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Automatic calculation of the energy levels of oneelectron diatomic moleculesPaul B. Bailey Citation: The Journal of Chemical Physics 69, 1676 (1978); doi: 10.1063/1.436744 View online: http://dx.doi.org/10.1063/1.436744 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/69/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Exact solvable model of a oneelectron diatomic molecule Am. J. Phys. 54, 567 (1986); 10.1119/1.14541 Energy levels of oneelectron atoms J. Phys. Chem. Ref. Data 6, 831 (1977); 10.1063/1.555557 Relativistic Effects in Diatomic Molecules: Evaluation of OneElectron Integrals J. Chem. Phys. 53, 1052 (1970); 10.1063/1.1674096 On a Continued Fraction Occurring in the Theory of OneElectron Diatomic Molecules J. Chem. Phys. 45, 2703 (1966); 10.1063/1.1727999 On the Electronic Energy of a OneElectron Diatomic Molecule near the United Atom J. Chem. Phys. 44, 3934 (1966); 10.1063/1.1726555
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Automatic calculation of the energy levels of one-electron diatomic molecules8
)
Paul B. Bailey
Applied Mathematics Division, Sandia Laboratories, Albuquerque, New Mexico 87185 (Received 25 April 1978)
A method is described for the calculation of energy levels of one-electron diatomic molecules which is completely automatic, does not have to have an initial guess supplied, and cannot accidentally compute a wrong energy level even when they are very close together.
I. INTRODUCTION . The problem of determining the energy levels of a
one-electron diatomic molecule is well known to be expressible in the form1
.E.. ( ~2 _ 1) dX) d~ d~
+(-C-p2W -1)- (l,~21) +R(Z1+Z2)~)X=O on (1,00)
.E.. (1 _ r/') dY) d1] d1]
(1)
+ (c _P2(1 - r/') - (1 ~~) +R(Z2 - Z1)1]) Y = 0 on( - 1,1) ,
(2) where only for a discrete (denumerably infinite) set of pairs of values of the eigenparameters p2 and C does there exist a pair of acceptable simultaneous solutions Xm, Y(1]).
Here C is an unknown separation constant,
p2= _jR2W , (3)
where W is the desired (but unknown) electronic energy level, Z1 and Z2 are the nuclear charges (with Z1~ Z2), R is the internuclear distance (in atomic units), and m is an integer (another separation constant) corresponding to the electron's orbital angular momentum.
The quantity of most interest, usually, is the electronic energy W.
Several different kinds of methods have been used for the computation of the eigenparameters, p2 and C, of this problem. Some, based on variational solutions to the electron Hamiltonian, 2.3 or on some kind of perturbation,4.5 can be effective in the case of the lower energy levels (or small or large internuclear separation) but tend to become unwieldy in the other cases. And in any event their effectiveness is limited by the skill and experience of the user.
When very high accuracy is required in the eigenvalues, the methods most commonly employed are based on some sort of analytical representation of the wave functions in (1) and (2), followed by numerical computation of series coefficients or recurrence relations. A method of this kind was used by Jaffe8 in 1934, before the advent of modern computers; and, even now, such methods are the ones most commonly adopted. 7-11
a)This work was supported by the U. S. Department of Energy (DOE) under Contract No. AT(29-1)-789.
A more recent method is that employed by Gordon, 12
in which the coefficients of the differential Eqs. (1) and (2) are approximated by piecewise linear functions and the resulting equations solved in terms of piecewise Airy functions. (See also Refs. 13 and 14.)
And then there are those other methods based on the integration of suitably chosen initial value problems for (1) and (2), as in Ref. 15. Until fairly recently, such methods have rarely, if ever, been adopted by serious workers in this area. No doubt this is partly due to the fact that suitable computers (and efficient initial value problem codes) have only recently become widely available. But even now it is not clear that these newer methods can be more efficient than the analytic representation methods when many digits (more than about eight) of accuracy are required. Nevertheless, the initial value problem methods do have some attractive features.
For example, the analytic representation methods can, and often do, suffer from the problem of accidentally computing the wrong eigenvalue. This problemof being uncertain which eigenvalue has been computedmost commonly occurs only when several different energy levels are very close together, but it does happen often enough to be a matter of serious concern.
On the other hand, it need not happen when USing initial value problem methods or Gordon's method. Basically, this is because with these methods it is easy to keep track of the numbers of nodes in the corresponding eigenfunctions.
The method to be outlined in this paper is one which is based on the numerical integration of certain initial value problems for (1) and (2). It has been designed to be as completely automatic as possible so that, unlike most other methods, no special knowledge or experience is required on the part of the user.
Even the usual requirement that the user supply a good initial guess for the pair of numbers (p2 ,c) being sought is not required here. If a good guess is not available, a satisfactory estimate is automatically self-generated. On the other hand, if a good guess for either p2 or C (or both) is made available, then that information is used in preference to generating an internal guess.
Perhaps it should be emphasized that the reason we can get by with internally generated initial guesses is not really because those guesses are so very good but
1676 J. Chem. Phys. 69(4), 15 Aug. 1978 0021-9606/78/6904-1676$01.00 © 1978 American Institute of Physics
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Paul B. Bailey: Energy levels of one-electron diatomic molecules 1677
rather because the remainder of the algorithm does not have to have terribly good guesses. In other words, initial guesses which are quite good enough for the method described here might be not nearly good enough for those other methods which can accidentally converge to eigenvalues different from the ones desired. That cannot happen here.
Actually, our scheme for solution of this two-parameter eigenvalue problem is based on the fact that we already have available a general purpose code, SLEIGN, lS for the automatic solution of ordinary one -dimensional Sturm-Liouville eigenvalue problems. And since a detailed description of the algorithms in that code has been published elsewhere, we need only describe here how that capability can be used to obtain the pair of eigenvalues in the present problem. This is done in the next chapter. Chapter 3 very briefly sketches the procedure employed for estimating the pair of eigenvalues, and then in Chapter 4 we list the results obtained on some representative one-electron problems.
II. METHOD OF SOLUTION
Very briefly, the method consists of treating each one of the two Eqs. (1) and (2) as a Sturm-Liouville eigenvalue problem for one of the two numbers, pZ and C, keeping the other fixed for the moment, and then using a Newton method for generating a new and (hopefully) better pair of numbers.
For notational convenience let us rewrite Eqs. (1) and (2) as
where
El=-C , plm=~2-1, r 1m=1
q1m = - Ez(1 - ~Z) - mZ/(~2 -1)+R(Zl + Z2)~
Ez=_p2, P2(1)=I-'/f, r 2('17)=I-1)2
q2(1) = -E1 - m2/(1-1)2)+R(Z2 -zl)11 .
(4)
Then El and E2 are the two eigenparameters which are to be found.
A convenient way of specifying which particular pair of eigenvalues E 1, E2 is wanted is by means of the familiar united atom charge and quantum numbers n = 1, 2, 3, ... ; 1=0,1,2,_ .. ; m=0,1,2, .... Thus givenn, 1, m we require that >Itl have n -1 -1 nodes in the interior of (1, + 00) and >It2 have 1 - m nodes in the interior of (-1,1)_
Since we already have available a general purpose code, SLEIGN, for the computation of eigenvalues and eigenfunctions of Singular and nonsingular Sturm-Liouville problems, it is a simple matter to obtain either El or E2 if all the other parameters of the corresponding equation are held fixed. However, since we want to use Newton's method in going from one pair of estimates of (Et> E 2) to the next, it is necessary to modify SLEIGN so as to obtain from it the necessary partial derivatives.
To be explicit, let us write the solutions >It1 , >Itz of (1),
(2) in polar form, using the usual Priifer transformation
>Itl = Pl sin61
Pl >It~ = Pl cos61
and
>It2 = P2 sin62
P2 >It~ = Pz cos6z
(A discussion of the Priifer transformation and its elementary properties is contained in Ref. 17, p. 209-213.) Substitution in (1) and (2) yields differential equations for 61 and 62 :
6~ = 1.. cos261 + (E1 r1 + q1) sin261 P1 (5i)
6~ = :2 cos262 + (E2 r 2 + Q2)sinz62 (5ii)
Corresponding to the appropriate boundary conditions which must be satisfied by >Itl and >It2 one easily deduces boundary conditions1S for 61 and 62 ; say
61(a 1) = a1, 61(b1) = (jl +n17r
62(a2) = a 2 , 62(b2) = ~ +n27r ,
where
and
nl =n -1-1
n2=1-m •
(6i)
(6ii)
If now one were to integrate (5i) from a1 to some interior point c1 starting with 61(a1) = aI' and again from b1 to ci starting with 61(bi) = ~ +nl 7r, with just any values for the E1 and E2 appearing in (5i) (E2 doesn't exactly "appear" there, but it of course is there), the two values of 61 at ~ = C 1 would not agree. They would if, and only if, E1 were the "eigenvalue" of (5i) corresponding to that fixed value of Ez•
Likewise if, for fixed E 1, one were to integrate (5ii) from a2 to some interior point, C2, and again from b 2 to C2' the two values of 62 at 1) = Cz would agree if, and only if, E2 were the "eigenvalue" of (5U).
So denoting by d1(E I , Ez) the difference in the two values of 61 at c1 , to emphasize the fact that this difference depends upon the values of E1 and Ez being used in (5i), and likewise denoting by d2(El> E2) the difference in the two values of 62 at c2, then the values of E1 and E2 wanted are those for which both
dj(EI' E 2)" 0 ,
i=land2.
(7)
Now it is easy to say what sort of Newton method is used to go from one estimate, (E~, Eg), of (E1, Ez) to another, (E~, E~): namely, set E2 = Eg in (5i) and find E'i such that d1(E'i , Eg) is small, and evaluate also the partial derivatives
J. Chern. Phys., Vol. 69, No.4, 15 August 1978
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1678 Paul B. Bailey: Energy levels of one-electron diatomic molecules
(These derivatives can incidentally be evaluated by integrating the variational equations obtained by differentiating (5i) with respect to E j or E 2.) Next set E t := Ej in (5ii) and find an E; such that dz(Ei,.e:) is small, and evaluate also (8dz/6E1) and (8dz/6Ez) there. Finally, linearize (7), i=l, about(~,Eg)and(7), i=2, about (E:: , .e:) and solve for the Newton iterate (Ef, E~).
.111. GENERATING AN INITIAL GUESS
Because of the fact that the code, SLEIGN, is able to generate (internally) reasonable estimates of the eigenvalues of Sturm-Liouville problems, very inexpensively, we can use the capability to generate an initial pair of numbers for the problem at hand. For starting with any value, Eg, we can thus obtain an estimate for the corresponding "eigenvalue," E7, say; and, along with it, an estimate of the value of d2(E~, E~). (Without going into the details here, those computations in SLEIGN only involve simple matrix operations.) Details can be found in Ref. 16. And since the sign of dz indicates whether Eg is too high or too low, a simple algorithm based on bisection or the secant method can quickly find an Eg, with companion E~, such that the estimated value of dz(E~, Eg) is small enough. Of course "small enough" here means that the estimated value of dz is as small as the estimated error, in which case there is no point in carrying the estimating process any further.
So the estimating process for (E~, Eg) consists of trying values of Eg (via an algorithm using bisection and the secant method) in (5i) with the estimating procedure in SLEIGN returning estimates of a correspoinding E7 and dz(E~, Eg).
Note that although this sort of estimating procedure works extremely well for the method described here, where it is easy to keep track of the number of nodes in the solutions of (1) and (2), much better estimates might be needed for some of the other methods to avoid accidentally computing a wrong energy level.
IV. NUMERICAL RESULTS
To obtain some indication of how the scheme outlined above performs in typical situations, a computer cod,e was constructed and several test problems run on a CDC 6600.
On the hydrogen molecular ion ls<T-state, the results obtained for a number of different internuclear separations (small, medium, and large) were compared with those in Ref. 9 and found to be obtainable, automatically with relative accuracy of at least 10-8 in all cases.
In the case of the hydrogen-helium molecular ion, the values obtained for the 2p7T-state were compared with those in ReI. 10. This state was chosen because the singularities of Eqs. (1) and (2) are different (worse) when m ~ 0 than when m = O. Even so, no difficulty was encountered in getting numbers at least as accurate as those of the cited reference.
Even the 5gu- and 6h<T-states of the hydrogen-carbon
TABLE I. Values of electronic energy, W, and separation constant, C, for the 5grr- and 6hu-states of the hydrogen-carbon molecular ion where they are most nearly equal. R is the internuclear separation in atomic units.
5gu 6hu
R -w -c -IV -c 21. 32 .7818294819 127.6371746 .7814149040 127.6461977
21. 34 .7815644968 127.7574373 .7813388341 127.7623529
21. 36 .7813029541 127.8776353 .7812600116 127.8785715
21. 38 .7811895561 127.9946111 .7810337321 127.9980112
24.0 .7727039676 143.2051868 .7502489667 143.7491700
molecular ion in the range of internuclear separations R = 21. 32 to R = 24.00 where the energy levels (and separation constants) are very close together (see Table I) were solved completely automatically with no outside help (no initial guesses given). Here again a relative accuracy of at least 10-8 was obtained. ("Exact" values for these states were obtained from unpublished work of Dr. J. M. Peek of Sandia Laboratories.)
Of course starting with good initial guesses can significantly reduce overall computing time, but even without initial guesses times on the CDC 6600 were generally of the order of 5-15 s. Such times are, of course, not competative with the most efficient methods, but they are not bad either. And the method is automatic.
ACKNOWLEDGMENT
The author would like to publicly acknowledge the very considerable help he has received from his colleague, Dr. J. M. Peek, in understanding a few of the computational problems associated with the one-electron diatomic molecule. His advice and encouragement have been invaluable.
tD• R. Bates and R. H. G. Reid, in Advances in Atomic and Molecular Physics edited by D. R. Bates and I. Estermann, (Academic Press, New York, 1968), Vol. 4.
2H• Eyring, J. Walter, and G. Kimball, Quantum Chemistry, (Wiley, New York, 1944), p. 196.
3S. Cohen, D. Judd, and R. J. Riddell, Jr., University of California Radiation Lab. Report No. UCRL-8802, 1959, (unpublished) •
'M. R. Woodward, Int. J. Quant. Chern. 6, 911 (1972). 5M. K. Ali and W. J. Meath, Int. J. Quant. Chern. 6, 949
(1972). 60. Jaffe, Z. Physik 87, 535 (1934). 7J • M. Peek and E. N. Lassettre, J. Chern. Phys. 38, 2392
(1963). 8D• R. Bates, K. Ledsham, and A. L. Stewart, Phil. Trans.
Roy. Soc. London A 246, 215 (1953). SM. M. Madsen and J. M. Peek, At. Data, Z, 171 (1971). taL. I. Ponomarevand T. P. Puzynlna, 1966 JINR-P4-3175
preprint, Joint Inst. Nuclear Research, Dubna. SC-T-67-0953, Sandia Labs., Albuquerque, New Mexico.)
l1J. D. Powers, Phil. Trans. Roy. Soc. LondonAZ74, 663 (1973).
12R. J. Gordon, J. Chern. Phys. 51, 14 (1969). 13J. Canosa and R. G. de Oliveira, J. Compo Phys. 5, 188
J. Chern. Phys., Vol. 69, No.4, 15 August 1978
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Paul B. Bailey: Energy levels of one-electron diatomic molecules 1679
(1970). 14S. Pruess, SIAM (Soc. Ind. Appl. Math) J. Numer. Anal. 10,
55 (1973). 15A. C. Allison, J. Compo Phys. 6, 378 (1970). 16p. B. Bailey, SAND77 - 2044, Sandia Laboratories, Albuquer
que, New Mexico, 1978. The code SLEIGN is freely available
from the author to any interested persons. l1E • A. Coddington and N. Levinson, Theory of Ordinary Dif
ferential Equations, (McGraw-Hill, New York, 1955), pp. 209-213.
lap. B. Bailey, SIAM (Soc. Ind. Appl. Math) J. Appl. Math. 14, 242 (1966).
J. Chern. Phys., Vol. 69, No.4, 15 August 1978
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