Was H2− observed in solid H2? A theoretical answer

Was H 2 observed in solid H2? A theoretical answer. H.U. Suter, B. Engels Institut fiir Physikalische und Theoretische Chemic, Universit/it Bonn, Wegelerstrasse 12, D-53115

Bonn, Germany

and S. Lunell Department of Quantum Chemistry, University of Uppsala, S-75120

Uppsala, Sweden

Abstract Density Funtional Theory (DFT) and ab initio CI cluster calculations were performed to simulate the hyperfine coupling constants of charged hydrogenic systems in solid hydrogen. Possible clusters are the complexated H+ and H2 ions. The isotropic hyperfine coupling constant in a H + cluster was found to be around 572 MHz, in excellent agreement with the experimentally observed 569 MHz. The convergence upon the size of the clusters studied was investigated with two different H~4 clusters. ©2oolbyAcademicPress.

Contents 1

Introduction

2

2

Methods of Calculation 2.1 AO basis set dependence . . . . . . . . . . . . . . . . . . . . . 2.2 Model systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Theoretical methods . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 4

3

Results

6

4

Summary and Conclusions

7

5

Acknowledgment

8

ADVANCES IN QUANTUM CHEMISTRY,VOLUME 40 Copyright ~12001 by Academic Press. All rights of reproduction in any form reserved. 0065-3276/01 $35.00

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Introduction

The investigation of solid H2 has been of growing interest in recent years. Solid H2 exist in the two forms para-H2 and ortho-H2, corresponding to the spin states of the two nuclei. The percentage of p-H2 varies from 99.8 % at 20 K to about 25 % at room temperature. The solid is a molecular crystal with strong covalent bonds and weak intermolecular bonds. The solid phase is hexagonal close packed with an a of about 3.6/~, although a Pa3 structure is also known at very low temperatures [1]. Recently Miyazaki et al. [2] did electron spin resonance (e.s.r.) experiments on v-irradiated solid H2. They argued that the measured signal at 12.4 Gauss is due to an electron trapped at a distance of 13.8 ft. with a p-H2 molecule. The distance was obtained from a formula due to Gordy and Morehouse [3]. Their conclusion was challenged by Symons [4], who claimed that they indeed may have observed H 2. From the observed splitting of 12.4 Gauss he derived a hyperfine coupling constant (hfcc) of about 203 Gauss=568.9 MHz. He excluded H + whose hfcc is well known (both experimentally [5] as well as theoretically [6]) to be at 912.3 MHz. Symons claimed that the difference to this value could not be due to shielding effects of the matrix. The same naturally holds for free hydrogen atoms. This argument was adopted in the recent e.s.r, experiments of Miyazaki and coworkers [7, 8]. In a recent paper, however, Symons and Woolley [9] have reexamined the experimental evidence by Miyazaki et al. and have suggested that the observed e.s.r, spectra should be assigned to a H + radical centre rather than H~. They also report electronic structure calculations on a variety of dihydrogen cluster radical species, that give strong, albeit indirect support to the revised interpretation. It is well-known that free H~ is not stable [10]. Therefore, if H~- was observed it was due to the stabilisation of the surrounding environment and one may better speak of a H~n species. The H + on the other hand will form H+n species. The question whether this kind of cluster remains stable as a free cluster is not the subject of this investigation. Instead, the aim of the present investigation is to directly verify or falsify the different suggested interpretations of the observed spectra by calculating the hfcc's of the possible clusters. The use of cluster models in the prediction of the hyperfine structure has turned out be highly successful, mainly due to the local nature of the hyperfine interaction. One example of such a correct prediction was the calculation of the hyperfine parameters of anomalous Muon [11] in diamond.

Was H;)Observed in Solid H2?A Theoretical Answer

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R1

R2

Figure 1: Structure of the cluster number 1.

2 2.1

M e t h o d s of C a l c u l a t i o n AO basis set dependence

Although the dependence of the hyperfine properties, such as the isotropic hfcc, on computational method and/or basis sets has been extensively studied in the literature [12], it is in the present connection instructive to examine specifically the performance of different types of hydrogen atom basis sets. This will also give us some insight in the possible errors of the reported calculations. H + has the advantage that the effect of electron correlation is not present in this system and hence the error of the Hartree-Fock calculation is identical with the error due to the AO basis set. 2.2

Model

systems

Three H2 clusters were used in this study. The first one was obtained by simply adding two hydrogen molecules normal to the ends of the central hydrogen molecule. This cluster is planar and depicted in Figure 1. Although this cluster seems inadequate to describe solid hydrogen, it will turn out that the hyperfine parameters are already described correctly with this model system. The next cluster is obtained by adding 4 additional hydrogen molecules normal to the axis defined by the central hydrogen molecule. The final cluster was inspired by the hexagonal-closed packed crystal structure, and represents one plane of the elementary cell. This cluster is depicted in Figure 2.

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Figure 2: Structure of the cluster number 3 (hcp structure). Due to the large flexibility of hydrogen molecules in the solid hydrogen, it was decided, contrary to the normal use of cluster models in solid state problems, to optimize the clusters fully, e.g. not to use the crystal parameters from Xray or similar experiments. 2.3

Theoretical

methods

Density functional calculations (DFT) using the B3LYP functional [13], and Moller-Plesset (MP) calculations were performed with the Gaussian94 program [14]. For some geometry optimizations the aug-cc-pVTZ basis set of Dunning and coworkers [15] was used. For the spin density calculations the AO basis set consists of the standard Chipman basis [16], augmented with a tight s function with an exponent of 85.087. To see the reasoning behind the use of the augmented Chipman AO basis set we have performed test calculations on the H + radical (cf. Table 1). The CI calculations were performed with the MELDF-X series of programs [17]. The selection of the reference space was done as usual [18, 19]. For H~-, to quote one example, this resulted in 25 reference determinants and 79180 configurations. Due to this relatively small number the diagonalisation was done without selection. For H~-4 and H+4 only CIS (CI with singles) was done.

Was H~Observed in Solid H2? A Theoretical Answer

AO basis set STO-3G STO-4G STO-5G STO-6G 3-21G 6-31G 6-31G** STO-6G 6-311++G** 6-31++G** 6-31++G(2df) DZ DZP Chipman Chipman+Peak cc-pVDZ aug-cc-pVDZ cc-pVTZ aug-cc-pVTZ cc-pVQZ aug-cc-pVQZ cc-pV5Z aug-cc-pV5Z exact [6]

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Fermi contact term 0.1641 0.1901 0.2063 0.2172 0.1739 0.2131 0.2109 0.2172 0.1919 0.2083 0.1906 0.2073 0.2048 0.1965 0.2063 0.1800 0.1762 0.1905 0.1906 0.1954 0.1957 0.2034 0.2034 0.2041

Table 1: The isotropic hfcc of H + for different AO basis sets at the bond distance of r=2 a.u. (0.2041 a.u. = 912.3 MHz).

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Molecule Method H~ MP2/aug-cc-pVTZ H6QCISD/aug-cc-pVTZ H6QCISD/Chipman+Peak H+ B3LYP/aug-cc-pVTZ H+ QCISD/Chipman+Peak H+ MP2/aug-cc-pVTZ

R2 2.955 2.957 2.646 1.707 1.662 1.678

R0 0.739 0.744 0.741 1.041 1.026 1.046

R1 0.744 0.749 0.760 0.793 0.787 0.785

Table 2: Geometry of H~

B3LYP CIS CISD MR-CISD

Aiso(H0) Tiso(H0) 562.13 564.90 69.07 574.62 68.09 571.61 68.09

Aiso(H1) 288.24 255.42 246.61 253.36

Table 3: Isotropic hfcc of hydrogen in H +

3

Results

H~ is, as mentioned above, unstable with respect to ionisation. The calculated isotropic hfcc will tend to zero with increasing basis set size. (The actual value for the Chipman AO basis set is 22 MHz, due to the missing Rydberg type funtions.) To obtain a stabilisation of the charged H~, cluster 1 consisting of 3 H2 moleules in a row was investigated. The H2 are thought to rotate freely in solid hydrogen. The cluster used is pictured in Figure 1. In Table 2 the optimized geometries of H~ and H + are depicted. The two outer H2 molecules were forced to be normal to the H2 chain, while the central H2 was chosen to be horizontal to the chain, as shown in Figure 1. The optimized geometry (with MP2/aug-cc-pVDZ) is more compact than expected from a comparison with solid hydrogen, where the center-center distance is around 3.5/~ [1]. For H +, we found r=1.7/~ and for H~ we found r=3.0 A. In the case of H +, our calculated geometry is very close to those reported by Kurosaki and Takayanagi [21] and by Symons and Woolley [9]. The hyperfine structure of H+ is depicted in Table 3. The value is converged to about 572 MHz, which is reasonably close to the experimental 569 MHz. The isotropic value of the hfcc for H~ is, on the other hand, close to zero for

Was H~ Observed in Solid H2?A Theoretical Answer

atom H0 H1 H2 H3

139

x y z 4-0.498201 0.0 0.0 4-1.378683 1.567917 4-0.372264 4-1.378683 -0.461568 4-1.543988 4-1.378683 -1.106349 4-1.171724

Table 4: Coordinates in/~ngstrSm for the optimized hcp structure (MP2).

both the two lowest lying states. We have further tested the effect of the size of the cluster by adding 4 more H2 molecules to the H~ cluster and found essentially the same difference between the positive and negative clusters, already with the CIS calculations. An attempt was also made to optimise H~4 clusters with a forced hcp structure (see Figure 2). The optimized geometry of the positive cluster is given in Table 4 and the hfcc's of both the positive and negative hcp clusters in Table 5. H~ surrounded by H2 in a hcp structure gave an Also of about 450 MHz, which is too small to account for the experimental observations. H+ in the hcp structure has an isotropic hfcc of about 800 MHz, which is too large compared to experiment, but smaller than the 912 of free H+. In this case, neither the positive nor the negative cluster gave a satisfactory agreement with experiment. It has, however, been found that an H+4 cluster which is optimized without constraints will distort to a configuration consisting of a central H+ cluster and four outer H2 weakly bound to the four corners of H+ [22]. The assumption of a hcp cluster is therefore unrealistic, considering the high mobility of the H2 molecules in the solid. Considering the results for all clusters it is evident that the anion has in all cases a too small isotropic hfcc. This is physically quite easy to understand, since the H~- is not stable and therefore the electron will be distributed among many H2 molecules. The cation on the other side has the tendency to build stable clusters which are more compact than the crystal structure. According to these calculations, the observed e.s.r, signals should be related to the cation instead of the suggested anion.

4

S u m m a r y and Conclusions

Density Functional Theory (DFT) and ab-initio CI calculations were performed on different ionized clusters of hydrogen (H2), in order to explain

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H~-(H2)6 UHF B3LYP MP2 QCISD CIS

Hnn

H

6.0 9.4 6.1 7.3 5.3

486.9 454.9 451.1 435.1 448.7

H +(H2)6 UHF B3LYP MP2 UQCISD CIS CISD MRDCI6 MRDCI7 MRDCI85

12.9 755.8 86.1 588.3 747.5 659.7 60.7 821.3 57.3 809.8 5 7 . 4 809.7 6 1 . 2 800.9 6 1 . 8 799.0

Table 5: Isotropic hfcc of hydrogen in hcp structure

the recent e.s.r, experiments of Miyazaki et al. [2]. These results could be explained by a small cluster containing H+ surrounded by two H2 molecules. The calculated Also of the H atoms in H+ is found to be around 570 MHz, for both DFT and MR-CI methods. The Aiso of the anion is much smaller, and to obtain a hfcc which is in agreement with the experiment it would be necessary to elongate the H~ bond distance considerably. The conclusion that can be drawn from the present calculations is that the e.s.r, signals of Miyazaki et al. [2] can be assigned to a H+ ion surrounded by H2 molecules, of which two are more tightly bound to the central ion than the rest, forming a clearly identifiable H+ unit, fully in accordance with the suggestion by Symons and Woolley [9].

5

Acknowledgment

H.U.S. wishes to thank the DAAD and for a travel grant under the Project 313/S-PPP-3/95. Some of the calculations were performed at the "Centro Svizzero di Calcolo Scientifico" (CSCS) in Manno (Switzerland).

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H2?A Theoretical Answer

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[16]

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[17]

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