Theoretical Study of Hyperfine Interactions in Small Arsenic

Article pubs.acs.org/JPCA

Theoretical Study of Hyperfine Interactions in Small Arsenic-Containing Radicals Ljiljana Stojanović* Faculty of Physical Chemistry, University of Belgrade, Studentski trg 12, 11 158 Belgrade, Serbia ABSTRACT: Various density functional theory (DFT) and ab initio MP2 and CCSD methods were employed in calculations of arsenic isotropic and anisotropic hyperfine parameters in three small radicals, AsH2, AsO2, and H2AsO. Convergent basis sets for these calculations were specially derived starting from Dunning’s correlation-consistent sets. DFT methods proved to be particularly appropriate choice, because the results obtained with the suitable functionals are in accordance with the available experimental values. Additionally, mechanisms of hyperfine couplings were investigated by examining individual orbital contributions to spin density. The spin polarization mechanism in AsH2 was studied by evaluating one- and two-electron integrals in spin-restricted and unrestricted case. Apart from nonrelativistic, scalar-relativistic Douglas−Kroll−Hess DFT calculations were performed to examine relativistic impacts on spin density distribution. first-order approximation) is proportional to electron spin density at the position of nucleus N,4 4π β β g g ⟨SZ⟩−1ρNα − β A iso(N ) = AFC(N ) = (1) 3 e N e N

1. INTRODUCTION Electronic paramagnetic resonance (EPR) is of fundamental importance for the study of radicals. The interaction between spin of unpaired electron and nuclear spins, known as hyperfine interaction, induces specific patterns in EPR spectra that are useful in investigations of geometric and electronic structure of radicals, as well as spatial distribution of unpaired electron spin density.1 For these reasons, there are numerous theoretical studies concerning hyperfine interactions in organic radicals. In the last two decades the interest in theoretical investigations of hyperfine interactions in molecules containing heavier nuclei emerges. A representative example is the benchmark study of hyperfine interactions in 3d transition metal complexes,2,3 which inspired the present one. However, to the author’s knowledge, there is a lack of systematic studies on hyperfine interactions in molecules containing post-3d elements. This is the basic motivation for the present one, which is concerned with hyperfine interactions in three small arsenic-containing radicals, AsH2, H2AsO, and AsO2, in their ground doublet electronic states. The aim of the present study is to investigate the mechanisms of hyperfine interactions and to determine the optimal methods for calculation of arsenic hyperfine parameters in given radicals.

where βe, βN, ge, and gN are the electron and nuclear g-factors and the Bohr and nuclear magnetons, respectively. ⟨Sz⟩ is the expectation value of the z-component of the total electronic spin angular momentum operator. Electron spin density at the position of the nucleus (contact density), Pα−β N , defined as the difference between α- and β-electron density, can be calculated using the δ-function centered at the position of the nucleus, RN, 1 1 ρNα − β = ∑ Pμα,−ν β⟨φμ|2Sẑ δ(RN )|φν⟩ = (ρNα − ρNβ ) n μ,ν n (2)

Pα−β μ,ν

where n is the number of unpaired electrons and is an element of the spin density matrix. The density operator is defined as Ne

ρ (r ) =

(3)

i=1

where Ne is the number of electrons and δ (r−ri) is threedimensional δ-distribution. Total electron density, and α- and β-spin densities can be calculated as the expectation values of the density operator in corresponding electronic and α- and β-spin states, respectively. (3)

2. THEORETICAL DETAILS Hyperfine interactions can be divided into isotropic and anisotropic (dipolar) part, which are represented by corresponding hyperfine coupling constants (hfccs).1,4,5 Isotropic hyperfine coupling constant of nucleus N, Aiso(N), (which is equal to the Fermi coupling constant, AFC, in the © 2012 American Chemical Society

∑ δ(3)(r − ri)

Received: May 17, 2012 Revised: July 9, 2012 Published: July 26, 2012 8624

dx.doi.org/10.1021/jp304786r | J. Phys. Chem. A 2012, 116, 8624−8633

The Journal of Physical Chemistry A

Article

3. COMPUTATIONAL DETAILS All computations of arsenic hyperfine coupling constants in AsH2, H2AsO, and AsO2 radicals are carried out using Gaussian 09 software.15 Geometries of the studied molecules are optimized by means of the UCCSD method with aug-cc-pVTZ basis set. Obtained molecular geometries are given in Table 1

Anisotropic hyperfine couplings are described by an anisotropic tensor, whose components in the first-order approximation are given by5 Tkl(N ) =

1 β β g g ⟨Sz⟩−1 ∑ Pμα,−ν β 2 e N e N μν × ⟨ϕμ|rN−5(rN 2δkl − 3rN, krN, l)|ϕν⟩

Table 1. Molecular Geometries of AsH2, H2AsO, and AsO2a

(4)

where rN = r − RN. Two main mechanisms induce spin density at a particular point. Those are direct effect of the unpaired electron and indirect effect of spin polarization. The magnitude of spin density can also be affected by electron correlation, spin−orbit interaction, relativistic effects, and molecular geometry.1,4,5 Contact density can be induced by unpaired electron only if its orbital is at least partly of s character (because all other orbitals have nodes at the position of nucleus). Classical explanation of spin polarization mechanism is that slightly attractive exchange interaction between the unpaired electron and same-spin electrons from doubly occupied orbitals reduces their repulsive Coulomb energy, inducing their approaching.6 According to this explanation, if the radial density maximum of the unpaired α-electron orbital is at a larger distance from nucleus compared to that of doubly occupied orbital, the α-component of the doubly occupied orbital will be attracted outward, leaving a little excess of the β-spin density at nucleus (negative polarization), and vice versa. Spin polarization can be treated on spin-unrestricted level of theory. Watson and Freeman confirmed the applicability of this model in calculations on 3d metal ions,7 where core 1s and 2s orbitals are polarized negatively by unpaired 3d electron, and valence 3s orbital (which has radial density maximum further from the nucleus than 3d orbital) has positive contribution to contact density (positive polarization). The mechanism of spin polarization has also been studied in detail in numerous cases of π-radicals containing second row elements,8−11 in which its classical description was confirmed. It is shown that in these cases core−shell polarization is negative, valence-shell polarization is positive, and sophisticated correlation methods are necessary for their correct treatment. The model of hyperfine interactions used in the present study neglects the effect of spin−orbit coupling. In the case of heavier nuclei there are significant relativistic effects on spin density distribution, particularly in the core region where electron kinetic energies are high. Therefore, hyperfine interactions are often treated using scalar-relativistic Hamiltonians, especially Douglas−Kroll−Hess Hamiltonian (DKH).12 Contrary to nonrelativistic and four-component relativistic cases where electronic density can be calculated as the sum of the squares of molecular orbitals, the squares of DKH orbitals (|ψL(r)|2) deviate from the squared spinors (ψ+(r) ψ(r)). They are related as13 ρ(r ) = ψ +(r ) ψ (r ) = |ψL(r )|2 + Δρ(r )

AsH2 H2AsO

AsO2

r(As−H), Å θ(H−As−H), deg r(As−H), Å r(As−O), Å θ(H−As−H), deg θ(H−As−O), deg θ(O−HAsH), deg r(As−O), Å θ(O−As−O), deg

1.5275b (1.518) 91.289 (90.746) 1.5074 (1.513) 1.6722 (1.672) 100.469 (101.8) 107.356 (106.6) 62.209 (63.1) 1.6243 (1.630) 126.712 (135.0)

a Available experimental values for AsH2 and H2AsO, and theoretical values for AsO2 are given in parentheses. bUCCSD/aug-cc-pVTZ values.

alongside the available experimental values for AsH216 and H2AsO17 and MP2 values for AsO218 obtained with Binning− Curtis valence double set with one set of polarization functions. Obtained geometries of AsH2 and H2AsO are in excellent agreement with the experimental ones. Though the bond lengths are in good agreement, the available MP2 optimized bond angle in AsO2 deviates from the corresponding CCSD result. To examine their sensitivity on molecular geometries, arsenic hyperfine parameters are computed at both optimized and experimentally derived geometries in the cases where the latter are available. 3.1. Methods. Unrestricted all-electron density functional calculations using Kohn−Sham formalism are carried out. Systematic investigation of various exchange−correlation functionals is performed. Due to particular importance of exchange interactions for the hyperfine coupling mechanism, three different kinds of exchange functionals are employed: Becke’s (B) generalized gradient approximation19 (GGA), Becke’s threeparameter hybrid functional20 (B3), and Becke’s half-and-half hybrid functional21 (BH). The last two exchange functionals include exact Hartree−Fock exchange (20% and 50%, respectively). These exchange functionals are combined with three GGA correlation functionals, LYP,22 P86,23 and PW91.24 Apart from nonrelativistic, scalar-relativistic Douglas−Kroll−Hess second-order (DKH2) DFT calculations are performed. All calculations were carried out using Finegrid option (75 radial shells and 302 angular points per shell), because increasing the integration grid density does not affect results significantly. Ab initio correlation methods are also employed: second-order Møller−Plesset correlation energy correction on UHF calculations (MP2), and coupled cluster with single and double excitations (CCSD) with UHF reference function. 3.2. Basis Set Study. One of the central issues of the present study is development of adequate basis sets for calculations of arsenic anisotropic and isotropic hfccs. Although there are no serious difficulties in calculation of anisotropic tensor components, calculation of the isotropic parameters still presents challenge for theoretical chemistry methods, because it is harder to reproduce the δ-function than the rN−5-shaped spatial distribution. Contact densities largely depend on the methods and approximations employed.25

(5)

where Δρ(r) presents the picture-change error (PCE). Because the approximation of electron density as the sum of squares of DKH orbitals is reasonable in valence region, relativistic effects on anisotropic hfccs can be sufficiently accurately treated on this level of theory. In the case of contact densities PCE is pronounced, and deteriorated isotropic hfccs are obtained with the approximated form of the density operator. This problem could be overcome by using correctly transformed density operator.14 8625



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Table 2. Effects of Basis Set Modification on Arsenic Hyperfine Parameters in AsH2 (in MHz) DZ unc. s unc. p unc. d +1s +2s +3s +4s QZ unc. s unc. p unc. d +1s +2s +3s +4s a

Aiso

Taaa

Tcc

88.690 28.300 32.267 34.560 34.973 35.241 35.301 35.349 13.481 13.360 15.147 16.222 16.278 16.297 16.302 16.321

−253.888 −253.395 −284.174 −284.266 −284.266 −284.266 −284.266 −284.266 −261.107 −261.019 −286.128 −288.356 −288.356 −288.356 −288.356 −288.356

498.021 497.072 556.687 556.688 556.687 556.687 556.687 556.687 511.175 511.010 559.997 564.431 564.431 564.431 564.431 564.431

TZ unc. s unc. p unc. d +1s +2s +3s +4s 5Z unc. s unc. p unc. d +1s +2s +3s +4s

Aiso

Taa

Tcc

−58.458 16.234 15.426 16.185 16.226 16.263 16.256 16.269 54.107 14.681 14.169 14.384 14.400 14.405 14.408 14.408

−267.025 −266.964 −285.964 −288.142 −288.142 −288.142 −288.142 −288.143 −280.489 −280.448 −286.912 −288.402 −288.402 −288.402 −288.402 −288.401

523.051 522.919 559.753 564.083 564.082 564.082 564.082 564.082 549.062 548.953 561.579 564.529 564.529 564.529 564.530 564.530

Two components of anisotropic tensor computed in its principal axis system are given.

description, but negligibly by s and d shell decontraction and addition of tight primitives. Thus, decontraction of s and p part of basis set and addition of two tight s primitives is sufficient to obtain reliable convergent anisotropic and isotropic hfccs. The choice of the exchange−correlation functional does not affect the convergence. Modified triple-ζ and quadruple-ζ basis converge toward similar hfccs values. Quintuple-ζ basis gives slightly different value for isotropic constant (Table 2). According to these results, at least triple-ζ quality basis is required to obtain reliable values of hyperfine parameters. It should be noted that aug-cc-pCVXZ sets were also examined, but results are not presented here. However, the main feature is that after modification they converge to close values of hfccs as corresponding aug-cc-pVXZ sets. In all DFT calculations, large ((27s 18p 14d 4f 3g 2h)/ [27s 18p 6d 4f 3g 2h] + 2s) set derived from aug-cc-pV5Z set by decontraction s and p part and addition of two tight s primitives is used. In the case of computationally demanding CCSD calculations the set ((21s 14p 10d 2f)/[21s 14p 4d 2f] + 2s) derived from aug-cc-pVTZ set is used. However, post-HF methods are sensitive to incompleteness of basis set. Diffuse and polarization functions from neighboring atoms can also contribute to spin density at nucleus. In the case of hydrogen and oxygen atoms EPR-III33 basis sets are used, because they are saturated in core region and augmented with diffuse and polarization functions.

Standard Gaussian sets have two drawbacks concerning calculation of isotropic hfccs. The first is related to the shape of spin density distribution in the vicinity of nucleustheir s functions have zero slope at the position of nucleus, being the reason that they are not able to reproduce the wave function cusp at the point of nucleus. The amplitude of the wave function, and consequently the spin density, at this point are underestimated. To overcome this, basis sets should be modified by introducing the additional flexibility in the core part, i.e., by addition of very tight s primitives.26,27 Another issue is that their contraction coefficients and exponents are optimized for energy calculations and they have flexibility only in the valence region. Description of core correlation, which affects hyperfine interactions, requires further improvement of the core part of basis set. Therefore, it is necessary to use large sets suitable for the core and valence region, which makes treatment of larger molecules with highly correlated methods difficult. As the starting basis set two Dunning’s correlation-consistent basis sets are used: aug-cc-pVXZ,28 and aug-cc-VXZ-DK29 (X = D, T, Q, and 5). Basis sets are overtaken from EMSL Basis Set Library.30 The first set is optimized for nonrelativistic, and the second for calculations with scalar-relativistic DKH Hamiltonian. The first part of basis set modification is its stepwise decontraction; s, p, and d shells are gradually decontracted. Additionally, the obtained sets are progressively augmented with tight (large exponent) s functions, to achieve the convergence of hyperfine parameters. Exponents for the additional tight functions are chosen such that they form geometric progression with the two tightest functions of the set, i.e., with the ratio of exponents equal to quotient of the exponents of the two tightest functions (it is approximately constant through the series of examined bases (≈6.7)). Such choice of the exponents provides fast convergence of hyperfine parameters.31,32 The effects of stepwise decontraction and addition of tight primitives to aug-cc-pVXZ sets on calculated arsenic hyperfine parameters in AsH2 obtained using B3LYP functional are shown in Table 2. The results for AsO2 and H2AsO, which follow the similar trend, are omitted for the sake of brevity. When isotropic parameters are considered, decontraction of the s part of basis set followed by the addition of tight s primitives has the largest effect. Negligible contributions come from decontraction of p and d shell. Anisotropic parameters are highly affected by p shell

4. RESULTS AND DISCUSSION 4.1. AsH2. Electronic configuration of the ground X2B1 electronic state of AsH2 is [core] 8a12 4b22 9a12 4b11. The SOMO (4b1), which is perpendicular to the molecular plane (yz-plane of the space-fixed coordinate system) is localized around the arsenic, and consists of arsenic 4px orbital. Such composition indicates that contact density is exclusively induced by spin polarization, which makes description of exchange−correlation crucial for correct treatment of hyperfine interactions. Nonrelativistic results for hyperfine constants obtained using different density functionals are given in Table 3. To facilitate comparison with the available experimental results, the anisotropic tensor is defined in the same way as in the studies of AsH2 radical rotational spectrum,16,34 in the principal axis 8626



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Table 3. Values of Arsenic Hyperfine Parametersa of AsH2, H2AsO, and AsO2 (in MHz) Obtained with Different Functionals and Experimental Values AsH2

BPW91

BP86

BLYP

Aiso

−74.5 −76.3 (−307.0) −270.9 −269.6 (−293.9) −283.3 −281.8 (−306.5) 554.1 551.4 (600.3) 0.7538 (0.7537) BPW91

−51.1 −52.8 (−229.8) −271.7 −270.4 (−294.7) −282.8 −281.4 (−306.0) 554.6 551.9 (600.7) 0.7532 (0.7531) BP86

−28.8 −30.1 (−148.6) −276.1 −274.9 (−299.2) −285.3 −284.0 (−308.5) 561.4 558.9 (607.7) 0.7524 (0.7523) BLYP

Taa

Tbb

Tcc

⟨S2⟩ H2AsO Aiso

Tx

Ty

Tz

⟨S2⟩ AsO2 Aiso Tx Ty Tz ⟨S2⟩

714.8 731.3 (2222.5) −105.1 −105.7 (−111.2) −101.9 −102.7 (−107.2) 207.0 208.4 (218.4) 0.7535 (0.7535) BPW91

726.4 742.9 (2257.9) −105.5 −106.0 (−110.9) −102.6 −103.4 (−107.3) 208.1 209.4 (218.2) 0.7532 (0.7532) BP86

915.7 815.4 −73.7 −77.5 137.6 144.4 −63.9 −66.9 0.7526

928.2 827.2 −72.3 −76.3 136.1 143.1 −63.7 −66.8 0.7524

746.4 763.7 (2321.5) −104.0 −104.4 (−109.8) −101.7 −102.5 (−106.8) 205.7 207.0 (216.6) 0.7532 (0.7532) BLYP 936.1 832.1 −69.9 −74.1 133.8 140.9 −63.8 −66.8 0.7524

B3PW91 −17.4 −18.9 (−144.3) −270.6 −269.2 (−294.0) −285.7 −284.2 (−309.3) 556.2 553.5 (603.3) 0.7546 (0.7544) B3PW91 669.2 688.4 (2065.1) −109.5 −110.7 (−116.0) −97.5 −98.7 (−102.1) 207.0 209.4 (218.6) 0.7580 (0.7579) B3PW91 905.2 798.0 −99.9 −103.0 164.6 170.8 −64.7 −67.8 0.7559

B3P86

B3LYP

−5.3 −6.8 (−104.0) −272.6 −271.2 (−296.0) −286.8 −285.2 (−310.6) 559.4 556.6 (606.6) 0.7542 (0.7540) B3P86

14.4 13.2 (−32.2) −275.5 −274.2 (−298.9) −287.6 −286.2 (−311.2) 563.0 560.4 (610.2) 0.7533 (0.7531) B3LYP

683.4 702.7 (2107.9) −110.2 −111.3 (−116.6) −98.6 −99.9 (−103.2) 208.8 211.2 (219.8) 0.7576 (0.7576) B3P86

693.7 713.4 (2142.6) −108.4 −109.5 (−114.7) −97.3 −98.5 (−101.8) 205.7 208.0 (216.5) 0.7575 (0.7575) B3LYP

924.2 815.2 −99.1 −102.7 164.0 170.6 −64.9 −67.9 0.7557

921.9 811.2 −96.5 −100.3 161.1 167.8 −64.6 −67.5 0.7557

BHPW91 57.7 56.4 (65.6) −266.4 −265.0 (−289.9) −287.9 −286.4 (−312.0) 554.4 551.4 (602.0) 0.7572 (0.7568) BHPW91 527.8 551.3 (1603.3) −110.9 −113.3 (−116.9) −82.3 −84.5 (−84.8) 193.2 197.8 (201.7) 0.7695 (0.7690) BHPW91 858.9 740.8 −138.3 −141.3 205.4 211.1 −67.1 −69.8 0.7644

BHP86 76.8 75.5 (129.5) −267.6 −266.2 (−291.2) −287.4 −285.8 (−311.4) 555.0 552.0 (602.6) 0.7564 (0.7560) BHP86

BHLYP 90.3 89.2 (181.9) −272.6 −271.1 (−296.3) −290.0 −288.5 (−314.0) 562.6 559.6 (610.3) 0.7552 (0.7549) BHLYP

exp 57.823b 57.820c −287.698b −287.694c −322.157b −322.150c 609.855b 609.844c

exp

542.8 566.2 (1650.1) −110.6 −112.9 (−116.7) −80.9 −85.0 (−85.6) 193.6 197.9 (202.3) 0.7687 (0.7682) BHP86

551.7 575.2 (1680.4) −108.5 −110.7 (−114.4) −81.7 −83.7 (−84.2) 190.2 194.4 (198.6) 0.7681 (0.7677) BHLYP

727.8d

873.3 754.6 −136.8 −140.0 203.3 209.2 −66.5 −69.2 0.7639

870.2 747.7 −135.4 −138.1 201.6 207.0 −66.2 −68.9 0.7641

937e

exp

−130e 171e −42e

The first value is computed at experimental and the second one at CCSD optimized geometry in the case of AsH2 and H2AsO. In the case of AsO2 the first value is computed at MP2, and the second at CCSD optimized geometry. DKH2 values are given in parentheses bReference 16. cReference 34. dReference 17. eReference 18. a

system of the moment of inertia. The values of hyperfine parameters at experimental and optimized geometries are very similar. The components of anisotropic hyperfine tensor depend slightly on the exchange−correlation functional (Tcc ≈ 560 MHz). Weak dependence on the exchange part is caused by the fact that spin polarization has miner role in anisotropic coupling, and that it is induced directly by spin density of the SOMO. Comparison with the experimental values (Tcc = 609.855 MHz16) shows that the anisotropic hfccs are underestimated, which could be explained by significant relativistic effects (section 4.4.). Gradient corrected Becke’s and hybrid B3 functional underestimate spin polarization and give negative values of arsenic isotropic hfccs. Reliable values are obtained with further inclusion of exact exchange, i.e., with the BH exchange functional.

The choice of correlation functional significantly affects isotropic hfccs. PW91 gradient-corrected functional is capable of producing more accurate results than widely used LYP correlation functional. The LYP functional, which treats correlation using a form fitted to the wave function of the helium atom, is incapable of describing important correlation effects in multielectron system. The best agreement with experimental value of isotropic hfcc (57.823 MHz16) is obtained with BHPW91 functional (57.7 MHz). Nonrelativistic orbital contributions to spin density at the position of arsenic obtained with unrestricted DFT methods using different functionals are examined (Table 4). Molecular orbitals of a1 symmetry composed predominantly of arsenic s orbitals have the largest contributions to contact density. For a given exchange functional, contact densities obtained with different correlation functionals change such as 8627



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Table 4. Orbital Contributions to Spin Density at Arsenic Position (in MHz) in AsH2a

a

MO

BLYP

BPW91

BP86

B3PW91

BHPW91

DKH2/BHPW91

UHF

CCSD′

CCSD″

1a1 2a1 4a1 5a1 core 8a1 9a1 val. 4b1 total

−32.59 10.07 9.88 0.14 −12.48 230.39 −246.71 −16.32 0.00 −28.8

−35.58 15.09 1.97 0.16 −18.36 236.07 −292.16 −56.14 0.00 −74.5

−34.31 15.09 1.97 0.16 −17.09 232.41 −266.41 −34.01 0.00 −51.1

−37.74 10.09 17.79 0.16 −9.70 333.68 −341.41 −7.70 0.00 −17.4

−45.60 10.07 45.27 0.16 9.90 459.47 −411.74 47.81 0.00 57.7

−156.24 44.51 146.86 6.14 41.27 1284.72 −1260.72 24.33 0.00 65.6

−63.87 20.18 88.39 0.19 44.89 698.59 −537.16 161.61 0.00 206.5

−48.07 12.60 71.98 0.16 36.67 581.56 −446.91 134.77 0.00 171.4

−46.90 15.43 64.60 0.15 33.27 533.51 −409.10 124.51 0.00 157.78

Only the largest contributions to spin density are presented.

ρ(PW91) < ρ(P86) < ρ(LYP). The treatment of electron correlation via correlation functional affects largely the valenceshell, particularly 9a1 orbital polarization (Table 4). The choice of exchange functional affects both core- and valence-shell polarization; for a given correlation functional contact densities increase in the order B, B3, and BH. It is well-known that GGA functionals underestimate, and UHF overestimates polarization. Underestimation of spin polarization with pure GGA functionals is compensated by its overestimation with UHF in hybrid functionals; i.e., isotropic hfccs increase upon inclusion of the exact exchange. According to the result obtained with BHPW91 functional, which is in agreement with the experimental one, the 1a1, 2a1, and 4a1 orbitals show the largest polarization among the core−shell orbitals (Table 4). The 1a1 orbital, which is composed of the 1s arsenic orbital, is negatively polarized, and 2a1 and 4a1 orbitals, which consist respectively of arsenic 2s and 3s orbitals, have the large positive contribution. The overall polarization of the core− shell is positive, contrary to expectations. Valence-shell polarization is dominant; the 8a1 orbital has large positive polarization (459.5 MHz). It is composed primarily of arsenic 4s and hydrogen 1s orbital with small admixture of arsenic 4pz and 5dz2 orbitals. Oppositely, the 9a1 orbital has large negative polarization (−411.7 MHz). Besides the arsenic 4s and hydrogen 1s orbital, it has large contribution of the arsenic 4pz orbital. The overall positive contact density is a consequence of large positive spin polarization of 8a1 orbital which surpasses negative contribution of 9a1 orbital, and smaller positive core−shell polarization. Radial density distribution of the individual MOs with significant spin polarization is investigated along the in-plane symmetry axis (Figure 1a). All the mentioned orbitals have maxima located closer to the nucleus than the SOMO. On the basis of this analysis, it would be expected that they have negative contribution to contact density (because the SOMO is α). However, as already stated, only the 1a1 and 9a1 orbitals show negative polarization, whereas the rest of them are positively polarized. The dependence of the differences between α- and β-spin density for 8a1 and 9a1 orbital on the distance from the nucleus along in-plane symmetry axis is shown in Figure 1b. Exchange interaction between these orbitals and the SOMO dominates in valence region, inducing approaching of their α-components toward SOMO’s radial density maximum, i.e., their decontraction. However, in the space closer to the nucleus, the exchange interaction with the SOMO decreases, and the 8a1 orbital is polarized oppositely. This illustrates that the classical model, which relies solely on exchange interaction between the

Figure 1. (a) ROHF radial density distributions of the SOMO and orbitals prone to spin polarization in AsH2. (b) Differences between α and β radial wave function squares in the cases of 8a1 and 9a1 orbitals. In both cases the direction of the in-plane symmetry axis of molecule (z-axis) is chosen.

unpaired electron and same-spin electrons, is inapplicable for molecules containing heavier nuclei, and that other types of interactions should be taken into account. Spin polarization induces changes in Coulomb, exchange, and nuclear attraction integrals. To explain the principles of spin polarization in AsH2, the exchange and Coulomb integrals between the SOMO and molecular orbitals prone to polarization, and nuclear attraction integrals are computed.3 The changes in the exchange and Coulomb interactions between doubly occupied orbitals are not examined here. In the first case the integrals are computed using ROHF orbitals, and in the second case with UHF orbitals. The grid of 0.2 au over a cube of 8628



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Table 5. Coulomb and Exchange Integrals between the SOMO and Occupied Orbitals, and Nuclear Attraction Integrals (in au) Coulomb integrals Ψ

1a1

⟨4b1rΨr|4b1rΨr⟩ ⟨4b1rΨαu |4b1rΨαu ⟩ ⟨4bα1uΨαu |4bα1uΨαu ⟩ ⟨4bα1uΨβu|4bα1uΨβu⟩

0.002787 0.002771 0.002799 0.002799

Ψ

1a1

⟨4b1rΨr|Ψr4b1r⟩ = Erex ⟨4b1rΨαu |Ψαu 4b1r⟩ ⟨4bα1uΨαu |Ψαu 4bα1u⟩ = Eαex Eαex − Erex

0.000000 0.000000 0.000000 0.000000

2a1 0.412235 0.412121 0.416317 0.416317 exchange integrals 2a1

4a1

8a1

9a1

0.408116 0.408166 0.412908 0.412908

0.354136 0.362394 0.367314 0.367314

0.309204 0.309527 0.312576 0.312576

4a1

0.000178 −0.000602 0.000184 −0.000638 0.000199 −0.000684 0.000021 −0.000082 nuclear attraction integrals

8a1

9a1

0.216405 0.238496 0.244499 0.028094

0.079469 0.073137 0.072038 −0.007431

Ψ

1a1

2a1

4a1

8a1

9a1

⟨Ψr|Z/r|Ψr⟩ = Ernuc ⟨Ψαu |Z/r|Ψαu ⟩ = Eαnuc ⟨Ψβu|Z/r|Ψβu⟩ = Eβnuc 2Ernuc − (Eαnuc + Eβnuc)

0.718504 0.714394 0.714416 0.008198

103.497 103.441 103.441 0.112000

60.1795 60.0869 60.0980 0.1741

18.5401 18.8666 19.1266 −0.9130

14.9396 15.0454 15.0090 −0.1752

edge length 20.6 au (arsenic is located in the center) is employed in the integration (obitals are extracted using Gaussian Cubegen option). Comparing the results in spin-restricted and unrestricted cases (Table 5), it can be concluded that exchange interaction between the 8a1 orbital and the SOMO increases upon spin relaxation, contributing to large energy gain (0.028094 Ha). The 1a1 orbital is very contracted toward nucleus, and its exchange interaction with the SOMO is negligible. Exchange interactions between other orbitals (except 2a1) and the SOMO decrease upon relaxation. Coulomb integrals between the SOMO and the considered orbitals slightly increase upon spin polarization, partly reducing large energy gain from exchange interaction. The largest contribution to energy gain upon spin polarization comes from enhanced nuclear attraction of 8a1 (0.9130 Ha) and 9a1 (0.1752 Ha) orbital. Because the exchange interaction with the SOMO intensely stabilizes the 8a1 orbital, it would be expected according to classical description that it has a negative contact density, but it is positively polarized. On the other hand, the 9a1 orbital is positively polarized, even though it is destabilized by exchange interaction with the SOMO (Figure 1b). The 1a1 orbital is polarized negatively, although its exchange interaction with the SOMO is negligible. The 8a1 and 9a1 orbitals polarized in this way are stabilized by a large energy gain from their enhanced nuclear attraction. According to this description, polarization of the 1a1, 8a1, and 9a1 orbitals in the core region cannot be explained only by their interaction with the SOMO, but exchange interactions with other doubly occupied orbitals should also be considered. At this point it is interesting to compare spin polarization mechanism in AsH2 with corresponding mechanisms in NH2 and PH2. Hyperfine couplings in the NH2 radical are studied in detail in several publications.8,35 The SOMO (1b1), which consists primarily of the nitrogen 2px orbital, induces significant polarization of 1a1, 2a1, and 3a1 orbitals. The orbital ordering according to radial density maxima positions, observed from the nucleus, is 1a1, 1b1, 2a1, and 3a1. Hence, it is expected that the unpaired α-electron attracts the α-density of the 2a1 and 3a1 orbitals toward the nucleus (positive polarization) and the α-density of 1a1 toward the valence region (negative polarization).

Table 6. Orbital Contributions to Nitrogen Isotropic hfcc in NH2 (in MHz) and to Spin Density at Phosphorous in PH2 (in au) NH2 PH2 a

MO Aisoa MO ρα−β

1a1 −51 1a1 −0.0690

2a1 60 2a1 −0.0473

3a1 12 4a1 0.2886

5a1 −0.1790

Reference 8.

The authors confirmed applicability of classical description by examining orbital contributions to contact density (Table 6). Similar analysis for PH2 molecule is carried out for the needs of the present study. The SOMO (2b1) is composed of the phosphorus 3px orbital and polarizes mostly the 1a1, 2a1, 4a1, and 5a1 orbitals. The SOMO occupies the same region of space as the valence 4a1 and 5a1 orbitals, and its radial density maximum is located on the furthest distance from nucleus compared to other orbitals. Individual MO contributions to contact density are given in Table 6. According to the classical model, it is expected that contributions of mentioned orbitals to contact density are negative, but the 4a1 orbital has a positive contribution. Similar to the case of AsH2, exchange interaction with the SOMO is not the main driving force of the isotropic hyperfine interactions. 4.2. H2AsO and AsO2. The H2AsO radical is a pyramidal molecule of Cs symmetry. The electronic configuration of the X2A′ state is [core] 12a′2 13a′2 5a″2 14a′2 15a′2 6a″2 16a′1. The SOMO is delocalized; the largest part of spin density is located around oxygen (≈50%) and arsenic (≈38%). It consists of oxygen 2p orbitals (lying in the plane parallel to HAsH plane), 4s, and in-plane (HAsH) arsenic 4p orbitals, and of hydrogen 1s orbitals. The most important mechanism of arsenic isotropic hyperfine interaction is the direct effect of unpaired electron density (Table 7), due to dominant contribution of arsenic 4s orbital to the SOMO. The effect of spin polarization is secondary, because of the small contribution of arsenic 4p orbitals to the SOMO. Because spin polarization has a minor effect on contact density, a relatively low dependence of arsenic isotropic hfccs on exchange functional is expected. However, isotropic hfccs decrease when exact exchange is included, especially in the case of BH functional (Table 3). Because the UHF drastically 8629



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Table 7. Orbital Contributions to Spin Densities (in au) at Arsenic in H2AsO and AsO2 H2AsO

BLYP

BPW91

B3PW91

BHPW91

2a′ 6a′ 2a″ core 12a′ 13a′ 5a″ 14a′ 15a′ 6a″ val. 16a′ total AsO2

0.00 8.00 × 103 1.66 × 10−3 9.66 × 10−3 −2.64 × 10−1 3.03 × 10−1 5.87 × 10−3 −8.82 × 10−3 −3.66 × 10−1 −2.93 × 10−4 −3.30 × 10−1 1.30 × 10+0 9.79 × 10−1 BLYP

1.00 × 10−2 2.00 × 10−3 1.39 × 10−3 1.34 × 10−2 −2.50 × 10−1 3.15 × 10−1 6.39 × 10−3 −2.31 × 10−4 −3.71 × 10−1 −2.03 × 10−4 −3.00 × 10−1 1.23 × 10+0 9.43 × 10−1 BPW91

1.00 × 10−2 1.80 × 10−2 1.27 × 10−3 2.93 × 10−2 −3.23 × 10−1 4.47 × 10−1 1.48 × 10−2 −1.55 × 10−2 −6.44 × 10−1 −1.61 × 10−3 −5.22 × 10−1 1.34 × 10+0 8.47 × 10−1 B3PW91

1.00 × 10−2 3.60 × 10−2 1.52 × 10−3 4.75 × 10−2 −4.23 × 10−1 5.71 × 10−1 1.62 × 10−1 −1.33 × 10−1 6.36 × 10−3 −1.00 × 10+0 −8.17 × 10−1 1.47 × 10+0 7.01 × 10−1 BHPW91

−4.00 × 10−2 4.50 × 10−2 5.00 × 10−3 −1.55 × 10−2 −3.30 × 10−1 −1.95 × 10−1 −5.40 × 10−1 9.32 × 10+0 8.78 × 10+0

1.00 × 10−2 1.59 × 10−1 1.69 × 10−1 −7.03 × 10−2 −3.10 × 10−3 −2.42 × 10−1 −3.15 × 10−1 8.87 × 10+0 8.72 × 10+0

6.00 × 10−2 3.32 × 10−1 3.92 × 10−1 −1.36 × 10−1 3.82 × 10−1 −2.77 × 10−1 −3.10 × 10−2 8.19 × 10+0 8.55 × 10+1

−3.00 × 10−2 5.50 × 10−2 2.50 × 10−2 −2.23 × 10−2 −4.02 × 10−1 −1.91 × 10−1 −6.15 × 10−1 9.46 × 10+0 8.87 × 10+0

2a1 5a1 core 9a1 10a1 11a1 val. 12a1 total

underestimates isotropic hfcc (Aiso = 174 MHz, Table 8), an admixture of the exact exchange significantly diminishes the value of Aiso compared to values obtained with GGA Becke’s exchange functional. Also, the spin contamination increases going from the B to BH functional, which might cause additional deterioration of the results. Therefore, Becke’s exchange functional is the most satisfying choice. Correlation functionals do not affect the results to a large extent, still P86 gives Aiso (726.4 MHz), which agrees best with the experimental one (727.8 MHz17). Because the isotropic hfccs are sensitive to molecular geometry in this case, those obtained at experimental geometry are used as more reliable. The AsO2 molecule has C2v symmetry. The electronic configuration of the X2A1 state is [core] 9a12 3b12 10a12 6b22 4b12 11a12 2a22 7b22 12a11. The SOMO is delocalized; the largest part of spin density is localized around oxygen atoms (≈46%), and the smaller part around arsenic (≈16%). The contribution of oxygen 2p orbitals to the SOMO is the largest, but also there is a significant contribution of arsenic 4s and out-of-plane 4p orbitals. The mechanism of arsenic hyperfine interactions is similar to the case of H2AsO. Isotropic hfccs also decrease as the exact exchange is included for the same reason as in the case of H2AsO. However, this effect is not so pronounced because UHF does not underestimate Aiso to such a large extent. Becke’s GGA functional combined with LYP functional gives Aiso (936.1 MHz), which agrees the most with the experimental value (937 MHz18). We discuss the results obtained at MP2 optimized geometry, because they are in better agreement with the experimental ones. The values obtained at CCSD geometry deviate approximately 11−16% from those at MP2 geometry. The components of the anisotropic tensor of AsO2 are given in principal axis system of anisotropic hyperfine tensor (Table 3) to compare it to the values from the ESR study18 where they are defined in the same manner. In this case applied DFT methods give results that deviate from the experimental values. The anisotropic hyperfine tensor is also defined in its principal axis

Table 8. Values of Arsenic Hyperfine Parameters (in MHz) in AsH2, H2AsO, and AsO2 Obtained with UHF, MP2, and CCSD Methodsa AsH2

UHF

MP2

Aiso

205.9 206.5 −250.8 −249.6 −286.8 −286.1 536.9 535.7 0.7659 UHF

99.2 97.6 −258.4 −258.3 −264.6 −264.3 523.0 522.7 0.7532 MP2

Taa Tbb Tcc ⟨S2⟩ H2AsO Aiso Tx Ty Tz ⟨S2⟩ AsO2 Aiso Tx Ty Tz ⟨S2⟩

174.5 193.0 −86.9 −89.3 −34.3 −35.1 121.2 124.4 0.7858 UHF 692.0 549.6 −192.1 −194.2 266.8 270.9 −74.8 −76.7 0.7868

595.4 627.4 −135.1 −140.0 −98.7 −102.5 233.8 242.5 0.7656 MP2 1108.2 1026.8 −100.3 −101.9 152.2 160.1 −51.9 −58.2 0.7625

CCSD 171.4 170.9 −252.6 −252.6 −266.2 −266.1 518.8 518.7 0.7524 CCSD 654.7 677.4 −109.3 −110.8 −91.7 −93.2 201.0 204.1 0.7532 CCSD

expb 57.823 −287.698 −322.157 609.855

expb 727.8

expb 937 −130 171 −42

a

The values in cells are given at two geometries in the manner defined for Table 3. bReferences are given in Table 3. 8630



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parameters obtained with UHF and MP2 (and CCSD) methods. In the case of H2AsO experimental values for anisotropic parameters are not available, but MP2 and CCSD values are in accordance with DFT results (Table 3). Anisotropic parameters in AsO2 proved to be hard to reproduce, because both ab initio and DFT results differ from the experimental ones, which could be a consequence of multireference character of the ground state wave function and relativistic effects. The effect of underestimation of inner-shell spin polarization by DFT methods might be present here, but it is covered by dominant contribution of the unpaired electron to contact density. The treatment of correlation affects significantly isotropic hfccs in the studied cases. Correlated motion in core−shell partly decreases the repulsion between opposite-spin electrons, leading to the contraction of spin density in this region. This leads to expansion of valence-shell and unpaired electron density, because nuclear charge becomes more screened by core electrons. In this way isotropic hfccs increase upon improving of correlation description, i.e., going from MP2 to CCSD results (Table 8). 4.4. Relativistic Effects. Arsenic hyperfine coupling constants in AsH2 and H2AsO obtained in scalar-relativistic DKH2 calculations using untransformed density operator (due to software facilities) are presented in parentheses in Table 3. In the case of H2AsO, unrealistically large values of isotropic hfccs are obtained. On the other hand, in the case of AsH2 isotropic hfcc computed with BHPW91 functional (65.6 MHz) is still reasonable comparing to the experimental value (57.823 MHz16), but the corresponding nonrelativistic result is more reliable. Scalar-relativistic anisotropic hfccs in AsH2 (Tcc = 602.0 MHz) are in much better agreement with the experimental value (Tcc = 609.855 MHz16) comparing to the nonrelativistic counterparts (Tcc = 554.4 MHz). The important feature of relativistic effects is the contraction of core s and p1/2 orbitals, and consequently, the orbitals composed of them, toward heavy nucleus. The SOMO composed of the arsenic 4px orbital becomes also contracted upon relativistic effects. The consequences of this are changes in spin density distribution. The difference between DKH2 and nonrelativistic spin densities in molecular plane of AsH2 (obtained with BHPW91) is presented in the Figure 2.

system in the case of H2AsO, but experimental vales are not available here. The absolute values of the anisotropic tensor components in H2AsO decrease, and in the case of AsO2 increase upon inclusion of the exact exchange. This can also be explained by the difference in results obtained with the pure GGA functionals and with UHF method (Table 8), i.e., going from B to BH functional, values of anisotropic tensor become closer to those obtained with UHF method. Inclusion of the exact exchange has two competing effects in the case of H2AsO. First, the composition of the SOMO changes: contribution of 3d orbitals decreases, whereas contribution of 4s and 4pz orbitals increases going from the B to BH functional. As the consequence of its increased 4s character, the orbital contribution of the SOMO to contact density increases. The second effect is significant enhancement of negative valence-shell spin polarization, which diminishes the overall contact density (Table 8). The latter effect is more pronounced when compared to the former, resulting in a lowering of isotropic hfccs. 4.3. Post-Hartree−Fock Results. The results for hyperfine coupling constants calculated with the UHF, MP2, and CCSD methods are given in Table 8. Arsenic anisotropic parameters in the case of AsH2 obtained with MP2 and CCSD methods are close to those obtained with the appropriate DFT method and do not deviate much from corresponding UHF values. This indicates that its ground state is dominated by one electronic configuration. There is a large discrepancy between CCSD (171.4 MHz at experimental geometry) and the experimental (57.823 MHz) value of isotropic hfcc. This could be a consequence of incompleteness of the basis set, which manifests in underestimation of correlation effects. This finding is in accordance with the early results by Sekino and Bartlett, who confirmed the sensitivity of hfccs obtained with finite-field coupled cluster (CC) method on the basis set quality in several radicals.36 Perera et al. came to similar findings in computations of isotropic hfccs of the first-row atoms by means of analytical method with relaxed CC densities, using a variety of extended sets.37 Convergent CCSD computations of hyperfine parameters can serve as benchmark computations for examination of the reliability of DFT results. In this case BHPW91 gives the correct value of Aiso, but because only the overall spin densities are measurable, it cannot be affirmed for sure whether the individual DFT contributions are correct. There are examples where DFT methods underestimate polarization of oppositely polarized orbitals, and as a consequence of the error cancellation, they give correct isotropic hfccs.38 Isotropic hfccs and orbital contributions to contact density obtained using the CCSD method with sets derived from aug-cc-pVTZ (CCSD′), and aug-cc-pVQZ basis (CCSD″) are given in Table 4. Overestimated isotropic hfccs are obtained even with the larger basis set (157.8 MHz). The polarization of the positively polarized 4a1 and 8a1 orbitals, and negatively polarized 9a1 orbital decreases with the set enlargement, but convergence has not been reached with used sets. Larger sets exceeded available computational resources. For this reason, it was not possible to examine the reliability of DFT orbital contributions to the contact density by their comparison with CCSD results. However, the success of the BHPW91 functional is the result of compensation of underestimation of spin polarization by GGA with its overestimation by UHF, which could also be considered as error cancellation.38 Ab initio results in the cases of H2AsO and AsO2 are characterized by large spin contamination even with the MP2 method (Table 8), which induces huge divergence between anisotropic

Figure 2. Contour diagram of difference between DKH2 and nonrelativistic spin density in molecular plane. Values of positive and negative differences are indicated by solid and dashed lines, respectively. Contour lines are shown at the following values of difference densities: ±10−9, ±10−8, ±10−7, ±10−6 and ±10−5. As the periphery is approached, differences decrease. 8631



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Compared to the nonrelativistic case, the excess of α-density in the vicinity of arsenic (positive difference densities), and the excess of β-density around hydrogen nuclei (negative difference densities) is located in the scalar-relativistic case. Relativistic effects, hence induce shifting of α-density toward arsenic (heavier nucleus), and intensify spin polarization. This is confirmed by examining orbital contributions to arsenic contact density in AsH2 obtained with the DKH2/BHPW91 method (Table 4). Polarization of core and valence orbitals of a1 symmetry is enhanced under the influence of relativistic effects. 4.5. Basis Sets of Adjacent Atoms. The choice of basis set of the neighboring atoms (H, O) is very important, because arsenic contact density can be induced by diffuse and polarization functions of adjacent atoms. Although this effect is often considered as secondary, it significantly influences the results. When EPR-II is used instead of the EPR-III basis set for neighboring atoms, arsenic hyperfine parameters are significantly underestimated (results are not given).

stabilization induced by their enhanced attraction with the nucleus.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Telephone: +381 64 3815137. Fax: +381 11 2187133. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work has been financialy supported by Ministry of Education and Science of Serbia (Contract No. 172040). The author thanks Professor Miljenko Perić for reading the manuscript and useful suggestions and her colleague Milan Senćanski for help concerning technical issues.

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5. CONCLUSIONS The performance of various density functional and postHartree−Fock approaches in calculation of arsenic hyperfine parameters in AsH2, H2AsO, and AsO2 radicals is investigated. The applicability of a particular method depends on mechanisms of hyperfine couplings in a concrete case. Isotropic hyperfine interactions in AsO2 and H2AsO are induced primarily by direct contribution of the SOMO, and by less pronounced spin polarization. Generalized gradient Becke’s exchange functional combined with PW91 and P86 correlation functionals gives reliable values of hfccs. Because UHF underestimates isotropic hyperfine parameters, functionals with the parts of the exact exchange also underestimate them. The main mechanism of isotropic hyperfine interactions in AsH2 is spin polarization. Functionals with half of the exact exchange are capable of producing satisfactory results for isotropic hfccs. Anisotropic hfccs are underestimated compared to experimental values, but improved values are obtained in scalar-relativistic calculations. Ab initio MP2 and CCSD methods have proved inappropriate for calculations of hyperfine parameters in given molecules. One of the reasons is their pronounced sensitivity to incompleteness of the basis set. On the other hand, they are computationally too demanding when large sets are used. In calculations with a smaller convergent set, dynamical correlation in AsH2 is underestimated, resulting in overestimated isotropic parameters. In the case of H2AsO, problematic spin contamination is not entirely removed even in CCSD calculations, inducing underestimated isotropic parameter. By inspecting molecular orbital contributions to contact density in AsH2, we conclude that positive valence-shell spin polarization is dominant. The polarization of the core−shell is also positive. The mechanism of spin polarization is complex, and it cannot be described by the classical model based on the exchange interactions between the unpaired electron and samespin electrons from occupied orbitals. This interaction is dominant only in the valence spatial region, where it determines the direction of polarization. In the region closer to the nucleus the polarization directions are mainly opposite to expectations. This indicates that exchange interactions between doubly occupied orbitals, which are not considered in the present study, should probably be taken into account to elucidate the mechanism of isotropic couplings. Described polarization of orbitals is favorable from the energy standpoint, because of large

REFERENCES

(1) Carrington, A. McLachlan, A. D. Introduction to Magnetic Resonance; Harper and Row: New York, 1967. (2) Munzarová, M.; Kaupp, M. J. Phys. Chem. A 1999, 103, 9966−9983. (3) Munzarová, M. L.; Kubácě k, P.; Kaupp, M. J. Am. Chem. Soc. 2000, 122, 11900−11913. (4) Weil, J.; Bolton, J. R.; Wertz, J. E. Electron Paramagnetic Resonance: Elementary Theory and Practical Applications; Wiley and Sons: New York, 1994. (5) Weltner, W. Magnetic Atoms and Molecules; Van Nostrand: New York, 1983. (6) Watson, R. E.; Freeman, A. J. In Hyperfine Interactions; Freeman, A. J., Frankel, R. B., Eds.; Academic Press: New York, 1967. (7) Watson, R. E.; Freeman, A. J. Phys. Rev. 1961, 123, 2027−2047. (8) Funken, K.; Engels, B.; Peyerimhoff, S. D.; Grein, F. Chem. Phys. Lett. 1990, 172, 180−186. (9) Ishii, N.; Shimizu, T. Phys. Rev. A 1993, 48, 1691−1694. (10) Engels, B.; Peyerimhoff, S. D. Mol. Phys. 1989, 67, 583−600. (11) Chipman, D. M. Theor. Chim. Acta 1992, 82, 93−115. (12) (a) Douglas, M.; Kroll, N. M. Ann. Phys. 1974, 82, 89−155. (b) Hess, B. A. Phys. Rev. A 1986, 33, 3742−3748. (13) (a) Reiher, M. Faraday Discuss 2007, 135, 97−124. (b) Eickerling, G.; Mastalerz, R.; Herz, H.-J.; Scherer, W.; Reiher, M. J. Chem. Theory Comput. 2007, 3, 2182−2197. (14) Mastalerz, R.; Lindh, R.; Reiher, M. Chem. Phys. Lett. 2008, 465, 157−164. (15) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; et al. Gaussian 09, Revision A.02; Gaussian, Inc.: Wallingford, CT, 2009. (16) Hughes, R. A.; Brown, J. M.; Evenson, K. M. J. Mol. Spectrosc. 2000, 200, 210−228. (17) He, S.-G.; Sunahori, F. X.; Yang, J.; Clouthier, D. J. J. Chem. Phys. 2009, 131, 114310 1−10. (18) Knight, L. B.; Jones, G. C.; King, G. M.; Babb, R. M.; McKinley, A. J. J. Chem. Phys. 1995, 103, 497−505. (19) Becke, A. D. Phys. Rev. A 1988, 38, 3098−3100. (20) Becke, A. D. J. Chem. Phys. 1993, 98, 5648−5652. (21) Becke, A. D. J. Chem. Phys. 1993, 98, 1372−1377. (22) (a) Lee, C.; Yang, W.; Parr, G. R. Phys. Rev. B 1988, 37, 785−789. (b) Miehlich, B.; Savin, A.; Stoll, H.; Preuss, H. Chem. Phys. Lett. 1989, 157, 200−206. (23) (a) Perdew, J. P. Phys. Rev. B 1986, 33, 8822−8824. (b) Perdew, J. P.; Wang, Y. Phys. Rev. B 1986, 34, 7406(E). (24) (a) Perdew, J. P. Physica B 1992, 172, 1−6. (b) Perdew, J. P.; Wang, Y. Phys. Rev. B 1992, 45, 13244−13249. (25) Chipman, D. M. In Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy; Langhoff, S. R., Ed.; Kluwer: Dordrecht, The Nederlands, 1995. 8632



Article

(26) Al Derzi, A. R.; Fau, S.; Bartlett, R. J. J. Phys. Chem. A 2003, 107, 6656−6667. (27) Fau, S.; Bartlett, R. J. J. Phys. Chem. A 2003, 107, 6648−6655. (28) Wilson, A. K.; Woon, D. E.; Peterson, A. K.; Dunning, T. H., Jr. J. Chem. Phys. 1999, 110, 7667−7676. (29) De Jong, W. A.; Harrison, R. J.; Dixon, D. A. J. Chem. Phys. 2001, 114, 48−53. (30) Schuchardt, K. L.; Didier, B. T.; Elsethagen, T.; Sun, L.; Gurumoorthi, V.; Chase, J.; Li, J.; Windus, T. L. J. Chem. Inf. Model. 2007, 47, 1045−1052. (31) Chipman, D. M. J. Chem. Phys. 1989, 91, 5455−5465. (32) Liang, L.; Rassolov, V. A. J. Phys. Chem C 2010, 114, 20648− 20658. (33) (a) Rega, N.; Cossi, M.; Barone, V. J. Chem. Phys. 1996, 105, 11060−11067. (b) Barone, V. In Recent Advances in Density Functional Methods, Part I; Chong, D. P., Ed.; World Scientific Publishing Co.: Singapore, 1996. (34) Fujiwara, H.; Kobayashi, K.; Ozeki, H.; Saito, S. J. Chem. Phys. 1998, 109, 5351−5355. (35) Engels, B.; Perić, M.; Reuter, W.; Peyerimhoff, S. D.; Grein, F. J. Chem. Phys. 1992, 96, 4526−4535. (36) Sekino, H.; Bartlett, R. J. J. Chem. Phys. 1985, 82, 4225−4229. (37) Perera, S. A.; Watts, J. D.; Bartlett, R. J. J. Chem. Phys. 1994, 100, 1425−1434. (38) Engels, B. In Calculation of NMR and EPR Parameters: Theory and Application; Kaupp, M., Bühl, M., Malkin, V. G., Eds.; Wiley-VCH: Weinheim, 2004.

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