NR2 and P3+: Accurate, Efficient Electron-Propagator Methods for

Article pubs.acs.org/JPCA

NR2 and P3+: Accurate, Efficient Electron-Propagator Methods for Calculating Valence, Vertical Ionization Energies of Closed-Shell Molecules H. H. Corzo,† Annia Galano,‡ O. Dolgounitcheva,† V. G. Zakrzewski,† and J. V. Ortiz*,† †

Department of Chemistry and Biochemistry, Auburn University, Auburn, Alabama 36849-5312, United States Departamento de Química, Universidad Autónoma Metropolitana, Iztapalapa San Rafael Atlixco 186, Col. Vicentina, Iztapalapa, C.P. 09340, México, D.F., Mexico

‡

S Supporting Information *

ABSTRACT: Two accurate and computationally efficient electron-propagator (EP) methods for calculating the valence, vertical ionization energies (VIEs) of closed−shell molecules have been identified through comparisons with related approximations. VIEs of a representative set of closed-shell molecules were calculated with EP methods using 10 basis sets. The most easily executed method, the diagonal, second-order (D2) EP approximation, produces results that steadily rise as basis sets are improved toward values based on extrapolated coupled-cluster singles and doubles plus perturbative triples calculations, but its mean errors remain unacceptably large. The outer valence Green function, partial third-order and renormalized partial thirdorder methods (P3+), which employ the diagonal self-energy approximation, produce markedly better results but have a greater tendency to overestimate VIEs with larger basis sets. The best combination of accuracy and efficiency with a diagonal self-energy matrix is the P3+ approximation, which exhibits the best trends with respect to basis-set saturation. Several renormalized methods with more flexible nondiagonal self-energies also have been examined: the two-particle, one-hole Tamm−Dancoff approximation (2ph-TDA), the third-order algebraic diagrammatic construction or ADC(3), the renormalized third-order (3+) method, and the nondiagonal second-order renormalized (NR2) approximation. Like D2, 2ph-TDA produces steady improvements with basis set augmentation, but its average errors are too large. Errors obtained with 3+ and ADC(3) are smaller on average than those of 2ph-TDA. These methods also have a greater tendency to overestimate VIEs with larger basis sets. The smallest average errors occur for the NR2 approximation; these errors decrease steadily with basis augmentations. As basis sets approach saturation, NR2 becomes the most accurate and efficient method with a nondiagonal self-energy.

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INTRODUCTION Ionization energies (IEs) are fundamental properties of molecules. Because they are closely related to electron-density distributions,1,2 spectra and reactivity that involve transfer of electronic charge,3,4 qualitative notions of chemical bonding such as electronegativity and hardness,2,5−8 and many other problems of molecular design, synthesis, and characterization,9−11 IEs are often the focus of highly developed methods of measurement.12−14 Several experimental techniques including X-ray spectroscopy,12,14,15 photoelectron spectroscopy,13−21 zero-electron kinetic energy spectroscopy,22 timeresolved photoionization mass spectrometry,23,24 and photoion−photoelectron coincidence spectroscopy25 can provide accurate measurements of IEs. Comparisons with experiments or a need for data that are difficult to obtain experimentally may require the evaluation of adiabatic or vertical IEs. Adiabatic IEs correspond to transitions from the lowest rovibronic level of a molecule to its counterpart for the molecular cation. Vertical IEs (VIEs) may be defined as © XXXX American Chemical Society

electronic energy differences between a molecule and a cation at the equilibrium geometry of the molecule. In this definition, zero-point energies of the molecule and vibrational spacings corresponding to the cation’s potential-energy surface are ignored. After ignoring rotational terms, the VIE (Iv) and the adiabatic IE (Ia) may be related to each other by I v = Ia + (UCM − UC) + (Z M − ZC)

(1)

where UC and UM C are the total energies of the cation at its equilibrium geometry and at the equilibrium geometry of the molecule, respectively, and ZM and ZC are the zero-point vibrational energies of the molecule and cation, respectively. The second term on the right side of the previous equation (i.e., the geometrical relaxation energy) is always positive. Neglect of the third term (the difference of zero-point energies) Received: February 2, 2015 Revised: July 29, 2015

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DOI: 10.1021/acs.jpca.5b00942 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

energy EN0 , ENr ± 1 and |ΨNr ± 1⟩, respectively, denote energies and states with N ± 1 electrons, and η is a positive infinitesimal constant that guarantees the convergence of the Fourier transform from the time-dependent expression. The EP has its physical content in its poles, energies where singularities lie, and residues, coefficients of the terms responsible for the singularities. In eq 2, the first term describes electron detachments from the ground state in which poles are negatives of IEs, whereas the second term describes electron attachments where poles are negatives of electron affinities. The usual approximation procedure to evaluate the EP involves a perturbation expansion. Poles of an unperturbed EP, G0(E), where

implies that Iv is an upper bound to Ia, but this assumption may not always be valid. In photoelectron experiments with ultraviolet or X-ray radiation, transitions to more than one electronic state of the cation usually are observed.15,20,25,26 In vibrationally resolved spectra, transitions to various levels corresponding to a given electronic state of the cation may be identified. (Preparation of molecular targets at low temperatures may eliminate so-called hot bands in which transitions from a vibrationally excited level of the molecule’s ground electronic state are represented.) When such resolution is not attained, the maximum of a broad peak that spans many vibronic transitions may be the only datum that is comparable to computations. In most cases, the chief task for the interpretation of spectra is the assignment of peaks, broad or narrow, to cationic final states. Accurate determination of the first few VIEs generally suffices for this purpose. Such data may be compared with vibrationally resolved peaks with maximum intensity (i.e., those with the largest Franck−Condon factors) for a given electronic state. Because these transitions are not vertical, discrepancies of 0.1 eV or more may occur (see below). Judgments on the accuracy of methods for evaluating VIEs therefore should be based on calculations of high quality. In this report, several ab initio electron-propagator (EP) methods used in the calculation of VIEs are compared. In EP theory,27−36 wave functions and energies for initial (molecular) and final (cationic) states are not constructed; only canonical Hartree−Fock (HF) orbitals for the initial state are needed. Instead, a search for the poles of the EP is undertaken, for these quantities are equal to the electron-binding energies (IEs and electron affinities) of the molecule. Correlation and relaxation corrections to the results of Koopmans’s theorem (KT), in which canonical HF orbital energies provide uncorrelated, frozen-orbital approximations to electron binding energies, are included in the self-energy matrix. The performance of various approximations that require evaluation of only the diagonal elements of the self-energy matrix in the canonical HF basis will be considered, as well as methods where the full self-energy matrix is employed. To identify the most efficient and accurate EP methods, the computational demands and predictive performance of several approximations will be compared. Arithmetic efficiency and storage requirements will be considered. To discern the intrinsic merits of each method, basis-set effects also will be examined in comparisons with reliable calculations of VIEs. This information suffices for recommendations on the use of EP methods. The Conclusions section ends with a discussion of additional methodological improvements that consider basis-set saturation.

[G0(E)]pq = δpq(E − ϵp)−1

equal canonical HF orbital energies, ϵp. The latter are eigenvalues obtained through self-consistent solution of the HF equations, which in canonical form read

FϕpHF = ϵpϕpHF

Ip = −ϵp

G−1(E) = G−0 1(E) − Σ(E)

r

(6)

Poles may be computed from the equivalent expression

35

[F + Σ(Er)]ϕrDyson = ErϕrDyson

(7)

where F is the Fock operator of eq 4 that is generated by the one-electron density matrix of the reference state, which may be correlated. Er is a pole of G(E) which satisfies the Dyson equation, and the eigenfunctions are known as Dyson orbitals. The latter orbitals are not necessarily normalized to unity. For an electron detachment energy (e.g., a VIE) from an Nelectron reference state that is associated with a final state r, the Dyson orbital reads ϕrDyson(x1) = N1/2

∫ dx2dx3 ... dxN ΨN(x1, x2 , x3 , ..., xN )

Ψ*N − 1,r(x 2 , x3 , ..., xN )

(8)

where xs is the space−spin coordinate for electron s. For electron affinities, where an electron is added to the reference state, the Dyson orbital is given by

⎧ ⟨Ψ0N |a p†|ΨrN − 1⟩⟨ΨrN − 1|aq |Ψ0N ⟩ ⎪ ⎨ Gpq (E) = lim ⎪∑ η→ 0 E − E0N + ErN − 1 − iη ⎩ r

∑

(5)

Results at the KT level usually are quantitatively inadequate and often produce erroneous orderings of cationic states because they neglect electron correlation and orbital relaxation in final states. With the introduction of an energy-dependent, nonlocal potential known as the self-energy, Σ(E), an improved EP can be obtained from G0(E) by employing the Dyson equation, which in its inverse, matrix form reads

ELECTRON-PROPAGATOR METHODS The p−q element of the EP matrix in its spectral form27,35 reads

+

(4)

where ϕHF is a canonical HF orbital. In the frozen-orbital, p single-determinant approximation employed in KT, IEs are related to occupied orbital energies by

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⟨Ψ0N |aq |ΨrN + 1⟩⟨ΨrN + 1|a p†|Ψ0N ⟩ ⎫ ⎪ ⎬ N+1 N E − Er + E0 + iη ⎪ ⎭

(3)

ϕrDyson(x1) = (N + 1)1/2

∫ dx2dx3 ... dxN +1ΨN +1,r

(x1 , x 2 , x3 , ..., xN + 1)Ψ*N (x 2 , x3 , ..., xN + 1) (9) (2)

The pole strength for a given Dyson orbital reads

where p and q are general spin−orbital indices, |ΨN0 ⟩is the exact non-degenerate ground state of an N-electron system with

πr = B

∫ |ϕrDyson(x)|2 dx

(10) DOI: 10.1021/acs.jpca.5b00942 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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methods used in this study are applicable when pole strengths are between 0.85 and unity. Table 1 summarizes the scaling of arithmetic operations and storage of transformed electron repulsion integrals (ERIs)

Pole strengths approach unity when correlation effects are weak and approach zero when such effects are strong. For electronbinding energies where KT provides a reasonable qualitative description of initial and final states, pole strengths are generally above 0.85. For final states with strong shake-up (two-hole, one-particle, or 2h1p) character, pole strengths between 0.8 and zero are typical. In the basis of HF spin-orbitals, the self-energy includes all relaxation and correlation corrections. In EP theory, the first correction to the self-energy matrix occurs in second order in the fluctuation potential. Poles of the electron propagator correspond to eigenvalues obtained self-consistently from the following form of the Dyson equation [F + Σ(E)]C(E) = EC(E)

Table 1. Scaling of Arithmetic and Storage Requirementsa method D2 OVGF P3 (P3+) 2p−h TDA 3+ ADC(3) NR2

(11)

a

where the energy argument of the self-energy matrix is equal to an eigenvalue. The corresponding eigenvector, C(E), provides a linear combination of orbitals that is proportional to the Dyson orbital. In some approximations, non-diagonal elements of the self-energy matrix in the canonical HF orbital basis are neglected. This diagonal, or quasiparticle, approximation assumes that KT provides a qualitatively reasonable description of an ionizing transition and leads to a simplified form of the Dyson equation that reads E = ϵp + Σpp(E)

(12)

(13)

In the diagonal approximation, Dyson orbitals become proportional to normalized, canonical HF orbitals such that ϕpDyson =

πp ϕpHF

(14)

where pole strengths are evaluated by the formula −1 ⎛ dΣpp(E) ⎞ ⎟⎟ πp = ⎜⎜1 − dE ⎠ ⎝

noniterative bottleneck

2

OV OV4 O3V2 OV4 OV4 O2V4 O2V3

O2V3 O2V3 O2V4 O2V4 O3V3

storage OV2 V4 OV3 V4 V4 V4 OV3

O = number of occupied orbitals, V = number of virtual.

required by each EP method. Iterative, arithmetic bottlenecks arise in the search for poles of the EP. Noniterative, arithmetic bottlenecks are created by the need for intermediate matrices that need be evaluated only once. Storage requirements pertain to ERIs. Because the number of virtual orbitals, V, is generally much larger than the number of occupied orbitals, O, methods that require ERIs with four virtual-orbital indices have significantly larger storage requirements unless arithmetically intensive semi-direct algorithms that regenerate these integrals as needed are employed.44,45 The D2 approximation requires a relatively small set of fourindex ERIs in the canonical HF basis where one index corresponds to a canonical orbital of interest for each VIE calculation. The other three indices correspond to 2h1p or 2p1h operators, and therefore, memory requirements are relatively small. The lengthiest calculation has an arithmetic scaling factor of OV2. In practice, the preceding ERI transformations to the HF basis constitute the bottleneck for D2 calculations. Outer valence Green function (OVGF) methods also assume a diagonal self-energy matrix and describe low energy transitions.29,30,46 All versions of OVGF are based on the third-order expansion of the self-energy and include estimates of higher order contributions. Three frequently applied procedures have been denominated as versions A, B, and C.29,30,47 An algorithm that introduces several numerical criteria for the choosing among these alternatives has been developed29,30,47 and implemented.48 P3, also known as partial third-order quasiparticle theory,49,50 is an approach in which several terms in the third-order selfenergy are omitted from the calculations. This approximation retains all second-order terms and has some 2h1p terms in third order. The renormalized partial third order, or P3+, method is obtained by introducing a renormalization factor in some terms.51 This simple procedure avoids exaggeration of final state relaxation effects with virtually no additional computational effort with respect to P3. In practice, the arithmetic bottleneck of P3 and P3+ calculations also occurs in the transformation of ERIs to the HF basis. Several methods in which the diagonal self-energy approximation has not been made also are considered presently. The two-particle, one-hole, Tamm−Dancoff approximation29 (2phTDA) is a renormalized method that employs a HF reference state and considers all first-order couplings among h, p, 2h1p, and 2p1h operators.

In this way, corrections to the results of KT may be calculated for each canonical HF orbital. (Note that correlation contributions to the matrix elements of the Fock operator in eq 7 are absorbed into Σ(E).) Poles strengths are obtained from ⎛ ⎞−1 dΣ(E) Cr (E)⎟ πr = ⎜1 − C†r (E) ⎝ ⎠ dE

iterative bottleneck

(15)

The derivative of Σ(E) or Σpp(E) with respect to E is evaluated at the pole (i.e., the self-consistent value of E in eq 11 or 12). These derivatives also serve to accelerate the search for poles by use of Newton’s method for finding the roots of a function of a single variable. Two or three iterations usually suffice to obtain convergence within 0.01 eV. Diagonal selfenergy approximations have been exploited in a wide variety of applications.37−41 Employment of eq 12 with the second-order self-energy defines the diagonal, second-order, or D2, approximation. Strong relaxation and correlation effects in the cation’s final states, especially those that occur at higher energies, usually generate a dispersion of photoionization intensity from the main spectral line to many satellites of low intensity. This breakdown of the orbital picture of ionization may occur when pole strengths are smaller than 0.85.42,43 The quasiparticle C



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RESULTS AND DISCUSSION Because band maxima in photoelectron spectra do not necessarily correspond to accurate VIEs, molecular and cationic total energies at the coupled-cluster singles and doubles with perturbative triples,63 or CCSD(T), level with correlationconsistent basis sets of double, triple, and quadruple ζ quality have been extrapolated via a cubic fit64 to produce alternative standards for 52 transitions where the final, cationic state is the lowest for a given Abelian irreducible representation. VIEs inferred from extrapolated CCSD(T) results differ significantly from band features reported in a compendium of photoelectron spectra.26 The mean unsigned error (MUE) for the 52 VIEs of Table 2 is 0.25 eV, and the mean signed error is −0.16 eV, i.e., reliable calculated VIEs tend to be larger than the data of the compendium. Such discrepancies indicate the importance of geometric relaxation and vibronic effects in determining band maxima in photoelectron spectra.26 Methods that underestimate VIEs (see below) may be in fortuitously close agreement with band maxima. Results for two typical molecules, H2CO and CO2, are displayed in Figures 1−4. For both molecules, the lowest IE and two of the higher IEs in Table 2 are displayed, respectively,

In the 3+ method, those higher order couplings that suffice to include all third-order terms in the self-energy are calculated.30 Certain types of higher order terms also are included in 3+. The third-order algebraic diagrammatic construction, or ADC(3), method retains these terms and also adds fourth-order and some higher order terms in the energy-independent part of the self-energy matrix.29 The nondiagonal, renormalized, second-order (NR2) method retains some third-order and higher order terms and is complete only in second order in the self-energy.52 NR2 is the least demanding of the nondiagonal methods examined below.

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COMPUTATIONAL METHODS All calculations were performed with a version of the Gaussian09 package of programs48 that has been modified to execute P3+ calculations or with the development version of Gaussian.53 Geometry optimizations and frequency calculations for the molecules listed in Table 2 have been carried out using the MP2/6-311G**method.54,55 Table 2. Ionization Energies (IEs)

a

molecule

no. of IEs

PGa

LAPGb

MOs

HCl HF CO2 CO H2CO C2H2 C2H4 N2 CH3SH CH3Cl CH3OH CH3COCH3 CH3CHO CH3F HCN C6H6 C5NH5 HCOOH CH3CH2CH3 SH2 CH4

2 2 4 3 4 2 4 3 5 3 5 8 6 3 3 5 8 6 7 3 1

C∞v C∞v D∞h C∞v C2v D∞h D2h D∞h Cs C3v Cs C2v Cs C3v C∞v D6h C2v Cs C2v C2v Td

C2itv C2v D2h C2v C2v D2h D2h D2h Cs Cs Cs C2v Cs Cs C2v D2h C2v Cs C2v C2v D2

2π,5σ 1π,3σ 1πg,1πu,3σu,4σg 5σ,1π 2b2,1b1,5a1 1πu,3σg 1b2u,1b2g,3ag,1b3u 3σg,1πu,2σu 3a″,10a′ 3e,7a1 2a″,7a′ 5b2,2b1,8a1,1a2 10a′,2a″ 2e 1π,5σ 1e1g,3e2g 1a2,2b1,9a1,9b2 10a′,2a″ 2b2,6a1,1a2 2b1,5a1,2b2 1tg

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Point group. bLargest Abelian point group.

Ten basis sets54−62 were used in combination with the D2, OVGF, P3, P3+, 2ph-TDA, 3+, ADC(3), and NR2 methods. To facilitate comparisons with future studies of photoelectron spectra in which shakeup states appear, augmented, correlationconsistent (aug-cc) basis sets have been included. These sets often provide alternatives of intermediate quality between the regular correlation-consistent (cc) bases (e.g., aug-cc-pvdz is more flexible and reliable than cc-pvdz but smaller than ccpvtz) and may facilitate more stable extrapolations to the complete-basis limit. All OVGF pole strengths for the 52 VIEs under consideration (see Table 2) were above 0.85 and confirm the qualitative validity of perturbative improvements to KT. A complete set of calculated results is provided in the Supporting Information.

Figure 1. Diagonal self-energy results (eV) for H2CO. D


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Figure 2. Nondiagonal self-energy results (eV) for H2CO.

Figure 3. Diagonal self-energy results (eV) for CO2.

in parts a, b, and c of each figure. Results for Pople, cc, and augcc basis sets are placed, respectively, on the left, middle, and right sectors of each graph in the order of the total number of basis functions. Extrapolated CCSD(T) results are denoted by broken red lines. For all methods, predicted VIEs increase with basis-set enlargement. Increases in VIEs between cc-pvdz and cc-pvtz are notably larger than those between aug-cc-pvdz and aug-cc-pvtz. When basis-set extrapolations of EP VIEs are appropriate, estimates based on aug-cc basis sets appear to be preferable to those based on regular cc basis sets. There is a rough equivalency between the two largest Pople basis sets, 6311G(2df,2p) and 6-311++G(2df,2p), and the cc-pvtz and augcc-pvtz basis sets. The D2 method tends to underestimate VIEs by a wide margin, even with the largest basis sets. For cc and aug-cc basis sets, the slopes of the D2 curves in Figures 1 and 3 resemble those of the other diagonal methods, especially P3 and P3+. OVGF, P3, and P3+ results more closely approximate the extrapolated CCSD(T) standards. P3+ VIEs are generally smaller than those of P3. The 2ph-TDA also has a marked tendency to underestimate VIEs. (See Figures 2 and 4.) The NR2, 3+ and ADC(3) methods are in much closer agreement with standard values. NR2 results are smaller than those of 3+

and ADC(3). The two latter methods tend to be in close agreement with each other. MUEs corresponding to the first VIEs of the 21 molecules under study or to all 52 VIEs are shown in Figures 5 and 6, respectively. The two figures are very similar and reveal no qualitative differences between the first and higher VIEs presently under consideration. The latter figures have no curves for KT, for the corresponding MUEs far exceed 1 eV for all basis sets. Histograms of signed errors obtained with the ccpvqz basis are displayed in Figure 7. For the OVGF method, the MUE declines as more basis functions are added. Basis saturation is nearly complete with the cc-pvqz or aug-cc-pvqz sets. There are more overestimates than underestimates in the histograms. P3’s MUE usually declines with larger basis sets. However, P3 also has a greater tendency to overestimate than underestimate VIEs with the cc-pvqz or aug-cc-pvqz sets. In fact, the MUE is larger for aug-cc-pvqz than for aug-cc-pvtz. P3+ retains the computational advantages of P3 while providing higher order estimates. Its MUE declines steadily with basis-set improvements. P3+ makes fewer overestimates (55%) than P3 (77%) and OVGF (77%) in Figure 7. (The mean signed errors in Figure 7 for P3+, P3, and OVGF are E


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Figure 5. MUEs (eV) for 21 first VIEs.

Despite the difference in their treatments of energyindependent, self-energy terms, the results of the two methods are often in close agreement and their MUE curves strongly resemble each other. In Figure 7, ADC(3) has a greater tendency than 3+ to overestimate VIEs. Slight increases in MUEs occur for ADC(3) and 3+ when cc-pvtz and aug-cc-pvtz are replaced respectively with the cc-pvqz and aug-cc-pvqz sets. NR2 predictions in Figure 7 also are in acceptable agreement with CCSD(T) standards. NR2 VIEs are higher than those of 2ph-TDA but are overestimates less often than those of 3+ or ADC(3). MUEs for the NR2, 3+, and ADC(3) methods lie between 0.1 and 0.2 eV with the cc-pvqz and aug-cc-pvqz basis sets. Whereas ADC(3) tends to produce overestimates (all 52 cases in Figure 7, mean signed error = 0.12 eV), NR2 has the opposite tendency (63% of 52 cases, mean signed error = −0.11 eV). Overestimates (81%) and underestimates (19%) by 3+ are more equally distributed (mean signed error =0.08 eV). As basis sets are saturated, the NR2, 3+, and ADC(3) methods are likely to produce more overestimates. However, only NR2’s MUE is likely to decrease, for its average prediction in Figure 7 underestimates the VIE.

Figure 4. Nondiagonal self-energy results (eV) for CO2.

−0.01, 0.10, and 0.01 eV, respectively.) As basis sets improve beyond the quadruple-ζ level, the MUE of P3+ will be less adversely affected than those of P3 and OVGF. Although D2 underestimates VIEs by unacceptable amounts, it improves as the basis set grows. D2 curves behave similarly to those of P3+, although the MUEs of D2 are larger. The dispersion of D2 columns in Figure 7 is relatively wide and indicates that the quality of this method’s results is less predictable. Acceptable MUEs of 0.25 eV or less are obtained for the OVGF, P3, and P3+ methods with basis sets of triple ζ plus double polarization or higher quality. A slight advantage for OVGF over P3+ appears for the cc-pvqz case but vanishes for aug-cc-pvqz. The quality of 2ph-TDA results improves steadily with larger basis sets, but this method also produces unacceptably large errors. Because 2ph-TDA is less efficient than D2, it is a less attractive alternative for the purpose of estimating basis set effects. The dispersion of 2ph-TDA columns in Figure 7 also is relatively wide. VIEs obtained with ADC(3) and 3+ are generally higher than those of 2ph-TDA and in closer agreement with CCSD(T) standards. Predicted VIEs increase as basis sets are enlarged.

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CONCLUSIONS NR2 is the best of the presently examined electron-propagator methods for calculating valence VIEs for closed-shell molecules when Koopmans’s theorem provides a reasonable qualitative F


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Figure 6. MUEs (eV) for 52 VIEs.

description. This method displays the best convergence to extrapolated CCSD(T) standards with respect to basis sets. Among its advantages over ADC(3) and 3+ are a noniterative arithmetic bottleneck with O3V3 scaling and independence of electron repulsion integrals with four virtual indices. The mean unsigned errors of the nondiagonal ADC(3) and 3+ methods increase with basis-set improvements beyond the triple ζ level. Both methods require contractions with O2V4 scaling, iterative OV4 steps and a full integral transformation. 2ph-TDA displays steady improvement with basis set enlargement, but is the least accurate and precise of the non-diagonal methods considered here. P3+ is the diagonal self-energy approximation with the best combination of accuracy and efficiency. P3 is a method of equal efficiency, but inferior accuracy, with respect to P3+. OVGF is less efficient than P3+ and has a somewhat greater tendency than P3+ to overestimate VIEs with larger basis sets. Although the A, B, and C versions of OVGF introduce no numerical parameters, the OVGF selection procedure does so. The highly efficient D2 method’s large errors decrease steadily as basis sets are improved but remain unacceptably large and dispersed. When lengthier calculations with large basis sets are feasible, the NR2 method should be employed. For a given basis, its need for storage of transformed integrals is no greater than that of P3+. Because of its treatment of couplings between 2hp operators that is complete through first order, the NR2 method is more likely than P3+ to give a correct description of higher

Figure 7. Error distibutions (eV) for 52 VIEs.

VIEs with lower pole strengths. P3+ provides an expedient alternative to NR2 for larger molecules. When only small basis sets are practical, the P3, OVGF, 3+, and ADC(3) methods may provide useful data. However, the success of these methods rests on a cancellation of underestimation from basis-set incompleteness and overestimation that is intrinsic to these self-energy approximations. The advantages of the NR2 and P3+ methods may be extended by employment of techniques that account for basisset saturation. Composite methods can take advantage of the parallel behavior of curves such as those of Figures 1 and 3 by performing D2 calculations with large basis sets and P3+ calculations with relatively small basis sets. Quasiparticle virtual orbitals65 that are adapted to specific VIEs may be used to increase the efficiency of calculations with large basis sets. Explicitly correlated methods for estimating the effects of basisset saturation have produced encouraging preliminary results.66 Implementation of all of these techniques is currently in progress.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.5b00942. G


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REFERENCES

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