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The reduction of the multi-dimensional schrödinger equation to a one-dimensional integral equation
Volume -10,number 4
CHEMICAL
PHYSICS LETTERS
15 August 1971
:
.~ERED~(=TION~FTHEMULTI-DIMENSI~NALSCHR~~DINOEREQUA~~N %'OAONE-DIMENSIONALI&GRALEQUATION C. M. ROSENTHk Theoretical Giemistj hdhdte, Uiiiverdy of kkcotzsin; Madison. Wisconsin 53706, USA
Received 29 June 1971
A method is proposed for reducing the multi-dimensionalSchriidingerequation to a one_dimensionaIintegral equation. The reduction is exact; and the resultingintegral equation although complicated, may be treated by any of a number of numerical methods. Two 24iniensional problems, the harmonic oscillator and the S-limitfor heIium and one 3dimensional problem, the helium ground state are treated in this way.
1. Introduction Ordinary differential equations and one-dimensional integral equations may be solved numerically to a high degree of accuracy in a variety of ways. Replacing derivatives with finite differences and integrals with quadrature sums are particularly well suited to one dimension.-In principle these methods are applicabIe to higher dimensions as well, but as is well known, the accuracy of the solutions obtained decreases as rapidly as the cost of computing them increases, simply because more differences or more points have to be used. Thus it wouEd certainly be desirable to redtice the multi-dimensional Schrkiinger equation to a orie-dimensional integrd equation. We describe here such a reduction which, in the examples studied, seems to be of practical as well as theoreticai importance.
2, Method of reduction
The method relies on two ideas, one very familiar to quantum chemists, the variational principfe, ar.d one more famiIiar to mathematicians, the existence of continuous space ftiing curves. The form of the wavefunction used in conjunction with the variational principie is’very si+ar in appearance to a form suggested by Somorjai [ 11: in his studies, $ is constructed from a continuous su@erposition of a set of bound function of n variables, indexed with n continuous new parameter variables. Thus
l
*(xl, X2’. . . ,x,),-JF(xl,x2,
. . .,xn;al,
9,.
. ., an)C(al,g,
. . -,an)dalQLt-
- .doL, -
A
of integratiok in paraineter space and F is a prescribed fun&on of the 2n .vari&es xl, . . -, xn; gezeral, the iepreser&tiori is exact-.(; is determined&y r?quiriig that $ satisfy the usual variation pr.&ple; fhat is.E=(~!HIz@/~J~ IT,@must be stationaj ,tith respect to v&k&ions of 9, i.e., .: .:. ‘_ : variations of G_ Thi~,gene&& ~~i&gral equation for G ofthe &me n&nl&of dir&n&Ons 3s @e~&igi&p&tial differential equ&on-fqi $; this is the princij’a’,disadvantag~ . .. _’ ^of the xtk+od.-
k k .the domain
q,. - - ‘9.c+ If F’is suffickntly
:. ,/ .: : -: ., .. .. .: -. _ ‘. :.:.:,: ., : : -.. .; : an.i G~IIIC &5&&2-0&~~ : ?Iris~~eke~~ was s&p&id by Gnuit GP-12832 froiii’the Nitiond-Scieg~.Eoau&ion . . _..‘. (, . ._ ,.. : : :
381.
Volntie
10, iumber
4
CHEMICAL PHYSICS LET’iERS
15 August 1971
..
4.Apptications L The first two-dimensional problem studied was the. two&mensionai fsotropic barntonic osciilator. The logical _ choice for F is a product of Prussians in x and y piith continuously indexed decay fact&s .:
F(x,y;~a,p)=exp[--a2]
exp[-fly:].
;
~OcuC~,~~
0<#3
=i
expk-40
x”l exp[-JX0u23
:
..*
.G(0 dt ,
0 where
of the previous section, the ex&t G(r) for the ground. state is degenerate. Each of the degenerate sohrtions is of the form 6 (t - fo) where to is any point such that iI = $, aa ~5. There are many such points to since the functions al(t) and 02(2(t)are not one to one. In order to make a less trivial test of the method, the alternative choice was made:
with p a scale factor with respect to which we-optimize at the end of the calculation. Thus we write
d&y)
=I
ewf--polC0
Ix 1 exp MW)
b rl G (0
dt ,
0 with a(t)
,1 = a#)’
: ..
P(rt’& :
. of the previous section. Large p ,@rs corresponds to a ma&r variational calculation (@:krepresented as the superposition of a smaller set of exponentials) and small p corresponds to.a larger v&.riational calculation. Of course ~=&dp~cu(J/= const., 9 e-0, respectitiely) correspond to no variational cafcuh&t at &so the .optimum p. will lie close to but not equal to 0. :.Jt is a-simple matter to calculate ($ I IJOand ($ W I$} needed in the var@tional calculation; For example .-. -. .1, ,. ._ : r 1 4‘_
(ILId) =JJ
G(r) G(t’) dtdt’ ;
.-_ 6 o~+[++-~(t’)~[~(~).+,p(tljl’.
‘.
:
After varying with re.$ect-to, G we o&r+ an ir@ral~equaticn ._
of the form:
‘.
..
.g’-Voliii&l~;,xilii&r ._ . ..
4 .
Cii&iICAL P~J’S~cS &ET’QRS~:..
IS Auguit !971
fro.f,the_over&p
tc&n: e.g.,
r-Z@far‘np-gpprb~ationshave:been made; (i-) is exact since the:set’e&+&p It r] exb[-fi Ip l(y)] for f=ed p, 4cE (1 ,.,.A);p g (1 5.~) is complete; At-&is point -ho&ever,. some approximation .&necessary to solve eq. (1). As the kernels K1, iui,.tid/S involve -the functions &t and a&r e&x% solution is not very likelyTo obtain .‘appr&im&~sblutions, there are-two alternative-routes: replace K, , K2,%nd S by. degenerate kernels offmite rank .. .;-:: : I. _
-,
..
.-
--.I
...and solve. the m&l&g k X N’secular e+ati& for E, or ripiace the integrals 3f (I) $ finite quad&we sums. The . &ond:issometihat easier and all the examples studied so far have been handled in this way. Both methods may
‘be expected to.yield good r&r&s since the kernels are, a@square integrable, symmetric functions,oF t and t’. .- ._.Tabic 1‘shoti:the results for .the .optimuui and two neighboring values of p. A 20.point Gauss-Legendre scheme ‘. was used-for evalu&ng the:integrals..The ground-state value came out well arrd theexcited states less well, as ,.with ahvariationai calctilations. Furthermore’ the exact two-Fold degeneracy of the second state and three-fold degeneracy..of the third state is’(approximately) predicted:&rely better rcsults,would have been.obtained with a _~garissianbasis, but these results were encouraging: .T&eriext tivo-dimensional~problcm studied was the S-limit of helium. UGinga better function ..
?(fl.
,.)i ; a, P)= exp[lpa(~). r< I ek6 l-Pp 0) r, 1 ,
where f<
7
d(q,
3% I> = max(tI, i2) the results listed in table 2 were obtained. Using a symmetrized
.:.gave results nearly as good. The value for p 4 1.2 is to be comp.&ed with a recent fmitedifference solution of the partial differential equation by Barraclough and Morrey (31 in which’s ground-state value of -2.878990 was obrtie’d. It should aisd be noted at this point +&t -Jlis method retains. the upper-bound character of variational :Ccalc$&ions eve? thou&the mtegral equation (1) is solved by quadrature. Table 3 lists the results for a 3~d$ner&on& $roblem;&ti groundstateof .hehum-using 1‘. : , -. ..:...
...
.-
series-of p-v&es-a.rid different numbers of quadrature $oi&s. The best result -2.90355 Idiffers from the-. .. - -Pekeris.value [4].:&.2.90372 by.about.1 part]in. 2Odoo.‘I31 additfon a v&e for agau&ian calculation is also ,-... . .._‘:, listed~%is $ of the form .. :‘_ :r&u& corres@orids to @nctiou &
.:. -.
‘38p,-
-=:
.-
Volume 10, kmbei
4.
CHEMICAL PHYSICS LETTERS :
Table 1 ‘. Harmonic osciIlatoi
Et E. E2 & E4
ES
p=O.6
P’O.8
3.0791 1.0067~ 3.1243 5.7978 -. 5.9374
3.1115 1.0036 3,i440 5.4248 5.6107 6.2593
_
..6.4551
L5 August 1971 :
-p=l.O..
p
TabIe 3 Helium ptind stite
:-
,.2opoints
i2 ..points
3.0604 1.0064 3.0804 5.7018 6.1039
’ 0.28 0.26 0.30 0.30516 0.32
-2.8973 -2;8967 -2.8969 -
--2.901774 -2.901878. T -2.901845
6.7708
051363
-2.8531
50
points .. 64
-290337
points
-290355
a)
a) G-aussian-typebasis. Table 2 Helium S-limit
p = 1.0
p7 1.2
p’i.4
-28789659
-2.8790025
-28789657
EO RI(t),
3 of R2($]. The value listed was obtained with 20 points. This.form for $ is of interest because all of the necessary integrals needed to compute K,, K2 and.S may be worked out explicitly for any molecular or atomic problem. Furthermore as R l(f) and R2(t) range over ail space the representation is competely general.
5. Conclusions A method
has been described which exactly
converts the multi-dimensional
Schro’dinger equation
of
to a one-
dimensional integral equation. Quadrature solution of this equation gave results practical importance in the examples studied. The degeneracy of the ground state for the harmonic oscillator which w& observed with the gaussian W,Y;
o(t), S(0) = expI-40
x21 ew I-B(Ou*l
is no doubt characteristic of this type of solution.. Even so, it causes no difficulty in obtaining a solution, either in those problems which may be solved exactly or those which must be solved numerically. For example, ina quadrature solution, two points tl; and fi might well map into the same point.a1, . . ., sir 4. In. such a case a solution could be obtained simply be deleting the ti or thejth column of the secular equation. This problem is clearly related to the matter of choosing points in the best manner to solve a two-dimensional-integral equation with specified kernels. Both this question and that of choosing the best space filling curve must be looked into in .-greater detail in the future. Mimic type calculations on multi-electrouic atoms may be carried out in’this way and ,the.res&s should be as good as for He. In order to go beyond the $-limit it may be necessary to use a gau&ian type representation of- the type suggested at the-end of the last section. To.@ good results with this approach,it may be necessary to use more points in solving the one-dimensional integralequation. Approxiniating the kern& as separable sums has .’ not yet.been attempted but v&l be in the near future..
in
: ~Gcn&vledgementS
like totha&~&:Phil&e
: .I would to acl&&edge
,de&ntgolfier for-may helpful d&us&&s re@li&g fhisl_&ork;I would like : .-.. the interest ejt$ressed~,by Dr,J.W-l&schfeldcr_i.n. thij~d_ofsoIutioni:-,’ I ‘I ..:I .. :._..-. -385..
,. .-.
j5 &ug
i ..
19*1.
: : ,,._
CHEMICAL
PHYSICS LETTERS
15 August 1971
:
.~ERED~(=TION~FTHEMULTI-DIMENSI~NALSCHR~~DINOEREQUA~~N %'OAONE-DIMENSIONALI&GRALEQUATION C. M. ROSENTHk Theoretical Giemistj hdhdte, Uiiiverdy of kkcotzsin; Madison. Wisconsin 53706, USA
Received 29 June 1971
A method is proposed for reducing the multi-dimensionalSchriidingerequation to a one_dimensionaIintegral equation. The reduction is exact; and the resultingintegral equation although complicated, may be treated by any of a number of numerical methods. Two 24iniensional problems, the harmonic oscillator and the S-limitfor heIium and one 3dimensional problem, the helium ground state are treated in this way.
1. Introduction Ordinary differential equations and one-dimensional integral equations may be solved numerically to a high degree of accuracy in a variety of ways. Replacing derivatives with finite differences and integrals with quadrature sums are particularly well suited to one dimension.-In principle these methods are applicabIe to higher dimensions as well, but as is well known, the accuracy of the solutions obtained decreases as rapidly as the cost of computing them increases, simply because more differences or more points have to be used. Thus it wouEd certainly be desirable to redtice the multi-dimensional Schrkiinger equation to a orie-dimensional integrd equation. We describe here such a reduction which, in the examples studied, seems to be of practical as well as theoreticai importance.
2, Method of reduction
The method relies on two ideas, one very familiar to quantum chemists, the variational principfe, ar.d one more famiIiar to mathematicians, the existence of continuous space ftiing curves. The form of the wavefunction used in conjunction with the variational principie is’very si+ar in appearance to a form suggested by Somorjai [ 11: in his studies, $ is constructed from a continuous su@erposition of a set of bound function of n variables, indexed with n continuous new parameter variables. Thus
l
*(xl, X2’. . . ,x,),-JF(xl,x2,
. . .,xn;al,
9,.
. ., an)C(al,g,
. . -,an)dalQLt-
- .doL, -
A
of integratiok in paraineter space and F is a prescribed fun&on of the 2n .vari&es xl, . . -, xn; gezeral, the iepreser&tiori is exact-.(; is determined&y r?quiriig that $ satisfy the usual variation pr.&ple; fhat is.E=(~!HIz@/~J~ IT,@must be stationaj ,tith respect to v&k&ions of 9, i.e., .: .:. ‘_ : variations of G_ Thi~,gene&& ~~i&gral equation for G ofthe &me n&nl&of dir&n&Ons 3s @e~&igi&p&tial differential equ&on-fqi $; this is the princij’a’,disadvantag~ . .. _’ ^of the xtk+od.-
k k .the domain
q,. - - ‘9.c+ If F’is suffickntly
:. ,/ .: : -: ., .. .. .: -. _ ‘. :.:.:,: ., : : -.. .; : an.i G~IIIC &5&&2-0&~~ : ?Iris~~eke~~ was s&p&id by Gnuit GP-12832 froiii’the Nitiond-Scieg~.Eoau&ion . . _..‘. (, . ._ ,.. : : :
381.
Volntie
10, iumber
4
CHEMICAL PHYSICS LET’iERS
15 August 1971
..
4.Apptications L The first two-dimensional problem studied was the. two&mensionai fsotropic barntonic osciilator. The logical _ choice for F is a product of Prussians in x and y piith continuously indexed decay fact&s .:
F(x,y;~a,p)=exp[--a2]
exp[-fly:].
;
~OcuC~,~~
0<#3
=i
expk-40
x”l exp[-JX0u23
:
..*
.G(0 dt ,
0 where
of the previous section, the ex&t G(r) for the ground. state is degenerate. Each of the degenerate sohrtions is of the form 6 (t - fo) where to is any point such that iI = $, aa ~5. There are many such points to since the functions al(t) and 02(2(t)are not one to one. In order to make a less trivial test of the method, the alternative choice was made:
with p a scale factor with respect to which we-optimize at the end of the calculation. Thus we write
d&y)
=I
ewf--polC0
Ix 1 exp MW)
b rl G (0
dt ,
0 with a(t)
,1 = a#)’
: ..
P(rt’& :
. of the previous section. Large p ,@rs corresponds to a ma&r variational calculation (@:krepresented as the superposition of a smaller set of exponentials) and small p corresponds to.a larger v&.riational calculation. Of course ~=&dp~cu(J/= const., 9 e-0, respectitiely) correspond to no variational cafcuh&t at &so the .optimum p. will lie close to but not equal to 0. :.Jt is a-simple matter to calculate ($ I IJOand ($ W I$} needed in the var@tional calculation; For example .-. -. .1, ,. ._ : r 1 4‘_
(ILId) =JJ
G(r) G(t’) dtdt’ ;
.-_ 6 o~+[++-~(t’)~[~(~).+,p(tljl’.
‘.
:
After varying with re.$ect-to, G we o&r+ an ir@ral~equaticn ._
of the form:
‘.
..
.g’-Voliii&l~;,xilii&r ._ . ..
4 .
Cii&iICAL P~J’S~cS &ET’QRS~:..
IS Auguit !971
fro.f,the_over&p
tc&n: e.g.,
r-Z@far‘np-gpprb~ationshave:been made; (i-) is exact since the:set’e&+&p It r] exb[-fi Ip l(y)] for f=ed p, 4cE (1 ,.,.A);p g (1 5.~) is complete; At-&is point -ho&ever,. some approximation .&necessary to solve eq. (1). As the kernels K1, iui,.tid/S involve -the functions &t and a&r e&x% solution is not very likelyTo obtain .‘appr&im&~sblutions, there are-two alternative-routes: replace K, , K2,%nd S by. degenerate kernels offmite rank .. .;-:: : I. _
-,
..
.-
--.I
...and solve. the m&l&g k X N’secular e+ati& for E, or ripiace the integrals 3f (I) $ finite quad&we sums. The . &ond:issometihat easier and all the examples studied so far have been handled in this way. Both methods may
‘be expected to.yield good r&r&s since the kernels are, a@square integrable, symmetric functions,oF t and t’. .- ._.Tabic 1‘shoti:the results for .the .optimuui and two neighboring values of p. A 20.point Gauss-Legendre scheme ‘. was used-for evalu&ng the:integrals..The ground-state value came out well arrd theexcited states less well, as ,.with ahvariationai calctilations. Furthermore’ the exact two-Fold degeneracy of the second state and three-fold degeneracy..of the third state is’(approximately) predicted:&rely better rcsults,would have been.obtained with a _~garissianbasis, but these results were encouraging: .T&eriext tivo-dimensional~problcm studied was the S-limit of helium. UGinga better function ..
?(fl.
,.)i ; a, P)= exp[lpa(~). r< I ek6 l-Pp 0) r, 1 ,
where f<
7
d(q,
3% I> = max(tI, i2) the results listed in table 2 were obtained. Using a symmetrized
.:.gave results nearly as good. The value for p 4 1.2 is to be comp.&ed with a recent fmitedifference solution of the partial differential equation by Barraclough and Morrey (31 in which’s ground-state value of -2.878990 was obrtie’d. It should aisd be noted at this point +&t -Jlis method retains. the upper-bound character of variational :Ccalc$&ions eve? thou&the mtegral equation (1) is solved by quadrature. Table 3 lists the results for a 3~d$ner&on& $roblem;&ti groundstateof .hehum-using 1‘. : , -. ..:...
...
.-
series-of p-v&es-a.rid different numbers of quadrature $oi&s. The best result -2.90355 Idiffers from the-. .. - -Pekeris.value [4].:&.2.90372 by.about.1 part]in. 2Odoo.‘I31 additfon a v&e for agau&ian calculation is also ,-... . .._‘:, listed~%is $ of the form .. :‘_ :r&u& corres@orids to @nctiou &
.:. -.
‘38p,-
-=:
.-
Volume 10, kmbei
4.
CHEMICAL PHYSICS LETTERS :
Table 1 ‘. Harmonic osciIlatoi
Et E. E2 & E4
ES
p=O.6
P’O.8
3.0791 1.0067~ 3.1243 5.7978 -. 5.9374
3.1115 1.0036 3,i440 5.4248 5.6107 6.2593
_
..6.4551
L5 August 1971 :
-p=l.O..
p
TabIe 3 Helium ptind stite
:-
,.2opoints
i2 ..points
3.0604 1.0064 3.0804 5.7018 6.1039
’ 0.28 0.26 0.30 0.30516 0.32
-2.8973 -2;8967 -2.8969 -
--2.901774 -2.901878. T -2.901845
6.7708
051363
-2.8531
50
points .. 64
-290337
points
-290355
a)
a) G-aussian-typebasis. Table 2 Helium S-limit
p = 1.0
p7 1.2
p’i.4
-28789659
-2.8790025
-28789657
EO RI(t),
3 of R2($]. The value listed was obtained with 20 points. This.form for $ is of interest because all of the necessary integrals needed to compute K,, K2 and.S may be worked out explicitly for any molecular or atomic problem. Furthermore as R l(f) and R2(t) range over ail space the representation is competely general.
5. Conclusions A method
has been described which exactly
converts the multi-dimensional
Schro’dinger equation
of
to a one-
dimensional integral equation. Quadrature solution of this equation gave results practical importance in the examples studied. The degeneracy of the ground state for the harmonic oscillator which w& observed with the gaussian W,Y;
o(t), S(0) = expI-40
x21 ew I-B(Ou*l
is no doubt characteristic of this type of solution.. Even so, it causes no difficulty in obtaining a solution, either in those problems which may be solved exactly or those which must be solved numerically. For example, ina quadrature solution, two points tl; and fi might well map into the same point.a1, . . ., sir 4. In. such a case a solution could be obtained simply be deleting the ti or thejth column of the secular equation. This problem is clearly related to the matter of choosing points in the best manner to solve a two-dimensional-integral equation with specified kernels. Both this question and that of choosing the best space filling curve must be looked into in .-greater detail in the future. Mimic type calculations on multi-electrouic atoms may be carried out in’this way and ,the.res&s should be as good as for He. In order to go beyond the $-limit it may be necessary to use a gau&ian type representation of- the type suggested at the-end of the last section. To.@ good results with this approach,it may be necessary to use more points in solving the one-dimensional integralequation. Approxiniating the kern& as separable sums has .’ not yet.been attempted but v&l be in the near future..
in
: ~Gcn&vledgementS
like totha&~&:Phil&e
: .I would to acl&&edge
,de&ntgolfier for-may helpful d&us&&s re@li&g fhisl_&ork;I would like : .-.. the interest ejt$ressed~,by Dr,J.W-l&schfeldcr_i.n. thij~d_ofsoIutioni:-,’ I ‘I ..:I .. :._..-. -385..
,. .-.
j5 &ug
i ..
19*1.
: : ,,._