J. Phys. Chem. 1993,97, 2461-2418
2461
Scaling and Interpolation for Dimensionally Generalized Electronic Structure M. Mpez-Cabrerrt Department of Physics, University of Michigan, Ann Arbor, Michigan 48109- 1055
A. L. Tan* and J. C. Loeser' Department of Chemistry, Oregon State University, Corvallis. Oregon 97331 -4003 Received: September 18, 1992
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Simple electronic structure problems generalized with respect to the spatial dimensionality D have two singular limits, D 1 and D =. Suitable scalings render the limiting solutions finite and reveal their complementary characters: D- 1 gives point (delta function) interactions between particles, while D- = yields point probability distributions between those particles. A single scaling that treats both limits simultaneously allows one to construct approximate D = 3 solutions by interpolation between the much simpler limits. In this paper we show how to construct and utilize such a uniform scaling. We construct the scaling, solve the limits, and interpolate to D = 3 for six model problems: Yukawa potential, hydrogen atom in a spherical cavity, Hz+, Hartrce-Focb H2,and HartreeFock two-electron atoms in weak magnetic and electric fields. The interpolated D = 3 energies and properties are typically accurate to within a few percent, except for cases where the D = limit gives multiple or unstable solutions. Extensions and improvements are also discussed.
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1. Introduction
It is straightforward to generalize an electronic structure problem with respect to the spatial dimensionality D. One simply needs to interpret the Laplacians Vi2 and distances rij in the Hamiltonian as D-dimensional quantities:
D -
- xju)2
rij2 = u=
I
The reason for considering such a generalization is that there will invariably be one or more values of D at which analytic simplifications occur, and problems become significantly easier to solve. For the problems treated in this paper, simplification occurs for D = 1 and =, These simplifications can be used in several ways. At the qualitative level, the solution obtained at some nonphysical value of D can be tken as a heuristic model. D = 1 solutions have frequently been used in this For reasons which will be reviewed below, in electronic structure problems D = 1 is actually a singular limit (better denoted D 1) which can be treated by replacing Coulomb interactions with delta function potentials. Solutionsto numerous delta function models of electronic structure have been ~ b t a i n e d though ,~ in most cases without recognizing that these constituted the D 1 limits of dimensionally generalized Coulomb problems. Dimensional continuation can be used in a more quantitative way by taking the solution for some nonphysical D as the starting point for a perturbation expansion. Here it is D 0 which has been most usefuL4 This is a second singular limit and can be treated by replacing all Laplacians with nondifferential (generalized centrifugal) operators. The natural perturbation expansions about thislimit (inpowersof 1ID) aregenerally divergent
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' Present address: Department of Biological Chemistry and Molecular Pharmacology, Harvard Medical School. 240 Longwood Avc.. Boston, MA 02115. Prcsent address: Department of Chemistry, University of Wyoming, Laramie, WY 8207 1 . f
0022-3654/93/2091-2461S04.00/0
but can be resummed by transform techniques. 1/D expansions have been obtained for several simple electronic structure pr0blems.5-~ There is also the possibility of using both of the dimensional limits simultaneously. This is the dimensional interpolation strategy introduced by Herschbach."J In its simplest form, the procedure utilizes as input only the D-. 1 and D = eigenvalues and approximates those at intermediate D as a linear function of 1/D. Thus, the D = 3 eigenvalue is approximated as E(3) = + '/,E("). This procedure has been applied to only a few electronic structure problems so far.l"I2 One reason why dimensional interpolation may not have been applied more widely is that it has as a prerequisite a suitable dimensional scaling, and it has not in general been clear how to construct these. Scaling is required to tame the singular limits, which in the unscaled problem are characterized by poles and/or zeros in the quantities of interest. Unlike the dimensional generalization, the dimensional scaling is not always obvious. The one that has been used in most previous applicationsis limited to a quite narrow class of problems and will not work outside this class. The primary purpose of this paper is to obtain the natural scaling appropriate for 1/D interpolation in a broader class of problems. This class is still fairly restricted (roughly speaking to problems dominated by Coulomb ground state behavior) but broad enough to treat several interesting problems. The scaling also takes a form which invites further generalization. We test out the scalingon six problems: (1) Yukawa potentials (exponentially screened Coulomb potentials); (2) hydrogen atoms confined to spherical cavities; (3) the hydrogen molecule ion; (4) the Hartree-Fock hydrogen molecule; (5) H F two-electron atoms in weak magnetic fields; ( 6 ) H F two-electron atoms in weak electric fields. For each problem we apply dimensional interpolation to obtain approximate D = 3 solutions from explicit dimensional limit solutions. In all cases we consider only ground states, and for the last two problems we focus on the atomic properties that describe the leading-order effects of the fields, namely the magnetic susceptibility and the electric polarizability. We begin in section 2 by using the hydrogen atom to show how electronic structure problems which have been generalized with respect to the spatial dimensionality D can be scaled so as to render the physics finite for all D. A simple prescription for writing down the Pscaled Hamiltonian for simple electronic
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Q 1993 American Chemical Society
Lbpez-Cabrera et al.
2460 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
structure problems is presented in section 3 and applied to the six model problems. Sections 4 and 5 are devoted to deriving exact solutions for these six problems in the D 1 and D limits. The limit results are used in section 6to obtainapproximate D = 3 solutions by interpolation. Finally, section 7 describes ways in which the methods described in this paper can be extended and improved.
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2. DScaling for Hydrogen
We consider first the hydrogen atom, which will serve as a guide. Using the dimensional generalizationof eq 1, the D-space Schrbdinger equation in atomic units is
Since the D-term sum appears upstairs in the Laplacian and downstairs in the Coulomb term, one can anticipate that any bound state solution will tend to become more compact as D is decreased and more diffuse as D is increased. Indeed, one finds for the ground state thatlo
J.o(r)= exp(-2r/(D - 1))
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Eo = -2/(D - 1)2 Thus, the atom implodes when D a.
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(3) 1 and falls apart when D
The dimensional limits can be rendered more intelligible and useful by means of suitable distance and energy scalings. Perhaps the most straightforwardscalingsare by (r)Oand IEol, the average distance and absolute energy in the ground state,I0
as one for the probability amplitude q = Jrl/V,where Jr 0: is the D-dimensional radial volume element or Jacobian factor:
(8) From this equation one can see explicitly how it is that the D-dependent coefficients manage to tame what would otherwise be divergent behavior. First consider D ~ 3 . Here the (3/D)2 coefficient of the Laplacian counteracts the quadratic divergence of the centrifugal potential. It also kills the second derivative, leaving for the kinetic energy only a finite centrifugal potential in the D limit:
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The ground-state solution is given by the minimum of the nondifferential Hamiltonian, which is at ro = and EO= -!/Z; formally, we can also write I*(r)12 = 6(r - )I2).Note that ( r ) o and lE01 agree with the D = 3 values, as they should. Now consider D 1. Here it is the (D - 1) factor in the Coulomb coefficient which is key. This term tames the Coulomb potential at just the rate needed to keep the atom from collapsing, namely at the rate of disappearance of centrifugal effects. As mentioned in the introduction, the D 1 limit of the scaled Coulomb potential is just a delta function. This can be seen from the fact that for D 1, the (D - 1) factor kills the potential for r # 0, but does not eliminate the singularity at r = 0:
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1imJ-1
D- 1
dDr = 1imJ-
2r
-1
D - 1 2rDl2
-P ' d r
=1
2r I'(D/2)
(10)
Thus It is convenient if the scalings have no effect at D = 3. Since the D = 3 values are 3/2 (bohrs) and I / 2 (hartrees), we will define the reduced variables by
D- 1 lim - 6(r) -1 2r Taking into account the remaining factors in eq 8, the D limit of the Schrbdinger equation is
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d2 - 36(r)] q ( r ) = E&) [--29 d?
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1
(12)
With these definitions, the expectation values of the reduced quantities will be numerically equal to the D = 3 values for any D. We now recast the Schrbdinger equation in terms of these reduced quantities. With the understanding that the variables in this and all succeeding equations are reduced, we do not label them explicitly as such:
The ground state (and only bound state) solution is given by *(r) = exp({d/3) and Eo = Again it may be checked that ( r ) O and lEol agree with the D = 3 values. The treatment of the dimensional limits in the last two paragraphs suggests an alternative and simpler route to the D-scaled Hamiltonian. Instead of scalingdistancesand energies, one can consider instead that it is the terms of the Hamiltonian that need to be scaled. For the hydrogen atom, for example, one could write
This is the D-scaled Schrbdinger for the hydrogen atom. Note that 1 / D (up to a constant) is a natural perturbation parameter in this equation. By its construction, we know that the ground-state solutions to eq 6 will be finite for 1 ID Ia. However, in preparation for the generalization to more complex problems, it is useful to see explicitly how the D-dependent coefficients in this equation do in fact manage to finitize the limits. We start by writing out the D-dimensional Laplacian in spherical coordinates. For S-states, the radical equation islo
and demand that the Laplacian and Coulomb coefficients tame any divergences associated with the dimensional limits. (Of course, we also require that the coefficients reduce to unity at D = 3.) From the analysis of the limits just given, we know that A Lmust counter the divergence of the centrifugal potential as D m and that Ac must balance thedisappearanceof thecentrifugal potential as D 1. Since there are no other singular effects to keep in check, the conditions on the coefficients are
The first derivative can be eliminated by rewriting the equation
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is finite for D- 1 -D-1 forD+l ~ , ( = l f o r D = 3 (14) ~ ~ ( - 1f o r D - 3 -l/D2 for D - c a, is finite for D These conditions can be met in many ways. However, if we limit
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Dimensionally Generalized Electronic Structure
The Journal of Physical Chemistry, Vol. 97, NG.10, 1993 2469
TABLE I: Dscrkd H.miltoaiurs for tbe Six Model Problems in Atomic Units' Yukawa potential
hydrogen in a spherical cavity
Hz+
HartrebFock H2
susceptibilities of Hartrec-Fock two-electron atoms polarizabilities of HartreGFock two-electron atoms
For the threc Hartrec-Fock problems, V(r) = Jlr
- r't1[p(r')]2
dr'.
ourselves to the same factors (namely D and D - 1 ) that occurred in the distance and energy scalings above, then there is a unique solution:
Inserting these into eq 13 gives eq 6, the same as before. The conditions on A L and Ac could have been met in other ways, for example by using factors of 4 / ( D + 1) in place of 3/D. Thus, the above choice is somewhat arbitrary. However, it happens to be the choice which coincides with the scaling based on (r)Oand lEol,as well as the one for which 1 / D emerges as the natural perturbation parameter, For these reasons, we will stick with it. (Because there is a connection between scaling factors and the natural interpolation parameter, the arbitrariness associated with alternative scalingsturns out not to be that significant. We will return to this in section 6 when we treat interpolation.) We have now described two routes to the same D-scaled Hamiltonian, namely scaling of variables and scaling of terms in the Hamiltonian. We recommend the term-scaling approach over thevariablescalingapproachfor two reasons. First, scalings are potentially confusing, and the term-scaling approach keeps their motivation and form as simple as possible. Second, termscaling is especiallyappropriate for perturbed electronicstructure problems, since the term-scalings for the perturbations generally turn out to be unity (whereas the corresponding variable-scalings do not), as we are about to show. 3. Dsc8linginceDerrl We now apply the term-scaling procedure to more general electronic structure problems. Thus, we write down the Schr& dinger equation with D-dependent coefficients in front of the various terms of the Hamiltonian (which areof courseunderstood to be D-dimensional operators) and then fix the coefficients by demanding that the physics remain finite for all D. If the Hamiltonian consists only of Laplacians and Coulombpotentials, then the conditions on the coefficients will be identical to those in eq 14, so their values will again be those in eq 15. I t is not hard to see that the conditions and coefficient for the Coulomb potential also apply to other potentials that have 1/r singularities
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at the origin (such as the Yukawa and Hulthtn potentials'), because the D 1 behavior is entirely determined by the singularity. For convenience, we will refer to all such potentials as 'Coulombic". For terms in the Hamiltonian other than Laplacians and Coulombic potentials, one can go through the same kind of analysis to find suitable coefficients. However, in most cases no singular behavior is encountered in either dimensional limit, so the conditions are just
is finite for D
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1 =1 forD=3 (16) is finite for D a, If we again stick to factors of D and D - 1 in constructing the coefficient, then we obtain A, = 1. In summary, the procedure for constructing a D-scaled Hamiltonian is to take the D = 3 Hamiltonian, insert the coefficients A Land AC from eq 15 in front of the Laplacians and Coulombicpotentials, and leave everything else alone. With the possible exception of one problem, these rules allow us to write down D-scaled Hamiltonians for the six model problems. The Hamiltonians for these problems are given in Table I. The one exception is the problem of magnetic susceptibilities. We consider this in some detail, since it affords an instructive example of a non-Coulombic potential for which dimensional limit considerationscome into play. In D = 3, the diamagnetic perturbation to an atom due a uniform magnetic field B is ordinarily written'3
(
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where c = 137 is the speed of light in atomic units (the inverse of the finestructureconstant e2/hc),andrland8,are thedistance from the nucleus and angle relative to the field of electron 1. For the D-scaled Hamiltonian, we need to interpret this as a Ddimensional quantity and insert an appropriate coefficient (possibly unity) so as to keep its effect finite for 1 ID Ia. We now consider two distinct Pspace interpretations of eq 17 but show that both lead to the Hamiltonian given in Table I. In
L6pez-Cabrera et al.
2470 The Journal of Physical Chemistry. Vol. 97, No. 10, 1993 preparation, note that in D-space an isotropic averaging gives (cos2 ei) = 1/D
(18)
(sin2 ei) = (D - 1 ) / D
(19)
This can be seen most easily by recognizing that cos Oi and sin Bi project into the subspaces parallel to and perpendicular to a specifieddirection and that the dimensions of these subspaces are respectively 1 and D - 1. Alternatively, one can compute the expectation values explicitly, using the fact that the D-space angular volume element integrated over all but the polar angle is Je = sinb2 0. Now consider the two interpretations. First, if one does not worry about where eq 17 comes from, then it seems natural to use eq 19 and write (sin2 ei) = (D - l)/D. This interpretation requires a coefficient AI satisfying
(
generalization. We choosehere to treat the internuclear potential in the same way as the other Coulomb potentials for simplicity and because some problems (in particular extended systemsl5) are much easier to treat if all Coulomb interactions are handled in the same way. By construction, the six D-scaled Hamiltonians in Table I will yield finite solutions for 1 ID I 01. In particular, they will yield finite solutions in the easily solved limits, D 1 and D -. The next two sections are devoted to obtaining the dimensional limit solutions for the six problems.
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It is straightforward to write down the D 1 limits of the D-scaled Hamiltonians. Using eq 1 1, one finds that the limiting forms of the D-scaled Laplacians and Coulomb potentials are
l ; / ( D - 1) for D-. 1 for D = 3 (20) is finite for D If we again construct the coefficient using factors of D and D 1, we obtain
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=
Q)
($)(&)
On the other hand, one might interpret eq 17 differently. Consideration of its derivation shows that ( sin2e,) is actually just a shorthand for (cos2 ffi cos2 f',), where 6') and ff', are angles with respect to wo directions defined by the uniform magnetic field.I4 (Recall that the magnetic field is really an antisymmetric second-rank tensor with components 3,"= aA,/ax, - aA,/ax,; it is only in D = 3 that one can use a single-subscript, axial-vector shorthand.) Evaluating the angular average in this interpretation using eq 18 gives (cos2 V, cos2 O",) = 2/D, so a coefficient A2 satisfying
+
+
is finite for D - 1 =1 forD-3 (22) A2{ -D forD-.is required. Again using only factors of D and D - 1, this yields
A, = ( W 3 )
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4. D - 1 SOluti01~9
D-1 lim(
D-1 1 -)( T), = 36(ri - r,)
3 D
(24)
Note that any other potential with a l / r singularity will also generate a delta function; specifically,f(r)/r will yield 3f(0)6(r). In eq 24, the potential terms are really simplified in two ways, namely by their reduction to delta functions and by the fact that for D 1 we can write rij = ri - rj. In other words, there are simplifications due to the fact that interelectron effects enter only as cusp conditions and due to the restriction to one degree of freedom. Note also that the coordinates ri are standard D = 1 variables with domain (-m, m). We point this out because of the potential confusion arising from the fact that for D > 1 the coordinates ri = lrd are nonnegative. For D = 1 we are really using a shorthand, combining the usual radial coordinate with the D 1 limit of angular space, which is the discrete set {-1, +l]. Now consider the six problems. For several of them, the D 1 limitsarereadily writtendownandsolved. EasiestistheYukawa problem, since for D 1 the Hamiltonian reduces to that for the hydrogen atom,
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(23)
Which interpretation should be used? In fact, it docs not matter, since upon evaluating the angular averages both AI(sin2 ei) and A~(cos* + cos2 Vi) yield the same result, namely 2/3, independent of D. (In retrospect, this had to happen, since the scaling procedure made use of the same factors, D and D- 1, that occurred in the expectation values, and neither of these factors could remain uncancelled in the finitized Hamiltonian.) Note that in Table I we have avoided the interpretation problem by giving the Hamiltonian in the final form generated by either interpretation. We make note of one other choice which is ultimately of no consequence. The general procedure described above scales all Coulomb potentials in the same way, including internuclear potentials. However, there is also an argument to be made for including the internuclear terms among the "other" potentials, sincetheconditionsineq 14for theformoftheCoulombcoefficimt were really dynamical in origin, and need not apply to the internuclear terms within the Bom-Oppenheimer approximation. Thus, we could in Table I have written the internuclear potentials for both H2+ and H2 as 1/R,independent of D. It turns out not to matter which form we use, however, because in the end we are interested in interpolated results at D = 3, and the interpolation procedure will give 1/R for either choice of the dimensional
As described in section 2, the solution is just &) = exp(dA/3) and E(') = - -1
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2
(26)
For the problem of a hydrogen atom in a spherical cavity, the
D
1 limit is given by
(Note that (r/rO)- = 0 if r < ro, and 0 if r > ro.) The boundary conditions at i r o are met by setting cp(r) = sinh k(ro - 14). Applying the cusp condition at the origin then gives the solution as 9 E(') = - - k2 where 3k = tanh kro (28) 2 This is valid for any ro, though the solution for k is imaginary if ro < 3. To avoid imaginary values, one can write
E ( ' )= for this regime.
+ -92 k2
where 3k = tan kr,,
(29)
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2471
Dimensionally Generalized Electronic Structure
D
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Next consider H2+. Using R to denote the bond length, the 1 Hamiltonian reads
For the ground state, symmetry and the boundary conditions at infinity allow us to set d r ) equal to cosh kr in the inner region (14 < I/$) and to a exp(-kld) in the outer regions (14 > II2R), whereu is a constant. Applyingcontinuity and thecuspcondition at either nucleus givesI7J8
E ( ' ) = -9/2k2 where 3k = 1 + exp(-kR)
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(31)
Before turning to the D 1 limit of Hartree-Fock H2, we review the solutionto the simpler problem of Hartree-Fock helium (or more generally, a two-electron atom or ion of nuclear charge 2).This solution serves as a check for H2 (of which it is the R 0 limit), and the starting point for the two-electron atom polarizability and susceptibility calculations below. Since the spatial Hartree-Fock wavefunction is of the form 9(rl)cc(r2),the H F potential of one electron on the other is
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This is of the same form as the solution for helium. The constants will be different, though the orbital energy is clearly still given by t = -9/2k2. For 14 < I/2R we cannot set C = 0. The solution to eq 38 with the correct symmetry for the ground state is
Any two of the three constants (a,K , p ) in this expression can be eliminated immediately. From the properties of elliptic functions, the squared derivative obtained from eq 40 is
+
(dq,/dr)2 = C Y - ~ K ~1( - p ) ( ~ < ~ ~ ( 2-1l)q,' ,
a2= 3 / 2 ( ~ 2- k2)
= l/2(1
347 -7) dr
k = I / J 2 2 - 1)
= - 9/2k2= - ' / 2 ( ~- '/2)2
E(') = 2 c - J =
36(r
-z2+ 1/2z-
+ 1/2R)- 36(r - I/2R)+ 3[q(r)I2)q(r)=
2 dr2 The solutions now turn out to be a hyperbolic function (csch) for the outer regions and a Jacobian elliptic function (nc) for the inner region. Tosee this, multiply through by dqldrand integrate. For r # fl/2R this gives
+C
(38) where C is an integration constant. For 14 > 1/2R we can set C = 0, since both C,Oand dcp/dr must vanish as 14 The solution to q 38 with the correct asymptotic behavior is then
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I/2R
(43)
(3
- Ksn dc)2 = k2 + T1( K ~- k2)nc2 1 + ~ K =E2k
+ 1 - k2)R $ K ~
(44)
Here it is understood that the arguments of the elliptic functions (sn, dc, and nc) and elliptic integral (E) are all ( I / ~ K R I I / ~ ( ~ k2/K2)). At any R, this pair of equationscan be solved numerically for k and K . By use of the same convention for the arguments, the total energy is then 2c - J = -9k2 - 3J44, which reduces to 9 4
E(') = -3k2 - 6 ( -~k2)nc2 ~ - - ( K ~ - k4)R
(45)
Finally, we return to H F two-electron atoms to consider the effects of uniform magnetic and electric fields. From Table I, the perturbations are given by
w(r) (37)
(dqldr)' = l/9p4 - 2 / 9 c q 2
dr
+
(36)
Now consider Hartree-Fock H2.21 The self-consistent molecular orbital cc(r) satisfies an equation similar to that above, except of course that there are two nuclear terms:
(--9 --d2
1/2R
Any of these equations (but most convenientlythe cusp condition) can be used to eliminate reference to hyperbolic functions, and therfore 6, in the other two. This yields two equations in two unknowns
b= ln(4Z - 1) (35) This is equivalent to the solution given by other^.^^.^^ Finally, the results for the orbital energy, the Coulomb integral, and the total energy are
J = 3J[Cp(r)I4d r = 1/2(Z-
(42)
0 = cp
