ХИМИЧЕСКАЯ ФИЗИКА, 2004, том 23, № 2, с. 71-76
ЭЛЕМЕНТАРНЫЕ ^^^^^^^^
ФИЗИКО-ХИМИЧЕСКИЕ ПРОЦЕССЫ
УДК 539.19
POTENTIAL SURFACES OF THE H3 SYSTEM
© 2004 r. M. Jungen*, M. Lehner, R. Guérout
Institut für Physikalische Chemie, Universtität Basel Klingelbergstr. 80, CH-4056 Basel, Switzerland Received 16.11.2002
We wish to draw the attention to a set of ab initio calculations for the potential energy surfaces of about 20 electronic states of H3 and H+ , carried out for a large range of nuclear arrangements corresponding to the dissociation of the metastable molecule. In the text we give a qualitative discussion of the properties of the surfaces and we present calculations for the vibrational levels of the 2sa1 excited state.
1. INTRODUCTION
It is well known that in contrast to the H+ positive ion the H3 molecule is not stable but dissociates to H2 + H. However more than 20 years ago Herzberg [1] detected emission spectra between long lived metastable excited
electronic states of H3, consisting of a H+ core and a Rydberg electron. Today the interest for H3 has two motivations: Spectroscopy of the metastable "Rydberg radical" [2-4] and the process of dissociative recombination when a H+ ion captures an electron and the resulting H3 intermediate subsequently dissociates [5-7]. The latter phenomenon which is still not fully understood [8] is of high astrophysical interest. In both cases dynamical processes such as predissociation and JahnTeller coupling must be treated and thus information about the electronically excited states of the H3 system is needed. The same is true for the three-body breakup of metastable H3 molecules observed by Müller et al. [9].
It is the purpose of the present study to provide potential energy surfaces for a considerable number of excited electronic doublet states of H3, of course including the ground state, in a large region of possible nuclear arrangements from the metastable Rydberg radical to dissociation. Quartets are omitted for the present because they seem to be of lesser importance for the process of electron capture by a stable H+. Our calculations should extend the geometrical range and the choice of excited states of our earlier work for H3 where H3 had been studied first only at and later also near the
H+ equilibrium geometry [4, 10, 11] and where the
work was restricted to the npa'2 Rydberg series [4] or to quartet Rydberg states [12].
In this paper we shall discuss qualitatively some important properties of the calculated surfaces. Although
* Corresponding author (Martin. Jungen@unibas.ch).
the figures and diagrams are restricted for convenience to C2v nuclear arrangements, the electronic states will be characterized, depending on the context, by their symmetry behavior, i.e. the appropriate representation in the D^h, D3h, C2v or Cs point groups. Furthermore we shall use the symmetry symbols for the many-electron states (upper case letters) as well as for orbitals (lower
case). Because the parent ion H+ is a small equilateral triangle in its equilibrium geometry we refer to the united atom Li for characterizing the electron orbitals. An nlx terminology will be used where n and l designate the principal and angular quantum numbers of the corresponding state of the Li atom and % is the group theoretical character of the Rydberg orbital. In the regions close to dissociation this will sometimes be replaced by a characterization of the electronic state of
the fragments H2 or H+ and H. Thus in different regions of the coordinate space different names will be used for the same electronic surface. This may be puzzling for the unprepared reader. We therefore recommend to consult the correlation diagram in Fig. 2 where the various designations are connected with each other.
2. CALCULATIONS
In contrast to recent studies for H+ [13] the surfaces should not be expanded in terms of a system of predefined functions, using only a modest number of theoretically determined energies. The values have been calculated point by point on a set of superimposed equidistant three dimensional grids with different spacings, which should allow reliable interpolation.
As the underlying problem is the dissociation of H3 to H2 + H, mass scaled Jacobi coordinates are an appropriate choice for describing the system. If s is the bound length of the diatom H2 and S the distance from the midpoint of the diatom to the third nucleus, the coordinates are R = 54(4/3), r = s4(3/4) and 9 is the angle
R ■■
1 2 3 4 5 6
r, a0
Fig. 1. Potential energy curves of H+ : Three sections through the potential surfaces for C2v symmetry (0 = 90°) and R = 0, 1.53ao, 10ao (see text).
between the two vectors r and R. With these coordinates the equilibrium geometry of the ion H+ (an equilateral triangle with bond lengths of 0.873 A) is at R = = r = 1.5355a0 and 0 = 90°.
We have calculated the surfaces of H3 and H+ for 2366 different geometries, using grids with
Table 1. Ab initio electronic energies (a. u.) of the Rydberg states of H3 at the equilibrium geometry of H+
State Present Ref. [19] Ref. [4]
2pe' -1.55913 -1.55518 -1.56197
2sa1 -1.47898 -1.47800
2p a2 -1.47528 -1.47123 -1.47778
3pe' -1.41233 -1.41217 -1.41751
3sa1 -1.39734 -1.39743
3P a2 -1.39834 -1.396 -1.40131
3de' -1.39518 -1.395
3de" -1.39596 -1.394
3d a1 -1.39339 -1.393
4pe' -1.37629 -1.375 -1.38162
H+ -1.34141 -1.34384a) -1.34315
a) Reference [13].
• R: 27 values between 0.0 and 9.9a0,
• r: 16 values between 0.9 and 5.9a0,
• 0: 7 values between 0° and 90°.
As usual in quantum chemistry the highly correlated wave functions are expanded in gaussians. For the representation of Rydberg orbitals we used the successful sets introduced by Kaufmann et al. [14]. The basis must be chosen in such a way that the dissociating system can consist of an excited H2 molecule and H(1s) or conversely of an excited H atom and H2 in the ground state. Thus there are two Rydberg basis sets located on different centers. As a consequence, for geometrical configurations where the three nuclei are close to each other this basis may be linearly dependent because of too many nearly coinciding diffuse functions. Therefore the choice of the gaussian basis is slightly different for
different geometries. As H+ and H3 have only a few electrons the method of choice is CI (configuration interaction) calculations using our selfmade valence CI program for the calculation of electronically excited
states. For H+ this comes close to full CI. For neutral H3 on the other hand some compromise had to be accepted in order to not too much slow down the computations. As a consequence the result accounts for part of the correlation energy only; typically the lowest two electronic states should be corrected by about 800 cm1, the highest calculated Rydberg states by 200 cm1. Therefore with a limited series of extensive CI calculations carried out on a coarse grid only, an additional set of reference energies is constructed which allows an additional scaling correction of the calculated surfaces.
For the H+ equilibrium geometry these reference energies are collected in Table 1.
15 electronic surfaces of H3 have been calculated; the number of reliable results depends slightly on the coordinates. For the equilibrium geometry of H+ they range from 2pe' up to 4pe'; this corresponds roughly to the combinations H2: X1 £+ + H(n) with n up to 3 and H2 + H(1s)
up to H2: a3 Z+ as well as H(1s) + H(1s) + H(1s) of the dissociating system. Similarly three singlet and two triplet surfaces of H+ have been treated, corresponding
at equilibrium to the 1A'1 ground state and two components of both the lowest 3E and 1E states.
Thus this study at present encompasses a total of more than 45 000 single quantum chemical results for different geometries and various electronic levels of
both H+ and H3. In the remaining sections we try to introduce this huge material by discussing qualitatively a series of graphical representations.
28200 4X+
14B2
,7 12A2, 3-42B1 7
6-92A1; 4-52B2
12A1
H2(X) + H(n = 3) 60200
26300 4pe' 24A11
22400 3da{ 62A1 ^.'/ - 32B1, 12A2 ■'/ / 52A1 J — H2(3V) + H(1s) 58600
22350 3de" 14A1, 14Br,
22300 3pa2 22B1 ~ ~ ~ - __ _ 42A1, 22B1 J — H2(3n„) + H(1s) 58500
22100 3de' 52A1, 32B2 ~ ___ - - 32B2 — H^V) + H(1s) 54400
21700 18900 3sa{ 3pe' 42A1 ^^ 32A1, 22B2 12B1, 22B2 1 ^ 2-32A1 J H2(X) + H(n = 2) +45000
7000 2pa2 - 12B^ " " 14B2 ,
+4700 2sa{ 22A1 ^^ . - 12B2^ — H2(3Z„+) + H(1s) 0
(-13000) 2pe' - 12A1, 12B2
Equilibrium of H3+ (D3h)
Energies: cm 1 with respect to H + H + H
H2(X%+) + H(1s) -37300
Dissociation to H2 + H
Fig. 2. Correlation diagram in C2 v symmetry for the electronic states of H3 between the equilibrium geometry of H+ (left, including
the lowest quartet state of linear H3) and the dissociation to H2 + H fragments (right). The estimated energies are given in cm-1 with respect to the H + H + H limit (-1.5 a.u.). With the circle a predicted conical intersection is marked (see text).
3. THE PARENT ION H+
A discussion of the potential energy surfaces of Ry-dberg states requires the knowledge of the potential of the correponding parent ion to which the Rydberg series converge. If several electronic states of the parent ion are energetically close to each other, the Rybderg states are likely to be mixtures of configurations with different ionic cores and Rydberg orbitals. As we shall see in the following this is the case only for large values of R or r.
Here we present a view of the lowest electronic surfaces of the ion H+. Figure 1 visualizes the potentials in C2v symmetry, showing sections along r for three different values of the dissociation coordinate R.
R = 0 means linear H+. It has been established theoretically that the lowest triplet state 3X+ has a stable linear minimum with a H-H distance of 1.3 A [15]; Ahlrichs et al. have estimated a dissociation energy (including the correction for zero point vibration) of 6.8 kca
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