THEO CHEM

ELSEVIER Journal of Molecular !hucture (Theochem) 389 (1997) 251-2.56

The determination of the equilibrium structures of oxygen, ozone, and hydrogen peroxide using the ab initio and density functional

theory methods

Branko S. Jursic

Department of Chemistry, University of New Orleans, New Orleans, LA 70148, USA

Received 14 September 1995; revised 22 February 1996

Abstract

The geometries and energies of small oxygen containing molecules are studied by both the ab initio and density functional theory (DFI’) methods. The RHF, MP2, and QCISD(T) ab initio methods, BHandH, BHandHLYP, Becke3LYP, Becke3P86 DFT hybrid methods, BLYP, and the BPS6 non-local DFT methods with the 3-21G*, 6-31G(d,p), 6-311 + G(2d,2p) and 6- 31 1 + + G(3df,3pd) basis sets were used for the computational study. The obtained results from the different methods were

compared to the experimental values. The suitability of the DFT methods for reproducing experimental data were discussed.

Keywords: Equilibrium structure; Density functional theory; Oxygen-hydrogen molecule

1. Introduction

Both the ab initio and density functional theory

(DFT) methods are becoming increasingly popular for predicting physical-chemical properties of molecules. Quantum chemists do not expect to gen-

erate highly accurate equilibrium structures using the non-correlated methods and minimal basis sets. For the majority of organic compounds, the structures are

reasonably well reproduced using the minimum basis SCF level of theory, with a typical error for the C-C bond distance of 0.01-0.01 [l]. This is not true for

small molecules that are mainly composed of highly

electronegative atoms [2]. In many cases, these struc- tures cannot be modeled using the ab initio methods

[2]. Today, the DFT methods are becoming increas- ingly popular tools for the modeling of chemical structures (31. We have investigated small inorganic

structures using both the ab initio and DFT methods.

An excellent agreement is obtained between the DFT hybrid generated structures for the NO dimer [4],

nitrogen oxides [5], nitrogen fluorides [6], sulfur fluorides [7] and the experimental data. These

methods have also been proven to generate reliable

results for the computational study of the Cope [8,9] and Claisen rearrangements [9]. Today, chemists are interested in determining how well the DFT methods

will reproduce structures of small oxygen containing molecules.

2. Theoretical methods

All of the calculations were performed using the GAUSSIAN~Z [lo] implementation of the density func- tional theory (DFT) methods. The optimizations were

0166-1280/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SOl66-1280(96)04528-9

252 B.S. JursiclJoumal of Molecular Structure (Theochem) 389 (1997) 251-256

carried out without any geometric restrictions using three parameter functional with the non-local correla- the Fletcher-Powell [ll] method and the default tion provided by the LYP expression. The Becke3P86 Gaussian convergence criteria. Three ab initio meth- is Becke’s three functional parameters with the non- ods were applied: restricted Hartree-Fock (RHF) self local correlation provided by the PERDEW~~ expres- consistent field [12], second order Moller-Plesset sion [18]. Two non-local DFT methods, Becke’s (MP2) perturbation theory [13] and the quadratic con- exchange with Lee, Yang and Parr correlation func- figuration interactions (single, doubles and triples); tional (BLYP) and Becke’s exchange with the QCISD(T) [14]. Several hybrid methods were used Perdew’s gradient corrected functional (BP86) were for the optimization of structures: Becke [15] 50-50 used. To demonstrate the effect of the basis sets on the method (BHandH), which includes the 50% HF computational accuracy, the calculations were per- exchange and 50% Slater [16] exchange without the formed by using 3-21G* [19], 6-31G(d,p), 6-311 + correlation functional; Becke 50-50 method with G(2d,2p) [20,21] and 6-311 + + G(3df,3pd) [21,22] LYP [ 171 correlation added (BHandHLYP); Becke’s basis sets.

Table 1

The parameters for oxygen and ozone calculated using the ab initio and DFI methods

Theory model r,/A rJA AEI rziA alI” AEEl1

RHF/3-21G*

RHF/6-31G(d,p)

RHF/6-311 + (2d,2p)

RHF/6-311 + + G(3df,3pd)

MP2/3-21G’

MP2/6-31G(d,p)

MP2/6-311 + (2d,2p)

QCISD(T)/3-21G*

QCISD(T)/6_31G(d,p)

BHandH/3-21G*

BHandH/6-31G(d,p)

BHandW6-311 + (2d,2p)

BHandW6-311 + + G(3df,3pd)

BHandHLYP/3_21G*

BHandHLYP/6_31G(d,p)

BHandHLYP/6-311 + (2d,2p)

BHandHLYP/6-311 + + G(3df,3pd)

Becke3LYP/3-21G*

Becke3LYP/6_31G(d,p)

Becke3LYP/6-311 + (2d,2p)

Becke3LYP/6-311 + + G(3df,3pd)

Becke3P86/3-21G*

Becke3P86/6-31G(d,p)

Becke3P86/6-311 + G(2d,2p)

Becke3P86/6-311 + + G(3df,3pd)

BLYP/3-21G*

BLYP/6-31G(d,p)

BLYP/6-311 + G(2d,2p)

BLYP/6-311 + + G(3df,3pd)

BP86/3-21G*

BP86/6_31G(d,p) BP86/6-311 + G(2d,2p)

BP86/6-311 + + G(3df,3pd)

Experimental [25]

1.242

1.166

1.157

1.153

1.487

1.275

1.257

1.370

1.246

1.261

1.180

1.172

1.167

1.272

1.178

1.305

1.299

1.335

1.242

1.329

1.234

1.216

1.237 42.70 1.308 117.0 90.11

1.163 40.95 1.204 119.0 85.31

1.155 40.30 1.196 119.3 82.42

1.151 39.80 1.191 119.4 80.96

1.392 25.04 1.389 113.5 47.13

1.247 31.28 1.300 116.3 35.55

1.231 29.97 1.289 116.8 29.98

1.333 26.43 1.431 114.9 33.04

1.230 25.06 1.299 116.8 40.35

1.258 44.01 1.316 116.2 58.70

1.181 44.16 1.219 118.2 56.23

1.174 43.73 1.210 118.6 54.50

1.169 43.43 1.206 118.7 53.46

1.269 42.43 1.333 116.1 64.35

1.189 42.95 1.232 118.3 64.57

1.183 42.65 1.224 118.6 63.24

42.5 1 1.219 118.6 62.57

1.301 39.46 I.367 116.1 44.53

1.214 39.33 1.264 117.9 46.21

1.208 38.66 1.257 118.3 45.05

1.203 38.56 1.251 118.3 44.70

1.295 39.75 1.359 116.2 43.21

1.207 39.70 1.255 118.0 43.71

1.201 39.10 1.247 118.3 42.31

1.196 38.98 1.242 118.3 41.90

1.330 37.98 1.401 116.7 30.41

1.240 37.39 1.298 117.9 34.08

1.234 37.83 1.291 118.2 33.16

1.228 38.48 1.287 118.1 33.11

1.324 38.26 1.393 116.8 29.08

1.231 37.75 1.286 118.0 31.72

1.225 38.21 1.280 118.2 30.52

1.220 39.41 1.274 118.1 30.36

1.207 37.05 1.278 116.8 34.11

r I - the bond distance for the singlet oxygen; rz - the bond distance for the triplet oxygen; AE, - energy difference between the singlet and

triplet oxygen; r3 = the bond distance in ozone; a, = the bond angle in ozone; till - enthalpy for the reaction 31202 H 203.

B.S. JursiciJoumal of Molecular Structure (Theochem) 389 (1997) 251-256 253

3. Results and discussion

Molecular oxygen, O2 is unique among gaseous

diatomic species because of its even number of elec-

trons and paramagneticity. The paramagnetic proper- ties of oxygen were first observed by Faraday in 1848. This property is explained by the molecular orbital

theory [23]. The two least-strongly bound electrons in O2 occupy degenerate orbitals of p symmetry and

have parallel spins. There are three states for the two

electrons in p*, the lowest energy electron configura-

tion of 02, these are 3C&0.0 kcal mol-‘), ‘Ag (22.0 kcal mol-‘) and ‘CL (37.05 kcal mol-‘) [24].

The two single states are extremely important in the

gas-phase oxidation. The excitation is accompanied

by a modest0 increase in the O-O bond distance from 1.207 A in the ground state, to 1.216 and

1.228 A in the excited states [25]. The ab initio mod-

eling of the oxygen molecules do not produce satis- factory results (Table 2). It is crucial that the chosen

basis set includes d-orbitals, even though it was pro-

ven that d-orbitals with a minimal basis set like 3- 21G* are not capable of reproducing the experimental data. This was also determined in our previous study

of the suitability of ab initio methods for modeling small polar molecules [4-71. The RHF calculation

does not predict the geometric parameters or the ener-

gies correctly [4-71. The bond distances are consider- ably shorter and the energies are higher when

compared to the experimental values. It is obvious that a correlation method should be applied. However, when the MP2 calculations are applied they again fail. The MP2 ab initio method overestimates the correla-

tion interaction. Now, the predicted bond distances

are too long and the predicted energies are too small (Table 1). Surprisingly, an even higher level of ab initio calculations, like the QCISD(T)/6-31G(d,p)

does not qualitatively produce better results than

those obtained using the MP2 calculations.

In cases where the HF and sometimes the MP2 ab initio calculations cannot reproduce the experi- mental results, the DFT methods are applied because

they usually produce results that are quite close to the experimental values [4-71. Here, the oxygen mole- cules were studied using the hybrid and non-local DFT methods. The correlation interactions are

necessary to reproduce both the structure and the energies. The BHandH hybrid method, which

contains the Slater’s exchange without the correlation

functional, can only produce geometries that are

slightly better than those obtained using the RHF cal-

culations. However, the estimated energies are worse. The computational values obtained using the hybrid DFT method, which includes the correlation func-

tional, are closer to the experimental values. As indi-

cated in many of our previous papers, the best

geometries are obtained using the Becke3 hybrid

DFT methods. Thus, the geometries obtained using

the Becke3LYP and Becke3P86 with the 6-31G(d,p)

and 6-311 + G(2d,2p) basis sets are in excellent

agreement with the experimental values (Table 1). Surprisingly, the highest basis sets applied do not

produce the best results. This might be explained

with the presence of the polarized functional found in the basis sets.

Similar to the MP2 calculations, the non-local DFT methods, BLYP and BP86, produce O-O distances

which are too long. However, the obtained energies

are more accurate than any other calculations per-

formed in this study. There was a considerable pro- blem with the optimization of the singlet oxygen

structure when using all of the DFT methods. The

SCF=QC method required time consuming calcu- lations. Even with this restriction, the convergence

was not achieved in the majority of the calculations.

In these cases, the single point energy was evaluated on the oxygen triplet geometries by slightly changing

the O-O bond distance (Table 1). Surprisingly, the optimization of the ozone struc-

ture using the ab initio and DFT methods were both

straightforward processes. The structure of the ozone

molecule is bent. The microwave measurements led to a bond distance of 1.278 2 0.003 A and a bond angle

of 116.8 + 0.5”. The structure has an interatomic

O...O distance of 2.18 A for the two terminal

oxygens, which is 0.62i shorter than the normal

van der Waals O...O distance. There is also a con-

siderable charge distribution judging from the dipole moment of 0.54 D.

Here again, the RHF calculations produce geome- tries that are considerably different than those obtained from microwave experiments [25]. The pre-

dicted bond distances are too small and the bond angle is too wide. The heat of the oxygen transformation to ozone is almost three times higher than the experi- mental value. Contrary to the structure of oxygen

254 B.S. JursiciJoumal of Molecular Structure (Theochem) 389 (1997) 251-256

that does not have polarity, the MP2 calculations reproduce a reasonable structure for ozone. For exam-

ple, when using thf MP2/6-311 + G(2d,2p), the bond distance is O.OllA longer and the predicted bond

angle is the same as the experimental structure. The

enthalpy of the oxygen transformation to ozone is underestimated (Table 1). The best reproducibility

of the experimental result among the DFT methods

is obtained using the non-local method, particularly the BLYP/6_31G(d,p). The predicted bond distance is

0.02 A longer, the bond angle is 1.1” wider and the

obtained energy is almost identical to the experimen- tal value (Table 1).

The structure of hydrogen peroxide is hard to model. In the gas phase, the molecule has a sketch configuration with the O-H bonds possessing a

dihedral angle of 111.5”. This is a result of the repul-

sion interactions between the O-H bonds and the

lone-pairs of each oxygen atom. This form persists

in liquid state and is modified by the presence of hydrogen bonding. In the crystal state, at - 163°C [26], the bond distances and bond angles are changed

due to the strong hydrogen bonding. Now, the H-O

and O-O bond distances are 0.988 and 1.458 A,

respectively. The bond angle is wider (101.9”) with a dihedral angle that is almost perpendicular (90.2”).

The rotation barriers are 7.04 and 1.01 kcal mall’ for

the trans and cis conformations, respectively. How-

ever, a very wide and shallow rotational energy sur-

face indicates that the hydrogen peroxide structure should be very hard to reproduce using the quantum

mechanic methods. This is well demonstrated by the

hydrogen peroxide structures generated using both the

ab initio and DFT methods (Table 2). All of the para- meters are far from the desired agreement between the

theory and the experiment. For example, the best

Table 2

The parameters for hydrogen peroxide calculated using the ab initio and DFT methods

Theory model HO/i 00/A HOO/deg HOOWdeg

RHF/3-21G* 0.970 1.473 99.4 179.5

RHF/6-31G(d,p) 0.946 1.396 102.3 116.3

RHF/6-311 + (2d,2p) 0.942 1.388 103.0 111.3

RHF/6-311 + + G(3df,3pd) 0.942 1.382 103.1 110.5

MP2/3-21G* 0.998 1.544 95.7 180.0

MP2/6-31G(d,p) 0.969 1.461 98.6 120.3

MP2/6-311 + (2d,2p) 0.963 1.460 99.4 114.9

BHandH/3-21G* 0.986 1.477 98.1 180.0

BHandH/6_31G(d,p) 0.959 1.396 101.6 115.8

BHandW6-311 + (2d,2p) 0.955 1.390 102.3 112.0

BHandHLYP/3-21G* 0.985 1.496 97.9 177.0

BHandHLYP/6_31G(d,p) 0.957 1.420 101.1 117.2

BHandHLYP/6-311 + G(2d,2p) 0.953 1.414 101.8 112.9

Becke3LYP/3_21G* 1.001 1.528 96.5 180.0

Becke3LYP/6_31G(d,p) 0.971 1.456 99.8 118.5

Becke3LYP/6-311 + G(2d,2p) 0.966 1.451 100.6 114.4

Becke3P86/3-21G* 0.996 1.518 96.6 180.0

Becke3P86/6_31G(d,p) 0.968 1.440 100.1 117.4

Becke3P86/6-311 + G(2d,2p) 0.964 1.435 100.8 112.9

BLYP/3-21G* 1.016 1.560 95.5 180.0

BLYP/6-31G(d,p) 0.982 1.494 98.5 119.8

BLYP/6-311 + G(2d,2p) 0.977 1.492 99.4 115.5

BP86/3-21G* 0.997 1.518 96.6 180.0

BP86/6-31G(d,p) 0.968 1.440 100.1 117.4 BP86/6-311 + G(2d,2p) 0.964 1.435 100.9 112.9

Experimental [25] 0.950 1.415 94.8 111.5

Contrary to the generated geometries, the predicted rotation barriers are in excellent agreement with the experimental values (Table 3). All of

the predicted energies differ less than 2 kcal mol.‘. It seems that the BLYP produces an energy that is closest to the experimental value for the

rotation barrier through the trans hydrogen peroxide.

B.S. JursiclJoumal of Molecular Structure (Theochem) 389 (1997) 251-256 2.55

agreement for the O-O bond distance was obtained using the RHF and BHandH with the minimal basis set (3-21G*). The agreement is coincidental because the optimized structures using this basis set are always trans regardless of the applied method (Table 2). Sur- prisingly, by applying the 6-311 + G(2d,2p) basis set, an excellent modeling of the dihedral angle was obtained. At the same time, the predicted bond angle and bond distance are not acceptable. Again, the best results are obtained using the MP2 ab initio and the BLYP DFT methods. For example, the BLYP/ 6-311 + G(2d,2p) predicts the bond distances that dif- fer from thf experimental structure by 0.027 (O-H) and 0.017A (O-O). The angle deviation is also considerable. For example, the bond angle and the dihedral angles differ by 4.6 and 4.0”, respectively (Table 2). As demonstrated, the agreement is not an impressive one.

Contrary to the generated geometries, the predicted rotation barriers are in excellent agreement with the experimental values (Table 3). All of the predicted energies differ less than 2 kcal mol-‘. It seems that the BLYP produces an energy that is closest to the experimental value for the rotation barrier through the trans hydrogen peroxide.

4. Conclusion

In conclusion, the BHandH hybrid DFT methods do not present a considerable improvement over the HF ab initio calculations. One can observe a monotonous change in every geometric parameter when using an increasing basis set. The generated geometries and

Table 3

The rotational barriers (kcal mol.‘) for hydrogen peroxide calcu-

lated by using the 6-31G(d,p) basis set

Theory method cis trans

RHF 8.61 0.88 MP2 8.78 0.60

BHandH 9.04 0.97

BHandHLYP 8.58 0.82 Becke3LYP 8.36 0.69 Becke3P86 8.62 0.80 BLYP 8.04 0.58

BP86 8.33 0.69 Experimental [25] 7.04 1.01

energies for oxygen, ozone and hydrogen peroxide using the Becke3 hybrid and the non-local DFT methods are of the same quality, like the MP2 ab initio calculations. An extraordinary improvement over the MP2 calculations using the Becke3LYP and the Becke3P86 hybrid DFT methods obtained earlier for the nitrogen oxides [4,5] was not observed in the case of modeling oxygen, ozone and hydrogen peroxide. Nevertheless, the DFT calculations might be an alter- native to the MP2 ab initio calculations when studying molecules containing mainly oxygen. It was also clear that satisfactory geometric parameters and energies for the studied systems can be obtained using the 6- 31G(d,p) basis set.

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