J. Phys. Chem. 1992,96, 6021-6027
6021
Spectroscopic Properties of an Isotropically Compressed Hydrogen Atom Saul Goldman* and Chris Joslin Department of Chemistry and Biochemistry, and the Guelph- Waterloo Centre for Graduate Work in Chemistry, University of Guelph, Guelph, Ontario, Canada N1 G 2 W1 (Received: November 1 1 , 1991; In Final Form: March 24, 1992) We use a numerical solution of the Schrcidinger equation for a hydrogen atom at the center of an inert, impenetrable, spherical cavity, to predict the influence of isotropic compression on spectroscopic properties. The angular momentum sublevel degeneracy that occurs for the free atom is removed under isotropic compression. We account for this splitting on the basis of the underlying distribution functions. We find that the principal lines in the Lyman and Balmer series are blue-shifted relative to the freeatom frequencies, but, depending on the cavity radius, the intensities of these lines can be either enhanced or diminished, relative to the free atom intensities. We find that the excited-state lifetimes are all reduced by compression, with the reduction being especially large for the 2s level. We also find that nuclear volume effects on the electronic energy are greatly enhanced by isotropic compression.
I. Introduction In this paper we calculate the spectroscopic properties of a hydrogen atom at the center of an inert and impenetrable spherical cavity. Our interest in this arose from the more general question of how the spectrum of a nonbonding atom changes, from its free state, when the atom is in a physically interesting environment (e.g., in a solid matrix or liquid, in a micropore, or near a wall). The results presented here form a prelude to this general study. The formal solution of the Sc-ger equation for a hydrogen atom at the center of an inert impenetrable spherical cavity has been known for many years. In an important early paper,' this model was used in a semiquantitative calculation to predict the effect of isotropic compression of a hydrogen atom on the energy of its Is, 2s, and 2p levels. Other previous applications of this model include predictions of the pressure and polarizability? hyperfine ~ p l i t t i n g , ~and - ~ the magnetic screening constant6 of compressed hydrogen. In subsequent work,' these functions were reevaluated for hydrogen in a spherical cavity with soft (penetrable) walls. Also,hard and soft cavity models have been used together with quantum chemical calculations on multielectron atoms,&1°and spheroidal cavities have been used with the hydrogen molecule, and the hydrogen molecule However, a model calculation on the optical spectrum of an isotropicallycompressed hydrogen atom has never been done. Of course, the cavity model (whether spherical or spheroidal, with either hard or soft walls) is a very simple one. Molecular environments are highly irregular and anisotropically polarizable. Therefore, in an attempt to make these calculations physically relevant, we picked the cavity radii by reference to known spectroscopic data. The 1s 2p transition of H in a solid Ar matrix has been measured and found to be 0.390 au,14 which corresponds to a blue shift of 0.015 au relative to the free atom. From Table I it is seen that at ro = 8% = 4.23 A (% = Bohr radius) the energy for this transition is -0.40 au, so that a cavity whose radius is 8 a. is a rough model for H in solid Ar at ordinary pressures. Cavities significantly smaller than this would correspond to more highly compressed H; larger ones would correspond to H that is less perturbed by its surroundings. Our results are given for cavity radii varying from 0.10 to 50 Bohr radii. This range was selected on the basis of the foregoing result, and much of our emphasis is for cavities whose radius is -4 to -10 a,,. The rest of this article is organized as follows. In section I1 we outline the solution of the schrixlinger equation for this problem and provide working equations for the wave function. The bulk of our results are given in section 111, where the principal spectroscopic consequences that result from isotropic compression of a hydrogen atom by an impenetrable cavity wall are presented and discussed.
-
-
11. Theory
A. Derivation. If electron spin and relativistic effects are neglected, and the nuclear mass is presumed infinite relative to 0022-3654/92/2096-6021$03.00/0
that of the electron, the radial part of the Schriidinger equation for a hydrogen atom takes the formlsJ6 d2R + -2 -dR dp2
P
dP
[
X + - ] R =1) -1 - O
4
P
(1)
p2
for which the general solution is (2)
R = e-P/*p'F(a,y,p)
where F(ci,y,p) is the confluent hypergeometric function:
These equations are based on atomic units (le1 = h = m, = 1) and on the substitutions p
=2 G r ;
x = I/&;
u(r)= -1/r
where U(r) is the potential energy and E is the total energy. In eqs 2 and 3, ci = -A + I 1 and y = 21 2. The solution represented by eqs 2 and 3 is general, in the sense that it applies both to a free hydrogen atom and to one at the center of a spherical cavity of finite size. The application of different radial boundary conditions results in different final forms of R. Consider fmt the free hydrogen atom. If R is to be a physically acceptable solution, it must remain finite everywhere. At the origin, by eqs 2 and 3, R is finite. But R will not remain finite in the limit p -, unless ci is zero or a negative integer, i.e., unless X is a positive integer greater than I. If this condition is not met, F(ci,y,p)takes the form of an infinite series that goes as e+' as p -,15*16 so that in this limit R would diverge as e+'12p'. The condition that X be an integer greater than I results in a truncation of the infinite sum in eq 3, to a polynomial of degree X - [I 11. -. When this Under this constraint, R remains finite as p polynomial is used for F(ci,y,p) in eq 2, R becomes the familiar solution of the radial equation for a hydrogen atom, written in terms of associated Laguerre functions, with X now identified as the principal quantum number n. Next, consider a hydrogen atom at the center of a spherical cavity of finite size. Here again, R remains finite at the origin. does not arise, so that R is nowhere Now, however, the limit p singular, and consequently X need not be an integer. In this case, the values of A, or equivalently E, are obtained numerically from the locations of the zeros of R . B. Determination of E and Working Equations for R . R is a function of the variables r, I, and E. If r is set equal to the cavity radius ro,J.(ro,e,+)must vanish for all (e,+) and hence R(ro)must vanish for all allowed values of I and E, to ensure continuity of the wave function. The spherical harmonic index I equals the number of angular nodes in the wave function, and can be 0 or
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0 1992 American Chemical Society
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Goldman and Joslin
6022 The Journal of Physical Chemistry, Vol. 96, No. 14, 1992
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The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 6023
Isotropically Compressed Hydrogen Atom
E 40
30
i
-0.3
I
3s
3 P l
-0.5 -0.6 -071 ' -1000
'
' ' ' ' ' ' ' ' ' 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100
-10
E Figure 1. Plot of the radial part of the wave function R against the energy E (in atomic units) for a cavity of radius 8 a, and angular momentum quantum number l = 0. The values of R were obtained using eqs 2-4 (see text). The first, second, third, intersections of the curve through the R = 0 line give the energies: E ( l s ) , E(2s), E(3s), respectively, for this cavity size.
...
...
a positive integer. Thus if RR*,where the asterisk means complex conjugate, is plotted against E for selected values of ro and 1, the values of E that correspond to nodes in RR* are the eigenvalues. Spbcifcally, for I = 0, the lowest-energy node in RR* gives E(ls), the next node, E(2s), etc. For 1 = 1, the lowest-energy node in RR* gives E(2p), the next node, E(3p), etc. As with the notation used for a free hydrogen atom, where the outermost radial node is at infinity, the "n"in ns,np, etc. is the total (radial plus angular) number of nodes in #. This procedure for determining the values of E is illustrated by the plot in Figure 1. Equations 2 and 3 were used to compute R for negative values of E. For E < 0, for all values of ro, I , and n studied (Table I), we found that 80 terms in the sum shown in eq 3 provided values for R with seven-figure accuracy. To guarantee at least sevenfigure accuracy, we always carried 120 terms in this sum. The reason such a large number of terms is sometimes needed is that the sequence in this sum involves terms of nearly equal size, but of opposite sign. By q s 2 and 3, it is seen that R is real for negative E. For E > 0, R is pure real for 1 even, and pure imaginary for 1 odd (seecq 4). Equations 2 and 3 can be used for a working expression for R,and to obtain E, for both negative and positive E, by going through the above prescription with R as a complex variable. We found however, that for positive E, it was simpler to use an alternate representation of R. This representation is obtained by transforming the radial equation for the hydrogen atom into the so-called"Coulomb wave" equation17*18and then using the solution of this equation (the "regular Coulomb wave function") for F(a,v,p) in eq 2. When this is done, one obtains17J8
'
0.0
1.0
2.0
3.0
4.0
5.0
6.0
Y
10
F i e 2. Illustration of the effect of isotropic compression on the energy
of the first three principal levels of a hydrogen atom. Both here and in subsequent figures, r, and r are in atomic units (Bohr radii).
I. As explained previously (section IIB), they were obtained numerically from eqs 2-4 by solving for the zeros in R. The zeros were located efficiently by a method based on a mixture of linear interpolation, extrapolation, and bisection (NAG routine COSADF). As seen from Table I, all the energy levels are raised relative to their free-atom values. The extent by which a level is raised increases as ro decreases, and it increases with the principal quantum number n, for a given ro and 1. Also,the energy levels for a given ro and n are split, such that the sublevel with the largest I has the lowest energy. The splitting of the angular momentum sublevel degeneracy is a consequence of the hard wall at r,. This degeneracy in the free hydrogen atom is a consequence of the central Coulombic field which the electron experiences everywhere (except very near the nucleus). The presence of a wall at finite ro results in a violation of the requirement that the potential be purely Coulombic everywhere. Of course, the fact that the compression is isotropic ensures that the magnetic sublevel degeneracy will not be split. The increase in energy of the levels with decreasing ro, and the splitting, are shown graphically for the first three principal levels, for medium-sized cavities, in Figure 2. (b) Splitting of the Angular Momentum Sublevel Degeneracy. Here we account for the fact that the energy of the sublevels, within a given principal level, decreases as 1 increases. For definiteness, we consider the 3s, 3p, and 3d sublevels in a hydrogen atom at the center of the cavity of radius 8 a ~ .As seen from Table I, E(3s) > E(3p) > E(3d). To account for this ordering, we first work out the expected potential and kinetic energies for these sublevels. These expectation values were obtained from (PE(31))
(4)
-(
!)31
1 = 0, 1, 2
(5)
= - &8R31(r)R3/*(r)rdr (KE(31))
q = -l/G
Thus we used eqs 2 and 3 for R for E I0, and eq 4 for R for E > 0, in all our calculations. Of course both representations give the same result when sufficient numbers of terms are carried in the sums. As with the calculation of R for E I0, we camed 120 terms in the sum in eq 4 for E > 0 and confiied that this always provided at least seven-figure accuracy in R.
III. Results and Discussion (a) h r g y Levels. The energy levels for 0.10 Iro I50, 1 5 n I6,O II In - 1 are given with seven-fwe accuracy in Table
E(3l) - (PE(31))
(6)
Our values of (KE(31)) were obtained from q 6 , wherein eqs 2-4 were used for E(30, and eq 5 was used for (PE(3c)). In these expressions, PE and KE stand for potential energy and kinetic energy, respectively, and R3,(r)is the normalized radial wave function for sublevel (31). Both here and below, the r-space normalization constants, NnI,were obtained from
(7)
where the primed quantities are the unnormalized radial functions obtained from eqs 2-4. The integral in eq 7 was done by numerical quadrature. The results from eqs 5 and 6 are given in Table 11, from which we see that the expected potential energies for these sublevels are
6024 The Journal of Physical Chemistry, Vol. 96, No. 14, 1992
Goldman and Joslin 8 ,
TABLE Ik Expected Values of the Potential a d Kinetic Eaergies for a Hydrogen Atom at the Center of a Spherical Cavity of Radius
1
level 3s 3p 3d
(PE)
WE)
-0.425 316 8 -0.3210030 -0.224 8490
0.671 808 8 0.4783712 0.270907 2
1
3s
8 no"
E 0.246 492 0 0.1573682 0.046 058 3
i
1
0 -
"All entries are in atomic units. 3P
(d
n
0.3 -
3s
40 -
:::A I 30
0.4
1
-
3d
,
,
,
15
18
21
00
00
03
06
09
p 1 2
2 4
Figure 4. Momentum distribution function for the 3s, 3p, and 3d orbitals of a hydrogen atom. Solid curves are for a hydrogen atom in a cavity (ro = 8); dotted curves are for an unconfined hydrogen atom. p is in atomic units h/ao.
3d
The distribution function that underlies the mean kinetic energy is the momentum-space wave function This is simply the Fourier transform of the ?-space wave function $(i),where ? ( r , O , ~ $ ) . lIn ~ ~atomic units it is
+e).
0 00 0
2
4
6
8
r
12
10
14
16
18
Figure 3. Radial distribution functions. (rR)2gives the probability per unit length that the electron will be found at a distance r from the nucleus regardless of direction. Solid curves are for a hydrogen atom at the center of a spherical cavity of radius 8 ao; dotted curves are for an unconfined hydrogen atom.
ordered as (PE(3s)) < (PE(3p)) < (PE(3d)). However, this potential-energy ordering is overridden by the expected kinetic energies that go as (KE(3s)) > (KE(3p))
> (KE(3d))
We note in passing that the well-known result'98 (8)
which is a consequence of the application of the virial theorem to a system described by a potential that is everywhere Coulombic is here not obeyed, because of the presence of the cavity wall. The appropriate application of the hypervirial theorem to systems in cavities has been the subject of much previous work.10*12*2@-22 To understand in a little more detail why the expected potential and kinetic energies follow the patterns they do, we worked out the underlying probability distribution functions for each. These are displayed in Figures 3 and 4. In Figure 3 we plot r21R3/I2,which is proportional to the probability of finding an electron in an annulus of thickness dr at a distance r from the nucleus. We also show the free-atom result for comparison. With the free atom the expected distance is the same for the 3s, 3p, and 3d levels, despite the fact that the number of peaks and their positions are different for these sublevels. For H in a cavity, the fact that the expected distances go as (43s)) < (43p)) < (r(3d)) is c o ~ e c t e dto the relative growth and movement inward of the inner peak(s), as we go through the sequence: 3d, 3p, 3s.
(9)
If the momentum p is represented by the polar coordinates (p, +,), then the momentum-space wave function can be written in terms of spherical harmonics:
,e,
+n/m@)
Fn/(P)V Y e p d p )
(10)
A working expression for F&) is obtained by substituting $(r,6,#)
= R ( r ) r ( 6 , + )for $(F) into eq 9, expanding the plane wave in spherical harmonics, and using the orthonormality properties of spherical h a r m 0 n i ~ s . I Doing ~ ~ this gives F ~ / ( P=) Jro+j/br)Rn/(r)dr
so that the total energies are ordered as E(3s) > E(3p) > E(3d) 2(KE) = -(PE)
+@) = ( 2 r ) - 3 / 2 J ~ - i p r $ ( i )dr
20
(1 1)
where j,( ) is a spherical Bessel function. The quantity IpFd(p)12is the momentum distribution function. The probability that the magnitude of the momentum lies between p and p + dp (independent of direction) is given by IpFd(p)12dp. This function is normalized through the condition S,mbFn/(p)P Q=1
(12)
Since the expected value of the square of the momentum given by
W ) n / = xmp2bFn/b)l2dp
(13)
the expected value of the kinetic energy of the electron is obtained from (KE)n/ =
i / , J m ~ 2 1 p ~ n / ~ ) 1dp 2
The momentum distribution function for the 3s, 3p, and 3d levels, obtained by eqs 11 and 12, with the R,,,(r)'scalculated by eqs 2-4, are shown in Figure 4. Here we also include the freeatom result for comparison. Analytic expressions for F A ) are available for the latter.Igb The integrals in eqs 11 and 12, for the case of
The Journal of Physical Chemistry, Vol. 96,No. 14, 1992 6025
Isotropically Compressed Hydrogen Atom
5,
1
4
41
'II:
3
3 2
I
l o . .
4.0
1
1
3.0 2.0
1 .oo
. .
-
0.0!& 0
'
. 20
10
0.75 -
I
50
40
30
'To Figure 5. (top) Illustration of how the frequency (relative to an unconfined H atom, and on a log scale) of absorption or stimulated emission is influenced by isotropic compression, for the principal lines in the Lyman series. (bottom) Intensity of the Lyman series transitions for a compressed atom, relative to the corresponding intensities of the unconfined atom. For both frequencies and intensities: (0)2p 1s; (0) 3p Is; (A) 4p Is.
-
A
a
0.50
-
0.25
-
0.00
*8 .
A 0
.
.
!
.
=
2
2
.
-
-
the cavity, were done by numerical quadrature. The basis for the sequence (KE(3s)) > (KE(3p)) > (KE(3d)) is clear from Figure 4. The momentum distribution function for the (3s) orbital has its outermost peak at p N 1.3 au, which is further out on the momentum scale than the position of the outermost peak for the 3p orbital (- 1 au), which in turn occurs at a higher p value than the maximum for the 3d orbital (-0.7 au). Since the mean kinetic energy is obtained from the integral over p21pFnr(p)12, the sequence KE(3s) > KE(3p) > KE(3d) is readily understood in terms of the ranges of the function IpF,&)12 for the three orbitals. We also note that the small damped high-momentum oscillations shown in Figure 4 (beyond p 1.9 for 3s, beyond p 1.6 for 3p, and beyond p 1.25 for 3d) are an artifact of the hard wall. Physically, they originate from the infinite impulse the electron experiences on contact with the hard wall; mathematically they are due to the discontinuity in the slope of $(r) at the wall. Their influence on the mean kinetic energy is qualitatively similar for the three orbitals and partially cancels in the comparison. (e) Frequency and InWty sbws for Absorption or Stimulnted JhWon, Our results on the frequency and intensity shifts, relative to the free atom, for the principal lines in the Lyman and Balmer series are shown in Figures 5 and 6, respectively. The frequencies are all blueshifted. This is as expected, because isotropic compression raises the energies of the upper levels more than it does those of the lower levels (Table I and Figure 2). The frequency ratios shown in Figures 5 and 6 were obtained by
-
-
-4ro) =-
4-1
Wro)
w-1
-
(14)
The cavity effect on intensities shown in Figures 5 and 6 is more interesting. The intensity ratios are seen to go through a maximum greater than 1 for physically relevant values of the cavity radius. The intensities vanish in the limit of infinitesimally small cavities, and of course approach the free-atom values in the limit of infinitely large cavities. Similar patterns for both the frequencies and intensities were obtained for the other lines in these series. Thus the effect of isotropic compression on these lines is to
Equation 17 is for G in atomic units. The integral in eq 17 was evaluated by Gauss quadrature. The functions R were obtained from eqs 2 4 , with each R separately normalized by eq 7. The Gnkdlil(m)values were taken from the l i t e r a t ~ r e . ' ~ . ~ ~ (a) Excited-State Lifetimes. The transition probability for spontaneous (as opposed to stimulated) emission, is given by23a 1+1
PnI.+/, = C ( A E ) 3 ~ G d N / , ( r o )if I' = I + 1 (18) 1
= C ( A E ) 3 z G n l , d r ( r o ) if 1'1 I - 1 For AE and G in atomic units, and P in s-l, the constant C = 2.1417 X 1O1Os-l. The half-life, T , of a level is obtained by23a 7-1
=
c
nl>n'/'
P+dp
(19)
where the sum is over all allowed transitions to lower energy levels. We used eqs 17-19 to obtain the effect of isotropic compression on T , for the 2s, 2p, 39, 3p, and 3d levels of H. Our results for the 2p, 39, 3p, and 3d levels are shown, relative to the free-atom
Goldman and Joslin
6026 The Journal of Physical Chemistry, Vol. 96, No. 14, 1992
1
A 8
0.6
A
i 0
A
0.0
0
30
20
10
50
40
r0
Figure 7. Effect of isotropic compression on the half-life r, for the 2p (0);3s (m);3p (A);3d (A) levels of a hydrogen atom. The unconfined atomvalues (T& in I r a s are19*23 2p (0.16), 3s (la), 3p (0.54), 3d (1.56).
,
5 , log(s/s)
-5
i 1 i
0
0 -
1
0
0
j
0 0
-’Ob -15
I1
-20
I
0
I
oo
TABLE III: Factor by Which Nuclear Volume Effects 011 the Electronic Energy Are Amplified, IS a Consequence of a Coustraining Cavity Wall at Distance ra from the Nucleus” n (principal quantum number)b r0 an 1 2 3 4 5 6 50 0.100+1 0.100+1 0.100+1 0.102+1 0.165+ 1 0.394+1 25 0.100+1 0.100+1 0.112+1 0.308+1 0.848+1 0.186+2 10 0.100+1 0.140+1 0.695+1 0.236+2 0.596+2 0.127+3 8 0.100+ 1 0.202+1 0.1 13+2 0.377+2 0.957+2 0.206+3 6 0.101+1 0.372+1 0.209+2 0.697+2 0.180+3 0.395+3 4 0.112+1 0.900+1 0.499+2 0.173+3 0.463+3 0.105+4 2 0.237+1 0.414+2 0.255+3 0.959+3 0.272+4 0.643+4 0.961+1 0.232+3 0.158+4 0.626+4 0.184+5 0.445+5 1 0.5 0.550+2 0.154+4 0.110+5 0.451+5 0.135+6 0.331+6 0.25 0.372+3 0.1 12+5 0.823+5 0.342+6 0.105+7 0.255+7 0.10 0.527+4 0.164+6 0.123+7 0.517+7 0.157+8 0.390+8 t
”Entries obtained by eq 23. bThe notation is the same as in Table I.
and its dielectric constant (in atomic units) equal to 1, then the energy shift, relative to a point-charge model for the nucleus, can be shown by first-order perturbation theory to be
Since RN is very small (o(10-5A)), R,,,(r) N R,,,(O). Also, since R,,(O) = 0 except for 1 = 0 (s states) the effect only arises for s states. Thus, from eq 20 the energy shift is
I 0
RN2
8
4
12
16
20
TO
8. Effect of isotropic compression on the half-life of the (2s) level of the hydrogen atom. The vertical axis is log,,. F i i
T in Figure 7, from which it is clear that the effect of isotropic compreasion is to always reduce the half-life of these levels, relative to the free-atom half-lives. This is easily understood from eqs 18 and 19. increases sufficiently rapidly with compression (Table I, Figure 2), to override the nonmonotonic behavior in C (Figures 5 and 6, bottom). Consequently, T drops with compression. The most dramatic reduction in half-life oocurs for the 2s level. This is shown in Figure 8. For the free H atom, the 2s level in the dipole approximati~n~’~ has an infinite lietime, Le., ~ ( 2 s = ) Q). This is because the 2s 1s transition is forbidden and the 2s 2p transition does not arise because E(2s) = E(2p). We have seen, however, that isotropic compression lifts this degeneracy, with E(2s) > E(2p). Consequently, the 2s 2p trnsition is allowed for the compressed atom, and this provides a path by which the 2s level can decay, in the dipole approximation. As shown in Figure 8, the half-life of the 2s level is very sensitive to cavity size; for cavity radii in the range 4-8 (-2.1 to -4.2 A) the and 4.4 X lo-’ lifetime of the 2s level varies between 7.5 X s. These values are typical of T for the lower energy levels in the free H atom (see caption to Figure 7 for values). (e) Nuclear Volume Isotope Effect. When the electron is very close to the nucleus, the electrostatic potential does not follow the r-’ law because, close to the nucleus, the nucleus cannot be taken to be a point charge. The actual potential depends on exactly how the charge is distributed over the nonzero volume of the nucleus, and on the permittivity of the n u c l e ~ s . ~The ~ J disturbance ~~ due to the nonzero nuclear volume is, for the unconfined H atom, an exceedingly small hyperfine effect. But a little thought will confirm that, if the electron is restricted from going far from the nucleus, the nuclear volume effect will be amplified. This is why we bnsider it here. We first obtain an approximate expression for the nuclear volume effect for the unconfined H atom. We follow the treatment given in ref 19c. If the nucleus is taken to be a sphere of radius RN, with its positive charge homogeneously distributed throughout its volume,
-
-
-
AE = ~ l R n o ~ 0 ~ 1 2
(21)
where the point-charge model has the lower energy. What is actually measured is the difference in energy shifts, 6E, between two isotopes, whose nuclei have radii RN and RN + 6RN. From eq 21, to first order, the isotope shift is RN 6E -IRM(0)l’SRN (22) 5 where the isotope with the larger nucleus has the higher energy. From eqs 21 and 22, we find
-a =,- = 6Er0 IRno(O)1,2 am6Em IRno(0)12m
(23)
where the subscripts ro and mean ‘for H in a cavity” and ‘for unconfimed H”, respectively. The amount by which either of these effects is amplified by having the H atom constrained to a finite cavity is obtained by the ratio shown at the extreme right of eq 23. The values of this ratio are given in Table 111, from which we see that it grows with decreasing rofor fmed n,and that it grows with increasing n for fixed ro. From the entries in this table, it is seen that the amplification effect of a constrainingcavity wall on nuclear volume effects can be very large. Specifically, the shift is amplified by 7 or 8 orders of magnitude for the 6s level in the smallest cavity shown. This enhancement suggests that studies on the electronic energies of compressed atoms may be useful for obtaining information on the size or charge distribution of the nucleus. Acknowledgment. We are grateful to the Natural Sciences and Engineering Research Council of Canada for financial assistance, and to Bernie Nickel for a helpful discussion. Registry NO. H, 12385-13-6.
References and Notes (1) Degroot, S.R.;Ten Scldam, C. A. Physicu (Urrechf) 194612,669.
(2) Michels, A.; de Boer, J.; Bijl, A. Physicu, 1937, 4, 981. (3) Suryanarayana, D.;Weil, J. A. J. Chem. Phys. 1976, 64, 510. (4) We& J. A.; J. Chem. Phys. 1979, 71, 2803. (5) Arteca, G. A.; Fernindez, F. M.;Castro, E. A. J . Chem. Phys. 1984,
80, 1569.
(6) Buckingham, A. D.; Lawley, K. P.Mol. Phys. 1960,3, 219. (7) Ley-Koo, E.; Rubinstein, S. J . Chem. Phys. 1979, 71, 351.
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Effectlve Dimensionality of Layered Diffusion Spaces Roberto A. Garza-Upez and John J. Kozak* Department of Chemistry, Franklin College of Arts and Sciences, University of Georgia, Athens, Georgia 30602 (Received: December 23, 1991)
The stochastic master equation dpi/dt = -&Cupj for the evolution of the probability pi, with transitions between nearest-neighbor sites governed by the transition matrix Gii' is solved for finite k X k X k cubic lattices, subject to the temporal boundary condition pi(0) =,,6 m being an interior slte of the lattice, and two sorts of spatial boundary conditions (periodic and strictly confining). From the regime where the entropy function S(t) = - [ z i p i ( ? ) In pi(?)] grows linearly with In t , we extract the spectral dimension d, of the lattice. It is found that the Euclidean dimension de of the defining k X k X k lattice effectively converges to de 3 when k = 11. Taken together with the results reported earlier on k X k square-planar lattices and insights drawn from calculations of the mean walk length ( n ) before trapping of a random walker on finite de = 3 , 4 cubic lattices subject to periodic boundary conditions, the study allows the identification of the minimal size of diffusion space such that, in studying diffusion-mediated processes in microheterogeneous media, the dynamics may be sensibly described by a Fickian equation with a Laplacian defined for integer dimension.
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I. Introduction In classical, continuum theories of diffusion-reaction processes based on a Fickian parabolic partial differential equation of the form
ac(T,t)
= DV2C(7,t)+ f [C(T,t)] at specification of the Laplacian operator is required.' Although this specification is immediate for spaces of integral dimension, the problem becomes rather more complicated for spaces of intermediate or fractal As examples of problems in chemical kinetics where the relevance of theoretical predictions using an approach based on eq 1 is open to question, one can cite the avalanche of work reported over the past decade on diffusion-reaction processes in microheterogeneous mediaH (zeolites, clays, and organized molecular assemblies such as micelles and vesicles) where the (local) dimension of the reaction space is (often) not clearly defined. It is to examine one aspect of this general problem that the work reported in this paper is directed, To introduce the problem considered here, it has been shown in earlier w ~ r k ~that . ' ~the entropy function S(t) = - [ z i p i In pi] constructed from the solution of the stochastic master equation (withthe initial boundary condition that the probability is localized on a single site) increases linearly with In t (t time) (see the discussion in section 11). The slope of the linear regime is dJ2, where d, is the spectral dimension of the underlying lattice. If attention is focused on Euclidean lattices, one is assured that the dimension de of the Euclidean space, the fractal (Hausdorff) dimension d,, and the spectral dimension d, are equal.11J2Consider, then, a squareplanar lattice (coordination number or valency v = 4); in calculations reported in ref 9, we showed numerically that d, = 2.00, already for lattices 21 X 21. Then, in ref 10, in our lattimtatistical study of diffusion-controlled processes in smectite clays, we showed by studying a series of layered square-planar lattices of increasing spatial extent that the Eu0022-3654/92/2096-6027$03.00/0
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clidean dimension of the expanded (two-layer) system remained effectively de 2. Now, intuitively, one expects that if square-planar lattice layers continue to be stacked one above the other (generating, eventually, a lattice of cubic symmetry, i.e., one for which each site has a valency v = 6), the dimensionality of the space should converge, eventually, to d, = d f = de = 3. The chief aim of the present work is to determine the (minimal) size of k X k X k cubic lattices such that an effective Euclidean dimension de 3 is realized. Our calculations are carried out assuming first periodic boundary conditions and then strictly c o n f i i g boundary conditions,the latter class of spatial boundary conditions defined operationally such that if the randomly diffwing particle is on a boundary site of the defining k X k X k lattice, its next displacement can be only back to the same site, to an adjacent boundary site, or to an internal site of the lattice. Once this "minimal" cubic lattice has been determined, the expectation is that one will then have a means of gauging the validity of theoretical studies of diffusion-reaction processes occurring in compartmentalized systems of finite spatial extent when analyzed using an approach based on eq 1, with the Laplacian defined for integer dimension.
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11. Formulation
In recent work, we have solved numerically the stochasticmaster equation dpi(t)/dt = - [ ? G , j ~ j ( t ) I
(2)
for various Euclidean and fractal lattices subject to periodic (or confining) spatial boundary conditions and to the temporal boundary condition pi(t=O) = 6im (3) where m is an interior site of the lattice?JO Here, the matrix Cij describes the transition rate of the probability p i ( t ) to the site i from a nearest-neighbor site j . Thus, if Mij is the probability of 0 1992 American Chemical Society
