Theory of energy shifts associated with deviations from Born-Oppenheimer behavior in 1Σ-state diatomic molecules

JOURNAL OF MOLECULARSPECTROSCOPY19, 305-324 (1966)

Theory of Energy Born-Oppenheimer

Shifts Associated with Deviations from Behavior in ‘Z-State Diatomic Molecules*

R. NI. HERMAN AND A. Department

of Physics,

The Pennsylvania

AsGHARrANt

State University,

I .niversity Park, Pennsylvanicl

The exact Hamiltonian operator for nuclei and electrons is considered for diatomic molecules. From it, using perturbation theory, an accurate linear, separable effective Schrddinger equation for nuclear motion in ‘2 state molecules is derived. This equation constitutes a refinement of the Born-Oppenheimer approximation, capable of describing the energy level structure exact to terms of order m/M (the ratio of the electronic to the nuclear reduced mass) times the Born-Oppenheimer values. From the radial equation, one can easily deduce the corrections to the more important molecular energy level parameters and to relationships between them, in terms of a relatively few functions which characterize the electronic states. Ultimately, these must be evaluated either empirically or numerically. Applications and numerical examples which bear on the present study are reviewed. INTRODUCTION

An approximation that is basic to almost all molecular structure theory is the Born-Oppenheimer (BO) approximation (1) . In this approximation, the nucleifor purposes of dealing with electronic motion-are considered to remain fixed while the electrons move about in the combined Coulomb fields of the nuclei and other electrons in a stationary eigenstate. The eigenenergy of the electrons for given internuclear separation, R, together with the electrostatic energy of mutual repulsion between the nuclei then constitutes a potential function, E(R j, which depends only upon the internuclear separation and, parametrically, upon the particular quantum numbers for the elect.ronic stat,?. The nuclear motion is then determined by solving a two-body Schrijdinger equation with E(R) as the potential function, leading to the familiar nucalear vibration-rotationeigenstates and eigenenergies. On the other hand, rigorously speaking, the diatomk molecule problem is 011~ of several electrons and two finite-mass nuclei at once, rather than two successivcx problems, one of many electrons followed by one of two nuclei. Consequently, the BO approximation gives rise to some discrepancies between the observed and * This research was supported in part by the National Science Foundation. t Present address: Science Faculty, National University of Iran, Tehran, Iran. 305

306

HERMAN

AND

ASGHARIAN

predicted behavior of diatomic molecules. By and large, however, the approximation is adequate, simply because the electrons are so much lighter than the nuclei. Generally speaking, molecular parameters which are nonzero as calculated through the BO approximation are accurate to within a factor m/Al, the ratio of the electronic to the molecular-reduced mass. In the present paper, a theoretical analysis of the deviations in the energy level constants of ‘Z-state diatomic molecules from those obtainable through the BO approximation is carried out. We concern ourselves only with the lowest order correction terms-of order m/M times the BO energy level parameters-higher order corrections being neglected. Of course, many specific problems of the type under consideration in the present paper have already been analyzed (g-11 ). The literature on the subject is varied, with the notation and conventions employed often differing to suit the needs of each investigator. The subject is further complicated by the fact that any of several different coordinate representations may be used in carrying out an analysis (1.2). It is partly the purpose of the current paper to present a complete and unified approach to the problem, therefore. Further, it is not known whether deviations from BO behavior cause large fractional changes in the higher spectroscopic constants (the latter of which often represent energies much smaller in magnitude than the actual energy level shifts associated with deviations from the BO approximation), which are important to the empirical determination of the anharmonic potential constants. Errors in these constants could then cause relatively large errors in I’o0 and the small correction terms relating the energy level parameters, Y ii , to the importa,nt molecular constants CAand B, , as occur in the theory of Dunham (1~). Hence it is intended, here, to develop a theory for ‘X-state diatomic molecules which in principle allows one to deduce the effects, associated with breakdown of the BO approximation, on any energy level parameter whatever. To do this, we shall develop a refined effective Schrodinger equation for nuclear motion, which takes into account all deviations from BO behavior to order m/N.’ The usefulness of this equation will be demonstrated by obtaining corrections for the more important energy level parameters, and the equation governing the interrelationships between the energy level parameters w, , B, and D, for any given molecular isotope. Deviations between actual and BO energy level parameters in X-state diatomic molecules can be experimentally are, along with some examples: 1. Deviations

between

tested through three types of observations.

measured

energy level parameters

These

and those predicted

1To the authors’ knowledge, this represents the first time such an equation has been published, and its consequences developed. Fisk and Kirtman (Reference 10) have developed a generalized internuclear potential operator which, within approximations, renders their effective SchrGdinger equation for J = 0 states equivalent to ours. The present equation is more general, however, in that its validity is not restricted to states for which J = 0.

ENERGY

SHIFTS

IN 1)TATOMIC MOLECIJLES

307

through accurate theoretical calculations which employ the BO approximation. There are a few examples which deserve stt’ention, all of which are limited to I 1117 simplest molecular electronic ground statq” those of Hz+ and H? . n. The dissociation energy of Hz+. Johnson (3) calculahed the dissociation energ) from the lowest vibrational level to be Do0 = 21 387 cm-’ in the HO approxim:ltion. Employing correction terms for 0,’ whose forms were first, given bv vm Vleck (2’), she found the finite nuclear mass corrections to be - 60 cn-‘I, ohtaining, as :1 refined value for Don, the value 21 327 f 20 cm-‘. The prescnhl? accepted experimental value (14) for t(his constant is 21 366 f 15 cnC1. Thr~ sit,uation here is somewhat confused, however, in that. Dalgarno and IVIcGrroll I 4 j, essentially repeating Johnson’s calculat8ion, found t,hal the finite nnclr:u, mass corrections to D,’ are +3 cm-‘, while the results of cnlculalions by (~ohcn, Hi&es, and Riddell (5) indicate a value of 213X2 cm-’ for I),:, including a finite> nuclear mass correction of - 3 cm-‘. 0. The dissociation energy of Hz . In calculations enlploying v:kat~ional w;:1v(’ functions conkLining up to 80 terms, Kolos and Wolnicwicz (8) have recc>ntly calculated the dissociation constant of HP as De0 = 3s 292.7 mldl in t,h(b B( ) :~pproximat,ion. Upon introducing relativistic corrections, this hecomcbs DP” = $8 292.2 cn-‘. They have also computed the ndiabat#ia finit’e nuclear mnss corrections, obt.aining t,he final result I), = 38 297.1 CII~-~, somewhat larger 01:lr1 Herzherg and Jlonfils’ (16) measured value, 38 292.:3 or 3S 292.9 f 0.5 cn1 ‘. I vnlue 36 1 l-Z.? cn-.‘, as comparrd t.o the value 36 113.0 f 0.3 or 36 113.6 f O.:%cnl--* mcnsurc(l b> Herzberg and Xlonfils (15). Although the agreemenl with cxperimenl~ is 11w mcricallg improved, t,he results of the present paper suggest. that the inq)rov+ ment is coincidental. The problem will be discussed at’ great’cr length below. V. The mtational constant of Hz. Fink, Wiggins, and Rank (16) recent,ly mcasuretl the rotat.ional constant of Hz as B, = 60.8679 f 0.008 ~111~~. Its VOW, hsccl on the equilibrium int,ernuclear separation calcul:~t,c~d by Kolos &Jld Wolnicwicz ( 81 in t,ho HO approximation, is 60.8937 cK1, relativist,ic c*orrect,ions h:Iving bc1111 :y,plied. lteccnt,ly, Tipping and Herman ( 171 have cah~ulated t’he finit’c nuclc:u mass corrections kng the results of the Tiolos and Wolnicwicz calcul:lt,ion t.o m:&c the adiabatic corrections, and the mugnetirt moment d:it.:L of Barnes, Rray, ;md ltnnlsay (~8) t,o determine the nomtdiahatic corrections. With these (‘orrcct ions, t,hc best theoretical value for this constant is R, = 6O.S.%i;i f 0.001 cn?,

308

HERMAN

AND

ASGHARIAN

representing reasonably close agreement with Fink, Wiggins, and Rank’s experimental value . In addition, the theoretical B, values for Hz , HD, and Dz show good agreement with Stoicheff’s (19) measurements. 2. The second type of observation concerns the relationships between spectral constants for different isotopes of the same molecule. In the BO approximation, observed molecular constants, as obtained through the Dunham theory (IS) from measured positions of the energy levels, obey the relationships

WeilWej

BC3,/B,

=

=

(Mj/Mi)"',

(lb)

C”_iIMi),

(lc)

etc., where subscripts i and j refer to isotopes i and j, Mi,j being the nuclear reduced mass for the molecular isotope in question. Deviations from BO behavior may then be found by noting discrepancies between these relationships and those found experimentally. Some examples follow : a. The dissociation energy of molecular hydrogen. Herzberg and NIonfds (15) have found 0,” to vary slightly from one isotope to another in hydrogen. Specifically, D:(Hz)

= 38 292.3 f

0.5 cm,

Dz(HD)

= 35 290.3 f

1.5 cm’,

D:(Dz)

= 38 290.8 f- 0.7 cm-l.

b. The rotational constant of carbon monoxide. In measurements of the rotational constants for C?O*’ and C1*O16,Townes and his collaborators (11) observed the deviation from Eq. (1~). They theorized that the difference between the actual, and the BO values of B, for any CO isotope consists primarily of a calculable nonadiabatic part, proportional to the (observed) magnetic moment for the isotope, and secondarily of an adiabatic correction term (the “wobble stretching”), whose magnitude could be determined through the above measurements on C12016and C14016.By assuming a l/M-dependence for the magnitude of the latter correction, and by accurately measuring the Be-values of other CO isotopes, they were able to determine isotopic mass ratios for the nuclear combinations C13-C1’, 018-016, and O”@. The present work shows [see Eq. (45)] that the adiabatic corrections do not have the assumed l/M -dependence, however. Although it is difficult to say how large the errors introduced by this assumption are, they could significantly reduce the accuracy of the 018_016 and O”-0” microwave mass ratio determinations. 3. In the third type of observation, one considers a single molecular isotope at a time. Although the molecule may be too complex to allow accurate computation of the energy level parameters, there nevertheless exist well-known relationships

ENERGY

SHIFTS IN DIATOMIC

MOLECULES

309

between some of the energy level parameters. The most important of these is the relationship between B, , CJ.and the centrifugal stretching constant, D, . In t,hct BO approximation, the relationship is a very simple one,

D, = -4B,3/~,', these constants being obtained from spectroscopic! measurements through Dun ham’s theory, for example. If this relationship is violated experimentally, therefore, one must attribute the discrepancy to the inadequacy of the BO approximation. An important example of this type of discrepancy has been observed in HCl by Rank, Rao, and Wiggins (SO). The experimental value of Yo~(--De)is rclported as -5.3193a X 10h4 cm’ while the Dunham theoretical value (~~~B~/w~' ) is -5.315g1 X 10e4 cm-‘. An analysis of this discrepancy, given elsewhere by Herman and Asgharian (21), reveals that the discrepancy does not appear to hi resolvable through the present theory. THEORY

The first systematic approach to the problem of deviat,ions from BO energies in ‘Z-state diabomic molecules appears to have been made by van Vleck (2). His theoretical results were obtained in a relatively straightforward manner, and appear to be correct. While many find his results useful, others, finding his no& tion cumbersome or his results difficult to work with directly, have preferred t,o work out specific problems through different mathematical representations morr suited to their specific needs. The present treatment’ is more complete than van Vleck’s in that all energy level shifts to order m/Jl times the BO energies are given accurately; direct, in that these effects are incorporated into a single separable linear differential equation governing nuclear motion; and is presented in a form that is convenient for numerical calculations or semiempirical-type estimates foi the magnitudes of pertinent quantities. The influence of the small terms on thr energy level structure can readily be derived from the resulting radial equation for nuclear motion pqs. (39), (42)l. The simplest coordinate representation in which to formulate the problem al,pears also to be that initially adopted by van Vleck, which is constructed in the following manner. The kinetic energy operator, for all part,icles (nuclei 1,2 having masses iJll , Af2 and electrons i = 3, 4, * . ’ ,‘n + 2, there being n electrons in the system),

is

where the coordinates RI’, Rz’, and ri’ measure particle positions relative to the laboratory frame. Let us define the nuclear relative coordinates and the nuclear center-of-mass (NCM) coordinates in the usual way,

310

HERMAN

so that in terms becomes

R

=

Rz’ -

R’NCM

=

(IMJG

of these

T=

AND

ASGHARIAN

RI’, +

M&)/(~I

and the ri’, the kinetic

coordinates

2

1 M1

+

M2

sz+4 c v:‘i

energy

operator

\

V&M

_g

(3)

JlZ),

+

+

7

L

M being the reduced mass of the two nuclei. By similarly defining the molecular center of mass, hM , and the position coordinates of the electrons relative to that of the nuclear center of mass, r, , R CM

=

[(MI

+

MdR

kCM

+

1)~ Cri’]i(Ml

+

MS

+

mn), (5)

ri = ri’ in the manner

RkCM

conventional

to atomic

physics,

is obtained where TCMrepresents the translational kinetic energy of the molecule as a whole. Inasmuch as the center-of-mass dependence of the Schrodinger equation is separable (T depends on bM only through TCM, while V is a function only of R and the ri), we shall not consider it further. The second term in the brace of Eq. (5) represents a mass polarization energy similar to that occurring in the kinetic energy operator for the many-electron atom, this operator containing the diagonal V? operators, among others. If one prefers, he can group these with the first team in the brace, giving the familiar form

P being

a reduced

electronic

mass,

P = 7n(M1+

Mz)l(m

+

Ml

+

Mz).

While the latter form is sometimes useful, Eq. (6) is more acceptable for our purposes, inasmuch as the first term in braces is then independent of the nuclear masses, deviations from BO behavior arising from the other terms by themselves. The exact Schrodinger equation is now f 7

Vi” + a&

F

Vi.Vj + A V$)

+ V(R;

ri) -E}

$J = 0.

(7)

In the BO approximation (1) , one assumes that #(R, r;) can be factored as an electronic state, qx(ri ; R), times a nuclear state xk’(R). One thus observes fhat the largest terms associated with differential operators acting on the electronic

ENERGY

SHIFTS IN DIATOMIC

:311

MOLECULES

state are t.he terms - (fi2/21n) xi Vi2px(ri ; R), the others being relatively insignificant. Neglecting these small differential operators as they act on ‘ph , therefore, t,he equation --&XV?+

V(ri,R)

L

cPh(ri; RI -

is obtained

for the BO approximation. -

2

V? +

7

$&, V&~(R)

px

Defining

= Eh”k PA XkX

EfO( R) through

V(ri , R) - L’:“(R)\

px(r; ; R)

the equation = 0,

(9)

1 we obtain,

by substitut,ion -

into Eq. (811, an equation

2G OR2 + E;O(R,

-

for nuclear

1

EXB, &R)

[

motion,

= 0.

(10)

We now wish to analyze the problem more exactly, keeping the small terms involving kinetic energy operators not retained in the BO approach. To do this, however, we find that simple product func%ions of form cpxxk’ are not sufficiently general for the description of t,he molecular states. Instead, we must represent, rC, by a linear rombination of functions (12) belonging to the complete orthonornral h set, ‘PXXL, # = 5 CMXXkh. Substituting

this expansion

into the Schrijdinger

equnt,ion,

(7 ),

(11)

is obtained. A. XDIABATIC CORRECTIOKS TO THE BORK-OPPEXHEIMER POTENTIAL Were it not for the t’erm on the right side of Eq. ( 11‘,, we vould redefine the PA 2~s follows :

!f- g F

0;”-

‘(dflli; nfJ

-_

2tl

g

Vn4 +

vi.4, (12) I’tr;

, Rj

pA’(r; ; R) = Fx’CRjph’.

312

HERMAN

AND

ASGHARIAN

Thus, in this approximation, Eq. (11) is again satisfied by simple product type wave functions. Because of the appearance of the VR2operator in Eq. (12)) this equation is strictly not an eigenvalue equation. Nevertheless, the equation is solvable in the sense that the operator - (@/2M)V,’ may be considered as a perturbation, in which case complete orthonormal eigenfunctions can be employed and corrections to these, associated with the perturbation, can then be determined through the usual techniques. (The corrected eigenfunctions obtained in this manner are no longer orthogonal, however.) For a determination of the lowest order changes in energies associated with the last term on the left-hand side of Eq. (11)) the approximate BO functions, cp~, are sufficiently accurate. We shall no longer make the distinction between the cp~and ‘pi’, therefore. To lowest order in m/M, the potential function &‘(R) is given by I&(R)’ = Ex(IQBO -

?(M,“; M,)

@

IC Vi’Vj 1A>i,j

& (A 1VR21A>- (13)

The latter terms constitute the adiabatic corrections to the BO potential, and lead to corresponding corrections in the diatomic molecular energy level structure. The term “adiabatic” refers to the fact that the simple product nature of the #-function is still preserved by the terms giving rise to the corrections, whereas (virtual) transitions into other nuclear-electronic states are not excited by the operators responsible for these corrections. For many purposes, the above form of the adiabatic corrections is satisfactory, and numerical computation may be carried out directly. However, there is another form in which these corrections can be written, which deserves mention. Referring to the transformation equations (3)) it follows that VI =

-VR + [MI/ (MI +

M~)I%c~ ,

while VNCM=

-7%

+ [(MI + Mdl(M1 + MB + mn)lVcx

follows from Eqs. (5). Since the electronic states, as we have written them, do not depend on RCM , for our purposes VI is equivalent to VI = -V,

- IMI/(MI+ Mdl CVi.

Hence, Ml

l +

M2

c i.j

v,.vj

can be obtained, implying that the adiabatic corrections to the BO potential can

ENERGY SHIFTS be written

IN DIATOMIC

MOLECULES

3 I:1

in the form

E:h’(R)= EXBO(R) -

”i &

(

x / VI” / x ) +

iv;* ( x I v2”I x > . )

(1-k)

Through tJransformations similar to those employed in deriving Eq. (14)) it is obvious that the adiabatic energy corrections reduce to the sum of isotope shifts for the resulting atoms, as R becomes large. B. NONADIABATIC ds previously t’he n0nadiabat.k

CORRECTIONS

stated, it is sufficient to use BO pi’s in determining the effects of terms. For this purpose, we take as the Schrodinger equation

~c+‘A[-2

0~’ + Ei(R)

- E

1

(15)

x,t(Rj

-5

= 0.

VR (F~.VR x/;“(R)

The form of this equation is standard to ordinary that the XkX’ssatisfy the eigenvalue equation

[This involves a minor redefinition tween Ep,'(R) and EFO(R) .]The written

> perturbation

of the x ‘s, associated effecotive perturbation

X’ = - (li”/nr)v,eV,“,

theory,

provided

with the difference beHamiltonian may b(t

(17)

the superscripts indicating that the differential operators apply only to the electronic, or nuclear, states. The first-order energy shift on the latter operator obviously vanishes. This is true because VR j X} represents the incremented varia,tion in (OXas R is changed to (R + AR), in which case cpx acquires first-order cont,ributions only from ot,her states cpi,( X’ # X), which are orthogonal to cp~. Hence, we have (x ( G, [ X) = 0. The second-order

energy

change

for t#he state kX is

If the Cartesian components of R = (X, Y, Z) are chosen so that the Z-axis instantaneously coincides with the molecular axis, the cylindrical symmetry of the Z-states guarantees that electronic matrix element produrts of the type

314

HERMAN

(X / Vx 1X’)(X’ j Vz 1X) are zero.....

AE$

= -_

;2,&

AND

ASGHARIAN

Hence the second-order energy reduces to (MC1ViX’Vi k’)(h’k’ 1ViXVi k) EW - Exlc

(19)

i=x,y,z

By neglecting molecular vibration-rotation energies against electronic energies in the denominators, which is justified on the basis of the fact that we are interested only in lowest order effects (the ratios (Ek! - Ek)/( EL, - Ek) are of order m/M, after all), we obtain

ii4

(2)

AEXk = --

0 ( Vi 1X’)Vi (A’ ( Vi ( X) Vi (Ey - Ex)

AZ i=x,y,z

&P

(2oj

’

where we have used the closure rule in summing on k’. By noting that the only nonvanishing derivatives Vi@ 1Vi ( X)are those for which i = 2, and by defining the functions $g:(R)

= $

c h’#h

(X I vx ( A’)@ ( ox I x> 6%~ - Ed (21) lx

I VY

IX’>(X’

(Ev

e!&;(R)

zz

$ c

-

I VY Ed

I X) ’

6 I vzI 0(X’ I vzI A> (Ev - Ex)

X’#h

(22)

’

and

2

g:(R)

= z

(A I vzI NW

c h’#X

I vzI v/am

(Ex~ - Ex)

(23)

’

AEE’ can be written g1”(RPR2 + [g;(R)

- g:(R)] & + 936(R)

where we have identified Vz with d/aR in the above equation. It is therefore the nuclear states were governed by an effective perturbation of the form3

d(RjV~”+ b2e(R)-

g:(R)]

$ + g;(R)

3 An analysis of the higher order (e.g., 3rd, 4th, . . .) perturbations, VR~ operator reveals that to good approximation, they are identical . . .) order effects on the effective interaction, Eq. (25).

.

as if

(25)

on the - (hz/IM)~g. to higher (e.g., 2nd,

ENER( :Y SHIFTS

We therefore

postulate

IN I ,IATOMIC

an effective Schrbtlinger

MOLECULES

:315

equation for nuclear motion, of form

based on the above reasoning. It is seen that’ the terms obtained by neglect,ing the nuclear energies against the elect,ronic are equivalent to velocity-dependent pot,ential energies (10). Alore correct, physically, is t’he view that these t,erms represent t.he kinetic energy associated with the fact that’ the electrons are pulled about by the nuclei as they perform their rotational and vibrational motions. We expect that inertial energies of this Oype should depend upon bhe squared nuclear velocity components, with no further dependence upon nuclear masses. That this is largely true4 can be seen ill Eqs. (21), (22) and (25). [The effects of the ga(R)-dependent term vanish in the approximation in which we are working. See Ey. (41 j, ff.] This may also be inferred from the form of the effective perturbation operator, Eq. (17). Actually, the latter is a mass polarization (or motion polarization) operator which has the effect of inducing the electrons to follow the nuclei in their mobion. Treating the nuclear motion classically, for t’he moment, t’he interaction ran be written SC’ rff = W.(fi/i,V,‘, 2, being the classical relative nuclear velocity. It is following the derivation of Eq. (15 ) , we had assumed which case the operator (fi/Jll) rR” would everywhere the resulting calassical Hamiltonian function governing have been

interest’ing to note that if, classicaal nuclear motion (,in have been replaced by W) , nuclear motion would then

with

The quantization of this equation yields Eq. (26) directly [provided that we again neglect the term in g3(R)]. The neglect of nuclear energies against elec+ronics, above, is therefore seen to be equivalent to making an int,erim classical-nuclearmotion approximat,ion directly. In this approximation, one does not hold t’he nuclei fixed as he does in the BO approximation. In&ad, he assumes that the nuclei follorr~ classical trajectories while the electrons find their eigenstates and energies. It remains to illuminate the g”-functions. It can be shown that gl”(R) is equal 4 Of course, gle and gze vary with the position

of the molecular

center of mass.

HERMAN AND ASGHARIAN

316

to the electronic contribution to the gyromagnetic ratio of the (hypothetical) rigidly rotating molecule with fixed internuclear separation R. Since this identification has been made elsewhere (11), only a brief outline will be given at the present time. The magnetic interaction energy for a k-state molecule in a magnetic field is composed of two parts. The first, that associated with the rigidly rotating nuclei, is given by X:,,

= - gln&H. J,

(27)

where pn is the nuclear magneton, and gin is given by

(28) The second contribution is associated with electronic rotation. In the BO approximation, there is no net angular momentum associated with ‘Z-states. On the other hand, we know that the electrons exhibit an inertial effect when the nuclear frame rotates, indicating that they do indeed possess a small amount of orbital angular momentum, obtained precisely through the - (fiz/M)VRe.Vgn interaction with the nuclei. The result is that the electrons then possess a small magnetic moment which in turn interacts with an applied field. Let us denote the electronic angular momentum by L, taking the origin to be coincident with the molecular center of mass. The interactions giving rise to the electronic magnetic energy are

(29) p being the Bohr magneton. Under our previous choice of coordinate axes, L, 1A) is zero for Z-states, so that xk,

= P(& L, + H, L,) - ;

0,” *vlzn.

(30)

Now by considering L, and L, as angular differential operators, it is readily verified that L, = (fi/i)RVY and

(31) L, = - (6/i)RVx”.

Moreover J is related to Vgn through the equations J7, = -(fi/i)RV,“, J, = (fi/i)RV,“, and

(32)

ENERGY SHIFTS IN i)IATOMIC

MOLECULES

3 17

J, = 0. Substitut,ing Eqs. (31) and (32) int’o Eq. (30)) and evaluating energy contribution which is linear in H and J, the result

the second-order

is obtained, the R-dependence arising from the fart that the eigenstates PA, cp~, and eigenenergies Ex , Ehr depend on R. The effective Schrodinger equation for nuclear mot,ion may finally be written

K

1 -t- $

P

gl(R)

+

>

VR2+ c

ggr’(R:)

P

;tti

P

igdR) - g,(R)‘) 3

1

+ &‘lR)

- IfXhl $(R)

the gyromagnetic

function

for the molecule

(33 ) = 0,

J

therefore, by noting that the reduced mass associated atoms is related t’o M through t’he equation

Here,

‘I

with

the pair of neutral

is

gl(R) = gl’(R’) + gin, while gz(R) is specaified through

gdR)

(37)

the relation

(3s)

- gl(R) = g2’C.R) - g1C(R)r

The function gz(R) is analogous to gl(R ), referring to an electronic enertial effect. with respect to vibration, as opposed to rot&ion. Unfortunately, there is no low order magnetic effect associated with g*(R), As a result, one is prevented from obtaining its magnitude through independent measurement, contrary to the case for gl(R). (A semiempirical method for estimat’ing gz(R) gl(R) has been suggested in Reference 21, however.) C. THE EFFECTIVE RADIAL EQUATION In order to apply Eq. ing the radial equat’ion

(35),

one must’ first separat’e

the angular

motion,

+,AR!)&j (-A[(l+~g,iR))~+2(1+~~!gl(R) + Ex’iR)

+

6% + (~~d~f,)g~(R)J.J(.J + 1) _ ~~~~~RiJ(R) 2JlAt p -____--

= 0

leav-

(3y)

318

HERMAN

AND

ASGHARIAN

By making the substitution

and multiplying the resulting equation by

R exp

x

[gl(R) - gdR) + g3e(R)l,

the equation

-_ m

(gl(R) - gz(R) +

MP + fi2[1+

>I(41)

g3e(R))+ d(gl(R) - B(R) + gt(R)) R dR

R2

+ 1) + Ex’(R) -

(m/M,)gOW(J 2M~t R’

&CR)

= 0

is obtained. The terms in [gl(R) - gz(R) + g:(R)] and its derivative behave as further adiabatic corrections to the nuclear potential function E&‘(R) . However, these corrections are of order (m/M,)gl(R) times a rotational energy, which is a factor m/M smaller than the previously calculated adiabatic corrections. Accordingly, these terms can be neglected. Equation (41) as it stands, is still not amenable to solution through Dunham’s WKB treatment however, which accompanies the operator because the function [l + (m/M,)gz(R)],

d2/clR2, varies with R. Division by { 1 + ( m/M,)[s2(R) - g2(Re)]}, R, being the equilibrium internuclear separat,ion, results in Dhe equation

+

fi*[l + (m/M,>(gl(R) - gz(R)+ dR~))l

J(J

+

2MAt R2 I -

+ gp

$

(gz(R) -

(gdR) -

(42)

gz(Re))

P

1)

Eh’(R) 1

S-?,(R) g&L))&,)

= EN J%(R).

This equation is amenable to a WBB treatment, provided that EhvI , as it appears on the left side, is treated as a known power series expansion in (V + x) and J(J + 1). (The BO expansion parameters, Y?F, provide sufficient accuracy.) DISCUSSION

AND

CONCLUSIONS

The ways in which Eq. (43)) t’ogether with Eq. (14) for IL’(R) can be used in actual calculation depend upon the specific molecular property to be examined.

ENERGY

SHIFTS

:319

IN J) IATOM IC MOLECULES

Some illustrations, drawn from examples of deviat,ions from BO behavior mclltioned earlier, follow. 1. IAscl*epancies betzreen observed spectwscopir constants awl those preclidd through awwate calculation in the BO apI)Iv,.ci,natio7r.Two t,ypes of cxnmple~ are known. These are : a. Dissociation energies. We assume that the Dunham espansion for molec~uln~ energies remains valid, i.e., Ex,, The only term influence either directly is t,he nificant, and it

= 3

lr:j(l'

+

1 i

)'[J(J

+

( 43 )

1 )]j.

in t#he expansion of E A,., on the left side of Eq. (42) which can the apparent potential minimum or the Dunham correction J,,,, proves to be insigterm Yt: itself. However, t,his contribution is easily seen from Eq. (#2), that

holds to good approximation,5 the const,ants being expressed in cm-‘. It. appears, therefore, that if the power series expansion for the energies in terms of il-1~~ coefficients 171j is valid, together with Dunham’s prediction for E7,1rj , the csperimental and t,he exact theoretical values of I),” must be equal. That Kolos and Wolniewicz (8, 9) did not obtain agreement with the experimental valuc~ of De” but did, largely, on the dissociation energy from the lowest vibrational state for Hz and D, , must be taken as :III indication that bhe magnitude of the ordinary zero point vibrational energy in their c&ulation is not accurate. The magnitudes and sign of the discrepancies between their cxlculated tlitierences between the energies of the v = 0 and zl = I vihration:~l st,ates and those mc~asurccl experimr~ntally tend to support, t,his supposition. It is the conclusion of 1111x present authors, t,herefore, that the dilemma assoc*iat(~tl wit,h the fact llrst) t hc calculut~ed (8) value of 0,” is larger t)hall that obsrrvc,tl i somewhat indirectly) rxperimentally (16) is not removed in l-11(>latter c~:dcul:tl ion i 9 ) l,y Kolos :m(l Wolnicwicz. h . Rotational constants. The J(J + 1) -:lependcnt cffccts arising from the tcrnl involving Ehl.r on the left side of Eq. (42) are twofold. I%&, the 1’:; tern) adds to the ot’her .J(J + l)-dependent) operators in Eq. (42). Upon evaluation at’ R, , however, this term vanishes. Secondly, the I’&(’ term, added to &,‘(IZ), shift:: the value of R,. Once again, however, the effect of this term on E7”1 is negligible. 5To be accurate, one should incorporate the ordinary relativistic corrections int,o t.he I30 nuclear potential (see References 8, 17).

electronic

energ-

HERMAN

320 Hence,

it is apparent

that

AND

ASGHARIAN

B, can be represented

by

while R, is given5 by [see cf Eq. (14)]

k being the force constant (BO approximation). From Eq. (45), it is seen that the adiabatic corrections to B, do not, in fact, vary as l/M as was assumed by Townes and his collaborators (II). Thus, in their determination of the adiabatic correction through measurements of the C12016 and C14016 B, values, they determined only the l/MC-dependent correction. This implies that the over-all refinements were made correctly in the C1’-C12 mass ratio determination (in opposition to Nesbet’s (22) conclusions), while at the same time, one is forced to conclude that the (adiabatic) corrections as applied to the oxygen mass ratio determinations are erroneous, are fortuitously equal. The magunless (d/dR)(X [ V? / X) and (d/dR)(X j Vl j X) nitudes of these errors, although difficult to assess without’ further study, could be as large as the quoted uncert’ainties in the microwave mass ratio determinations. 2. Deviations from simple isotopic relations predicted through the BO approximation. From Eq. (42), it follows that the molecular (Dunham) constants, corrected for deviations from BO behavior, each can be written in a form such that the isotope relations R”(i)/De”(j)

(1 + %I,

=

= (1 +

[We(i)/Oe(j)l[~~A,(i)/~~At(~)l””

[B,(i)/B,(j)l[~At(i)ln~At(j)l

=

Sij),

(1 + Xi>,

. . .

are valid,

where

c, 6, y, . . - have the mass dependencies

~ij = ~1L+-fl(A -

Ml(i)lM, +

E2lM4.d

Jfl(j)Mdi) etc. The values

of cl , e2, - - . can be determined

- M2(4M,

Mz(j)Mn(i) experimentally

’ by comparing

ENERGY

SHIFTS

IN DIATOMIC

MOLECULES

321

ratios of experimentally determined (Dunham) constants. The theoretical forms of El , t2, -yl, and yz can be inferred from Eqs. (44), (4:5). Another important constant should be discussed in this connection, however: a. The vibrational constant. Once again, it can be shown that the energy ,73~~.~ on the left side of Eq. (42) is insignificant. The vibrational caonst’ant is therefore seen to be

withfi

3. Deviations _from predicted j.elationships between th,e spectroscopic constants [I./ a single isotope. As already stated, the most important of these is the relationship between D, , B, , and ae. To make the comparison, observe first that if g*(R) = gl(R) = gI(R,) held, Eq. (42) would have a fon?l identical t,o that of a BO equa-

tion, in spite of the fact that correction terms to the actual BO equation for the molecule had been included.’ Hence, the relationship between D, , B, , and wy would then hold, as in the BO case. The terms in Eq. (42) leading to BO relationships intrinsic to a given isotope are those which break the ~Lathe/nafical form of the BO-type equation. Specifically, let the spectral constants which would be obtained if gz(R) = gl(R) = gl(R,) held, be denoted by w,(O), B,(O), etc. In terms of these ronstants, the actual rotational and vibrational constants are t,hen given by B, = B,(O) and w, = we(O){1 + (m/‘%U,)[gdReJ

- g1(Re)l).

The calculation of D, is slightly more complicated. Specifically, the J(J + 1) dependent part of the energy ExvJ on the left side of Eq. (42) is important, so t,hat effectively, the radial equation is

gdRe)

1+ ;

1

P

fi2 + ZaGi

g2

1 +

(m/Mp)(gl(R)

- e(R)

+ ti(Rp))

R2

fi It is important to evaluate the derivative in Eq. (48) at f1,, rather than R?. 7 This contradicts the usual assertion that if D, is accurately given by-4B,3/~,1, oscillator is necessarily a BO oscillator.

the

322

HERMAN

Now the constant tion 1 + f

D, is proportional

(??L/M,)[g,(R)

R2

AND ASGHARIAN

to the square

SP(R) + g&L)1

+

of the derivative

(74J~,)bz(R)

-

of the func-

dW1

R,Z

1

9

evaluated at R = R, , divided by t,he force constant. The ratio De/De(O) is therefore given by the ratio of the squared derivative to its value obtained by assuming g2(R) = gI(R) = gl(RF). Accordingly, D, = D,(O){ 1 is obt’ained, with the result’ that D, , B, , and ue are related through

(l)z/n~,)R,[clgl(R)/dRlH=R,~ the Dunham the equation

gz(Re)

-

corrected

spectral

constants

g1(R,) - R,

As previously mentioned, Herman and Asgharian (21) have applied this expression to the spectral data of Rank, Rao, and Wiggins (20) for HCI. For this molecule, there is difficulty in obtaining accurate values for gS(Re), gl(R,), are sufficient to render accurate and [dg~(R,)/dRl~=~, . These uncert,ainties predict,ions for the correct’ions to D, impossible. In spite of this, there is a discrepancy between theory and experiment, in that the theoretically predicted correction factor in Eq. (50) is found to be -0.00029, its probable upper limit being +0.00037, whereas experimentally, this fact’or is (20) 0.00065 f 0.00007. Without further experimental data, there appears to be no way of resolving this discrepancy. 4. Dunham corrections in the e.rpeGnental determination of Di, tie , B, , D,, it is necessary to make the etc. In obtaining De’, we , B, , D, , etc. experimentally, Dunham corrections (1 S) on the experimentally determined energy level parameters, 0,” and Y ~j (I, j # 0, 0). To do this, I’oo along wit.h small contributions to and Yrz arising from the anharmonicity of the potential must be obY10 ) YOl ) tained, which implies that the higher coefficients al , a2 , . 1. in the potent’ial function expansion must he known with precision. One obtains these experimentally from a knowledge of the higher I’lj’s. This, in turn, requires that the contributions to the higher Yli’s associated with deviations from BO behavior must either be known, or negligible. Order of magnitude estimates on the fractional changes introduced into t’he higher Yli’s, made with t,he help of Eq. (42)) indicate that they are not larger t,han in the cases of the lower Y Ij’s (~0.1% ), leading to uncertainties in the Dunham corrections sufficient to cause uncertainties of the order 10-5-10-4 (see References 17, 21) in the Dunhamcorrected parameters for hydrides. Although small, these uncertainties are not entirely negligible, however.

ENERGY

SHIFTS IN DIATOMIC

MOLECULES

:123

SUMMARY

Sm:dl tmns in the exact Schr6dinger equat.ion for tlw elwtrons nnd nwlci of 1x7 ~. ,-~tj:ltc diatomic molecules, which are ignored in t hc BO ,zpp~oximatioll, 11:tvc bwrl retained and their effects on the energy levc~l p:umncttw 11:~~ bcw tlc$erminetl to order m/M times t#he I30 vnlucs. Hy c+Ycc*tively introducing :t (~lassi(*al path qproximntion (which corrcupontls to nr~glcct.ing corrcc*t,iou terms of order ( 111 !JZ )’ in treating the nonndi:ibat ica BO-breaking ilitwwtiow J :I linmr, sepand~lr effect’ive Schr6dinger equation for nuc~lcnr motion h:ts bwtl derived, tllc eigcnvnlues of which specify t,he r~~olec~da~~cmergy Icvcls to :I lligtl tkgwe of :wcm:wy. Using pert~urhntiou t hcorct.iwl :wgunic~rils, one (x11 inb lucyliately :ISSCSBthe changes in individual n~olccwlu~~ pmtnlctm, or ill rcl:lt ioIlships bctwwn p:muneters associnkd with deviation-: from HO bch:lvior. Thw;c~ changes :w related to only :L few func+ions, simple in fornl, \vhicbh c~h:uxc~tc.rizc~ the &cstronic* stat,es. The latkr are often obt:linable through direct ~lruncric4 wlml:~tion, or through empirical methods, Spwifics cwrrcc*tious whicah nlust tw :rpplietl to sonw of the more important energy Icwl wnstmlts mtl r&f ionshilw bet wean these mnstj:tnts have been derived, with the fr:rnwworl< of f I)(, Thmhn tllwy. ACKNOWLEI>(

:bll
The itrlthors wish to thank Professors D. 11. I:nnk xntl ‘1’. A. Wiggins for hcalpflll tlisc,lwsions throughout, the course of this research. RECEIVED:

Sovemher

16, 1965 REFERENCES

1. ?(I. Boon .ISD J. 1~. OPPENHEIMER, dnn. Phys. 84, 457 (1927); T1.E~1trr;c:, J. WSTEH, A\ND(:.E. KIUB.\LL,“QIIL~II~U~I Chemistry.” Wiley, New York, 19-l-l; "'l'llr Jlolec~~lur Theory of ( ~nws J. 0. HIRSC'HFELDEK, C. F. CURTIS, AND 1<.H. RX-I(I), nld Liquids.” Wiley, New York, 1954. 2. J. H. VAN \'LECK, J. Chem. Phys. 4, 327 (1936). S. 1.. JOHXSON, Phys. Rev, 60, 373 (1941). 4. _I. I).\LG.\RNo .\NDIt.MCC~RR~LL, Proc. Roy. SW. A237, 383 (1956). 5, S. COHES, J. 1:. HISKEG, IND R. J. RIDDELL,Phys. Rev. 119, 1025 (1900). 6. A. hijUAN, J. Chem. Phys. 36, 1490 (19(i2). 7. W. KOLOS .INDL. WOLNIETVICZ,dictaPhIIs. Polon. 20, 129 (1961); Null. A1ul. I’l~ys. Xu,. [ai, 9, 103 (1964). 8. W. KOLOS .\XDIi.WOLNIEWICZ, J. Chem. Phys. 41, 3663 (1904). 9. W. KOLOS .\NDI,.W~LNIE\?;ICZ, J. Chew Phys. 41, 3671 (19fiJ). 10. Ci, A. FISK SD B. KKRTMAN, J. Chetn.Phys. 41, 351li (19G4). 11. B. I
324 15. 16. f7. 18. 19. 80. Rf . 22.

HERMAN

AND

ASGHARIAN

G. HEHZBERG AND A. MONFILS, J. Mol. Spectry. 6, 482 (1960). UWE FINK, T. A. WIGGINS, AND 11. H. RANK, J. Mol. Spectry. 18, 384 (1965) R. TIPPING AND R. M. HERM.IN, J. Chem. Phys. (to be published). R. G. BARNES, P. J. BRAY, AND N. F. RAMSEY, Phys. Rev. 94, 893 (1964). B. P. STOICHEFF, Can. J. Phys. 36, 730 (1957). D. H. RANK, B. S. Rao, AND T. A. WIGGINS, J. Mol. Spectry. 17, 122 (1965). R. M. HERMAN AND A. ASGHARIaN (to be published). R. K. NESBET, J. Chem. Phys. 40, 3619 (1964).