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Unconditional convergence in SCF theory: a general level shift technique
Volume 47, number 3
CHEMICAL PHYSICS LETTERS
1 May 1977
UNCONDITIONAL CONVERGENCE XNSCF THEORY: A GENERAL LEVEL SHIFT TECHNLQUE R CA-6 Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada
and J-A. HERNhDEZ
and F. SANZ
Seccih de Quimica Cudntica, Institute Barcelona 17. Spain
Quimico
de Sam%,
Received 4 February 1977
A level shift procedure is described in the framework of one-operator SCF formalisms. This procedure permits one to obtain in any case unconditional convergence to a stationary energy.
1. Introduction
shift procedure.
the virtual orbitals), preserving the initial ordering of the orbital eigenvalues, so the SCF iterations, through an invariant “aufbau” principle, converge to an “a priori” chosen wavefunction. Then, a SCF desideratum can be transformed into a principle to be followed by any potential user off&e level shift technique. - “Given an initial ordered set of MO’s, the ordering and symmetry of each MO subset shall be maintained until self-consistency is achieved”. The ordering principle is a sine-qua-non condition which, if not fulfilled, brings about that the auffiau principle cannot be invariant throughout SCFiterations_ As a consequence, energy fluctuates between two different values and, in some cases, MO symmetry is lost.
2. The ordering principle
3. Level shift technique in a general SCF procedure
Nonconvergence in SCF calculations is mainly due to the mixing among MO’s belonging to the various subsets of the molecular basis, as long as the iterative SCF path is followed. A simple traceback analysis of any reluctant case can make this clear. In the original level shift technique [2] , the main purpose OS to separate the eigenvalues of each MO within each subset (for example: the occupied MO and
The principal drawback of many SCF formalisms consists in the definition of a pseudosecular equation for each shell [5]. The level shift technique has been described up to now for this kind of formalisms [4] _ It has been recently proved that a general procedure can be built-up, within a unique formalism, for closed, open and MC problems, which brings any SCF caIcuIation to a unique pseudosecular equation @I _ This
It is well known that Hartree-Fock procedures do not converge in some cases [l ] _ Doubtless, a good deal of computation time has been lost due to this remarkable question, but few solutions have been published. The work of Saunders and Hillier seems to be, up to date, the only way to cope with the problem [2]. Hillier and Saunders procedure has been proved convenient through many calculations [3] and it can be used in closed or open shell computational algorithms
[41This note intends
to add new aspects to the level
Volume 47, number 3
CHEMICAL
formalism is based on a coupling operator Cx,, whose eigenvectors are the desired MO’s, that is: %?Oli>=q(i>,
(1)
the set {Ii>], being the MO obtained solving (I), gives a stationary energy and preserves in every case the hermitean condition of the Lagrange multipliers matrix, but the result of eq. (1) does not necessarily follow the
ordering principle. It is interesting to note here that cases in which, for some formalisms, the SCF process diverges (for instance, thr: OH radical in Roothaan’s formalism as quoted by Sleeman [I ] ), a smooth convergence is found using a unique equation such as (1). Other cases diverge in any formalism and the principa1 examples of this can be found in molecular structures very far from equilibrium, where the tendency of orbital reordering is very strong. In every case the way to follow the ordermg principle can be achieved through previous experience in eigenspace manipuIation of SCF solutions [7]. The shift operator can be defined as V=
+-/iXij
(2)
by means of the %?, eigenspace projectors {[i>Cil) and an arbitrary set of real numbers @}. A new coupling operator, written as %=%@v
(3)
has the same set of eigenvectors as ??+,, with a secular equation 32(i) = (+ + &)li> ,
(4)
that is with eigenvalues {ei + pi}- As a consequence, the electronic energy, associated to the MO set (Ii>} remains invariant. A complete separation of each MO or entire subsets of the MO manifold can be easiIy obtained choosing the appropriate shifts. In this way the ordering principle can be fulfilled, and one can obtain any desired wavefunction. From this basic ploperty of the coupling operator %, a flexible algorithm can be constructed, which can be .adapted to each case, even at each iteration, without extra effort. In this context, Saunders and Hillier’s IeveI shift technique
can be made equivalent
operator written as
to a level
shift
1 May 1977
PHYSICS LETTERS
the fast sum running over some MO subsets previously defined within each case. It should be said, finally, that the ordering principle is not only essential for the general SCF computational behaviour, but also for the application of accereration or extrapolation procedures [83. The LCAO form for an operator used as in eq. (2), will be simply a mdtI-iX defined by V = SITOicicF]
s 9
(6)
where the {Ci} are the MO expansion coefficients and S the overlap matrix.
4. Koopmans’ theorem and level shifted closed she11 Fock operators
Koopmans’ theorem [9] has been a celebrated and widely used argument in order to have an estimate of ionization potentials for closed sheff molecuIes. In this case, use of level shifted ciosed shell Fock operators written as F= Fo f C&li>
permits one to obtain, through the usual SCF procedure, the proper canonical orbitals. It is evident that all the mathematical facts which apply to Fo can be used on F. Then, using Koopmans’ result ri = -(i( FI i) = -(ei + pi) or in this context using the arbitrar-iness of pi we can have an ionization potential taylor made. We feel that this can add more information on recently quoted “breakdowns” and inconsistencies [lo] of the theorem. We have shown that an arbitrary reordering of MO’s can be obtained with the level shift technique, in this manner one can adequate the MO energies to experimental spectra, and as a consequence one can safely conclude that, even in a cIosed shell level, MO eigenvalues have little physical meaning. A reflection on this restit should be made. MO energies are not observables, then any discussion involving them is in general meaningless. Ionization potentials should be associated to energy differences between ionized and neutral states.
Volume 47, number 3
CHEMICAL
PHYSICS
5. Conclusions
A level shift technique properIy used in a unique coupling operator framework can give a SCF process with unconditional convergence. Monoconfigurational closed or open shell calculations and multiconfigurational procedures are transformed in this manner in stable computational algorithms. In the level shift context, Koopmans’ theorem looses any significance even in the closed shell case.
References [l]
D.H. Sleeman, Theoret. Chim. Acta ll(l968) 135; S.W. Harrison, G.A. Henderson, L.J. Massa and P. Solomon, Astrophys. J. 189 (1974) 605. [ 21 V.R. Saunders and 1-H. Hillier, Intern. J. Quantum Chem. 7 (1973) 699; A.I. Dement’ev, N-F. Stepanov and S.S. Yarovoi, Intern. J. QuantumChem. 8 (1974) 107. [3] A. Veiiiard, in: Computational techniques in quantum chemistry and molecular physics, NATO Advanced Study Institutes Series (Reidel, Dordrecht, 1975). [4] M.F. Guest and V.R. Saunders, MoL Phys 28 (1974) 819; M.H. Wood and A. Veillard, hfol. Phys. 26 (1973) 595; 0. Matsuoka, Mol. Phys. 30 (1975) 1293;
LETTERS
i May L977
J.E. Grabenstetter and F. Grein, &foL Phys 31(1976) 1469. [S] J. Hendekovig Theoret. Chim. Acta (1976) L93; K. Faegri Jr. and R. Manne, Mol. Phys. 31(1?76) 1037; I-f. Hsu, E.R Davidson and RX Pitzer, J. Chem. Phyr 65 (1976) 609; . J.W. CaIdwell and hf.S. Gordon,Chem. Phys. Letters 43 (2976) 493. [6] K. Hio and H. Nakatsuji, J. Chem. Phys 59 (1973) L45i K. Hiio, J. Chem. Phys 60 (1974) 3215; R Caballol, R GaIIifa, J.&f. Riera and R. Carb6, Intern J. Quantum Chem. 8 (1974) 373; R. Garbo, R. Gaiiifa and J-M. Riera, Chem. Phys. Letters 30 (1975) 43; 33 (1975) 545; Y. fshikawa, Chem. Phyr Letters 37 (1976) 597. (71 S. Huzinaga and C. Amau, Phyr Rev. AL (1970) 128.5; C. Arnau, R Carbb and S. Huzinaga, Afinidad 28 (L971) 1147; S. Huzinaga, D. h!cWtlliims and AA. Cant& Advan Quantum Chem. 7 (1973) 187; R. Garbo, Intern. I. Quantum Chem. 8 (1974) 423. [8] I. Rdeggen, Chem. Phys Letters 22 (1973) L40;
W.B. Neilsen, Chem. Phys. Letters 18 (1973) 225; LC Chans, Chem. Phys. Letters 36 (19%) 611; R Barr and H. Basch, Clnem. Phys Letters 32 (L975)
537.
[9] T. Koopmans, Physica l(1934) 104. [IO] M. Jungen. Chem. Phys Letters 21 (1973) 68; L.S. Cederbaum, Chem. Phys Letters 15 (1974) 562; R.L. DeKock, Chem. Phys. Letters 27 (1974) 297; G. Dogett, Mol. Phys. 29 (1975) 313.
583
CHEMICAL PHYSICS LETTERS
1 May 1977
UNCONDITIONAL CONVERGENCE XNSCF THEORY: A GENERAL LEVEL SHIFT TECHNLQUE R CA-6 Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada
and J-A. HERNhDEZ
and F. SANZ
Seccih de Quimica Cudntica, Institute Barcelona 17. Spain
Quimico
de Sam%,
Received 4 February 1977
A level shift procedure is described in the framework of one-operator SCF formalisms. This procedure permits one to obtain in any case unconditional convergence to a stationary energy.
1. Introduction
shift procedure.
the virtual orbitals), preserving the initial ordering of the orbital eigenvalues, so the SCF iterations, through an invariant “aufbau” principle, converge to an “a priori” chosen wavefunction. Then, a SCF desideratum can be transformed into a principle to be followed by any potential user off&e level shift technique. - “Given an initial ordered set of MO’s, the ordering and symmetry of each MO subset shall be maintained until self-consistency is achieved”. The ordering principle is a sine-qua-non condition which, if not fulfilled, brings about that the auffiau principle cannot be invariant throughout SCFiterations_ As a consequence, energy fluctuates between two different values and, in some cases, MO symmetry is lost.
2. The ordering principle
3. Level shift technique in a general SCF procedure
Nonconvergence in SCF calculations is mainly due to the mixing among MO’s belonging to the various subsets of the molecular basis, as long as the iterative SCF path is followed. A simple traceback analysis of any reluctant case can make this clear. In the original level shift technique [2] , the main purpose OS to separate the eigenvalues of each MO within each subset (for example: the occupied MO and
The principal drawback of many SCF formalisms consists in the definition of a pseudosecular equation for each shell [5]. The level shift technique has been described up to now for this kind of formalisms [4] _ It has been recently proved that a general procedure can be built-up, within a unique formalism, for closed, open and MC problems, which brings any SCF caIcuIation to a unique pseudosecular equation @I _ This
It is well known that Hartree-Fock procedures do not converge in some cases [l ] _ Doubtless, a good deal of computation time has been lost due to this remarkable question, but few solutions have been published. The work of Saunders and Hillier seems to be, up to date, the only way to cope with the problem [2]. Hillier and Saunders procedure has been proved convenient through many calculations [3] and it can be used in closed or open shell computational algorithms
[41This note intends
to add new aspects to the level
Volume 47, number 3
CHEMICAL
formalism is based on a coupling operator Cx,, whose eigenvectors are the desired MO’s, that is: %?Oli>=q(i>,
(1)
the set {Ii>], being the MO obtained solving (I), gives a stationary energy and preserves in every case the hermitean condition of the Lagrange multipliers matrix, but the result of eq. (1) does not necessarily follow the
ordering principle. It is interesting to note here that cases in which, for some formalisms, the SCF process diverges (for instance, thr: OH radical in Roothaan’s formalism as quoted by Sleeman [I ] ), a smooth convergence is found using a unique equation such as (1). Other cases diverge in any formalism and the principa1 examples of this can be found in molecular structures very far from equilibrium, where the tendency of orbital reordering is very strong. In every case the way to follow the ordermg principle can be achieved through previous experience in eigenspace manipuIation of SCF solutions [7]. The shift operator can be defined as V=
+-/iXij
(2)
by means of the %?, eigenspace projectors {[i>Cil) and an arbitrary set of real numbers @}. A new coupling operator, written as %=%@v
(3)
has the same set of eigenvectors as ??+,, with a secular equation 32(i) = (+ + &)li> ,
(4)
that is with eigenvalues {ei + pi}- As a consequence, the electronic energy, associated to the MO set (Ii>} remains invariant. A complete separation of each MO or entire subsets of the MO manifold can be easiIy obtained choosing the appropriate shifts. In this way the ordering principle can be fulfilled, and one can obtain any desired wavefunction. From this basic ploperty of the coupling operator %, a flexible algorithm can be constructed, which can be .adapted to each case, even at each iteration, without extra effort. In this context, Saunders and Hillier’s IeveI shift technique
can be made equivalent
operator written as
to a level
shift
1 May 1977
PHYSICS LETTERS
the fast sum running over some MO subsets previously defined within each case. It should be said, finally, that the ordering principle is not only essential for the general SCF computational behaviour, but also for the application of accereration or extrapolation procedures [83. The LCAO form for an operator used as in eq. (2), will be simply a mdtI-iX defined by V = SITOicicF]
s 9
(6)
where the {Ci} are the MO expansion coefficients and S the overlap matrix.
4. Koopmans’ theorem and level shifted closed she11 Fock operators
Koopmans’ theorem [9] has been a celebrated and widely used argument in order to have an estimate of ionization potentials for closed sheff molecuIes. In this case, use of level shifted ciosed shell Fock operators written as F= Fo f C&li>
permits one to obtain, through the usual SCF procedure, the proper canonical orbitals. It is evident that all the mathematical facts which apply to Fo can be used on F. Then, using Koopmans’ result ri = -(i( FI i) = -(ei + pi) or in this context using the arbitrar-iness of pi we can have an ionization potential taylor made. We feel that this can add more information on recently quoted “breakdowns” and inconsistencies [lo] of the theorem. We have shown that an arbitrary reordering of MO’s can be obtained with the level shift technique, in this manner one can adequate the MO energies to experimental spectra, and as a consequence one can safely conclude that, even in a cIosed shell level, MO eigenvalues have little physical meaning. A reflection on this restit should be made. MO energies are not observables, then any discussion involving them is in general meaningless. Ionization potentials should be associated to energy differences between ionized and neutral states.
Volume 47, number 3
CHEMICAL
PHYSICS
5. Conclusions
A level shift technique properIy used in a unique coupling operator framework can give a SCF process with unconditional convergence. Monoconfigurational closed or open shell calculations and multiconfigurational procedures are transformed in this manner in stable computational algorithms. In the level shift context, Koopmans’ theorem looses any significance even in the closed shell case.
References [l]
D.H. Sleeman, Theoret. Chim. Acta ll(l968) 135; S.W. Harrison, G.A. Henderson, L.J. Massa and P. Solomon, Astrophys. J. 189 (1974) 605. [ 21 V.R. Saunders and 1-H. Hillier, Intern. J. Quantum Chem. 7 (1973) 699; A.I. Dement’ev, N-F. Stepanov and S.S. Yarovoi, Intern. J. QuantumChem. 8 (1974) 107. [3] A. Veiiiard, in: Computational techniques in quantum chemistry and molecular physics, NATO Advanced Study Institutes Series (Reidel, Dordrecht, 1975). [4] M.F. Guest and V.R. Saunders, MoL Phys 28 (1974) 819; M.H. Wood and A. Veillard, hfol. Phys. 26 (1973) 595; 0. Matsuoka, Mol. Phys. 30 (1975) 1293;
LETTERS
i May L977
J.E. Grabenstetter and F. Grein, &foL Phys 31(1976) 1469. [S] J. Hendekovig Theoret. Chim. Acta (1976) L93; K. Faegri Jr. and R. Manne, Mol. Phys. 31(1?76) 1037; I-f. Hsu, E.R Davidson and RX Pitzer, J. Chem. Phyr 65 (1976) 609; . J.W. CaIdwell and hf.S. Gordon,Chem. Phys. Letters 43 (2976) 493. [6] K. Hio and H. Nakatsuji, J. Chem. Phys 59 (1973) L45i K. Hiio, J. Chem. Phys 60 (1974) 3215; R Caballol, R GaIIifa, J.&f. Riera and R. Carb6, Intern J. Quantum Chem. 8 (1974) 373; R. Garbo, R. Gaiiifa and J-M. Riera, Chem. Phys. Letters 30 (1975) 43; 33 (1975) 545; Y. fshikawa, Chem. Phyr Letters 37 (1976) 597. (71 S. Huzinaga and C. Amau, Phyr Rev. AL (1970) 128.5; C. Arnau, R Carbb and S. Huzinaga, Afinidad 28 (L971) 1147; S. Huzinaga, D. h!cWtlliims and AA. Cant& Advan Quantum Chem. 7 (1973) 187; R. Garbo, Intern. I. Quantum Chem. 8 (1974) 423. [8] I. Rdeggen, Chem. Phys Letters 22 (1973) L40;
W.B. Neilsen, Chem. Phys. Letters 18 (1973) 225; LC Chans, Chem. Phys. Letters 36 (19%) 611; R Barr and H. Basch, Clnem. Phys Letters 32 (L975)
537.
[9] T. Koopmans, Physica l(1934) 104. [IO] M. Jungen. Chem. Phys Letters 21 (1973) 68; L.S. Cederbaum, Chem. Phys Letters 15 (1974) 562; R.L. DeKock, Chem. Phys. Letters 27 (1974) 297; G. Dogett, Mol. Phys. 29 (1975) 313.
583