Asymptotic behavior of atomic bound state wave functions*

Reinhart Ahlrichs

Institute of Physical Chemistry, University of Karlsruhe, Karlsruhe, Germany (Received 24 October 1972; revised manuscript received 19 March 1973)

In the present paper we investigate the asymptotic properties of an exact bound state wave function <p of an n -electron atomic system within the infinite nuclear mass approximation and neglecting relativistic effects. An explicit upper bound is derived for Ilr r <pi I, where r; denotes the distance of the ith electron from the nucleus and /.I. = 1,2,3, .. ·. We are then able to derive upper bounds for expressions like Ilh (r ;)<pII, where h (x) is an exponentially increasing function. We finally indicate an exponentially decreasing pointwise bound for <p.

I. INTRODUCTION

Two types of singularities occur in the Schrodinger eigenvalue equation of atomic and molecular systems. The first one is due to the singularities of the Coulomb potentials at points in configuration space where the coordinates of two or more particles coincide. The second one is concerned with the singular boundary con­dition, i.e., the case of one or more particle coordinates going to infinity.

The consequences of the Coulomb singularities on the wave function are well understood in the case that just two particles coincide 1 ,2 and lead to the well-known cusp conditions.

Concerning the asymptotic behavior of bound state eigenfunctions of SchrOdinger operators, it is generally presumed that they vanish exponentially. So far, this has been rigorously proven only for one-electron mole­cular ions by Fox and Bazley 3 and for three-particle systems, e.g., helium like ions, by Slaggie and Wichmann. 4

It has been proven by Schnol5 that the eigenfunctions of an n-particle Schrodinger operator vanish exponen­tially in absolute value provided the potential is bounded from below and the corresponding eigenvalue is isolated and has finite multiplicity. Schmincke 6 has shown that the same result can also be obtained for certain classes of unbounded potentials, which do not include the case of Coulomb potentials, however.

In the present paper we consider bound state eigen­functions-for the precise meaning of this term see Sec. II-of an n-electron atomic ion in the infinite nu­clear mass approximation, neglecting relativistic effects. The Hamilton operator is then spin independent and we have to consider only spatial wavefunctions o:p(rl"" ,rn )·

ill Sec. IT we first define various quantities which will be used frequently in the following derivations. We also give a brief account of some basic mathematical concepts. The asymptotic properties of a bound state eigenfunction cp(rl' ••• ,rn) are then discussed in Sec. III in deriving upper bounds for expressions like IIrlfCPIl. The main results can be summarized as follows.

(a) In subsection lIlA we first derive an explicit upper bound for IlrlfCPII, see (17), (18). It is shown that this bound cannot be improved considerably in general.

(b) In subsection IIIB it is proven that cp vanishes exponentially in the mean, i.e.,h(ri)cp is square in­tegrable where h(x) is an exponentially increasingfunc­tion. The upper bound obtained for Ilh(ri)cpll is also shown to be about as close as possible. We finally de­rive an exponentially decreasing pointwise bound for

1860 J. Math. Phys., Vol. 14, No. 12, December 1973

I cp(rv '" ,rn) I ,which appears to be relatively poor, however.

The various expliCit bounds obtained are of a simple form and depend only on the ionization potential of the state under consideration and on the nuclear charge.

II. NOTATION AND BASIC CONCEPTS

We consider solutions of the eigenvalue equation (1) in the Hilbert space JC == L2(n 3n ):

(H - E)cp == 0, cp ;c O.

For convenience cp is required to be normalized to unity:

IIcpll == 1.

The Hamiltonian H is given by

H = T + V,

T==-t 6 6; i~l,n

V = - z 6 (l/r;) + .6 (l/r;j)' i=l,n '<J

(1)

(2)

(3)

(4)

where r; denotes the coordinate vector of the ith elec­tron with respect to the nucleus, Ir;1 = ri'r j " =rj -rj ,

6 is the Laplacian in three-dimensional sparie,z the nuclear charge, and n the number of electrons. Whenever confusion is possible, we will write H (n) instead of H to indicate the dependence on n. A comment is appropriate here on the definition of H, since it is not obvious that the formal expression (3), (4) defines a self-adjoint operator on JC. It has,however,been shown by Kat07 that a self-adjoint operator iI = T + V exists, which is uni­quely determined by (3) and (4). T is a generalization of T in the following sense: If T/,/ E X,exists locally al­most everywhere and T.f E JC.l. then T/ = 1'/.1,7 The domains DT and Dil of T and H COinCide, which is due to the fact that each single term in V and hence V itself is relatively bounded with respect to l' with bound zero:

z 6 1I(1!ri)/1l + 6 II (l/r ij )/1I ~ all/II + bIIT/II, i ;<j

/ EDT' a, b > 0, (5)

where b > 0 can be chosen arbitrarily small. 7

We further note that any eigenfunction cp of ii [i.e., a solution of (ii - E)cp = 0, cp ;c 0], is (equivalent to) an analytical function and satisfies the differential equation (1) in any region of the configuration space R 3n where V is regular.l,S cp is still continuous at the singular points of V and has only a rather mild singularity.

In the following we have to deal with sufficiently well­behaved functions only for which T/ = 1'/ and hence

Copyright © 1973 by the American Institute of Physics 1860

1861 Reinhart Ahlrichs: Asymptotic behavior

Hf = Hf. Having this in mind, we will fre,quently sup­press the tilde in order to keep the notation simple.

Throughout this paper we consider only bound state eigenfunctions. To give a proper definition of this term we have to consider the implications of statistics and spin. As the Hamiltonian is invariant with respect to a permutation of the particles, the eigenfunctions of H can be classified according to the irreducible representa­tions of the symmetric group Sn' Because of the Pauli prinCiple, only some of these are physically realizable,9 which in the nomenclature of Wigner (Ref. 9, pp. 129-33), can be classified by the irreducible representations 15 (k), k = 0,1, ... ,[n/2]. The parameter k is related to the total spin S through S = n/2 - k. If the eigenfunction cp under consideration belongs to /5 (k) , it can only be ionized into (n - I)-particle states which transform according to the irreducible representations'/5 (k) or

'/5 (lrl) of Sn-l'

Let E c,.-I)denote the lowest eigenvalue of H (n-l) O,k _

belonging to ' D (k) and let

E (II-I) = min (E (n-l) E (n-l») o O,k, O,k-l •

The (lowest) ionization potential E of the state des­cribed by cp is then given by E = E ~n-l) - E.

In the following we consider only states for which

E = E ~n-l) - E > 0,

which are usually called bound states.

(6)

In the subsequent considerations we will use frequently the quantities '11 and y defined as

'11 =.J2E, (7)

y = Z/'I1. (8)

It is further convenient to introduce the n-electron operator H';

H'; =H +Z/r;. (9)

For the subsequent derivations we need an estimate of the following kind:

(10)

where f is a sufficiently well-behaved fun~ion which belongs to the irreducible representation D (k) of Sn' and Q; denotes an operator which depends on r; only. We decompose Q if with respect to the irreducible repre­sentations of the subgroup Sn-l (of Sn) of permutations that do not affect r '. The only nonvanishing contributions belong to the irreducible representations 'Ii (k-l) and 'Ii(k) of Sn_l.91f we further note that

H'i = H(n-l)(rl"" ,r;-1,ri+ 1 ,··. ,rn) - iLl.i

+ I) ~ ~ H(P-l), j("i) r;j

the ,inequality (10) follows immediately from (6) and the definition of E (n-1).

o We finally define the differential operator Pr. ,

(11)

It is easily shown that Pr . is relatively bounded with respect to T: I

lip r/ 112 "" - (j, LI. J) "" 0' II f 112 + (1/0') II Tf 11 2 ,

o < 0' < 00,

J. Math. Phys., Vol. 14, No. 12, December 1973

1861

which then proves that PrJ is well defined and square integrable if this is the d.se for T f.

The subsequent derivations are considerably Simpli­fied by a result due to Combes,10 who Showed that under the present conditions rfcp E JC and further rll.cp E DB for arbitrary IJ. = 1,2,3, .... '

III. RESULTS A. Explicit bounds for IIrl;l <p11

I

The derivations presented in this subsection are essentially based on the relationship (10). We then obtain a few steps the desired bound for IIr~cpll; see (17).

We start from the following identity:

which is easily proven if we use the eigenvalue equation in the form (H'i - E) cp = (Z/r;)cp, and apply standard differentiation techniques. It should further be noted that all terms occuring in Eq. (12) are well defined in the sense of Hilbert space theory, as has been shown in a paper by Combes10 we have mentioned briefly at the end of Sec. II, above.

From the inequality (10) one obtains the following estimate for the lhs of Eq. (12):

(r~cp, (H'i - E)r~cp) ~ dr~cpIl2. (13)

The term (r~cp,PT.r~-1cp),occuring on the rhs of Eq. (12), can be integrated by parts (for an almost identi­cal derivation, see Ref. 11, p. 345):

(14)

Equation (14) is valid only if the lhs of (14) is real, which is easily verified from (12). Combination of (12), (13),and (14) yields

IIr~cpll2 "" 2yTJ-ll1r~cplI'lIr~-1cpli + 1J.2'11-2 I1rr1cpIl2, (15)

where y and '11 are defined in (7) and (8). The inequality (15) can be solved with re!,pect to Ilr~cpll

IIr~cpli "" [y + (1J.2 + y2)1/2]'I1-lIIrr-1cpll, IJ. = 1,2, .... (16)

Iterating (16), we obtain the desired explicit bound for Ilr~cpll :

IJ. = 1,2, ... , (17)

where II cp II can be ommitted since cp was required to be normalized to unity.

In order to make the J.I dependence of (17) more trans­parent, we will now derive another bound which is some­what poorer than (17),however. Using the easily veri­fied inequality

y + (;\2 + y2)1/2 "" (;\ + y) [1 + y2/2(;\ + y/2)2],

one obtains

n [y + (;\2 + y2)1/2] "" [r(1J. + y + 1)/r(y + 1)]

x n [1 + y2/2(;\ + y/2)2], )..= 1,~

1862 Reinhart Ahlrichs: Asymptotic behavior

where r(x) denotes the gamma function. The product on the rhs of the last equation is a slowly varying func­tion of IJ. If we insert the product representation of sinh (x) (see Ref. 12, p. 29), this term can be further approximated by an expression independent of jJ.. A simple manipulation finally yields

IIr~cpll ~ C(y)r(/J + y + l)1rJl , f.L =: 0,1,2, ..• , (18)

where

C(y) = -./2: sinh (1Ty/J2)(r(y + 1hTy f1 (1 + y2 /2,\ 2»-1 A:1,0 (19)

and {) = [y/2]. The product occuring in (19) has to be set equal to unity if {) < 1. The bound (18) was proven only for J.1. ~ 1, but it is easily verified that it holds also for f.L =0.

In the case of one-electron atomic ions the eigenvalue equation (1) is solvable and the bound (18) can be com­pared with the exact result. Let us, e.g., consider the eigenfunction CPn ,n-1 (r), where n is the principal quan­tum number and 1 =: n - 1 the angular momentum. (The magnetiC quantum number m does not enter the follow­ing formulas.) The radial partfn,n_1(r) of CPnm-l is

fn,n-l (r) = Nrn-1e(- Zr/n),

where N is a normalization constant. The ionization po­tential € is given by € = z2/2n 2,which yields 1/ Z /n and y = n. ExpreSSing IlrI'CPn,n-111 in terms of y and 1/, we obtain

IlrJlCPn,n-lll2 = r(2J.1. + 2y + 1) [r(2y + 1)'(21/)2Jl]-1,

which gives asymptotically for large J1.

IIrJlCPn,n_lll~(r(y + 0.5)r(y + 1)]-1/2(J.1. + y)-1/4 X r(J.1. + y + 1)1/-1'. (20)

Comparison of this expression with the bound (18) shows indeed that (18) has, besides the slowly varying term (J1. + y )-114, the correct dependence on J.1. and 1/. The factor C(y), however, is not too good for large y. Con­Sidering the rather rough approximations that had to be made to derive (18), this bound for IIrJlcp11 is surprisingly good.

For the following considerations it is convenient to have a bound of the kind (18) for noninteger powers of r:

IIr~+pcpli ~ C(y)r(J1. + y + 1) (J.1. + y + l)p1/-rp, (18') •

where p is a real number ° <:; p <:; 1. The estimate (18') is valid for p = ~ , since by virtue of Schwarz f inequality

Ilrll+1/2cp112 = (yflcp, rv+1cp) <:; IIr~cpll.llr~+lcpli ii' "

"" [C(y)r(J.1. + y + 1) (f.L + Y + 1)1/2 1/-r1/2 )2.

Repeating the same procedure we see that (18') holds for arbitrary dual fractions p = j·2- m,j = 0,1, .•• , 2m , and hence by continuity for any real number 0 <:; p <:; 1.

B. Exponential behavior of <p

The f.L and 11 dependence of IIrI'CPn,n-lll displayed by (20) is a direct consequence of the exponentially de­creasing behavior of cP n ,n-l' From the Similarity of (18) and (20) one might expect that any bound state eigenfunction cP vanishes exponentially. This is indeed the case in a sense to be specified below. Let us before­hand define the function hfl,v(x) for real /3, integer v, with v> /3 :". O,X ;. 0, through

J. Math. Phys., Vol. 14, No. 12, December 1973

hfl,v(x) = ~ [xA-fl/r(,\ + 1)], ·A= v ,00

which can also be written as

hfl,v(x) = x-/l (ex - ~ [xAjr(,\ + 1)]). A:O,v-1

he,v(x) behaves asymptotically like x-fleX:

lim xfle-x he v (x) = 1. %--+00 ,

We can now phrase Theorem 1.

1862

(21)

(22)

Theorem 1: For any electronic bound state eigen­function cP of the Hamiltonian (3), he v(1/r)<p is square integrable if (3 > y + 1, where 11 and; are defined in Eqs. (7) and (8).

Proof: By virtue of (18') we have

1/A-Bllr. A- fl <p1I IIh e,v(1/rj )cpl!", ~ r( • 1)

A~V,OO A +

<:; C(y) ~ r(,\ + y - [{3]) (,\ + Y _ [(3])[S]+l-a, A: v,oo rCA + 1)

where [13] denotes the largest integer contained in fl. To evaluate the sum on the rhs, we take advantage of

r(x)xb/r(X + b) "" 1, if b ;. 1,x > 0,

which follows immediately from the logarithmic con­vexity of the gamma function; see Ref. 13, p. 4. Putting x = A + Y - [(3] and b = 1 + [/3] - y, we get immediately

I!h e,v(1JTi)cpli "" C(y) ~ (,\ + 'Y - [{3])r-a, (23) A: v,oo

where the rhs does in fact converge for any (3 > y + 1. QED

Theorem 1 cannot be improved much in general as is seen from a consideration of the particular one-electron eigenfunctions CPn n-l(r) already discussed in the pre­ceeding section. Taking into account the asymptotic be­havior of he,v(x) [see (22)],we easily verify that hfl v(1/r)cpn n-1(r) is square integrable if {3 > y + t,which is 'only slightly less restrictive than the requirement of Theorem 1: {3 > y + 1.

We shall now prove that any bound state eigenfunction cP of H vanishes exponentially if r i ~ 00. For this pur­pose it is convenient to write x = (r l' r 2' ••• ,r n)' It has been shown by Kato1 that any eigenfunction <p of the Hamiltonian (3) is HBlder continuous for €I < 1, i.e., for any e with 0 < e < 1, there exists a parameter B depend­ing on e but independent of x and x such that

I cp(x) - cp(x)k Blx -xl e.

The inequality (24) implies

Icp(X}12;. (1<p(x)I-Blx _xl e)2

for Ix -xl"" R o = (lcp(x)I/B]l/e.

(24)

(25)

We can now use (25) to verify the follOWing chain of inequalities:

;. (Il!,in he, v (Wi) + 1\ 2 ~ri-Til$Ro J

x B-3n1ew3n I cp(x) I (3n ie+2)'[282 /3n(3n + 9)(3n + 21/)], (26)

1863 Reinhart Ahlrichs: Asymptotic behavior

where w = 21T m / 2 /r(m/2) is the surface area of the unit sph:re in R m. In order to get rid of the parameter Ro = [lcp(x)IB]l/e occuring in (26), we note that Icp(x)1 is bounded1 and hence Ro < Rmax. Since ha v is essen­tially the exponential, see (22), there exists a constant K > 0 such that

mln [h a,v(1rr) + 1] '" mln [ha,V(rrYi' + 1] ITCTil5Ro I TCTt I <Rmax

'" K[h a,v(1]r i ) + 1].

Combining the last inequality with (26), we finally obtain

13 > y + 1, (27)

where K = Gn/e + 1)-1 and K is a suitable chosen para­meter independent of x. By virtue of the asymptotic behavior of h a, v' see (22), (27), yields for sufficiently large r i

Icp(x)I.;;Kr/8e-T/KTi. (28)

If we put s = m~xr., the following inequality (28') is an immediate con~eq~ence of (28), provided s is suf­ficiently large:

(28')

IV. DISCUSSION

In the preceding sections we have derived bounds for Ilrfcpll and IIha,v(w)cpll,where cp denotes as before. an atomic bound state eigenfunction. The correspondlng bounds (18) and (23) provide information about the asym­ptotic behavior of cp in the mean for the case r; -7 <Xl.

It has further been shown that (18) and (23) cannot be improved considerably in general.

One can also use the results presented above to obtain bounds for expressiOns involving many-electron opera­tors. Let us, e.g., consider R defined as

R = (. I) ri 2) 112. ,=1,n

It is then easily proven that

R21' .;; nl'-l I) r~1' i=l,n z_

which in combination with (18) yields

IIRl'cpll.;; c(y)r({J. + y + 1)n1'1P

and further, in analogy to (23),

(29)

Ilh a,v<"1]R/n)cpll .;; C(y) 6 (A + Y - [13])),-a, (30) >..=; v,oo

provided v > 13 > y + 1.

The bounds (29) and (30) are rather poor,however. The relationship (30) proves that cp vanishes in the mean at least like e-T/R/n, whereas one would expect a decay like e-T/R. It has in fact recently been shown by O'Connor14 that cpeT/'R E Jeif 1]' < 1]. From O'Connor's analYSis it follows, of course,that cpeT/'Ti E JC,if 1]' < 1],

this result is, however, weaker than the estimate (23)

J. Math. Phys., Vol. 14, No. 12, December 1973

1863

which includes the case 1]' = rl. We, furthermore, note that O'Connor's treatment does not furnish explicit bounds like (23) or (18) derived in the present paper.

The exponentially decreasing pointwise bound (28) is rather poor too. Partly stimulated by the present study,Simon15 has recently shown that Icpl < A(1]')e- n'R, if 1]' < 1]. The proof of Simon is based on the treatment of O'Connor.

We finally make some comments concerning the gen­eralization of the results obtained in this study. It could appear that the derivations given in Sec. III are restrict­ed to the particular Hamiltonian specified in Eqs. (3) and (4). This is not the case, however. The considerations presented in this work can in fact be extended to Hamil­tonians of the form H = T + W, where W is a so called Kato potential. 1We then define H; as

H; =H + T/lr; - r;,ol,

where the parameters T and r ;,p have to be chosen such that the relationship (31) holds Lwhich is analogous to (10) above],

(31)

for dT) > O. The present method breaks down, however, if (31) cannot be fulfilled. Under the presupposition that (31) holds, all bounds derived above are obviously valid if we replace r; by Ir; -ri 0 1 ,Z by T and E by dT). The parameters T and r i 0 can then be chosen to optimize the various bounds. 'In the atomic case the optimal choice is, of course, r; 0 = 0 and T = Z, which in turn has the advantage that'E can be interpreted as the ioni­zation potential.

ACKNOWLEDGMENTS

The author is indebted to Dr. L. Thomas for valuable comments concerning the manuscript and to Dr. W. Kutzelnigg and Dr. V. Staemmler for numerous dis­cussions. I further thank Professor B. Simon for bring­ing Ref. 10 to my attention and Dr. A. J. O'Connor for allowing me to see his work before publication.

* A preliminary note on the pertinent results of this study has been published in Chern. Phys. Lett .. 1S, 609 (1972).

IT. Kato, Commun. Pure Appl. Math. 10, 151 (1957). 2W. A. Bingel, Z. Naturforsch. A 18, 1249 (1963). 3N. W. Bazley and D. W. Fox, Int. J. Quantum Chern. 3, 581 (1969). 4E. L. Slaggie and E. H. Wichmann, J. Math. Phys. 3, 946 (1962). 'E. E. Schnol, Mat. Sbornik (N.S.) 42 (84),273 (1957) (in Russian); see also I. M. Glazman, Direct methods of qualitative spectra analysis of singular differential operators (Moscow, 1963) (English translation, Israel Program for Scientific Translations, Jerusalem, 1965).

6U._W. Schmincke, Math. Z. 111, 267 (1969). 7T. Kato, Trans. Am. Math. Soc. 70, 195 (1951). 8E. Hopf, Math. Z. 34, 194 (1932). 9E. P. Wigner, Group theory and its application to the quantum

mechanics of atomic spectra (Academic, New York, 1959). IOJ. M. Combes, Nuovo Cimento A 64, 111 (1969). liT. Kato, Perturbation theory for linear operators (Springer, Berlin,

1966). 12K. Knopp, Funktionentheorie (de Gruyter, Berlin, 1955), Vol. II. 13Jahnke-Emde-Losch, Tables of higher functions (Teubner, Stuttgart,

1960). 14A. J. O'Connor, to be published. 15B. Simon, private communication.