Second-order Polarization Propagator Approximation Research Articles

Performance of range-separated long-range SOPPA short-range density functional theory method for vertical excitation energies.

In this paper, benchmark results are presented on the calculation of vertical electronic excitation energies using a long-range second-order polarization propagator approximation (SOPPA) description with a short-range density functional theory description based on the Perdew-Burke-Ernzerhof (PBE) functional. The excitation energies are investigated for 132 singlet states and 71 triplet states across 28 medium-sized organic molecules. The results show that overall SOPPA-srPBE always performs better than PBE and that SOPPA-srPBE performs better than SOPPA for singlet states, but slightly worse than SOPPA for triplet states when CC3 results are the reference values.

On the performance of HRPA(D) for NMR spin–spin coupling constants: Smaller molecules, aromatic and fluoroaromatic compounds

In this study, the performance of the doubles-corrected higher random-phase approximation [HRPA(D)] has been investigated in calculations of nuclear magnetic resonance spin–spin coupling constants (SSCCs) for 58 molecules with the experimental values used as the reference values. HRPA(D) is an approximation to the second-order polarization propagator approximation (SOPPA) and is, therefore, computationally less expensive than SOPPA. HRPA(D) performs comparable and sometimes even better than SOPPA, and therefore, when calculating SSCCs, it should be considered as an alternative to SOPPA. Furthermore, it was investigated whether a coupled-cluster singles, doubles and perturbative triples [CCSD(T)] or Møller-Plesset second order (MP2) geometry optimization was optimal for a SOPPA and a HRPA(D) SSCC calculation for eight smaller molecules. CCSD(T) is the optimal geometry optimization for the SOPPA calculation, and MP2 was optimal for HRPA(D) SSCC calculations.

Exploring Alternate Methods for the Calculation of High-Level Vibrational Corrections of NMR Spin-Spin Coupling Constants.

Traditional nuclear magnetic resonance (NMR) calculations typically treat systems with a Born-Oppenheimer-derived electronic wave function that is solved for a fixed nuclear geometry. One can numerically account for this neglected nuclear motion by averaging over property values for all nuclear geometries with a vibrational wave function and adding this expectation value as a correction to an equilibrium geometry property value. Presented are benchmark coupled-cluster singles and doubles (CCSD) vibrational corrections to spin-spin coupling constants (SSCCs) computed at the level of vibrational second-order perturbation theory (VPT2) using the vibrational averaging driver of the CFOUR program. As CCSD calculations of vibrational corrections are very costly, cheaper electronic structure methods are explored via a newly developed Python vibrational averaging program within the Dalton Project. Namely, results obtained with the second-order polarization propagator approximation (SOPPA) and density functional theory (DFT) with the B3LYP and PBE0 exchange-correlation functionals are compared to the benchmark CCSD//CCSD(T) and experimental values. CCSD//CCSD(T) corrections are also combined with literature CC3 equilibrium geometry values to form the highest-order vibrationally corrected values available (i.e., CC3//CCSD(T) + CCSD//CCSD(T)). CCSD//CCSD(T) statistics showed favorable statistics in comparison to experimental values, albeit at an unfavorably high computational cost. A cheaper CCSD//CCSD(T) + B3LYP method showed quite similar mean absolute deviation (MAD) values as CCSD//CCSD(T), concluding that CCSD//CCSD(T) + B3LYP is optimal in terms of cost and accuracy. With reference to experimental values, a vibrational correction was not worth the cost for all of the other methods tested. Finally, deviation statistics showed that CC3//CCSD(T) + CCSD//CCSD(T) vibrational-corrected equilibrium values deteriorated in comparison to CCSD//CCSD(T) attributed to the use of a smaller basis set or lack of solvation effects for the CC3 equilibrium calculations.

Indirect nuclear spin-spin couplings with third-order contributions added to the SOPPA method.

In this article, a modification of the second-order polarization propagator approximation (SOPPA) method is introduced and illustrated for the calculation of the indirect nuclear spin-spin couplings. The standard SOPPA method, although cheaper in terms of computational cost, offers less accurate results than the ones obtained with coupled cluster methods. A new method, named SOPPA+A3-3, was therefore developed by adding the terms of the third-order A matrix that rely on the second-order double amplitudes. The performance of this third-order contribution was studied using the coupled cluster singles and doubles method as a reference, calculating the spin-spin couplings of molecules of diverse sizes and compositions, and comparing them to the SOPPA method. The results show that inclusion of this third-order contribution gives more accurate results than the standard SOPPA method with a level of accuracy close to that of the coupled cluster method with only a small increase in the computational cost of the response calculation that dominates the computational cost for small- to medium-sized molecules. The implementation of the first contributions to the third-order polarization propagator approximation in the Dalton program, thus, already shows a significant change in these molecular properties over those obtained with the standard SOPPA method.

The Importance of the Density Functional Theory Exchange-Correlation Hartree-Fock Term in Magnetic Resonance: Application to an Aqueous Environment.

Within the framework of Density Functional Theory (DFT), the relevance of the term Hartree-Fock exchange (HFE) for a variety of molecular properties is a critical point. For this reason, we spend efforts to understand these relationships in nuclear magnetic resonance (NMR) parameters in a water solvent. This work takes advantage of the appropriate aug-cc-pVTZ-J basis set and the Minnesota family of DFT methods, which consider different portions of HFE contributions. With regard to solvent participation, the results are based on a sequential Monte Carlo/Quantum Mechanics procedure, which builds the structures of the liquid under realistic thermodynamic conditions. Compared to the accurate results of second-order polarization propagator approximation (SOPPA) and experimental data, all NMR parameters show a huge dependence on the size of the HFE contribution. For instance, the inclusion of this term in 1JOH and 2JHH indirect spin-spin couplings does vary with 49.661 and 25.459 Hz, respectively. The M06-HF method accounts for 100% of HFE and better matches the σO and σH shielding constants. On the other hand, 1JOH and 2JHH demand a medium contribution (54% of HFE), the best description being associated with the M06-2X method. Thus, the dependence varies regarding the phenomenology of the property in focus and the order for independent treatments. For elements that participate in hydrogen bonds simultaneously as donor and acceptor actors, the results indicate that explicit solvent molecules must be considered in the quantum mechanical calculations for better modeling of paramagnetic shielding constants.

Benchmarking Correlated Methods for Static and Dynamic Polarizabilities: The T145 Data Set Evaluated with RPA, RPA(D), HRPA, HRPA(D), SOPPA, SOPPA(CC2), SOPPA(CCSD), CC2, and CCSD.

Due to the importance of predicting static and dynamic polarizabilities, the performance of various correlated linear response methods including random phase approximation (RPA), RPA(D), higher-order random phase approximation (HRPA), HRPA(D), second-order polarization propagator approximation (SOPPA), SOPPA(CC2), SOPPA(CCSD), CC2, and CCSD has been evaluated against CCSD(T) (static case) and CCSD (dynamic cases) for the T145 set of 145 organic molecules. The benchmark reveals that the HRPA(D) method has the best performance for both static and dynamic polarizabilities apart from CCSD. RPA(D) ranks second for the dynamic cases and third for the static case. Using coupled-cluster amplitudes in SOPPA(CCSD) and SOPPA(CC2), the SOPPA results are significantly improved. The HRPA method has the largest deviations from the reference values for both cases. In general, according to the performance and computational cost of the methods, the HRPA(D) and RPA(D) methods are proposed for calculations of static and dynamic polarizabilities of this and similar sets of molecules.

Basis set dependence of 1H–X spin–spin coupling constants in non-empirical pure DFT framework, X = 1H, 13C, 19F, 35Cl: Case of CHCl=CH–CF3 stereoisomers

A collection of indirect spin–spin coupling constants (SSCCs) including 3J(H,H), nJ(C,H) (n = 1–3), nJ(H,F) (n = 3, 4), and nJ(H,Cl) (n = 2, 3) were assessed in the non-empirical density functional theory (DFT) framework for the stereoisomers of CHCl=CH–CF3. The present paper puts on view the DFT method, the basis set, and the stereoisomer dependences of the SSCCs. The basis sets cc-pVXZ and aug-cc-pVXZ (X = D, T, Q, and 5) were used to handle these nuclear parameters. The DFT scheme through the non-empirical exchange–correlation functionals Perdew–Burke–Ernzerhof, Perdew–Kurth–Zupan–Blaha exchange–correlation functional, Tao, Perdew, Staroverov, and Scuseria (PBEPBE, PKZBPKZB, and TPSSTPSS) was benchmarked by the computationally expensive second-order polarization propagator approximation method to provide a very good correlation of the nuclear SSCC parameters. This method was corrected by the relativistic, vibrational, and solvent effects. The findings show that the SSCCs are highly sensitive to the basis set. The convergence of SSCCs to the basis set limit is achieved for some coupling constants and tends to achieve for the others, except for 2J(H,Cl) and 3J(H,Cl). The functionals PBEPBE and PKZBPKZB appear as the excellent deal to assess all the coupling constants of this study. nJ(H,F) (n = 3, 4) and nJ(H,Cl) (n = 2, 3) are very well described by all the three competitive functionals and the different quality X of the basis set in regard to the experimental results and the reference values.

Efficient J‐oriented tin basis sets for the correlated calculations of indirect nuclear spin–spin coupling constants

New J-oriented tin basis sets, acvXz-J (X=2, 3, 4), have been developed at the level of the second-order polarization propagator approximation with the coupled-cluster single and double amplitudes, SOPPA (CCSD), for the purpose of correlated calculations of indirect nuclear spin-spin coupling constants involving tin nucleus. High-quality coupled-cluster calculations of several tin-proton and tin-carbon spin-spin coupling constants, performed with one of the newly developed basis sets, namely, the acv3z-J, taking into account relativistic, solvent, and vibrational corrections showed that the acv3z-J basis set is capable to provide reliable results, as compared with the experimental data.

Karplus relationships of the 2 JHNα and 3 JΗΝβ couplings in organic azides.

Density functional theory (DFT) and second-order polarization propagator approximation (SOPPA) computations in model organic azides revealed a Karplus-like dependence not only of the vicinal 3 JH-C-Nα-Nβ coupling but also of the geminal 2 JH-C-Nα one, with the H-C-Nα Nβ dihedral angle. Karplus equations were derived from the DFT computations on the isopropylazide model system. In light of these stablished relationships, natural abundance 1 H-15 N couplings obtained for the azide group of the zidovudine antiviral helped to probe its conformation around the C-Nα bond as being of the synclinal type.

How large a 3 JHCCH coupling can be? Prediction of giant vicinal couplings in quinonoid systems.

Second-order polarization propagator approximation (SOPPA) and density functional theory (DFT) B3LYP computations revealed the presence of giant 3 JHCCH couplings, up to more than 60 Hz, in conjugated quinonoid systems, very far above the typical limit of ~15 Hz. Strong hyperconjugative interactions of the C-H σ orbitals with the quinonoid 8-e- π system, which allows effective propagation of the spin polarization, seem to be responsible for those extraordinarily large couplings. Computation on several model systems showed the additive nature of the contributions from the available coupling paths to the total coupling.

RPA(D) and HRPA(D): calculation of carbon–carbon spin–spin coupling constants for saturated cycloalkanes

This study investigates the performance of two approximations to the popular second-order polarization propagator approximation (SOPPA), the doubles-corrected methods RPA(D) and HRPA(D), in calculating carbon–carbon spin–spin coupling constants (SSCCs) in 39 saturated carbocycles, totalling 188 unique coupling constants. RPA(D) scales an order below SOPPA in computational complexity while HRPA(D) differs from SOPPA in the leading coefficient. These methods may therefore prove beneficial in predictions of coupling constants of large molecules. It was found that HRPA(D) performs similarly to SOPPA in terms of accuracy for all coupling constants as well as significantly improves on RPA(D). With a roughly 55% reduction in computation time from SOPPA to HRPA(D), the latter shows great promise for the calculation of nuclear indirect carbon–carbon SSCCs in saturated carbocycles.

The Second-Order-Polarization-Propagator-Approximation (SOPPA) in a four-component spinor basis.

A theoretical framework for understanding molecular structures is crucial for the development of new technologies such as catalysts or solar cells. Apart from electronic excitation energies, however, only spectroscopic properties of molecules consisting of lighter elements can be computationally described at a high level of theory today since heavy elements require a relativistic framework, and thus far, most methods have only been derived in a non-relativistic framework. Important new technologies such as those mentioned above require molecules that contain heavier elements, and hence, there is a great need for the development of relativistic computational methods at a higher level of accuracy. Here, the Second-Order-Polarization-Propagator-Approximation (SOPPA), which has proven to be very successful in the non-relativistic case, is adapted to a relativistic framework. The equations for SOPPA are presented in their most general form, i.e., in a non-canonical spin-orbital basis, which can be reduced to the canonical case, and the expressions needed for a relativistic four-component SOPPA are obtained. The equations are one-index transformed, giving more compact expressions that correspond to those already available for the four-component RPA. The equations are ready for implementation in a four-component quantum chemistry program, which will allow both linear response properties and excitation energies to be calculated relativistically at the SOPPA level.

Noniterative Doubles Corrections to the Random Phase and Higher Random Phase Approximations: Singlet and Triplet Excitation Energies.

The second‐order noniterative doubles‐corrected random phase approximation (RPA) method has been extended to triplet excitation energies and the doubles‐corrected higher RPA method as well as a shifted version for calculating singlet and triplet excitation energies are presented here for the first time. A benchmark set consisting of 20 molecules with a total of 117 singlet and 71 triplet excited states has been used to test the performance of the new methods by comparison with previous results obtained with the second‐order polarization propagator approximation (SOPPA) and the third order approximate coupled cluster singles, doubles and triples model CC3. In general, the second‐order doubles corrections to RPA and HRPA significantly reduce both the mean deviation as well as the standard deviation of the errors compared to the CC3 results. The accuracy of the new methods approaches the accuracy of the SOPPA method while using only 10–60% of the calculation time. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc.

Performance of SOPPA-based methods in the calculation of vertical excitation energies and oscillator strengths

We present two new modifications of the second-order polarization propagator approximation (SOPPA), SOPPA(SCS-MP2) and SOPPA(SOS-MP2), which employ either spin-component-scaled or scaled opposite-spin MP2 correlation coefficients instead of the regular MP2 coefficients. The performance of these two methods, the original SOPPA method as well as SOPPA(CCSD) and RPA(D) in the calculation of vertical electronic excitation energies and oscillator strengths is investigated for a large benchmark set of 28 medium-sized molecules with 139 singlet and 71 triplet excited states. The results are compared with the corresponding CC3 and CASPT2 results from the literature for both the TZVP set and the larger and more diffuse aug-cc-pVTZ basis set. In addition, the results with the aug-cc-pVTZ basis set are compared with the theoretical best estimates for this benchmark set. We find that the original SOPPA method gives overall the smallest mean deviations from the reference values and the most consistent results.

Investigation of the resonance-assisted hydrogen bond in model β-diketones through localized molecular orbital analysis of the spin-spin coupling constants related to the O-H···O hydrogen bond.

The resonance-assisted hydrogen bond (HB) phenomenon has been studied theoretically by a localized molecular orbital (LMO) decomposition of the spin-spin coupling constants between atoms either involved or close to the O-H···O system of some β-diketones and their saturated counterparts. The analysis, carried out at the level of the second-order polarization propagator approximation, shows that the contributions in terms of LMO to the paramagnetic spin orbital and the spin dipolar Ramsey terms proof the importance of the delocalized π-electron structure supporting the idea of the existence of the resonance-assisted HB phenomenon phenomenon. The LMO contributions to the Fermi contact term indicate mainly the presence of the HB that may or not be linked to the π-electrons.

Nonempirical calculations of the one-bond (29)Si-(13)C spin-spin coupling constants taking into account relativistic and solvent corrections.

The computational study of the one-bond (29)Si-(13)C spin-spin coupling constants has been performed at the second-order polarization propagator approximation (SOPPA) level in the series of 60 diverse silanes with a special focus on the main factors affecting the accuracy of the calculation including the level of theory, the quality of the basis set, and the contribution of solvent and relativistic effects. Among three SOPPA-based methods, SOPPA(MP2), SOPPA(CC2), and SOPPA(CCSD), the best result was achieved with SOPPA(CCSD) when used in combination with Sauer's basis set aug-cc-pVTZ-J characterized by the mean absolute error of calculated coupling constants against the experiment of ca 2 Hz in the range of ca 200 Hz. The SOPPA(CCSD)/aug-cc-pVTZ-J method is recommended as the most accurate and effective computational scheme for the calculation of (1)J(Si,C). The slightly less accurate but essentially more economical SOPPA(MP2)/aug-cc-pVTZ-J and/or SOPPA(CC2)/aug-cc-pVTZ-J methods are recommended for larger molecular systems. It was shown that solvent and relativistic corrections do not play a major role in the computation of the total values of (1)J(Si,C); however, taking them into account noticeably improves agreement with the experiment. The rovibrational corrections are estimated to be of about 1 Hz or 1-1.5% of the total value of (1)J(Si,C).

The Second-Order Polarization Propagator Approximation (SOPPA) method coupled to the polarizable continuum model

We present an implementation of the Polarizable Continuum Model (PCM) in combination with the Second-Order Polarization Propagator Approximation (SOPPA) electronic structure method. In analogy with the most common way of designing ground state calculations based on a Second-Order Møller–Plesset (MP2) wave function coupled to PCM, we introduce dynamical PCM solvent effects only in the Random Phase Approximation (RPA) part of the SOPPA response equations while the static solvent contribution is kept in both the RPA terms as well as in the higher order correlation matrix components of the SOPPA response equations. By dynamic terms, we refer to contributions that describe a change in environmental polarization which, in turn, reflects a change in the core molecular charge distribution upon an electronic excitation. This new combination of methods is termed PCM–SOPPA/RPA. We apply this newly defined method to the challenging cases of solvent effects on the lowest and intense electronic transitions in o-, m- and p-nitroaniline and o-, m- and p-nitrophenol and compare the performance of PCM–SOPPA/RPA with more conventional approaches. Compared to calculations based on time-dependent density functional theory employing a range-separated exchange–correlation functional, we find the PCM–SOPPA/RPA approach to be slightly superior with respect to systematicity. On the other hand, the absolute values of the predicted excitation energies are largely underestimated. This – however – is a well-know feature of the SOPPA model itself and is not connected to its combination with the PCM.

Assessment of charge-transfer excitations with time-dependent, range-separated density functional theory based on long-range MP2 and multiconfigurational self-consistent field wave functions

Charge transfer excitations can be described within Time-Dependent Density Functional Theory (TD-DFT), not only by means of the Coulomb Attenuated Method (CAM) but also with a combination of wave function theory and TD-DFT based on range separation. The latter approach enables a rigorous formulation of multi-determinantal TD-DFT schemes where excitation classes, which are absent in conventional TD-DFT spectra (like for example double excitations), can be addressed. This paper investigates the combination of both the long-range Multi-Configuration Self-Consistent Field (MCSCF) and Second Order Polarization Propagator Approximation (SOPPA) ansätze with a short-range DFT (srDFT) description. We find that the combinations of SOPPA or MCSCF with TD-DFT yield better results than could be expected from the pure wave function schemes. For the Time-Dependent MCSCF short-range DFT ansatz (TD-MC-srDFT) excitation energies calculated over a larger benchmark set of molecules with predominantly single reference character yield good agreement with their reference values, and are in general comparable to the CAM-B3LYP functional. The SOPPA-srDFT scheme is tested for a subset of molecules used for benchmarking TD-MC-srDFT and performs slightly better against the reference data for this small subset. Beyond the proof-of-principle calculations comprising the first part of this contribution, we additionally studied the low-lying singlet excited states (S1 and S2) of the retinal chromophore. The chromophore displays multireference character in the ground state and both excited states exhibit considerable double excitation character, which in turn cannot be described within standard TD-DFT, due to the adiabatic approximation. However, a TD-MC-srDFT approach can account for the multireference character, and excitation energies are obtained with accuracy comparable to CASPT2, although using a much smaller active space.

Algebraic-diagrammatic construction polarization propagator approach to indirect nuclear spin–spin coupling constants

A new polarization propagator approach to indirect nuclear spin-spin coupling constantans is formulated within the framework of the algebraic-diagrammatic construction (ADC) approximation and implemented at the level of the strict second-order approximation scheme, ADC(2). The ADC approach possesses transparent computational procedure operating with Hermitian matrix quantities defined with respect to physical excitations. It is size-consistent and easily extendable to higher orders via the hierarchy of available ADC approximation schemes. The ADC(2) method is tested in the first applications to HF, N(2), CO, H(2)O, HCN, NH(3), CH(4), C(2)H(2), PH(3), SiH(4), CH(3)F, and C(2)H(4). The calculated indirect nuclear spin-spin coupling constants are in good agreement with the experimental data and results of the second-order polarization propagator approximation method. The computational effort of the ADC(2) scheme scales as n(5) with respect to the number of molecular orbitals n, which makes this method promising for applications to larger molecules.

On the importance of excited state dynamic response electron correlation in polarizable embedding methods

We investigate the effect of including a dynamic reaction field at the lowest possible ab inito wave function level of theory, namely the Hartree-Fock (HF) self-consistent field level within the polarizable embedding (PE) formalism. We formulate HF based PE within the linear response theory picture leading to the PE-random-phase approximation (PE-RPA) and bridge the expressions to a second-order polarization propagator approximation (SOPPA) frame such that dynamic reaction field contributions are included at the RPA level in addition to the static response described at the SOPPA level but with HF induced dipole moments. We conduct calculations on para-nitro-aniline and para-nitro-phenolate using said model in addition to dynamic PE-RPA and PE-CAM-B3LYP. We compare the results to recently published PE-CCSD data and demonstrate how the cost effective SOPPA-based model successfully recovers a great portion of the inherent PE-RPA error when the observable is the solvatochromic shift. We furthermore demonstrate that whenever the change in density resulting from the ground state-excited state electronic transition in the solute is not associated with a significant change in the electric field, dynamic response contributions formulated at the HF level of theory manage to capture the majority of the system response originating from derivative densities.