- Email: [email protected]
On a double minimum vibrational potential
ON A DOUBLE
MINIMUM
VIBRATIONAL
POTENTIAL
CARL W. DAVU) Department of Chemistry, University of Connecticut, Stotrs, CT 06268, U.S.A.
(Received 29April 1976) Abstract-The FORMAC program used to generate the matrix elements of a harmonic oscillator perturbed by a Gaussian is presented, and the resultant expressions validated by comparison with pencil and paper results, and by direct quantum-mechanical computations of relevant energy levels in the ammonia case.
Investigations of double minimum potentials have been carried out for many years. Not having a theoretical model for the shape of a vibrational potential having two or more minima, authors have invented various functional forms which, for reasons usually of tractability, have then been used for solving the Schrodinger Equation (Chan & Stelman, 1%3; Chan et al. 1964; Coon et al., 1966; Danielis et nL. 1975; Dennison & Uhlenbeck, 1932; Hand, 1927; Jones & Coon, 1969; Mannings, 1935; Morse & Stuckelburg, 1931; and Somorjai & Homig, 1962). One excellent candidate for such a double minimum potential is a standard Hooke’s Law force, perturbed by a Gaussian barrier U = (kx*)/2 + A, exp (+x2).
OUTER: DO N = NSTART(1) TO 20 BY 2; DOM=NTO20BY2: LET(PREMULT = ((I/( 1 + BE+A)) * *(l/2)) * (FAC(N)*FAC(M)/((2**N)*(2**M)))**(1/2)); LET(SUM = 0); INNER: DO L = LSTART(I) TO N BY 2; LET(L = “L”; SUM = SUM + ((-THETA) * * (((N t M)/2) - L) * (k * * L))/ (FAC(L) * FAC((M - L)/2) * FAC((N - L)/2))); END INNER; LET(SUM = SUM * PREMULT); LET(SUM = REPLACE(SUM,THETA.BETA/(I + BETA))); LET(SUM = REPLACE(SUM,k,2/( 1+ BETA))): END OUTER; END EVEN-ODD;
(1)
Chan & Stelman (1%3) have obtained matrix elements for the Gaussian part of this potential in a simple harmonic oscillator representation, using a special connectivity formula to bypass evaluating a horrible expansion, viz., (exp (-fi.2)).,n = (l/(1 + p))“+z !m !/2” 2”)1’z KI(_
I2 I!((m I= 0,2,4,6,
The procedure is easily altered to obtain a different subset of matrix elements. If evaluation of a matrix element is desired, this may be effected by means of the statement:
(j)(n+mw-l
1)/2)!((n - 1)/2)!
(2)
a
or 1=1,3,5,7,...
LET(var = REPLACE(SUM,BETA,value));
where K = 2/(1+ 8) and 8 = ,3(1+ @). With the appearance of symbol manipulation languages, specifically FORMACt, it is no longer necessary to avoid such expansions, and, in fact, a direct evaluation of the matrix elements given in Eq. (2) can be carried out in an astonishingly small amount of time (both human and machine) using no special (from the user’s point of view) multiple precision arithmetic. The following excerpt from a FORMAC program obtains all non-zero matrix elements of the perturbation up to 20th order:
The FORMAC variable var will contain the representation of the value of the matrix element. If this value is to be used in another part of the program, as a value for a PL /I variable, then conversion can be effected by means of a statement:
DCL NSTART(2) INIT(O,I) LSTART(2) INIT(O,l);
which would be reasonable for the above program. Presumably, after assembling the entire matrix, G, the program would diagonalize the matrix. The N = 0, M = 6 (and N = 6, M = 0) value generated by the above program was
G(N,M),G(M,N) = ARITH(var); where G has been declared as a matrix, e.g. DCL G(O :20,0: 20)BIN FLOAT;
EVEN - ODD: DOI=lTO2;
SUM = -0.55901699 BETA3/(BETA + 1)“’
(3)
which evaluated, in units of (l/(1 +@))**(1/2), at p = 1, to a (FORMAC default precesion) value of -0.06987712,
t(a) Tobey, R., Baker, I., Crews, R., Marks, P. & Victor, K., PL/I Format Interpreter, IBM Publication No. 3600 03.0.304. 93
CARL W.
94
which is one significant figure less than the Chan & Stelman (1%3) result. At the exit from loop INNER, a typical “value” of SUM is, (for example the N = 8, M = 16 element): 56 THETA2K6 + 140 THETA’K4 + 80 THETA6K2 + 5 THETA* +5 K”) where we have, for clarity, gathered statement PRINT-OUT
terms
via the
(SUM = CODEM(SUM));
In units of hro, the results of a calculation of the energy Table 1 V = (Q’12 + LIe-*@)hvo L1= 14.7 @=0.07 v0 = 885.6 cm-’ (Benedict & Phyler,
1957; Swslen & Ibers, 1962)
Swalen & Ibers (1962)
Present
nl
Observed
{4 variable)
work
0, O*
0 O.ooO896 1.05297 1.69341 1.80398
L
1, 2s 2* 3s 3, 4s 4, 5s 5* 6s 6,
2.69135 3.2695
0 0.001107 1.04788 1.09192 1.8078 2.1273 2.6954 3.2667
o.ooo746 1.064088 1.095675 1.84637 2.12577 2.69073 3.85665 4.50697 5.19287 5.90191 6.70106 7.44717
t(a) To&y, R., Baker, J., Crews, R., Marks, P., Victor, K., PL/I Format Interpreter, IBM Publication No. 3600 03.0.304. (b) An extended version, FORMAC73 is now available, due CO K. A. Bahr, Gesellschaft fiir Mathematik und Datenverarbeitung MBH Bonn, Bereich Darmstadt, IFV, 6100 Darmstadt, Rheinstrasse 75, West Germany. (c) Knoble, H. D., private comnmnication; a FORTRAN FORMAC (SHARE No. 36ODO33013)is also available. Write SHARE Program Library Agency, Triangle University Computation Center, P.O. Box 12076, Research Triangle Park, NC 27709, U.S.A. For the interested reader, a copy of desired matrix elements can be sent in either printed form or on punched cards or magnetic tape.
DAVID
levels of NH, are shown in Table 1, compared to the results of Swalen & Ibers (1962). The calculation displayed is not one for a best fit of a and /3, but is purely illustrative. The computation of ail non-zero elements with N I 20 took 1 mitt of IBM 360/65-370/155 time and required 138K of core storage. ReFERENCEs B&r, K. A. (1976). private communication. Benedict, W. S. & Plyler, K. K. (1957). Can 1. Phys. 35, 1235. Chan, S. I. & Stelmaa, D. (1%3), J. ‘Chem. Phys. 3, 545. Chan, S. I., Stelman, D. 81:Thompson, L. E. (1964), L Chem. Phys. 41, 2828. Coon, I. B., Naugle, N. W. & McKenzie, R. D. (1%6), J. Mol. Spectrosc. 29, 107. Danielis, V., Papousek, D., Spirko, V. & Horak, M. (1975), J. Mol. Specirosc. 54, 339. Dennison, D. M. & Uhlenbeck, G. E. (1932). Phys. Rev. 41,3 13. Hund, F. (1927), 2. Physik 43, 805. Jones, V. T. & Coon, J. B. (1%9), J. Mol. Spectrosc. 31, 137. Mannings. M. F. (1935), J. Chem. Phys. 3, 136. McCarthy, J., Abrahams, W., Edwards, D. J., Ha&T. P. & Levia. M. I. (1%2), LISP 1.5 Programmer’s Manual, Cambridge, Mass., M.I.T. Press. Morse., P. M. & Stuckelburg, E. C. G. (1931), Helv. Phys. Acta 4, 337. Sammet, J. E. (1%9), Programming Languages, Prentice-Hall, Englewood Cliffs, N.J., p. 474. Somorjai, R. L. & Wornig, D. F. (1962), J. C/rem. Fhys. 36,1980. Swalen, J. D. & Ibers, J. A. (1%2), J. Chem. Phys. 36, 1914. (LISP 1.5 Primer by Clark Weissman) (1967) Dickenson. Belmont, CA. ArTENDtX Of the two best known non-numerical computing languages (Sammet, 1969) LISP (McCarthy et cl., 1962) and FORMACt FORMAC is by far the easier to learn for most scientists. FORMAC allows the user to manipulate algebraic expressions in virtually the same manner that the user would employ at his/her desk using pencil and paper. Additionally, as demonstrated here, PORMAC allows, after formula manipulation, substitutions which then allow the host language (PL/I in our case here) to continue numerical computations in the traditional manner. As a tool, FORMAC allows a large amount of tedious substitutions to be carried out in an error free method on formulae so that greater confidence can be had in the results of the computations based on these formulae. Unfortunately, a potential user of FORMAC must learn a bit of PL/I (ii he wishes to use the newest versions) and he must learn FORMAC itself. No text exists for FORMAC learners, comparable to (LISP 1.5 Primer (by (Clark Weissman))) (Weissman, 1967). The original Tobey manualt is my only source of information. Dr. Bahr (Bahr, 1976) has promised both a new manual and a FORMAC version which operates under the IBM Optimiiing Compiler for PL/I; inquires should be directed to hi.
MINIMUM
VIBRATIONAL
POTENTIAL
CARL W. DAVU) Department of Chemistry, University of Connecticut, Stotrs, CT 06268, U.S.A.
(Received 29April 1976) Abstract-The FORMAC program used to generate the matrix elements of a harmonic oscillator perturbed by a Gaussian is presented, and the resultant expressions validated by comparison with pencil and paper results, and by direct quantum-mechanical computations of relevant energy levels in the ammonia case.
Investigations of double minimum potentials have been carried out for many years. Not having a theoretical model for the shape of a vibrational potential having two or more minima, authors have invented various functional forms which, for reasons usually of tractability, have then been used for solving the Schrodinger Equation (Chan & Stelman, 1%3; Chan et al. 1964; Coon et al., 1966; Danielis et nL. 1975; Dennison & Uhlenbeck, 1932; Hand, 1927; Jones & Coon, 1969; Mannings, 1935; Morse & Stuckelburg, 1931; and Somorjai & Homig, 1962). One excellent candidate for such a double minimum potential is a standard Hooke’s Law force, perturbed by a Gaussian barrier U = (kx*)/2 + A, exp (+x2).
OUTER: DO N = NSTART(1) TO 20 BY 2; DOM=NTO20BY2: LET(PREMULT = ((I/( 1 + BE+A)) * *(l/2)) * (FAC(N)*FAC(M)/((2**N)*(2**M)))**(1/2)); LET(SUM = 0); INNER: DO L = LSTART(I) TO N BY 2; LET(L = “L”; SUM = SUM + ((-THETA) * * (((N t M)/2) - L) * (k * * L))/ (FAC(L) * FAC((M - L)/2) * FAC((N - L)/2))); END INNER; LET(SUM = SUM * PREMULT); LET(SUM = REPLACE(SUM,THETA.BETA/(I + BETA))); LET(SUM = REPLACE(SUM,k,2/( 1+ BETA))): END OUTER; END EVEN-ODD;
(1)
Chan & Stelman (1%3) have obtained matrix elements for the Gaussian part of this potential in a simple harmonic oscillator representation, using a special connectivity formula to bypass evaluating a horrible expansion, viz., (exp (-fi.2)).,n = (l/(1 + p))“+z !m !/2” 2”)1’z KI(_
I2 I!((m I= 0,2,4,6,
The procedure is easily altered to obtain a different subset of matrix elements. If evaluation of a matrix element is desired, this may be effected by means of the statement:
(j)(n+mw-l
1)/2)!((n - 1)/2)!
(2)
a
or 1=1,3,5,7,...
LET(var = REPLACE(SUM,BETA,value));
where K = 2/(1+ 8) and 8 = ,3(1+ @). With the appearance of symbol manipulation languages, specifically FORMACt, it is no longer necessary to avoid such expansions, and, in fact, a direct evaluation of the matrix elements given in Eq. (2) can be carried out in an astonishingly small amount of time (both human and machine) using no special (from the user’s point of view) multiple precision arithmetic. The following excerpt from a FORMAC program obtains all non-zero matrix elements of the perturbation up to 20th order:
The FORMAC variable var will contain the representation of the value of the matrix element. If this value is to be used in another part of the program, as a value for a PL /I variable, then conversion can be effected by means of a statement:
DCL NSTART(2) INIT(O,I) LSTART(2) INIT(O,l);
which would be reasonable for the above program. Presumably, after assembling the entire matrix, G, the program would diagonalize the matrix. The N = 0, M = 6 (and N = 6, M = 0) value generated by the above program was
G(N,M),G(M,N) = ARITH(var); where G has been declared as a matrix, e.g. DCL G(O :20,0: 20)BIN FLOAT;
EVEN - ODD: DOI=lTO2;
SUM = -0.55901699 BETA3/(BETA + 1)“’
(3)
which evaluated, in units of (l/(1 +@))**(1/2), at p = 1, to a (FORMAC default precesion) value of -0.06987712,
t(a) Tobey, R., Baker, I., Crews, R., Marks, P. & Victor, K., PL/I Format Interpreter, IBM Publication No. 3600 03.0.304. 93
CARL W.
94
which is one significant figure less than the Chan & Stelman (1%3) result. At the exit from loop INNER, a typical “value” of SUM is, (for example the N = 8, M = 16 element): 56 THETA2K6 + 140 THETA’K4 + 80 THETA6K2 + 5 THETA* +5 K”) where we have, for clarity, gathered statement PRINT-OUT
terms
via the
(SUM = CODEM(SUM));
In units of hro, the results of a calculation of the energy Table 1 V = (Q’12 + LIe-*@)hvo L1= 14.7 @=0.07 v0 = 885.6 cm-’ (Benedict & Phyler,
1957; Swslen & Ibers, 1962)
Swalen & Ibers (1962)
Present
nl
Observed
{4 variable)
work
0, O*
0 O.ooO896 1.05297 1.69341 1.80398
L
1, 2s 2* 3s 3, 4s 4, 5s 5* 6s 6,
2.69135 3.2695
0 0.001107 1.04788 1.09192 1.8078 2.1273 2.6954 3.2667
o.ooo746 1.064088 1.095675 1.84637 2.12577 2.69073 3.85665 4.50697 5.19287 5.90191 6.70106 7.44717
t(a) To&y, R., Baker, J., Crews, R., Marks, P., Victor, K., PL/I Format Interpreter, IBM Publication No. 3600 03.0.304. (b) An extended version, FORMAC73 is now available, due CO K. A. Bahr, Gesellschaft fiir Mathematik und Datenverarbeitung MBH Bonn, Bereich Darmstadt, IFV, 6100 Darmstadt, Rheinstrasse 75, West Germany. (c) Knoble, H. D., private comnmnication; a FORTRAN FORMAC (SHARE No. 36ODO33013)is also available. Write SHARE Program Library Agency, Triangle University Computation Center, P.O. Box 12076, Research Triangle Park, NC 27709, U.S.A. For the interested reader, a copy of desired matrix elements can be sent in either printed form or on punched cards or magnetic tape.
DAVID
levels of NH, are shown in Table 1, compared to the results of Swalen & Ibers (1962). The calculation displayed is not one for a best fit of a and /3, but is purely illustrative. The computation of ail non-zero elements with N I 20 took 1 mitt of IBM 360/65-370/155 time and required 138K of core storage. ReFERENCEs B&r, K. A. (1976). private communication. Benedict, W. S. & Plyler, K. K. (1957). Can 1. Phys. 35, 1235. Chan, S. I. & Stelmaa, D. (1%3), J. ‘Chem. Phys. 3, 545. Chan, S. I., Stelman, D. 81:Thompson, L. E. (1964), L Chem. Phys. 41, 2828. Coon, I. B., Naugle, N. W. & McKenzie, R. D. (1%6), J. Mol. Spectrosc. 29, 107. Danielis, V., Papousek, D., Spirko, V. & Horak, M. (1975), J. Mol. Specirosc. 54, 339. Dennison, D. M. & Uhlenbeck, G. E. (1932). Phys. Rev. 41,3 13. Hund, F. (1927), 2. Physik 43, 805. Jones, V. T. & Coon, J. B. (1%9), J. Mol. Spectrosc. 31, 137. Mannings. M. F. (1935), J. Chem. Phys. 3, 136. McCarthy, J., Abrahams, W., Edwards, D. J., Ha&T. P. & Levia. M. I. (1%2), LISP 1.5 Programmer’s Manual, Cambridge, Mass., M.I.T. Press. Morse., P. M. & Stuckelburg, E. C. G. (1931), Helv. Phys. Acta 4, 337. Sammet, J. E. (1%9), Programming Languages, Prentice-Hall, Englewood Cliffs, N.J., p. 474. Somorjai, R. L. & Wornig, D. F. (1962), J. C/rem. Fhys. 36,1980. Swalen, J. D. & Ibers, J. A. (1%2), J. Chem. Phys. 36, 1914. (LISP 1.5 Primer by Clark Weissman) (1967) Dickenson. Belmont, CA. ArTENDtX Of the two best known non-numerical computing languages (Sammet, 1969) LISP (McCarthy et cl., 1962) and FORMACt FORMAC is by far the easier to learn for most scientists. FORMAC allows the user to manipulate algebraic expressions in virtually the same manner that the user would employ at his/her desk using pencil and paper. Additionally, as demonstrated here, PORMAC allows, after formula manipulation, substitutions which then allow the host language (PL/I in our case here) to continue numerical computations in the traditional manner. As a tool, FORMAC allows a large amount of tedious substitutions to be carried out in an error free method on formulae so that greater confidence can be had in the results of the computations based on these formulae. Unfortunately, a potential user of FORMAC must learn a bit of PL/I (ii he wishes to use the newest versions) and he must learn FORMAC itself. No text exists for FORMAC learners, comparable to (LISP 1.5 Primer (by (Clark Weissman))) (Weissman, 1967). The original Tobey manualt is my only source of information. Dr. Bahr (Bahr, 1976) has promised both a new manual and a FORMAC version which operates under the IBM Optimiiing Compiler for PL/I; inquires should be directed to hi.