Electron Propagator Methods for Vertical Electron Detachment

Article pubs.acs.org/JCTC

Cite This: J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

Electron Propagator Methods for Vertical Electron Detachment Energies of Anions: Benchmarks and Case Studies Manuel Díaz-Tinoco, H. H. Corzo, and J. V. Ortiz* Department of Chemistry and Biochemistry, Auburn University, Auburn, Alabama 36849-5312, United States

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S Supporting Information *

ABSTRACT: Ab initio electron propagator methods are efficient and accurate means of calculating vertical electron detachment energies of closed-shell, molecular anions with nuclei from the first three periods. Basis set extrapolations enable definitive comparisons between electron propagator results and benchmarks defined by total energy differences obtained with coupled-cluster, single, double, plus perturbative triple substitution theory. The best compromises of accuracy and efficiency are provided by the renormalized, partial third-order, diagonal (P3+) self-energy and by the nondiagonal, renormalized, second-order (NR2) approximation. The outer-valence Green function, the two-particle-one-hole Tamm− Dancoff approximation, the third-order algebraic diagrammatic construction, and the renormalized third-order methods also are examined. A detailed analysis of errors for small anions is performed. Case studies include F−(H2O) and Cl−(H2O) complexes, C5H−5 , two P2N−3 pentagonal rings, and a superhalide, Al(BO2)−4 , whose electron detachment energy is more than double those of the halide anions. These applications illustrate the versatility of electron propagator methods, their utility for interpreting negative-ion photoelectron spectra, and their promise in the discovery of unusual properties and patterns of chemical bonding. Composite methods, which combine basis set effects calculated at the relatively efficient diagonal, second-order level and higher correlation effects calculated with small basis sets, provide excellent estimates of basis setextrapolated P3+ or NR2 results and facilitate applications to large molecules. In the P3+ and NR2 methods, a judicious choice of low-order couplings between hole operators that correspond to the assumptions of Koopmans’s theorem and operators that describe final-state relaxation and polarization and initial-state correlation leads to predictive accuracy, computational efficiency, and interpretive lucidity.

1. INTRODUCTION The discovery of molecules and molecular ions with novel patterns of chemical bonding has often advanced through an alliance between anion photoelectron spectroscopy and computational quantum chemistry. Exemplary discoveries include extensions of traditional aromaticity concepts,1 planar, fourcoordinate carbon atoms,2 covalency involving f-orbitals in lanthanide borides,3 spectroscopic confirmation of the superhalogen concept,4 control of radical formation in the photoactive yellow protein chromophore,5 coexisting diffuse and valence anions in pyrimidine complexes with inert-gas atoms,6,7 and doubleRydberg anions with ground-state, extra-valence electron pairs.8,9 Although photoelectron experiments may determine electron detachment energies of anions with error bars below 0.1 eV, responsible structures and state assignments may be difficult to infer from spectral data. Calculations based on reliably optimized molecular geometries may facilitate structural interpretations of spectra, provided that electron detachment energies are predicted accurately. Such capabilities become especially useful when more than one isomer of the initial-state anion is present in an experimental sample. Methods of quantum chemistry that predict vertical electron detachment energies (VEDEs) of anions to an accuracy of 0.1−0.2 eV are © XXXX American Chemical Society

potentially powerful tools for assigning peaks and other features in photo-detachment spectra on anions. Agreement between theory and experiment becomes more consequential when reliable data are related to concepts of chemical structure and bonding that identify subtle similarities and differences between molecules, deepen understanding of periodic trends, recognize relationships between spectra, structure, energetics, and reactivity, or stimulate design of subsequent experiments and calculations. Simple concepts of chemical bonding sometimes suffice for this purpose, but their connection to quantitatively accurate calculations may be tenuous. Qualitative trends pertaining to more than one factor may reinforce or oppose each other, and their relative effects therefore must be judged quantitatively. Computational methods that rigorously connect numerical predictions to concepts of chemical bonding can impart improved terminology and more refined understanding to the dialogue between experiment and theory. Electron propagator (EP) theory10−12 provides a framework for fulfilling the predictive, quantitative aims and the interpretive, Received: July 17, 2018 Published: September 25, 2018 A

DOI: 10.1021/acs.jctc.8b00736 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

Article

Journal of Chemical Theory and Computation

Table 1. Scaling of Arithmetic and Storage Requirementsa

qualitative promise of computational quantum chemistry in a single stroke. The former role is founded on the rigor and systematically improvable character of an ab initio, many-body theory. To accomplish the other mission, it is necessary to perform a perspicuous reduction of the copious information (e.g., amplitudes occurring in configuration-interaction or coupled-cluster wave-functions) that such theories generate. EP theory offers an especially direct means to this end, for its Dyson quasi-particle equation generalizes the ab initio foundation of molecular-orbital theory, the Hartree−Fock equations, to an exact limit that takes full account of many-body effects. In doing so, it retains the most interpretable features of its frozenorbital, uncorrelated antecedent by producing physically meaningful eigenvalues and eigenfunctions of an operator wherein all many-body effects reside. Algorithms based on the Dyson equation also enable calculations on large, complicated systems through their relatively low requirements for memory and arithmetic operations. Several generations of approximate EP methods have been developed chiefly for calculating vertical ionization energies of main-group, closed-shell molecules and interpretation of their gas-phase, photoelectron spectra. Doll and Reinhardt performed pioneering EP calculations on He and Be.13 Electron binding energies were calculated ab initio with low-order selfenergy approximations by Cederbaum,14 Simons,15 and Purvis and Ö hrn16 in the early 1970s. Pickup and Goscinski laid a foundation for the derivation and interpretation of self-energy approximations by identifying second-order relaxation, polarization, and correlation terms.17 Progress in EP methodology has been recorded in many reviews10−12,18−36 and two books.37,38 Predictive capabilities of these techniques have been evaluated, and recommendations for computational practitioners have been published recently.39 With programs that perform these calculations being available to many users for over two decades,40 applications to VEDEs of anions also have been widespread.4,5,7,9,41−48 Such acceptance notwithstanding, a systematic test of the most commonly employed EP methods for anionic initial states and uncharged final states is needed for two reasons. First, computational practitioners may profit from methodological recommendations, especially those that pertain to unfamiliar methods. Second, deficiencies of these methods with respect to predictive accuracy, computational efficiency, and interpretive lucidity may be identified, with a view toward subsequent improvements. This study will focus on VEDEs of closed-shell anions. In such cases, Hartree−Fock theory may provide a reasonable point of departure for treatments of many-body effects. Unambiguous comparisons with benchmark data are facilitated by considering only VEDEs. A critical examination of EP results versus benchmark calculations for small molecules will concentrate on statistical measures of reliability. A series of applications that illustrate the capabilities of EP methods will follow. Halide−water complexes, organic and inorganic rings with five members, and a superhalide provide a representation of typical calculations. Conclusions will include methodological recommendations for general users that may enable identification of useful extensions to other classes of molecules and to transitions between specific vibrational levels.

methods

iterative

D2 D3 and OVGF P3 and P3+ 2phTDA NR2 ADC(3) 3+ MP2 CCSD CCSD(T)

OV2 OV4 O3V2 OV4 O3V2 O2V4 OV4

noniterative 2 3

OV O2V3 O3V3 O2V4 O2V4 O2V2

O2V4 O3V4

integral storage OV2 V4 OV3 V4 OV3 V4 V4 O2V2 V4 V4

a

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

detachment or attachment. For uncharged initial states, poles equal either ionization energies or electron affinities. For anionic initial states, they are equal to energies of electron detachment or attachment, with uncharged or dianionic final states, respectively. Element pq of the EP matrix reads:10−12,38

l o ⟨Ψ0N |aq†|ΨrN − 1⟩⟨ΨrN − 1|ap|Ψ0N ⟩ o Gpq(E) = lim+o ∑ m o η→ 0 o E − E0N + ErN − 1 − iη o n r o ⟨Ψ0N |ap|ΨrN + 1⟩⟨ΨrN + 1|aq†|Ψ0N ⟩ | o o +∑ } N+1 N o o E E E i − + + η o r 0 r (1) ~ N where p and q are general spin−orbital indices, |Ψ0 ⟩ is the exact nondegenerate ground state of an N-electron system with energy EN0 , ENr ± 1 and |ΨNr ± 1⟩, respectively, denote energies and states with N ± 1 electrons, and η is a positive infinitesimal constant that assures the convergence of the Fourier transform from the time-dependent expression. Electron-binding energies are negative numbers in the usual cases when removal of an electron requires energy or addition of an electron releases energy. Residues are numerators that correspond to terms with vanishing denominators. Matrix elements of creation and annihilation operators occurring in residues of the EP are coefficients of corresponding spin−orbitals in Dyson orbitals (vide infra). Poles and residues may be obtained using the Dyson equation:10−12,24

G−1(E) = G−0 1(E) − Σ(E)

(2)

Poles of the zeroth-order EP matrix, G0(E), defined as G0(E)pq = δpq(E − εp)−1

(3)

are equal to orbital eigenvalues, εp, obtained with the Hartree− Fock method. The self-energy matrix, Σ(E), describes orbital relaxation in final states with N ± 1 electrons and differential correlation contributions to electron-binding energies. One way to search for poles of the EP matrix is to require that the determinant of G−1(E) be equal to zero. Under such conditions, eigenvalues obtained from the Dyson quasi-particle equation: [F + Σ(E)]C = EC

(4)

correspond to EP poles. Because E occurs on both sides of this equation, an iterative search may be undertaken. After making a guess for E and diagonalizing the sum of the Fock and self-energy matrices, F + Σ(E), one of the eigenvalues may be reinserted

2. THEORY Poles of the EP are defined as energies that correspond to a vanishing denominator. These values equal energies of electron B


Article

Journal of Chemical Theory and Computation Table 2. Point Groups, Bond Lengths (Å), and Angles (deg) of Test Anions XY− −

BO BS− AlO− AlS− CN− CP− SiN− SiP− OF− OCl− SF− SCl−

PG C∞v C∞v C∞v C∞v C∞v C∞v C∞v C∞v C∞v C∞v C∞v C∞v

rXY 1.24 1.69 1.65 2.11 1.18 1.61 1.74 2.13 1.52 1.71 1.74 2.13

XH−x

PG

θHXH

rXH

−

F Cl− OH− SH− NH2− PH2− CH−3 SiH−3 BH−4 AlH−4

C∞v C∞v C2v C2v C3v C3v Td Td

0.96 1.34 1.03 1.43 1.10 1.54 1.24 1.64

101.2 91.7 108.0 95.1

HXY−

PG

rHX

θHXY

rXY

HBN− HBF− HAlP− HAlCl− HCC− HCO− HSiSi− HSiS− HNB− HNF− HPAl− HPCl− HOO− HSS−

C∞v Cs C∞v Cs C∞v Cs Cs Cs C∞v Cs Cs Cs Cs Cs

1.19 1.27 1.61 1.68 1.07 1.22 1.51 1.56 1.00 1.03 1.45 1.43 0.96 1.35

180.0 104.4 180.0 95.4 180.0 109.7 154.0 102.9 180.0 95.3 72.5 92.9 97.3 101.3

1.28 1.39 2.12 2.30 1.25 1.24 2.13 2.06 1.29 1.55 2.24 2.21 1.53 2.12

Figure 1. Error distributions (eV) for small anions: diagonal methods.

into the left side. Such a process may be continued until selfconsistency with respect to E is achieved. At convergence, E is

a pole, and its eigenvector (C) gives a linear combination of spin−orbitals that is proportional to the Dyson orbital. C


Article


Figure 2. Error distributions (eV) for small anions: nondiagonal methods.

Newton’s method usually suffice to converge E to 0.01 eV. In the diagonal approximation, Dyson orbitals are proportional to canonical HF orbitals:

The Dyson orbital for an electron detachment energy that connects an initial state to the rth final state, ϕDyson (x1), is r defined as ϕrDyson(x1) = N1/2

ϕpDyson(x1) =

∫ dx 2 dx 3

... dxN Ψ0N (x1 , x 2 , x3 , ..., xN )ΨrN − 1 *(x 2 , x3 , ..., xN )

dΣpp(E) zy ji zz πp = jjjj1 − zz d E k {

−1

(9)

Methods with diagonal self-energies include diagonal second-order (D2), diagonal third-order (D3), partial thirdorder (P3), renormalized partial third-order (P3+), and four versions of the outer valence Green function (OVGF-X, X = A, B, C, N). In D2 and D3, all terms occurring through second- or third-order, respectively, of the Møller−Plesset fluctuation potential are included. For ionization energies of closed-shell molecules, D2 and D3 tend, respectively, to overestimate and underestimate corrections to Koopmans results when basis set effects are exhausted.39,49 Because the chief deficiency of occupied Hartree−Fock orbital energies is neglect of orbital relaxation in final states, these values usually lead to overestimated ionization energies. In typical applications, D2 and D3, respectively, yield underestimates and overestimates. Fortuitous agreement with experiment is often obtained with D3 when basis sets of double or triple ζ quality are employed. To take advantage of this bracketing pattern, two approaches to making final estimates have been employed. Ratios of second-order and third-order self-energy terms may be used to estimate higher-order effects by assuming geometric series. Three such schemes, OVGF-A, OVGF-B, and OVGF-C, have been widely applied. On the basis of extensive experience from

−1

(6)

Pole strengths may vary between zero and unity. When they approach unity, the one-electron picture exemplified by Koopmans’s theorem (KT) is validated. Strong correlation effects (such as those occurring in shake-up final states) are accompanied by low pole strengths. For electron binding energies in which the assumptions of Koopmans’s theorem are qualitatively reasonable (i.e., single Slater determinants built from initial-state spin−orbitals dominate the initial and final states), neglect of off-diagonal elements of Σ(E) usually has negligible effects on converged poles. The diagonal self-energy (or quasiparticle) approximation produces a simplified form of the Dyson equation that reads: εp + Σpp(E) = E

(8)

where πp is the pole strength given by

(5)

where xs is the space−spin coordinate for electron s. In general, Dyson orbitals are not normalized to unity. Their norms are known as pole strengths, πr, which may be calculated with i dΣ(E) yz Cr zzz πr = jjjj1 − C†r dE k {

πp ϕpHF(x1)

(7)

All relaxation and correlation corrections to a canonical, Hartree− Fock orbital energy are provided by a diagonal element of Σ(E) with the same spin−orbital index. A few iterations that employ D


Article

Journal of Chemical Theory and Computation Table 3. F−(H2O) VEDEs, eV Diagonal Methodsa b

state

basis

KT

D2

D3

82A′ 82A′ 82A′ 72A″ 72A″ 72A″

aTZ aQZ CBS aTZ aQZ CBS

6.43 6.43 6.43 6.72 6.72 6.72

3.32 3.43 3.51 3.59 3.70 3.78

6.79 6.83 6.86 6.86 6.90 6.93

state 2

8 A′ 82A′ 82A′ 72A″ 72A″ 72A″

A

B

5.38 5.90 5.39 5.92 5.40 5.93 5.53 5.98 5.55 6.00 5.56 6.01 Nondiagonal Methods

C

P3

P3+

5.69 5.74 5.78 5.82 5.88 5.92

5.23 5.37 5.47 5.44 5.57 5.66

4.89 5.02 5.12 5.10 5.22 5.31

basisc

TDA

NR2

ADC(3)

3+


3.89 3.98 4.05 4.02 4.11 4.18

4.64 4.76 4.85 4.80 4.93 5.02

5.40 5.46 5.50 5.63 5.69 5.73

5.84 5.87 5.89 5.93 5.96 5.98

ΔCCSD(T)

4.99

5.38 ΔCCSD(T)

4.99

5.38

a

A, B, and C stand for OVGF-A, OVGF-B, and OVGF-C, respectively. OVGF-N (recommended value) is shown in bold. baXZ = aug-cc-pVXZ. CBS = (43IE(aQZ)-33IE(aTZ))/(43-33). caXZ = aug-cc-pVXZ. CBS = (43IE(aQZ)-33IE(aTZ))/(43-33).

Table 4. Cl−(H2O) VEDEs, eV Diagonal Methodsa b

state

basis

KT

D2

D3

82A″ 82A″ 82A″ 72A′ 72A′ 72A′


4.82 4.82 4.82 4.84 4.84 4.84

3.83 3.94 4.02 3.85 3.96 4.04

4.54 4.63 4.70 4.56 4.65 4.72

state 2

8 A″ 82A″ 82A″ 72A′ 72A′ 72A′

A

B

4.21 4.42 4.28 4.51 4.33 4.58 4.23 4.44 4.30 4.53 4.35 4.60 Nondiagonal Methods

C

P3

P3+

4.43 4.52 4.59 4.45 4.54 4.61

4.18 4.30 4.39 4.20 4.32 4.41

4.13 4.25 4.34 4.15 4.27 4.36

basisc

TDA

NR2

ADC(3)

3+


3.80 3.89 3.96 3.83 3.91 3.97

4.06 4.18 4.27 4.09 4.20 4.28

4.36 4.45 4.52 4.38 4.47 4.54

4.41 4.50 4.57 4.43 4.52 4.59

ΔCCSD(T)

4.39

4.41 ΔCCSD(T)

4.39

4.41

a

A, B, and C stand for OVGF-A, OVGF-B, and OVGF-C, respectively. OVGF-N (recommended value) is shown in bold. baXZ = aug-cc-pVXZ. CBS = (43IE(aQZ)-33IE(aTZ))/(43-33). caXZ = aug-cc-pVXZ. CBS = (43IE(aQZ)-33IE(aTZ))/(43-33).

calculations performed with basis sets of approximately triple ζ quality, a selection procedure (OVGF-N) identifies which of the three OVGF formulas is most likely to have a convergent geometric series.21,24 This procedure includes numerical parameters and was optimized for ionization energies of molecules. An alternative strategy (P3) identifies a subset of thirdorder terms that provide more reliable corrections than D3 while reducing arithmetic bottlenecks and memory requirements.27,50 A renormalized extension (P3+)51 employs arguments similar to those invoked in OVGF-B. Nondiagonal self-energies include the two-particle−one-hole, Tamm−Dancoff approximation (2phTDA),21 the renormalized, third-order method (3+),24 the third-order algebraic, diagrammatic construction, or ADC(3),21 and the nondiagonal, renormalized, second-order method (NR2).52 All third-order terms in the self-energy are retained in the 3+ and ADC(3) methods. NR2 may be regarded as a nondiagonal extension of P3 and P3+ that explicitly evaluates certain terms in fourth and higher orders. Arithmetic scaling factors and storage requirements for diagonal and nondiagonal methods are summarized in Table 1.

Most of the methods have fifth-power dependence in their iterative steps. (Noniterative steps pertain to intermediates that need be calculated only once.) Evaluation of constant selfenergy terms in ADC(3) has a sixth-power arithmetic bottleneck53 that is similar to that of coupled-cluster singles and doubles iterations.54 OVGF introduces only trivial additional calculations beyond D3. P3+ also improves on P3 with minimal effort. For electron detachment energies, P3, P3+, and NR2 do not require transformed electron repulsion integrals with four virtual indices, but D3 and OVGF do. These integrals are involved in the OVGF, 3+, 2phTDA, and ADC(3) arithmetic bottlenecks, which are larger than their counterparts for P3+ and NR2.

3. COMPUTATIONAL METHODS The present calculations were executed with a version of Gaussian 0955 that was modified to perform P3+ and nondiagonal calculations, with Gaussian 16,40 or with the development version of Gaussian.56 Geometry optimizations and harmonic frequency calculations were performed with the coupled-cluster single and double plus perturbative triple substitution, or CCSD(T) E


Article

Journal of Chemical Theory and Computation Table 5. C5H−5 VEDEs, eV Diagonal Methodsa b

c

state

basis

122E″1 122E1″ 122E1″ 112A″2 112A″2 112A2″


KT

D2

D3

1.97 1.98 1.98 7.66 7.66 7.66

1.62 1.69 1.74 5.57 5.63 5.67

2.18 2.18 2.18 6.71 6.71 6.71

A

B

1.75 1.88 1.79 1.88 1.82 1.88 6.14 6.25 6.16 6.24 6.18 6.24 Nondiagonal Methodsd

C

P3

P3+

2.01 2.03 2.04 6.36 6.38 6.40

2.16 2.24 2.30 6.22 6.30 6.36

2.06 2.14 2.20 6.06 6.14 6.20

state

basisc

TDA

NR2

ADC(3)

3+

12 E″1 122E1″ 122E1″ 112A″2 112A2″ 112A2″


1.45 1.50 1.54 4.97(0.67) 5.00(0.67) 5.02(0.67)

1.95 2.03 2.09 5.71(0.66) 5.78(0.66) 5.83(0.66)

2.02 2.06 2.09 6.21(0.68) 6.25(0.68) 6.28(0.68)

1.95 1.95 1.95 6.17(0.68) 6.17(0.69) 6.17(0.68)

2

ΔCCSD(T)

2.04

6.12 ΔCCSD(T)

2.04

6.12

a

A, B, and C stand for OVGF-A, OVGF-B, and OVGF-C, respectively. OVGF-N (recommended value) is shown in bold. bAll pole strengths for the 112A2″ state are between 0.79 and 0.82. caXZ = aug-cc-pVXZ. CBS = (43IE(aQZ)-33IE(aTZ))/(43-33). dPole strength in parentheses when lower than 0.85.

Figure 3. Relative timings of post-integral transformation calculations versus the number of valence basis functions in augmented, correlation-consistent basis sets for the lowest vertical electron detachment energy of C5H−5 .

Figure 4. Relative timings of post-Hartree−Fock calculations versus the number of valence basis functions in augmented, correlationconsistent basis sets for the lowest vertical electron detachment energy of C5H−5 .

Table 6. N2PNP− and P2N−3 Pentagonal Structures and Relative Energiesa

method,57 and the 6-311++G(2df,2pd) basis set, except when noted otherwise.58−61 All vibrational frequencies are real, positive numbers. The initial test set included 36 small, closed-shell, atomic and molecular anions from the first three periods (see Table 2). The 12 diatomic anions in the first column have 10 or 14 valence electrons. The two diatomic hydrides and six polyhydrides are isoelectronic with the two halides, F− and Cl−. Triatomic hydrides with second or third period atoms may be linear or bent. HXY− anions for which the ground state is a triplet were discarded. In 19 cases, two final states per anion are considered. Augmented, correlation-consistent double, triple, and quadruple ζ basis sets62−66 were used in combination with EP methods. Complete basis set (CBS) effects were estimated by extrapolating triple and quadruple ζ results with an inverse cubic function.67 However, when the result for triple ζ is higher than that for quadruple ζ, the latter result is taken as the CBS value. All pole strengths under consideration exceeded 0.85 and confirm the qualitative validity of perturbative

N2PNP− relative energy (eV) adiabatic detachment energy (eV) point group distance (Å) distance (Å) distance (Å) angle (deg) angle (deg) a

0.00 3.21 (2B2) C2v P−N P−N′ N−N P−N′−P N−N−P

P2N−3 0.77 3.68 (2A1)

1.675 1.658 1.355 109.4 113.8

C2v P−P P−N N−N P−P−N N−N−N

2.100 1.701 1.323 93.1 119.4

CCSD(T)/6-311+G(2df).

improvements to Koopmans results, except when specifically noted. Results for 55 VEDEs of small anions are provided in the Supporting Information. CCSD(T) benchmarks were obtained by extrapolating the correlation energy with triple and quadruple ζ basis sets and adding the quadruple ζ Hartree−Fock energy. In the case of Al(BO2)−4 , the benchmark was obtained F


Article

Journal of Chemical Theory and Computation Table 7. P2N−3 VEDEs, eV, with Diagonal Methodsa state

basisb

KT

D2c

D3

A

B

C

P3

P3+c

ΔCCSD(T)

EOMd

2

aTZ aQZ CBS aTZ aQZ CBS aTZ aQZ CBS aTZ aQZ CBS aTZ aQZ CBS

4.10 4.10 4.10 5.19 5.19 5.19 5.22 5.22 5.22 5.76 5.76 5.76 6.37 6.37 6.37

4.18 4.25 4.30 4.20 4.28 4.34 4.39 4.47 4.53 3.41 3.49 3.55 3.56 3.65 3.72

4.29 4.31 4.32 4.87 4.91 4.94 5.41 5.42 5.43 5.46 5.49 5.51 6.16 6.19 6.21

4.17 4.24 4.29 4.48 4.52 4.55 4.75 4.77 4.78 4.36 4.39 4.41 4.80 4.83 4.85

4.13 4.15 4.16 4.53 4.56 4.58 4.90 4.90 4.90 4.47 4.48 4.49 4.97 4.98 4.99

4.26 4.30 4.33 4.67 4.73 4.77 5.04 5.07 5.09 4.67 4.71 4.74 5.11 5.15 5.18

4.45 4.52 4.57 4.76 4.85 4.92 5.29 5.38 5.45 4.89 5.00 5.08 5.28 5.38 5.45

4.40 4.48 4.54 4.65 4.73 4.79 5.09 5.18 5.25 4.51 4.60 4.67 4.82 4.92 4.99

4.35 4.40 4.43 4.50 4.59 4.65 4.92 4.98 5.03 4.30 4.36 4.41 4.64 4.71 4.76

4.38

13 B1 132B1 132B1 122A1 122A1 122A1 112A2 112A2 112A2 102A1 102A1 102A1 92B2 92B2 92B2

4.70

4.97

4.22

4.71

a

A, B, and C stand for OVGF-A, OVGF-B, and OVGF-C, respectively. OVGF-N (recommended value) is shown in bold. baXZ = aug-cc-pVXZ. CBS = (43IE(aQZ)-33IE(aTZ))/(43-33). cD2 values in italics have 0.80 < PS < 0.83. P3+ values in italics have PS = 0.84. dIP-EOM-CCSD/aug-ccpVTZ//CCSD/aug-cc-pVTZ.73

Table 8. P2N−3 VEDEs, eV, with Nondiagonal Methods state

basisa

TDA

NR2

ADC(3)

3+

ΔCCSD(T)

EOMb

132B1 132B1 132B1 122A1 122A1 122A1 112A2 112A2 112A2 102A1 102A1 102A1 92B2 92B2 92B2


3.90 3.94 3.97 4.07 4.12 4.16 4.20 4.25 4.29 3.41 3.46 3.50 3.63 3.69 3.73

4.27 4.35 4.41 4.60 4.68 4.74 4.92 5.00 5.06 4.40 4.49 4.56 4.74 4.83 4.90

4.23 4.28 4.32 4.64 4.70 4.74 4.97 5.02 5.06 4.78 4.83 4.87 5.22 5.28 5.32

4.18 4.20 4.21 4.63 4.66 4.68 4.97 4.98 4.99 4.77 4.79 4.80 5.26 5.28 5.29

4.35 4.40 4.43 4.50 4.59 4.65 4.92 4.98 5.03 4.30 4.36 4.41 4.64 4.71 4.76

4.38

4.70

4.97

4.22

4.71

a

aXZ = aug-cc-pVXZ. CBS = (43IE(aQZ)-33IE(aTZ))/(43-33). bIP-EOM-CCSD/aug-cc-pVTZ//CCSD/aug-cc-pVTZ.73

Improving the basis set shifts results toward higher VEDEs for all methods, in a manner similar to that observed for molecular ionization energies.39,49 Fortuitously small errors may emerge from calculations with aug-cc-pVDZ or aug-cc-pVTZ. The best results in the CBS limit for diagonal methods are obtained with OVGF-A (μ = −0.01, MUE = 0.14, σ = 0.19 eV) and the less computationally demanding P3+ method (μ = 0.09, MUE = 0.13, σ = 0.16 eV). OVGF-N’s selection procedure, which was created chiefly for ionization energies of molecules, usually recommends the OVGF-B alternative, and it never recommends OVGF-A. Extrapolated P3+ results tend to slightly overestimate VEDEs. Among nondiagonal methods, NR2 obtains the best results: μ = −0.05, MUE = 0.11, σ = 0.13 eV. (Note that μ is negative only for 2phTDA, NR2, and OVGF-A.) Results for 2phTDA and ADC(3) display two maxima. 3+ has four cases with errors larger than 0.8 eV. NR2’s σ is notably lower than those of the other nondiagonal methods before and after basis set extrapolation. Extrapolated NR2 results have a less pronounced tendency to overestimate VEDEs than P3+ and have slightly lower σ and MUE values. 4.2. Microsolvated Halide Anions. Microsolvated anions provide molecular-level information about chemical interactions

with an MP2 correlation energy extrapolation with triple and quadruple ζ and a δCCSD(T) extrapolation with double and triple MP2 ζ. Core orbitals were omitted from self-energy summations.

4. RESULTS AND DISCUSSION The capabilities of EP methods for the calculation of VEDEs of anions may be tested in calculations on small, closed-shell, molecular anions of the first three periods. After a statistical analysis of these results, the performance of EP methods on anions that typify current computational and experimental research is considered. Halide−water complexes, the cyclopentadienyl anion encountered frequently in organometallic chemistry, two pentagonal P2N−3 isomers, and the Al(BO2)−4 superhalide constitute a representative sample. 4.1. Small Anions. 55 VEDEs for the 36 test anions listed in Table 2 were predicted with ΔCCSD(T) and EP methods. The distribution of errors with respect to basis set-extrapolated ΔCCSD(T) results is shown in Figures 1 and 2 for diagonal and nondiagonal EP methods, respectively. Each histogram displays the mean error (μ), the standard deviation (σ), and the mean unsigned error (MUE). TDA stands for 2phTDA in the legends of these histograms and subsequent tables. G


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between solvents and solutes, and the study of these species can serve as a benchmark for modeling ionic solvation processes.42,43,68 Experimental spectra68 indicate first VEDEs of 4.80 and 4.29 eV for F−(H2O) and Cl−(H2O), respectively. These results are in good agreement with CCSD(T) estimates (4.99 and 4.39 eV, see Tables 3 and 4) and NR2 values (4.85 and 4.27 eV). OVGF methods fail to accurately predict the F−(H2O) complex’s first VEDE, with errors higher than 0.4 eV. However, P3+ and NR2 show a better description of correlation effects. Similar results are obtained for the isolated F− anion, where the ΔCCSD(T) result of 3.44 eV is in close agreement with the experimental value of 3.40 eV, but the OVGF results are off by about 1 eV (see the Supporting Information). Correlation effects for F− are so complicated that only NR2 gives close agreement, with a prediction of 3.31 eV. The remaining diagonal EP methods (D2 and D3) also have large discrepancies with experimental results. Diagonal methods perform notably better for Cl−(H2O); errors for OVGF-B and P3+ are 0.19 and −0.05 eV, respectively. For the second lowest VEDE of F−(H2O), P3+ gives a smaller error (−0.07 eV) than does NR2 (−0.36 eV). ADC(3) and 3+ also give relatively high errors. Of the OVGF methods, only the A version gives a useful result that is too high by 0.18 eV. For the Cl case, OVGF-B, P3+, and NR2 yield errors of 0.19, −0.05, and −0.13 eV, respectively. OVGF-A has a lower error than OVGF-B, −0.06 eV. ADC(3) and 3+ yield overestimates of 0.13 and 0.18 eV, respectively. 4.3. Cyclopentadienyl Anion. The cyclopentadienyl (C5H−5 ) anion is ubiquitous in organometallic chemistry and binds with atomic metal ions in several hapticity (e.g., η = 1, 3, 5) patterns.69 D2 and D3 results on the lowest VEDE produce the bracketing pattern typical for molecules with respect to the CCSD(T) value of 2.04 eV (previous experimental results70 provide a VEDE of 1.808 eV), and OVGF methods yield good agreement (see Table 5). P3+ also performs well, as do the nondiagonal NR2, ADC(3), and 3+ methods. All pole strengths are above 0.85. The first VEDE of C5H−5 appears to be a relatively simple case, where the Koopmans description of the final state is qualitatively valid. Previous predictions for the second VEDE70 are close to 6.1 eV. Results reported in Table 5 with ΔCCSD(T) (6.12 eV) and EP methods (6.20, 6.18, 6.24, 5.83, 6.28, and 6.17 for P3+,

Figure 5. Comparison between the spin density (isovalue = 0.02) and canonical molecular orbitals (isovalue = 0.05) for P2N−3 .

Table 9. N2PNP− VEDEs, eV, with Diagonal Methodsa state

basisb

KT

D2c

D3

A

B

C

P3

P3+c

ΔCCSD(T)

2


3.44 3.44 3.44 4.94 4.94 4.94 6.16 6.17 6.18 6.82 6.82 6.82 7.14 7.14 7.14

3.56 3.63 3.68 2.82 2.90 2.96 5.00 5.09 5.16 4.76 4.84 4.90 4.06 4.17 4.25

3.84 3.86 3.87 4.66 4.69 4.71 6.21 6.23 6.24 6.46 6.50 6.53 6.85 6.87 6.88

3.52 3.57 3.61 3.68 3.70 3.71 5.49 5.52 5.54 5.56 5.60 5.63 5.43 5.46 5.48

3.64 3.64 3.64 3.80 3.81 3.82 5.64 5.65 5.66 5.70 5.72 5.73 5.59 5.60 5.61

3.76 3.79 3.81 3.97 4.01 4.04 5.77 5.81 5.84 5.85 5.89 5.92 5.75 5.80 5.84

3.95 4.02 4.07 4.12 4.22 4.29 5.99 6.09 6.16 5.91 6.01 6.08 5.90 6.02 6.11

3.89 3.96 4.01 3.80 3.88 3.94 5.77 5.87 5.94 5.63 5.72 5.79 5.41 5.52 5.60

3.79 3.83 3.86 3.64 3.68 3.73 5.64 5.70 5.74

13 A2 132A2 132A2 122B2 122B2 122B2 112B1 112B1 112B1 102B2 102B2 102B2 92A1 92A1 92A1

5.27 5.33 5.38

a

A, B, and C stand for OVGF-A, OVGF-B, and OVGF-C, respectively. OVGF-N (recommended value) is shown in bold. baXZ = aug-cc-pVXZ. CBS = (43IE(aQZ)-33IE(aTZ))/(43-33). cD2 values in italics have 0.80 < PS < 0.83. P3+ values in italics have PS = 0.84. H


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Journal of Chemical Theory and Computation Table 10. N2PNP− VEDEs, eV, with Nondiagonal Methods state

basisa

TDA

NR2

ADC(3)

3+

ΔCCSD(T)

2


3.38 2.74 2.74 2.69 2.74 2.78 4.78 4.84 4.88 4.90 4.94 4.97 4.25 4.32 4.37

3.77 3.84 3.89 3.66 3.75 3.82 5.53 5.62 5.69 5.55 5.62 5.67 5.36 5.47 5.55

3.70 3.74 3.77 4.01 4.06 4.10 5.66 5.72 5.76 5.82 5.88 5.92 5.83 5.90 5.95

3.70 3.72 3.73 4.01 4.03 4.04 5.64 5.67 5.69 5.84 5.87 5.89 5.87 5.90 5.92

3.79 3.83 3.86 3.64 3.68 3.73 5.64 5.70 5.74

13 A2 132A2 132A2 122B2 122B2 122B2 112B1 112B1 112B1 102B2 102B2 102B2 92A1 92A1 92A1

5.27 5.33 5.38

a

aXZ = aug-cc-pVXZ. CBS = (43IE(aQZ)-33IE(aTZ))/(43-33).

OVGF-A, OVGF-B, NR2, ADC(3), and 3+, respectively) are in apparently good agreement. However, the low pole strengths (0.66−0.69 for nondiagonal and 0.79−0.82 for diagonal methods) for the 2A″2 final state indicate that the Koopmans picture is not qualitatively valid; that is, shake-up character is not negligible. Diagonal methods may have given fortuitously good predictions, but nondiagonal methods are needed for proper interpretation of results. Whereas the 3+ and ADC(3) results are larger than the ΔCCSD(T) values, NR2 yields an underestimate. Because couplings between 2hp operators are treated only through first-order and 3h2p operators are absent in 3+, ADC(3), and NR2, only semiquantitative predictions can be made with these methods for final states with pole strengths below 0.85. Relative timings for the lowest VEDE are displayed in Figures 3 and 4 as a function of augmented correlation-consistent basis sets. Partial transformations that do not generate electron repulsion integrals with four virtual indices were performed, for limitations on storage most frequently determine the feasibility of calculations on large systems. The slowest calculation, ΔCCSD(T) with the aug-cc-pVQZ basis, is chosen to have a value of unity on the vertical axis of Figure 3, which considers only post-transformation steps. After ΔCCSD(T), the longest calculations pertain to ADC(3), 3+, and 2ph-TDA, which may be accelerated by performing a full integral transformation when sufficient disk storage is available. Curves for OVGF and NR2 lie below, followed by P3+ and, finally, D2. The logarithmic scale of the vertical axis spans 6 orders of magnitude and reveals the computational advantages of the D2, P3+, and NR2 methods. Approximately parallel curves indicate that the scaling factors of Table 1 provide a reasonable, qualitative guide to the relative efficiency of the EP methods and the present CCSD(T) standard. In Figure 4, post-Hartree− Fock timings that include the integral transformation are displayed. An additional curve for ΔMP2 calculations closely resembles its D2 and P3+ counterparts and also lies close to the NR2 curve. These results suggest that the feasibility of P3+ or NR2 is likely to be comparable to that of ΔMP2 for larger systems. Because these EP methods for VEDEs require only one, partial integral transformation with ON4 scaling (where O is the number of occupied orbitals and N is the number of basis functions), their advantages over indirect methods that repeat this fifth-power step (e.g., ΔMP2) increase when more

Figure 6. Comparison between the spin density (isovalue = 0.02) and canonical molecular orbitals (isovalue = 0.05) for N2PNP−.

than one final state is requested. In practice, integral transformations are often slower than subsequent P3+ or NR2 steps. 4.4. P2N−3 Pentagons. Recently, a salt containing diphosphatriazolate (P2N−3 ) anion, a planar, pentagonal ring with adjacent P nuclei, was prepared by Velian et al.71 (See results of geometry optimizations in Table 6, which indicate the existence and relative stability of another pentagonal isomer.) The experimental result of Hou et al. for the adiabatic electron detachment energy is 3.76 eV.72 Calculations indicate that there is a large, symmetry-lowering relaxation energy for P2N3.72 VEDEs calculated at the ΔCCSD(T) level by Hou et al.72 and with IP-EOM-CCSD by Jin et al.73 are 4.27 and 4.22 eV, respectively. Both of these calculations were performed with an aug-cc-pVTZ basis set on a 2A1 final state, that I


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Journal of Chemical Theory and Computation Table 11. Al(BO2)−4 VEDEs, eV, with Diagonal Methodsa,b state

basisc

KT

D2

D3

A

B

C

P3

P3+

ΔCCSD(T)

2

TZ1 TZ2 aTZ aQZ CBS TZ1 TZ2 aTZ aQZ CBS TZ1 TZ2 aTZ aQZ CBS

9.95 9.93 9.92 9.92 9.92 10.16 10.14 10.13 10.13 10.13 10.18 10.15 10.14 10.14 10.14

7.58 7.79 7.79 7.90 7.98 7.88 8.05 8.06 8.16 8.23 7.90 8.07 8.08 8.18 8.25

9.42 9.62 9.64 9.69 9.73 9.73 9.91 9.94 9.98 10.01 9.75 9.93 9.96 10.00 10.03

8.70 8.83 8.85 8.88 8.90 8.99 9.10 9.12 9.15 9.17 9.01 9.12 9.14 9.17 9.19

8.90d 9.09d 9.11 9.15 9.18 9.20 9.38 9.40 9.43 9.45 9.23 9.40 9.42 9.45 9.47

8.88 9.08 9.10 9.16 9.20 9.18 9.36 9.38 9.44 9.48 9.20 9.38 9.40 9.46 9.50

8.75 8.94 8.95 9.07 9.16 9.03 9.20 9.21 9.33 9.42 9.04 9.22 9.23 9.35 9.44

8.55 8.74 8.75 8.86 8.94 8.82 9.00 9.01 9.12 9.20 8.84 9.02 9.03 9.14 9.22

8.36d 8.60 8.62

30 T1 302T1 302T1 302T1 302T1 282E 282E 282E 282E 282E 252T2 252T2 252T2 252T2 252T2

8.76e 8.69 8.91 8.93 9.06e 8.72 8.94 8.96 9.09e

a Structure obtained from ref 44. bA, B, and C stand for OVGF-A, OVGF-B, and OVGF-C, respectively. OVGF-N (recommended value) is shown in bold. caXZ = aug-cc-pVXZ. CBS = (43IE(aQZ)-33IE(aTZ))/(43-33). TZ1 = 6-311+G(d) and TZ2 = 6-311+G(3df). dValues reported previously /a(DT)Z. in ref 44. eMP2/a(TQ)Z+δCCSD(T) MP2

is, a radical with a σ hole. Excited states of P2N3 also were accessed in the IP-EOM-CCSD calculations, which produced VEDEs listed in Tables 7 and 8. The order of final states predicted by IP-EOM-CCSD is 2A1, 2B1, 2A1, 2B2, and 2A2, with σ, π, σ, σ, and π holes, respectively. To provide standards for the present study, basis setextrapolated ΔCCSD(T) calculations were performed. The order of the final states is the same as that in the IP-EOMCCSD calculations, but the separation between the lowest VEDE (4.41 eV) and the second lowest VEDE is only 0.02 eV. UHF reference states for the CCSD(T) calculations on doublets are obtained from initial determinants built with anionic, restricted HF orbitals (see Figure 5). In the final UHF wave functions, substantial spin contamination occurs in most cases. Differences between UHF and CCSD spin densities reveal substantial correlation effects and qualitative contrasts with the anionic, restricted HF orbital amplitudes. Despite these caveats, there is close agreement between IP-EOM-CCSD and ΔCCSD(T) results with the aug-cc-pVTZ basis. Although the order of final states is not definitive with ΔCCSD(T), it is likely that its VEDEs are accurate to within 0.1−0.2 eV, in accord with the aims stated in the Introduction. Extrapolated P3+ and ΔCCSD(T) results are in close agreement for the three lowest VEDEs, although P3+ reverses the order of the first two states. They have slightly larger discrepancies of 0.2−0.3 eV for the two highest VEDEs. Agreement with IP-EOM-CCSD for the aug-cc-pVTZ basis also is close, except for the lowest 2A1 cases, where the discrepancy is almost 0.2 eV. The OVGF-N procedure produces reasonable VEDEs, but the predicted order of states is completely different: 2B1, 2A1, 2A1, 2A2, 2B2. For the nondiagonal methods, NR2 gives the best results, with discrepancies of 0.15 eV or less when basis set extrapolations are performed. Geometry optimizations disclose that the pentagonal N2PNP− isomer is lower in energy than the P2N−3 structure by 0.77 eV (see Table 6). The VEDE for this isomer was calculated to be 3.73 eV (see Tables 9 and 10) with extrapolated ΔCCSD(T). Several other final states also were considered, but in one case, the second 2B2 state, the UHF self-consistent field converged to the first 2B2 state’s total energy. Figure 6 shows that in this isomer, there are also strong correlation effects on spin

densities and considerable spin contamination in UHF wave functions employed in CCSD(T) calculations. For the remaining four states, P3+ and OVGF-N results are within 0.2 eV of the ΔCCSD(T) values. Among the nondiagonal methods, NR2 gives the best results. The present results on the more stable isomer may prove useful should alternative methods of anion synthesis be employed, especially if both isomers are present in the experimental sample. 4.5. Superhalide Al(BO2)−4 . VEDEs of the superhalide Al(BO2)−4 are very large and imply that the corresponding molecule has an electron affinity that is more than double those of the most electronegative halogen atoms. (Superhalogens are defined as molecules with electron affinities that are larger than those of the halogen atoms, the largest of which, 3.6 eV, pertains to Cl.4,74,75) Gutsev et al. found discrepancies between ΔCCSD(T) and OVGF results of about 0.4 eV.44 The data of Table 11 confirm these conclusions and also disclose more satisfactory agreement between ΔCCSD(T) and the P3+ method. The latter results are about 0.1 eV higher than the benchmarks. The superior efficiency of the NR2 method makes it a feasible approach to the VEDEs of larger superhalogens: excellent agreement between NR2 and ΔCCSD(T) is shown in Table 12. P3+, NR2, and ΔCCSD(T) agree that the VEDE of Al(BO2)−4 is 8.8 eV, the ground state of the radical is 2T1, and that two excited states, 2E and 2T2, lie 0.3 eV higher. Because the energy separation between the second and third VEDEs is so small, the order of these states remains in doubt at the CCSD(T) and NR2 levels of theory. 4.6. Composite Methods. When execution of a higherlevel calculation with a large basis set is infeasible, composite methods that assume the additivity of basis set and higher-level corrections may be employed. For example, combining the difference between D2/aTZ and D2/CBS results (i.e., the basis set effect) with P3+ or NR2 values obtained with the aTZ basis provides an estimate of P3+/CBS or NR2/CBS VEDEs.46 Two methods of this kind are denoted CP3+/a34 and CNR2/a34, respectively, for they require only D2/aTZ, D2/aQZ, and either P3+/aTZ or NR2/aTZ data. Such estimates introduce discrepancies of 0.03 eV or less for all case studies. A similar, but less precise, pair of methods, CP3+/a23 and CNR2/a23, require D2/aDZ, D2/aTZ, and either P3+/aDZ or NR2/aDZ J


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Journal of Chemical Theory and Computation Table 12. Al(BO2)−4 VEDEs, eV, with Nondiagonal Methodsa state

basisb

NR2

ΔCCSD(T)

2

TZ1 TZ2 aTZ aQZ CBS TZ1 TZ2 aTZ aQZ CBS TZ1 TZ2 aTZ aQZ CBS

8.41 8.60 8.61 8.72 8.80 8.70 8.87 8.88 8.99 9.07 8.72 8.89 8.90 9.01 9.09

8.36c 8.60 8.62

30 T1 302T1 302T1 302T1 302T1 282E 282E 282E 282E 282E 252T2 252T2 252T2 252T2 252T2 a

0.01 eV, occurs for the least computationally intensive method, CP3+/a23, but the other values are only slightly larger. The histogram of CNR2/a34 has an especially well-behaved profile. Averages over the case studies shown in Figure 8 give rise to similarly encouraging statistics. CP3+ methods produce more overestimates than underestimates. CNR2/a23 has a tendency to underestimate. CNR2/a34 has a more symmetric profile. 4.7. Physical Insights. The success of the P3+ and NR2 methods has several foundations. (1) Final-state orbital relaxation and configuration interaction (i.e., polarization in the terminology of Pickup and Goscinski17) effects are determined chiefly by couplings between hole (h) and two-hole−one-particle (2hp) operators. Self-energy terms with 2hp denominators are generated by these couplings. Because their bare, first-order expressions are too large, relaxation effects are overestimated in second-order and 2ph-TDA calculations. Inclusion of second-order corrections provides an overly screened description of double substitutions into virtual orbitals of the final state. Because these terms enter with a factor of one-half in P3, P3+, and NR2, a balanced description results. (2) Third-order and higher-order relaxation and polarization effects in Koopmans-like final states also require a full, firstorder description of couplings between 2hp operators. Corresponding ring and ladder terms in the self-energies with 2hp denominators are produced thereby and respectively describe long-range and short-range correlation effects in final states. Estimates of higher-order effects are provided by the renormalization procedure of P3+. Explicit calculation of higherorder terms occurs in NR2.

8.76d 8.69 8.91 8.93 9.06d 8.72 8.94 8.96 9.09d

b

Structure obtained from ref 44. aXZ = aug-cc-pVXZ. CBS = (43IE(aQZ)-33IE(aTZ))/(43-33). TZ1 = 6-311+G(d) and TZ2 = 6-311+G(3df). cValue reported previously in ref 44. dMP2/a(TQ)Z /a(DT)Z. +δCCSD(T) MP2

calculations. VEDEs predicted with composite methods have been compared to basis set-extrapolated ΔCCSD(T) for the test anions (see Table 2) and for the case studies. Encouraging results with μ, σ, and MUE below 0.15 eV are obtained for all four composite methods. The distributions of errors are shown in Figures 7 and 8. For small anions, the four methods have approximately equal values of σ and MUE. The smallest |μ|,

Figure 7. Error distributions (eV) for small anions: composite methods. K


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Figure 8. Error distributions (eV) for large anions: composite methods.

doublets may be calculated reliably with electron propagator methods that have relatively modest computational requirements and thereby enable employment of flexible basis sets that negatively charged species usually require. Tests on an unbiased set of small, closed-shell anions of the first three periods that include the effects of basis set saturation indicate that the OVGF-A, P3+, and NR2 methods have the lowest errors with respect to benchmark ΔCCSD(T) results. P3+ is the least computationally demanding of these methods, has no numerical parameters and employs the diagonal self-energy approximation. Of the nondiagonal methods, NR2 is the most efficient and accurate, imposing only slightly greater arithmetic work relative to P3+. Timing data and storage requirements indicate that P3+, NR2, and ΔMP2 calculations may be executed with approximately equal facility. OVGF methods are more demanding than P3+ with respect to arithmetic operations and require more kinds of transformed electron repulsion integrals than P3+ or NR2. These data usually require more disk storage or the use of less efficient algorithms that recalculate the additional integrals when they are needed. For vertical electron detachment energies, the OVGF-N selection procedure, which employs numerical criteria, usually prefers the OVGF-B method and often gives larger errors than OVGF-A. Case studies on halide−water complexes, organic and inorganic pentagonal rings, and a superhalide illustrate the potential of electron propagator methods. The failures of the OVGF methods for fluoride and its complex with water suggest a preference for P3+ when diagonal methods are required and, if time allows, for NR2 in the study of similar systems. The chloride-containing cases are less problematic for the OVGF methods. P3+ and NR2 perform well for the first VEDE of

(3) Electron-pair energies in initial states that are missing after electron detachment are described chiefly by couplings between h and 2ph operators. For closed-shell, single-reference molecules, a treatment of these effects that is balanced with respect to items 1 and 2 is provided by first-order h−2ph couplings. These initial-state correlation terms with 2ph denominators converge much more slowly with respect to basis sets than those with 2hp denominators. Many self-energy terms that occur in third-order may be numerically inconsequential and relatively difficult to evaluate. Because Dyson orbitals for valence electron detachments are usually proportional to canonical Hartree−Fock orbitals, there is little need to include p couplings with other operators. Such terms are ignored completely in the diagonal self-energy methods. Energy-independent terms in the self-energy occur in the OVGF, 3+, and ADC(3) methods and arise from corrections to the initial state’s one-electron density matrix. Corresponding order-by-order expansions often converge slowly.53 These terms are omitted in P3+ and NR2. The P3+ and NR2 models therefore provide the following insights into VEDEs of closed-shell anions. First, relaxation and polarization effects in all orders arising from h and 2hp operators should be estimated or explicitly calculated. Second, the second-order self-energy provides a balanced treatment of initial-state correlation effects. Third, Dyson orbitals are nearly proportional to canonical Hartree−Fock orbitals.

5. CONCLUSIONS Vertical electron detachment energies (VEDEs) of closed-shell anions that pertain to ground and excited states of uncharged L


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Journal of Chemical Theory and Computation C5H−5 . For the next final state, where a π hole occurs at higher energy, NR2 is the more prudent choice, for pole strengths, which are considerably lower than 0.85, indicate the need for a nondiagonal self−energy method. OVGF methods, ADC(3), and 3+ also provide useful data. Consistent assignments of doublet final states and accurate VEDE results on P2N−3 are obtained from P3+ and NR2 calculations. Predictions on a more stable, pentagonal isomer with nonadjacent P nuclei show a similar pattern of reliability for P3+ and NR2 relative to benchmarks. NR2 pole strengths show the advent of shake-up character for two final states. P3+ and NR2 are the most reliable methods for evaluating the very large VEDEs of the Al(BO2)−4 superhalide and are promising tools for the discovery of even greater molecular electron affinities. These results augur well for predictions of VEDEs of closedshell anions of the first three periods with the P3+ and NR2 methods. The predictive accuracy and computational efficiency of these approximate self-energies may be extended to species where calculations with the most flexible basis sets are infeasible by employing composite models49 in which basis set extrapolations are performed at the more efficient D2 level. None of the diagonal methods has arithmetic requirements that exceed that of the partial integral transformation that precedes an MP2 calculation. Only the evaluation of secondorder couplings between occupied-orbital (h) and shake-up (2hp) operators necessitates the noniterative O3V3 bottleneck in NR2 calculations. By neglecting off-diagonal elements in the NR2 self-energy matrix, an O3V2 bottleneck identical to that of P3+ can be procured with little loss of NR2’s accuracy for Koopmans-like final states. The success of the P3+ and NR2 methods for calculating VEDEs of closed-shell anions rests on a balanced treatment of initial-state correlation and final-state orbital relaxation and polarization effects. Low-order couplings between simple h operators that correspond to the assumptions of Koopmans’s theorem with 2ph and 2hp operators capture, respectively, the dominant initial-state and final-state effects. Couplings between 2hp operators also are important in describing relaxation and polarization. The relative unimportance of p operators and non-Koopmans h operators for a typical final state provide an additional reason to simplify NR2 calculations by neglecting off-diagonal elements of the self-energy matrix and thereby to recover Dyson orbitals that equal the product of the square root of the pole strength and a canonical, Hartree−Fock orbital. Work in this direction is in progress.

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Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank O. Dolgounitcheva and V. G. Zakrzewski for technical support and F. Pawłowski for useful criticisms of the manuscript.

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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.jctc.8b00736. Tables S1−S36 (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*Tel.: (334) 844-6998. E-mail: [email protected]. ORCID

J. V. Ortiz: 0000-0002-9277-0226 Funding

The National Science Foundation supported this research through grant CHE-1565760 to Auburn University. M


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Electron Propagator Methods for Vertical Electron Detachment

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