Theoretical Study of Solid Hydrogens Doped with Atomic Oxygen, Lab Reports of Probability and Statistics

Download Theoretical Study of Solid Hydrogens Doped with Atomic Oxygen and more Lab Reports Probability and Statistics in PDF only on Docsity! A theoretical study of solid hydrogens doped with atomic oxygen Zhiming Li and V. A. Apkarian Department of Chemistry, University of California, Irvine, California 92697-2025 Lawrence B. Harding Theoretical Chemistry Group, Argonne National Laboratory, Argonne, Illinois 60439 ~Received 18 July 1996; accepted 8 October 1996! Structure and reaction dynamics in solid H2/D2 doped with O~ 3P , 1D , 1S! is investigated through simulations based on accurate ab initio potential energy surfaces. The ab initio calculations are performed at MCSCF level, with neglect of spin–orbit interactions. The dynamical simulations rely on nonadditive effective potentials, taking into account the anisotropy of the open shell atom by using diabatic representations for the globally fitted potential energy surfaces of O–H2. The ground state of the doped solid is well described as O~3P! isolated in para-H2~J50! since the atom– molecule interaction anisotropy is not sufficient to orient H2. O~ 3P! atoms radially localize the nearest-neighbor shell, and lead to a linear increase in the density of the solid as a function of impurity concentration. The doped solid is stable at cryogenic temperatures, with a free energy barrier for recombination of next nearest-neighbor O~3P! atoms of 120 K. The solid state O~1D!1H2 reaction is considered in some depth. While in high symmetry sites the reaction is forbidden, even at 4 K, thermal fluctuations are sufficient to promote the insertion reaction. © 1997 American Institute of Physics. @S0021-9606~97!00103-7#I. INTRODUCTION Solid hydrogens doped with atomic oxygen have long been identified as potential cryogenic propellants with en- hanced specific impulse.1–4 Yet, little is known about the cryo-chemistry or the physical properties of such doped sol- ids. The relevant chemistry in this case is the reactivity of the guest with the host, O1H2, and the recombination of guest atoms, O1O, after detrapping. As one of the more funda- mental prototypes in chemistry, the O1H2 reaction has been extensively studied in the gas phase.5,6 However, transferring of knowledge from these high temperature studies to cryo- genic conditions is not straightforward. The requirements for characterization of chemistry at the cryogenic temperatures relevant to solid hydrogens, T,10 K, are severe with respect to the demand on accuracy of potential surfaces. As an ex- ample, it is well established that the reaction of O~1D! with H2 has a gas kinetic cross section, 7 and most calculations agree that if a barrier exists on this surface it is of the order of kT or less.8 Yet, evidence has been given that O~1D! atoms vertically prepared by the radiative O~1S!→O~1D! transition in solid hydrogen at 4 K, do not react with the host.9 The evidence for this conclusion is based on the ob- servation that the atomic fluorescence does not bleach. In- deed if the barrier to this reaction were of the order of 100 K, it would be perfectly ignorable in high temperature studies, and would be outside the reliability of most potential energy surface calculations. Yet, at the temperatures of relevance to solid H2, a 100 K barrier would be insurmountable. A care- ful exploration on state-of-the-art potential energy surfaces was therefore deemed necessary for making reliable predic- tions with respect to chemistry, and to rationalize the recent experimental studies. The second reaction of concern, the O1O recombination, would be expected to be controlled by the mobility of the guests in the solid. The mobility of a942 J. Chem. Phys. 106 (3), 15 January 1997 0021-9606/9 Downloaded¬12¬Feb¬2004¬to¬128.200.47.19.¬Redistribution¬subjeclassical impurity in a quantum host,10 of which solid hydro- gen is a good example; or for that matter, any molecular level description of dynamics in quantum many-body sys- tems, presents serious conceptual challenges for which there are few tried theoretical tools. This consideration is another prime motivation for our studies. In recent years, there has been considerable experimental and theoretical progress in studies of chromophores isolated in quantum hosts, in solid H2, 11 in solid and superfluid He,12 and in large clusters of He.13 A major aim of these studies is the elucidation of structural and dynamical peculiarities of such media. As far as solid hydrogens are concerned, one of the better investigated systems has become the electronic spectroscopy of Li atom doped solid hydrogens, which was studied experimentally by Fajardo et al.14 and subsequently scrutinized theoretically by several groups.15,16 Whaley et al. studied the ground states of several large doped quantum clusters by quantum Monte Carlo methods.17 Diffusion Monte Carlo methods pioneered by Anderson,18 have been used to calculate the ground state of doped clusters, and by taking advantage of the symmetry of the node in the first excited state, the wave function for the vibrationally excited state and therefore infrared spectra of impurities isolated in clusters has been obtained.19 The same methodology had ear- lier been used by Buch in calculations of mixed ortho and paraclusters of hydrogen.20 Such simulations can be used to determine approximate line shapes for electronic excitation, and the calculations of this property were pursued for Li impurities in para-H2 clusters, 15 and in solid hydrogen.16 The Monte Carlo methods, in effect, are equilibrium calculations of stationary states, usually limited to the ground state.21 A rather simple method for dynamical simulations in solid hy- drogen, which incorporated the effect of zero-point energies through Gaussian convolution of pair potentials was imple-7/106(3)/942/12/$10.00 © 1997 American Institute of Physics ct¬to¬AIP¬license¬or¬copyright,¬see¬http://jcp.aip.org/jcp/copyright.jsp 943Li, Apkarian, and Harding: Solid hydrogens doped with atomic oxygenmented earlier by us, and was applied for obtaining structural and dynamical information on O2 doped solid hydrogens. 22 Although we introduced our method in an ad hoc fashion, it finds justification in the centroid dynamics rigorously devel- oped by Cao and Voth,23 who have given an alternate method for constructing pseudo potentials,24 and used them in simulations of liquid para-hydrogen.25 Despite the ap- proximate nature of our treatment, good agreement was found with the structural determinations of pure and doped hydrogens obtained by the more accurate path integral methods,15,16 and correct predictions were made for the fluo- rescence polarization of O2 isolated in solid hydrogens. 26 A further refinement of the method, based on the construction of effective potentials that reproduce not only structure but also the correct phonon spectrum, was more recently ad- vanced in the treatment of impurity rotations in quantum solids.22~b! Here, we combine the method of effective poten- tials with a treatment of the many-body interactions of the open shell O atom, which was previously implemented in the study of oxygen atom mobilities in rare gas solids.27 The latter treatment is based on the construction of adiabatic po- tential energy surfaces from experimentally determined an- isotropic pair potentials.28 The present treatment is similar in spirit, the major difference being that in the case of O–H2 interactions, in addition to the angular anisotropy of the elec- tronic distribution on the O atom, the orientational anisot- ropy of H2 molecules needs to be explicitly considered. As we will argue, based on analysis of the O–H2 pair, angularly averaged pair potentials are justifiable in this application. Limitation of the treatment to the J50 free rotor states of H2, renders the description of the many-body O–H2 interac- tions to be identical in form to that of O–rare gas interac- tions. With the method justified, we proceed to examine the effects of the O impurity on the structure of solid H2, mo- bility of O~3P! in the lattice, and reactivity of the excited O~1D! atom with the host. The rest of the paper is organized as follows. Section II describes the ab initio potential surfaces, and details of the fitting procedure in a diabatic representation. Based on these surfaces, in Sec. III, we present a simple particle-in-a-ring model to establish the validity of a free rotor description of the host molecules in the presence of O~3P!. With that justi- fication, and noting that in the presence of the paramagnetic impurity the host will convert to its ground state, in the rest of the paper we treat H2 as spherical particles. The method employed for the simulations in this limit is outlined in Sec. IV. The structural properties of O~3P! in solid D2 and the reactivity of O~1D! in the same host are discussed in Sec. V. Conclusions are given in Sec. VI. II. POTENTIAL ENERGY FUNCTIONS A. Ab initio calculations The calculations reported here are of the multireference, configuration interaction ~MR–CI! variety, in which orbitals are first optimized using a state-averaged, multiconfigura- tion, self-consistent field ~MCSCF! calculation and then these orbitals are used in a subsequent MR–CI calculation.J. Chem. Phys., Vol. 106, Downloaded¬12¬Feb¬2004¬to¬128.200.47.19.¬Redistribution¬subjeTo optimize the orbitals, separate, singlet and triplet, state averaged, 10 orbital, 8 electron, complete active space self- consistent field ~CASSCF! calculations were carried out us- ing the MOLPRO29,30 program package. For the singlet sur- faces, the state averaging was done over the five lowest states @corresponding to the five components of the O~1D!1H2 asymptote#. For the triplet states the state aver- aging was done over the three lowest states @corresponding to the three components of the O~3P!1H2 asymptote#. In all cases, all states were weighted equally in the averaging. In these calculations the 10 active orbitals consist of the oxygen 2s , 2px , 2py , 2pz , 3s , 3px , 3py , and 3pz orbitals and the HH, s , and s* orbitals. Tests using smaller active spaces were found to be unsatisfactory due to problems resolving the 1s , 2s , and 2p orbitals and/or to inequivalent p orbitals at large RO–HH separations. For the singlet surfaces the above CASSCF wave function includes 13 860 configurations. This is far too many configurations to use as a reference space for the CI calculations. In order to make the CI calculations feasible a smaller reference space was used consisting of a 6 orbital, 8 electron CAS in which the four lowest occupation natural orbitals from the 10 orbital CAS ~corresponding to the oxygen n53 orbitals! are removed from the active space. Thus the active space for the CI calculations consists of the following orbitals: oxygen, 2s , 2px , 2py , and 2pz and HH, s , and s*. Tests using the MOLPRO29,31 internally contracted CI method to do the CI calculations proved to be unsatisfac- tory because at large RO–HH separations the five components of the O~1D!1H2 asymptote were predicted to be signifi- cantly nondegenerate ~apparently an artifact of the contrac- tion scheme!. Consequently it was deemed necessary to do uncontracted CI calculations using the COLUMBUS32 program package. The effects of higher order excitations ~beyond singles and doubles! were estimated using the normalized, multireference Davidson33 correction. The calculations were carried out on a 10 processor, IBM-SP computer. Calculations were done using three different basis sets to test the sensitivity of the results to the size of the basis set. The basis sets used are the Dunning,34 correlation consistent, augmented, polarized double zeta, triple zeta, and quadruple zeta ~aug-pvdz, aug-pvtz, and aug-pvqz! basis sets. These basis sets all include diffuse functions to improve the accu- racy at large distances. All of the dynamical results reported here employ fits to results obtained with the aug-pvtz basis set. Calculations with the aug-pvdz and aug-pvqz basis sets were used only to check the convergence of the calculated relative energies with respect to changes in the basis set size. In the long-range regions of interest here only small differ- ences were found between the aug-pvdz and aug-pvtz results. Calculations with the aug-pvqz basis set were only be done in regions of high symmetry. For the C2v approach the aug- pvqz calculations were found to yield interaction energies ;2 cm21 less attractive than the aug-pvtz calculations for O–H2 distances between 5 and 8 au. For shorter distances, the aug-pvqz results become substantially more attractive than the aug-pvtz results. However, these regions are not relevant to the dynamics studied here. Contour plots of the six singlet potential surfaces areNo. 3, 15 January 1997 ct¬to¬AIP¬license¬or¬copyright,¬see¬http://jcp.aip.org/jcp/copyright.jsp 946 Li, Apkarian, and Harding: Solid hydrogens doped with atomic oxygenIII. ROTATIONALLY ADIABATIC POTENTIAL ENERGY SURFACES FOR O(3P)–H2 In the ground electronic state of the system, magnetic dipole coupling between O~3P! and H2, will lead to spin relaxation of the nearest-neighbor molecules within ;0.1 s.40,41 All subsequent studies of the doped solid will therefore be limited to O atoms isolated in a cage of para-H2 ~ortho-D2!, i.e., to molecules with only even rotational states.40 The anisotropy of the O~3P!–H2 interaction with respect to the H2 orientation will determine whether the nearest-neighbor H2 molecules can be regarded as free ro- tors, or strongly hindered rotors. In the former case, in the J50 ground state, only the spherically averaged potentials have any meaning, while in the case of strong hindering, the angle dependence should be explicitly taken into account. The anisotropy of the O~3P!–H2 interaction can be gauged from Fig. 4, where the adiabatic potential energy curves are shown, for collinear, VS(R), and T-shape, VP(R), approaches. The spherically averaged potential V j50(R) is also shown in the same figure. This potential supports two stretching states of the O~3P!–H2 complex, as shown in Fig. 5. The dissociation energy is approximately 50 cm21, and the equilibrium distance is 3.2 Å. Note, that the v50 wave func- tion spans the minima in both ‘‘T’’ and linear geometries. Fixing the distance of the O–H2 complex at the minimum of the orientationally averaged potential, we solve the one- dimensional rotational Hamiltonian H52 \2 2I ]2 ]u2 1V~R0 ,u!, ~8! in which I is the moment of inertia of H2 and V(R0 ,u) is the rotational potential at R053.2 Å, by expanding the wave function in the even plane harmonics FIG. 4. The lowest adiabatic potential curves of O~3P!–H2 for linear, T-shaped, and spherically averaged geometries ~see the text for details!.J. Chem. Phys., Vol. 106, Downloaded¬12¬Feb¬2004¬to¬128.200.47.19.¬Redistribution¬subjec~u!5( m am A2p e2imu; m50,62,64,.. . . ~9! The angular potential V(R0 ,u) is shown in Fig. 6, along with the angular probability uc~u!u2. The barrier to rotation of H2 in the complex is approximately 25 cm21, significantly smaller than the vibrational zero-point energy of 45 cm21. While the preferred orientation is the linear geometry, u50°, there is substantial probability at u590°. The probability ra- tio between T-shaped vs linear geometries of the complex, uc~90°!u2/uc~0°!u2, are 70% for H2 and 50% for D2. More meaningful are the squared coefficients of the expansion in Eq. ~9!, which are a measure of the purity of the rotational states of the molecule. In the case of H2, a0 250.935 while (a2 21a22 2 )50.0646, i.e., the ground state has 93% free rotor FIG. 5. Potential energy curve for O~3P!–para-H2 stretching coordinate and the probability density uC(R) u2 of the two bound eigenstates. FIG. 6. Rotational barrier of H2 in O~ 3P!–H2 complex and the probability density uC(u ,R5R0) u2 of the ground rotational state.No. 3, 15 January 1997 ct¬to¬AIP¬license¬or¬copyright,¬see¬http://jcp.aip.org/jcp/copyright.jsp 947Li, Apkarian, and Harding: Solid hydrogens doped with atomic oxygenJ50 character, with only ;6% admixture from the first ex- cited rotational level, J52; and negligible contribution from higher states. In the case of deuterium, the excited state ad- mixture is larger. Now, J50 character is 83.8%, and the first excited state contribution is 17.1%. Two considerations fur- ther reduce this orientational localization, inclusion of the vibrational zero-point amplitude and the consideration that in the many-body system, due to the H2–H2 interactions, the orientational anisotropy will be further reduced. Thus, it is quite clear that even in the case of D2, little error will be made by assuming the free rotor J50 state to represent the ground state of the molecule. In short, in the ground state of the system, H2 molecules can be regarded as spherical, and the use of orientationally averaged potentials are well justi- fied. We adhere to this assumption in the rest of this paper. IV. SIMULATION METHOD We carry out semiclassical molecular dynamics simula- tions of O~3P , 1D! atoms in D2 solid based on the parametri- zation of the global potentials, and after integrating out their u dependence. The simulations do not assume additivity of potentials, but rather incorporate the anisotropic interactions of the open shell atom with the lattice by diagonalization of the potential matrix. The detailed description of the method has been given in Refs. 27 and 35, here we only present a brief outline. The interaction Hamiltonian for an oxygen atom in the field of n close shell spherical D2 molecules, in first order perturbation, can be expressed as42 H int5VO–D2~r ,R1 ,R2 ,. . . ,Rn!1VD2–D2~R1 ,R2 ,. . . ,Rn!, ~10! in which r and R represent, respectively, the electronic co- ordinate on the O atom and the coordinate of the spherical D2 molecule as measured from the origin centered on the oxy- gen core. Note O has two valence holes which are treated as coupled in the atomic limit. The single variable r is used to represent the electronic degree of freedom as an effective charge distribution. Accordingly, we will denote its orbital angular momentum as l[L5l11l2 , and m[ML . The an- gular dependence of VO–D2 is then expanded in Legendre polynomials, PL(r•Rk) VO–D2~r ,R1 ,R2 ,. . . ,Rn!5 (k51 n ( L50 ` VL~r ,Rk!PL~Rk•r !, ~11! and the electronic eigenenergies are obtained by diagonaliz- ing VO–D2 in the uncoupled basis sets, ulm.0 uns . D2; l51 for O~3P!; l52 for O~1D!. The closed shell ortho-D2 molecu- lar functions uns . D2, will be dropped from further explicit notation. Using the addition theorem for the expansion of the Legendre polynomial, H int can be evaluated as the product of spherical harmonicsJ. Chem. Phys., Vol. 106, Downloaded¬12¬Feb¬2004¬to¬128.200.47.19.¬Redistribution¬subjeVmm5 ( k51 n ( L50 ` 4p 2L11 K Y lm~u ,w!U (M52L L YLM~u ,w! 3UY lm~u ,w!L VL~Rk!YLM* ~uk ,wk!, ~12! in which the subscript k refers to the D2 molecules, of which there are n; the unsubscripted coordinates are for the O atom electron; and L is the order of the Legendre polynomial. The conditions l1L1l5even, and l1L1l1>0, limit the sum- mation over L . Only two terms, L50,2, contributed in the case of O~3P!, and only three terms, L50,2,4, contribute in the case of ~1D!. The individual matrix elements Vmm8, using the real 1D basis set, are given explicitly in Appendix A. Classical molecular dynamics simulations will be per- formed on the minimum energy surface of the diagonalized interaction Hamiltonian of the system. For the host, only ortho-D2~J50! is considered. Pseudopotentials obtained by convolution of the zero-point wave function, as outlined in Ref. 22, are used to describe the D2–D2 interactions. V. RESULTS AND DISCUSSIONS A. Structure and stability of ground state O(3P) atom in solid D2 First we carry out simulations of an fcc D2 solid, with a single O~3P! atom as a substitutional impurity. Periodic boundary conditions are used and the system is equilibrated at 4 K for approximately 10 ps prior to sampling of the trajectories. Figure 7 shows the radial distribution function of D2–D2 and O~ 3P!–D2 in this system. The comparison makes FIG. 7. Radial distribution functions of O–D2 and D2–D2 for a system of O~3P! occupying a single substitutional site in an fcc D2 solid at 4 K.No. 3, 15 January 1997 ct¬to¬AIP¬license¬or¬copyright,¬see¬http://jcp.aip.org/jcp/copyright.jsp 948 Li, Apkarian, and Harding: Solid hydrogens doped with atomic oxygenit clear that the first shell of D2 molecules are localized around O~3P! atoms, with ;10% increase in local density. In considering solids with high concentration of dopants, it is crucial to establish the stability with respect to recom- bination of atoms. To this end, we consider the free energy barrier for recombination of a pair of O atoms initially placed in the lattice as second nearest neighbors. The total energy of the system as a function of separation between O atoms, and by allowing the lattice to fully relax at every step, is shown in Fig. 8. The mean-field reaction barrier is ap- proximately 80 cm21. Two terms are expected to contribute to the barrier height, the reduced coordination number per oxygen atom and the work associated with the activation volume. These contributions are separated by a reference cal- culation, namely the mean potential for recombination of O atoms in a pair of O~3P!~D2!12 clusters in the absence of the extended lattice. For this purpose, two icosahedral O~3P!~D2!12 clusters were considered, with an initial O–O separation same as that of the second nearest neighbors in the host. The calculated mean-field barrier for recombination in the clusters is 40 cm21, nearly half that in the lattice. Quite clearly, the activation volume is positive, and the solid state recombination barrier can be expected to increase as a func- tion of external pressure. Moreover, it is clear that at the relevant temperatures of 4 K, the solid can be doped at the few % level, without substantial recombination. We there- fore calculate the molar volume of the solid as a function of O~3P! atom concentration. To ensure that the solid remains at zero external pressure, for each doping level, the volume of the cell is adjusted to minimize the total energy. The par- tial molar volume of the solid, dV/dn , is negative and linear. This is illustrated in Fig. 9, where it can be seen that the molar volume of a solid doped at the 4% level is ;2% less FIG. 8. Mean-field reaction barrier for next nearest-neighbor O~3P!1O~3P! reaction in the fcc D2 lattice.J. Chem. Phys., Vol. 106, Downloaded¬12¬Feb¬2004¬to¬128.200.47.19.¬Redistribution¬subjethan that of the undoped solid. In effect the impurities stabi- lize the lattice. B. Reactivity of excited state O(1D) atom in D2 solid Given the fact that the present potentials show that in the T approach the O~1D!–H2 potential is strictly reactive, the experimental observation of nonreaction on this surface in the solid state is rather intriguing. We examine whether this is a many-body effect, or whether the reaction barrier is eliminated in the J50 state through rotational averaging. The rotationally averaged radial functions VL(R) are extracted from the rotationally averaged ab initio pair potentials using the following well-known relations28 V05 1 5~VS12VP12VD!, V25~VS2VP!1~VP2VD!, ~13! V45 9 5~VS2VP!1 3 5~VD2VP!. The required pair potentials VS ,VP ,VD and their parametri- zation are given in Appendix B. ~Note there are some typo- graphical errors in Ref. 27!. Figure 10 shows the ab initio excited state potentials of O~1D! interacting with a spherical D2 molecule. The three Legendre components, V0 , V2 , and V4 , are plotted as a func- tion of distance between the O atom and the center of mass of D2. Note, only V0 is nonreactive, and therefore, only in spherically symmetric sites is the atom rigorously unreactive with its host. In Oh sites, V2 may be ignored since its de- composition under the crystal field does not contain the to- tally symmetric representation. However, V4 does, and con- tributions from V0 and V4 lead to the T2g and Eg splitting of the 1D surface. These surfaces are bound for an octahedral cluster, as shown in Fig. 11 for the breathing sphere coordi- FIG. 9. Change of the molar volume as a function of O atom concentration in solid D2 .No. 3, 15 January 1997 ct¬to¬AIP¬license¬or¬copyright,¬see¬http://jcp.aip.org/jcp/copyright.jsp 951Li, Apkarian, and Harding: Solid hydrogens doped with atomic oxygenfrom the stable initial configuration to the entrance channel for reaction a free energy barrier, that can be surmounted at 4 K, exists. VI. CONCLUSION We have carried out ab initio electronic structure calcu- lations of the adiabatic potential energy surfaces of O(1S ,1D ,3P)1H2 and produced global fits to the surfaces using a limited basis diabatic representation. The formalism allows the calculation of energetics and dynamics on the minimum energy surfaces arising from each electronic term. It was established that description of the lattice as composed of spherical H2 molecules is well justified for the ground state of the system. All calculations were therefore limited to solids of spherical ortho-D2 or para-H2 in J50. In the O~3P! ground state, at the relevant cryogenic tem- peratures, the doped solid is stable with respect to recombi- nation of atoms. The activation energy for recombination is 120 K, and the activation volume is positive. The latter im- plies that the lattice can be further stabilized by the applica- tion of pressure. Moreover, it was shown that doping stabi- lizes the lattice by increasing its local density around the impurity. These conclusions are in qualitative agreement with recent experiments, where O atoms were photolytically generated in solid D2, and it was observed that recombina- tion only occurs under conditions where the entire solid starts to melt.9 Although somewhat stabilized in highly symmetric envi- ronments, O~1D! is found to be reactive in the solid even at 4 K. This result remains in contradiction with the experimen- tal suggestion that the O~1D!1D2 reaction does not occur in the solid state.9 The origin of this disagreement is not clear at present. The possibility that the observed O(1S→1D) emis- FIG. 16. Potential energy curves for O~1D! and spherical H2 .J. Chem. Phys., Vol. 106, Downloaded¬12¬Feb¬2004¬to¬128.200.47.19.¬Redistribution¬subjesion in the experiment is due to O atoms complexed with O2 has been suggested. However, even in that case it is not clear why O~1D! atoms do not disappear by reaction either with O2 or H2. Alternatively, it may be argued that the present treatment is not adequate for the analysis of chemistry under the experimental conditions. Indeed the spin–orbit Hamil- tonian was left out of the ab initio calculations, and this may have a profound effect on chemistry at 4 K. The calculations were also limited to the minimum energy surfaces of each term state, and therefore, did no include intersystem crossing possibilities. However, it is highly doubtful that such a pro- cess could compete with the observed ps reaction time scale. In short, the issue of reactivity of O~1D! in solid H2 remains unresolved in our minds. ACKNOWLEDGMENTS This research was made possible through an AFOSR grant, F49620-1-0251, under the University Research Initia- tive ~Z.L. and V.A.A.! and the U.S. Department of Energy, Division of Chemical Sciences under Contract No. W-31- 1109-Eng-38 ~L.B.H.!. APPENDIX A: MATRIX ELEMENTS FOR THE INTERACTION HAMILTONIAN IN THE D BASIS SET The explicit expression of matrix element Vmm8 for d basis set ~l52! have been given in a previous paper,27 be- cause there are some typographical errors there, we present the correct ones here. For computational purposes, we find it most convenient to use Cartesian coordinates, in a real basis set. For the d basis set dz2 u1&5Y 2,0 , dxy u2&5~1/& !~Y 2,21Y 2,22!, dx22y2 u3&5~ i/& !~2Y 2,21Y 2,22!, ~A1! dxz u4&5~ i/& !~Y 2,11Y 2,21!, dyz u5&5~1/& !~2Y 2,11Y 2,21!, the interaction matrix V is real and symmetric (Vi j5V ji) with elements V115V02 1 7 F123 z 2 r2GV21F 3282 1514 z 2 r2 1 5 4 z4 r4GV4 , V225V01 1 7 F123 z 2 r2GV21F 1562 528 z 2 r2 2 5 24 ~x416x2y22y42z4! r4 GV4 , V335V01 1 7 F123 z 2 r2GV21F 1562 528 z 2 r2 1 5 24 ~2x416x2y22y41z4! r4 GV4 , No. 3, 15 January 1997 ct¬to¬AIP¬license¬or¬copyright,¬see¬http://jcp.aip.org/jcp/copyright.jsp 952 Li, Apkarian, and Harding: Solid hydrogens doped with atomic oxygenV445V02 1 14 F113 ~x 22y22z2 r2 GV21F 114 2 5 42 ~x22y216z2! r2 1 5 6 z2~x22y21z4! r4 GV4 , V555V02 1 14 F123 ~x 22y21z2 r2 GV21F 114 1 5 42 ~x22y226z2! r2 2 5 6 z2~x22y22z4! r4 GV4 , V125@~2x 21y2!/)r2#F37 V21S 5282 54 z 2 r2DV4G , V135@xy /)r 2#F2 67 V22S 5142 52 z 2 r2DV4G , V145@yz/)r 2#F37 V22S 15142 52 z 2 r2DV4G , V155@xz/)r 2#F37 V22S 15142 52 z 2 r2DV4G , V235 5 6 xy~x22y2! r4 V4 , V245F yzr2 GF2 37 V22S 5282 15x 225y215z2 12r2 DV4G , V255Fxzr2 GF37 V21S 5281 5x 2215y225z2 12r2 DV4G , V345Fxzr2 GF37 V21S 5281 25x 2115y225z2 12r2 DV4G , V355F yzr2 GF37 V21S 5281 15x 225y225z2 12r2 DV4G , V455Fxyr2 GF37 V22S 5212 5z 2 3r2DV4G , where r25x21y21z2 and V0 , V2 , and V4 are the radial potential energy functions associated with the Legendre ex- pansion in Eq. ~10! of the text. APPENDIX B: POTENTIAL ENERGY FUNCTIONS FOR O(3P, 1D) INTERACTING WITH A SPHERICAL D2 The pair interaction potential energy functions between O~3P , 1D! and a spherical H2 or D2 are obtained by averag- ing the interaction potential functions over the orientation of the H2 molecule. There are two potential energy curves for O~3P!–D2 shown in Fig. 15, and three for O~ 1D!–D2 as shown in Fig. 16. In the first case, VS is for pz and VP for px ,py where the subscripts identify the orientation of the doubly occupied orbital. This explains the characters of these two curves, VS is shallower, because of the earlier onset of repulsion between the filled atomic orbital and the closedJ. Chem. Phys., Vol. 106, Downloaded¬12¬Feb¬2004¬to¬128.200.47.19.¬Redistribution¬subjemolecular shell. In the case of O~1D!, VS is for dz2, VP for dxy ,dvz , and VL for dxy ,dx22y2 ~VP and VD are doubly de- generate!. 1A. M. Bass and H. P. Broida, Formation and Trapping of Free Radicals ~Academic, New York, 1960!. 2G. J. Minkoff, Frozen Free Radicals ~Interscience, New York, 1960!. 3M. E. Fajardo, S. Tam, T. L. Thompson, and M. E. Cordonnier, Chem. Phys. 189, 351 ~1994!. 4Proceedings of the High Energy Density Matter ~HEDM! Contractors’ Conference, edited by T. L. Thompson ~Woods Hole, MA 1993!. 5D. C. Chatfield, R. S. Friedmen, G. C. Lynch, D. G. Truhlar, and D. W. Schwenke, J. Chem. Phys. 98, 342, ~1993!, and references therein. 6K. Mikulecky and K. H. Gericke, J. Chem. Phys. 96, 7490 ~1992!, and references therein. 7 ~a! G. H. Dieke and H. M. Crosswhite, J. Quant. Spectrosc. Radiat. Trans- fer 2, 97 ~1962!; ~b! J. A. Davidson, C. M. Sadowski, H. I. Schiff, G. E. Streit, C. J. Howard, D. A. Jennings, and A. L. Schmeltekopf, J. Chem. Phys. 64, 57 ~1976!; ~c! R. F. Heidner and D. Husain, Int. J. Chem. Kinet. 5, 819 ~1973!; L. J. Steif, W. A. Payne, and R. B. Klemm, J. Chem. Phys. 62, 4000 ~1975!; ~d! R. A. Young, G. Black, and T. G. Slanger, ibid. 49, 475 ~1986!. 8 ~a! P. A. Whitlock, J. T. Muckerman, and E. R. Fisher, J. Chem. Phys. 76, 4468 ~1982!; ~b! G. Durand and X. Chapuisat, Chem. Phys. 96, 381 ~1985!; ~c! M. S. Fitzcharles and G. C. Schatz, J. Phys. Chem. 90, 3734 ~1986!; J. K. Badenhoop, H. Koizumi, and G. C. Scahtz, J. Chem. Phys. 91, 142, ~1989!; ~d! E. M. Goldfield and J. R. Wiesenfeld, ibid. 93, 1030 ~1990!; ~e! P. J. Kuntz, B. I. Niefer, and J. J. Sloan, ibid. 88, 3629 ~1988!; Chem. Phys. 151, 77 ~1991!. 9A. Danilychev, S. Tanaka, H. Kajihara, S. Koda, and V. A. Apkarian ~in preparation!. 10The quantum nature of a host is characterized by the de Boer parameter, J. de Boer, Physica 14, 139 ~1948!. 11 ~a! M. E. Fajardo, S. Tam, T. L. Thompson, and M. E. Cordonnier, Chem. Phys. 189, 351 ~1994!; ~b! T. Momose, M. Uchida, N. Sogoshi, M. Miki, S. Masuda, and T. Shida, Chem. Phys. Lett. 246, 583 ~1995!; T. Momose, M. Miki, M. Uchida, T. Shimizu, I. Yoshizawa, and T. Shida, J. Chem. Phys. 103, 1400 ~1995!. 12 ~a! Q. Hui, J. L. Persson, J. H. M. Beijersbergen, and M. Takami, Laser Chemistry, 15, 221 ~1995!; ~b! S. I. Kanorsky, M. Arndt, R. Dziewior, and A. Weis, Phys. Rev. B 49, 3645 ~1994!; ~c! B. Tabbert, M. Beau, H. Gunther, W. Haubler, C. Honninger, K. Meyer, B. Plagemann, and G. zu Putlitz, Z. Phys. B 97, 425 ~1995!. 13 ~a! R. Frochtenicht, J. P. Toennies, and A. Vilesov, Chem. Phys. Lett. 229, 1 ~1994!; A. Scheidemann, B. Schilling, J. P. Toennies, and J. A. Northby, Physica B 165, 135 ~1990!; ~b! F. Stienkemeier, J. Higgins, W. E. Ernst, and G. Scoles, Z. Phys. B-Condensed Matter, 98, 413 ~1995!; Phys. Rev. Lett. 74, 3592 ~1995!; J. Chem. Phys. 102, 615 ~1995!; S. Goyal, D. L. Schutt, and G. Scoles, J. Phys. Chem. 97, 2236 ~1993!; S. Goyal, D. L. Schutt, G. Scoles, and G. N. Robinson, Chem. Phys. Lett. 196, 123 ~1992!. 14M. E. Fajardo, J. Chem. Phys. 98, 110 ~1993!. 15D. Scharf, G. J. Martyna, and M. L. Klein, J. Chem. Phys. 99, 8997 ~1993!. 16D. Scharf, G. J. Martyna, D. Li, G. A. Voth, and M. L. Klein, J. Chem. Phys. 99, 9013 ~1993!. 17K. B. Whaley, Inter. Rev. Phys. Chem. 13, 41 ~1994!; R. N. Barnett and K. B. Whaley, J. Chem. Phys. 94, 2953 ~1992!; 99, 9730 ~1993!. 18 J. B. Anderson, J. Chem. Phys. 63, 1499 ~1975!. 19D. Blume, M. Lewrenz, F. Huisken, and M. Kaloudis, J. Chem. Phys. ~submitted!. 20V. Buch, J. Chem. Phys. 100, 7610 ~1994!. 21Monte Carlo Methods in Statistical Physics, Topics in Applied Physics 71, edited by K. Binder ~Springer, New York, 1992!. 22 ~a! M. Sterling, Z. Li, and V. A. Apkarian, J. Chem. Phys. 103, 5679 ~1995!; ~b! Z. Li and V. A. Apkarian, J. Chem. Phys. ~submitted!. 23 J. Cao and G. A. Voth, J. Chem. Phys. 99, 10070 ~1993!; 100, 5106 ~1994!; 101, 6157 ~1994!. 24 J. A. Cao and G. A. Voth, J. Chem. Phys. 101, 6168 ~1994!. 25M. Pavese and G. A. Voth, Chem. Phys. Lett. 249, 231 ~1996!. 26A. V. Danilychev, V. E. Bondybey, V. A. Apkarian, S. Tanaka, H. Kaji- hara, and S. Koda, J. Chem. Phys. 103, 4292 ~1995!.No. 3, 15 January 1997 ct¬to¬AIP¬license¬or¬copyright,¬see¬http://jcp.aip.org/jcp/copyright.jsp 953Li, Apkarian, and Harding: Solid hydrogens doped with atomic oxygen27A. V. Danilychev and V. A. Apkarian, J. Chem. Phys. 100, 5556 ~1994!. 28V. Aquilanti, R. Candori, and F. Pirani, J. Chem. Phys. 89, 6157 ~1988!. 29MOLPRO is a package of ab initio programs written by H.-J. Werner and P. J. Knowles, with contributions from J. Almlof, R. D. Amos, M. J. O. Deegan, S. T. Elbert, C. Hampel, W. Meyer, K. Peterson, R. Pitzer, A. J. Stone, and P. R. Taylor. 30H.-J. Werner and P. J. Knowles, J. Chem. Phys. 82, 5053 ~1985!; P. J. Knowles and H.-J. Werner, Chem. Phys. Lett. 115, 259 ~1985!. 31H.-J. Werner and P. J. Knowles, J. Chem. Phys. 89, 5803 ~1988!; P. J. Knowles and H.-J. Werner, Chem. Phys. Lett. 145, 514 ~1988!. 32R. Shepard, I. Shavitt, R. M. Pitzer, D. C. Comeau, M. Pepper, H. Lis- chka, P. G. Szalay, R. Ahlrichs, F. B. Brown, and J.-G. Zhao, Int. J. Quantum Chem. Symp. 22, 149 ~1988!. 33S. R. Langhoff and E. R. Davidson, Int. J. Quantum Chem. 8, 61 ~1974!; D. W. Silver and E. R. Davidson, Chem. Phys. Lett. 52, 403 ~1978!. 34T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 ~1989!; R. A. Kendall, T. H.J. Chem. Phys., Vol. 106, Downloaded¬12¬Feb¬2004¬to¬128.200.47.19.¬Redistribution¬subjeDunning, Jr., and R. J. Harrison, ibid. 96, 6796 ~1992!; D. E. Woon and T. H. Dunning, Jr., ibid. 98, 1358 ~1993!. 35W. G. Lawrence and V. A. Apkarian, J. Chem. Phys. 101, 1820 ~1994!. 36M. H. Alexander and M. Yang, J. Chem. Phys. 103, 7956 ~1995!. 37M. H. Alexander, J. Chem. Phys. 99, 6014 ~1993!. 38M. E. Rose, Elementary Theory of Angular Momentum ~Wiley, New York, 1957!. 39The potential parameters are available from the authors. 40 I. F. Silvera, Rev. Mod. Phys. 52, 393 ~1980!. 41The order of magnitude for the relaxation times is the same as for any paramagnetic center, of which T and H induced conversions are particu- larly well characterized, see for example: J. D. Sater, J. R. Gaines, E. M. Fearon, P. C. Souers, F. E. McMurphy, and E. R. Mapoles, Phys. Rev. B 37, 1482 ~1988!; V. Shevtsov, A. Frolov, I. Lukashevich, E. Ylinen, P. Malmi, and M. Punkinnen, J. Low Temp. Phys. 95, 815 ~1994!. 42See, for example, W. E. Baylis, J. Phys. B 10, L477 ~1977!.No. 3, 15 January 1997 ct¬to¬AIP¬license¬or¬copyright,¬see¬http://jcp.aip.org/jcp/copyright.jsp