Chukyo University Repository for Academic Resources / 中京大学学術情報リポジトリ

齊藤, 史郎

1
Large-scale nonrelativistic multireference 
configuration interaction calculation 
of the ground state 3P of the carbon atom
Saito Shiro L.
Abstract
The ground state 3P energy of the carbon atom which is given from large-scale nonrelativistic 
multireference configuration interaction (MRCI) calculations is presented. The maximum angular 
momentum of orbitals used is 9 (i.e., m-symmetry). The orbitals used are the Hartree-Fock orbitals 
for occupied orbitals and natural orbitals for the correlating orbitals. The number of expansion terms 
in the largest MRCI wave function is 6,235,756. The MRCI energy obtained is –37.844698 hartree. 
The full configuration interaction (FCI) energy limit is estimated: the estimated energy is –37.84475 
hartree. Moreover, the energy contribution from the orbitals having higher angular momentum is 
estimated. The final estimated energy is –37.84487 hartree.
1. INTRODUCTION
The electronic structure of atoms is described by the wave function Ψ, which is given from the 
Schrödinger equation:
HΨ = EΨ . (1)
Here, H is the nonrelativistic Hamiltonian:
H =   (–   ∆i – ri–1) +   rij–1, (2)
where ri is the distance between the i-th electron and the nuclei, and rij is the distance between 
the i-th electron and the j-th electron. Eq.(1) is solved by various methods, e.g., configuration 
interaction (CI), coupled cluster (CC), many-body perturbation theory (MBPT), or explicitly 
N
�
i = 1
1
2
N
�
i< j
2 CHUKYO UNIVERSITY Faculty of Liberal Arts and Sciences 2023 Vol. 4 No. 1
correlated function (ECF) methods. In particular, the ECF method is powerful and gives highly 
accurate results. Unfortunately, it is very difficult to apply the ECF method to many-electronic 
systems because of appearance of complicated Hamiltonian matrix elements. Even the calculation 
for the carbon atom with 6 electrons is difficult.
The CI method can be applied to many-electronic atoms and is extensively used because the CI 
wave function is simple form. The CI wave function is expressed by a linear combination of excited 
configuration state functions (CSFs) from a reference configuration Φ: 
Ψ = Φ +   Ci
SΦi
S +   Ci
DΦi
D +   Ci
TΦi
T +   Ci
Q Φi
Q + ..., (3)
where Φi
S, Φi
D, Φi
T, and Φi
D are singly, doubly, triply, and quadruply excited CSFs from Φ, respectively. 
Actual calculations use truncated CI wave functions because the full CI (FCI) wave function is 
very large. The simplest truncated CI wave function is a linear combination of singly and doubly 
excited configurations from a single reference configuration: this CI wave function is called a ‘single 
reference singly and doubly excited CI’ (SRSDCI) one. The SRSDCI wave function only considers 
two-body effects. It is important to further incorporate many-body effects into the CI wave function. 
Most important many-body effect is described by quadruply excited configurations. To effectively 
consider the many-body effects, multiple reference configurations are employed. Such the CI 
wave function is called a ‘multireference singly and doubly excited CI’ (MRSDCI) one. Since the 
MRSDCI wave function can incorporate the many-body effects beyond the two-body effect, the 
MRSDCI energy converges rapidly to the FCI one. Therefore, the MRSDCI method is superior about 
consideration of the many-body effects.
A defect of the CI wave function is to restrict incorporation of the higher partial wave component 
of the electron pair. Correct wave function must include inter-electronic distances. The inter-
electronic distance between the electron 1 and 2, r12, is expanded as
r12 =       (–        +        )     Yl–m (θ1φ1) Ylm(θ2φ2), (4)
where r> is the greater of r1 and r2, r< is the lesser, and Yl
m is the spherical harmonics. As seen 
from Eq.(4), the CI wave function must be used orbitals having infinite angular momentum. In actual 
CI calculations, however, only the orbitals with finite angular momentum can be used. The maximum 
angular momentum used is less than 10 usually because the CI wave function becomes large size. 
Our aim is to obtain the ground state energy of the carbon atom from large-scale nonrelativistic 
�
i
�
i
�
i
�
i
∞
�
l = 0
4π
(2l+1)
1
2l – 1
r<
l
r>
l–1
1
2l – 3
r<
l+2
r>
l+1
l
�
m = –l
Large-scale nonrelativistic multireference configuration interaction calculation of the ground state 3P of the carbon atom 3
MRSDCI calculations. Furthermore, the FCI energy limit and the higher λ effect were estimated by 
extrapolation techniques. To this end, the Hartree-Fock (HF) calculation for the ground state 3P of 
the carbon atom was first performed. Subsequently, SRSDCI and MRSDCI calculations were carried 
out. The present CI calculations employed the HF orbitals for occupied ones and the natural orbitals 
(NOs) for the correlating ones.
The orbital is usually expanded by Slater-type functions (STFs) and Gauss-type functions (GTFs). 
This work adopted a set of B-splines1,2. The B-splines of order K {Bi,K (r)} are piecewise polynomials 
of degree K – 1 on a knot sequence in a cavity of radius R. The knot sequence {ti} is a set of points 
defined on an interval [0, R]. Bi,K (r) is nonzero in the interval [ti, ti+K ). The advantage of B-splines is 
that they are very flexible and are relatively free from computational linear dependence. Thus, it is 
possible to apply a single B-spline set to the expansion of orbitals, irrespective of orbital symmetry, 
without loss of accuracy. Atomic calculations with the B-spline set will give highly accurate results. 
The atomic HF calculations with the B-spline set gives highly accurate energies.3,4
The next section explains basis functions used, NO construction procedures, and extrapolation 
procedures. The third section summarizes the ground state energy of the carbon atom and discusses 
comparison with the other CI energies and the estimated exact value. 
2. COMPUTATIONAL ASPECT
The B-spline set used for the present calculations were constructed on the exponential knot 
sequence5, with endpoints of K-fold multiplicity. The knot sequence {ti}i=1, 2, …, N+K+2 can be written as
{ti}i=1, …, K = 0, (5)
{ti}i=K+1, …, N+2 = R1 (   )    , (6)
{ti}i=N+3, …, N+K+2 = R, (7)
where R1 is the initial interval. Since the first and last terms of the B-splines with K-fold multiplicity 
are nonzero at r = 0 and at r = R, respectively, the N-term B-spline set was constructed omitting 
them, in order to satisfy the boundary conditions. This work employed the parameters, N = 100, 
R = 60 (bohr), K = 9, and R1 = 10
–4 (bohr).
Our CI calculations used the HF orbitals for occupied orbitals and the NOs for the correlating 
orbitals. The NOs were generated by a series of SRSDCI calculations with the reference space 
consisting of the HF configurations.
Let us now explain the generating procedure for NOs. First, the SRSDCI calculation was carried 
out with the HF orbitals in the sp-space, to obtain sp-NOs. Subsequently, d-NOs were generated 
R
R1
i – K – 1
N – K + 2
4 CHUKYO UNIVERSITY Faculty of Liberal Arts and Sciences 2023 Vol. 4 No. 1
by the SRSDCI calculation with the sp-NOs. The NOs with higher angular momentum than d-NOs 
were generated in the same way, step by step. The spdf-NOs were truncated until 30-terms for each 
symmetry. NOs with higher symmetry greater than f were truncated so that the energy loss was less 
than –10–6 hartree at each step. Finally, we obtained an orbital set (30s30p30d30f30g28h27i25k25l2
4m). SRSDCI and MRSDCI calculations were carried out using this orbital set. 
To estimate the FCI limit for the energy, a further series of MRSDCI calculations was carried out 
using the orbital set, increasing the reference configurations which were selected for the largest 
weight in the previous MRSDCI wave function. The FCI limits of the MRSDCI energies (ECI) 
were estimated by extrapolating ECI to wref=1, where wref is the weight of reference space. The 
convergence pattern of ECI, ECI+Q, and EAV were analyzed with respect to wref. ECI+Q is ECI plus the 
quadruple correction6. EAV is the average of ECI and ECI+Q. Although EAV has no physical meaning, EAV 
is very useful for estimating the FCI limit. 
The higher λ effect for the MRSDCI energies was also estimated by extrapolating the energy 
contributions due to the respective λ-NOs to λ→∞. To treat the systems on equal footing, we 
extrapolated the CI energies to λ→∞ using ECI with wref ≈ 0.998. The energy contribution ΔEλ is 
plotted on a log-log scale, and it was fitted by a linear relation. The energy contribution from the NOs 
having angular momentum greater than 9 (Eλ>9) was estimated as 
Eλ>9 =    ΔEλ. (8)
The FCI energy limit including the higher λ effect was finally obtained.
The HF calculations with the B-spline set were carried out using our atomic self-consistent field 
program code based on the algorithm of Roothaan and Bagus7. All CI calculations were performed by 
the program ATOMCI8,9.
3. RESULTS AND DISCUSSION
Table 1 summarizes the CI energies and the quadruple corrected energies of SRSDCI and MRSDCI 
calculations, together with other CI works. Each CI calculation is denoted as SRSDCI(λ) and 
MRSDCI(λ), where λ is the maximum angular momentum of the orbital set used. The MRSDCI(m) 
energy is –37.844698 hartree. The difference between this energy and the SRSDCI(m) energy 
–37.837750 hartree is large. This shows that the many-body effects are large. The quadruple 
corrected SRSDCI(m) energy is –37.844757 hartree. This energy is very near to the MRSDCI(m) 
∞
�
λ = 10
Large-scale nonrelativistic multireference configuration interaction calculation of the ground state 3P of the carbon atom 5
energy. Table 2 lists the weights of principal reference configurations of the MRSDCI(m) wave 
function. Although the main configuration of the MRSDCI(m) function is the HF configuration, the 
weights of doubly excited configurations from the HF configuration are also large. These doubly 
excited configurations induce quadruply excited configurations from the HF configuration. This 
shows that quadruply excited configurations are important.
Table 1.  CI energies (ECI) and quadruple corrected energies (ECI+Q) of SRSDCI, MRSDCI, and other CI 
works (in hartree). NCI is the number of expansion terms in CI wave function. 
Method Orbital set NCI ECI ECI+Q
This work
SRSDCI(p) (30s30p) 7,483 –37.777320 –37.780176
SRSDCI(d) (30s30p30d) 13,438 –37.828576 –37.835389
SRSDCI(f) (30s30p30d30f) 19,333 –37.834933 –37.841916
SRSDCI(g) (30s30p30d30f30g) 25,228 –37.836558 –37.843559
SRSDCI(h) (30s30p30d30f30g28h) 30,590 –37.837172 –37.844178
SRSDCI(i) (30s30p30d30f30g28h27i) 35,477 –37.837457 –37.844464
SRSDCI(k) (30s30p30d30f30g28h27i25k) 39,777 –37.837608 –37.844616
SRSDCI(l) (30s30p30d30f30g28h27i25k25l) 43,877 –37.837696 –37.844703
SRSDCI(m) (30s30p30d30f30g28h27i25k25l24m) 47,753 –37.837750 –37.844757
MRSDCI(p) (30s30p) 150,780 –37.779846 –37.779965
MRSDCI(d) (30s30p30d) 961,515 –37.834542 –37.834671
MRSDCI(f) (30s30p30d30f) 2,262,542 –37.841772 –37.841873
MRSDCI(g) (30s30p30d30f30g) 3,519,573 –37.843481 –37.843586
MRSDCI(h) (30s30p30d30f30g28h) 3,889,442 –37.844110 –37.844218
MRSDCI(i) (30s30p30d30f30g28h27i) 4,559,555 –37.844401 –37.844509
MRSDCI(k) (30s30p30d30f30g28h27i25k) 5,796,821 –37.844555 –37.844664
MRSDCI(l) (30s30p30d30f30g28h27i25k25l) 5,710,780 –37.844643 –37.844751
MRSDCI(m) (30s30p30d30f30g28h27i25k25l24m) 6,235,756 –37.844698 –37.844806
Other works
DCI a (4s3p2d2f) 40 –37.77888
SDCI b (6s6p6d4f) 181 –37.7507
FCI c (7s6p5d3f) 234 –37.83378
SDTQCI d (10s9p8d8f6g4h2i) 993 –37.8393
selected CI f (7s6p5d4f3g) 1,080 –37.835365
a A.W. Weiss (Ref.10).
b H.F. Schaefer et al. (Ref.11).
c A. Bunge et al. (Refs.12 and 13).
d F. Sasaki and M. Yoshimine (Ref.14).
f M.B. Ruiz and R. Tröger (Ref.15).
6 CHUKYO UNIVERSITY Faculty of Liberal Arts and Sciences 2023 Vol. 4 No. 1
Table 2. Weight of principle reference configurations
wRef Configuration
0.970720 1s2 2s2 2p2
0.141121 1s2 2s1 2p2 3d1
0.132360 1s2 2p4
0.071498 1s2 2s1 2p1 3s1 3p1
Let us compare with other CI works: the doubly excited CI (DCI) energy of Weiss10, the singly 
and doubly excited CI (SDCI) energy of Schaefer et al.11, the FCI energy of Bunge et al.12,13, the 
singly, doubly, triply, and quadruly excited CI (SDTQCI) of Sasaki and Yoshimine14, and the selected 
CI energy of Ruiz and R. Tröger15. All other CI works listed in Table 1 used STFs. The FCI energy 
–37.83378 hartree is smaller than MRSDCI( f) energy –37.841772 hartree. The SDTQCI energy is 
also smaller than the MRSDCI(i) energy. This reason is that the FCI and the SDTQCI calculations 
used small orbital set. 
The CI energy –37.844698 hartree was obtained from the largest MRSDCI(m) calculation. The FCI 
limit energy was estimated by extrapolation ECI to wref = 1. Fig. 1 shows the convergence pattern of 
ECI, ECI+Q, and EAV against the weight of reference space. This extrapolations estimated FCI energy 
limit –37.84475 hartree. The rest many-body effect (~ –0.00005 hartree) is very small because the 
weight of reference space of the MRSDCI wave function is very near to 1: wref ≈ 0.998. 
Fig. 1. Convergence of CI energy with respect to the weight of the reference space.
Large-scale nonrelativistic multireference configuration interaction calculation of the ground state 3P of the carbon atom 7
The higher λ effect for the CI energies was also estimated by extrapolating the energy 
contributions due to the respective λ-NOs to λ→∞. Fig.2 shows the energy contribution to the CI 
energy from the respective λ-NOs, on a log-log scale. It reveals that the energy contribution by adding 
λ-NOs (–ΔEλ) converges linearly at λ ≥ 4. This correlation can be expressed as 
–ΔEλ = aλ
– b, (9)
where a and b are fitting parameters. The energy contribution from the NOs having angular 
momentum greater than 9 are estimated as Eq.(8). The energy contribution –0.00012 hartree was 
obtained. Thus, the energy –37.84487 hartree was obtained. Table 3 lists the energies of this work 
and the estimated exact value. The two extrapolations estimated the energy lowering ~ 0.00017 
hartree. The total energy of FCI limit + higher λ is quite near to the estimated exact value: the 
difference is ~ –0.0001 hartree. 
Fig. 2 Contributions to CI energy from the respective λ orbitals.
Table 3. Extrapolated energies and estimated exact value (in hartree).
Method Reference Energy
MRSDCI(m) This work –37.844698
FCI limit This work –37.84475
FCI limit + higher λ This work –37.84487
Estimated exact value Ref.16 –37.8450
8 CHUKYO UNIVERSITY Faculty of Liberal Arts and Sciences 2023 Vol. 4 No. 1
In summary, the ground state energy of the carbon atom –37.844698 hartree was given from 
the MRSDCI(m) wave function expanded with 6,235,756 CSFs. Furthermore, the FCI limit energy 
–37.84475 hartree and the FCI limit + higher λ energy –37.84487 hartree were estimated by the 
extrapolation techniques. Although the MRSDCI(m) energy is very near to the estimated exact 
value, the remaining many-body and higher λ effects are still large. The present calculation is 
probably the limit of the CI calculation for the carbon atom.
REFERENCES
1 C. de Boor, A practical guide to splines, (Springer, Berlin, Heidelberg, New York, 2001). 
2 J. Sapirstein and W.R. Johnson, J. Phys. B 29, 5213 (1996).
3 S.L. Saito, Theor. Chem. Acc. 109, 326 (2003).
4 S.L. Saito, At. Data and Nucl. Data Tables 95, 836 (2009).
5 T. L. Gilbert and P. J. Bertoncini, J. Chem. Phys. 61, 3026 (1974).
6 S.R. Langhoff and E.R. Davidson, Int. J. Quantum Chem. 8, 61 (1974).
7 C.C.J. Roothaan and P.S. Bagus, Meth. Comput. Phys. 2, 47 (1963).
8 F. Sasaki, Int. Quantum Chem. 8, 605 (1974).
9  F. Sasaki, M. Sekiya, T. Noro, K. Ohtsuki, and Y. Osanai, In: E. Clementi (ed) Modern techniques in 
computational chemistry, p.181 (MOTECC. ESCOM, Leiden, 1990).
10 A.W. Weiss, Phys. Rev. 162. 17 (1967).
11 H.F. Schaefer, III, R.A. Klemm, and F.E. Harris, Phys. Rev. 181, 137 (1969).
12 A. Bunge and C.F. Bunge, Phys. Rev. A1, 1599 (1970).
13 A. Bunge, J. Chem. Phys. 53, 20 (1970).
14 F. Sasaki and M. Yoshimine, Phys, Rev. A9,17 (1974).
15 M.B. Ruiz and R. Tröger, Adv. Quantum Chem. 76, 223, (2017).
16  S.J. Chakravorty, S.R. Gwaltney, E.R. Davidson, F.A. Parpia, and C. Froesse Fischer, Phys. Rev. A47, 3649 
(1993).


English

Large-scale nonrelativistic multireference configuration interaction calculation of the ground state 3P of the carbon atom

 中京企業研究 No.45, 2023.12　　　　　　もくじ
〔研究論文〕 
1970 年代相互銀行業界の戦略的迷走
　　  “相互”名称削除論議と信金業界の
　　  ｢ 信金らしさ ｣追求との比較史  　　 …………………………… 　由里　宗之　　　1
〔研究ノート〕 
気候危機・民主主義の危機と労使関係
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文化の経済社会論
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プロジェクトに関係した研究成果　……………………………………………………………… 　187


Japanese

目次　45

山田　誠治

ノンサーベイ法による地域産業連関表の精度の検証

論 文
小特集
奄美島唄という文化生産：録音メディア編・セントラル楽器を軸に
加　藤　晴　明   （1）
〈移動〉を描く
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自閉スペクトラム青年の余暇支援アプリケーションを
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曽我部　哲　也
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辻　井　正　次
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若者の社会参加と地域に対する愛着の関係
　　　A 町少年議会経験者の語りから
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片　山　寛　信
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中京大学
第 17 巻2023 第 1 号
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現代社会学部紀要
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【第Ⅰ部　講演等記録】1．2020年度中京大学教養教育研究院経験交流会　「オンライン授業における工夫と課題」

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中京大学社会科学研究所
一般論文
近代日本における伝染病予防法制の形成と展開
　 　　　愛知県西加茂郡三好村を事例として　 　
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