturbation Theory for Intermolecular Interaction Energies. Mol. Phys. 2007, 105(8), 1073–1083. https://doi.org/10.1080/00268970701283781. 8. Møller, C.; Plesset, M. S. Note on an Approximate Treatment for Many-Electron Systems. Phys. Rev. 1934, 36, 618–622. 9. Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. A Fifth-Order Perturbation Comparison of Electron Correlation Theories. Chem. Phys. Lett. 1989, 157(6), 479–483. https://doi.org/10.1016/S0009-2614(89)87395-6. 10. Jung, Y.; Lochan, R. C.; Dutoi, A. D.; Head-Gordon, M. Scaled Opposite-Spin Second Order Møller-Plesset Correlation Energy: An Economical Electronic Structure Method. J. Chem. Phys. 2004, 121(20), 9793–9802. https://doi.org/10.1063/1.1809602. 11. Grimme, S. Accurate Calculation of the Heats of Formation for Large Main Group Compounds With Spin-Component Scaled MP2 Methods. J. Phys. Chem. A 2005, 109(13), 3067–3077. https://doi.org/10.1021/jp050036j. PMID: 16833631. 12. Gerenkamp, M.; Grimme, S. Spin-Component Scaled Second-Order Møller-Plesset Perturbation Theory for the Calculation of Molecular Geometries and Harmonic Vibrational Frequencies. Chem. Phys. Lett. 2004, 392(1–3), 229–235. https://doi.org/10.1016/j.cplett.2004.05.063. http://www.sciencedirect.com/science/article/pii/S000926140400764X. 13. Maurer, S. A.; Kussmann, J.; Ochsenfeld, C. Communication: A Reduced Scaling J-Engine Based Reformulation of SOS-MP2 Using Graphics Processing Units. J. Chem. Phys. 2014, 141(5), 051106. https://doi.org/10.1063/1.4891797. 14. Song, C.; Martı´nez, T. J. M. Atomic Orbital-Based SOS-MP2 With Tensor Hypercontraction. II. Local Tensor Hypercontraction. J. Chem. Phys. 2017, 146(3), 034104. https://doi.org/10.1063/1.4973840. 15. Hill, J. G.; Platts, J. A. Spin-Component Scaling Methods for Weak and Stacking Interactions. J. Chem. Theory Comput. 2007, 3(1), 80–85. https://doi.org/10.1021/ ct6002737. 16. Grabowski, I.; Fabiano, E.; Della Sala, F. A Simple Non-empirical Procedure for SpinComponent-Scaled MP2 Methods Applied to the Calculation of the Dissociation Energy Curve of Noncovalently-Interacting Systems. Phys. Chem. Chem. Phys. 2013, 15, 15485–15493. https://doi.org/10.1039/C3CP51431E. 17. Fabiano, E.; Della Sala, F.; Grabowski, I. Accurate Non-Covalent Interaction Energies Via an Efficient MP2 Scaling Procedure. Chem. Phys. Lett. 2015, 635, 262–267. https://doi. org/10.1016/j.cplett.2015.06.082. http://www.sciencedirect.com/science/article/pii/ S0009261415005138. 18. Tan, S.; Barrera Acevedo, S.; Izgorodina, E. I. Generalized Spin-Ratio Scaled MP2 Method for Accurate Prediction of Intermolecular Interactions for Neutral and Ionic Species. J. Chem. Phys. 2017, 146(6), 064108. https://doi.org/10.1063/1.4975326. 19. Szabados, A´. Theoretical Interpretation of Grimme’s Spin-Component-Scaled Second Order Møller-Plesset Theory. J. Chem. Phys. 2006, 125(21), 214105. https://doi.org/ 10.1063/1.2404660. 20. Fink, R. F. Spin-Component-Scaled Møller-Plesset (SCS-MP) Perturbation Theory: A Generalization of the MP Approach With Improved Properties. J. Chem. Phys. 2010, 133(17), 174113. https://doi.org/10.1063/1.3503041. 21. Grabowski, I.; Hirata, S.; Ivanov, S.; Bartlett, R. J. Ab Initio Density Functional Theory: OEP-MBPT(2). A New Orbital-Dependent Correlation Functional. J. Chem. Phys. 2002, 116(11), 4415–4425. https://doi.org/10.1063/1.1445117.
292
Mateusz Witkowski et al.
22. Bartlett, R. J.; Grabowski, I.; Hirata, S.; Ivanov, S. The Exchange-Correlation Potential in Ab Initio Density Functional Theory. J. Chem. Phys. 2005, 122(3), 034104. https:// doi.org/10.1063/1.1809605. 23. Kozuch, S.; Martin, J. M. L. DSD-PBEP86: In Search of the Best Double-Hybrid DFT With Spin-Component Scaled MP2 and Dispersion Corrections. Phys. Chem. Chem. Phys. 2011, 13, 20104–20107. https://doi.org/10.1039/C1CP22592H. 24. Yu, F. Spin-Component-Scaled Double-Hybrid Density Functionals With Nonlocal van der Waals Correlations for Noncovalent Interactions. J. Chem. Theory Comput. 2014, 10(10), 4400–4407. https://doi.org/10.1021/ct500642x. PMID: 26588137. 25. Kozuch, S.; Martin, J. M. L. Spin-Component-Scaled Double Hybrids: An Extensive Search for the Best Fifth-Rung Functionals Blending DFT and Perturbation Theory. J. Comput. Chem. 2013, 34(27), 2327–2344. https://doi.org/10.1002/jcc.23391. 26. Bremond, E.; Savarese, M.; Sancho-Garcia, J. C.; Perez-Jimenez, A´. J.; Adamo, C. Quadratic Integrand Double-Hybrid Made Spin-Component-Scaled. J. Chem. Phys. 2016, 144(12), 124104. https://doi.org/10.1063/1.4944465. 27. Mezei, P. D.; Csonka, G. I.; Ruzsinszky, A.; Ka´llay, M. Construction of a Spin-Component Scaled Dual-Hybrid Random Phase Approximation. J. Chem. Theory Comput. 2017, 13(2), 796–803. https://doi.org/10.1021/acs.jctc.6b01140. PMID: 28052197. 28. Grabowski, I.; Fabiano, E.; Della Sala, F. Optimized Effective Potential Method Based on Spin-Resolved Components of the Second-Order Correlation Energy in Density Functional Theory. Phys. Rev. B 2013, 87, 075103. https://doi.org/10.1103/ PhysRevB.87.075103. 29. Buksztel, A.; S´miga, S.; Grabowski, I. The Correlation Effects in Density Functional Theory Along the Dissociation Path. In: Electron Correlation in Molecules—Ab Initio Beyond Gaussian Quantum Chemistry; Hoggan, P. E., Ozdogan, T., Eds.; Advances in Quantum Chemistry; Vol. 73; Academic Press: Oxford, 2016; pp. 263–283. https:// doi.org/10.1016/bs.aiq.2015.07.002 (chapter 14). 30. Grabowski, I.; Fabiano, E.; Teale, A. M.; S´miga, S.; Buksztel, A.; Della Sala, F. OrbitalDependent Second-Order Scaled-Opposite-Spin Correlation Functionals in the Optimized Effective Potential Method. J. Chem. Phys. 2014, 141(2), 024113. https://doi. org/10.1063/1.4887097. 31. S´miga, S.; Della Sala, F.; Buksztel, A.; Grabowski, I.; Fabiano, E. Accurate Kohn-Sham Ionization Potentials From Scaled-Opposite-Spin Second-Order Optimized Effective Potential Methods. J. Comput. Chem. 2016, 37(22), 2081–2090. https://doi.org/ 10.1002/jcc.24436. 32. Handy, N. C.; Schaefer, H. F., III. On the Evaluation of Analytic Energy Derivatives for Correlated Wave Functions. J. Chem. Phys. 1984, 81(11), 5031–5033. 33. Bartlett, R. J. Analytical Evaluation of Gradients in Coupled-Cluster and Many-Body Perturbation Theory. In: Geometrical Derivatives of Energy Surfaces and Molecular Properties, Jørgensen, P., Simons, J., Eds.; Reidel, Dordecht: The Netherlands, 1986; pp. 35–61. 34. Salter, E. A.; Trucks, G. W.; Bartlett, R. J. Analytic Energy Derivatives in Many-Body Methods. I. First Derivatives. J. Chem. Phys. 1989, 90(3), 1752–1766. https://doi.org/ 10.1063/1.456069. 35. Jørgensen, P.; Helgaker, T. Møller-Plesset Energy Derivatives. J. Chem. Phys. 1988, 89(3), 1560–1570. 36. Koch, H.; Jensen, H. J. A.; Jørgensen, P.; Helgaker, T.; Scuseria, G. E.; Schaefer, H. F., III Coupled Cluster Energy Derivatives. Analytic Hessian for the Closed-Shell Coupled Cluster Singles and Doubles Wave Function: Theory and Applications. J. Chem. Phys. 1990, 92(8), 4924–4940. 37. Hald, K.; Halkier, A.; Jørgensen, P.; Coriani, S.; H€attig, C.; Helgaker, T. A Lagrangian, Integral-Density Direct Formulation and Implementation of the Analytic CCSD and CCSD(T) Gradients. J. Chem. Phys. 2003, 118(7), 2985–2998.
Density-Based Analysis of Spin-Resolved MP2 Method
293
38. Valiev, M.; Bylaska, E. J.; Govind, N.; Kowalski, K.; Straatsma, T. P.; Van Dam, H. J. J.; Wang, D.; Nieplocha, J.; Apra, E.; Windus, T. L.; de Jong, W. A. NWChem: A Comprehensive and Scalable Open-Source Solution for Large Scale Molecular Simulations. Comput. Phys. Commun. 2010, 181(9), 1477–1489. https://doi.org/10.1016/j.cpc. 2010.04.018. http://www.sciencedirect.com/science/article/pii/S0010465510001438. 39. Dunning, T. H., Jr. Gaussian Basis Sets for use in Correlated Molecular Calculations. I. The Atoms Boron Through Neon and Hydrogen. J. Chem. Phys. 1989, 90(2), 1007–1023. https://doi.org/10.1063/1.456153. 40. Woon, D. E. Benchmark Calculations With Correlated Molecular Wave Functions. V. The Determination of Accurate Ab Initio Intermolecular Potentials for He2, Ne2, and Ar2. J. Chem. Phys. 1994, 100(4), 2838–2850. https://doi.org/10.1063/1.466478. 41. Kendall, R. A.; Dunning, T. H., Jr.; Harrison, R. J. Electron Affinities of the First Row Atoms Revisited. Systematic Basis Sets and Wave Functions. J. Chem. Phys. 1992, 96(9), 6796–6806. https://doi.org/10.1063/1.462569. 42. Woon, D. E.; Dunning, T. H., Jr. Calculation of the Electron Affinities of the Second Row Atoms: Al-Cl. J. Chem. Phys. 1993, 99(5), 3730–3737. https://doi.org/ 10.1063/1.466148. 43. Johnson, R. D., III., Ed. NIST Computational Chemistry Comparison and Benchmark Database, NIST Standard Reference Database Number 101 Release 18, 2016, http://cccbdb.nist.gov/. 44. Jankowski, K.; Nowakowski, K.; Grabowski, I.; Wasilewski, J. Coverage of Dynamic Correlation Effects by Density Functional Theory Functionals: Density-Based Analysis for Neon. J. Chem. Phys. 2009, 130(16), 164102 (pages 9). 45. Jankowski, K.; Nowakowski, K.; Grabowski, I.; Wasilewski, J. Ab Initio Dynamic Correlation Effects in Density Functional Theories: A Density Based Study for Argon. Theor. Chem. Acc. 2010, 125(3–6), 433–444. 46. Grabowski, I.; Teale, A. M.; S´miga, S.; Bartlett, R. J. Comparing Ab Initio DensityFunctional and Wave Function Theories: The Impact of Correlation on the Electronic Density and the Role of the Correlation Potential. J. Chem. Phys. 2011, 135(11), 114111. 47. S´miga, S.; Buksztel, A.; Grabowski, I. Density-Dependent Exchange-Correlation Potentials Derived From highly Accurate Ab Initio Calculations. In: Proceedings of MEST 2012: Electronic Structure Methods With Applications to Experimental Chemistry; Hoggan, P., Ed.; Vol. 68 Elsevier Academic Press Inc, 2014; pp. 125–151. 48. Grabowski, I.; Teale, A. M.; Fabiano, E.; S´miga, S.; Buksztel, A.; Della Sala, F. A Density Difference Based Analysis of Orbital-Dependent Exchange-Correlation Functionals. Mol. Phys. 2014, 112(5–6), 700–710. https://doi.org/10.1080/00268976.2013.854424.
Density-Based Analysis of Spin-Resolved MP2 Method Mateusz Witkowski, Szymon Śmiga1, Ireneusz Grabowski Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Torun, Poland 1 Corresponding author: e-mail address: [email protected]
Contents 1. Introduction 2. Computational Details 3. Methodology and Results 3.1 Proportionality of the Spin-Resolved MP2 Densities 3.2 Two-Dimensional Scan 3.3 Dipole Moments 4. Final Remarks Acknowledgments References
279 281 282 282 284 288 288 290 290
Abstract The extensive study of the spin-resolved second-order Møller–Plesset method in the context of the electron density is performed. It was found the well-defined proportionality of the same- and opposite-spin parts of the MP2 correlated electronic density. We have rationalized the value of the scaling parameter used in the foundation of the SOS-MP2 (Jung et al., 2004) method from the density point of view. Our analysis is complemented by the calculations of the dipole moments using differently parameterized spin-resolved MP2 methods.
1. INTRODUCTION Since 2003, when Grimme proposed the spin-component-scaled MP2 method,1 a rapid development of spin-resolved (SR) methods is observed in the field of quantum chemistry. Many useful variants have been proposed, analyzed, and implemented2 in different levels of approximations and different application areas.3–7 However, those based on the second-order Møller–Plesset
Advances in Quantum Chemistry, Volume 76 ISSN 0065-3276 https://doi.org/10.1016/bs.aiq.2017.05.004
#
2018 Elsevier Inc. All rights reserved.
279
280
Mateusz Witkowski et al.
perturbation theory (MP2)8 energy expression are the most interesting ones from point of view of a simplicity and computational cost. Rescaling of the same-spin (SS) and opposite-spin (OS) part of the MP2 correlation energy proposed in Ref. 1 leads to a spin-resolved MP2 method (SR-MP2) EcSRMP2 ¼ cOS E MP2ðOSÞ + cSS EMP2ðSSÞ
(1)
with EcMP2ðOSÞ ¼ EcMP2ðSSÞ ¼
1 X X jðiσ aσ j jτ bτ Þj2 2 σ , τ6¼σ ijab Eiσ + Ejτ Eaσ Ebτ
1 X X jðiσ aσ jjσ bσ Þj2 2 σ ijab Eiσ + Ejσ Eaσ Ebσ
1 X X ðiσ aσ jjσ bσ Þðaσ jσ jbσ iσ Þ , 2 σ ijab Eiσ + Ejσ Eaσ Ebσ
(2)
(3)
where cOS and cSS are the OS and SS coefficients, respectively. In the above a, b, and i, j label the virtual and occupied indexes, respectively, while σ and τ label the spin indexes. Such simple modification of the MP2 energy expression results in significant improvement in the accuracy of calculated properties which, in some cases, can be compared to those obtained from a more advanced methods such as the coupled-cluster single double and perturbative triple [CCSD(T)]9 approach. At the same time the SR-MP2 method retains the computational simplicity of its original counterpart as well as other important features like size-consistency and orbital-rotational invariance. In the initial work1 the cOS ¼ 6/5 and cSS ¼ 1/3 scaling parameters have been optimized with respect to a training set of reaction energies. Defined in this manner the “spin-component scaled” MP2 method (SCS-MP2) gives rather good improvement in the quality of atomization energy, barrier heights,1,10 heats of formation,11 or molecular geometries.1,10,12 Another interesting variant of SR-MP2 method is the one, proposed by Jung et al. dubbed “scaled-opposite spin” MP2 (SOS-MP2).10 In this simplified method, the SS contribution to the correlation energy in Eq. (1) is modeled by proper selection of the cOS ¼ 1.3 parameter, while cSS ¼ 0. In general, SOS-MP2 leads to a similar quality of results as the SCS-MP2 method. However, due to neglect of the SS term in Eq. (1), computational efficiency
Density-Based Analysis of Spin-Resolved MP2 Method
281
increases significantly, leading to reduction of the numerical cost from OðN 5 Þ to OðN 4 Þ or even less.13,14 Many other useful parameterizations of Eq. (1) have been proposed, based on semiempirical7,15–18 as well as theoretical considerations.19,20 Furthermore, the spin-resolved concept was also applied in the context of the density functional theory (DFT) and ab initio DFT,21,22 where Eq. (1) was utilized as the correlation energy functional in the double-hybrid23–27 and second-order optimized effective potential (OEP) method,28–31 respectively. The latter studies28,30 are especially interesting because they have revealed that their exists almost linear proportionality between the OS and SS part of the correlation energies, Kohn–Sham correlation potentials and correlated electronic densities (although, the proportionality factors are different for different quantities). This suggests that similar proportionality of densities may also exist in the SR-MP2 method. To date, however, such an analysis was not carried out. Thus to fill this gap, in this chapter we devote special attention to the density analysis of spin-resolved methods. The work is organized as follows. In Section 3.1 we discuss the proportionality of the spin-resolved densities and its implications. Next, in Section 3.2 we extend this analysis by presenting two-dimensional scans of the SR-MP2 densities with respect to the CCSD(T). Finally, in Section 3.3 we investigate the impact of these results on the dipole moments which are especially sensitive to the quality of the electron density.
2. COMPUTATIONAL DETAILS The main quantity used in our analysis is the electron density calculated obtained from the MP2 and CCSD(T) methods. The densities are calculated from the relaxed density matrices32–34 constructed using the Lagrangian approach.35–37 In the case of SR-MP2 method the SS and OS parts of correlated densities have been obtained by a scaling of adequate parts in the Lagrangian. Calculations have been carried out in NWChem software package.38 The basis sets employed are the spherical cc-pVXZ and aug-cc-pVXZ basis sets, where X¼(D,T,Q).39–42 All electrons were included in the calculation of correlated quantities. The calculations were performed for several atomic: Ne, Ar, Kr, Zn, and molecular systems, i.e., Cl2, CS, F2, He2, N2, OCS, P2, and those listed in Table 1. For the latter ones, the experimental geometries were taken from Ref. 43. As in our previous studies16,17,28,30 the CCSD(T)9 results were used as the reference benchmark data.
282
Mateusz Witkowski et al.
3. METHODOLOGY AND RESULTS In this section we discuss several aspects related to the SR-MP2 method in the context of electron density. In particular we analyze the proportionality of the spin-resolved densities by examining its relative parts and the possibility of reconstruction of the CCSD(T) density by proper scaling of the SR-MP2 density. Due to large qualitative and quantitative resemblance of densities obtained within all basis sets, for the sake of clarity of the discussion, we report only those obtained with aug-cc-pVQZ basis set.
3.1 Proportionality of the Spin-Resolved MP2 Densities In order to demonstrate all features of second-order spin-resolved densities we utilize the concept of difference total densities (DTD) distribution, defined with respect to the total Hartree–Fock (HF) density [ρHF(r)] ðrÞ ¼ ρmethod ðrÞ ρHF ðrÞ, ρmethod c
(4)
where ρmethod(r) is the total density gained from correlated (e.g., MP2 or CCSD(T)) calculations. The DTD was successfully used as a testing tool in the development of new methods, especially in the context of DFT.28,44–48 We note that in case of atoms, the ρc(r) is closely related to another quantity, namely the difference radial-density (DRD) distributions,44 defined as ðrÞ: DRDmethod ðrÞ ¼ 4πr 2 ρmethod c
(5)
Thus, the latter one is used for atoms. For the clarity of discussion, henceforth we will refer to both quantities as correlated densities. In Fig. 1 we present the SS and OS parts of MP2 correlated densities obtained for Ne and Ar atoms and for the CO molecule. Similar results have been observed in the case of other systems (not reported here). One can immediately observe the clear point-by-point proportionality of the spinresolved parts of the MP2 correlated densities. All minima and maxima positions are almost ideally aligned with each other, even in the molecular case. This in turn suggests that SS correlation effects can be effectively modeled by properly rescaling the OS contributions only, which is more favorable from the computational point of view. We remark, however, that the relative density scaling factor γ which links SS and OS correlated densities SS ρOS c ðrÞ ¼ γρc ðrÞ
(6)
283
4pr 2rc (a.u.)
4pr 2rc (a.u.)
Density-Based Analysis of Spin-Resolved MP2 Method
0.04 0.03 0.02 0.01 0 −0.01 −0.02 −0.03 0.02
Ne, g = 2.3 OS SS γ SS
0 −0.02 −0.04
Ar, g = 2.1
−0.06 0
1
2 r (a.u.)
rc (a.u.)
0.05
3
4
0 −0.05 −0.1 −3
CO, g = 2.1 −2
−1
0 z (a.u.)
1
2
3
Fig. 1 Correlated density distributions for Ne (top), Ar (middle), and CO (bottom, plot along the molecular axis) calculated for SS and OS parts of MP2 density. On each plot, the value of γ factor rescaling SS correlated density has also been shown. See the text for details.
is not a general constant. It slightly depends on the choice of basis set as well as the molecular system itself. For example, in the case of the CO molecule for the aug-cc-pVDZ and aug-cc-pVQZ, γ changes from 2.3 to 2.1, respectively. Nevertheless, in general, for most systems and basis sets, the proportionality parameter ranges between the 1.7 and 2.5. For the systems presented in Fig. 1, the γ values are γ ¼ 2.3 for Ne and γ ¼ 2.1 for Ar and CO. We underline here that proportionality of resolved density is the general feature observed in all considered systems in our set, which is the first important finding of this study. Furthermore, above results confirm the previous findings,2,16–18 based mostly on the analysis of SS and OS parts of the SR-MP2 correlation energy expression. The density-based analysis presented here is more demanding than those based on the correlation energy, because we scale not just one number (like in the correlation energy case) but the scaling holds for the whole space where the electron density is calculated. Hence, this kind of analysis can be crucial for understanding and rationalizing the behavior of spin-resolved-based methods.
284
Mateusz Witkowski et al.
3.2 Two-Dimensional Scan In the previous section we have shown that SS and OS parts of the MP2 correlated densities display a very high degree of proportionality, and then that proper scaling of one term (e.g., the SS one) can reproduce general spatial behavior of another one (OS correlated density), with relatively good accuracy. In order to better understand and make possible use of this issue in its practical application, we perform more quantitative analysis. To this end, we conducted a full two-dimensional scan of the space spanned by the cOS and cSS parameters, analyzing, in each point of space the error indicator function (the integrated density differences—IDD30) defined as Z IDDðcOS , cSS Þ ¼ drρSRMP2 ðr; cOS ,cSS Þ ρCCSDðTÞ ðrÞ: (7) c c where ρcCCSDðTÞ is the correlated CCSD(T) density. The first term in Eq. (7) is the correlated spin-resolved density calculated for the fixed pair of scaling coefficients as ρSRMP2 ðr;cOS , cSS Þ ¼ cOS ρOSMP2 ðrÞ + cSS ρSSMP2 ðrÞ: c c c
(8)
The cOS and cSS coefficients have been varied in the range between 0.0 and 2.0 with the 0.1 interval. The integration in Eq. (7) runs over whole threedimensional space on a very accurate grid of spatial points. One can note that the indicator defined by Eq. (7) tells explicitly how accurately the SR-MP2 method can reproduce the CCSD(T) reference ρSRMP2 ), for a fixed pair of coefficients. density, i.e., we obtain (ρCCSDðTÞ c c It also shows implicitly, the global qualitative relative proportionality of OS and SS parts of MP2 correlated densities. As before the scans have been done for all investigated systems. However, in Fig. 2, we report only the scan obtained for a CO molecular system which presents general trend observed in our results. Inspection of the figure confirms the observations made in Section 3.1, i.e., the proportionality of the OS and SS parts of the MP2 correlated densities. Furthermore we see that the SR-MP2 method can reproduce the reference CCSD(T) densities with quite good accuracy. We note that similar findings have been observe in Ref. 28, however, in the context of ab initio DFT. Nonetheless, several other features may be observed (a) The IDD indicator shows the minima for a continuous set of parameter pairs, which can be obtained by a simple linear relation cOS ¼ αcSS + β, with α and β being system-dependent constants. This relation indicates the evident proportionality of the spin-resolved densities.
285
Density-Based Analysis of Spin-Resolved MP2 Method
0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05
0.5 IDD (a.u.)
0.4 0.3 0.2 0.1 0 0
0.5
1 COS
1.5
2 0
0.5
1
1.5
2
CSS
Fig. 2 The IDD scan in the function of cOS and cSS parameters for CO molecule.
(b) The performance of standard SCS-MP2 and SOS-MP2 methods, which yield almost identical IDD’s, i.e., 0.07 and 0.065, respectively, is better than in the MP2 case where IDD ¼ 0.09. The global minimum, with lower IDD ¼ 0.06, is found for cOS ¼ 1.1 and cSS ¼ 0.1 pair of coefficients. (c) The absolute minimum value of IDD is nonzero. This is due to the fact that the IDD is an integrated quantity which describes how correlation effects are reproduced in the density for all points in the space. This, in turn, means that the reference CCSD(T) density cannot be reproduced by a simple scaling of second-order spin-resolved densities. This may lead to the conclusion that wherever the MP2 method fails badly (e.g., in systems with multireference character) also the simple rescaled methods will lead to similar predictions. (d) For the case where SS part of the density is neglected (cSS ¼ 0), it is always possible to find a value of the cOS parameter such that the IDD error is minimized and its value is close to the global absolute minimum. This is a direct consequence of the proportionality found here and described in Section 3.1. In fact, the SS compensation term is proportional to the inverse of γ (present in Eq. (6)) and gives ðrÞ: ρcCCSDðTÞ ðcOS + γ 1 cSS ÞρOSMP2 c
(9)
This, in turn, justifies assumed proportionality used in the foundation of the SOS-MP2 method. According to SOS-MP2 definition,10 in order to compensate the lack of SS part of the correlation, one needs to
286
Mateusz Witkowski et al.
increment the OS scaling factor of SCS-MP21 method by about 1/9. This means that the proportionality coefficient γ for SOS-MP2 method equals 10/3 3.3 which is slightly larger than those obtained in the present work (γ ¼ 1.7 2.5). Due to the latter feature we have also performed the one-dimensional scans along the cOS axis with cSS ¼ 0. The plots obtained for Ne, Ar, and CO are presented in Fig. 3. Note that all plots have a parabolic profile with the clear minimum (indicated by blue-dotted vertical line) obtained at cmin OS ¼ 1.22 for min Ne, cmin ¼ 1.48 for Ar and c ¼ 1.14 for CO. Similar results (not reported) OS OS have been obtained for the rest of the systems with the minima spreads in the range between 1.0 1.5. This feature shows evidently that the cOS parameter for which the minimum IDD value is obtained (cmin OS ) is strictly system-dependent quantity. This, in turn, leads to the conclusion that the universal value of the parameter, which works with the same accuracy for all systems cannot be found. Thus, as in the SOS-MP2 method, it seems reasonable to consider an average optimal copt OS parameter which in our case is 0.1 0.08
Ne
0.06 0.04 0.02 0 0.06 0.05 0.04 0.03 0.02 0.01 0 0.25 0.2 0.15 0.1 0.05 0
Ar
CO
0
0.2
0.4
0.6
0.8
1 COS
1.2
1.4
1.6
1.8
2
Fig. 3 The IDD scans along the cOS axis (cSS ¼ 0) for Ne (top), Ar (middle), and CO (bottom). Blue (dotted) line indicates the minimum value of IDD error (Ne—cmin OS ¼ 1.22, Ar— min cmin OS ¼ 1.48, CO—cOS ¼ 1.14) of the IDD error. Red (dash-dotted) line indicates the average optimal value of copt OS ¼ 1.25.
287
Density-Based Analysis of Spin-Resolved MP2 Method
4pr 2rc (a.u.)
4pr 2rc (a.u.)
obtained as an overall average value of all cmin OS parameters minimizing IDD opt error, what gives cOS ¼ 1.25 (indicated in Fig. 3 by dash-dotted red line). Interestingly, this value is relatively close to the one proposed in SOSMP2 method (i.e., cOS ¼ 1.3).10 However, the correlation energies calculated with the copt OS ¼ 1.25 coefficient (optimized for the correlated densities) are slightly worse than those obtained with the cOS ¼ 1.3 one, which was optimized to reproduce the CCSD(T) correlation energies. For the systems considered in this chapter, the MAE errors of correlation energies with respect to CCSD(T) data are 0.037 for SOS-MP2 and 0.054 for the SR-MP2 with copt OS ¼ 1.25. To conclude this section, in Fig. 4 we report the correlated densities calmin culated with optimal copt OS and minimizing cOS parameter compared to the reference CCSD(T) ones. Again, we see that for the cmin OS one can reproduce the reference CCSD(T) data with a very good accuracy. The correlated densities calculated using the copt OS parameter also perform very well, being slightly min worse than in the case of cOS value. The SOS-MP2 correlated densities (not reported in Fig. 4) are in line with our optimal ones, which is not surprising due to relatively small difference in the cOS parameter value. 0.02 0 −0.02 −0.04 −0.06 −0.08
min
cos
opt cos
0.04 0.02 0 −0.02 −0.04
min
Ar, cos = 1.48 0
1
2 r (a.u.)
0.05 rc (a.u.)
min
Ne, cos = 1.22
CCSD(T)
3
4
0 −0.05 min
−0.1 −3
CO, cos = 1.14 −2
−1
0 z (a.u.)
1
2
3
opt Fig. 4 The correlated densities calculated for OS-MP2 density with cmin OS , cOS ¼ 1.25 scaling parameters for Ne (top), Ar (middle), and CO (bottom). For comparison the CCSD(T) correlated densities are also plotted.
288
Mateusz Witkowski et al.
3.3 Dipole Moments To link the results of the density analysis conducted in the previous sections, with a more tangible quantity we present in Table 1 the assessment of the dipole moments calculated using different SR-MP2 methods (SCS-MP2, SOS-MP2, and with the copt OS coefficient). The dipole moment is an important quantity characterizing the charge distribution of a molecular electronic system being directly related to the electronic density. Thus, it can be also used as a kind of measure of the quality of electron density. For comparison, several other standard methods have been considered, i.e., HF and MP2. To assess the quality of the results, we report in Table 1 the root mean square errors (RMS), the mean absolute errors (MAE) and finally the mean absolute relative errors (MARE) calculated with respect to the CCSD(T) reference data. Note that only systems with nonzero dipoles are considered and the aug-cc-pVTZ basis set was used in the calculations. At first glance one can note that all SR-MP2 methods show a reduction of all types of error compared to the standard MP2. The results obtained with optimized cOS parameter are comparable with those from the SOSMP2 method, yielding MAE and MARE of 0.26, 4.1% and 0.25, 3.6%, respectively. Slightly larger errors are observed for the SCS-MP2 method which yields MAE of 0.035 and MARE of 6.7%. Nevertheless, we note that the performance of all SR-MP2 methods is good, being roughly twice better than the standard MP2. It must be noted, that in the case of NNO molecule the MP2 method predicts a qualitatively incorrect dipole moment 0.052 D compared with +0.115 D for CCSD(T) (the experimental value is +0.167 D43). The error is reduced in all SR-MP2 methods for which the correct sign and quantitatively accurate predictions of dipole moment are obtained. The errors obtained for the different SR-MP2 methods are minor, which can be a result of the pretty flat minimum in the density scan (see Fig. 3). So, all the parameterizations used in SCS-MP2, SOS-MP2, and with our copt OS parameter seem to work similarly and give the dipole moments of almost the same quality. Certainly because of the lower computational cost of the opposite part of the MP2 method, the one parameter variant (i.e., with cSS ¼ 0) should be used in practical calculations.
4. FINAL REMARKS The density-based analysis presented in this work indicated several aspects related to SR-MP2 methods. First of all, we have shown the evident
Table 1 Dipole Moments (in Debye) Obtained for Several Spin-Resolved Methods SR-MP2 CCSD(T)
HF
MP2
SCS-MP2 SOS-MP2 This Worka
CH2CF2
1.416
1.559
1.352
1.398
1.420
1.426
ClF
0.893
1.114
0.893
0.909
0.917
0.925
CH3CN
4.517
4.918
4.533
4.513
4.503
4.519
H2S
0.973
1.091
0.999
0.988
0.982
0.986
HCOOH 1.379
1.641
1.351
1.358
1.363
1.373
HNO
1.615
1.945
1.617
1.618
1.621
1.634
HNO3
2.194
2.558
2.122
2.138
2.148
2.164
CHF3
1.466
1.635
1.473
1.480
1.483
1.489
SiO
3.048
3.713
2.955
3.014
3.042
3.068
SO
1.372
1.763
1.312
1.339
1.352
1.368
SO2
1.653
2.026
1.571
1.614
1.635
1.650
N2 O 3
2.440
3.566
2.378
2.408
2.426
2.470
SOCl2
1.493
1.943
1.411
1.457
1.492
1.509
FCN
2.191
2.333
2.226
2.195
2.180
2.186
H2CO
2.360
2.841
2.355
2.372
2.380
2.398
H2CS
1.674
2.165
1.632
1.642
1.646
1.666
H2 O
1.838
1.980
1.851
1.851
1.850
1.855
HCCF
0.713
0.895
0.683
0.727
0.748
0.754
HCl
1.089
1.209
1.120
1.107
1.101
1.105
HCN
3.007
3.293
3.030
3.029
3.028
3.038
HF
1.803
1.937
1.814
1.811
1.809
1.814
NH3
1.507
1.611
1.521
1.524
1.525
1.528
CO
0.138
0.254
0.287
0.247
0.228
0.209
NNO
0.115
0.618
0.052 0.052
0.103
0.123
0.063
0.035
0.026
0.025
0.046
0.026
0.019
0.020
6.7
4.1
3.6
Relative to RMSb CCSD(T) MAEc d
MARE [%] 12.6
SOS-MP2 method with copt OS ¼ 1.25. Root mean square error. c Mean absolute error. d Mean absolute relative error. The HF and CCSD(T) data are also included for comparison. The calculations have been performed in aug-cc-pVTZ basis set. a
b
290
Mateusz Witkowski et al.
proportionality of the SS and OS parts of the MP2 correlated densities. This physical feature is important because it provides a rationale for the wellknown SOS-MP2 method, which up to now have been based mainly on the empirical observations focusing only on the correlation energy. Moreover, we have shown that the cOS is a system-dependent parameter which optimized for particular system leads to the significant improvement of the representation of correlation effects in density. This means that all optimal-parameter-based SR-MP2 methods (e.g., SCS-MP2, SOS-MP2) benefit rather from the statistical error cancelation than the actual better representation of correlation effects. Nonetheless, despite this small drawback, the spin-resolved methods can lead to relative good increase of accuracy of the results. Furthermore, due to relatively small numerical cost (especially the SOS variant), the method can be applied to relatively large molecular systems. In conclusion, the analysis provides a valuable insight for practical and deeper understanding of the SR-MP2 methods what can be essential for the further development of SR-MP2 type approximation for different applications (e.g., noncovalent interaction as in Refs. 16 and 17). Additional work can be foreseen. For example, a similar type of analysis can be performed for other spin-resolved methods, e.g., SCS-CCSD.3
ACKNOWLEDGMENTS This work was partially supported by the National Science Center under Grant No. DEC2016/21/D/ST4/00903 and DEC-2013/11/B/ST4/00771. Conflict of interest: The authors declare no competing financial interest.
REFERENCES 1. Grimme, S. Improved Second-Order Møller-Plesset Perturbation Theory by Separate Scaling of Parallel- and Antiparallel-Spin Pair Correlation Energies. J. Chem. Phys. 2003, 118(20), 9095–9102. https://doi.org/10.1063/1.1569242. 2. Grimme, S.; Goerigk, L.; Fink, R. F. Spin-Component-Scaled Electron Correlation Methods. WIREs Comput. Mol. Sci. 2012, 2(6), 886–906. https://doi.org/10.1002/ wcms.1110. 3. Takatani, T.; Hohenstein, E. G.; Sherrill, C. D. Improvement of the Coupled-Cluster Singles and Doubles Method Via Scaling Same- and Opposite-Spin Components of the Double Excitation Correlation Energy. J. Chem. Phys. 2008, 128(12), 124111. https:// doi.org/10.1063/1.2883974. 4. Hellweg, A.; Grun, S. A.; Hattig, C. Benchmarking the Performance of SpinComponent Scaled CC2 in Ground and Electronically Excited States. Phys. Chem. Chem. Phys. 2008, 10, 4119–4127. https://doi.org/10.1039/B803727B. 5. Pitonak, M.; Rezac, J.; Hobza, P. Spin-Component Scaled Coupled-Clusters Singles and Doubles Optimized Towards Calculation of Noncovalent Interactions. Phys. Chem. Chem. Phys. 2010, 12, 9611–9614. https://doi.org/10.1039/C0CP00158A.
Density-Based Analysis of Spin-Resolved MP2 Method
291
6. Lochan, R. C.; Jung, Y.; Head-Gordon, M. Scaled Opposite Spin Second Order Møller-Plesset Theory With Improved Physical Description of Long-Range Dispersion Interactions. J. Phys. Chem. A 2005, 109(33), 7598–7605. https://doi.org/10.1021/ jp0514426. PMID: 16834130. 7. Distasio, R. A., Jr.; Head-Gordon, M. Optimized Spin-Component Scaled SecondOrder Møller-Plesset Perturbation Theory for Intermolecular Interaction Energies. Mol. Phys. 2007, 105(8), 1073–1083. https://doi.org/10.1080/00268970701283781. 8. Møller, C.; Plesset, M. S. Note on an Approximate Treatment for Many-Electron Systems. Phys. Rev. 1934, 36, 618–622. 9. Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. A Fifth-Order Perturbation Comparison of Electron Correlation Theories. Chem. Phys. Lett. 1989, 157(6), 479–483. https://doi.org/10.1016/S0009-2614(89)87395-6. 10. Jung, Y.; Lochan, R. C.; Dutoi, A. D.; Head-Gordon, M. Scaled Opposite-Spin Second Order Møller-Plesset Correlation Energy: An Economical Electronic Structure Method. J. Chem. Phys. 2004, 121(20), 9793–9802. https://doi.org/10.1063/1.1809602. 11. Grimme, S. Accurate Calculation of the Heats of Formation for Large Main Group Compounds With Spin-Component Scaled MP2 Methods. J. Phys. Chem. A 2005, 109(13), 3067–3077. https://doi.org/10.1021/jp050036j. PMID: 16833631. 12. Gerenkamp, M.; Grimme, S. Spin-Component Scaled Second-Order Møller-Plesset Perturbation Theory for the Calculation of Molecular Geometries and Harmonic Vibrational Frequencies. Chem. Phys. Lett. 2004, 392(1–3), 229–235. https://doi.org/10.1016/j.cplett.2004.05.063. http://www.sciencedirect.com/science/article/pii/S000926140400764X. 13. Maurer, S. A.; Kussmann, J.; Ochsenfeld, C. Communication: A Reduced Scaling J-Engine Based Reformulation of SOS-MP2 Using Graphics Processing Units. J. Chem. Phys. 2014, 141(5), 051106. https://doi.org/10.1063/1.4891797. 14. Song, C.; Martı´nez, T. J. M. Atomic Orbital-Based SOS-MP2 With Tensor Hypercontraction. II. Local Tensor Hypercontraction. J. Chem. Phys. 2017, 146(3), 034104. https://doi.org/10.1063/1.4973840. 15. Hill, J. G.; Platts, J. A. Spin-Component Scaling Methods for Weak and Stacking Interactions. J. Chem. Theory Comput. 2007, 3(1), 80–85. https://doi.org/10.1021/ ct6002737. 16. Grabowski, I.; Fabiano, E.; Della Sala, F. A Simple Non-empirical Procedure for SpinComponent-Scaled MP2 Methods Applied to the Calculation of the Dissociation Energy Curve of Noncovalently-Interacting Systems. Phys. Chem. Chem. Phys. 2013, 15, 15485–15493. https://doi.org/10.1039/C3CP51431E. 17. Fabiano, E.; Della Sala, F.; Grabowski, I. Accurate Non-Covalent Interaction Energies Via an Efficient MP2 Scaling Procedure. Chem. Phys. Lett. 2015, 635, 262–267. https://doi. org/10.1016/j.cplett.2015.06.082. http://www.sciencedirect.com/science/article/pii/ S0009261415005138. 18. Tan, S.; Barrera Acevedo, S.; Izgorodina, E. I. Generalized Spin-Ratio Scaled MP2 Method for Accurate Prediction of Intermolecular Interactions for Neutral and Ionic Species. J. Chem. Phys. 2017, 146(6), 064108. https://doi.org/10.1063/1.4975326. 19. Szabados, A´. Theoretical Interpretation of Grimme’s Spin-Component-Scaled Second Order Møller-Plesset Theory. J. Chem. Phys. 2006, 125(21), 214105. https://doi.org/ 10.1063/1.2404660. 20. Fink, R. F. Spin-Component-Scaled Møller-Plesset (SCS-MP) Perturbation Theory: A Generalization of the MP Approach With Improved Properties. J. Chem. Phys. 2010, 133(17), 174113. https://doi.org/10.1063/1.3503041. 21. Grabowski, I.; Hirata, S.; Ivanov, S.; Bartlett, R. J. Ab Initio Density Functional Theory: OEP-MBPT(2). A New Orbital-Dependent Correlation Functional. J. Chem. Phys. 2002, 116(11), 4415–4425. https://doi.org/10.1063/1.1445117.
292
Mateusz Witkowski et al.
22. Bartlett, R. J.; Grabowski, I.; Hirata, S.; Ivanov, S. The Exchange-Correlation Potential in Ab Initio Density Functional Theory. J. Chem. Phys. 2005, 122(3), 034104. https:// doi.org/10.1063/1.1809605. 23. Kozuch, S.; Martin, J. M. L. DSD-PBEP86: In Search of the Best Double-Hybrid DFT With Spin-Component Scaled MP2 and Dispersion Corrections. Phys. Chem. Chem. Phys. 2011, 13, 20104–20107. https://doi.org/10.1039/C1CP22592H. 24. Yu, F. Spin-Component-Scaled Double-Hybrid Density Functionals With Nonlocal van der Waals Correlations for Noncovalent Interactions. J. Chem. Theory Comput. 2014, 10(10), 4400–4407. https://doi.org/10.1021/ct500642x. PMID: 26588137. 25. Kozuch, S.; Martin, J. M. L. Spin-Component-Scaled Double Hybrids: An Extensive Search for the Best Fifth-Rung Functionals Blending DFT and Perturbation Theory. J. Comput. Chem. 2013, 34(27), 2327–2344. https://doi.org/10.1002/jcc.23391. 26. Bremond, E.; Savarese, M.; Sancho-Garcia, J. C.; Perez-Jimenez, A´. J.; Adamo, C. Quadratic Integrand Double-Hybrid Made Spin-Component-Scaled. J. Chem. Phys. 2016, 144(12), 124104. https://doi.org/10.1063/1.4944465. 27. Mezei, P. D.; Csonka, G. I.; Ruzsinszky, A.; Ka´llay, M. Construction of a Spin-Component Scaled Dual-Hybrid Random Phase Approximation. J. Chem. Theory Comput. 2017, 13(2), 796–803. https://doi.org/10.1021/acs.jctc.6b01140. PMID: 28052197. 28. Grabowski, I.; Fabiano, E.; Della Sala, F. Optimized Effective Potential Method Based on Spin-Resolved Components of the Second-Order Correlation Energy in Density Functional Theory. Phys. Rev. B 2013, 87, 075103. https://doi.org/10.1103/ PhysRevB.87.075103. 29. Buksztel, A.; S´miga, S.; Grabowski, I. The Correlation Effects in Density Functional Theory Along the Dissociation Path. In: Electron Correlation in Molecules—Ab Initio Beyond Gaussian Quantum Chemistry; Hoggan, P. E., Ozdogan, T., Eds.; Advances in Quantum Chemistry; Vol. 73; Academic Press: Oxford, 2016; pp. 263–283. https:// doi.org/10.1016/bs.aiq.2015.07.002 (chapter 14). 30. Grabowski, I.; Fabiano, E.; Teale, A. M.; S´miga, S.; Buksztel, A.; Della Sala, F. OrbitalDependent Second-Order Scaled-Opposite-Spin Correlation Functionals in the Optimized Effective Potential Method. J. Chem. Phys. 2014, 141(2), 024113. https://doi. org/10.1063/1.4887097. 31. S´miga, S.; Della Sala, F.; Buksztel, A.; Grabowski, I.; Fabiano, E. Accurate Kohn-Sham Ionization Potentials From Scaled-Opposite-Spin Second-Order Optimized Effective Potential Methods. J. Comput. Chem. 2016, 37(22), 2081–2090. https://doi.org/ 10.1002/jcc.24436. 32. Handy, N. C.; Schaefer, H. F., III. On the Evaluation of Analytic Energy Derivatives for Correlated Wave Functions. J. Chem. Phys. 1984, 81(11), 5031–5033. 33. Bartlett, R. J. Analytical Evaluation of Gradients in Coupled-Cluster and Many-Body Perturbation Theory. In: Geometrical Derivatives of Energy Surfaces and Molecular Properties, Jørgensen, P., Simons, J., Eds.; Reidel, Dordecht: The Netherlands, 1986; pp. 35–61. 34. Salter, E. A.; Trucks, G. W.; Bartlett, R. J. Analytic Energy Derivatives in Many-Body Methods. I. First Derivatives. J. Chem. Phys. 1989, 90(3), 1752–1766. https://doi.org/ 10.1063/1.456069. 35. Jørgensen, P.; Helgaker, T. Møller-Plesset Energy Derivatives. J. Chem. Phys. 1988, 89(3), 1560–1570. 36. Koch, H.; Jensen, H. J. A.; Jørgensen, P.; Helgaker, T.; Scuseria, G. E.; Schaefer, H. F., III Coupled Cluster Energy Derivatives. Analytic Hessian for the Closed-Shell Coupled Cluster Singles and Doubles Wave Function: Theory and Applications. J. Chem. Phys. 1990, 92(8), 4924–4940. 37. Hald, K.; Halkier, A.; Jørgensen, P.; Coriani, S.; H€attig, C.; Helgaker, T. A Lagrangian, Integral-Density Direct Formulation and Implementation of the Analytic CCSD and CCSD(T) Gradients. J. Chem. Phys. 2003, 118(7), 2985–2998.
Density-Based Analysis of Spin-Resolved MP2 Method
293
38. Valiev, M.; Bylaska, E. J.; Govind, N.; Kowalski, K.; Straatsma, T. P.; Van Dam, H. J. J.; Wang, D.; Nieplocha, J.; Apra, E.; Windus, T. L.; de Jong, W. A. NWChem: A Comprehensive and Scalable Open-Source Solution for Large Scale Molecular Simulations. Comput. Phys. Commun. 2010, 181(9), 1477–1489. https://doi.org/10.1016/j.cpc. 2010.04.018. http://www.sciencedirect.com/science/article/pii/S0010465510001438. 39. Dunning, T. H., Jr. Gaussian Basis Sets for use in Correlated Molecular Calculations. I. The Atoms Boron Through Neon and Hydrogen. J. Chem. Phys. 1989, 90(2), 1007–1023. https://doi.org/10.1063/1.456153. 40. Woon, D. E. Benchmark Calculations With Correlated Molecular Wave Functions. V. The Determination of Accurate Ab Initio Intermolecular Potentials for He2, Ne2, and Ar2. J. Chem. Phys. 1994, 100(4), 2838–2850. https://doi.org/10.1063/1.466478. 41. Kendall, R. A.; Dunning, T. H., Jr.; Harrison, R. J. Electron Affinities of the First Row Atoms Revisited. Systematic Basis Sets and Wave Functions. J. Chem. Phys. 1992, 96(9), 6796–6806. https://doi.org/10.1063/1.462569. 42. Woon, D. E.; Dunning, T. H., Jr. Calculation of the Electron Affinities of the Second Row Atoms: Al-Cl. J. Chem. Phys. 1993, 99(5), 3730–3737. https://doi.org/ 10.1063/1.466148. 43. Johnson, R. D., III., Ed. NIST Computational Chemistry Comparison and Benchmark Database, NIST Standard Reference Database Number 101 Release 18, 2016, http://cccbdb.nist.gov/. 44. Jankowski, K.; Nowakowski, K.; Grabowski, I.; Wasilewski, J. Coverage of Dynamic Correlation Effects by Density Functional Theory Functionals: Density-Based Analysis for Neon. J. Chem. Phys. 2009, 130(16), 164102 (pages 9). 45. Jankowski, K.; Nowakowski, K.; Grabowski, I.; Wasilewski, J. Ab Initio Dynamic Correlation Effects in Density Functional Theories: A Density Based Study for Argon. Theor. Chem. Acc. 2010, 125(3–6), 433–444. 46. Grabowski, I.; Teale, A. M.; S´miga, S.; Bartlett, R. J. Comparing Ab Initio DensityFunctional and Wave Function Theories: The Impact of Correlation on the Electronic Density and the Role of the Correlation Potential. J. Chem. Phys. 2011, 135(11), 114111. 47. S´miga, S.; Buksztel, A.; Grabowski, I. Density-Dependent Exchange-Correlation Potentials Derived From highly Accurate Ab Initio Calculations. In: Proceedings of MEST 2012: Electronic Structure Methods With Applications to Experimental Chemistry; Hoggan, P., Ed.; Vol. 68 Elsevier Academic Press Inc, 2014; pp. 125–151. 48. Grabowski, I.; Teale, A. M.; Fabiano, E.; S´miga, S.; Buksztel, A.; Della Sala, F. A Density Difference Based Analysis of Orbital-Dependent Exchange-Correlation Functionals. Mol. Phys. 2014, 112(5–6), 700–710. https://doi.org/10.1080/00268976.2013.854424.