Hylleraas-configuration-interaction study of the 2 2

S ground state of neutral lithium

and the first five excited 2

S states

James S. Sims

National Institute of Standards and Technology, Gaithersburg, Maryland 20878-9957, USA

Stanley A. Hagstrom

Department of Chemistry, Indiana University, Bloomington, Indiana 47405, USA

Received 15 September 2009; published 19 November 2009

High-precision Hylleraas-configuration-interaction Hy-CImethod variational calculations are reported for

the 2 2

S ground state of neutral lithium. The nonrelativistic energy is calculated to be

−7.478 060 323 451 9 hartree, demonstrating that the Hy-CI technique is capable of sub-nanohartree accuracy

for three-electron systems. A Hylleraas expansion without linked products of odd powers of rij gives

−7.478 060 323 452 hartree, showing the relative unimportance of such terms for lithium at the nanohartree

level of accuracy. Hy-CI calculations are also reported for the 3 2

S, 4 2S, 5 2S, 6 2S, and 7 2S lithium excited

states.

DOI: 10.1103/PhysRevA.80.052507 PACS numbers: 31.15.p

I. INTRODUCTION

Hylleraas-configuration-interaction Hy-CI1 attempts

to speed up the convergence of configuration interaction CI

by directly introducing a correlation factor rmn

, odd, into

the wave function. In Hy-CI a wave function is constructed

as a linear combination of so-called configuration state functions CSFs,

 =

i

ciCSFi. 1

The CSFs are given by

CSFi =

j

dijrmn

i 

k=1

N

j

k

kj

k

k , 2

where  is the symmetry adaptation operator explained later

Eq. 4, N is the number of electrons, and j

k

kand j

k

k

denote functions of the space and spin coordinates of the kth

electron, respectively.

We have previously pointed out the close relationship between Hy-CI and Hylleraas Hymethod calculations for two

electrons 2. For three electrons, the methods diverge and

direct comparisons are not possible because products of rmn

raised to odd powers so-called odd-odd termsmay exist in

Hy expansions but not in the corresponding Hy-CI expansions Hy method even powers of rmn may be represented in

Hy-CI by higher spherical harmonics in the basis set 2.

However, for three electrons there should still be a close

relationship between the two methods provided the Hy expansion terms contain at most a single odd power of rmn in

the rmn products see Eq. 14.

Therefore, one issue addressed here is whether linked

terms By linked products we mean rmn products with a common index, like rij

mrikn . Unlinked terms like rijmrkln , no common

indexdo not occur in Li. like r12

m r13n with odd-odd powers

are necessary in Hy. This is an important point since it is the

odd-odd rmn products that lead to some very difficult integrals and which are presumably the reason there have been

no really accurate Hy calculations for atoms with more than

three electrons. Pipin and Bishop 3 achieved microhartree

accuracy in an Hy-CI calculation on lithium, showing that

such terms are unimportant at the microhartree level of accuracy. Since odd-odd linked products are by definition excluded in Hy-CI, the accuracy obtainable in Hy-CI without

them is also an issue. In this work we are able to achieve

sub-nanohartree accuracy for the Li ground state using

Hy-CI without such terms, and we further show that with the

Hy method these factors are only important at the subnanohartree level of accuracy. We believe that in general,

from an Hy perspective, the important linked products of rij

factors are those that have a single rij factor to an odd power

which can then easily be represented in the Hy-CI method by

a single rij factor times appropriate CI terms CSFs. While

we have demonstrated the unimportance of linked odd-odd

products of rij, it remains to be seen how important unlinked

products like r12r34 will turn out to be for systems with more

than three electrons.

II. METHOD OF CALCULATION

For three electrons, the Hy-CI wave function we use is

Li =

K

CKK, 3

where

K = 	rij

Ks=1

3


Ks

rsK = OasOL,ML

OS,MS	rij

Ks=1

3


KsrsK 4

denotes the Kth antisymmetrized spin and angularmomentum projected CSF. OL,ML and OS,MS are idempotent

orbital and spin angular-momentum projection operators of

the Löwdin type 4 for a state of total quantum numbers

PHYSICAL REVIEW A 80, 052507 2009

1050-2947/2009/805/0525078052507-1 ©2009 The American Physical Society
L,ML ,S,MS Russell-Saunders LScoupling is assumed.

K is a primitive spin product function for term K and

Ksrsrepresents the sth basis orbital in the Kth term. The

basis orbitals are taken to be un-normalized Slater-type orbitals. Oas is the idempotent antisymmetry projection operator. For three-electron doublet states there exist two linearly

independent primitive spin functions 1=	
	 and 2

=		
. It is possible to converge on the exact wave function

by employing only one primitive spin function, which for us

is 1=	
	. Similar observations have been made by Larsson 5 for Li. Cencek and Rychlewski 6 gave the general

proof that only one primitive spin function is needed to ensure convergence of eigenvalues to the exact root of the

Hamiltonian. As has been pointed out by King in his review

article on Li 7, the second spin function can be important

for computing precise values of properties other than the

energy, such as hyperfine coupling constants 5,8–10 and

spin-dependent expectation values like the Fermi contact

term 5.

The wave function given by Eq. 3is a linear combination of terms K, where the coefficients CK are those which

minimize the total energy, E, given by

E = 
H


 = KLCKCLHKLKLCKCLSKL

, 5

where

HKL = 
KHL;SKL = 
KL. 6

The nonrelativistic Hamiltonian H is in atomic unitsthe

atomic unit of energy is chosen as e4 /2= 1 hartree, where

=memN/me+mN

H =

i=1

N

Hi +

i
j

rij

−1. 7

Hi=Ti+Vi

, where Hi is a one-electron operator electron

iconsisting of a kinetic-energy part Ti=−1/2i

2 and a

nuclear attraction part Vi=−Z/ri

. The condition for the energy to be an extremum, E= 0, is the well-known matrix

eigenvalue secularequation



L

HKLCL =

L

SKLCL. 8

Solving this equation is equivalent to solving the

N-dimensional generalized eigenvalue problem

HC = SC, 9

where H and S have matrix elements HKL and SKL given by

Eq. 6. Hence the coefficients CK in Eq. 3are found by

solving the Kmax-dimensional generalized eigenvalue problem using the familiar inverse iteration method. Quadruple

precision and parallel processing were used throughout the

calculation. We show in the appendix how we handle the

antisymmetrization, spin, and angular-momentum projections involved in computing the Hamiltonian and overlap

matrix elements.

III. RESULTS AND DISCUSSION

A. Hy-CI calculation of the 2 2

S Li ground state

The CSFs given by Eq. 4can be written as

K = FKr1,r2,r3K 10

in terms of spatial and spin functions FKr1 ,r2 ,r3and K.

In this work we use only the first spin function K=1

=	
	 and the spatial part of the wave function FKr1 ,r2 ,r3

is given by a particular choice of rij factor and Hartree orbital product

FKr1,r2,r3= rij

Ks=1

3


Ksrs 11

where K is restricted, without loss of generality, to be either

0 or 1.

In Table I we list our best results for the lithium ground

state at various levels of truncation, where we use different

orbital exponents for s, p, d, and f orbitals, and we use different orbital exponents within each symmetry for K and L

shell orbitals. In column five are shown the energy improvements for each CSF term type added. In column one are

listed the basis orbitals that are used to generate the CSFs for

each block type in the order electron 1 	 spin, electron 2 


spin, electron 3 	 spin. For example, in the first line

1:9sKs means the basis orbitals are 1sKs through 9sKs orbitals

Ks denotes an orbital exponent appropriate for a K shell

electron. All of the listed basis orbitals are used to generate

all of the CSFs that are unique for this basis set selection.

The choice of terms is highly regular, there having been no

attempt to cut down on the number of terms. The number of

unique terms CSFsin a block can be easily computed from

the listed basis orbitals. Since the orbital exponent for an L

shell orbital is different from a K shell orbital and the K shell

pair of electrons have different spins, the number of terms is

just the product of the number of different orbitals for each

electron times 4 for 1,r12 ,r13 ,r23. For example, for 1: 9sKs

1:9sKt 1:9sLs R there are nine sKs orbitals, nine sKt orbitals,

and nine sLs orbitals, so the total number of CSF terms of

type sKs sKt sLs R becomes 9994= 2916.

The wave function in Table I can be thought of in the

following way. The first three term types are essentially an

Li+ core plus valence electron CI picture, with each orbital

product multiplied by 
1,r12 ,r13 ,r23, what one might call an

“explicitly correlated CI” picture. ffsR terms on the next to

last line in the table completes this picture. The next four

CSF blocks after the first threeincorporate K-L intershell

CI correlation. Beyond this point CI is not much help in

coming up with appropriate orbital products. For the remaining term types we use as our guide the expansion of likely

important Hy rij products see Eq. 14 to see what types of

Hy-CI terms arise. For example, consider the Hy term s1s2s3

r12

2 r132 only even power products need be considered. Since

rij

2 = ri2 + rj2 − 2ri

rj cos ij 12

and

JAMES S. SIMS AND STANLEY A. HAGSTROM PHYSICAL REVIEW A 80, 052507 2009

052507-2
cos ij = 4/3 

m=−1

m=+1

Y1

m

i,iY1

mj,j, 13

expansion of s1s2s3 r12

2 r132 gives rise to sss terms, psp terms,

pps terms, and p1p2p1p3terms. Expanding this last

term gives spp and dpp terms. Similarly, r12

2 r232 gives rise to

the new term pdp and r13

2 r232 to ppd. None of the dpp, pdp,

or ppd products are obvious CI orbital promotions. Of course

in the same way the earlier terms in Table I arise from suitable Hy terms, e.g., r12

2 leads to pps, r132 to psp, r124 to dds,

and so on. Indeed all orbital products have an Hy equivalent

rij product term. Of course all such orbital products are only

candidate terms and must be tested individually for importance essentially by trial and error.

In our early calculations we did in fact overlook the

dppR, pdpR, and dppR terms, as did previous workers,

yet were still able to get surprisinglyto within 10 nanohartree of nanohartree accuracy. After adding ppdR, pdpR,

and dppR terms we ran into another hard energy limit,

preventing us from getting closer than 2 nanohartree of nanohartree accuracy. The problem in this case turned out to be

the absence of the ddd and fpd, pfd,...term types shown

in Table I. ddd arises from the expansion of r12

4 r134 while fpd

comes from the expansion of r12

2 r134 and so on for the other

permuted forms of fpd. As can be seen, maybe only two or

three of the fpdR term types are in fact needed. Finally, we

added ffsR to acknowledge that f f 
1,r12 is of importance in the description of the Li+ ion at the nanohartree

accuracy level. fsf R- and sff R-type terms were also

tested and found not to be important. Also no instance of an

orbital product including g orbitals was found to be important, as for example, ggsR. The last set of expansion terms

in Table I introduces additional correlation into the K shell in

the amount of 0.836 nanohartree, indicating a less than perfect description of the sssR part of the wave function. We

attribute this to be a result of not having sufficient flexibility

in the s-orbital basis sets, arising from the limit of just eight

orbital exponents permitted for all orbitals, this limit imposed by a not easily remedied problem in the integral codes.

Orbital exponent optimization was rather carefully done

at the N= 10 044 expansion level by hand no analytical derivatives. The sK orbital exponent splitting shown gained

only 1 nanohartree. Due to limitations in the integral codes it

was not possible to give every orbital type its own orbital

exponent. Although with our sequential codes we could work

around this problem, it was not practicable to implement

similar changes in our parallel codes without major changes

to the code itself. In a purely practical sense this was not a

serious problem since the dependence of the final energy on

the orbital exponents turned out to be very flat. As has been

pointed out by Pulchalski and Pachucki 12 and Yan,

Nörtershäuser, and Drake 13, the number and choice of the

nonlinear parameters is very important at the nanohartree

level. This is true for both the Hy method and Hy-CI.

At this point we decided to call a halt to the 2 2

S calculations, having reached our goal of sub-nanohartree accuracy,

because to do substantially better would involve substantial

TABLE I. Hy-CI calculations of the Li ground-state energy in hartrees. In the table

R=
1,r12,r13,r23, N is the number of terms added, and Ntot is the cumulative number of terms.

Terms added N Ntot ENtotin hartreesa

,

b −E in nanohartrees

1:9sKs 1:9sKt 1:9sLs R 2916 2916 −7.477 634 670 861

2:9pKp 2:9pKp 1:7sLs R 1792 4708 −7.478 059 148 015 424477.

3:8dKd 3:8dKd 1:6sL R 864 5572 −7.478 059 958 638 810.623

2:8pKp 1:7sKt 2:8pLp R 1372 6944 −7.478 060 162 571 203.933

1:7sKs 2:8pKp 2:8pLp R 1372 8316 −7.478 060 224 321 61.750

3:8dKd 1:6sKt 3:8dLp R 864 9180 −7.478 060 245 033 20.712

1:6sKs 3:8dKd 3:8dLp R 864 10044 −7.478 060 252 297 7.264

2:7pKp 2:7pKp 3:8dLp R 864 10908 −7.478 060 314 870 62.573

2:7pKp 3:8dKd 2:7pLp R 864 11772 −7.478 060 317 298 2.428

3:8dKd 2:7pKp 2:7pLp R 864 12636 −7.478 060 318 157 0.859

3:8dKd 3:8dKd 3:8dLp R 864 13500 −7.478 060 320 454 2.297

4:7fKf 2:5pKp 3:6dLp R 256 13756 −7.478 060 321 720 1.266

2:5pKp 4:7fKf 3:6dLp R 256 14012 −7.478 060 321 979 0.259

4:7fKf 3:6dKd 2:5pLp R 256 14268 −7.478 060 322 140 0.161

2:5pKp 3:6dKd 4:7fLp R 256 14524 −7.478 060 322 183 0.043

3:6dKd 4:7fKf 2:5pLp R 256 14780 −7.478 060 322 236 0.053

3:6dKd 2:5pKp 4:7fLp R 256 15036 −7.478 060 322 245 0.009

4:9fKf 4:9fKf 1:6sLs R 864 15900 −7.478 060 322 637 0.392

1:6sKa 1:6sKa 1:6sLs R 864 16764 −7.478 060 323 451 9 0.815

Estimated exact 11 −7.478 060 323 910 1032

a

K-shell orbital exponents are Ks= 4.40, Kt= 3.60, Kp= 4.65, Ka= 7.65, Kd= 5.30, and Kf = 5.50. b

L-shell orbital exponents are Ls= 1.05 and Lp= 1.30.

HYLLERAAS-CONFIGURATION-INTERACTION STUDY OF ... PHYSICAL REVIEW A 80, 052507 2009

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modifications to our codes to accommodate more orbital exponents, analytical derivatives, and mechanisms for shortening the expansion, like putting a limit on the sum of the

powers of ri as was done in the Hy method work recall that

inverse iteration is an ON3process.

In Table II we compare our results with previous

Li ground-state calculations. We have added two results to

the table to facilitate this comparison. One is an s, p rij

energy of E8316=−7.478 059 573 327 hartree obtained

using term types 1,2,4,5, and 19 in Table I. Notice that microhartree accuracy is achievable with just s and p orbitals.

For comparison, Pestka and Woźnicki 17 required a basis

of s, p, d, and f and powers of rij through 3 to get

−7.478 060 1 hartree. Pipin and Bishop 3 reported significant contributions from ffs and ggs, but we believe this just

reflects an inadequate ss-, pp-, and dd-product basis.

The second result entered into Table II is an s, p,d rij

energy limit obtained using terms 1–11 and 19

from Table I which gave an energy of E14 364

=−7.478 060 321 379 hartree, no rij powers higher than the

first power. Adding f orbitals we obtained an s, p,d, f rij

energy of E16 764=−7.478 060 323 451 9 hartree. This is

slightly better than the explicitly correlated Gaussian ECG

result of Pachucki and Komasa 19 and better than all but

the most accurate Hy calculations of Pulchalski and Pachucki 12, Yan, Nörtershäuser, and Drake 13, and Pulchalski, Kedziera, and Pachucki 11, coming to within 0.56

nanohartree of the estimated exact energy of Li.

B. Hy calculation of the 2 2S Li ground state

Next we used the code of Pulchalski and Pachucki 12 to

determine the importance of odd-odd and odd-odd-odd

power rijrik and rijrikrkj terms as these are the only ones for

which there is no correspondence with Hy-CI terms. The

Pulchalski and Pachucki method of generating their wave

functions is the multiple basis set method originally developed by Yan and Drake 21. The key idea is to start with a

fully correlated Hy variational basis set of the form

r23

n1r31n2r12n3r1n4r2n5r3n6e−	1r1−
2r2−3r31, 14

where 1 is a spin function with spin angular momentum 1/2.

The basis set is then replicated several times with different

nonlinear scale parameters 	, 
, and , with the scale parameters fully optimized for each sector. For the analysis given

here, we used the 9576 term basis set of Pulchalski and Pachucki 12. The basis is divided up into five sectors as follows:

1all n3, n1=0, n2=0;

2all n3, n1=0, n20;

3all n3, n10, n2=0;

4n3=0, n10, n20;

5n30, n10, n20.

Table III gives the results of dropping odd-odd and oddodd-odd terms where, in the table, N is the cumulative number of terms, and N is the number of terms surviving after

odd-odd and odd-odd-odd terms have been eliminated.

Hence the N column is the N-column wave function filtered

to contain only at most a single odd rij power.

The most striking observation fifth line in the table,

E6170vs E9576 is that products of odd powers of rij

contribute only at the sub-nanohartree level for accurate

wave functions E6170is E9576with odd-odd and oddodd-odd rij products removed.

We also did runs with only one odd rij power and the

maximum rij power in any term restricted to 6 and then 5.

The results are also tabulated in Table III.

TABLE II. Comparison of theoretical Li ground-state nonrelativistic energies in hartrees.

Technique Author No. of terms Energy Ein hartrees

MCSCFa Tong, Jönsson, and Fischer 199314 −7.477 968 61

CI Jitrik and Bunge 199715 l
 = 13 −7.478 025 4

Hy King and Bergsbaken 199016 296 −7.478 059 53

Hy-CI This work, s, p rij basis 8316 −7.478 059 573 327

Hy-CI Pestka and Woźnicki 199617 386 −7.478 060 1

Hy-CI Pipin and Bishop 19923 1618 −7.478 060 1

Hy Lüchow and Kleindienst 199418 1420 −7.478 060 320 8

Hy-CI This work, s, p,d rij basis 14364 −7.478 060 321 379

ECGb Pachucki and Komasa 200619 7000 −7.478 060 323 2

Hy-CI This work, s, p,d, f rij basis 16764 −7.478 060 323 451 9

Hy This work, one odd rij 6170 −7.478 060 323 452

Hy Yan, Tambasco, and Drake 199820 3502 −7.478 060 323 618 9

Hy Pulchalski and Pachucki 200612 9576 −7.478 060 323 889 7

Hy Yan, Nörtershäuser, and Drake 200813 9577 −7.478 060 323 892 4

Hy Pulchalski, Kedziera, and Pachucki 200911 13944 −7.478 060 323 909 560

Hy Pulchalski, Kedziera, and Pachucki 200911 Exactest.−7.478 060 323 910 1032

a

Multiconfigurational self-consistent field.

b

Explicitly correlated Gaussians.

JAMES S. SIMS AND STANLEY A. HAGSTROM PHYSICAL REVIEW A 80, 052507 2009

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Note that one does not need to go beyond rij

6 to achieve

sub-nanohartree accuracy for lithium in Hy calculations, and

since rij

5 in Hy-CI corresponds to ddrij, one doesn’t need

ffsrij in Hy-CI until one reaches the nanohartree level of

accuracy. This agrees perfectly with what we showed in Sec.

III A.

C. Excited states

Following King 22, we calculate energies for the 3 2S,

4 2

S, 5 2S, 6 2S, and 7 2S states of the lithium atom. The 3 2S

state provided another valuable test of the Hy-CI method as

there exists an accurate Hy energy value 23 to compare

with. As Table IV shows, selection of expansion terms for

this state mirrors those for the ground 2 2

S state except that

we dropped the pdf term types which contribute only at the

sub-nanohartree level. We use more L-shell orbitals the

K-shell ionic core should be similar to the ground 2 2

S state,

but the L-shell orbitals should reflect the higher, more energetic state. The last two lines in Table IV suggest we have as

many as are required for this state. ffsR and sKasKasLsR

terms contribute like they do for the 2 2

S state, indicating

that these are truly K-shell core effects. As in the case of the

2 2

S state, here and for the higher states we make no attempt

to filter or otherwise weed out unimportant terms. Note that

the 3 2

S state is within 0.494 nanohartree of the Pulchalski,

Moro, and Pachucki 23 calculation, demonstrating that

Hy-CI can achieve sub-nanohartree accuracy for Li excited

states as well as the ground state.

Table V shows energies for various truncations of the

wave function for the remaining excited states. We have

dropped dddR since it contributes only 0.5 nanohartree for

the 3 2

S stateand have changed the basis set description for

the L shell to hopefully be more flexible we use two different Ls orbital exponentsfor these even more diffuse states.

As one might expect, the importance of 7sLs and 5sLa increases as we go to higher excited states. The contributions

of the various term types scale inversely roughly with the

excited electron principle quantum number with some irregularities due probably to the less than perfect optimization of

the orbital exponents in the K shell. The improvement of our

results over the previously best calculations of King 22 is

consistent until one reaches the 6 2

S state, where we are

about 128 nanohartree better. For the 7s state it is evident

from the contributions of the last two term types that the

sssR part needs some further work, the wave function not

being sufficiently flexible for this highly excited state a

change over to hydrogen-like orbitals is probably called for

as the excited electron becomes more Rydberg-like. Despite

TABLE III. Hy calculations of the Li ground-state energy by section in hartrees. max is the maximum

allowed rij power.

Section N ENin hartrees N ENin hartrees

1 1036 −7.476 983 290 850 1036 −7.476 983 290 850

1–2 3222 −7.478 060 320 124 2696 −7.478 060 305 902

1–3 5404 −7.478 060 323 278 4340 −7.478 060 321 990

1–4 6951 −7.478 603 233 29 5327 −7.478 060 322 893

1–5 9576 −7.478 060 323 889 6170 −7.478 060 323 452

1–5,

max= 6 5740 −7.478 060 323 348

1–5,

max= 5 5124 −7.478 060 322 122

YNDa 9577 −7.478 603 238 92

PKPb 13944 −7.478 060 323 909 560

a

Yan, Nörtershäuser, and Drake 200813. b

Pulchalski, Kedziera, and Pachucki 200911.

TABLE IV. Energies in hartreesfor various truncations of the

wave function of the 3 2

S excited state of lithium. In the table

R=
1,r12,r13,r23.

N Terms added 3 2

S a,

b

4000 1: 10sKs 1: 10sKt 1: 10sLs R −7.353 707 280 732

7240 2: 10pKp 2: 10pKq 1: 10sLs R −7.354 097 937 123

8808 3: 9dKd 3:9dKd 1:8sLs R −7.354 098 336 936

9816 2: 7pKp 1:7sKs 2:7pLp R −7.354 098 375 949

10824 1: 7sKs 2:7pKq 2:7pLp R −7.354 098 399 926

11688 3: 8dKd 1:6sKs 3:8dLp R −7.354 098 401 484

12552 1: 6sKs 3:8dKd 3:8dLp R −7.354 098 402 948

13416 2: 7pKp 2:7pKq 3:8dLp R −7.354 098 416 384

13916 2: 6pKp 3:7dKd 2:6pLp R −7.354 098 418 002

14416 3: 7dKd 2:6pKq 2:6pLp R −7.354 098 418 592

14916 3: 7dKd 3:7dKd 3:7dLp R −7.354 098 419 084

15516 4: 8fKq 4:8fKq 1:6sLs R −7.354 098 419 557

16380 1: 6sKa 1:6sKa 1:6sLs R −7.354 098 420 690

16780 1: 10sKs 1: 10sKt 11: 11sLs R −7.354 098 420 909

17180 1: 10sKs 1: 10sKt 12: 12sLs R −7.354 098 420 933

Exact

est.24 −7.354 098 421 42619

a

K-shell orbital exponents are Ks= 3.90, Kt= 2.80, Kp= 4.40, Kq

= 5.80, Kd= 4.65, and Ka= 8.0. b

L-shell orbital exponents are Ls= 0.635 and Lp= 0.86.

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these deficiencies, Table VI which gives a comparison of

our excited-state results with previous theoretical work

shows that our 4 2

S, 5 2S, 6 2S, and 7 2S state results improve substantially upon previously published work. Included in the table are our best estimates of the exact energies of these states. These estimates are in line with the

sensitivity analysis for the ssss part of the wave function,

which is probably where most of the error lines. The error in

the 7 2

S state is somewhat bigger due to the inadequate valence shell basis discussed above.

IV. CONCLUSION

In this work we have demonstrated the ability of Hy-CI

calculations to achieve sub-nanohartree accuracy for lithium.

We have also demonstrated that Hy calculations can achieve

sub-nanohartree accuracy for three-electron systems using

only a single odd rij in any term in the wave function, showing the relative unimportance of linked products of odd rij in

the Hy method at the nanohartree level of accuracy.

These calculations suggest that Hy-CI should also be capable

of achieving sub-nanohartree accuracy for three-electron

systems, and we have demonstrated that in these calculations. While we have shown the relative unimportance of

linked products of rij for on the order of less than nanohartree accuracy, it remains to be seen how important unlinked

products like r12r34 will turn out to be for systems with more

than three electrons. We intend to explore this point in a

future publication. We have also shown how to pick expansion terms in an Hy-CI calculation both from CI considerations and by using the expansion of likely Hy rij products as

a guide. Finally, we point out that there still is room for a

more systematic effort to determine a really accurate Hy-CI

value for Li and its excited states including P and Dnow

that we understand the problem in the Hy-CI context.

ACKNOWLEDGMENTS

We are deeply indebted to Krzysztof Pachucki for making

his Hylleraas code available to us, without which we would

not have been able to do the Hy calculations reported in this

paper. We also thank Arne Lüchow for kindly informing us

of the activities of the late Professor H. Kleindienst’s group

and Maria Belen Ruiz for filling us in on the work of Professor J. Rychlewski’s group and also of some of Jacek Komasa’s work.

APPENDIX: PROJECTION REDUCTION

Using Eq. 4, matrix element HKL becomes

HKL = 
KHL = 
K

PHLP

, A1

where K

P denotes the Kth primitive unprojectedfunction

rij

Ks=1

3 
Ks

rsK. SKL is similar, with H replaced by the

unit operator 1.

Now notice that

TABLE V. Energies in hartreesfor various truncations of the wave function of the 4 – 7 2S excited states of lithium. In the table

R=
1,r12,r13,r23.

N Terms added 4 2

S a,

b 5 2

S c,

d 6 2

S e,

f 7 2

S g,

h

4000 1: 10sKs 1: 10sKt 
1:6sLs,1:4sLa R −7.318 144 893 422 −7.303 167 312 632 −7.295 475 782 289 −7.291 010 447 021

7240 2: 10pKp 2: 10pKq 
1:6sLs,1:4sLa R −7.318 530 329 869 −7.303 550 963 613 −7.295 858 767 247 −7.291 391 343 337

9200 3: 9dKd 3:9dKd 
1:6sLs,1:4sLa R −7.318 530 775 712 −7.303 551 347 519 −7.295 859 122 525 −7.291 391 712 630

10208 2: 7pKp 1:7sKs 2:7pLp R −7.318 530 802 894 −7.303 551 444 715 −7.295 859 339 642 −7.291 392 085 840

11216 1: 7sKs 2:7pKp 2:7pLp R −7.318 530 835 014 −7.303 551 568 082 −7.295 859 494 639 −7.291 392 239 327

12080 3: 8dKd 1:6sKs 3:8dLp R −7.318 530 835 967 −7.303 551 569 884 −7.295 859 500 882 −7.291 392 257 397

12944 1: 6sKs 3:8dKd 3:8dLp R −7.318 530 836 739 −7.303 551 571 169 −7.295 859 503 301 −7.291 392 262 718

13808 2: 7pKp 2:7pKq 3:8dLp R −7.318 530 841 199 −7.303 551 573 566 −7.295 859 505 242 −7.291 392 264 598

14308 2: 6pKp 3:7dKd 2:6pLp R −7.318 530 842 944 −7.303 551 575 028 −7.295 859 506 393 −7.291 392 265 574

14808 3: 7dKd 2:6pKq 2:6pLp R −7.318 530 843 272 −7.303 551 575 322 −7.295 859 506 709 −7.291 392 265 851

15408 4: 8fKq 4:8fKq 1:6sLs R −7.318 530 843 884 −7.303 551 576 190 −7.295 859 507 594 −7.291 392 266 942

16272 1: 6sKq 1:6sKq 1:6sLs R −7.318 530 845 168 −7.303 551 577 933 −7.295 859 509 441 −7.291 392 268 343

16672 1: 10sKs 1: 10sKt 7:7sLs R −7.318 530 845 287 −7.303 551 578 188 −7.295 859 509 733 −7.291 392 272 268

17072 1: 10sKs 1: 10sKt 5:5sLa R −7.318 530 845 331 −7.303 551 578 291 −7.295 859 509 943 −7.291 392 273 116

a

K-shell orbital exponents are Ks= 3.50, Kt= 2.70, Kp= 4.40, Kq= 5.80, and Kd= 4.65. b

L-shell orbital exponents are Ls= 0.33, La= 0.75, and Lp= 0.55. c

K-shell orbital exponents are Ks= 3.50, Kt= 2.70, Kp= 4.40, Kq= 5.80, and Kd= 4.65. d

L-shell orbital exponents are Ls= 0.25, La= 0.37, and Lp= 0.53. e

K-shell orbital exponents are Ks= 3.50, Kt= 2.70, Kp= 4.40, Kq= 5.80, and Kd= 4.65. f

L-shell orbital exponents are Ls= 0.18, La= 0.30, and Lp= 0.53. g

K-shell orbital exponents are Ks= 4.00, Kt= 3.60, Kp= 3.40, Kq= 5.80, and Kd= 4.65. h

L-shell orbital exponents are Ls= 0.14, La= 0.28, and Lp= 0.53.

JAMES S. SIMS AND STANLEY A. HAGSTROM PHYSICAL REVIEW A 80, 052507 2009

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N ! Oas =

p

− 1

p

P = u− 1

u

Pu

	



− 1

P






+



− 1

P

	
 = A	A
 + A	
 A2

where N is the number of electrons, P	 refers to those permutations which permute only 	 spins, P
 refers to those

permutations which permute only 
 spins, and P	
 refers to

only those permutations which permute an 	 spin and a 


spin. If we rewrite the bra and ket functions in Eq. A1in

terms of a single spin function, the HKL matrix element will

end up involving only A	A
 since permutations between 	

and 
 give zero results when integrating over spin.

In our case K=L=1=	12	3=	
	, so we can

write

K

P = OL,ML

OasOS,MSgKfK1, A3

where gK=rij

K and fK is the hartree product s=1

3 
Ks

rs.

Using the quantum-mechanical “turnover rule” 33, the

TABLE VI. Comparison of theoretical Li n 2S excited-state nonrelativistic energies in hartrees.

State Technique Author No. of terms Energy Ein hartrees

3 2

S Hy Perkins 197225 22 −7.353 5

Hy Larsson 197226 59 −7.353 917

MCSCF Jönsson, Fischer, and Bierón 199527 13306 −7.354 014

Hy-CI Pipin and Woźnicki 198328 170 −7.354 030

Hy King 199129 447 −7.354 076

Hy Lüchow and Kleindienst 199230 898 −7.354 097 8

Wang, Zhu and Chung 199231 −7.354 098 0

Hy-CI Pestka and Woźnicki 199617 392 −7.354 098 04

Hy King 200722 1900 −7.354 098 355

Hy Lüchow and Kleindienst 199418 1398 −7.354 098 369

Hy-CI This work, s, p,d, f rij basis 17180 −7.354 098 420 933

Hy Yan and Drake 200032 3502 −7.354 098 421 082

Hy Pulchalski, Moro and Pachucki 200623 9576 −7.354 098 421 380

Exact est.Pulchalski and Pachucki 200824 −7.354 098 421 42619

4 2

S Hy Perkins 197225 18 −7.317 5

Hy Larsson 197226 59 −7.318 366

MCSCF Jönsson, Fischer, and Bierón 199527 13306 −7.318 451

Hy King 199129 501 −7.318 491

Hy Lüchow and Kleindienst 199230 898 −7.318 525

Hy-CI Pestka and Woźnicki 199617 241 −7.318 529 38

Wang, Zhu and Chung 199231 −7.318 530 3

Hy Lüchow and Kleindienst 199418 1398 −7.318 530 665

Hy King 200722 1900 −7.318 530 816

Hy-CI This work, s, p,d, f rij basis 17072 −7.318 530 845 331

Exact est.This work, s, p,d, f rij basis −7.318 530 8461

5 2

S Hy Larsson 197226 59 −7.303 392

Hy King 199129 450 −7.303 439

Hy Lüchow and Kleindienst 199230 898 −7.303 547

Wang, Zhu and Chung 199231 −7.303 550 8

Hy King 200722 1900 −7.303 551 551

Hy-CI This work, s, p,d, f rij basis 17072 −7.303 551 578 291

Exact est.This work, s, p,d, f rij basis −7.303 551 5791

6 2

S Hy Lüchow and Kleindienst 199230 898 −7.29 583

Hy King 200722 1900 −7.295 859 38

Hy-CIThis work, s, p,d, f rij basis 17072 −7.295 859 509 943

Exact est.This work, s, p,d, f rij basis −7.295 859 5112

7 2

S Hy-CI This work, s, p,d, f rij basis 17072 −7.291 392 273 116

Exact est.This work, s, p,d, f rij basis −7.291 392 2763

HYLLERAAS-CONFIGURATION-INTERACTION STUDY OF ... PHYSICAL REVIEW A 80, 052507 2009

052507-7
commutivity of  and H, and the idempotency condition

†

=, HKL reduces to

HKL = 
K

PHLP

. A4

Projecting on 1 with the Löwdin spin projection operator

4 OS,MSS= 1/2,MS= 1/2, we get

OS,MS1 = OS,MS	
	 = 2

3

	
	 − 1

3


		 − 1

3

		
 A5

where the subscripts not shownon the 	 and 
 are always

in the order 1,2,3.

This can be written

OS,MS

1 = 	

2

3 − 13

P12

 − 1

3

P23

 1. A6

Using the identity

Oas = − 1

p

OasPrP

−1, A7

HKL becomes

HKL = 
gKfK1HOL,ML

Oas	

2

3

+

1

3

P12

r +

1

3

P23

r gLfL1 .

A8

Now we can integrate over spin, effectively removing spin

from the matrix element:

HKL = 
gKfKHA	A
OL,ML

BrgLfL. A9

We have used the fact that OL,ML commutes with Oas in the

above equation. Here the “reduced” antisymmetrizer A	A


A	A
 = 1 − P13A10

and

B = 2

3

+

1

3

P12 +

1

3

P23 A11

operate on the spatial coordinates. r has been dropped from

the terms in B since only spatial coordinates remain. Using

Eq. A10, we obtain for our final expression for HKL

HKL = 
gKfKH1 − P13OL,MLBgLfL. A12

In Eq. A12one can apply OL,ML either before or after applying B. We first apply the permutations B to gLfL, then we

project on the resulting terms with OL,ML

. In doing the OL,ML

projection, we use the fact that gL=rij

L commutes with OL,ML

1,34 and only apply OL,ML to the orbital products.

Depending on the structure of gLfL, 1−P13BgLfLmay

further simplify, and this should be checked before application of OL,ML we in fact do this in working out the righthand side of H. In practice we routinely swap the bra and

kets if doing so will reduce the number of terms resulting

from Eq. A12.

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JAMES S. SIMS AND STANLEY A. HAGSTROM PHYSICAL REVIEW A 80, 052507 2009

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