J. Phys. Chem. 1983, 87, 1432-1437
1432
Polarizability Densities within Atoms. 1. Simple One-Electron Systems William H. Orttung’ and Dariush Vosooghl Department of Chemishy, University of California,Riverside, Caiifornia 9252 1 (Received: September 18, 198 1: I n Final Form: June 28, 1982)
The classical dielectric formalism is often used to model atomic interactions with electric fields. Although the formalism can be applied to atomic interiors in a straightforward manner, the underlying physical model of a continuum dielectric does not encompass nonlocal effects of electronic momenta. In addition, the concept of a local field needs to be carefully considered for points within an atom. To explore these questions, we have investigated three simple systems (particle in a box, harmonic oscillator, and hydrogen atom). Field-induced charge increments (derived by quantum or classical mechanics) were identified with the formal bound-charge density. Integration then led to a formal dielectric susceptibility (or polarizability density). The latter was found to be a function of the spatial variation of the applied field. This result suggests that the classical formalism must be extended to include the effects of electronic momenta before it can be used with confidence.
Introduction Simple models of a molecule and its environment have often been based on the classical dielectric formalism. Typically, an atom or ion is represented by a uniform dielectric sphere, and the surroundings beyond a sharply cut cavity radius are represented by a continuous dielectric. A uniform dielectric ellipsoid is often a convenient choice for a molecule. When this type of oversimplified model fails to predict observed properties, it is usually not clear whether the failure is due to the model or to an inadequacy of the classical dielectric formalism itself. Methods are now available for the evaluation of more realistic dielectric models,lS2and analysis of the formalism itself has become desirable. We begin with a review of the mathematical and physical aspects of the classical formalism in the context of intraatomic applications. Quantum-mechanical and classical mechanical results are derived for simple one-electron systems in uniform and nonuniform applied fields by using the framework of the continuum dielectric theory to deduce formal polarizability densities. The results are then examined and interpreted. Since the original work of Oxtoby and Gelbart3s4applying the dielectric formalism within atoms, a series of related papers has appeared. This work has recently been referenced by O ~ t o b y . ~ Classical Dielectric Formalism We begin with a brief review of essential results. Some of the interesting subtleties bearing on our application are discussed in the next section. The theory may be developed in equivalent differential or integral f ~ r m s , each ~J deduced from the Coulomb law of force between charge elements. In the differential formulation, the Coulomb law transforms into the Poisson partial differential equation Y.(t.Vf#)) = -4rp
where
f#)
(1)
is the electrostatic potential, p is the free-charge
density, and t is the dielectric constant tensor. All other electrostatic quantities may be calculated from 4 (the solution of eq l) and t. If E is the electric field at a point of space, and P is the polarization (dipole moment per unit volume, or induced dipole density), then E = -Vp, (24 P = (1/4r)(t - l ) * E= X-E (2b) where X is the classical dielectric susceptibility tensor. Since polarizability is defied as the ratio of induced dipole moment to applied field, it is clear from eq 2b that the susceptibility tensor X is a polarizability density tensor in the classical dielectric formalism. In macroscopic applications, X is related to the ease of separation of elastically bound positive and negative charges in a physically infinitesimal region. Since induced dipoles are cut by the surface of such a region, a net bound charge will be formed within the region if P is inhomogeneous. The magnitude of the enclosed charge is given by the surface integral of the polarization. Use of the divergence theorem on this integral gives 0.P = (3) for the bound-charge density. In the equivalent integral formulation, the polarization replaces 4 as the basic function. (We assume no “freecharge” density.) The field a t a point r is given by E(r) = Ek + I T ( r - r’).P(r’)du’ V
where Ekis the applied field of spatial type k (defined in the following derivations) and T(r) is the dipole propagator tensor given by T(r) = (3rr - r21)/r5 (5) The integral of P over a volume V is the dipole moment M of the contents of V. If P arises from a uniform applied field E,, then the polarizability tensor (Y of the contents of V is defined by
M = S P du = WE, (1)W. H. Orttung, Ann. N. Y. Acad. Sci., 303, 22 (1977). (2) W. H. Orttung, J . Am. Chem. SOC.,100, 4369 (1978). (3) D. W. Oxtoby and W. M. Gelbart, Mol. Phys., 29, 1569 (1975). (4) D. W. Oxtoby and W. M. Gelbart, Mol. Phys., 30, 535 (1975). (5) D. W. Oxtoby, J. Chem. Phys., 72, 5171 (1980). (6) W. F. Brown, Jr., Handb. Phys., 17, l(1956). (7) C.J.F. Bottcher, “The Theory of Electric Polarization”, 2nd ed., Vol. I, Elsevier, Amsterdam, 1973.
(4)
V
(6)
Dielectric Model for Atomic Interiors In a typical classical dielectric, there is no free-charge density, and the bound-charge density is zero in the absence of an applied field. Within an atom, on the other hand, each volume element contains only time-averaged
0022-3654/83/2087-1432$01.50/00 1983 American Chemical Society
Polarizability Densities within Atoms
negative charge (except at the nucleus). The average charge density at a point may be calculated by quantum mechanics. It results from a balance between dynamic effects and the electrostatic interactions of electrons and nuclei. Since the atomic charge distribution is stable (in the sense of the preceding sentence) in the absence of an applied field, this initial (zero-field) charge density can be ignored in the dielectric model of distributed polarizability. When a field is applied to an atom, the average charge density is altered at each point and a new balance is achieved. The atomic charge density increment due to an applied field is closely analogous to the bound-charge density induced in a dielectric by a field. Hence, we conclude that our polarizability model deals with “bound” rather than “free” charge in the dielectric context and drop the right-hand side of eq 1. (The same assumption has already been made in eq 4. Note that free-charge terms are needed if we are also interested in polarity, as in the case of molecules.) In the mathematical sense, an atomic charge density increment in a particular volume element can be described in terms of inhomogeneous charge shifts in the surroundings. The picture is the same as that for a classical dielectric, in which an inhomogeneous polarization yields a bound-charge density (eq 4). If we think in terms of charge shifts (rather than the mathematically equivalent charge increments), we can then imagine that each shifted element forms a dipole with an equal and opposite charge at its initial position. The electric field of these dipoles, summed over the atom, will be the same as the electric field of the charge increments. This field can then be used to calculate the change of electrostatic interactions when a field is applied, in close analogy to eq 4 and 5 (or 2a). In the preceding paragraphs, we have indicated that the response of an atom to a static applied field can be formally described by the model of an uncharged continuum dielectric. Physically, however, there are differences in the nonelectrostatic forces that determine the response to an applied field. In a dielectric, the nonelectrostatic forces are local (within a “physically infinitesimal” region), but, in an atom, we have nonlocal dynamic forces. The following sections of this paper explore the effects of this physical difference in simple cases where the formal dielectric properties are found to be quite unlike those of a classical dielectric. One other subtle point needs to be clarified. We have defined an electric field in eq 2a and 4. If the field point is within the dielectric, then we are obliged to imagine that a small cavity is excavated in the dielectric about the field point.6 It is well-known that the field within the cavity remains finite as the cavity size is reduced to zero, but the value of the field depends on the shape of the cavity. Equations 2 and 4 both implicitly assume a needle-shaped cavity parallel to the field direction. The uncompensated bound-charge density on the surface of a cavity of this shape contributes nothing to the field within it. It is only for this assumption that eq 2 is valid without an added factor. Similarly, a complicating term is conveniently unnecessary in eq 4. It can also be shown that this field is the average electric field within the dielectric (in the sense of a spatial average at one instant). If we are thinking of an atomic or molecular fluid, we usually want to know the “local field” at a particular atom or molecule. The Lorentz local field is appropriate if the atom is on a lattice point of a cubic array. Onsager found a better formula for the field at a point in a spherical cavity moving with an atom in a fluid. It is an interesting coin-
The Journal of Physical Chemistry, Vol. 87, No. 8, 7983 1433
cidence that, for a single atom in a vacuum between charged plates (uniform applied field), the Lorentz local field at a point within the atom is the same as the applied field at a point in the surrounding vacuum if the atom is represented as a uniform dielectric sphere. (For an inhomogeneous dielectric sphere ( t = t ( r ) ) ,the preceding statement is only approximately true.) Since an electron cannot interact with itself electrostatically at atomic-scale distances, the average charge density of a one-electron atom should not interact with itself, and the Lorentz internal field should not be a bad approximation to the correct local field (which, in this case, is simply the applied field itself). In the preceding paragraph, we argue that the correct local field in a one-electron system is simply the applied field. In the limit of many electrons, the self-interaction aspect becomes of minor importance,and we ask what local field is appropriate in this limit. A little reflection shows that it does not really matter what choice we make. If we do not use the Maxwell average field, E, of eq 2 and 4, then we will probably use a result of the form, F = f ( e ) E . Such a choice will introduce a complicating factor in eq 2b, and an additional term in eq 4. The values of X will also be altered, but nothing of observable significance (such as P = XnF) is affected if the calculations are carried through consistently. We therefore prefer the simplest choice, f (t) = 1,leading to the standard formulation given in eq 2 and 4. The assumption of a needle-shaped cavity at points within atoms involves only the shortest range interactions, which appear to be of little or no significance for the effect of applied fields on atoms or interacting atoms. One-Electron Systems in Applied Fields The effects of uniform and nonuniform applied fields on three one-electron quantum systems are considered. Polarizability densities and other properties are evaluated from first-order perturbed wave functions and secondorder perturbed energies. All expressions are in atomic units. The limits corresponding to classical mechanics are also obtained in most of the cases. If and Wn0 are solutions of the unperturbed Schroedinger equation, Hoc: = W2{2,where H o = -1/2V2 + VO, and if H’is the perturbation due to the applied field, then the first-order correction to the nth nondegenerate eigenfunction, c,,’, may be expanded in the unperturbed basis functions:
cno
m
(7) where C,,
= H’,,,,/(Wn0 - Wmo)
if m # n and C,, = 0 (8)
HL,, is the matrix element of H‘, and the second-order correction to the nth eigenvalue, Wno,is W / = mCf n IH&n12/(W,o- Wmo) Alternatively, an analytic expression for the first-order correction may be obtained by substituting cn’
=
f(r)cn0
(10)
in the first-order perturbation equation and solving for The second-order energy correction is then given by the expectation value
f(r).s99
(8) A. Dalgarno, Ado. Phys., 11, 281 (1962). (9) J. O., Hirschfelder, W. B. Brown, and S. T. Epstein, Ado. Quantum Chem., 1, 255 (1964).
Orttung and Vosooghi
The Journal of Physical Chemistry, Vol. 87, No. 8, 7983
1434
Wn" = (fno,Hffn')
(11)
Dalgarno's model8 of a one-electron system at the origin, perturbed by a point charge of magnitude -2, at x >> 0, is convenient for present purposes. For the one-dmensional systems (electron in a box, harmonic oscillator), the perturbing potential H' of the point charge at xp on the electron near the origin is m
+ 2, 1 ~ ~ / ~ p k +(12) '
H'= Z,/(X, - X) = Z,/X,
k=l
If we ignore the constant initial term of eq 12, we may write m
H'=
EkXk/k!
(13a)
k=l
For k = 1 (uniform applied field E,), nonzero values of eq 20 are H',,, = -E,/L(8/n2)nm/(n2- m2)2,if n i m is odd. We then evaluate eq 9 as m
Wl/ = E12L4(128/as)C ' n 2 m 2 / ( n -2 m2I5 ( 2 1 ) m=l
where single (or double) primed summations include only terms with odd (or even) n f m and m # n. The total polarizability, an,is obtained from eq 15 and 21. It may also be calculated from the total induced dipole moment, pn (=anEl),given by the expectation value of -x for the first-order corrected wave function of' the nth state. The first-order correction is obtained from eq 7 and 8 as m
fl,,' = -E1L3(2/L)'i2(16/n4)C ' [ n m / ( n 2-
where Ek = Z,k!/X,k+'
m=l
(13b)
E , is a uniform applied field, E2 is a uniform applied field gradient, and so on. If the first-order corrections to j-2 are denoted by f k i , then
m2)3]sin m d y
+ y2) ( 2 2 )
For n = 1, flo = (2/L)'I2cos x y , and eq 22 reduces to fll'
= m
- m2)3]sin mxy E1L3(2/L)1/2(16/7r4)' [m(-1)m/2/(1
m
m=2
(23)
and each term of the perturbation may be considered separately to first order in Ek. We will also make use of the following relations: Wn" = -Y 2 nE 12 -V'Pkn =
b k n
-(IfknI2
(15) -
1fn0l2)
(16)
where El is a uniform applied field and a, is the total polarizability of the electron in the nth state. It is of interest to compare the quantum results with the predictions of classical mechanics. For the one-dimensional examples, the classical probability, S(x) dx,that the electron is between x and x + dx, may be taken as the fraction of time that the particle is in dx.'O If the period of the motion is llf, where f is the frequency, then S(x) dx = f dt = 2f dx/u(x), where u ( x ) is the speed of the particle, and we obtain S(X) = 2 f / u ( x )
(174
l / f = $dx/u(x)
(17b)
For a conservative potential
w = T ( x )+ V ( x )=
Y2U(X)2
+ V(x)
= E1L3(2/L)'i2(1/2x2)[y cos ny
(11'
+ (y2 - y4)x sin x y ] ( 2 4 )
Equation 23 is the Fourier expansion of eq 24. Special methods are required to obtain the analogue of eq 24 for excited states." If eq 24 for n = 1, k = 1 is used in eq 11, and the result compared with a related result" for k = 1, L = n, and arbitrary n , we obtain the useful analytic result
Wl/ = -E12(L/?r)4(15/n2 - x2)/(24n2)
(25)
which is much easier to use than eq 21. The right-hand side of eq 16 may be evaluated by using eq 24 to obtain dPll/dx. Integration of eq 16 gives P,,(x). (The integration constant was determined by setting Pll = 0 at y = k1/2.)We then obtain a formal expression for Xll as Pll/E1according to eq 2b:
Xl,(X) = (L3/x4)[cos2 sry
+ xy sin 2ny + x 2 b 2- y4) sin2 sry] (26)
(18)
Since U(0)= 0 in our one-dimensional examples, we have W = 1/2u(0)2, and u ( x ) = u(0)[1- 2U(x)/u(0)2]'~2
The result of the Dalgarno method Jf eq 10 is
Xlnmay be obtained for arbitrary n by the usual expansion method if eq 23 is used instead of eq 24:
(19) ( - ~ ) ( ~ + ~ p -cos ~ ) /( ~n
Derived Results Electron in a Box. When 2, = 0 in eq 13, the potential is zero within the interval -L/2 5 x I L / 2 and infinite
n
+ mp
+ mp)ry
(27)
elsewhere. The solutions of the unperturbed Schroedinger equation are f2 = ( 2 / L ) l J Z sin n d y + and Wno= ( n n / L Y / 2 ,where n = 1, 2, ... and y = x / L . The matrix element in eq 8 is then
Equation 27 is numerically identical with eq 26 when n = 1. It can also be shown that
Hkmn =
Since alnis negative for n > 1 (from eq 25), we expect Xln also to be predominantly negative for n > 1. Examination of eq 21 suggests that this "undielectric" behavior is related to the fact that the energy level spacing increases with n
x:;l
(2EkLk/k!)
y k sin mn(y
+ y2) sin n a b + 7') dy (20)
(10)L. Cohen, J. Chem. Phys., 70, 788 (1979), and references cited therein, provide a more complete discussion of the classical formulation.
(11) W. B. Brown and J. 0. Hirschfelder, Proc. N a t l . Acad. Sci. L'. S.A., 50, 399 (1963).
The Journal of Physical Chemistry, Vol. 87, No. 8, 1983 1435
Polarizablllty Densities within Atoms
for the particle in a box. The positive terms with m n have small denominators in eq 21 and are therefore larger than the negative terms with m > n.12 For k = 2 (uniform field gradient E2),the expansion method gives
c2;
m
= E2L4(2/L)1/2(8/7r4)C ” [nm/(n2 m=l
m2)3]sin m d y
+ y2) (29)
The polarizability is wo2, independent of n, in exact agreement with the classical result for a static field. The first-order correction to the wave function is obtained from eq 7 and 8 as
tin' = Eia3Nn[nHn-i(y)- s/2Hn+l(y)Iexp(-y2/2)
(38)
Equation 16 may then be integrated for Pl,(x) and divided by El to obtain the formal expression for Xl,(x), as in the electron in a box case (except that the integration constant was determined by setting P1, = 0 at y = km): X l n ( x ) = a41tn0(~)I2
(39)
For k = 2 (uniform field gradient), we obtain l2,’
Equation 30 clearly differs from eq 27. Now consider the problem from the point of view of classical mechanics. In the absence of an applied field, u(x) = uo = 2fL, f = u0/2L, and So(x)= 1/L. If we equate the unperturbed classical and quantum energies uo2 = (nn/L)2
(31)
to obtain a relation between uo and n, we then find -ASl = -Ely/vo2 = -E1,(L/n~)2
(32a,b)
correct to first order in El. Here -ASl is analogous to the first-order quantum Apln. Following the same procedure as that leading to eq 26, we obtain the classical result
x , ( ~= )( ~ 3 / 2 n w ) ( y 2- y4)
(33)
if we take X1= 0 at y = h1/2. For k = 2 (uniform field gradient), we follow a procedure like that for k = 1 to obtain (to first order in E,)
-AS2 = -E2L3(y2- 1/12)/(2n27r2)
(40) XZn(x) = (a4/4)Ilno(x)12
classical polarizability densities of eq 33 and 35 are easily understood in terms of the slowdown of the electron as it reaches regions of higher potential energy in each cycle of motion. Harmonic Oscillator. We proceed in the same manner as with the electron in a box, except that the infiiite square well is replaced by a quadratic potential, k ’x2/2, corresponding to a resonant frequency fo = w0/2, where wo = k”/, in atomic units. The solutions of the unperturbed = N , exp(-y2/2)H,(y) and Schroedinger equation are lno W,O = (n + 1/2)w0, where n = 0, 1,2, ...,y = x / a , a = w{l/z = k’-1/4 in atomic units, H,(y) is a Hermite polynomial, and N , = (2“n!~’/~a)-’/~. Using Dalgarno’s model again, with xp >> a, the matrix element of eq 8 is
H a m , = (Ekak/k!)N,N,S__H,(y)ykH,(y)exp(-y2) dx (36) Results for specific k values are obtained by using the orthonormality of the tn0 and the recursion relation, yHn = nHn-l + l/zHn+l, where H, = 0 if n < 0. For k = 1 (uniform applied field), nonzero values of eq 36 are H’,,, = Ela(n/2)1/2or Ela[(n + 1)/2]1/2 if m = n - 1 or n + 1, respectively. We then obtain
(41)
From eq 39 and 41, we see that X 2 , = X1,/4 for all values of n. When the harmonic oscillator is treated by classical mechanics, ASl(x) and AS2(x) are obtained to first order in the applied field as in the electron in a box case. It is then found that X l ( x ) and X 2 ( x )are given by the limiting forms of eq 39 and 41 for large n, in which lf2I2is replaced by Sob). Hydrogen-likeAtom. The problem is three-dimensional and VO is -Z/r in atomic units. In spherical polar coordinates (r,O,r$)with charge -Zp at zp or (rp,O,O),eq 12 and 13 become m
H’ = Zp/lrp - rl = Zp/rp + 2, C [rk/r,k+l]Pk(cos6) k=l
(42)
(34)
X ~ ( X=) [L3/(6n2r2)](y2 - 74) (35) Thus, classically, X 2 is 1/3 of X1at each x . The negative
Win" = (-wo-2/2)E12
= &a4N,[n(n - l)Hn-Z(y) - f/4Hn+2(y)]exp(-y2/2)
m
H ’ = C (Ek+/k!)Pk(cos 6)
(43)
Ek = ZPk!/rpk+l
(43)
k=l
where
Ek is still defined as in eq 13, but for k > 1 the more complex off-axis spatial dependence is now involved. We will consider only the ground state, for which Pnlm= Ploo = (Z3/rr)1/2e-zrand W2 = Wlo = -2/2. Dalgarno’s results for this case,8 in our slightly modified notation, are rk,1m = -(Ek/Zk!)[l.k+’/(k + 1) + +/kz]pk(cos 6)’Ploo (44)
Equation 45 gives the well-known result, 9/2, for the dipole polarizability when k = 1 (uniform applied field) and Z = 1 (H atom). For lkk,lm,the right-hand side of eq 16 is
and the corresponding left-hand side of eq 16 is V.Pk =
(37)
(12) A. Dalgarno in “Quantum Theory. I. Elements”;D. R. Bates, Ed., Academic Press, New York, 1961, Chapter 5.
Since our problem has axial symmetry, dPk,/a4 = 0. The applied electric field at any point within the one-electron atom is given by eq 2a and 43 as
1436
The Journal of Physical Chemistty, Vol. 87, No. 8, 1983
E = OH’ = e$,
1 + e&, = e,dH’/dr + e,-aH’/a6 r
Orttung and Vosooghi
l*Rl
(48)
I
where e, and eoare unit vectors along the r and 6 directions. We obtain
Ekr = (kEk/k!)+lPk(COS 6) EkO
= -(Ek/k!)+’
Sin 8 Pk’(C0s 6)
(49)
(50)
F-7
From eq 2b, P k and Ek are formally related by the polarizability density Xk. Because of the atomic symmetry, Xk depends on r but not on 6 or 4, and it may be anisotropic in the radical vs. angular directions. Thus, Xk will be diagonal in a local coordinate system aligned with the spherical polar coordinates, and, in general, Xkr# xke = Xk@ Substitution of eq 2b, 49, and 50 into eq 47 with Apk from eq 46 then gives the following for eq 16:
t
(~1,100=
2rJTJm(X1, cos2 6 + XISsin2 6)r2sin 6 dr d6 (52)
where qloO = (9/2)Z4 from eq 45. If we make the simplest assumption that X is isotropic in the ground state of the H-like atom, then Xl,,,&) = Xl,(r) = Xle(r), and eq 51 gives the solution 1 = -(3 + 6Zr + (53) 47rz
1
1t
i
IO^,^,
M
F-i
4
Equation 51 involves two unknown functions, Xk,(r)and Xk@(r),and cannot be solved in general unless another differential or local relation between the functions is available. Consider the case of a uniform applied field (12 = 1). Then the solutions must satisfy the integral relation
t
.--
1 G
0
- l/2
G
1/2
Y
Flgure 1. Electron In a box. The first row shows the electron denstis for n = 1, 2,3,and the classlcal limit. The second row is the change of charge density in a uniform applied field, k = 1 (-), and in a uniform applied field gradient, k = 2 (- - -); the n values on the classical plot indicate energies equal to the corresponding quantum states. The third row shows the formal polarizability densities. All three rows are for L = 1, and the second is for ,Ek = 1.
4”mpJ 2
G
Equation 53 satisfies eq 52, which suggests that our assumption of an isotropic X is at least a useful working hypothesis. For k 2 1, assuming isotropic X, eq 51 gives
(54) which differs for each value of k , i.e., for each type of spatial variation of the field.
Numerical Results The results of the preceding derivations are displayed in Figures 1-3. In all cases, classical and quantal, X2, differs from X l w The ratio, X,,/X,, depends on position for the electron in a box (Figure 1)and for the hydrogen atom (Figure 3). However, in the classical limit for the electron in a box (Figure l),the ratio becomes 1/3. For the harmonic oscillator (Figure 2), the ratio is independent of position, for all quantum and classical states. Negative values of X,, and X2, occur for excited states of the electron in a box, but not for the harmonic oscillator or for the hydrogen atom ground state (Figures 2 and 3). The subtle differences between Figures 1 and 2 are interesting consequences of the different form of the potential curves in the two cases. The steep wall at y = in Figure 1 reduces the average electron response to the field in this region, and the effect seems to propagate to
IGX XLn
-2
-I
0
I Y
2
G
G
-2 - I
G
I
2
Y
Figure 2. Harmonic oscillator. The quantities shown are analogous to those in Figure 1, except that the ground state is n = 0 rather than n = 1. I t is also assumed that a = 1 and Ek = 1. The axes have been scaled to make qualitative comparison with Flgure 1 convenient.
the central region. For the harmonic oscillator, Xkn is proportional to (at least for lZ = 1 , 2 ) , suggesting an equal charge shift at each position. However, this property cannot be a completely local effect since X,,# X Z n ;i.e.,
The Journal of Physical Chemistry, Vol. 87, No. 8, 1983 1437
Polarizability Densities within Atoms 0.4 I
I
I
0.2 I
I
1
I
field and for each value of the energy. In the context of a one-dimensional system, the response of a classical dielectric to a small field is described by a relation of the form
I
I
I
P(x) = X(x) E ( x ) (55) The results for one-electron systems are in the form m
Pn(x) =
..
'%
0
I
0
I
I
I
2 R (A.U. 1
3
4
Figure 3. Hydrogen atom ground state. Electron density (top); charge density change (middle) for uniform applied field, k = 1 (-), and field gradient, k = 2 (- - -); and formal polarizabiltty density (bottom).
X depends on the spatial distribution of field intensity. For the hydrogen atom, the weak potential in the outer regions allows larger shifts of the electron density. Our result for X,(r) in Figure 3 and eq 53 is in agreement with the corresponding result of 0xt0by.l~ Gibbs14has considered the case of strong applied fields for the total polarizability of the particle in a box. Discussion The preceding analysis clearly shows that, for a oneelectron system, a different polarizability density function is required for each type of spatial variation of the applied (13) D. W.Oxtoby, J. Chem. Phys., 69,1184 (1978). (14)J. H.Gibbs, Phys. Rev., 94,292 (1954).
k=l
Xkn(X)[Ekxk-'/(k- I)!]
(56)
where the energy level is indicated by subscript n and the square bracket on the right is the field of spatial type k. Equation 56 may be viewed as a formal generalization of eq 55, in which X(x) is replaced by a set of functions Xkn(X), d of which become equal to a single function X(x) in the limit of a system behaving like a classical dielectric. We conclude that the main difficulty of the classical dielectric formalism within atoms is its failure to allow for the nonlocal dynamic effects of the electrons. It remains to be seen whether a simple and practical remedy can be devised. A final comment regarding the possible anisotropy of the hydrogen atom polarizability density is also in order. If the polarizability density (at a point of the atom) is isotropic, then the magnitude of the polarization (at that point) will be independent of the direction of the applied field, whether in the radial or angular direction. A spherical shell at a given radius should then be shifted without distortion by a uniform applied field. However, it has been suggested' that, if this were true, the shifted shells would exert no restraining force on the nucleus (unless the nucleus passed through the shell). A clue to the resolution of this dilemma is seen in eq 46, in which the bound-charge density induced in a shell by a uniform applied field is seen to have a cos 0 dependence. Such a result is inconsistent with the assumption of a uniform polarization. The fallacy of the earlier argument was in the assumption of a uniformly shifted shell of charge. It can be shown that, when an isotropic polarizability density is inhomogeneous (spatially varying), an initially spherical shell of charge becomes nonspherical as it is polarized by a uniform applied field. Thus, since inhomogeneity is sufficient to allow shells at all radii to participate in restraint of the nucleus, we conclude that the present considerations (and eq 52) do not require more than the simplest assumption of an isotropic polarizability density. Registry No. H, 12385-13-6.
