Low-Lying Potential Energy Surfaces - ACS Publications

Chapter 12 2

2

The X ( Π ) and X ( Π ) Potentials of the Halogen Monoxides: A Comparison of RKR and Ab Initio Results 1

3/2

2

1/2

Charles E. Miller

Downloaded by COLUMBIA UNIV on September 14, 2012 | http://pubs.acs.org Publication Date: August 14, 2002 | doi: 10.1021/bk-2002-0828.ch012

Department of Chemistry, Haverford College, Haverford, PA 19041-1392

2

X(Π )

Experimentally determined R K R potentials for the 1 3/2 2 and 2 1/2 states of the halogen monoxides F O , C l O , B r O and IO are compared to available ab initio potentials. The results suggest that fully relativistic ab initio calculations have the capability to reproduce the experimental bond lengths, harmonic vibrational frequencies and fine structure intervals of the X O series with reasonable accuracy. The quest for spectroscopically accurate X O potentials will provide an excellent benchmark for future theoretical methods.

X(Π )

This paper addresses the question of what it means to have a spectroscopic quality potential energy surface. The literature of the last few years contains numerous references to spectroscopic quality ab initio molecular potentials, yet the definition of spectroscopic quality necessarily depends on the resolution of the experimental spectrum and the theoretical calculation. The accuracy of a molecular potential is perhaps better characterized by asking how well it reproduces experimental observables. This topic is explored by comparing the ^ι( Π3/2> and X2( Îlm) potentials of the halogen monoxides determined from high-resolution spectroscopic studies with potentials computed using ab initio methods. The R K R potentials determined from spectroscopic data provide 2

260

2

© 2002 American Chemical Society

In Low-Lying Potential Energy Surfaces; Hoffmann, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002.


261 faithful representations of the X O experimental information and serve as valuable benchmarks for ab initio work. The halogen monoxides, X O (where X = F, C l , Br, I), represent a fundamental series of main group inorganic oxides that should have interesting chemical bonding due to the competition between the two highly electronegative atoms and the presence of an unpaired valence electron. They are diatomic molecules, so that, in principle, one may obtain "exact" quantum mechanical potentials from spectroscopic data inversions. Similarly, determining the potentials from ab initio methods should prove a tractable problem. Examination of the entire X O series enables one to investigate periodic trends, comment on structure/reactivity relationships and begin to understand the bonding likely to occur in larger halogen oxides. The high-resolution spectroscopic data used to derive the R K R potentials for F O , CIO, B r O and 10 has been described in a series of recent publications (1-4). The potentials include transitions between molecular eigenstates with up to 10,000 cm" of internal energy (-40% of the X - 0 bond dissociation energy) measured with microwave accuracy. The X O ground state possesses the electronic configuration (ζσ) (νσ*) (χσ) (π>π) (νπ*) where the values of v, w, x, y and ζ vary with the identity of the halogen atom. The unpaired π* electron gives rise to an inverted Π state with the Χι( Π /2) lying below the X ( TLm) state. The fine structure splitting between Xi( n^ri) and Xt^Tlm) scales with the ? 3 / 2 - Pi/2 splitting of the halogen atom. A l l four members of the X O series examined here are good examples of Hund's case (a) coupling, the limit in which the fine structure splitting is much larger than the rotational constant, A/B » 1 . 1

2

2

2

4

3

2

2

2

3

2

2

2

2

FO 2

2

The Xi( n /2) and Χ2( Πι ) potentials of F O were derived from a combined fit to the existing microwave, L M R and high-resolution FTIR spectroscopic data (2). The data included transitions for vibrational levels up to ν = 7 and direct measurements of X ( n ) - Xi( Tlm) fine structure transitions. The R K R potentials and associated vibrational intervals are shown in Figure 1. Figure 2 compares the Χ Π R K R potential with available ab initio potential energy 3

/2

2

1

2

3/2

2

points (5-7). A broader comparison with ab initio work is made in Table I where r , ω and (ûeX values are collected. One notes that all of the calculations in Table I report effective Π equilibrium properties. Additionally, the majority of calculations are unable to reproduce simultaneously the equilibrium bond length and the harmonic vibrational frequency. The MRCI/aug-cc-pVQZ calculations e

e

2


β


100

110

120

130

140

150

160

170

r/pm 2

2

Fig. 1. FO n and Π RKR potentials and vibrational levels (2). 3/2

100

110

120

1/2

130

140

150

160

170

r/pm

Fig. 2. A comparison of the FO RKR TIpotential with several ab initio potentials (2). 2


263 2

of Leonard et al. (7) provide the most accurate characterization of F O Χ ( Π ) but closer examination of Figure 2 shows that these calculations yield a potential that is too anharmonic for energies above 2000 cm" . 1

2

Table I. A b Initio Characterization of F O Χ ( Π )


Level of Theory B3LYP/6-311++G(3df,3pd) MP2/6-31G(d) UMP4/5s4p2d AUMP2/6-31G* QCISD(T)/6-31+G* QCISD(T)/ 6-311++G(3df) CCSD(T)/cc-pVTZ UCCSD(T)/5s4p2d UGA-CCSD/6-31G* UGA-CCSD/TZ2P MRCI/aug-cc-pVQZ MRCI/extended gaussian Experiment ( Il ) 2

eff

Tf/pm 134.6 134.4 135.2 136.53 138 135.6 137.33 137.0 138.3 135.4 135.5 136.1 135.411

(Oe/cm Reference 1120 (8) 1542 (9) 1468 (10) 1092 (11) 1017 (6) (12) 1027 1072 (13) 1060 (10) 1075 (14) 1085 (14) 1047 (7) 1

992 1053.01

(5) (2)

CIO 2

2

The Χ ι ( Π ) and Χ ( Π ι ) potentials of CIO were derived from a combined fit to the existing microwave, FIR and high-resolution FTIR spectroscopic data (3). The data included transitions from · α 0 isotopomers in vibrational levels up to ν = 2. Figure 3 shows the C 1 0 potentials and their associated vibrational levels. Pettersson et al. (75) reported a tabulated set of CCI+Q potential energy points and these are plotted against the R K R X( TÏ) potential in Figure 4. The agreement between the experimental and ab initio potentials is very good. The small discrepancies for energies above 4000 c m may be due to the extrapolation of the R K R potential to energies not included in the inversion. Table II compares the experimental Χ( Π) parameters with available ab initio values; again no ab initio calculations report resolved Π equilibrium properties. The theoretical characterizations of CIO agree much better with experiment than the corresponding F O calculations, probably reflecting difficulties with the anomalous properties of the F-O bond (see below). Methods employing sophisticated electron correlation effects perform very well, with the 3/2

2

/2

3 5

35

3 7

1 6 , 1 8

16

2

- 1

2

2



140

150

160

170

180

R(CI-0)/pm 2

2

Fig. 3. CIO Χι( Π ) and X ( Tl ) RKR potentials and vibrational levels 3/2

140

150

2

160

m

170

180

R(Cl-0)/pm

Fig. 4. A comparison of the RKR and CCI+Q potential for CIO Χ ( Π). 2


265 2

Table II. Ab Initio Characterization of CIO Χ ( Π)

Level of Theory HCTH/TZ2P QCISD(T)/ 6-311++G(3df) CCI(9)+Q CCI(13)+Q+rel EHFACE2U MRCI/aug-cc-pV5Z Experiment ( II ) 2

eff


1

(ûjcrti

T/pm 157.0 158.0 157.2 158.5 157.0 157.9 157.01

1

a^Xe/cm'

840.9 845 810 848.5 852.3 852.3

MRCI/aug-cc-pV5Z (7) and E H F A C E 2 U spectroscopic accuracy.

6.7

4.74 4.9 5.528

(16) calculations

Reference (17) (12) (15) (15) (16) (7) (3) approaching

BrO 2

2

The Xi( U ) and Χ ( Π ι ) potentials of B r O were derived from a combined fit to the existing microwave, L M R and high-resolution FTIR spectroscopic data (1). The data included transitions from Br 0 isotopomers in vibrational levels up to ν = 8. Figure 5 shows the B r O potentials and their associated vibrational levels. The MRCI+Q/aug-cc-pVQZ potential (no spin-orbit coupling) recently reported by L i et al.(7S) is compared to the X( IIeff) potential in Figure 6. Note that the ab initio potential has been shifted by -1.0 pm for better agreement with the R K R potential. The two potentials exhibit minimal differences for energies up to 8000 cm" . This suggests that M R C I calculations with sufficiently large basis sets and explicit inclusion of spin-orbit coupling should be able to reproduce the X O R K R potentials accurately. Comparisons of the experimental and ab initio Χ( Π) characterizations of the potential minimum are given in Table III, including the spin-orbit state resolved bond lengths and vibrational frequencies determined by L i et al. (18). The M R C I calculations do an outstanding job of reproducing the experimental harmonic vibrational frequency. They also reproduce the relative values of r in both Π states, although they systematically overestimate the B r - 0 bond length by 1.0 pm. In fact, all of the theoretical methods overestimate the B r - 0 bond length. Analysis of the magnetic hyperfine structure and quadrupole coupling parameters (see below) indicates that relativistic effects make a measurable impact on the B r O electronic structure; the discrepancy between the experimental and theoretical bond lengths may reflect a small relativistic contraction. m

2

/2

7 9 , 8 1

1 6 , l 8

2

1

2

e

2



160

180

200

r/pm

Fig. 6. A comparison of the RKR and MRCI potential for BrO X ( n ). 2

eff


267

Table III. Ab Initio Characterization of BrO X CTÎ) Level of Theory

xjpm

1

(ùjcrn

UMP2/TZ2P MP2/AREP-311G** QCISD(T)/6-311+G(3df) CISD/AREP-311G** CCSD(T)/AN04 MRCI/aug-cc-pVQZ n MRCI+Q/aug-cc-pVQZ Experiment

172.4 175.0 173.1 176.9 172.7 172.6

813 705.8 741 728 728.5

172.7 171.72

MRCI+Q/aug-cc-pVQZ Experiment

173.3 172.41

a^x^/cm' Reference 1

3.5

(22) (23) (12) (23) (22) (7)

730 732.88

4.649

(18) (1)

719 717.95

4.658

(18) (1)

6.5

2


3/2

IO 2

2

The Xi( n / ) and Χ ( Π ) potentials of IO were derived from a fit to the extensive microwave data reported by Miller and Cohen(4), augmented by the M O O D R transitions measured by Bekooy et al.(i9) and the fine structure 3

2

2

1/2

1

splitting of 2091 ± 40 cm" measured by Gilles et al.(20) The data included transitions from I 0 isotopomers in vibrational levels up to ν = 13. The IO potentials and their associated vibrational levels are shown in Figure 7. The R K R and M R D - C I / R E C P potentials of Roszak et al.(2i) are plotted in Figure 8. The ab initio potentials have been shifted by -5.0 pm in this plot for a better comparison with experiment. The M R D - C I calculations overestimate the 1-0 bond length due to freezing of the iodine 4d electrons at the configuration interaction step (27). The diffuse 4d electrons in IO are significantly polarized by the electronegative oxygen atom, resulting in stronger bonding than reflected in the calculations. The relative agreement of the experimental and ab initio potentials for both the Χ ι ( Π ) and Χ ( Πι/ ) states is impressive given the difficulty of treating the large number of electrons in IO with an explicit inclusion of spin-orbit coupling in the ab initio code. The ab initio potentials l 6 1 8

2

2

3/2

2

2

1

become too anharmonic above 3000 c m " but this behavior was noted for the theoretical potentials of all of the lighter halogen monoxides and appears to be a systematic difficulty.



268

160

180

200

220

r/pm 2

2

Fig. 7.10 Χι( Π ) and Χ ( Π ) RKR potentials and vibrational levels. 3/2

160

180

2

200

1/2

220

240

r/pm Fig. 8. The RKR and MRD-CI potentials for 10 Χ( Π). The solid lines mark the RKR potentials while the lines with symbols mark the adjusted ab initio potentials from ReK21). 2


269 2

Table IV. Ab Initio Characterization of ΙΟ Χ ( Π)


Level of Theory

xjpm

1

(ùjcrri

ftt *e /cm'

1

MP2/6-311+G(3df) CASPT2(9,6) CCSD(T)/6-311+G(3df) MRCI/aug-cc-pVQZ QCISD(T)/6-311+G(3df)

188.8 187.99 189.39 187.9 190.0

724 652 664 694.5 644.4

4.76 3.7 4.6 5.5

MRD-CI/RECP Experiment

192.2 186.769

650 681.7

4.35

193.9 188.475

626 645.3

4.31

2

n

Reference (24) (25) (25) (7) (12)

(21) (4)

m

MRD-CI/RECP Experiment

(21) (4)

2

Experimental and ab initio characterizations of the Χ( Π) potential minima are given in Table IV. It is difficult to compare the calculations without resolved spin-orbit components to the experimental data since changing from Χι( Π /2) to ^2( Πι/2) increases the 1-0 bond length by 1.7 pm and decreases the harmonic vibrational frequency by 36 cm \ These results suggest that the two spin-orbit states may have significant differences in their electron configurations. This hypothesis is explored further in the treatment of relativistic effects given below. 2

3

2

Relativistic Effects The intrinsically relativistic nature of the electronic structure of the halogen monoxides is dictated by the presence of two spin-orbit components in the ground electronic state. Any accurate characterization of the X O X ! Tly and X2 Π ι / potentials must therefore treat relativistic effects explicitly. However, this requirement greatly increases the cost and complexity of the ab initio effort(26), resulting in few relativistic potential surface calculations such as the 10 study by Roszak et al(21) One more frequently finds relativistic effects incorporated as corrections to non-relativistic energies or treated through the use of effective core potentials (23). The fine structure splittings of the X O series provide the most direct insight into the relativistic contributions in these molecules and the role of the halogen 2

2

2

2


2

Table V. X O Χ ( Π) Fine Structure Splittings

Level of Theory

1

At/cm'

Reference

FO MCSCF/6-31G(d,p)/SBK RELCCSD-T/aug-cc-pVTZ Experiment

-185.7 -194.6 -196.6

(27) (26) (2)

-290.0 -314.5 -321.8

(27) (26) (3)

-726 -975.4

(18) (1)

-1683 -2091

(21) (20)


OO MCSCF/6-3 lG(d,p)/SBK RELCCSD-T/aug-cc-pVTZ Experiment

BrO MRCI+Q/aug-cc-pVQZ Experiment

IO MRD-CI/RECP Experiment

atom spin-orbit coupling in defining their electronic structures. The fine structure interval separating the Χ! Π and X Π states increases as halogen atom spin-orbit coupling increases (Table V ) . The R E L C C S D calculations reported by Visscher et al.(26) perform remarkably well for F O and CIO, reproducing the experimental fine structure splittings within a few wavenumbers. These calculations approach the accuracy one would hope to achieve for spectroscopic quality ab initio potential. It would be very interesting to learn if the R E L C C S D calculations perform as well for values of r away from the r position, especially if they can reproduce the observed vibrational state dependence of the fine structure splitting. It is also unclear whether the performance of this method extends to BrO and IO where the impact of the relativistic effects is significantly larger. The performance of multi-reference configuration interaction (MRCI) methods for BrO(78) and 10(27) illustrate the difficulties associated with capturing the correct form of the Π potentials as well as an accurate characterization of the fine structure splitting. Figures 6 and 8 show that the shifted M R C I calculations describe the contours of the BrO and IO potentials quite well. However, the calculated fine structure splittings are systematically low, differing by -26% for BrO and -20% for IO. Despite these discrepancies, it appears that M R C I methods are approaching the accuracy necessary to calculate high quality ab initio potentials for the X O series. 2

2

3 / 2

2

1 / 2

e

2


271 The magnetic hyperfine structure and quadrupole coupling parameters provide detailed information on the electron distributions of the X O series. The relationships between these parameters and various expectation values reflect contributions from only those electrons which contribute to the orbital angular momentum (designated L ) , only those electrons which contribute to the spin angular momentum (designated S), and all electrons (designated T). The variable θ defines the angular orientation of the electron distribution relative to the internuclear axis.


a=

2g W„(r-*) N

L

2

3

/3cos 0-l^

2

\

r

j (1)

eQq =eQ[ x

β

β

9 2

-

3

)

=-3ββ(—5-

The experimentally derived expectation values for all four X O molecules are summarized in Table VI. The spin density on the halogen atom is derived by the ratio of the molecular and relativistic atomic radial and angular expectation values. The spin density on the F atom in F O is anomalously low, 20%, a characteristic that is also reflected in the unusually small dipole moment for this molecule. (5, 7) The spin densities found on the halogen atoms in CIO, B r O and IO are between 37 and 39% and exhibit little dependence on the identity of the halogen atom. The spin densities calculated using non-relativistic radial integrals for the evaluation of the halogen atomic expectation values lead to an unphysical increases in the calculated spin density on the halogen atom, reaching 62% for IO. Another indication of the importance of relativistic contributions to the 2

bonding in the X O series is given by the trend in ^ Ψ ( θ ) ^ , the probability of finding the spin inducing electron at the halogen nucleus. Table V I shows that


272 there is a clear periodic decrease in this expectation value, passing through zero and reaching a negative value for 10. Examination of the expression for the Fermi contact term, b , from which this expectation value is derived shows that should be a positive number since it is the product of nominally positive parameters. Many authors attribute such unphysical behavior to spin polarization but it appears quite consistent with the other results from the X O series, demonstrating the importance of relativistic corrections. Note that the angular distribution of the electrons which contribute to the molecular spin, F


, shows a very reasonable decreasing trend through the series without any unphysical behavior, most likely due to the contraction of the halogen atom ρ-π* orbital to optimize overlap with the oxygen ρ-π* orbital.

2

Table VI. Derived X O Χ ( Π) Electron Distribution Expectation Values

Molecular Values

(

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