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JOURNAL OF MOLECULAR SPECTROSCOPY 55, 374406 (1975)

Calculation of Rotation-Electronic Energies and Relative

Transition Intensities in Diatomic Molecules

J. W. C. JOHNS

Division of Physics, National Research Cowwil of Canada, Ottawa, Ontario, KIA OR6

AND

D. W. LEPARD

IfepartmenL of Physics, Brock University, St. Catharines, Onturio, LZS 3A1

Two computer programs have been developed to facilitate the calculation of line positions

and relative intensities in the spectra of diatomic molecules. The programs were designed

to handle transitions to and between Rydberg “complexes” but can also be used for other

electronic transitions as well as for the calculation of vibration-rotation and Raman spectra.

A general method is described for the construction of the relevant energy matrices directly

in the computer in such a way as to include all terms up to those involving sixth powers of

the angular momentum operators. Spin-orbit, spin-rotation, spin-spin as well as rotation-

electronic interactions are included in the treatment. Examples of the application of the

programs to spectra of NP., OS, and BH are given.

I. INTRODUCTION

In this paper we describe two computer programs which were originally developed

as an aid to the analysis of band spectra involving Rydberg state complexes of diatomic molecules. Subsequently, it was found that the computations could be generalized to include spectra arising from states of any multiplicity, Rydberg or non-Rydberg,

and the projection of I along the internuclear axis, A. The I and the X values effec-

rules (3~ tf +) as well as the usual electric dipole selection rules (3~ +-+ F). As a consequence the general usefulness of the programs has been greatly increased. They have

been used, for example, by Jungen and Miescher (1) to help in the analysis of the f-complexes of nitric oxide and by one of us [Lepard (Z)] to treat the Raman effect in nitric oxide and oxygen.

In applying the programs to the study of Rydberg states we limit ourselves to molecules in which each Rydberg state is formed from a molecular ion core in a Z* electronic state, of any multiplicity, and a single outer Rydberg electron. Each Rydberg state is therefore primarily characterized by the quantum numbers of the outer electron: the

principal quantum number n, the orbital angular momentum quantum number 1

and the projection of 1 along the internuclear axis, X. The I and the X values effec- tively represent the electronic orbital angular momentum of the molecule and we will consequently denote them by L and A in the remainder of this paper. For any given

374

Copyright 0 1975 by Academic Press. Inc.

All rights of reproduction in any form reserved.

DIATOMIC MOLECULES 375

value of L, jlij can take on the values L, L - 1, L - 2,. . , 0. In an atomic Rydherg

series these levels are, of course, degenerate but the deviation from spherical symmetry

of the molecular ion core removes this degeneracy giving rise to L doubly degenerate

and one nondegenerate distinct molecular electronic states. The splitting of these

(L + 1) states is generally small in Rydberg states, particularly if PL is large, so that

for each value of ti and L there is a close group of states. Such a group of states is known

as an “L-complex,” or more specifically as a p-, ti-, or f- comples according as L = 1,

2, or 5. The various states within a complex become mixed as a result of electronic-

rotation interaction, otherwise known as “L-uncoupling.” This often causes gross

perturbations to the energy level pattern and to the intensity distribution within the

rotational branches. These perturbations sometimes make it difficult or impossible

to analyze the bands by traditional methods based on combination differences because

one of a pair of branches needed for the formation of a set of combination differences

may become too weak to be observed. In the limit where the states within one group

become degenerate, the energy level pattern becomes simple. This situation is known

as Hund’s case (d) [see for example (3, p. 229)]. 0 ur calculations, however, start from

a case (a) basis where CL is still a good quantum number, since the matris elements

are well known and the formulation of the theory is relatively simple. Nevertheless,

diagonalization of the effective rotational Hamiltonian yields energies and wavefunctions

corresponding to Hund’s case (d) for those physical situations actually represented b>

this case.

Examples of p- and d-complexes of singlet states (and also of some triplet states

with either unresolved or very close triplet structure) have been known for a long

time in the spectra of Hz and Hez. References to early work on these molecules can be

found in the book by Herzberg (3). Theoretical interpretation of the energies of the

rotational levels and of the intensities of the transitions has been understood for a long

time. Among early work was that by Hill and Van Vleck (4) and MacDonald (5).

Despite the fact that the theory of L-uncoupling is well understood, the calculations

become more complex as L increases and as multiplet states are considered, so that it

becomes necessary to use an electronic computer in order to calculate the energies of the

levels and the intensities of the transitions between them. At the same time the use

of the computer has made it relatively easy to include the effects of centrifugal dis-

tortion so that relatively light molecules can be treated with some hope of success.

To illustrate another use of the programs, it is interesting to consider spin forbidden

transitions such as %‘I;. These transitions may be considered as made allowed t)j.

contamination of the Y&* and Cli states with IX’ and ‘IIF states, respectivel): Formall\-

we may replace S and Z with L and il. Frequencies and intensities of transitions may then

be calculated by treating the triplet state as a singlet state p-complex. It is interesting

to note that our calculations of the intensities of the lines in CZ’-lZ* transitions do

not agree with the expressions given by Schlapp (6) but do agree with those given b\-

Watson (7) who corrected an error in the relative phases of the transition moments used

by Schlapp.

Although the two computer programs were primarily developed as an aid to the

analJ.sis of transitions in diatomic molecules in which one or both of the electronic

states may be regarded as molecular “complexes,” the preceding paragraph illustrates

their applicability to a wide variety of problems with diatomic and possibly linear

376 JOHNS AND LEPARD

polyatomic molecules. One of the programs determines molecular constants by a non-

linear least-squares fit to observed frequencies (or to combination differences) and quantum number assignments. The other program takes known or assumed molecular

constants and calculates the transition frequencies and relative intensities of the transitions in either tabular or profile form for comparison with an observed spectrum. The

least-squares analysis program is somewhat more general than the transition calculation

program. For example, the 3Z-*2: problem mentioned above can be handled directly by this program, whereas the transition calculation program requires replacement of the

37 state by the analogous p-complex. The reason is simply that, unlike “forbidden” transitions, “allowed” transitions are governed by the selection rules AS = AZ = 0: These restrictions lead to only one type of transition moment and it was not considered profitable to include in the program the variety of types of transition moments resulting

from relaxation of the AZ = 0 restriction. In any event Whiting, Patterson, Kovacs,

and Nicholls (8) have recently described a computer program which can perform these calculations.

In many respects our approach to the calculation of rotational fine structure in elec-

tronic transitions of diatomic molecules is very similar to that described recently by

Zare, Schmeltekopf, Harrop and Albritton (9). However, we believe that we have achieved a useful simplification in the calculation of matrix elements involving centri-

fugal distortion parameters. At the same time our treatment of rotation-electronic interaction effects, such as A-type doubling, is somewhat more direct because of our in-

terest in Rydberg transitions where they can no longer be treated by second order

perturbation theory. In Section II we describe the basic theory that is used and indicate the way in which

centrifugal distortion has been taken into account. Some details of the program are described in Section III. In Section IV examples are given of the application of the

program to spectra of Nz, 02 and BH.

II. BASIC THEORY

In a previous paper by one of us (Z), subsequently referred to as I, an approach similar

to that of Kopp and Hougen (10) was used to describe the basis set wavefunctions and matrix elements. The wavefunctions were written as either

[2s+lA,; JQ; p) = IL, A)/& Z)lJ, Q), p = +

when A = I: = D = 0, or the linear combinations

(la)

~‘s+l~~;,n;p)=~~ll_,A),S,z),J,~)+PlLI --h)lS, -z>lJ, +I, P=* (lb)

for all other cases. The symbol p enables one to obtain the parity of the level being considered ; it will be defined below. The wavefunctions appearing on the right-hand

side of Eq. (1) are appropriate to Hund’s case (a) and contain an electronic orbital part IL, A), an electronic spin part IS, 2), a total angular momentum part IJ, 0) and a vibrational part ID), not explicitly denoted. Associated with these wavefunctions are the quantum numbers of angular momentum L, S, and J, and their corresponding

projections on the internuclear axis A, 2, and !YI = A + Z. The quantity I, represents the

molecular electronic orbital angular momentum; it is often not a good quantum number

However, for our model of Rydberg states, matrix elements involving the molecular electronic orbital angular momentum reduce to those involving only the electronic orbital angular momentum of the single Rydherg electron. Tn this case I, ran be con-

sidered as well defined. Although the projection quantum numbers are signed quantities, the right-hand side

of Eq. (lb) includes both signs, since the wavefunctions given in Eq. (1) are the result of a Wang type transformation. As a consequence, we shall adopt the convention that

D = ri + 2 where A 3 jiil and Z = Z but, when ‘1 = 0, S = is/. This convention pro-

vides a unique label for each symbol, 2N+1A~, representing the electronic state that

appears on the left-hand side of Eq. (l), and also unambiguously defines the projection

quantum numbers that appear on the right hand-side of Eq. (lb). The rovibronic parity of levels is determined by appl!-ing the reflection CT,,, in the rz

plane of the molecule fixed reference s!-stem, to the wavefunctions in Eq. (1). Following

I we first choose phases of the wavefunctions so that

lT,,lr,, 11) = f (-1)*/L, --3), (2a)

C7, Is, S) = (-l)=lS, -Z), (2b)

CJJ, 62) = (- l),‘-‘!jJ, ---n), (2c)

and we note that these transformations result in real and positive matrix elements of

the ladder operators. The sign choice in Eq. (2a) depends on the type of Z states assumed present as part of a complex of electronic states. That is, for all A, we must use + or -

according as there are Z+ or Z- states present.’ Then using Eq. (2) we find that the

rovibronic parity of the lzs+lLIn; JQ; p) in Eq. (1) may be written

P,,, = Pv(-l)J-” (3)

where P, = &p according as il = 0 = O* (2% states). For ‘S* states P, = =tp = =t,

since p = + only by Eq. (la), and therefore P, is the vibronic parity. Then for all

other cases we define P, as the vibronic parity, since it characterizes the lIZ* like be-

havior of the two linear combinations of wavefunctions used to describe the state.

Note that this is simply a generalization of the definition of e and f levels introduced by Kopp and Hougen (10) to describe states of even multiplicity. Our + and - vibronic parity labels can be respectively identified with their e and f labels when S = 3, 5, I) 3, . . .) and with theirf and e labels when S = 3, 3, y, . . . . Note also that our definition of the parity is similar to that recently proposed by Zare et al. (9).

Following Van Vleck (12) the electronic rotational Hamiltonian may be written

Ei,,,t = c G(r)& (4)

1 Assuming the /I,, A) are built up from the atomic orhitals 1 I,, Xc) of the electrons, Eq. (2a) could

he replaced hy o”,& Li) = [_l)L-“+“l”lL.--.G

[see for example (II)]. However this does not appear to be a useful substitution since even if the Ii

are known, the molecular electronic orbital angular momentum L is often poorly defined.


where the Ci(r) represent the electronic rotation constants and are functions of the A internuclear distance, and the Oi represent the associated operators. The various

constants and operators considered by us have been collected together in Table I. The first five of these terms are conventional in the sense that they are commonly used

in the description of the energy levels of diatomic molecules. The sixth and eighth terms

have been added by analogy with the fourth, and the seventh by analogy with the fifth. It may be noted that the eighth term simply provides a mechanism for allowing a different value of the spin-orbit parameter (A (r), the second term) to be used in off

diagonal matris elements from that used on the diagonal. It may also be noted that the fourth and sixth terms may be completely ignored in any model since they are independent of J and the effects they give rise to can be incorporated in the electronic

energy differences (first term). The advantage of including such terms separately arises in cases where the electronic energy differences may be explained without the

use of the first term. For esample, the energies of the two components 3Zo and 3& of a

3Z state can be described equally well by two electronic energies or, as is more usually done, by the inclusion of a spinspin interaction term ASS.

Up to this point the vibrational aspect of the problem has been ignored. The simple

harmonic oscillator will be our zeroth order vibrational Hamiltonian H,ib”. The zeroth

order rovibronic wavefunctions will be taken as products of those in Eq. (1) with simple harmonic oscillator wavefunctions Iv), although this product will not be explicitly

denoted. In order to consider the effects of anharmonicity and the mixing of vibrational states by the C,(r) we now espand each as a Taylor series about the equilibrium posi-

tion, using the parameter i = (r-r,). That is.

Ci, = Ci(y,), and 1/ is the nuclear potential. The Hamiltonian of Eq. (5) has matrix elements off diagonal in the vibrational

quantum number 7~. However, we shall assume that the difference between any two matrix elements diagonal in all quantum numbers and separated by one vibrational interval is approximately equal to the vibrational frequency w,. Then we may use a Van Vleck transformation [see for example (13)] to obtain an effective electronic rotational Hamiltonian for each vibrational state. To third order we can write the matrix of this Hamiltonian in our case (a) basis as

~,~pf = Hi - ~~2 + +[H?H~ + H~H~H~ + H~H~~] (6)

where Ho, HI, and Hz are obtained as follows. The matrix Ho is obtained from the

matrix of the operator Hrot, defined by Eq. (4), by replacing C,(T) with Ci, where

Ci, = Cj, - ari(v + fr), CQ = 2x[ (6&x/w,) - fi] (7)

and x = h/8$pcw, is a constant involving Planck’s constant h, the reduced mass of the molecule p, the velocity of light c, and the vibrational frequency we. The matrix HI is


TAHLE I

Effective Rotational Hamiltonian

* %S $Ii‘_--& Spin-cp;n interaction effects

5 y.7 (J-i-S).S Spin-rotation interactian effects

t hLL

$3L,'-L;) Orbit-t,r'bit interac?ion efrects

7 YL (J-L-S).L Orbit-rotation interaction effects

hLS $3LzSz-L.S) Sgin-orbit i:;teraction effects

obtained from Hrut by replacing C<(r) with

D,l’Z = di(X/W,)1,‘3, (8)

and the matrix Hz is obtained from Hrot by replacing C,(r) with

Hi/D; = I~jlii - (2.~tli~/~,)]/~,. (9)

The significance and notation of the constants appearing in Eqs. (7), (8), and (9)

is best explained by looking at the special case of Ci(r) = B(Y). Lsing B(r) = /1/8?r*&,

the right-hand sides of these equations yield expressions which may be compared to the usual ones, such as the famiiiar Kratzer relation for D, given, for example, by Herzberg

(3). This allows us to identify CiV, Cie, CY~, D;, and H, as B,, B,,(Y, D, and H, respectivel)-. If in addition we assume that B(r) is the only nonvanishing C,(Y) in H,,t, then the right

hand side of Eq. (6) is readily seen to reduce to the matrix of the operator B,(J-L-S)* - D,(J-L-S)4 + H,(J-L-S)6. It is because of these results that we chose to use a

square root notation in Eq. (8) and a quotient notation in Eq. (9) to define the constants appearing in HI and He. Before proceeding further it would perhaps be wise to introduce a more general notation for the higher order constants like the familiar D and H men-

tioned above. In our formulation we consider eight operators and their associated

parameters. The constants D and H are themselves associated with the parameter B and we will henceforth call them DR and HH. This notation will allow us to distinguish

readily between any of the sixteen higher order terms which can be included in the calculation. When B(r) is not the only nonvanishing C;(r) in H,,,t there are effects not analogous to those described above. As an example, suppose there is only one other such

Ci(r). Then the second term in Eq. (6) reduces to the matrix of the operator

-DR(J-L-S)~ - Dl;dLz - (D#2)(Di1’2)[(J-L-S)26j + 6,(J-L-S)2].

In addition to the term involving D,, which is analogous to DO, there is now a cross term involving the product of D$2 and Dill2 Consider, as a more specific example, the

382 JOHNS AND LEPARU

in expressing pi in terms of spherical components

(14)

associated with the molecule fixed reference system, and these components transform

under uv as

CUPI = P-t. (15)

Succeeding steps are almost identical to those used in I for magnetic dipolar scattering.

As a consequence we finally obtain

] (#‘(J’, 9’) I/G I#“(,“, p”)) 1 2 = ;[G]P.~~~“,P”

= 51 c UAJ n,u,,~~,,,(2s+1A’~~; J’O’; p’l c f--t~L1 1 2S+1A”e,,; J”Q”;$“) 1 2 (16) bJR)(A”nll t

which is the line strength. The so-called polarization coefficients CijR of I do not appear in Eq. (16) since here they all have the value unity. Also, GR, fR, and pR have been

TABLE III

Transition Matrix Elements

<“‘n, IA,, z P+lA ,,;J'a';~'If_~v~ IzS+‘A,;JW a,b

<A+t,+tlAQ> = ~<L',A+tl~tlL,A><J',n+tlf_tlJ,R, for A+t,O c,b

cn1-n IE,> = T~'<L',~III~IL,o><J',Q-~~~~~J,~> for DO d

‘El-n1 “,,’ = p'<L',O/~_lIL,l><J',n-llfllJ,n> for a<1 e

Notes: a)

b)

C)

d)

e)

The transition matrix elements are obtained by

ranging over all AR and t for each element listed.

Initial and final states have the ?,ame vibronic

parity when AJ = fl,'the opposite vibronic parity

when AJ = 0.

AR f Zi is present in the t vibronic parity

matrix, absent in the ? vibronic parity matrix.

In general n=l; the exception is il=fi if

either Kn = I0 and Ata, # Z0

or An # Lo and A',, = ZO.

The upper op lowed sign is used according as

initial and final states are governed by the same

op opposite sign choice in Eq. (2a). The complete

sign (: p') for an Initial Et state reduces to

r P", where P" is the vibronic parity of the final

state.

The complete sign (p') for a final L' state

reduces to ? Pv where Pv is the vibronic parity

of the final state.

383

J

3+1

J - I

-- _

written as G, f and p, since R = 1. ‘IYhef_t are of course the same reduced operators as

those used in I for magnetic dipolar scattering. It is also important to note that the

p, do not transform under uU in the same way as the (Y? of I (compare Eq. (15) here

with Eq. (8) in Z), since the pL1 are the components of a polar rather than an axial vector.

The resulting change in sign of Zf_,pl under u,. is actually a consequence of the fact t

that pi (space fixed) changes sign under the space fixed inversion operation and leads

to the usual rovibronic parity selection rules, =k w F only, for electric dipole absorp-

tion. Eq. (16) satisfies the usual restrictions for electric dipole absorption; that is

/AJI<l, lAfil<l, J = O+++ J = 0, AS = AZ = 0 and as a consequence A,l = An.

In addition, AQ = t, and the rovibronic parity selection rule appears in the form that p”P’ = - (-I)-\” *r (_ 1)A.J according as initial and final states are governed by- the

same or opposite sign choice in Eq. (2a). The nonvanishing matris elements of Eq.

(16) which satisfy these conditions are listed in Table III. Each term in Table III

represents the product of a vibronic transition moment with a rotational matrix element.

The rotational matrix elements are listed in Table IV. The transition moments can be

considered as adjustable parameters, which may be obtained from the observed sljec-

trum after one or two trial calculations, or they may be calculated from assumed values

of L’, A’, I,” and A”. It is important to note, however, that these transition moments

as well as those in I are all real.

The reality of the transition moments may be established from time reversal con-

siderations. The time reversal operator, 0, performs complex conjugation and reverses

the sense of angular momentum. This results in the same transformation laws under

uV and 0 for the it, the atR of I, the angular momentum ladder operators which in

fact also transform in the same way as the cztl of I and, apart from a question of sign,

the wavefunctions. To answer the sign question, we assume all wavefunctions are

represented by ladder operators acting on the state with highest projection quantum


number. We now multiply all wavefunctions by a phase factor which we fix by requiring

the state of highest projection quantum number to transform under 13 and cu in the same manner. Then, using the fact that the transformations given in Eq. (2) for crv resulted in real and positive matrix elements of the ladder operators, we find that the other

projection quantum number states must also transform under 0 and u,, in the same manner.4 We now apply 0 and uv in succession to the transition moments and find, since

flu does not change the sign of an integral, that all transition moments are real.

III. PROGRAM DETAILS

The program SCLS determines the molecular constants of diatomic states from

observed transitions by a nonlinear least-squares technique. For each state the spin multiplicity, 2S + 1, the value of L for a “complex state” (A = L, L - 1, . . . , 0) or the value of A for a “single state” (only one value of A), and the + or - character

assumed for Z states, must be specified as input data. The rovibronic parity selection rules, assumed obeyed by all transitions, must also be given. For each vibronic parity of a state, all of the pertinent constants of the type described here in part II are as-

sumed possible. In addition, constants of type AB and ADB* may also be associated with each component zs+lAa of a state in order to allow for different effective B and

Dg) values; off-diagonal matrix elements use the average of the two associated diagonal values. Provision is also made to take into account L and S uncoupling that is not

satisfactorily explained by the model (which assumes integral values of L and S) by multiplying the matrix elements (A’IA”) and @‘IX”) of Table II by factors of the type

(1 + 6). Values for all constants mentioned above must be specified as input data,

along with corresponding control codes. Constants associated with zero control codes retain their input values during the least-squares refinement procedure. Nonzero codes specify trial values required for the least-squares process. Constants may be mathe-

matically equivalenced by assigning the same nonzero control code. For example, the constants B+ and B- associated with the two vibronic parities of a state may be con-

sidered represented by a single parameter, B. The data input for each transition consists of the frequency, a label specifying the

appropriate pair of energy levels, and a weight which can be assumed inversely pro- portional to the square of the accuracy of measurement. The energy levels are labelled with the vibronic parity, the J value, and values of Q and A appropriate to one of the

basis functions. The latter part depends only on the relative order of the energy levels obtained after diagonalization of H,,teff and could be arbitrarily assigned. The actual

criterion used by the program orders the energy eigenvalues to correspond to the order of the diagonal matrix elements (AalA,) of HO, listed in Table II. The molecular transi-

tions are now labelled as hAAs AnAJrr, h(J) (17)

where the f sign is the vibronic parity of the lower state, P,, and Q, A, and J are numerical values of lower state quantum numbers. The upper state level is indirectly specified by M = A’ - A”, A52 = 0’ - Q”, and AJ = J’ - J”, in the usual notation.

4 This result is not correct for the IJQ) since the states are actually IJQM) and, unlike (r., 0 also reverses the sense of M. This results in an additional factor, ( -l)J-M, when Eq. (2~) is used

for 8.


Forexample, . . . . 0, P,Q,R,S, . . . . specifiesAJ= . . . . -2, -l,O, 1, 2, . . . . For

half-integral quantum numbers the program uses z = I; - i, fi = D - 3, and

J = 3 - 3, in place of Z, 0, and J, for both input and output. The vibronic parity of

the upper state level, not specified by Eq. (17), is internally determined with the aid

of the rovibronic parity selection rules. When combination differences are used in

analysis the observed transitions may be given as data and the program requested to form all possible differences for one of the two states. Combination differences may

also be formed externally by subtracting the frequencies of appropriate pairs of transitions. When used as input data these differences are treated as frequencies of transitions, and the transition label must specify the two energy levels involved.

The least-squares process used here may be described in terms of the usual linear case [see, for example (16)]. In matrix form we start by defining the theoretical relation

1’ = JA (18)

connecting the frequencies or combination differences, yi = Ei’ - B,:i” (the E’s are

the energies of the appropriate pairs of levels), and the molecular constants, aj, that are to be determined; the matrix elements of J, the Jacobian, are the derivatives

ay;/daj. The solution is represented by

-4 (talc) = I!-rJIVY (obs) (19)

where 1’ = J It-J and It- is a diagonal matrix in which wi represents the weight as-

signed to yi (obs). The back calculation uses Eq. (18) to find Y (talc). Standard deviations are given by the square root of the diagonal elements of the variance matrices

J’ c-4 (talc)] = a21~-1

k’ [Y(calc)] = 2JI*-‘_?

and the variance uZ is given by

where tl = E’ (obs) - Y (talc),

U2 = dWd/ (iv - ?I.) (21) K is the number of observational equations, and n

(20)

is the number of constants to be determined. For nonlinear least-squares, assumed and

trial values of constants, represented by A (trial), are used to form Hroteff which is then diagonalized to yield Y (trial). The Jacobian in the defining relation, Eq. (18), is evaluated by repeating this calculation for each aj with aj replaced with aj + &zi. .I (talc) - ;I (trial) is then determined from Eq. (19) with Y (obs) replaced with I’ (obs) - I’ (trial). The solution is iterated by using A (talc) as a new A (trial). The criterion used for a satisfactory tinal solution is that the corrections to all constants must be less than a fraction, specified at input, of their standard deviations. Note that

the value &a, b!r which a constant is to be varied is some multiple, specified at input, of the corresponding standard deviation; for the first iteration an internally determined fraction of the constant itself is used. Note also that during the course of the analysis

lines included in the solution may be rejected, and lines not included may be added. The criterion used is that a satisfactory ldi/ must be less than a fraction, specified at input, of r/w,‘j2.

The output consists of a table of molecular constants and standard deviations, a table of correlation coefficients, and a table of transitions. The correlation coefficient


for constants ai and aj is defined as (U-l),j/( (U-l)ii(U-l)j~)*‘z. The transition table con-

tains the label, observed and calculated frequency or combination difference, the weight, and the standard deviation. These calculated frequencies or combination differences are obtained from diagonalization of Hroteff, using the final values of the constants,

instead of from the back calculation. The weights of data not included in the analysis are listed as negative quantities.

The program SCIT computes the frequencies and relative intensities of transitions. Equation (12) is used for electric dipole absorption, along with an additional factor (I _ &Zcv/kT ) to take account of stimulated emission. Electric dipole emission uses

the same formula except that the frequency factor y4 is used in place of V, and the upper state energy replaces the lower state energy in the exponential factor. Magnetic dipole

and electric quadrupole transitions are calculated using G(l) and G@), respectively, defined by Eq. (lob) in I, in pl ace of the G given here in Eq. (16). Stokes Raman intensities are calculated using Eq. (6) and Eq. (10) of I. For anti-Stokes scattering the frequency

factor (~0 + Av)~ is used in place of (~0 - Av)~, where vo is the frequency of the exciting line, and the upper state energy is used in place of the lower state energy in the exponen-

tial factor. The depolarization ratios are also calculated for Raman transitions. The cases considered are linearly polarized or natural light where the incident and scattered light directions and electric vectors are each along some space fixed axis. The polarization

coefficients defined in I yield

GI = (1/6)G(” + (l/lO)G’“‘, (22a)

G,, = (1/3)G((‘) + (2/15)G@) (polarized light), (22b)

G,, = (1/6)G(“) + (1/12)G(‘) + (7/60)G@) (natural light), (22c)

for these cases, and the depolarization ratio is equal to GI/GII; the complete intensity

for a transition is given by using (GI + GII) in the relative intensity expression. Note, however, that the geometries considered also include the case of two identical line

strengths given by Eq. (22a) for polarized light and Eq. (22~) for natural light. The molecular constants are assumed known and given as input data along with the

parameters defining each state, the nuclear statistical weights, the temperature, the Raman exciting line and a code specifying the appropriate Gl and GII, if applicable. The vibronic transition moments needed to determine the transition matrix elements

G are also assumed known and given as input data. These moments are just the matrix elements between basis state functions of the appropriate components of the electric dipole moment, magnetic dipole moment, electric quadrupole, or polarizability tensor. The output may take any of the following forms; tables of energy levels, frequencies and intensities of transitions, and a relative intensity profile. For profiles, the relative intensities may be broadened by either a Gauss or Lorentz curve. The frequency region, the spacing between successive points in the profile, and the width at half intensity of the broadening curve must be specified as input data. A lower intensity limit must also be specified; calculations are regarded complete when no transitions above this limit are found for three successive J values. A second intensity limit is given for the table of transitions; intensities below this value are not included in the table. The abscissae may be plotted as an increasing or decreasing frequency scale, or as


chart or plate readings if the appropriate calibration curve is input; a scaling factor

should also be specified to match these values to the physical scale of the drafting paper.

For infrared absorption it is important to note that there is no provision to convert the relative intensities to absorption or transmission, by exponentiation followed bJ, rebroadening to allow for instrumental slit-width. For emission or Raman scattering

there is no provision to convert the relative intensities to optical densities recorded b>, photographic plates, by taking logarithms of relative intensities. On the other hand,

we have used the program largely for absorption spectra recorded on photographic

plates. In this case the logarithmic recordin g of the plate essentially converts the transmission back to the relative intensity, and observed and calculated intensities

may therefore be expected to agree when the model is satisfactory.

A more complete description of the programs, including input, output, and descrip- tions of main programs and subroutines may be obtained from the authors on request.

IV. EXAMPLES

In this section we give some simple esamples of the application of the programs. We

have chosen to consider (a) the Worley-Jenkins Rydberg series of nitrogen, (b) the ground state of oxygen, and (c) the 3d Rydberg complex of BH. These examples have

been chosen because they illustrate respectively application to a simple Rydberg series, to a single molecular state and to a Rydberg complex which is “contaminated” b!

another Rydberg state. These systems have been studied previously bp others; for N? and BH we have made some new observations which have allowed us to obtain what

we feel is a better set of molecular parameters and for 02, while we have no new measurements to report, the microwave and optical data have been treated simultaneous11 to obtain a good set of ground state parameters.

As mentioned in the introduction, the programs have also been used by Jungen and Miescher (I) to help in the analysis of thef-complexes of NO, by Jungen (17) to calculate line intensities in the (l-complexes of SO and by Gewurtz and Lew (18) to fit the

rl%+-PTI~ band system of HF+. One of us (2) has also used the programs to treat the Raman effect in NO and Op. In addition we have also tested the programs b\r fitting

data in the literature on the A2&X2Zf band system of BeCl (19), on the K, L+-S

p-complex band system of CaH (20) and on the vibration-rotation fundamental of NC) (21). In each case the least-squares program yielded a set of parameters substantial11 in agreement with those already published.

(a) The Morley-Jenkins Rydbwg Series c,f IVn.

This Rydberg series was first reported by Worley and Jenkins (Z) in 1938. Successive investigations at ever higher resolution (23-25), showed that each member con- sisted of a pair of bands, one 2-Z; and one n-2, which could be described as fzp-coni- plexes. At low values of ?z, the two bands are too far apart to be readily recognised as being complexes and the bands are often quite severely perturbed by non-Rydberg states as discussed by Carroll and Yoshino (24). At intermediate values of n the pairs have a striking appearance which can be readily understood by the theory described in Section II. At high values of IZ the separation of the two components becomes ver>’


small and, as a consequence of L-uncoupling, the rotational structure collapses and is no longer resolved. The series converges to a limit near 796 A.

At about the time that Carroll and Yoshino were studying the bands, the 10 meter

vacuum spectrograph at the National Research Council [see Douglas and Potter (B)]

TABLE V Vacuum wave numbers and assignments in the Worley-Jenkins Rydberg Series of N2.

a) Band near 8351. n = 5.

J

0

1

2

3 4

5

6

7

3

9

1"

11

12

13

14

15

lb

17 It:

19

20

21

J

0

1

2

3 4

5

6

7 p,

9

10

1L

1'

13

14

15

16

17

19

19

20

+Q,Q, 0.0

(o-c) +Qho o

119944.30 -0.19

948.38 -0.09

952.34 -0.11

960.30 -0.08

964.33 -0.01

968.24 -0.03

972.20 to.02

967.13 to.06

980.00 to.08

983.82 to.08

991.34 to.09

119936.00

932.56

928.55

924.56

920.58

916.62

912.64

908.63

904.GO

900.58

396.52

e92.37

888.23

883.95

879.62

875.07

871.00

866.85

863.00

858.71

tQ.Q, 050

(0-C 1 +Q'QP~ o

121870.02 -0.09

874.27 -0.04

878.59 -0.01

382.99 +0.01

887.50 to.08

892.07 to.16

896.55 +O.ll

901.36 to.33

905.53 to.01

909.89 -0.15

914.18 -0.35

121861.94

858.09

854.33

850.69

847.13

843.62

e40.23

836.'?3

533.41

879.92

326.37

822.61

818.63

814.35

809.61

804.10

797.46

(0-C) t-R 0.0

(O-C) tRJqo o

1

-0.52

to.03

0.0

0.0

to.01

to.05

to.07

to.07

to.07

f0.10

+0.11

to.05

to.04

-0.05

-0.20

-0.51

-0.28

-0.08

to.47

to.66

19742.95

746.40

749.53

752.49

755.16

-0.03

-0.03

-0.06

0.0

757.52 to.04

759.63 to.06

761.43 to.03

762.89 -0.09

768.39'1 -0.91

771.39 tb.02

768.39' t2.21

7b8.39h t1.61

768.3911 t1.28

768.39 t1.17

763.39h t1.30

(0-C) +R*RR" u (0-C)

121774.06 to.06

-0.09 777.09 -0.06

-0.06 779.89 0

-0.05 782.27 +0.03

-0.02 784.16 -0.04

+0.01 785.69 -0.11

to.02 786.93 -0.12

to.09

to.11

to.09

-0.01 788.39h -0.16

-0.5"

-1.01

-1.78

-2.95

-4.81

-7.74

(0-C) +R*% 090

119737.52 to.22

736.50 -0.01

735.57 -0.02

734,45 -0.08

733.28 -0.07

731.87 -0.16

730.62 to.04

728.91 -0.08

727.13 -0.14

725.34 -0.07

723.40 -0.02

7x.49* to.19

719.10 -0.28

716.66* +o.oa

714.09 -0.02

711.39" -0.04

7Op.68* to.05

+“‘“qo o

121769.21 +0.14

768.68 +0.30

767.61 to.05

766.72 to.10

765.58 to.06

764.30 0.0

762.91 -0.02

761.48 +0.04

759.80 0

757.95 -0.08

756.08 -0.05

754.03 -0.07

751.913 -0.01

749.68 +0.06

746.80' -0.37

(0-C)

llq726.68 to.17

721.49x -0.21

716.66' +0.03

711.39' to.09

705.83 to.12

699.92 to.07

693.80 to.05

687.27 -0.12

679.94* -0.85

679.94X t‘z.00

669.02 t2.16

661.07 t1.51

653.31 t1.29

645.53 t1.17

637.44 +1.15

629.08 to.97

+%, o

121751.91* -0.09

746.80X to.42

740.37 -0.02

734.03 0.0

727.32 -0.02

720.29 -0.02

712.94 -0.04

705.41 to.04

697.51 +0.04

689.43 to.11

680.91 -0.01

(0-C)

o-c

741.91 to.02


TfiBLE Y (writ. )

: +Q*Q, 0.0

(‘3-c) +Q% o,u 123663.50 to.09

; 668.32 -0.02

3.L.

:i 679.;!m -0.05

684.Y> -0.02

690.6~ -0.08

; 696.59 -0.06

; 702.55 f0.02

708.,2 to.15

-3

::

-5

123654.q8

651.53

clia.;la

645.73

653.32

641.06

633.35

fG3G.90

634.84

632.78

G30.66

628.46

626.03

+:.‘.l

tc.,,!!

+ 1: ” 2

- 0 li i

- I: L1 .;

-0.oi

_C.,‘1,

-r i ‘,

-0 ‘1 3

1’10.91

cll3,Lll

: 7 ;; It i

1 1 4 6 ;

'1 ? 3 c I.3 -. C91J7 069.87

038.23

374.7:

071.69

069.48

068.07

066.0 I

oc3.93

060.84

P'R.74

was being set up for use in the 700 to 900 w region. During the course of focussing this instrument, some excellent plates of the Worley-Jenkins Rydberg series of Nz were obtained. Since these plates seemed to show somewhat higher resolution than had been obtained hitherto we felt that it would be of some value to repeat the analysis with our

program. The plates that we measured were taken in the fourth order of the grating


(30 000 lines per inch) with the helium continuum as background. Calibration was supplied by the spectrum emitted by a neon filled iron hollow cathode lamp photo- graphed in the first order. The full width at half height of the nitrogen lines was about

0.07 cm-‘, indicating a resolving power of about 160 000. Accuracy of our lines is of the order of 0.03 cm-’ and wavenumbers agree with those given by Carroll and Yoshino

to within 0.1 cm-‘. The line frequencies together with their assignments are presented in Table V.

As already mentioned the bands at low values of 98 are perturbed by non-Rydberg

states. Simple L-uncoupling theory cannot be applied to these states until after the

effects of these other perturbations are taken into account. This has already been done by Carroll and Yoshino (24) and we will not repeat it here. The first Rydberg series

member to show obvious p-complex structure occurs for n = 5 near 83.5 A”. Spectro- grams of this complex and of the one for IZ = 6 near 821 A are presented in Fig. 1. Both of these complexes are very clearly composed of two transitions; one 2-Z and one

n-2. The bands are somewhat degraded to longer wavelengths but otherwise appear to have a relatively normal structure (apart from a local perturbation near J’ = 11 in the n-Z: band with tt = 5). The separation between the two components of the

PZ = 5 comples is 2 198 cm-’ (103B) and drops to =92 cm-l (48B) for the n = 6 complex. The most readily visible consequence of the L-uncoupling is the effect on the branch intensities. It is easy to see that the R branch in the Z-Z transition is weaker than the P branch and that the R branch in the n-Z transition is stronger than the P branch. In other words it is the branches closest to the center of the complex which are enhanced in intensity. This intensity perturbation is even more obvious in the IZ = 8

complex shown in the top spectrum of Fig. 2. Here the separation between the two components is only about 40 cm-’ (21B) and L-uncoupling has proceeded even further

so that the weak R branch associated with the Z-Z transition and the weak P branch

associated with the H-2 transition have all but disappeared. It is now also easy to see the effect that L-uncoupling has on the energy levels which causes the spacing in the

strong P branch and the strong R branch to decrease making these branches more compact. In the case (d) limit these branches will look like Q branches with all the lines piled on top of each other; the case (d) selection rule for them is AR = OK. The two weak branches become like S and 0 branches with selection rules AR = i-2 and AR = -2, respectively.

The bottom spectrum in Fig. 2 shows two higher complexes, for fz = 9 and $8 = 10,

in which the transition towards Hund’s case (d) is even more advanced. Note particularly the compact appearance of the band with PZ = 10; this is because all the branches with significant intensity are of Q form.

Except for the small local perturbation already mentioned and another larger scale perturbation at high J in the Z component of the II = 6 complex, the theory described

6 Note that we have denoted the pair of bands which lie close together and form a complex as having the same value of n despite the fact that, as Mulliken has pointed out, the nodal behavior of the wave- fu’nctions require that a pair of bands be described as np~~‘lI. and (n + l)po,‘Z.+, However from the point of view of the interpretation of the rotational structure it does not matter if the pair of bands interact by “pure precession” and have the same value of n or by “accidental pure precession” and have different values of n but nearly the same energy. See also discussion by Carroll (25).

6 In Hund’s case (d) the energy levels are given by F(K) = BR(R + 1) and for each value of R, except R = 0, there are three nearly degenerate levels with J = R - 1, J = R, and J = R + 1.

1123

614 c

m-l

n =

10

n=9

FIG

. 2.

Spe

ctro

gram

s of

the

n =

8

(top

) an

d o

f th

e 7t

=

9 an

d v

z =

10 (

bott

om)

p-co

mpl

exes

of

NP.

I~IATOMIC MOLISCULES 393

11 n ‘Jo

B+ B- ilpXIO h

5 I 119739.42(O) 0 119936.94(12) 1.9245(151b 1.9272(12j' _'.1(3) -U.(ljii(?)

0 1 121770.69(Y) " 121662.71(9) :.9153(9)' 3.P247(l^)b l.ES(3) -L.CYl(jj

:. 1 123614.74(Y) 3 123655.12(q) ~.!wO(lP l.wG3(ls)L 1.?1(3) -0.0052("1)

9" 1 124079.1(12) 0 124104.2(3)

1.921(21 1.9d SO.0

m 125666' 1.92P 0.6'

in Section II fits the data very well. The numerical results of the fine structure analyses

are presented in Table VI which lists the parameters obtained in the least-squares fits. The quality of the fits are illustrated in Table V which includes either residuals (ob-

served-calculated) or the calculated frequencies for comparison with those observed. The quality of the fits are also illustrated in Fig. 3 in which densitometer traces of the

n = 6 (top) and n = 8 (bottom) complexes (spectrograms of which have already been given in Figs. 1 and 2) are compared with computer profiles generated by the intensity

program. In order to calculate the line strengths needed to generate these profiles, an assumption had to be made concerning the wavefunctions of the orbitals involved in

the transition. There are two matrix elements of the molecule fixed components of the dipole moment operator to be considered. They are defined [cf. Eq. (14)]

/*II = (L’, Ol/&IL”, O),

1 Pl = - (L’, 11 pz + i/J, ) L”, 0).

dl

(23a)

(23b:)

In these equations the primes refer to the upper state and the double primes to the lower state in the usual manner.There can be little doubt about assigning L’ = 1 but the value of L” is not so easy to determine. The orbital in question is the 3a, orbital of Nz. It can be regarded, for our purposes, as a linear combination of s and d atomic orbitals (a p atomic orbital need not be considered since it will give rise to a rU molecular or-


bital). If L” = 0 (s) and L” = 2 (d) are substituted into Eq. 23 one finds

for L” = 0 p,, = 41, pr = -1,

for I,” = 2 /.4lI = +2, /kL = +1.

Figure 3 was obtained on the assumption that L” = 0. With L” = 2 we find that the

S and 0 form branches become stronger than the Q form branches. That is, that the

R branch of the Z-2 transition is stronger than the P branch and the P branch of the

PIG. 3. Densitometer traces of the II = 6 (top) and n = 8 (bottom) p-complexes of Nz. In each case he lower of the two traces is a computer generated plot using the parameters Listed in Table VI.

DIATOMIC MOLECULES 30.5

II-S transition is stronger than the R branch. There can thus be little doubt that the

3u, orbital of NZ is predominantly su in character. Indeed this orbital is expected to be

.iso, in the united atom approximation [see for example (3, p. 329, Fig. 157)]. It should

Ixrhaps be pointed out that (‘arroll (25) used espressinns derived by Kov6cs (27) 10

calculate line intensities in the \Vorlcy -Jenkins series of Nr. These espressions were

obtained on the assumption of a p + su type transition and KovLcs does not appear

to have considered the possibility that such a transition could be of p +- (1~ type.

(b) The Ground State of Oxygew

Until recently by far the best optical value for the rotational constant B,, of the

ground state of oxygen was that obtained by Babcock and Herzberg (28). Early micro-

wave results [see for example (_‘9)] were obtained b>r fitting the triplet spin splittings

which give rise to transitions centered for the most part near 60 GHz. These transitions

depend only indirectly on the value of Ho so that optical spectroscopists continued to

accept the value obtained bJ- Babcock and Herzberg (28) despite the fact that it did

not agree well with the microwave results. Later kIcKnight and Gordy (30) observed

several transitions involving a change in the quantum number N and consequentI?

very much improved the microwave value for HO and made it clear that Babcock and

Herzberg had under-estimated their experimental uncertainties. Recently Butcher,

\Villetts and Jones (31) have recorded the Raman spectrum of oxygen under conditions

of extremely high resolution and obtained a value for Ho in agreement with the micro-

wave value. Finally, Albritton, Harrop, Schmeltekopf and %are (32) have shown that

the experimental results of Babcock and Herzberg are compatible with the latest

microwave work provided that all the photographic data are used.

None of the recent investigations have attempted to include all the available data,

that is microwave and photographic, at the same time; we thus felt it would be of some

value to repeat the calculations. U’e have included the transitions listed by Welch and

Mizushima (33), combination differences obtained from Babcock and Herzberg (&Y),

the Raman data of Butcher et al. (31), two transitions recently measured by Evenson

and Tomutsu (34) using the technique of laser Zeeman spectroscopy (laser magnetic

resonance) and a line given by Steinbach and Gordy (35).

Each datum was weighted inversely as the square of the estimated precision. The

lxogram was made to fit these transitions simultaneously by simply requiring that the

upper and lower state constants be identical and that Raman type, or magnetic dipole

selection rules were operative. Since the ground state of oxygen is electronically “z-

we also set 1, = 0 and 2S + 1 = 3.

The results are included in Table VII which lists our final constants together with

those of Welch and Mizushima (33). When we attempted to determine a value for the

centrifugal distortion constant H, the program returned a value with a standard devia-

tion larger than itself. m:e therefore set H = 0, repeated the fit and found that there

was no significant increase in the overall standard deviation. There thus seems to be no

esperimental justification for the inclusion of this parameter. Indeed, theoretical

CdCUkLtkJnS by Albritton, Harrop, Schmeltekopf, and Zare (36) predict the value of 11

to be nearly two orders of magnitude smaller than that reported by Welch and Mizu-

shima. Our calculations would seem to indicate that the error limits on B, D, and 11


TABLE VII Rotational constants Of the Ground state of oxywna

Welch and Mizushima (33)

RO 43100.5180!45)

u, 144.9629(14) i

B2 -0.157(17)

A0 59501.342(11)

__----

Al 58.470(45)

present Work

ir 43100.460(17) MHZ 1.43767638(55) cm-'

145.07(27) HI:: J1.8390(92) 10-6 cm-1 Db

HB -_-__ HZ

h 59501.3398(69) AlIZ 1.98475078(23) cm-l

DA 5.889(13) HZ 1.9645(43) 10-7 cm-l

D h,Bb 58.460(33) kHZ 1.9500(11) -6 -1

i0 cm

Yo -252.5865(15) Y -252.5858(10) MHz -8.425353(34) 1O-3 cm-l

---__- D 0.1049(22) HZ 3.500(73) 10-12 an-1 Y

Yl -246.4(30) D Y,BC

-246.7(26) HZ -8.230(86) 10-9 cm1

w a) Error limits, given in parentheses, are three standard deviations and are in units of the least significant digit quoted.

b) Dh,B = -2&y.'%

c) DY B = -2ay.q

given by Welch and Mizushima are optimistic by up to two orders of magnitude. To

some extent this conclusion is confirmed by the later results of Evenson and Mizushima (37) who give B = 43 100.518 f 0.06 MHz, D = 144.96 f 0.90 kHz, and H = -0.17

f3 Hz. Our error limits on B and D (given in Table VII) are somewhat smaller than

these. Thiswis to be expected since we have included a larger amount of data which

extends to higher values of J. The reason for the overly optimistic estimates given by Welch and Mizushima seems to lie in the fact that only three pieces of data [one line measured by McKnight and Gordy (30) and the two laser Zeeman lines measured by Evenson and Mizushima (37)] strongly dependent on the overall rotation of the molecule were used. It is thus not too surprising that these three transitions are reproduced exactly by their parameters, which include B, D, and H. At the same time

these authors have not allowed for the fact that the laser Zeeman data has a precision up to four orders of magnitude worse than much of the microwave data. We believe

that the parameters obtained from our fit and given in Table VII are the best available at this time and that the error limits (quoted as 3~) are realistic.

The extent to which our parameters fit the data is illustrated in Table VIII which lists all the microwave transitions and a selection of the optical data. As can be seen, the fit is essentially perfect since the residuals are almost all of the same order of mag-

nitude as the precision of measurement; the few exceptions probably indicate bad measurements. The data of Babcock and Herzberg (28) seem to be particularly good. As Albritton et al. (32) have pointed out the original discrepancy was probably due to the method of data reduction.

Veseth and Lofthus (38) have also very recently reported a fit to the ground state of oxygen. They have treated the optical data of Babcock and Herzberg (28) and compare this with the microwave data although they did not fit the two sets at the same time.

DIATOMIC MOT,ECULES 307

‘I’RBLE VIII Observed and Calculated Transitions and Combination

Differences in the Ground State of Oxygen.

Transitiona Observed

-Q'RPo o (2)

(2)

(4)

(4)

(6)

(8)

(10)

(12)

(12)

(14)

(16)

cis,

(20)

(22)

(22)

(24)

(26)

(28)

(30)

(4)

(6)

(8)

(10)

(12)

(14)

(16)

(18)

(20)

(22)

(24)

(26)

(28)

(30)

(32)

(34)

(36)

(38)

(40)

56264.770

56264.766

58446.600

58446.580

56590.978

60434.776

61150.570

61800.169

61800.155

62411.223

62996.6

'63568.520

64127.7.77

64670.2

64678.9

65224.12

65764.744

2.206

2.228

62466.255

60306.044

59164.215

58323.885

57611.4

56968.180

56363.393

55783.819

55221.372

54671.145

54129.4

53599.4

53066.8

1.760

1.740

1.721

1.687

1.688

1.645

1.658

PIYCiSiOll w. o_c

to.01 56264.760 to.010

fO.O1 56264.760 +0.002

to.01 58446.586 +0.014

fO.O1 58446.586 -0.006

20.01 56590.981 -0.003

f0.01 60434.777 -0.001

to.01 61150.560 +0.006

to.01 61800.161 to.008

to.01 61i300.161 -0.006

fO.O1 62411.225 -0.002

to.2 62997.991 -1.4

fO.O1 63568.532 -0.012

20.01 64127.777 0.0

*0.2 64678.903 -0.7

+0.2 64678.903 0.0

+0.01 65224.058 to.06

to.01 65764.740 +0.004

to.005 2.212 -0.006

f0.005 2.229 -0.001

fO.O1 62406.260 -0.005

20.01 60306.064 -0.020

fO.O1 59164.211 +0.004

fO.O1 58323.887 -0.002

+0.2 57612.494 -1.1

?O.Ol 56960.219 -0.039

20.01 56363.403 -0.010

kO.01 55783.811 +0.008

to.01 55221.367 +0.005

to.01 54671.146 -0.001

io.4 54129.967 -0.6

iO.8 53595.688 +3.7

*0.8 53066.807 0.0

to.005 1.753 +0.007

to.005 1.735 to.005

f0.005 1.718 to.003

*0.005 1.701 -0.014

*0.005 1.6'34 +0.004

to.005 1.667 -0.022

to.005 1.650 +o.ooLl

IJni t

MIiE

Mfiz

MHZ

MHZ

MHZ

MHZ

MHZ

MHZ

MHZ

MHZ

MHZ

MHZ

MM2

cm -1

cm -1

cm -1

cm -1

cm -1

-1 cm

cm -1


TABLE VIII (Cont.1

Transitiona Observed

+Q*Qs, Q (1) 14.381

(3) 25.876

(5) 37.372

(7) 48.859

(9) 60.336

(11) 71.809

(13) 2496283.0

(13) 83.270

(15) 2839400.6

(15) 94.711

(17) 106.146

(19) 3524220.8

(19) 117.565

(21) 3865810.

(21) 128.953

(23) 140.322

(25)

(27) 163.025

(29) 174.284

(311 185.5.42

(33) lg6.760

-QJRQO o (2) 424763.207

(2) 424763.80

(4) 25.816

(6) 37.382

(8) 48.921

(10) 60.452

(12) 71.965

(14) 83.469

(16) 94.948

(18) 106.421

(20) 117.881

(22) 129.304

(24) 140.711

(26) 152.096

(28) 163.469

(30) 174.824

(32) 186.104

i 0.005

i 0.005

t 0.005

i 0.005

i 0.005

? 0.005

*30.

i 0.005

t10.

t 0.005

i 0.005

?lO.

f 0.005

i30.

t 0.005

t 0.005

f 0.005

t 0.005

f 0.005

f 0.005

20.06

io.20

io.005

10.005

*0.005

f0.005

fO.005

io.005

io.005

io.005

?0.005

io.005

10.005

to.005

10.005

to.005

'0.005

14.376 +0.005

25.875 +0.001

37.369 to.003

48.057 +0.002

60.337 -0.001

71.808 to.001

2496278.1 t4.s

63.267 to.003

2839405.1 -4.5

94.712 -0.001

106.142 +0.004

3524224.5 -3.7

117.555 to.010

3565505.7 +4.3

128.949 +a.004

110.j22 0.0

151.673

362.998 +0.0:7

174.297 -0.013

185.568 -C.O26

196.809 -0.043

424763.189 t0.018

1424763.189 + .61

25.813 +0.003

37.383 -0.001

48.927 -0.006

60.455 -0.003

71.969 -0.004

83.469 0.0

94.953 -0.005

106.421 0.0

117.371 +0.010

129.301 +0.003

140.710 to.001

152.096 0.0

163.457 +0.012

174.792 -10.032

1'36.097 to.007

CI" -1

-1 cn

Mt!!-

-1 cm

MHZ

cm -1

MHZ

-1 cm

cn -1

-1 cm

-1 cm

-1 cm

-1 cm

cm -1

MHZ

MHz

-1 cm

cm -1

-1 om

-1 cm

-1 cm

-1 cm

cn, -1

-1 cm

-1 cm

-i cI1

cm-1

C"l -1

cm-:

cm-i

-1 cm

DIATOMIC h~fOLECULF,S 309

_,!,i. -‘Nl,i,

(0)

(2)

(4)

(6)

(8)

(10)

(12)

(14)

(15)

(18:

(10)

(2’)

(:!I)

(26)

( ‘2 )

(ill)

(?.‘) J

(1)

(3)

(5)

(7)

(9)

<II)

(13)

(15)

(17)

(I!))

(21)

(23)

(25)

(77)

(29)

(31)

(33)

Ncte: a, “ee srctirn III for cl?:


Their theoretical approach starts from what may be described as a case (c) basis and, as a consequence, is not easily compared with that reported here. Regardless of which approach yields a more physically meaningful set of parameters the case (a) starting point described here fits the optical data of Babcock and Herzberg (28) (U = 0.0046)

much better than the approach described by Veseth and Lofthus (38) (a = 0.017). From the point of view of representing the data the traditional case (u-b) starting point

would appear to be much better. One last comment concerns the Raman data of Butcher et al. (31). As these investi-

gators have pointed out, each observed line is actually an unresolved triplet. We have, just as they did, corrected the measured line positions by calculating an intensity pro-

file for each unresolved triplet. This was done very easily with the aid of the profile

program using a set of parameters close to those quoted in Table VII. The calculated relative intensities agree well with those already given by Renschler, Hunt, McCubbin

and Polo (39).

(c) The 3d-4~ Complex of BH

The 3d-complex Rydberg state of BH gives rise to strong absorption near 1510 A.

The band was first observed by Bauer, Herzberg, and Johns (40) during the flash

photolysis of borine carbonyl (BHKO). They showed that the upper state could be described as a “complex” consisting of 2, ff, and A components and that, contrary to

expectations, these components did not have an energy dependence quadratic in A.

Bauer et al. allowed for this by treating the origins of the three components as separate parameters. The only other parameter used in their treatment was the rotational

constant B. Despite the simplicity of the model the agreement between observed and calculated energy levels was quite good but it was, nevertheless, considerably worse than the experimental errors.

We have obtained some new spectra of this transition by the flash discharge technique

as described by Balfour and Douglas (41) using decaborane (BroHra) as starting ma-

terial. These new spectra were taken with the same spectrograph that was used to

obtain the spectra of Nz described above. They were taken in the seventh order of a 1.5 000 lines per inch grating which gives higher resolution than that reported by Bauer et al. The new spectra also extend to higher values of J. We thus felt that it would be

of some value to try and fit these new data using our programs. The measured line positions together with the assignments are given in Table IX.

In many respects the initial results were very disappointing. We found that many

empirical parameters had to be added before the residuals (O-C) approached the estimated accuracy of the measurements. Jungen (17) has recently shown that mixing between s and d Rydberg series is important in the spectrum of NO. Indeed this is

expected to be a general phenomenon since s Rydberg series usually have quantum defects about 1.0 greater than d Rydberg series. As a consequence (n + 1)s Rydberg states lie close to nd Rydberg states. Bauer et aZ. showed that the 4s Rydberg state of BH, I%+, lies at 67394 cm-‘, just 1000 cm-’ above the 3d complex consisting of FQ+

at 66 006 cm-‘, GrII at 66 325 cm-‘, and HIA at 66 470 cm-l. It may be noted that in this d-complex the II and A components lie close together about 3.50 cm-l above the z component. We feel that this departure from the quadratic dependence of the energy

J

0

1

2

3

4

5

6

7

a

9

10

11

12

13

14

15

J

0

1

2

3 4

5

6

7 a

9

10

11

12

13 14

DIATOMIC MOLECULES

TABLE IX Vacuum wave numbers in the 3d-4s com~lcr of BH.

o-c +s,sRo o I II

a) ii'd-X'Z+

+S,*Q 0.0

I II

66524.52' -18.13 +o.o4

583.4Sa'P -13.36 -0.92

643.20aBP -12.38 -3.89

o-c +Ri% 0.0 I II

66407.46 +5.22 -2.30

421.66 -1.44 -1.38

c) FIP+-x’s+ o-c

d) I’S+-X’Z+a o-c

J +QD$,~ I II +Q*QP, ~ , I II +%, G II +% II 0.0

66472.27 -19.78 +0.77

499.49 -15.99 +0.71

527.25 -13.18 +0.61

555.71p -10.63 to.56

587.OOP - 5.93 +2.69

615.73 - 4.30 t1.62

646.26 - 1.23 +1.72

677.44 + 2.22 +1.&j

709.21 + 6.09 t2.07

741.45 ClO.37 l 2.18

774.00 t14.96 t2.08

839.75 t25.20 t1.06

870.93 +20.99 -1.83

b) G'li-X'Z+ o-c

+“‘“Qo o I II

- 66374.57

351.13

329.50

308.95

289.47

271.13

253.86

237.61

222.57

208.57

195.65

184.13

173.56

t5.90 -1.31

+0.43 -1.13

-1.56 -0.66

-2.16 -0.24

-1.89 +0.17

-0.93 +0.66

to.53 t1.14

+2.39 +1.56

+lr.ao t2.09

+7.50 t2.52

t10.78 +2.89

+14.71 +3.49

t18.96 +3.85

401

66406.60 -18.06 co.11

418.638 -13.23 -0.79

43l.5ap -12.24 -3.75

452.aOp - 6.36 -0.63

471.76p - 5.20 -1.32

491.91p - 4.68 -1.89

513.16 - 4.41 -2.00

534.85 - 4.71 -1.97

556.74 - 5.54 -1.73

578.54 - 6.97 -1.30

599.97 - 9.11 -0.69

620.84 -11.97 +0.13

o-c +RJPo o I II ,

- -

66336.59 303.29 274.46

249.13

226.36

205.91

186.91

169.19

154.w

139.39

126.00

113.69

102.39

t5.21 -2.31

-1.82 -1.76

-5.69 -1.65

-7.64 -1.59

-8.48 -1.66

-a.29 -I..52

-7.75 -1.71

-6.94 -2.18

-4.06 -1.09

-2.34 -1.66

CO.26 -1.86

t3.21 -2.25

i6.49 -2.90

0 66090.41 +7.91 -0.33 67420.45 +0.25 66053.10 ~3.47 -0.51 446.25 +0.27 673Z8 +0.30 019.38 +i'.73 -0.50 473.36 +0.56 349.56 to.20

65977.74 t7.11 +0.23 500.73 to.05 328.07 to.08 929.39. +5.92 +0.73 876.68 +4.60

307.77 -0.05 +1.27 280.77 -0.15

821.35 +3.39 +I.91 270.91 -0.44 763.96 +1.71 tl.99 705.97 +0.25 t2.11

476.19 -4.53 t1.01 420.67 -5.40 to.17 366.28 -6.09 -0.95 313.08 -6.63 -2.43 261.61 -6.55 -3.85

18 922.10 -4.07 -14.34 19 914.17 -2;94 -18.53 20 -1.32

w a Taken from ref. (2).

21 ',",% +0.61 :;;*;: P Denotes lines believed to be perturbed, 22 093:90 t2.68 see Fig. 4. :4' 887.30 +4.77 r&,";

881.85 t7.81 -50:56 l Denotes lines used more than once in

:5 074.30 H3.00 .-61.83 the analysis.

869.31 t12.45 -72.05

402

on A is largely due to s-d mixing, that is, to a homogeneous interaction between IQ+ and F’Z+. Accordingly we have included I%+ in our fit.

In order to do this, some small changes had to be made to the program. This was done by “pretending” that a triplet d-complex was to be treated in order to increase the size of the basis set. The 3d states and the 4s state were respectively represented by

the corresponding triplet states and the 3110 state. Unwanted matrix elements were set equal to zero and those involving the 4s state were altered. Since this is in effect a non

standard use of the program we include the resulting energy matrices here for reference

in Table X. Note that we have, for the + levels, a 4 X 4 matrix which contains the 3 X 3 matrix for the d-complex which can be compared to that given by Bauer et al.

The first row and column describes the new %+ state which interacts with the YZ+

state of the complex by means of a J independent matrix element (i.e., a homogeneous

interaction) and with the ‘II state by means of a J dependent matrix element (i.e., a

heterogeneous interaction). We have also required that the energies of the d-complex be described by a single band origin plus a quadratic term XLL(~A’ - 4). There are also two empirical factors &I and Flz which allow for the fact that L is not a good quan-

tum number. For the - levels we have a 2 X 2 matrix which, apart from the addition of the parameter 6r2, is identical with that given by Bauer et al. We have, of course, added centrifugal distortion terms as indicated at the bottom of Table X using the

method described in Section II. Two comparable fits have been carried out. In the first, the I%+ state was not in-

cluded and each component of the d-complex was allowed to have a separate band origin. In the second, the fit was carried out as indicated above. The ground state

constants were held fixed at the values found by Johns, Grimm, and Porter (42). In each fit we eliminated all lines with J 2 1.5 since the Z component of the complex

was the only one to be observed at higher values of J. In addition, several lines thought to involve perturbed levels were excluded ; these lines are indicated in Table IX. The parameters obtained in the two tits are given in Table XI. The extent to which these parameters fit the data is indicated both in Table IX, which lists residuals (obs-talc), and also in Fig. 4 in which these residuals are plotted against J. It is easy to see that


TABLE XI Constants (in cm? .) for the 3d-4s (complex of BHa.

a) The 3d-complex (P IL+, G 'l, and H 'A)

Constant Calculation I Caiculation II

B

m

H/Db

v.

hLL

E_ L

%I

EA

12.14 t 3.1: 12.255 k 0.033

0.0350 t 0.0090 0.0353 i 0.002c

1.0 x 10 -4 1.0 x 10 -4

d 66373.4 + 1.7

-d 23.68 f 0.87

65995.4 f 9.9 - vu-4h;L = 66278.66

66319.5 * 11.0 I uo-2h;L = 66326.02

66495. 2 !P. : vk,+4~Li = 66468.10

calculation II, with u = 1.9 cm-‘, is a considerable improvement over calculation I,

with cs = 9.4. This improvement has been obtained without adding too many extra parameters despite the inclusion of more data. The final fit still does not reproduce

the experimental results to within the accuracy of measurement. This is partly due to

the local perturbations mentioned above but is also certainly due to further inadequacies

in the model. There is no doubt that an improved fit could be obtained by adding extra

parameters but the value of such a procedure seems doubtful to us. The results presented here do show, quite convincingly, that s-d mixing is important in the 3d complex of BH.

CONCLUSION

In this paper we have described a method for the calculation of rotation-electronic energy levels of diatomic molecules. There are no limitations, except those set by the size of a given computing facility, to the value of L, A or to the multiplicity of the states which can be treated. The method is particularly convenient when the computations are to be carried out in a digital computer since only a small number of matrix elements of the zeroth order Hamiltonian need to be stored (see Table II). All other matrix elements which are required as a consequence of introducing centrifugal distortion effects are generated automatically by the computer programs. While this may not


+8- l 6- +4- +2-

-8. -IO-

-6: , , , , I , --OS_ 1 , 0 5 IO J

FIG. 4. Residuals (obs-talc) obtained from least-squares fits to the 3d-complex of BH. Residuals obtained from calculation I have been plotted as squares and have been joined by dashed curves. Re- siduals obtained from calculation II (that is including the effects of s-d mixing) have been plotted as

circles joined by full curves. The suspected presence of several local perturbations has been indicated by breaks in the curves.

result in the most efficient program for a specific situation, it does allow almost any conceivable situation to be handled by a single program.

The programs were developed originally to treat Rydberg series in diatomic molecules. They have been found to be particularly useful in the calculation of the apparently anomalous intensity distributions often found in the rotational structure associated with such Rydberg series. At the same time they have also been found to be helpful in calculations involving transitions between what might be described as “normal” states. Such transitions can be governed by magnetic dipole and electric quadrupole selection rules as well as the more usual electric dipole selection rules. As a consequence Raman spectra can also be treated.


The programs are not, however, completely general since the model assumes that

Rydberg states have ion cores with no angular momentum. It allows therefore neither

for the inclusion of all the states which can occur, for example when a 3d orbital is added to a % core, nor for the spin orbit interaction of that core which can mix the triplet and singlet Rydberg series. The programs are thus not suitable for dealing with the Rydberg series of molecules such as HF or HCl. Further work on this type of situa-

tion needs to be done.

Note added in proof. Brown et OZ. have recently proposed a useful extension to the Kopp and Hougen

[lo] e, j scheme for the labeling of parity doublets. For half integral spin their scheme is the same as

that of Kopp and Hougen and for integral spin, levels with parity +(-l)J are labeled e and those

with - (-l)J are labeled j. The difference between the scheme described in this paper and that pro-

posed by Kopp and Hougen for half integral spin has already been discussed in Section 2. For eeen integral spin our P, = + corresponds to e whereas for odd spin our P, = + corresponds to j.

It is easy to see that the Brown et al. scheme has an obvious advantage over ours when perturbations between states of different multiplicities are considered. Thus in a singlet-triplet perturbation one tinds that levels in the e submatrix of the singlet perturb levels in the e submatrix of the triplet. In our

scheme one would find that the P, = + submatrix of the singlet interacts with the P, = - submatrix

of the triplet.

RECFXVEID: .4ugust 22, 19i4

REFERENCES

1. CH. JUNGEN AND E. MIESCHER, Can. J. Pkys. 47, 1769 (1969).

2. D. W. LEPARD, Can. J. Phys. 48, 1664 (1970). 3. G. HERZBERG, “Spectra of Diatomic Molecules,” Van Nostrand, Princeton, N. J., 1950. 4. E. L. HILL AND J. H. VAN VLECK, Phys. Rev. 32, 250 (1928). 5. J. K. L. MACDONALD, Proc. Roy. Sot. London, Ser. A 138, 183 (1932). 6. R. SCHLAPP, Pkys. Rev. 39,806 (1932). 7. J. K. G. WATSON, Can. J. Phys. 46, 1637 (1968).

8. E. R. WHITING, J. A. PATTERSON, I. Kov.&cs, AND R. W. NICHOLLS, J. Mol. Spectrosc. 47,84 (1973). 9. R. N. ZARE, A. L. SCHYELTEKOPF, W. J. HARROP, AND D. L. ALBRITTON, J. MoZ. Spectrosc. 46,37

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29. M. MIZUSHIMA AND R. M. HILL, Phys. Rev. 93, 745 (1954). 30. J. S. MCNIGHT AND W. GORDY, Phys. Rev. Lett. 21, 1787 (1968). 31. R. J. BUTCHER, D. V. WILLETTS, AND W. J. JONES, Proc. Roy. Sot. London, Ser. A 324,231 (1971). 32. D. L. ALBRITTON, W. J. HARROP, A. L. SCHMELTEKOPF, AND R. N. ZARE, J. Mol. Spectrosc. 46, 103

(1973). 33. W. M. WELCH AND M. MIZUSHIMA, Phys. Rev. A5, 2692 (1972). 34. K. M. EVENSON AND L. TO~TSU, private communication.

35. W. STEINBACH AND W. GORDY, Phys. Rev. 8, 17.53 (1973). 36. D. L. ALBRITTON, W. J. HARROP, A. L. SCHMELTEKOPF, AND R. N. ZARE, J. Mol. Spectrosc. 46, 25

(1973). 37. K. M. EVENSON AND M. MIZUSHIMA, Phys. Rev. A6, 2197 (1972). 38. L. VESETH AND A. LOFTHUS, Mol. Phys. 27, 511 (1974). 39. D. L. RENSCHLER, J. L. HUNT, T. K. MCCUBBIN, JR., AND S. R. POLO, J. Mol. Spectrosc. 31, 173

(1969). 40. S. H. BAUER, G. HEKZUERG, AND J. W. C. JOHNS, J. Mol. Spectrosc. 13, 256 (1964).

41. W. J. BALFOUR AND A. E. DOUGLAS, Can. J. Phys. 46, 2277 (1968). 42. J. W. C. JOHNS, F. A. GRIMM, AND R. F. PORTER, J. Mol. Spectrosc. 22, 43.5 (1967). 43. J. M. BROWN, J. T. HOUGEN, K.-P. HUBER, J. W. C. JOHNS, I. KOPP, H. LEFEBVRE-BRION, A. J.

MERER, D. A. RAMSAY, J. ROSTAS, AND R. N. ZARE. The labeling of parity doublet levels in

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