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

小木曽, 千代子

23526

Ogiso, Chiyoko

23527

Japanese

Teikin-no-sho owned by Toyo bunko (The Oriental Library)　－Reprinting (1)－

播磨良紀先生主な研究業績

酒井, 敏

23534

ある忘れられた画家と森鴎外　―安田稔との出会いと縁―

Professor Piccolo's Seminar
The students in our 2022 fourth-year Seminar wrote gradua-
tion essays on a wide-ranging choice of topics, from Disney ani-
mation, LGBTQ issues, and Japan's Takarazuka Revue to
Salvador Dali, Esports, and pop music.
In her essay, "Changing Gender Roles and Disney Princess
Movies", Hana Sakaguchi wrote about the popular sub-genre of
Disney animation: Disney Princess movies. She focused on how
the characteristics of the female protagonists have changed con-
siderably from the past to the present, just as the roles expected
of women in society have changed. By looking at three "genera-
tions" of Disney films, Hana was able to conclude that the most
important change is that the process leading up to the ending of
the movies has changed to one in which the princess herself takes
hold of her happiness instead of waiting for a prince to rescue
her.
Saki Sakakibara, in "A Comparison of Same-sex Marriage
Status in Japan and Canada", focuses on the history of same-sex
marriage in Japan and Canada, the current situation and chal-
lenges of Canada and Japan regarding legalizing same-sex mar-
riage, and the possibility of legalizing same-sex marriage in
Japan. Although Japan is the only G7 country that does not
recognize same-sex marriage, Saki concludes by noting that
groups such as #Equality Act Japan suggest a more positive fu-
ture by working to enact laws banning discrimination against
73
LGBTQ people.
Yuka Esaki chose to write about Shakespeare's Romeo and
Juliet in her graduation thesis, but with a unique focus: the cur-
rent musical production of the play by the Takarazuka Revue. In
her essay, "The Effect of the Characters of 'Love' and 'Death' in
the Takarazuka Revue Production of Romeo and Juliet", Yuka
focuses on the effect of the Revue's inclusion of two additional
characters in the play: personifications of the concepts of Love
and Death. In the essay, Yuka carefully works through the
Takarazuka's production of the play to show how this unconven-
tional approach manages to remain true to the original while
also providing a new and interesting way for audiences to view
the play.
In her essay, "The Impact of Dali's Trauma", Arisu Terasaki
considers Salvador Dali's art in relation to biographical elements
and events in his life, especially particular "traumas" that influ-
enced content and method of his artistic production. In addition,
Arisu considers the most important relationships in Dali's life
(his older brother; his mother; and his wife, Gala. By looking at
particular paintings produced by Dali, she show how these
events and relations in Dali's life were very influential in his
work and were not a negative thing, but rather a factor in the
creation of Dali's great works.
Inspired by his own experience with esports (electronic
sports), Yuta Okamoto chose this as the topic for his graduation
essay: "Why is Japan Behind in Esports?" He begins by giving
an overview of the history of esports in Japan, compares esports
74
in Japan with esports in South Korea, critiques why Japan is be-
hind in esport competition compared with other countries. In
particular, Yuta discusses why improved English ability of
Japanese players is necessary if Japan is to compete successfully
in esports at the international level.
In "The Difference between Western Pop Music and Japanese
Pop Music" Ayana Higashida looks at the relationship and influ-
ences between Western and Japanese pop music. She discusses
the differences between Japanese and Western pop music by fo-
cusing on four points: melody, tempo, rhythm, and chord pro-
gression. In conclusion, Ayana sees Western and Japanese
influencing each other even more than now.
Finally, this year it was particularly difficult to choose a best
essay from among some very well-written essays. However, due
to its in-depth research and well-argued thesis Hana Sakaguchi's
graduation thesis has been chosen as the best essay in our ourth-
year Seminar for 2022.
2022 Graduation Essay Awards 75


Professor Piccolo's Seminar

2022 年度活動報告
○ 2月 22 日 (水) 秋季特別講演会
講師：白井澄子氏 (白百合女子大学 元教授)
演題：『赤毛のアン』 においてアン・シャーリーは何を
体現するのか？
○ 2023 年 3 月 『中京英文学』 (第 43 号) 発行
〇 2023 年 3 月 25 日
International Conference "Writing Climate / Changing
Fictions: A one-day symposium organized by the the British
and American Cultural Studies Program, Faculty of World
Englishes, Chukyo University, with funding from Chukyo
University Society of British and American Cultural Studies
and the Ministry of Education, Culture, Sports, Science and
Technology (MEXT)"
Keynote Lecture: Dr, John Clement Ball (University of New
Bransuwick)
"Gun Island v. Blaze Island: Improbability, Risk and Eco-
Cosmopolitanism in Two Recent Climate-Change Novels."
151


2022年度活動報告

中村, 将人

23411

Nakamura, M.

23414

Costing of South Manchuria Railway’s Factories

齊藤, 史郎

23548

SAITO, Shiro L.

23549

1
Multireference configuration interaction study of 
positronium-alkali atom complexes
Saito Shiro L.
Abstract
Positronium (Ps) binding energies of positronium-alkali atom complexes, PsLi, PsNa, PsK, and 
PsRb, which were calculated by multireference configuration interaction (MRCI) method and 
extrapolation procedures are presented. In this work, a frozen-core approximation is applied. The Ps 
binding energies of PsLi and PsNa are good agreement with accurate values given from the stochastic 
variational method (SVM) with explicitly correlated Gaussian function. The Ps binding energy of PsK 
is larger than the SVM value. For PsRb, the Ps binding energy of 0.093 eV is obtained. To analyze 
accuracy of the Ps binding energies, electron affinities for Li, Na, K, and Rb are also calculated.
1. INTRODUCTION
Positronium (Ps) is a bound state of one electron (e–) and one positron (e+). The positronium 
forms bound states with various atoms. These are called positronium-atom complexes. There 
are many quantum mechanical studies for various positronium-atom complexes.1–22 It is revealed 
from those studies that several positronium-atom complexes are energetically stable. Quantum 
mechanical calculations for positronium-atom complexes require explicitly correlated functions, 
e.g., Hylleraas-type function or explicitly correlated Gaussian (ECG) function. Calculation methods 
with the explicitly correlated functions give very accurate results. Unfortunately, such methods 
are unsuitable for application to many-electronic systems because many-electron integrals appear. 
Therefore, the explicitly correlated functions are applied to only few-body systems.
Configuration interaction (CI) methods are extensively applied to quantum mechanical 
calculations for many-electronic systems. To obtain accurate CI wave functions, orbitals having high 
angular momentum are required for construction of CI wave functions. However, orbitals having 
2 CHUKYO UNIVERSITY Faculty of Liberal Arts and Sciences 2023 Vol. 3 No. 2
angular momentum less than ~10 are actually used because CI wave functions become very large. CI 
calculations of positronium-atom complexes require orbitals having quite high angular momentum. 
Hence, the energy contribution from orbitals having higher angular momentum has to be estimated 
with the help of an extrapolation procedure. Moreover, consideration of many-body effect is also 
important. In our previous works, we have performed multireference singly and doubly excited 
configuration interaction (MRSDCI) calculations for positronium hydride7,19, positronium halide17,18,20, 
positronium-carbon atom19, and so on.
Mitroy and co-workers have applied the stochastic variational method (SVM) with ECG to several 
positronium-atom complexes which have two valence electrons using a model potential.16 Their 
model potential is considered polarization effects. Their calculation method is called FCSVMpol. 
Positronium-atom complexes calculated by them are positronium hydride, positronium-alkali atoms, 
positronium-copper atom, and so on.16 Mitroy and co-workers have given highly accurate Ps binding 
energies for those systems.
In this work, we performed MRSDCI calculations for positronium-alkali atom complexes, PsLi, 
PsNa, PsK, and PsRb, and estimated the full CI (FCI) energy limits and higher angular momentum 
effect using extrapolation procedures. The Ps binding energies of PsLi, PsNa, and PsK given from our 
calculation method are compared with those by Mitroy and co-workers.16
  The Ps binding energy (EB) of positronium-atom complexes (PsA) is calculated by 
 (1)
where EPs, EA, and EPsA are the total energies of Ps, A, and PsA, respectively. The Ps binding energy 
can also be calculated using the electron affinity (EEA) of A and the positron ionization energy (EPI) 
of PsA as follows:
 (2)
From Eq.(2), one can see that accurate electron affinities and positron ionization energies are 
required, to calculate accurate Ps binding energy. Accuracy of the Ps binding energy can be analyzed 
by accuracy of the electron affinity of A at least, because there are experimental values of the 
electron affinity for various atoms. To analyze accuracy of Ps binding energy, we calculated the 
electron affinities and the positron ionization energies of PsA. 
In this work, we carried out Hartree-Fock (HF) calculations for the ground state (2S) of neutral 
alkali atoms Li, Na, K, and Rb, the ground state (1S) of anions of the alkali atom Li–, Na–, K–, and Rb–, 
and the ground state (2,1S) of the positronium-alkali atom complexes PsLi, PsNa, PsK, and PsRb. 
MRSDCI calculations with a frozen-core approximation were carried out for Li–, Na–, K–, Rb–, PsLi, 
PsNa, PsK, and PsRb using the natural orbitals (NOs). Hence, the reference configurations of the 
Multireference configuration interaction study of positronium-alkali atom complexes 3
MRSDCI calculation are [core]ns2 and [core]ns2 1s+
1 for A– and PsA, respectively. Here, the subscript 
‘+’ means positronic orbital.
The next section explains basis functions, NO construction procedures, and extrapolation 
procedures. The third section summarizes the Ps binding energies and the positron ionization 
energies for PsLi, PsNa, PsK, and PsRb, and discusses them in comparison with the works of other 
researchers. The accuracy of the Ps binding energies is also analyzed by calculating the electron 
affinities for Li, Na, K, and Rb.
2. COMPUTATIONAL ASPECT
The basis set used is B-splines described below. The B-spline set is one of piecewise polynomials, 
because this set is free from computational linear dependence and is very flexible.23,24 Therefore, 
all atomic orbitals of each system can be expanded using a common B-spline set regardless of 
the symmetry of the atomic orbitals. The present basis set consists of N Kth-order B-splines on 
a knot sequence defined on an interval [0,R]. A knot sequence was used with endpoints of K-fold 
multiplicity:
 (3)
where R1 is the initial interval and β is the parameter characterizing the distribution of the knots. 
Here β is decided to satisfy the following condition:
 (4)
Since the first and last terms of the B-splines with K-fold multiplicity are nonzero at r=0 and r=R, 
respectively, the N-term B-spline set was constructed omitting them. We adopted N=40, R=60 
(bohr), K=9. β was optimized by HF calculations for each system individually. 
Our MRSDCI calculations used the NOs with angular momentum λ up to 8 (i.e. l-symmetry). The 
NOs were generated by a series of MRSDCI calculations with reference spaces consisting of principal 
configurations of a single reference SDCI wave function (a ‘minimal reference space’). The minimal 
reference configurations of PsA are listed in Table 1.
Let us now explain the generating procedure for NOs. First, MRSDCI calculations were carried 
out with the HF orbitals, to obtain spdf-NOs. Subsequently, g-NOs were generated by MRSDCI 
calculations with the spdf-NOs. The NOs with higher λ than g were generated in the same way, step 
by step. Those NOs whose occupation number was less than 10-7 were truncated at each step because 
a wave function constructed with non-truncated NOs is very large. 
To estimate the FCI energy limits and the higher λ effects for the total energies, a further series 
4 CHUKYO UNIVERSITY Faculty of Liberal Arts and Sciences 2023 Vol. 3 No. 2
of MRSDCI calculations was carried out using obtained NOs, increasing the reference configurations 
which were selected for the largest weight in the previous MRSDCI wave functions. These 
calculations were continued until wref≈0.99, where wref is the weight of reference configurations. We 
refer the reference space finally constructed as a ‘maximum reference space’.
Table 1. Minimal reference configurations of PsLi, PsNa, PsK, and PsRb.
PsLi PsNa PsK PsRb
2s21s+ 3s
21s+ 4s
21s+ 5s
21s+
2s3s2s+ 3s4s2s+ 4s5s2s+ 5s6s2s+
2s2p2p+ 3s3p2p+ 4s4p2p+ 5s5p2p+
2s3p3p+ 3s4p3p+ 4s5p3p+ 5s6p3p+
2s3d3d+ 3s3d3d+ 4s3d3d+ 5s4d3d+
2s4d4d+ 3s4d4d+ 4s4d4d+ 5s5d4d+
2s4f4f+ 3s4f4f+ 4s4f4f+ 5s4f4f+
The FCI limits of the MRSDCI energies (ECI) were estimated by extrapolating ECI to wref=1. To this 
end, the convergence pattern of ECI, ECI+Q, and EAV were analyzed with respect to wref. ECI+Q is ECI plus 
the Davidson correction20. EAV is the average of ECI and ECI+Q. 
The higher λ effect for the total 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 total energies to λ→∞ using ECI with wref≈0.99. The energy contribution ΔEλ was 
plotted on a log-log scale, and it was fitted by the following relation with four parameters α, β, γ, and 
ε26:
 (5)
where λc is the point at which the derivatives of the two functions (5)                            are equal, and 
is expressed as
 (6)
The first function of Eq.(5) is equal to the second function of Eq.(5) in case of α=4, β=1, and γ=log10ε. 
The contribution from the NOs having λ greater than 8 (Eλ>8) was estimated as 
 (7)
Multireference configuration interaction study of positronium-alkali atom complexes 5
We finally obtained the FCI limits including the higher λ effect.
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 Bagus.27,28 All CI calculations were performed 
by the program ATOMCI29,30 modified for atomic systems containing positrons.
3. RESULTS AND DISCUSSIONS
Table 2 summarizes the results of the MRSDCI calculations with the minimal reference space and 
the maximum reference space for PsLi, PsNa, PsK, and PsRb. The loses of the MRSDCI energies due 
to the NO truncation are very small: the maximum lose is 0.00002 hartree (~0.0005 eV). The energy 
loses may not affect the positron ionization energies and the Ps binding energies. The weights of the 
MRSDCI wave function with the maximum reference configuration are very close to 1. 
Table 2.  Results of MRSDCI with minimal and maximum reference spaces. Δε is loss of total energies due 
to the NO truncation procedure and NCI is the number of terms of MRSDCI wave function.
Reference spaces NCI wref ECI (hartree) Δε (hartree)
PsLi minimal 26515 0.897407 –7.683650 0.000020
maximum 144581 0.991153 –7.683797
PsNa minimal 24728 0.884815 –162.103970 0.000016
maximum 138796 0.991190 –162.104139
PsK minimal 22175 0.845244 –599.402598 0.000009
maximum 118264 0.988084 –599.402879
PsRb minimal 21298 0.829349 –2938.592397 0.000008
　 maximum 116153 0.987129 –2938.592728
Table 3 lists the weights of principal configurations of the MRSDCI wave functions with the 
maximum reference space for PsLi, PsNa, PsK, and PsRb. The weight of [core] ns21s+ configuration 
for all the systems is quite small: it is less than 0.5. The weights of npn'p+ and ndn'd+ excited 
configurations are large. For PsRb, the weight of 5p2p+ exited configuration is almost same as that of 
[core] 5s21s+ configuration. This shows that PsRb has larger the diatomic molecule-like structure Ps-
Rb than the atomic structure e+-Rb–. The structure of the positronium-alkali atom complexes mainly 
is the diatomic molecule-like character. This situation is larger with an increase of the atomic number 
for the alkali atoms.
6 CHUKYO UNIVERSITY Faculty of Liberal Arts and Sciences 2023 Vol. 3 No. 2
Table 3. Weights of principal configurations of wave functions for PsLi, PsNa, PsK, and PsRb.
PsLi PsNa PsK PsRb
weight configuration weight configuration weight configuration weight configuration
0.419 2s21s+ 0.386 3s
21s+ 0.299 4s
21s+ 0.274 5s
21s+
0.282 2s2p2p+ 0.279 3s3p2p+ 0.285 4s4p2p+ 0.280 5s5p2p+
0.102 2s3d3d+ 0.117 3s3d3d+ 0.143 4s3d3d+ 0.153 5s4d3d+
0.039 2s4f4f+ 0.048 3s4f4f+ 0.062 4s4f4f+ 0.068 5s4f4f+
0.024 2s3p3p+ 0.023 3s4p3p+ 0.028 4s5g5g+ 0.031 5s5g5g+
0.016 2s5g5g+ 0.020 3s5g5g+ 0.022 4s5p3p+ 0.021 5s6p3p+
We estimated FCI limits of ECI by extrapolating ECI to wref=1. Fig. 1 shows the convergence pattern 
of ECI, ECI+Q, and EAV against the weight for PsLi, PsNa, PsK, and PsRb. ECI for all the systems have 
no change at wref≈0.99. Hence, all the FCI limits can be regarded as the total energy given from the 
MRSDCI calculations with the maximum reference configuration. 
Fig. 1.  Convergence of CI energies for the positronium-alkali atom complexes with respect to the weight 
of the reference space.
Multireference configuration interaction study of positronium-alkali atom complexes 7
Fig. 2 shows the energy contributions from the respective λ-NOs, on a log-log scale. The 
parameters of Eq.(5) were decided by fitting –ΔEλ with Eq.(5). The energy contributions from the 
NOs having λ greater then 8 (Eλ>8) were estimated as Eq.(7). The resulting FCI limits and the higher λ 
contributions for the total energies are summarized in Table 4. As can be seen in Fig. 2, as the atomic 
number of the alkali atoms is larger, –ΔEλ is larger. The functional form is similar to each other. 
Table 4. Full CI limits (EFCI) and higher λ contributions (EL>8) for total energies (in hartree).
EFCI EL>8 EFCI+EL>8
Li– –7.455360 –0.000001 –7.455361
PsLi –7.683797 –0.010299 –7.694096
Na– –161.878906 –0.000001 –161.878907
PsNa –162.104139 –0.012369 –162.116508
K– –599.183042 –0.000002 –599.183044
PsK –599.402879 –0.016533 –599.419412
Rb– –2938.374897 –0.000006 –2938.374903
PsRb –2938.592728 –0.018142 –2938.610870
Fig. 2. Contributions to the total energies from the respective λ orbitals.
8 CHUKYO UNIVERSITY Faculty of Liberal Arts and Sciences 2023 Vol. 3 No. 2
Table 5 summarizes the positron ionization energies and the Ps binding energies for PsLi, PsNa, 
PsK, and PsRb, together with the works of other researchers. The most accurate Ps binding energies 
are given from FCSVMpol calculations presently. PsLi has been calculated by several methods. 
However, their Ps binding energies are no good agreement with the FCSVMpol values except 
for diffusion Monte Carlo (DMC) performed by Bressanini et al.10 The second-order variational 
perturbation theory (VPT) calculation8 slightly improves the HF value. However, the Ps binding 
energy given from VPT is negative value. The Ps binding energies given from density functional 
theory (DFT) calculation performed by Harrison3 are almost same as the HF value. DFT performed 
by Kanhere et al.4 gives the Ps binding energies which are inconsistent with the FCSVMpol value. Our 
Ps binding energy of PsLi are good agreement with the FCSVMpol value. The higher λ effect greatly 
increases the Ps binding energy for PsLi. This situation is similar for the Ps binding energies of the 
other positronium-alkali atom complexes.
Table 5. Positron ionization energies (EPI) and Ps binding energies (EB) (in eV).
PsLi PsNa PsK PsRb
EPI EB EPI EB EPI EB EPI EB
This work
HF 2.766 –4.159 2.618 –4.288 2.247 –4.634 2.135 –4.737
minimal MRSDCI 6.212 0.025 6.124 –0.134 5.974 –0.332 5.918 –0.410
FCI limit 6.216 0.029 6.129 –0.130 5.982 –0.324 5.927 –0.401
FCI limit + higher λ 6.496 0.309 6.465 0.207 6.432 0.126 6.421 0.093
Other works
HFa 2.776 –4.163
VPTb 3.990 –2.190
DFTc 2.776 –4.027
DFTd 8.844 2.667
DMCe 6.97 0.78
DMCf 6.506 0.261
VMCf 0.160 –6.117
FCSVMpol 
g 6.517 0.336 6.487 0.229 6.399 0.089
a Patrick and Cade (Ref.2).
b Saito (Ref.8).
c Harrison (Ref.3).
d Kanhere et al. (Ref.4).
e Yoshida and Miyako (Ref.9).
f Bressanini et al. (Ref.10).
g Mitroy and Ryzhikh (Ref.16).
Multireference configuration interaction study of positronium-alkali atom complexes 9
In comparison with the FCSVMpol values, our Ps binding energies for PsLi and PsNa are 
underestimated. However, our Ps binding energy for PsK is larger than that of FCSVMpol: this 
difference is 0.037 eV. To analyze this difference, we calculated the electron affinity of K together 
with that of Li, Na, and Rb and list them in Table 6. Our electron affinities are good agreement with 
the FCSVMpol
16 and experimental values32. Our electron affinity of K is larger than the FCSVMpol value 
by 0.005 eV. This difference may not affect the Ps binding energy for PsK. The higher λ effect makes 
the Ps binding energy of PsK higher than the FCSVMpol value. If the FCSVMpol value for PsK is correct, 
the energy contribution for λ>8 must be small value. Namely, the fitting curve of PsK in Fig. 2 must 
be steeper than that of PsLi and PsNa. The fitting curve form for PsK is similar to that of PsLi and 
PsNa. Therefore, our estimation of the higher λ effect could be correct. Conclusively, the FCSVMpol 
calculation of PsK is not convergent yet, and the Ps binding energy of PsK must be much large. 
Table 6. Electron affinities of alkali atoms (in eV).
Li Na K Rb
This work
SDCI 0.616 0.544 0.497 0.475
Other works
FCSVMpol 0.619a 0.545a 0.492b
Experimentc 0.618 0.548 0.501 0.486
a Mitroy and Ryzhikh (Ref.15).
b Mitroy and Ryzhikh (Ref.31).
c Hotop and Lineberger (Ref.32).
We obtained the Ps binding energy of PsRb 0.093 eV. The electron affinity of Rb which we 
calculated is smaller than the experimental value by 0.011 eV. Hence, our Ps binding energy of 
PsRb may is small by ~0.01 eV. To improve the Ps binding energy of PsRb, MRSDCI calculations 
considering the core-valence interaction have to be carried out. 
REFERENCES
1. D.M. Schrader and Y.C. Jean (ed), Positron and positronium chemistry, (Elsevier, Amsterdam, 1988).
2. A.J. Patrick and P.E. Cade, J. Chem. Phys. 75, 1893 (1981).
3. J.G. Harrison, J. Chem. Phys. 84, 1659 (1986).
4. D.G. Kanhere, A. Kshirsagar, V. Bhamre, Chem. Phys. Lett. 160, 526 (1988).
5. D.M. Schrader, T. Yoshida, and K. Iguchi, Phys. Rev. Lett. 68, 3281 (1992).
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6. D.M. Schrader, T. Yoshida, and K. Iguchi, J. Chem. Phys. 98, 7185 (1993).
7. S.L. Saito and F. Sasaki, J. Chem. Phys. 102, 8040 (1995).
8. S.L. Saito, Chem. Phys. Lett. 245, 54 (1995).
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23. C. de Boor, A practical guide to splines, (Springer, Berlin, Heidelberg, New York, 2001). 
24. J. Sapirstein and W.R. Johnson, J. Phys, B 29, 5213 (1996).
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29. F. Sasaki, Int. Quantum Chem. 8, 605 (1974).
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32. H. Hotop and W.C. Lineberger, J. Phys. Chem. Ref. Data 14, 731 (1985).


Multireference configuration interaction study of positronium-alkali atom complexes

多田, 哲

23550

TADA, Satoshi

23551

Creation of the Cathedral Quarter in Lyon: A Historical Approach

小峰, 総一郎

23552

KOMINE, Soichiro

23553

Patel, Kiran Klaus: „Die »Volkserziehungsschule« ―Der Arbeitsdienst für Männer und Frauen“ (2011)

中村, 努

23554

NAKAMURA, Tsutomu

23555

Rob Kitchin: Civil liberties or public health, or civil liberties and public health? Using surveillance technologies to tackle the spread of COVID-19

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