INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 37 (2004) 2933–2942 PII: S0953-4075(04)79661-3

A configuration interaction approach to bosonicsystems

D Sundholm and T Vanska

Department of Chemistry, University of Helsinki, FIN-00014 Helsinki, Finland

Received 13 April 2004Published 1 July 2004Online at stacks.iop.org/JPhysB/37/2933doi:10.1088/0953-4075/37/14/007

AbstractA direct configuration interaction approach for studies of bosonic systemshas been developed and implemented. The methods have been applied onmodel systems consisting of charged bosons in a central Coulomb field. Byincreasing the size of the one-particle basis sets for a fixed number of bosons,the computational efforts increase quadrically as a function of the number ofconfigurations, whereas by increasing the number of bosons for a fixed basisset, the methods scale almost linearly with the size of the configuration space.The test calculations also show that a truncated configuration interaction modelconsidering only single and double replacements from the reference accountsfor nearly 100% of the correlation energy. The present configuration interactionprogram can easily be modified for computational studies of other kinds ofbosonic systems because the only quantities containing information about thesystem under consideration are the interaction strength integrals.

1. Introduction

Computational approaches to treat fermionic systems at correlated levels of theory have duringthe past 40 years experienced radical improvements, whereas hardly any efforts have beenmade on the development of similar ab initio correlated methods for studies of bosonic systems.The recent discovery of Bose–Einstein condensates (BEC) [1] has awakened a vast interest incomputational studies of interacting bosons. Computational studies of bosonic systems wereinitially performed at the Hartree–Fock level [2–4], thus omitting the boson–boson correlationeffects, whereas in more recent work different quantum Monte Carlo (QMC) approacheshave mainly been employed for considering boson–boson correlations [5–17]. Computationalapproaches such as configuration interaction (CI) methods, which are extensively used inatomic and molecular electronic structure calculations, are rarely applied on bosonic systems.In the literature, we have found two papers describing advanced CI methods for studiesof bosonic systems [18, 19]. An alternative method to solve the Schrödinger equationapproximately, called the reduced Hamiltonian interpolation approach, has recently been

0953-4075/04/142933+10$30.00 © 2004 IOP Publishing Ltd Printed in the UK 2933

2934 D Sundholm and T Vanska

employed in the calculations of the correlation energy for bosonic systems [20, 21]. It hasbeen argued that CI methods are computationally too demanding and impractical for most BECapplications [9]. However, since QMC methods have not yet replaced ab initio approachesin molecular electronic structure calculations, the notion that CI methods for bosonic systemsare inferior to the QMC ones is probably not correct. Path-integral Monte-Carlo (PIMC)methods have proved very useful in the evaluations of critical temperatures, Tc [6]. On theother hand, Monte Carlo approaches have severe difficulties to provide accurate one-bodydensity matrices and related properties [11], whereas at the CI level, a variety of properties canbe deduced from the one- and two-body density matrices [22] which are easily evaluated fromCI wavefunctions [23]. In studies of bosonic systems, the use of similar ab initio correlationapproaches as developed for atoms and molecules will also open the avenue for the calculationof many different properties [24].

In molecular electronic structure theory, efficient algorithms have been developed whichhave made it feasible to solve very large CI problems [23, 25]. Full configuration interaction(FCI) is not necessary for accurate solutions of the Schrödinger equation; truncated CI modelsconsidering only a small fraction of all possible configurations take most of the correlationeffects into account. However, one has to be aware of some limitations of truncated CIapproaches such as the lack of the size-extensivity property, which can affect the outcomeof the computations. Coupled-cluster methods would avoid the size-extensivity problem, buttheir implementation is somewhat more involved.

Here, we present the theory and implementation of full and truncated CI methods. Thetheory is presented using a second-quantization formulation closely related to that used inatomic and molecular electronic structure calculations as recently proposed by Esry [18]. It isclear from a previous implementation of CI methods for bosonic systems [19] that there arehardly any well-documented data in the literature for correlation calculations of many-bosonsystems, therefore we chose a well-defined model system for testing the correctness of ourmethods and their implementation. The model system consists of charged bosons in a centralCoulomb field potential, i.e. an atomic nucleus surrounded by spin-free electrons. Anotheradvantage with this model system is that basis sets are already available from atomic andmolecular electronic structure calculations. With the present formulation, the computationalmethods are easily adapted to other kinds of multi-boson systems, since the information aboutthe system appears only in the interaction strength parameters, i.e. the one- and two-bodyintegrals.

In section 2, the theory, equations and implementation are reviewed. The computationalmethods and algorithms are discussed in section 3, and in section 4, we present some resultsincluding timings of the calculations.

2. Theory

2.1. Second quantization

In this section, we briefly repeat the second-quantization formalism for bosonic systems. Agiven configuration �I of Nbosons bosons can be expressed as a product of creation operatorsa†

α operating on the vacuum state

�I = a†1a

†2 · · · a†

N |0〉. (1)

The a†i operator creates a boson in the specified one-particle state (orbital i). The product

of creation operators yields a boson configuration, also called an occupation-number stringcontaining the orbital occupation numbers of the configuration. The transpose of the creation

A configuration interaction approach to bosonic systems 2935

operator (ai) removes a particle from the given position in the occupation-number string andis, therefore, called an annihilation operator. The creation and annihilation operators areconstructed to satisfy the boson commutation relations as

a†α|n1, n2, . . . , nα−1, nα, nα+1, . . .〉 =

√nα + 1|n1, n2, . . . , nα−1, nα + 1, nα+1, . . .〉 (2)

and

aα|n1, n2, . . . , nα−1, nα, nα+1, . . .〉 = √nα|n1, n2, . . . , nα−1, nα − 1, nα+1, . . .〉. (3)

Thus ai and a†i fulfil the boson commutation rules[

a†α, a

†β

] = 0; [aα, aβ ] = 0; [aα, a

†β

] = δαβ. (4)

2.2. The Hamiltonian

The Hamiltonian considering one- and two-body interactions expressed in the second-quantization formalism for bosonic systems can be written as

H =∑pq

hpqa†paq +

1

2

∑pqrs

gpqrsa†pa†

r asaq, (5)

where hpq and gpqrs represent the strengths of the one- and two-body interactions. The one-particle interaction (here given in atomic units) contains the kinetic energy operator and theinteraction with external potentials

hpq =∫

χ∗p(r)

(−1

2∇2 + Vext(r)

)χq(r) dr (6)

χ(r) are orthonormal single-particle states (orbitals). The two-body interaction strengths cananalogously be written as

gpqrs =∫ ∫

χ∗p(r1)χ

∗r (r2)V (r1, r2)χs(r2)χq(r1) dr1 dr2, (7)

where V (r1, r2) is the interaction potential between the bosons. Since the aim of this workis to develop a new computational approach we apply the methods on a model system. Dueto our previous experiences from atomic structure calculations [26, 27], the obvious choice ofmodel system is atomic electronic structure potentials including a mutual Coulomb repulsionbetween the bosons. This model system is well defined and can be used as a check of thecorrectness of the program code. In this model, the external one-particle interaction potentialis represented by a Coulomb potential created by a particle of the opposite charge

Vext(r) = − Z

|r| , (8)

where Z is the charge and r is the distance from the charge. The two-body interaction cananalogously be assumed to consist of the interaction potential between equally charged bosons

V (r1, r2) = 1

|r1 − r2| . (9)

Thus, our model is an atomic system with the fermionic electrons replaced by negativelycharged bosons with a charge of −1 in atomic units. This is a convenient choice, since theorbitals can be expanded in standard Gaussian basis functions used in atomic and molecularstructure calculations.

2936 D Sundholm and T Vanska

2.3. Configuration interaction

In configuration interaction calculations, the wavefunction is expanded as a linear combinationof configurations obtained by permuting the particles among the orbitals. When allcombinations are taken into account, the model is called full configuration interaction(FCI), whereas in restricted configuration interaction models, only those configurations areconsidered which are obtained by exciting one, two, three, . . . , etc, particles from a referenceconfiguration to the non-occupied space (virtual orbitals) of the mean field calculation. Forexample, the CI singles and doubles (CISD) model can be obtained by exciting at most twoparticles from the reference. For bosonic systems, the reference is usually chosen to be theconfiguration that has all particles in the energetically lowest orbital. The ansatz for the CIwavefunction can then be written as

|�CI 〉 =∑

I

CI |ψI 〉 (10)

and the total energies of the ground and excited states can be obtained by minimizing theenergy function

E = 〈�CI |H |�CI 〉; 1 = 〈�CI |�CI 〉 (11)

for normalized wavefunctions. This procedure involves construction and diagonalization ofthe Hamilton matrix, the elements of which are

HIJ = 〈ψI |H |ψJ 〉. (12)

The main drawback with such a strategy is that the number of configurations increases rapidlywith the size of the basis set and number of bosons. Serious computational problems beginto appear for cases with more than 104 configurations. In molecular structure calculationsone faced this problem 30 years ago, though for much smaller matrices, and it was shownthat an iterative procedure involving only linear transformations of trial vectors with theHamilton matrix is sufficient when only the few lowest eigenvalues are desired. However, fora practical implementation, the linear transformations must be carried out efficiently; i.e. inthe configuration representation directly from the hpq and gpqrs integrals [28]. One can furtherspeed up the computations by exploring ‘Condon–Slater’ rules to identify contributions fromnonvanishing matrix elements and by using an efficient configuration numbering scheme.

The matrix–vector multiplication |σ 〉 = H |c〉 can, in this context, be written as

σI =∑

J

(∑pq

hpq〈I |a†paq |J 〉 +

1

2

∑pqrs

gpqrs〈I |a†pa†

r asaq |J 〉)

CJ , (13)

where σI is the component of the product vector corresponding to the I th configuration. Inthe code the summations over single-particle states can, effectively, be reduced to a singlesummation for the one-body part and a double summation for the two-body part. This isaccomplished by first summing over the indices corresponding to the annihilation operators,allowing to identify the vanishing matrix elements (annihilating the vacuum). The summationover the creation indices is then performed only for the nonvanishing cases.

2.4. Addressing of configuration

To achieve fast configuration interaction algorithms, it is important to have a unique andefficient addressing scheme of the configurations. We have generalized to bosonic systems thegraphical addressing strategy previously used for spin strings [23, 29]. For bosons, each pathcontains three types of arcs. Vertical arcs indicate unoccupied one-particle states (orbitals),

A configuration interaction approach to bosonic systems 2937

diagonal arcs appear for singly occupied orbitals and for orbitals with occupation number oftwo and larger horizontal arcs are also present. The vertex weights, whose sum yields theconfiguration addresses, are stored in a matrix with the dimensions Nactive × Norbitals, whereNactive is the number of active bosons, depending on the CI truncation level. The number ofoperations to obtain a configuration index from the occupation-number string is proportionalto Nactive.

3. Computational methods

The methods described in the previous section have been implemented. The program is writtenin Fortran 90 with the exception of a few subroutines which were coded using Fortran 77.Timings of the program revealed that in some situations the Fortran 90 code was, independentof the compiler used, almost a factor of 10 slower than the corresponding code written inFortran 77. By avoiding some F90-specific techniques in the computationally most demandingroutines, the Fortran 90 code can, however, be written in such a way as to be on par with F77.

We have studied atom-like systems with nuclear charge from 2 to 10, i.e. from ‘Helium’to ‘Neon’1 using the triple-zeta valence quality basis sets augmented with double polarizationfunctions (TZVPP) [30] from the quantum chemical program Turbomole [31]. The basis setswere completely uncontracted and the f -functions were omitted. The number of one-bodybasis states increase from 17 for He to 41 for Ne.

The uncontracted TZVPP basis sets are not perfectly suited to accurate studies of bosonicatoms. In the calculations of the excitation energies, we employed the atomic even-temperedbasis sets of Schmidt and Ruedenberg [32]. To obtain excitation energies close to the basis-set limit, their even-tempered basis sets were further augmented by diffuse functions andalso higher angular-momentum functions were added. The even-tempered basis sets used inthe calculation of the excitation energies consisted of 21s4p basis functions. The recipe ofSchmidt and Ruedenberg was employed to obtain a basis set of 20s functions. This set wasaugmented by three diffuse s functions (with the same spacing), and two were removed fromthe steep end. The p functions were obtained by generating a set of six p functions and deletingthe two steepest ones. For the Ne spectrum in figure 2(b), the two d functions from the TZVPPbasis set were added.

Since the number of configurations, which translates into the size of the CI-matrix to bediagonalized, rapidly grows with the number of basis functions and particles, our calculationswere mainly performed at the CI singles and doubles (CISD) level. However, as mentionedabove, iterative techniques can be used for obtaining the lowest few eigenstates of large, sparsematrices. These methods scale almost linearly with the number of configurations, in contrastto the N3 scaling of full diagonalization. The maximum number of configurations suitable forfull diagonalization of the Hamiltonian matrix is around 15 000, whereas by using the directtechnique we have obtained the lowest eigenstate for a system with 735 471 configurations in3.5 h using an ordinary personal computer (PC). The timings are discussed in more detail insection 4.

The reference state needed for the truncated CI calculations was obtained by performinga Hartree–Fock level calculation. Convergence was typically reached in 10–20 iterations.For improved convergence we have used the direct inversion of the iterative space (DIIS)convergence acceleration method [33], which, indeed, in some cases was found to be necessaryfor Hartree–Fock convergence to be reached.

1 From here on we will refer to the fictitous atoms without quotes.

2938 D Sundholm and T Vanska

Table 1. Ground-state energies (in au) of He–Ne calculated at the HF and CISD levels. The lastcolumn lists the relative weight of the reference (HF) state in the CI calculation.

Correlation energyH–L gap Hartree–Fock CISD Ref. state occ.

Systema (au) (au) (au) (au) (%) (%)

He (2) 0.904 53 −2.859 90 −2.899 53 −0.039 62 1.37 99.21Li (3) 0.892 26 −8.545 95 −8.650 06 −0.104 10 1.20 99.28Be (4) 1.175 65 −19.017 26 −19.220 35 −0.203 08 1.06 99.30B (5) 1.485 40 −35.742 11 −36.114 39 −0.372 28 1.03 99.37C (6) 1.814 32 −60.179 47 −60.737 50 −0.558 02 0.92 99.43N (7) 2.164 50 −93.794 80 −94.575 73 −0.780 93 0.83 99.50O (8) 2.536 04 −138.052 06 −139.092 54 −1.040 48 0.75 99.54F (9) 2.929 03 −194.414 61 −195.752 17 −1.337 55 0.68 99.58Ne (10) 3.343 49 −264.346 20 −266.018 24 −1.672 04 0.63 99.61

a The number of bosons is given within parentheses.

Table 2. Estimated ground-state and correlation energies (in au) of neutral fermionic He–Ne atomscalculated by C Froese Fischer et al [34, 35]

Correlation energyTotal energy

Atom (in au) (in au) (in %)

He −2.903 724 −0.042 044 1.47Li −7.478 06 −0.045 33 0.61Be −14.667 36 −0.094 34 0.65B −24.653 91 −0.124 85 0.51C −37.845 0 −0.156 40 0.42N −54.589 2 −0.188 31 0.35O −75.067 3 −0.257 94 0.35F −99.733 9 −0.324 53 0.33Ne −128.937 6 −0.390 47 0.30

4. Results

All calculations described below were performed on a PC with a 1400 MHz AMD athlonprocessor. The results of the CISD calculations on the series of neutral atomic systems aresummarized in table 1. The total energy of the neutral bosonic atoms decreases much fasterwith increasing nuclear charge than for fermionic atoms. This is easily understood sincein the bosonic case mainly the 1s orbital is occupied. The correlation energies are also inabsolute value significantly larger than for fermionic atoms [34, 35] (see tables 1 and 2). Thecomparison is not completely fair since the correlation energies of the fermionic atoms havebeen extrapolated and are very close to the basis-set limit [34, 35], whereas for the bosonicsystem we have used basis sets without f and higher angular momentum functions which areimportant for systems with Coulombic interactions. But even with the basis sets used, therelative correlation energies of the bosonic system are much larger than the correspondingextrapolated correlation energies of the fermionic atoms. The relative magnitude of thecorrelation energy decreases with increasing particle number, and for the heavier systems itis about a factor of 2 larger than for the corresponding fermionic atoms. One can also seethat the correlation energy increases approximately linearly with the number of boson pairs.The linear trend is also expected since in atoms the 1s correlation energy is almost constantand independent of the nuclear charge [34].

A configuration interaction approach to bosonic systems 2939

SV SVP TZP TZVPP−2.90

−2.89

−2.88

−2.87G

roun

d S

tate

Ene

rgy

for

He

Basis

−266.2

−265.8

−265.4

−265.0

Gro

und

Sta

te E

nerg

y fo

r N

e

(a)

CIS CISD FCI−8.66

−8.64

−8.62

−8.60

−8.58

−8.56

−8.54

Gro

und

Sta

te E

nerg

y fo

r Li

CI Method

−19.25

−19.20

−19.15

−19.10

−19.05

−19.00

Gro

und

Sta

te E

nerg

y fo

r B

e

(b)

Figure 1. (a) The ground-state energy as a function of basis set for He (thin line) and Ne (thickline) calculated at the CISD level. (b) The ground-state energy for Li (thin line) and Be (thickline) calculated at different CI levels. The FCI calculation on Be was performed using the directiterative CI method.

Table 3. Gaussian basis sets used in the He and Ne calculations, respectively.

Number of basis functions Details

Basis set He Ne He Ne

SVa 4 19 4s 7s4pSVPb 7 25 4s1p 7s2p1dTZPc 8 28 5s1p 10s6pTZVPPd 17 41 5s2p1d 11s6p2d

a Uncontracted split valence basis sets.b Uncontracted split valence basis sets augmented with one polarization function.c Triple zeta valence basis sets augmented with one polarization function.d Triple zeta valence basis sets double polarization functions.

The gap between the highest occupied orbital and the lowest unoccupied orbital (the H–Lgap) increases with increasing nuclear charge. The huge H–L gap of about 3.3 au for Nesuggests that the wavefunction is dominated by one single configuration. Indeed, the relativecontribution to the wavefunction from the HF reference is more than 99% for all systemsconsidered.

Figure 1(a) shows the ground state energy for selected systems as a function of basis set(see table 3 for details on the basis sets). The energy decreases monotonously with increasingbasis-set size. For Ne a steep decrease is noted, with no sign of saturation. Larger single-particle basis sets would be needed to achieve energies close to the basis-set limit. However,one should keep in mind that the basis sets are optimized for atoms with fermionic electrons,and may therefore not be well suited for bosonic systems. For He, where the bosonic andfermionic cases coincide in that the ground state has both particles occupying the lowestsingle-particle state, we see a clear convergence. Figure 1(b) displays the ground state energyas a function of the level of CI truncation for CIS (CI singles), CISD and FCI. It is clearlyseen that the largest part of the correlation energy can be obtained on the level of single anddouble replacements from the reference. Indeed, for Li and Be only around 1 per cent of thecorrelation energy is contained within the higher excitations.

2940 D Sundholm and T Vanska

−3

−2.5

−2

−1.5

−1

−0.5

1 s

2 s1 p 3 s4 s 5 s

6 s 2 p

7 s

8 s

Ene

rgy

(a.u

.)

−266

−264

−262

−260

−258

−256

−254

1 s

2 s3 s 4 s1 p 5 s

6 s2 p

1 d

7 s

Ene

rgy

(a.u

.)

Figure 2. The low-energy part of the excitation spectrum for He and Ne calculated at the CISDlevel.

Table 4. Ground-state excitation energies (in au) of the two lowest excited states of bosonic atomscalculated at the CISD level using even-tempered sp basis sets.

System (Nbosons) 1s → 2s 1s → 2p

He (2) 0.753 693 0.804 009Li (3) 1.254 546 1.301 098Be (4) 1.880 061 1.971 374B (5) 2.522 424 2.601 840C (6) 3.360 608 3.478 262N (7) 4.324 238 4.456 130O (8) 5.456 980 5.597 132F (9) 6.583 510 6.732 448Ne (10) 7.900 260 8.059 138

Table 4 lists the ground-state excitation energies of the two lowest excited states ofHe–Ne. The two lowest excitation energies are found to increase linearly with the nuclearcharge. The first excited S and P states are nearly degenerate. This can also be seen infigure 2 which displays the low-energy part of the excitation spectrum of He and Ne.

We have also investigated charged states of He–Ne and found that all singly chargedbosonic atoms are stable, even He− which is in contrast to fermionic He. Doubly chargedanions were also studied. However, the electron affinity of He− was found to be negative;He2− lies higher in energy than He−. For the heavier doubly charged anions, we were notable to converge the Hartree–Fock equations, but FCI calculations on Li2− and calculationswith non-integer nuclear charges indicate that the doubly charged anions are unstable for theHe–Ne bosonic systems, even though the Coulomb potential for, e.g., Ne is very deep and onewould expect it to be able to accommodate a few extra electrons.

Figure 3 shows some results on the performance of the iterative direct CI technique. FullCI calculations were carried out on two different systems, and the number of configurationswas increased in two different ways: first by keeping the number of particles constant andincreasing the basis-set size and then by keeping the basis set constant and increasing the

A configuration interaction approach to bosonic systems 2941

2600 5984 11480 196000

200

400

600

800

1000

1200T

otal

CP

U ti

me

(s)

Number of configurations

(a)

969 74613 245157 7354710

500

1000

1500

2000

2500

3000

Number of configurations

CP

U ti

me/

itera

tion

(s)

(b)

Figure 3. Timings for the direct iterative CI algorithm. (a) FCI keeping the number of particlesconstant (Nbosons = 3) and increasing basis-set size. The number of single-particle levels is 24,32, 40 and 48. (b) FCI keeping the basis set constant (17 single-particle levels) and increasing thenumber of particles (3, 6, 7, and 8).

number of particles. In the latter case the two-body interaction was scaled by dividing itby the number of particles in order to enforce system stability. As seen in figure 3, neitherapproach leads to a strictly linear scaling, a fact which is evident from the inspection of thematrix-vector multiplication algorithm in equation (13); an increase in the number of particlesor in the number of single-particle states leads to additional computations in the innermostloops over the basis functions, also when ‘Condon–Slater’ rules are used for selecting thecontributions. Linear regression fits yield a leading power of Nconf , which is 1.96 for(a) and 1.25 for (b).

5. Conclusions and outlook

In this work, we have developed and implemented a direct configuration interaction approachfor bosonic systems. The test calculations on bosonic atoms show the applicability of themethods. The program is written quite generally. The interaction parameters, i.e. the one- andtwo-electron integrals, are the only quantities containing information about the system underconsideration. This means that the same CI code can be used in studies of different bosonicsystems such as Bose–Einstein condensates, helium droplets, and molecular vibrations. Inthe present calculations, the CISD method is able to provide nearly 100% of the correlationenergy which shows promise for future studies of the above mentioned bosonic systems, sincethe scaling of the CISD calculations is dependent only on the number of one-particle functionsand thus independent of the number of bosons. It remains to be seen, however, whether CISDis as adequate for other systems as it appears to be for the present ones.

Acknowledgments

We acknowledge the support from the Academy of Finland (53915, 200903 and 206102)and Magnus Ehrnrooth’s Foundation. We also thank Markus Lindberg and Mats Brasken fordiscussions and Jonas Juselius for invaluable help with F90.

2942 D Sundholm and T Vanska

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