17

Click here to load reader

The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

  • Upload
    paul-c

  • View
    217

  • Download
    1

Embed Size (px)

Citation preview

Page 1: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

The Asymmetric Rotor I. Calculation and Symmetry Classification ofEnergy LevelsGilbert W. King, R. M. Hainer, and Paul C. Cross Citation: J. Chem. Phys. 11, 27 (1943); doi: 10.1063/1.1723778 View online: http://dx.doi.org/10.1063/1.1723778 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v11/i1 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 2: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

THE ASYMMETRIC ROTOR 27

liquid is added, the gel adsorbs considerable amoun ts of ammonia which cannot be removed either by rinsing with water, or by heating in vacuum to iS0ac.

However, a great part of this ammonia can be removed by heating the gel (before the dye is added) in an open porcelain container for several hours at SOO°C. The gas rising from the surface of the gel contains enough ammonia to color an indicator paper; the ammonia can also be de­tected by its smell. Excessive heating of the gel changes its structure; hence gel treated as just described was used in only a few of the experi­ments here reported. Ammonia contained in a gel prepared in the usual way can (like oxygen) be partially dislodged by water vapor. If the apparatus described above is used to demon­strate the evolution of oxygen, the bulk of the ammonia set free by water vapor is caught together with water in the second air trap. If the oxygen is pumped off and the liquid-air bath

then removed from this trap for a short time, some of the ammonia together with a small amount of the water evaporates. When this gas is compressed into a small volume in the capillary of the McLeod gauge, pressures which much surpass the vapor pressure of water at room temperature may be measured. An electric discharge through this gas causes an expansion, due to the decomposition of ammonia into nitrogen and hydrogen.

Ammonia is more difficult than oxygen to remove from the gel. It therefore seems justified to assume that oxygen can be dislodged from the gel by ammonia in a manner similar to that in which it is liberated by water. We therefore conclude that the quenching action of ammonia observed by Kautsky like the apparent quench­ing by water vapor was probably due to oxygen. Since we were mainly interested in the influence of oxygen or water we performed no direct ex­periments to test this hypothesis.

JANUARY. 1943 JOURNAL OF CHEMICAL PHYSICS VOLUME 11

The Asymmetric Rotor

I. Calculation and Symmetry Classification of Energy Levels

GILBERT W. KING, Arthur D. Little, Inc., Cambridge, Massachusetts, R. M. HAINER, AND PAUL C. CROSS, Brown University, Providence, Rhode Island

(Received September 30, 1942)

A table of energy level patterns for rigid asymmetric rotors is given, by means of which this approximation to the rotational energies of all molecules up to J = 10 may be readily evaluated. The symmetry classification of each level is determined and expressed in terms of the K values of the limiting prolate- and oblate-symmetric rotors. A simple method is developed for calcu­lating the transformation which diagonalizes the energy matrix and is applied to the derivation of perturbation formulas.

I. INTRODUCTION

T HE fundamental theory of the asymmetric rotor has been discussed many times, and

the equations defining the energy levels derived by various methods.1- s The selection rules have

1 E. E. Witmer, Proc. Nat. Acad. Sci. 13, 60 (1921). 2 S. C. Wang, Phys. Rev. 34, 243 (1929). 3 H. A. Kramers, and G. P. Ittmann, Zeits. f. Physik

53,553 (1929); 58,217 (1929); 60, 663 (1930). 4 O. Klein, Zeits. f. Physik 58, 730 (1929). 6 H. B. G. Casimir, Rotation of a Rigid Body in Quantum

Mechanics 0. B. Wolter's, The Hague, 1931). 6 H. H. Nielsen, Phys. Rev. 38, 1432 (1931).

been satisfactorily stated, but no complete classification of the symmetry properties has been given to date. Part of the discussion to follow is accordingly devoted to the characteriza­tion of the levels by the representations of the Four-group to which their wave functions belong. The notation will be consistent with that recently proposed by Mulliken. 9 The remainder of this

7 D. M. Dennison, Rev. Mod. Phys. 3, 280 (1931). 8 B. S. Ray, Zeits. f. Physik 78, 74 (1932). 9 R. S .. Mulliken, Phys. Rev. 59, 873 (1941).

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 3: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

28 KING, HAINER, AND CROSS

paper considers in some detail the methods of calculating the energy levels. Tables of E(<<) vs. « are given to J = 10, where « is the convenient parameter of asymmetry introduced by Ray.s General methods for obtaining perturbation formulas are discussed, and the numerical coefficients for the expansions about the limiting symmetric rotors (<< = ± 1) and the "most asym­metric" rotor (<<=0) are given. The determina­tion of the latter necessitated the evaluation of the transformations which diagonalize the energy matrix for «=0. These transformations will later be applied to the evaluation of the Einstein coefficients of the asymmetric rotor.

II. ENERGY AND ASYMMETRY PARAMETERS

The energy of a rigid asymmetric rotor is expressed as

E(Ia, h Ic)=t(Pa2/Ia+PNIb+P/jIc), (1)

where Pa, P b , Pc are the components of the angular momentum along the principle inertial axes a, b, c of the rotor with Ia~ Ib~ Ie as the corresponding moments of inertia. For con­venience, define

a=h,2j2I., b=h2/2h c=h2/2Ic, (2)

whence

E(a, b, c) = (aPa2+bPb2+cPc2)/h2• (3)

The matrices Pa, P b, Pc may be any set of angular momentum matrices which satisfy the Poisson bracket relations. To examine the general properties of such a set of matrices, let us take a right-handed system of Cartesian axes x, y, z fixed in the rotor with the origin at the center of mass. (The 3! ways in which Pa, P b, Pc may be identified with P x , P y , P. are discussed in Sections III and IV, where the symmetry classification of the asymmetric rotor energy levels is determined.) The commutation rules are:

P",Py-PyPx = -ihP., PyP.-P.py = -ihP"" PzP",-PxPz = -ihPy •

A solution of these matrix equations is

(Py)J.K; J,K+l= -i(Px)J,K; J,K+l

(4)

= (h/2)[J(J + 1) -K(K + 1)J!, (5)

(P.)J,K; J,K=hK, where J~K.

The representation used here is that which diagonalizes P. and p2 P x2+Py2+Pz2, and which corresponds to the wave functions chosen by Wang,2 MulIiken,9 and Van Vleck,l° The phase factor of these functions is such that P II is real and positive, and Px is imaginary, which is just the reverse of the choice made by Klein4

and later by Dennison 7 in their matrix algebra treatment.

It follows from (5) that the squares of the angular momenta, which appear in (3), are

(Pi)J,K; J,K= (P:/)J,K; J,K

= (h2/4)[{J(J + 1) -K(K+l)}

x (J(J+l)-(K+l)(K+2)}]t, (6)

(P Z2)J,K; J,K=h2K2,

whence

(7)

The calculation of the energy levels is greatly facilitated by the change of variables proposed by Ray.s This is given here in some detail since his paper contains several misprints. Let (f and p be scalar factors. Then

E(ua+p, O'b+p, uc+p) = [(ua+p)Pa2+ (Ub+p)Pb2+ (uc+p)Pc2J/h2

= [u(aPa2+bP1?+cPc2)+p(P,,2+ P b2+P c2) J/h2,

which by (3) and (7) reduces to

=uE(a, b, c)+pJ(J+l). (8)

10]. H. Van Vleck, Phys. Rev. 33, 467 (1929). We may, therefore, make use of the symmetry determinations of Mulliken in our work. The actual phases of the symmetric rotor wave functions do not affect the energy, but must be clearly defined when symmetry is concerned, The confusion resulting from the many arbitrary choices of signs, phases, and "Eulerian" angles in the different quantum mechanical formulations has been cleared up by Van Vleck, especially in footnotes 9,20,21,25, and 29. The phase to be determined here has only to be con­sistent with that of previous work which we wish to use.

From the rigid phase relation between P", and Py given by (5), we see that P z has the same real, imaginary, or complex character as P,,+iPy , This quantity is very simply expressed in the Eulerian angles used by Mulliken

Px+iPy = lie-i<P[csC o(alax) -cot o(ala</» -i(alaO)]. (Sa)

Applying this operator to the symmetric rotor wave functions used by Mulliken and first clearly defined by Van Vleck, we find P,,+iPy, and hence p", to be imaginary. The complex cor.jugate must be taken on the first factor.

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 4: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

THE ASYMMETRIC ROTOR 29

10 1.0

0.9 0.8

08 0.6

0.7 Oil

0.6 0.2

60.5 0.0 K 0.4 -0.2

0.3 -0.4

02 -0.$

01 -o.S

0.0 -10 0 30 60 90 120 ISO 180

29

FIG.!' Variation of Ii and of " with the valence angle of symmetrical triatomic molecules. The cases illustrated are m=M(N-!) and m«M(N .... l).

Ray now chooses

u=2/(a-c), p= - (a+c)/(a-c),

(9)

so that ua+p= 1, ub+p= (2b-a-c)/(a-c),· (10) O'c+p= -1.

He then defines the parameter of asymmetry K as

K= (2b-a-c)/(a-c), (11)

so that -1:::;; K~ 1. This parameter K results naturally from Ray's choice of diagonalizing Pb, the angular momentum about the intermediate axis of inertia. The chief value of using K is apparent from the relation, proved by Ray.s

E/(K) = -E_/(-K), (12)

which gives the energies for positive K from the energies for negative K. The limit K -1 or b = c is Mulliken's prolate-symmetric rotor, while the limit K= +1 or b=a is his oblate-symmetric rotor.

Substituting (9), (10). (11), into (8) we obtain

E(l, K, -1)=E(K)=[2/(a-c)]E(a, b, c) - [(a+c)/(a-c)]J(J+ 1),

which on rearrangement yields for the energy of any asymmetric rotor

2E(a, b, c)=(a-c)E(K)+(a+c)J(J+1). (13)

This relation, and that of (12), show that if the energy levels E(K) for values of K between -1 and 0 are determined, once and for all, the energy levels of any asymmetric rotor can be calculated by simple multiplication and addition.

It is often more convenient to employ a parameter of asymmetry 0, defined as

0= (K+1)/2= (b-c)/(a-c). (14)

It follows that O~ 0:::;; 1 and K=20-1.

Rigid Planar Molecule's

For rigid planar molecules 0 has a very simple form. The axes of Ia and Ib must be in the plane of the molecule and the moments of inertia must satisfy the relation

Ic=Ia+lb •

Combination of (2), (14). and (15) gives

0= Ia2/ Ib2 = b2/a2•

Rigid Symmetrical Triatomic Molecules

(15)

(16)

For a symmetrical triatomic molecule, 0 is easily expressed in terms of the structural parameters of the molecule (assuming a model of point masses in a rigid configuration)_ Let the central mass be lv1, each attached mass m, the valence angle 28, and the bond distance r. Also introduce N=M/(M+2m). Then

I,,=2mNr2 cos2 8, I..=2mr2 sin2 0, (17) IJ. = 2mr2(N cos2 0+sin2 0).

As 20 is varied from 0 to 11"'. 0 varies from 0 to 1 to O. The principal axes of inertia in the plane of the molecule are respectively parallel (11"') and perpendicular (0") to the symmetry axis. Further­more, Ia=Ib=Iu=I"and 0= 1 when O=tan-1 Nt. Since we require Ia~ Ib~ Ie. I,,=L if 0>tan-1 N\ and 1,.=1 .. if 8 <tan-1 Nt. The angles cor­responding to the most asymmetric rotors, o=!, are 0= cot-l (2N2)-t > tan-1 Ni and O=tan-1 (N2/2)I<tan-1 Ni. The variation of 0 with 0 is illustrated in Fig. 1. From the above identification of the least moment of inertia with I ~ or I .. one obtains from (16) and (17)

0=N2 cot4 0 for 0>tan-1 Nt, 0=N-2 tan4 0 for o <tan-1 Ni.

(18)

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 5: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

'i(1 2 3 4

o l' 3' 6' 10' 15' 420 168 108 45

6

21' 840 360 270 165 66

7

28'

8

36' 2520 1155 990 780 546 315 120

9

45' 3960 1848 1638 1365 1050 720

10

56' 6940 2808 2548 2205 1800 1360 918 513 190

11

66' 8580 4095 3780 3360 2856 2295 1710 1140

J 12 80 180 1512 675 550 396 234 91

2 15 63 3 28 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

t""

~:: Il>/{l '" <'ll :::.-

.. 00 ~ '-<",

(j~ 0"1l>

'" 9 9", ~.fii' 0" .... '<0" l"<'ll

.CO Sl ~~ "'0 ~-.

~Z S?

o ~ C1 Jil Il> a

Values of

408 153

630 231

I(J, n)=i [J(J+l)-n(n+l)] [J(J+1)-n(n-l)] =t(J-n)(J -n+l)(J+n)(J+n+l).

The factor 2 which always occurs with I(J, 1) in the factored secular equations has been included in calculating the entries in the table. These values are therefore set apart by the use of bold face type.

~ .... ntllo..n~o.. <: • ::r- 0 aq' ..... "''' ('1) '" ~('1)"'::l::+l::l''O..oj_ .... @ ~0"~('1)::r-': t:l -. til 0 ('1) "" ::: (l) tl5 (1q'O ...... 0 ..... ",::r-o ::l~~ <0 X '"" ::l .... ..':ri" ... " 0 '" ...... ('1) • o..::r- '-. '" _ ........ til '" ('1) ..... ::l ~ ::l .: ,.-., '::l'"" ::l(1q;:;' (l) ..... '" 0.. 0 ::l '"" (l) ::r- '"" til • ::j .... o::r-o..~::r-o~ ro -:::.r-::S (tI _. \J (b ........, ~ '" ('1) (? A til ::l ~ III o..;:+, ..... _...... ..... '<! • til iii' (1q 0 ~ '0 • ~J @

::r- -. (1q::l "" '" o..",X ,"","1 (1q (\) ::l '0 g a, ~ ~ ct • .0 C;:' ::t>0..0'" 0" n '" ::l(\)fjl-:::t;i16' S~~ (\) 0. ..... ('1) III '"" ..oj '" '"'-i_. 0. "" 0" -::l til ::r- "1 _.::l Ill· ~l:!i0.0(\) ::l~, en p ~ tfJ 1*"'0 ~

<:t' til (\) :::s-.... ><' 0. '"" III :1.' ..§::l(l)::r-"1~~0" ~('\(l)[email protected]~ .... III ..::l (\) (l) ..: ::t> til ..... • X "" ..oj 0 (\) - n • go tl5 ~ ::r-::l III ct. ~ g, ::l ;:;. (\) - n ::l _. O"O .... ·tIl'<!::r-(1qO'" ('1) ::l 0 (\) -. ~ ...... ::l 0. ::l ::l::r- 0 til 0 ...... til 116 '"" _.;::; "I ...... ::r- 0 '"" ::r- n ;;> 0 ....... (l)-.\Jl('1)::r-en-.o.

........ ~ ~ ......... II ~ ;-, 1 ~ ......... II X ... 1.-

~~ ~~ ++ .......... '-" ......... I I ~ ~ ,-...,--.. ~ ~

1+ .......... '-" '-" L;--J L..J

---­N .... ...,..,

12

782 12012

5775 5400 4896 4284 3591 2850 2100 1386 759 276

~ :::s­(\)

@ .: 116

iii' :3 ill 0. ('1)

o ...... ,.... ::r­('1)

5-0" @ < Pi' ..... o· ::l

13

91' 16380

7920 7480 6885 6156 5320 4410 3465 2530 1656

900 325

t:t'J >1

t II

t:t'J ~ '" >1 II :::t;

S ...... ~ + .... ......... L..J . ...

r-.. N

.$

14

105' 21840 10608 10098 9405 8550 7560 6468 li313 4140 3000 1950 1053 378

II

~ ~ + .... + ~ I ~

~ r-.. .... '0 .........

15

120' 2S560 13923 13338 12540 11550 10395 9108 7728 6300 4875 3510 2268 1218 435

TABLE 1.

16

136' 36120 17955 17290 16380 15246 13915 12420 10800 nco 7371 5670 4060 2610 1395 496

17

153' 4£512 22800 22050 21021 19734 18216 16500 14625 12636 10584 8526 6525 4650 2976 ]584 561

0.-'13~

18

In' 58140 28560 27720 26565 25116 23400 21450 19305 17010 14616 12180 9765 7440 5280 3366 1785 630

t:t'J >1

_. @:: '" ~ ..oj Jcio::t~:::s-o ::: .... ' IJ (l)

:>i 1/ ~ r-I ..... ~ + ..... ......... I ~ l-l

+ ~

t:3 ~. ~ ~

e:.~~~::: _.' :::-r ~ ::l(1q~(1q~ .....·m ('1) :;:;.~ • ::P ('1) ('1) til

("""'!- "'1 - rn ::r- ill ('1) ..... 0 ('1) - S X ..... "",,"- ('t> '""t _. ::l ...... 0 ::l ('1) 0. 0. ... fij '0 ('1) ('1) ~. @ ::l X ;:) ~ rn t:t. ...... Q ~ -<' '. U'l .......... l ..... ill ill ::l -<::l """(1q _. n (1q. ""0' 00 ('1) < ~ 0. ('1) ~ " 3g:.::l o • 'O('1) ........... ~ '0 ...... ;; (\) (l) ill :::s­o.::l .., ('1) ~ • ('1) (\) ('1) "

~ a, ~ ~ '<! ..... "1 _. ::r- (1q .....

iii' (J) '<! :::s-

19 20

190' 2162 71820 87780 35343 43263 34408 42228 33120 40800 31500 39000 29575 36855 27378 34398 24948 31668 22330 28710 19575 25575 16740 22320 13888 19008 11088 15708 8415 12495 5950 9450 3780 6660 1998 4218 703 2223

~ ~~. ('1) I»

ail ::s '<~

a:~ I» ~ ~;;> ..... ('1)

~ a. (/.I (/.I

S-

f

780

... ~

I gj 2:

~ 2: (Jl

o "'l

~

~ t:>.l ~ ....."

21

231' 106260 52440 51300 49725 47736 45360 42630 39585 36270 32736 29040 25245 21420 17640 13986 10545 7410 4680 2460 861

22

2532 127512 63000 61750 60021 57834 55216 52200 48825 45136 41184 37026 32725 28350 23976 19684 15561 11700 8200 5165 2709 946

23

2762 151800 75075 73710 71820 69426 66555 63240 59520 55440 51051 46410 41580 36630 31635 26676 21840 17220 12915

9030 5676 2970 1035

24

3Q02 179400 88803 87318 85260 82650 79515 75888 71808 67320 62475 57330 51948 46398 40755 35100 29520 24108 18963 14190 9900 6210 3243 1128

25

325' 210600 104328 102718 100i85 97650 94240 90288 85833 80920 75600 69930 63973 57798 51480 45100 38745 32508 26488 20790 15525 10810 6768 3528 1225

26

351' 245780 121800 120060 117645 114576 110880 106590 101745 96390 90576 84360 77805 70980 63960 56826 49665 42570 35640 28980 22701 16920 U760 7350 3825 1326

27

378' 285012 141375 139500 136396 133584 129591 124950 119700 113886 107559 100776 93600 86100 78351 70434 62436 54450 46575 38916 31584 24696 18375 12750 7956 4134 1431

28

406' 328860 163215 161200 158400 154836 150535 145530 139860 133570 126711 119340 111520 103320 94815 86086 77220 68310 59455 50760 42336 34300 26775 19890 13780 8586 4455 1540

29

435' 377580 187488 185328 182325 178500 173880 168498 162393 155610 148200 140220 131733 122808 113520 103950 94185 84318 74448 64680 55125 45900 37128 28938 21465 14850 9240 4788 1653

J/ ~n 4652 0

431520 1 214368 2 212058 3 208845 4 204750 5 199800 6 194028 7 187473 8 180180 9 172200 10 163590 11 154413 12 144738 13 134640 14 124200 15 113505 16 102648 17 91728 18 80850 19 70125 20 59670 21 49608 22 40068 23 31185 24 23100 25 15960 26 9918 27 5133 28 1770 29

.§ ~ ~ ::;-: ~ tl5 g:. ~ 0' ~ iii' g:. ~ g ~ a, g:. Z ,o .... ('1)13-i16(\) ":nn":rj(l)I"'I' '0 ('1)0 ::;-:.:"0 ;:;'~::l 0.:+ (l) 0 0 S ~ rl- ('1) ..... '" @" El pj g S· n ct. ~;;; ::r- rl- ~ '"I 0 - ~ 5. _. ~ rl-rl- - 0. (1q (\) e:. n ..... '0 0 ..... S ;"l ('1) ('1)';"l::r-..... '0 ('1) _Ill -rl-~o 0. 0 ;::J ::l ..... ::l rc;::J '"

Q '" '"I -'::l '<! (l)"" ":rj (\) til m (\) ::l rl- - S rl-.... "1 '" ::l (1q rl- til <; aq' pj ..... ;::. ... @" no ('1) ...... o"'ggJ_tIl::r- ~. O",...,.tIl(1q ... §::r- ..... o ..... 3 (\) ..,. i}l '<! '" ::t' - ::l. 0 '<! '" - (\) ~ ... '0 ('1) J, S rl- 0 S· ..... '<! '" ..... _ 0 '<! til (\) no 0 .... '" ~::) no _. ('1);"l ('1) '0 '<! .., ('1) ...... "'.., ;::J 3 "'" ____ til (1q 0::l < p 0::l '" ;"l ..... ... 0. 3 (\) ('1) .... ::r- @ 3 ('1) ('1) ..... ::l (\) .., ;::J

.: 0 3 ::t ~ N 00 0 '" _.... iii' n ..... "1 '" 3 a- ...... ~ ('1) S ri· ..... S' ~ ~ rl- n ct. o.::r- ~ S (\) '" '" e:.,...,. ~ ::l c::n ~ S III t:t'J iii' ('1) ~::t g. ~ ~ g. rl 3 5'. ,I.. @" g:. 1"'1' ~ ~ ~ S' ~ ~ ~ ri' ::lStIln=O~~::l~gonO'~::l~~c::ne:. ..... 13::lill'<!"10.-0'- ::lS"""- ~::l-/""'o.rl-::r- (\)tIl tIlS til (\)::r-" n 0"1 ('1)c ~ ~ .rc 0. ~ ..... "-'..... ~ N til (\) '-" .... (\) _= "1 Pi'

... ..... w n ~ 0 rl- "1 • ill v '" rl-"''<!NN::+l~ ~ ",S'::r-~~ ::l"''-''O ':< ...... ('1)<:!:>01::l~tIlN ::l('1)O(\)O(1q~""'S

o ... II ('1) P 0l... I <... ...... - ('1) til ..... "1 0 ... 0 ill 0 (\) .... M ('1)...... ('1)... _. n - "'" ..... - n til ... ..: n ::l::l

..:SooM'::r-~n'"l.:~~O::lON .0.::1 til III 0 ..... 0 O"'-_::l"""n.:<:!:> .('1);;> (\) ".°3': SE'9 ..... • n;-t ct.(jJill '0 e.. _. ('1) -::1 C!S.. ('1) ..., ",' 0 ...,... (\) 0 (\) ::l t:t'J ('1) (\) ::ltll -p .... tIlfij:::: ...... ,....ill ::l' o.--;;'::ln .....;::. • ..., .......... "'_'<! ..... ~::l tIl ..... ::l'-"~S ::r- ::r- ::r- ::r-... 0 '" :::s- 0l ~ o:::s- 0 _.::l i}l ('1) ill ('1) ('1) ('1) ('1) "1 til ('1) • ('1) ...... (J) rt en ......

VJ o

~

z o

::r:: >-Z tTl ;;0

> Z o (j

;;0

o 1.Jl 1.Jl

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 6: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

THE ASYMMETRIC ROTOR 31

difference in sign of non-diagonal elements does not affect the roots of Hermitian matrices, but the sign of H must be retained to assign the symmetry properly. To maintain consistency in the definition of H, the positive square root of f is used in (20).

The identifications of a, b, c with x, y, z are given in the first three rows of Table II. These are labeled Ir, IIr, IIIr for the three permutations of right-handed, a, b, c axes and P, Ill, IIIl for the corresponding permutations (based upon the assignment to the unique, or z, axis) of left­handed a, b, c axes. By this notation the matrices displayed by RayS in his Eqs. (22a), (22b), (22c) and those corresponding to the Mulliken 9 prolate Ka and oblate Ke cases are to be identified as shown in Table II. All the above types give, of course, the same secular equations and energy levels. Each type has its own particular use for special ranges of K.

The expressions for P, G, H in terms of K or 0 for the r types are found in Table III. To obtain the matrices of the corresponding I types, the sign of H must be changed.

Limiting Cases

In the case of the limiting oblate spheroid, Ib-t1a, b-ta, K-t1, o-t1; and with z=c, i.e., Type III, the energy matrix becomes diagonal, H-tO. This corresponds to Mulliken's case K =Ke, the elements of the matrix E(K) =E(1) become

Substitution into (13) yields the usual expression for the energy levels of the oblate-symmetric rotor,

E(a, b, c) =E(a, a, c) =aJ(J+1)-(a-c)Ke2• (23)

For K or 0 in the neighborhood of +1, Type III

TABLE II.

x b c c a a b y c b a c b a z a a b b c c Type IT I I IIr III IIIr IIII Ray 22a 22b 22c Mulliken Ka K, Wang 1,<1, Iz>lx Dennison *

TABLE III.

Type I' II' III' I' II' IIIr F HK-I) 0 HK+!) 0-1 0 0 G I -I I 20-1 -1 H -HK+!) HK-I) -0 0-1 G-F -HK-3) -HK+3) -(0-2) 20-1 -(0+1) F+G-H K+I ...:-1 0 20 2(0 -1) 0 F+G+H 0 K+I K-l 0 20 2(0-1)

is nearly diagonal and thus is the most con­venient type to use in determining the energy levels.

For the limiting prolate spheroid, Ib-t1e, b-tc, K-t-1, O-tO. With z=a, i.e., Type I, the energy matrix is again diagonal corresponding to Mulliken's K=Ka , the elements being

(24)

E(a, b, c) =E(a, c, c) =cJ(J +1) + (a-c)Ka2• (25)

Type I, being nearly diagonal for K nearly -1, or for 0 nearly 0, is the most convenient form to use under these conditions.

Type II never becomes diagonal. It is useful for the case of maximum asymmetry, K=O, when its main diagonal elements are all zero.

IV. SYMMETRY PROPERTms

Irreducible Representations of the Asym­metric Rotor Wave Functions

The energy of a rotor is invariant under the operations of an external rotation group which has infinitely many representations characterized by the quantum numbers J and M. In the absence of external fields, one need not examine these any further than to note the resulting 2J + 1 degeneracy of the energy levels.

The wave functions of the symmetric rotor, which are used as the basis functions in the calculations of the asymmetric rotor energies, belong also to an internal rotation group Doo. This group has infinitely many representations characterized by the quantum numbers J and K. They are denoted by ~I(K =0, J even), ~2(K = 0, J odd), n(K=l, ... ), t:..(K=2, ... ), etc. On the other hand, the wave functions of the asymmetric rotor belong to the Four-group V(a, b, c), defined by the three rotation operators Cza, C2b, C2c. The character table for this group

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 7: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

32 KING, HAINER, AND CROSS

TABLE IV.

E C,' C,b C2a

A 1 1 1 Be 1 -1 -1 Bb -1 1 -1 Ba -1 -1 1

TABLE V. Correlation of the species classification of sym-metric rotor and asymmetric rotor wave functions.

Parity of Rep. of Representations of V(a, b, c)

K DooCz) K J+y Vex, y, z) I' I' II' II' III' III'

0 ~1 e e A A A A A A A 0 ~. e 0 B. Ba Ba Bb Bb Be Be 1 II 0 e By "Be Bb Ba Be Bb Ba 1 II 0 0 Bx Bb Be Be Ba Ba Bb 2 Ll e e A A A A A A A 2 Ll e 0 B, Ba Ba Bb Bb Be Be

is given in Table IV. The notation for the representations shOws directly the axis of rota­tion for which the character is + 1.

However, the symmetric rotor basis functions 12

commonly used, 1/;x(J, K, M), do not belong to this Four-group, but, as pointed out by M ulli­ken,9 the Wang2 linear combinations of them do belong to the Four-group.

S(J, K, M,'Y)=2-![V(J, K, Ai) +(-1)'Y1/;x(J, -K, M)J, (26)

where l' is odd or even, say 1 or O. For K =0, only l' even (1' = 0) exists and

S(J, 0, M, 0) =1/;X(J, 0, M). (27)

In general these new basis functions S(J, K, M, 1') have been constructed relative to arbitrary axes x, y, z and not relative to the axes a, b, c of the molecule as here defined. They are, therefore, characterized by the representations A, B .. By, Bx of the Four-group Vex, y, z). Here, too, the representations have been labeled to show directly the axis of rotation for which the character is + 1. This makes very easy the correlation of the representations of V(a, b, c) with those of Vex, y, z). A always corresponds to A, and B a, B b, Be correspond to B x, By, B z, according to the same permutation as that iden­tifying a, b, c with x, y, z. These correlations are

12 See Mulliken, reference 9, Eq. (3), and Van Vleck, reference 10, footnote 25.

given on the right side of Table V. The left side of this table gives Mulliken's identification of the representations of the Group Doo(z) (to which the S(J, K, M,'Y) also belong) with the repre­sentations of the group Vex, y, z).13

Factors of the Energy Matrix E(K) in the S(J, K, M, y) Representation

For a given J the energy matrix E(K) in any representation based upon the 1/;x(J, K, Af)'s, which may be readily obtained from (19), (20), (21) and Table III, is of the order 2J + 1 since14

- J~ K ~ J. The transformation to a representa­tion based upon the S(J, K, M, 'Y)'s enables further factoring of E(K) into four submatrices, i.e.,

. . . X'E(K)X =E++E-+O++O-, (28)

where

-1 1 -1 1

X=X'=2-! 2! (29) 1 1

1 1

is the Wang transformation,1/;x=XS. The orders from the top left corner are 1/;x(J, - J, M), 1/;x(J, -J+1, M), .. . 1/;x(J, J, M), and S(J, J, M, 1), S(J, J-1, M, 1), .. ·S(J, 1, M, 1), S(J, 0, M, 0), ... S(J, J, j1;1, 0). These sub­matrices, in terms of the original elements of E(K), may be displayed in the form

Eoo 2!Eo2 0 ..................

E+= 2!Eo2 E22 E24 (30) 0 E24 E44

E- has the same elements as E+ after removal of the first row and first column, as indicated by

13 See Mulliken, reference 9, Appendix I and Table III. The identifications are made by examining the behavior of the S(J, K, M, 'Y) under the operations C.X, C2Y , C2z of the group V(x, y, z)

C.'S= (_l)KS, C2"S= (-l)J+'YS, C2xS= (-l)K+J+'YS.

14 It may, however, be displayed as two submatrices whose indices involve, respectively, only even and only odd K's.

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 8: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

THE ASYMMETRIC ROTOR 33

the dotted lines, and

(Ell±E_ll)

(31)

Use has been made of the relations

and (32)

EK, K+2 = EK+2, K = E_K, -K-2 = E-K- 2, -K. (33)

These submatrices are identified by the sym­bols,5 E+, E-, 0+, 0- in which E and 0 refer to the even- or oddness of the K values in the matrix elements and + and - to the even- or oddness of "I. Thus, these four species of sub­matrices together with the six original repre­sentations of E(K) give twenty-four different submatrices

Iro-110-

III 10-

which belong to the four symmetry species. Each submatrix of one type, e.g., IrE+, corresponds to the same set of energy levels as some sub­matrix of each of the other types. This correlation is accomplished by determining the symmetry classification of each submatrix.

Symmetry of the Submatrices

The Wang functions S(J, K, M, "I) which occur in any submatrix must all belong to the same irreducible representation of the Four­group Vex, y, z). These representations may therefore be used to classify the submatrices. From the submatrix designation we know the parity of K and J +"1 for a given J. Thus by reference to Table V one can construct Table VI. The corresponding classification of E+, E-, 0+, 0- for the Types Ir, ... IIIl under the group V(a, b, c) follows at once from Table V.

The symmetry species of any given energy level can be found from the classification of the submatrix from which the level was obtained. To distinguish one level from the other levels of a given J it has been found very convenient to label it by the values of K to which it corresponds in the limiting cases K = -1, prolate-symmetric

rotor, and K = + 1, oblate-symmetric rotor. This notation has the added attraction that the sym­metry classification under V(a, b, c) may be found directly from the odd- or evenness of the two K's in question. That is, the four species A, Be, B b ,

Ba are indicated, respectively, by the bipartite indices ee, oe, 00, eo in which the first symbol gives the parity of K for K= -1 and the second for K= +l,l5

The classification of the submatrices to the symmetry species A, Be, B b , Ba in terms of the parities of the limiting K_l' K 1 symmetric rotor cases is gi ven in Table VII.

The prevalent method of labeling energy levels is by J" where J specifies the J-set and T

takes on the 2J + 1 values - J ~ T ~ J; the lowest energy level being J -J and T increasing with increasing energy to give the highest energy level as J J. This is satisfactory since there are no crossings of energy levels in the entire range -1 ~ K~ 1, but it gives no indication of sym­metry. The proposed method of labeling by two subscripts; the first K_l! being16 0,1,1,2,2··· from lowest to highest energy levels and the second, K 1, being 0,1,1,2,2··· from highest to lowest energy levels, gives not only the symmetry through the parity of the indices but also T, or the rank, through the relation

(34)

The submatrix which contains any designated level in any type of representation, Ir ... IIIl,

TABLE VI. Symmetry classification of the submatrices in V(x, y, z). (Note that as J-J±l, the ± matrices interchange symmetry.)

Species J+l' Representation

Submatrix K 'Y J even J odd J even J odd

E+ e e e 0 A B. E- e 0 0 e B, A 0+ 0 e e 0 By Bx 0- 0 0 0 e Bx By

15 That the symmetry classification in V(a, b, c) is given uniquely in terms of the parity of K_l, KI follows from the assignments of a, b, c to x, y, z for these cases (z = a, and z=c, respectively). From Tables VI and V, K_l even requires the symmetry A or B.-A or Ba, K_I odd, Bx or By-Bb or Be. On the other hand, Kl even requires A or B.-A or Be, KI odd, Bx or By-Ba or B b. Hence it must follow that A=ee, Be=oe, Bb=OO, and Ba=eo. It is immaterial whether right- or left-handed Types I and III are chosen for the prolate and oblate representations.

16 We are using K_l and Kl as the absolute magnitude of the corresponding limiting K values, as is customary in the labeling of symmetric rotor energy levels.

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 9: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

34 KING, HAINER. AND CROSS

TABLE VII. Symmetry classification of the submatrices in Yea, b, c) by the parity of K_\, K\. ee=A, oe=Bc• oo=Bb• eo=B •.

[I [[I II[1

Sub- J J J J J J J J J J J J mat. even odd even odd even odd even odd even odd even odd

E+ ee eo ee eo ee 00 ee 00 et oe ee oe E- eo ee eo ee ao ee 00 ee oe ee ae ee 0+ oe 00 00 oe eo De oe eo {)O eo eo 00

0- 00 oe oe 00 oe eo eo De eo 00 00 eo

TABLE VIII. Species classification.

Grou p theory KK Denniso;' Mulliken (K_,K,) Mecke Ray J+y

A ee ++ (ABC) abc e Be oe +- (AB) c 0

Bb 00 (A C) b e Ba eo -+ (BC) a 0

is readily determined from the parity of the subscripts and Table VII. All the levels from a given submatrix of a given type of representation may be listed by writing the first subscripts as a descending series of numbers having the parity of the first index (from Table VII), J~ K?; 0, then writing the second subscripts as an ascend­ing series having the parity of the second index (O~ K~ J), with the exception that the zero must be omitted from an even series unless it is paired with K-l (or K 1) J. See also Table XI.

To illustrate the foregoing, the level 743 has the symmetry eo or Ba; 7=4-3=1, so this level is seventh-high for J = 7. If, for example, a IIIr representation is used, Table VII shows that the level is in the 0+ submatrix. The levels from the E- submatrix of J = 7. Type III', which has the symmetry ee, are 762 , 744 , 726 • The zero subscript does not occur here, since ee cannot contain the maximum K for J odd.

The correlation of the numerous notations for specifying the symmetry species is given in Table VIII. The parity of J +'Y is included. Note that it is the parity of the sum, K_1+K1•

Sub matrix Symmetry by Order and Trace Equivalence

The identification of the symmetries of the submatrices in the KK notation, as displayed in Table VII, may also be obtained from the following considerations.

One of the four submatrices fora.givenJ.has an order different from that of the other three.

as shown in Table IX. For all types of repre­sentation, this submatrix of unique order must obviously belong to the same symmetry class and be either E+ (J even) or E- (J odd). Hence, in the KK notation it must belong to the symmetry class ee.

For a given J all the submatrices of Types Ir .. . II II, having a given symmetry, must have the same roots, and hence, the same trace. This fact may be employed to complete the identifi­cation of the symmetries. The traces of the submatrices have the values shown in Table X. The explicit expressions for the different Types Ir ... IIP are readily obtained by combination with Table III.

One sub matrix has the same trace as the submatrix of symmetry ee for all values of K.

This follows from the vanishing of (F+G+ll), F, and (F+G-H) for Types I, II, and III, respectively. For Types I and III, this submatrix is always o± and hence its KK symmetry is 00.

In the limits K=±l, three of the submatrices have the same trace and one has a unique trace. The unique trace for K = -1 is E± for Type I representation and o± for Type III, and hence the submatrix has the symmetry eo.

The remaining sub matrix must have sym­metry oe. This can be determined directly by noting that the unique trace at K= + 1 is o± for Type I and E± for Type III.

V. ENERGY CALCULATIONS

Energy Equations

The energy levels of the rigid rotor are the roots of the characteristic equations of the four

Submatri"

Suhmatri"

E+ E-0+ 0-

TABLE IX. Order of submatrices.

J even

!(J+2) V V V

J odd

!(J+1) HJ-l) HJ+l) !(J+l)

TABLE X. Traces of submatrices.

J even Jodd

Kr Kr'+3F Kr-3F Kr' Kr-!3(F+G-H) Kr'+!3(F+G+H) Kr-!3(F+G+H) Kr'+!3(F+G-H)

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 10: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

THE ASYMMETRIC ROTOR 35

submatrices of any Type Ir ... lIP, chosen for

convenience. By inspection of (30) and (31),

it is seen that these submatrices are of the

Jacobian form

FS

E±; D= 2JI2f(J,1) o

(35)

1

4(G-F)+FS JI2f(J,3)

A more convenient form is

(36)

where L is the transformation given by (51). The values of the k/s and b;'s for each sub­matrix may be obtained from the proper forms of (37) and (38).

o

1 16(G-F)+FS

(37)

(G-F)+FS±HJ!(J,O) H2f(J,2)

o

1 9(G-F)+FS

JI2f(J,4)

o 1

25(G-F)+FS (38)

where S=J(J+1). Values of f(J, n) and F, G, and H are found in Tables I and III, respectively.

Continued Fraction Form of the Energy Equations

The solution of a secular determinant ID-XII = 0 is greatly facilitated by the fact that the determinant is a continuant equivalent to the continued fraction

--------=0. (39) (kl-X) - [b 2/(k 2-A) - ... ]

By the use of (37), (38), Table I and Table III, the continued fraction form of any submatrix in terms of J and K may be written by inspection.

In general, the secular equations must be solved by approximation methods. It is impor­tant, therefore, to choose the Type I, II, or III representation which gives the most rapid con­vergence of the successive approximations, as discussed in Section III. Furthermore, it is advantageous to approximate the mth root of a given equation by a form of the continued fraction which has km as the leading term, as

given by (40) and as suggested by the method of Crawford and CrossY

(40)

The relation ~Xm = ~km may be used to obtain the root for which the convergence is poorest or to check the numerical accuracy of the solutions.

The T value of the level Am can be determined from the K-l and Kl values. In Type Ir, the representation used for most of the calculations here reported, the K-l value is that of the K which enters as K2 in the main diagonal element km determined from (37) or (38). The values of Kl and of T for K_l=n are given in Table XI.

Energy Table

By the use of the above outlined procedure, the table of characteristic roots E(K) given in Appendix I has been computed. From this table

17 B. L. Crawford and P. C. Cross, ]. Chern. Phys. 5, 621 (1937).

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 11: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

36 KING, HAINER, AND CROSS

TABLE XI.

Representation

Submatrix J even J odd K,

IT E+ A ee Ba eo J-n 2n-J IT E- B. eo A ee J-n+l 2n-J-l IT 0+ Be oe Bb 00 J-n+l 2n-J-l IT 0- Bb 00 Be oe J-n 2n-J

In the first row. n =0.2 . ... J (or J -1); in the second row. n =2.4. ... J (or J -1); in the last two rows. n =1.3 . ... J -1 (or J).

the energy levels of rigid asymmetric rotors up to J = 10 may be readily evaluated by (13) for I(

values from -1 to ° by intervals of 0.1. Values of E(I() for I( values from ° to 1 with similar intervals are readily obtained from the relation given in (12). For other values of I( a fairly accurate estimation of the energies may be obtained by interpolation. The calculations were carried to five decimal places in order to permit the use of the rigid approximation to a molecular rotor as the first approximation in a more com­plete treatment of its dynamical properties.

Although in general E(K) cannot be given in explicit form, roots which are derived from linear or quadratic factors may be expressed explicitly and are given in Table XII. .

Polynomial Form of the Energy Equations

The usual polynomial form of a characteristic equation is obtained most easily by expanding the determinant I D - AI I = ° in terms of the first principal minors Pi, of order i+ 1, by means of the recursion formulas17a

PO=kO-A, Pl=(k 1 -A)Po-b1 ,

Pi = (k i - A)Pi-l - biPi-2'

If the order of D is n+ 1,

( -l)nHPn=O

(41)

(42)

is the usual form of the characteristic equation expressed as a polynomial in A, with A nH the leading term:

A-ko=O, A2_A(kl+ko)+klko-bl = 0, }" 3 - },,2(k2+kl +ko) + A(k2kl +k2ko+klko

-b2-bl)-k2klko+k2bl+kob2=0, etc. (43) ----

17a J. J. Sylvester, Phil. Mag. [4J 5, 446 (1853).

With the aid of (36) and (37) or (38) the chat­acteristic equations of any submatrix may be systematically develope':!. Type II representa­tions, having the simplest forms of ki and bi ,

give these equations most readily. The true energies of a given rotor are then determined by the substitution of the roots AT (or E(K)) of these equations into (13).

The numerical coefficients of these character­istic equations in the parameter K can be obtained from those given by Nielsenl8 by the substitution of A/ K for his Wand of 1/ K for his b.

VI. DIAGONALIZATION OF THE ENERGY MATRICES

To determine transition intensities, or to apply perturbation theory to the energy calcu­lations over small ranges of the parameter K, it IS necessary to find a transformation T which

TABLE XII. Explicit solutions of E(K).

JK_IKI A(K) 000 0 110 K+l 111 0 lor K-l 220 2[K+ (K'+3)!] 221 K+3 211 4K 212 K-3 202 2[K- (K'+3)!] 330 5K+3+2(4K'-6K+6)! 331 2[K+(K'+15)!J 321 5K-3+2(4K'+6K+6)! 32, 4K 31, 5K+3 - 2(4K'-6K+6)1 3 13 2[K- (K'+ 15)1] 303 5K-3 - 2(4K'+6K+6)! 440 441 5K+5+2(4K'-10K+22)1 431 10K+2(9K'+ 7)1 432 5K-5+2(4K'+ lOK+22)1 422 423 5K+5 - 2 (4K'-10K+ 22)1 413 lOK-2(9K'+7)! 414 5K-5 -2(4K'+ lOK+22)! 404

5" 10K+6(K'+3)1 52, 10K-6(K'+3)1

18 H. H. Nielsen, Phys. Rev. 38, 1432 (1931), corrected and extended through J = 11 by H. M. Randall, D. M. Dennison, Nathan Ginsburg, and Louis R. Weber, Phys. Rev. 52, 160 (1937).

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 12: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

THE ASYMMETRIC ROTOR 37

diagonalizes each energy submatrix. Since E is Hermitian, Tcan bechosenorthonormal,18asothat

T'ET= II ATII =A, (44)

where A is diagonal. Here T is not a running index, but refers to the T values occurring in the submatrix E under consideration. The asym­metric rotor wave functions, A (J, T, M, 'Y) are expressed as linear combinations of the Wang functions by

A (J, T, M, 'Y) = LK tKTS(J, K, M, ,,). (45)

The summation over K is over only those K's which occur as squares in the main diagonals of the submatrix, as determined by (37) and (38). The value of 'Y is fixed for each submatrix.

The orthonormal matrix T can be found in a fairly simple way if one first examines the diagonalization of D. There is a matrix V, diagonalizing D, i.e.,

V-IDV=A, (46)

which can be computed from the last columns of adjID-ATII. Let such a column, which is the column of V corresponding to the root An be denoted by vr. The components of Vr can be easily evaluated by substituting Ar in the re­cursion formula (41). Denoting Pi(A r) as Pin

1

T=LVN= b l -!

bl-!bz-!

VII. ENERGY CALCULATION BY PERTURBATION METHODS

1 -pOr

Plr

Perturbation theory may be applied to the calculation of the roots E(K+dK), provided the roots E(K) are known. For a given value of K, the transformation T, which diagonalizes a given submatrix, may be evaluated by the methods of Section VI. The diagonalized submatrix A (0) has

18a The term "orthonormal" as used here means that the rows and columns are orthogonal and normalized. The term "orthogonal" as used here means that the columns are orthogonal but not normalized (hence the rows are not orthogonal).

one obtains

(47)

Thus if ViT is the (i+ 1)th element of this column, then

V=IIViTII, i=Oton, (48) with

Vo r =1, Vir= (-l)ipi_l. r' (49)

The relation between V and T can be found from the relation between the original Hamil­tonian E, (35), and the continuant matrix D of (36), i.e.,

(SO)

with

1 o o L= Illiill = 0 b l - i 0 (51)

o 0 bl-!bz-!

Substitution of (SO) into (46) gives

V-IL-IELV=A. (52)

The transformation L V is orthogonal18a and can be normalized by post-multiplication by the diagonal matrix

N = II n TT II = II [Li(liiviT)2]-i II· (53)

The transformation T required in (44) is then

1 I n TT

-pOr' - POr" n-r'r' (54) Plr' Plr" nr"r"

for main diagonal elements the roots E(K), which for convenience we have called i\r (0).

Application of this transformation to a sub­matrix E for a slightly different asymmetry parameter K+dK gives

T'ET=A (O)+E'dK= IIAr(O)11 + Ile~r'lldK, (55)

where E'dK is the perturbation matrix. In general, for K~±l, none of the elements of E' is zero, and (55) no longer has the continuant form.

The second-order perturbation formulas giving approximate expressions for the perturbed ener-

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 13: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

38 KING, HAINER, AND CROSS

gies A/2) may now be written by inspection:

The above procedures have been applied in the evaluation of the second-order perturbation around the most asymmetric case, K=O, i.e., dK-+K. The coefficients of the expansion of E(K) as a power series in K are given in Appendix II.

Perturbed Symmetric Rotors

In the limits K=-1 and K=+1, diagonal energy matrices are obtained in Type I and

Type III representations, respectively, thus eliminating the preliminary evaluation of the diagonalizing transformation T. The application of perturbation theory to the calculation of approximate energies for nearly symmetric rotors is therefore greatly simplified. In addition, the use of (12) eliminates the necessity of expanding about both K= -1 and K= 1. The following derivation yields the fifth-order perturbation formulas for the perturbed symmetric rotor 8",0 or K"'-l.

The continued fraction form of the submatrix secular equations of a Type I representation in terms of the asymmetry parameter 8 is

8~jm+l i\.m = km 0+ 8km ' ---------------------­

kO +8k' -i\. -(02j /ko +8k' -i\. - ... ) m+l m+l m m+2 m+2 m+2 m

(57) kO +8k' -i\. -(82j /ko +8k' -A - ... )'

m-I m-I m m-I m-2 m-2 m

where the kO's, k"s, andf's19 for each submatrix are obtainable from (37) and (38) and Tables I and III. The first-order approximation to Am is

(58)

The third-order approximation is readily obtained by substitution of the Am(l) for i\.m on the right side of (57) and expanding the denominators as far as the linear term in 8.

(59)

Substituting Am(3) for Am on the right side of (57) and expanding the denominators to include all cubic terms in 8, one obtains for the fifth-order approximation:

~) }

(60)

19 Note that the f's used here with running subscript indices for each submatrix are not thej(J, n)'s of Table I, although they are closely related.

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 14: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

THE ASYMMETRIC ROTOR 39

where !m+2 )

kO -ko ' m+2 m

(61)

kO !m-l )

-ko ' (62)

m-2 m

(63)

Appendix III gives the numerical coefficients of ~m(3).

VIII. CONCLUSION

In the course of the approximate calculation of the energies E(dK) of the perturbed most asymmetric rotor, the matrices of the transfor­mations which diagonalize the energies of a Type II representation for K=O were evaluated. Some time in the future, we plan to apply these transformations to the evaluation of the elements of the direction-cosine matrices for the case K = O.

These in turn will enable the calculation of the Einstein coefficients which appear in the expres­sions for the intensities of rotation transitions. The evaluation of the rigid rotor approximation to the intensities for H 20, K'" - 0.436, and for H 2S, K",0.5 are partially completed.

APPENDIX I. TABLE OF

The authors wish to express their appreciation for the care with which Mr. E. Howard, Jr., Mr. J. E. Whitney, Mr. R. D. Mair, Mr. N. R. Larson, Mr. E. N. Marvell, Mr. W. Davis, Jr., and Miss E. Leoni performed certain numerical calculations here reported.

RIGID ROTOR ENERGY LEVEL PATTERNS E(K)

The asymmetry parameter K= (2b-a-c)/(a-c), where a, b, c, equal h,2/2Ia, h2/2Ib, h2/2Ie, respectively, and where the condition Ia ( h (Ie is applied in assigning the moments of inertia.

Energy level patterns for 0 ( K ( 1 may be readily obtained from the table by the use of the relation

E,J(K) = - E_,J( - K).

Rotational energy levels are given by

E,J(a, b, c) = [(a+c)/2]J(J+ 1)+ [(a-c)/2JE,J(K).

Symmetries are included in terms of the JK_l. K, notation.

~6 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 :/{,. JK ..... K, .-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

00 •• 0 0 0 0 0 0 0 0 0 0 0 Oe

h .• 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1. II" 0 0 0 0 0 0 0 0 0 0 0 I. 10,1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 1_1

22,0 2 2.10384 2.21576 2.33631 2.46606 2.60555 2.75528 2.91568 3.08712 3.26987 3.46410 2, 22,1 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 21 21" -4 -3.6 -3.2 -2.8 -2.4 -2.0 ~1.6 -1.2 -0.8 -0.4 0 20 21" -4 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3 2_1

20,2 -6 -5,70384 -5.41576 -5.13631 -4.86606 -4.60555 -4.35528 -4.11568 -3.88712 -3.66987 -3.46410 2_,

3.,. 6 6.15245 6.31027 6.47424 6.64530 682496 7.01332 7.21314 7.42586 7.65364 7.89898 3. 3 •• 1 6 6.15236 6.30949 6.47147 6.63837 6.81025 6.98717 7.16917 7.35629 7.54855 7.74597 3, 3'.1 -4 -3.58082 -3.12186 -2.62186 -2.08082 -1.5 -0.88175 -0.22917 0.45421 1.16476 1.89898 31 3,., -4 -3.6 -3.2 -2.8 -2.4 -2.0 -1.6 -1.2 -0.8 -0.4 0 3,

31,2 -10 -9.15245 -8.31027 -7.47424 -6.64530 -5.82496 -5.01332 -4.21314 -3.42586 -2.65364 -1.89898 3_1 31" -10 -9.75236 -9.50949 -9.27147 -9.03837 -8.81025 -8.58717 -8.36917 -8.15'629 -7.94855 -7.74597 3_, 30 •• -12 -11.41918 -10.87814 -10.37814 -9.91918 -9.5 -9.11825 -8.77083 -8.45421 -8.16476 -7.89898 3_.

4< •• 12 12.20299 12.41230 12.62852 12.85233 13.08461 13.32641 13.57912 13.84441 14.12449 14.42221 4. 44,1 12 12.20299 12.41227 12.62834 12.85173 13.08301 13.32279 13.57174 13.83056 14.10000 14.38083 43 43,1 -2 -1.43958 -0.85577 -0.24426 0.40000 1.08276 1.81033 2.58928 3.42587 4.32541 5.29150 4, 43" -2 -1.44022 -0.86120 -0.26350 0.35224 0.98528 1.63481 2.30000 2.97998 3.67388 4.38083 4,

42,2 -12 -11.14570 -10.18216 -9.11536 -7.95859 -6.72860 -5.44216 -4.11432 -2.75776 -1.38320 0.00000 4. 4." -12 -11.20299 -10.41227 -9.62834 -8.85173 -8.08301 -7.32279 -6.57174 -5.83056 -5.10000 -4.38083 4_, 41,3 -18 -16.56042 -15.14423 -13.75574 -12.40000 -11.08276 -9.81033 -8.58928 -7.42587 -6.32541 -5.29150 L, 41,4 -18 -17.55978 -17.13880 -16.73650 -16.35224 -1598528 -15.63481 -15.30000 -14.97998 -14.67388 -14.38083 4_.

40,. -20 -19.05729 -18.23015 -17.51316 -16.89374 -16.35601 -15.88425 -15.46480 -15.08665 -14.74129 -14.42221 4_.

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 15: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

40

""- 8 0 J K_I,K. ",,-. -1

5". 20 5,,1 20 5." 2 5.,2 2

60" 77,. 77,1 76.1 7 ..

75,2 75,3 74,3

7."

73," 73,5 72,0

72,6

71,6

7.,7 7.,7

B.,. &,. &,. 87,2

&,2 &,3 8" 85,4

8.1.4 84,5

&,' s.,. 82,6 &,7 8,.7 81,.

&,8

90.0 90,. 90,. 98 ••

!h. !h. 90, 90,.

95, .. 95,5 94,6 94,6

9a,6 90.7 9,,7 !h.8

!h,. !h._ 90,_

1010,0

Hit ••• 10. .• 10.,2

-12 -12 -22 -22

-28 -28 -30

30 30 8 8

-10 -10 -24 -24

-34 -34 -40 -40

-42

42 42 16 16

-6 -6

-24 -24

-38 -38 -48 -48

-54 -54 -56

56 56 26 26

o o

-22 -22

-40 -40 -54 -54

-64 -64 -70 -70

-72

72 72 38 38

8 8

-18 -18

-40 -40 -58 -58

-72 -72 -82 -82

-88 -88 -90

90 90 52 52

0.05 -0.9

20.25361 20.25361 2.71155 2.71153

-10.92537 -10.92796 -20.57904 -20.71153

-25.82824 -27.32565 -28.63251

30.30423 30.30423 8.86322 8.86322

-8.67533 -8.67542

-22.30159 -22.30932

-31.86770 -32.12881 -36.96163 -39.05391

-40.16120

42.35486 42.35486 17.01500 17.01500

-4.42312 -4.42313

-21.95606 -21.95636

-35.56370 -35.58293 -44.99890 -45.45864

-49.96804 -52.74881 -53.66004

56.40550 56.40550 27.16683 27.16683

1.82951 1.82951

-19.60431 -19.60432

-37.12871 -37.12961 -50.70574 -50.74775

-59.96219 -60.70539 -64.85679 -68.41477

-69.14410

72.45613 72.45613 39.31868 39.31868

10.08234 10.08234

-15.25136 -15.25136

-36.67904 -36.67907 -54.19095 -54.19330

-67.72006 -67.80326 -76.75116 -77.87402

-81.63937 -86.05614 -86.62521

90.50677 90.50677 53.47056 53.47056

0.1 -0.8

20.51481 20.51481 3.44756 3.44727

-9.79317 -9.81475

-18.92805 -19.44727

-23.72164 -26.70006 -27.51951

30.61738 30.61738 9.75433 9.75431

-7.29809 -7.29951

-20.48431 -20.54826

-29.33128 -30.31787 -34.07002 -38.20605

-38.78801

42.71997 42.71997 18.06163 18.06162

-2.78954 -2.78962

-19.81731 -19.82250

-32.90450 -33.06125 -41.38121 -43.03912

-46.22593 -51.66910 -52.06311

56.82257 . 56.82257 28.36913 28.36913

3.72124 3.72123

-17.11206 -17.11240

-34.10050 -34.11595 -47.02209 -47.35653

-55.08643 -57.62785 -60.23498 -67.10020

-67.35688

72.92518 72.92518 40.67675 40.67675

12.23284 12.23284

-12.40015 -12.40016

-33.20766 -33.20883 -50.18492 -50.17453

-62.80114 -63.44066 -70.47091 -74.10206

-76.14922 -84.50853 -84.67077

91.02780 91.02780 54.98444 54.98444

KING, HAINER, AND CROSS

0.15 -0.7

20.78427 20.78426 4.21049 4.20892

-8.58944 -8.66510

-17.09089 -18.20892

-21.69483 -26.11916 -26.61960

30.94020 30.94020 10.67572 10.67561

-5.86179 -5.86955

-18.51309 -18.73335

-26.53537 -28.57065 -31.36263 -37.44226

-37.74304

43.09619 43.09619 19.14256 19.14255

-1.09415 -1.09480

-17.56836 -17.59639

-29.95229 -30.47630 -37.47756 -40.74616

-42.1>1975 -50.72509 -50.89664

57.25221 57.25221 29.60991 29.60991

5.68052 5.68047

-14.51381 -14.51661

-30.88066 -30.96275 -42.84192 -43.91096

-49.97556 -54.76993 -56.25418 -65.98234

-66.07651

73.40825 73.40825 42.07753 42.07753

14.45731 14.45731

-9.43737 -9.43761

-29.56890 -29.57856 -45.75802 -45.96402

-57.13828 -59.06273 -64.10763 -70.67590

-71.65838 -83.22427 -83.27451

91.56430 91.56430 56.54530 56.54530

0.2 -0.6

21.06273 21.06267 5.00354 4.99818

-7.29895 -7.48387

-15.12406 -16.99818

-19.76369 -25.57880 -25.87948

31.27357 31.27357 11.63026 11.62975

-4.35735 -4.38365

-16.35698 -1688024

-23.61166 -26.88992 -28.87328 -36.74951

-36.90456

43.48153 43.48153 20.26091 20.26087

0.66943 0.66638

-15.18606 -15.27966

-26.6ta.15 -27.86342 -33.51139 -38.58120

-39.88051 -49.88749 -49.96344

57.69554 57.69554 30.89263 30.89263

7.71361 7.71330

-11.79659 -11.80954

-27.41611 -27.68384 -38.18998 -40.47535

-44.94942 -.12.12499 -52.90606 -65.00774

-65.04361

73.90658 73.90658 43.52486 43.52486

16.76247 16.76244

-6.35217 -6.35370

-25.73632 -25.78034 -40.95556 -41.60316

-50.92760 -54.76962 -58.08158 -67.56800

-68.00513 -82.11906 -82.13555

92.11765 92.11765 58.15739 58.15739

0.25 -0.5

21.35107 21.35092 5.83082 5.81665

-5.90702 -6.27602

-13.07747 -15.81665

-17.94405 -25.07490 -25.25335

31.61849 31.61847 12.62142 12.61966

-2.77237 -2.84099

-14.00000 -15.00321

-20.64988 -25.27748 -26.62142 -36.11645

-36.19624

43.88613 43.88613 21.42032 21.42013

2.50949 2.49918

-12.63901 -12.87783

-23.09517 -25.25166 -29.61371 -36.54230

-37.30015 -49.13365 -49.16761

58.15386 .18.15386 32.22133 32.22131

9.82803 9.82667

-8.94181 -8.98198

-23.61282 -24.30158 -33.21092 -37.09670

-40.18548 -49.67895 -50.06860 -64.13963

-64.15359

74.42165 74.42165 45.02321 45.02321

19.15619 19.15603

-3.13080 -3.13740

-21.66848 -21.81221 -35.63960 -37.14601

-44.48261 -50.62458 -52.60633 -64.73980

-64.92675 -81.14089 -81.14648

92.68947 92.68947 59.82560 59.82560

0.3 -0.4

21.65036 21.64990 6.69763 6.66583

-4.40221 -5.01632

-10.98945 -14.66583

-16.24815 -24.60358 -24.70818

31.97615 31.97610 13.65350 13.64861

-1.09071 -1.24202

-11.44897 -13.11511

-17.71113 -23.73408 -24.60153 -35.53350

-3.5.57431

41.30236 44.30235 22.62515 22.62446

4.43711 4.40873

-9.88997 -10.40000

-19.29241 -22.66461 -25.87353 -34.62446

-35.01706 -48.44617 -48.46165

58.62871 58.62871 33.60075 33.60066

12.03308 12.02844

-5.92261 -6.03922

-19.50903 -20.84414 -28.07787 -33.80861

-35.79447 -47.41301 -47.60027 -63.35283

-63.35829

74.95515 74.95515 46.57781 46.57780

21.64787 21.64717 0.24515 0.22283

-17.30456 -17.68086 -29.81240 -32.65039

-38.07036 -46.66453 -47.75172 -62.15024

-62.22810 -80.25693 -80.25884

93.28164 93.28164 61.55565 61.55564

0.35 -0.3

21.96188 21.96077 7.61079 7.54704

-2.77918 -3.79947 -8.88912

-13.54701

-14.68270 -24.16130 -24.22167

32.34798 32.34782 14.73201 14.72012

0.70756 0.41154

-8.73201 -11.22733

-14.81213 -22.25936 -22.80000 -34.99279

-35.01311

44.73481 44.73179 23.88066 23.87866

6.46744 6.39965

-6.90390 -7.85829

-15.35524 -20.12130 -22.35690 -32.82037

-33.01701 -47.81314 -47.81986

59.12189 59.12189 35.03654 35.03622

14.34073 14.32726

-2.70132 -2.97225

-15.00742 -17.34256 -22.94146 -30.63410

-31.82321 -45.3D659 -45.39376 -62.62987

-62.63199

75.50910 75.50910 48.19485 48.19481

24.24891 24.24648 3.80150 3.73803

-12.56784 -13.40321 -23.58911 -28.17155

-31.89729 -42.90671 -43.46093 -59.76129

-59.79288 -79.44566 -79.44631

93.89638 93.89638 63.35435 63.35434

0.4 -0.2

22.28725 22.28477 8.57900 8.46136

-1.04053 -2.53993 -6.80000

-12.46136

-13.24672 -23.74484 -23.77900

32.73569 32.73527 15.86409 15.83798

2.64470 2.11663

-5.88916 -9.35010

-12.08230 -20.85190 -21.17463 -31.48788

-31.49809

45.18543 45.18537 25.19317 25.18825

8.62113 8.47556

-3.66093 -5.26703

-11.36816 -17.63693 -19.10561 -31.12122

-31.23810 -47.22100 -47.22692

59.63558 59.63557 36.53557 36.53460

16.76690 16.73241 0.77076 0.21113

-10.18985 -13.82839 -17.92750 -27.58741

-28.25352 -43.33959 -43.37883 -61.95832

-61.95912

76.08592 76.08591 49.88181 49.88164

26.97372 26.96631 7.57555 7.41715

-7.38796 -9.00616

-17.14554 -23.75826

-26.11873 -39.35371 -39.61925 -57.51053

-57.55295 -78.69235 -78.69257

94.53636 94.53636 65.22987 65.22984

0.45 -0.1

22.62851 22.62338 9.61304 9.40961

0.80359 -1.27204 -4.74287

-11.40961

-11.93210 -23.35134 -23.37017

33.14140 33.14039 17.05926 17.00616

4.74200 3.86897

-2.95181 -7.49261

-9.46491 -19.50936 -19.69145 -34.01355

-34.01816

45.65657 45.65638 26.57205 26.55961

10.92575 10.63855

-0.16798 -2.6420!

-7.40649 -15.22320 -16.13111 -29.51757

-29.57583 -46.67173 -46.67296

60.17242 60.17238 38.10638 38.10367

19.33369 19.25327 4.54704 3.50079

-5.15070 -10.33195 -13.14274 -24.67532

-25.02598 -41.49370 -41.51068 -61.32914

-61.32943

76.68854 76.68853 51.64782 51.64726

29.84111 29.82081 11.62086 11.26493

-1.74182 -4.52481

-10.66402 -19.45078

-20.82173 -35.99791 -36.11803 -55.46141

-55.46610 -77.98656 -77.98663

95.20481 95.20481 67.19213 67.19202

0.5 5 / o • /Jr

22.98829 5, 22.97825 5. 10.72586 5, 10.39230 52

2.73757 0.00000

-2.73757 -10.39230

-10.72586 -22.97825 -22.98829

33.56782 33.56554 18.33030 18.22865

7.01437 5.66311 0.00000

-5.66311

-7.01437 -18.22855 -18.33030 -33.56554

-33.56782

46.15116 46.15066 28.02767 28.00000

13.41608 12.88860 3.53956 0.00000

-3.53957 -12.88860 -13.41608 -28.00000

-28.02167 -46.15066 -46.15116

60.73566 60.73555 39.75991 39.75292

22.07216 21.89858 8.66888 6.88121

0.00000 -6.88121 -8.66888

-21.89857

-22.07216 -39.75292 -39.75991 -60.73555

-60.73566

77.32058 77.32056 53.50433 53.50266

32.87689 32.82576 16.00923 15.27959

4.31610 0.00000

-4.31610 -15.27959

-16.00923 -32.82576 -32.87689 -53.50266

-53.50133 -77.32056 -77.32058

95.90573 95.90573 69.25347 69.25308

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 16: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

THE ASYMMETRIC ROTOR 41

"-. B 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0,45 0.5 :/.{, JK_,.K,"-.K-I -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

lOs., 18 20.33528 22.74489 25.23517 27.81341 30.48810 33.26924 36.16881 39.20139 42.38529 45.74432 10, lOs., 18 20.33527 22.74489 25.23517 27.81341 30.48808 33.26914 36.16840 39.19991 42.38053 45.73037 lOs Hit., -12 -8.89789 -5.68593 -2.35463 1.10694 4.71202 8.47676 12.42191 16.57584 20.97829 25.68964 10. Un •• -12 -8.89789 -5.68593 -2.35468 1.10677 4.71111 8.47295 12.40876 16.53640 20.87367 25.43593 10,

10. .• -38 -34.22662 -30.29839 -26.20121 -21.91700 -17.42058 -12.67461 -7.62217 -2.18038 3.75474 10.27364 10, 10..5 -38 -34.22663 -30.29846 -26.20216 -21.92298 -17.44611 -12.7.5906 -7.85527 -2.73819 2.57523 8.05291 10, 105., -60 -55.64584 -51.06976 -46.24088 -41.10481 -35.57362 -29.53245 -22.88752 -15.6452.3 -7.94325 0.00000 1(1, 105., -60 -55.64594 -51.07323 -46.26924 -41.23155 -35.97438 -30.53144 -24.95372 -19.30370 -13.64837 -8.05291 I()_.

Uk •• -78 -73.14002 -67.90347 -62.14553 -55.70142 -48.53948 -40.84180 -32.89913 -24.99790 -17.38959 -10.27364 10_. 10,.1 -78 -73.14548 -67.99406 -62.60314 -57.06078 -51.47118 -45.92680 -40.50444 -35.20085 -30.23225 -25.43593 10_. 10,.1 -92 -86.59725 -80.20810 -72.83972 -64.98059 -57.11839 -49.60411 -42.67271 -36.41928 -30.79761 -25.68964 10_, 103.8 -92 -86.74965 -81.32404 -75.96389 -70.80642 -65.92196 -61.33674 -57.04805 -53.03606 -49.27319 -45.73037 10_.

10,.8 -102 -95.36420 -87.56942 -79.96900 -73.07902 -67.07495 -61.87841 -57.28235 -53.13293 -49.31110 -45.74432 10_< 10, .• -102 -96.96993 -92.48017 -88.49517 -84.94730 -81.76026 -78.86492 -76.20507 -73.73723 -71.42832 -69.25308 1(1.., 101 •• -108 -100.32958 -94.02065 -89.11005 -85.17893 -81.84560 -78.89585 -76.21603 -73.74100 -71.42956 -69.25347 10_. 10..10 -108 -105.67708 -103.W124 -102.4.';750 -101.22019 -100.14037 -99.16041 -98.26134 -97.42648 -96.64413 -95.90573 10_.

100.10 -110 -106.11121 -104.00IH -102.48373 -101.23362 -100.14256 -99.16106 -98.26154 -97.42654 -90.64415 -95.90573 lO_lO

APPENDIX II. COEFFICIENTS OF THE SECOND-ORDER APPROXIMATION TO THE ENERGIES E('K) OF PERTURBED "MOST ASYMMETRIC" ROTORS 'K"'O

J, ,,0 J{I ,,2 Jr ,,0 "I ,,2

00 0 0 0 9.9 ±77.32058 6.48187 ± 1.69428 9'8 ±77.32056 6.48161 ± 1.69279

1±l ± 1 1 0 9.7 ±53.50433 19.06330 ± 5.24270 10 0 0 0 . 9.6 ±53.50266 19.04556 ± 5.15451

2,2 ± 3.46410 2 ± 0.57735 9.6 ±32.87689 31.31083 ±1O.19637

2.1 ± 3 1 0 9.4 ±32.82576 30.85743 ± 8.36221

20 0 4 0 9.3 ±16.00923 45.87258 ±21.28125 9.~ ±15.27959 40.95446 ± 7.86816

3'3 ± 7.89898 2.55049 ± 1.02061 9.1 ± 4.31610 62.27144 ±14.63618 3.2 ± 7.74597 2 ± 0.25820 9. 0 45.32189 0 3.1 ± 1.89898 7.44951 ± 1.02061 30 0 4 0 10±10 ±95.90573 7.18725 ± 1.86883

4.- ±14.42221 3.07691 ± 1.06343 10.9 ±95.90573 7.18721 ± 1.86846

4.3 ±14.38083 2.86800 ± 0.61053 10,,8 ±69.25347 21.15760 ± 5.71802

4.2 ± 5.29150 10 ± 3.40355 10,,7 ±69.25308 21.15295 ± 5.69155

4.1 ± 4.38083 7.13200 ± 0.61053 10.6 ±45.74432 34.56504 ±10.32836

40 0 13.84615 0 10 •• ±45.73037 34.42074 ± 9.63341 10 •• ±25.68964 48.92328 ±19.69299

5.5 ±22.98829 3.70146 ± 1.09887 10±> ±25.43593 46.78521 ±11.74885 5.4 ±22.97825 3.63637 ± 0.90921 10.2 ±10.27364 68.24765 ±30.39124 5.3 ±10.72586 11.57003 ± 4.66868 10.1 ± 8.05291 55.45391 ± 5.93853 5.2 ±10.39230 10 ± 1.73208 100 0 79.83824 0 5.1 ± 2.73757 19.72850 ± 3.56981 50 0 12.72720 0

l1'l1 ±116.49101 7.89303 ± 2.04411 6.6 ±33.56782 4.37860 ± 1.20617 11"10 ±116.49101 7.89302 ± 2.04401 6.5 ±33.56554 4.36049 ± 1.14046 11.9 ±87.00484 23.26159 ± 6.22114 6.4 ±18.33030 13.14291 ± 4.63150 11,8 ±87.00476 23.26049 ± 6.21376 6.3 ±18.22865 12.50987 ± 2.91242 11.7 ±60.63695 37.96101 ±10.86339

6'2 ± 7.01437 23.62143 ± 8.97291 11.6 ±60.63335 37.91851 ±1O.62714

6'1 ± 5.66311 18.12993 ± 1.77196 11 •• ±37.57816 52.80410 ±18.19006 60 0 29.71418 0 11%4 ±37.49896 52.01224 ±14.56648

11,.3 ±18.52846 71.40516 ±35.23697 7 .7 ±46.15116 5.07485 ± 1.35549 11.2 ±17.59094 64.18842 ±12.69943 7.6 ±46.15066 5.07021 ± 1.33530 11.1 ± 5.07342 92.6,7506 ±23.73327 7 •• ±28.02767 14.98862 ± 4.60001 110 0 69.45454 0 7 .4 ±28 14.77564 ± 3.85550 7 .3 ±13.41608 25.96144 ±11.24935

12,12 ±139.07639 8.59911 ± 2.21982 7 .2 ±12.88860 22.92976 ± 4.23885 7 .1 ± 3.53957 37.97507 ± 8.00483 12"1l ±139.07639 8.59910 ± 2.21979

70 0 26.44871 0 12'10 ±106.75754 25.37029 ± 6.73525 12 •• ±106.75752 25.37001 ± 6.66084

8"8 ±60.73566 5.77734 ± 1.52190 12.8 ±77.54083 41.41539 ±11.58290 8", ±60.73555 5.77621 ± 1.51620 12.7 ±77.53995 41.40366 ±11.50884 8.6 ±39.75991 16.99343 ± 4.83960 12,6 ±51.56206 57.21806 ±17.89676 8>5 ±39.75292 16.92944 ± 4.56773 12 •• ±51.53914 56.95433 ±16.55247 8,.. ±22.07216 28.37652 ±10.70308 12,.. ±29.20920 74.88820 ±31.98796 8"'3 ±21.89857 27.10072 ± 6.58510 12.3 ±28.86916 71.66763 ±18.51034 8"2 ± 8.66888 43.00657 ±17.81406 12,2 ±11.83982 99.41162 ±47.13339 8d ± 6.88121 34.19363 ± 3.53356 12'1 ± 9.18884 82.00526 ± 9.02566 80 0 51.69231 0 120 0 114.19496 0

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 17: The Asymmetric Rotor I. Calculation and Symmetry Classification of Energy Levels

42 KING, HAINER, AND CROSS

APPENDIX III. COEFFICIENTS OF THE THIRD-ORDER APPROXIMATION TO THE ENERGIES E(o) OF PERTURBED SYMMETRIC ROTORS o~O, OR ,,~-l

JK_I,KI ,,0 "I ,,2 ,,3 JK-I,KI ,,0 "I ,,' ,,3

00,0 0 0 0 0 88,0-1 56 8 2.14286 1.07143 87,1-2 26 23 6.5625 3.28125 8~2_3 0 36 11.50714 5.75357 85,3-4 -22 47 17.8125 8.90625

11,0 0 2 0 0 84.4-& -40 56 27.6 13.8 h.1 0 0 0 0 83• 5 -54 63 47.8125 186.32813 10, I -2 2 0 0 83,6 -54 63 47.8125 -138.51563

8',6 -64 68 273.75 136.875 82,7 -64 68 -41.25 -20.625

22,0 2 2 1.5 0.75 81,7 -70 107 -72.1875 -198.51563 81,8 -70 35 -72.1875 126.32813

2,,1 2 2 0 0 80, s -72 72 -315 -157.5 21,1 -4 8 0 0 21, , -4 2 0 0 99,0-1 72 9 2.39063 1.19532 20,' -6 6 -1.5 -0.75 .98,1-2 38 26 7.28571 3.64286

97,,_3 8 41 12.60937 6.30468 96,3-4 -18 54 18.96429 9.48214

33,0 6 3 0.9375 0.82031 96,4_5 -40 65 27.65625 13.82813 94,&-6 -58 74 42 21

33,1 6 3 0.9375 0.11719 93.6 -72 81 72.84375 361.26562 32,1 -4 8 7.5 3.75 93,7 -72 81 72.84375 -288.42188 32" -4 8 0 0 9,,7 -82 86 426.75 213.375 31,2 -10 17 -0.9375 -0.82031 92,8 -82 86 -68.25 -34.125 31,3 -10 5 -0.9375 -0.11719 91,8 -88 134 -115.5 -382.59375 30,3 -12 12 -7.5 -3.75 91,9 -88 44 -115.5 267.09375

90,9 -90 90 -495 -247.5

44,0_1 12 4 1.16667 0.58333 1010,0-1 90 10 2.63889 1.31945 109,1_, 52 29 8.01563 4.00782

43,1 -2 11 3.9375 4.42969 lOs, '-3 18 46 13.75397 6.87698 43,2 -2 11 3.9375 -0.49219 107,3_4 -12 61 20.31770 10.15885 4,,2 -12 16 21.33333 10.66667 106,4-& -38 74 28.60714 14.30357 4,,3 -12 16 -1.16667 -0.58333 105,5-6 -60 85 40.57292 20.28646 41,3 -18 29 -3.9375 -4.42969 10.,6-7 -78 94 61.16667 30.58333 41,4 -18 9 -3.9375 0.49219 103,7 -92 101 106.59375 656.57812 40,4 -20 20 -22.5 -11.25 103,8 -92 101 106.59375 -549.98438

10,,8 -102 106 636.33333 318.16666 10,,9 -102 106 -106.16667 -53.08334

5&,0-1 20 5 1.40625 0.70313 101,9 -108 164 -175.5 -691.03125

5.,1_2 2 14 4.5 2.25 10,,10 -108 54 -175.5 515.53125

53, , -12 21 9.09375 14.39062 100.10 -110 110 -742.5 -371.25

53,3 -12 21 9.09375 -5.29688 11 11,0_1 110 11 2.8875 1.44375 52,3 -22 26 48 24 11\0,1_2 74 32 8.75 4.375 52,4 -22 26 -4.5 -2.25 11 9,2-3 32 51 14.925 7.4625 51,4 -28 44 -10.5 -15.09375 118,3_4 -4 68 21.78571 10.89286 51,5 -28 14 -10.5 4.59375 11 7,4-5 -34 83 30 15 50• 5 -30 30 -52.S -26.25 116, &-6 -60 96 41.08929 20.54464

116,6-7 -82 107 57.18750 28.59375 11.,7_8 -100 116 85.875 42.9375

66,0_1 30 6 1.65 0.825 113,8 -114 123 150.9375 1131.21094

65,1_' 8 17 5.15625 2.57813 11 3,9 -114 123 150.9375 -980.27344

6,,2_3 -10 26 9.6 4.8 112,9 -124 128 915 457.5

63,3 -24 33 17.34375 38.20312 11,,10 -124 128 -157.5 -78.75

63,4 -24 33 17.34375 -20.85938 11 1,10 -130 197 -255.9375 -1183.71094

62,4 -34 38 93.75 46.875 11 1,11 -130 65 -255.9375 927.77344

62,& -34 38 -11.25 -5.625 110,1l -132 132 -1072.5 -536.25

61,& -40 62 -22.5 -40.78125 1212,0-1 132 12 3.13636 1.56818

61,6 -40 20 -22.5 18.28125 60,6 -42 42 -105 -52.5 1211,1_2 86 35 9.4875 4.74375

12\0,2-3 44 56 16.11364 8.05682 129,3_4 6 75 23.325 11.6625 126,4-6 -28 92 31.64285 15.82143

77,0-1 42 7 1.89583 0.94792 127, &-6 -58 107 42 21 76, 1-2 16 20 5.85 2.925 12 6,6_7 -84 120 56.20715 28.10357 75,2-3 -6 31 10.47917 5.23958 12&,7_8 -106 131 78.1875 39.09375 74.3_. -24 40 17.06667 8.53333 12.,8_9 -124 140 117.9 58.95 73,4 -38 47 29.8125 88.03125 123,9 -138 147 207.9375 1863.53906 7 .. -38 47 29.8125 -58.92188 123,10 -138 147 207.9375 -1655.60156 7,: & -48 52 166.08333 83.04166 122,10 -148 152 1276.5 638.25 72,6 -48 52 -22.91667 -11.45834 122,11 -148 152 -225 -112.5 71,6 -54 83 -42.1875 -94.21875 121,11 -154 233 -360.9375 -1940.03906 71,7 -54 27 -42.1875 52.73438 121,12 -154 77 -360.9375 1579.10156 70,7 -56 56 -189 -94.5 120,12 -156 156 -1501.5 -750.75

Downloaded 21 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions