Calculation of the spectral transition frequencies of matrix-isolated

J. Phys. Chem. 1987, 91, 1623-1628

1623

Calculation of the Spectral Transition Frequencies of Matrix-Isolated H,, Dp, and HD Impurities in Solid Kr and Xe under Pressure V. Chandrasekharan, M. Chergui, Laboratoire des Interactions Moleculaires et Haute Pressions, CNRS, Universite Paris Nord F-93430, Villetaneuse, France

B. Silvi, Laboratoire des Spectrochemie Moleculaire, Universite et Pierre et Marie Curie, 75230 Paris, Cedex 05, France

and R. D. Etters* Physics Department, Colorado State University, Fort Collins, Colorado 80523 (Received: March 3, 1986; In Final Form: September 29, 1986)

The vibrational, rotational, and vibrational-rotational spectra of dilute H2, D2, and HD molecules matrix-isolated in solid Kr and Xe are calculated at zero temperature and at pressures 0 5 P 5 570 kbar. The impurity local mode frequencies are also calculated vs. pressure. It is found that the pure vibrational and rotational-vibrational transition frequencies are strongly red shifted in the solid at P = 0 with respect to their gas-phase values, and the pure rotational transitions are weakly red shifted. The agreement with Raman and infrared results is generally but not uniformly good. As the pressure is increased, the transition frequencies of all the modes in solid Xe and some in solid Kr red shift with respect to their values in the solid at P = 0. But at higher pressures all modes are strongly blue shifted. The local mode frequencies strongly blue shift for all P > 0.

Introduction The purpose of this work is to calculate the transition frequencies of the low-lying vibrational, rotational, and rotationalvibrational modes of dilute H2, D2, and H D impurities, substitutionally located in solid Kr and Xe, over a wide range of pressures. Figure 1 identifies the transitions investigated in this work, where v, and J correspond to the vibrational and rotational quantum numbers that characterize the states of the molecule. Thus, Figure l a identifies those transitions allowed for para-H2 and ortho-D2, and Figure l b is for ortho-H2 and para-D2. Since H D is not homonuclear, all transitions (a, b) are allowed. In this work we benefit from a recent refinement’ of the interaction between the impurity molecule and the host lattice atoms that has been accurately determined from spectroscopic data. The lattice distortion is determined by relaxing to equilibrium the first 16 shells of atoms around the impurity within the context of a Hartree approximation.2 In addition, the intramolecular potential is expanded out to seventh order in a power series about the gas-phase equilibrium bond length. Also, the impurity local mode frequencies are calculated vs. pressure. Raman scattering3q4 and infrared absorption studiess have measured some of the transition frequencies of H2 in solid Kr and Xe at zero pressure, but no results have been reported for Dz or H D impurities and there are no data at pressure P > 0. The pure vibrational and vibrational-rotational transition frequencie~’~are strongly red shifted from their gas-phase values.6 InfraredS and the Raman measurements of Jodl and Bier3 show small red shifts for the pure rotational transitions, but other Raman data4 give small blue shifts, except for the So( 1) transition for H2-Xe. This difference is outside the experimental error, as are the quantitative (1) Carley, J. Ph.D. Thesis, University of Waterloo, Waterloo, Ontario, 1977. LeRoy, R. J. In Resonances in Electron-Molecule Scattering, van der

Waals Complexes, and Reactive Chemical Dynamics: Truhler, D. G . , Ed.; American Chemical Society: Washington, DC, 1984; ACS Symp. Sec. No. 263, pp 231-265. (2) Hartree, D. R. Proc. Phil. SOC.1928, 24, 111. (3) Jodl, H.; Bier, K. In Vibrational SDectra and Structure; Durig, J . R., Ed.; Elsevier: Amsterdam, 1984; Vol. 13, p 285. (4) Prochaska, G.; Andrews, L. J . Chem. Phys. 1977, 67, 1139. (5) Warren, J.; Smith, G.; Guillory, W. J . Chem. Phys. 1980, 7 2 , 4901. (6) Stoicheff, B. P. Can. J . Phys. 1957, 35, 730.

0022-3654/87/2091-1623$01.50/0

differences in many of the other reported transition frequencies. The source of this error is not known. In an earlier calculation Vitko and Coll’ determined the frequency shift of the Q,(O)transition of H2 impurities in solid Kr and Xe a t zero pressure, using a somewhat different approach than that employed in this work. They used an earlier versions of the H2-host lattice atom potential, the orientational dependence was neglected, and the host lattice was assumed to be static and undistorted by the presence of the impurity. Their results were in good agreement with experiment.

Method In an earlier work on hydrogen impurities in solid argon, a detailed description of the calculational methods used here is given. Consequently, only a condensed review of the theory is presented in this section, and further details may be found in ref 9. For a single molecule A (H2, HD, or D2) isolated in a solid rare gas matrix B (Kr or Xe) the Hamiltonian can be expressed as H = HB HA + HAB (1) where

+

N

(4) where m, M , and p are the A, B, and reduced A masses, respectively. The rotational kinetic energy operator of the A molecule is .%,and { = ( r - re)/re,where r is the instantaneous bond length of the A molecule in the solid lattice at pressure P and re is its gas-phase value.I0 The index i = 1 always refers to (7) Vitko, J.; Coll, C. F. J . Chem. Phys. 1978, 69, 2590. (8) LeRoy, R. J.; van Kranendonk, J. J . Chem. Phys. 1974, 61, 4750. (9) Silvi, B.; Chandrasekharan, V.; Chergui, M.; Etters, R. D. Phys. Rev. B Condens. Matter 1986, 33, 2749. (10) Herzberg, G.Spectra of Diatomic Molecules; Van Nostrand Reinhold: New York, 1950.

0 1987 American Chemical Society

Chandrasekharan et al. J=2

J=3 v = 2

%(O) J

S,(I) J= I

=O

Q,( I )

O,(O) J =2

s, (0)

J =3 V ' I

J

S,(I)

states are very close to the free rotor states of the isolated molecule. As for the intramolecular degrees of freedom, an inspection of eq 7 shows that the terms dependent on {are small compared to V,,,,(.t) and thus represent a perturbation of the isolated molecule potential. Similarly, since the {-dependent terms in eq 7 are small compared to the term k = 0, the CM degrees of freedom are nearly decoupled from the orientational and intramolecular degrees of freedom. This suggests that the ground state of the system can be accurately described by the product function

J = I

=O

I{fiiI*o,{)

Q,(O)

Q,(I)

J =2

%CO)

J=3 v

J 20

=o

So( I ) J= I

=

I(fiiI)Ie)I.t)

(8)

where the (Xi) locate the instantaneous C M positions of the particles. Because of the relatively strong coupling of the A molecule to the B atoms, and between B atoms, the center of mass motion can be accurately described by the Hartree expression2 (9) where

and

where the PAare Legendre polynomials characterizing the angle B I j between the molecular axis and the vector connecting the charge center of the molecule to thejth host lattice atom. The Kr-Kr pair potential has been taken from the work of Aziz'* where a term C9/r9has been added to account for the three-body interactions. The quantity C9 = 0.1 11 032 8, in units of the well depth t for that potential and r = R/R,, where R is the distance between mass centers and R, is the position of the potential minimum. Similarly, the Xe pair potential X2 is taken from ref 13, where C9 = -0.1398. An important element in simplifying the calculation is to recognize that the rotational, translational, and intramolecular vibrational degrees of freedom are only weakly coupled to one another. For example, eq 7 shows that the angular dependence in the A-B potential appears as P2(cos e). However, the sum of this term over a static lattice with cubic symmetry is well-known to be zero. Thus, only noncubic distortions of the crystal around the impurity can lead to nonzero orientational contributions, which are most assuredly small at low pressures. This is supported by experimentail e ~ i d e n c e ~ - ~ *which ' ~ - ~ *shows that the rotational (11) van Kranendonk, J. Rev. Mod. Phys. 1976, 40, 531. (12) Aziz, R. A. Mol. Phys. 1979, 38, 177. (13) Barker, J.; Watts, R.; Lee, J. K.; Schafer, T.; Lee, Y .T. J . Chem. Phys. 1974, 61. 3081.

Equation 11 is clearly appropriate since the rotational states are nearly free rotors for which the spherical harmonics YjM are eigenstates. The ai and CJMare variational coefficien? to be determined. The equilibrium positions of the B atoms {R?),i = 2, 3, ..., N , must also be treated as variational parameters because of the distortion of the lattice around the impurity. Similar to le), the state 1.t) can be described by an expansion using a harmonic oscillator basis set. The first step in the calculation is to determine the distortion of the host lattice around the A impurity. Because of the very weak coupling of the C M coordinates to the orientational and intramolecular degrees of freedom, the lattice distortion is calculated by using only the overwhelmingly dominant k , X = 0 term in eq 7. The expectation value of the Hamiltonian H with respect to the oGhogonal product states, eq 9, then gives the lattice energy E({ai],(Rp]), where the one-dimensional integrals are evaluated by using a Hermite quadrature routine and the lattice sums are taken out to a distance of five nearest-neighbor lengths. It is explicitly characterized by eq 13 in ref 9. The lattice energy is then minimized with respect to the 2 N quantities (ai),(R", thereby determining their values. The problem is simplified to a tractable level by treating only a1and a2,associated with the molecule and the first-nearest-neighbor B atoms, respectively, as independent variables. The ai associated with more distant shells are assumed to be the same and identical with that determined from a subsidiary calculation on the pur2 host lattice. Moreover, the equilibrium C M lattice vectors (Rp)are reduced by allowing only the first 16 shells of B atoms around the impurity to relax to equilibrium. Because of the uncorrelated nature of the product states, given by eq 8, the distortion preserves the Oh symmetry of the lattice around the impurity, which is characterized by only 30 symrfletry coordinates. thus, the multidimensional function E((a,),(Rp]) is minimized with respect to these symmetry coordinates and ( a I , a 2using ) a pattern recognition optimization strategy at each of 30 different molar volumes. The details of this strategy are given e1se~here.I~ As will be shown in the next section, the lattice distortions are small and have only a minor effect on the predicted vibrational and rotationtal transition Kriegler, R.; Welsh, H. Can. J . Phys. 1968, 46, 1181. DeRemigis, J.; Welsh, H. Can. J . Phys. 1970, 48, 1622. Smith, G.; Warren, J.; Guillory, W. J . Chem. Phys. 1976, 65, 1591. LeRoy, R.; Carley, J. In Advances in Chemicol Physics; Lawley, K . P., Ed.; Wiley: New York, 1980; p 353. (18) McKellar, A.; Welsh, H. Can. J . Phys. 1972, 50, 1458. (19) Pan, R. P.; Etters, R. D. J . Chem. Phys. 1980, 72, 1741. (14) (15) (16) (17)

The Journal of Physical Chemistry, Vol. 91, No. 6, 1987

H2, DZ, and H D Impurities in Solid Kr and Xe frequencies. The equilibrium position of the center of charge of the molecular impurity is assumed to be at a substitutional site in the host lattice which, except for HD, coincides with the position of the center of mass. The analysis of the internal vibrational and rotational spectra of the matrix-isolated molecule is simplified by first taking the expectation value of the-total relaxed Hamiltonian with respect O). of theoretical and exto the product states ~ { R i ) ) ~ Because perimental information presented earlier, the small anisotropies in the AB potential are negelected so the rotational states are given by those of a free rotor. Thus, the only perturbation of the rotational states, caused by the crystal field of the host lattice, is due to the change in the rotational constant B ( { ) , given by eq 5. If we expand B ( 0 in powers of {up to cubic terms, the resulting effective Hamiltonian, except for ignorable constants, becomes

1625

9 00 I I I

_ilI 700

5

I 4

500

$ 1

v

400

where the b,((_V,,),a,J)_are given by eq 15 of ref 9, and ( Von(RI,?) = (Rl,R,lVo,,lRl,R,) is the Hartree2 integral over the C M motion of particles 1 and j . In this work, the eigenvalues EVJof eq 12 are calculated by using the WKB solution formulated by Dunham.2o To test the accuracy of this standard method, the Schrodinger equation was directly solved by using the Numerov-Cooley method. Also, a perturbation calculation was conducted where the exact eigenfunctions IvJ) of the free molecule with an internal potential, given by eq 6, were used as zeroth-order states. All of these methods gave very similar results except at very high pressures where the perturbation expansion was not as accurate and the calculated frequency shifts differ from the WKB and Schriidinger equation results by as much as 10%. H D impurities present a special calculational problem in that the charge center and the C M are displaced from one an_other by a distance r/6. However, the single-particle orbital IR,) is properly centered about the C M equilibrium position RlO,but the interaction between the molecule and the j t h host lattice atom depends on the distance &_tween fhe charge center and the C M of the j t h atom. That is, R,,' = R1, (r/6)ii, where ii is a unit vector giving the direction between the molecular C M and the center of charge. In this work it was assumed that because the rotational states are nearly isotropic, the charge center uniformly samples a spherical surface of radius r/6 about the center of mass, so ii is averaged over this surface by integrating over the s+id angle doo. This ca_n be accomplished by either expressing lRl) in terms of ii and Rl'or tr2nsforming eq 7 to a function of the as was done in this calcucenter of mass position V(R,,,ii,OIJ,{), lation. Translational oscillations of the molecular impurity in the cage of the host lattice atoms produce a local mode with frequency

+

vCM

(cm-I) = aI(h/2?rmC)

(13)

where C is the velocity of light. This expression can be understood by recognizing that the product states given by eq 10 characterize a harmonic oscillator ground state, from which the frequency can be readily determined.

Results The first stage of the calculation involved relaxing to equilibrium the first 16 shells of atoms around the impurity and evaluating the wave function parameters a l , a2,and a3which minimize the crystal energy. At V = 27.1 cm3/mol and P zz 0, the ratios of the H2-Kr bond lengths of the first 16 neighboring Kr shells to those for the pure Kr lattice are 0.9932, 1.0012, 0.9990, 0.9989, 1.0001, 0.9996,0.9998, 1.0001,0.9997, 1.0000, 1.0000, 0.9999, 0.9999,0.9999, 1.0000, and 1.oooO. For Xe at V = 34.7 cm3/mol and P zz 0, they are 0.9920, 1.0014,0.9989,0.9987, 1.0002,0.9995, 0.9997, 1.0001, 0.9997, 1.OOOO,1.0000, 0.9999, 0.9999, 0.9999, 1.OOOO,and 1.OOOO. Thus, the distortion is clearly small, a result (20) Dunham, J. L. Phys. Rev. 1932, 41, 721.

-I

t

300

2001

IO0

1

36

32

28

v

24 20 ( cm3/ mole

16

12

Figure 2. Dashed lines are the H,, HD, and D, local mode frequencies vs. molar volume in solid Xe, in order of decending frequency. Solid lines are the local mode frequencies in solid Kr, with the same format. The solid square is the measured frequencyI5 for H,-Kr at zero pressure.

which can be deduced by recognizing that VAB is considerably weaker than V B ~ If . one compares the location of the potential minima, it is evident the VAB should act to diminish the nearest-neighbor AB bond, whereas, the additional A zero-point energy will act to slightly expand that bond. The accumulation of these effects results in a negligible distortion for Ha, Dz, and H D in Kr or Xe at low pressures, but at the highest pressures investigated, the nearest-neighbor AB bond length is reduced by about 3%. The variational parameters al, az and a3strongly increase with pressure and depend on the impurity species. The dashed lines in Figure 2 give the calculated local mode frequencies vcM(cm-') at zero temperature vs. molar volume for H2, HD, and D2 in solid xenon, in order of descending frequency, and the solid lines give the same results for these molecular impurities in solid krypton. The solid square is the experimental valueI5 for Hz-Kr at T = 115 K. The variational parameter a I can be determined from these curves via eq 13. It is noted that all the calculated impurity local mode frequencies show strong monotonic blue shifts with increasing pressure. The calculated frequency for H2-Kr at P = 0 is in good agreement with experiment," but no observed data have been reported for Hz-Xe or for Hz, D,, H D in Kr or Xe at P > 0. The calculated vibrational, rotational, and vibrational-rotational transition frequencies of Hz, DZ,and H D impurities in solid Kr and Xe are shown in Tables I and 11, respectively, vs. molar volume. The corresponding pressures, which range from 0 to 570 kbar, are determined by differentiating a polynomial fit of the energy-volume relation. These pressures are the same for each of the isotopic impurities in solid Kr and are therefore listed only with the Hz-Kr results in Table I, and similarly for the solid X e host lattice results in Table 11. The listed frequencies represent the shift from gas-phase values, given by Stoicheff.6 Useful experimental data exist only for H2 impurities in Kr and Xe at P = 0, and the Raman ~ c a t t e r i n gand ~ , ~infrared absorption data5 are presented in Table 111. Discussion and Conclusions Zero-Pressure Results. The calculated frequencies of the pure vibrational and rotational-vibrational transitions of HZ,D2. and

1626 The Journal of Physical Chemistry, Vol. 91, No. 6, 1987

Chandrasekharan et ai.

TABLE I: Frequency Shifts (cm-') with respect to Gas-Phase Values vs. Molar Volume (cm3/mol), for Vibrational, Rotational, and Vibrational-Rotational Transitions of H2,D2,and HD Impurities in Solid Krypton (Pressures in kbar)

P

V

SdO)

SO(])

Ql(0)

0.1 1 1.64 3.71 6.51 15.43 22.36 31.78 62.23 86.55 120.47 168.20 236.32 334.8 479.66

27.1 26.1 25.1 24.1 22.1 21.1 20.1 18.1 17.1 16.1 15.1 14.1 13.1 12.1

-2.23 -2.27 -2.28 -2.26 -2.06 -1.84 -1.49 -0.23 0.84 2.38 4.57 7.70 12.17 18.59

-3.71 -3.78 -3.79 -3.76 -3.42 -3.05 -2.47 -0.37 1.42 3.99 7.64 12.85 20.29 30.98

27.1 26.1 25.1 24.1 22.1 21.1 20.1 18.1 17.1 16.1 15.1 14.1 13.1 12.1

-1.17 -1.19 -1.21 -1.20 -1.13 -1.03 -0.87 -0.29 0.22 0.96 2.01 3.52 5.69 8.82

27.1 26.1 25.1 24.1 22.1 21.1 20.1 18.1 17.1 16.1 15.1 14.1 13.1 12.1

-1.64 -1.66 -1.65 -1.62 -1.42 -1.21 -0.90 0.20 1.12 2.42 4.27 6.89 10.61 15.91

Sl(1)

Q2(0)

Q2(1)

SdO)

-31.35 -3 1.32 -30.78 -29.55 -23.93 -18.77 -1 1.27 14.54 35.72 65.37 106.87 165.09 247.05 362.94

QI(~) SI(O) A. H2-Kr -31.35 -33.60 -31.32 -33.59 -30.77 -33.04 -29.53 -3 1.76 -23.88 -25.85 -18.70 -20.41 -1 1.17 -12.48 14.71 14.83 35.96 37.28 65.69 68.70 107.29 112.72 165.66 174.49 247.81 261.48 363.94 384.52

-35.09 -35.09 -34.53 -33.20 -27.07 -21.41 -13.17 15.23 38.58 71.27 117.07 181.31 271.87 399.90

-6 1 -6 1 -59 -56 -44 -32 -16 39 83 146 233 354 525 766

-6 1 -6 1 -59 -56 -43 -32 -16 39 84 146 234 356 527 768

-63 -6 3 -6 1 -58 -4 5 -34 -17 40 86 150 240 366 542 791

-65 -64 -63 -60 -46 -34 -17 41 88 154 246 374 555 809

-1.94 -1.98 -2.01 -2.00 -1.87 -1.71 -1.45 -0.47 0.38 1.60 3.35 5.87 9.49 14.71

-23.72 -23.91 -23.77 -23.16 -19.87 -16.64 -1 1.84 5.09 19.20 39.11 67.18 106.76 162.77 242.30

B. D2-Kr -23.73 -24.90 -23.92 -25.1 1 -23.77 -24.97 -23.16 -24.35 -19.86 -20.95 -16.62 -17.61 -11.82 -12.63 5.14 4.97 19.28 19.66 39.22 40.38 67.31 69.61 106.95 110.85 163.02 169.22 242.64 252.13

-25.68 -25.90 -25.76 -25.14 -21.66 -18.23 -13.12 4.95 20.04 41.34 71.37 113.77 173.77 259.02

-47 -47 -47 -45 -38 -3 1 -2 1 15 44 85 143 225 340 504

-47 -47 -47 -45 -38 -3 1 -2 1 15 44 85 144 225 341 505

-48 -48 -48 -46 -39 -32 -22 15 45 87 146 230 348 515

-49 -49 -49 -47 -40 -33 -22 15 45 88 148 233 353 523

-2.73 -2.76 -2.75 -2.69 -2.35 -2.01 -1.49 0.34 1.88 4.05 7.14 11.50 17.69 26.53

-26.14 -25.81 -24.94 -23.36 -17.00 -11.45 -3.55 22.98 44.44 74.20 115.53 173.05 253.35 365.84

C. HD-Kr -26.14 -27.78 -27.44 -25.80 -24.93 -26.55 -23.34 -24.91 -16.95 -18.27 -12.47 -1 1.39 -4.19 -3.47 23.65 23.14 44.64 46.18 77.45 74.47 115.89 120.88 173.52 181.36 265.83 253.98 366.66 384.20

-28.86 -28.52 -27.60 -25.91 -19.07 -13.07 -4.52 24.26 47.56 79.90 124.83 187.40 274.79 397.26

-5 1 -50 -48 -44 -30 -18 -1 56 101 164 250 371 538 172

-5 1 -50 -48 -44 -30 -18 -1 56 101 164 25 1 372 539 773

-52 -5 1 -49 -4 5 -3 1 -19 -1 57 103 168 257 381 553 793

-53 -52 -50 -46 -32 -19 -1 58 106 171 262 388 564 808

H D in solid Kr and Xe at zero pressure exhibit large red shifts with respect to their gas-phase values, which are displayed in Tables I and 11. This feature is understood to be predominantly a consequence of the attractive force on the impurity molecules due to the crystal field of the host lattice, which acts to stretch the intramolecular bond length and thus lower the natural frequency of the molecules. It is noted that the frequency shifts Ql(0) and Q 1 ( l ) are nearly identical, as are the pairs [Q2(0), Qz(l)], [S,(O), S,(l)], and [S,(O), Sz(l)]. Thus, it appears that these shifts are insensitive to ortho-para differences of the impurity species, but they do depend strongly on the isotopic impurity mass and the nature of the host lattice. The pure rotational transitions S,(O) and So(l),for H2, D2, and H D show small calculated red shifts in both solid Kr and Xe. The paucity of observed data and the divergence in reported results, as shown in Table 111, limit comparison with experiment to H,(Kr) and H,(Xe) at zero pressure. For these systems the pure vibrational shifts Ql(0) and Q l ( l ) are in good agreement with e ~ p e r i m e n t ,and ~ . ~ the Raman data of Jodl and Bier3 show no difference between them, which is consistent with our calculation that shows the shifts are insensitive to the ortho-para difference in the impurity species. Both theory and experiment also give large red shifts for the rotational-vibrational transitions S,(O) and S l ( l ) , but the quantitative agreement, while good for S I (1), is not so good for Sl(0). The calculated pure rotational transitions S,(O) and So(1) show small red shifts in good agreement

with the Raman measurements of Jodl and Bier3 and the infrared data of Warren et al.: but the Raman measurements of Prochaska and Andrews4 give blue shifts for these transitions except So(l) of H2-Xe, where the red shift is -0.1 cm-'. These comparisons are interesting because the calculated pure rotational shifts result from the change in the molecular bond length from gas to solid, which alters the centrifugal perturbation via a change in the rotational constant B ( { ) . The only other possible perturbation of the rotational states is from the angular-dependent P,(cos 0) terms. These terms would produce a splitting of the rotational levels that has not been observed, and the preponderance of evidence supports our conclusion that they are negligibly small. Unfortunately, the uncertainty in measured data renders comparisons with our results inconclusive as a means to judge the validity of our approximation in this regard. Finally, the calculated local mode frequency for H2-Kr is in good agreement with experiment,15 as shown in Figure 2. The previous calculations by Vitko and Coll' of Ql(0) for H,-Kr and H-Xe at P = 0 gave -31 3= 2 and -43 2 cm-I, respectively, which compare very well with our results of -31.35 and -40.8 cm-I. This mutual agreement is not readily understood since the theoretical approach and the approximations used by them and us are somewhat different, as are the HI-B potentials. However, the leading terms in f of both potentials are similar and they dominate the prediction of Ql(0) at zero pressure, which may account for the agreement. It is unfortunate that contact between these two

*

The Journal of Physical Chemistry, Vol. 91, No. 6, 1987

H2, D2, and H D Impurities in Solid Kr and Xe

1627

TABLE 11: Frequency Shifts (cm-') with respect to Gas-Phase Values vs. Molar Volume (cm3/mol), for Vibrational, Rotational, and Vibrational-Rotational Transitions of HB D2, and H D in Solid Xenon (Pressures in kbar)

P

V

SdO)

SO(^)

QI(O)

QI(1)

SI(1)

Q2(0)

Q2(1)

S2(O)

S2(1)

-46.39 -47.77 -49.90 -50.67 -48.99 -42.99 -29.52 -3.09 46.00 135.30 297.13 591.99 821.81

-8 3 -8 5 -88 -89 -85 -73 -47 4 98 266 570 1122 1550

-8 3 -8 5 -88 -89 -85 -73 -46 5 98 267 572 1125 1553

-86 -88 -92 -92 -88 -76 -48 4 100 275 589 1159 1601

-88 -90 -94 -94 -90 -77 -50 4 103 28 1 603 1187 1639

A. H2-Xe 0.71 2.01 5.55 10.84 18.64 29.99 46.41 70.10 104.57 156.40 241.58 409.61 570.21

34.7 33.7 31.7 29.7 27.7 25.7 23.7 21.7 19.7 17.7 15.7 13.7 12.7

-2.75 -2.84 -3.00 -3.1 1 -3.10 -2.88 -2.31 -1.10 1.25 5.63 13.77 28.97 41.08

-4.58 -4.73 -5.00 -5.17 -5.15 -4.80 -3.84 -1.82 2.09 9.40 22.94 48.24 68.39

-41.61 -42.83 -44.73 -45.37 -43.80 -38.31 -26.07 -2.10 42.37 123.19 269.56 536.09 743.69

-41.65 -42.87 -44.76 -45.40 -43.80 -38.30 -26.01 -1.96 42.63 123.65 270.34 537.38 745.32

-44.47 -45.78 -47.83 -48.55 -46.92 -41.15 -28.20 -2.79 44.38 130.18 285.64 568.90 789.65

34.7 33.7 31.7 29.7 27.1 25.7 23.7 21.7 19.7 17.7 15.7 13.7 12.7

-1.41 -1.46 -1.55 -1.62 -1.63 -1.55 -1.29 -0.73 0.38 2.49 6.45 13.89 19.84

-2.34 -2.43 -2.58 -2.69 -2.7 1 -2.57 -2.15 -1.21 0.65 4.16 10.74 23.14 33.06

-30.46 -3 1.47 -33.13 -34.00 -33.41 -30.22 -22.48 -6.77 22.95 77.67 177.69 361.19 504.85

-30.48 -31.48 -33.14 -34.01 -33.41 -30.22 -22.46 -6.73 23.03 77.82 177.94 361.62 505.39

-31.91 -32.96 -34.71 -35.64 -35.05 -31.75 -23.72 -7.38 23.58 80.61 184.90 376.36 526.33

-32.88 -33.97 -35.78 -36.74 -36.14 -32.77 -24.52 -7.73 24.09 82.72 189.97 386.90 541.18

-6 1 -63 -66 -68 -66 -59 -4 3 -10 52 165 370 746 1040

-6 1 -63 -66 -68 -66 -59 -4 3 -10 52 165 371 747 1041

-62 -64 -68 -69 -68 -6 1 -44 -10 53 168 378 763 1063

-63 -65 -69 -70 -69 -62 -45 -1 1 53 171 384 775 1080

34.7 33.7 31.7 29.7 27.7 25.7 23.7 21.7 19.7 17.7 15.7 13.7 12.7

-2.05 -2.1 1 -2.22 -2.27 -2.23 -2.01 -1.48 -0.42 1.57 5.24 11.96 24.34 34.10

-3.41 -3.52 -3.70 -3.78 -3.70 -3.34 -2.46 -0.70 2.63 8.75 19.94 40.56 56.80

-35.66 -36.55 -37.73 -37.61 -35.20 -28.82 -15.68 9.08 53.90 133.86 276.23 530.75 726.10

C . HD-Xe -35.68 -37.77 -36.57 -38.72 -37.75 -40.00 -37.62 -39.91 -35.19 -37.40 -28.80 -30.74 -15.61 -16.96 9.20 9.03 54.12 56.15 134.24 140.26 276.86 290.09 531.77 558.14 727.37 764.02

-39.19 -40.18 -41.52 -41.43 -38.86 -3 1.98 -17.73 9.15 57.89 144.91 299.96 577.39 790.51

-7 1 -7 2 -74 -74 -68 -54 -26 27 121 288 583 1109 1511

-7 1 -7 3 -7 5 -74 -68 -54 -26 27 121 288 585 1111 1513

-73 -75 -77 -76 -70 -56 -27 27 124 295 599 1140 1553

-74 -76 -78 -78 -72 -57 -27 28 126 301 61 1 1163 1584

B. D2-Xe

TABLE Ilk Measured Raman and Infrared Frequency Shifts from Gas-Phase Values (cm-') at Zero Pressure for H, Impurities in Solid Kr and Xe at Temperatures 12-15 KC ref system SdO) SO(1) Sl(1) "CM Ql(1) sm Qi(0) -31.8 -25.2 -19.7 -31.1 4.6 0.9 H2-Kr -2.98 -4.06 -36.7 -36.7 -1.38 -4.06 106d

H2-Xe

1.6 -4.38

-0.1 -4.86

-40.1 -45.1

-34.2 -45.2

-28.7

-37.8

"Reference 4. bReference 3. cReference 5. dReference 15. CImpuritylocal mode frequency measured at 115 K. works cannot be extended to other transitions and to higher pressures. Finite-Pressure Results. As the pressure of the solid is increased from zero, several interesting features appear. The transition frequencies of H2, D2, and H D in solid Xe decrease with increasing pressure until P 11 kbar ( V z 29.7 cm3/mol). At higher pressures they monotonically increase with pressure, as is evident from Table 11. This is unlike the behavior previously calculated9 for these modes in solid Ar, which monotonically blue shift at all pressures P 1 0. As shown in Table I some modes in solid Kr also initially decrease in frequency as the pressure is increased from zero, but it is less pronounced and the pressure above which blue shifts occur is much lower than in Xe. These results can be understood as follows. The dominant term in eq 7 that perturbs the states of the molecular impurity is (A&) = (O,l), and in solid Ar, the equilibrium nearest-neighbor bond length locates the molecule almost exactly at the minimum of this term in the atom-molecule A-B potential. Thus, with increasing pressure

=

the molecule is farther up the repulsive wall of this interaction, giving rise to an increasingly larger repulsive force on the molecule, reducing its bond length and increasing the frequency of the modes. Thus, a monotonic blue shift with increasing pressure above P = 0 is expected, as was previously c a l ~ u l a t e d .However, ~ the equilibrium position of the molecule in solid Xe at zero pressure is on the long-range attractive tail of the (A&) = (0,l) term in V,, which produces an attractive force on the molecule, stretching the bond and decreasing the frequency. With increasing pressure this attractive force increases until the molecular position is at the inflection point of the potential, at which point increasing pressure causes the force to become less attractive and finally repulsive, resulting in an increase in frequency. Based upon this argument, it is therefore expected that the modes of the isotopic impurities red shift as the pressure in the solid is increased from 0 to N 11 kbar and further increases in pressure produce monotonically increasing blue shifts. In solid Kr the situation is intermediate between that of solid Ar and Xe. It is noted that the

1628

J . Phys. Chem. 1987, 91, 1628-1634

blue shift dv/dP at higher pressures is about 3 times larger than for H,, 02,N2, C 0 2 , F,, et^.,^'-^^ in their respective bulk solids. This is a consequence of the strong repulsive core of V , , which produces a relatively large crystal-field perturbation on the molecule at high pressures. Figure 2 shows that the local mode frequencies, calculated from eq 13, are strongly and monotonically blue shifted with decreasing molar volume (increasing pressure). Observations and Summary. It was observed in this work that the CM zero-point oscillations of the impurity molecule make an important contribution to the frequency shifts. This is manifested in the Hartree averages of the lattice sums EjVol(R,,’),etc., in b,. To examine this feature, these sums were also calculated classically, where both the hydrogen molecule and the host lattice atoms were statically located at their equilibrium lattice sites. In addition, they were calculated in the Einstein approximation, where the wave function of the molecular impurity was determined by minimizing the energy of the crystal with respect to ai,with the host lattice atoms fixed at their static equilibrium lattice sites. The results for the Ql(0) frequency shift in solid Kr and Xe at P = 0 are (-38.5, -31.74, -31.35 cm-’) and (-46.28, -41.69, -41.61 cm-I) for the classical, Einstein, and Hartree approximations, respectively. It is evident that averaging these lattice sums over the C M zero-point fluctuations of the impurity makes an important contribution to the predicted Ql(0) frequency shift, whereas the zero-point fluctuations of the Kr and Xe atoms are unimportant since they are small and nearly classical in character. This is in contrast to earlier workg on hydrogen impurities in solid Ar, where the greater zero-point motion of the host lattice atoms

was influential. Similar results were found for frequency shifts of the other transitions. The Kr and Xe pair potentials used in this workI2J3 have previously been shown to accurately predict gas-phase and solid properties at low pressures. However, there is some evidence24 that these expressions are less accurate representations of the interactions in the solid at high pressures, P L 75 kbar. Thus, we assign a possible error of as much as 15-20% to the calculated pressures above this value. Fortunately, this is only a minor difficulty here since the predicted frequency shifts depend only weakly on the details of the host lattice interactions and, instead, depend strongly on the accurate atom-molecule potential VAB. Consequently, the volume dependence of the results reported in this work is quite accurate, and the data have been presented accordingly. It is unfortunate that no experimental data or theoretical results have previously been reported at P > 0, but it appears that sapphire high-pressure cells with sample volumes large enough to give acceptable Raman or infrared intensities can provide data up to 100 kbar. It would also be desirable to have measured data on many of the transition frequencies calculated in this work but as yet not observed. In summary, the calculated results are generally in good agreement with the limited experimental data, but there are some exceptions. However, these comparisons are hampered by a divergence in the reported measurements.

(21) Etters, R. D.; Helmy, A. Phys. Rev. B: Condens. Matter 1983, 27, 6439 and papers referenced therein. (22) Sharma, S.; Mao, H.; Bell, P. Phys. Rev. Lett. 1980, 44, 886. Silvera, I. F.; Wijngaarden, R. Phys. Rev. Lett. 1981, 47, 39. (23) Kirin, D.; Etters, R. D. J . Chem. Phys., in press.

7439-90-9; Xe, 7440-63-3.

Acknowledgment. We thank Robert LeRoy for his assistance on this problem and the Cray GC, VR group for providing computational facilities. Registry No. H,, 1333-74-0; HD, 13983-20-5; D,, 7782-39-0; Kr,

(24) Asaumi, K. Phys. Rev. B Condens. Mutter 1984,29, 7026. Schiferl, D.: Mills, R. L.; Trimmer, L. E. Solid State Commun. 1983, 46, 783.

Dinaphthylpropane as a Probe of the Behavior of Bile Salt-Lecithin Mixtures Frank Reda and Charles Spink* Chemistry Department, State University of New York, Cortland. New York 13045 (Received: April 9, 1985:

In Final Form: October 22, 1986)

The fluorescent probe 1,3-di-@-naphthylpropanewas used as a probe of the behavior of bile salt-lecithin mixtures. Making use of the changes in the monomer-to-excimer ratio of the probe, the internal environment of micelles and vesicles of egg lecithin or dipalmitoylphosphatidylcholine that form when mixed with taurocholate or taurodeoxycholate have been studied. The probe fluorescence behavior and light scattering from the mixtures near the micelle-vesicle boundary correlate with previous studies on these systems, so it is possible to determine at what compositions micelles prevail in the solutions. It was found that at 1 mg/mL lecithin concentrations micelles prevail when the mole ratio of bile salt to lecithin is greater than 1.5/1 for taurodeoxycholate and 3/1 for taurocholate mixtures. The dinaphthylpropane probe also revealed that at higher mole ratios and at lower temperatures changes occur in the bile salt-lecithin micelles. It is hypothesized that the variations are associated with a lateral phase separation of lecithin domains within the disk-shaped micelles that prevail in these solutions.

Introduction The physicochemical behavior of bile salt-lecithin mixed micelles has received considerable attention because of the importance of the micelles in several fundamental physiological processes. These aggregates are implicated in the transport of cholesterol through the biliary tract,’ as well as in the movement of fat hydrolysis products to the intestinal wall for absorption.2 Changes (1) Carey, M. C.; Small, D. M. J . Clin. Invest. 1978, 61. 998. Shaffer, E. a.; Small, D. M. J . Clin. Invest. 1977, 59, 828. Holzbach, R. T.; Marsh, M.; Olszenski, M.; Holan, K. J . Clin. Invest. 1973, 52, 1467. (2) Carey, M. C.; Small, D. M.; Bliss, C. M. Annu. Reo. Physiol. 1983, 45, 651.

0022-3654/87/2091-1628$01.50/0

in the solubilizing characteristics of the micelles for cholesterol are thought to be involved in the pathogenesis of gallstone disease, an affliction affecting millions of persons each year.3 Studies of the properties of bile salt-lecithin mixed micelles indicate that thermodynamically and structurally the aggregates are quite c o m p l e ~ . ~ Light - ~ scattering experiments have shown (3) Bouchier, I . A. D. Annu. Rev. Med. 1980, 31, 59. (4) Mazur, N. A,; Benedik, G. B.; Carey, M. C. Biochemistry 1980, 19, 601. ( 5 ) Muller, K. Biochemisfry 1981, 20, 404. (6) Spink, C. H.; Muller, K.; Sturtevant, J. M. Biochemistry 1982, 24, 6598.

0 1987 American Chemical Society