PHYSICAL REVIEW A VOLUME 45, NUMBER 9 1 MAY 1992

Molecular final states after P decay of tritium-substituted molecules

T. A. Claxton, * S. Schafroth, and P. F. MeierPhysik-Institut der Uniuersitat Zu'rich-Irchel, CH-805? Zurich, Switzerland

(Received 30 October 1991)

Calculations on the ground state of HT and CH3T and the excited states of HHe+ and CH3He+ havebeen carried out within the sudden approximation to test alternative approaches to existing problems,with the ultimate aim of studying the effects of a molecular environment on the detailed energetics of tri-tium P decay. By using an appropriate projection operator most of the transition probability has beenaccounted for. Commonly used sum rules are shown to have limited utility. The inaccuracies intro-duced by describing an excited state by a single spin-adapted configuration are discussed. The problemof determining transition probabilities from configuration-interaction wave functions is highlighted andsome solutions are proposed.

PACS number(s): 31.20.Di, 31.20.Ej, 31.20.Tz, 31.50.+w

I. INTRODUCTION

Experiments designed to determine the rest mass m, ofthe electron antineutrino have recently been reviewed [1].In some experiments [2—5] the shape of the P spectrumclose to the end point is measured; this spectrum arisesfrom the decay of tritiated molecules, denoted by RT, ac-cording to the reaction

RT~(RHe+)*+e +v, .

Here v, is the electron antineutrino and RHe+ is themolecular complex remaining after the P electron hasbeen ejected from the nucleus of the tritium atom. It isassumed (because of the very short P-decay time, about10 ' s} that R He+ has the same geometry as R T. Thisassumption is commonly known as the sudden approxi-mation. Although RT is reasonably expected to be in itsground state the energy of the ejected particles, hence theshape of the P spectrum close to the end-point region,will be affected by the final state of RHe+ from energyconservation laws. The asterisk in (RHe+)' emphasizesthat RHe+ could be in one of its excited states. The Pdecay is a multichannel process that will be dominated byan averaged energy of the R He+ final states. IfbE„=E„(RHe+ )

—Eo(R T), that is, the energydifference between the parent RT and child R He+ mole-cules on just one channel, the averaged energy differenceis hE, where each AE„ is weighted according to the prob-ability 8'0„ that RT wi11 decay and leave RHe+ in thenth excited state. The diagram which graphically relatesE„(RHe+) and Wo„will be called the excitation spec-trum. An example is given in Sec. VIII.

Energy is transferred to (b,E negative) or subtractedfrom (b,E positive) the ejected particles. The current lev-el of reliability of the experiment of Fritschiet al. [5], which uses a tritiated OTS [CH3(CHz —CHT—CHT}3(CH2)7SiC13] source, is now sufficiently high towarrant the calculation of Eo(RT), Eo(RHe+), and bEto an accuracy better than 1 eV. The fitting process ofthe experimental data also requires an accurate value of

Woo. The excitation spectrum, E„(RHe+) together withWp, is not so critical [6] but it is generally accepted that

Wp should be as close to 1.0 as possible; an accept-able target is for a value larger than 0.995. Previous cal-culations on various tritiated organic molecules [7,8] in-dicate that the model system, isopropyl tritium,(CH3)2CHT, is adequate for this purpose.

The previous calculations on isopropyl derivativesR =(CH3)2CH [7,9] used methods which concentratedon trying to estimate the excited states of RHe+ fromconfigurations constructed from the self-consistent-field(SCF) virtual orbitals or soine judicious unitary transfor-mation of the virtual orbitals. It was calculated that theprobability that R He+ will end up in its ground state orone of its singly excited configurations is about 95% andover a 3% chance that the configuration will be doublyexcited. The transition probability not accounted for hasbeen associated [9] with contributions from triply (orhigher) excited configurations, ionization, neglect of elec-tron correlation, and an incompleteness in the basis set.

In this paper we describe an attempt to tackle theproblem from a slightly different point of view, usingsmaller molecules R =H and CH3. The case forR =(CH3)2CH will be dealt with elsewhere. The paper isbroadly divided into six sections.

(i) We show that the first-moment sum rule, alreadydiscussed by Kaplan, Smutnyi, and Smelov [7], has anequivalent form for wave functions that are not eigen-functions of the RT Hamiltonian operator.

(ii) We emphasize the inadequacy of using spin-adapted configurations to represent excited states.

(iii) We demonstrate how the total transition probabili-ty can be completely collected, that is, Q„Wo„=1, if thecalculations on RT and iVHe use the same finite basisset of atomic orbitals. In acldition this enables us to gen-erate easily corresponding orbitals [10] to overcomenonorthogonality problems [11,12].

(iv) We present results for calculations on the decay ofHT and CH3T using a triple-g plus polarization-functionbasis set and a double-g basis set, respectively [16]. Thesecalculations not only closely satisfy the first-moment sum

45 6209 Qc1992 The American Physical Society

6210 T. A. CLAXTON, S. SCHAFROTH, AND P. F. MEIER

rule but also collect essentially all the transition probabil-ity.

(v) We also discuss the usefulness of higher-momentsum rules [9].

(vi) Lastly, the problems of calculating the transitionprobability for configuration-interaction wave functionsare highlighted and some solutions are proposed.

~q&o(RT) ) = g(y„(R He+) ~(po(RT)) ~y„(RHe+ ) ), (10)

from which we easily obtain

( yo(R T)~H(R He+ ) ~yo(R T) )

= y((Ip„(RHe+)(q7O(RT))~

B„(RHe+)

II. THE FIRST-MOMENT SUM RULE = g%'0„@„(RHe+)

The first-moment sum rule which seems to have beenintroduced by Kaplan, Smutnyi, and Smelov [7] to pro-vide an easy way of calculating the average energydifference AE between the ground state of RT and thefinal states of RHe+ reads

but only if we introduce the further requirement that

(y„(RHe+)~H(RHe+)~p (RHe+)) =5„8„(RHe+) .

(12)

where

bH=H(R He+ )—H(RT)

(2)

(3)

If Eq. (11) is substituted into (8) the required equation (5)is reproduced. This may be regarded as a derivation ofthe sum rule for finite basis sets. However, the crucialcondition that must be fulfilled is expressed by Eq. (12).

III. REPRESENTATION OF EXCITED STATES

for p =2, 3, . . . .We will show that an equivalent form [Eq. (5)] of the

first-moment sum rule exists for suitably chosen wavefunctions y for the ground state of R T and the groundand excited states of R He+, which are expressed in termsof a finite basis set. Thus we intend to show that

To make the distinction quite clear we use the symbol 8to represent the expectation values

60(R T ) = ( po(R T )~H(R T ) yo(R T ) )

and

6 „(RHe+ ) = (y„(RHe ) ~II(R He+ ) ~p„(R He+ ) ), (7)

with 66„=6„(RHe+)—6'o(RT) and "Vo„ to representthe transition probability from yo(RT) to y„(RHe+).The left-hand side of (5) can therefore be written as

(Ipo(R T)~EH ~IpQ(R T) )

=(yo(RT)~IH(RHe+)~Ipo(RT)) —60(RT) . (8)

If the q&„(RHe+) form a complete set, described by thesame finite basis set of orbitals which is used to describeIpo(R T), and

(y„(RHe+)~y (RHe+)) =6„ (9)

we can expand yo(R T) in terms of q&„(RHe+ ) as follows:

is the difference of the Hamiltonian operators of the twomolecules, &o(RT) is the eigenfunction of lowest energyof H(RT), and Wo„ is the probability that the final stateof RHe+ will be n of energy E„(RHe+).

Higher-moment sum rules have also been proposed ofthe type

(4o(RT)~(bH) ~4O(RT)) = g (bE„ ) Wo„=bE~ (4)

One of the most convenient ways to construct excitedstates is to move one or more electrons from the SCF oc-cupied orbitals to the SCF virtual orbitals. It may benecessary to take linear combinations of suchconfigurations to preserve the spin state, in which casethey are called spin-adapted configurations (SAC's), andgiven the symbol y„(RHe+). Huzinaga and Arnau [13]argued that the effective Hamiltonian operator in theSCF method produced virtual orbitals which are moresuited to constructing spin-adapted configurations for anegatively charged molecule. A new operator was pro-posed which maintained the SCF occupied orbitals butmodified the virtual orbitals. The method was shown tobe useful in predicting the low-energy excited states ofmolecules of chemical interest. However, the methoduses different virtual orbitals depending upon which oc-cupied orbitals supply the excited electrons and so as ageneral method for obtaining the excitation spectrum,these SAC's will not in general be orthogonal. Calcula-tions that use the method of Huzinaga and Arnau to gen-erate the excited states of RHe+ will never satisfy thefirst-moment sum rule no matter how many states aregenerated.

The same arguments do not apply if the same virtualorbitals (SCF or modified) are used to construct all theSAC's, since orthogonality as in (9) will be preserved.However, SAC's will not in general satisfy the crucialcondition described by Eq. (12). The Ip„(RHe+) whichsatisfy all the necessary conditions for the sum rule canbe derived from the g„(R He+) by taking suitable linearcombinations of them. %'e set up the matrix M, with ele-ments

(R He+ ) IH(R He+ ) ly„(R He+ ) &

and obtain the coefficients of the linear combinationsfrom the matrix C by means of the eigenvalue equationMC = DC where 6 is diagonal with elements D„„which is

the energy of the wave function Ip„(R He+ )

45 MOLECULAR FINAL STATES AFTER P DECAY OF TRITIUM-. . . 6211

C „y (RHe+ ).Of course the number of excited states that can be han-

dled by this procedure is small compared to the numberof SAC's which can be generated from the SCF orbitals.It is obviously necessary to invoke a very effective selec-tion procedure before embarking on such calculations.

h (RHe+)b =e (RHe+)b

h (RHe+)b =e,"(.RHe+)b;"

(16)

where, for example, the b,-' are the SCF virtual orbitals ofRHe+ and the a,' are the SCF occupied orbitals of RT.We form the projection operator

IV. CALCULATIONOF THE TRANSITION PROBABILITY

(18)

h (R T)a,'= e,'(R T)a,.',h (RT)a,"=e,"(RT)a,',

(14)

The essential expansion we must discuss is illustratedin Eq. (10). This is only valid if the finite basis set used togenerate y0(RT) is also used to construct the y„(RHe+).This means that the selected atomic basis set must be thesame for RT and R He+.

Fortunately, the doubling of the charge on the tritiumatom after P decay involves a new set of basis orbitalswhich do not seriously overlap with the original set in ei-ther double-g or triple-g with polarization functions.This prompted us to use both basis sets of the H atomand the He atom on the methylene protons. No problemswith linear dependence were experienced. The advantageof choosing this unusual basis set is that any wave func-tion of R T can be expressed in terms of the complete setof states of RHe+. If continuum states are expected tobe important they should also be built into the basis set.This has not been attempted here. Electron correlation isnot applicable here nor is any incompleteness of the basissets. This is not to say that electron correlation or basis-set completeness should be ignored, but they have noth-ing to do with the first-moment sum rule. In the case ofpropane, the proposed model system, where the numberof basis orbitals is 98 (double g) or 120 (triple g) this isclearly too large to construct suitable states y„(RHe+)which will satisfy the first-moment sum rule withoutsome drastic reduction in the number of virtual orbitals.Let us assume that y0(R T) is a single-determinantal wavefunction (SCF). There are 13 occupied orbitals. For adouble-g basis there will be over 10 singly excited spin-adapted configurations and this number is of the order of10 if double excitations from the ground state are al-lowed. This number is clearly too large to allow theconfigurations to be manipulated to obtain suitable excit-ed states of RHe+ which, as we have indicated above, isa necessary operation if the first-momentum sum rule isto be obeyed.

The number of configurations can be reduced by reduc-ing the number of virtual orbitals. The number of occu-pied orbitals depends only on the number of electrons inthe molecule but the number of virtual orbitals is largelychosen by the size of the basis set. Previous attemptshave been made [13] at reducing the number of virtualorbitals by excluding those which are ineffective, mainlyin the calculation of the correlation energy. Here our ob-jective is only to collect as much of the transition proba-bility as possible.

If h is the appropriate effective SCF Hamiltonianoperator other terms we need can be defined as follows:

and form the matrix M with elements M „=( b '~P

~

b„' ) .%'e obtain the coeScients C „ from the matrix C in theeigenvalue equation MC =AC where A is a diagonal ma-trix, A;; is the sum of the squares of the overlap that eachoccupied SCF orbital of RT makes with the new modifiedvirtual orbital (MVO), 8, of R He+, where 8,'=g C, b"It is clear that the maximum number of nonzero diagonalelements of A cannot be greater than the number of occu-pied orbitals. Only those virtual orbitals 8," with A, ; &0will contribute to the transition probability. The problemof satisfying the first-moment sum rule has now beendrastically reduced since the number of excited states isno longer dependent on the size of the basis set. In fact,the necessary configuration-interaction (CI) calculationwhich must be done is no larger than that if only aminimum basis of orbitals were used. In the case ofisopropyl tritium the number of modified virtual orbitalsis only 13 for either the double-g or triple-g basis set.

No attempt has been made to include that part of thevirtual space which is important in constructing the"true" excited states. It follows that satisfying the first-moment sum rule is a necessary but not sufFicient cri-terion. On the other hand, and it is worth repeating, thecalculations described below have seriously attempted tosatisfy the requirements of the first-moment sum rule.A11 other calculations have not. In this sense, if improve-ments should be necessary, these calculations may be re-garded as the starting points. But although the antineu-trino experiment (see Introduction) is relatively insensi-tive to the finer details of the excitation spectrum it is stillnecessary to use good quality basis sets to get reliablevalues for E0(RT), E0(RHe ), and bE. Improvementsin the excitation spectrum may prove to be unnecessary.

V. TRANSITION PROBABILITYAND CI WAVE FUNCTIONS

The transition probability between two single-determinantal configurations is given by the square of thedeterminant of the overlap between the molecular orbit-als (MO's) in each configuration. If the MO's of eachconfiguration are not orthogonal to each other the calcu-lation becomes far from trivial. In such cases if there aren terms in each of the CI wave functions, n /2 such eval-uations have to be made. On the other hand, if the sameMO's are used for all the configurations of both mole-cules RT and R He+ the transition probabilities betweeneach pair of configurations is either one (if they are thesame) or zero. The calculation is now just the sum of theproducts of the coefFicients of identical configurations inthe two CI wave functions, just n products. The

6212 T. A. CLAXTON, S. SCHAFROTH, AND P. F. MEIER 45

difference between these two situations is so enormousthat every effort has been directed at obtaining suitableCI wave functions for both RT and R He+ using the sameset of MO's.

It is well known that complete CI calculations giveground-state wave functions which are independent ofthe MO's (as long as the atomic basis orbitals are thesame in all cases). Any unitary transformation can be ap-plied to the SCF MO's without altering the result. Thatmeans that the SCF MO's from RT could be used as thebasis for a complete CI calculation of RHe+ and RTfrom which %'00 could be easily calculated. The sameanswer must be obtained if the R He+ SCF MO's wereused. Unfortunately, such calculations are rarely possi-ble. But at least it does indicate a way forward.

VI. HIGHER-MOMENT SUM RULES

The accuracy of the spectrum of excited states can beassessed by using higher-moment sum rules than the first.The first-moment sum rules does not test the distributionof R He+ energy states but the higher-moment sum rulesdo. Unfortunately when p) 1 we have concluded that(@o(RT)i(&H) i@o(RT)) will only reproduce the sum

g„hb %Vo„ in that form if 4o(R T) is an eigenfunction ofH(RT). Although comParison of g„BEDWo„ from thecalculated spectra of states is useful, there does not seemto be any point in evaluating

(+o(RT)~(bH)~ 40(RT)),

particularly if @o(RT) is assumed to be a SCF wave func-tion since we cannot reproduce the expressionsg„b,A'"'No„, even for p =2. Martin and Cohen [14] haveused negative moments (p (0) to discuss the electronshakeoff probabilities for tritium P decay in the T2 mole-cule.

VII. TRITIUM HYDRIDE, HT

This is the simplest molecule that we can discuss. Ithas been the subject of a number of theoretical investiga-tions within the sudden approximation. We have used atriple-g basis set including polarization functions. Thesame basis set was used for both HT and HHe+ by in-cluding both hydrogen and helium basis sets at bothatoms. The calculations were performed using the pro-gram package MELD [15]. The SCF values are given inTable I where the sum rule has been used to obtain hBand the average excitation energy AD* of the child mole-cule defined by

TABLE I. Theoretical data for SCF wave function for HTusing a triple-g basis set with polarization functions. All ener-gies are in hartrees.

TABLE II ~ Theoretical results for the SCF wave functionsfor HT and HHe+ including the states of HHe+ obtained fromone virtual orbital. All energies are in hartrees.

State Energy

—2.940 816—1.564 644—0.215 908

b 6 = —1.1023526@ =2. 131 332A6* =0.705 466

bb*=bo(RT) —Co(RHe+) —b8 . (20)

TABLE III. Results of sudden approximation calculationsusing the SCF ground state of HT and the ground and excitedstates of HHe+. All energies are in hartrees. NSAC meansnumber of spin-adapted configurations.

NSAC 38 145

As described in Sec. IV we can find a unitary transforma-tion of the virtual orbitals of RHe+ such that the wholetransition probability is collected in the same number ofmodified virtual orbitals as there are occupied orbitals.Interestingly in the case of HT this means only onemodified virtual orbital. Only two states of HHe+ can beformed from one virtual orbital and so the spectrum ofstates for HHe is fully described by just these twostates. The results are given in Table II.

b, A' is the same as calculated previously because thecalculation is so designed to satisfy the sum rule if allterms are included. There is every possibility that h6may be small because there are only two excited states.However, as shown previously, there is no point in calcu-lating it except from the excitation spectrum. Our in-terest here lies in whether b, 8 is sensitive to change asthe number of excited states increases. If it is, then itmay be useful to compare b, C which have been obtainedfrom different calculated excitation spectra. 'No„ is thetransition probability that the SCF ground state makeswith the nth multideterminantal state of HHe+. It isclear that this calculation does not attempt to give excit-ed states of a realistic energy, it only seeks to satisfy thefirst-moment sum rule. This is obviously not sufticient.This is easily remedied here since with only two electronsthe number of excited spin-adapted configurations whichcan be generated is relatively small, even with triple-gbasis sets plus polarization functions. However, to follownormal practice in this type of work we will also considerincomplete calculations, that is, a configuration-interaction calculation where some of the configurationsare excluded by some criterion. All SCF virtual orbitalsof HHe+ were used, but only those states which overlap)0.000001 with the SCF ground state of HT were

c~o {HT)60(HHe+)

~00

—1.132 998—2.929 756—1.102 352

= 0.694406= 0.620987

Do(HHe )

%00g„"Ko„

—2.941 3110.613 7630.980 0

—1.137 3470.676 3482.468 539

—2.968 1010.601 8711.000 0

—1.102 3520.732 7512.572 530

45 MOLECULAR FINAL STATES AFTER P DECAY OF TRITIUM-. . . 6213

0(HT)0&&He

= —1.170084= —2.968 101=0.588 2

TABLE IV. Results of complete CI calculations on HT andHHe+. All energies are in hartrees.

pro(R T)= gc; X, (R T), (21)

It is formally very simple but not easy to accomplish. Ify are the SAC's, typically generated from SCF canonicalorbitals, we can write the ground-state wave functions as

selected to give 38 SAC's. The 145-SAC calculation in-cludes all configurations. The ground state of HT is stillrepresented by just the SCF wave function (one deter-minant) whereas to satisfy the sum rule all HHe+ statesare represented by linear combinations of SAC's, so thatEq. (5) is true (see Sec. III). The results are collected inTable III.

Comparison of the two calculations in Table III showsthat the selection criterion was not a good one from anenergy point of view. 'Ntm is consistently less than theSCF value (Table I). The 38 states selected do not im-

prove the energy over that obtained for just one modifiedvirtual orbital clearly showing that criteria used to max-imize the transition probability value do not simultane-ously assist the selection of states to improve the energy.The 145-state calculation in Table III is the best resultthat we can obtain using just the SCF wave function torepresent the ground state of HT. Significantly '%00 isthe lowest value of any of the calculations so far. Thefirst-moment sum rule is obeyed of course and althoughb, 6 is larger than the two-state calculation (using justone modified virtual orbital, Table II) as expected it is notthat different from the 38-state calculation which is thefirst indication that this value is not a sensitive enoughparameter to test the quality of the excitation spectrumof the transition HT —+(HHe+ )". The b 8" result can becompared with the value 0.72063, which has been calcu-lated from the numbers for the transition probabilitiesobtained with a higher-quality wave function by Fackleret al. [17].

The problem of calculating 'Moo, the transition proba-bility from the ground state of HT to the ground state ofHHe+, does not arise for a two-electron system sinceconfiguration-interaction calculations involving all singleand double replacements of the root function are com-plete and the result depends only on the basis set. Sincethe basis set is the same for HT and HHe+ we can useany orthogonal set of molecular orbitals made up fromthe basis set to obtain not only the energies of HT andHHe+ but also "lVOO. They are listed in Table IV.

It is significant that 'Noo, the best estimate for thechosen basis set, is the smallest value yet calculated. Thevalues of 'Noo) 0.6 arise from the fact that in those cal-culations the RT wave function yo(RT) was just a single-determinant SCF. This dominates the values of 'lVOO.

However, if g&0(RT) is now a linear combination ofSAC's, the coefficient of the SCF determinant will be re-duced tending to lower %'Oo. Of course the other termsnow contribute but not enough to compensate.

VIII. METHYL TRITIUM CH3T

As described in Sec. V the major problem is to calcu-late %Vo„, and Too in particular, from CI wave functions.

yo(R He+ ) = g c;" ' y;(R He+ ) .

Since, in general, (X;(RT)~X~(RHe+) )%5," the evalua-tion of (tpo(RT) ~qo(R He+ ) ), necessary for 'Moo, is nottrivial owing to the large number of terms in the expres-sion. As stated earlier, if n is the number of SAC s in theCI wave function there will be n /2 terms. Ifpo(RT)=go(RT) the calculation requires only n termsand can be calculated directly. This is why the problemhas not arisen before.

In principle we can use any set of SAC's X;(RX) toconstruct the CI wave functions of both RT and RHe+since both use the same basis set. Let X denote a nucleusof any charge which replaces T or He. However, if X isgiven a charge of 1 it is simply the RT molecule. The CIwave functions are

q)0(RT) = gd, X;(RX), (23)

yo(RHe+)= gd; 'X, (RX) (24)

where now the evaluation of (po(RT)~tpo(RHe+)) is

comparatively easy. The diSculty now is to know how toselect the terms in these CI expansions to approximatelyreproduce the wave functions of Eqs. (21) and (22), wherenormally just single and double replacements of the SCFroot function [single and double replacements in aconfiguration-interaction calculation (SDCI)] aresufficient. Not only do (23) and (24) have to introduceelectron correlation into the wave function they also haveto use many of the SAC's simply to reconstruct the densi-ty of the SCF wave function on either RT or R He+. %eknow that it can be done since if the complete set ofSAC's, X;(RX) for any X, are included, that is a completeCI calculation, go=go. However, such calculations arenormally impracticable.

Initially we tried to restrict the calculations to theSDCI type and studied the effect of varying the charge onX. These were done with a Slater-type orbital (STO) 36(three Gaussians) basis since these were initial test calcu-lations. The results are given in Table V.

The value for 'Woo, using SCF wave functions on bothRT and RHe+, is 0.6472, much larger than the full CIvalue of 0.6209 for the minimal STO 36 basis set as givenby the full CI value (any X can be chosen as indicated forthis calculation). The SDCI results using the same set ofSAC's for both molecules gives values which are clearlytoo high even in this very small basis set. It was original-ly conjectured that a wave function from an "intermedi-

6214 T. A. CLAXTON, S. SCHAFROTH, AND P. F. MEIER 45

TABLE V. STO 3G calculations on methyl tritium. All energies in hartrees. The MRSDCI calcula-tions have six reference configurations, the two electrons from the highest-energy SCF MO being pro-moted to each virtual orbital in turn.

Calculationtype

y;(RX)RT R He+ @o(RT) 6,(RHe+)

SCFSDCISDCISDCIMRSDCI,6MRSDCI, 6MRSDCI, 6SDTQCISDTQCISDTQCIFull CI

1.01.01.52.01.01.52.01.01.52.0X

2.01.01 ' 5

2.01.01.52.01.01.52.0X

0.64720.63280.64070.63530.62580.63420.63250.62110.62110.62130.6209

—39.704 926—39.775 456—39.768 167—39.750 251—39.776 014—39.771 163—39.757 368—39.777 342—39.777 252—39.776 696—39.777 366

—41.483 257—41.526 687—41.540 323—41.548 512—41.538 008—41.542 932—41.548 670—41.549 524—41.549 715—41.549 818—41.549 836

—1.1695

—1.1431

—1.1315

—1.1315

ate" molecule, RX where X has a charge 1.5, might be thebest compromise. Certainly if the sum Co(R T }+ho(RHe+) is used as a criterion it is the lowest valuefound when X has a charge 1.5. However, this gave thehighest result for 'Woo save for the SCF value.

It was decided to extend the CI wave functions to in-clude also triple and quadruple replacements (SDTQCI}of the reference function. The similarity to the full CIvalues, not just for 'Non but also the energies, was en-

couraging. It was postulated that the triply and quadru-ply excited SAC's were effective in helping reproduce thecorrect densities on each molecule rather than emphasiz-ing the need for that level of electron correlation. Thisled to the SDCI calculations which use more than onereference configuration. We will use the termMRSDCI, n to refer to multiple-reference (MR) CI calcu-lations which have n referenc|; functions from which allsingle and double replacements are made. In Table V wehave listed the results for calculations which used sixreference configurations obtained by promoting two elec-trons from the highest-energy SCF canonical orbital toeach of the virtual orbitals in turn, five functions and theSCF function. Although the energies are generally im-proved over the corresponding SDCI calculations theeffect on 'lVoo is disappointing.

The result was not significantly improved if all the pos-sible reference functions were used by promoting the pairof electrons from each occupied orbital in turn to each ofthe virtual orbitals, 26 reference functions in total. Thisresult could be rationalized by concluding that the triplyand quadruply excited SAC's with all spins unpairedwere significant in helping to reproduce the correct densi-ty on each molecule, RT and RHe+, from the "inter-mediate" molecular orbitals. If "intermediate" moleculeSCF single-deterrninantal wave function is expanded interms of, say, the RT SCF orbitals the importance of theunpaired electron configurations is apparent.

It became increasingly clear that the only way forwardwas to pursue the SDTQCI calculations, but at the sametime to look for ways of reducing their size so that it will

be possible to carry out calculations for larger basis setsand, more importantly, the model system isopropyl triti-um. It was already noted in Sec. IV that the electrondensity of the RT molecule can be expressed entirely bythe occupied orbitals of RHe+ and the same number ofcarefully selected modified virtual orbitals (MVO's), inthe case of CH3T this is five occupied and five MVO's. Asimilar procedure was used but the matrix that was diag-onalized was simply the sum of the SCF density matricesof RT and RHe+, that is, p„T+p~H+. The result is

TABLE VI. Calculations on methyl tritium using a double-g basis set. All energies are in hartrees. F, 0, and V denote frozen, oc-cupied, and virtual orbitals.

SDCISDTQCISDTQCISDTQCISDTQCISDTQCISDTQCIFull CI

F 01—52—5

2—5

2—5

2—5

2 —5

2 —5

1 —5

6—206—106—126—136—146—15

6-10,16-206—10

@o(RT)

—40.282 354 9—40.220 0140—40.246 098 3—40.252 820 9—40.260 191 3—40.260 766 3—40.237 805 2—40.220 393 7

@0(RHe+)

—42. 103458 5—42.053 467 8—42.066 136 8—42.075 128 4—42.084 8109—42.085 367 2—42.068 621 3—42.053 782 5

—1.1634—1.1260—1.1168—1.1185—1 ~ 1202—1 ~ 1204—1.1240—1.1256

0.62200.57890.57900.57850.57810.57830.57850.5787

45 MOLECULAR FINAL STATES AFTER P DECAY OF TRITIUM-. . . 6215

TABLE VII. Results of theoretical calculations on methane derivatives (CH3T and CH3He+) using adouble-g basis set. NSAC means number of spin-adapted configurations. All energies are in hartrees.The first two rows refer to SCF calculations and the next two rows represent the results using 376SAC's for CH3He+. In the last two rows the energies determined with SDCI calculations taking 2926SAC s into account are given. The indicated value '%DO=0. 578 was calculated as described in the text.It is included here as it is the best value we have obtained by other means.

Molecule

CH3TCH3He+CH3TCH3He+CH3TCH3He+

Energy

—40. 173 350—42.002 936—40. 173 350—42.088 664—40.297 045—42. 117394

NSAC

1

1

1

37629262926

0.6167

0.5904

0.578

g„'lVO„

0.9983

—1.1288

—1.1288—l. 1403—l. 1141

0.7009

0.70090.75070.6795

analogous to obtaining the natural orbitals with the ei-genvalues being analogous to the occupation numbers.Only five MVO's have nonzero occupations and this setwas used in exactly the same way as the minimal basis setof orbitals above. The occupancy of one of the occupiedorbitals is so close to 2 that it was decided to "freeze" it,that is, the electrons did not participate in formingSAC's. The results are collected in Table VI. The firstthing to notice is that %Voo is lower for this double-g basisset than the STO 36 basis set. Secondly, the SDCI valueis even higher than the SCF value. Any calculationwhich includes triply and quadruply excited SAC's seemsto immediately lower "Voo to about 0.578 and is relativelyconstant. In order to use more virtual orbitals than thenumber of occupied orbitals the SCF density, pzz, withthe charge on X equal to 1.5, was fractionally added tothe above density, that is, p&T+p&H ++0.01p». Thiscompound density was diagonalized in the usual way andnow a further five modified virtual orbitals were obtained.This process can be continued if further modified virtualorbitals are required, for example,

pwT+pqH ++c g pox, (25)

0.06—

0.05—04D

0 04

0.03

nf0.02

0 I

0 40 60 eoExci tation energy (eV)

FIG. 1. Excitation spectrum (transition probability vs excita-tion energy) of the transition CH4~(CH3He+)* obtained froma 376-state CI calculation. The ground-state transition proba-bility (not plotted) is 0.590, the sum over all calculated probabil-ities is 0.998.

where pzz is for differently charged "intermediate" mol-

ecules and c is some small value we have arbitrarilychosen to be 0.01. Computer resources restricted our cal-culations to five occupied and ten virtual orbitals (labeledfrom 6 up to 15 Table VI). The most significant result inTable VI is the observation that "lVoo only varies in thefourth decimal place, as more MVO's are used to gen-erate SAC's. To check that the five ignored MVO's didnot contain a large contribution to 'Moo a calculation wascarried out using MVO's from 6 to 10 and from 16 to 20(these last five all have zero occupation numbers). Theresult is satisfactory. It is therefore concluded that %'oofor methyl tritium using a double-g basis set is 0.578.

The excitation spectrum, which closely satisfies thefirst-moment sum rule, was generated as follows. Firstly,the unitary transformation described in Sec. IV wasfound so that the five MVO's of CH3He+ could be ob-tained. Together with the five SCF occupied orbitals theSAC's were constructed which had the largest values of'No„In fact. the criterion used selected 376 most impor-tant SAC's. The SAC's were used to form excited statesof CH3He+ using the procedure described in Sec. III.The spectrum of states which has been calculated withthe help of a special purpose program [18] is diagram-matically displayed in Fig. 1. The spectrum closely obeysthe first-moment sum rule. The results are presented inTable VII where the total transition probability is seen tobe only deficient by 0.17%. Also the value of b @ is onlyin error by about 1%.

As described in Sec. I the analysis of the experimentaldata in the experiment of Fritschi et al. places differentemphasis on the relevant results (Table VII). The detailsof the excitation spectrum seem less important than thefact it should obey the first-moment sum rule.

Of more importance are the energies of CH3T andCH3He+. These have been calculated from their respec-tive SCF wave functions followed by standard SDCI cal-culations taking 2926 SAC's into account. It has notbeen possible to obtain '&00 directly from these wavefunctions but the methods discussed at the beginning ofthis section indicate that 'Noo has a value of 0.578 tothree decimal places.

6216 T. A. CLAXTON, S. SCHAFROTH, AND P. F. MEIER 45

IX. CONCLUSIONS

The excitation spectrum of both HHe+ and CH3He+has been calculated to satisfy the first-moment sum rule,a necessary but not a sufficient condition. The higher-moment sum rules have been shown to have a limitedutility. The transition probability 'Noo has been estimat-ed to be 0.578 for methyl tritium using a double-g basis.

We anticipate that the procedures developed here will besuitable to calculate 'N~ for isopropyl tritium.

ACKNOWLEDGMENTS

This work has partly been supported by the Swiss Na-tional Science Foundation. One of us (T.A.C.) is indebtedto the Research Board of Leicester University for makingavailable travel grants. We would like to thank E.Holzschuh and W. Kundig for helpful discussions.

*Permanent address: Dept. of Chemistry, The University,Leicester, LE1 7RH, UK.

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