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Comment on the energy variation method
CHEMICAL PHYSICS LElTERS
Volume 100, number 1
26
August 1983
COMMENT COMMENT ON THE ENERGY VARIATION
METHOD
H.G. MILLER and T. GEVECI National Research Institute for Mathematical Sciences, CSIR. P.O.Box 395. Pretoria 0001. South Afn-ca Received 15 January 1983; in f&
form 1 March 1983
It is pointed out that the classical minmax principle plays a significant role in the derivation of HendekoC%s results. Secondly, we conjecture that HendekoviE’sresults may be generalized.
Configuration mixing calculations with basis states obtained from variational methods appear, at present, to be a useful means of obtaining a more than adequate description of the lower-lying eigenstates of a many-particle system [l-5] _This two-step hierarchf cal strategy is, in essence, a Rayleigh-Ritz procedure executed in a subspace, SH, of the full Hilbert space which has been constructed in an optimal manner. The optimization results from the fact that the set of basis states {I@} which span SH are obtained from a non-linear variational procedure_ If the set of basis states form an orthonormal set. the diagonal elements of Rayleigh-Ritz secular equation de&
- E$
I = 0,
(1)
with JIM = (~~11Hi@~I),
{IQ)3E SH,
have been minimized which implies for bound-state calculations that these matrix elements are as large as possible in the negative sense. The matrix, k, is therefore by construction as diagonally dominant (again in the negative sense) as possible for the particular choice of basis states designated by the variational procedure_ Hence with relatively few varia-
tionally determined basis states the Rayleigh-Ritz procedure may be expected to yield, in a democratic manner, a reasonably good approximation to the lower-lying eigenvalue spectrum_ Recently this two-step procedure has been examined in some detail by Hendekovid [6] who has formulated two new energy variation theorems which 0 009-2614/83/0000-0000/S
03.00 0 1983 North-Holland
support such a non-linear variational description of excited states and their manifolds_ We should like to point out the significance of the minmax principle of PoincarE [7] (which is equivalent to the CourantWeyl principle [8] ) in the derivation of HendekoviZs results. For the finite-dimensional subspace SH of the full Hilbert space, the minmax principle states: If the Rayleigh quotient,R(G), is maximized over an Z-dimensional subspace, S1 C SH , then the minimum possible value for this maximum is the Zth root E; of (1): 5 H = min max R(Q), SI 19E SI where R(Q) = <$]&+)/<@&~1,and the domain offi is the full Hilbert space. Here Sz ranges over all Z-dimensional subspaces of the subspace SE1_ The minmax principle also yields
El = min max R(Q), SI I#E SI
where El is the Zth exact eigenvalue of fi, and S1 ranges over all Z-dimensional subspaces of the full Hilbert space. Thus, it is obvious [9] that e; > El_
(3
It is well known that the eigenvectors I@> corresponding to ey are orthonormal and uncoupled:
115
Volume 100. number 1 Hendekovif’s
CHEMICAL PHYSICS LETTERS
theorem 3 follows readily from (2) and
WSecondly. we suggest an alternative variational procedure for generating
a set of basis states which need not be orthonormal. but in most cases will be linearly independent. Choose as basis states those solutions of the variational equations whose variational energies
are the smallest. Since the variational equations are in general non-linear_ these solutions are in general not orthogonal to one another. but are. in most cases. linearly independent 13 ] _ Basis states obtained in this manner are in most instances_ not trivially related to those generated from constrained variational principles which guarantee the orthogonality of the basis states_ again. because of the non-linearities inherent in both variational procedures. Furthermore. by not requiring the solutions of the variational equations to be orthogonal to one another the generation of the set of basis states is considerably simplified [3] _ Since the basis states generated in this manner are non*rthogonal, the sum of the variational energies is not the manifold energy as defined by Hendekovid. However, an orthononnal set of basis states may be constructed quite simple from a set of IV linearly independent variationally determined basis states and the manifold energy of these N orthogonal states is an upperbound to the manifold energy theNlowest eigenstates of hi. This. of course, follows theorem 4 in Hendekovid’s paper. Furthermore. we conjecture that the present choice of basis states results in some degree of minimization of the manifold energy. as defined above. and hence represents a means of optimizing the choice of the
116
26 August 1983
subspace in which the Rayleigh-Ritz procedure is to be applied_ Because the orthogonal and nonorthogonal sets of variationally determined basis
states are not trivially related it is difficult to ascertain a priori, which set represents the better choice_ Preliminary calculations [3] of the lowest eigenstates in a simple system indicate no marked preference for either orthogonal or non-orthogonal variationally determined basis states. For the same number of basis states. the manifold energies do not differ significantly which supports our present conjecture_ We gratefully acknowledge the referee’s clarifying remarks about the role of the minmax principle_
[l]
H.G. Miller and R.M. Dreizler. Nucl. Phys. A316 (1979)
[2] zk.
Miller, RN. Dreizler and G. Do Dang, Phys. Letters 778 (1978) 119. (31 H.G. Miller and H.P. Schriider, Z. Physik A304 (1982)
273. [4] CF. Jackels and E.R. Davidson, J. Chem. Phys. 64 (1976) 2908. [S] R.J. Buenker, V. BonaX&Kouteck$ and L. Pogliani. J. Chem. Phys. 73 (1980) 1836. [6] J. HendekoviS, Chem. Phys. Letters 90 (1982) 198. [7] H. Poincar6, Am. J. Math. I? (1980) 211. [8] R. Courant, Math. 2.7 (1920); H. Weyl, Math. Ann. 71 (1912) 441; H.F. Weinberger, Variational methods for eigenvalue approumdtion. Regional Conference Series in Applied Mathematics. Vol. 15 (SIAM. Philadelphia, 1974) pp_ 55-57. [9] G. Strang and GJ. Fix. An analysis of the finite element method (Prentice Hall, Engiewood Cliffs, 1973) pp_ 216-223.
Volume 100, number 1
26
August 1983
COMMENT COMMENT ON THE ENERGY VARIATION
METHOD
H.G. MILLER and T. GEVECI National Research Institute for Mathematical Sciences, CSIR. P.O.Box 395. Pretoria 0001. South Afn-ca Received 15 January 1983; in f&
form 1 March 1983
It is pointed out that the classical minmax principle plays a significant role in the derivation of HendekoC%s results. Secondly, we conjecture that HendekoviE’sresults may be generalized.
Configuration mixing calculations with basis states obtained from variational methods appear, at present, to be a useful means of obtaining a more than adequate description of the lower-lying eigenstates of a many-particle system [l-5] _This two-step hierarchf cal strategy is, in essence, a Rayleigh-Ritz procedure executed in a subspace, SH, of the full Hilbert space which has been constructed in an optimal manner. The optimization results from the fact that the set of basis states {I@} which span SH are obtained from a non-linear variational procedure_ If the set of basis states form an orthonormal set. the diagonal elements of Rayleigh-Ritz secular equation de&
- E$
I = 0,
(1)
with JIM = (~~11Hi@~I),
{IQ)3E SH,
have been minimized which implies for bound-state calculations that these matrix elements are as large as possible in the negative sense. The matrix, k, is therefore by construction as diagonally dominant (again in the negative sense) as possible for the particular choice of basis states designated by the variational procedure_ Hence with relatively few varia-
tionally determined basis states the Rayleigh-Ritz procedure may be expected to yield, in a democratic manner, a reasonably good approximation to the lower-lying eigenvalue spectrum_ Recently this two-step procedure has been examined in some detail by Hendekovid [6] who has formulated two new energy variation theorems which 0 009-2614/83/0000-0000/S
03.00 0 1983 North-Holland
support such a non-linear variational description of excited states and their manifolds_ We should like to point out the significance of the minmax principle of PoincarE [7] (which is equivalent to the CourantWeyl principle [8] ) in the derivation of HendekoviZs results. For the finite-dimensional subspace SH of the full Hilbert space, the minmax principle states: If the Rayleigh quotient,R(G), is maximized over an Z-dimensional subspace, S1 C SH , then the minimum possible value for this maximum is the Zth root E; of (1): 5 H = min max R(Q), SI 19E SI where R(Q) = <$]&+)/<@&~1,and the domain offi is the full Hilbert space. Here Sz ranges over all Z-dimensional subspaces of the subspace SE1_ The minmax principle also yields
El = min max R(Q), SI I#E SI
where El is the Zth exact eigenvalue of fi, and S1 ranges over all Z-dimensional subspaces of the full Hilbert space. Thus, it is obvious [9] that e; > El_
(3
It is well known that the eigenvectors I@> corresponding to ey are orthonormal and uncoupled:
115
Volume 100. number 1 Hendekovif’s
CHEMICAL PHYSICS LETTERS
theorem 3 follows readily from (2) and
WSecondly. we suggest an alternative variational procedure for generating
a set of basis states which need not be orthonormal. but in most cases will be linearly independent. Choose as basis states those solutions of the variational equations whose variational energies
are the smallest. Since the variational equations are in general non-linear_ these solutions are in general not orthogonal to one another. but are. in most cases. linearly independent 13 ] _ Basis states obtained in this manner are in most instances_ not trivially related to those generated from constrained variational principles which guarantee the orthogonality of the basis states_ again. because of the non-linearities inherent in both variational procedures. Furthermore. by not requiring the solutions of the variational equations to be orthogonal to one another the generation of the set of basis states is considerably simplified [3] _ Since the basis states generated in this manner are non*rthogonal, the sum of the variational energies is not the manifold energy as defined by Hendekovid. However, an orthononnal set of basis states may be constructed quite simple from a set of IV linearly independent variationally determined basis states and the manifold energy of these N orthogonal states is an upperbound to the manifold energy theNlowest eigenstates of hi. This. of course, follows theorem 4 in Hendekovid’s paper. Furthermore. we conjecture that the present choice of basis states results in some degree of minimization of the manifold energy. as defined above. and hence represents a means of optimizing the choice of the
116
26 August 1983
subspace in which the Rayleigh-Ritz procedure is to be applied_ Because the orthogonal and nonorthogonal sets of variationally determined basis
states are not trivially related it is difficult to ascertain a priori, which set represents the better choice_ Preliminary calculations [3] of the lowest eigenstates in a simple system indicate no marked preference for either orthogonal or non-orthogonal variationally determined basis states. For the same number of basis states. the manifold energies do not differ significantly which supports our present conjecture_ We gratefully acknowledge the referee’s clarifying remarks about the role of the minmax principle_
[l]
H.G. Miller and R.M. Dreizler. Nucl. Phys. A316 (1979)
[2] zk.
Miller, RN. Dreizler and G. Do Dang, Phys. Letters 778 (1978) 119. (31 H.G. Miller and H.P. Schriider, Z. Physik A304 (1982)
273. [4] CF. Jackels and E.R. Davidson, J. Chem. Phys. 64 (1976) 2908. [S] R.J. Buenker, V. BonaX&Kouteck$ and L. Pogliani. J. Chem. Phys. 73 (1980) 1836. [6] J. HendekoviS, Chem. Phys. Letters 90 (1982) 198. [7] H. Poincar6, Am. J. Math. I? (1980) 211. [8] R. Courant, Math. 2.7 (1920); H. Weyl, Math. Ann. 71 (1912) 441; H.F. Weinberger, Variational methods for eigenvalue approumdtion. Regional Conference Series in Applied Mathematics. Vol. 15 (SIAM. Philadelphia, 1974) pp_ 55-57. [9] G. Strang and GJ. Fix. An analysis of the finite element method (Prentice Hall, Engiewood Cliffs, 1973) pp_ 216-223.