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Reply to the comment on the energy variation method
Volume 100, numk
CHEMICAL PHYSKS I.ETX-ERS
1
26 August 1983
COMMENT REPLY TO THE COMbfENT ON THE ENERGY VARIATION METHOD Josip HENDEKOVIe
and Jasna KUCAR
Rudjer Bo?kovZ Institute, 41001 Zagreb. ihat&
Yugoslavia
Received 5 April 1983: in fiiai form 14 June 1983
The assertion that the energy variation theorem for exited states follows from the classical minmas principle is further reinforced. The problem of minimizingthe manifold energy in the non-orthogonal basis is also solved.
In the preceding comment [1] Miffer and Geveci make two remarks on the recent paper by one of us (JH) [Z f , which we shall here elaborate_ First: Miller and Geveci comment on the following theorem of ref. [2] : An upper bound to the kth eigenvalue Ek of the hamiltonian is given by &k = (bk IHittbx_) 2 Ex-
(I)
if an arbitrary admissible normalized function kpk) is orthogonal and uncoupled to the first k - 1 approximate mutually orthogonal and uncoupled normalized functions I$& such that &k > 4- We say that J$J~)is uncoupled to I@$ if c~~t~l#j) = 0. Miller and Geveci correctly point out that this theorem follows from the classical minmax principle_ In fact, closer inspection reveals that the minmax principle leads to an even stronger formulation of this theorem. The theorem as it stands provides only sufficient conditions for the upper bound ofEk. We shall prove, however, that those are also the izeeesmv conditiom. As discussed by Miller and Geveci [I 1, the minmax principle provides an upper bound elH to the exact eigenvalues Et of the hamiltonian, if an Z-dimensional manifold S’ is sweeping over a certain subspace K of the full Hilbert space H. q are stationary values of the Rayleigh quotient R(e) in K_ Various approximation schemes differ mostly in the algorithm defining the subspace K, usually given by certain parameterization. Thus, the generalized Euler angles of
orthogonal orbital transformations serve as parameters in the SCF method, and parametrically define
The construction of approximate excited states, to which the quoted theorem refers, consists in the following aIgorithmr Step I : Find the lowest stationary point of R(Q) by minimizing this functional in some parametrically dependent subspace K of %I_ Step 2: Find the next stationary point of R(Q) in K, keeping r&e~rel~~~~sIy fott~td points stff~-oizff~. Since relations (3) in ref. El] characterize all stationary points of R(Q) [3], these conditions must be exacted in the search of every new stationary point. Thus, these constraints are both necessary and sufficient in the non-linear variational description of excited statesSecond: In comment [ 1 J Miller and Geveci also treat the problem of optimizing the manifold of several states, to which theorem 4 of ref. [2] refers_ (See also ref. [4] .) They examine the significant question of relaxing the orthogonality constraints in the manifold optimization, and susest an independent optimization of individual vectors. They do realize that this procedure does not minimize the manifold energy, as defined in ref. [2] _ However, theorem 4 of ref. 121 does not require that the basis vectors of the manifold be orthogonal. If the particuIar parameter&a&on of a manifold of states dlows non-orthogonal vectors, the manifold energy should be computed in such a basis and minimized_ It is, in fact, easy to derive the appropriate formula for the manifold energy in the non-orthogonal basis. Let {l#k)), k = 1,2, . . . . x’?, be a set of N linearly in-
the space K. 0 009-2614/83/0000-0000/$03.uu
Q 1983 North-Holland
117
Volume 100, number 1
CHEMICAL
PHYSICS LETTERS
dependent generally non-orthogonal vectors spannmg an N-dimensional manifold_ We introduce an equivalent orthogonal set {]$J}.obtained by the Liiwdin orthogonahzation procedure [5]
From eq. (6) in ref. [2] we obtain energy 17%~ :
References
(3)
118
expression
fold optimization within the parameterization leading to non-orthogonal vectors. The state-by-state optimization without constraints, proposed by Miller and Geveci, may result in a good manifold but need not. Although the individual energies may turn out to be low, if a common component of overlapping vectors is large the manifold energy E>f might still be inferior_
for the manifold
A’
This
26 August 1983
should serve as a basis for the mani-
(11 H-G. Miller and T. Geveci, Chem. Phys. Letters 100 (1983) 115. [2] J. HendekoviE. Chem. Phys. Letters 90 (1982) 198. [3] G. Strang and GJ_ I%.., An analysis of the finite element method (Prentice Hall, Endewood Cliffs, 1973) ch. 6. 141 R. Courant and D. Hilbert, Methoden der mathcmatischen Physik, Erster Band (Springer, Berlin, 1931) p. 399. [5] P--O. LBwdin, J. Chem. Phys. 18 (1950) 365.
CHEMICAL PHYSKS I.ETX-ERS
1
26 August 1983
COMMENT REPLY TO THE COMbfENT ON THE ENERGY VARIATION METHOD Josip HENDEKOVIe
and Jasna KUCAR
Rudjer Bo?kovZ Institute, 41001 Zagreb. ihat&
Yugoslavia
Received 5 April 1983: in fiiai form 14 June 1983
The assertion that the energy variation theorem for exited states follows from the classical minmas principle is further reinforced. The problem of minimizingthe manifold energy in the non-orthogonal basis is also solved.
In the preceding comment [1] Miffer and Geveci make two remarks on the recent paper by one of us (JH) [Z f , which we shall here elaborate_ First: Miller and Geveci comment on the following theorem of ref. [2] : An upper bound to the kth eigenvalue Ek of the hamiltonian is given by &k = (bk IHittbx_) 2 Ex-
(I)
if an arbitrary admissible normalized function kpk) is orthogonal and uncoupled to the first k - 1 approximate mutually orthogonal and uncoupled normalized functions I$& such that &k > 4- We say that J$J~)is uncoupled to I@$ if c~~t~l#j) = 0. Miller and Geveci correctly point out that this theorem follows from the classical minmax principle_ In fact, closer inspection reveals that the minmax principle leads to an even stronger formulation of this theorem. The theorem as it stands provides only sufficient conditions for the upper bound ofEk. We shall prove, however, that those are also the izeeesmv conditiom. As discussed by Miller and Geveci [I 1, the minmax principle provides an upper bound elH to the exact eigenvalues Et of the hamiltonian, if an Z-dimensional manifold S’ is sweeping over a certain subspace K of the full Hilbert space H. q are stationary values of the Rayleigh quotient R(e) in K_ Various approximation schemes differ mostly in the algorithm defining the subspace K, usually given by certain parameterization. Thus, the generalized Euler angles of
orthogonal orbital transformations serve as parameters in the SCF method, and parametrically define
The construction of approximate excited states, to which the quoted theorem refers, consists in the following aIgorithmr Step I : Find the lowest stationary point of R(Q) by minimizing this functional in some parametrically dependent subspace K of %I_ Step 2: Find the next stationary point of R(Q) in K, keeping r&e~rel~~~~sIy fott~td points stff~-oizff~. Since relations (3) in ref. El] characterize all stationary points of R(Q) [3], these conditions must be exacted in the search of every new stationary point. Thus, these constraints are both necessary and sufficient in the non-linear variational description of excited statesSecond: In comment [ 1 J Miller and Geveci also treat the problem of optimizing the manifold of several states, to which theorem 4 of ref. [2] refers_ (See also ref. [4] .) They examine the significant question of relaxing the orthogonality constraints in the manifold optimization, and susest an independent optimization of individual vectors. They do realize that this procedure does not minimize the manifold energy, as defined in ref. [2] _ However, theorem 4 of ref. 121 does not require that the basis vectors of the manifold be orthogonal. If the particuIar parameter&a&on of a manifold of states dlows non-orthogonal vectors, the manifold energy should be computed in such a basis and minimized_ It is, in fact, easy to derive the appropriate formula for the manifold energy in the non-orthogonal basis. Let {l#k)), k = 1,2, . . . . x’?, be a set of N linearly in-
the space K. 0 009-2614/83/0000-0000/$03.uu
Q 1983 North-Holland
117
Volume 100, number 1
CHEMICAL
PHYSICS LETTERS
dependent generally non-orthogonal vectors spannmg an N-dimensional manifold_ We introduce an equivalent orthogonal set {]$J}.obtained by the Liiwdin orthogonahzation procedure [5]
From eq. (6) in ref. [2] we obtain energy 17%~ :
References
(3)
118
expression
fold optimization within the parameterization leading to non-orthogonal vectors. The state-by-state optimization without constraints, proposed by Miller and Geveci, may result in a good manifold but need not. Although the individual energies may turn out to be low, if a common component of overlapping vectors is large the manifold energy E>f might still be inferior_
for the manifold
A’
This
26 August 1983
should serve as a basis for the mani-
(11 H-G. Miller and T. Geveci, Chem. Phys. Letters 100 (1983) 115. [2] J. HendekoviE. Chem. Phys. Letters 90 (1982) 198. [3] G. Strang and GJ_ I%.., An analysis of the finite element method (Prentice Hall, Endewood Cliffs, 1973) ch. 6. 141 R. Courant and D. Hilbert, Methoden der mathcmatischen Physik, Erster Band (Springer, Berlin, 1931) p. 399. [5] P--O. LBwdin, J. Chem. Phys. 18 (1950) 365.