Resolution of a paradox in the calculation of ... - Science Direct

Volume 203, number 2,3

CHEMICAL PHYSICS LETTERS

19 February 1993

Resolution of a paradox in the calculation of exchange forces for H2+ T.C. Scott ‘, J.F. Babb a, A. Dalgamo a and John D. Morgan III a,b a Institutefor Theoretical Atomic and Molecular Physics,Harvard-SmithsonianCenter for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA b Departmentof Physics and Astronomy, UniversityofDelaware, Newark DE 19716, USA Received 17 July 1992; in final form 3 December 1992

Tang, Toennies and Yiu have shown that despite the inherent symmetry of H2+,wavefunctions obtained from a combination of the unsymmetrized polarization expansion and the 1/R expansion can be used in the Holstein-Herring formula to calculate for large internuclear distances R the leading 8(eeR) terms in the exchange energy between the lowest pair of states. However, the associated claim by Tang and Toennies that the polarization expansion of the wavefunction converges not to the gerade molecular wavefunction, but to an asymmetric function localized about a single nucleus, conflicts with other numerical and analytical results. We show by a limiting procedure that use of the infinite polarization expansion for the wavefunction in the Holstein-Herring formula provides a result that is not equal to the exact exchange energy, although it has the correct leading 0 (ebR) behavior and is impressively close to the exact exchange energy for large R.

R ) !P=,where ‘Pais the unperturbed eigenfunction of

1. Introduction In this article we continue our examination of the Holstein-Herring method for calculating the exchange energy AE between quasidegenerate electronic states of molecules and of molecular ions [ 1,2] which arises from electronic tunneling between two identical potential wells. If @*and @,,are respectively the gerade and ungerade electronic eigenfunctions, and &= ( cD~+CD,,) /t/z is a wavefunction “localized” on atom a, then the exchange energy AE is exactly given by [ 1]

(1) The integral in the numerator is a surface integration over the median symmetry plane, denoted M, and the integral in the denominator is a volume integration over the half-space containing nucleus b. Herring showed that for large internuclear distances R, the leading term (4/e)Re-R of M(R) for the lowest pair of states of Hz could be obtained by replacing Cp,in the integrals of eq. (1) with an approximate “localized” wavefunction, xr =gH(z/

atom a, the z axis runs from nucleus a at z= 0 through nucleus b at z= R, and the prefactor g”(z/R) = (1 -z/R)-‘e-Z’R is obtained by solving a simple first-order ordinary differential equation [ 21. Since recent developments in the use of the Holstein-Herring method make heavy use of two expansions, the polarization expansion and the 1/R expansion, which have a number of subtleties, especially if one tries to combine them, we carefully define them to establish our notation and to draw attention to pitfalls. For Hz, the Schriidinger equation (in au) which describes the perturbing effect of nucleus b on atom a can be written as [(-~~z-~)+~(-~+I)~=P

(2)

where r, and rb are the distances of the electron from nuclei a and b, respectively, and Ais an ordering parameter which equals 1 for the physical case of identical nuclei. The eigenvalue E and the eigenfunction x are implicitly functions of both A and R. For any

0009-2614/93/$ 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.

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fixed finite value of R, the eigenvalue e(n) and the eigenfunction x(n) can be used to generate Taylor series in powers of ;1:

series are called the (infinite) polarization expansions of an eigenvalue and an eigenfunction, in

which atom a is the unperturbed system. We shall also have occasion to refer to thejnite polarization expansions truncated at a particular order AN,which are denoted by ezcN’(2) and x;(N) (A). The perturbing potential energy due to the interaction of atom a with nucleus b can be expanded as a multipole series in powers of l/R as (4) where 13,is the angle between the z axis and the position vector r, of the electron relative to nucleus a. For any given value of 1, one can perform formal Rayleigh-Ritz power series expansions of the eigenvalue and the eigenfunction in powers of 1/R, where the zeroth-order Hamiltonian is that for atom a and the zeroth-order wavefunction is the exact atomic eigenfunction for atom a. One thus obtains formal 1/ R expansions of the perturbed atomic eigenvalue and the corresponding eigenfunction given for 3,= 1 by

XlIR(R)a

C

j=o

Aa,i RJ -

(5)

Truncation of these expansions at some order ( 1/R )” yields finite 1/R expansions, which are denoted by c~/~‘~)(R) and x:/“‘“‘(R). Recently, Tang, Toennies, and Yiu [ 31 proposed to calculate the exchange energy by first writing out the recursive equations for the coefficients of the polarization series x:(A) for the wavefunction, solving these equations to leading order in powers of l/R, setting r’= fR, summing the results (with ;i = 1) and then using the resulting sum in the numerator of the Holstein-Herring surface integral. Assuming the de176

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nominator was unity to within order O(e-R), they found that this procedure yielded the correct leading term (4/e)RemR for the exchange energy splitting M(R) between the two lowest states of HZ. They also found numerically that including progressively higher-order terms in the polarization series seemed to generate a sequence of approximants to the higherorder terms (of the form (l/R)“R emR) in the asymptotic expansion of the dominant exponential contribution to the exchange energy. Their results are based on the 1/R expansion of the polarization expansion of the wavefunction truncated at order AN and evaluated at 1= 1, denoted x!(““)(A= 1). This yields a wavefunction which agrees, term by term, with the 1/R expansion of the perturbed wavefunction xb’R to within some Jinite order ( 1/R)“‘. Extending their approach to N’+co yields the 1/R expansion x:‘~(~’ as M’+co which, in this limit, is identical to the 1/R expansion for the wavefunction as specifically written in eq. (A.2) for Hz in the work of Morgan and Simon [4]. The method of Tang, Toennies, and Yiu [ 31 was partially elucidated when it was shown by Scott, Dalgamo, and Morgan [l] that the function they obtained by summing over the leading-order (in powers of I/R) terms of each coefficient of the polarization series is exactly the leading term of the Holstein-Herring function x! concentrated about nucleus a. Scott et al. also showed how to extend the Holstein-Herring method to excited states of H2+and to the lowest pair of states of HZ, and thus indicated that the Holstein-Herring method is of considerably greater general utility than had previously been appreciated. These results revived interest in the Holstein-Herring method because of its potentially useful application, in particular the formulation of a straightforward procedure for calculating the exchange energy splittings between states of multi-electronic molecules at large internuclear distances. Nonetheless, some important issues remained unsettled. In particular, the fact that the correct dominant 0(e-“) terms of M(R) can be calculated by replacing in the Holstein-Herring formula (eq. ( 1) ) the true localized wavefunction @awith a function obtained from the polarization series might seem to suggest that the infinite polarization expansion ~a’(A= 1) of the wavefunction is in some sense a good approximation to be true localized wavefunction Qa,.

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CHEMICAL PHYSICS LETTERS

Indeed, on the basis of some impressive numerical results obtained from low-order polarization expansions, Tang and Toennies [5] have subsequently claimed that the polarization expansion of the wavefunction about nucleus a converges not to the ground gerade state of Hz , but to a function localized about nucleus a. However, the analysis of Whitton and Byers Brown [ 61 on a model problem consisting of a scaled onedimensional limit of the Hz system [ 7)) namely the double-well Dirac delta function model, indicated that the finite polarization expansion of the wavefunction xLcN1 (I = 1) does converge as N-+co to the gerade ground state function. They also inferred that this property would hold for the physically relevant H: system in three dimensions. This result was further vindicated by the numerically based analysis of Chalasidski et al. [ 81 and the analytical examination by Kutzelnigg [9] of the structure of the perturbation theory in the vicinity of A= 1. In this Letter, we present the resolution of the paradox between the claims of Tang et al. and the conventional wisdom about the polarization expansion. It will be shown that, just as in the case of the model problem [ lo], although the polarization expansion 2 of the wavefunction can be used in the HolsteinHerring formula to obtain the dominant 0(e-“) terms of the exchange energy AE(R ) and to generate numerical results which are impressively close to the true U(R) for large R, it cannot generate the subdominant terms in AE(R) of o(ee2”), 0(e-3R), etc. Many of the defined quantities and results contained in this work were initially derived from a detailed analysis of the simpler one-dimensional model problem [lo]. A key issue is the realization that the 1/R expansions of eq. (5) and the functions in eq. ( 3) are distinct, as can be seen by a careful examination of their definitions [lo]. For the one-dimensional model problem, it was found [lo] that although Xap (A= 1) is the gerade ground state wavefunction QgB, the truncated 1/R expansion dfRtM) of the perturbed wavefunction converges to (in fact is exactly equal to) the unperturbed atomic wavefunction ‘Paand not to a symmetric molecular functions. d’R(M1is close to the true localized wavefunction & = ( Og+ Qu) /$, and when used in the Holstein-Herring formula it yields the correct leading term 2e-R in the exchange energy

19 February 1993

splitting, but not the correct sub-dominant terms of 0 (eczR) and higher. As shown in section 2, this distinction carries over to Hz, where the convergence properties of the 1/R expansion d/R are examined in the vicinity of nucleus a. The infinite polarization expansion g (,I= 1) of the wavefunction converges to the gerade ground state wavefunction Qg (or symmetric wavefunction @+), so its direct substitution into the HolsteinHerring formula yields a ratio of the form “O/O”. As shown in section 3, the only meaningful result left to consider is the substitution of the polarization expansion Xap (I) of the wavefunction for R# 1 into the Holstein-Herring formula, where the limit as I.+ 1 is taken using l’H6pital’s rule. In this case, a finite value denoted A&(R) (or equivalently A_!?+(R)) is obtained, which for large R is very close to but is not equal to M(R). A similar result denoted A&(R) (or equivalently AE_ (R) ) is obtained if the corresponding excited state whose ,I= 1 limit is the ungerade wavefunction @,, (or anti-symmetric wave function a._) is instead substituted into the Holstein-Herring formula. A&(R) and A&(R) differ from the true A/Z(R) for large R only by small subdominant terms of Co( eeZR) because of two previously unrecognized facts [ lo]. The first fact is that the properties of the ratio of the Holstein-Herring formula are such that the resulting mathematical structures of AZ?*obtained by the limiting process are almost identical to that of the true AE, the only difference being that the derivatives #l a@, (A, R )/ dA )A=, in AEk play the roles of Dr respectively. The second fact is that the normalized a@, (A,R ) /iU 11=1 differ from $ (R ) by only exponentially small terms of fJ(e-“). A numerical demonstration of the consequence of these properties is provided in section 4, where it is also shown that AE,(R) and A&(R) lie, respectively, below and above the correct AE( R) for large R. In section 5 our conclusions and some final comments are presented.

VI For 1# 1 we use g (or t ) and u (or - ) 10label the asymmetric electronic states of this molecule which map onto the gerade and ungerade states in the limit d+ 1.

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2. l/R expansion of the wavefunction As was observed by Morgan and Simon [ 41, the same techniques which they used to prove that the gerade and ungerade energies Eg and E,, (and hence the polarization energy 4(E,+E,,) ) have asymptotic series in powers of 1/R can be used to prove that the localized wavefunction has a norm asymptotic series in powers of 1/R. Furthermore, as stated elsewhere [ 11, the divergence of the 1/R expansion of the energy t:/“(R) implies that the L2 norm of the l/R expansion d’“, computed using intermediate normalization, also diverges, as shown below. Projecting an unperturbed atomic eigenfunction Y,,onto the exact molecular Schrbdinger equation [&fUR)l~(R)=-UR)!f’(R) for the molecular energy E(R) and wavefunction Y(R) and the use of Hermiticity and H,Yo=E,!J’o followed by re-arrangement leads to the identity E(R)_E

=

0

(%lW)I@W)) (f&l@(R))

.

The replacement of Q(R) with the Mth-order truncated l/R expansion J&‘~(~‘(R ) in this identity yields correctly the Mth-order truncated asymptotic series EcM) (R) for E(R). By the Schwarz inequality, I~~ol~(R)l~:‘~~~~(R)~l

with

SO

intermediate

normalization

W’ol~x:‘~(“‘)UW=lh

Since 11 V(R) Y& has a convergent l/R expansion, one readily sees that if the series for E(R) diverges, so does the series for lk:“‘m(R)lh=(I

d3t’ l~~‘R’“‘(R;r’)~2)1’z, R’

where r’ is the spatial vector of the electron. However, this in itself does not exclude the possibility that the 1/R expansion xi’” (R, r) could converge on a compact subset C of the whole space [R3,so that (

d’r’ (~:‘~(~)(R;r’)l~

C

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could have a convergent l/R series (e.g., the divergence in the l/R expansion for the norm of the wavefunction could conceivably arise from misbehavior of the series for the wavefunction only in a neighborhood of nucleus b). However, our numerical findings indicate that the l/R series for the wavefunction probably diverges in all space. The truncated 1JR expansion l:IR(M) was examined using both intermediate normalization and normalization to unity with respect to the L2 norm. It was evaluated at the midpoint, i.e. z= fR on the z axis joining the two nuclei, for R= 1, 2, 3, 4, 5 and 10. The 1/R expansion for the wavefunction was also examined at various points about atom a, and in all cases we obtained a divergent series. Not surprisingly, as R becomes small, the divergent nature becomes more “violent”. Note that just like the leading term of the Herring function [l] forH2+, i.e.g(r,)=‘Y,exp(-S,),the infinite 1jR expansion of the first-order polarization expansion I!(‘) is locally normalizable in any region of space which excludes a neighborhood of nucleus b. We have also verified that this holds for the next term in the Herring function, g(rp)= Y, x exp ( -S, - SZ), but graphical analysis shows that the effect of the singularity at nucleus b is more pronounced. Presumably, a similar scenario applies to the polarization expansion of the wavefunction of Hz : as one includes more terms in the finite polarization expansion xI(w’and then forms the infiniteorder 1/R expansion of x:(N), the effect of the singularity at nucleus b becomes more pronounced, so that in the limit of infinite order, the resulting wavefunction is no longer normalizable in any region of space. Exact results for the gerade and ungerade wavefunctions, 9 and @“, were calculated at various points along the z axis by Bates et al. [ I 1, p. 2 18J. For example, at R = 4 and z = 1, $=0.175,

@“=0.146,

@,=0.227.

The truncated l/R expansions ~1’~‘~’using intermediate normalization for M=O, 1, 3, .... 15 are shown in table 1. As expected from the asymptotic nature of this series [ 41, the partial sums hover about the value of the localized wavefunction (p, before eventually diverging.

CHEMICALPHYSICSLETTERS

Volume 203, number 2,3

Table 1 Truncated 1/R expansion y.‘G+‘~for H: forR=4 evaluated using intermediate normalization at (x, y, t ) = (0,0, 1), where nucleus a is at (0, 0,O) and nucleus b at (0, 0, R). The sequence hovers about the exact value of @.=0.227... before eventually diverging M

IIR140 X*

0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.20755 0.20755 0.22701 0.22971 0.22784 0.23024 0.23001 0.22906 0.22923 0.22794 0.22582 0.22252 0.2I520 0.20071 0.16971 0.09966

Thus, just as in the case of the one-dimensional model problem, the 1/R expansion of the perturbed wavefunction xi/” of HZ is distinct from the polarization series of the wavefunction xi. However, unlike the model problem, the l/R expansion 2:‘” seems to diverge in all space.

3. Holstein-Her&g formnlae If the true localized wavefunction @ais replaced with the gerade wavefunction 0g in the HolsteinHerring formula of eq. ( 1), both the numerator and the denominator go to zero because of symmetry. In this case, the assumption that the denominator is unity to within LD(eeR)is no longer valid, and one must consider both the numerator and denominator of the Holstein-Herring formula. The question arises: what would the HolsteinHerring formula yield if @awere instead replaced by the truncuted series of the polarization expansion x;(N)(A) at A= 1 in the limit as N-too? This question can be answeredby consideringa modification of the Holstein-Herring formula (eq. ( 1) ) by taking its limit

19 February 1993

(6) using l’H8pital’srule, where @ais replaced by a function of L denoted @(A). For @(A), we use the polarization function, which is equal to the ground state eigenfunction d&,(A).Q+(A)is an analytic function of d. Thus the use of I’Hbpital’srule as in expression (6) is equivalent to getting the value of the infinite Taylor series of the Holstein-Herring formula in powers of R evaluated at A= 1. For the H: problem, xz (A) is analytic about I= 1 for any finite R [ 9, lo]. Consequently, the Taylor series is unique, the limit at A= 1 exists, and it is also unique. This ensures a meaningfulresult when approximatingthe limit with a sequence of rational functions of 1 given by the ratios of the numerator to the denominator using the truncated series for the wavefunction xzcN’in the Holstein-Herring formula. This analysis can also be repeated using instead the excited state solution by setting Y(A)= oU(A). The behavior of the Holstein-Herring formula is similar. In this limit, @,,(A)goes to zero at the midplane z=fR, and so does the denominator in the Holstein-Herring formula of eq. (1) because of symmetry. In accordance with our chosen notation, AEg (or AE,) denotes the limit in (6) for the case @(A)=xi(A)=Q8(A) as A+l, AE, (or AE_) denotes the corresponding limit for the case @(A)= &,(A) as 131 N2. First, it is straightforward to show using the definition @a,=($+ d$)/& of a localized wavefunction that the Holstein-Herring formula of eq. (1) can be rewritten entirely in terms of the gerade wavefunction 4 and the ungerade wavefunction 4, (7) Further, it can be shown that Al&(R), which results from a “O/O”limit of the Holstein-Herring formula, is given by [lo]

ff2 Note that the quantities A.E*calculated using Qi (A) in the Holstein-Herringformula are distinct from the true (and unique) energy splitting AE, even for 1= I.

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Similarly, it can also be shown that

This property also holds for the n-dimensional double well problem [ 101.

We note that the exact expression (7) and the approximate expressions (8) and (9), which result from taking “O/O” limits by I’Hopital’s rule, are very similar in structure. Eq. (8) for A&(R) can be obtained from eq. (7) for AR(R) by replacing QU(R) witha0#,R)/a;iI,=,. Similarly,eq. (9) forA& can be obtained from eq. (7) for A.Y(R) by replacing Qg(R) with &D,,(&R)/aIll=,. The next step is to compare a@, (2, R) /aAiI=, with QT (R). We start with a perturbative expansion using the symmetric (,I= 1) wave equation. The quantities a@, (A, R)/anl,+,. which are the first-order corrections to the symmetric and anti-symmetric wavefunctions @+, can be expressed as

4. Numerical results

where by construction ( @? 16, ) = 0. For the general N-dimensional double-well problem, it can be shown that [IO]

(11) where Vis the perturbing potential due to nucleus b. For Hz, V=l/R-l/r,, and (w+jl/R]~~)=O by orthogonality. Approximating @Joby a normalized linear combination of atomic orbitals (LCAO), we find that c,(R)=+

C@,I (l/b)lQJt WR)

>

(12) In section 4 we demonstrate numerically that eq. ( 12) remains unchanged when the exact eigenfunctions are used. Clearly, for large R the term with c* (R) in eq. ( IO) dominates and

a@, (A R) aA

180

a%(R). A=1

(13)

Let t&Jr, R) and @,(r, R) be, respectively, the lso* and 2~0, ground and first excited electronic states of Hz calculated in the Born-Oppenheimer approximation, where r is the position of the electron measured relative to the midpoint between the nuclei whose separation is R. In the calculations discussed below, the coordinate r was expressed in prolate spheroidal coordinates ({, q, 4) as defined in ref. [ 121. Integrations over q or @were done in closed form, while integrations over r were done using numerical quadrature. With Z, fixed to 1, the electronic wavefunctions for an electron bound to two nuclei depend on the charges of the nuclei only through the charge ratio 1, where A= (2,/Z,) and Z,, Z, are, respectively, the charges of nucleus a at z=O and b at z=R. For H:, A= 1. The wavefunctions @gand 0” were obtained in semi-analytic form to an accuracy of order lo-l4 using the matrix method of Hunter and Pritchard [ 131.The HolsteinHerring formula, eq. (I), was evaluated using the exact localized wavefunction & = (4 + @“)/& to obtain the exchange energy AE( R). As eq. ( 1) is an identity, the value of Al?(R) so calculated must be equal to the difference between the exact 2pa, and lso, eigenenergies. With 18 terms in t and 17 terms in q in the semi-analytical wavefunctions, our numerical evaluation of eq. ( 1), which is given in column 3 of table 2 for 1