Local energy calculations employing importance sampling: Positronium hydride wavefunctions as severe tests of the method

Volume j 1, numb3

L.Shfnrch 1975

CHEMICAL PHYSICS LETTERS

3

LOCAL ENERGY dLCiJLATIONS EMPLOYINGIMPORTANCESAMPLING: POS~RONILIMHYDRIQE WAVEFUNCTIONSAS SEVERETESTS OF THE METHODi S. EHRENSON Chemistry Department, Brookhaven National Laboratory, Received I8 November

Upton, New York 11973, USA

1974

Ground+tate positronium hydride wavefunctions are generated by least squares solutions of the Schriidinger equation and compered with variationd results. Effects on wavefunction quality of configuration space umpling and of low-lying virtual excited states are detailed.

Wavefunction

calculation

by local energy

methods

be described

by eigenfunctions

(trial

Functions,

TP)

the knportance sampling teclutiyue appears from recent results to be promising [I ,2] _This is despite expressed doubts as to the’accuracy ultimately attainable through what amounts to numerical integration with modest numbers of sample points. The latter question as it applies to any sampling techrrque is reviewed in recent papers examining simple atomic and molecular systems [3-S]. The present work undertook to further test the efficacy of importance sampling (which is a modification of the Monte Carlo method [6] ), and of the local energy method it&f, by what was conceived as zr controlled yet substantial change of the extensively studied molecular system, H,. Positronium hydride (NJ), composed of a proton, two electrons and a positron, bears obvious similarities to HZ, but differs importantly, for present purposes, in the Born-Oppenheimer dimensionality of its wavefunction and location and nature of excited states. Considerable attention of late has been paid to PsH, as regards generation by Ritz (variational) me?hods of a variety of correlated wavefunctions, with the main purpose being accurate calculation of the armihilative probabili-. ty of two photon events [7-lo]. R’esults of these

The primrny weighting (point .selection, WI?) function employed is the simplest closed-shell TF, i.e., the l-term (constant) function with C, = 0.5. In anal-.ogy to the procedure for Hz (ref. [2] ,.appendix r>,

studies

this weighting

employing

will be used in comparisons

with

the local

energy. r&ults to be presented here. .. The ground state for Psfi, of 2JS symmetry, may f Research.performed under the auspices of the U.S. Atomic Etfxgy Commission.

of the form, F(r n+e; 3rn+e; Pm+,+>x$@a),

(I)

with the spin function, x, factorable [7]. A suitable expansion of the spatial function F, remmrscent of the James and Coolidge function for Hf,, and of the form adopted in the studies of refs. [7-9] is,

Subscript

numbers

the positron,

refer to the electrons,

the nuclear

index

p indicates

is suppressed.

super-

scripts are integers implicitly dependent on expansion term, k, and $, the exponential term, may be either of the open- or closed&$ form, $” ‘=exp C--Ly(rl+r2) - Wr,, +rQ],

(3)

rj” = exp {-cwl --Pr2- Grp - brZp] + exp C-CL,-~z-~r,,-rr&I.

function

is CZS; in

(4)

prolate

spheroidal

cadirmtes ant!6), and 0,; the ~imuthal a&es of the electrons, relative to proton at the origin and positron along the z-axis, are immediately separatile and chosen from uniform distributions over the~intervals 0 to 2n from the pseudo-rdndom numbers (PRK);Rer 529

- V&me

,Tsble 1 1 Results for sin&term and closed-shell expanded Points’.

C.S. (-1.45,0.6108)b) 200

-.

.. :

-.

. 400

OS. (0.228,1)

0.0136 0.0215 Oi3668

hydride wavefunctions

0.0219 0.2747

b.c)

a)

‘4.term

1Merm

C.S. .’ (-0.193,C.6S18)b)

C.S. (0.604,0.6604)

0.0085

c.qosz

0.0064 O.iS38

0.0176 0.1025

0.0121 Cl.0136

-0.0004

-0.0027

0.0138 0.0232

-0.0090 0.0146 0.2401

c.0012 0.0140 0.0729

-0.0007 0.0063 0.0890

0.0068 0.0084 0.0222

0.0062 0.0092 0.3’36

0.0031 0.0075 0.1072

0.0031 0.0063 0.0948

0.0076 0.0031 0.0248

1600.

0.0039 0.0104 0.2963

0.0017 C.0035 0.1008.

0.0011 0.005 3 0.0964

0.0045 0.0055 0.0251

2000

0.0039 0.0053 0.2554

0.0021 0.0064

-0.0010 0.0022

0.0904

0.0972

12OC .,.

.

2400 :

‘.

b,

0.0099

0.0831

800

‘.

0.0220 0.0196 0.0212

0.0216 0.0966

..

:

positronimn

l-term

..

15 March 1975.

CHEhIICALPMYSICS~LEl-f~RS

31 I nuriibcr 3

0.0046 0.0041 0,025a

0.0040 0.0052

c.0017

0.0001

0.0050

0.0052

0.2749

0.9870

0.0014 0.0976

0.0017 0.0262

a) Each entry block cmtains in column orde:, all in hartrees; (~)~~-Evar~ti~~~, the mean deviations and the rms varionccs over three runsgith the specified number of configurstion$ Ah points generated from the l-term C.S. function with scales (o = 0.6108, p = 0;3054) as variationslly optimized IS]. Successively larger samples of a given run include the points of smaller samples; the same sample configurations apply to all TF runs. b) In order, the variationally obtained binding energy in eY, relative to Hand e*e-, and the snle factor [7]. Cl Exponents from ref. [9].

2nd Rez, for each configuration. Analytical integration over i and q for both electrons provides the mar: ginal probability

disribution

ue (for each configuratior~

for rp, ad

hence

its

val-

fm\l the PRN, R,). With

this value of r-n, choice of the electronic coordinates follows closely that outlined for H,. Both ,$and 77 are obtained from conditional probability cirstribution functions, independently for each electron be.cause of the separability of their coordinates in this YF, under generation’of the Pm!, in the order, qr, &, 72 and k2. It should be noted that a 9-di.mensional space.(2 electrons and the positron) is covered in the sampling; motion of the proton is ignqred. Other WF dimksed below are similsr!,y tre.ated. Variance minirnhation procedures follow ref. [2] .or alternatively,

a lagrangian n~armalization procedure [3] which under minimum constraint conditions (respectively, invtiance

of thr constoot

expansion

coefficient,

nor-

mabation) always,yielded tl!e sameenergyresults at convergence. Columns 1,3 and 4 of table 1 contain results for. the I-, 4- and 13.linear term closed-shell TF, employing the WF discussed above, optimally scaled (variationally) for i-is own energy, In all (4- and 13-term) cases, convergencein eneigy variance to 10m6 (AV/v) was rapid from either the variationalb..obtained, coefficietits sets; or those modestly changed (tested, by = + 10%). Th,: converged energies were uniformly better than those obtained with the variational coefficiints. In all respects; &th one possible exception,

CHEMICALPHYSICS LETTERS

Volume 31, number 3

expectations are met upon comparison of results witlti a given TF as well as among the different TF. That is, increased .sampling is accompanied by increased accuracy and diminished uncertainty. Neglecting small sample results, the variance is also seen to increase, toward bounds, with increase in sample size [2] . Increase in TF expansion size is accompanied by dramatic improvement in the energy variances. Only in comparisons of accuracy, relative to the variational results, is the general monotonic behavior noted above missing. The 13-term function cannot generafly apa proximate its variational energy any better than the l-term function can; substantial increase in sample size (up to 30 000 points) does not improve agreement. it should be noted, however, that this result is not unique. Poorer accuracy for the 12- than the 5term function for Hz [2] and for the 6-term than the 3-term He function [l] have been demonstrated at some sainpling levels. Not shown in table 1 are results employing modified scale factors in the primary WF. Fcr the.l- and 4-term TF, variation in scale ((Y)up to 0.814 and down to OS produce

results

which

are in all respects

15 !darch 19X

pling levels, the l-term open-shell function results rethose of the 4- and 13-term closed-shell TF both on the basis of variance and accuracy as well as wavefunction “goodness” (bind&@. Only in terms of uncertainty (scatter) do they at all resemble the 1-term closed-shell function. As before, irrconsequential effects accompany change in WF scale. The 4- and higher-term open-shell TF present 2 considerably different picture, however. in alI cases studied, the energy results are found to be-satisfactory at the zeroth-order level(wherethe variational coefficients are used), but all exhibit a signiIicant worsenitlg upon minimally constrained vari~ca minimization when the primary WF (Q = 0.6108) samphng is employed. Moreover, these functions me rekively unstable in the variance minimization procedure, often requiring mere cycles and damping to reach convergence and, occasionally, adjusted initki guesses for starting coefficients. Fig. 1 presents Mav and variance results for the 4ter-m function, employing ‘the same PRN sets as semble

statis-

tiAIy indistitwirltible. Q), values vary by only small fractions of the uncertainties, often, but not always, toward greater binding for the 1-term TF and lesser, binding for the 4-term TF upon increase in the scale. The 13-term TF varies more strongly (toward greater binding); above the 1200 point level, the improvement in accuracy with increase in scaleto 0.8 14 is on the order of the size of the reported uncertainties. Satisfactory results were also obtained upon other suspectedly severe modifications JG-I sampling. Choosing rndistributions from quasi-1s hydrogenic functions @I-, t e’, 5 = :--z) produces acceptable local energy estimates for these TF, iis does a rcduct;on (by up to a factor of 5 tested) in the number of distinct r,-values generated. Based on criteria of variance as well as energy [8], these closed-shell TF are all rather crude, compared to expansions of.comparable length for ground state HZ. The open-shell functions are considerably better; the single-term function is bound by 0.228 ev.[9]. The local energy results for this function appear in the second column of table 1. These are seen to be unremarkable in essentially every respect compared to the closed&eU’results. At all but the smallest sam-

211

Fig. I_ Results for the &term open-,%b..elI PsH f&ion ated under I-tern1 closed-shell function sampling-

gener-

531

..

.Volu.mc 31; number 3

CHEMICAL PKYSICS LETTERS.

previously, which ari typical both as regards other ENsets hnd larger.open-shell TF. Bejrosd the ob: vious k&tire of the variake minimization method, and the tentative implications of this result regarding the +erational utility of Such extraporation procedures as “doeXcient spoiling” [11,12j , other features of these results a;e noteworthy. Whereas the zerothorder energies are relatively ukffected by \VF scale -itia:ion, the converged results akstrongly dependent, Increases in scale are seen to be accompanied by substantial decreases (greater bindi!g)‘in WBv and incrases

. . :

in the rms variances.

1 ~IHWIW%~

is

Further,

if the variance

-energy (as it does for the l-term TF and for the exparided fupction employing variational coefficients), 41ay so skew the distribution of points as to unbalance the second moment, i.e., k2L This would itiply a functions, albeit

Table 2 Results of open- 5nd closed-sheU samp& for the d-term open&ll positronium hydride wavefukiiona) Pointi

400

800

subjectedto the additional constraint

of normalization and fixed lead term coefficient, much more, satisfactory energies (relative to the variationai results) a’re obtained*. Taken together, these results suggest either or both of ihe following phenomena may bz involved. The closed-shell WF (cr = 0.6108);while capable of proiidting satisfactory samplink for the first moment in

geater Xnsitivity to election in

correlation

part kn&Gtly,

than

1200

in the

In order to test these hypotheses, the following studies uiese undertaken.

2000

important difference betwecr. this and the closedshell funciion,‘empIoyed zissampling functions, resides in tlie interdependence (correlation) of electron positions in the open-shell function. This feaiure, which may be, easily verified fro-m examination of the’ strcn@y

complicates

the sampling

pra-

” Both conditions, of course, imply previous knowledge of ”

the.variationxl rcsulis, and hence wotild not be applicable in B ptioi iencration of.wav&unctions. tt has a useful'di-

-- agnotiic role h’ere, hdwever, a$ regards sampling efficiency _..aiitiekited

:332,

state?ffecis ar@y+is;

‘-, .; _’ : :.__‘.”

- ‘, ..:

:,

converged.

O.OOi6 .0.0087 0.0534 -0.0136

0.0339

0.0220 0.0218 0.0028

0.0027 0.0057 0.0409 -0.0039

0.0165 0.0063 0.0142 0.0108

-0.003c 0.004?.

0.0229 0.0082

~0.0020

0.0184 0.0038 Cl.0181

..

0.0229

0.0073 0.0800

-0.0110

0.0102

-0.0044

0.0137

-0.0021. 0.004;.

0.0189

0.0013 0.0038 0.061? -0.0043

0.0178 o.oqo2 0.0164 0.0130

0.0026 0.0031

0.0199 0.0002

0.0746 -0.0044

0.0172 0.0137

0.0020 0.00!9 0.0667

0.0186 0.0013 0.0173 0.0127

-0.002; 0.003; 0.0525 -0.0100 -0.0047 0.0050

0.0493 -0.0073

-:-% 0:049 1 -0.0095

0.0080 0.0253 0.0088 0.0137 0.0066 0 0266 0.0054 0.0196

0.0106 0.0253 0.0154

0.0197 0.0110 0.0256 0.0111

-0.0043

0.0027 0.0024 0.0622 -0.0042

0.0192 0.0009 0.0171 0.0132

3) Footnote a) of table 1 pertains. The variational binding energy of the 4-term opensheli TF is 0.487 eV. The additional entries, last in each column black, are tie expecCationvalue dif%recces (3

The l-term open-shell function of Neamtan et al. [9]. was emjloyed as sampling fd;7ction. The most

of @+,

“0” b)

2400

or lower than that of the ground state, or a mixture of bound and excited states, may be accessible !ocd energy minimization procedure.

converged

-0.009&

1600

Open-shell

YY b).

0.0531

cd open-shell in &ed-

Closed-shell

0,047c;

sheU functions (here entirely through rl2-expansion tenk). Alternatively, a virtual excited state of the same symmetry with energy variance comparable to

fo”

15 March 1975

:

z~>i~Ziz~~,,ii(&),

ing function

- &v

).The open-shell weight1

from ref. [9].

b, With the variational coefficients,

ref. [7].

cedures, as regards prog&rning analysis and generation tiine (per point). The results for the 4-term TF -for bqth sampling procedures are contained in table 2. It is imn-kdiately, apparent that the open-shell samp&.ng procedure: is no more satisfactory as regards the converged results thtiriwasthe simplei closed-.sheU procedrrre. Thi:: is despite ukifo&dy better distribu.;ions of mobikparticle configurations t!!ou& open-, shell sampling (lrelative t,o the variational TF), which .-was expgcted, :lnd may be verified bj the fiotential :

CHEMICAL

Volume 31, number 3 energy

expectation

difference

PHYSICS

entries

in table 2. In fact, the open-shell sampling results appear at Ieast as complicated as those obtained from ciosed-shell sampling by level mixing, if this is indeed the main reason for departure frcm the variational coefficient results. Uniformly smaller variances are reached in a more stable fashion in these runs; the energies do not dif-fer appreciably, however. Furthermore, less fluctuation is noted in the open-shell sampling results, both among Sets of equal numbers of points and with increase in ~a.n~ple&e. TILLSdecreased uncertainty argues against explanations invoking errant sampling or approaches to singularities uncontrolled by the WF. The open-shell PsH functions of 12 and more terms are known to have second-lowest roots (of their secular equations, in the variational calculation) which are on the order of only 0.03 I-! above the lowest. While the relative positions of these roots are not reported for the 4-term function, the separation is probabIy not much greater*. Closeness of the roots, plus the expectation that the ground state contains significant Gontfibutions of positron excited with respect to the rluCleus, as \vel.l as excited Virtual positronium contributions, both of which are well represented in the open-shell functions, support the level mixing hypothesis. Further support is found in the following observations; Scaling of the TF functions [7,13] at significant sampling levels (above 800 points) to achieve minimurn energy reveals a substantial imbalance (R 15%) in the virial ratios associated with the converged energies, which is consistent with level mixing. In contrast, the closed-shell and one-term openshell TF, with closed-shell WF, typically scale by 1.00 + 0.02.That the results obtained upon imposition of the normalization and constant lead-coefficient constraints are significantly closer tothe variational results likewise supports this contention, as do the previously observed effects acsompanying scaling. of the WE;. Recognitiori that the lowest tirtual excited-state

LETTERS

wavefunction of this symmetry is much more sensitive to the presence of explicit r expansion terms than the ground-state function P731 aLlows an even more convincing test of mixing. Upon removal of rpdependence from the TF polynomial expansions, through pendent

division by rs+f+u+u+‘v+x, leaktig terms deonly upon t Re appropriate t and 77combina-

tions, much more satisfactory local energy results are obtained. For the 4-term open-shell function so altered, the rms ((I+~- E,,) over the 6 sampling levers from 400 to 2400 points is olnly 0.0084 H, compared to 0.0168 H for the original rpcontaining expansion*. More=r, scale factors for the altered TF are found to be much cIoscr.tounity (rms w 1.OE), again sug gesting better local energy estimation and, indirectly that the original function suffers from appreciable variational state mixing. In summary, choice of the PsH system appears to have satisfactorily fulftiled its anticipated role as a severe test of importance sampling and of the local energy method for generation of wavefunctions. The power of the selection method, and simultaneouslythe expected insensitivity to degree of matching between trial and selection fun‘ctions are confiied. At the same time, the blurring effects of low-lying virtual excited statesof the same symmetry on ground-state energy and wavefunctio!l canlpasitian are aaplrically demonstrated. Preliminary efforts toward the obvious remedy of orthogonalizatinn, employing the importance sampled overlap matrix (Egp L&$~[2]) have proven.gznerally unsuccessful, both for the wellbehaved H, functions as well as in the present case [14] . Orthogonalization employing the overtap integral matrix, similar to a method recently suggested by Scott for SCF functions [IS] , is under current in-

vestigation, although whether the analysis and subsequent effort invoIved for functions contaiping explicit correlation terms (r12 and rp> would ultimately be justified is questionable.

*Both aver

results from

as for the

From inform&ion in :ef. [7], a separation of < 0.07 H would be expected for a three-term function containing rli- and $-terms

in the expansion,

WF witi

Q = Cl.6108 and

results ir.tabIe 2. Note es WC& the-ariadouI

ori&.!TF (-0.7679 K). It probably is slightly higher, which would make the altered

energy

*

the dosed-shell

the same 3 PRrU’ sets. One of these sets is not the =me

:

employed is that of the

function fit even better, approaching in quality the results for the TF reported in table 1.

533

Volume 3i, number 3

‘.

keferenbes [l]

+.L_ k,

CHEMICAL PHYSICS LETTERS

Chkn.

Phys. Letters 13 (1972)

504.’

.:gJ. SiEhrmon ond G,D, ilw-p, htern, J, Quantum Chem, _-: 7 (p73jio97. i3] hf.3. Llcyd and i&l. Delves, Intern. J. Quantum Chem. 3’[1969j 163_ :[‘1] T:A. Rburke and E.T. Stwart, Can. J. Phys. 45 (1957) 2755;46.(i968) lGO3. k [$I RiG. Clark aid E.T. Stewart, Mol. Phys. 21 (1971) 745; ‘. 22 (1971) 341. (61 H. Khhn, RAND Report No. RM-1237-AEC (The Rand Corp., Santa Monica, Cal., 1956) pp. 100-104,130143,unpublished.

-’

15 hlarch 1975

171 C.F. Leberla and DM. Schrad&, Phys. cev. 178 (19G9) 24, and re:‘erences therein. 181P.B. NW-& D-M. Schrader and C-F. Lcbeda, Phus. Rev. A9 (1974) 2248. [Y] 8.M. Nnmtan, G, Darcech and G. Oczkowski, Phys. Rev. 126 (1962) 193. [lO] S.K. Houston and RJ. Drachrnan, Phys. Re;. A7 (19?3) 819. [ll] H.~onroy,J.Chem.Phys.41(196~)1336. [12] H.H. Michels, 3. Chem. Phys. 44 (1966) 3834. [13] A. Fro’mw and G.G. Hall, J. Mol. Spectry. 7 (1961) 410. (141 S. Ehrenscln, unpublished work. [15] M.W. Scott, J. Chem. PI&. 60 (1974) 3875.