Effect of removing the no-virtual pair approximation on the correlation energy of the He isoelectronic sequence. II. Point nuclear charge model.

Abstract

The correlation energies (CEs) of the He isoelectronic sequence Z=2-116 with a point nuclear charge model were investigated with the four component relativistic configuration interaction method. We obtained CEs with and without the virtual pair approximation which are close to the values from Pestka et al.'s Hylleraas-type configuration interaction calculation. We also found that the uniform charge and point charge models for the nucleus differ substantially for Z > or = 100.

Full text

Page 1

Effect of removing the no-virtual pair approximation on the correlation
energy of the He isoelectronic sequence. II. Point nuclear charge model
Yoshihiro Watanabe,1Haruyuki Nakano,1,2and Hiroshi Tatewaki3,a?
1Department of Chemistry, Faculty of Sciences, Kyushu University, Fukuoka 812-8581, Japan
2CREST, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan
3Graduate School of Natural Sciences, Nagoya City University, Nagoya, Aichi 467-8501, Japan
?Received 25 December 2009; accepted 22 February 2010; published online 24 March 2010?
The correlation energies ?CEs? of the He isoelectronic sequence Z=2–116 with a point nuclear
charge model were investigated with the four component relativistic configuration interaction
method. We obtained CEs with and without the virtual pair approximation which are close to the
values from Pestka et al.’s Hylleraas-type configuration interaction calculation. We also found that
the uniform charge and point charge models for the nucleus differ substantially for Z?100. © 2010
American Institute of Physics. ?doi:10.1063/1.3359857?
I. INTRODUCTION
The correlation energy ?CE? is defined as the difference
between the total energy ?TE? calculated with electronic cor-
relations included and that calculated by the Hartree–Fock
method. The nonrelativistic CE of the He isoelectronic se-
quence is almost constant for atoms heavier than6C.1–4In
contrast, the relativistic CE of these strongly depends on the
atomic number Z. Pestka et al.5discovered this using the
relativistic Hylleraas-type configuration interaction ?Hy-CI?
method. Pestka et al.,6Tatewaki et al.,7and Watanabe et al.8
found similar Z dependence, using multiconfiguration Dirac–
Fock, third-order Douglas–Kroll CI, and four component
Dirac–Fock–Roothaan CI ?DFR-CI? calculations with DFR
1s+spinor and plural numbers of the s, p, d, and f primitive
Gaussian type functions ?pGTFs?, respectively. The Z depen-
dence of the CE in the latter6–8calculation was stronger than
that of Hy-CI.5We shall use the notations TE?I? and CE?I?,
where I in the parentheses refers to the DFR or the correlated
method for calculating the TE and CE: I=DFR-CI or Hy-CI.
We recently showed9that the over estimates of CEs for
heavier atoms in the DFR-CI ?Ref. 8? were due to the no-
virtual pair approximation ?NVPA?,10–12where excitations to
the Dirac negative sea were prohibited. Pestka et al.13,14per-
formed unprojected and projected Hy-CI calculations, where
the latter uses the Hy-type basis sets giving positive kinetic
energies. The projected Hy-CI ?Ref. 14? corresponds to the
DFR-CI with NVPA, and the unprojected Hy-CI ?Ref. 13?
corresponds to the DFR-CI with VPA; we shall use the sym-
bol “VPA” when the calculations are performed without
NVPA. We abbreviate the TE and CE given by the projected
Hy-CIandunprojectedHy-CI
CENVPA?Hy-CI?, TEVPA?Hy-CI?, and CEVPA?Hy-CI?, respec-
tively. Using the uniform charge ?UC? model for the nucleus,
we have found that CEVPA?DFR-CI? ?Ref. 9? is reasonably
parallel to CEVPA?Hy-CI?,13but the difference between the
toTENVPA?Hy-CI?,
two increases as the nuclear charge increases; for example,
the difference is 0.4 mhartrees at40Zr and 4.5 mhartrees at
116Uuh.
The aim of the present work is to obtain the exact TEs
under the Dirac–Coulomb Hamiltonian which is widely used
in the atomic and molecular electronic calculations, and the
quantum electrodynamical terms are not included. The new
pGTF basis set is developed for the point nuclear charge
model to attain this objective. The TEs by DFR and by
DFR-CI are calculated and the resulting CEs are discussed.
Almostperfectagreement
CEVPA?Hy-CI? and the present CEVPA?DFR-CI?, and the dif-
ference between the CENVPA?Hy-CI? and the present
CENVPA?DFR-CI? is analyzed. Since the ground state of the
He-like ions are Feshbach resonance state15–17of the elec-
tronic states having negative kinetic energies, we discuss the
validity of the present results with the stabilization
method.18–21Throughout this work we adopt the atomic
units.
isfoundbetweenthe
II. METHOD
The Dirac–Coulomb Hamiltonian is used, where we take
the nucleus to be a point charge ?PC?. The calculation pro-
cedure is as follows.
We first determined a universal GTF basis set. An accu-
rate basis set that gives the numerical Dirac–Fock ?NDF?
limit is needed, since the CE is defined as
CE = TE?DFR-CI? − TE?DFR?.
?1?
Next, we performed NVPA DFR-CIs with DFR 1s+
spinor and s, p, d, f, and g pGTFs, using exponents with
coefficients greater than 1?10−3or 1?10−2in the 1s+
spinor. The numbers of pGTFs selected vary from one atom
to another. For example, these are ?29+1??2, 29?6, 29
?10, 29?14, and 29?18 for s, p, d, f, and g pGTFs for
116Uuh if the threshold is 1?10−3, and ?18+1??2, 18?6,
18?10, 18?14, and 18?18 for s, p, d, f, and g pGTFs if
the threshold is 1?10−2?“+1” means that the DFR 1s+
spinor is added in case of the s basis set?. From the selected
a?Electronic mail: htatewak@nsc.nagoya-cu.ac.jp.
THE JOURNAL OF CHEMICAL PHYSICS 132, 124105 ?2010?
0021-9606/2010/132?12?/124105/7/$30.00 © 2010 American Institute of Physics
132, 124105-1

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pGTFs, we construct an equal number of orthogonalized
spinors for the CI calculations. The total number of spinors
for the CI calculations are 1452 and 902, including the 1s+
DFR spinor for the smaller and larger threshold, respec-
tively. Five types of CI calculations were performed:
s-CI
?1s2→s?s??,
spd-CI ?1s2→?s? or p? or d???s? or p?or d???, spdf-CI
?1s2→?s? or p? or d? or f???s? or p? or d? or f???, and
spdfg-CI
?1s2→?s? or p? or d? or f? or g???s? or p?
or d? or f? or g???. The CE of the respective CIs is given
by
sp-CI
?1s2→?s? or p???s? or p???,
CEi
NVPA= TENVPA?i-CI? − TE?DFR?,
?2?
?i = s,sp,spd,spdf,spdfg?.
The numbers of dimensions for the respective CIs in the case
of116Uuh are 900, 8585, 29 726, 39 083, and 70 480 for the
s-, sp-, spd-, spdf-, and spdfg-CI calculations, where the
thresholds for selecting the pGTFs are 1?10−3for the first
three CI calculations, and for spdf-CI the thresholds are
1?10−2for s, p, and d pGTFs and 1?10−3for f pGTFs; for
spdfg-CI the thresholds are 1?10−2for s, p, d, and f pGTFs
and 1?10−3for g pGTFs. The accuracy of the spdf- and
spdfg-CI can be questioned, but we can safely discuss the
resulting CEs resulting from this CI calculation ?see below?.
To clarify the correlation effects from the s, p, d, f, and
g spinors and obtain an accurate TE, we define the partial CE
of the s, p, d, f, and g symmetries8,9via the following equa-
tions:
CEp
NVPA= CEsp
NVPA− CEs
NVPA,
CEd
NVPA= CEspd
NVPA− CEsp
NVPA,
?3?
CEf
NVPA= CEspdf
NVPA− CEspd
NVPA,
CEg
NVPA= CEspdfg
NVPA− CEspdf
NVPA.
When we calculate CEf
with a selection threshold of 1?10−2except for the f and g
pGTFs, as noted before. For116Uuh, where the truncation
error is expected to be greatest, we calculated two CEf
one of which is obtained as defined above and the other from
the basis set with a threshold of 1?10−3used for all the
symmetries s−f. The difference between the two CEf
0.0000 36 mhartrees, confirming that the error in evaluating
CEf
small. We expect the calculated CEg
racy, as in the case of CEf
TENVPA?DFR-CI? = TE?DFR? + CENVPA,
NVPAand CEg
NVPA, we use the pGTFs
NVPAs,
NVPAs is
NVPAif the truncated s, p, and d basis sets are used is
NVPAto have high accu-
NVPA. From these equations we have
where
CENVPA=?
i=s
g
CEi
NVPA.
?4?
The TENVPA?DFR-CI? is obtained from Eq. ?4?, where
CEi=f,g
Thirdly DFR-CI calculation without NVPA is performed
using a modified DFR-CI program,22where DFR spinors
NVPAis calculated using smaller basis sets.
TABLE I. TE by DFR and CEs, CENVPAand CEVPA, from CI with the s, p,
d, f, and g spinors in hartrees.
Z
TE?DFR?
NVPA VPA
2
3
4
5
6
7
8
9
?2.861 813
?7.237 206
?13.614 001
?21.993 149
?32.375 989
?44.764 201
?59.159 794
?75.565 105
?93.982 800
?114.415 873
?136.867 655
?161.341 806
?187.842 329
?216.373 565
?246.940 200
?279.547 271
?314.200 165
?350.904 626
?389.666 763
?430.493 051
?473.390 336
?518.365 844
?565.427 188
?614.582 370
?665.839 791
?719.208 260
?774.696 997
?832.315 648
?892.074 289
?953.983 434
?1018.054 051
?1084.297 567
?1152.725 881
?1223.351 374
?1296.186 924
?1371.245 915
?1448.542 253
?1528.090 380
?1609.905 287
?1694.002 532
?1780.398 253
?1869.109 190
?1960.152 698
?2053.546 771
?2149.310 059
?2247.461 889
?2348.022 289
?2451.012 013
?2556.452 560
?2664.366 208
?2774.776 033
?2887.705 947
?3003.180 723
?3121.226 026
?3241.868 455
?3365.135 570
?3491.055 938
?0.041 838
?0.043 221
?0.043 947
?0.044 386
?0.044 679
?0.044 886
?0.045 039
?0.045 155
?0.045 246
?0.045 317
?0.045 376
?0.045 423
?0.045 462
?0.045 494
?0.045 522
?0.045 546
?0.045 567
?0.045 585
?0.045 602
?0.045 618
?0.045 633
?0.045 648
?0.045 663
?0.045 678
?0.045 694
?0.045 711
?0.045 729
?0.045 749
?0.045 770
?0.045 793
?0.045 818
?0.045 846
?0.045 875
?0.045 907
?0.045 941
?0.045 979
?0.046 019
?0.046 063
?0.046 109
?0.046 159
?0.046 212
?0.046 269
?0.046 331
?0.046 395
?0.046 464
?0.046 537
?0.046 615
?0.046 697
?0.046 784
?0.046 876
?0.046 973
?0.047 074
?0.047 182
?0.047 294
?0.047 414
?0.047 538
?0.047 669
?0.041 838
?0.043 221
?0.043 946
?0.044 386
?0.044 677
?0.044 883
?0.045 033
?0.045 146
?0.045 232
?0.045 299
?0.045 349
?0.045 388
?0.045 417
?0.045 437
?0.045 450
?0.045 457
?0.045 460
?0.045 457
?0.045 449
?0.045 439
?0.045 424
?0.045 407
?0.045 386
?0.045 363
?0.045 338
?0.045 309
?0.045 279
?0.045 245
?0.045 211
?0.045 175
?0.045 136
?0.045 095
?0.045 054
?0.045 011
?0.044 968
?0.044 925
?0.044 876
?0.044 824
?0.044 775
?0.044 726
?0.044 685
?0.044 624
?0.044 564
?0.044 511
?0.044 457
?0.044 402
?0.044 348
?0.044 294
?0.044 232
?0.044 175
?0.044 119
?0.044 064
?0.044 010
?0.043 954
?0.043 892
?0.043 838
?0.043 785
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
124105-2 Watanabe, Nakano, and TatewakiJ. Chem. Phys. 132, 124105 ?2010?

Page 3

with negative kinetic energies are utilized; for example, we
performed CI with 1s2→s?s?, where s? and s? include
spinors with positive and negative kinetic energies. The total
number of GTFs used is twice that of the NVPA calculations;
in the case of116Uuh there are 2904 and 1804 GTFs for the
smaller and larger thresholds, respectively. Equations parallel
to ?2? and ?3? are obtained for the case of VPA. Here, we
only write the equation corresponding to Eq. ?4?
TEVPA?DFR-CI? = TE?DFR? + CEVPA
and
CEVPA=?
i=s
g
CEi
VPA.
?5?
To obtain CEVPAit is necessary to perform s-, sp-, spd-,
spdf-, and spdfg-CI calculations with VPA. We were able to
handle s- and sp-CI calculations having dimensions 3600
and 34 300, respectively, but could not perform spd-, spdf-,
and spdfg-CI calculations since the dimension of the Hamil-
tonian matrix becomes large. We can obtain CEs
CEp
CEi=d,f,g
?CEd= CEd
VPAand
VPAbut not CEi=d,f,g
VPA, we introduce ?CEi=d,f,gas given below
VPA− CEd
VPA. Instead of directly calculating
NVPA
= CEspd
? CEsd
VPA− CEsp
VPA− ?CEspd
NVPA− CEsp
VNPA?
VPA− CEs
VPA− ?CEsd
NVPA− CEs
VNPA?
= CEsd
VPA− CEsd
NVPA− ?CEs
VPA− CEs
VNPA?
= CEsd
VPA− CEsd
NVPA− ?CEs.
?6?
Likewise we have
?CEf= CEsf
VPA− CEsf
NVPA− ?CEs,
and
?CEg= CEsg
VPA− CEsg
NVPA− ?CEs.
?7?
Here, we calculate sd-, sf-, and sg-CIs instead of spd-,
spdf-, and spdfg-CIs; for116Uuh we found that the approxi-
mation in the third row of Eq. ?6? gives an error of 0.018
mhartrees relative to ?CEdgiven by VPA sp- and spd-CIs
with a truncation threshold of 1?10−3. We can disregard this
small error of the order 0.01 mhartrees when ?CEdis calcu-
lated. We also expect smaller errors than this for ?CEi=f,g.
The dimensions of sd-, sf-, and sg-CIs in the space VPA are
47 796, 61 252, and 74 708, whereas those in NVPA are
11 949, 15 313, and 18 677. We obtain CEd
CEg
CEi
Using Eqs. ?5? and ?8?, we now obtain TEVPAwithout calcu-
lating spd-, spdf-, and spdfg-CI.
VPA, CEf
VPA, and
VPAin Eq. ?5? as
VPA= CEi
NVPA+ ?CEi?i = d,f, and g?.
?8?
III. RESULTS AND DISCUSSION
A. Basis set
The CE of the He-like ions was considered up to116Uuh.
We first applied the previous universal-like GTFs8,9for the
UC model to calculate the He-like ions with the PC model.
TABLE I.
?Continued.?
Z
TE?DFR?
NVPA VPA
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
?3619.659 168
?3750.975 957
?3885.038 136
?4021.878 716
?4161.531 939
?4304.033 336
?4449.419 779
?4597.729 550
?4749.002 397
?4903.279 611
?5060.604 098
?5221.020 454
?5384.575 056
?5551.316 147
?5721.293 932
?5894.560 687
?6071.170 860
?6251.181 199
?6434.650 872
?6621.641 606
?6812.217 837
?7006.446 863
?7204.399 017
?7406.147 856
?7611.770 352
?7821.347 116
?8034.962 629
?8252.705 496
?8474.668 726
?8700.950 032
?8931.652 160
?9166.883 251
?9406.757 237
?9651.394 272
?9900.921 208
?10 155.472 123
?10 415.188 900
?10 680.221 868
?10 950.730 516
?11 226.884 284
?11 508.863 446
?11 796.860 100
?12 091.079 274
?12 391.740 175
?12 699.077 592
?13 013.343 489
?13 334.808 819
?13 663.765 581
?14 000.529 188
?14 345.441 188
?14 698.872 397
?15 061.226 553
?15 432.944 569
?15 814.509 526
?16 206.452 571
?16 609.359 921
?17 023.881 245
?17 450.739 770
?0.047 806
?0.047 950
?0.048 100
?0.048 258
?0.048 423
?0.048 596
?0.048 776
?0.048 965
?0.049 161
?0.049 368
?0.049 583
?0.049 807
?0.050 042
?0.050 286
?0.050 542
?0.050 809
?0.051 087
?0.051 378
?0.051 681
?0.051 997
?0.052 328
?0.052 673
?0.053 033
?0.053 409
?0.053 802
?0.054 213
?0.054 643
?0.055 092
?0.055 561
?0.056 053
?0.056 568
?0.057 108
?0.057 673
?0.058 265
?0.058 887
?0.059 542
?0.060 228
?0.060 949
?0.061 709
?0.062 511
?0.063 356
?0.064 247
?0.065 189
?0.066 187
?0.067 243
?0.068 364
?0.069 554
?0.070 822
?0.072 172
?0.073 613
?0.075 154
?0.076 808
?0.078 583
?0.080 495
?0.082 560
?0.084 795
?0.087 221
?0.089 868
?0.043 733
?0.043 687
?0.043 639
?0.043 586
?0.043 539
?0.043 497
?0.043 423
?0.043 377
?0.043 334
?0.043 306
?0.043 245
?0.043 218
?0.043 195
?0.043 184
?0.043 152
?0.043 134
?0.043 113
?0.043 096
?0.043 106
?0.043 105
?0.043 088
?0.043 101
?0.043 121
?0.043 149
?0.043 187
?0.043 195
?0.043 201
?0.043 275
?0.043 391
?0.043 486
?0.043 585
?0.043 679
?0.043 815
?0.043 923
?0.044 094
?0.044 275
?0.044 445
?0.044 649
?0.044 863
?0.045 194
?0.045 453
?0.045 751
?0.046 088
?0.046 531
?0.046 990
?0.047 428
?0.047 914
?0.048 556
?0.049 292
?0.049 929
?0.050 615
?0.051 423
?0.052 374
?0.053 351
?0.054 544
?0.055 755
?0.057 130
?0.058 641
124105-3Effect of removing the no-virtual-pair approx.J. Chem. Phys. 132, 124105 ?2010?

Page 4

The errors in TE?DFR? with the PC model are less than
0.408 hartrees, but they are large if we compare them with
those ??0.01 ?hartrees? from the UC model. They are not
sufficiently accurate to calculate the CE given by Eq. ?1?,
where TE?DFR? is required to have the NDF accuracy. We
therefore develop new even-tempered basis sets. After sev-
eral test calculations, the range of exponents was found to lie
between 4.00?10−2and 1.60?1022, where the largest expo-
nent is 4.7?1010times larger than that of the UC model
?exponents for the UC model are between 5.88?10−2and
3.37?1011?. The test basis set was composed of 90 functions
where even-tempered exponents are assumed. The errors in
TE?DFR? calculated with this set from TE?NDF? are less
than 1.0 ?hartrees for all atomic ions considered. We found
that a plot of CE versus the atomic number Z has anomalies,
however: a sharp increase and a sharp decrease in the CE are
observed at several atoms. We suspect that this is due to
insufficiency in the number of the basis set. The exponent
parameters in even-tempered basis sets are determined by
Eq. ?9?. We increase the number of a basis set by changing ?
where we fix ?=0.04. We finally settle on a universal set
composed of 136 s-type pGTFs, where ? is close to 1.5 and
near to the value used in the Refs. 8 and 9
?n= ??n−1?? = 0.04, = 1.495 650, n = 1, ... ,136?.
?9?
The errors in TE?DFR? calculated with this set from
TE?NDF? are again less than 1.0 ?hartrees, but we found no
difficulties in calculating CEs as in the case of 90 expansion
terms. This sequence is also used for the p, d, f, and g
spinors.
B. CE with NVPA
We performed NVPA DFR-CIs with DFR 1s+spinor and
s, p, d, f, and g pGTFs, with exponents having coefficients
greater than 1?10−3or 1?10−2in the 1s+spinor, following
the discussion relating to Eqs. ?3? and ?4?. The TE?DFR? and
CENVPA?DFR-CI? given by Eq. ?4? with s, p, d, f, and g
spinor
CEVPA?DFR-CI?, which will be discussed below. Figure 1
also shows CENVPA?DFR-CI? and CENVPA?Hy-CI? together
with two CEVPAs.
We see that the present two CENVPA?DFR-CI?s and
CENVPA?Hy-CI? decrease sharply as the atomic number Z
increases. The difference between the spdfg-CI and the
Hy-CI CENVPAs begins from 0.20 mhartrees at
reaches 3.46 mhartrees at116Uuh. We now discuss these dif-
ferences. The partial CEs in NVPA, CEi
Figs. 2?a? and 2?b?. The CEs
creases; it has a maximum of ?14.7 mhartrees at16S and a
minimum of ?54.7 mhartrees at116Uuh. The changes in
CEs
mum of ?21.5 mhartrees at2He and a minimum of ?29.1
mhartrees at116Uuh, where the change in CEp
mhartrees. On the other hand, CEi?2
when Z increases: ?1? CEd
mhartrees at
116Uuh, where the change in CEd
CEs
at2He and a minimum of ?1.3 mhartrees at116Uuh; and ?3?
the CEg
minimum of ?0.6 mhartrees at116Uuh, where the change in
CEf
that thedifference between
CENVPA?Hy-CI? reaches 3.46 mhartrees for116Uuh. Variation
in CEi
count of the CEs from the correlating orbitals with beyond g,
it becomes difficult to explain the differences between
CENVPA?DFR-CI? and CENVPA?Hy-CI?; geometrical extrapo-
lation gives
setsaresetoutin TableI, togetherwith
2He and
NVPAs are set out in
NVPAchanges sharply as Z in-
NVPA?DFR-CI? reach 40.0 mhartrees. CEp
NVPAhas a maxi-
NVPAis 7.6
NVPAchanges moderately
NVPAhas a maximum of ?2.3
2He and a minimum of ?4.3 mhartrees at
NVPAis 2.0 mhartrees, 5% of
NVPA; ?2? the CEf
NVPAhas a maximum of ?0.6 mhartrees
NVPAhas a maximum of ?0.2 mhartrees at2He and a
NVPAis 0.4 mhartrees, 1% part of CEs
NVPA. We recall
CENVPA?DFR-CI?
and
NVPAas Z increases flatters out. If we further take ac-
CEs,p,d,...,?
NVPA
= CEs
NVPA+ CEp
NVPA+ CEd
NVPA+ CEf
NVPA/
?1 − CEg
NVPA/CEf
NVPA?.
?10?
CEs,p,d,...,?
a discrepancy of 3.05 mhartrees ?CENVPA?Hy-CI?: ?93.33
mhartrees?.
Since the changes in CEs
the truncated s spinor set appears likely to be responsible for
thelargedifferences between
CENVPA?Hy-CI?. We, tested this, however, when we adopted
the threshold for truncation of the basis set; for116Uuh,
wherethemaximum
TE ?s-CI with truncated 30?2 spinors? = −17450.794428
and alsoTE?s-CI with full 136?2 spinors?
=−17450.794428, both of which give CEs
mhartrees. The same is true for all of the atoms. For CEp
we had calculated two CEp
these is obtained by CI with TE?s 30?2 spinors with thresh-
old 1?10−3and p 29?6 spinors with threshold 1?10−3?,
and the other is CI with TE?s 30?2 spinors with threshold
1?10−3and p 53?6 spinors with threshold 1?10−5?. The
former gives CEp
?29.077 mhartrees. Enlarging the p basis set scarcely
NVPA
is ?90.28 mhartrees at116Uuh, and we still have
NVPAversus Z are large, use of
CENVPA?DFR-CI?
and
differenceis expected,
NVPAof ?54.658
NVPA,
NVPAvalues. For116Uuh, one of
NVPA=−29.066 mhartrees and the latter
FIG. 1. CEs in hartrees. Solid lines: CENVPA?DFR-CI? and CEVPA?DFR-CI?
with s, p, d, f, g pGTFs; dashed lines: CENVPA?Hy-CI? and CEVPA?Hy-CI?
calculated from Refs. 14 and 13, respectively.
124105-4Watanabe, Nakano, and TatewakiJ. Chem. Phys. 132, 124105 ?2010?

Page 5

changes the CEp
cation in the s and p sets is not responsible for the large
difference between CENVPA?DFR-CI? and CENVPA?Hy-CI?
which is 1–5 mhartrees beyond80Hg. We also infer that the
correlating spinors with higher angular momentum ??h? are
not responsible for the difference between the two CENVPAs.
Using numerical multiconfiguration self-consistent-field with
higher angular momentum spinors ?l=7? and the extrapola-
tion, Parpia et al.23have obtained CENVPAfor He-like ions
?Z=1–26? with infinite spinors. Their extrapolated value for
26Fe24+is ?45.8 mhartrees which is close to our CEspdfg
value of ?45.7 and ?45.9 mhartrees of the extrapolated
value given by Eq. ?10?, confirming the accuracy of the
present CENVPAvalues.
Sapirstein et al.24have shown that TENVPAs obtained by
CI calculations depend on the electronic potential adopted,
namely, depend on the resulting spinors and resulting CSFs;
the size of the CI space for NVPA practically changes ac-
cording to the electronic potential adopted. Their result also
suggests that CENVPAs depend on the functional form to con-
struct the NVPA CI space. The disagreement between
CENVPA?Hy-CI? and CENVPA?DFR-CI? is therefore accepted,
even though the two calculations use fairly large basis sets,
since the NVPA CI spaces spanned by the spinors or Hy
functions with positive kinetic energies are expected to be
different. Without NVPA, the disagreement between CE?Hy-
CI? and CE?DFR-CI? should be smaller, since the spaces
generated with the full basis functions with the positive and
negative kinetic energies ?the complete CI? are used.
NVPAvalues by mhartrees. We see that trun-
NVPA
C. CE without NVPA
We calculate VPA DFR-CI, where three kinds of CSFs
are constructed from a pair of spinors with positive or nega-
tive kinetic energy. Upon adapting the Davidson’s diagonal-
ization method,25,26we directly obtained the solution for
which the main CSF is the DFR 1s+
To simplify the larger calculations involved in consider-
ing spinors with negative kinetic energies, we used Eqs.
?5?–?8?. In calculating ?CEd, the errors caused by the ap-
proximation at third line in Eq. ?6? were less than 0.018
mhartrees for116Uuh, as discussed. We expect the error in
?CEi?ddue to this approximation to be smaller still.
The calculated CEVPAs, the sums of the CEi
been set out in Table I and in Fig. 1. Figure 1 shows that
CEVPA?DFR-CI? by spdfg-CI are almost in agreement with
CEVPA?Hy-CI?, indicating the good choice of the approxima-
tions in Sec. II. The agreement of CEVPA?DFR-CI? with
CEVPA?Hy-CI? also indicates that the CEVPAs are indepen-
dent of the methodology to determine the CI space.
Let us discuss the details of CEVPA. The partial CEs
CEi
CEi
CEs
CEs
and a minimum of ?44.4 mhartrees at116Uuh. We recall that
CEs
minimum of ?54.7 mhartrees at116Uuh.
On the basis of the second order perturbation theory,
CEi
2.
VPAs have
VPAs are shown in Figs. 2?a? and 2?b? together with
NVPAs. Figure 2?a? shows that CEs
NVPA, but it also shows that CEs
NVPA. CEs
VPAis greater than
VPAdecreases sharply as
VPAhas a maximum of ?14.7 mhartrees at16S
NVPAhas a maximum of ?14.7 mhartrees at16S and a
VPAis given approximately by
CEi
VPA?
−
?
??,??
I1:??????0,??????0
????DFR − 1s2??H???1s2→ ??????2/?Ei???,???− E0?
+ ?
??,??
I2?I1
????DFR − 1s2??H???1s2→ ??????2/?E0− Ei???,????,
?11?
where i denotes the symmetry to which the spinors ?? and ??
belong. The first term gives the approximate value of
CEi
tive kinetic energies, and the second term gives the approxi-
mate value for the partial virtual pair correction ??CEi
arising from the configurations with the spinors, at least, one
of which has negative kinetic energy. ?CEi
that the CEi
virtual pair correction ?CEVPAas ?i?CEi
the coupling between the positive and negative states ?often
called the Brown–Ravenhall continuum? through electron-
electron interaction.
In contrast to CEs
then decreases as Z increases, the other CEi?1
NVPAwhere the spinor energies are calculated with posi-
VPA?
VPAis positive so
VPAis greater than CEi
NVPA. We define the ?total?
VPAwhich denotes
VPA, which has a maximum at16S and
VPAs have a mini-
mum between15P and18Ar and then increase monotonically,
indicating that ?CEi
CEi
values except for CEp
1.5, and 1.3 mhartrees for i=p, d, f, and g respectively, com-
pared to CEs
governedbyCEs
and
=−58.6 mhartrees for
=−89.9 mhartrees. The partial virtual pair corrections,
?CEi
and 1.9 mhartrees for i=s, p, d, f, and g.
We recall that CENVPAand CEVPAvalues for116Uuh by
Hy-CI were, respectively, ?93.3 and ?60.7 mhartrees com-
pared to the corresponding values of ?89.9 and ?58.6 mhar-
VPAis important for i?p. The ?absolute?
VPAvalues are fairly small compared to ?absolute? CEs
VPA. At116Uuh, CEi
VPA
VPAare ?18.1, 1.0,
VPA=−44.4 mhartrees. We realize that CEVPAis
VPA
CEp
116Uuh
VPA.Intotal,
the
CEVPA
CENVPA
and
VPAgiven by CEi
VPA−CEi
NVPAare 10.3, 10.9, 5.3, 2.8,
124105-5Effect of removing the no-virtual-pair approx.J. Chem. Phys. 132, 124105 ?2010?