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Variational calculations on a correlated helium wave function using maple
Compw Chem. Vol. 12, No. 3. pp. 22%23
Printedin Great Britain
VARIATIONAL HELIUM
1, 19813
OOP7-8485/88 $3.00 + 0.00 Person Press plc
CALCULATIONS WAVE FUNCTION
ON A CORRELATED USING MAPLE
CARL W. DAVID Department of Chemistry. University of Connecticut, U-60 Rm. 161, 215 Glenbrook Road, Storrs, CT 06268. U.S.A. (Receiwd 8 Seprember 1987) An extraordinary amount of effort has been expended in obtaining the ground state electronic energy of helium. To date, the best computations are due to Freund et al. (1984) who obtained an energy of -2.9037243770340, believed to be accurate to “a few parts in 1013*‘, which they compared to the (at that time) current “best computation” due to Frankowski & Pekeris (1966). The Freund ef al. computation employed a wave function of 230 terms, and employed logarithmic terms similar to those suggested by Fock (1954). With the introduction of symbolic computational methodology, it becomes reasonable to attempt nonlinear variational computations on helium wave functions of higher complexity than have been attempted in the past. We have computed the “best” energy for a trial wave function of the form: f=exp[-(Z--)(r,+r,)+br,,]
i5 := subs(s=O,i4); n2 := int(nl,t=o..u); n3 := int(n2,u=O..s); n4 := int(n3.s); n5 := subs(s=O.n4); energy energy
subs(Z=Z .energy): normal(anargy): diff(energy,sigma): normal(densig): numer(densig): denb := diff(enargy.b): denb := normal(danb): denb := numerldenb): sigma1 := .3125: for i from 1 to 10 do j :=i+l;
f := subs(sigma= sigma.i,denb); b.i := solve(f=O,b); g := subs(b=b.i,densig); sigms.j := solve( g=O,sigma); energy.i := subs( slgma=sigma.j,b=b.i,anergy); od;
(I)
where 0 and 6 are constants whose values are to be found by minimizing the function E(a,
6) =
(2)
(we restrict ourselves to Y real), and Z is the atomic number of the nucleus (Z = 2 in our case). Following Hylleraas (1928, 1929) the setting up of the proper integrals has been reduced to an algorithm (Bethe & Salpeter, 1957). We have employed that algorithm in a symbolic rather than a numerical setting using MAPLE (Char er al., 1985), a relatively new symbolic mathematical processor which has been implemented on both VAX and IBM systems (IBM 370/3084 here). The following program was used: psi := exp I-(z-s&mal’*s
:= 15/ns;
:= energy := densig := densig := densig :=
+ b-1;
tl :A diff(psi,s);
t2 := diff(psi,t); t3 := diff(psi,u); il := U*(s**2 - twc2 ) * ( t1**2 + t2**2 + t3**2 - t*2)"tl + t*(s~Z-U*2)*t2) + 2 * t3 * (s*(u**2
)
-psi"2 *(4*z*s*u - s**2 + t**2) ; nl := u*(s"z - t&2) * psi"2; 12 := ht(il,t=O..u); 13 := int(i2,u=0..s); 14 := int(i3.s);
This program has used several conventions known intimately to quantum chemists. The energy is computed in Hartrees, the distance variables are in units ofa0.Thevariabless=r,+r,,a=r,-r,andu=r,, have been employed. Further, although our example does not include r in the wave function, the program has been written to include the effect of having I in the list of variables for the wave function. In this way, the expression for i 1 in line 5 of the program can be recognized as a direct transcription of Bethe and Salpeter’s (1957) equation. Only line 1 need be changed to include a function of t if desired. The first result of this program’s execution was an expression for “energy” of the form energy (a, b). This energy was then minimized. The non-linear variation part of the program begins at line 18, where the derivative of the energy with respect to 0 is computed.? For u = 0.141978 and b = 2546523, the energy was a minimum, with a value of - 2.889618 (more decimal places are available to MAPLE, but are not necessarily significant here) after 10 cycles of simultaneous solving of the equations: denergycO db a energy = (). ao
statements ending in a colon are not printed while those ending in a semicolon are printed.
TMAFLE
229
(3)
CARL W. DAVID
230
The expressions for the partial derivatives were obtained and the equations were cyclically solved until self-consistency was achieved. This value compares very favorably with the “optimal” value of -2.89116 (Roberts, 1969) for a wave function of the form y = exp[--+I
+
dfh)
(4)
which miniwhere f is that (undetermined) faction mizes the energy variationally. For b = 0.0 and m = 5/16, the energy (using the same program) was - 2.847656, which is correct. The introduction of minimal correlation has resulted in u dropping from 0.3125 to 0.1423, while the b value has achieved about one-half its optimal value [for fulfilling the helium Cusp condition (Kato, 1951, 1957)]. Although MAPLE (and analogous languages like MACSYMA) should now be included in the repertoire of tools available to theoretical chemists, it is important to note limitations to symbolic mathematical systems which must be taken into account when preparing to use them. In the case of variational computations such as those discussed here, it is important to make several observations which have influenced the computation in various ways. (1) The line which reads: i4: = int (i3, s) which calls for the computer to integrate an exprcssion over the variable ‘s’, should have been (and originally was): i4: = int (i3, s = 0. . . infinity) which is the instruction to carry out the definite integral rather than the indefinite integral actually used. The reason for this change was that MAPLE can not evaluate the upper limit for this particular integral properly. For example, MAPLE evaluates limit (E**( -x),
x = infinity)
as zero, but limit ( - E**( -x),
x = infinity)
remains unevaluated instead of being evaluated as zero. It is the same in our case; i.e., the integrals whose upper limit actually were zero were unevaluated by MAPLE. Therefore, our program substitutes only the lower bound (s = 0). (2) Originally, the computation (above) proceeded to take the partials: d energy
au
(5)
aenergy au
a energy p= ab
o
but MAPLE was unable to solve for the values of b and D which minimized the energy. This would have been the most automated computation possible, but MAPLE stumbled over the last part with no error messages, only a plaintive statement that it couldn’t do the job requested. Instead, the expressions for the partial derivatives were obtained, and u = 5/16 substituted into equation (6) which allowed us to solve for a temporary value of 6, which could then be substituted into equation (5). This resultant approximation for b was re-substituted into equation (6) to allow solution for an improved value for 6, etc., etc., etc. The code starting at sigmal: =
.3125
displays this primitive approach_ (3) Potential users of MAPLE under CMS in an IBM environment should be aware that (a) MAPLE (in common with other symbolic systems) generates enormously long expressions which sometimes are bewilderingly incomprehensible, and (b) MAPLE is very expensive to run. It is impossible to compare MAPLE running on our IBM 370 to, say, MAXIMA running on a VAX, given two diierent computers, and presumably two different charging algorithms, but running MAPLE under CMS has proven to be quite costly. (c) Just as with other mathematical programming languages, it is possible to convert MAPLE output statements into compilable FORTRAN (or other high level language) statements for use in numerically intensive computational settings. The potential MAPLE user should be aware however, that MAPLE is useful at arbitrary levels of arithmetical significance, so such conversions to “foreign” languages should not be carried out when the interest is accuracy. Rather, such conversions are only useful when repeated evaluation of expressions (reduction to numerical values) is desired. (4) Finally, although MAPLE can integrate many functions, it can not integrate all functions. Specifically, MAPLE was unabte to handle wave functions which contained logarithmic (Fock type) terms. With the introduction of effective computer mathematical systems, the theoretical chemist’s toolbox is being expanded with a tool which, when used carefully, will allow computations which were previously inconceivable by all except for experts. REFERENCES
and 6 energy ab
(6)
with the intention of generating the expressions for these partial derivatives, and employing the MAPLE ‘solve’ facility to solve the set of equations:
Bethe H. A. & Salpeter E. E. (1957) Quantum Mechanics of One- and Two-Electron Atoms, p. 147. Academic Press, New York. Char B. W.. Gecldes K. O., Gonnet G. H. & Watt S_ M. (1985) MAPLE User’s Guide, 4th edition. WATCOM
k5y$$Fns
Ltd, 415 Phillip Street, Waterloo, Ontario.
Variational
calculations
on a correlated
Fock V. A. (1954) Izv. Akad. Nauk SSSR, Ser. Fiz. 18,161. Frankowski K. & Pekeris C. L. (1966) P&s. Rev. 146, 46 150,366E. Freund D. E., Huxtable B. D. 81 Morgan III J. D. (1984) Phys. Rev. A29, 930.
helium wave function
231
Hylleraas E. A. (1928) Z. Phys. 48, 469. Hylleraas E. A. (1929) Z. Phys. 54, 347. Kate T. (1951) Trans. Am. marh. Sot. 70, 195. Kato T. (1957). Comm. Pure Appl. Math. 10, 151. Roberts P. J. (1969) Thcorer. chim. Acra (Berl.) 13, 340.
Printedin Great Britain
VARIATIONAL HELIUM
1, 19813
OOP7-8485/88 $3.00 + 0.00 Person Press plc
CALCULATIONS WAVE FUNCTION
ON A CORRELATED USING MAPLE
CARL W. DAVID Department of Chemistry. University of Connecticut, U-60 Rm. 161, 215 Glenbrook Road, Storrs, CT 06268. U.S.A. (Receiwd 8 Seprember 1987) An extraordinary amount of effort has been expended in obtaining the ground state electronic energy of helium. To date, the best computations are due to Freund et al. (1984) who obtained an energy of -2.9037243770340, believed to be accurate to “a few parts in 1013*‘, which they compared to the (at that time) current “best computation” due to Frankowski & Pekeris (1966). The Freund ef al. computation employed a wave function of 230 terms, and employed logarithmic terms similar to those suggested by Fock (1954). With the introduction of symbolic computational methodology, it becomes reasonable to attempt nonlinear variational computations on helium wave functions of higher complexity than have been attempted in the past. We have computed the “best” energy for a trial wave function of the form: f=exp[-(Z--)(r,+r,)+br,,]
i5 := subs(s=O,i4); n2 := int(nl,t=o..u); n3 := int(n2,u=O..s); n4 := int(n3.s); n5 := subs(s=O.n4); energy energy
subs(Z=Z .energy): normal(anargy): diff(energy,sigma): normal(densig): numer(densig): denb := diff(enargy.b): denb := normal(danb): denb := numerldenb): sigma1 := .3125: for i from 1 to 10 do j :=i+l;
f := subs(sigma= sigma.i,denb); b.i := solve(f=O,b); g := subs(b=b.i,densig); sigms.j := solve( g=O,sigma); energy.i := subs( slgma=sigma.j,b=b.i,anergy); od;
(I)
where 0 and 6 are constants whose values are to be found by minimizing the function E(a,
6) =
(2)
(we restrict ourselves to Y real), and Z is the atomic number of the nucleus (Z = 2 in our case). Following Hylleraas (1928, 1929) the setting up of the proper integrals has been reduced to an algorithm (Bethe & Salpeter, 1957). We have employed that algorithm in a symbolic rather than a numerical setting using MAPLE (Char er al., 1985), a relatively new symbolic mathematical processor which has been implemented on both VAX and IBM systems (IBM 370/3084 here). The following program was used: psi := exp I-(z-s&mal’*s
:= 15/ns;
:= energy := densig := densig := densig :=
+ b-1;
tl :A diff(psi,s);
t2 := diff(psi,t); t3 := diff(psi,u); il := U*(s**2 - twc2 ) * ( t1**2 + t2**2 + t3**2 - t*2)"tl + t*(s~Z-U*2)*t2) + 2 * t3 * (s*(u**2
)
-psi"2 *(4*z*s*u - s**2 + t**2) ; nl := u*(s"z - t&2) * psi"2; 12 := ht(il,t=O..u); 13 := int(i2,u=0..s); 14 := int(i3.s);
This program has used several conventions known intimately to quantum chemists. The energy is computed in Hartrees, the distance variables are in units ofa0.Thevariabless=r,+r,,a=r,-r,andu=r,, have been employed. Further, although our example does not include r in the wave function, the program has been written to include the effect of having I in the list of variables for the wave function. In this way, the expression for i 1 in line 5 of the program can be recognized as a direct transcription of Bethe and Salpeter’s (1957) equation. Only line 1 need be changed to include a function of t if desired. The first result of this program’s execution was an expression for “energy” of the form energy (a, b). This energy was then minimized. The non-linear variation part of the program begins at line 18, where the derivative of the energy with respect to 0 is computed.? For u = 0.141978 and b = 2546523, the energy was a minimum, with a value of - 2.889618 (more decimal places are available to MAPLE, but are not necessarily significant here) after 10 cycles of simultaneous solving of the equations: denergycO db a energy = (). ao
statements ending in a colon are not printed while those ending in a semicolon are printed.
TMAFLE
229
(3)
CARL W. DAVID
230
The expressions for the partial derivatives were obtained and the equations were cyclically solved until self-consistency was achieved. This value compares very favorably with the “optimal” value of -2.89116 (Roberts, 1969) for a wave function of the form y = exp[--+I
+
dfh)
(4)
which miniwhere f is that (undetermined) faction mizes the energy variationally. For b = 0.0 and m = 5/16, the energy (using the same program) was - 2.847656, which is correct. The introduction of minimal correlation has resulted in u dropping from 0.3125 to 0.1423, while the b value has achieved about one-half its optimal value [for fulfilling the helium Cusp condition (Kato, 1951, 1957)]. Although MAPLE (and analogous languages like MACSYMA) should now be included in the repertoire of tools available to theoretical chemists, it is important to note limitations to symbolic mathematical systems which must be taken into account when preparing to use them. In the case of variational computations such as those discussed here, it is important to make several observations which have influenced the computation in various ways. (1) The line which reads: i4: = int (i3, s) which calls for the computer to integrate an exprcssion over the variable ‘s’, should have been (and originally was): i4: = int (i3, s = 0. . . infinity) which is the instruction to carry out the definite integral rather than the indefinite integral actually used. The reason for this change was that MAPLE can not evaluate the upper limit for this particular integral properly. For example, MAPLE evaluates limit (E**( -x),
x = infinity)
as zero, but limit ( - E**( -x),
x = infinity)
remains unevaluated instead of being evaluated as zero. It is the same in our case; i.e., the integrals whose upper limit actually were zero were unevaluated by MAPLE. Therefore, our program substitutes only the lower bound (s = 0). (2) Originally, the computation (above) proceeded to take the partials: d energy
au
(5)
aenergy au
a energy p= ab
o
but MAPLE was unable to solve for the values of b and D which minimized the energy. This would have been the most automated computation possible, but MAPLE stumbled over the last part with no error messages, only a plaintive statement that it couldn’t do the job requested. Instead, the expressions for the partial derivatives were obtained, and u = 5/16 substituted into equation (6) which allowed us to solve for a temporary value of 6, which could then be substituted into equation (5). This resultant approximation for b was re-substituted into equation (6) to allow solution for an improved value for 6, etc., etc., etc. The code starting at sigmal: =
.3125
displays this primitive approach_ (3) Potential users of MAPLE under CMS in an IBM environment should be aware that (a) MAPLE (in common with other symbolic systems) generates enormously long expressions which sometimes are bewilderingly incomprehensible, and (b) MAPLE is very expensive to run. It is impossible to compare MAPLE running on our IBM 370 to, say, MAXIMA running on a VAX, given two diierent computers, and presumably two different charging algorithms, but running MAPLE under CMS has proven to be quite costly. (c) Just as with other mathematical programming languages, it is possible to convert MAPLE output statements into compilable FORTRAN (or other high level language) statements for use in numerically intensive computational settings. The potential MAPLE user should be aware however, that MAPLE is useful at arbitrary levels of arithmetical significance, so such conversions to “foreign” languages should not be carried out when the interest is accuracy. Rather, such conversions are only useful when repeated evaluation of expressions (reduction to numerical values) is desired. (4) Finally, although MAPLE can integrate many functions, it can not integrate all functions. Specifically, MAPLE was unabte to handle wave functions which contained logarithmic (Fock type) terms. With the introduction of effective computer mathematical systems, the theoretical chemist’s toolbox is being expanded with a tool which, when used carefully, will allow computations which were previously inconceivable by all except for experts. REFERENCES
and 6 energy ab
(6)
with the intention of generating the expressions for these partial derivatives, and employing the MAPLE ‘solve’ facility to solve the set of equations:
Bethe H. A. & Salpeter E. E. (1957) Quantum Mechanics of One- and Two-Electron Atoms, p. 147. Academic Press, New York. Char B. W.. Gecldes K. O., Gonnet G. H. & Watt S_ M. (1985) MAPLE User’s Guide, 4th edition. WATCOM
k5y$$Fns
Ltd, 415 Phillip Street, Waterloo, Ontario.
Variational
calculations
on a correlated
Fock V. A. (1954) Izv. Akad. Nauk SSSR, Ser. Fiz. 18,161. Frankowski K. & Pekeris C. L. (1966) P&s. Rev. 146, 46 150,366E. Freund D. E., Huxtable B. D. 81 Morgan III J. D. (1984) Phys. Rev. A29, 930.
helium wave function
231
Hylleraas E. A. (1928) Z. Phys. 48, 469. Hylleraas E. A. (1929) Z. Phys. 54, 347. Kate T. (1951) Trans. Am. marh. Sot. 70, 195. Kato T. (1957). Comm. Pure Appl. Math. 10, 151. Roberts P. J. (1969) Thcorer. chim. Acra (Berl.) 13, 340.