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Applications of double perturbation theory to microwave spectroscopy; the molecular dipole moment and… Maso, A.C.P. 1971

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APPLICATIONS OF DOUBLE PERTURBATION THEORY TO MICROWAVE SPECTROSCOPY j THE MOLECULAR DIPOLE MOMENT AND THE NUCLEAR QUADRUPOLE COUPLING CONSTANT OF METHYL IODIDE by A. C. P. MASO B.Sc, University of Sao Paulo, B r a z i l , 1 9 6 4 B.A.Sc,, University of Sao Paulo, B r a z i l , 1 9 6 6 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the department of Chemistry We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1 9 7 1 In present ing th i s thes is in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the Un i ve r s i t y of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make i t f r e e l y a va i l ab l e for reference and Study. I fu r ther agree that permission for extens ive copying of th is thes is for s c h o l a r l y purposes may be granted by the Head of my Department or by his r epresen ta t i ves . It is understood that copying or pub l i c a t i on of th i s thes is for f i n a n c i a l gain sha l l not be allowed without my wr i t ten permiss ion . Department of Chemistry The Un i ve r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date October 12, 1971 i i ABSTRACT Double p e r t u r b a t i o n theory i s a p p l i e d to the problem o f measuring the molecular d i p o l e moment o f a l i n e a r o r symmetric top molecule i n the presence o f a s i n g l e quadrupolar nucleus, A computer program i s developed and a new v a l u e f o r the d i p o l e moment o f methyl i o d i d e i s r e p o r t e d . i i i TABLE OF CONTENTS Page CHAPTER ONE: Introduction 1 1.1 Perturbation Theory 2 1.1.1 Single Perturbation Theory 3 1.1.2 Double Perturbation Theory 4 1.1.3 Extension of Double Perturbation Theory 5 1.2 Dipole and Nuclear Quadrupole Moments 6 CHAPTER TWOj Theoretical Considerations 11 11.1 The Matrix Elements 11 11.1.1 The Rigid Rotor Hamiltonian 11 11.1.2 The Quadrupole Hamiltonian 11 11.1.3 The Dipole Hamiltonian 12 11.2 Description of the Program 14 CHAPTER THREE j Experimental Procedures and Results 19 I I I . l Description of the Apparatus 19 III 02 Sample Handling 23 111.3 Accuracy of the Measurements and the Cali -bration of the Cell 24 III.3.1 Calibration with Carbonyl Sulfide 25 111.4 Experimental Results 30 III. 5 Conclusions and Comments 40 BIBLIOGRAPHY 42 APPENDIX LIST OF TABLES iv TABLE I TABLE II TABLE I I I TABLE IV TABLE V Calibration of the c e l l using the J=0-*1 transition of OCS 27 DC voltage = 300 volts 27 DC voltage = 600 volts 28 DC voltage = 900 volts 29 J=0—1, F = 5/2-^5/2, MF = 5/2 transition of Methyl Iodide at 14 695.22 MHz 32 J=C—*1, F = 5/2-»3/2, Mp = 1/2 transition of Methyl Iodide at 15 275.87 MHz 35 J=l -*2, F = 7/2—9/2, M_ = 1/2 transition of Methyl Iodide at 30 123 MHz 38 Summary of the Results 39 LIST OF FIGURES Page Figure 1, Block Diagram o f the Main Program 17 F i g u r e 2. L i s t i n g o f the Main Program, Subroutines, and F u n c t i o n s , 44 Figure 3. Schematic Diagram o f the C e l l Showing the Septum and the Approximate Shape o f the F i e l d L i n e s . 20 F i g u r e 4. Block Diagram o f the Spectrometer 21 F i g u r e 5. S t a r k C e l l 22 ACKNOWLEDGMENTS I want to express my special thanks to Professor D. P. Chong for the numerous suggestions and discussions; without his help and encour-agement this work would not have been possible, I am also indebted to Dr. M. C. L. Gerry for his help and advice throughout this thesis, I must mention Mr. J. Sallos and his group from the Electronic Shop of this Department for their continuous help with the equipment. I am very grateful both to the National Research Council (Canada) and the Conselho Nacional de Pesquisas (Brazil) for their support in the year 1968-69 i n terms of the Brazil-Canada cultural agreement. 1 CHAPTER ONE Introduction This work i s intended to be a f i r s t example of the a p p l i c a t i o n of double perturbation theory to the s o l u t i o n of some p r a c t i c a l problems of microwave spectroscopyt r o t a t i o n a l spectra are dependent upon the existence of a molecular dipole moment, but are affected also by the presence of one or more quadrupolar n u c l e i , c e n t r i f u g a l d i s t o r t i o n , i n t e r n a l r o t a t i o n , and other e f f e c t s . Often one needs to consider the effects of two of these simultaneously! one may, f o r example, need to measure the effects of two quadrupolar n u c l e i , or the effects of the molecular dipole moment and p o l a r i z a b l l i t y , or the dipole moment of a molecule whose r o t a t i o n a l spectrum i s affected by a nuclear quadrupole coupling. As a prototype we have chosen the l a s t of these examples, and have considered the simple case of symmetric tops and l i n e a r mole-cules with one quadrupolar nucleus to demonstrate the a p p l i c a b i l i t y of the theory. The molecular dipole moment and the quadrupole coupling constant are obtained from the experimental data by an extended Newton-Raphson method, to the use of which perturbation theory lends i t s e l f i n a very natural way. We begin with a review of those parts of perturbation theory that are relevant to our problem and with a b r i e f look i n t o the work done pre-viously about the Stark e f f e c t i n molecules containing one quadrupolar nucleus. In chapter two the matrix elements of the Hamiltonian are re-calculated i n a form suitable f o r the a p p l i c a t i o n of the theory. This i s followed by a description of our computer program designed to calcu-l a t e the dipole moment jx. and the nuclear quadrupole coupling constant 2 "y z z . The t h i r d chapter i s a d e s c r i p t i o n o f the theory (and the program). F i n a l l y some experimental r e s u l t s are g i v e n and d i s c u s s e d , 1.1 P e r t u r b a t i o n Theory Atoms and molecules are m i c r o s c o p i c systems and t o study t h e i r behavior the methods o f quantum mechanics must be a p p l i e d . I n t h i s theory a system i s completely d e s c r i b e d by a s t a t e v e c t o r i n a g e n e r a l i z e d H i l b e r t space whose dynamical v a r i a b l e s are h e r m i t i a n l i n e a r operators over t h i s space. The v a l u e s these v a r i a b l e s can assume are t h e i r r e s p e c t i v e e i g e n -v a l u e s . The energy o r Hamiltonian o p e r a t o r i s very r e l e v a n t s i n c e i t determines the time e v o l u t i o n o f the system. Moreover, i f the time i s uniform i n r e l a t i o n t o the system, the energy eigenvalues are constants o f the motion; the system i s s a i d t o be s t a t i o n a r y . The s o l u t i o n o f eigenvalue equations i s , t h e r e f o r e , a main concern from the p r a c t i c a l p o i n t o f view, but i n most problems exact s o l u t i o n s are d i f f i c u l t t o o b t a i n w i t h our present mathematical techniques and approximation methods are used. Sometimes s m a l l terms may appear i n the Hamiltonian and by n e g l e c t i n g them we are l e f t w i t h a s i m p l i f i e d equation t h a t d e s c r i b e s the system as a z e r o t h order approximation. Once i t s e i g e n f u n c t i o n s and eigenvalues are determined the e f f e c t s o f the neglected terms can be brought i n as a s e r i e s o f approximations ( p e r t u r -b a t i o n terms). D i f f e r e n t p e r t u r b a t i o n methods have been developed, to d e a l w i t h time-dependent and time independent p e r t u r b a t i o n s r e s p e c t i v e l y . I n what f o l l o w s we w i l l be concerned only about time Independent p e r t u r -b a t i o n i n the R&yleigh-Schrodinger form''", and a l s o w i l l assume the s t a t e s to be d i s c r e t e and non-degenerate. 3 1,1.1 Single Perturbation Theory 1 I f the Hamiltonian H f o r the system can be put i n the formj H = H 0 + A V (1) where A V i s a small perturbation to HQ, a soluti o n to the equationj H |q> = E(q) | q> (2) can be found i n the form of a power series i n A t U/=Jo* nU«n> (3) and E(i) = N ! O A N e ( , I ' n ) M In t h i s notation q i s a l a b e l to indicate a l l quantum numbers necessary to define the state and n (or N) IS the order of approximation. Putting H* = H c - E(qjO) and V* = V - E ( q j l ) and <q | q> =1 a f t e r some mathematical manipulation we get i Ho|q;0>=0f (5) which upon so l u t i o n y i e l d s the zeroth-order approximation. The higher approximations to the energies aret E ( q j l ) = <q|0| V |q{0>, (6) E(q;2) = <q|0| V*| q f l > , E(qi2N) = <q|N-l | V'| q fN>- I E(q,k) ^  < qjN+j-k | q;N-j> , fc=2 j=0 N = 2,3 t i -TJ • . fc-1 . E(q }2N+l) = <q|N | V 1 q;N>- Z E(qjk) Z <q;N+l+j-k |qjN-j) , K=2 j=0 N = 2,3, The higher order corrections to the wave-functions can be expressed i n terms of an a r b i t r a r y complete set. I f the set of unperturbed eigen-functions [ j q j O ^ j i s used great s i m p l i f i c a t i o n i s achieved! |q,n>= Z |k,0>< k|0 |q,n> = ^ | k,0> (?) 4 As we go higher i n o r d e r , the c a l c u l a t i o n of the wavefunction c o r r e c t i o n s becomes very lengthy and t e d i o u s ; the f i r s t three approximations a r e i l q j l > = J ( 8 ) £ q " €. / ( t . V* V I k;0> 1 ? ' V . V . J q ; 0 > |q;2>= "J k K j J (*' - — j q j Jq ' ^ (c5q-fj)(6q-£k) 2 ( V £ j ) 2 |q;3>= f 1 'f ^ Vtt I 1 » ° > . iHVjqVkqlk'°> q' (^ q-£j)(f q-2 k)(£ q^l) 2 ( € q - f j )2( f q - ? k ) S ' v . I j;0> 2 , 2 ' V .V* M V . I q;0\ . E ( Q I 2 ) T_aiiLr - f t * J ^ k q ' > do) ( f q - £ j )2 ( V £ j ) 2 ( W where the primes',denote summations o m i t t i n g terms o f energy E(q;0) = £ , Examination o f the energy s e r i e s shows th a t to obtain the c o r r e c -t i o n s up to the order 2N or 2N + i , a knowledge o f the wavefunction ap-proximations up t o the order N i s necessary. Therefore, we can c a l c u l a t e the energy approximations up t o the seventh order i f we c a l c u l a t e the wavefunction approximations up t o the t h i r d o r d e r , 1.1.2 Double P e r t u r b a t i o n Theory.* I t may happen th a t 'the Hamiltonian can be put i n the form: H = H 0 + X V + / t f (11) where \ V and^uW are two independent p e r t u r b a t i o n s . The theory i s e x t e n -ded by u s i n g double power s e r i e s i n "X and y i | q> = 1 f V V | q j l . m ) , <q | q> = 1 (12) 1=0 m=0 ^ E(q) = 1 S E<d } L' M) (13) 1=0 M=0 where ag a i n q l a b e l s the s t a t e and l,m (or L,M) are the orders o f approximation i n V and W r e s p e c t i v e l y . 5 The perturbation equations for I q|l,0^  and E(q|Lt0) are found to be the same as those for Iqtn) and E(qfN) i n the single perturbation series. Those for Iqj0,m^  and E(qjO,M) are obtained from those for |q,n/> and E(q,N) by substituting W for V and W for V*. The mixed terms, however, are new and the energies are given byj E(qjL,M) = <q{0,0 1 V\qj L-ltM>+ <qjO,0 U |q;LtM-l> (14) " >0 klo ^.J.^C 1- ^ j o V <jL *W < q j ° ' ° W»L-J,M-k> The wavefunctions are similarly expanded i n terras of the complete set (|q>0,0)) i lq|l,m)= £ |k,0,0> < k{0,o| qjl.m) = | Cqk* |k;0,0> (15) Another important result i s Dalgarno's interchange theorem. I t permits the calculation of the mixed energy terms of the form E(q;i,M) entirely i n terms of the functions lq|0,m) (or E(q,L,l) i n terms of ]qjl.O> ) i M E(q|i,M) = jZ^qjO.M-kl v| q;0,k> (l6) E(q;L,l) = X <q;L-k,0 U Ulk,0> (17) k=0 The results above allow us to calculate the energy approximations up to the total order L + M = 4 i f we calculate the wavefunction series up to the total order 1 + m = 3« So i n the case of double perturbations we need more wavefunction approximations than i n the case of a single perturbation, which makes i t s application more d i f f i c u l t and tedious, 1,1.3 Extension of Double Perturbation Theory, A new perturbation ex-2 pression, recently published by Chong , i s quite useful i n calculations where approximations to higher orders may be necessary. The result i s g i v e n belowi E(qjL,M) = ( i - ^ L 0 ) q j l . A - 1 I V I q | L - l - l f B-L+l+1 > + + (1-<^ M 0) ^  < q; A-m,ra |w \ q> B-M+m+1, M-m-1?-- « - % i h  E ( q i v p -  k - w p + r ) * x<q;P,A+l+j-k-p I q ; r , B - j - r > ( l 8 ) where the l i m i t s o f summation a r e j = max (O.L-B-l) N = L + M ^ 1 Ig = min (A,L-1) A - N - B - 1 . M1 = max (0,M-B-1) B = N/2 f o r N even M"2 = min (A,M-1) B = (N - l ) / 2 f o r N odd Pj^ = max (0, L-k-B+j) P 2 = min (L,A+l+j-k) = max (0 tL-k-p) ^2 = min (B-j,L-p) Now we can c a l c u l a t e the energy s e r i e s up t o the t o t a l order L + M u s i n g wavefunction approximations o f t o t a l order not h i g h e r than (L+M+l)/2, T h i s s i t u a t i o n i s analogous t o t h a t o f the s i n g l e p e r t u r -b a t i o n c a s e , where the energy can be c a r r i e d up t o the order 2N+1 w i t h the knowledge o f the wavefunctions up t o the Nth order o n l y . 1.2 D i p o l e and Nuclear Quadrupole Moments. Measurements o f molecular d i p o l e moments have been made i n gases 4 5 and i n substances i n s o l u t i o n . Two methods were developed by P. Debye i n the f i r s t method the d i p o l e moment can be obtained from a study o f the temperature dependence o f the d i e l e c t r i c constent o f a gas; the second method, which can be a p p l i e d t o substances i n s o l u t i o n uses both the d i e l e c t r i c constant ( a t a s p e c i f i e d temperature) and the r e f r a c t i v e 7 index. The accuracy obtainable by these methods i s dependent on the purity of the sample used but the measurements i n solution are also affected by possible solvent effects. More recently with the development of microwave techniques and the attainment of high resolution i n the spectra of gases at low pressures ( ~ lOyUHg) i t has become possible to use samples with a limited degree of purity. This i s due to the fact that the microwave spectrum of a gas i s composed of a set of very narrow lines whose frequencies are not affected by small quantities of other gases i n the sample. I t also allows us to discriminate between different vibrational states of the molecule, whereas the previous methods yield only average values. In ideal conditions a precision better than ,01% ^  can be obtained, which makes the dipole moment useful i n the testing of molecular wave-functions and the molecular geometry since i t i s due to a distribution of charges. v Some molecules besides having a dipole moment also contain one or more nuclei with electric quadrupole moments. Nuclei are known to have 7 no e l e c t r i c a l dipole moment , but may have a quadrupole moment which when subject to the inhomogeneous internal electric f i e l d of a molecule finds an equilibrium orientation i n relation to the molecular framework and rotates with the molecule as a whole, thus producing the coupling of the nuclear spin with the molecular angular momentum. As seen from g the interaction Hamiltonian the coupling i s made through the f i e l d gradient rather than the f i e l d i t s e l f , and these effects are observed i n rotational spectra as a hyperfine s p l i t t i n g of the lines. The rotational Hamiltonian for a molecule placed i n an electric f i e l d can be written ast H " H 0 + HS + " Q 09) 8 where H Q i s the r i g i d r o t o r H a m iltonian, Hg i s the d i p o l e c o u p l i n g term and HQ i s the nuc l e a r quadrupole i n t e r a c t i o n term. These two terms are sm a l l i n r e l a t i o n to the spacing between the r i g i d r o t o r energy l e v e l s and can be t r e a t e d as p e r t u r b a t i o n s . I n t h i s work we r e s t r i c t o urselves to symmetric tops and l i n e a r molecules w i t h only one quadrupolar n u c l e u s . Consequently, the e i g e n f u n c t i o n s and eigenvalues o f the Hamiltonian H are c h a r a c t e r i z e d by the q u a n t i t i e s I , J , K,F,Mp or a l t e r n a t i v e l y by I , J , K, 9 Mj, Mj , where I i s the n u c l e a r s p i n o f the quadrupole n u c l e u s , J i s the molecular angular momentum, K i s the p r o j e c t i o n o f J along the molecular a x i s , F i s the v e c t o r sum o f I and J , Mp i s the p r o j e c t i o n of F along the d i r e c t i o n o f the e l e c t r i c f i e l d , Mj i s the p r o j e c t i o n o f J along the d i r e c t i o n o f the e l e c t r i c f i e l d , Mj i s the p r o j e c t i o n o f I along the d i r e c t i o n o f the e l e c t r i c f i e l d . Before the present work, i n order t o determine a d i p o l e moment i n the presence o f a nuc l e a r quadrupole c o u p l i n g , three p o s s i b i l i t i e s had to be considered! (a) the weak f i e l d case (HgC^Hq), ( B ) the st r o n g f i e l d case (HQ<<Hg), and (c) the in t e r m e d i a t e f i e l d case ( H q ~ H g ) . I n the weak f i e l d case (Hg<<HQ) Hg was t r e a t e d as a p e r t u r b a t i o n t o the Hamil-t o n i a n H Q + H Q , whose e i g e n f u n c t i o n s are best d e s c r i b e d by the r e p r e s e n -t a t i o n I , J , K, F, Mp. The f i r s t c o r r e c t i o n to the energy due t o Hg was found t o b e ^ i M) - . ^EKMpD , (20) *• ~ 2J(J+1)F(F+1) where D = F(F+1) + j ( j + l ) - l ( l + l ) . The valu e o f the quadrupole constant c o u l d be obtained from the h y p e r f i n e s p l i t t i n g o f the l i n e s when no f i e l d i s a p p l i e d , and the d i p o l e moment could be c a l c u l a t e d from a g r a p h i c 9 of the change i n frequency of a suitable l i n e as a function of the Stark f i e l d . In the strong f i e l d case (H^CHg) the quadrupole energy HQ was treated as a perturbation to the Hamiltonian H Q + Hg, Now the set I, J, K, Mj, Mj i s the most convenient to use since F no longer i s a constant of the motion. Analogously, the f i r s t order correction to the energy due to HQ was found to be**i (21) = * " f - i ) [ 3 M l 2 - l ( l + l ) ] [ 3M 2 rJ(J + D] 4l(2I-l)(2J-l)(2J+3) W(J+1) V where ~XZZ i s the nuclear quadrupole coupling constant. As i n the weak f i e l d case the quadrupole constant could be obtained from the hyperfine s p l i t t i n g of the lines when no f i e l d was applied. For the dipole moment, however, a graphic of the average frequency of a multiplet as a function of the Stark f i e l d was used and the calculation proceeded as i f there was no quadrupole s p l i t t i n g . The intermediate case (H Q ~ H Q ) was more d i f f i c u l t and required a special treatment, A direct diagonalization of the secular determinant had to be done*2j (iJMpMjJ Hg| IJMpMj.) - * / | =0 (22) for chosen values of I, J, K and Mp, The determination of the constants was thus more d i f f i c u l t than i n the other two cases. In this thesis we apply double perturbation theory to the inter-mediate case i n such a way as to make possible a systematic f i t t i n g of the constants by a least squares method. In doing so we also reduced a l l three cases to a single one from the computational point of view. In anticipation of a possible slow convergence of the perturbation series we carried the calculations up to the seventh order i n the energy using 10 Chong's extension of the theory. An extended Newton-Raphson method J was applied to obtain a least squares f i t of the dipole moment and the nuclear quadrupole coupling constant simultaneously. The extended Newton-Raphson method, which requires the use of perturbation theory for i t s application to our problem, has the advantage of a faster convergence than the linear Newton-Raphson method and a very low tendency to remain on local minima thus increasing the r e l i a b i l i t y of i t s results. 11 CHAPTER TWO Theoretical Considerations 11,1 The Matrix Elements ' To calculate the perturbations to the r i g i d rotor energy levels the matrix elements of each term of the Hamiltonian are best expressed using an unperturbed wave function representation which i s the same as that for the weak f i e l d case, i.e., ||lJKFMp)j . As we mentioned before any complete set of functions could be used, but that would unnecessarily complicate the calculations, 11.1.1 The Rigid Rotor Hamiltonian. The only nonzero elements for the symmetric top are functions of J and K alone. So they can be written 14 down directly t < IJKFMp I H 0 \ IJKFMp>= BJ(j+l) + (A-B)K 2, i f A >B = C, (23) <IJKFMp | HQ | IJKFMp)= BJ(j+l) + (C-B)K2, i f C< B = A, (24) with H = AP 2 + BP 2 + CP 2, where A, B. C are the rotational constants O X y Z expressed i n frequency units (MHz) and P , P , P are the angular momentum y z operators along the principal i n e r t i a l axes xyz of the molecule. We also found i t necessary to include the effects of centrifugal distortion which are dependent only on J and Ki Hd = - D J J 2 ( J + 1 ) 2 - D J KJ(J+1)K 2 - DjK 4 (25) Henceforth we w i l l refer to H Q ( i e , , H Q + H^ ) as the unperturbed Hamil-tonian or as the r i g i d rotor by an extension of language. 11.1.2 The Quadrupole Hamiltonian . The nuclear quadrupole coupling i s expressed by the Hamiltonian^^t (26) Q . . and V E are symmetric tensors of the second rank, the quadrupole i j coupling tensor and the f i e l d gradient tensor respectively. To get the 12 matrix elements the coupled representation ||lJKFMp)j i s factored 1^ into the molecular and nuclear coordinate systems! <IJ.K.F'MF|HQ| IJKFMF> = I ^ MpMp t^W™"* X (2?) x W(l,I»,J,J«;2tF) [(2I+1)(2J*+1)] ^ <I||"Q|| I><J'K'IIUE|| JK > where W i s a Racah coefficient and the double bars are used to indicate reduced matrix elements. On the other hand the nuclear quadrupole moment can be defined byi eQ = V 2 / 3 <I, Mj = I \X\ I, Mj = I> ( 2 8 ) 17 and using the Wigner-Eckart theorem ' to factor out the projection quantum numbers, we get, eQ= V273 C ( I , 2 , I | I , 0 , I ) < I )1*Q11 I> , 17 where C i s a Clebsh-Gordon coefficient i n the Rose notation. Similarly the electric f i e l d gradient V Z Z can be defined i n the laboratory system XYZ byj ^J'.K'.Mj = JJV ^ I J . K . M J = J> =V273<J ,,K ,,M J = J|VE1J,K,MJ = J> = V 2 ? 3 C(j,2,J',J,0tj)<J«K' H ^ E H J K > . ( 3 0 ) Hence: <IJ'K'FMp 1 H^ 1 IJKFMp> = ) I + J - F W(l,I,J,J»}2,F) X x [( 2I+1)(2J-+1)] > x ^ ^ ^ ^ < 3 1 ) (JK'J ) V Z Z I JKJ^ has been evaluated*® and the only non-vanishing elements are found to be» <JKJ| V z z l JKJ> = V . . 3K2 - J(J+1) (32) ( J + l ) ( 2 J + 3 ) , . „ , , . . (J+l)(J+ 2 ) \ 2 J + 3 < J + 1 , K , J ] V z z 1J,K,J> = V _ . _ 3K ( ( J + 1 ) 2 - K 2 ^ * ( 3 3 ) <J+2,K,J| V„ 7) J,K,J> = 3V^ . f [ ( J + l ) 2 - K 2][(j+2) 2-K 23 Z Z (J+2K2J+ 3 ) (J+l)(2J+5 where xyz i s the molecule fixed system. * This was found to be wrong i n ref, 1 8 , 13 The product* e < ^ z z = ^ z z x s t n e nuclear quadrupole coupling constant, one of the molecular parameters to be determined by the computer program. And f i n a l l y we get : <IJKFMplHJlJKFlO = £ ( - l ) I + J " F x W(l,I,J,J?2,F) eQ,[(2I+l)(2J+l)\ 2  H C ( j ; 2 ; j ; j f 0 , j ) X C(I,2 fI}I,0,I) v V Z Z [ 3 K 2 - J(J+1)] (35) (J+1K2J+3; J+J-F <I,J+l,KFM plH 0l IJKFMp) = i ( - l ) I + J " F W(I,I,J,J+1;2,F) eQ[(2I+l )(2J+3)1 2 C(J,2tJ+l»J,0,j) C(I,2,I|1,0,I) x V Z 2 3K / ( j + l ) 2 - ! ^ * (36) (J+1XJ+2) V ,(2J+3)j J x <I,J+2,KF«-l|H0l IJKFMp> i ( - l ) I + J _ F W(l,I,J,J+2;2.F) eQftel+l)(2J+5)~1 2 - C(J I2,J+2|J I0,J) C(I,2,IjI,0,I) 3 f [ ( J - H ) 2 - K 2 ] [ ( j - f 2 ) 2 - K 2 ] 1 ^ (37) (J+2)(2J+3) [ ( J + l ) ( 2 J + 5 ) j V x zz 11.1,3 The Dipole Hamiltonian. The Stark i n t e r a c t i o n i s given by the expression Hg = E = - y* Ecos 6 (38) whereytt i s the molecular dipole moment, E i s the uniform e l e c t r i c f i e l d applied and © i s the angle between the Z axis (laboratory system) and the z axis (molecular system). Thus, the matrix elements are: <IJ'K»FfMp)Hgl IJKFMF> = -^ <E <IJ»K,F»MF\jZ5Zz\ IJKFMp> (39) where fb„ = oos8 i s an Irreducible tensor operator of the f i r s t rank. Again by the Wigner-Eckart theorem: <IJtK»F«MFlHs\ IJKFMp) = -yuE C(F,1,F» ;MpV0,Mp) < IJ«K«F«l0Zz\ IJKF> (40) and transforming the elements of fiZz to the uncoupled representation : <IJ»K»F»MF)Hg\lJKFMp> = -yuE C(F,1,F» |Mp,0,Mp) (-1 ) I + 1 - J , " F x {(ZJk-l)(2F+1)] 2 x W(j,J« fF fF»jl,l)<J»K'lljZ5 Z zl|jK> (41) But <J'K 'll^zJljK>= <J'K'Mj|^ Z zlJKMj> (42) CCJ.I.J'IMJ.O.MJ) and 2 0<J»K'Mi^ Z z\jKM> = < J ' 1 0 Z z U > < J»K«1 0 Z z \ JK> <J'Mj^ Z z\ JMj> (43) * 2 Other notations: eQq or e Qq 14 The only non-vanishing elements are given byi i 2 v-2 1 2 (44) <J+1,K l|0zJ J,K> = f ( J + l ) 2 - K: L (J+i)(2J+3) <JtKt|jZ5z_l|j,K>= K Z Z [J(J+1)1 i (45) And f i n a l l y s u b s t i t u t i n g i n the i n i t i a l expression, we gett <IJ+iKF ,M F|H sllJKFM F> = - yUEC(F,l,F ,jM F,0,M F) x (-1)I"J"P x x [(2J+3)(2P+1)] 2 x V ( J l J + l , F t F , | l , l ) x \2 v 2 1 (j+l)(2J+3) * (46) <IJKF«MF|HsllJKFMF> = ^EC(F,1,F» iMp.O.My) x ( - l ) I + 1 - J - F x [(2J+1)(2F+1)]^ x W(J,J,F,P«|1.I) x K [JTJ+ITP (47) Now that we have a l l the elements of the Hamiltonian expressed i n matrix form we proceed to apply double perturbation theory by w r i t i n g a computer program, II . 2 Description of the Computer Program The program written i n the Fortran IV language, consists of a main program and a number of subroutines and functions. Double precision has been used throughout because of the very large number of operations involved and the consequent possible accumulation of errors. The objective i s to make a l e a s t squares f i t over a number of t r a n s i t i o n frequencies at any Stark f i e l d s using the molecular dipole moment and the nuclear quadrupole coupling constant as the parameters to be ad-justed . From the computational point of view we are i n i t i a l l y faced with three problems! the i n f i n i t e basis set, the degeneracy of the states to be l i f t e d by the perturbations, and the large number of parameters (l,J,K,F,Mp) used to describe the states. The i n f i n i t e number of r o t a t i o n a l energy l e v e l s or i n f i n i t e basis set leads to i n f i n i t e matrices. The procedure generally adopted i s to choose a convenient f i n i t e subset and assume that the neglected elements 15 are i r r e l e v a n t since l e v e l s that are too f a r apart are not mixed by the perturbations. Also i n our problem i f J*. }J+2 or J * ^ J-2 the matrix elements are zero. This way i f we are interested i n a t r a n s i t i o n from the ground to the f i r s t excited state a perturbation c a l c u l a t i o n up to the seventh order w i l l not involve l e v e l s higher than the tenth excited state. However,these assumptions can be tested by using sets of d i f f e r -ent dimensions i n the calc u l a t i o n s . The degeneracy of the energy l e v e l s can be circumvented by using a mathematical a r t i f i c e t i f y U 0 and "X are the i n i t i a l guessed values of jx and X the t o t a l Hamiltonian can be written as t H = H 0 +yUHg +7^*= (H Q +jU oH s + X Q H Q ) + C U - j U ^ H g + ( "X - * 0 ) H Q ( 4 8 ) where we wrote^UHg and "XHQ f o r Hg and H^ to show those parameters e x p l i c i t l y . Now H Q = H Q +/A0Hg + XH^ i s non-degenerate and the whole Hamiltonian H can be transformed to a basis i n which H£ i s diagonal. The overabundance of quantum numbers leads to excessively large matrices the elements of which are determined by two sets of numbers I,J,K,F,Mp| , The nuclear spin I does not change during a r o t a t i o n a l t r a n s i t i o n and can, therefore, be treated as a constant parameter. Due to s e l e c t i o n rules and the construction of our instrument (where the dir e c t i o n s of the Stark f i e l d and the microwave e l e c t r i c f i e l d vector are p a r a l l e l ) the states with d i f f e r e n t values of the projection quantum numbers K and Mp are not connected by any t r a n s i t i o n . This way we can consider only those matrix elements which are diagonal i n I and K and Mp. This w i l l only l i m i t us i n each computer batch to consider t r a n s i -tions f o r which these numbers are the same. The remaining quantum numbers J and F are associated with the set of natural numbers ( i d e n t i -f i c a t i o n l a b e l s ) by a single-valued monotonic function i n order of * We write X f o r XZ 2 ^o s i m p l i f y the notation. 16 increasing J*s, and f o r each J i n order of increasing F's. A s i m p l i f i e d diagram of the main program i s shown i n Figure 1, I t s t a r t s by reading the input datat the dimension of the matrices (N), the t o t a l number of frequencies (NFQ) to be used, the r o t a t i o n a l constants of the molecule (A,B ,C ) , the values of the f i x e d quantum numbers (l,K,Mp), the maximum value of J (MAXJ), the i n i t i a l " g u e s s e d values of the dipole moment (AMU) and the quadrupole moment (AKZZ), Next the subroutine BASIS i s c a l l e d to get the matrix elements and the i d e n t i f i c a t i o n labels of the states. Then the experimental frequencies (FQOBS) are read together with t h e i r respective Stark voltages and quantum numbers, A new basis i s found which diagonalizes H Q and the whole Hamiltonian i s transformed to the new basis. The perturbation c a l c u l a t i o n i s done to each state involved by c a l l i n g the subroutine APER a f t e r which the adjustment ofthe parameters (AMU and AKZZ) i s done by c a l l i n g the subroutine NEWLQ. I f the corrections to the two parameters i s small enough the r e s u l t s are printed out. I f not the corrections are added to the old values of AMU and AKZZ and the c a l c u l a t i o n i s done again beginning with a new c a l c u l a t i o n of H^ = H Q +>UHg + "XHQ, This i s the i t e r a t i o n loop of the Main Program, Figure 2 i s a l i s t i n g of the Main Program and a l l the subroutines and functions, Ke have written the subroutine BASIS to calculate the matrix elements and the i d e n t i f i c a t i o n matrix whereby the overabundance of quantum numbers i s circumvented. This subroutine would have to be extensively revised i f generalizations to asymmetric tops or to more than one quadrupole moment were considered. The functions QUAD, DIPOLE and INDEX were programmed to calculate common factors i n the matrix elements of the quadrupole and dipole term and the i d e n t i f i c a t i o n l a b e l s 17 Figure 1, Block Diagram of the Main Program ( Start ~~) Readi N, NFQ, A,B,C,I,K,MF, MAXJ, AMU, AKZZ Writei N, NFQ, A,B,C,I,K,M MAXJ, AMU, AKZZ 1 C a l l BASIS Read: FQOBS(l) J,J ,,F,F ,,VOLT(l) i Diagonalize and Transform Hg and H to New Basis ^ C a l l APER (Pert, calc.) C a l l NEWLQ (least sqs, f i t ) to get DELTA(l) (correction) (stop ^y+- P r i n t : AKZZ, AMU, FQOBS.y FQCAL, Standard errors respectively. BASIS also c a l l s the functions W (Racah c o e f f i c i e n t s ) and C318 (Clebsh-Gordon c o e f f i c i e n t s ) and the subroutine EPRINT to pr i n t out the matrices. The perturbation c a l c u l a t i o n i s done by c a l l i n g the subroutine APER*. We programmed i t using Chong's extension of perturbation theory to calculate the energy series up to the t o t a l order L + M = 7. APER ?1 works i n conjunction with the subroutine PERT2 and PERI . The former calculates the double perturbations up to the t o t a l order L + M = 4 i n the energies and 1 + m = 3 i n the wave*?unctions. In t h i s work PERK which has previously done calculations f o r the ground state alone, was generalized to include excited l e v e l s as w e l l , PERI does single per-turbation calculations up to the seventh order i n the energy and up to the t h i r d order i n the wavefunctions. The subroutine NEWLQ has been written do do the l e a s t squares f i t by an extended Newton-Raphson method, and also calculate the standard errors f o r the quadrupole constant and the dipole moment. This sub-routine was designed to use the r e s u l t s of APER. In generalizing the program to more than two parameters, NEWLQ would have to be revised. * D.P. Chong, and A.C.P. Maso, This program has been sent to Q.C.P.E. (Quantum Chemistry Program Exchange) f o r publication, 1 9 7 1 . 19 CHAPTER THREE Experimental Procedures and Results I I I . l Description of the Apparatus The experiments were done using the 100 kHz Stark modulated micro-wave spectrometer 2** assembled i n our laboratory with commercial compon-ents. In t h i s instrument the c e l l was a J'ft, Hewlett-Packard 8425B Stark c e l l consisting of an X-band wave guide; i t contains a septum i n i t s i n -t e r i o r , as shown schematically on Figure 3t to which a voltage was applied while the c e l l was kept grounded, thus producing an approximately uniform e l e c t r i c f i e l d . In the region from 8 GHz to 18 GHz the microwaves were produced by a Hewlett-Packard 8400B Microwave Spectroscopy Source, using backward wave o s c i l l a t o r s with t h e i r frequencies automatically determined i n r e l a t i o n to a reference o s c i l l a t o r and displayed by a counter; a block diagram i s shown i n Figure 4. Klystrons (OKI E l e c t r i c Industries Co,, Ltd,) were used from 18 GHz to 3° GHz and the frequencies were measured i n r e l a t i o n to that of a MICR0-N0W model 101C Frequency M u l t i p l i e r Chain which has a s t a b i l i t y of one part per m i l l i o n over a 24 hour period and produces signals at i n t e r v a l s of 50 MHz; the differences i n frequency be-tween the klystron and these signals were measured with a calibrated Ham-marlund model SP600 receiver. F i n a l l y the microwaves were r e c t i f i e d by a c r y s t a l detector at the end of the wave-guide and the absorption pat-terns were displayed by an oscilloscope or a chart recorder a f t e r passing through a pre-amplifier and a phase-sensitive (lock-in) a m p l i f i e r (Prince-ton Applied Research, Model 121). In addition to the standard equipment we decided to introduce two modifications to Improve the accuracy and the r e l i a b i l i t y of the Stark 20 Figure 3. Schematic Diagram of the C e l l Showing the Septum and The Approximate Shape of the F i e l d Lines. Teflon Insulators 21 Figure 4. Block Diagram of the 'Spectrometer to pump and gas I n l e t Detector FT" Dig. Volt. Preamp. Lock-in Amplifier Chart Recorder C e l l :*AHben.. DC Power Supply 100 kHz SW Generator Synchroniser •MS Ssmrce & Sweeper Iteferer&ce Oscillator T Coeuster Oscilloscope r BURGL55 -=f" U200 BATTE.rUE.5 ~=f^  900 v 600v 30Ov . O U f 3YM o O V M 0 IN) 23 f i e l d . The reason was that at high voltages the square waves from the 100 kHz generators ( I n d u s t r i a l Components, Inc., of Beaverton, Ore.) were dis t o r t e d somewhat thus adding to the inhomogeneity of the Stark f i e l d (Figure 3). A s a t i s f a c t o r y s o l u t i o n , also used by previous workers 2^, was found i n the introduction of a DC voltage superposed on the square waves of variable amplitude not higher than 200,0 v o l t s . The DC power supply, b u i l t i n the Electronic Shop of the Department, was connected to the septum i n the c e l l and the 100 kka generator as shown i n Figure 4. I t was designed to y i e l d constant voltages of 300* 600 and 900 v o l t s and i t s i n t e r n a l c i r c u i t i s depicted i n Figure 5. The voltages were measured using a d i g i t a l voltmeter (Electro Instruments, San Diego, model 4010) which gives four d i g i t s up to 1000 v o l t s . The exact values of the DC voltages were not necessary except to keep a check on t h e i r s t a -b i l i t y , since a c a l i b r a t i o n of the instrument could be made as a function of the amplitude of the square waves f o r each value of the DC voltage. The DC power supply allowed superposition of the square wave voltage on top of the DC voltage, III.2 Sample Handling The samples of permanent gases were transferred from the s t e e l cylinders i n which they are availab l e to bulbs connected to the vacuum l i n e . The small amount of a i r that may have got into the bulb i n t h i s process was removed by freezing the gas to -195.8°C i n an external bath of l i q u i d nitrogen while the bulb was open to the vacuum l i n e . Substances that are l i q u i d at room temperature were kept i n small glass containers that are e a s i l y connected to the vacuum l i n e by a ground glass j o i n t . As before, the a i r was removed from the sample by freezing i t with l i q u i d 24 nitrogen while the container was open to the vacuum l i n e . Once the microwave c e l l was completely evacuated a small amount of gas was transferred i n t o i t by closing the stopcock that leads to the pumps and opening a connection between the sample container or bulb and the c e l l . The excess that eventually got into the c e l l was removed by the pumping u n t i l a suitable pressure was reached (^lOyuHg). The sample was now ready to have i t s microwave spectrum observed, 111,3 Accuracy of the Measurements and the Cal i b r a t i o n of the C e l l The t r a n s i t i o n frequencies and t h e i r corresponding Stark voltages were the raw experimental data obtained from the Instrument, The f r e -quencies were read from the frequency counter which i s accurate to + ,01 MHz, but the actual accuracy of the measurement was dependent upon the half-width of the l i n e and i t s signal-to-noise r a t i o . I t i s estimated that the center of the l i n e can be determined with an accuracy better than one tenth of i t s half-width. Thus, f o r the J = 0-*-l t r a n s i t i o n of carbonyl s u l f i d e , which i s a sharp l i n e (half-width ^ 0,5 MHz), the maximum accuracy of + 0.01 MHz was attained while f o r i t s Stark lobe at very high f i e l d s +0,02 MHz may be considered good. The Stark f i e l d was much more d i f f i c u l t to measure: i t requires i n p r i n c i p l e the measurement of the Stark voltage and the separation between the septum and the walls of the c e l l . However, undulations i n the septum compounded with d i s t o r t i o n s of the f i e l d on the edges contribute to make i t s value more elusive. To overcome these d i f f i c u l t i e s the c e l l was calibrated using carbonyl s u l f i d e , whose dipole moment has been measured 22 accurately by other methods , The d i g i t a l voltmeter gives four figures up to 1000 v o l t s , but the fluctuations of the amplitude of the square waves were of about + 0,2%i 25 the DC c e l l had fluctuations of le s s than about 3 v o l t s i n 900 v o l t s , i . e . + 0.16^, and the o v e r a l l accuracy of the Stark voltage was + 0.17$. III.3.1 C a l i b r a t i o n with Carbonyl Sulfide . This substance was chosen f o r several reasons» i t i s very easy to handle and has a very simple spectrum, i . e . that of a l i n e a r molecule; i t s J = 0-»-l t r a n s i t i o n at 12 162.97 MHz i s a very strong and sharp l i n e whose only Stark lobe can be measured e a s i l y and accurately up to very high f i e l d s . The dipole moment has been measured by several workers the most recent and accurate value to date being O.7I521+0.00020 Debyes 2 2. This way a l l dipole moments are measured i n r e l a t i o n to that of carbonyl s u l f i d e . The c a l i b r a t i o n consisted i n measuring the displacement of the Stark lobe as a function of the amplitude of the square wave voltage f o r each value of the DC c e l l voltage. The value of the Stark f i e l d i s calculated from the displacement of the Stark lobe, and thus we have the Stark f i e l d as a function of the square wave voltage. The second order correction to the energy of a l i n e a r molecule i s given byj J2) _ _ _ _ _ ________ 2hBJ(J+l) (2J-l)(2J+3) We can use i t here to calculate the change i n frequency of the l i n e when the f i e l d E i s applied. For the upper l e v e l J = 1 and (2) = JX2 E 2 (50) ^ ,r- lOhB while fo r the lower l e v e l J = 0 and the r e s u l t obtained by a l i m i t i n g process 1st <2> n2 E n = - ^. & ( 5 1) 0 ^ h B ' (2)_ J*2 E 2 . J(J+1) - 3M2 (49) - —zr^rrTTT-1 '-1)1 26 Now the displacement i n frequency i s i l5hB (52) and E = .; = h l/^hBAV = (419.44 + 0.12 JAV2" (53) 2JLL » where the l a t e s t published values of the constants have been used and h the lar g e s t contribution to the error i n the c o e f f i c i e n t J^J-i s due to the dipole moment of CCS, The expression above i s s u f f i c i e n t l y accurate f o r our purposes since higher order corrections including the effe c t s of the p o l a r i z a b i l i i t y i n OCS («< = 5.3^ x 10" 2^cm 3) 2 3 are neg-o v o l t s l i g i b l e f o r the J = 0—*1 t r a n s i t i o n and f o r f i e l d s o f about 10 J c m - , As pointed out i n reference 23 such effects can only be studied w e l l i n higher J t r a n s i t i o n s ( i e , J = 4—*5) where they are at l e a s t one order of magnitude larger. The measurements were done with the c e l l cooled to -78.5°C with dry ice to improve the si g n a l to noise r a t i o at very low pressure (•~5/'Hg). Although t h i s i s not important f o r OCS, whose J = 0 — * l t t r a n s i t i o n i s very strong, the c a l i b r a t i o n had to be done i n the same conditions i n which the c e l l was to be used with other gases. The resu l t s are shown i n Table 1, The smallest displacement was A\?~4 MHz and the largest Av>~30 MHz} the percentage error i n these two extreme cases i s given by r 2 I E x 10 = E 0.12 + 0.01 429.44 2 £ v . .2 x l ( f (54) * I f we write E = AA.v>2 with A = 419.44 + 0.12 then {E__ ft _£A + 1 /( AV) (55) E A 2 Av> to obtain the percentage errors. 2? TABLE I Cal i b r a t i o n of the c e l l using the J = 0 —*i t r a n s i t i o n of OCS DC voltage = 300 v o l t s Square Wave Amplitude Frequency of the Stark Stark F i e l d (Volts) Lobe (MHz) (volts/cm) 100.0 12 166.31 767 110.0 12 166.42 779 120.0 12 166.51 789 130.0 12 166.59 798 140.0 12 166.70 810 150.0 12 166.80 821 160.0 12 166.90 823 170.0 12 167.00 842 180.0 12 167.09 851 190.0 12 167.1? 860 1 TABLE I (continued) DC voltage = 600 v o l t s Square Wave Amplitude Frequency of the Stark Stark F i e l d (Volts) Lobe (MHz) (volts/cm) 100.0 12 174.89 1448 110.0 12 175.07 1459 120.0 12 175.24 1469 130.0 12 175.42 1480 140.0 12 175.59 1490 150.0 12 175.76 1500 160.0 12 175.94 1511 170.0 12 176.12 1521 180.0 12 176.30 1531 190.0 12 176.47 1541 TABLE I (concluded) DC voltage = 900 v o l t s Square Wave Amplitude Frequency of the Stark Stark F i e l d (Volts) Lobe (MHz) (volts/cm) 100.0 12 188.7? 2130 110.0 12 189.03 2141 120.0 12 189.28 2152 130.0 12 189.54 2162 140.0 12 189.78 2172 150.0 12 190.05 2182 160.0 12 190.29 2192 170.0 12 190.54 2202 180.0 12 190.80 2213 190.0 12 191.07 2223 30 The values obtained are +0,16% and +0,05% respectively; they are better than the r e s e t a b i l i t y of the voltages ( i . e . , +0.17%). III.4 Experimental Results We now describe some experiments that were done to check our computer program. The purpose, as we mentioned before, was to obtain a l e a s t squares f i t of the dipole moment and the nuclear quadrupole constant i n the i n t e r -mediate f i e l d region. In the f i r s t place i t was necessary to choose a suitable molecule which could be e a s i l y measured i n the region of interestj since the c a l i b r a t i o n of the c e l l with OCS i s imprecise at low voltages (because of the small displacement of the Stark lobe), we had to work with a molecule having a large quadrupole moment so that the intermediate f i e l d case occurred at higher voltages. Methyl iodide, which i s reported 26,27 i n the l i t e r a t u r e as having a quadrupole moment of -1933.99 + 0.25 MHz and a dipole moment of 1.647 + 0,014 Debyes was chosen and measured. The experimental procedure was the same as i n the c a l i b r a t i o n of the c e l l . The c e l l was cooled with dry i c e to -78.5°C to increase the s i g n a l -to-noise r a t i o while keeping a low c o l l i s i o n broadening of the l i n e s . The pressure was kept very low (~"5 J*H&) and the power j u s t enough to prevent saturation of the l e v e l s . For each value of the DC power supply voltage measurements were made f o r 10 values of the square wave amplitude while the instrument was swept manually over a very narrow region (~lMHz). The J = 0 -* 1 t r a n s i t i o n of methyl iodide i s s p l i t i n t o a number of 26 l i n e s the ones of higher i n t e n s i t y being at t 14 695.22 MHz (F = 5/2—»5/2) 15 100.74 MHz (F = 5/2 »7/2) 15 275.87 MHz (F = 5/2—»3/2) 31 The f i r s t and the l a s t ones were singled out f o r measurement since t h e i r Stark lobes were f a s t e r and e a s i l y resolvable at low f i e l d s ; the ex-perimental measurements and the t h e o r e t i c a l c a l c u l t i o n s are shown i n Tables I I and I I I , 26 The J = l - * 2 (K=l) t r a n s i t i o n i s s p l i t i n to the following l i n e s j 29 753.71 MHz (F = 5/2 »7/2) 29 782.71 MHz (F = 5/2 '5/2) 29 923.50 MHz (F = 5/2 »3/2) 29 939.87 MHz (F = 7/2 *7/2) 29 986.84 MHz (F = 7/2 *5/2) 30 123.64 MHz (F = 7/2 »9/2) 30 215.95 MHz (F=3/2 *3/2) We chose the l i n e at 30 123.64 MHz (F = 7/2—>9/2) which i s the most intense of the group and has e a s i l y i d e n t i f i a b l e Stark components. The r e s u l t s and calculations are shown i n Table IV, III.3.1 Discussion of the Results. The examination of Tables I I , I I I , and TV shows a very good agreement between the observed and t h e o r e t i c a l frequencies; i f f a c t , a l l the differences i n frequency are of the same order of magnitude as the experimental error i n t h e i r measurement. This can be considered a good check on the correctness of the matrix elements as w e l l as on the program as a whole. The r e s u l t s presented i n Tables I I , I I I , IV and V were obtained using matrix dimensions of 20 x 20, that i s , with matrix elements up to J = 4 i n t h i s case. Tests were made on the influence of the matrix s i z e by running the program with 5 x 5» 10 x 10, 15 x 15, 20 x 20, and 30 x 30 for the J = 1 —* 2 and J = 0—*1 t r a n s i t i o n s of methyl iodide. 32 TABLE I I J = 0-»l, F = 5/2->5/2, M p = 5/2 Transition of Methyl Iodide at 14 695.22 MHz Stark F i e l d Observed Line Theoretical Line Difference Between (v/cm) Frequencies Frequencies Obs, & Theor, squencies flrequenc (MHz) (MHz) Frequencies 767 14 707.47 14 707.462 0.008 779 14 707.85 14 707.850 0 . 0 0 0 789 14 708.18 14 708.177 0 .002 798 14 708.51 14 708.476 0 . 0 0 3 810 14 708.89 14 7O8.878 0.001 821 14 709.22 14 709.253 - 0 . 0 3 2 832 14 709.58 14 709.632 - 0 . 0 5 2 842 14 7 0 9 . 9 ^ 1^ 709.981 -0.041 85I 14 709.29 14 710.299 - 0 . 0 0 9 860 14 710.70 14 710.621 0.079 Av> = I2-I5 MHz " X - -1933,28 + 1.12 MHz = 1.6362 + 0.0008 Debyes 33 TABLE I I (continued) Stark F i e l d (v/cm) 1448 1459 1469 1480 1490 1500 15U 1521 1531 1541 Observed Line Frequencies (MHz) 14 738.90 14 739.59 14 740.22 14 740.81 14 741.48 14 742.00 14 742.6? 14 743.33 14 744.00 14 744.70 Theoretical Line Frequencies (MHz) 14 738.907 14 739.565 14 740.168 14 740.835 14 741.446 14 742.061 14 742.741 14 743.364 14 743.991 14 744.621 Difference Between Obs. & Theor. Frequencies -0.007 0.025 0.052 -0.025 ,0.034 -0.061 -0.071 -0.034 0.009 0.079 -V _ 43.50 MHz X = -1932.85 + 2.68 MHz = 1.6389 + 0.0003 Debyes 34 TABLE I I (concluded) Stark F i e l d (v/cm) 2130 2141 2152 2162 2172 2182 2192 2202 2213 2223 Observed Line Frequencies (MHz) 14 788.67 14 789.5^ 14 790.48 14 79L27 14 792.20 14 793.11 14 79^.00 14 794.90 14 795.77 14 796.65 Theoretical Line Frequencies (MHz) 14 788.616 14 789.55^ 14 790.496 14 791.356 14 792.220 14 793.O87 1* 793.957 14 79^.83l 14 795.796 14 796.678 Difference Between Obs, & Theor. Frequencies 0.054 -0.014 -0.016 -0.086 -0,020 0.023 0.043 0.069 -0.026 -0.028 Av> = 93-101 MHz X = -I93I.63 + 3.89 MHz >U. = 1.6392 + 0.0001 Debyes 35 TABLE I I I J = 0-+1, F = 5/2-^3/2, M p = 1/2 Transition of Methyl Iodide at 15 275.87 MHz Stark F i e l d Observed Line Theoretical Line Difference Between (v/cm) Frequencies Frequencies Obs. & Theor. (MHz) (MHz) Frequencies 767 15 285.21 15 285.199 0.010 779 15 285.4? 15 285.491 -0.021 789 15 285.75 15 285.738 0.012 798 15 286.00 15 285.963 0.037 810 15 286.27 15 286.268 0.002 821 15 286.51 15 286.551 -0.040 832 15 286.81 15 286.838 -0.028 842 15 28?.09 15 287.102 -0.012 851 15 287.36 15 287.343 0.017 860 15 287.61 15 287.586 0.024 Av» =9-11 MHz X - -1934.22 + 0.78 MHz = 1.6353 + «0006 Debyes 1 3 6 TABLE I I I (Continued) Stark F i e l d (v/cm) 1448 1^59 1469 1480 1490 1 5 0 0 1 5 1 1 1 5 2 1 1 5 3 1 1541 Observed Line Frequencies (MHz) 1 5 3 0 9 . 6 7 1 5 310.18 1 5 3 1 0 . 7 0 1 5 3 H . 2 1 1 5 3 1 1 . 7 2 1 5 3 1 2 . 1 9 1 5 3 1 2 . 7 2 1 5 3 1 3 . 2 3 1 5 3 I 3 . 7 9 1 5 3 1 ^ . 2 9 Theoretical Line Frequencies (MHz) 1 5 309.664 1 5 3 1 0 . 1 9 2 1 5 3 1 0 . 6 7 7 1 5 3 H.214 1 5 3 1 1 . 7 0 6 1 5 3 1 2 . 2 0 2 1 5 3 1 2 . 7 5 2 1 5 3 1 3 . 2 5 6 1 5 3 1 3 . 7 6 3 1 5 3 1 ^ . 2 7 ^ Difference Between Obs. & Theor, Frequencies 0.062 -0.012 0 . 0 2 3 -0.004 0.014 -0.012 - 0 . 0 3 2 -0.026 0 . 0 2 7 0.016 Av> = 3 4 - 3 8 MHz "X _ - 1 9 3 4 . 2 8 + 1 . 1 3 MHz <M = 1,6401 + ,0002 Debyes 3? TABLE I I I (Concluded) Stark F i e l d (v/cm) 2130 2141 2152 2162 2172 2182 2192 2202 2213 2223 Observed Line Frequencies (MHz) 15 35L38 15 352.19 15 353.00 15 353.81 15 354.57 15 355.35 15 356.11 15 356.92 15 357.66 15 358.53 Theoretical Line Frequencies (MHz) 15 351.378 15 352.203 15 353.034 15 353.793 15 354.557 15 355.326 15 356.098 15 356.875 15 357.735 15 358.521 Difference Between Obs. & Theor. Frequencies 0.002 -0.013 -0,034 0.017 0.013 0.024 0.012 0.045 -0.075 0.009 A V = 75-82 MHz " X = -1936.75 + 2.58 MHz A = I.6368 + 0.0001 Debyes 38 TABLE IV J = 1~*2, F = 7/2-»9/2, M F = 1/2 Transition of Methyl Iodide at 30 123.64 MHz Stark F i e l d Observed Line Theoretical Line Difference Between (v/cm) Frequencies Frequencies Obs. and Theor. (MHz) (MHz) Frequencies 1448 30 157.55 30 157.547 0.003 1459 30 157.66 30 157.670 -0.010 1469 30 157.79 30 157.783 0.007 1480 30 157.93 30 157.910 0.020 1490 30 158.02 30 158.027 -0.007 1500 30 158.14 30 158.145 -0.005 1511 30 158.25 30 158.278 -0.028 1521 30 158.40 30 158.400 ,0.000 1531 30 158.54 30 I58.523 0.017 1541 30 158.65 30 158.648 0.002 Av> = 34-35 MHz "X = -1933.80 + 0.97 MHz JX = I.6345 + 0.0004 Debyes 39 TABLE IV (Concluded) Stark F i e l d (v/cm) 2130 2141 2152 2162 2172 2182 2192 2202 2213 2223 Observed Line Frequencies (MHz) 30 168.67 30 168.89 30 169.14 30 169.35 30 169.57 30 I 6 9 . 8 I 30 170.00 30 170.22 30 170.44 30 170.67 Theoretical Line Frequencies (MHz) 30 168.674 30 168.905 30 I69.I38 30 169.351 30 169.565 30 169.780 30 169.997 30 170.215 30 1 70.457 30 I7O.678 Difference Between Obs. and Theor, Frequencies -0.004 -0.015 0.002 -0.001 0.005 0.030 0.003 - 0.005 -0.017 -0.008 Av> = 45-57 MHz X - -I934.9O +1.80 MHz P- = 1.6345 + 0.0002 Debyes TABLE V Summary of the Results F i e l d Region Quadrupole Dipole Moment (v/cm) Constant (MHz) (Debyes) 767 - 860 -1933.28 + 1.12 I.6362 + 0.0008 1448 - 1541 -1932.85 + 2.68 1.6389 + 0.0003 2130 - 2223 -1931.63 + 3.89 1.6392 + 0.0001 767 - 860 -1934.22 + 0.78 I.6353 + 0.0006 1448 - 1541 -1934.28 +1.13 1.6401 + 0.0002 2130 - 2223 -1936.75 + 2.58 1.6368 + 0.0001 1448 - 1541 -1933.80 + 0.97 1.6345 + 0.0004 2130 - 2223 -1934.90 +1.80 1.6345 + 0.0002 Unbiased Average Values and t h e i r Standard Deviations! X = -1933.96 + 0.53 MHz /* - I.6369 + 0.0008 Debyes 40 As expected the 5 ^ 5 and 10 x 10 runs showed unreliable r e s u l t s , and the 15 x 15 showed only a small deviation, while the 20 x 20 (up to J = 4) and 30 x 30 (up to J. = 6) gave i d e n t i c a l r e s u l t s within the standard deviations. In view of t h i s fact a l l calculations were done using matrices 20 x 20 i n order to save computer funds. The values of the nuclear quadrupole constant agree very well with 27 the published value which i s inside the standard error i n t e r v a l i n a l l cases. The values of the dipole moment also agree with the published 3 value-' within the range of i t s accuracy i n a l l cases, but our values tended to c l u s t e r on the lower side of the i n t e r v a l . We examined reference 3 i n d e t a i l and found that t h e i r measurements were made at r e l a t i v e l y low f i e l d s (140-3^ v/cm) where the c a l i b r a t i o n of the c e l l was not as precise as ours due to the small displacement of the Stark lobe of the J = 0—^1 t r a n s i t i o n of 0GS (~1 MHz)j since i t i s an early publication the value of the dipole moment of OCS used i n the c a l i b r a t i o n of t h e i r c e l l (O.7O85 + 0,004 Debyes) d i f f e r s from the l a t e s t published value used i n t h i s thesis ( c f , j section 111,2), For these reasons we think that our re s u l t s are more accurate and r e l i a b l e than t h e i r s . F i n a l l y we conclude that the average i n Table V i s the most accurate value to date of the molecular dipole moment of methyl iodide, 111,5 Conclusions and Comments This thesis has i l l u s t r a t e d the use of double perturbation theory i n microwave spectroscopy. As an example, we have successfully applied double perturbation theory to the problem of determining the molecular 1 dipole moment i n the presence of a nuclear quadrupole coupling. We have been able to obtain an accurate value f o r the dipole moment of methyl iodide, and our re s u l t s could yet be improved by using a better Stark 41 c e l l and more r e l i a b l e and accurate Stark f i e l d s . This work can be extended i n several directions i n a very natural way. The f i r s t p o s s i b i l i t y i s to apply the theory to larger classes of molecules such as asymmetric tops with one quadrupolar nucleus. Besides more complexity i n the matrix elements, the number of parameters to be determined i s increased» now three components are needed f o r the dipole moment and two parameters to describe, appropriately the quadrupole coupling tensor. Another a p p l i c a t i o n would be the determination of the nuclear quadrupole coupling constants of a molecule containing two quadrupolar nuclei i n the absence of a Stark f i e l d . 42 BIBLIOGRAPHY 1. J . 0. Hirschfelder, W. Byers Brown and S. T, Epstein, Adv. Quantum Chem. 1, Academic Press, New York, 1964, p. 255 2. D. P. Chong, J. Phys. Chem., 75_, I549 (1971). 3. R. G. Shulman, B. P, Dalley and C, H, Townes, Phys. Rev., Vol, 78, 1^ 5 (1950). 4. C, K. Ingold, Structure and Mechanism i n Organic Chemistry, Cornell Univ. Press, New York, 1953# P. 94. 5. P. Debye, Polar Molecules. Dover Publ., Inc., New York, 1929, 6. C, H, Townes and A, L. Schawlow, Microwave Spectroscopy, MacGraw-H i l l , New York, 1955, P. 268. 7. E, Segre, Nuclei and P a r t i c l e s , Benjamin, Inc., New York, 1965, p, 221. 8. J. E. Wollrab, Rotational Spectra and Molecular Structure, Academic Press, New York, I967, p. 117. 9. F. Coester, Phys. Rev,, 77, 454 (1950)| U. Fano, US Nat. Bur. Standards , 40, RP1866 (1948)j J , Bardeen and C. H. Townes, Phys. Rev. 6, 627 (1948)t W, Low and C, H. Townes, Phys. Rev. 9, 1295 (1949). 10. C. H. Townes and A. L. Schawlow, op, c i t , , p. 259. 11. J. E. Woolrab, op. c i t . , p. 260. 12. J . E. Woolrab, op, c i t . , p. 261j W. Low and C, H, Townes, Phys. Rev,, 76, 1295 ( W ) . 13. J. A. Hebden, Ph.D. Thesis, U.B.C., 1970, Appendix A. 14. J . E, Woolrab, op. c i t , , p. 15. 15. J , E. Woolrab, op. c i t , , p. 117. 16. M. E, Rose, Elementary Theory of Angular Momentum, John Wiley & Sons, New York, 1966, p. 117. 17. M. E, Rose, op. c i t . , p. 119. 18. M, W, P. Strandberg, Microwave Spectroscopy, Methuen,"London, 1 9 5 \ P. *3. 19. M. E. Rose, op, c i t , , p. 119. 20. M. W. P. Strandberg, op, c i t , , p, 4. 43 21, D. P. Chong, QCPE 139. 22, J. S. Muenter, J. Chem. Phys., 48, 4544 (1968). 23, L. H, Sharpen, J, S, Muenter, and V, W, Laurie, J, Chem. Phys., 46, 2431 (1967) 24, C, H, Townes and A. L. Schawlow, op, c i t . , Chapter 15. 25, J, Muenter, Ph.D. Thesis. Stanford Univ., 1966. 26, Microwave Spectral Tables. National Bureau of Standards Monograph 70, Volume TV, Washington, D.C., 1968, p. 92. 27, F, Sterzer and Y. Beers, Phys. Rev. 100, 1174 (1955). 44 APPENDIX Figure 2, L i s t i n g o f the Main Program, Subroutines, and Functions, C ALL ARRAYS IN THE SUBROUTINES CONTAIN A VARIABLE DIMENSION ND IN C ORDER TO SAVE SPACE C N IS THE SIZE OF THE BASIS SET ACTUALLY USED (N.LE.ND) C ND AMD N MUST BE SET IN T HF MAIN PROGRAM .  C IF(N ALL.EQ. 1 ) ALL STATES WILL BE COMPUTED, AND NWANT AND KWANT C NEED NOT BE S PEC 1 FIFO C IF(NALL.NE. 1) THEN KWANT SPECIFIES THE TOTAL NUMBER OF STATES C WANTED C KWANT SPECIFIES THE STATES WANTED, E.G. NALL = 0 NWANT = 2 C KWANT(1) = 3 KWANT(?)=7 MEAN THAT STATES 3 AND 7 ONLY WILL BE C CALCULATED C IF(NWF.EQ.O) PRINTING OF WAVEFUNCTIONS WILL BF SUPRESSED C IF(NSUPl.EQ.O) PRINTING OF PERI WILL BE SUPRESSED C I F ( NSU P2 . E 0.0 ) PRINTING OF EE (PEPT2) WILL BE SUPRESSED C IF(NSUP3.EQ.O) PRINTING CF FN (APER) WILL BE SUPRESSED _C IF(NSUM.EQ.Q) PRINTING OF PERTURBATION SUMS WILL BE SUPRESSED  C IF(NTR.NE.l) WEIGHTS WT(I) ARE MADE EQUAL TO UNITY C IMPLICIT REAL*8(A-H,0-Z) 01 MENS ION HQ (40 ,40 ) , HQ Ml (40, 40.) , HS (40 , 40 ), HSMK 40, 40 > ,0R ( 10,2) , 1EB(40),EP1(40),EN(8,8,20),ENG(40) ,EPMAT(40,40) ,U(4G ,40 ) , 2DEL(40),C(4,4,40),EE(8,8,20),VP(40,40),WP( 40,40) ,AV(4n) ,BV(4^) , 3CV(40) ,E ( 20 ,7) , KWANT (40) , X(40 ) , IQ(40 ) ,NID(40 ,40 ) , SNJ (60 ), SNF ( 60) DIMENSION UR(2,2),IR(2),XR(2),EM(10,2,2,2,2),AN(10,2,2,2,2,2), 1FQCALI10),FQOBS(10),NU(10),NL( 10 ),EPSON( 10),DELTA( 2) ,WT( 10) , 2V0LT(10) ,R(2 ,2) ,GEY(10,2),CK(10,2,2),EL(10,2,2,2),HMGT(40,40), 3SIGMAC2),EP(40,10),HQM(40,40,10),HSM(40,40,10) COMMON M,Q  C0MMON/X1/ NSUP1 COMMON/X 2/ NSUP2,NWF CCMMON/X 3/ NSUP3 COMMON/N W/ NTR INTEGER C ND=40  NALL=0 READ(5,9) N,NWF,NSUP 1,NSUP2,NSUP 3,NSUM,NTR,NFC PRINT 11 11 F0RMAK/5X, 1N NWF NSUP1 NSUP2 NSUP3 NSUM NT R NFQ •) WRIT E(6,9) N,NWF,NSUP1,NSUP 2,NSUP 3,NSUM,NTR,NFQ 9 F0RMAT(8I6)  READ(5,1) A,B ,CI,SI,SK,SMF,MAXJ 1 FORMAT(3D13.0,308.0,14) READ(5,2) AKZZ,AMU,DJ,DJK 2 FORM AT(4D10.0 ) WRITE(6,61) AKZZ,AMU,DJ,DJK fel FORMA T ( 1H0,//1X,'AKZZ = « ,F14.5,5X,«AMU = ' , F1.0 .5 ,  15X,«DJ =»,F14.6,5X,'DJK =,,F14.6) WRITE(6,3) A,BtCI , S I , S K , SMF , M AX J 3 FORMAT ( 1H0, *A = • ,F 10. 2, 5X, »13 = • , F 1 0 .2 , 5 X , • C = • , F 10 .2 , / / ] X , 1M =• , F 5 . 2 , 5 X , • K = ' , F5. 2, 5X, «MF = » , F5.2 , 5X,•MA XJ = • , I 3) CALL FACTOR _ MAXJ = MAXJ + 1  45 5 C A L L B A S I S ( A , B , C I , S I , S K , S M F , M A X J f N D , H Q , H S , E P M A T , E B , N I D , S N F , 1 S N J , D J , D J K ) C C I N P U T O F E X P E R I M E N T A L F R E Q U E N C I E S - F Q O B S C 5 9 W R I T E ( 6 , 6 0 )  6 0 F O R M A T ( / / 5 X U P P E R J * , 6 X , ' U P P E R F ' , 6 X , " L O W E R J , , 6 X f , L 0 W E R F « , 7 X , 1 ' F Q O B S » , 7 X , • I » , 7 X , ' V O L T A G E • ) D O 6 6 1 = 1 , N F Q R E A D ( 5 , 8 6 ) U J , U F , D J , D F , F Q O B S ( I ) , V O L T { I ) 8 6 F O R M A T ( 4 D 8 . 0 , 2 D 1 3 . 0 ) N U ( I ) = I N D E X ( N C , N I D , U J , U F , S I )  N L ( I ) = I N D E X ( N O , N I D , O J , D F , S I ) W R I T E ( 6 , 8 7 ) U J , U F , 0 J , D F , F Q O B S ( I ) , I , V O L T ( I ) 8 7 F 0 R M A T I 5 F 1 3 . 5 , I 6 , F 1 3 . 5 ) „ 6 6 C O N T I N U E C 0 0 6 7 1 = 1 , N F Q  K W A N T ( I ) = N U < I I 6 7 C O N T I N U E I M I N = N F C + 1 N W A N T = 2 * N F Q 0 0 6 8 J = I M I N , N W A N T K K = J - I M I N + l  K W A N T ( J ) = N L ( K K ) 6 8 C O N T I N U E V . R I T E ( 6 , 7 2 ) 7 2 F O R M A T { 1 H O , 6 H K W A N T ) W R I T E ( 6 , 6 9 ) ( K W A N T < K ) , K = 1 , N W A N T ) 6 9 F O R M A T ( I 0 1 4 )  C C T R A N S F O R M A T I O N T O C I A G O N A L B A S I S C . N C Y L 1 0 = 1 6 5 D O 9 5 1 = 1 , N F Q 1 M I N U S = I - 1  I F ( 1 M I N U S . L T . 1 ) G O T O 8 4 D O 6 3 K = l , I M I N U S I F ( D A B S ( V O L T ( I ) - V O L T ( K ) ) . L T . l . D - 5 ) G O T O 6 4 . G O T O 6 3 6 4 K A P P A = K G O T O 7 5  6 3 C O N T I N U E 8 4 C Q E F F = 0 . 5 0 3 4 0 2 D 0 * A M U * V O L T { I ) D O 7 0 K = 1 , N _ D O 7 0 J = 1 , N H N O T ( K , J ) = E P M A T ( K , J ) + A K Z Z * H Q ( K , J ) + C G E F F * H S ( K , J ) 7 0 C O N T I N U E  C A L L H D I A G ( H N C T , N O , N , 0 , U , N R , X , I Q ) D O 7 9 M = 1 , N D O 7 9 J = M , N . . . _ -T E R M Q = 0 . 0 D 0 T E R M E = 0 . C D O D O 8 0 L = 1 , N  D O 8 0 K = l , N T E R M Q = T E R M Q + U ( L , M ) * H Q ( L , K ) * U ( K , J ) T E R M E = T E R M E + U ( L , M ) * H S ( L , K ) * U ( K , J ) * 0 . 5 0 3 A 8 D 0 * V O L T ( I ) 8 0 C O N T I N U E H Q M ( M , J , I ) = T E R M Q H O M ( J , M , I ) = T E R M Q '  46 H S M ( M , J , I ) = T E R M E H S M ( J , M , I ) = T E R M E 7 9 C O N T I N U E D O 8 5 K = 1 , N E P ( K , I ) = H N O T ( K , K ) 8 5 C O N T I N U E :  C C P E R T U R B A T I O N S E R I E S C . . K=I G O T O 7 6 7 5 K = K A P P A 7 6 D O 7 3 J = 1 , N D O 7 3 L = 1 , N E P 1 ( L ) = E P ( L , K ) H Q M 1 ( J , L ) = H Q M ( J , L , K ) H S M 1 ( J , L ) = H S M ( J , L , K ) 7 3 C O N T I N U E  Q = N U ( I ) I F ( Q . E Q . N U ( K ) . A N D . I . N E . K ) G O T O 3 2 C A L L A P E R ( N D , N W A N T , E P 1 , H Q M 1 , H S M 1 , I , E N , C , E E , D E L , V P , W P , A V , 8 V , C V , E ) G O T O 3 0 3 2 D O 3 3 J = l , 8 D O 3 3 L = l , B  E N ( J , L , I ) = E N ( J , L , K ) 3 3 C O N T I N U E 3 0 Q = N L ( I ) . . . . . I W = I + N F Q I F ( Q . E G . N L ( K ) . A N D . I . N E . K ) G O T O 3 4 C A L L A P E R ( N D , N W A N T , E P 1 , H O M l , H S M I , I W , E N , C , E E , D E L , V P , W P , A V , B V , C V , E ) G O T O 3 6 3 4 K W = K + N F U D O 3 5 J = l , 8 . „ . D O 3 5 L = l , 8 E N ( J , L , I W ) = E N ( J , L , K W ) 3 5 C O N T I N U E  C C C A L C U L A T E D F R E Q U E N C I E S ( F Q C A L ) C 3 6 N U P = N U ( I ) N L P - N L ( I ) F Q C A L { I ) = E P 1 ( N U P ) - E P 1 ( N L P )  9 5 C O N T I N U E C C C A L C U L A T I O N O F E P S O N . . . . . _ P R I N T 1 0 2 1 0 2 F O R M A T ( / / I X I E P S O N ' J D O 1 . 0 0 1 = 1 , N F Q  E P S C N ( I ) = F Q O B S ( I ) - F Q C A L ( I ) W R I T E ( 6 , 1 0 3 ) I , E P S O N U ) 1 0 3 F O R M A T ( 1 3 , F 1 5 . 7 ) 1 0 0 C O N T I N U E -C C L E A S T S Q U A R E S F I T  C C A L L N E W L Q ( N , N W A N T , N F O , E N , N U , N L t E P S C N , D E L T A , G E Y , C K , E L , E M , A N , 1 Q R , W T , R ) T E S T - D E L T A ( 1 ) * * 2 + D E L T A ( 2 ) * * 2 W R I T E < 6 , 1 0 1 ) T E S T » N C Y L 1 0 1 0 1 F O R M A T ( 1 H O , ' T E S T M P R O G L O O P = ' , D 1 5 . 7 , 5 X , ' N C V I . 1 . 0 = , , I 4 )  4 / I F ( T E S T . L T . 1 . 0 D - 9 . O R . N C Y L 1 0 . E Q . 1 0 ) G O T C 1 1 0 A K Z Z = A K Z Z + D E L T A ( 1 ) A M U = A M L + D E L T A ( 2 ) N C Y L 1 0 = N C Y L 1 0 + 1 G O T O 6 5 1 1 0 S = 0 . 0 D 0  D O 2 9 0 1 = 1 , N F Q S = S + E P S O N ( I ) * W T ( I )* E P S O M I ) 2 9 0 C O N T I N U E C A L L H D I A G ( R , 2 , 2 , 1 , U R , M R , X R , I R ) D O 2 9 5 J = l , 2 I F ( N F Q . L E . 2 ) G O T O 2 9 2  S I G M A T J ) = D S Q R T { S / ( D A B S ( R ( J , J ) ) * ( N F Q - 2 ) ) ) G O T O 2 9 5 2 9 2 S I G K A ( J ) = 1 . 0 D 1 5 2 9 5 C O N T I N U E P R I N T 9 6 , A K 7 Z , S I G M A ( 1 ) , A M U , S I G M A ( 2 ) 9 6 F O R M A T ( / / I X , ' A K Z Z = ' , F 1 5 . 7 , I X , 0 1 5 . 7 , 1 O X , ' A M U = » , F 1 5 . 7 , 1 X , Dl 5 . 7 ) P R I N T 9 7 9 7 F 0 R M A T J / / 3 X , ' I ' , 6 X , « F G C A L ' , 6 X , ' F G O B S ' ) D O 1 2 0 1 = 1 , N F Q P R I N T 9 8 , I , F Q C A L ( I ) , F Q O B S ( I ) 9 8 F O R M A T ( I 3 , 2 F 1 5 . 7 ) 1 2 0 C O N T I N U E  S T O P F N D S U B R O U T I N E B A S I S ( A , B , C I , S I , S K , S M F , M A X J , N D , H Q , H S , E P M A T , E B , 1 N M , S N F , S N J , D J , D J K ) C B A S I S C A L L S O T H E R S U B R O U T I N E S A N D E V A L U A T E S T H E M A T R I X C E L E M E N T S F O R S Y M M E T R I C T O P S ( U N P E R T U R B E D H A M I L T O N I A N ,  C D I P O L E C O U P L I N G A N D Q U A D R U P O L E C O U P L I N G ) C I N P U T P A R A M E T E R S A R E : A , B , C , S I , S K , S M F , M A X J , N D , D J , D J K c O U T P U T P A R A M E T E R S A R E : E P . M A . T , H S , H Q , N I D , S N F , S N J . I M P L I C I T R E A L * 8 ( A - H , C - Z ) D I M E N S I O N H £ ( N D » N D ) , H S ( N D , N D ) , E B ( 3 0 ) , E P ! « A T ( N D , N D ) , N I D ( N O , N D ) , 1 S N J ( 4 0 ) , S N F ( A O )  I D = 1 D O 5 K = 1 , M A X J S J = K - 1 ... _ . . ; _ .. . I F ( S J - S I ) 1 0 , 1 0 , 1 5 1 0 M A X F = 2 . 0 D 0 * S J + 1 . O D O G O T O 2 0 1 5 M A X F = 2 . 0 D 0 * S I + 1 . O D O 2 0 DO 5 L = 1 , M A X F F D C = L . _ „ _ _ F = D A B S ( S J - S I ) + F D C - 1 . 0 D 0 A B = C A B S ( A - B ) 1 F ( A B . L T . 1 0 . O D - 9 ) G O T O 2 5  E B C I D ) = B * S J * ( S J + l . O D O ) + ( A - B ) * S K * * 2 G O T O 3 0 2 5 E B ( I D ) = B * S J * ( S J + 1 . O D O ) + ( C I - B ) * S K * * 2 3 0 E B ( I D ) = E B ( I D ) - D J * ( S J * ( S J + l . O D O ) ) * * 2 - D J K * S J * ( S J + 1 . 0 D 0 ) * S K * * 2 N I D ( K , L ) = I D S N J ( I D ) = S J S N F ( I D ) = F I D = I D + 1 5 C O N T I N U E . . . . . „ . . . _ 5 9 9 W R I T E ( 6 , 6 0 0 ) 6 0 0 F O R M A T ( 1 H 0 , • D I C T I O N A R Y • . / l X . ' I D ' ^ X ^ J ' j l O X . ' F 1 ) D O 7 0 0 I = 1 , N D  48 WRITE (6 ,800 ) I i S N J ( l ) , S N F ( I ) 800 F O R M A T ! / I X , 1 2 , F 1 0 . 2 , F l 0 . 2 ) 700 CONTINUE DO 35 L=1,ND DO 3 5 M= 1,ND EPMAT( L , M ) = 0 .OCO  H0 (L ,M)=C.ODO HS( L ,M ) = 0.ODO 35 CONT INUE DO 40 K=1,ND EPM AT ( K , K ) = EB ( K ) 40 CONTINUE  DO 45 J=1,ND DO 45 K=J ,ND S J=S NJ (J ) . .. . F = SNF ( J) SJP = SNJ{ K) FP=SNF IK ) ; NDIF=SJP-SJ NDI FF= DABS ( FP-F ) I F {NDIF .NE.0 ) GO TO 50 I F ( N D I F F . G T . 1 . 0 D - 3 ) GO TO 42 TEQ= ( 3 .0D0*SK**2-S. J* { S J+1.0D0) )/( ( S J+ 1. CDO) * ( 2 . ODO* S J + 3 . 0 DO ) ) H Q ( J , K ) = Q U A O t S I , S J , S J P , F ) * T E Q  42 I F ( S J . L T . l . O D - 3 ) GO TO 41 H S ( J , K ) = - D I P O L E ( S I , S J , S J P , F , F P , S M F ) * S K / D S Q R T ( S J * ( S J + 1.ODO) ) GO TO 56 .. • _ 41 HS( J , K ) = C . 0 D 0 GO TO 56 50 I F ( N D I F . N E . 1 ) GO TO 55  1 F I N D I F F . G T . l . O D - 3 ) GO TO 43 TEQ = 3 . 0 D 0 * S K / ( ( S J + 1 .0DO) * ( S J+2 .ODO ) ) TEQ= TE Q* DSORT{( ( S J + 1 . O D O ) * * 2 - S K * * 2 ) / ( 2 . 0 DO *S J + 3•0 DO ) ) HQ(J,K ) = OUAf:-( S I , S J , S J P , F ) * T E Q 43 TST = DSQST(( (SJ + 1 . 0 0 0 ) * * 2 - S K * * 2 ) / ( (SJ + 1 . 0 D 0 ) * ( 2 . 00 0 * SJ + 3.OD C)) ) H S ( J , K ) = - D I P O L E ( S I , S J , S J P t F , F P , S M F ) * T S T  GO TO 56 55 I F ( N D I F . N E . 2 ) GO TO 45 I F ( N D I F F . G T . 1 . O D - 3 ) GO TO 56 TEQ=3.0 00/ ( ( S J+2 .0 DO)* (2 ,ODO*SJ + 3 .ODO)) TEQ=TEQ*DSQRT(( (SJ+ 1 . O D O ) * * 2 - S K * * 2 ) * ( ( S J + 2 . O D O ) * * 2 ~ S K * * 2 ) / ( ( S J + 1 1 . 0 D 0 ) » ( 2 . 0 D 0 * S J + 5 . 0 D 0 ) ) )  H G ( J , K ) = G U A G ( S I , S J , S J P , F ) * T E Q 56 HO(K , J )=H0( J ,K ) H S ( K , J ) = H S ( J , K ) 45 CONTINUE WR ITE (6 ,62 ) 62 F O R M A T ( / / I X , 3 0 H QUACRUPOLE PERTURBATION INPUT )  CALL E PR IN T(H G ,ND ,ND ,ND ,ND) W R I T E ! 6 , 6 3 ) 63 FORMAT ( / / I X ,30H UNPERTURBED HAMILTONIAN INPUT) CALL E PR I NT(E PMAT ,ND,ND,NO,ND) WR ITE (6 ,64 ) 64 F O R M A T ( / / I X , 2 6 h DIPOLE PERTURBATION INPUT)  CALL E P R I N T ( H S , N D , N D , N D , N D ) RETURN END SUB ROUT INE APER(ND,NWA NT ,E P ,V ,W, L O G , E N , C , E E , D E L , V P , W P , A V , B V , C V , t ) C C APER CALLS CTHFR SUBROUTINES AND EVALUATES F ( L,M) p p p  4y C 1 . L E . ( L + M ) . L E . 7 C E V A L U A T I O N O F E ( L , M ) F O R 1 . L E . ( L + M ) . L E . 4 I S A C H E C K O N P E R T ? C I N P U T P A R A M E T E R S A R E N D » N W A N T , E P , V , W C O U T P U T P A R A M E T E R I S E N C A L L O T H E R C A L L I N G P A R A M E T E R S A R E D U M M I E S c : i I M P L I C I T R E A L * 8 ( A - H , O - Z ) D I M E N S I O N C ( 4 , 4 , N O ) , V ( M D , N D ) , W ( N D » N D ) , E P ( N D ) , E N ( 8 , 8 , N W A N T ) , 1 E E ( 8 , 8 , N W A N T ) , D E L ( N D ) , V P ( N D , N D ) , W P ( N D , N D ) , A V ( N D ) , 2 B V ( N D ) , C V ( N D ) , E ( N W A N T , 7 ) C O M M O N N , Q C C M V C N 7 X 3 / N S O P 3  I N T E G E R 0 R D E R , P , P 1 , P 2 , Q Q , Q 1 , Q 2 » A , B , Q C C E N ( L , M ) = E ( L - 1 t M - 1 ) C D G 1 2 I K = 1 , 8 D O 1 2 I J = 1 , 8 -E N ( I K , I J , L G C ) = 0 . O D O 1 2 C O N T I N U E C A L L P E R T 2 ( N D , N W A N T , V , W , E P , C , E E , V P , Vi P , 0 E L , A V , B V , C V , F ) E N ( 1 , l T L O C ) = E P ( Q ) D O 1 0 M = l , 8 M A X = 9 - M  D O 1 0 L = 1 , M A X I F ( L . E G . l . A N D . M . E Q . l ) G O T O 1 0 O R D E R = L + M B = O R D E R / 2 A = O R D E R - B - l C C F I R S T T E R M C T E R M I N G . O D O . . . . . J F ( L . E Q . l ) G O T O 2 0 L 1 = M A X 0 ( 1 , L - B ) L 2 = M I N 0 ( A , L - 1 )  D O 1 L L = L 1 , L 2 S U M = O . O D O D O 2 J = 1 » N D O 2 K = 1 , N S U M = S U M + C ( L L , A - L L + 1 , J ) * V ( J , K ) * C ( L - L L , B - L + L L + 1 , K ) 2 C O N T I N U E  T E R M 1 = T E R M 1 + S U M 1 C O N T I N U E C C S E C O N D T E R M C 2 0 T E R M 2 = C . 0 D 0  I F ( M . E C . l ) G O T O 3 0 M 1 = M A X C ( 1 , M - B ) M 2 = M I N O ( A , M - 1 ) D O 3 M M = M l , M 2 S U M = 0 . O D O D O 4 J = 1 , N  D O 4 K = l , N S U M = S U M + C ( A - N M + 1 , M N , J ) * W ( J , K ) * C ( B - M + M M + 1 , M - M M , K ) 4 C C N T 1 N U E T E R M 2 = T E R M 2 + S U N 3 C O N T I N U E C 50 C T H I R D T E R M C 30 T E R M 3 = 0 . 0 D 0 I F ( O R D E R . E Q . 3 . O R . O R D E R . E Q . 4 ) G O T O 4 0 DO 5 K = 2 , B K A P P A = K - 1  T E N J = 0 . 0 D 0 DO 6 J = l , K A P P A P 1 = M A X 0 ( 1 , L - K - B + J + l ) _ . . • P 2 = M I N 0 ( L , A + J - K + l ) T E N P = 0 . 0 D 0 D O 7 P = P 1 , P 2  Q 1 = M A X 0 ( l , l . - K - P + 2 ) Q 2 = M I N 0 ( B - J + l , L - P + 1 ) T E N Q = 0 . 0 D 0 _ . _ DO 8 Q Q = Q 1 , 0 2 S U M = 0 . 0 D 0 DO 9 J J = 1 , N  S U M = S U N + C ( P t A + J - K - P + 2 , J J ) * C ( G Q » B - J - g Q + 2 , J J ) C O N T I N U E T E N Q = T E N Q + S U M * E N ( L - P - Q Q + 2 , K - L + P + Q Q - 1 , L O C ) C O N T I N U E T E N P = T E N P + T E N G C O N T I N U E  T E N J = T E N J + T E N P C O N T I N U E T E R M 3 = T E R M 3 + T E N J _ C O N T I N U E E N ( L » M , L O C ) = T E R M 1 + T E R M 2 - T E R M 3 C O N T I N U E ' I F ( N S U P 3 . E Q . 0 ) G O T O 5 7 P R I N T 5 6 , Q C A L L F P R I N T ( E N , 8 , 8 , N W A N T , 8 , 8 , L 0 C ) . _ . F O R M A T ( / / I X , • E N - P E R T U R B A R T I O N E N E R G I E S F O R S T A T E = ' , 1 2 , 1 3 X , ' L O C A T I O N = ' , 1 2 ) R E T U R N  E N D S U B R O U T I N E P E R T 2 ( N D , N W A N T , V , W , E P , C , E E , V P , W P , D E L , A V , B V , C V , E ) C . . C P E R T 2 C A L L S P E R I A N D G E T S P S I ( L , M ) F O R 1 . L E . ( L + M ) . L E . 3 C A N D A L S O E ( L » M ) F O R 1 . L E . ( L + M ) . L E . 4 C I N P U T P A R A M E T E R S A R E N D , N W A N T , E P , V , V>  C O U T P U T P A R A M E T E R S A R E C , E E C A L L O T H E R C A L L I N G P A R A M E T E R S A R E D U M M I E S C . I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) D I M E N S I O N C ( 4 , 4 , N D ) , V ( N D , N D ) , W ( N D , N D ) , V P ( N D , N D ) , W P ( N D , N D ) , 1 D E L ( N D ) , A V ( N D ) , B V ( N D ) , C V ( N O ) , E ( N W A N T , 7 ) , E E ( 8 , 8 , N W A N T ) , E P ( N O ) C O M M O N N , Q C 0 M M 0 N / X 2 / N S L P 2 , N W F I N T E G E R Q C C P S I ( N , K , Q ) = S U M O F P H I ( J ) * C ( N + 1 , K + 1 , J , Q ) C D O 1 K = 1 , N C ( 1 * 1 , K ) = G . O D O D O 2 L . = 1 , N V P ( K , L ) = V ( K , L ) W P ( K , L ) = W ( K , L ) 2 C O N T I N U E  9 8 7 6 5 4 0 1 0 5 6 5 7 V P ( K , K ) = V ( K , K ) - V ( Q , Q ) W P ( K , K ) = W ( K , K ) - W ( 0 , 0 ) D E L ( K ) = E P ( Q ) - E P ( K ) 1 C G N T I N U E C ( 1 , 1 , Q ) = 1 . 0 D 0 C A L L P E R I ( N O , N W A N T , V , V P , A V , B V , C V , E , D E L ) D O 3 K = l , N C ( 2 , 1 , K ) = A V ( K ) C ( 3 , 1 , K ) = B V ( K ) . .. C ( 4 , 1 , K > = C V ( K ) 3 C O N T I N U E D O 4 K = l , 8  D O 4 L = l , 8 E E ( K , L , Q ) = 0 . O D O 4 C O N T I N U E E E ( 1 , 1 , Q ) = E P ( Q ) D O 5 K = 2 , 8 L = K - 1 E E ( K , 1 , Q ) = E ( Q , L ) 5 C G N T I N U E C A L L P E R I ( N D , N W A N T , W , W P , A V , B V , C V , E , D E L > O O 6 K = 1 , N C ( 1 , 2 , K ) = A V ( K ) C ( 1 , 3 , K ) = B V ( K )  C ( 1 , 4 , K J - C V ( K ) 6 C O N T I N U E D O 7 K = 2 , 8 . L = K - 1 E E < 1 , K , Q » = E ( Q , L ) 7 C O N T I N U E  C C E ( 1 , K ) = E E ( 2 , K + 1 ) C D O 2 2 K = 2 , 4 T E M P = 0 . O D O D O 2 3 J = 1 , K  S U M = 0 . 0 D O D O 2 4 L = l , N D O 2 4 M = 1 , N . . . „ . . . _ L L = K - J + 1 S U M = S U M + C ( 1 , L L , L ) * V ( L , M ) * C ( 1 , J , M ) 2 4 C O N T I N U E  T E M P = T E M P + S U M 2 3 C O N T I N U E E E ( 2 , K , Q ) = T E M P _ 2 2 C O N T I N U E C C F ( K , 1 ) = E E ( K + 1 , 2 )  C D O 8 K = 3 , 4 T E M P = O . O D O _ . D O 2 5 J = 1 , K L L = K - J + 1 S U M = Q . C D Q  D O 9 L = 1 , N D O 9 M = l , N S U M = S U M + C ( L L , 1 , L ) * W ( L , M ) * C . ( J , l , M ) 9 C O N T I N U E T E M P = T E M P + S U M 2 5 C O N T I N U E  E E ( K , 2 , Q ) = T E M P 8 r C O N T I N U E C T H E R E S T O F T H I S S U B R O U T I N E I S T O C A L C U L A T E C c E( 2 , 2 ) t T H A T I S E E ( 3 , 3 ) c E E ( 3 , 3 , Q ) = V P 1 2 0 0 + W P 2 1 0 0 - T E R M S c T E R M S = T 1 + T 2 + T 3 c T 1 = E 0 2 * S 2 0 0 0 c T 2 = E 1 1 * S 1 1 0 0 c c T 3 = E 2 0 * S C 2 0 0 c P S I ( 1 , 1 ) c S U M = 0 . 0 D 0 D O 1 0 K = l , N S U M = S U M + C ( 1 , 2 , K ) * C ( 2 , 1 , K ) 1 0 C O N T I N U E C ( 2 , 2 , 0 ) = - S U M D O 1 1 K = 1 , N I F < K . E C . C ) G O T O 1 1 S U M = 0 . O D O D O 1 2 L = 1 , N S U M = S U M + V P ( K , L ) * C ( 1 , 2 , L ) + W P ( K , L ) * C ( 2 , 1 , L ) 1 2 C O N T I N U E C ( 2 , 2 , K ) = S U M / D E L ( K ) 1 1 r C O N T I N U E _ _ •— • - • c T E R M S c S 1 = 0 . 0 D 0 S 2 = C . 0 D 0 D O 1 3 K = 1 , N . S 1 = S 1 + C ( 1 , 2 , K ) * C ( 3 , 1 , K ) + C ( 2 , 1 , kT*c"( 2,2,k) S 2 = S 2 + C ( 2 , 1 , K ) * C ( 1 , 3 , K ) + C ( 1 , 2 , K ) * C ( 2 , 2 , K ) 1 3 C O N T I N U E S 2 1 0 0 - - S 1 S 1 2 0 0 = - S 2 S 1 1 0 0 = C ( 2 , 2 , 0 ) S 2 0 0 0 = C ( 3 , 1 , Q ) S 0 2 0 0 = C ( 1 , 3 , Q ) T 1 = E E ( 1 , 3 , 0 ) * S 2 0 0 0 T 2 = E E ( 2 , 2 , Q ) * S 1 1 0 0 T 3 = E E ( 3 , 1 , Q ) * S C 2 0 0 , T E R M S = T 1 + T 2 + T 3 c c P S I ( 1 , 2 ) c C ( 2 , 3 , Q ) = S 1 2 0 0 D O 1 4 K = 1 , N I F ( K . E Q . O ) G O T O 1 4 S U M = 0 . 0 D 0 D O 1 5 L = 1 , N S U M = S U M + V P ( K , L ) * C ( 1 , 3 , L ) + W P { K , L ) * C ( 2 , 2 , L ) 1 5 C O N T I N U E S U M = S U M - E E ( 1 , 3 , 0 ) * C ( 2 , 1 , K ) - F E ( 2 , 2 , Q ) * C ( 1 , 2 , K ) C ( 2 , 3 , K ) = S U M / D E L ( K ) 1 4 r C O N T I N U E L C P S I ( 2 , 1 ) 53 C ( 3 , 2 , Q ) = S 2 K 0 D O 1 6 K = 1 , N I F ( K . E Q . O ) G O T O 1 6 S U M = 0 . O D O D O 1 7 L = 1 , N  S U M = S U M + W P ( K , L ) * C ( 3 , I , L ) + V P ( K , L ) * C ( 2 , 2 , L ) 1 7 C O N T I N U E S U M = S U M - E E ( 3 , 1 , 0 ) * C ( 1 , 2 , K ) - E E ( 2 , 2 , Q ) * C ( 2 , 1 , K ) C ( 3 » 2 ? K ) = S U M / D E L ( K ) 1 6 C O N T I N U E C C V P 1 2 0 0 A N D W P 2 1 0 0 C S 1 = 0 . O D O . „ _ . S 2 = O . C D O D O 1 8 K = l ,N S 1 = S 1 + V P ( Q > K ) * C ( 2 , 3 , K )  S 2 = S 2 + W P ( Q , K ) * C ( 3 , 2 , K > 1 8 C O N T I N U E V 1 2 0 0 = S 1 . . . _ . . . W 2 1 0 0 = S 2 E E ( 3 , 3 ? Q ) = V 1 2 0 0 + W 2 1 0 0 - T E R M S I F ( N S U P 2 . E Q . 0 ) G O T O 2 0 0  P R I N T 5 1 , 0 C A L L F P R I N T ( E E , 8 , 8 , N W A N T , 5 , 5 , Q ) 2 0 0 C O N T I N U E . I F ( N W F . E G . O ) G O T O 2 0 1 P R I N T 5 6 , 0 P R I N T 5 2  D O 1 9 K = l , N P R I N T 5 3 , K , C ( 1 , 2 , K ) , C ( 2 , 1 , K ) , C ( 1 , 3 , K ) , C ( 2 , 2 , K ) , C ( 3 , 1 , K ) 1 9 C O N T I N U E . _ ' . . P R I N T 5 4 D O 2 0 K = 1 , N P R I N T 5 5 , K , C ( 1 , 4 , K ) , C ( 2 , 3 , K ) , C ( 3 , 2 , K ) , C ( 4 , 1 , K )  2 0 C O N T I N U E 2 0 1 C O N T I N U E 5 1 F O R M A T ( / / I X , 3 8 H E E - P E R T U R B A T I O N E N E R G I E S F O R S T A T E , 1 2 ) 5 2 F 0 R M A T ( / / 1 X , 9 1 H K P S K C 1 ) P S I ( 1 , 0 ) P S I I C , 1 2 ) P S I ( 1 , 1 ) P S I ( 2 , 0 ) ) 5 3 F O R M A T ( / I X , 1 2 , 1 P 5 D 1 8 . 7 )  5 4 F G R M A T ( / / 1 X , 7 2 H K P S I { 0 , 3 ) P S I ( 1 , 2 ) P S K 2 , 1 1 ) P S K 3 , 0 ) ) 5 5 F 0 R M A T ( / 1 X , 1 2 , 1 P 4 D 1 8 . 7 ) 5 6 F O R M A T ( / / l X , 2 5 H W A V E F U N C T I G N S F O R S T A T E , 1 2 ) R E T U R N E N D  S U B R O U T I N E P E R I ( N D , N W A N T , V M A T , W M A T , A V E C , B V E C , C V E C , E P E R T , D E L ) C C P E R I I S F O R S I N G L E N O N - D E G E N E R A T E P E R T U R B A T I O N A N D I S I D E N T I C A L C T O T H E M O S T P A R T O F D P E R T O C I N P U T P A R A M E T E R S A R E N D , N W A N T , D E L , V M A T , W M A T S C U T P U T P A R A M E T E R S A R E A V E C » B V E C , C V E C , E P E R T  I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) D I M E N S I O N V M A T ( N C , N D ) , W M A T ( N D , N D ) , D E L ( N D ) , A V E C ( M D ) , B V E C ( N D ) , l C V E C ( N D ) , E P E R T ( N W A N T , 7 ) C O M M O N N , 0 C C M M C N / X 1 / N S U P 1  I N T E G E R Q E 1 = V M A T ( 0 , Q ) C C F I R S T O R D E R W A V E F U N C T I O N C A V E C ( Q ) - = 0 . 0 0 0  E 2 = 0 . O D O E 3 = 0 . 0 D 0 S 1 1 = 0 . O D O D O 3 K = 1 , N I F ( K . E Q . C ) G O T O 3 A V E C ( K ) = V M A T ( K , Q ) / O E L ( K )  E 2 = E 2 + V M A T ( Q » K ) * A V E C ( K ) S 1 1 = S 1 1 + A V E C ( K ) * A V E C ( K ) 3 C O N T I N U E DO 4 K - l i N D O 4 L = l , N 4 E 3 = E 3 + A V E C ( K . ) » W M A T ( K , L ) * A V E C ( L ) C C S E C O N D O R D E R W A V E F U N C T I O N C . B V E C ( Q ) = - S 1 1 * 0 . 5 D 0 E 4 = 0 . O D O E 5 = 0 . 0 0 0  S 2 0 = B V E C ( C ) S 2 1 = 0 . 0 D C . S 2 2 = 0 . 0 D 0 . _ D O 1 8 K = l , N I F ( K . E Q . O ) G O T O 1 8 T E M P = 0 . O D O  D O 5 L = l , N 5 T E M P = T F M P + W M A T ( K , L l * A V E C ( L ) B V E C ( K ) = T E M P / C E L ( K ) . 1 8 C O N T I N U E D O 6 K = 1 , N S 2 1 = S 2 1 + A V E C < K ) * B y E C ( K )  S 2 2 = S 2 2 + B V E C ( K ) * B V E C ( K ) D O 7 L = 1 , N E 4 = E 4 + A V E C ( K ) * W M A T ( K , L ) * B V E C ( L ) E 5 = E 5 + B V E C ( K ) * W M A T ( K , L ) * B V E C ( L ) 7 C O N T I N U E 6 C O N T I N U E  E 4 = E 4 + E 2 * S 2 0 E 5 = E 5 - 2 . 0 D 0 * E 2 * S 2 1 C C T H I R D O R D E R F U N C T I O N C S 3 1 = 0 . C D 0  S 3 2 = 0 . 0 D 0 E 6 = 0 . 0 D 0 E 7 = 0 . O D O C V E C ( Q ) = - S 2 1 D O 8 K = l , N I F ( K . E Q . Q ) G O T O 8  T E M ( - ' = 0 . 0 D O D O 9 L = l , N 9 T E M P = T F M P + W M A T ( K , L ) * B V E C ( L ) C V E . C ( K ) = ( T E M P - E 2 * A V E C ( K ) ) / D E L ( K ) 8 C O N T I N U E D O 1 0 K = l t M  55 S 3 1 = S 3 1 + A V E C ( K ) * C V E C ( K ) S 3 2 = S 3 2 + B V E C ( K ) * C V E C ( K ) D O 1 1 L = 1 , N E 6 = E 6 + B V E C ( K ) * W M A T ( K , L ) * C V E C ( L ) E 7 = E 7 + C V E C ( K ) * W M A T ( K , L I * C V E C ( L ) 1 1 C O N T I N U E  1 0 C O N T I N U E E 6 = E 6 - E 2 * ( S 3 1 + S 2 2 ) - E 3 * S 2 1 E 7 = F 7 - 2 . 0 D 0 * E 2 * S 3 2 - E 3 * ( 2 . O D O * S 3 1 + S 2 2 > C C P E R T U R B A T I O N E N E R G I E S C E P E R T J Q , 1 ) = E 1 E P E R T ( 0 , 2 ) = E 2 E P E R T ( Q , 3 ) = E 3 . . . . E P E R T C Q , 4 ) = E 4 E P E R T ( Q , 5 ) = E 5 E P E R T t Q , 6 ) = E 6  E P E R T { Q , 7 ) = E 7 I F ( N S U P 1 . E O . C ) G O T O 8 0 P R I N T 6 4 , Q D O 1 9 K = 1 , 7 P R I N T 6 5 , E P E R T ( Q , K ) 1 9 C O N T I N U E  C C P E R T U R B A T I O N W A V E F U N C T I O N S C . . . . '_ _ . _ P R I N T 5 5 P R I N T 5 6 D O 1 3 K = I , N  N Z = K P R I N T 5 7 , N Z , A V E C ( K ) , B V E C ( K ) , C V E C I K ) 1 3 C O N T I N U E _ 80 C O N T I N U E 5 1 F O R M A T ( 1 H O » / 1 X » 3 1 H T E R M W I S E P E R T U R B A T I O N E N E R G I E S ) 5 5 F O R M A T ( 1. H 0 , / 1 X , 3 9 H P E R T U R B A T I O N W A V E F U N C T I O N S Q = , 1 2 )  5 6 F O R M A T ( 1 H O , I X 5 8 H F I R S T O R D E R S E C O N D O R D E R T H I 1 R D O R D E R ) 5 7 F O R M A T ( / I X , I 2 , 1 P 1 D 1 7 . 7 , 1 P 2 0 1 9 . 7 ) 6 4 F O R M A T ( 1 H 0 , / / I X , 1 3 H E P E R T Q = , 1 2 ) 6 5 F O R M A T ( / I X , 1 P 1 D 1 8 . 7 ) R E T U R N  E N D S U B R O U T I N E H C I A G ( H , N D , N , I E G E N , U , N R , X , I Q ) C C M I H D I 3 , F O R T R A N I V D I A G O N A L I Z A T I O N O F A R E A L S Y M M E T R I C M A T R I X B Y C T H E J A C O B I M E T H O D . C P R O G R A M M E D B Y C O R B A T O A N D M . M E R W I N O F M I T  C C A L L I N G S E Q U E N C E F O R D I A G O N A L I Z A T I O N C C A L L H D I A G ( H , N , I E G E N , U , N R ) C W H E R E H I S T H E A R R A Y T O B E D I A G O N A L I Z E D . C N I S T H E O R D E R O F T H E M A T R I X , H . C I E G E N M U S T B E S E T U N E Q U A L T O Z E R D I F O N L Y E I G E N V A L U E S A K E T O B E C C O M P U T E D .  C I E G E N M U S T B E S E T E Q U A L T O Z E R O I F E I G E N V A L U E S A N 0 E I G E N V E C T O R S C A R E T O B E C O M P U T E D . C U I S T H E U N I T A R Y M A T R I X U S E D F O R F O R M A T I O N O F T H E E I G E N V E C T O R S . C N R I S T H E N U M B E R O F R O T A T I O N S . C A D I M E N S I O N S T A T E M E N T M U S T B E I N S E R T E D I N T H E S U B R O U T I N E . C D I M E N S I O N H ( N , N ) , U ( N , N ) , X ( N ) , I Q ( N )  C C O M P U T E R M U S T O P E R A T E I N F L O A T I M G T R A P M O D E C T H E S U B R O U T I N E O P E R A T E S O N L Y O N T H E E L E M E N T S O F H T H A T A R E T O T H E C R I G H T O F T H E N A I N D I A G O N A L . T H U S , O N L Y A T R I A N G U L A R C S E C T I O N N E E D B E S T O R E D I N T H E A R R A Y H . I M P L I C I T R E A L * 8 { A - H , O - Z ) D I M E N S I O N H ( N D , N C ) , U ( N D , N D ) , X ( N D ) , I Q ( N D )  I F ( I E G E N ) 1 5 , 1 0 , 1 5 1 0 D O 1 4 1 = 1 , N D O 1 4 J = l , N . . . . . I F ( I - J ) 1 2 , 1 1 , 1 2 1 1 U ( 1 , J ) = 1 . 0 G O T O 1 4  1 2 U ( I , J ) = 0 . 0 . 1 4 C O N T I N U E 1 5 N R = 0 I F ( N - 1 ) 1 0 0 0 , 1 0 0 0 , 1 7 C S C A N F O R L A R G E S T O F F D I A G O N A L E L E M E N T I N E A C H R O W C X ( I ) C O N T A I N S L A R G E S T E L E M E N T I N I T H R O W  C I Q ( I ) H O L D S S E C O N D S U B S C R I P T D E F I N I N G P O S I T I O N O F E L E M E N T 1 7 N M I 1 = N - 1 D O 3 0 I = 1 , N M I 1 X ( I ) = 0 . 0 I P L 1 = 1 + 1 D O 3 0 J = I P L 1 , N  I F ( X ( I ) - D A B S ( H ( I , J ) ) ) 2 0 , 2 0 , 3 0 2 0 X ( I ) = D A B S ( H ( I , J ) ) I Q ( I ) = J . . . . . . . . _ _ 3 0 C O N T I N U E C S E T I N D I C A T O R F O R S H U T - C F F . R A P = 2 * * - 2 7 , N R = N O . O F R O T A T I O N S R A P = 0 . 7 4 5 0 5 8 0 5 9 D - C 3  H D T E S T = 1 . 0 0 3 8 C F I N D M A X I M U M O F X ( I ) S F O R P I V O T E L E M E N T A N D C T E S T F O R E N D O F P R O B L E M . _ _ . / 4 0 D O 7 0 1 = 1 , N M I 1 I F ( I - 1 ) 6 0 , 6 C , 4 5 4 5 I F ( X M A X - X ( I ) ) 6 0 , 7 0 , 7 0  6 0 X M A X = X ( I ) I P I V = I J P I V = I G ( I ) . _ 7 0 C O N T I N U E C I S M A X . X ( I ) E Q U A L T O Z E R O , I F L E S S T H A N H O T E S T , R E V I S E H D T E S T I F ( X M A X ) 1 0 0 0 , 1 0 0 0 , 8 0  8 0 I F ( H D T E S T ) 9 C , 9 0 , 8 5 8 5 I F ( X M A X - H D T E S T ) 9 0 , 9 0 , 1 4 8 9 0 H O I M I N = D A B S . ( H ( 1 » 1 ) ) . _ . . D O 1 1 0 1 = 2 , N I F ( H D I M I N - D A B S ( H ( I , I ) ) ) 1 1 0 , 1 1 0 , 1 0 0 1 0 0 H P I M I M = [) A B S ( H ( I , I ) )  1 1 0 C O N T I N U E H D T E S T = H D I M I N * R A P C R E T U R N I F M A X . H d , J ) L E S S T H A N ( 2 * * - 2 7 ) A B S F ( H { K , K ) - M I N > I F ( H D T E S T - X M A X ) 1 4 8 , 1 0 C 0 , 1 0 0 0 1 4 8 NR= N R + 1 C C O M P U T E T A N G E N T , S I N E A N D C O S I N E , H ( I , I ) , H ( J , J )  X 0 I F = H ( I P I V , 1 P I V ) - H ( J P I V , J P I V ) X 0 = D S I G N ( 2 . 0 D 0 , X D I F ) * H ( I P I V , J P I V ) X S = X D I F * * 2 + 4 . 0 * H ( I P I V , J P I V ) * * 2 1 5 0 T A N G = X C / ( D A B S ( X D I F ) + D S Q R T ( X S ) ) C O S I N E = 1 . 0 D 0 / D S G R T ( 1 • O D O + T A N G * * 2 ) S I M E = T A N G * C O S I N E H I I = H ( I P I V , I P I V ) H ( I P I V , I P I V ) = C G S I N E * * 2 * ( H I I + T A N G * ( 2 . 0 * H ( I P I V , J P I V ) + T A N G * H ( J P I V , J P I V ) ) ) H ( J P I V , J P I V l = C O S I N E * * 2 * < H ( J P I V , J P I V ) - T A N G * ( 2 . 0 * H ( I P I V , J P I V ) - T A N G * t I I I ) ) H ( I P I V , J P I V ) = 0 . 0 • C P S E U D O R A N K T H E E I G E N V A L U E S C A D J U S T S I N E A N D C O S F O R C O M P U T A T I O N O F H U K ) A N D U ( I K ) C I F ( H ( I P I V , I P I V ) - H ( J P I V , J P I V ) ) 1 5 2 , 1 5 3 , 1 5 3 C 1 5 2 H T E M P = H ( I P 1 V , I P I V ) C H U P I V . I P I V ) = H ( J P I V , J P I V ) C H ( J P I V t J P I V ) = H T F M P C R E C O M P U T E S I N E A N D C O S C H T E M P = D S I G N ( 1 , O D O , - S I N E ) * C O S I N E C C O S I N E = D A B S ( S I N E ) . . . . . C S I N E = H T E M P C 1 5 3 C O N T I N U E C I N S P E C T T H E I P S B E T W E E N 1 + 1 A N D N - 1 T O C E T E R M T N F  C W H E T H E R A N E W M A X I U M V A L U E S H O U L D B E C O M P U T E S I N C E C T H E P R E S E N T M A X I M U M I S I N T H E I O R J R O W . D O 3 5 0 1 = 1 , N M I 1 I F ( I - I P I V ) 2 1 0 , 3 5 0 , 2 0 0 2 0 0 I F ( I - J P I V ) 2 1 0 , 3 5 0 , 2 1 0 2 1 0 I F ( I Q ( I ) — I P I V ) 2 3 0 , 2 4 0 , 2 3 0  2 3 0 I F { I Q ( I ) - J P I V ) 3 5 0 , 2 4 0 , 3 5 0 2 4 0 K = I Q U ) 2 5 0 H T E M P = H ( I , K ) . •_ _ _ . _ H ( I , K ) = 0 . 0 I P L 1 = I + 1 X ( I ) = c.0  C S E A R C H I N D E P L E T E D R O W F O R N E W M A X I M U M D O 3 2 0 J = I P L 1 , N I F ( X ( I ) - D A B S ( H { I , J ) ) ) 3 0 0 , 3 0 0 , 3 2 0 _ 3 0 0 X U ) = D A B S ( H ( I , J ) ) I 0 ( I ) = J 3 2 0 C O N T I N U E  H ( I , K ) = H T E M P 3 5 0 C O N T I N U E X ( I P I V ) = 0 . 0 : X ( J P I V ) = 0 . 0 C C H A N G E T H E O R D E R E L E M E N T S O F H D O 5 3 0 1 = 1 , N , I F ( I - I P I V ) 3 7 0 , 5 3 0 , 4 2 0 3 7 0 H T E M P = H I I , I P I V ) H ( I , I P I V ) = C O S I N E * H T E M P + S I N E * H ( I , J P I V ) I F ( X ( I ) - D A B S ( H ( I , I P I V ) ) ) 3 8 0 , 3 9 0 , 3 9 0 3 8 0 X ( I ) = D A B S ( H { I , I P I V ) ) I Q ( I ) = I P I V  3 9 0 H ( I , J P I V ) = - S I N E * H T E M P + C O S I N E * H ( I , J P I V ) I F ( X ( 1 ) - D A B S ( H ( I , J P I V ) ) ) 4 C 0 , 5 3 0 , 5 3 0 4 0 0 X ( I ) = D A B S ( H ( I , J P I V ) ) I 0 ( I ) = J P I V G O T O 5 3 0 4 2 0 I F ! I - J P I V ) 4 3 0 , 5 3 0 , 4 8 0  4 3 0 H T E M P = H ( I P I V f I ) h ( I P I V , I ) = C C S I N E * H T E M P + S I N E * H ( I , J P I V ) I F ( X ( I P I V ) - C A B S ( H ( I P I V , I ) ) ) 4 4 0 , 4 5 0 , 4 5 0 4 4 0 X ( I P I V ) = D A B S ( H ( I P I V , I ) ) I 0 ( I P I V ) = I 4 5 0 l t d , J P I V ) = - S I N E * H T E M P + C O S I N E * H ( I t J P I V ) DO I F ( X { I )-DABS ( H( I, J P I V ) ) ) 40 0, 5 30, 530 480 HTEMP = H ( I P I V , I ) H ( I P I V , I ) - C0SINE*HTEMP + SINE*H(JPI V,I) I F ( X ( I P I V ) - C A B S ( H ( I P I V , I ) ) ) 490,500,500 490 X U P I V)=DABS( H( IPI V,I ) > IQ( IP IV ) - I  500 H ( J P I V , I ) = - SINE*HTEMP + COS INE*H(JPIV, I ) I F ( X ( J P I V ) - D A B S ( H ( J P I V t I ) ) ) 510 »530,530 510 X( JPIV ) = DABS (H( J P I V , I ) ) _„ IQ(JPIV) = I 530 CONTINUE C TEST FCR COMPUTATION OF EIGENVECTORS  IF(IEGE N) 40,540,40 540 DO 550 1=1,N HTEMP=U( I, IPIV) _ U(I,IPIV)=COSINE*HTEMP+SINE*U(I,JPIV) 550 U ( I , J P I V ) = -SINE*HTEMP+COSINE*U(I ,JPIV) GO TO 40  1000 RETURN END SUBROUT INE NEW.LCHN, NWANT ,N FQ , EN, NU, NL ,EPSON,DELTA,GEY,CK,EL, 1EM,AN,QR,WT,R) C NEWLQ DOES A LEAST SQUARES FIT ON TWO PARAMETERS BY AN C EXTENDED NEWTGN-RAPHSON METHOD  C INPUT PARAMETERS ARE: N , NWANT,NFQ,EN,EPSON C OUTPUT PARAMETERS ARE: DELTA,R,WT IMPLICIT RE AL*8(A-H,0-Z) DIMENSION EN( 8, 8, NWANT) , NU ( NF Q ) , NL ( NF Q) , E PS ON (IMF 0 ) , D ELT A ( 2 ) , 1GEY(NFQ,2),CK(NFQ,2,2),EL ( N FQ , 2, 2 , 2 ) ,R ( 2 , 2 ) , R I ( 2 , 2 ) , F A C T ( 6 0 ) , 2DELT(2) ,WT(NFG) ,EM(NFQ,2,2,2,2),AN(NFQ,2,2,2,2,2),QR(NFQ,2) COMMON/NW/ NTR COMMON/SU/ FACT IF(NTR.EQ.l) GG TO 200 DO 205 1=1,NFQ WT I I )=1 .ODO 205 CONTINUE  GO TO 210 200 READ(5,201) (WT(I ) , 1 = 1 ,NFQ ) 201 FORMA T { D 1 3 • 0) •: _ •_ _ C C GEY, CK, EL, EM, AN C 210 DO 215 M=1,NFQ MU=M ML=M+NFQ GE Y ( M , 1) =E N { 2 , 1 , MU) -E N ( 2 , 1 , ML F " " GEY (M, 2) = EN (1,2, MU)- EN ( 1, 2, ML ) DO 212 1=1,2  DO 212 J = 1,2 I2=I+J-1 I 1 = 4-12 CK(H,I,J) = FACT( I 1)*FACT(12)*(EN(I 1,12,MU)-EN(II,12,ML) ) DO 213 K=l,2 I 2M+J+K-2 11=5-12 'EL(M,I,J ,K)=FACT( II )*FACT(I2)*(EN( 1 1, I2.MU )-EN( I 1,I 2,ML) ) DO 214 L = 1,2 I2=I+J+K+L-3 11=6-12 EM.( M , I , j tK,L)=FACT ( I l)*FACT ( I?)*(EN ( I l,I2,MU)-EN ( I l,I2,ML)). 59 DO 216 MM = 1 ,2 12=1+J+K+L+MM-4 11=7-12 AN( M , I ,J,K,L,NM) =F ACT ( 11 ) *F ACT ( 12 ) * ( EN ( 11, I 2, MU )-EN ( 11, I 2, M L ) ) 216 CONTINUE 214 CONTINUE  213 CONTINUE 212 CONTINUE 215 CONT INUE ... . _ . _ . C C CALCULAT ION OF R ( I , J) C :  281 DO 225 1=1,2 DO 225 J = l , 2 DELT A( I )=0 .OOO _ . .. RU , J) =0. ODO 225 CONTINUE NCYL20 = 1  226 DO 230 M = l,NF Q DO 2 30 1 = 1,2 TRM=O.ODO _ „ _„ TRN=O.ODO TRO=0.OD0 TRP=0. ODO  DO 235 J=l,2 T RM=T R M+CK(M, I,J J * DEL T A(J >/2.ODO DO 235 K = l ,2 . ; _ TRN= TRN+EL(M,I,J,K)*DELTA(J)*DELTA(K)/6.0D0 DO 235 L=l,2 TRO= TR 0+ E M ( M , I , J , K , L ) * DE LT A ( J ) »DE LT A ( K ) *D EL T A ( L ) / 2 4 .0 DO  DO 235 MM=1,2 TRP=TRP + AN(M, I , J ,K,L ,MM)*DELT A(J)*DELTAIK)*DELTA(L)* IDELTAt MM )/120.D0 _ . .. 235 CONTINUE QR(M,I)= GEY(M,I)+TRM+TRN+TRO+TRP 230 CONTINUE  DQ 240 1=1,2 DO 240 J=l,2 . TERM=0.000 _ DO 245 M=1,NFQ TERM=TERM+OR(f,I )*WT(M)*QR(M,J) 245 CONTINUE  R(1,J)=TERM 240 CONTINUE DO 242 1= 1, 2 DO 242 J=l,2 RI( I , J)=R(I , J) 242 CONTINUE  CALL D1NVRTIRI,2,2,DET,CCND) IF(DET) 250, 255, 250 255 PRINT 260 260 FI)RMAT(//1X,» DETERMINANT OF R IS NULL') GO TO 280 _C , C CALCULATION OF DELTA C 250 DO 265 J=l t 2 DELTlJ)=0.ODO 265 CONTINUE DO 270 1 = 1,7  2 7 0 D O 2 7 0 J = l , 2 D O 2 7 0 K = 1 , N F Q D E L T I I ) = D E L T { I ) + R I ( I , J ) * Q R ( K , J ) * W T ( K ) * E P S O N ( K ) C O N T I N U E T E S T = D A B S ( ( D E L T l l ) * * 2 + D E L T ( 2 ) * * 2 + D E L T A ( 1 ) * * 2 + D E L T A ( 2 ) * * 2 ) / 2 . O D O -1 ( D E L T ( 1 ) * D E L T A ( 1 ) + D E L T ( 2 ) * D E L T A ( 2 ) ) ) 2 7 2 2 7 1 W R I T E ( 6 , 2 7 2 ) T E S T , N C Y L 2 0 , C O N D F O R M A T ( 1 H C , ' T E S T = « , D l 5 . 7 , 1 0 X , • N C Y L 2 0 = • , 1 4 , 1 0 X , • C O N D = ' , 0 2 0 . 7 ) W R I T E ( 6 , 2 7 1 ) ( D E L T ( I ) , 1 = 1 , 2 ) F O R M A T ( I H O , ' D E L T 1 = 1 , D 1 5 . 7 , 1 5 X , ' D E L T 2 = « , D 1 5 . 7 ) D O 2 7 5 1 = 1 , 2 D E L T A t I ) = D E L T ( I ) 2 7 5 2 8 0 C O N T I N U E I F ( T E S T . L T . 1 . 0 D - 4 . O R . N C Y L 2 0 . E Q . 2 0 ) G O T C 2 8 0 N C Y L 2 0 = N C Y L 2 0 + 1 G O T O 2 2 6 R E T U R N E N D S U B R O U T I N E F P R I N T ( V M A T , M I , M J , M Q , N I , N J , N Q ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) D I M E N S I O N V M A T ( M I , w J , M Q ) _ . _ . D O 1 K = 1 , N I N M A X = N J - K + 1 W R I T E ( 6 , 5 1 ) ( V M A T ( K , J , N Q ) , J = 1 , N M A X ) 1 5 1 C O N T I N U E F O R M A T ! I H O , ( I P 5 0 1 5 . 7 ) ) R E T U R N E N D S U B R O U T I N E E P R I N T ( V M A T , M I , M J , N I , N J ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) 1 5 1 D I M E N S I G N V M A T ( M I , N J ) D O 1 K = 1 , N I W R I T E ( 6 , 5 1 ) ( V M A T ( K , J ) , J = 1 , N J ) C O N T I N U E F O R M A T ( I H O , ( 1 P 5 D 1 5 . 7 ) ) R E T U R N E N D D O U B L E P R E C I S I O N F U N C T I O N Q U A D < S I , S J , S J P , F ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) . _ _ _ _ _ N X P N = S I + S J - F P R G O = D S Q R T < ( 2 . 0 D 0 * S I + l . O D O ) * ( 2 . 0 D 0 * S J P + 1 . 0 0 0 ) ) C L E B = C 3 1 8 ( S 1 , 2 . 0 0 0 , S I , S 1 , 0 . 0 0 0 , S I ) * C 3 1 8 ( S J , 2 . O D O , S J P , SJ , 0. 000 , SJ) 1 5 0 1 0 0 0 1 5 0 0 I F ( C L E B ) 1 5 0 , 1 0 0 0 , 1 5 0 Q U A D = ( - 1 . O D D ) * * N X P N * W ( S I , S I , S J , S J P , 2 . O D C , F ) * P R O D / ( 4 . O D O * C L E B ) G O T O 1 5 0 0 _ _ . Q U A D = 0 . 0 D 0 R E T U R N E N D D O U B L E P R E C I S I O N F U N C T I O N D I P O L E ( S I , S J , S J P , F , F P , S M P ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) N X P 0 = S I + 1 . O D O - S J P - F P R O D = D S Q R T ( ( 2 . O D O * S J P + 1 . O D O ) * ( 2 . 0 P 0 * F + 1 . 0 0 0 ) ) P A R C = C 3 1 8 ( F , 1 . O D D , F P t S M F , 0 . 0 0 0 , S M F ) * W ( S J » S J P » F , F P , l . O D O , S I ) D I P O L E = P R O D * P A R C * ( - 1 . O D O * * N X P O R E T U R N E N D F U N C T I O N I N D E X ( N D , N I O , S J , F , S I ) D O U B L E P R E C I S I O N S J , F , S I , D A B S D I M E N S I O N N I D ( N D , N D ) J = S J + 1 . 0 D 0 61 L = F - D A B S ( S J - S I ) + l . 0 D O I N D E X = N I D ( J , L ) R E T U R N E N D S U B R O U T I N E F A C T O R _ C F A C T O R U S I N G D O U B L E P R E C I S I O N  C K F A C T O R I A L I S S T O R E D I N F A C T ( K + 1 ) I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) D O U B L E P R E C I S I O N F A C T ( 6 0 ) C O M M O N / S U / F A C T F A C T ( 1 ) = 1 . O D O D O 2 K = l , 3 9  F K = K F A C T ( K + 1 ) = F K * F A C T ( K ) 2 C O N T I N U E . . . _ ... . . .. _ R E T U R N E N D D O U B L E P R E C I S I O N F U N C T I O N P E L T A R ( A , B , C ) C C T R I A N G L E C O E F F I C I E N T I N D O U B L E P R E C I S I O N C K F A C T O R I A L I S F A C T ( K + 1 ) C I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) D O U B L E P R E C I S I O N F N , F D , F A C T ( 6 0 ) , D S Q R T D I M E N S I O N A l ( 3 ) , 1 I ( 3 ) C O M M O N / S U / F A C T C . C H E C K T H A T T R I A N G L E O B E Y E D ' A I ( 1 ) = A + B - C A I ( 2 ) = A - B + C A l ( 3 ) = B + C - A  D E L T A R = l . O D O D O 5 K = l , 3 I F ( A l ( K ) . L T . ( - O . l ) I D E L T A R = O . O D O 2 5 I I ( K ) = A I ( K ) + l . D - 5 5 C O N T I N U E I F ( O A B S ( D E L T A R ) . L T . l . C - 9 ) G O T O 3 0 1 1 = I I ( 1 ) + 1 1 2 = I I ( 2 ) + l 1 3 = I I I 3 1 + 1 . . . . . . . . K l = A + B + C + 2 . C 0 C 0 1 D 0 F N = F A C T ( 1 1 ) / F A C T < K l ) F N = F N * F A C T ( 1 2 ) * F A C T ( 1 3 ) D E L T A R = D S O R T ( F N ) 30 C O N T I N U E R E T U R N E N D D O U B L E P R E C I S I O N F U N C T I O N W ( A , B , C , D , E , F ) C C R A C A H C O E F F I C I E N T S U S I N G D O U B L E P R E C I S I O N C R A C A N C O E F F S B Y R O S E E O N ( 6 . 7 ) C N I S K A P P A + 1 . . . C K F A C T O R I A L I S F A C T ( K + 1 ) C I M P L I C I T R E A L * 8 ( A - H , 0 - Z )  D O U B L E P R E C I S I O N F A C T ( 6 0 ) , S U M , F N , F D , D E L T A R C O M M O N / S U / F A C T W = D E L T A R ( A , B , E ) * D E L T A R ( C , D , E ) * D E L T A R ( A , C , F ) * D E L T A R ( B , D , F ) I F ( C A B S ( W ) . L J . 1 . C - 9 ) G 0 T O 1 0 L I = A + B + C + D + l . D - 4 L 2 = A + D + E + F + l . D - 4 L 3 = B + C + E + F + l . D - 4 L L = M I N O ( L 1 , L 2 , L 3 ) + 1 N i l = A + B + E + l . C - 4 N 2 = C + D + E + l . C - 4 N 3 = A + C + F + 1 . 0 - 4 N 4 = B + D + F + l . C - 4 N N = M A X O ( N l , I S 2 , N 3 , N 4 ) + l S U M = O.ODO CO 8 N = N N , L L . . . . N R = A + B + C + D + l . D - 4 N E X = N - 1 + N R c N E X P = 1 I F N E X E V E N , = 2 I F N E X ODD N E X P = 1 + N E X - 2 * ( N E X / 2 ) G O T O ( 3 , 4 ) , N E X P 3 S I G = l . O D O G O T O 5 4 S I G = - l . O D O 5 K l = N - N l K 2 = N - N 2 K 3 = N - N 3 K 4 = N . - N 4 K 5 = L l - N + 2 K 6 = L 2 - N + 2 K 7 = L 3 - N + 2 1 1 1 = N + 1 F N = S I G * W / F A C T ( K l ) F N = F N * ( F A C T ( I I 1 ) / F A C T ( K 2 ) ) F N = F N / F A C T ( K 3 ) F N = F N / F A C T { K 4 ) F N = F N / F A C T ( K 5 ) F N = F N / F A C T < K 6 ) F N = F N / F A C T ( K 7 } . 8 S U M W = = S U M + F N S U M -1 0 C O N T I N U E R E T U R N  E N D D O U B L E P R E C I S I O N F U N C T I O N C 3 1 8 ( J 1 , J 2 , J 3 , M 1 , M 2 , M 3 ) C C L E B S C H G O R C A N . C O E F F S U S I N G D O U B L E P R E C I S I O N I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) D O U B L E P R E C I S I O N F A C T { 6 0 ) , F N , F D , C O , S U M , D S Q R T , J 1 , J 2 , J 3 , M l , « 2 , H 3 C O M M O N / S U / F A C T  D I K E N S I C N A ( 3 ) C C L E B S C H - G O R C A N C O E F F B Y R O S E E Q 1 3 . 1 8 ) C N I S N U + 1 C K F A C T O R I A L I S F A C T ( K + 1 ) C C C H E C K T H A T T R I A N G L E O B E Y E D  C A ( 1 ) = J 3 - J 2 + J 1 A ( 2 > = J 3 + J 2 - J 1 A ( 3 ) = J 1 + J 2 - J 3 C 3 1 8 = l . O D O C O 5 K = 1 , 3 ' 1 F ( A ( K ) . L T . ( - 0 . 1 D O ) J C 3 1 8 = O . O D O 5 C O N T I N U E A 1 = J 1 - D A B S ( M l ) A 2 = J 2 - D A B S ( M 2 ) A 3 = J 3 - D A B S ( M 3 J 1 F ( A 1 . L T . 0 . 0 0 0 . Q R . A 2 . L T . O . O D O . P R . A 3 .1 . T . C . O D O ) C 3 1 8 = 0 . 0 0 0  I F ( D A B S ( C 3 1 8 ) . L T . 1 . D - 9 ) G 0 TO 10 I I 1 - J 3 - J 2 + J 1 + 1 . C 0 0 1 D 0 I 12 = J3-J1 + J 2 + 1 . 0 0 0 1 0 0 11 3=J1 + J2-J3+ 1.0C01D0 I I4=J3+M3+1.0C01D0 I I 5 = J3->M3+ 1. 000 1 DO Kl= J 1+ J2+ J3+2 .CC01 K2= Jl-Ml + 1 .0001 DO K3= Jl+Ml + 1 . C001 DO K4= J2-M2+1 .0C01DO K5= J2+M2 + 1 .0 00 ICO FFl = 2 . 0 * J 3 + l . O D O FN = F F 1 * ( F A C T ( I I 1 ) / F A C T ( K 1 ) ) FN = FN * ( F A C T ( 1 1 2 J / F A C T ( K 2 ) ) FN = FN * ( F A C T ( I I 3) / F A C T ( K 3 ) ) FN = FN * ( F A C T ( I I 4 ) / F A C T ( K 4 ) ) FN = FN * ( F A C T U I 5 ) / F A C T I K 5 ) ) CO = DSORT(FN) L1=J2+J3+M1+i .D-4 L2= J3- J1+ J2+1 .D-4 L3=J 3+M3 + 1 .D-4 LL=MI N O ( L 1 » L 2 , L 3 ) + l N1=J1-M1+1.D-4 B= J1- J2-M3 IF ( B . L T . O . O D O ) GO TO 11 N2 = B+l .D-4 GO TO 12 11 N2 = B - l . D - 4 12 CONTINUE NN=-MINC(CtN2J+l SUM=0.0D0 DO 8 N=NN,LL NR= J2+M2+1.D-4 NEX=N-1+NR NNN=NE X/2 N ?\IN = NNN*2 IF (NEX .EQ .NNN) GO TO 2 SIG=-1 . ODO GO TO 4 :  2 SIG=1.ODO 4 11 1 = L l-N+2 I I2=N1+N Kl= L2-N+2 K2= L3-N+2 K3= N2+N FN = S I G * ( F A C T ( I 11 ) /FACT IK1 ) ) FN = FN * ( F A C T ( I I 2 ) / F A C T ( K 2 ) ) FN = F N/ F ACT ( K3 ) FD = FACT (N ) 8 SUM=SUM+FN/FD C318=C0*SUM 10 CONTINUE RETURN END 

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