PHYSICAL REVIEW A VOLUME 31, NUMBER 6 JUNE 1985

Boundary conditions and channel-coupling-array calculation for the H2 ungerade triplet state

J. Shertzer, * E. Bernstein, and F. S. LevinPhysics Department, Brown University, Providence, Rhode Island 02912

(Received I October 1984)

It is shown that, in order to yield physically well-behaved energy curves E(R), the wave-functioncomponents used in channel-coupling-array analyses of the ungerade states of H2 must obey a boun-dary condition which is derived herein. A very simple ansatz, obeying this new boundary condition,is constructed for the components corresponding to the lowest-lying triplet H2 ungerade state. Eventhough the ansatz is very crude, the resulting E(R) differs from that of an exact calculation by atmost 0.04 a.u. and is well behaved as the internuclear separation approaches zero. Since previousH& ungerade channel-coupling-array calculations yielded energy curves that were infinitely attrac-tive as R ~0, this paper provides the theoretical basis for a practical means of eliminating this un-

physical behavior.

I. INTRODUCTION

In previous studies, the channel-coupling-array (CCA)theory of multiparticle scattering' " has been successful-ly applied to bound-state calculations. ' ' The CCA ap-proach differs from standard bound-state calculations us-ing the Schrodinger equation in one major respect: theCCA analog of the Schrodinger Hamiltonian is a non-Hermitian matrix operator. Consequently, there is a setof CCA channel component wave functions; their sum isequal to the single Schrodinger wave function. ' ' Byintroducing trial channel components into the CCA equa-tion, one can obtain an approximate energy which is, . ingeneral, not equal to the energy obtained when the sum ofthe components is used as a trial Schrodinger wave func-tion in a variational calculation. In particular, the use ofsimple, almost crude, approximate channel componentsfor H2+, H2, and HeH+ ground states have produced en-ergy curves which are significantly more accurate thanthose obtained variationally from the equivalentSchrodinger function. This has led to a physical pictureof bonding for these systems that is absent from theequivalent Schrodinger analysis.

However, the earliest attempts to generate ungerade(excited-state) energy curves for Hz and FI2 were unsuc-cessful: at small internucleon separation, the energycurves became i.nfinitely attractive, implying a collapse ofthe system. An analysis of this unphysical result for theH2+ system led to the introduction of an additional boun-dary condition on the ungerade channel components, andan ansatz satisfying this boundary condition was shown toyield an accurate H2+ energy curve. '

In this paper we' extend the analysis for the Hz+ case tothat of the ungerade states of Hz. We show that the typeof additional boundary condition which gave physicallywell-behaved ungerade results for H2+ does not suffice forH2. A new boundary condition is therefore derived andits validity is confirmed via a calculation based on a verysimple approximation to the CCA channel componentwave function for the lowest-lying ungerade triplet state.As in previous CCA calculations on atomic and molecular

systems, the resulting energy curve is remarkably accurategiven the crudity of the approximation. Our main resultsare given in Sec. III, following a brief review of the CCAapproach in Sec. II. A short discussion (Sec. IV) con-cludes the paper.

II. REVIEW OF THE CCA THEORY

Consider a system of n distinguishable particles whichobey the Schrodinger equation

(H —E)4=0,where the total Hamiltonian is

(2.1)

H=Hp+ g V~ .(j&i)

(2.2)

H=HJ+ V~ for all j . (2.4)

In the CCA theory, the Schrodinger wave function isexpanded into channel components P~' '"

(2.5)J

where g. includes at least all two-cluster channels.JSubstituting Eqs. (2.4) and (2.5) into the original

In Eq. (2.2), Ho is the n-particle kinetic energy operatorand VJ is the two-body interaction between particle i andJ.

In multiparticle scattering theory, a channel is definedas a unique way of partitioning the n particles into clus-ters, where a cluster consists of a bound system of parti-cles (or a lone particle). A channel Hamiltonian HJ is de-fined to be the sum of the kinetic energy operator Ho andall interactions internal to the clusters, denoted by VJ.

Hj ——Hp+-VJ . (2.3)

The remaining interactions (between particles in differentclusters) are contained in the external channel potential,denoted VJ. The sum of the channel Hamiltonian and theexternal potential is the n-particle Hamiltonian:

3570 1985 The American Physical Society

31 BOUNDARY CONDITIONS AND CHANNEL-COUPLING-ARRAY. . . 3571

wjk ——l, (2.7)

is introduced into Eq. (2.6) via the external channel poten-tial. Interchanging the order of summation, one obtains'

g(E —H, )y, =g W,,V'y, . (2.8)J J, /

For the theory used in this paper, ' ' the array 8"is de-fined by

WJk=Sk, +, k =i .

This choice of W generates the set of equations

(2.9)

Schrodinger equation (2.1) and summing over all chan-nels, one obtains

g (HJ + V~ E—)Qq——0 . (2.6)

J

A channel-couphng array P, which has the property * '

Pi2% +—(1,2) =+4+—(1,2) . (2.13)

The corresponding symmetry relations for the channelcomponents of Eq. (2.11) are

Pi24i*(1»)=Pi*(»1)=—+42*(1 2) (2.14a)

P12$2 (1~2)=fp (2, 1)=+$3 (1,2), (2.14b)

In order to calculate scattering amplitudes or bound-state eigenvalues, the Schrodinger wave function andCCA channel components must satisfy the symmetry re-lations with respect to particle exchange. In the presentcase of two identical particles, the spatial wave functioneither is even (+ ) or odd (—) with respect to particle ex-change. Let N be the number of two-cluster channels andP&2 the operator that exchanges the particles 1 and 2.The Schrodinger wave function must satisfy the symme-try relation

g [(E HI )g~ —V—J+'/I+ i]=0 . (2.10) Pi2it'iv(1») =IN~(»1) = —+iIt'i (I ». (2.14c)

The CCA channel components are those QJ. which satisfythe individual equations'

(E HJ )QJ.———VJ+'/~+i . (2.11)

This set of coupled equations can be expressed in vectornotation as

For systems consisting of different groups of identicalparticles, symmetry relations are defined within eachgroup. In this paper, the symmetry of the electrons willbe denoted by + (even) and —(odd), and the symmetryof the protons will be denoted by g (gerade) andu (ungerade) .

(EI—H)/=0,(H);~ =H;5J+5;+i 1 VJ,

(2.12a)

(2.12b)III. H URGER ADE, BOUNDARY CONDITION

(f);=/; . (2.12c)

Subject to scattering boundary conditions, the solutionof the CCA matrix equations yield exact transition ampli-tudes. If bound-state boundary conditions are imposed,one obtains either the Schrodinger eigenvalues and eigen-states, or spurious solutions which satisfy g.1tJ ——0.%Phile the n-particle scattering case, n &2, requires thesolution of a set of coupled equations like (2.12), abound-state calculation can always be performed bydirectly solving the single Schrodinger equation. Hence,for exact bound-state calculations, the CCA theory neednot be a practical formalism. ' However, the Schrodingerequation is exactly solvable for only two realistic systems,the hydrogen atom and the hydrogen molecular ion. Forsystems of two or more electrons, only approximate ener-gies can be obtained, e.g., by introducing trial wave func-tions into the Schrodinger equation. The minmaxtheorem guarantees that the approximate energy liesabove the exact eigenvalue. Approximate calculations canalso be carried out in a similar fashion with the CCAequations. Trial channel components can be constructedand approximate energies calculated. The advantages ofthe CCA approach have been relative computational easeand relatively accurate energy curves obtained using sim-ple trial components. ' ' The major disadvantages are aconsequence of the non-Hermiticity of the CCA Hamil-tonian: the minmax principle is not applicable and rnono-tonic convergence is not guaranteed as the basis set is in-creased.

The hydrogen molecule consists of two protons(a and b) and two electrons (1 and 2). The system can bepartitioned irito six labeled-particle two-cluster channels, '"

channel 1: (a, 1)+(b,2)channel 2: (a,2)+(b, 1 )

channel 3: (a, 1,2)+b H H+channel 4: (b, 1,2)+a

channel 5: (a,b, 1)+2 I H +channel 6: (a,b, 2)+1 2 +

Based on asymptotic energy considerations, the majorcontribution to the molecular ground state is expected tocome from the two H+ H channels, which are the onesretained in this analysis. The relevant channel Harniltoni-ans and external channel potentials are given by'

Hi Hp —1/r, i —1/r——b2

H2 ——Hp —I/r, g 1/ri, i, —V' = —1/r, 2 1/r~ i+ 1/8 + 1/r—iq,

V = —1Ir, i—1/rip2+ 1/R + 1Ir i2,

(3.1a)

(3.1b)

(3.1c)

(3.1d)

where Hp is now the kinetic energy operator of the twoelectrons, R is the internuclear separation, and, e.g., r ~ isthe distance between proton a and electron I. The CCAequations for the truncated two-channel system are givenby

3572 J. SHERTZER, E. BERNSTEIN, AND F. S. LEVIN

Hi —E p 2

=0H, —E (3.2)

The first and second excited states of the hydrogenmolecule are the triplet ungerade state X+ and the sing-let ungerade state 'X+. The triplet ungerade energy curvehas no minimum, and approaches —1.0 a.u. asymptotical-ly, corresponding to the separated atom picture of two hy-drogen atoms in the ground state. This behavior is pro-duced by a molecular orbital (MO) wave function con-structed from (ls) hydrogenic states which is odd withrespect to electron or proton exchange. " The singletungerade state has an energy minimum E (R,q=2.43ao)= —0.757, where R,q is the equilibrium valueof R. Asymptotically, the energy approaches —0.625a.u. , corresponding to a hydrogen atom in the groundstate ( —0.5 a.u. ) and one in the first excited state ( —0.125a.u.). The MO wave function for the singlet ungeradestate is constructed from a combination of (ls) and (2s)hydrogenic states; the resulting MO energy curve is wellbehaved if the total wave function is odd with respect toproton exchange and even with respect to electron ex-change.

In order to generate energy curves for the ungeradestates using the CCA formalism, the trial channel com-ponents must obey the proper symmetry relations. Thesewill be most easily exploited using elliptical coordinates x;and y;, where x; =(r„+rb; )/R and y; =(r„rb; )/R. —De-fining exchange operators P,b for the protons and P12 forthe electrons, the ungerade components obey

that vanish when either yi or y2, or both are zero. In nei-ther case was the resulting approximate ungerade energycurve well behaved for R approaching zero.

A solution to this problem is obtained through exam-ination of (3.5). We assume that the exact $1 or our ap-proximate pl is finite and normalizable, so that one canform the inner product of both sides of (3.5) with $1.This leads to

E(R)&$1 ~ fl )~(1/R)&$1~

P,bg—l ), R ~0 . (3.7)

In the limit that R —+0, &pl~

P,bt/il )—must be greaterthan zero in order to guarantee that the energy approaches+ oo. Requiring that &pl vanish when either (or both) elec-tron is on the midplane is not sufficient to guarantee that&pl ~

P,baal )—is positive. It therefore follows that in or-der to generate physical ungerade energy curves for H2,one must impose the additional boundary condition

& $1 ~P,bg"1 ) )—0, R ~0 . (3.8)

Just as with the H2+ case, we tested the validity of thisconclusion using simple trial components. A simple an-satz which obeys the symmetry relations for the tripletungerade state and satisfies the boundary condition (3.8) is

=&@11~1I @1&+&01II'21 P.bf—l& .

For small 8, the dominant term on the right-hand sideof (3.6) is the proton-proton interaction in the matrix ele-ment of the external channel potential:

Pabst 1(x1 3 1 x2 3 2) 42(x1 3 1 x2 3'2)

Pab02(x 1 3 I x2 3 2) 41(x 1 yl x2 3 2) .

(3.3a)

(3.3b)

Similarly, the symmetry of the singlet (+ ) and triplet( —) states is defined via the electron exchange operator

=(yl —y»exp[ —R (xl+y1 )/2]

Q exp[ —R (x2 —y2)/2],——(yl —y2)exp[ —R (x2+y2)/2]

&& exp[ —R (xl —yl )/2] .

(3.9a)

(3.9b)

1241 (xl yl x2 3 2) —f2 (xl yl x2 3 2)

P12 I('2 (x 1 3 1 x 2 3 2 ) —41 (x 1 3 I x2 3 2 )

(3.4a)

(3.4b)

These relations must be obeyed by both exact and ap-proximate components. The use of these conditions aloneis insufficient to produce well-behaved ungerade energycurves. The analysis of the H2+ ungerade CCA equa-tions in Refs. 16—19 led to the imposition of the addition-al boundary condition that requires individual channelcomponents to vanish on the midplane in the limit as theinternuclear separation R approaches zero: g,". (x,y=0)~0, as R~O. However, this type of boundary con-dition cannot be straightforwardly applied to the Hz case,as indicated by inspecting the relevant CCA wave-function component equation. The CCA equation for theH2 ungerade component pl (singlet or triplet) is given by

Ef1(xl 3 1 x2 3 2)

~101(x1 yl x2 3 2 )+ ~ (t'2(xl yl x2 3 2)

and since the channel component is a function of both y]and y2, (3.5) as it stands yields no conclusion concerningthe behavior on the internuclear midplane. Indeed, wehave tested this by constructing approximate components

We used Eqs. (3.9) because these channel componentsasymptotically approach hydrogenic 1s states, which isconsistent with a separated atoms picture. The approxi-mate ungerade energy corresponding to these componentsis obtained by substituting (3.9) into (3.6). As in previousCCA calculations, there are fewer integrals to evaluate

I I I I I I I I I I

-0I5

~ ~~ ~

f f f I I f I I I I I I f I I I I

3 4 6 8a (aalu Ref,dh)

FIG. 1. CCA (line) and Kolos-Wolniewicz (dotted line) ener-

gy curves for the H2 ungerade triplet states.

31 BOUNDARY CONDITIONS AND CHANNEL-COUPLING-ARRAY. . . 3573

than in the case where a variational wave function equalto the sum of (3.9a) and (3.9b) is used in the Rayleigh-Ritz formulation. The resulting CCA triplet, ungeradeenergy curve is shown in Fig. 1, where it is compared witha much more exact result obtained by Kolos and Wol-niewicz. The CCA curve is physically well behaved forall R, thus confirming the validity of our new boundarycondition, Eq. (3.8). The maximum difference betweenthe exact and the CCA curves of 0.04 a.u. is remarkablysmall given the crudity of the ansatz (3.9). The fact thatthe CCA curve lies lower than the exact one is a conse-quence of the lack of a minmax theorem for the non-Hermitian CCA equations.

IV. DISCUSSION

This paper has had a threefold purpose: first, to under-stand the origin of the unphysical ungerade energy curves

obtained in previous CCA calculations on the H2 mole-cule "' second, to show how to eliminate these curves;and third, to demonstrate that the elimination procedureleads to physically well-behaved results. The first twogoals were accomplished by establishing that the CCAungerade channel components must obey a boundary con-dition not previously, employed. The third purpose wasachieved by constructing a simple ansatz for the lowest-lying triplet H2 ungerade channel components that satis-fied the new condition, and then using them in the CCAequations to generate a well-behaved energy curve.

ACKNOW'LEDGE MENTS

This work is part of the Ph.D. thesis of J. Shertzer.Portions of it form the Sc.B. thesis of E. Bernstein. Thisresearch has been supported in part by the U. S. Depart-ment of Energy. We gratefully acknowledge computingfacilities made available by Brown University.

*Present address: Physics Department, Holy Cross College,Worcester, MA 01610.

Present address: Physics Department, Yale University, NewHaven, CT 06511.

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