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New applications of the least-squares method: Unbound states of helium and asymptotically correct bound states
Volume 12, number 4
NEW kPPLICATIONS OF THE LEAST-SQUARES METHOD: UNBOUND STATES OF HELIUM AND ASYMPTOTICALLY CORRECT BOUND STATES F.H. READ PhjjsicsDepartmenr. Schuster Laboratory, Manchester Received 5 November
University,
UK
1971
Two new applkations of the least-squares method are desuibed, namely (i) in the ca!cuIation of the energies and wavefunctions of the unbound 2sz *S and 2p * ‘S s’ates of He, and (ii) in the -use (for the 2’S, 3’S and 4LS states of He) of basis set functions which are asymptotically correct to a high order.
1. Introduction The least-squares method of solving Schrtidinger’s equation was first applied [ 1 ] to the ground state of H;, and has since (see i2] for recent
references)
been
used on various two and three electron systems such as He, H-, H2 and Li, for which it has given results of comparable accuracy [2] to those obtained by more conventional variational procedures. It is the purpose of the present work to show that there are two applications for which the least-squares method may have distinct advantages over ccnventional methods, namely: (i) in the ca!culation of the energies and wavefunctions of unbound (i.e., autoionizing) states and (ii) in the use of b asis set functions which are asymptotically correct to a high order. We start by defining
Mlx)G
ix w(xi)e’(xi)x(xi) i
(1)
for *any functions B and x, where the summation is taken over spe&jk pointsxi in the total space of the
system, and where the weighting function \v have been chosen in such a way that @lx,’is an appioximation to the conventional continuous integral @lx>.The present version of the least-‘squares method consists of expanding the desired wavefunction J/ in terms of a basis set $ (i.e., $ = c n a,@“) and then determin ing fhe coefficients Q,, .by minimizing
subject to the normaking ($I$)‘=
! .
condition (3)
H is the hamiltonian of the system. The energy E may
either (i) be put equal to the true energy Et af a state if ttJs is known, or (ii) be varied about the neighbourhood of a suspected eigenenergyJthis is discussed beIowj. The mean energy E of the resuking wavefunction may
be calculated from EA=’ .
(4)
It can be shown analytically that, as in the usurrl variational methods, this energy is correct COthe second order when the wavefunction itself is carrect to the first order.
The f-act that we are minimi& g ((H-E)J/KH--E)ti)’ means that we obtain energies and waveiimctions of excited states without having to orthogonal& to, or ot!ain, the wavefunctions of lower energy states of rhe same symmetry. Also since no integrations are performed there are no extra difficuIties in using awkward basis functions such as those which contain conintegral powers of radii, or powers of inter-particle distances ‘in etc.
.’
Volume 12, number 4
CHEMICALPHYSICSL!ZTTERS
15 January 1972
I.
2. Uiibound statei
table’1. The,re is good agreement v$$ the work of other authors. AU the computations were done on the Atlas Computer at Manchester University,and the to. tal computing time for each of the results shown in. the table was about 5 min. It should,be noted that the energy we have caiculated is essentially the unshiftedresonance energy f4], and we have ignored (as have the other authors
To show that the least-squares method may be used
for unbound states we have studied the autoionizing 2s” “S and 2p2 IS states of helium. We expand the spatirii part of their wavefunctions as $=Q-=+gou-,
(5)
quoted in fable 1.)the energyshift mused by the
where both pa& 4 are of the form
coupling between the discrete state and the ionization continuum. In effect we have cumulated eigenvalues of the operator @IQ, where Q is the operator which projects out that part of Hilbert space containing functions which decay exponenti~]y to zero as ri or r2 go tc) infmity.
S is the spatial symmetrization operator. The functions r#Outerare designed for large vdues of rf or t2: they have small values of cyI and LYE(0.8 and 0.4, respectivelyj, and l, M, n run from 0 to 2, I and 3, respectively. CJn the other hand the functicns p” are designed to reproduce the high!y correlated regions in which both electrons are near the nucleus: hese or and cr2 are larger (2.5 and 1.0) and I, m, n run from 0 to 2, I and I, respectively, The inclusion of #tier always seems to give a greatly improved accuracy. The fact that_ the basis functions are not orthogona! did not give rise to any computational difficulties. The total number of basis set functions was 36 for both the 2s2 and 2p2 states. The number of discrete points in the space (rl, r2) is shown in table 1: these points were most dense near the nucleus, and were, made up frdm about 40 different radial iengths and 5 different valutis of the angle 612 between the vectors r1 and r2. The quoted values of at and a2 were found by a preliminary approximate minimization of S wit&respect to them. The energy E kas put equal to’the experimentally measured value gt for each state, and the resulting mean energies E are shown in
3. Asynptotimlly
torrest
bound
states
As x second new application we have calcufated the ~avef~n~tio~s and energiesof the bound 2lS, 3lS and 4% states of helium and the I ‘S state of H- , with @Outer‘replaced by the single function 4qrn f which is chosen to give an accurate representation of the wavefunctitin when one electron is far fron: the nucleus. Thiswas done by using the ~ou~ornb approximation with the inclusion of a correction for the pokzizability of the inner He+ or H core. For example the asymp totic function used for the 4% state is r#v
= S exp(-2rl-0.2592r2)[~~858
- 2i.78r;858
Table 1 State ,He2s21S Hd 2p2‘rS Hels2.1sS. 1, Hels2s2$. ?&;335 3% HeIsr)sb’S
-0.777 131, -0.778 -0.595 [3], -0.619 -2.903724 -2.145974
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-2.9033$7 -&I45831
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-2.061284..
-2,033$87 (81 -0527750 [ 71
k
IO” * 2 x 30-3 la-? ~-2, x W3
-0.776,
[S], -0.777 * 0.002 [6] IS];-0.619 t 0.002 [6] f?] 18)
'-2.061272[8]
:
Points ,/zFx 102 I3asis set
%-Et
Et (au)
,
,’ 0,60
' 1.7 X:u_I"4
0.37.
36 1661 36 ‘. 1823 29 .612 2$ 2486 25 2206 ‘25 2614
31 ‘. ‘I586
. ~. ‘,
.’ ,_. .’,_. ;.:, ., -
..’
,’ ).
:
,_ ,. . . ,,
‘,
:
1s ranuary 1972
CHEMICALPHYSICSLE’IX’ERS
Volume 12, number 4
(2). If these energies had not been known then it would have been possible to fmd *&emby treating E as a variable. This isiJlustrated by fig. 1, which shows
I -a07
I -1C-S
t
L
-5of
-a704
-aw
0
E
Fig. 1. The mean energy 2 (full curve) and root-mean-square (broken curve) as a fun&cm of the input energy E for atomic helium. The horizontal lines show the known enererrorSffz
gies of the 3’S and 4lS stat%s, and the diagonal line gives E = E.
(7) where f-iis a function designed to remove singuiarities at the nucleus, and is given by
the dependence of E (full curve) and 8”” (broken curve) on the variable E. This figure iUustrates (i) that 2 = Et when E = Et, (ii). that S is then a ~n~um, and (iii) that 2 is correct to a higher order (actu~y to the second order, as mentionedabove)than the computed wavefunction itself. Fig. 1 was obtained in a comparatively short corn. puting time since a basis set of the form of eqs. (4) and (5) was used, and once matrix efements such as G#+, I@,,>‘,@,,iB$,.$ and (HQ,K@Q had been evaluated for one point they could be used again for the other points in fig. 1. Graphs such as that of fig. 1, showing a step at each eigenenergy, may prove to be z useful way of mapping eigenenergies. We have used this technique to show that there are no IS states iying between the 2s2 and 2p 2 *S states of helium, Further work is in progress on these uses of the least-squares method, as well as OR possible advantages of other uses (such as scattering processes, and the variation of the weighting function w).
Acknowledgement f _i= 1 - exp(-2r2)
2 (2r2)n/n! = O(C1)l ( ZZ=0 >
(8)
‘I’& author is grateful
The terms in ‘12 are due to :he polarization correction, and the function (p”sy” satisfies Schrtidinger’s equation to a high order when one eIectron is much farther from thb nucleus than the other. @ma [as in eq. (6)] is still added to give a correction for the regions near the nucleus. The calculated energies (see table I) compare favourably with the results of calculations by conventional methods. For completeness we include the ground staie of helium in the table. Here the basis set was of the form given by eqs. (4) and (5) above. Again there is good agreement wit31the known energy.
4. Ab titio
arzd F.
References
determination of eigenenergies
We.have used the known energies Et for B in eq.
to Drs. N. Bard&y
Mandlfor their encouragementat a ctiticaI stageof the work.
”
[I) A.A.Frost, 3. Chem. Phys. 10 (1942) 240. [2] M.H.Uoyd and L.hI.Delves. Intern. J. Quantum Chem. 3 (1969) 169. (31 PGBurke and D.D.hlcV&r, Pro& Phys. Sot. (London) 85 f 1965) 989. [4 J U.Fa.no, Phys Rev. 124 (1961) 1866. fSj T.F.O%faUey and S.Geftman.Phys. Rev. 137 /1965) A 1344. (61 AlBordenave-Montesquieu,P.BenoitCattin and D. BIanc, Abstracts of Vllth Intern. Conf. on Physics of E!ectronic ad Atomic Colikiok, Eds. L.hi.Btanscomp et al., CNorth-Holland, Amsterdam, 1971) p. 1041. [7] fthmtowski and C.t.%&&s, Phys. Rcw. L46 (1966) 46. [8] Y.Atcad, C.L.Pekeris and B.S&iff, Phys Rev, A4 (147f) 516, 551
,’
:
‘(
NEW kPPLICATIONS OF THE LEAST-SQUARES METHOD: UNBOUND STATES OF HELIUM AND ASYMPTOTICALLY CORRECT BOUND STATES F.H. READ PhjjsicsDepartmenr. Schuster Laboratory, Manchester Received 5 November
University,
UK
1971
Two new applkations of the least-squares method are desuibed, namely (i) in the ca!cuIation of the energies and wavefunctions of the unbound 2sz *S and 2p * ‘S s’ates of He, and (ii) in the -use (for the 2’S, 3’S and 4LS states of He) of basis set functions which are asymptotically correct to a high order.
1. Introduction The least-squares method of solving Schrtidinger’s equation was first applied [ 1 ] to the ground state of H;, and has since (see i2] for recent
references)
been
used on various two and three electron systems such as He, H-, H2 and Li, for which it has given results of comparable accuracy [2] to those obtained by more conventional variational procedures. It is the purpose of the present work to show that there are two applications for which the least-squares method may have distinct advantages over ccnventional methods, namely: (i) in the ca!culation of the energies and wavefunctions of unbound (i.e., autoionizing) states and (ii) in the use of b asis set functions which are asymptotically correct to a high order. We start by defining
Mlx)G
ix w(xi)e’(xi)x(xi) i
(1)
for *any functions B and x, where the summation is taken over spe&jk pointsxi in the total space of the
system, and where the weighting function \v have been chosen in such a way that @lx,’is an appioximation to the conventional continuous integral @lx>.The present version of the least-‘squares method consists of expanding the desired wavefunction J/ in terms of a basis set $ (i.e., $ = c n a,@“) and then determin ing fhe coefficients Q,, .by minimizing
subject to the normaking ($I$)‘=
! .
condition (3)
H is the hamiltonian of the system. The energy E may
either (i) be put equal to the true energy Et af a state if ttJs is known, or (ii) be varied about the neighbourhood of a suspected eigenenergyJthis is discussed beIowj. The mean energy E of the resuking wavefunction may
be calculated from EA=
(4)
It can be shown analytically that, as in the usurrl variational methods, this energy is correct COthe second order when the wavefunction itself is carrect to the first order.
The f-act that we are minimi& g ((H-E)J/KH--E)ti)’ means that we obtain energies and waveiimctions of excited states without having to orthogonal& to, or ot!ain, the wavefunctions of lower energy states of rhe same symmetry. Also since no integrations are performed there are no extra difficuIties in using awkward basis functions such as those which contain conintegral powers of radii, or powers of inter-particle distances ‘in etc.
.’
Volume 12, number 4
CHEMICALPHYSICSL!ZTTERS
15 January 1972
I.
2. Uiibound statei
table’1. The,re is good agreement v$$ the work of other authors. AU the computations were done on the Atlas Computer at Manchester University,and the to. tal computing time for each of the results shown in. the table was about 5 min. It should,be noted that the energy we have caiculated is essentially the unshiftedresonance energy f4], and we have ignored (as have the other authors
To show that the least-squares method may be used
for unbound states we have studied the autoionizing 2s” “S and 2p2 IS states of helium. We expand the spatirii part of their wavefunctions as $=Q-=+gou-,
(5)
quoted in fable 1.)the energyshift mused by the
where both pa& 4 are of the form
coupling between the discrete state and the ionization continuum. In effect we have cumulated eigenvalues of the operator @IQ, where Q is the operator which projects out that part of Hilbert space containing functions which decay exponenti~]y to zero as ri or r2 go tc) infmity.
S is the spatial symmetrization operator. The functions r#Outerare designed for large vdues of rf or t2: they have small values of cyI and LYE(0.8 and 0.4, respectivelyj, and l, M, n run from 0 to 2, I and 3, respectively. CJn the other hand the functicns p” are designed to reproduce the high!y correlated regions in which both electrons are near the nucleus: hese or and cr2 are larger (2.5 and 1.0) and I, m, n run from 0 to 2, I and I, respectively, The inclusion of #tier always seems to give a greatly improved accuracy. The fact that_ the basis functions are not orthogona! did not give rise to any computational difficulties. The total number of basis set functions was 36 for both the 2s2 and 2p2 states. The number of discrete points in the space (rl, r2) is shown in table 1: these points were most dense near the nucleus, and were, made up frdm about 40 different radial iengths and 5 different valutis of the angle 612 between the vectors r1 and r2. The quoted values of at and a2 were found by a preliminary approximate minimization of S wit&respect to them. The energy E kas put equal to’the experimentally measured value gt for each state, and the resulting mean energies E are shown in
3. Asynptotimlly
torrest
bound
states
As x second new application we have calcufated the ~avef~n~tio~s and energiesof the bound 2lS, 3lS and 4% states of helium and the I ‘S state of H- , with @Outer‘replaced by the single function 4qrn f which is chosen to give an accurate representation of the wavefunctitin when one electron is far fron: the nucleus. Thiswas done by using the ~ou~ornb approximation with the inclusion of a correction for the pokzizability of the inner He+ or H core. For example the asymp totic function used for the 4% state is r#v
= S exp(-2rl-0.2592r2)[~~858
- 2i.78r;858
Table 1 State ,He2s21S Hd 2p2‘rS Hels2.1sS. 1, Hels2s2$. ?&;335 3% HeIsr)sb’S
-0.777 131, -0.778 -0.595 [3], -0.619 -2.903724 -2.145974
HT.132l’s-
..:. .. .
.:
: ;.
:
,.
; ,.
:
;.
5,24 3-70 2.76 2.02
.
1O-5
I.19
-1.0.X lo* :'.
,.
.
I ..
'11.2~
-2.033597 -0.527685~~
.
i
3.7 x IO? 2.4 X’to”
-2.9033$7 -&I45831
:
,550 ,’
1,.
-0.6@
-2.061284..
-2,033$87 (81 -0527750 [ 71
k
IO” * 2 x 30-3 la-? ~-2, x W3
-0.776,
[S], -0.777 * 0.002 [6] IS];-0.619 t 0.002 [6] f?] 18)
'-2.061272[8]
:
Points ,/zFx 102 I3asis set
%-Et
Et (au)
,
,’ 0,60
' 1.7 X:u_I"4
0.37.
36 1661 36 ‘. 1823 29 .612 2$ 2486 25 2206 ‘25 2614
31 ‘. ‘I586
. ~. ‘,
.’ ,_. .’,_. ;.:, ., -
..’
,’ ).
:
,_ ,. . . ,,
‘,
:
1s ranuary 1972
CHEMICALPHYSICSLE’IX’ERS
Volume 12, number 4
(2). If these energies had not been known then it would have been possible to fmd *&emby treating E as a variable. This isiJlustrated by fig. 1, which shows
I -a07
I -1C-S
t
L
-5of
-a704
-aw
0
E
Fig. 1. The mean energy 2 (full curve) and root-mean-square (broken curve) as a fun&cm of the input energy E for atomic helium. The horizontal lines show the known enererrorSffz
gies of the 3’S and 4lS stat%s, and the diagonal line gives E = E.
(7) where f-iis a function designed to remove singuiarities at the nucleus, and is given by
the dependence of E (full curve) and 8”” (broken curve) on the variable E. This figure iUustrates (i) that 2 = Et when E = Et, (ii). that S is then a ~n~um, and (iii) that 2 is correct to a higher order (actu~y to the second order, as mentionedabove)than the computed wavefunction itself. Fig. 1 was obtained in a comparatively short corn. puting time since a basis set of the form of eqs. (4) and (5) was used, and once matrix efements such as G#+, I@,,>‘,@,,iB$,.$ and (HQ,K@Q had been evaluated for one point they could be used again for the other points in fig. 1. Graphs such as that of fig. 1, showing a step at each eigenenergy, may prove to be z useful way of mapping eigenenergies. We have used this technique to show that there are no IS states iying between the 2s2 and 2p 2 *S states of helium, Further work is in progress on these uses of the least-squares method, as well as OR possible advantages of other uses (such as scattering processes, and the variation of the weighting function w).
Acknowledgement f _i= 1 - exp(-2r2)
2 (2r2)n/n! = O(C1)l ( ZZ=0 >
(8)
‘I’& author is grateful
The terms in ‘12 are due to :he polarization correction, and the function (p”sy” satisfies Schrtidinger’s equation to a high order when one eIectron is much farther from thb nucleus than the other. @ma [as in eq. (6)] is still added to give a correction for the regions near the nucleus. The calculated energies (see table I) compare favourably with the results of calculations by conventional methods. For completeness we include the ground staie of helium in the table. Here the basis set was of the form given by eqs. (4) and (5) above. Again there is good agreement wit31the known energy.
4. Ab titio
arzd F.
References
determination of eigenenergies
We.have used the known energies Et for B in eq.
to Drs. N. Bard&y
Mandlfor their encouragementat a ctiticaI stageof the work.
”
[I) A.A.Frost, 3. Chem. Phys. 10 (1942) 240. [2] M.H.Uoyd and L.hI.Delves. Intern. J. Quantum Chem. 3 (1969) 169. (31 PGBurke and D.D.hlcV&r, Pro& Phys. Sot. (London) 85 f 1965) 989. [4 J U.Fa.no, Phys Rev. 124 (1961) 1866. fSj T.F.O%faUey and S.Geftman.Phys. Rev. 137 /1965) A 1344. (61 AlBordenave-Montesquieu,P.BenoitCattin and D. BIanc, Abstracts of Vllth Intern. Conf. on Physics of E!ectronic ad Atomic Colikiok, Eds. L.hi.Btanscomp et al., CNorth-Holland, Amsterdam, 1971) p. 1041. [7] fthmtowski and C.t.%&&s, Phys. Rcw. L46 (1966) 46. [8] Y.Atcad, C.L.Pekeris and B.S&iff, Phys Rev, A4 (147f) 516, 551
,’
:
‘(