Particle-particle Random Phase Approximation Research Articles

Accurate Excitation Energies of Point Defects from Fast Particle-Particle Random Phase Approximation Calculations.

We present an efficient particle-particle random phase approximation (ppRPA) approach that predicts accurate excitation energies of point defects, including the nitrogen-vacancy (NV-) and silicon-vacancy (SiV0) centers in diamond and the divacancy center (VV0) in 4H silicon carbide, with errors of ±0.2 eV compared with experimental values. Starting from the (N + 2)-electron ground state calculated with density functional theory (DFT), the ppRPA excitation energies of the N-electron system are calculated as the differences between the two-electron removal energies of the (N + 2)-electron system. We demonstrate that the ppRPA excitation energies converge rapidly with a few hundred canonical active-space orbitals. We also show that active-space ppRPA has weak DFT starting-point dependence and is significantly cheaper than the corresponding ground-state DFT calculation. This work establishes ppRPA as an accurate and low-cost tool for investigating excited-state properties of point defects and opens up new opportunities for applications of ppRPA to periodic bulk materials.

Linear Scaling Calculations of Excitation Energies with Active-Space Particle-Particle Random-Phase Approximation.

We developed an efficient active-space particle-particle random-phase approximation (ppRPA) approach to calculate accurate charge-neutral excitation energies of molecular systems. The active-space ppRPA approach constrains both indexes in particle and hole pairs in the ppRPA matrix, which only selects frontier orbitals with dominant contributions to low-lying excitation energies. It employs the truncation in both orbital indexes in the particle-particle and the hole-hole spaces. The resulting matrix, whose eigenvalues are excitation energies, has a dimension that is independent of the size of the systems. The computational effort for the excitation energy calculation, therefore, scales linearly with system size and is negligible compared with the ground-state calculation of the (N - 2)-electron system, where N is the electron number of the molecule. With the active space consisting of 30 occupied and 30 virtual orbitals, the active-space ppRPA approach predicts the excitation energies of valence, charge-transfer, Rydberg, double, and diradical excitations with the mean absolute errors (MAEs) smaller than 0.03 eV compared with the full-space ppRPA results. As a side product, we also applied the active-space ppRPA approach in the renormalized singles (RS) T-matrix approach. Combining the non-interacting pair approximation that approximates the contribution to the self-energy outside the active space, the active-space GRSTRS@PBE approach predicts accurate absolute and relative core-level binding energies with the MAEs around 1.58 and 0.3 eV, respectively. The developed linear scaling calculation of excitation energies is promising for applications to large and complex systems.

Particle-Particle Collective Excitations of Sn isotopes

In this paper, energy-level schemes and reduced electric transition strengths of neutron-rich Tin isotopes 102, 110, 116, 120, 122Sn (Z=50) are studied using collective models, that is, particle-particle Tamm-Dancoff Approximation and particle-particle Random Phase Approximation. According to these models, the excited states of closed-core A+2 systems with multipolarity J and isospin T can be described as a linear combination of particle-particle pairs. In our investigation, the low-lying states of the investigated isotopes 102, 110, 116, 120, 122Sn are described by acting two-particle operators on a correlated core 100Sn, 108Sn, 114Sn, 118Sn, and 120Sn, respectively. The Hamiltonian is diagonalized within the model space include {1g7/2, 2d5/2, 2d3/2, 3s1/2 and 1h11/2} orbits, using the matrix elements of neutron-neutron interaction and modified surface delta interaction. The calculated values are checked by using the resultant eigenvalues and eigenvectors to calculate the excitation energies and reduced electric transition strengths. Our calculated results are compared to the available experimental data, and these comparisons led to reasonable agreements. Effective charges are also used to account for the core polarization effect.

Renormalized Singles Green's Function in the T-Matrix Approximation for Accurate Quasiparticle Energy Calculation.

We combine the renormalized singles (RS) Green's function with the T-matrix approximation for the single-particle Green's function to compute quasiparticle energies for valence and core states of molecular systems. The GRST0 method uses the RS Green's function that incorporates singles contributions as the initial Green's function. The GRSTRS method further calculates the generalized effective interaction with the RS Green's function by using RS eigenvalues in the T-matrix calculation through the particle-particle random phase approximation. The GRSTRS method provides significant improvements over one-shot methods G0T0 and G0W0 as demonstrated in calculations for GW100 and CORE65 test sets. It also systematically eliminates the dependence of G0T0 on the choice of density functional approximations. For valence states, the GRSTRS method provides excellent accuracy, which is better than that of G0T0 and G0W0. For core states, the GRSTRS method identifies correct peaks in the spectral function and significantly outperforms G0T0 on core-level binding energies (CLBEs) and relative CLBEs.

Comparing particle-particle and particle-hole channels of the random phase approximation

We present a comparative study of particle-hole and particle-particle channels of random-phase approximation (RPA) for molecular dissociations of different bonding types. We introduced a direct particle-particle RPA scheme, in analogy to the direct particle-hole RPA formalism, whereby the exchange-type contributions are excluded. This allows us to compare the behavior of the particle-hole and particle-particle RPA channels on the same footing. Our study unravels the critical role of exchange contributions in determining behaviors of the two RPA channels for describing stretched molecules. We also made an attempt to merge particle-hole RPA and particle-particle RPA into a unified scheme, with the double-counting terms removed. However, benchmark calculations indicate that a straightforward combination of the two RPA channels does not lead to a successful computational scheme for describing molecular dissociations.

Charge transfer excitation energies from ground state density functional theory calculations.

Calculating charge transfer (CT) excitation energies with high accuracy and low computational cost is a challenging task. Kohn-Sham density functional theory (KS-DFT), due to its efficiency and accuracy, has achieved great success in describing ground state problems. To extend to excited state problems, our group recently demonstrated an approach with good numerical results to calculate low-lying and Rydberg excitation energies of an N-electron system from a ground state KS or generalized KS calculations of an (N - 1)-electron system via its orbital energies. In the present work, we explore further the same methodology to describe CT excitations. Numerical results from this work show that performance of conventional density functional approximations (DFAs) is not as good for CT excitations as for other excitations due to the delocalization error. Applying localized orbital scaling correction (LOSC) to conventional DFAs, a recently developed method in our group to effectively reduce the delocalization error, can improve the results. Overall, the performance of this methodology is better than time dependent DFT (TDDFT) with conventional DFAs. In addition, it shows that results from LOSC-DFAs in this method reach similar accuracy to other methods, such as ΔSCF, G0W0 with Bethe-Salpeter equations, particle-particle random phase approximation, and even high-level wavefunction methods like CC2. Our analysis shows that the correct 1/R trend for CT excitation can be captured from LOSC-DFA calculations, stressing that the application of DFAs with the minimal delocalization error is essential within this methodology. This work provides an efficient way to calculate CT excitation energies from ground state DFT.

Spin-state energetics of iron(II) porphyrin from the particle-particle random phase approximation

The particle-particle random phase approximation (pp-RPA) has been deployed to study the spin-state energetics of transition metal (TM) complexes for the first time in this work. Namely, we designed and implemented a non-canonical reference pp-RPA protocol that is capable of capturing the singlet low-spin (LS) – triplet intermediate-spin (IS) excitation process of iron(II) complexes; herein we applied this method to iron-porphyrin related heme derivatives with clearly defined LS and IS electronic states. Coupled to the CAM-B3LYP functional and to Dunning-type basis sets, we utilized both the active-space and Davidson methods to solve the pp-RPA equation effectively to obtain vertical singlet–triplet excitation energies. Correcting these vertical metrics with a structural relaxation factor for each species, we evaluated the relative stability of LS and IS electronic states. Comparison of the pp-RPA results to established ab initio data revealed that pp-RPA describes well excitation energies and related relative spin state stabilities if the transition is based on non-bonding d-orbitals, such as complexes without an axial ligand in the investigated set of molecules. But it notably overestimates the stability of the singlet LS state to the triplet IS state in complexes, where the d-orbitals at which the excitation is centered have bonding or antibonding character.

Single, Double Electronic Excitations and Exciton Effective Conjugation Lengths in π-Conjugated Systems.

The 21Ag and 11Bu excited states of two prototypical π-conjugated compounds, polyacetylene and polydiacetylene, are investigated with the recently developed particle-particle random phase approximation (pp-RPA) method combined with the B3LYP functional. The polymer-limit transition energies are estimated as 1.38 and 1.72 eV for the 21Ag and 11Bu states, respectively, from an extrapolation of the computed excitation energies of model oligomers. These values increase to 1.95 and 2.24 eV for the same transitions when ground-state structures with ∼33% larger bond length alternation are adopted. Applying the pp-RPA to the vertical excitation energies in oligodiacetylene, the polymer-limit transition energies of the 21Ag and 11Bu states are computed to be 2.06 and 2.28 eV, respectively. These results are in good agreement with experimental values or theoretical best estimates, indicating that the pp-RPA method shows great promise for understanding many photophysical phenomena involving both single and double excitations.

Accurate Treatment of Charge-Transfer Excitations and Thermally Activated Delayed Fluorescence Using the Particle-Particle Random Phase Approximation.

Thermally activated delayed florescence (TADF) is a mechanism that increases the electroluminescence efficiency in organic light-emitting diodes by harnessing both singlet and triplet excitons. TADF is facilitated by a small energy difference between the first singlet (S1) and triplet (T1) excited states (Δ E(ST)), which is minimized by spatial separation of the donor and acceptor moieties. The resultant charge-transfer (CT) excited states are difficult to model using time-dependent density functional theory (TDDFT) because of the delocalization error present in standard density functional approximations to the exchange-correlation energy. In this work we explore the application of the particle-particle random phase approximation (pp-RPA) for the determination of both S1 and T1 excitation energies. We demonstrate that the accuracy of the pp-RPA is functional dependent and that, when combined with the hybrid functional B3LYP, the pp-RPA computed Δ E(ST) have a mean absolute deviation (MAD) of 0.12 eV for the set of examined molecules. A key advantage of the pp-RPA approach is that the S1 and T1 states are characterized as CT states for all of experimentally reported TADF molecules examined here, which allows for an estimate of the singlet-triplet CT excited state energy gap (Δ E(ST) = 1CT - 3CT). For experimentally known TADF molecules with a small (<0.2 eV) Δ E(ST) in this data set, a high accuracy is demonstrated for the prediction of both the S1 (MAD = 0.18 eV) and T1 (MAD = 0.20 eV) excitation energies as well as Δ E(ST) (MAD = 0.05 eV). This result is attributed to the consideration of correct antisymmetry in the particle-particle interaction leading to the use of full exchange kernel in addition to the Coulomb contribution, as well as a consistent treatment of both singlet and triplet excited states. The computational efficiency of this approach is similar to that of TDDFT, and the cost can be reduced significantly by using the active-space approach.

Excitation energies from particle-particle random phase approximation with accurate optimized effective potentials.

The optimized effective potential (OEP) that gives accurate Kohn-Sham (KS) orbitals and orbital energies can be obtained from a given reference electron density. These OEP-KS orbitals and orbital energies are used here for calculating electronic excited states with the particle-particle random phase approximation (pp-RPA). Our calculations allow the examination of pp-RPA excitation energies with the exact KS density functional theory (DFT). Various input densities are investigated. Specifically, the excitation energies using the OEP with the electron densities from the coupled-cluster singles and doubles method display the lowest mean absolute error from the reference data for the low-lying excited states. This study probes into the theoretical limit of the pp-RPA excitation energies with the exact KS-DFT orbitals and orbital energies. We believe that higher-order correlation contributions beyond the pp-RPA bare Coulomb kernel are needed in order to achieve even higher accuracy in excitation energy calculations.

Multireference Density Functional Theory with Generalized Auxiliary Systems for Ground and Excited States.

To describe static correlation, we develop a new approach to density functional theory (DFT), which uses a generalized auxiliary system that is of a different symmetry, such as particle number or spin, from that of the physical system. The total energy of the physical system consists of two parts: the energy of the auxiliary system, which is determined with a chosen density functional approximation (DFA), and the excitation energy from an approximate linear response theory that restores the symmetry to that of the physical system, thus rigorously leading to a multideterminant description of the physical system. The electron density of the physical system is different from that of the auxiliary system and is uniquely determined from the functional derivative of the total energy with respect to the external potential. Our energy functional is thus an implicit functional of the physical system density, but an explicit functional of the auxiliary system density. We show that the total energy minimum and stationary states, describing the ground and excited states of the physical system, can be obtained by a self-consistent optimization with respect to the explicit variable, the generalized Kohn-Sham noninteracting density matrix. We have developed the generalized optimized effective potential method for the self-consistent optimization. Among options of the auxiliary system and the associated linear response theory, reformulated versions of the particle-particle random phase approximation (pp-RPA) and the spin-flip time-dependent density functional theory (SF-TDDFT) are selected for illustration of principle. Numerical results show that our multireference DFT successfully describes static correlation in bond dissociation and double bond rotation.

Charge transfer excitations from particle-particle random phase approximation-Opportunities and challenges arising from two-electron deficient systems.

The particle-particle random phase approximation (pp-RPA) is a promising method for studying charge transfer(CT) excitations. Through a detailed analysis on two-electron deficient systems, we show that the pp-RPA is always able to recover the long-distance asymptotic -1/R trend for CT excitations as a result of the concerted effect between orbital energies and the pp-RPA kernel. We also provide quantitative results for systems with relatively short donor-acceptor distances. With conventional hybrid or range-separated functionals, the pp-RPA performs much better than time-dependent density functional theory (TDDFT), although it still gives underestimated results which are not as good as TDDFT with system-dependent tuned functionals. For pp-RPA, there remain three great challenges in dealing with CT excitations. First, the delocalized frontier orbitals in strongly correlated systems often lead to difficulty with self-consistent field convergence as well as an incorrect picture with about half an electron transferred. Second, the commonly used density functionals often underestimate the energy gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital (LUMO) for the two-electron deficient species, resulting in systems with delocalized orbitals. Third, the performance of pp-RPA greatly depends on the energy difference between the LUMO and a higher virtual orbital. However, the meaning of the orbital energies for higher virtual orbitals is still not clear. We also discuss the performance of an approximate pp-RPA scheme that uses density functional tight binding (pp-DFTB) as reference and demonstrate that the aforementioned challenges can be overcome by adopting suitable range-separated hybrid functionals. The pp-RPA and pp-DFTB are thus promising general approaches for describing charge transfer excitations.

Accurate and efficient calculation of excitation energies with the active-space particle-particle random phase approximation.

An efficient method for calculating excitation energies based on the particle-particle random phase approximation (ppRPA) is presented. Neglecting the contributions from the high-lying virtual states and the low-lying core states leads to the significantly smaller active-space ppRPA matrix while keeping the error to within 0.05 eV from the corresponding full ppRPA excitation energies. The resulting computational cost is significantly reduced and becomes less than the construction of the non-local Fock exchange potential matrix in the self-consistent-field (SCF) procedure. With only a modest number of active orbitals, the original ppRPA singlet-triplet (ST) gaps as well as the low-lying single and double excitation energies can be accurately reproduced at much reduced computational costs, up to 100 times faster than the iterative Davidson diagonalization of the original full ppRPA matrix. For high-lying Rydberg excitations where the Davidson algorithm fails, the computational savings of active-space ppRPA with respect to the direct diagonalization is even more dramatic. The virtues of the underlying full ppRPA combined with the significantly lower computational cost of the active-space approach will significantly expand the applicability of the ppRPA method to calculate excitation energies at a cost of O(K4), with a prefactor much smaller than a single SCF Hartree-Fock (HF)/hybrid functional calculation, thus opening up new possibilities for the quantum mechanical study of excited state electronic structure of large systems.

Nature of ground and electronic excited states of higher acenes

Higher acenes have drawn much attention as promising organic semiconductors with versatile electronic properties. However, the nature of their ground state and electronic excited states is still not fully clear. Their unusual chemical reactivity and instability are the main obstacles for experimental studies, and the potentially prominent diradical character, which might require a multireference description in such large systems, hinders theoretical investigations. Here, we provide a detailed answer with the particle-particle random-phase approximation calculation. The (1)Ag ground states of acenes up to decacene are on the closed-shell side of the diradical continuum, whereas the ground state of undecacene and dodecacene tilts more to the open-shell side with a growing polyradical character. The ground state of all acenes has covalent nature with respect to both short and long axes. The lowest triplet state (3)B2u is always above the singlet ground state even though the energy gap could be vanishingly small in the polyacene limit. The bright singlet excited state (1)B2u is a zwitterionic state to the short axis. The excited (1)Ag state gradually switches from a double-excitation state to another zwitterionic state to the short axis, but always keeps its covalent nature to the long axis. An energy crossing between the (1)B2u and excited (1)Ag states happens between hexacene and heptacene. Further energetic consideration suggests that higher acenes are likely to undergo singlet fission with a low photovoltaic efficiency; however, the efficiency might be improved if a singlet fission into multiple triplets could be achieved.

Conical Intersections from Particle-Particle Random Phase and Tamm-Dancoff Approximations.

The particle-particle random phase approximation (pp-RPA) and the particle-particle Tamm-Dancoff approximation (pp-TDA) are applied to the challenging conical intersection problem. Because they describe the ground and excited states on the same footing and naturally take into account the interstate interaction, these particle-particle methods, especially the pp-TDA, can correctly predict the dimensionality of the conical intersection seam as well as describe the potential energy surface in the vicinity of conical intersections. Though the bond length of conical intersections is slightly underestimated compared with the complete-active-space self-consistent field (CASSCF) theory, the efficient particle-particle methods are promising for conical intersections and nonadiabatic dynamics.

Singlet-triplet energy gaps for diradicals from particle-particle random phase approximation.

The particle-particle random phase approximation (pp-RPA) for calculating excitation energies has been applied to diradical systems. With pp-RPA, the two nonbonding electrons are treated in a subspace configuration interaction fashion while the remaining part is described by density functional theory (DFT). The vertical or adiabatic singlet-triplet energy gaps for a variety of categories of diradicals, including diatomic diradicals, carbene-like diradicals, disjoint diradicals, four-π-electron diradicals, and benzynes are calculated. Except for some excitations in four-π-electron diradicals, where four-electron correlation may play an important role, the singlet-triplet gaps are generally well predicted by pp-RPA. With a relatively low O(r(4)) scaling, the pp-RPA with DFT references outperforms spin-flip configuration interaction singles. It is similar to or better than the (variational) fractional-spin method. For small diradicals such as diatomic and carbene-like ones, the error of pp-RPA is slightly larger than noncollinear spin-flip time-dependent density functional theory (NC-SF-TDDFT) with LDA or PBE functional. However, for disjoint diradicals and benzynes, the pp-RPA performs much better and is comparable to NC-SF-TDDFT with long-range corrected ωPBEh functional and spin-flip configuration interaction singles with perturbative doubles (SF-CIS(D)). In particular, with a correct asymptotic behavior and being almost free from static correlation error, the pp-RPA with DFT references can well describe the challenging ground state and charge transfer excitations of disjoint diradicals in which almost all other DFT-based methods fail. Therefore, the pp-RPA could be a promising theoretical method for general diradical problems.

Analytic gradients, geometry optimization and excited state potential energy surfaces from the particle-particle random phase approximation.

The energy gradient for electronic excited states is of immense interest not only for spectroscopy but also for the theoretical study of photochemical reactions. We present the analytic excited state energy gradient of the particle-particle random phase approximation (pp-RPA). The analytic gradient formula is developed from an approach similar to that of time-dependent density-functional theory (TDDFT). The formula is verified for both the Hartree-Fock and (Generalized) Kohn-Sham reference states via comparison with finite difference results. The excited state potential energy surfaces and optimized geometries of some small molecules are investigated, yielding results of similar or better quality compared to adiabatic TDDFT. The singlet-to-triplet instability in TDDFT resulting in underestimated energies of the lowest triplet states is eliminated by pp-RPA. Charge transfer excitations and double excitations, which are challenging for most adiabatic TDDFT methods, can be reasonably well captured by pp-RPA. Within this framework, ground state potential energy surfaces of stretched single bonds can also be described well.

Proton-neutron pairing vibrations inN=Znuclei: Precursory soft mode of isoscalar pairing condensation

L=0 proton-neutron ($pn$) pair-addition and pair-removal strengths in $^{40}$Ca and $^{56}$Ni are investigated by means of the $pn$ particle-particle random-phase approximation employing a Skyrme energy-density functional. It is found that the collectivity of the lowest $J^\pi = 1^+$ state in the adjacent odd-odd nuclei becomes stronger as the strength of the isoscalar (T=0) pairing interaction increases. The results suggest the emergence of the T=0 $pn$-pairing vibrational mode as a possible critical phenomenon toward the T=0 pairing condensation.

Excitation energies from particle-particle random phase approximation: Davidson algorithm and benchmark studies.

The particle-particle random phase approximation (pp-RPA) has been used to investigate excitation problems in our recent paper [Y. Yang, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 224105 (2013)]. It has been shown to be capable of describing double, Rydberg, and charge transfer excitations, which are challenging for conventional time-dependent density functional theory (TDDFT). However, its performance on larger molecules is unknown as a result of its expensive O(N(6)) scaling. In this article, we derive and implement a Davidson iterative algorithm for the pp-RPA to calculate the lowest few excitations for large systems. The formal scaling is reduced to O(N(4)), which is comparable with the commonly used configuration interaction singles (CIS) and TDDFT methods. With this iterative algorithm, we carried out benchmark tests on molecules that are significantly larger than the molecules in our previous paper with a reasonably large basis set. Despite some self-consistent field convergence problems with ground state calculations of (N - 2)-electron systems, we are able to accurately capture lowest few excitations for systems with converged calculations. Compared to CIS and TDDFT, there is no systematic bias for the pp-RPA with the mean signed error close to zero. The mean absolute error of pp-RPA with B3LYP or PBE references is similar to that of TDDFT, which suggests that the pp-RPA is a comparable method to TDDFT for large molecules. Moreover, excitations with relatively large non-HOMO excitation contributions are also well described in terms of excitation energies, as long as there is also a relatively large HOMO excitation contribution. These findings, in conjunction with the capability of pp-RPA for describing challenging excitations shown earlier, further demonstrate the potential of pp-RPA as a reliable and general method to describe excitations, and to be a good alternative to TDDFT methods.

Occupation numbers in strongly polarized Fermi gases and the Luttinger theorem

We study a two-component Fermi gas that is so strongly polarized that it remains normal fluid at zero temperature. We calculate the occupation numbers within the particle-particle random-phase approximation, which is similar to the Nozieres-Schmitt-Rink approach. We show that the Luttinger theorem is fulfilled in this approach. We also study the change of the chemical potentials which allows us to extract, in the limit of extreme polarization, the polaron energy.