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The calculation of the energy levels of nearly symmetric rotors
JOTiRNAL
OF
MOLECULAR
SPECTROSCOPY
The Calculation
4,
93-98 (1960)
of the Energy Symmetric SAMUIXL
It~ternationaE Business
C.
Levels of Nearly
Rotors WAIT,
Machines,
JR.*
United h’ingdom,
Ltd., London
A method is given for extending the King et al. tables of coefficients for calculation of the energy levels of nearly symmetric rotor molecules. The method is general and may be applied to any secular determinant of continuant form. In addition, it may be used for any desired order of approximation and is not limited by the size of the secular det#erminant. Values of the coefficients up to t,he seventh-order approximat.ion and for J I 51 have been obtained and will he available on request.
At1 asymmetric rotor has 2J + 1 energy levels for each value of t,he rot,ationnl yuant,um number J. The energy levels for any value of J can be obtained as functions of the eigenvalues of four secular equations, each of order approsimately .1,/Z. Thus, in going t,o a high J value, a formidable calculating problem is creat’ed. If intensity calculations are to be done, the eigenvectors must also be obt,ained. Cert,ain approximate methods have been used in cases where the accuracy desired was not too great. These have been discussed in detail by Hainer ct al. ( f ). The validity of these methods is limit,ed, however, t,o a small range of the asymn~et~ry parameter.’ Nevertheless, t,hey do prove very useful in their range of ~~~pli~a~}ilitysince they eliminate direct; solutioi~ of the secular equatiou. How-
* Fulbright~ Scholar 195G57: Present Address: Depxrtment of Chemistry, Carnegie Irlstitute of Technology, Pittsburgh, Pennsylvania. 1 Present Address: Department of Physics, Massachusetts Institute of Twhnology, Cambridge, Massachusetts. I The asymmetry parameter is defined in terms of the rotational constants of the molrcule, :I, B, and C, which in turn are related simply t,o the moments of inertia, e.g.. .I = h&r”I,
93
WAIT
94
AND
BARNETT
ever, for rot’ors which are “very” asymmetric, one must still resort to direct solut8ion of the secular equations. For rotors which are nearly symmetric (either prolate or oblate), it is possible to express the reduced energy, i.e., eigenvalues of the secular equations, in polynomial form. This has been done by King et al. (2) for J 5 12 to the third degree terms in the asymmetry parameter 6 = (B - C)/(A - C). The resulting energy levels are satisfactory for low J values, but if the treatment is extended, it is found that the third-order approximation is not satisfactory at high J. Recently Schwendeman (3) has given a tabulation of coefficients to the fifth-degree terms for J 5 40 in terms of a parameter b (&late = (C - B)/ (2il - B - C) ; bob,ate= (A - B)/(2C - A - B). These give considerably more accurate results than the original King et al. approximation.* In the present work, a method is given for extending the King et al. approximation in terms of 6 to any desired degree of approximation. The method described below is general for any secular determinant in continuant form. METHOD
OF CALCULATION
The energy level matrices for the asymmetric rotor are of continuant form, i.e., have nonzero elements only on the diagonal and immediate super- and subdiagonals. The secular determinants formed from these matrices have the property that an eigenvalue can be expressed as a continued fraction. This gives one important method of obtaining the eigenvalues, namely iteration of the finite continued fraction. The procedure to be described below was developed by reference to the continued fraction form of the secular determinant. However, it could equally well have been derived from the original determinant. This method is applicable to any matrix which can be expressed as a continuant, and will prove useful for solving secular equations of this form. It should be noted, however, that the method is an approximation and gives the most accurate results when the matrix is already nearly diagonal, i.e., when the subdiagonal elements approach zero or are small in comparison with the diagonal elements. The treatment for “Type I” representations of King et al. (2) is typical. It follows directly from Eq. (57) of their work. This equation is repeated here as Eq. (1). A, = k:f
Skii,
-
&+I + &a -
Am -
@W&+1 b+.fm+,/(k:+z + 6k;+,+z - A, s’fm
k:-I +
A, -
[$fm+J(k:-,
6kk
-
+ 6k;-,-2 -
h,
-
. . ->I
-
.-
.)]
(1)
’
* It has been pointed out (4) that b (e in Ref. 4) is preferable to 6 as an asymmetry parameter since the secular determinant is an even function of b, but not of 6. The present communication will, therefore, be limited to a discussion of the general method of calculation.
NEARLY
SYMMETRIC
ROTORS
!)5
A, is the m’th root for a particular value of J, 6 is the asymmetry parameter, and the fm, k, , and k: depend in a simple way on J. The continued fracbion terminates after J/2 or (J - 1)/2 partial quotients (for J even and odd, respect,ively) . Equation ( 1) may be rewritten in the form: X, = k: + Sk; -
6’(I,,r
+ J,,l),
(2)
where
s1113x= J/2,
J even
or
(J
-
1)/2, J odd.
The roots A,,, , and the quantities I,,, power series in the parameter 6.
and J,,,
may be expanded formallp as a
An iterative procedure can now be developed for computing the aCi’js in alt,ernation with the b’s and d’s for increasing i values. The quantities that correspond to the truncation of the series in Eq. (s) and in Eqs. (6) and (7) after (27 + 1)) and after (2r - 2s + 1) terms, respectively, may be denoted by X2’+” and by I$’ , .I$’ . x(zr+l) 111
=
2r+1 z
dl%
(8)
(10)
WAIT
96
AND
BARNETT
It is also convenient to put
Xow the eoe~eients I$: , that contrib~lte to the summation (9), are determined only by those a$: with j 5 (2~ - 2s + 1). In a similar manner the dg,,)8, that contribute to Jgi for a fixed va lue o f r, depend only on the cz! with j 6 (2r 2s+ l).Theu$~andc~~forj4 (2r - 2s + 1) can, however, be found from the &) with i 5 2r - 1 and the bgi_1 and dgi-1 with i 5 (2r - 2.5 - 1). The iterative process is initiated by linearizing Eq. (I ) to give x (0) = k:+&, nl
(13)
and each iteration cycle is completed by the calculation for the current value of r of the two coefficients azr) and a,?‘+” in the expansion of h, . The iteration formulas follow from the binomial expansion of the reciprocal of a power series. In general, if
-I =J,(2) +0(2n+1),
(14)
Jn(z> = Jn-2(x) - {[(an-l/UObn-1+
i (anlao>- i2alan-llao2)l~“ll~“~ + O(x”+t).
It follows at once from Eqs. (Q), (ll), p
= p&1’ w&s
(16)
and (16) that:
_ fm+,[ ~~~-2~)/(~~~z~~*~-~s +
ja$;-2”+“/@W)2
_ 2ox’, ~~:-2~‘/(~~)“t82r--Zs+l]
(17)
so that = -~~~~~,~-z~)/(u~~)~, b127-z81 %B b
(2r-zstl) m,*
_ -
-fnL+,ta~:-2"t"/(~~1)2
(1% -
2~~~~~~~-~s)/(~~)3~.
A comparison of Eqs. (3a) and (11) shows, however, that
(19
NEARLY
S~M~TRI~
ROTORS
07
Corresponding to Eqs. (18) and (19) there are the further results (22) and (23) obtained in a comparable manner from Eqs. (lo), ( 12)) and (16), while a comparison of Eqs. (4a) and (12) leads to Eqs. (21) and (25). d(zr--28) rn.8 = -fm&+l cC,-z6J/(c;:)2, ,j$,;-‘“+” = -fm-,*(e~~-2x+‘)/(c~~>B - 2c~:c~~:-2”‘/(c~~)3~, (21--“a> CZF-2S) - d(2r-2*-4’ Cm,s = --a, m,sil I ~‘?r--26+lf (a--2s+11_ dg:;;s-l I = --a, cm,,
(22) (23) @4) (23
The iternt.ive cycle is completed by the calculation of ~2~’ and ~2”~~’ using Eqs. (Xi) which are direct consequences of Eqs. (21, (8), (9 ), and (10). (2r1 = +b$-“’ %7&
+ dg;-“‘,
(27fll = +bz,;-1’ a,
+ dg;-‘).
(26)
The expressions quoted by King et al. (9) for the coefficients &’ to a:’ can be derived rapidly by use of the above formulas. A corresponding analysis can be made for t,he other types of representation leading to the same recurrence fnrm&as, but to different expressions for the k:, k,:, , and fW factors. A cletailed error analysis on the above method has not been carried out. However, from t,he results me~~t,ionedbelow, it is apparent that the magnitude of the sub-diagonal elements plays an inlportant part in determining the accuracy of the results. Thus, for the first three or four eigenvalues of a given determinant, t!he region of usefulness is limited to t,he range of 0 5 6 5 0.025 (K 5 -0.95). However, for subsequent eigenvalues it is possible t’o use them for 0 5 6 5 0.05 (K 2 -0.9), and in many cases a good result is obtained for 6 I 0.1 (K = -0.8 ). In any application of this method, however, care must be taken in sel&ing the “range of usefulness” of the results. Using the above method of calculation, a program was written for the IRM A.30 digital computer. This program is general and allows one to go to a,ny desired degree of approximat,io~l for J 5 100. The coefficients a$’ of Eq. (5) have been calculat.ed up through i = 7, i.e., t,he seventh-order approximati(~n. Tables of t,hese eoeficients will not, he published because of their volume, but will he av:tiIahle on request. , ACKNOWLEUGMENT We wish t.o thank generosity ait,h time in programming and Thanks are also due RWEIVED:
International Business Machines, United Kingdom, Ltd. for their on the computer. Especially, we would like to acknowledge the help preparation of results given by Wss D. A. Emery and Mrs. J. Wieher. to Professor D. P. Craig for many discussions aiding this work.
JXVUARY 19, 1959
98
WALT
AND
BARNETT
REFERENCES 1. M. R. HAINER,P. C. CROSS,ANDG. W. KING, J. Chem. Phys. 17, 826 (1949). 2. G. W. KING, R. M. HAINER,AND P. C. CROSS,J. Chem. Phys. 11, 27 (1943). 9. R. H. SCHWENDEMAN, “A Table of Coefficients for the Energy Levels of a Near Symmetric Top”, Research Report, Department of Chemistry, Harvard University, 1957. 4. 6. R. POLO, Can. J. Phys. 36, 880 (1957).
OF
MOLECULAR
SPECTROSCOPY
The Calculation
4,
93-98 (1960)
of the Energy Symmetric SAMUIXL
It~ternationaE Business
C.
Levels of Nearly
Rotors WAIT,
Machines,
JR.*
United h’ingdom,
Ltd., London
A method is given for extending the King et al. tables of coefficients for calculation of the energy levels of nearly symmetric rotor molecules. The method is general and may be applied to any secular determinant of continuant form. In addition, it may be used for any desired order of approximation and is not limited by the size of the secular det#erminant. Values of the coefficients up to t,he seventh-order approximat.ion and for J I 51 have been obtained and will he available on request.
At1 asymmetric rotor has 2J + 1 energy levels for each value of t,he rot,ationnl yuant,um number J. The energy levels for any value of J can be obtained as functions of the eigenvalues of four secular equations, each of order approsimately .1,/Z. Thus, in going t,o a high J value, a formidable calculating problem is creat’ed. If intensity calculations are to be done, the eigenvectors must also be obt,ained. Cert,ain approximate methods have been used in cases where the accuracy desired was not too great. These have been discussed in detail by Hainer ct al. ( f ). The validity of these methods is limit,ed, however, t,o a small range of the asymn~et~ry parameter.’ Nevertheless, t,hey do prove very useful in their range of ~~~pli~a~}ilitysince they eliminate direct; solutioi~ of the secular equatiou. How-
* Fulbright~ Scholar 195G57: Present Address: Depxrtment of Chemistry, Carnegie Irlstitute of Technology, Pittsburgh, Pennsylvania. 1 Present Address: Department of Physics, Massachusetts Institute of Twhnology, Cambridge, Massachusetts. I The asymmetry parameter is defined in terms of the rotational constants of the molrcule, :I, B, and C, which in turn are related simply t,o the moments of inertia, e.g.. .I = h&r”I,
93
WAIT
94
AND
BARNETT
ever, for rot’ors which are “very” asymmetric, one must still resort to direct solut8ion of the secular equations. For rotors which are nearly symmetric (either prolate or oblate), it is possible to express the reduced energy, i.e., eigenvalues of the secular equations, in polynomial form. This has been done by King et al. (2) for J 5 12 to the third degree terms in the asymmetry parameter 6 = (B - C)/(A - C). The resulting energy levels are satisfactory for low J values, but if the treatment is extended, it is found that the third-order approximation is not satisfactory at high J. Recently Schwendeman (3) has given a tabulation of coefficients to the fifth-degree terms for J 5 40 in terms of a parameter b (&late = (C - B)/ (2il - B - C) ; bob,ate= (A - B)/(2C - A - B). These give considerably more accurate results than the original King et al. approximation.* In the present work, a method is given for extending the King et al. approximation in terms of 6 to any desired degree of approximation. The method described below is general for any secular determinant in continuant form. METHOD
OF CALCULATION
The energy level matrices for the asymmetric rotor are of continuant form, i.e., have nonzero elements only on the diagonal and immediate super- and subdiagonals. The secular determinants formed from these matrices have the property that an eigenvalue can be expressed as a continued fraction. This gives one important method of obtaining the eigenvalues, namely iteration of the finite continued fraction. The procedure to be described below was developed by reference to the continued fraction form of the secular determinant. However, it could equally well have been derived from the original determinant. This method is applicable to any matrix which can be expressed as a continuant, and will prove useful for solving secular equations of this form. It should be noted, however, that the method is an approximation and gives the most accurate results when the matrix is already nearly diagonal, i.e., when the subdiagonal elements approach zero or are small in comparison with the diagonal elements. The treatment for “Type I” representations of King et al. (2) is typical. It follows directly from Eq. (57) of their work. This equation is repeated here as Eq. (1). A, = k:f
Skii,
-
&+I + &a -
Am -
@W&+1 b+.fm+,/(k:+z + 6k;+,+z - A, s’fm
k:-I +
A, -
[$fm+J(k:-,
6kk
-
+ 6k;-,-2 -
h,
-
. . ->I
-
.-
.)]
(1)
’
* It has been pointed out (4) that b (e in Ref. 4) is preferable to 6 as an asymmetry parameter since the secular determinant is an even function of b, but not of 6. The present communication will, therefore, be limited to a discussion of the general method of calculation.
NEARLY
SYMMETRIC
ROTORS
!)5
A, is the m’th root for a particular value of J, 6 is the asymmetry parameter, and the fm, k, , and k: depend in a simple way on J. The continued fracbion terminates after J/2 or (J - 1)/2 partial quotients (for J even and odd, respect,ively) . Equation ( 1) may be rewritten in the form: X, = k: + Sk; -
6’(I,,r
+ J,,l),
(2)
where
s1113x= J/2,
J even
or
(J
-
1)/2, J odd.
The roots A,,, , and the quantities I,,, power series in the parameter 6.
and J,,,
may be expanded formallp as a
An iterative procedure can now be developed for computing the aCi’js in alt,ernation with the b’s and d’s for increasing i values. The quantities that correspond to the truncation of the series in Eq. (s) and in Eqs. (6) and (7) after (27 + 1)) and after (2r - 2s + 1) terms, respectively, may be denoted by X2’+” and by I$’ , .I$’ . x(zr+l) 111
=
2r+1 z
dl%
(8)
(10)
WAIT
96
AND
BARNETT
It is also convenient to put
Xow the eoe~eients I$: , that contrib~lte to the summation (9), are determined only by those a$: with j 5 (2~ - 2s + 1). In a similar manner the dg,,)8, that contribute to Jgi for a fixed va lue o f r, depend only on the cz! with j 6 (2r 2s+ l).Theu$~andc~~forj4 (2r - 2s + 1) can, however, be found from the &) with i 5 2r - 1 and the bgi_1 and dgi-1 with i 5 (2r - 2.5 - 1). The iterative process is initiated by linearizing Eq. (I ) to give x (0) = k:+&, nl
(13)
and each iteration cycle is completed by the calculation for the current value of r of the two coefficients azr) and a,?‘+” in the expansion of h, . The iteration formulas follow from the binomial expansion of the reciprocal of a power series. In general, if
-I =J,(2) +0(2n+1),
(14)
Jn(z> = Jn-2(x) - {[(an-l/UObn-1+
i (anlao>- i2alan-llao2)l~“ll~“~ + O(x”+t).
It follows at once from Eqs. (Q), (ll), p
= p&1’ w&s
(16)
and (16) that:
_ fm+,[ ~~~-2~)/(~~~z~~*~-~s +
ja$;-2”+“/@W)2
_ 2ox’, ~~:-2~‘/(~~)“t82r--Zs+l]
(17)
so that = -~~~~~,~-z~)/(u~~)~, b127-z81 %B b
(2r-zstl) m,*
_ -
-fnL+,ta~:-2"t"/(~~1)2
(1% -
2~~~~~~~-~s)/(~~)3~.
A comparison of Eqs. (3a) and (11) shows, however, that
(19
NEARLY
S~M~TRI~
ROTORS
07
Corresponding to Eqs. (18) and (19) there are the further results (22) and (23) obtained in a comparable manner from Eqs. (lo), ( 12)) and (16), while a comparison of Eqs. (4a) and (12) leads to Eqs. (21) and (25). d(zr--28) rn.8 = -fm&+l cC,-z6J/(c;:)2, ,j$,;-‘“+” = -fm-,*(e~~-2x+‘)/(c~~>B - 2c~:c~~:-2”‘/(c~~)3~, (21--“a> CZF-2S) - d(2r-2*-4’ Cm,s = --a, m,sil I ~‘?r--26+lf (a--2s+11_ dg:;;s-l I = --a, cm,,
(22) (23) @4) (23
The iternt.ive cycle is completed by the calculation of ~2~’ and ~2”~~’ using Eqs. (Xi) which are direct consequences of Eqs. (21, (8), (9 ), and (10). (2r1 = +b$-“’ %7&
+ dg;-“‘,
(27fll = +bz,;-1’ a,
+ dg;-‘).
(26)
The expressions quoted by King et al. (9) for the coefficients &’ to a:’ can be derived rapidly by use of the above formulas. A corresponding analysis can be made for t,he other types of representation leading to the same recurrence fnrm&as, but to different expressions for the k:, k,:, , and fW factors. A cletailed error analysis on the above method has not been carried out. However, from t,he results me~~t,ionedbelow, it is apparent that the magnitude of the sub-diagonal elements plays an inlportant part in determining the accuracy of the results. Thus, for the first three or four eigenvalues of a given determinant, t!he region of usefulness is limited to t,he range of 0 5 6 5 0.025 (K 5 -0.95). However, for subsequent eigenvalues it is possible t’o use them for 0 5 6 5 0.05 (K 2 -0.9), and in many cases a good result is obtained for 6 I 0.1 (K = -0.8 ). In any application of this method, however, care must be taken in sel&ing the “range of usefulness” of the results. Using the above method of calculation, a program was written for the IRM A.30 digital computer. This program is general and allows one to go to a,ny desired degree of approximat,io~l for J 5 100. The coefficients a$’ of Eq. (5) have been calculat.ed up through i = 7, i.e., t,he seventh-order approximati(~n. Tables of t,hese eoeficients will not, he published because of their volume, but will he av:tiIahle on request. , ACKNOWLEUGMENT We wish t.o thank generosity ait,h time in programming and Thanks are also due RWEIVED:
International Business Machines, United Kingdom, Ltd. for their on the computer. Especially, we would like to acknowledge the help preparation of results given by Wss D. A. Emery and Mrs. J. Wieher. to Professor D. P. Craig for many discussions aiding this work.
JXVUARY 19, 1959
98
WALT
AND
BARNETT
REFERENCES 1. M. R. HAINER,P. C. CROSS,ANDG. W. KING, J. Chem. Phys. 17, 826 (1949). 2. G. W. KING, R. M. HAINER,AND P. C. CROSS,J. Chem. Phys. 11, 27 (1943). 9. R. H. SCHWENDEMAN, “A Table of Coefficients for the Energy Levels of a Near Symmetric Top”, Research Report, Department of Chemistry, Harvard University, 1957. 4. 6. R. POLO, Can. J. Phys. 36, 880 (1957).