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High sectors in the Fock space coupled-cluster method
Volume 194, number 1,2
CHEMICAL PHYSICS LETTERS
19 June 1992
High sectors in the Fock space coupled-cluster method S.R. Hughes and Uzi Kaldor School of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel Received 24 March 1992
The coupled-cluster method with single and double excitations (CCSD or full SUBZ) is applied to systems with n electrons outside a closed shell, nd 6. The CCSDTl scheme, which includes triple excitations approximately, is also employed. The systems of interest are the F atom and its ions. F5+ ISthe reference state, and the ionization potentials and excitation energies involving the 2p and 3s electrons are calculated. The CCSD results are good for n= 1 and 2, but deteriorate with increasing n. The CCSDTl scheme gives satisfactory results in the n = 3 and 4 sectors, but overestimates the effects for higher n.
1. Introduction The coupled-cluster method (CCM) [ l-31 has been widely used for ab initio electronic structure calculations and has proved to be very successful for single reference state, closed shell systems [ 41. It has also been successfully applied to multiple reference state, open shell systems [ 5- 15 ] with m valence holes and n valence particles added to the core determinant, m + n < 2. We are interested in pursuing applications to higher sectors for two reasons. First, there are many atomic systems of interest in which more than two electrons lie outside a closed shell. The second reason relates to the problem of calculating potential curves for molecular systems. If a single bond is to be stretched to its breaking point, then the n = 2 sector will usually suffice for a treatment of the situation [ 16 1. However, if double or triple bonds are to be described, then the sectors n = 4 and n = 6 respectively will be required. In particular, the n= 6 sector will be needed for the complete potential curve of the N2 molecule. In this Letter, an open shell system is considered using the Fock space formulation of the CCM due to Lindgren [ 171. The sectors investigated are those with n valence particles, nG6, denoted (0, n). The case of the fluorine atom is studied using two apCorrespondence CO: U. Kaldor, School of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel.
proximation schemes. The first of these is the natural CCSD (coupled-cluster singles and doubles), also known as SUB2. This sets to zero all m-body correlation amplitudes for m > 2, and then iterates to convergence the remaining linked, non-linear equations for the one- and two-body amplitudes. The second approximation is the CCSDTI scheme of Bartlett and co-workers [ 18 1. This includes the most important three-body terms in the iteration cycle. For the complete model spaces used here, the Lindgren formulation satisfies the linked cluster theorem in that only linked terms appear when the method is interpreted by means of Goldstone diagrams. This makes the description of the system properly size-extensive. In a later paper, incomplete model spaces will be required, specifically, the quasicomplete spaces used to treat the (2, 1) and ( 1, 2) sectors of the Fock space. It will then be necessary to consider the recently developed formulation of Mukherjee which furnishes a linked description of a general model space [ 191.
2. Method In Lindgren’s method the normal-ordered operator is given by Q=expS.
0009-2614/92/$ 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.
wave
(1)
99
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1,2
CHEMICAL
PHYSICS
S is the excitation operator describing connected single, double, ... excitations, s=s, =
5
+s, + ... )
(2)
(3)
{"?~j}~~+f~~{ntCIfala~}~&+...,
where sJ, si,, ... are excitation amplitudes and the curly brackets denote normal order with respect to a core (reference) determinant. The equations for these excitation amplitudes in a complete model space are determined from [S, HoI =IQf'f-J--Q~H,,L,n,
,
(4)
where H eff = PHQP
(5)
and H=H,+V
(6)
in the usual way. P and Q are the projection operators which describe the valence space and its orthogonal complement, respectively. In the natural SUBn truncation scheme the amplitudes S,,, for m > n are set to zero. The p-body equations for p
... .
(7)
Haque and Mukherjee have shown 1201 that this partitioning allows for a partial decoupling of the open-shell CCM equations. The equation for Scn) involves only Scrn) elements with m < n, so that the very large system of non-linear equations is separated into smaller subsystems which are solved consecutively. First, the equations for S(O) are iterated to convergence, then the S(i) are obtained using the known S(O), and so on. This separation, which is exact, reduces the computational effort significantly. The S amplitudes may also be written in terms of the total number of electrons involved in an excitation of a P space configuration, for example P’=SJ2’ 100
+sJ2’ +... .
(8)
LETTERS
19 June 1992
The working equations used for computations were obtained from (4) in a completely topological fashion. In any given case all possible Goldstone diagrams satisfying the particular set of constraints were constructed. These were then directly encoded into the programs used for the solution of the equations with no intervening algebraic stage. The process was greatly aided by figs. 7 and 9 of ref. [21]. These facilitate the drawing of all possible Hugenholtz structures with n external lines for both one and two body operators. Once this is done, it is a relatively straightforward matter to expand these structures into the relevant set of Goldstone diagrams. The diagrams for the one- and two-body equations in the (0, 0) sector, which is the closed shell case, were given by Cullen and Zerner in figs. 6 and 8 of ref. [ 221. It is easy to obtain the corresponding equations for the (0, n) and (n, 0) sectors for n < 2. In the (0, n) case each diagram is taken and exactly n hole lines bent down in all possible ways in order to form n valence particle lines. Similarly, in the (n, 0 ) case exactly n particle lines are bent down to form n valence hole lines. The highest sector with iterative equations is n = 2 when the CCSD approximation is used. For the sectors beyond this it is necessary to use the building blocks available to construct all diagrams contributing to the various Hew matrices. These building blocks are the one and two body potentials and amplitudes. Thus, for the (0, 3 ) sector all diagrams with six external lines are required. All of these lines are arranged so as to be valence particle lines. For the 3, 4, 5 and 6 sectors there are respectively 26, 15,4 and 1 such He, diagrams. If we are limited to at most twobody amplitudes and potentials then no HeEdiugrum can be constructed for any higher sector. However, it is possible to construct an Herr matrix by using the available Her diagrams together with an appropriate number of spectator lines. The CCSDT 1 approximation is defined in the first place by the addition of all diagrams containing an S, amplitude to the RHS of the one- and two-body equations. The S, amplitude is in turn defined to equal all diagrams of the form VS,. It is not defined in terms of itself and so the highest sector for which the equations are iterative continues to be n = 2, although there is a static contribution to the S$‘) amplitude. The CCSDTl scheme has only been applied
Volume 194, number 1,2
CHEMICAL PHYSICS LETTERS
to single-reference systems [ 181. Its present extension to multi-reference problems raises the question of the definition of the folded diagrams. The two possibilities are of the forms S, V and Sz H$-), and there is one such diagram for each of the n = 2 and n = 3 sectors. S, V is selected in the present work, because this is consistent with the definition of the CCSDT 1 scheme. CCSDT 1 includes all second-order perturbation theory diagrams in the open and closed sectors and so, following ( 5 ) , all third-order Hefl diagrams are included in the open sectors. All the diagrams peculiar to CCSDT 1 described in this section are to be found in figs. l-3 of ref. [ 23 1.
19 June 1992
3. Results The method described here has been applied to the fluorine atom using the ( 19s/14p/4d/3f) basis of Gaussian-type orbitals contracted to (8s/7p/4d/3f) as specified in table 1. This consists of seven s and six p orbitals taken from table LXXX1 of ref. [ 241, to which have been added one s, one p, four d and three f orbitals. Five spherical d and seven f orbitals were used as opposed to six Cartesian d and ten f. The ACES II #’ program package was used to calculate integrals and Hat-tree-Fock orbitals. The closed shell F5+ served as the reference state, and up to six xl ACES II, written by J.F. Stanton, J. Gauss, J.D. Watts, W.J. Lauderdale and R.J. Bartlett. Quantum Theory Project, University of Florida, 1990.
Table 1 Specification of the basis (see text) Exponent
Coefficient
Coefficient
2641.133 625.1424 203.1062 77.73855 32.73855 14.90071 7.082732 3.48635 1 1.740970 0.866502 0.426099 0.206097 0.095982 0.045
0.000016 0.000141 0.0008 11 0.003544 0.012376 0.035842 0.086781 0.167378 1.0 1.0 1.0 1.0 1.0 1.0
p orbitals
s orbitals 1
Exponent
2115112.0 316699.7 72075.71 20416.83 6661.458 2405.188 938.2595 389.2710 169.8499 77.24367 36.32874 17.57387 8.693530 4.321769 1.959520 0.903185 0.410405 0.182024 0.075
0.000002 0.0000 11 0.000060 0.00025 1 0.000916 0.002987 0.008882 0.024232 0.060031 0.131135 0.238563 0.324308 1.0 1.0 1.0 1.0 1.0 1.0 1.0
1
2 3 4 5 6 7 d orbitals 1 2 3 4
2.0 0.7 0.25 0.08
1.0 1.0 1.0 1.0
0.7 0.25 0.08
1.0 1.0 1.0
f orbitals 1 2 3
3 101
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CHEMICAL PHYSICS LETTERS
19 June 1992
Table 2 Ionization potentials and excitation spectra of the fluorine atom and its ions (all values given in eV with errors in parentheses) Experimental a)
CCSD
CCSDT 1
114.00 (-0.21) 64.81 (-0.25) (0.23)
113.95 (-0.26) 64.80 (-0.26) (0.26)
F4+
IP 2~~3s 2S average absolute error
114.21 65.06
F’+
IP 2s22p2 ‘D 2sz2pz ‘S 2~~2~3s ‘P 2~~2~3s ‘P average absolute error
81.23 3.13 6.64 51.63 52.52
86.86 3.19 6.22 51.33 52.20
(-0.37) (+0.06) (-0.42) (-0.30) (-0.32) (0.29)
86.92 3.26 6.57 51.43 52.31
(-0.31) (+0.13) (-0.07) (-0.20) (-0.21) (0.18)
F2+
IP 2s22p’ ‘D 2s22p’ *P 2s22p23s 4P 2s22p23s 2P 2s22p23s *D 2s22p23s ‘S average absolute error
62.65 4.23 6.39 39.26 40.23 42.65 46.20
62.27 4.36 6.17 38.96 39.96 42.41 45.48
(-0.38) (+0.13) (-0.22) (+0.30) (-0.27) (-0.24) (-0.72) (0.32)
62.59 4.41 6.39 39.16 40.05 42.71 45.99
(-0.06) (+0.18) (0.00) (-0.10) (-0.18) (+0.06) (-0.21) (0.11)
F+
IP 2s22p4 ‘D 2sz2p4 ‘S 2s22p’3s ‘S 2s22p33s ‘S 2s22p33s ‘D 2s22p33s ‘D 2s22p’3s ‘P 2s22p33s ‘P average absolute error
34.98 2.59 5.51 21.90 22.61 26.21 26.66 28.17 28.46
34.07 2.63 5.05 21.47 22.25 25.93 26.32 27.76 28.15
(-0.92) (+0.04) (-0.52) (-0.43) (-0.42) (-0.34) (-0.34) (-0.41) (-0.31) (0.41)
35.25 2.59 5.20 21.97 22.57 26.52 26.82 28.46 28.83
(+0.27) (0.00) (-0.37) (-0.07) (-0.10) (+0.25) (+0.16) (+0.29) (+0.37) (0.21)
F
IP 2s22p43s 4P 2s22p33s 2P 2s22p33s 2D average absolute error
17.42 12.70 12.98 15.37
15.54 12.03 12.09 14.63
(-1.88) (-0.67) (-0.89) (-0.74) (1.04)
19.03 (+1.61) 13.74 (+ 1.04) 13.78 (+0.80) 16.35 (-+0.98) (1.11)
F
IP
3.45
-0.82
(-4.31)
6.60 (3.15)
ai)Ref. [25].
electrons were added. The CCSD and CCSDTl results are given in table 2. Haque and Kaldor [ 231 have applied their CCSD + T approximation to the (0, n) sectors of the 0 atom for n < 4. This scheme includes a static P’S, contribution to the S, amplitudes in addition to CCSD. They found that the S, contribution to the n = 1 and n = 2 sectors is small, but that the contribution to n = 3 and n= 4 is considerably more pronounced, the reason being that S, contributes significantly to the three- and four-body effective 102
Hamiltonians. This is clearly borne out by the present results. For n = 1 and n = 2 there is little difference between CCSD and CCSDTl. For n = 3 and n=4 there is in general a marked improvement of CCSDT 1 over CCSD. Once we arrive at n = 5 and n =6 the change is certainly drastic. However, CCSDTl greatly overestimates the S3 effects, giving errors almost as big as those of CCSD but with opposite sign. The substantial effect of S, can be explained by the high electron density in the valence shell of first-row atoms. This makes it likely that three
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CHEMICAL PHYSICS LETTERS
electrons will mutually interact and so cause large contributions to the three body amplitudes. For the n= 5 and n=6 sectors the calculation of each five- and six-body Z&r matrix element requires, amongst other things, the addition of 10 and 20 contributions, respectively, from the three-body He, matrix. Now, the equation for S, used here is a truncation of the natural three body CCSDT equation and so relatively small systematic errors in the contributions to the three-body Hef, matrix may accumulate and spoil the results.
4. Summary and conclusion The (0, n ) sectors for n < 6 of the open-shell CCM Fock space in both the CCSD and the CCSDTl approximations have been investigated. The latter of these schemes makes approximate inclusion of S, amplitudes in the iteration cycle. The system of interest has been the F atom and its various ions. Their several excitation spectra and ionization potentials have been calculated. For the sectors n = 1 and n = 2 the CCSD approximation is good, and S, effects are small. For n = 3 and n = 4 the CCSD results become progressively worse, but the S, corrections included in CCSDTl are sufficient to yield good results. CCSDT 1 improves markedly over CCSD. For n = 5 and n = 6 the CCSDT 1 contribution is large, but both approximations are badly wrong. Clearly it is necessary in these latter cases to follow Ktimmel’s dictum that the CCM is smarter than we are and to try to develop a better sub-approximation to the natural CCSDT or full SUB3 scheme. Work is in progress to apply such an approximation scheme. Further applications to other atomic and molecular systems are in progress and the results will be published shortly together with a full description of the diagrams employed.
Acknowledgement
The research reported here was supported by the US-Israel Binational Science Foundation. SRH thanks the Committee for Planning and Budgeting
19 June 1992
of the Israeli Council on Higher Education for a fellowship.
[ 1 ] F. Coester, Nucl. Phys. 1 (1958) 421; F. Coester and H. Kilmmel, Nucl. Phys. 17 ( 1960) 477. [2] H. Ktlmmel, K.H. Ltlhrmann and J.G. Zabolitzky, Phys. Rept. 36 (1978) 1. [3] J. Ciiek, J. Chem. Phys. 45 (1966) 4256; Advan. Chem. Phys. 14 (1969) 35; J. Paldus, J. Click and I. Shavitt, Phys. Rev. A 5 ( 1972) 50; J. Paldus, J. Chem. Phys. 67 (1977) 303; B.G. Adams and J. Paldus, Phys. Rev. A 20 ( 1979) 1. [ 41 R.J. Bartlett, Ann. Rev. Phys. Chem. 32 ( 1981) 359; J. Phys. Chem. 93 (1989) 1697. [ 5 ] B. Jeziorski and H.J. Monkhorst, Phys. Rev. A 24 ( 1981) 1668. [6] A. Banerjee and J. Simons, Intern. J. Quantum Chem. 19 (1981) 207. (71 W.D. Laidig and R.J. Bartlett, Chem. Phys. Letters 104 ( 1984) 424; W.D. &dig, P. Saxe and R.J. Bartlett, J. Chem. Phys. 86 (1987) 887. [ 81 L. Meissner, S.A. Kucharski and R.J. Bartlett, J. Chem. Phys. 91 (1989) 6187; L. Meissner and R.J. Bartlett, J. Chem. Phys. 92 ( 1990) 561. [9] D. Mukherjee, R.K. Moitra and A. Mukhopadhyay, Mol. Phys. 30 (1975) 1861; 33 (1977) 955; A. Mukhopadhyay, R.K. Moitra and D. Mukherjee, J. Phys. B 12 (1979) 1. [lo] R. Offermann, W. Ey and H. Kgmmel, Nucl. Phys. A 283 ( 1976) 399; R. Offermann, Nucl. Phys. A 273 (1976) 368; W. Ey, Nucl. Phys. A 296 ( 1978) 189. [ 111 I. Lindgren, Intern. J. Quantum Chem. Symp. 12 (1978)
[ 121 G:Kutzelnigg and S. Koch, J. Chem. Phys. 79 (1983) 4315. [ 131 I. Lindgren and D. Mukherjee, Phys. Rept. 151 (1987) 93. [ 141 D. Mukherjee and S. Pal, Advan. Quantum Chem. 20 (1989) 292.
[ 15 ] A. Haque and U. Kaldor, Chem. Phys. Letters 117 ( 1985) 347; U. Kaldor and A. Haque, Chem. Phys. Letters 128 ( 1986) 45; U. Kaldor, J. Comput. Chem. 8 (1987) 448; J. Chem. Phys. 87 (1987) 467; Phys. Rev. A 38 (1988) 6013. [ 161 S. Ben Shlomo and U. Kaldor, J. Chem. Phys. 89 ( 1988) 956; U. Kaldor, Chem. Phys. 140 ( 1990) 1. [ 171 I. Lindgren and J. Morrison, Atomic many-body theory (Springer, Berlin, 1982).
103
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[ 18 ] Y.S. Lee, S.A. Kucharski and R.J. Bartlett, J. Chem. Phys. 81 (1984) 5906; J. Noga, R.J. Bartlett and M. Urban, Chem. Phys. Letters 134 (1987) 126; M. Urban, J. Noga, S.J. Cole and R.J. Bartlett, J. Chem. Phys. 83 (1985) 4041. [ 191 D. Mukherjee, Chem. Phys. Letters 125 (1986) 207; D. Sinha, S. Mukhopadhyay and D. Mukhetjee, Chem. Phys. Letters 129 ( 1986) 369; S. Pal, M. Rittby, R.J. Bartlett, D. Sinha and D. Mukhejee, Chem. Phys. Letters 137 (1987) 273; J. Chem. Phys. 88 (1988) 4357; I. Lindgren and D. Mukhejee, Phys. Rept. 151 (1987) 95.
104
19 June 1992
[20] A. Haque, Ph.D. Thesis, Calcutta University ( 1983); A. Haque and D. Mukhejee, J. Chem. Phys. 80 ( 1984) 5058. [21]J.S.Arponen,Ann.Phys. 151 (1983) 311. [22] J.M. Cullen and M.C. Zemer, J. Chem. Phys. 77 (1982) 4088. [ 23 ] A. Haque and U. Kaldor, Chem. Phys. Letters 120 ( 1985 ) 261. [24] H.J. Partridge, J. Chem. Phys. 90 (1989) 1043; NASA Technical Memorandum 101044 (1989). [ 251 C.E. Moore, Atomic energy levels, NSRDS-NBS, Circular No. 467 (US GPO, Washington, 1949).
CHEMICAL PHYSICS LETTERS
19 June 1992
High sectors in the Fock space coupled-cluster method S.R. Hughes and Uzi Kaldor School of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel Received 24 March 1992
The coupled-cluster method with single and double excitations (CCSD or full SUBZ) is applied to systems with n electrons outside a closed shell, nd 6. The CCSDTl scheme, which includes triple excitations approximately, is also employed. The systems of interest are the F atom and its ions. F5+ ISthe reference state, and the ionization potentials and excitation energies involving the 2p and 3s electrons are calculated. The CCSD results are good for n= 1 and 2, but deteriorate with increasing n. The CCSDTl scheme gives satisfactory results in the n = 3 and 4 sectors, but overestimates the effects for higher n.
1. Introduction The coupled-cluster method (CCM) [ l-31 has been widely used for ab initio electronic structure calculations and has proved to be very successful for single reference state, closed shell systems [ 41. It has also been successfully applied to multiple reference state, open shell systems [ 5- 15 ] with m valence holes and n valence particles added to the core determinant, m + n < 2. We are interested in pursuing applications to higher sectors for two reasons. First, there are many atomic systems of interest in which more than two electrons lie outside a closed shell. The second reason relates to the problem of calculating potential curves for molecular systems. If a single bond is to be stretched to its breaking point, then the n = 2 sector will usually suffice for a treatment of the situation [ 16 1. However, if double or triple bonds are to be described, then the sectors n = 4 and n = 6 respectively will be required. In particular, the n= 6 sector will be needed for the complete potential curve of the N2 molecule. In this Letter, an open shell system is considered using the Fock space formulation of the CCM due to Lindgren [ 171. The sectors investigated are those with n valence particles, nG6, denoted (0, n). The case of the fluorine atom is studied using two apCorrespondence CO: U. Kaldor, School of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel.
proximation schemes. The first of these is the natural CCSD (coupled-cluster singles and doubles), also known as SUB2. This sets to zero all m-body correlation amplitudes for m > 2, and then iterates to convergence the remaining linked, non-linear equations for the one- and two-body amplitudes. The second approximation is the CCSDTI scheme of Bartlett and co-workers [ 18 1. This includes the most important three-body terms in the iteration cycle. For the complete model spaces used here, the Lindgren formulation satisfies the linked cluster theorem in that only linked terms appear when the method is interpreted by means of Goldstone diagrams. This makes the description of the system properly size-extensive. In a later paper, incomplete model spaces will be required, specifically, the quasicomplete spaces used to treat the (2, 1) and ( 1, 2) sectors of the Fock space. It will then be necessary to consider the recently developed formulation of Mukherjee which furnishes a linked description of a general model space [ 191.
2. Method In Lindgren’s method the normal-ordered operator is given by Q=expS.
0009-2614/92/$ 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.
wave
(1)
99
Volume 194, number
1,2
CHEMICAL
PHYSICS
S is the excitation operator describing connected single, double, ... excitations, s=s, =
5
+s, + ... )
(2)
(3)
{"?~j}~~+f~~{ntCIfala~}~&+...,
where sJ, si,, ... are excitation amplitudes and the curly brackets denote normal order with respect to a core (reference) determinant. The equations for these excitation amplitudes in a complete model space are determined from [S, HoI =IQf'f-J--Q~H,,L,n,
,
(4)
where H eff = PHQP
(5)
and H=H,+V
(6)
in the usual way. P and Q are the projection operators which describe the valence space and its orthogonal complement, respectively. In the natural SUBn truncation scheme the amplitudes S,,, for m > n are set to zero. The p-body equations for p
... .
(7)
Haque and Mukherjee have shown 1201 that this partitioning allows for a partial decoupling of the open-shell CCM equations. The equation for Scn) involves only Scrn) elements with m < n, so that the very large system of non-linear equations is separated into smaller subsystems which are solved consecutively. First, the equations for S(O) are iterated to convergence, then the S(i) are obtained using the known S(O), and so on. This separation, which is exact, reduces the computational effort significantly. The S amplitudes may also be written in terms of the total number of electrons involved in an excitation of a P space configuration, for example P’=SJ2’ 100
+sJ2’ +... .
(8)
LETTERS
19 June 1992
The working equations used for computations were obtained from (4) in a completely topological fashion. In any given case all possible Goldstone diagrams satisfying the particular set of constraints were constructed. These were then directly encoded into the programs used for the solution of the equations with no intervening algebraic stage. The process was greatly aided by figs. 7 and 9 of ref. [21]. These facilitate the drawing of all possible Hugenholtz structures with n external lines for both one and two body operators. Once this is done, it is a relatively straightforward matter to expand these structures into the relevant set of Goldstone diagrams. The diagrams for the one- and two-body equations in the (0, 0) sector, which is the closed shell case, were given by Cullen and Zerner in figs. 6 and 8 of ref. [ 221. It is easy to obtain the corresponding equations for the (0, n) and (n, 0) sectors for n < 2. In the (0, n) case each diagram is taken and exactly n hole lines bent down in all possible ways in order to form n valence particle lines. Similarly, in the (n, 0 ) case exactly n particle lines are bent down to form n valence hole lines. The highest sector with iterative equations is n = 2 when the CCSD approximation is used. For the sectors beyond this it is necessary to use the building blocks available to construct all diagrams contributing to the various Hew matrices. These building blocks are the one and two body potentials and amplitudes. Thus, for the (0, 3 ) sector all diagrams with six external lines are required. All of these lines are arranged so as to be valence particle lines. For the 3, 4, 5 and 6 sectors there are respectively 26, 15,4 and 1 such He, diagrams. If we are limited to at most twobody amplitudes and potentials then no HeEdiugrum can be constructed for any higher sector. However, it is possible to construct an Herr matrix by using the available Her diagrams together with an appropriate number of spectator lines. The CCSDT 1 approximation is defined in the first place by the addition of all diagrams containing an S, amplitude to the RHS of the one- and two-body equations. The S, amplitude is in turn defined to equal all diagrams of the form VS,. It is not defined in terms of itself and so the highest sector for which the equations are iterative continues to be n = 2, although there is a static contribution to the S$‘) amplitude. The CCSDTl scheme has only been applied
Volume 194, number 1,2
CHEMICAL PHYSICS LETTERS
to single-reference systems [ 181. Its present extension to multi-reference problems raises the question of the definition of the folded diagrams. The two possibilities are of the forms S, V and Sz H$-), and there is one such diagram for each of the n = 2 and n = 3 sectors. S, V is selected in the present work, because this is consistent with the definition of the CCSDT 1 scheme. CCSDT 1 includes all second-order perturbation theory diagrams in the open and closed sectors and so, following ( 5 ) , all third-order Hefl diagrams are included in the open sectors. All the diagrams peculiar to CCSDT 1 described in this section are to be found in figs. l-3 of ref. [ 23 1.
19 June 1992
3. Results The method described here has been applied to the fluorine atom using the ( 19s/14p/4d/3f) basis of Gaussian-type orbitals contracted to (8s/7p/4d/3f) as specified in table 1. This consists of seven s and six p orbitals taken from table LXXX1 of ref. [ 241, to which have been added one s, one p, four d and three f orbitals. Five spherical d and seven f orbitals were used as opposed to six Cartesian d and ten f. The ACES II #’ program package was used to calculate integrals and Hat-tree-Fock orbitals. The closed shell F5+ served as the reference state, and up to six xl ACES II, written by J.F. Stanton, J. Gauss, J.D. Watts, W.J. Lauderdale and R.J. Bartlett. Quantum Theory Project, University of Florida, 1990.
Table 1 Specification of the basis (see text) Exponent
Coefficient
Coefficient
2641.133 625.1424 203.1062 77.73855 32.73855 14.90071 7.082732 3.48635 1 1.740970 0.866502 0.426099 0.206097 0.095982 0.045
0.000016 0.000141 0.0008 11 0.003544 0.012376 0.035842 0.086781 0.167378 1.0 1.0 1.0 1.0 1.0 1.0
p orbitals
s orbitals 1
Exponent
2115112.0 316699.7 72075.71 20416.83 6661.458 2405.188 938.2595 389.2710 169.8499 77.24367 36.32874 17.57387 8.693530 4.321769 1.959520 0.903185 0.410405 0.182024 0.075
0.000002 0.0000 11 0.000060 0.00025 1 0.000916 0.002987 0.008882 0.024232 0.060031 0.131135 0.238563 0.324308 1.0 1.0 1.0 1.0 1.0 1.0 1.0
1
2 3 4 5 6 7 d orbitals 1 2 3 4
2.0 0.7 0.25 0.08
1.0 1.0 1.0 1.0
0.7 0.25 0.08
1.0 1.0 1.0
f orbitals 1 2 3
3 101
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19 June 1992
Table 2 Ionization potentials and excitation spectra of the fluorine atom and its ions (all values given in eV with errors in parentheses) Experimental a)
CCSD
CCSDT 1
114.00 (-0.21) 64.81 (-0.25) (0.23)
113.95 (-0.26) 64.80 (-0.26) (0.26)
F4+
IP 2~~3s 2S average absolute error
114.21 65.06
F’+
IP 2s22p2 ‘D 2sz2pz ‘S 2~~2~3s ‘P 2~~2~3s ‘P average absolute error
81.23 3.13 6.64 51.63 52.52
86.86 3.19 6.22 51.33 52.20
(-0.37) (+0.06) (-0.42) (-0.30) (-0.32) (0.29)
86.92 3.26 6.57 51.43 52.31
(-0.31) (+0.13) (-0.07) (-0.20) (-0.21) (0.18)
F2+
IP 2s22p’ ‘D 2s22p’ *P 2s22p23s 4P 2s22p23s 2P 2s22p23s *D 2s22p23s ‘S average absolute error
62.65 4.23 6.39 39.26 40.23 42.65 46.20
62.27 4.36 6.17 38.96 39.96 42.41 45.48
(-0.38) (+0.13) (-0.22) (+0.30) (-0.27) (-0.24) (-0.72) (0.32)
62.59 4.41 6.39 39.16 40.05 42.71 45.99
(-0.06) (+0.18) (0.00) (-0.10) (-0.18) (+0.06) (-0.21) (0.11)
F+
IP 2s22p4 ‘D 2sz2p4 ‘S 2s22p’3s ‘S 2s22p33s ‘S 2s22p33s ‘D 2s22p33s ‘D 2s22p’3s ‘P 2s22p33s ‘P average absolute error
34.98 2.59 5.51 21.90 22.61 26.21 26.66 28.17 28.46
34.07 2.63 5.05 21.47 22.25 25.93 26.32 27.76 28.15
(-0.92) (+0.04) (-0.52) (-0.43) (-0.42) (-0.34) (-0.34) (-0.41) (-0.31) (0.41)
35.25 2.59 5.20 21.97 22.57 26.52 26.82 28.46 28.83
(+0.27) (0.00) (-0.37) (-0.07) (-0.10) (+0.25) (+0.16) (+0.29) (+0.37) (0.21)
F
IP 2s22p43s 4P 2s22p33s 2P 2s22p33s 2D average absolute error
17.42 12.70 12.98 15.37
15.54 12.03 12.09 14.63
(-1.88) (-0.67) (-0.89) (-0.74) (1.04)
19.03 (+1.61) 13.74 (+ 1.04) 13.78 (+0.80) 16.35 (-+0.98) (1.11)
F
IP
3.45
-0.82
(-4.31)
6.60 (3.15)
ai)Ref. [25].
electrons were added. The CCSD and CCSDTl results are given in table 2. Haque and Kaldor [ 231 have applied their CCSD + T approximation to the (0, n) sectors of the 0 atom for n < 4. This scheme includes a static P’S, contribution to the S, amplitudes in addition to CCSD. They found that the S, contribution to the n = 1 and n = 2 sectors is small, but that the contribution to n = 3 and n= 4 is considerably more pronounced, the reason being that S, contributes significantly to the three- and four-body effective 102
Hamiltonians. This is clearly borne out by the present results. For n = 1 and n = 2 there is little difference between CCSD and CCSDTl. For n = 3 and n=4 there is in general a marked improvement of CCSDT 1 over CCSD. Once we arrive at n = 5 and n =6 the change is certainly drastic. However, CCSDTl greatly overestimates the S3 effects, giving errors almost as big as those of CCSD but with opposite sign. The substantial effect of S, can be explained by the high electron density in the valence shell of first-row atoms. This makes it likely that three
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electrons will mutually interact and so cause large contributions to the three body amplitudes. For the n= 5 and n=6 sectors the calculation of each five- and six-body Z&r matrix element requires, amongst other things, the addition of 10 and 20 contributions, respectively, from the three-body He, matrix. Now, the equation for S, used here is a truncation of the natural three body CCSDT equation and so relatively small systematic errors in the contributions to the three-body Hef, matrix may accumulate and spoil the results.
4. Summary and conclusion The (0, n ) sectors for n < 6 of the open-shell CCM Fock space in both the CCSD and the CCSDTl approximations have been investigated. The latter of these schemes makes approximate inclusion of S, amplitudes in the iteration cycle. The system of interest has been the F atom and its various ions. Their several excitation spectra and ionization potentials have been calculated. For the sectors n = 1 and n = 2 the CCSD approximation is good, and S, effects are small. For n = 3 and n = 4 the CCSD results become progressively worse, but the S, corrections included in CCSDTl are sufficient to yield good results. CCSDT 1 improves markedly over CCSD. For n = 5 and n = 6 the CCSDT 1 contribution is large, but both approximations are badly wrong. Clearly it is necessary in these latter cases to follow Ktimmel’s dictum that the CCM is smarter than we are and to try to develop a better sub-approximation to the natural CCSDT or full SUB3 scheme. Work is in progress to apply such an approximation scheme. Further applications to other atomic and molecular systems are in progress and the results will be published shortly together with a full description of the diagrams employed.
Acknowledgement
The research reported here was supported by the US-Israel Binational Science Foundation. SRH thanks the Committee for Planning and Budgeting
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of the Israeli Council on Higher Education for a fellowship.
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