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ECCC3 Paper 32

OPTIMIZED GEOMETRIES and ENERGIES.

The ground state equilibrium geometry pertains to the C3v group, and is pyramidal. The ground state reference electronic configuration is : (1A1)�, (2A1)�, (1E)�, (2E)�, (3A1)�, (4A1)1. The (1E) and (2E) molecular orbitals (MO) are degenerate.

In table 1, are indicated the compositions of the various basis sets we employed. We started with known valence basis sets which we completed by diffuse gaussians on the oxygen and the hydrogen atoms. We employed three small basis sets : 4-31G/R1, 4-31G/R2, D95**/R3 and one medium-sized D95**/R4 base, and four larger basis sets : VD/R5, VD/R6, VD/R7, DH/R8.


Table 1. Basis set composition.

Code name Valence base Diffuse gaussians
4-31G/R1 4-31G O s 0.10 0.050 0.025 p 0.025
4-31G/R2 4-31G O s 0.10 0.061 0.024 p 0.0845 H 0.040
D95**/R3 D95** O s 0.061 0.024 p 0.059 0.028 d 0.015 H s 0.08 p 0.08
D95**/R4 D95** O s 0.10 0.061 0.024 0.0066 p 0.059 0.028 d 0.015
H s 0.08 p 0.08 s 0.025
VD/R5 O VD [13s/8p --> 8s/4p] 4 s ; 4 p even-temp. ; 4 d (2.0 0.6 0.15 0.007)
H VD [8s --> 5s] 2 s even-temp. 2 p 1.4 0.25
VD/R6 O VD [13s/8p --> 8s/4p] 4 s ; 4 p even-temp. ; 4 d (2.0 0.6 0.15 0.007)
H VD [8s --> 5s] 2 s even-temp. ; 4 p (4.15 1.06 0.27 0.07)
VD/R7 O VD [13s/8p --> 8s/4p] 4 s ; 4 p even-temp. ; 4d (2.0 0.6 0.15 0.007)
H VD [8s --> 5s] 2 s even-temp.; 4 p (4.15 1.06 0.27 0.07) ; 2d ( 2.27;0.56 )
DH/R8 O DH [10s/5p --> 3s/2p] 4 s ; 3 p even-temp. ; 2d (2.0 0.33)
H DH [8s --> 2s] 2 s even-temp. + 2 p (1.4 0.25)


In table 2, one can find the optimized geometries (Degree and Angstrom), UHF energies (Hartree), ionization potentials (IP in eV), and dipole moments (DIP in Debye) of H3O., obtained with various basis sets.
Spin contamination was found to be small, for example < S� > =0.752 for D95**/R3 ,DH/R8 and VD/R7.

Table 2. Optimized geometries

Basis set Species Symmetry H-O-H (degree) O-H ( �) Energy (A.U) IP (eV) Dip. moment (Debye)
4-31G/R1 H3O. C3v 118.9 0.995 -76.375 4.84 0.249
4-31G/R2 H3O. D3h 120 0.997 -76.378 4.86 0.009
D95**++ H3O. C3v 108.6 0.986 -76.504 4.89 0.471
D95**/R3 H3O. C3v 108.67 0.986 -76.505 4.89 0.806
D95**/R4 H3O. C3v 108.64 0.987 -76.506 4.915 0.7947
D95**/R4 H3O+ C3v 113.66 0.963 -76.331
VD/R5 H3O. C3v 108.19 0.983 -76.519,071 4.89 0.837
VD/R6 H3O. C3v 108.18 0.982 -76.519,326 4.99 0.857
VD/R7 H3O. C3v 108.56 0.982 -76.521,061 4.88 0.847
DH/R8 H3O. C3v 108.58 0.984 -76.506,763 4.89 0.783
DH/R8 H3O. D3h 120.00 0.981 -76.501,290 4.77 0.0


From these tables, it is possible to acertain the influence of the basis set quality. Polarization functions are absolutely necessary for obtaining a pyramidal geometry. When considering pyramidal geometries, with almost the same angles, the dipole moment varies from 0.4709 Debye to 0.857 Debye. Extra diffuse functions have a dramatic effect on the dipole moment value for example considering D95++** and D95**/R3, while the total energy varies little. The ionization potentials (IP) were obtained via the Koopmams' theorem, and correspond to a good approximation to the vertical detachment energy (VDE) of the outer electron. The adiabatic detachment energy (ADE) was also computed, taking into account the fully relaxed H3O+ cation geometry and is lower than the VDE. For the D95**/R4 basis set, since we optimized also the parent hydronium cation, it is possible to compare the electron affinity (EA = 4.637 eV) of the cation, the vertical detachment energy (VDE=IP= 4.89 eV) and the adiabatic detachment energy (ADE= 4.76 eV)i of the hydronium radical. Gellene and Porter [Gellene 84] estimated that the electron affinity (EA) of H3O+ should be equal to the IP of H3O.. We can see that there is however a difference of 0.278 eV between the computed IP and the computed EA. One part of the difference may due to geometric changes, an another part may be related to electronic relaxation which is not taken into account within the framework of Koopmans' theorem. The IP is not necessarily higher, with a better basis set ( VD/R6 versus VD/R7). The more economical DH/R8 basis set features an IP value very similar to the IP value 4.88 eV of our most expansive basis set.

The inversion barrier, corresponding to the umbrella motion, was estimated around 3.43 Kcal/mole by optimizing the H3O. geometry under a D3h symmetry constraint. It appears therefore that this molecule is more floppy than ammonia (inversion barrier 5.8 Kcal/mole).

In relation to a Fermi contact point ESR analysis, we list the atomic spin densities in table 3. The spin densities on hydrogen atoms belonging to the H3O. radical are small but positive. It is interesting to notice that the spin densities on the hydrogen atoms seems to be quite sensitive to the geometry. In D3h planar or nearly planar geometries, the spin density value nearly triples. We report also the spin densities at the dehydrogenation transition state geometry, on the H2O and H. fragments.

Table 3. Atomic spin densities (Atomic units)

Basis set Geometry Oxygen Hydrogen
4-31G/R1 C3v 0.768 0.016
4-31G/R2 D3h 0.798 0.018
D95++** C3v 0.847 0.005
D95++**/R4 C3v 0.836 0.005
VD/R6 C3v 0.806 0.005
VD/R7 C3v 0.809 0.005
DH/R8 C3v 0.760 0.005
D3h 0.926 0.012
TST H2O 0.807 0.001
TST H. 0.071



It is crucial to examin the MO graphic of the outer electron in order to grasp its physical nature. In our specific case the symmetry-adapted (4A1) HF MO (fig. 1) or the corresponding natural orbital (NO) (fig. 2) are similar,so we may consider the graphical plot of one or the other.


Figure 1.


Figure 2.

This NO wavefunction (cf fig 2) features two clearly distinguishable nodal surfaces ( - . - . -) which separate out regions of alternative phase sign. So it is possible to assimilate grossly this NO and HF MO to a 3s-type supermolecular orbital. However the supermolecular perspective, though well adapted to spectroscopy, does not give any insight if the outer MO is a non-bonding MO or an antibonding MO.

We can see that the outer nodal surface goes between the oxygen atom and the hydrogen atom , therefore the outer MO is an antibonding orbital and as such cannot be assimilated to a pure Rydberg orbital . On the other hand, the outer orbital is somewhat special, by comparison to other valence bonding or antibonding orbitals, because it is so diffuse.


Figure 3.


On the total electron density map (cf. fig 3 ), we can clearly distinguish a high electron density region pertaining to the H3O+ core, and a low density, diffuse region related to the unpaired outer electron. One way to assess this diffuse character is to compare with orbitals of isolated atoms. The H3O. radical is isoelectronic to sodium (Na), which comprises a lone electron in a 3 s orbital (cf fig 4).



Figure 4.


Remarkably enough the 3s electron of the sodium atom is the most diffuse outer electron of all atoms. One can easily check that the various Na basis sets feature the smallest gaussian exponents, by consulting a handbook [Poirier 85] of gaussian basis sets. For convenience, we are going to designate, when we want to stress the diffuse character, the outer MO as the low-lying Rydgerg orbital, or the outer electron as the Rydberg electron, although we must keep in mind the antibonding character.

An antibonding character does not contribute to molecular binding, but it is yet another question to determine if this antibonding character is going to strongly undermine the electronic glue built by inner bonding MOs. We suggest that an easy and intuitive way to judge of the dissociative or the non-dissociative character of an antibonding molecular orbital, is to inspect MO graphics with a sufficient contour resolution. A totally nondissociative antibonding MO occurs when the nucleus position coincides with an electronic distribution barycenter. However, in the present case, we can measure on the graphics plot, that the maximum of the electronic density created in the third phase region of the NO or the 4A1 wavefunction, is situated approximately 0.14 � to the exterior of the hydrogen atom or 1.12 � from the central oxygen atom. (cf fig. 2, fig 3). In order to lower its energy within the specific outer wavefunction, the Hydrogen atom will tend to be propelled outside, towards this electron density maximum. As we shall see later, it is quite interesting to see that precisely this distance of 1.12 � corresponds to the O--H. distance in the transition state geometry towards dissociation.

Considering now an isolated hydronium radical in condensed phase, we might expect that, because of electronic repulsion due to surrounding electrons from neighbour molecules, the barycenter in the third phase region in the 4A1 MO might be displaced towards the proton, therefore reducing the dissociative character of the 4A1 orbital, and would contribute to increase the dissociation barrier of this radical in condensed phase.
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