Complete basis set limit ionization potentials of O3 and NO2 using the multiconfigurational spin tensor electron propagator method (MCSTEP)

Chemical Physics 238 Ž1998. 1–9

Complete basis set limit ionization potentials of O 3 and NO 2 using the multiconfigurational spin tensor electron propagator method žMCSTEP / Alexander J. McKellar a , Dodi Heryadi a , Danny L. Yeager a

a,)

, Jeffrey A. Nichols

b

Chemistry Department, Texas A & M UniÕersity, P.O. Box 300012, College Station, TX 77842-3012, USA b Pacific Northwest Laboratory, Battelle BouleÕard, P.O. Box 999, Richland, WA 99352, USA Received 29 May 1998

Abstract We have calculated low-lying principal vertical ionization potentials ŽIPs. of O 3 and NO 2 with the multiconfigurational spin tensor electron propagator method ŽMCSTEP. using several different basis sets. We obtain an estimate of complete basis set limit ŽCBS. MCSTEP IPs. This is the first time CBS estimates have been used with MCSTEP. We show that MCSTEP is accurate and reliable compared with experiment at the CBS limit for obtaining low-lying vertical IPs for open shell molecules such as NO 2 and highly correlated molecules such as O 3. Our results confirm previous assignments of photoelectron peaks based on calculations made using less accurate methods. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction Low-lying ionization potentials of small molecules are still of considerable interest both experimentally and theoretically. For example, in a recent lengthy paper Holland et al. presented new experiments and discussed in detail the photoabsorption, photodissociation and photoelectron spectroscopies of C 2 H 4 and C 2 D4 w1x. With MCSTEP, as with other electron propagator or single-particle Green’s function methods, low-lying vertical ionization potentials ŽIPs. and electron affinities ŽEAs. are calculated directly from an initial state w2,3x. This is in contrast to more traditional methods such as configuration interaction )

Corresponding author.

ŽCI. in which the initial and final state total energies are found, and the initial energy Ža large value. subtracted from the final energy Žanother large value. to determine the IP Ža relatively small value.. MCSTEP is applicable, accurate, and reliable for closed-shell systems with relatively small correlation effects, but was specifically formulated to accurately handle systems with initial states that are open-shell andror where non-dynamical correlation is important w4x. In this paper we determine the low-lying ionization potentials of O 3 and NO 2 . We determine the complete basis set ŽCBS. limit ionization potentials ŽIPs.. This is the first time CBS formulas have been used with MCSTEP. We examine several basis sets to study the effects of additional corerinner valence basis functions, diffuse basis functions, and the re-

0301-0104r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 2 8 6 - 9

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A.J. McKellar et al.r Chemical Physics 238 (1998) 1–9

moval of higher angular momentum functions on low-lying IPs determined with MCSTEP. Comparisons are made with experimentally determined vertical IPs w5,6x.

2. Theory 2.1. MCSTEP In this subsection some of the theory relevant to MCSTEP is very briefly discussed. The intent is not to be complete, but instead to be indicative of the methodology. For a more complete discussion interested readers are directed to the original papers on MCEP w7x and MCSTEP w4x. For a summary on MCEP and MCSTEP longer than given here interested readers should examine Ref. w8x. The poles of the single particle Green’s function are the ionization potentials and electron affinities of a system w2,3x. In general, these poles cannot be obtained exactly. Therefore, approximations have to be made. There are two principal approximations: for the initial or reference state and for the complete operator manifold that describes the ionization or attachment processes. The single particle Green’s function equations for closed shell atoms and molecules are traditionally solved by approximating the reference state by a single determinant Hartree–Fock state corrected by Møller–Plesset perturbation theory w9–24x. In these techniques operators are included in the operator manifold to assure the solution of the resulting equations correctly through a certain order in the electron–electron interaction. To obtain reliable ionization potentials for outer valence principal IPs for closed shell systems that have initial states well described by first order Møller–Plesset perturbation theory, it was found that for ionization potentials the equations needed to be solved at least through third order in the electron–electron interaction. Some higher order terms are sometimes required in the approximate Green’s function solution in order to obtain accurate IPs. Although these third-order and higher-order perturbative Green’s function methods have been very successful, they are somewhat limited in applicability. Perturbative approaches usually cannot handle

reliably Žor handle at all. systems with initial states that are open shell andror highly correlated Žnon-dynamical correlation. for either IPs or EAs. With MCSTEP these problems are solved by using a multiconfigurational reference state and explicitly coupling tensor ionization and attachment operators to a tensor initial state. For calculational simplicity an MCSCF reference state is usually used in MCSTEP, although in principle a CI state or other nonperturbatively correlated state could also be used. In the calculations reported here a complete active space MCSCF state is used as the reference state. The operator manifold chosen for MCSTEP includes simple electron addition and destruction operators, N y 1 transfer operators which remove an electron from the occupiedrpartially occupied orbitals and allow all possible rearrangements of the remaining electrons in the partially occupied space, and N q 1 transfer operators which add an electron to the unoccupiedrpartially unoccupied orbitals and allow all possible rearrangements of the remaining electrons in the partially occupied space w4,7x. MCSTEP IPs and EAs are obtained from the generalized eigenvalue equation MXf s v f NXf

Ž 1.

where Mr ,p s Ý Ž y1 .

S o y G yS fy g r

G

=W Ž gr gp So So ; G Sf . Ž 2 G q 1 .

1r2

G

5 NS0 :: Ž 2 . =²² NSo 5  hq r Ž gr . , H ,h p Ž gp . 4 and Nr ,p s Ý Ž y1 .

S o y G yS fy g r

G

=W Ž gr gp So So ; G Sf . Ž 2 G q 1 . G

1r2

5 NS0 :: =²² NSo 5  hq r Ž gr . ,h p Ž gp . 4

Ž 3.

v f is an IP or EA to the final ion tensor state < N " 1 Sf :: which has spin Sf . W is the usual Racah Ž . coefficient, h p Žgp . and hq r gr are tensor operator versions of members of the operator manifold with

A.J. McKellar et al.r Chemical Physics 238 (1998) 1–9

ranks gp and gr , respectively.  ,4 is the anticommutator

 A, B 4 s AB q BA

Ž 4.

and  ,,4 is the symmetric double anticommutator 1 2

1 2

 A, B,C 4 s  A, w B,C x 4 q  w A, B x ,C 4 .

3

We apply a 3-parameter exponential fitting function first described in Ref. w36x to estimate values at the complete basis set ŽCBS. limit: A Ž x . s A CBS q BeyC x

Ž 6.

Ž 5.

We have demonstrated for many atoms and molecules that MCSTEP is very accurate and reliable for lower-lying Žin energy. IPs w24–34x. These are the IPs corresponding to processes that are primarily simple electron removal Ži.e., the principal IPs.. MCSTEP IPs are not as accurate when important contributions from processes that are simple electron removalq excitation of the remaining electrons to diffuse orbitals are necessary for an accurate description. This is true regardless of the number of diffuse functions in the basis set since transfer type operators included in MCSTEP do not allow for excitations of electrons to unoccupied orbitals which may be diffuse. wThe orbitals used in MCSTEP are from a small CAS MCSCF for the initial Žneutral. state.x These IPs are usually higher in energy and may include most inner valence and core principal IPs as well as most of the shake-up IPs. It has previously been shown that these higher-lying IPs can be accurately determined by an extension of MCSTEP known as the repartitioned multiconfigurational spin tensor electron propagator ŽRMCSTEP. w4x. RMCSTEP is more complicated than MCSTEP and it is not used in the calculations reported in this paper. 2.2. Basis sets We use cc-pVXZ, aug-cc-pVXZ, and cc-pCVXZ basis sets where X s D,T,Q Ži.e., double, triple, quadruple. w35x. From now on for convenience we will drop the cc from the names of the basis sets. For all basis sets used for the calculations reported here we retained only the spherical components, i.e., five d-components, seven f-components, and nine g-components. In O 3 and NO 2 the pVDZ Ž²3s2p1d: on each nucleus. contains 42 basis functions, the pVTZ Ž²4s3p2d1f: on each nucleus. contains 90 basis functions, and the pVQZ Ž²5s4p3d2f1g: on each nucleus. contains 165 basis functions.

where x s 2, 3, and 4 for DZ, TZ, and QZ basis sets. In our case, the AŽ x . values are the MCSTEP IPs obtained using pVDZ, pVTZ and pVQZ basis sets. This is the first time we have used an estimate of the CBS limit with MCSTEP. For the aug-pVXZ basis sets, a diffuse function of each type of angular momentum present in the pVXZ is added w37x. For example, the aug-pVDZ basis set includes diffuse additional s, p, and d functions added to the standard pVDZ Ž²3s2p1d:. to give the Ž²4s3p2d:. aug-pVDZ basis set. The aug-pVXZ basis sets were formulated to describe the more diffuse character of anions, but these basis sets are also suitable for molecular systems in which long-range interactions are important. These basis sets were included in this study to explore the effects of including diffuse functions to aid in the description of the outer valence space on MCSTEP IP results. The pCVXZ basis sets were formulated to take into account core and core–valence electron correlation w38x. In this paper, we explore the effects of taking the core and core–valence effects into account with the MCSTEP method. The pCVXZ basis sets include additional fairly tight functions added to the corresponding pVXZ basis sets. The pCVDZ basis set doubles the core as well as the valence space and adds a tight polarization p function Žcore 1 s ™ 2 s1 p ., while for the pCVTZ basis set, the core 1 s is tripled and appropriate polarization functions added Žcore 1 s ™ 3s2 p1d .. For example, basis functions added to the pVDZ to form the pCVDZ are, for oxygen, an s function with an exponent of 8.215 and a p function with an exponent of 26.056.

2.3. Complete actiÕe space (CAS) choices We have previously discussed optimal CAS choices for the MCSCF reference state and MCSTEP calculations w32–34x. The CASs we use are relatively small and several calculations have demonstrated

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A.J. McKellar et al.r Chemical Physics 238 (1998) 1–9

that these choices give accurate low-lying principal ionization potentials. These CAS choices are ‘balanced’ in the sense that there are usually the same number of strongly occupied and weakly occupied partially occupied orbitals. With the SCF balanced choice, the three highest Lagrangian eigenvalue occupied and three lowest Lagrangian unoccupied eigenvalue SCF orbitals are chosen for the initial guess for the MCSCF reference state w32,34x. ŽNote that the Lagrangian eigenvalues are the SCF canonical orbital energies when the calculation is for a closed shell single determinant wavefunction.. Occasionally the SCF balanced choice results with one or more of the resulting MCSCF partially occupied orbitals having Lagrangian eigenvalues lower than some of the doubly occupied orbitals. That, of course, is not a desired result for a reference state to be used for outer IPs and leads to generally poorer quality MCSTEP IPs. When this situation arises, we choose either an SDCI balanced CAS or restrict rotations in the MCSCF. With the SDCI balanced CAS the occupation numbers of the natural orbitals obtained from a configuration interaction calculation involving all singles and doubles excitations from the SCF state are examined w34x. The symmetries of the three smallest occupation number strongly occupied orbitals and three largest occupation number weakly occupied orbitals are used to determine the orbitals in the MCSCF reference state CAS. Alternatively, by restricting MCSCF rotations the SCF balanced CAS can also be used in the reference MCSCF state even when the Lagrangian eigenvalues for the exact MCSCF state are in an undesired order w30x. The resulting stationary point is, of course, not the true MCSCF state but is only an approximation to it. This method works for MCSTEP only if no more than two or three MCSCF rotations are neglected and if the resulting approximate MCSCF stationary point has the desired Lagrangian eigenvalue order. Because only three strongly occupied orbitals are present in the MCSCF CAS used with MCSTEP, we expect that only the three outermost MCSTEP IPs will be accurately calculated. Higher-lying principal IPs that involve removal of an electron from orbitals that are not explicitly correlated in the MCSCF

reference state will not be determined as accurately. Furthermore, higher-lying MCSTEP IPs that require a significant mixing of shake-up processes involving excitation to diffuse orbitals will not be accurately predicted with MCSTEP. This is because there are no diffuse orbitals present in the MCSCF CAS orbitals.

3. Results and discussion 3.1. O3 The geometry we used in this calculation is the same as that used by Barysz et al. w39x, i.e., C 2v geometry with bond angle 117.48 and bond length 2.387 a.u. This is the experimental geometry. The ground state principal configuration of O 3 is 1a211b 22 2a21 3a21 2b 22 4a21 5a211b12 3b 22 4b 22 6a211a22 . Inclusion of the configuration 1a211b 22 2a21 3a21 2b 22 4a21 5a211b 12 3b 224b 22 6a21 2b12 is also necessary for adequately describing the neutral 1A1 ground state since it is necessary to properly describe the important contribution of biradical structure to the ground state w40–44x. Thus, perturbative Green’s function approaches Žwhich are based on a single determinant ground state corrected by perturbation theory. are not adequate for describing the IPs of O 3 w5,45x. However, since MCSTEP uses a multiconfigurational initial state, MCSTEP IPs should be accurate and reliable. The complete active space we used for these MCSTEP calculations was chosen by a method described previously in detail w32–34x — it is composed of the highest three Žin energy. occupied orbitals from the SCF calculation and the next three Žin energy. unoccupied orbitals regardless of symmetry, i.e., it is the SCF balanced CAS. This CAS has been previously demonstrated to give accurate and reliable MCSTEP IPs for several systems tested to date. Thus the CAS for these MCSTEP calculations is composed of all possible configurations of six electrons in the Ž4b 2 6a 11a 2 2b 17a 1 5b 2 . orbitals. For the neutral 1A1 ground state there are only 104 determinants. Note that this CAS choice along with the 5a 1 ,1b 1 , and 3b 2 orbitals represents all the orbitals that can be obtained from the 2p atomic or-

A.J. McKellar et al.r Chemical Physics 238 (1998) 1–9

bitals and that the CAS includes both the 1a211b 22 2a213a21 2b 22 4a21 5a211b12 3b 22 4b 22 6a211a22 and the 1a211b 22 2a213a21 2b 22 4a21 5a211b12 3b 22 4b 22 6a21 2b12 configurations Žas well as others. in the 1A1 ground state. We report in this subsection MCSTEP IPs to hundredths of eV in order to compare with experimental results for the vertical IPs which are reported to this accuracy w5x. From the observed photoelectron spectroscopy peaks it is somewhat unclear, however, where the exact position of the maxima occur since the three lowest IPs are fairly close in energy and these peaks overlap. There has also been some controversy about their assignment w5,45x. In Table 1 we report the lowest three MCSTEP IPs using the pVDZ, pVTZ, and pVQZ basis sets as well as the complete basis set limit ŽCBS. IPs. With all of the basis sets used in Table 1, MCSTEP gives IPs in very good to excellent agreement with experiment. The pVDZ MCSTEP IPs in Table 1 differ from the vertical IPs reported experimentally by y0.02 eV, y0.16 eV, and y0.24 eV and the pVTZ MCSTEP IPs differ from experiment by 0.12 eV, y0.08 eV, and y0.19 eV for the 2A1 Ž6a 1 .y1 , 2 B2 Ž4b 2 .y1 , and 2A 2 Ž1a 2 .y1 states respectively. The pVQZ MCSTEP IPs in Table 1 differ from experiment by 0.14 eV, y0.02 eV, and y0.12 eV for the 2 A1 Ž6a 1 .y1 , 2 B2 Ž4b 2 .y1 , and 2A 2 Ž1a 2 .y1 states respectively. The CBS limit MCSTEP IPs differ from experiment by 0.14 eV, 0.16 eV, and y0.01 eV for the 2A1 Ž6a 1 .y1 , 2 B2 Ž4b 2 .y1 , and 2A 2 Ž1a 2 .y1 states respectively. Other low-lying principal IPs for O 3 include the 2 A1 Ž5a 1 .y1 , the 2 B2 Ž3b 2 .y1 , and the 2 B2 Ž1b1 .y1 .

Table 1 O 3 MCSTEP vertical IPs using several different standard basis sets a Ion state 2

pVDZ b pVTZ b pVQZ b CBS c Experiment d

A1 Ž6a 1 .y1 12.71 B2 Ž4b 2 .y1 12.84 2 A 2 Ž1a 2 .y1 13.30 2

a

12.85 12.92 13.35

12.87 12.98 13.42

12.87 12.73 13.16 13.00 13.53 13.54

All results in eV. The MCSTEP IPs are reported to hundredths of an eV for comparison with experimentally reported vertical IPs. b Basis sets from Ref. w35x. c Complete basis set limit from Eq. Ž6.. d Ref. w5x. The authors report experimental photoelectron spectroscopy ŽPES. values for vertical IPs to 0.01 eV.

5

Table 2 O 3 MCSTEP vertical IPs using pCVXZ basis sets a Ion state

pCVDZ b

pCVTZ b

Experiment c

2

12.72 12.85 13.31

12.85 12.91 13.36

12.73 13.00 13.54

.y1

A1 Ž6a 1 B2 Ž4b 2 .y1 2 A 2 Ž1a 2 .y1 2

a All results in eV. The MCSTEP IPs are reported to hundredths of an eV for comparison with experimentally reported vertical IPs. b Basis sets from Refs. w35,38x. c Ref. w5x. The authors report experimental photoelectron spectroscopy ŽPES. values for vertical IPs to 0.01 eV.

Since the 5a 1 , 3b 2 , and 1b 1 orbitals are outside of the CAS, these principal IPs calculated with MCSTEP are not expected to be as accurate as those for the three lowest IPs and so they are not listed in the tables or reported in this paper. In Table 2 we give the 2A1 Ž6a 1 .y1 , 2 B2 Ž4b 2 .y1 , and 2A 2 Ž1a 2 .y1 MCSTEP IPs using pCVDZ and pCVTZ basis sets. These IPs differ from the corresponding pVDZ and pVTZ MCSTEP IPs by at most "0.01 eV. In Table 3 we report the aug-pVDZ and aug-pVTZ MCSTEP IPs. The aug-pVDZ MCSTEP IPs differ from experiment by 0.22 eV, 0.03 eV, and y0.03 eV for the 2A1 Ž6a 1 .y1 , 2 B2 Ž4b 2 .y1 , and 2A 2 Ž1a 2 .y1 states respectively. The aug-pVTZ MCSTEP IPs differ from experiment by 0.23 eV, 0.01 eV, and 0.08 eV for the 2A1 Ž6a 1 .y1 , 2 B2 Ž4b 2 .y1 , and 2A 2 Ž1a 2 .y1 states respectively. In Table 3 we also report the MCSTEP IPs using the pVTZ basis set with the f-functions removed ŽpVTZ-f. and the pVQZ basis set with the f and g functions removed ŽpVQZ-f,g.. Comparing these results with the aug-pVDZ and aug-pVTZ MCSTEP IPs shows that the effect of the extra diffuse functions fairly well mimic the pVTZ and pVQZ sets without f and g functions, respectively. However, the pVTZ-f results in Table 3 almost exactly mimic the pVTZ results of Table 1 differing by only y0.03 eV, y0.05 eV, and 0.00 eV for the 2 A1 Ž6a 1 .y1 , 2 B2 Ž4b 2 .y1 , and 2A 2 Ž1a 2 .y1 states respectively; and the pVQZ-f,g results almost exactly mimic the pVQZ results of Table 1 differing by only y0.02 eV, y0.05 eV, and y0.01 eV for the 2A1 Ž6a 1 .y1 , 2 B2 Ž4b 2 .y1 , and 2A 2 Ž1a 2 .y1 states respectively.

A.J. McKellar et al.r Chemical Physics 238 (1998) 1–9

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Table 3 O 3 MCSTEP vertical IPs using aug-pVXZ and pVXZ-fŽ,g. basis sets a Ion state

aug-pVDZ b

pVTZ-f c

aug-pVTZ b

pVQZ-f,g c

Experiment d

2

12.95 13.03 13.51

12.82 12.87 13.35

12.96 13.01 13.46

12.85 12.93 13.41

12.73 13.00 13.54

.y1

A1 Ž6a 1 B2 Ž4b 2 .y1 2 A 2 Ž1a 2 .y1 2

a

All results in eV. The MCSTEP IPs are reported to hundredths of an eV for comparison with experimentally reported vertical IPs. Aug basis sets from Refs. w35,37x. c Basis sets from Ref. w5x with the f-functions removed from the pVTZ and pVQZ sets and the g-functions removed from the pVQZ set. d Ref. w5x. The authors report experimental photoelectron spectroscopy ŽPES. values for vertical IPs to 0.01 eV. b

3.2. NO2 The experimental geometry we used in this calculation is the same as that used by Schirmer et al. w46x, i.e. C 2v geometry with ONO bond angle 134.078 and NO bond length 2.255 a.u. The ground state principal configuration of NO 2 is 1a211b 22 2a21 3a21 2b 22 4a21 5a211b12 3b 22 4b 22 1a22 6a11. Since this molecule has an open shell ground state, it is not expected that perturbational Green’s function approaches that rely on Žoften unreliable. open shell perturbation theory will be particularly successful. However, since MCSTEP does not use perturbation theory, MCSTEP IPs should be reliable. The CAS we used for these MCSTEP calculations was chosen by a method described previously in detail — it is composed of the highest three Žin energy. occupied orbitals from the SCF calculation and the next three Žin energy. unoccupied orbitals regardless of symmetry except that the 3b1 orbital replaces the 5b 2 w30,32,33x. Thus the CAS for these Table 4 NO 2 MCSTEP vertical IPs using several different standard basis sets a Ion state

pVDZ b pVTZ b pVQZ b CBS c Experiment d

1

11.08 12.57 13.18 13.58 14.51

A1 Ž6a 1 .y1 B2 Ž4b 2 .y1 3 A 2 Ž1a 2 .y1 1 A 2 Ž1a 2 .y1 1 B2 Ž4b 2 .y1 3

a

11.22 12.82 13.39 13.74 14.72

11.26 12.88 13.45 13.79 14.77

11.28 12.90 13.47 13.81 14.79

11.23 13.02 13.61 14.08 14.53

All results in eV. The MCSTEP IPs are reported to hundredths of an eV for comparison with experimentally reported vertical IPs. b Basis sets from Ref. w5x. c Complete basis set limit from Eq. Ž6.. d Ref. w6x. The authors report experimental photoelectron spectroscopy ŽPES. values for vertical IPs to 0.01 eV.

MCSTEP calculations is composed of all possible configurations of five electrons in the Ž4b 2 6a 11a 22b 17a 1 3b1 . orbitals. For the neutral 2A1 ground state there are only 76 MS s 1r2 determinants. We also report in this subsection MCSTEP IPs to hundredths of eV in order to compare with experimental results which are reported to this accuracy w6x. In Table 4 we report the lowest five principal MCSTEP IPs using the pVDZ, pVTZ, and pVQZ basis sets as well as the complete basis set limit ŽCBS. IPs. The results in Table 4 clearly show that the pVDZ basis set is not adequate while the pVTZ and pVQZ basis sets give IPs similar to each other that are also in very good to excellent agreement with experiment. The pVDZ MCSTEP IPs in Table 4 differ from the vertical IPs reported experimentally by y0.15 eV, y0.45 eV, y0.43 eV, y0.50 eV, and y0.02 eV and the pVTZ MCSTEP IPs differ from experiment by y0.01 eV, y0.20 eV, y0.22 eV, y0.34 eV, and 0.19 eV for the 1A1 Ž6a 1 .y1 , 3 B2 Ž4b 2 .y1 , 3A 2 Ž1a 2 .y1 , 1A 2 Ž1a 2 .y1 , and 1 B2 Ž4b 2 .y1 states respectively. The pVQZ MCSTEP IPs in Table 5 NO 2 MCSTEP vertical IPs using pCVXZ basis sets a Ion state

pCVDZ b

pCVTZ b

Experiment c

1

11.10 12.60 13.20 13.59 14.52

11.22 12.78 13.36 13.74 14.72

11.23 13.02 13.61 14.08 14.53

.y1

A1 Ž6a 1 B2 Ž4b 2 .y1 3 A 2 Ž1a 2 .y1 1 A 2 Ž1a 2 .y1 1 B2 Ž4b 2 .y1 3

a All results in eV. The MCSTEP IPs are reported to hundredths of an eV for comparison with experimentally reported vertical IPs. b Basis sets from Refs. w35,38x. c Ref. w6x. The authors report experimental photoelectron spectroscopy ŽPES. values for vertical IPs to 0.01 eV.

A.J. McKellar et al.r Chemical Physics 238 (1998) 1–9

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Table 6 NO 2 MCSTEP vertical IPs using aug-pVXZ and pVXZ-fŽ,g. basis sets a Ion state

aug-pVDZ b

pVTZ-f c

aug-pVTZ b

pVQZ-f,g c

Experiment d

1

11.30 12.84 13.48 13.85 14.80

11.22 12.74 13.34 13.72 14.69

11.28 12.88 13.46 13.83 14.80

11.26 12.80 13.40 13.77 14.73

11.23 13.02 13.61 14.08 14.53

.y1

A1 Ž6a 1 B2 Ž4b 2 .y1 3 A 2 Ž1a 2 .y1 1 A 2 Ž1a 2 .y1 1 B2 Ž4b 2 .y1 3

a

All results in eV. The MCSTEP IPs are reported to hundredths of an eV for comparison with experimentally reported vertical IPs. Aug basis sets from Refs. w35,37x. c Basis sets from Ref. w5x with the f-functions removed from the pVTZ and pVQZ sets and the g-functions removed from the pVQZ set. d Ref. w6x. The authors report experimental photoelectron spectroscopy ŽPES. values for vertical IPs to 0.01 eV. b

Table 4 differ from experiment by 0.03 eV, y0.14 eV, y0.16 eV, y0.29 eV, and 0.24 eV for the 1A1 Ž6a 1 .y1 , 3 B2 Ž4b 2 .y1 , 3A 2 Ž1a 2 .y1 , 1A 2 Ž1a 2 .y1 , and 1 B2 Ž4b 2 .y1 states respectively. The CBS limit MCSTEP IPs differ from experiment by 0.05 eV, y0.12 eV, y0.14 eV, y0.27 eV, and 0.26 eV for the 1A1 Ž6a 1 .y1 , 3 B2 Ž4b 2 .y1 , 3A 2 Ž1a 2 .y1 , 1A 2 Ž1a 2 .y1 , and 1 B2 Ž4b 2 .y1 respectively. Other low-lying principal IPs for NO 2 include the 1,3 A1 Ž5a 1 .y1 , the 1,3 B2 Ž3b 2 .y1 , and the 1,3 B2 Ž1b1 .y1 . Since the 5a 1 , 3b 2 , and 1b 1 orbitals are outside of the CAS, these principal IPs are not expected to be calculated as accurately with MCSTEP as are those for the three lowest IPs and so they are not listed in the tables or reported in this paper. In Table 5 we give the MCSTEP IPs using pCVDZ and pCVTZ basis sets. These IPs differ from the corresponding pVDZ and pVTZ MCSTEP IPs by at most y0.04 eV. In Table 6 we report the aug-pVDZ and aug-pVTZ IPs. The aug-pVDZ MCSTEP IPs differ from experiment by 0.07 eV, y0.18 eV, y0.13 eV, y0.23 eV, and 0.27 eV for the 1A1 Ž6a 1 .y1 , 3 B2 Ž4b 2 .y1 , 3A 2 Ž1a 2 .y1 , 1A 2 Ž1a 2 .y1 , and 1 B2 Ž4b 2 .y1 states respectively. The aug-pVTZ MCSTEP IPs differ from experiment by 0.05 eV, y0.14 eV, y0.15 eV, y0.25 eV, and 0.27 eV for the 1A1 Ž6a 1 .y1 , 3 B2 Ž4b 2 .y1 , 3 A 2 Ž1a 2 .y1 , 1A 2 Ž1a 2 .y1 , and 1 B2 Ž4b 2 .y1 states respectively. In Table 6 we also report the MCSTEP IPs using the pVTZ basis set with the f-functions removed ŽpVTZ-f. and the pVQZ basis set with the f and g function removed ŽpVQZ-f,g.. Comparing these results with the aug-pVDZ and aug-pVTZ MCSTEP IPs shows that the effect of the extra diffuse func-

tions almost exactly mimic the pVTZ and pVQZ sets without f and g functions, respectively. However, the pVTZ-f results even more closely mimic the pVTZ results of Table 4 differing by only 0.00 eV, y0.08 eV, y0.05 eV, y0.02 eV, and y0.03 eV for the 1A1 Ž6a 1 .y1 , 3 B2 Ž4b 2 .y1 , 3A 2 Ž1a 2 .y1 , 1A 2 Ž1a 2 .y1 , and 1 B2 Ž4b 2 .y1 states respectively. The pVQZ-f,g MCSTEP IPs also very closely mimic the pVQZ IPs of Table 4 differing by only y0.00 eV, y0.08 eV, y0.05 eV, y0.02 eV, and 0.04 eV for the 1A1 Ž6a 1 .y1 , 3 B2 Ž4b 2 .y1 , 3A 2 Ž1a 2 .y1 , 1A 2 Ž1a 2 .y1 , and 1 B2 Ž4b 2 .y1 states respectively.

4. Summary and conclusions We have calculated low-lying principal IPs of O 3 and NO 2 with MCSTEP using several different basis sets and compared with reported experimental vertical IPs. We obtained complete basis set limit IPs for MCSTEP by extrapolating the pVDZ, pVTZ, and pVQZ basis set results. We also looked at MCSTEP IPs with pCVXZ basis sets for any core and core– valence effects and aug-pVXZ basis sets for contributions from diffuse functions. The MCSTEP pVTZ, pVQZ, aug-pVDZ, augpVTZ, pCVTZ basis sets and CBS limit IPs are in very good to excellent agreement with experiment for both O 3 and NO 2 . Our results confirm previous assignments based on calculations previously made using less accurate methods w5,6,45,46x. The apparent accuracy of our IPs, however, indicates that some small reassessments of the exact position of the ‘experimental’ vertical IPs may be justified. The

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A.J. McKellar et al.r Chemical Physics 238 (1998) 1–9

pVDZ and pCVDZ basis set IPs for NO 2 demonstrate that pVDZ basis sets cannot be trusted to consistently give chemically accurate IPs. The addition of additional core and core–valence basis functions to pVXZ basis sets Žthe pCVXZ basis sets. had little effect on MCSTEP outer valence IPs. The addition of diffuse functions Žthe aug-pVXZ basis sets. for both O 3 and NO 2 had more effect on the MCSTEP IPs. However, it appears that this may be mostly due to having more s, p, and d functions mimicking somewhat the next larger pVXZ basis set. This is also shown by examining MCSTEP IPs using pVTZ sets with the f-functions removed ŽpVTZ-f. and pVQZ sets with both the f- and g- functions removed ŽpVQZ-f,g.. These MCSTEP IPs are essentially the same as those obtained with the corresponding pVTZ and pVQZ basis sets respectively. Of course, in performing quantum chemistry calculations it is always best to choose the largest standard basis set that can be used with given computational resources. However, these calculations show that for MCSTEP low-lying vertical IPs an economical choice for large systems andror where computational resources are limited is a large standard basis set such as pVTZ or pVQZ without additional core, core–valence, or diffuse functions but also with all higher angular momentum functions above d-removed, e.g., pVTZ-f or pVQZ-f,g. For O 3 and NO 2 these Žspherical component. basis sets have 69 functions ŽpVTZ-f. verses 90 functions ŽpVTZ. and 96 functions ŽpVQZ-f,g. verses 165 functions ŽpVQZ.. Yet, pVTZ-f MCSTEP IPs are almost identical to pVTZ MCSTEP IPs and pVQZ-f,g MCSTEP IPs are almost identical to pVQZ MCSTEP IPs. MCSTEP vertical IPs are highly accurate and reliable when used with large basis sets such as pVTZ, aug-pVTZ, pVQZ and aug-pVQZ for O 3 and NO 2 . However, another significant result of this paper is to demonstrate that CBS formulas can be used with confidence with MCSTEP for both highly correlated Že.g., O 3 . and open shell Že.g., NO 2 molecules. to give accurate IPs compared with reported experimental values. Acknowledgements AJM, DH, and DLY would like to acknowledge support for this research from The Robert A. Welch

Foundation Grant No. A-770. JAN wishes to acknowledge support of the Office of Health and Environmental Research, which funds the Environmental Molecular Sciences Laboratory Project, D-384, performed under Contract DE-ACO6-76RLO 1830 with Battelle Memorial Institute, which operates the Pacific Northwest Laboratory for the U.S. Department of Engergy. We also especially thank Jack Simons from the University of Utah for providing the MESSKIT suite of codes which we used to obtain MCSCF.

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