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Lower bounds for the energy levels of the lithium atom
Volume 26. number 2
CHEhI ICAL PHYSICS LETTERS
LOWER BOUNDS
FOR THE ENERGY
15 May 1974
LEVELS OF THE LITHWM
ATOM
Charles E. REID Quantum Theory Project and Department of Chemistry. University of Florida, Gainesrille, FIorida 32601. USA Received 13 March 1973
A recent modifktion of the intermediate hamiltonian method, intended to circumvent the problem associated with a low-lying continuum in the operator Ho, has been applied to the lithium atom with a basis of discrete hydrogenie functionr The intended raising of the continuum occurs. but not to the extent that the beginning of the continuum is raised above the ground state of the atom; indications arc that no basis of discrete hydrogenic cigenfunctions can accomplish this. Since these calculations were performed on the full lithium atom hamiltonian. they arc not strictly comparable to those of Fox and Sigillito, in which a simplified hamiltonian was used.
Lower bounds for the energy levels of the helium atom and its isoelectronic series have been calculated by several methods. One of these is the method of intermediate hamiltonians, which was originated by Weinstein, adapted to operators of the type of eq. (1) by Aronszajn, and applied to quantum mechanics by Bazley and Fox [I] *. Better bounds have been obtained by the bracketing-function method of Lijwdin [?I, as applied to this problem by Choi and Smith [3] and by Wilson and Reid [4]; the reason appears to be that this method accounts better for continuum effects. These methods both encounter serious difficulties when applied to the lithium atom or the lower members of its isoelectronic series, and as yet no satisfactory lower bounds are available for these systems. Recently a modification of the intermediate hamiltonian, intended to circumvent this difficulty, has been published by the author [S], independently of a previous derivation by Fox [6] **. This paper presents some preliminary calculations with this technique, and also indicates how it may prove valuable in conjunctionwith the bracketing;function method. The first step in applying ths intermediate hamiltonian method is to separate the hamiltonian into two parts:
H?Ho
+ v,
(1)
where #has known eigenfunctions and V is positive definite. V is then replaced by an operator V’ < V, con-strutted as described by Bazley and Fox [7] (originally by Aronszajn [l]) or equivalently by Lowdin’s “inner projection” formalism [2]. In either case V’ is of fiite rank and so is a compact operator. It then follows by Weyl’s theorem [S] that the intermediate hamiltonian Ho + V’ has the same limit points as Ho. For the lithium atom this means that the intermediate hamiltonian has a continuum beginning below the ground state of the complete hamiltonian, and this is the reason for.the failure of the method for this systemThe modification relies on the fact that though v is a three-electron operator (or an N-electron operator for .atoms generally), it ’ is a sum of two-electron operators. When each of these is treated by the inner projectiorrtechnique, the resulting’ .operaior is compact in &space of two-electron functions, but not.in that ofl\r-electron functions. ._ :.. .. :. : ‘. .: -. ,..: ;~Tks bookgivesextensiverefere&esto the early work-on this method.-, : “-m Brief mentiori of this method may be found in Notices Am. &lath Soc.:l6 (19k9j. .800, and Abstacts, .’ hfe+ll~(1970).-. :..~ : ~- .: : .. : : ‘.; ; -. . : .. .-_. : .-. ._:I.,..r: .- .. _., ;-. ..-,. .._.:,. ._. .-.__ _..
-.
l
- :,,.., ._ .-.: .. , ‘.. ,: .. :, ., ,. ’ -243;; .
SIAM Nationat: t ..-.
..
CHEMICAL
Volume 26, number 2
PHYSICS
LET-l-ERS
15 May 1974
Details needed in applying this technique are found in ref. [S]; in the present work hydrogenic s-functions with Z = 3 were used as the one-electron functions - a choice which simplifies some of these details. A few calculations including 2p functions were also carried out. Since this set is orthonormal, the intermediate hamiltonian is represented by a symmetric matrix, and a Givens-Householder subroutine [9] was used to fiid the eigenvalues. For IZ one-electron functions the first type of invariant space mentioned in ref. [5] is of rz(r? - 1)/3 dimensions, while the second type is of~~(n + I)/2 dimensions. It was found that the first type of subspace yielded 12- 1 low eigenvalues (below -7.2 bartree), while the nth was about -5.4 hartree. This is understandable from the fact that there are (II - 1) Slater determinants of the form A9 l(l) 9 l(2) $k(3), and all other configurations are multiplyexcited ones of high energy. Results are given in table 1. It will be noted that these results are slightly lower than those reported by Fox and Sigillito [IO]. This was initiahy a source of puzzlement, since at first glance the calculations appear to treat the same problem. Actually, however, Fox and Sigillito calculated lower bounds to the eigenvalues of the operatorPHP, where P is the projector onto the space spanned by functions of the radial coordinates alone [ 1 l] - For upper bound calculations by the variation theorem, the use of this operator is equivalent to treating the fuli hamiltonian with an all-s basis, but for lower bounds this is not true. The eigenvalues of PHP are the quantities commonly called “radial limit approximations’* to the eigenvalnes of H, and it is lower bounds to these radial limits that Fox and Sigillito have calculated_ By contrast, the operator treated here is the complete lithiu_m atom hamiltonian in the usual nonrelativistic fixed-nucleus approximation, and so has a much closer relation to the physical problem of the lithium atom. Computationally, the difference between the two treatments appears in the expressions for the integrals used in constructing the operator V’. If the normalized s-functions are written 9i(r, 8, ~) =Ri(r)/(4~)“*
)
then the integrals over ‘I2 are given by (JIl(l)~j(2)lr,2i~lk(l)~1(2))
This may be derived by any of several methods, such as that of Calais and L.iiwdin [ 121. The present work uses this entire expression; that of Fox and Sigillito replaces r r2 by max (rr, r2) and as a result uses only the first two integrals, omitting those that appear with the coefficient l/3. This difference also applies to a later calculation of Fox and Sigillito [ 131. Calculation of integrals by eq. (2) leads to serious loss of numerical accuracy. Because of this the extendedprecision facility of the IBM 370 (equivalent to about 33 decimal digits) was used for all these calculations.To eliminate any possible program errors, all integrals involving the fust five functions were also calculated by an entirely independent method (a double Gauss-Laguerre integration)_ This was accurate to only about seven figures, but Within this accuracy complete agreement was obtained. Finally, two separate and independently written programs were used to set up the matrices; there was exrtct agreement between them. Since the bracketing-function method has yielded better bounds for the helium atom than the intermediate hamiLtoni+ method, it is desirable to apply_it to the lithium atom. However, n&ericaLappLication of this method requires that a0 should~have onl$fmitbly m&yeigenvalues belo& the desired. eigerivalue of Ei, and so the usu-al division of H is not suitable. If an inteririedi&e hamiltonianti’ can be constructed with only a finite number, .. &4
:
.-
.;
;
_
y_,-.. ..
:_ ;_
:
.. ..
._ :
Volume 26, number 2
CHEMICAL
PHYSICS LETTERS
15 May 1974
Table 1 Lowest eigenvalucs for the Li atom intermediate hamiltonian for various basis sets, with the type of subspace ((1) to (2)) as described in ref.
[ 51 (the
degree polynomial
Inner projection basis
in
same as types 3 and 2 in ref. [6] ). Where estrapolated the values for n = 4 through 11= 10
values are given. they were found by fitting 3 sisth-
1/n to
Start of
Four lowest eIgenvaIues, with type of subspace
continuum
Is-tlS
with tz =
1 2 3 4 5 6 7 8 9 10
-8.75357 -8.09434
(2) (2)
-8.12857 -7.s7.559
(2) (2)
-7.86730 -7.79939 -7.79773 -7.79678 -7.79617 -7.79577 -7.79549 -7.79528
(2) ( 1) (1) (1) (1) (1) (1) (1)
-7.80266 -7.76275 -7.70609
i I) (2) (2)
-7.67198 -7.65764 -7.65724 -7.65696 -7.65676
(2) (I) (1) (1) (1)
-7.90982 -7.77434 -7.76605 -7.70775 -7.67293 -7.65823 -7.64986 -7.63471
(2) (2) (2) (2) (2) (1) (2) (2)
-7.62377 -7.61871
(2) (1)
-7.80857
(2)
-7.74668
(1)
-7.62857 -7.59434
-7.71105 -7.67458
(2) (2)
-7.65915 -7.65046 -7.63510
(1) (2) (2)
-7.58605 -7.58275 -7.58109 -7.580 14 -7.57955
-7.62415 -7.61891 -7.61586
(2) (1) (1)
-7.579 15 -7.57887 -7.57867
extrqiolated to -
-7.79438
Is, 2s and all 2p’s
-8.08033
(2)
-7.86158
(2)
-7.76033
(2)
-7.71890
(1)
-7.58033
-7.85375
(2)
-7.78147
(1)
-1.75250
(2)
-7.69750
(2)
-7.57250
Is, Zs, 3s and all 2p’s
-
-
-
-7.57767
(preferably only a small number) of eigenvalues below the ground state of H, then this H’ can be used as the Ho, and H-H’ as the V, in a new division of Hof the type in eq. (1). With this division of H the br,acketing-function method is usable in principle, and does not appear to offer any serious difficulty in application_ This seems likely to be a more practical means of improving lower bounds than increasing the basis set of the intermediate harniltbnian, which quickly leads to excessively large matrices. The results of these calculations indicate that for the lithium atom an intermediate hamiltonian with the necessary property cannot be constructed with a basis set of discrete hydrogen% s-functions and, in view of the small improvement from the inclusion of 2p functions, probably not from any discrete hydrogenic set. However, better intermediate hamiltonians should be constructible with other basis sets, and the necessary formalism is available [S, 5]_
References [l] A. Weinstein and W. Stenger, Methods of intermediate problems for eigenvalues (Academic Press, New York, 1972). (21 P--O. LEwdin, J. Chem. Phys. 43 (1965) Sl75. [31 J-H. Choi and D-W_ Smith, J. Chem. Phys. 45 (1966) 4425. (41 T.W. Wilson and C.E. Reid, J. Chem. Phys. 47 (1967) 3920. [S] C-E. Reid, Intern_ J. Quantum Chem. 6 (1972).793. (61 D-W. Fox, SIAM J. Math. Anal. 3 (1972) 617. [7] N.W. Bazley and D.W. Fox, J. Res. NatL Bur; Std. 65B (1961) 105; Phys. Rev. 124 (1971) 483. (81 F. Riesz and B. Sz.-Nagy. Functional analysis (Frederick Ungar, New York. 1955) p. 367; H. Weyl. Rend. Clrc. Mat. Palermp 27 (1909) 373. [91 F. Prosser.and H. Michels, Eigenvalues and Eigenvectors by the Givens Method, Pr&ram 62.3, Quantum Chemistry Program Exchange (modified fdr double-precision use on IBM.360 macfimes). [lo]
[II]
D-W. Foxand V-G. S&l&, D.W. Fpx &ii V.G. Sigiito,
Chem- Phy& Fetters 13.(19723 J:AppL hfat~:Pl;yr 23 (i972)
85; 392.
[I21 J.L..Calais&ld P.-O. Liiwdjn;J.
1131D-W. FOXyf
Mol. Spectry. 8 (19$?).203., V_G:Sigiito;~Chetq.-Phy$ Letters 14 (1972)
-_
583. ‘.. -.
-~ .I:
‘. 245 ‘.
CHEhI ICAL PHYSICS LETTERS
LOWER BOUNDS
FOR THE ENERGY
15 May 1974
LEVELS OF THE LITHWM
ATOM
Charles E. REID Quantum Theory Project and Department of Chemistry. University of Florida, Gainesrille, FIorida 32601. USA Received 13 March 1973
A recent modifktion of the intermediate hamiltonian method, intended to circumvent the problem associated with a low-lying continuum in the operator Ho, has been applied to the lithium atom with a basis of discrete hydrogenie functionr The intended raising of the continuum occurs. but not to the extent that the beginning of the continuum is raised above the ground state of the atom; indications arc that no basis of discrete hydrogenic cigenfunctions can accomplish this. Since these calculations were performed on the full lithium atom hamiltonian. they arc not strictly comparable to those of Fox and Sigillito, in which a simplified hamiltonian was used.
Lower bounds for the energy levels of the helium atom and its isoelectronic series have been calculated by several methods. One of these is the method of intermediate hamiltonians, which was originated by Weinstein, adapted to operators of the type of eq. (1) by Aronszajn, and applied to quantum mechanics by Bazley and Fox [I] *. Better bounds have been obtained by the bracketing-function method of Lijwdin [?I, as applied to this problem by Choi and Smith [3] and by Wilson and Reid [4]; the reason appears to be that this method accounts better for continuum effects. These methods both encounter serious difficulties when applied to the lithium atom or the lower members of its isoelectronic series, and as yet no satisfactory lower bounds are available for these systems. Recently a modification of the intermediate hamiltonian, intended to circumvent this difficulty, has been published by the author [S], independently of a previous derivation by Fox [6] **. This paper presents some preliminary calculations with this technique, and also indicates how it may prove valuable in conjunctionwith the bracketing;function method. The first step in applying ths intermediate hamiltonian method is to separate the hamiltonian into two parts:
H?Ho
+ v,
(1)
where #has known eigenfunctions and V is positive definite. V is then replaced by an operator V’ < V, con-strutted as described by Bazley and Fox [7] (originally by Aronszajn [l]) or equivalently by Lowdin’s “inner projection” formalism [2]. In either case V’ is of fiite rank and so is a compact operator. It then follows by Weyl’s theorem [S] that the intermediate hamiltonian Ho + V’ has the same limit points as Ho. For the lithium atom this means that the intermediate hamiltonian has a continuum beginning below the ground state of the complete hamiltonian, and this is the reason for.the failure of the method for this systemThe modification relies on the fact that though v is a three-electron operator (or an N-electron operator for .atoms generally), it ’ is a sum of two-electron operators. When each of these is treated by the inner projectiorrtechnique, the resulting’ .operaior is compact in &space of two-electron functions, but not.in that ofl\r-electron functions. ._ :.. .. :. : ‘. .: -. ,..: ;~Tks bookgivesextensiverefere&esto the early work-on this method.-, : “-m Brief mentiori of this method may be found in Notices Am. &lath Soc.:l6 (19k9j. .800, and Abstacts, .’ hfe+ll~(1970).-. :..~ : ~- .: : .. : : ‘.; ; -. . : .. .-_. : .-. ._:I.,..r: .- .. _., ;-. ..-,. .._.:,. ._. .-.__ _..
-.
l
- :,,.., ._ .-.: .. , ‘.. ,: .. :, ., ,. ’ -243;; .
SIAM Nationat: t ..-.
..
CHEMICAL
Volume 26, number 2
PHYSICS
LET-l-ERS
15 May 1974
Details needed in applying this technique are found in ref. [S]; in the present work hydrogenic s-functions with Z = 3 were used as the one-electron functions - a choice which simplifies some of these details. A few calculations including 2p functions were also carried out. Since this set is orthonormal, the intermediate hamiltonian is represented by a symmetric matrix, and a Givens-Householder subroutine [9] was used to fiid the eigenvalues. For IZ one-electron functions the first type of invariant space mentioned in ref. [5] is of rz(r? - 1)/3 dimensions, while the second type is of~~(n + I)/2 dimensions. It was found that the first type of subspace yielded 12- 1 low eigenvalues (below -7.2 bartree), while the nth was about -5.4 hartree. This is understandable from the fact that there are (II - 1) Slater determinants of the form A9 l(l) 9 l(2) $k(3), and all other configurations are multiplyexcited ones of high energy. Results are given in table 1. It will be noted that these results are slightly lower than those reported by Fox and Sigillito [IO]. This was initiahy a source of puzzlement, since at first glance the calculations appear to treat the same problem. Actually, however, Fox and Sigillito calculated lower bounds to the eigenvalues of the operatorPHP, where P is the projector onto the space spanned by functions of the radial coordinates alone [ 1 l] - For upper bound calculations by the variation theorem, the use of this operator is equivalent to treating the fuli hamiltonian with an all-s basis, but for lower bounds this is not true. The eigenvalues of PHP are the quantities commonly called “radial limit approximations’* to the eigenvalnes of H, and it is lower bounds to these radial limits that Fox and Sigillito have calculated_ By contrast, the operator treated here is the complete lithiu_m atom hamiltonian in the usual nonrelativistic fixed-nucleus approximation, and so has a much closer relation to the physical problem of the lithium atom. Computationally, the difference between the two treatments appears in the expressions for the integrals used in constructing the operator V’. If the normalized s-functions are written 9i(r, 8, ~) =Ri(r)/(4~)“*
)
then the integrals over ‘I2 are given by (JIl(l)~j(2)lr,2i~lk(l)~1(2))
This may be derived by any of several methods, such as that of Calais and L.iiwdin [ 121. The present work uses this entire expression; that of Fox and Sigillito replaces r r2 by max (rr, r2) and as a result uses only the first two integrals, omitting those that appear with the coefficient l/3. This difference also applies to a later calculation of Fox and Sigillito [ 131. Calculation of integrals by eq. (2) leads to serious loss of numerical accuracy. Because of this the extendedprecision facility of the IBM 370 (equivalent to about 33 decimal digits) was used for all these calculations.To eliminate any possible program errors, all integrals involving the fust five functions were also calculated by an entirely independent method (a double Gauss-Laguerre integration)_ This was accurate to only about seven figures, but Within this accuracy complete agreement was obtained. Finally, two separate and independently written programs were used to set up the matrices; there was exrtct agreement between them. Since the bracketing-function method has yielded better bounds for the helium atom than the intermediate hamiLtoni+ method, it is desirable to apply_it to the lithium atom. However, n&ericaLappLication of this method requires that a0 should~have onl$fmitbly m&yeigenvalues belo& the desired. eigerivalue of Ei, and so the usu-al division of H is not suitable. If an inteririedi&e hamiltonianti’ can be constructed with only a finite number, .. &4
:
.-
.;
;
_
y_,-.. ..
:_ ;_
:
.. ..
._ :
Volume 26, number 2
CHEMICAL
PHYSICS LETTERS
15 May 1974
Table 1 Lowest eigenvalucs for the Li atom intermediate hamiltonian for various basis sets, with the type of subspace ((1) to (2)) as described in ref.
[ 51 (the
degree polynomial
Inner projection basis
in
same as types 3 and 2 in ref. [6] ). Where estrapolated the values for n = 4 through 11= 10
values are given. they were found by fitting 3 sisth-
1/n to
Start of
Four lowest eIgenvaIues, with type of subspace
continuum
Is-tlS
with tz =
1 2 3 4 5 6 7 8 9 10
-8.75357 -8.09434
(2) (2)
-8.12857 -7.s7.559
(2) (2)
-7.86730 -7.79939 -7.79773 -7.79678 -7.79617 -7.79577 -7.79549 -7.79528
(2) ( 1) (1) (1) (1) (1) (1) (1)
-7.80266 -7.76275 -7.70609
i I) (2) (2)
-7.67198 -7.65764 -7.65724 -7.65696 -7.65676
(2) (I) (1) (1) (1)
-7.90982 -7.77434 -7.76605 -7.70775 -7.67293 -7.65823 -7.64986 -7.63471
(2) (2) (2) (2) (2) (1) (2) (2)
-7.62377 -7.61871
(2) (1)
-7.80857
(2)
-7.74668
(1)
-7.62857 -7.59434
-7.71105 -7.67458
(2) (2)
-7.65915 -7.65046 -7.63510
(1) (2) (2)
-7.58605 -7.58275 -7.58109 -7.580 14 -7.57955
-7.62415 -7.61891 -7.61586
(2) (1) (1)
-7.579 15 -7.57887 -7.57867
extrqiolated to -
-7.79438
Is, 2s and all 2p’s
-8.08033
(2)
-7.86158
(2)
-7.76033
(2)
-7.71890
(1)
-7.58033
-7.85375
(2)
-7.78147
(1)
-1.75250
(2)
-7.69750
(2)
-7.57250
Is, Zs, 3s and all 2p’s
-
-
-
-7.57767
(preferably only a small number) of eigenvalues below the ground state of H, then this H’ can be used as the Ho, and H-H’ as the V, in a new division of Hof the type in eq. (1). With this division of H the br,acketing-function method is usable in principle, and does not appear to offer any serious difficulty in application_ This seems likely to be a more practical means of improving lower bounds than increasing the basis set of the intermediate harniltbnian, which quickly leads to excessively large matrices. The results of these calculations indicate that for the lithium atom an intermediate hamiltonian with the necessary property cannot be constructed with a basis set of discrete hydrogen% s-functions and, in view of the small improvement from the inclusion of 2p functions, probably not from any discrete hydrogenic set. However, better intermediate hamiltonians should be constructible with other basis sets, and the necessary formalism is available [S, 5]_
References [l] A. Weinstein and W. Stenger, Methods of intermediate problems for eigenvalues (Academic Press, New York, 1972). (21 P--O. LEwdin, J. Chem. Phys. 43 (1965) Sl75. [31 J-H. Choi and D-W_ Smith, J. Chem. Phys. 45 (1966) 4425. (41 T.W. Wilson and C.E. Reid, J. Chem. Phys. 47 (1967) 3920. [S] C-E. Reid, Intern_ J. Quantum Chem. 6 (1972).793. (61 D-W. Fox, SIAM J. Math. Anal. 3 (1972) 617. [7] N.W. Bazley and D.W. Fox, J. Res. NatL Bur; Std. 65B (1961) 105; Phys. Rev. 124 (1971) 483. (81 F. Riesz and B. Sz.-Nagy. Functional analysis (Frederick Ungar, New York. 1955) p. 367; H. Weyl. Rend. Clrc. Mat. Palermp 27 (1909) 373. [91 F. Prosser.and H. Michels, Eigenvalues and Eigenvectors by the Givens Method, Pr&ram 62.3, Quantum Chemistry Program Exchange (modified fdr double-precision use on IBM.360 macfimes). [lo]
[II]
D-W. Foxand V-G. S&l&, D.W. Fpx &ii V.G. Sigiito,
Chem- Phy& Fetters 13.(19723 J:AppL hfat~:Pl;yr 23 (i972)
85; 392.
[I21 J.L..Calais&ld P.-O. Liiwdjn;J.
1131D-W. FOXyf
Mol. Spectry. 8 (19$?).203., V_G:Sigiito;~Chetq.-Phy$ Letters 14 (1972)
-_
583. ‘.. -.
-~ .I:
‘. 245 ‘.