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Examples of known SCF procedures which do not satisfy all necessary conditions for the energy to be stationary
Volume 18, number 4
CHEMICAL PHYSICS LETTERS
15 February
1973
I?XAMPLES OF KNOWN SCF PROCEDURES WHICH DO NOT SATISFY ALL NECESSARY CONDITIONS FOR THE ENERGY TO BE STATIONARY R. ALBAT and N. GRUEL Institut fir Theoretische Aysik, Justus Liebig-Universifiit, Giessen, Germany Received 6 October 1972 Revised manuscript received 10 November
1972
it is shown that some open-shell SCF procedures exist in the literature, which do not always fulfill all necessary conditions for the energy to be stationary. The source of the error is the incorrect trentment,of the off-diagonal Lagrange multipliers. The consequences of this incorrectness are demonstrated by two simple examples.
The open-shell SCF equations for many-electron systems developed by Roothaan [I] and extended by Huzinaga [2] are well-known and widely used in practice. An SCF procedure starting from a more general energy expression was derived by Birss and Fraga [3,4]. With Jhis procedure one should be able to handle more general electron configurations. Unfortunately, their main expression in general does not take into account all necessary conditions for the energy to be stationary. A further method set up by Hunt et al. [5] .also has this deficiency. As far as we how, these mistakes have not been corrected for up to now. We merely want to show them up and demonstrate the consequences in two simple examples. We consider an open-shell state of an N-electron system. ,The SCF wavefunctions of the degenerate components of this state are, in general, linear combinations of Slater determinants with coefficients fEed by symmetry. Each Slater determinant is built of symmetry adapted doubly occupied and singly occupied oneselectron spatial functions {&, G2, . ... Qn), which are out of a complete orthononnal set {$I. The energy expression to be varied is taken to be the average’vaiue of the energy expectation values of all degenerate components. Then it is possible to write the condition for the ener,v to be stationary under variatiqns S$ of 4 in the form [l-3] n
with the constraints i,j
Gqi,l~j)+~@i16#j)=O;
(2)
where 6E is the variation of the energy_In general, the explicit forms of the so-called Hartree-Fock operators Hi defined by (1) depend on the special energy expression of the state [I -3, 51. We now follow the work of Dahl et al. [6], who derived the necessary conditions for the spatial orbitals in an elegant and easy way. In ref. [6] it is shown that the conditions (1) and (2) are equivalent to
and (QjlH”l fix-’= (?JHil Gkk>; i, k < n . Condition (3) is equivalent to k<,t.
(5)
To satisfy (4) the Hartree-Fock equations (5) must be solved with *Lhecondition This expresses the herrniticity of the Lagrange multipliers Ekj. We can write (5) in the form Ekj
=
$,,.
(6a>
Volume 18, number 4
CHEMICALPHYSJCSLETTERS.
or with (4) alternatively as
(6bj Up to this point we have followed the derivation of Dahl et al. Now, it is evident that a solution of (6b) is also a solution of (6~9, but the inverse may not be true. That is to say: a solution of (6a) can, but need not satisfy condition (4) and therefore, can lead to an energy value which is not stationary. It is possible to couple (6a) and (6b) by multiplying (6b) with hf; and (6a) with 1-A, and adding the results. This is due to Huzinaga [7] who did it for another purpose. This yields
For X, # 0 a solution of (7) always satisfies (4) and therefore leads to a stationary energy. The formalism of Roothaan [l] and Huzinaga [2] is identical with putting h, = l/( 1 -f), ho = -fl(l -f) for each symmetry species (c stands for the closed shell orbitals, o for the open shell orbitals and fis the fractional occupation number). If unitary transformations arc possible between the Qi without altering the energy, then all or at least some of the conditions (4) are automatically fulfilled. From the foregoing it should be clear that the correct equations one has to solve are (6b) or (7) with hk + 0. In the formalism of Birss and Fraga (BS) [3], the starting point is an equation of the form (6a). Therefore, this formalism in general does not contain condition (4) or the hermiticity of the Lagrange multipliers. The BS equations yield a variety of solutions which do not all iead to a stationary energy. In ref. [4] an application of the BS formalism is given for the three lowest S-states of He using a basis set of three. STO’s: ls(2.0), ls’(O.8) and 2s(O.575). The resuits are correct for JS (1~)~ and 3S (Is) (3s). The former case is just a closed shell case, the latter is correct because a unitary transformation is possible. For IS (Is) (2s) the correct equations (6b) or (7) (with X # 0) yield unique SCF orbitals which give an energy of -2.16916 au. The use of (6a) results in a variety of solutions. We obtained some of them by starting the iteration procedure with different orbitals. Starting
I.5 February 1973
with Schmidt-orthogonahzed STO’s yields -2.15563 au. A start with orbitals which diagonake. (-$A-2/r) yields -2.1583Cl au. Starting with the solution for 3S(X = 3) gives -2.15541 au. FinaIIy, starting with the‘correct.solution of (6b) naturally yields -2.16916 au. In ref. [4] a value of -2.1376 au is reported, which gives a further solution. The OCBSE method due to Hunt et 21. 1.51was proposed to avoid the explicit appearance of the off-diagonal Lagrange multipliers in (5) by allowing onIy a restricted mixture of the $k. In going from one iteration step to the next, only the occupied orbit& $1 for ‘which H” = H’ and the unoccupied orbit& are allowed to improve bk_ The OCBSE method is in principie identical with Huzinagas “second method” [2], which was developed to handle open shells with equaI symmetry. We see that these two methods do not take into account condition (4) for H” f: HI. To demonstrate the failure of the OCBSE procedure, we present some simple calculations for *S (ls)*(2s) of Li using a basis set of three STO’s: ls(2.435), ls’(4.5) and 2s(O.67). With the correct procedure of Roothaan we obtain -7.43207 au for the energy. Following the OCBSE formalism, again the final result depends on the orbitak used for the start of the iteration. Starting with Schmidt-orthogonalized STO’s yields -7.43 1.52 au. Starting with orbitals which diagonalize (-*A-3/r) yields -7.42749 au. Finally, starting with the exact solution of Roothaan gives again -7.43207 au. So it seems that no correct SCF procedure for more than one open-shell with equal symmetry exists. It could be possible to derive a formalism in the same way as Birss and Fraga did, but with the use of eqs. (6b) or (7) instead of (6a). Another way is to satisfy the necessary conditions (3) and (4) directly in a matrix formulation. This was done with the help of a Green furxtion method by the authors [S/ . References [ 11 C.C.J. Roothaan, Rev. Mod. Phys. 32 (1960) 179. [ 31 S. Huzinaga, Phys Rev. 120 (1960) 866. [3] F.W. Birssand S. Fraga, 1. Chem. Phys. 38 (1963) 2552. 141 F.W. Birssand S. Fraga, J. Chcm. Phyc 40 (1964) 3203. (51 W.S. Hunt T.H. Dunnung and W.A. Goddard III, Chem. Phys. Letters 3 (1969) 606. [6] J.P. Dahl; H. Johansen, D.R. Truax and T. Ziegler, Chem. Phys. Letters 6 (1970) 64. f7] S. Huzinaga, J. Chem. Phys 51 (1969) 3971. [S] R. Albat and N. Gruen, to be published.
573
CHEMICAL PHYSICS LETTERS
15 February
1973
I?XAMPLES OF KNOWN SCF PROCEDURES WHICH DO NOT SATISFY ALL NECESSARY CONDITIONS FOR THE ENERGY TO BE STATIONARY R. ALBAT and N. GRUEL Institut fir Theoretische Aysik, Justus Liebig-Universifiit, Giessen, Germany Received 6 October 1972 Revised manuscript received 10 November
1972
it is shown that some open-shell SCF procedures exist in the literature, which do not always fulfill all necessary conditions for the energy to be stationary. The source of the error is the incorrect trentment,of the off-diagonal Lagrange multipliers. The consequences of this incorrectness are demonstrated by two simple examples.
The open-shell SCF equations for many-electron systems developed by Roothaan [I] and extended by Huzinaga [2] are well-known and widely used in practice. An SCF procedure starting from a more general energy expression was derived by Birss and Fraga [3,4]. With Jhis procedure one should be able to handle more general electron configurations. Unfortunately, their main expression in general does not take into account all necessary conditions for the energy to be stationary. A further method set up by Hunt et al. [5] .also has this deficiency. As far as we how, these mistakes have not been corrected for up to now. We merely want to show them up and demonstrate the consequences in two simple examples. We consider an open-shell state of an N-electron system. ,The SCF wavefunctions of the degenerate components of this state are, in general, linear combinations of Slater determinants with coefficients fEed by symmetry. Each Slater determinant is built of symmetry adapted doubly occupied and singly occupied oneselectron spatial functions {&, G2, . ... Qn), which are out of a complete orthononnal set {$I. The energy expression to be varied is taken to be the average’vaiue of the energy expectation values of all degenerate components. Then it is possible to write the condition for the ener,v to be stationary under variatiqns S$ of 4 in the form [l-3] n
with the constraints i,j
Gqi,l~j)+~@i16#j)=O;
(2)
where 6E is the variation of the energy_In general, the explicit forms of the so-called Hartree-Fock operators Hi defined by (1) depend on the special energy expression of the state [I -3, 51. We now follow the work of Dahl et al. [6], who derived the necessary conditions for the spatial orbitals in an elegant and easy way. In ref. [6] it is shown that the conditions (1) and (2) are equivalent to
and (QjlH”l fix-’= (?JHil Gkk>; i, k < n . Condition (3) is equivalent to k<,t.
(5)
To satisfy (4) the Hartree-Fock equations (5) must be solved with *Lhecondition This expresses the herrniticity of the Lagrange multipliers Ekj. We can write (5) in the form Ekj
=
$,,.
(6a>
Volume 18, number 4
CHEMICALPHYSJCSLETTERS.
or with (4) alternatively as
(6bj Up to this point we have followed the derivation of Dahl et al. Now, it is evident that a solution of (6b) is also a solution of (6~9, but the inverse may not be true. That is to say: a solution of (6a) can, but need not satisfy condition (4) and therefore, can lead to an energy value which is not stationary. It is possible to couple (6a) and (6b) by multiplying (6b) with hf; and (6a) with 1-A, and adding the results. This is due to Huzinaga [7] who did it for another purpose. This yields
For X, # 0 a solution of (7) always satisfies (4) and therefore leads to a stationary energy. The formalism of Roothaan [l] and Huzinaga [2] is identical with putting h, = l/( 1 -f), ho = -fl(l -f) for each symmetry species (c stands for the closed shell orbitals, o for the open shell orbitals and fis the fractional occupation number). If unitary transformations arc possible between the Qi without altering the energy, then all or at least some of the conditions (4) are automatically fulfilled. From the foregoing it should be clear that the correct equations one has to solve are (6b) or (7) with hk + 0. In the formalism of Birss and Fraga (BS) [3], the starting point is an equation of the form (6a). Therefore, this formalism in general does not contain condition (4) or the hermiticity of the Lagrange multipliers. The BS equations yield a variety of solutions which do not all iead to a stationary energy. In ref. [4] an application of the BS formalism is given for the three lowest S-states of He using a basis set of three. STO’s: ls(2.0), ls’(O.8) and 2s(O.575). The resuits are correct for JS (1~)~ and 3S (Is) (3s). The former case is just a closed shell case, the latter is correct because a unitary transformation is possible. For IS (Is) (2s) the correct equations (6b) or (7) (with X # 0) yield unique SCF orbitals which give an energy of -2.16916 au. The use of (6a) results in a variety of solutions. We obtained some of them by starting the iteration procedure with different orbitals. Starting
I.5 February 1973
with Schmidt-orthogonahzed STO’s yields -2.15563 au. A start with orbitals which diagonake. (-$A-2/r) yields -2.1583Cl au. Starting with the solution for 3S(X = 3) gives -2.15541 au. FinaIIy, starting with the‘correct.solution of (6b) naturally yields -2.16916 au. In ref. [4] a value of -2.1376 au is reported, which gives a further solution. The OCBSE method due to Hunt et 21. 1.51was proposed to avoid the explicit appearance of the off-diagonal Lagrange multipliers in (5) by allowing onIy a restricted mixture of the $k. In going from one iteration step to the next, only the occupied orbit& $1 for ‘which H” = H’ and the unoccupied orbit& are allowed to improve bk_ The OCBSE method is in principie identical with Huzinagas “second method” [2], which was developed to handle open shells with equaI symmetry. We see that these two methods do not take into account condition (4) for H” f: HI. To demonstrate the failure of the OCBSE procedure, we present some simple calculations for *S (ls)*(2s) of Li using a basis set of three STO’s: ls(2.435), ls’(4.5) and 2s(O.67). With the correct procedure of Roothaan we obtain -7.43207 au for the energy. Following the OCBSE formalism, again the final result depends on the orbitak used for the start of the iteration. Starting with Schmidt-orthogonalized STO’s yields -7.43 1.52 au. Starting with orbitals which diagonalize (-*A-3/r) yields -7.42749 au. Finally, starting with the exact solution of Roothaan gives again -7.43207 au. So it seems that no correct SCF procedure for more than one open-shell with equal symmetry exists. It could be possible to derive a formalism in the same way as Birss and Fraga did, but with the use of eqs. (6b) or (7) instead of (6a). Another way is to satisfy the necessary conditions (3) and (4) directly in a matrix formulation. This was done with the help of a Green furxtion method by the authors [S/ . References [ 11 C.C.J. Roothaan, Rev. Mod. Phys. 32 (1960) 179. [ 31 S. Huzinaga, Phys Rev. 120 (1960) 866. [3] F.W. Birssand S. Fraga, 1. Chem. Phys. 38 (1963) 2552. 141 F.W. Birssand S. Fraga, J. Chcm. Phyc 40 (1964) 3203. (51 W.S. Hunt T.H. Dunnung and W.A. Goddard III, Chem. Phys. Letters 3 (1969) 606. [6] J.P. Dahl; H. Johansen, D.R. Truax and T. Ziegler, Chem. Phys. Letters 6 (1970) 64. f7] S. Huzinaga, J. Chem. Phys 51 (1969) 3971. [S] R. Albat and N. Gruen, to be published.
573