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Analysis of atomic correlation energies
Volume 12, number 3
CHEMICAL PHYSICS LETTERS
ANALYSIS
OF ATOMIC
CORRELATION
ENERGIES*
J. KATRIEL NuclearResearch
Centre-Negev,
f’. O.B. 9001, Beer-Sileva, Israel
Received 5 November 197 1 Analysis of atomic correlation energies in terms of kinetic energy, nuclear attraction and interelectronic repulsion is carried out for the isoelectronic sequences of two-to nineelectron atoms. Reliminary observations concerning the monotonicity with respect to the number of electrons and the inadequacy of the restricted Hartree-Fock scheme for the low nuclear charge numbers of the isoelectronic sequences are made.
1. Introduction
repulsion contributions SO that E, = T,+L,+C,
‘Theevaluationand comprehensionof atomic and molecular correlation energies have been attempted by a broad range of approaches [ 11. Interpretations involving the partitioning of the correlation energy into additive pair contributions have been particularly appealing. According to the analysis made by Lowdjn [2], these are based on transition values of the hamiltonian rather than on expectation values. It was therefore suggested that a physically meaningful analysis be carried out by evaluating the expectation values of the various components of the hamiltonian with both an exact and a Hartree-Fock wavefunction. The contribution
of each component
to the correlation
energy
is thus obtained. This was carried out by Gruninger et al. [3] for the He and Li isoelectronic sequences. This type of ab initio evaluation of the various components of the correlation energy is severely restricted by the lack of sufficiently accurate wavefunctions for any but the simplest atomic and molecular systems. It is, however, possible to obtain rigorous estimates of these contributions by an analysis, suggested by Liiwdin [4] , of the available experimental correlation energies for a large number of atomic isoelectronic sequences’ [5]. Let the correlation energy be denoted by EC and the.kinetic energy, nuclear attraction and interelectronic * Based on a section of a thesis to be submitted to the senate of the Technion-Israel Institute of Technology, in partial fulfilment of the requirements for the D.Sc. degree.
by Tc, L, and Cc respectivety, From the vi&l
and HelImann-
Feynmantheoremswe obtain [4] T, = -E,,L,
Z iX,/aZ
=
and thus Cc = 2Ec-15,. In the present contribution we present the results of this type of analysis for a number of atomic systems. The results seem to clarify some aspects of the correlation problem and should be of use in the analysis of ab initio resuhs as well.
2. Results
)‘,
The experimental isoelectronic
correlation
energies for the various
sequences IS] were subjected to both
graphical and least-squares type smoothing
procedures after which the first derivative with respect to the nuclear charge was obtained. The remaining steps are straightforward. In table 1 we present the nuclear attraction evaluated in this way for the Hartree-Fock energy of the 3P state of the carbon isoelectronic sequence along with. the exactly computed values, from the data in ref. [6] . The aim of this tabIe is to demonstrate the accuracy obtainable by our approach, which after all has some intrinsic limitations due to its numerical nature, apart from those resulting from possible inaccuracies in the available values of the experimental correlation energies. The resuits of table 1 seem a convincing demonstration of the reliability of the procedure. The interelectronic repulsion contributions to the
4.57
Volume12, number 3
CHEkCAL
correlation
Table 1 Ha&e-Fock
nuclear attiction
for the ground
state of the
carbon isdtiectronic sequence, Z
aElaz
-Z(cX!l/rj)
-88.139
-88.138 -123.951 -165.73’1
z
6 7 8 -
IntereIe&nic
-123.949
-165.729
sequences
1 January I972
energies of the He to F isoelectronic in table 2. These are all
are present@
negative, zs one would expect. To save space, the corresponding nuclear attractions, which are believed to be much less mteresting from the point of view of the c&relation problem, are not given. They can, however, be extracted from the values of the interelectronic repulsion along with those of the total correlation energy. The smmth dependence of the experimental
Table 2 repulsion contribution to the correlation energy for atomic isoelectronic sequences with 2 to 9 electrons (in 2-1s
z
PHYSICS LETTERS
3%
4-1s
5-Q
6-3P
6JD
-0.0948
-0.113 -0.142 -0.164
-0.175 -0.199
-0.257
-0.27
au).
G’S
2 3 4 5 6
-010800 -0.0837 -0.0860 -0.0876 -0.0887
;
-0.0894 -0.0900
-0.0968 -0.0988
-0.184 -0,203
-0.219 -0.237
-0.275 -0.290
-0.285 -0.298
-0.328 -0.304
-0.0902 -0.0904
-0.0999 -0.1007
-0.220 -0.236
-0.270 -0.247
-0.316 -0.304
-0.3 15 -0.306
-0.353 -0.370
-0.251 -0.264
-0.284 -0.298 -0.309 -0.320 -0.33 1 -0.339
-0.327 -0.338 -0.347 -0.356 -0.366 -0.374
-0.329 -0.340 -0.354 -0.367 -0.380
1’0
-0.0840 -0.0886
-0.0923
11 12 13 14 15 16
1
-0.273
Z
7-4s
7-2D
73
8-3P
8-‘D
S-IS
9JP
7 8 9 10 11 12 13 14 15 16 17
-0.335 -0.35 1 -0.362 -0.372 -0.380 -0.385 -0.39 1 -0.396 -0.401 -0.405
-0.381 -0.386 -0.389 -0.394 -0.399 -0.406 -0.411 -0.416 -0.423 -0.428
-0.387 -0.398 -0.409 -0.422 -0.435 -0.45 1 -0.464 -0.480 -0.490 -0.500
-0.450 -0.452 -0.464 -0.476 -0.486 -0.497 -0.509 -0.520 -0.530
-0.476 -0.481, -0.495 -0.509 -0.523 -0534 -0.543 -0.555 -0.567
-0.405
-0.4
-0.517
-0.438
-0,527
-0539
-0574
-0.546
-0.581
-0.655
-0.566 -0.571 -0.583 -0.597 -0.608 -0.624 -0.639 -0.651 -0.664
18
-0.410
-0.494 -0.504 -0.523 -0.543 -0.562 -0.575 -0592 -0.607 -0.626 -0.643
19
-0.415
-0.445
-0.541
-0.554
-0.587
-0.672
-0.687
-0.450
-0.554
-0.560
-0.592
-0.686
-0.698
20
33
-0.674
“olume 12, number 3 correlation
CHEMICAL PHYSICS LETTERS
energy on the nuclear
charge for the He,
Li, Be, B, C(3P) and N(4S) sequences
is believed
to
1 January 1972
It is comforting to note that the second disturbing feature noted with respect to the correlation energy
guara@e the reiiability of the corresponding results up to an uncertainty in the last digit presented. For
is not observed in the interelectronic part. For a given Z, the absolute value of Cc is a monotonically in-
the other sequences the least reliable results are those corresponding to the neutral atoms for which the experimental correlation energies seem to indicate a particularly low value of the correlation contribution to the nuclear attraction, hence a particularly high value of the correlation contribution to the interelectronic repulsion. This feature of the data was smoothed out in our procedure. This amounts to not more than a ten percent increase of the values presented in table 2 for the corresponding neutral atoms. The slight discrepancies between our results for the He and Li isoelectronic sequences and the ab initio calculations of Gruninger et al. [3] are accounted for by the deviations of their correlation energies from the exact values. The value obtained for the interelectronic
creasing function of the number of electrons, as one would expect. It has been noted in the previous section that in some of the sequences considered the neutra1 atom exhibits a particularly high value of the correlation energy (in absolute value). This is most IikeIy an indication that the Hartree-Fock energy is particularly poor, namely, that the difference between the restricted Hartree-Fock scheme used for the definition of the corre!ation energies and the correct unrestricted Hartree-Fock energy is particularly large for the neutral atoms. This observation agrees with the resuits of Kaplan and Kleiner [lo] concerning the influence of the relaxation of the symmetry restrictions on the Hartree-Fock energies of negative atomic ions.
repulsion
Smoothing
contribution
to the correlation
ener=
of the
out the singular
behaviour
of the neutral
He atom agrees with that calculated from Pekeris’
atoms correlation energy is therefore believed to result
results [7] to within
in better estimates of the contribution of the interelectronic repulsion to the true correlation energy.
one percent.
Acknowledgement
3. Discussion A comprehensive discussion of the results presented, including such aspects as the formulation of an additive scheme of pair contributions to the interelectronic repulsion part of the correlation energies, or the more general question of the transferability of the various contributions to the correlation energy, is not attempted here. Two closely rel;ted disturbing features of the experimental correlation energies are some strong 2 dependences and the fact that in a number of cases the correlation energy is reduced (in absolute value) by the addition of a further electron to a given ion. The first of these two features has been the subject of an extensive study [8] whose main conclusion is
the strong Z dependence is due to near-degeneracy of the 2s and 2p orbitals. Actually, it is clear that tha!
correlation energies defined with respect to the unrestricted Hartree-Fock energy would be only slightly
Z dependent. This is due to the relation-X,/aZ and the fact that L,, being the correlation
=L,/Z
correction to an expectation vaiue of a one-electron local operator, should be of second order in the correlation error [9].
Most helpful discussions with Professor R. Pauncz are gratefully acknowledged. Discussions of the correlation problem by Professor P.-C). Lijwdin and Professor R. Pauncz at the Winter Institute in Quantum Chemistry, Solid State Physics and Quantum Biology (Florida 1970/7 1) were highly inspiring.
References [ I] R. Lefebvre and C. Moser, eds., in: Advances ~IIchemical Physics, Vol. 14 (Interscience. New York, 1969). [2] P.-O. Lijwdin, in: Advances in chemiwl physics, Vol. 14, eds. R. Lefsbvre
and C. Moser (Interscience.
New York,
1969) p. 283.
[3] J. Gruninger, Y. iihm and P.Q. LEwdin, I. Chern. whys. 52 (1970) 5551. [4] P.-O. Lijwdin, J. hloi. Spectry. 3 (19593 46. [S] E. Clementi; J. Chem. Phys. 38 (1963) 2248. G. Verhaegen and C.M. Moser, J. Phys. B3 (1970) 478. [6) E. Clementi, IBM J. Res. Dw.9 (1965) 2. suppl. [7] C.L. Pekeris, Phys. Rev. 112 (1958) 1649. [ 81 E. Clementi, J. Chem. Phys. 44 (1966) 3050. (91 C. MpUer and h1.S. PIesset,Phys. Rev. 46 (1934) 618. [IOj T.A. Kaplan and W.H. Kleiner,Phys. Rev. 1.56 (1967) 1.
459
CHEMICAL PHYSICS LETTERS
ANALYSIS
OF ATOMIC
CORRELATION
ENERGIES*
J. KATRIEL NuclearResearch
Centre-Negev,
f’. O.B. 9001, Beer-Sileva, Israel
Received 5 November 197 1 Analysis of atomic correlation energies in terms of kinetic energy, nuclear attraction and interelectronic repulsion is carried out for the isoelectronic sequences of two-to nineelectron atoms. Reliminary observations concerning the monotonicity with respect to the number of electrons and the inadequacy of the restricted Hartree-Fock scheme for the low nuclear charge numbers of the isoelectronic sequences are made.
1. Introduction
repulsion contributions SO that E, = T,+L,+C,
‘Theevaluationand comprehensionof atomic and molecular correlation energies have been attempted by a broad range of approaches [ 11. Interpretations involving the partitioning of the correlation energy into additive pair contributions have been particularly appealing. According to the analysis made by Lowdjn [2], these are based on transition values of the hamiltonian rather than on expectation values. It was therefore suggested that a physically meaningful analysis be carried out by evaluating the expectation values of the various components of the hamiltonian with both an exact and a Hartree-Fock wavefunction. The contribution
of each component
to the correlation
energy
is thus obtained. This was carried out by Gruninger et al. [3] for the He and Li isoelectronic sequences. This type of ab initio evaluation of the various components of the correlation energy is severely restricted by the lack of sufficiently accurate wavefunctions for any but the simplest atomic and molecular systems. It is, however, possible to obtain rigorous estimates of these contributions by an analysis, suggested by Liiwdin [4] , of the available experimental correlation energies for a large number of atomic isoelectronic sequences’ [5]. Let the correlation energy be denoted by EC and the.kinetic energy, nuclear attraction and interelectronic * Based on a section of a thesis to be submitted to the senate of the Technion-Israel Institute of Technology, in partial fulfilment of the requirements for the D.Sc. degree.
by Tc, L, and Cc respectivety, From the vi&l
and HelImann-
Feynmantheoremswe obtain [4] T, = -E,,L,
Z iX,/aZ
=
and thus Cc = 2Ec-15,. In the present contribution we present the results of this type of analysis for a number of atomic systems. The results seem to clarify some aspects of the correlation problem and should be of use in the analysis of ab initio resuhs as well.
2. Results
)‘,
The experimental isoelectronic
correlation
energies for the various
sequences IS] were subjected to both
graphical and least-squares type smoothing
procedures after which the first derivative with respect to the nuclear charge was obtained. The remaining steps are straightforward. In table 1 we present the nuclear attraction evaluated in this way for the Hartree-Fock energy of the 3P state of the carbon isoelectronic sequence along with. the exactly computed values, from the data in ref. [6] . The aim of this tabIe is to demonstrate the accuracy obtainable by our approach, which after all has some intrinsic limitations due to its numerical nature, apart from those resulting from possible inaccuracies in the available values of the experimental correlation energies. The resuits of table 1 seem a convincing demonstration of the reliability of the procedure. The interelectronic repulsion contributions to the
4.57
Volume12, number 3
CHEkCAL
correlation
Table 1 Ha&e-Fock
nuclear attiction
for the ground
state of the
carbon isdtiectronic sequence, Z
aElaz
-Z(cX!l/rj)
-88.139
-88.138 -123.951 -165.73’1
z
6 7 8 -
IntereIe&nic
-123.949
-165.729
sequences
1 January I972
energies of the He to F isoelectronic in table 2. These are all
are present@
negative, zs one would expect. To save space, the corresponding nuclear attractions, which are believed to be much less mteresting from the point of view of the c&relation problem, are not given. They can, however, be extracted from the values of the interelectronic repulsion along with those of the total correlation energy. The smmth dependence of the experimental
Table 2 repulsion contribution to the correlation energy for atomic isoelectronic sequences with 2 to 9 electrons (in 2-1s
z
PHYSICS LETTERS
3%
4-1s
5-Q
6-3P
6JD
-0.0948
-0.113 -0.142 -0.164
-0.175 -0.199
-0.257
-0.27
au).
G’S
2 3 4 5 6
-010800 -0.0837 -0.0860 -0.0876 -0.0887
;
-0.0894 -0.0900
-0.0968 -0.0988
-0.184 -0,203
-0.219 -0.237
-0.275 -0.290
-0.285 -0.298
-0.328 -0.304
-0.0902 -0.0904
-0.0999 -0.1007
-0.220 -0.236
-0.270 -0.247
-0.316 -0.304
-0.3 15 -0.306
-0.353 -0.370
-0.251 -0.264
-0.284 -0.298 -0.309 -0.320 -0.33 1 -0.339
-0.327 -0.338 -0.347 -0.356 -0.366 -0.374
-0.329 -0.340 -0.354 -0.367 -0.380
1’0
-0.0840 -0.0886
-0.0923
11 12 13 14 15 16
1
-0.273
Z
7-4s
7-2D
73
8-3P
8-‘D
S-IS
9JP
7 8 9 10 11 12 13 14 15 16 17
-0.335 -0.35 1 -0.362 -0.372 -0.380 -0.385 -0.39 1 -0.396 -0.401 -0.405
-0.381 -0.386 -0.389 -0.394 -0.399 -0.406 -0.411 -0.416 -0.423 -0.428
-0.387 -0.398 -0.409 -0.422 -0.435 -0.45 1 -0.464 -0.480 -0.490 -0.500
-0.450 -0.452 -0.464 -0.476 -0.486 -0.497 -0.509 -0.520 -0.530
-0.476 -0.481, -0.495 -0.509 -0.523 -0534 -0.543 -0.555 -0.567
-0.405
-0.4
-0.517
-0.438
-0,527
-0539
-0574
-0.546
-0.581
-0.655
-0.566 -0.571 -0.583 -0.597 -0.608 -0.624 -0.639 -0.651 -0.664
18
-0.410
-0.494 -0.504 -0.523 -0.543 -0.562 -0.575 -0592 -0.607 -0.626 -0.643
19
-0.415
-0.445
-0.541
-0.554
-0.587
-0.672
-0.687
-0.450
-0.554
-0.560
-0.592
-0.686
-0.698
20
33
-0.674
“olume 12, number 3 correlation
CHEMICAL PHYSICS LETTERS
energy on the nuclear
charge for the He,
Li, Be, B, C(3P) and N(4S) sequences
is believed
to
1 January 1972
It is comforting to note that the second disturbing feature noted with respect to the correlation energy
guara@e the reiiability of the corresponding results up to an uncertainty in the last digit presented. For
is not observed in the interelectronic part. For a given Z, the absolute value of Cc is a monotonically in-
the other sequences the least reliable results are those corresponding to the neutral atoms for which the experimental correlation energies seem to indicate a particularly low value of the correlation contribution to the nuclear attraction, hence a particularly high value of the correlation contribution to the interelectronic repulsion. This feature of the data was smoothed out in our procedure. This amounts to not more than a ten percent increase of the values presented in table 2 for the corresponding neutral atoms. The slight discrepancies between our results for the He and Li isoelectronic sequences and the ab initio calculations of Gruninger et al. [3] are accounted for by the deviations of their correlation energies from the exact values. The value obtained for the interelectronic
creasing function of the number of electrons, as one would expect. It has been noted in the previous section that in some of the sequences considered the neutra1 atom exhibits a particularly high value of the correlation energy (in absolute value). This is most IikeIy an indication that the Hartree-Fock energy is particularly poor, namely, that the difference between the restricted Hartree-Fock scheme used for the definition of the corre!ation energies and the correct unrestricted Hartree-Fock energy is particularly large for the neutral atoms. This observation agrees with the resuits of Kaplan and Kleiner [lo] concerning the influence of the relaxation of the symmetry restrictions on the Hartree-Fock energies of negative atomic ions.
repulsion
Smoothing
contribution
to the correlation
ener=
of the
out the singular
behaviour
of the neutral
He atom agrees with that calculated from Pekeris’
atoms correlation energy is therefore believed to result
results [7] to within
in better estimates of the contribution of the interelectronic repulsion to the true correlation energy.
one percent.
Acknowledgement
3. Discussion A comprehensive discussion of the results presented, including such aspects as the formulation of an additive scheme of pair contributions to the interelectronic repulsion part of the correlation energies, or the more general question of the transferability of the various contributions to the correlation energy, is not attempted here. Two closely rel;ted disturbing features of the experimental correlation energies are some strong 2 dependences and the fact that in a number of cases the correlation energy is reduced (in absolute value) by the addition of a further electron to a given ion. The first of these two features has been the subject of an extensive study [8] whose main conclusion is
the strong Z dependence is due to near-degeneracy of the 2s and 2p orbitals. Actually, it is clear that tha!
correlation energies defined with respect to the unrestricted Hartree-Fock energy would be only slightly
Z dependent. This is due to the relation-X,/aZ and the fact that L,, being the correlation
=L,/Z
correction to an expectation vaiue of a one-electron local operator, should be of second order in the correlation error [9].
Most helpful discussions with Professor R. Pauncz are gratefully acknowledged. Discussions of the correlation problem by Professor P.-C). Lijwdin and Professor R. Pauncz at the Winter Institute in Quantum Chemistry, Solid State Physics and Quantum Biology (Florida 1970/7 1) were highly inspiring.
References [ I] R. Lefebvre and C. Moser, eds., in: Advances ~IIchemical Physics, Vol. 14 (Interscience. New York, 1969). [2] P.-O. Lijwdin, in: Advances in chemiwl physics, Vol. 14, eds. R. Lefsbvre
and C. Moser (Interscience.
New York,
1969) p. 283.
[3] J. Gruninger, Y. iihm and P.Q. LEwdin, I. Chern. whys. 52 (1970) 5551. [4] P.-O. Lijwdin, J. hloi. Spectry. 3 (19593 46. [S] E. Clementi; J. Chem. Phys. 38 (1963) 2248. G. Verhaegen and C.M. Moser, J. Phys. B3 (1970) 478. [6) E. Clementi, IBM J. Res. Dw.9 (1965) 2. suppl. [7] C.L. Pekeris, Phys. Rev. 112 (1958) 1649. [ 81 E. Clementi, J. Chem. Phys. 44 (1966) 3050. (91 C. MpUer and h1.S. PIesset,Phys. Rev. 46 (1934) 618. [IOj T.A. Kaplan and W.H. Kleiner,Phys. Rev. 1.56 (1967) 1.
459