Spin conserving natural orbital functional theory M. Piris, J. M. Matxain, X. Lopez, and J. M. Ugalde Citation: J. Chem. Phys. 131, 021102 (2009); doi: 10.1063/1.3180958 View online: http://dx.doi.org/10.1063/1.3180958 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v131/i2 Published by the AIP Publishing LLC.
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THE JOURNAL OF CHEMICAL PHYSICS 131, 021102 共2009兲
Spin conserving natural orbital functional theory M. Piris,a兲 J. M. Matxain, X. Lopez, and J. M. Ugalde Kimika Fakultatea, Euskal Herriko Unibertsitatea, and Donostia International Physics Center (DIPC), P.K. 1072, 20080 Donostia, Euskadi, Spain
共Received 4 June 2009; accepted 27 June 2009; published online 14 July 2009兲 The natural orbital functional theory is considered for spin uncompensated systems, i.e., systems that have one or more unpaired electrons. The well-known cumulant expansion is used to reconstruct the two-particle reduced density matrix. A new condition to ensure the conservation of the total spin is obtained for the two-particle cumulant matrix. An extension of the Piris natural orbital functional 1 共PNOF1兲, based on an explicit form for the cumulant, to spin uncompensated systems is also considered. The theory is applied to the calculation of energy differences between the ground state and the lowest lying excited state with different spins for first-row atoms 共Li, Be, B, C, N, O, and F兲 and diatomic oxygen molecule 共O2兲. The values we obtained are very accurate results as compared to the CCSD共T兲 method and the experimental data. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3180958兴 The electronic energy for N-electron systems is an exactly and explicitly known functional of the one-particle and two-particle reduced density matrices 共1- and 2-RDM兲, denoted hereafter as ⌫ and D, respectively, E关⌫,D兴 = 兺 hik⌫ki + 兺 具ij兩kl典Dkl,ij . ik
共1兲
ijkl
In Eq. 共1兲, hik denotes the one-electron matrix elements of the core-Hamiltonian and 具ij 兩 kl典 are the two-electron matrix elements of the Coulomb interaction. The 2-RDM can be expressed in terms of the 1-RDM by means of a reconstruction functional D关⌫兴, which once used in Eq. 共1兲 yields a density matrix functional, E关⌫兴, for the energy.1 Further, one can express the 1-RDM in terms of the natural orbitals, 兵i共x兲其, and their occupation numbers, 兵ni其, using the spectral expansion of the 1-RDM, ⌫共x1⬘兩x1兲 = 兺 nii共x1⬘兲ⴱi 共x1兲,
共2兲
i
with x ⬅ 共r , s兲 being the composite space-spin coordinate for a single electron. This transforms the density matrix functional, E关⌫兴, into the natural orbital functional E关兵ni , i其兴. A detailed account of the state of the art of the natural orbital functional theory 共NOFT兲 can be found elsewhere.2,3 In essence, given the reconstruction functional, one has to minimize the resulting energy expression with respect to both the natural orbitals and their occupation numbers under the appropriate constrains. Among them, the so-called N-representability stands prominently. One advantage of NOFT is that restricting the occupation numbers 兵ni其 into the range 0 ⱕ ni ⱕ 1 constitutes the necessary and sufficient easily implementable condition for the N-representability of the 1-RDM.4 Nevertheless, it is worth emphasizing that this does not overcome the N-representability problem of the energy a兲
Electronic mail:
[email protected].
0021-9606/2009/131共2兲/021102/4/$25.00
functional,5,6 for the latter is related to the N-representability problem of the 2-RDM,7 via the reconstruction functional. In spite of the above difficulties, promising developments of NOFT have been achieved.8–13 Most relevant for the current investigation is the recent successful implementation of an efficient iterative diagonalization procedure for the orbital optimization, which speeds up the converge of the solution considerably.14 While this approach has been tested satisfactorily for a number of spin compensated systems, the correct description of spin-uncompensated remains unattained. Earlier attempts to solve the problems include a restricted formulation of NOFT,15 which has the virtue of avoiding the spin contamination, but underestimates the total energy since the open shells are not allowed, by construction, to contribute to the correlation energy. Additionally, an intermediate formulation of the GU functional,16 considering spin-independent natural orbitals but spin-dependent occupation numbers, was developed and applied to the first-row atoms.17 The major shortcoming of this approach is that it does not conserve, in general, the total spin. The aim of the present research is to formulate and to develop a computationally efficient implementation of NOFT for spin-uncompensated systems, which conserves the expectation values of the total spin, 具Sˆ2典, and its projection, 具Sˆz典. The spin-orbital set 兵i共x兲其 is usually split into two subsets: 兵␣p 共r兲␣共s兲其 and 兵p 共r兲共s兲其. Consequently, the expectation value of the operator Sˆz is given by the formula18 具Sˆz典 ⬅ M S =
冕
dr1Qz共r1兩r1兲 =
N␣ − N , 2
共3兲
where N is the number of electrons with spin, M S denotes the quantum number that describes the z spin component of the eigenstate 兩SM S典 with total spin S, whereas Qz represents the spin density, namely, Qz共r1兩r1兲 = 21 关⌫␣␣共r1兩r1兲 − ⌫共r1兩r1兲兴. 19
According to Loewdin’s expression, 131, 021102-1
共4兲
it follows that
© 2009 American Institute of Physics
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J. Chem. Phys. 131, 021102 共2009兲
Piris et al.
N␣ + N 共N␣ − N兲2 = S共S + 1兲 + N . + 4 2
N共N − 4兲 具Sˆ2典 ⬅ S共S + 1兲 = − 4 +
冕
dx1dx2D共r1s1,r2s2兩r1s2,r2s1兲,
共5兲
where D共x1⬘,x2⬘兩x1,x2兲 = 兺
pq
,␣ ␣  D␣ pq,rt S pt Sqr ,
共6兲
pqrt
⬘ ⬘ where S pt = 具 p 兩 t 典 is the overlap matrix. The spin contamination effects can straightforwardly be avoided by the use of a single set of orbitals 兵 p共r兲其 for ␣ and  spins ˆ2 共S␣ pt = ␦ pt兲 so that the expectation value 具S 典 becomes
N共N − 4兲 ,␣␣ , ␣,␣ + 兺 共D␣␣ 具Sˆ2典 = − pq,pq + D pq,pq − 2D pq,qp 兲. 共7兲 4 pq The matrix elements of the 2-RDM can conveniently be expressed in terms of the cumulant expansion20 as , D pq,rt =
np nq , 共␦ pr␦qt − ␦ pt␦qr兲 + pq,rt , 2
,␣ D␣ pq,rt =
n␣p nq ,␣ ␦ pr␦qt + ␣ pq,rt , 2
共8兲
1兲␦ pr,
np = n p ,
共13兲
where n p and m p must fulfill the following constrains: 2 兺 n p = N − 2S, p
兺p mp = 2S,
兺
␣,␣ = q pq,rq
共14兲
,␣ ␣  2 兺 ␣ pq,qp = n p − n p n p = n ph p − n pm p ,
共9兲
0.
Using Eq. 共8兲 and taking into account the above sum rules, the expectation value of Sˆ2 reads as N␣ + N 共N␣ − N兲2 ,␣ 具Sˆ2典 = + − 兺 n␣p np − 2 兺 ␣ pq,qp . 4 2 p pq 共10兲 Recall that for a given value S there are 共2S + 1兲 energy degenerate spin multiplet states. We will focus on the high-spin multiplet state, i.e., 兩SM S典 = 兩SS典. Let us consider, without loss of generality, that N␣ ⬎ N. Then, using Eq. 共3兲 with M S = S and N = N␣ + N, it is easily shown that
共15兲
q
where h p = 1 − n p denotes the hole in the natural orbital p. Finally, one can use a reconstruction functional D关⌫兴 to arrive at the specific constrains that must be imposed to the energy functional in order to conserve the spin. We shall use the Piris reconstruction functional,21 PNOF, which has the following form for the two-particle cumulant: , pq,rt = −
⌬ pq 共␦ pr␦qt − ␦ pt␦qr兲, 2 共16兲
⌬␣ ⌸ pq ,␣ ␣ = − ␦ pr␦qt + pt ␦ pq␦rt , pq,rt 2 2
where ⌬⬘ are real symmetric matrices and ⌸ is a spinindependent Hermitian matrix 共symmetric for real orbitals兲.2 By combining the sum rule 共15兲 with the ansatz 共16兲, one arrives at the following diagonal elements: 2 ␣  ⌬␣ pp = n p n p = n p + n pm p,
where is the cumulant matrix. It can be easily shown, from the sum rules of the 2-RDM blocks,2 that the spin components of the cumulant matrix must fulfill the following sum rules: 2兺
n␣p = n p + m p,
which allow us to simplify the spin conserving sum rule of Eq. 共12兲, namely,
N共N − 4兲 ,␣␣ , 具Sˆ2典 = − + 兺 共D␣␣ pq,pq + D pq,pq 兲 4 pq
np 共np −
共12兲
p
We assume further that n␣p ⱖ np , so that
is the 2-RDM in the coordinate representation. The 2-RDM has six nonzero spin blocks, but only three of them are independent: D␣␣,␣␣, D␣,␣, and D,. From Eq. 共5兲, expanding D by its spin components and taking into account the orthonormality conditions of the natural orbitals, one obtains2
, = q pq,rq
Equations 共10兲 and 共11兲 and the conservation of the total spin, namely, 具Sˆ2典 = S共S + 1兲, imply the following sum rule for the cumulant: ,␣ ␣   2 兺 ␣ pq,qp = N − 兺 n p n p .
Dij,kli共x1⬘兲 j共x2⬘兲ⴱk 共x1兲ⴱl 共x2兲
ijkl
− 2兺
共11兲
⌸ pp = n p ,
共17兲
which constitute the constraints to guarantee the conservation of the total spin for the PNOF. Notice that for real natural orbitals, the energy functional 共1兲 reads as
共
兲
˜ J E = 兺 p n p共2h pp + J pp兲 + 2 兺 ⬘pq nqn p − 41 ⌬ qp pq
关
兴
␣␣  + 兺 ⬘pq ⌸qp + 21 共⌬qp + ⌬qp 兲 − nqn p K pq
共
兲
+ 兺 ⬘pq 共nqm p + mqn p兲 J pq − 21 K pq + 兺 p m ph pp +
1 2
兺⬘pq mqmp共Jpq − Kpq兲,
共18兲
where J pq = 具pq 兩 pq典 and K pq = 具pq 兩 qp典 are the usual Coulomb ˜ denotes the spinless and exchange integrals, respectively. ⌬ qp ⌬ matrix, ˜ = ⌬␣␣ + ⌬␣ + ⌬␣ + ⌬ . ⌬ qp qp qp qp qp
共19兲
The prime in Eq. 共18兲 indicates that the q = p term is omitted from the summations. The N-representability conditions of
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J. Chem. Phys. 131, 021102 共2009兲
Spin conserving NOFT
TABLE I. Total energies for the ground state 共GS兲 and the lowest-lying excited state 共ES兲 of different spin, in Hartrees. Relative energies between these states, in eV. The aug-cc-pVTZ basis set 共Ref. 23兲 was used. Experimental data from Ref. 24 and MCHF values from Ref. 25. Total energies GS GS Li Be B C N O F
2
S S 2 P 3 P 4 S 3 P 2 P 1
Total energies ES
PNOF1
MCHF
⫺7.453 593 ⫺14.626 054 ⫺24.591 041 ⫺37.762 963 ⫺54.482 412 ⫺74.967 165 ⫺99.624 578
⫺7.478 567 ⫺14.668 383 ⫺24.616 349 ⫺37.811 590 ⫺54.554 039 ⫺75.047 520 ⫺99.738 313
ES 4
P P 4 P 1 D 2 D 1 D 4 P 3
PNOF1
MCHF
PNOF1
CCSD共T兲
MCHF
Exp.
⫺5.358 349 ⫺14.527 534 ⫺24.460 530 ⫺37.735 900 ⫺54.408 444 ⫺74.894 551 ⫺99.141 502
¯ ⫺14.567 693 ⫺24.484 105 ⫺37.764 802 ⫺54.465 769 ⫺74.974 059 ⫺99.271 869
57.016 2.681 3.551 0.736 2.013 1.976 13.145
56.944 2.722 3.549 1.427 2.716 2.209 13.343
¯ 2.740 3.599 1.273 2.402 1.999 12.693
57.469 2.725 3.579 1.264 2.384 1.967 12.697
the 2-RDM impose several bounds on the matrix elements of ⌬⬘. Adopting the same approach as in Ref. 21 for closedshell systems, a natural extension of the PNOF1 approximation to spin uncompensated systems affords the following energy functional: E = 2 兺 p n ph pp + 兺 p 关n2p + 47 n pm p − 83 m phmp 兴J pp + 兺 ⬘pq 关2nqn pJ pq − ⌳qpK pq + nqm p共2J pq − K pq兲兴 + 兺 p m ph pp +
1 2
兺⬘pq mqmp共Jpq − Kpq兲,
共20兲
where hmp = 1 − m p is the magnetization hole and the offdiagonal elements of ⌳ can be cast as ⌳qp = 关1 − 2共 21 − nq兲共 21 − n p兲兴冑nqn p + 共nq −
1 2
兲共np − 21 兲冑hqhp ,
Relative energies
共21兲
with 共x兲 being the well-known Heaviside step function. The solution in NOFT is commonly established optimizing the energy functional with respect to the occupation numbers and to the natural orbitals separately. A novel procedure was recently introduced,14 which yields the orthogonal natural orbitals by an iterative diagonalization method. Bounds on the occupation numbers are easily enforced by setting n p = cos2 ␥ p and m p = sin2 ␥ p sin2 ␦ p. Then, the SQP technique22 is used for performing the optimization of the energy with respect to the auxiliary variables ␥ p and ␦ p constraining the occupancies only to the sum rules of Eq. 共14兲. In order to illustrate the performance of the proposed procedure, we collected in Table I the calculated total energies for the ground state and the lowest-lying excited state of different spin, along with their corresponding energy differences, for the first-row elements 共Li–F兲. We calculated the energies at the PNOF1 and CCSD共T,full兲 levels of theory, combined in both cases with the correlation-consistent valence triple- basis set including polarization and diffuse functions 共aug-cc-pVTZ兲 developed by Dunning.23 The
PNOF1 calculations were performed with the code PNOFID,26 whereas the CCSD共T兲 ones were carried out using the 27 GAUSSIAN 03 program package. The almost exact multiconfigurational Hartree–Fock 共MCHF兲 values28 were taken from Ref. 25. Inspection of the total energies shown in Table I reveals that the calculated PNOF1 energies are bounded from below by the almost exact reference MCHF energies, for both the ground and the excited states. This is a challenging task not met by most current energy functionals, like current DFT implementations.29 With respect to the atomic spin-splitting energies, the data collected in Table I suggest that our PNOF1 implementation for spin uncompensated systems performs quite satisfactorily for all the atoms investigated. The outstanding agreement obtained for the calculated 3 P − 1D spin-splitting energy of the oxygen atom, relative to its corresponding experimental mark, and the poorer matching found for the carbon atom are remarkable. However, on the overall, the performance of PNOF1 matches satisfactorily that of the highly accurate MCHF method and outperforms that of the single reference, but accurate, CCSD共T兲 method. Finally, we calculated the triplet-singlet spin-splitting energy of the biologically relevant oxygen molecule. This is known to be a challenging task, which requires costly ab initio multiconfigurational methods to describe it correctly. Table II shows the results obtained with a variety of methods along with the experimental value of the spin-splitting energy between the 3⌺ ground state and the lowest-lying 1⌬ excited state. Calculations have been carried out at the experimental geometry taken from Ref. 30. As found for the spin-splitting energies of the atoms shown in Table II, the PNOF1 outperforms CCSD共T兲 and yields a reasonable spin-splitting energy of 0.83 eV, which must be compared with the experimental mark of 0.98 eV. The multiconfigurational calculations of Sevin and McKee31 give a value of 1.07 eV, surprisingly close the experimental
TABLE II. Relative energy, ⌬E in eV, between the 3⌺ GS and the lowest-lying 1⌬ ES of the oxygen molecule. The aug-cc-pVTZ basis set 共Ref. 23兲 was used for the PNOF1, B3LYP, and CCSD共T兲. CASPT2 and experimental values are from Refs. 31 and 30, respectively. Method ⌬E
PNOF1
CCSD共T兲
B3LYP
CASPT2
Exp.
0.83
1.26
0.44
1.07
0.98
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021102-4
result, in spite of the poor basis set, 6-31 g共d兲, used. DFT/ B3LYP badly fails to reproduce such a spin-splitting energy. In summary, we formulated the precise constrains that the two-particle cumulant matrix must fulfill, Eq. 共12兲, in order to conserve the expectation values of the total spin and its projection. Additionally, we derived the resulting corresponding constrains for the natural orbitals and their occupation numbers, Eq. 共17兲, for the Piris reconstruction functional, which, once combined with our iterative diagonalization procedure, yields an efficient implementation for spin uncompensated systems as demonstrated by the preliminary illustrative calculations, shown in this letter, which highlights the reliability of the proposed method. These developments constitute, in our opinion, a bold step toward an accurate and practical implementation of the natural orbital functional theory. Financial support comes from Eusko Jaurlaritza and the Spanish Office for Scientific Research. The SGI/IZO–SGIker UPV/EHU is gratefully acknowledged for generous allocation of computational resources. T. L. Gilbert, Phys. Rev. B 12, 2111 共1975兲; M. Levy, Proc. Natl. Acad. Sci. U.S.A. 76, 6062 共1979兲; S. M. Valone, J. Chem. Phys. 73, 1344 共1980兲; A. M. K. Müller, Phys. Lett. A 105, 446 共1984兲. 2 M. Piris, Adv. Chem. Phys. 134, 387 共2007兲. 3 D. R. Rohr, Ph.D. thesis, Vrije Universiteit Amsterdam, 2008. 4 A. J. Coleman, Rev. Mod. Phys. 35, 668 共1963兲. 5 J. M. Herbert and J. E. Harriman, J. Chem. Phys. 118, 10835 共2003兲. 6 P. W. Ayers and S. Liu, Phys. Rev. A 75, 022514 共2007兲; E. V. Ludena, F. Illas, and A. Ramirez-Solis, Int. J. Mod. Phys. B 22, 4642 共2008兲 共and references therein兲. 7 D. A. Mazziotti, Adv. Chem. Phys. 134, 21 共2007兲. 8 K. Pernal, O. Gritsenko, and E. J. Baerends, Phys. Rev. A 75, 012506 共2007兲. 9 M. Piris, X. Lopez, and J. M. Ugalde, J. Chem. Phys. 126, 214103 共2007兲; 128, 134102 共2008兲. 1
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