J . Phys. Chem. 1985,89, 3444-3447
3444
TABLE I. Dependence of ( d ’ ) (the Average Pseudodimensionality) and the Transfer Time re on the Acceptor Concentration P Aand Lengthwidth Ratio L/W“ LIW 5
(d’)
2.2
6 2.1
rc/r,
4.7 X
1.4 X
10-3
10-5
10
20 2 2.6 X 10-5
2 2.15
6
4.4 X 10-3
5.6 X 10-5
1.7
20 1.6 1.5 X 10-5
the time for which the donor intensity has decayed to 1/e of its initial value in units of r,, the characteristic time for a transfer between a donor and an acceptor separated by Ro, the radius of the sphere. T~ is
calculated intensity has been plotted vs. time in a In-In plot for different acceptor concentrations (see Figure 1). In such plots, the slopes of the curves are equal to d76. Although Figure 1 shows that d76 depends on the time, it also shows that it is approximately constant for a given range of time, concentration, and L:W. This allows us to use eq 3 to determine an average value of d’, ( d ’ ) , for structures such as the one we are presently using. It is then possible that, on Vycor-like structures, the decay can be described with a pseudoconstant exponent. For instance, most experimental results are obtained during the time required for the initial intensity to decrease by at most a factor of 100. During this time range, the value (d’) is apparently constant. From Figure 1, the best value of (d’) is determined for the different time and concentration range of interest. With these values, eq 3 is then tested, and the results are shown in Figure 2 which gives a straight line for In Z vs. t(d’)ll. Table I gives the values of ( d ’ )for the different concentrations and L:W for the time range required for the donor intensity to decay to 1/ 100 of its initial value. In this table, the transfer decay times ( r e )in units of a critical transfer time ( T , ) , the time of transfer at a donor-acceptor distance of the sphere radius, Ro,
are also given for different concentrations and L:W values. As expected, as the L:W ratio increases, (d’) decreases, and, eventually, one-dimensional behavior is observed. Also, as the concentration decreases (or the time increases), the effects of the excluded volume and surface disappear and three-dimensional behavior is recovered. The approximate simulation above shows that (d’) = 1.7 (the value determined experimentally)15can be obtained for an acceptor concentration of 6 molecules/sphere and an L:W ratio of 10. These values have to be compared to the experimental ones: if all of the pores between the spheres in Vycor are interconnected, the L:W ratio should be 200:40 = 5 . The concentration used by Even et al.Is was 2-3 molecules/sphere, if one assumes a random distribution of donors and acceptors within the Vycor glass. Our model oversimplifies the Vycor structure. The silicate spheres probably are randomly packed, and the microstructures of the pores are not known. Thus, one should not take our comparison with experiments too seriously. It is not the purpose of this study, however, to attempt to describe the Vycor structure, nor to compare our results with experimental observation. The main purpose of this work is to show that, if one studies the one-step trapping process on a nonfractal (e.g., cubic) lattice with excluded volumes of length scales equal or greater than the characteristic energy transfer length, an apparent fractal behavior can be obtained over limited range of time (or acceptor concentrations). Of course, one can try to distinguish between a true fractal and an apparent fractal behavior if the experimental study is carried out over a long time scale. Most unfortunately, this is not possible due to the limitation imposed on the experimentalist by the finite lifetime of the excited state of the donor. Extensions of this work to simulations of other interesting structures are now in progress, and the details of these simulations will be published later. Acknowledgment. We thank the support of the Office of Naval Research, P.E. would like to thank NATO for a travel grant.
Dimensional Interpoiatlon of Correlation Energy for Two-Electron Atoms J. G. Loesert and D. R. Herschbach* Department of Chemistry, Harvard University, Cambridge, Massachusetts 021 38 (Received: April 25, 1985)
Accurate ground-state energies for two-electron atoms parameterized by the spatial dimension D and the nuclear charge Z have been computed by generalizing the Pekeris and Hartree-Fock-Roothaan algorithms. In units scaled in proportion to ( D - 1)-2, a hydrogenic factor, the correlation energy is found to be just a linear function of 1 / D to a remarkably good approximation (within -1% for Z 2 2). As a fraction of the total energy, the correlation energy for Z = 2 varies from 2.28% at the D -,1 limit to 0.99% at the D -, limit. For Z 2 2, linear interpolation of the correlation energy between exact values available in the two limits gives total energies accurate to better than 0.005%, comparable to the best configuration interaction calculations.
Introduction Physical problems not amenable to ordinary perturbation methods can sometimes be usefully treated by regarding the spatial dimension D as a parameter which can assume arbitrary real values.’ This has been exemplified in particle physics, nuclear physics, critical phenomena, and quantum optics.* Typically the problem is solved analytically at some D # 3 where the physics is simpler, and then perturbation theory is used to obtain an approximate result for D = 3. Most often the analytic solution is obtained in the D limit, and 1 / D is employed as the
-
+Present address: Society of Fellows, Department of Chemistry, University of Michigan, Ann Arbor, MI 48109.
perturbation parameter. Here we apply this dimensional continuation approach to correlation energies for the ground state of two-electron atoms. This problem can be solved exactly in two limits, D w and D 1 . We find that dimensional interpoZation3 between these limits yields remarkably accurate correlation energies. Previous studies treating the total energies of two-electron atoms have provided several results employed in our work. Herrick and 1 limit of a two-electron atom Stillinger4 showed that the D +
-
-
(1) Witten, E. Phys. T d u y 1980, 33 (7), 38. (2) Yaffe, L.G. Phys. Today 1983, 36 (8), 50. (3) Herschbach, D. R. J. Chem. Phys., in press. (4) Herrick, D. R.; Stillinger, F. H. Phys. Reu. A: 1975, A l l , 42.
0022-3654/85/2089-3444$01 .SO10 0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 16, 1985 3445
Letters is equivalent to the one-dimensional 6 function model for the atom already solved by Ro~enthal,~ and performed simple variational calculations for several higher dimensions. Others6 have treated the 6 function model in the Hartree-Fock approximation and obtained an analytic formula for the energy. Mlodinow and Papanicolaou' showed that the D m limit is a classical one characterized by fmed interparticledistances, and obtained analytic results for this limit and the next two terms of the 1 / D perturbation expansion. In order to examine the dependence on D of the correlation energy between the D 1 and D limits, we require accurate approximations to the exact and Hartree-Fock energies at intermediate dimensionalities. These we obtained by generalizing the Pekeris* and the Hartree-Fock-Roothaan9 algorithms to arbitrary real values of D and carrying out numerical computatiomlo for a wide range of D and the nuclear charge Z . Over most of the range, the accuracy attained was better than 1 part in lo8, corresponding typically to better than 1 part in lo6 in the correlation energy. From these results, we find that between the D 1 and D m limits the correlation energy (in suitably scaled units) proves to be a nearly linear function of 1/D; the maximum deviations from linearity for Z 1 2 are only about 1%. Accordingly, linear interpolation in 1 / D between the exactly known limits gives total energies for D = 3 and Z 1 2 accurate to 0.005% or better, comparable to the best available configuration interaction calculations. +
+
+
TABLE I: Total and Correlation Energies' z = 2
z=3
Z = 6
-2.737 769 14 -2.71078644
-7.03211267 -1.003 619 66
-31.912 214 61 -31.882 359 31
-3.155 39251 -3.083 333 33
-7.657 952 05 -7.583 333 33
-33.16042621 -33.083 333 33
t3HF
-2.903 724 38 -2,861 68000
-7.27991341 -7.23641520
-32.406 246 60 -32.361 192 81
Ae3, by CI At3, by DI At3, exact
-0.04202 -0.042008 19 -0.042 044 38
-0.043 03 -0.043 828 25 -0.043 498 21
-0.043 45 -0.045 601 18 -0.045 053 73
tD em
€.HF €1
€,HF €3
- -
- - z=1
Dimensionality-Scaled Energies For a hydrogenic atom in D-dimensional space, the total energy (in hartree atomic units) is readily shown to be -1/222q2, where q = 2 / ( D - 1 ) . For a two-electron atom, the total energy except for electron repulsion is - 2 2 ~ thus, ~ ; the energy becomes singular as D 1 and vanishes as D m. For interpolation, it is convenient to adopt scaled units4 which make the energy finite in both limits: Pq2hartrees for energy and 271q-2 bohr radii for distance. In these dimension-scaled units, which we use throughout this paper, the Hamiltonian for the two-electron atom is
-
-
-0.08 0.0
where X = 1 / Z and the Laplacians and distances correspond to dimensionality D , which denotes the number of Cartesian coordinates specifying the position vector for each electron relative to the atomic nucleus fixed at the origin. Formally, the scale parameter q in the Hamiltonian appears where Planck's constant h occurs for ordinary, unscaled coordinates. The role of q is actually somewhat different, because it is also implicitly present in the sums which define the Laplacians and distances. Nonetheless, in terms of the dynamics generated, q 0 and q m (Le., D m and D 1) may still be described as classical and hyperquantum limits, respectively. The resulting probability distribution in configuration space is nonvanishing on a set of measure zero in the D m limit and classically allowed on a set of measure zero in the D 1 limit. The correlation energy, denoted by AtD, is given by the difference between the exact energy eD (excluding relativistic effects) and the corresponding Hartree-Fock energy:
-
-
-
+
0.4
0.8
1.0
Figure 1. Correlation energy AtD for the ground states of several twoelectron atoms, as a function of 6 = 1/D, the reciprocal of the space dimension. The correlation energies of real atoms (He, Li+, C4+)may be read off at 6 = 1/3 (indicated by arrow). For nuclear charge Z 2 2, the maximum deviations from linearity are of the order of 1%. (The because the system of proton plus curve for 2 = 1 terminates at 6 = two electrons is unbound in the Hartree-Fock approximation, and correlation energies are therefore not well-defined for D > 2.)
-
Closed expressions for these energies have been obtained7*"in the D m limit: e,
5 = -1 - -A2 32 c_HF
= -1
+ X2)3/2 + X4 + X(128 2048
(3)
+ 2-1P-X - X2/8
(4)
Corresponding results6J2 in the D cl=-l+-2 (5) Rosenthal, C. M. J . Chem. Phys. 1971,55, 2474. (6) Nogami, Y.; Vallieres, M.; van Dijk, W. Am. J . Phys. 1976, 44, 886; 1977,45, 1231. Foldy, L. L. Am. J . Phys. 1976,44, 1192; 1977,45, 1230. (7) Mlodinow, L. D.; Papanicolaou, N. Ann. Phys. (N.Y.) 1981, 232, 1 . See also: van der Menve, P. du. T. J. Chem. Phys. 1984, 81, 5976. (8) Pekeris, C. L. Phys. Rev. 1958, 212, 1649. (9) Roothaan, C. C. J. Reu. Mod. Phys. 1951, 23, 68. (IO) h e r , J. G. Ph.D. Thesis, Harvard University, 1984. Computations were made for X = 1-12 and Y = 1-6 in integer steps, with X = 12/0 and Y = 6 / Z for certain cases such as Z = 2 up to 30 more values of D were treated. A preliminary report of this work was presented at the Proctor and Gamble Award Symposium, 188th ACS National Meeting, Philadelphia, Aug 1984.
0.6
6 = 1/D
+
+
0.2
['8
'1
3?r
-
p+
1 limit are
[L-+~-... 6~
128
(5)
A closed formula for e l has not yet been found, despite heuristic evidence3that it should exist; however, the first 20 terms in the
power series expansion have k e n computed by Ro~enthal.~ Table ~
~
~~~~
(1 1) Goodson, D. Z.; Herschbach, D. R.. submitted for publication in J .
Chem. Phys. (12) White, R. J.; Stillinger, F. H. J. Chem. Phys. 1970, 52, 5800.
’
3446 The Journal of Physical Chemistry, Vol. 89, No. 16, 1985
Letters
0.000 1
I
I
1
I
1
h
0.0°5
I
1
Correlation energy
Y)
01
y“
-0.025
Dimensional Interpolation
L
-0.005
’ d
0.010
-
-0.075
‘
1
I
i
0
1
-0.015
3
Hartree-Fock -
i i
\energy
3
9
t:
aeries 1 (two terms), 1/D
-0.100
* I 1
..
U
-0.125 ~0.020
0.0
0.2
0.4 0 . 6
x -0.025
’-
-0.030
0.0
0.2
0.4
0.6
0.8
1.0
6 = 1/D Figure 2. Deviation from linearity with respect to 1/D for the ground
state of helium: exact energy (solid curve, from Pekeris calculations of ref lo), Hartree-Fock energy (dashed curve, from ref lo), and correlaand D tion energy (dot-dashedcurve). The limiting values at D 1, which define the straight lines from which these deviations are measured, are included in Table I. Drastic cancellation between the nonlinear components of the exact and Hartree-Fock energies results in the nearly linear dependence on 1/D of the correlation energy.
-- -
I lists for Z = 2,3,6 the energies given by these expressions and corresponding D = 3 results. For Z = 2, the correlation energy represents 0.99% of the total energy for D a, whereas it is 1.45% for D = 3 and 2.28% for D 1. In Figure 1, correlation energies obtained from our numerical Pekeris and Hartree-Fock calculations are plotted as a function of 1/D for several values of Z. The striking degree of linearity must be attributed to nearly complete cancellation of sizable nonlinear terms in the exact and Hartree-Fock energies. This is made clear in Figure 2, where the nonlinear components of Ac, e, and eDHFare plotted for the 2 = 2 case. Although the nonlinear parts of eD and eDHF are not large (less than about 1%of the total energy), they are about 2 orders of magnitude greater than the nonlinear portion of de, the correlation energy. The high degree of linearity exhibited by AcD as a function of 1/D suggests that, a t least for Z 2 2, a quite accurate approximation to the “real-world” value at D = 3 can be obtained by linear interpolation between the exactly known limits at D and at D 1; thus, we use
- -
=
0.8
1.0
1/z
Figure 3. Correlation energy AC,for two-electron atoms, as a function of X = 1/Z, the reciprocal of the nuclear charge: exact correlation energy (solid curve, from numerical calculations of ref 10) compared with approximations obtained from dimensional interpolation (dot-dashed curve) and from the 1/D expansion (dashed curve). Two terms are used in the 1/D expansion, since for typical Z this is the optimal truncation of the asymptotic series. The exact curve ends near X = 0.969, where the Hartree-Fock energy exceeds the first ionization threshold. The approximate curves end at A,, = 0.814, which marks the transition to an asymmetrical geometry for D and the limit of validity of eq 3. A square root branch point that results from this symmetry breaking accounts for the poor accuracy of the 1/D expansion.
-
Discussion A,in the case of DI calculations for the total energy,3 the high accuracy obtained from DI for the correlation energy results largely from taking into account the D 1 limit. This is seen from a comparison with the usual of the 1/D expansion, which deals only with the D m limit. The expansion takes the form cD = e, + cmf/D+ 1 / 2 e , ’ f / b + ..., where the primes denote differentiation with respect to 1/D. In general, this series is asymptotic but not convergent. The leading term e, is given in eq 3; it corresponds to the minimum of an effective potential comprised of centrifugal and Coulombic contribution^.',^ In this classical limit, the electronic configuration becomes rigid. There are two regimes, determined by a critical nuclear charge parameter
-
-
Xo = 25/2[(4/3)1/4- (3/4)’14] = 0.8144 ...
-
(8)
For X < A,, (or Z > Z , = 1.228...), the minimum energy configuration of the two-electron atom a t D corresponds to a symmetrical water-like structure. At Z = 2, for example, the two electron-nucleus vectors have equal lengths and an angle of 95.3O between them. For 1 > X > A,,, however, the symmetrical configuration becomes a saddle point and the minimum corresponds to an asymmetrical structure with one electron closer to the nucleus than the other. Although this asymmetrical regime is narrow (1 (7) < Z < 1.228), it proves to have repercussions elsewhere. The higher order terms in the 1/D series represent a semiwith the limiting values given by eq 3-6. Table I includes a classical expansion about the classical limit. The term linear in comparison of the correlation energies predicted by this dimen1/D can be obtained in closed It corresponds to harmonic sional interpolation (designated as DI) with those obtained from of the electrons about the classical minimum. Thus, ~ the best available configuration interaction (CI) c a l c u l a t i ~ n s . ~ ~ ~oscillations e,’ contains zero-point energies of the three vibrational normal Only for Z = 2 is the best CI value slightly more accurate. This modes. These are in turn proportional to the square roots of the utilizing piecewise polynomial came from a novel respective force constants. Because the force constant for the orbitals; the next best CI calculation on helium16 is an order of asymmetric stretching mode passes through zero at the transition magnitude less accurate than the DI result for the correlation to an asymmetrical minimum, emf suffers a square root branch energy. The total energy obtained by adding the DI estimate for point at X = A,,. For the Hartree-Fock energy, the 1/D expansion Ae3 to the Hartree-Fock energy is only 0.001 25% above the exact has the same character, but the branch point occurs in a more energys for Z = 2, and 0.004 53% and 0.001 69% below for Z = remote region (AoHF = 25/2/5 = 1.131 ...) and thus is relatively 3 and 6, respectively. insignificant. Terms involving higher powers of 1 / D correspond to anhar(13) Carroll, D. P.;Silverstone, H.J.; Metzger, R. M. J . Chem. Phys. monic vibrations of the electrons. However, in fact these terms 1979, 71, 4142. prove not relevant for the calculation of correlation energies. From (14) Brown, R. T.;Fontana, P.R.J. Chem. Phys. 1966, 45, 4248. our numerical calculations,10 we have determined the first six to (15) Weiss, A. W. Phys. Reu. 1961, 122, 1824. (16)Tycko,D.H.;Thomas,L.H.;King,K.M.Phys.Rev.1958,109,369. eight terms of the correlation energy expansions in 1/D for several
-
-
J . Phys. Chem. 1985,89. 3447-3449 two-electron atoms. These results reveal that the optimal asymptotic truncations consist of just the first two terms. Whereas the DI method interpolates linearly in 1/D between Ac.,, and A q , the truncated 1/D expansion extrapolates linearly from Ae, and A€-'. This extrapolation gives 1 A c ~= Ac, -A€-' 3
+
Hence, the 1/D extrapolation cannot agree with the dimensional interpolation unless Ael = Ac- AE-'. In Figure 3 we compare as functions of A = 1/Zthe accurate correlation energies obtained from our numerical calculations'0 with those predicted from DI and from the 1/D extrapolation. Both approximations terminate at A = &., The drastic effect of the square root branch point in em' is apparent. Except for small A, where the two approximations become comparable, this branch point renders the 1/D extrapolation much less accurate than the dimensional interpolation. Some insight into why the DI method works as well as it does has come from analysis of the dimensional singularities that occur in the D 1 limit." This analysis replaces the Taylor expansion in 1/D with a Laurent series; the contributions from the dominant
+
-
(17) Doren, D. J.; Herschbach, D. R. Chem. Phys. Lett., in press.
3447
-
singularities at D = 1 are then accounted for to all orders in 1/D. In effect, knowledge of the D 1 singularities provides quantum corrections which greatly improve the accuracy of the semiclassical 1/D expansion. The singularity analysis also yields a nearly linear dependence of the correlation energy on 1/D. From this perspective, the success found with the DI procedure owes much to the use of energy units scaled by (D - 1)-2 and to inclusion of the D 1 limit. The dimensional continuation approach seems likely to be feasible for larger atoms and molecules. Other systems already studied as a function of D are immediately accessible. Among these, spin lattices parameterized by the dimensionality of the spin space are especially inviting, because again both the D 1 limit (Ising models) and the D m limit (spherical models) can be solved exactly.'8
-
-
-
Acknowledgment. We thank Doug Doren and David Goodson for many enjoyable asymptotic discussions. We are grateful for support of this work by the Air Force Office of Scientific Research (under Contract No. F 49620-80-C-0017) and by the Exxon Corporate Research Science Laboratories. Registry No. He, 7440-59-7. (18) Baxter, R. J. "Exactly Solved Models in Statistical Mechanics"; Academic Press: London, 1982.
Temperature Jump Studies through Phase Boundaries in Lyotroplc Liquid Crystals P. Knight, E. Wyn-Jones,* and G. J. T. Tiddyf Department of Chemistry and Applied Chemistry, University of Salford, Salford M5 4WT, UK (Received: April 30, 1985)
-
-
-
The Joule heating temperature jump technique has been employed to monitor the kinetics of the hexagonal (H,) isotropic (LJ, isotropic (L,) lamellar (La),and bicontinuous cubic (VI) lamellar (L,) phase transitions in poly(oxyethy1ene) surfactantjwater systems. The formation of the new phase was monitored by using birefringence and turbidity measurements. Typically the phase transitions were complete in 0.5-3 s according to type. All showed S-shaped birefringence vs. time curves with an apparent initial "induction" period where no change was observed. The data are consistent with a nucleationfgrowth mechanism for the phase transition.
Introduction Lyotropic liquid crystals' or mesophases as they are sometimes called occur in many aqueous systems which contain amphiphilic molecules. The most common mesophase-forming systems consist of surfactants in aqueous solution at concentrations well above the critical micelle concentration. It has been shown2 that such systems can exist in a number of different phases, these being the isotropic micellar solution (designated L,) or one of several liquid crystalline structures, the lamellar, hexagonal, and bicontinuous cubic phases (designated L,, HI, and VI). The transition from a micellar solution to a mesophase or between the various mesophases can arise as a result of concentration and/or temperature changes. Lyotropic mesophases are stable over well-defined ranges of composition and temperature. These are usually represented on phase diagrams, which show phase boundaries as a function of composition and temperature. For a constant-composition mixture the system may undergo several changes in phase as a consequence of increasing temperature. The purpose of this investigation is to study the kinetics of the formation and breakdown of lyotropic liquid crystalline phases. The particular systems chosen for study are based on surfactants Unilever Research, Port Sunlight Laboratories, Wirral, Merseyside L62 4XN, UK.
of the poly(oxyethy1ene) type [n-C,H2n+l(OCH2CH2),0H, C,EO,]. The reasons for selecting these nonionic surfactants are threefold: firstly their phase behavior is well-known and covers many types of phase transition, secondly either the phase transitions occur in an experimentally accessible temperature range or they can be shifted to within this range by the addition of electrolyte, and thirdly such materials are readily available in the pure state. There has been no previous investigation to study the kinetic processes of mesophase/solution or mesophase/mesophase transitions. We have employed the Joule heating temperature jump technique3 to induce the various phase transitions (Figure 1).
Experimental Section A modified Hartley Instruments Research temperature jump (Model No. 122) was used to provide the heating pulse, giving temperature rises of up to 4 OC in a few microseconds. This was coupled with an optical detection system capable of monitoring changes in birefringence and turbidity. The data capture system ( 1 ) Tiddy, G. J. T.; Walsh, M. F. Srud. Phys. Theor. Chem. 1983,26, 151. (2) Winsor, P. A. Chem. Rev. 1968, 68, 1. (3) Eigen, M.; de Maeyer, L. "Techniques of Organic Chemistry", Part
2; Wiley-Interscience, New York, 1963; Vol. VIII.
0022-3654/85/2089-3447%01 SO10 0 1985 American Chemical Society
