J. Phys. Chem. 1983,87,5202-5206
5202
Polarizability and Quadrupole Moment of a Hydrogen Molecule in a Spheroidal Box R. LeSar’ University of California, Los Aiamos National Laboratory, Los Alamos, New Mexico 87545
and D. R. Herschbach Department of Chemistry, Harvard University, Cambridge, Massachusetts 02 138 (Received: June 20, 1983)
The polarizability and quadrupole moment of a hydrogen molecule enclosed within an infinite-walled spheroidal box are calculated as functions of the hox size, for a range corresponding to nominal pressures up to lo5 kbar. A five-term James and Coolidge variational wave function is employed and the Kirkwood approximation is used for the polarizability. Comparison is made with similar calculations for the H2+molecule-ion and the II atom, and also with recent density-functional calculations. The box model proves to greatly exaggerate the compression of the H2 bond with pressure but nonetheless yields a realistic correlation of vibrational frequency with internuclear distance. Renormalizing the pressure scale for the box model using results of the densityfunctional calculation is found to decrease markedly the predicted variation with pressure for several properties, including the polarizability, quadrupole moment, bond length, vibrational frequency, and electronic kinetic energy. When combined with the Herzfeld criterion the predicted volume dependence of the polarizability fails to give a transition to the metallic phase.
Introduction Changes in electronic and vibrational properties of molecules subjected to high pressures can be readily estimated by merely modifying the boundary conditions for variational calculations to make the wave function vanish on a surface of appropriate shape. This “molecule-in-abox” approach has been applied previously to the H2+ molecular i ~ n l and - ~ to the H2m ~ l e c u l e ~to- ~examine pressure-induced changes in the equilibrium bond length, vibrational force constant, total energy, ionization potential, molecular surface area, electronic kinetic energy, and correlation energy. Here we extend these calculations to include changes in the molecular polarizability and quadrupole moment. Although the box model is unrivaled in the variety of properties that can be treated, for most it appears to drastically overestimate the effects of compression. In this paper we also consider a renormalization of the pressure scale. This is carried out by comparing the bond lengths obtained from the box model with those obtained from a recent density-functional calculation for compressed molecular hydrogen.6 The box-model results are also used to evaluate the role of pressure-induced polarizability changes in the Herzfeld criterion for metallization and the correlation of bond length with vibrational force constant. The latter proves remarkably similar to the correlation for free gas-phase molecules known as Badger’s rule.7 Computational Procedure As in our previous s t ~ d yfor , ~ H, we use the five-term Jmes-Coolidge8 variational wave function. This includes a term proportional to the interelectronic distance and gives a more accurate energy for a free H2molecule than a comparable SCF-CI treatment.g For H2+,we again use (1) Cottrell, T. L. Trans. Faraday SOC.1951, 47, 337. (2) Singh, K. K. Physica 1964, 30, 211. (3) Ley-Koo, E.; Cruz, S. A. J. Chem. Phys. 1981, 74, 4603. (4) LeSar, R.; Hesschbach, D. R. J.Phys. Chem. 1981,85, 2798. Note that “vibrational” in the first line of section I1 should read “variational”. (5) Rabinovitch, A,; Thieberger, R. Proc. Int. Conf. Low Temp. Phys., l4th, 1975,1975, 4, 407. (6) Chakravarty, S.; Rose, J. H.; Wood, D.; Ashcrott, N. W. Phys. Reu. B 1981,24, 1624. (7) Herschbach, D.R.; Laurie, V. W. J . Chem. Phys. 1961,35,458 and work cited t,herein. (8) James, H. M.; Coolidge, A. S. J . Chem. Phys. 1933, I , 825. (9) Ree, F. H.;Bender, C. F. J. Chem. Phys. 1979, 71, 5362. 0022-365418312087-5202$0 1.5010
the variational function proposed by Cottre1l.l The calculations are carried out in prolate spheroidal coordinates (A, p , 4) centered on the nuclei. Surfaces of constant X = Xo are spheroids with major axis RXo, where R is the internuclear distance. The boundary conditions required by the presence of an infinite-walled spheroidal box and formulas for the matrix elements are given in ref 4. The nominal pressure scale is again obtained from the derivative of the total energy with respect to the box volume, P = dE/dV. Unless indicated otherwise, we give all formulas and numerical quantities in atomic units.1° The polarizability is evaluated by adapting the Kirkwood approximationll for atoms to a diatomic molecule.12 This gives for the parallel and perpendicular components
a1
a11 azz = 4n[(z2) + ( n - 1)(2122)]2
(1)
= axx= ay3 = 4 n [ ( x 2 ) + ( n - l)(X1X2)]2
(2)
Here n = 1for H2+and n = 2 for H2. Each electron in H2 has the same average value of ( z 2 ) and ( x2), whereas the terms ( z1z2) and ( x1x2) involving both electrons are negative as a consequence of the Coulombic repulsion. The average polarizability a and anisotropy K are defined as a = (all + 2~i,)/3
(3)
= (all - “~)/3Cu
(4)
K
The quadrupole moment13 is given by 8 = Y2R2- n[(z2)- ( x 2 ) ]
(5) The Cartesian coordinates are related to the prolate spheroidal coordinates by x = l/zR COS +[(A2 - 1)(1- p 2 ) ] 1 / 2 y = 1/R sin #[(A2 - 1)(1 - p2)]1/2
z = Y2RXk (10) Energy: 1 hartree = 2 Ry = 27.21163 eV. Distance: 1 bohr = 0.529177 A. Force constant: 1 au = 1.55692 X lo6 dyn cm-’. Polarizability: 1au = 0.148185 X lomucm3. Quadrupole moment: 1au = 1.34504 x esu cm2. (11) Kirkwood, J. G. Phys. 2 1932, 33, 57. (12) Hirschfelder, J. 0.;Curtis, C. F.; Bird, R. B. “Molecular Theory of Gases and Liquids”; Wiley; New York, 1954;pp 941-7. Note sign error in the second term of eq 13.2-18a. (13) Buckingham, A. D. Q. Reu. 1959, 13, 183.
0 1983 American Chemical Society
Hydrogen Molecule in a Spheroidal Box
The Journal of Physical Chemistry, Vol. 87, No. 25, 1983
5203
TABLE I: Expectation Values for H, in a Spheroidal Boxa Rho m
12 10 8 7 6 5 4 3 2
Re
E,
(rz)
(Z?
(X?
(r,,z)
(Z,Z,)
(X,X,)
1.403 1.403 1.395 1.355 1.301 1.208 1.068 0.893 0.686 0.455
-1.1716 -1.1685 -1.1638 -1.1440 -1.1136 -1.0441 -0.8800 -0.4749 0.6474 4.5947
2.466 2.363 2.267 2.027 1.812 1.531 1.189 0.836 0.509 0.242
0.984 0.951 0.915 0.819 0.7 29 0.611 0.466 0.321 0.190
0.741 0.706 0.675 0.604 0.542 0.460 0.361 0.258 0.159 0.077
5.540 5.346 5.141 4.595 4.099 3.447 2.649 1.835 1.095 0.508
-0.166 -0.170 -0.167 -0.148 -0.128 -0.102 -0.069
-0.069 -0.070 -0.068 -0.061 -0.054 -0.045 -0.033 -0.021 -0.011 -0.003
-0.040 -0.018 -0.005
0.088 a All quantities are in atomic units.I0 RA, is the major axis of the spheroid; R h o = m indicates results for a free, gas-phase molecule calculated with the same variational function. R e is the equilibrium bond length; E , is the total electronic energy. Brackets denote an expectation value of the indicated coordinates; origin for the coordinates is the center of mass and the z axis lies along the molecular b o n d ; r l , is the interelectron distance. Note that (r2)= (2’)t 2(x2)and (r12?= 2(r? - 2(2,2,) 4(X,XZ). TABLE 11: Expectation Values for H,’ in a Spheroidal Boxa
Rho m
12 10 8 7 6 5 4 3 2
TABLE IV: Polarizability and Quadrupole Moment for
H,’ in a Spheroidal Boxa
Ee
(r?
(2’)
(X2)
R
2.024 2.024 2.012 1.955 1.874 1.731 1.518 1.248 0.936 0.601
-0.6022 -0.6021 -0.6010 -0.5937 -0.5800 -0.5455 -0.4587 -0.2369 0.3867 2.5901
2.472 2.399 2.328 2.123 1.911 1,611 1.248 0.872 0.528 0.250
1.164 1.134 1.103 1.010 0.912 0,770 0.596 0.415 0.249 0.116
0.654 0.632 0.613 0.556 0.500 0.420 0.326 0.229 0.140 0.067
12 10 8 7 6 5 4 3 2
a Notation, units, and conventions as in Table I. Note that Table I1 of ref 4 has a misprint; E , a t Rho = 5 should read -0.4587.
m
12 10 8 7
6 5 4 3 2
P
0.0 ( 0 ) 1 . 6 (0) 7.4 ( 0 ) 5.9 (1) 1.7 (2) 5.5 ( 2 ) 1.8 ( 3 ) 7.7 ( 3 ) 3.5 ( 4 ) 3.8 ( 5 )
ff
I1
5.353 4.874 4.479 3.595 2.892 2.072 1.264 0.632 0.237 0.054
ff1
3.617 3.236 2.951 2.355 1.899 1.374 0.860 0.447 0.177 0.043
a
4.196 3.782 3.460 2.769 2.230 1.606 0.994 0.509 0.197 0.047
K
0.138 0.144 0.147 0.149 0.149 0.145 0.135 0.121 0.101 0.078
a II
0.0 ( 0 ) 2.7 (-1) 2.4 (0) 2.4 (1) 8.0 (1) 2.7 ( 2 ) 9.8 (2) 4.1 ( 3 ) 2.2 ( 4 ) 2.0 ( 5 )
m
a
@.L
5.424 5.140 4.864 4.083 3.328 2.372 1.421 0.688 0.247 0.054
1.709 1.600 1.501 1.238 0.998 0.707 0.425 0.209 0.078 0.018
a
K
0
2.947 2.780 2.622 2.186 1.775 1.262 0.757 0.369 0.134 0.030
0.420 0.424 0.428 0.434 0.438 0.440 0.438 0.432 0.420 0.397
1.556 1.547 1.534 1.457 1.343 1.149 0.882 0.593 0.329 0.132
Notation and units as in Table 111.
TABLE V:
Comparison with Free Molecules”
H,
TABLE 111: Polarizability and Quadrupole Moment for H, in a Spheroidal Boxa
RA,
P
Re
0
0.499 0.494 0.493 0.490 0.471 0.427 0.360 0.272 0.174 0.082
All quantities in atomic units” except the pressure P (in kbar; to be multiplied by the power of 1 0 indicated in the parentheses). a
The integrals required to evaluate the average values are readily expressed in standard forms.4 Results and Discussion Tables I and I1 list the pertinent average values determined from the variational parameters obtained previ0us1y.~Also listed are the equilibrium bond length Re and the corresponding total energy E, as functions of the major axis Rho of the spheroidal box. The ratio ( z 2 ) / ( x 2 ) is a measure of the deviation from sphericity of the electron distribution. As the box size shrinks, this ratio decreases toward unity. This occurs because the box is nearly spherical a t the equilibrium distance (cf. Figure 2 in ref 4) and for small boxes the repulsive interaction with the box walls becomes much more important than the Coulombic interaction of the nuclei and electrons. Polarizability and Quadrupole Moment. Tables I11 and IV give results for the polarizability components and quadrupole moment as functions of the box size. The
H,
property
best
% dev
best
a I1
6.831 4.743 5.439 0.128 0.492
-22 - 24 -23 +8
5.014 1.755 2.841 0.382 1.531
01 a K
0
+1
% dev
tl
-
-3 14 t10 +2
a Best available values for H,are from ref 1 3 (polarizability) and 1 4 (quadrupole m o m e n t ) ; for H,’from ref 1 5 (polarizability) and 1 6 (quadrupole moment). “Percent deviation” refers t o comparison with infinite box results (Rho= - ) from Tables I11 and IV.
values for an infinite box (Rho = m) correspond to a free molecule and thus permit a test of the Kirkwood approximation. Table V lists the best available values for the free H2 and H2+ molecule^.^^-'^ For H2,the Kirkwood approximation is seen to underestimate the polarizability quite appreciably, although it does better for the anisotropy. The percentage error in the polarizability obtained from the Kirkwood formula is similar to the H and He atoms12as well as early calculations for the H2 molecule.18 The results with the Kirkwood approximation on H2+, however, are remarkably accurate. The quadrupole moment does not involve the Kirkwood approximation; however, the higher accuracy obtained for the quadrupole moment results in large part from the major contribution of the bond length in eq 5 and the quadratic rather than quartic dependence on electronic coordinates. As the box (14) Langhoff, P. W.; Gordon, R. G.; Karplus, M. J. Chem. Phys. 1971,
55, 2126.
(15) Stogryn, D. E., Stogryn, A. P. Mol. Phys. 1966, 11, 371. (16) Kim, H.; Hameka, H. F.; Svendson, E. N. Chem. Phys. Lett. 1976, 41, 213 (a theoretical value). (17) Karl, G.; Nickel, B.; Poll, J. D.; Wolniewicz, L. Phys. Reu. Lett. 1975, 34, 1302 (a theoretical value). (18) Hirschfelder, J. 0. J . Chem. Phys. 1935, 3, 555.
5204
The Journal of Physical Chemistry, Vol. 87, No. 25, 1983
LeSar and Herschbach
102
loo 10'
loo lo-'
lo-'
lo-' 5 a = RXo
2
10-3
10
Flgure 1. Variation of the polarizability a, quadrupole moment 8,and polarizability anisotropy K as a function of box size. Given are the ratios to the free-molecule values for H, (solid curves) and H,+ (dashed curves). Also given are the results for the polarizability of the H atom in a rigid box (chain dash) and a box with zero potential at the wails (chain dot). The box size is parametrized by RX,, the major axis of the spheroid. For the H atom, a is the diameter of the spherical box.
size decreases, the equilibrium internuclear distance Re and the extent of the electronic density distribution decrease. The polarizability a and the quadrupole moment 8 likewise decrease. The anistropy K remains nearly constant (over the range RXo = 12 2) but for sufficiently small box sizes it decreases also, due mainly to the dominant role of the repulsive interaction with the box walls. The variation in these quantities is shown in Figure 1, where we plot the ratio of a, K , and 8 to their free-molecule values as a function of box size, specified by the major axis a = RXo. As well as the present results for H2 and H2+, we plot the variation in the polarizability of an H atom in a spherical plotted against the box diameter, for a box with an infinite potential barrier and one with a zero potential a t and beyond the wall. The latter curve shows an upturn a t small box size due to the electron escaping from the box. The similarity of the results for Hz and H,+ are striking, especially for the polarizability. The variation in ala, for the H atom in a rigid box is quite similar to that for Hz and H2+, but the H atom in the zero-potential box shows much smaller variation. Figure 2 compares the variation with box size of the electronic kinetic energy T,,the equilibrium bond length Re, the mean square of the electronic distribution (r z ) ,8, and a. On dimensional grounds, for small boxes these quantities might be expected to vary with appropriate powers of a, whereas for sufficiently large boxes all properties become constant. Indeed, we find that Re/a for both H2 and H2+approaches a constant value as a 0 and varies as a-1 for a m. Also shown is the variation of Re (rescaled to give the correct value for the isolated molecule) from a recent density-functional calculation6 on solid H2 where a is taken as twice the Wigner-Seitz radius. This curve is distinctly less flat than the box-model result. The kinetic energy for a particle in a box goes as and we find that a t small boxes Tea2approaches a constant, indicating that the effects of the box environment dominate in determining the electronic energy. The ratio ( r 2 ) / a 2
-
-
-
10-4
2
5 a = Rho
10
Flgure 2. Variation of the electronic kinetic energy T,, equilibrium bond length R e , mean square of the electronic distribution ( r ' ) , quadrupole moment 8,and polarizability a with a , the major axis of the spheroid. The parameters are scaled on dimensional grounds by appropriate powers of a , as described in the text. The solid curves are for H, and the dashed for H". Also plotted are the results for the bond length from a density-functional calculation on solid H, (ref 6), where a is twice the Wigner-Seitz radius.
exhibits similar behavior and is nearly the same function of a for both H2and H2+. The ratio @ / a 2differs substantially, as the quadrupole moment is much larger for H2+than H2 Most striking is the ratio a l a 3 ,which shows a marked decrease as a 0. That this deviation is due to an error in the Kirkwood approximation is seen by the behavior of ala4, which shows only a slight increase a t small a. Thus, the Kirkwood approximation is incorrect on dimensional grounds. Pressure Variation. Nominal values for the pressure, calculated by P = -dE/dV, are given in Tables I11 and IV. A t about 1 Mbar, the polarizabilities of H2 and H2+are reduced by about 70% and 74%, respectively, and the quadrupole moments by about 22 % and 43 70,respectively, relative to the free-molecule values. From our previous comparisons of box-model predictions to experimental results: we expect that the calculated effects of pressure are greatly exaggerated. For the polarizability, we can test the box model by comparing the results of ten Seldam and de Groot20 for an Ar atom in a box with a recent calculation,?-I which also uses the Kirkwood approximation but treats solid Ar using a perturbation model with the crystal-interaction potential found with local-density functionals. The box-model calculations show an 8% decrease in a, for Ar at about 3 kbar,2O which is roughly comparable to the present results for H2. The perturbation treatment finds that an 8% reduction does not occur until almost 500 kbar. Thus, the calculated box-model variation of polarizability with pressure seems to be much exaggerated. Figure 3 plots P = -dE/dV vs. box size for our calculations on Hz and H2+and similar results for the H atom.lg
-
(20) ten Seldam, C. A.; de Groot S. R. Physica 1952, 18, 905.
(21) LeSar, R. Phys. Reu. E , in press.
(19) Key-Loo, E.; Rubinstein, S. J. Chern. Phys. 1979, 71,351.
(22) Wijngaarden, R. J.; Lagendijl, A.; Silvera, I. F. Phys. Reu. E 1982, 26, 4957.
The Journal of Physical Chemistry, Vol. 87, No.
Hydrogen Molecule in a Spheroidal Box
io6 105 104
103
10' 10'
loo
\
'I,
\
lo-'
I
2
I
5 a = RXo
I '
I
10
Figure 3. Pressure as a function of a , the major axis for H, (solid) and H,' (dashed): the diameter of sphere for H atom in a rigid box (chain dash) and for H atom in a zero-potential box (chain dot): and twice the Wigner-Seitz radius of a localdensity-functionalcalculation for H, (dot). Also shown is the quantity P% for each case. P , is the revised pressure for H, in a box described in the text.
The dependence of P on a for these three systems is very similar over the entire range of box sizes. Also shown is P for the H atom in a box with nonrigid walls (zero potential outside). The dependence on a is qualitatively similar to that for rigid walls, but the pressure is about 1 order of magnitude lower over the range of box sizes. An appreciably steeper dependence of P on a is obtained from density-functional calculations on solid molecular hydrogen (plotted against twice the Wigner-Seitz radius).6 For a free particle in a rigid box, the pressure varies as the inverse fifth power of the box size. For confined atoms or molecules, the electronic kinetic energy becomes dominant as the box size decreases4and thus the quantity P1I5a should approach a constant. As shown in Figure 3, all of the box-model calculations, including the H atom in a zero-potential box, yield remarkably constant results for P1I5aover the entire range of box sizes. Likewise P'I5a for the density-functional calculation on solid H,, while showing more variation at large volumes, also approaches a constant value at small molecular volumes. This seemingly universal behavior of P1I5a may be useful in predicting properties of materials a t high pressures. Renormalization of Pressure Scale. Although the box model with rigid-wall boundary conditions exaggerates the pressure dependence of molecular properties, the pressure dependence appears to be a t least qualitatively correct. Thus, it seems worth trying to revise the pressure scale without otherwise tampering with the model. Here we simply renormalize the box-model pressures. Experimental data on molecular properties at very high pressures are too scarce for this purpose. Hence, we use results from the more exact local-density-functional (LDF) theoretical calculations of Chakravarty et a1.6 on solid H,. The equilibrium bond length Re offers a convenient means to link the box-model results to the LDF pressure scale. While the bond length may not be as sensitive as other molecular properties to perturbations of the electronic
25, 1983 5205
density far from the nuclei, changes in Re do provide a measure of the forces acting on the molecule in the highpressure crystal. As shown in Figures 2 and 3, for a given size a, the LDF bond length is larger than that found with the box model and the LDF pressure is much lower, but the qualitative variation is quite similar. The renormalized pressure, PR,is shown in Figure 3. For a given box size with equilibrium bond length Re, the corresponding PR is taken as the LDF pressure for that bond length. While for large box size (low pressures) the box-model pressures and the renormalized pressures converge, for small boxes PR is about 1 order of magnitude higher than the boxmodel pressure. The dependence of molecular properties on box size displayed in Figures 1 and 2 is readily converted to pressure dependence by using the PR scale of Figure 3. With the revised pressure scale, the Hz polarizability and quadrupole moment have decreased by 43% and 4 % ,respectively, by 1 Mbar. With the box-model pressure scale the corresponding decreases were 70% and 22%, respectively. Although the renormalization softens the pressure dependence, it still appears unrealistically large for the polarizability. This may be due in part to the Kirkwood approximation. For instance, the revised scale gives a 33% decrease in the polarizability by 500 kbar, a much larger change than the 8% reduction found for Ar at that pressure.21 More satisfactory results might be expected for other properties that are less dependent than the polarizability on the electronic distribution far from the nuclei. However, because of the similarity of the pressure scales a t large box size, results for such properties as the molecular surface area and electronic kinetic energy remain unchanged. These properties, discussed in our earlier work4, indicate that even at low pressures the box model exaggerates the effects of compression. A t high pressures, use of the PR scale does bring the electronic kinetic energy into better agreement with the extrapolated low-pressure experimental results (Figure 3 of ref 4). A quantitative comparison is not feasible, since the extrapolation extends over 3 or 4 decades in pressure. Finally, Raman spectra show that the Hz stretching vibrational mode increases by 130 cm-I by increasing the pressure to 500 kbar. Without renormalization the box model predicts4 a 2100-cm-l increase by 300 kbar; with renormalization the increase is reduced to 730 cm-l, still far larger than the experimental result. One of the more striking results found with the box model4 is a well-defined semilogarithmic correlation between the bond length, Re, and the vibrational force constant, hz. For free, gas-phase molecules this relation is known as Badger's rule.7 If such a rule holds for molecules compressed in a solid, then one can infer changes in bond length from changes in the vibratonal frequency. The LDF calculations6 on solid H2 also show that this relation holds, where the vibrational frequency can be calculated from the reported optic mode energy. The slope, d(1og k,)/dR,, from the box-model calculations is about 15% greater than that from the LDF calculations, again indicating that the box-model perturbation is too strong. Transition to Metallic Hydrogen. The pressure dependence of the polarizability is of interest in application of the Herzfeld criterionz3 for the insulator metal transition. In its simplest form, this presumes that the transition occurs when the atomic or molecular volume in a solid is related to the corresponding polarizability by = (47r/3)a
-
v
(23) Herzfeld, K.
F.Phys. Rev.
1927, 29, 701.
5206
J. Phys. Chem. 1983,87,5206-5213
where V and a are in atomic units. This criterion has been found to give remarkably good predictions for metallic transitions, when used with free-atom or -molecule pol a r i ~ a b i l i t i e s . ~ For ~ hydrogen, the Herzfeld criterion predicts a metallic transition a t about 2.0 Mbar, which is close to that predicted for band closure in the molecular solid.6 With the volume dependence of the polarizability available, more refined predictions can perhaps be made. However, in our box-model calculations we find that the Herzfeld criterion is not fulfilled for any box size studied. Since the V-a relation is independent of pressure scale, the box model cannot be used to predict a metallic transition in this way. In our previous discussion of the metallic transition: we inferred a transition a t 12 Mbar from the crossing of the free energy curves of H2 and H2+. There are two errors in this analysis. The volumes of the boxes around the H2 and H2+molecules were underestimated by about 15-20% for the curves in Figure 4 and free energies in Figure 5 of ref 4; thus, the molecular curves should be somewhat steeper. However, this has little effect on the intersection points of the H2 and H2+curves (which shift to V = 20.5 (24) Ross, M. J. Chem. Phys. 1972,56,4651.Ross, M.;McMahon, A. K. “Physics of Solids Under High Pressure”; Schilling, J. S.; Shelton, R. N., Eds.; North-Holland Publishing Co.: Amsterdam, 1981; pp 161-8.
instead of 21a03 in Figure 4 and to P = 12 instead of 14 Mbar in Figure 5 ) . More serious was the omission of the energy of the electron given up on ionization of hydrogen. If this energy (approximated by the spherical-box result, E = a2/2a2)is added to that of H2+,the resultant curve does not cross that for H2 a t any pressure; thus, again the model fails to predict a metallic transition. Assessment. In spite of the renormalization obtained by utilizing the pressure-bond length relation from a more accurate theoretical study,6 the box model still greatly exaggerates the effects of compression for several molecular properties. Apparently this occurs because the pressure dependence of properties such as the electronic kinetic energy differs considerably from that for the bond length. A box with rigid walls probably cannot simulate these differences well enough to make a scaling correction adequate. A box model with nonrigid walls may prove more satisfactory; such a treatment is already available for the H atom.lg
Acknowledgment. This study was sponsored by the Los Alamos National Laboratory Center for Materials Science and performed under the auspices of the Department of Energy and its Division of Materials Science of the Office of Basic Energy Sciences. Registry No. H,, 1333-74-0; H2+,12184-90-6; H, 12385-13-6.
Fluorescence Quenching of Liquid Alkylbenzenes Excited by Nonionizing and Ionizing Ultraviolet Radiation and by ,6 Radiation Frederick P. Schwarr’ and Mlchael Meot-Ner (Mautner) Chemical Thermodynamics Division, National Bureau of Standards, Washington, D.C. 20234 (Received: September 23, 1982, In Final Form: March 22, 1983)
The fluorescence yields and quenching by chloroform of eight liquid alkylbenzenes and 1-methylnaphthalene were measured at wavelengths corresponding to excitation into the S1, S2,and S3states; into the photoionization region (-1900-1250 A); and at higher excitation energies achieved by fl radiation (2.2 MeV) from a wSr source. With excitation into the S1state, the quenching constants vary widely with structure from 0.6 f 0.1 M-’ for benzene to 19 f 1 M-l for p-xylene. Excitation into the S2and S3states results in negligible quenching of these states by CC13H and a decrease in the fluorescence yield. A t energies above the photoionization onset the yield recovers by up to 80% of the S, fluorescence yield due to aromatic cation-electron ion-pair recombination. In the recovery range (-1900-1450 A) the quenching constants of the ion pairs increase in parallel with the increase in fluorescence yield to a constant value of 2.4 f 0.4 M-l (1450-1250 A) for all the benzene derivatives except indane and the quenching efficiency is a linear function of chloroform concentration. This is in contrast to the quenching of the ion pairs produced by the fl radiolysis which follows a square root dependence on [CHCl,]. The quenching constants of the radiolytic ion pairs are larger (6.3 f 0.5 M-’ for decylbenzene to 19 f 1 M-’ for p-xylene) than the quenching constants of the photolytic ion pairs by almost an order of magnitude. In methylnaphthalene the quenching is negligible, which indicates that the electron is trapped to form methylnaphthalene anions.
Introduction Recently, the S1(1B2u.-lAl,) fluorescence yield of liquid benzene excited in the photoionization region 1750-1150 A) was determined as a function of the excitation wavelength and of the electron scavenger concentration.’ The to from charge fluorescence in this is recombination between the correlated benzene cation and ( 1 ) F. P. Schwarz and M. Mautner, Chem. Phys. Lett., 85,239 (1982).
This
electron ion pairs to directly regenerate the S1 state.2 It was observed that the charge recombination fluorescence yield and the ion-pair quenching constants increase from 1900 to 1400 A and level Off below 1400 A.’ Below the ionization wavelength of 1750 A (at energies > Ip), the ion-pair quenching constants of the electron scavengers CC13H9 CH3C13 and C2H5C1 were, as expected, shown to be (2) C. Fuchs, F. Heisel, and R. Voltz, J.Phys. Chem., 76,3867 (1972).
article not subject to US. Copyright. Published 1983 by the American Chemical Society
