Stark effect, polarizabilities and the electric dipole moment of heteronuclear diatomic molecules in 1Σ states

-Chemical Physics 89 (1984) 275-295 Fjc@lt--Holland, Amsterdam .-

_

-_

_

_

m-T

5

275 z

_-

,-

_-

-_

_

‘STARK EFFECT;PO-~RlZABILITIES’ANb THE ELECI’RId DIPOLE OF HEI&ONtJCL_EAR DIATOMIC MOLECULES IN ‘Z STATES

MOME-&‘

-

M. BRI-EGER ’ Znstttul fiiiiStrahiungs- und Kemphysrk. Technische Universrtdt Berhn. RondelLsrrasse5. D-loo0 Berhn 37. FRG Recetved 13 February 1984

The theory of second-order Stark effect m ‘Z states of heteronuclear diatormc molecules IS thor&ghly reviewed. The rigorous treatment given demonstrates that by introducing rotational, vlbrahonal and electronic branch polarizabiht~es. the intrmsic character of the second-order Stark effect in diatomic molecules can be shown to be related more closely to polarizabthties than to dipole moments. The well-known expression for the Stark shift in ‘Z levels which is dominated by the square of the dipole moment is only a crude, though sufficient approximation whenever large dipole moments are involved. For small dipole moments. however, this approximation is hkely to fail. leading to an erroneous determination of such dipole moments. In the limiting case of negligible influence of the molecular rotation on the vlbronic matrix elements, the arithmetic mean of the electronic branch polar&abilities turns out to be equal to the well-known static electronic polarizabilities Q,, and Q~. The results are applied to the mterpretation of the Stark splitting tn the A’Z+. I determined by Stark quantum-beat spectroscopy.

v’= 5. J’=l

level of “LIH. recently

1. Introduction In F -eceding Stark quantum-beat experiments we investigated electric dipole moments in different vibratio_lal levels of the A’Z+ state of ‘LlH and ‘LiD [1,2]. In the case of the vibrational level u’ = 5 of ‘LiH w\ were confronted with the problem of determining a very small electric dipole moment from the measure-d field-induced splitting of energy sublevels_ Our first attempt [l] to determine the dipole moment was or-y partially successfull: biased by the knowledge of the first experimental result [3] privately _ commuticated prior to publication, we were not prepared to apply electric field strengths above 20 kV/cm. From the absence of any distortion of the decay curve at 20 kV/cm we postulated an upper limit of 0.06 au. For the absolute value of the dipole moment this upper limit still was twice the value predicted theoretically [4-71, but only half the one of ref. [3]. Due to the predicted [4-71 smallness of this dipole moment electric field strengths of the order of 100 kV/cm had to be applied [2] for obtaining a sufficiently developed quantum beat (see e.g. ref_ [8]). The smallness of this dipole moment is related to the fact that the dipole moment function, which is identical for LiH and LiD in the Born-Oppenheimer and in the adiabatic approximation, has a zero-crossover at 5.8 a,, and that for the levels U’ = 5 of ‘LiH and u’ = 7 of ‘LiD large positive and negative co&butions can&l almost &mpletely in the co&se of each vibrational period. This behaviour is not unique: a similar situation-can be encountered in the ‘I? ground state of the FO radical [9]. Due to the very smallness of the dipole moment the response of a ‘LiH molecule in the investigated level to the presence of an external electric field partly resembles the one a homonuclear molecule would show, i.e. polarizability terms becdme very importani. Being a heteronuclear molecule,

* Present address- Department of Physics, Texas A&M

University, College Station, Texas 77843, USA.

0301-0104/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

M. Brieger /

276

Srork effect in ‘B scores of dioromic molecules~

however, the well-known electronic polariziability terms are not the only ones that have to be taken into account. There are additional terms, which we call rotational and vibrational branch polarizabilities. In the limit of vanishing influence of the molecular rotation the electronic polarizabilities will turn out to be equal to the arithmetic mean of the electronic branch polarizabilities. Thus, all polarizability terms have a common origin. Neglecting all other terms the rotational branch polarizabilities may be crudely apbroximated by the well-known expression for the second-order Stark shift in rotational leveb of ‘Z states. For analyzing the measured splitting. we will start by thoroughly reviewing the theory of second-order Stark effect in molecular ‘I: states. In section 2.2 rotational. vibrational and electronic branch polarizabilities will be introduced_ Section 3 will present the application of the theory to the determination of the A’P, u’ = 5 dipole moment of ‘LiH including a more detailed interpretation of the measured splitting than that given in ref. [2].

2. Theory 2-i.

The Stark effect m ‘2

states of &atomic molecules

If diatomic molecules were composed of classically behaving particles. taking a snapshot of such a many particle system we would find the two nuclei with charges eZ, at the positions R,( K = 1, 2), whereas the N electrons, each with charge e, could be located at c( I = 1, _ _ _, N) relative to the origin of a common coordinate system. Following the usual procedure when turning over to quantum mechanics the operator for the dipole moment is defined by (see. e.g. ref. [lo]) p=e(CZ,R,-_Cq).

Since a diatomic molecule possesses an axis of cylindrical symmetry, a clear distinction should be made between the external or laboratory (L-) frame and the internal or molecular (M-) frame: in the molecular frame whose z axis, called 1 for distinction, is chosen to be identical with the line connecting both nuclei, the Cartesian components of the electric dipole moment operator are given by

(2) where 5,. ‘I,, 5, and cK refer to the origin of the M-frame. To take advantage of the tensor calculus, spherical coordinates are introduced as defined by hl _ -+2-‘/Z (PC f icL?), P&l--

PY = i+

(3)

where the superscript M refers to the molecular frame_ When the molecule is exposed to an external electric field of strength E, the charged particles it is composed of will interact also with this field, which in this way perturbs the internal structure of the molecule_ If the applied field is uniform in the region of interest only one term in the multipole expansion survives and gives rise to the additional hamiltonian H’=

-(f.&*E).

(4)

Being a scalar, this operator is invariant against changing the reference system. Identifying the direction of the uniform external electric field with the = direction in the L-frame, the scalar product reduces to the following expression relative to the L-frame H’ = -p,E,

= y&E;

= --j&E.

- -(5)

_ -M.

_ .

Brieger 7 Stark effecf m ‘B states of diaromic thoIecuI~

In the M-frame, however, it reads

-

_-

-. 277

r -

The constituents of the perturbing hamiltonian possess different, so to speak “natural” frames: the uniform electric field is generated in the,laboratory by e.g. a pair of condenser plates. So the L-frame is the “natural” frame for-the electric field and a natural choice of the quantixation axis is the field direction as imposed by the axis of the condenser plates. The operator of the electric dipole moment, however, consists primarily of the coordinates of the nuclei and electrons. Therefore, the M-frame is the “natural” frame for this operator_ _ Whatever frame is chosen for evaluating the scalar product, in each case it will be necessary to transform the components of one of its constituents to the other reference system: if, as in (5) for instance, the scalar product is evaluated in the L-frame which is the “natural” frame for the electric field strength, then the components of the dipole moment operator which are to act on the nuclear and electronic coordinates, must be transformed to the M-frame. This is provrded for by- the eulerian transformation which for spherical vector components reads 11L~=c(-l)p--4Dq(:~(apy)Ir~. P

(7)

This relation shows how a spherical vector component y: of the dipole moment in the L-frame is composed from a linear combination of its spherical vector component $ in the M-frame which is rotated relative to the L-frame by the eulerian angles a, p, y. The eulerian angles are chosen in accord with the convention adopted for instance by refs. [lO,ll]. D~‘~(cY/~Y) being an element of the transformation matrix, is the well-known D-function for an angular momentum J = 1, associated with a vector. Being a unitary orthogonal transformation the eulerian transformation can be reversed yielding

Here use has been made of [lO,ll] (9) An appropriate form for the perturbing hamiltonian therefore is

H’=

-EC(-l)PD;~;(CX/3y)&“! P

In the weak field case, which means that the shifts of the energy levels induced by the external electric field are small compared with the level spacings, the influence of the electric field on the molecular structure may be treated by perturbation theory_ For applying this theory the wavefunctions for the unperturbed molecule must be defined_ Usual first approaches to describing a diatomic molecule theoretically are the Born-Oppenheimer and the adiabatic approximations. Being of zeroth and first order, respectively. with respect to the energy, these approximations assume the radial and angular motions of the nuclei to be negligibly slow compared with the motion of the electrons, thereby splitting the total hamiltonian of the molecule into the dominant electronic part for fixed centers and the perturbative nuclear kinetic part. Meanwhile the fixed-centers part has been solved numerically for many molecules. Especially for light diatomic molecules like ‘LiH 14-71 the results are very accurate. The resulting electronic wavefunctions and electronic energy eigenvalues depend on-the separation H oEthe nuclei for.which the electronic Schr6diiiger equation has been solved. _ ‘In the generaIcase as well, the electronic problem for clamped nuclei is assumed to be solved yielding a

278

M. Briepr / Stark effect in ‘1 stales of dratomic mole&es

complete set of electronic wavefunztions.. This set is used for expanding the total wavefunction of the molecule into a superposition of fixed-centers electronic wavefunctions. The expansion coefficients in this series are arranged to depend on the i=rternuclea.r distance R. The substitution of this expansion for the wavefunction in the total SchrGdinger equation for the hamiltonian yields a system of coupled differential equations for the expansion coefficients. Bein,= approximations to the general case, the Born-Oppenheimer and the adiabatic approximation neglect this coupling. This is equivalent to keeping only one term in the expansion for the wavefunction and omitting all others. In this case the expansion coefficient turns out to be the wavefunction of the nuclear motion, and the separability of the electronic from the nuclear motion is achieved. in the Born-Oppenheimer approximation any influence of the nuclear motion on the electronic motion is ignored. Then the R-dependent electronic energy eigenvalue plays the role of the separation constant and can directly be identified with the potential energy curve on which the nuclei move. Taking account of the influence of the nuclear motion on the electronic motion in first order with respect to the energy. the adiabatic approximation adds R-dependent adiabatic corrections to the electronic BomOppenheimer energies and thus to the potential energy curves only, but leaves the electronic wavefunctions unaffected. Because of this. both the Born-Oppenheimer and the adiabatic approximation have identical electronic wavefunctions, but due to the non-identical potential energy curves differ in the vibrational wavefunctions. In both cases the rotation of the molecule-fixed reference system is the same and can be separated from the radial motion by taking for the wavefunction [lo] ~nAu(J)..IMA>

= ‘&,(&.q,.{,:

R)R-‘cD,,,,,,,,(R)(-1)“‘-.‘[(2J+

1)/8t’]“2D::~,(apy).

(11)

Being free of spin this wavefunction can give an approximate description of rotationally unperturbed molecular singlet states. Therefore. it is the appropriate one for small rotational quantum numbers. Since we aim at describing ‘2 states, we need not care about whether the state i1 is degenerate or not. Specifying in detail. \k,,(.$,. T,. S,: R) is the electronic wavefunction for clamped nuclei with 11 denoting the S-component of the electronic angular momentum and PZreferring to other quantum numbers necessary for distinguishing the electronic states. @n.,r(,j (R) is the vibrational wavefunction of the vibrational level u(J) in the potential energy curve associated wrth the clectromc state n, A and the total angular momentum J of the rotating molecule_ Due to factoring the wavefunction of the nuclear motions into the vibrational part @” ,c (,)( R) and the rotational part D::‘,( arpy), the rotational eigenvalue tr’[J(J + 1) - .i’] becomes the separation constant of the differential equations for the radial and angular motions of the nuclei and thus enters the Schriidinger equation for the vibrating nuclei by the centrifugal barrier. Therefore, the “effective” Born-Oppenheimer and adiabatic potential energy curves by this part depend on the quantum number J of the total angular momentum, and so does the vibrational wavefunction. This is indicated by the J in parentheses in the subscript u(J). As mentioned aboLe, 4::: (a/3y) being the wavefunction of a symmetric top, describes the rotation of the molecule by giving the relative orientation of the L- and the M-frame. Any rotational matrix element then reflects the averaged change in these orientations as induced by the operator. The subscript M attached to the D-function denotes the z component of the angular momentum J in the L-frame, whereas A is its 5 component in the M-frame. The square root is the normalization constant for the D function. Using this kind of wavefunction the matrix elements of the perturbin g Stark hamiltonian (10) will always involve the Integration [(2J’f

over three D-functions 1)(2Jt 8%’

= [(2J’+

I)]“’

1)(2J+

/,

z-

/,

7

which yields. with the help of (9) [lO,ll]

/,

l)]l”l(-l)“--bf’

z,-

4:I’(~((yP~)~~‘~(cTp~)~~~~~(~~~)

da sin B dB dy

(32)

M_ Brieger / Stark effect in ’ H states of diatom.% molecuh

279

Here the phases of the leading 3j-symbols have- been arranged artificially in order to make this result comparable with the Wigner-Eckart theorem which in the molecular case can be interpreted in such a way that a molecular matrix element is reduced with respect to both the L-_ and the M-frame. Having anticipated this result it is easy to quote the general form of the matrix elements of the Stark operator

= - f%t ,_nr E[(2J’

+ 1)(2J

X(_I)~*-.*’

1

J’ --A’

+ l)]“*( J A

A’-A

-l)J’-n’ (&Yu’(

( J’)

_$

I&?_/,

1nAu(

J)).

03)

(nX’u’(J’) [ &?_ A 1nAo(J)) is the matrix element that is left over when the integration over the eulerian angles has been performed. It involves the integration over the electronic position coordinates t,, T,, 5, relative to the rotating M-frame and the integration over the vibrational coordinate R. Using the electronic wavefunctions for fixed centers, the integration over the electronic coordinates either yields the dipole moment function, as is the case for electronically diagonal matrix elements. or, as in the non-diagonal case, the transition moment functions, also called dipole strength functions. All these are functions of the internuclear separation coordinate R, smce the dependence of the electronic wavefunctions and, in the case of the dipole moment function, of the molecular dipole moment operator on this coordinate is transferred to these quantities. Thus a dipole moment function is given by

IL,,_,@)=~I*,&,~ ?I,,

S,;

R)I*&‘(5,,

‘II.

3,; R) d-C,h, d3zv

while for a transition moment function the following

04)

expression holds

Due to the orthogonality of electronic wavefunctions belonging to different electronic states, only those parts of the dipole moment operator, as given by (1) and (2), are active in the integral for the transition moment function that involve the electronic position coordinates_ For indicating this situation we have omitted the nuclear coordinate R in the argument of the dipole moment operator in the integrand of expression (15). Characterizing the vibrational wavefunction used in the bra and ket notation by u,_,(J) and beginning with the same electronic state the residual matrix elements become in the diagonal case

@Ad

J)

Id? I nAu’(J')) = -

For the electronically

non-diagonal

(nAu( J) I&,+‘nu’(

J'))

(16)

matrix elements we obtain

= Onid J)

I Q,.,-,;,,(R) I~~_Y( J'b

07)

Like the ones in (17), the matrix elements in (16) are transition matrix elements for rovibrational transitions unless they are diagonal in all quantum numbers. Then, such a matrix element which is diagonal in all quantum numbers represents the expectation value for the observable dipole moment of that level

P::?= b,,d

J) I P,,(R) I um,(Jb

(18)

This expression demonstrates that in the Born-Oppenheimer and in the adiabatic approximation the observable dipole moment is the vibrational average of the dipole moment function_ In the case of ‘X states which we are interested in, each rotational level is non-degenerate and has definite parity. Therefore, there is no first-order Stark effect, which is accounted for in the matrix element

M_ Brieger /

280

Stark ejfect in tX srates of diatomic molecutes

(13) by the vanishing of the second 3j-symbol for J’ = J and A’ = A = 0. Thus for a ‘Z state the lowest non-vanishing order in the perturbation expansion of the Stark hamiltonian (10) leads to the second-order Stark effect- Due to the selection rules for dipole matrix elements,

the next non-vanishing

order would be

the fourth. This one will not be dealt with here since there is no evidence for it within the accuracy of the experiment [2] to be discussed below. In second-order Stark effect the shift of an energy level labelled “n” is generally given by W,“‘=

c

](n 1H’I

k)l’/(

wy-

w-p’).

(19)

k+n

where n and k denote sets of quantum numbers (13) for A = 0 we end up explicitly with

like the ones used in eq. (13). Inserting

the matrix

element

2

p’

n’Tc(J)JW

=E”xx

2

x(2J’+1)(2J+l)

J’ -A’

n’ .I’ L,‘(J’) /’

xl(n’2u(J)

’ A’

J 0

Wm(&JJ,- Wn$&,,,,.)-

I~*~,,~ln'~u'(J'))l*/(

(20)

L4ore convemence

in distmguishm g the drfferent contnbutions to the energy shift can be achieved by over all possible II’ states. i.e. ‘2 and ‘TI states, and extracting all those vibronic *S matrix elements from the summation over n’ that are electronically diagonal. These ones are further

performing

the summation

subdivided into the vibrationally diagonal one and into the sum over all vrbrationally elements of the investrgated ‘2 state:

non-diagonal

matrix

2

Ys,

(,),U

=

;

E”x(2J’+1)(2J+l) J’

I(~z*SU(J)I~~~~Z~~U(J’))~~

x

G,“:<,, +

c

c

(1)

( J’)

I(~‘Z~(J)III~~I~‘SU’(J’))I’

O’fiL

+

( J ) - G,&

G”L:f,,,(

c

ICn

J) - G:::l,,(J’)

‘Zu( J) I&* 1n”Zu’(

n*+ II c’ T” ‘X -

T“.I\‘ + G$;,

-

J’))I’ Gt:;;;;

Here T,_, denotes the electronic energy at the minimum of the potential energy curve for the state nA. G,“,^,,l( J) or G::{,? is the energy of the vibrational level u in the electronic state nli when the molecule is rotating with an angular momentum

J_

the first three of which contributions from interactions within the same ‘L: state (I, II) or with other ‘Z states (III) while part describes the influence of ‘II states. Being the most important usualIy, the first part generated by the interaction with the adjacent “rotational” levels u( J’ = J & 1). The second part As a resuh of this procedure

four parts can be distinguished

in (21)

consist of the fourth in (21) is originates

M. Brieger / Stark efffect in ‘B states of dtarbmic mo?ecides

-_

281

from interactions withal1 allowed rovibrational levels of the same_‘Z state, while the third part accounts for interactions -with rovibronic- levels of. all other ‘Z states. As -is--obvious from exl&ssion (21), anyexperimental technique that uses the second-order Stark effect inherently determines an energetically weighted sum over squares of transition- matrix elements and- actually has nothing to do with dipole moments. This is also true for the usually dominating first term which in a crude-approximation (see below) may be related to the dipole moment of the level. The applicability of this approximation which leads to the familiar expression for the Stark shift, has to be checked for the following reasons: As is also obvious from (21), the proportionate contributions of all terms do not depend on the strength of the electric field. but are given by the internal structure of the molecule. This expresses the fact that the molecule as a whole is perturbed. Therefore, any deduction of molecular electric dipole moments from measured energy level splittings actually requires a rather complete knowledge of the internal structure of the molecule. This is why care should be taken when using only the first part of expression (21) along with the usual approximation for evaluating an electric dipole moment from a measured level splitting since this would be equivalent to attributing the observed splitting solely to the interaction between the dipole moment of a specific level and the external field, possibly on the basis of a crude approximation_ As parts II through IV in (21) represent polarizability terms, omitting their influence may lead to an erroneous value for the dipole moment. Especiahy in cases where the knowledge of the internal structure of a molecule is rather poor and where the measured splitting indicates a small dipole moment and, consequently, a possibly non-negligible influence of the polarizability, it would often make more sense to give field-reduced splittmgs AW/Ez rather than evaluate questionable values for dipole moments_ Such field-reduced splittings are the quantities that are actually measured, and that can provide a test of theory, because they correspond directly to the sum over all perturbing levels. As will be discussed in section 2.2 all parts of (21) are closely related to polarizability terms. 2.2. Polanzabrluy The concept of polarizability is based on the idea that the charged particles an atom or a molecule with no intrinsic electric dipole moment is composed of, are displaced according to the sign and magnitude of their charges in opposite directions and differently, respectively, by their interaction with a uniform external electric field. Being constituents of a dynamical system, the charged particles are forced to change their motions for giving a net displacement. This corresponds to a polarization of the atom or molecule and is equivalent to an induced dipole moment whose magnitude and direction is determined by the polarizability tensor and the electric field strength. The induced dipole moment then is thought to interact with the field by itself bringing about the E’ dependence for the energy of such an object in the field: W= - SE- o(* E. This is accounted for quantum mechanically by the requirement of second-order perturbation theory which not only gives the correct E2 dependence, but also describes the perturbed motion of the particles by mixing their wavefunctions. This concept is very clear for atoms in non-degenerate states of definite parity which because of their spherical symmetry cannot possess an intrinsic electric dipole moment as long as parity is conserved_ The interaction of such atoms with a uniform external electric field is only possible because of their polar&ability and thus they react on the presence of an external electric field only in second order. Due to the field-induced anisotropy the spherical symmetry of the system is destroyed_ Therefore, even for such a symmetric object as an atom the polarizabihty is not a purely scaiar quantity, but is composed of a scalar and a spherical tensor part of rank 2 [12]. Another system which behaves verysimilarly to an atom in an electric field, is a homonuclear diatomic molecule. Due to its internal symmetry such a molecule cannot possess an intrinsic electric dipole moment, since its dipole moment function equals zero for alI R. Therefore, it is polarized in an electric field very analogously to an atom. This may be why the term polarizability is preferentially used in connection with homonuclear diatomic molecules. It wih be shown below, however, that thd interaction of. slowly rotating

282

A-K Bneger

/

Stark effect in *Z stares of draromic molecules

heteronuclear molecules in ‘Z states with a uniform external electric field can be attributed to terms that are closely related to polarizability. These terms which we will call branch polarizabilities are analogous to the well-known electronic polarizabilities and are defied in the very same way. For making this point as clear as possible we recall the definition of the Cartesian components of the polarizability tensor for homonuclear diatomic molecules (see, e.g., refs. [13,14])

(224

(22b)

WC) This tensor is symmetric and has only two independent elements because of the cylindrical symmetry. Anaiogously to eqs. (22a)-(22c) we define the parallel and perpendicular electronic branch polarizabilities a,,“‘\“(‘)( J, J’) and al ‘XL’(‘)(J, J’), respectively, for ‘Z states n’\‘w)(J,Jy= aii

_

c

c

2j(n’Zu(J)

@I”“ZU’(J’))I ,

(234

ll’#R u’ T,lr - Tn-,x -I- GLn_& - G,7Tj;; ap’

l-y J, J’) = - c

c

~(n’Zu(J)

IpLhfipz”ITu’(J’))j’ (2W

The electronic branch polarizabilities (23a) and (23b), respectively, are the dominant features of the third and fourth part of expression (21) for the Stark shift and primarily reflect the contribution due to the perturbation of the electronic motions. In the limiting case of negligible influence of the molecular rotation on the vibromc matrix elements the arithmetic mean of these branch polarizabilities will be shown to coincide with the static electronic polarizabilities of expressions (22a)-(22c). Being mathematically of the same kind like (23a) the parallel vibrational and rotational branch polarizabilities Y; I\“(“(J, J’) and pi’rU(J)(J, J’), respectively, are defined by rr’Ic(/)(J__Tr)=

_

c

yii U’PU

“‘=(l’( f+I

J, J')

W~‘~4J)

Gi”&(

= _ 21(&u(J) GC:$(

I& I~‘w-0~1’, J>

- G$i$(

(24)

J'>

~&n%(J’))Iz 9

J) - G,“::,,( J’)

(25)

bf_ Brieger / Stark effect m .‘E states of &atomtc moiedes

283

(24) and (25) represent the main features of the first -and second part of (21), respectively; and express the contributions due to the perturbation of the nuclear motions. Inserting (23) through (25) into (21) we obtain for the second-order Stark shift w”’ n’ro(J)JM

=-&5”~(2J’+1)(2J-t-l)

(

J’

“‘=)( + vII

$’

_“M

;

’

J’ *

1 I(

J, J’) + a;; ‘Tr(J)( J, J’)] + ( _;’

J *

1 ’ *

‘&(a’)(

J

J’) 9

PII

I[

;:

;)‘~cY$~~(“(

J, J’)}.

(26)

.Although being mathematically similar ~;i’~~(‘)( J, J’) and ,o~‘~~(~)(J, J’) differ _from CY(;‘\‘~(~)(J, J’) physically by the fact that they depend on the rovibrational matrix elements of the drpole moment fhzcrion n’xo(J)(J,rJ’) the rovibronic matrix elements of the n’Z-n”Z transhon of the n ‘X state, whereas for CY,, momerzt f~mctrons are involved_ This difference is essential for the degree of approximation that can be achieved when the dependence on J’ of the matrix elements is neglected: in contrast to dipole moment functions, transition moment functions in most cases are flat curves, i.e. their dependence on R is not strong since it is only inferred from the one of the electronic wavefunctions. For transition moment functions this R dependence usually only has to join the limitmg transition moments for united and separated atoms by a smooth curve as opposed to dipole moment functions where the R dependence of the electronic wavefunctions has to overcome the explicit R dependence of the electric dipole moment operator for making it approach zero in the limiting cases. Even if the R dependence of a transition moment function is more pronounced due to an avoided curve crossing (see section 3) it seldom is probed over the whole R range since in many cases the overlap between the vibrational wavefunctions of the upper and the lower state is well localized_ This is just the behaviour that justifies the R-centroid approximation. Therefore, molecular transition matrix elements are rather insensitive to the slight changes that arise from the substitution of the rovibrational wavefunction u’( J’) by neighbouring ones u’(J), where 1J’ - Jj G 2. Since for CY;‘\‘~‘(~)(J,J’) and CY”;’“(J)(J J’) also the rotational dependence of the energy denominator is only of the order 10m4 or less the electronic branch polarizabilities of one kind do not differ appreciably for the different branches_ This is not true for the vibrational branch polarizabilities v;““(~)( J, J’) nor for the rotational branch polarizabihties p;;‘xr(J)( J, J’). F or exploiting this different behaviour we investigate the contribution of the electronic branch polarizabilities to the second-order Stark shift separately z w(Zkl

18%

(J)JM

=

-&5’~(2J’+1)(2J+

1)

;

1

1 CY~;‘~‘(~)(J, J’) +

1’

2~~5~)(

J,

J’)

_

(27)

We now simplify the notation for the branch polarizabilities by dropping the cumbersome upper indexes. For getting partially rid of the J’ dependence in the electronic branch polarizabilities we add and subtract the complementary electronic branch polarizabilities. Since for a,,(J, J’) no Q branch exists, we have

a&J,

J’)=$

I

J-cl c

n,,(J,J’)+ J”=

a,,(J, J’)-

J-

(1 -8, J,s)(l -6J._J.~)cu,,(J,J”) 1

‘2’ (1 -S,_,r,)(l

-~,=,.+Y,,(J,

J”=

J-

1

I J”)

1 .

-

(28)

M

254

Brteger /

for al (J, J’) aI three branches

Stark effect in ‘H sates of diatomic molecules

P, Q and R occur Ii-l

a,(J,J’)=$

(l-L$,+l(J.J”)

c

a,(J,J’)+

J”=/-

I

1

1_

I

J-i-1 ++

c

2a,(J.J’)-

[

(1 -S,._,.+Y1(J,J”)

l”-l-l

For reasons that will become obvious immediately of one kind are summed up, with the well-known

(29)

we identify those parts where electronic polarizabilities

all branch

polarizabilities

8,,_,.,)a,,( J.J”) J”==-J1 1 J-i-1 C

=f

[~(J.J-l)+a,,(~.

(1 -

6,_,,,)(1

-

(304

Jtl)].

1+1

C1(1-8,.J..)a,(J.J”) ,“=,_ 1 =$[a,(J.J-l)+aL(J_J)+aL(J.J+l)]_ This

procedure

3j-symbols.

enables

Using

(30b)

us to perform

the properties

the summation

of the D-functions

over

J’

for the algebraic

it can be shown

expressions

that the followmg

with

relation

the

generally

holds (see appendix)

ccct-lY-pt2J’+ J’

l)PJ+

;J( _;

_;$ ;

1) i

.%I’ .I’

=C(

-1y-

’

t=+1)(2h-+

1)

*

i

;

_;

;)(;

_f

$

l

P

;)(

-if

;

$)(

_;

0”

;)(31)

For the special

cases of interest

here we obtain

~(2J’+1)(2J+-l)

(33)

J’

Mixmg

ail

together

second-order M"""'

n’\‘t(J)Jlf

we

end

up with

the

following

expression

for

the electronic

Stark shift

=

-

-f

+

i[a,,(J)+2a,(J)]

-%[q(J)--a,(J)]

I-4M’ (2 J - 1)(2 J + 1)(2J

+ 3)

f[a,,(J,J-l)-a,,(J.J+l)]

~~~~~~2~+~j)~’

contribution

to

the

-_-

;

-_

. - : 285 -

-

M-.Brieier i Starkeffectin !Jfstares of diatomic~molecules

+

$!A)[-41(

J,J-1)42a,(J,J)-q(J,J+l)]

(J+l)2--2

+ (2J+1)(2J+3)

-J+2[_a,(J J+I

’

J_;)_

“i(J,J)+2cx,(J,

J+l)]

E’_

(34)

The first term in this expression is exactly what we previously knew for homonuclear diatomic molecules about the contribution of the electronic polariz%bilities to the Stark energy [lo]. The linear combinations of a,, and a, given above are just the irreducible representations; the reducible cartesian polarizability tensor can be split into: the scalar polarizability (35) which is also often called the mean polarizability. and the tensor polarizability a,,,(J)

=f[a,,(J)

-%(J)],

(36)

which is also called the polarization anisotropy. The further terms in (34) which are dominated by the differences of the electronic branch polarizabilities, have been obtained from (27) by performing the summation over J’ and substituting the 3j-symbols by their equivalent algebraic expressions. For the reasons quoted above the differences in the electronic branch polarizabilities are very small. Therefore, only a negligible error is admitted when these terms are truncated in expression (34) which then simplifies to the usual contribution of the electronic polarizabilities

(37) Turning back to expression (26) we have to investigate the intrinsic characters of the vibrational and rotational branch polarizabilities. Although Y,,(J, J’) and p,,( J, J’) are mathematically analogous to a,,( J, J’), the same treatment as for a,,(J, J’), which is expressed by relations (28) through (37), is not applicable to Y,,(J, J’) and p,,( J, J’) in general because of the reasons quoted above. Nevertheless, individual cases are concetvable where due to a sufficiently flat course of the dipole moment function the above procedure may also apply to Y,,(J, J’) without running into serious errors. In such isolated cases a,, and Y,, could be joined in an effective parallel polarizability. In the case of the level AtZ+, v’ = 5, J’ = 1 of ‘LiH the vibrational P- and R-branch polarizabilities differ by 54.5% relative to the one for the P branch. Therefore, the procedure outlined above for the electronic branch polarizabilities is not suited for the vtbrational branch polarizabilities of the Investigated level. 2.3. DipoIe moments In p,,( J, J’), however, we cannot get rid of the rotational dependence at all, since this is its main feature. We can only suppress part of this rotational dependence. This is the apprortiation which will lead to the familiar expression for the Stark shift. For this purpose We go back io part I of (21) and carry but the

2S6

summation over J’. This yields P and R branches; the Q branch being suppressed by-the second -3JGsymbol because of parity. After insertin,0 the algebraic- expressions the 3j-symbols stand for. and making use of : (16), we have

+ (Jt

l)'- M'

2Jt3

I(u,,I~ (J) Gt”:,,(

IPL,~R)

I sdJ+

J) - G;z,,(

WI’

(38)

J + 1)

As has been mentioned above. the J dependence of the vibrational wavefunctions a,,,, CJ,( R) is inferred from the centrifugal barrier A’[ J( J + 1) - A2)]/2pR’ incorporated into the effective potential energy curve. This additional term affects especially the repulsive part of the potential energy curve by making it slightly more (J’ = J + 1) or less (J’ = J - 1) steep relative to the one associated with the angular momentum J. In the ftrst case (J’ = J + 1). in the environment of the inner turning point the vibrational vvavefunction @n ,r(,+,j( R) is sltghtly shifted towards larger R and diminished in amplitude_ This is compensated for constant normalization by ,I gain in amplitude near the outer turning point where the deviation of the potential energy curves is the lesser the higher the vibrational excitation_ Therefore, in the vibrational matrix element of a dipole moment function such a wavefunction favours a little bit more the outer part at larger R. The opposite is true for the case J’ = J - 1 by the same kind of arguments_ Quantitative comparison, however. of the resulting integrals is only posstble after numerically solving the vrbrational Schrodinger equations for the rotational quantum numbers J - 1. J and J + 1 and then using the relevant vibrational wavefunctions in the vibrational average of the dipole moment function. Even a qualitative treatment using effective deviations requtres the knowledge of the course of the dipole moment function_ As an example. we will treat the case of a dipole moment function that is negative for small R and positive at large internuclear separations. This case applies to ‘LtH in the first electronically excited state A’Z’ and will be discussed in detail in section 3. Assuming a vibrational level where the vibrational average of the dipole moment function yields a negative value. the influence of the centrifugal barrier on the matrix elements may be qualitatively described as follo\\s

I(wr(J)

l~,tlxRR)

I~,~~(J+~)>l~=l(~,:r(J)

IP,,I\-(R)

lu,,~\-(J))l’-~;+~

=&,-&,_

(39b)

The choice of the signs may irritate at first glance, but it should be remembered that the value of the matrix element is assumed to be negative_ Therefore. due to the difference between e.g. the wavefunctions CD as described above. the absolute value of the (J, J + 1) rotationally ,, l,cJ+l,(R) and @n.Iltl)(R)7 non-diagonal matrix element in (39b) is smaller than the absolute value of the diagonal one i.e. the dipole moment. This makes a difference of cz+, in t& squares of the matrix elements_ Analogous arguments hold for expression (39a). Approximating the denominators in (38) by the differences in the rotational energies which implies an error of the order of 10-s or less

G,“::,2(J)-Gc,“:$2(J-

l)=

+2B,nlxJ,

G”‘.+T;r_(J)-GF:Tp(J+

1) =

-2B,““(J+

(404

1)

Cab)

M

Brteger / Stark effect m ‘B states of diaromic n~o&tdes

and substituting the matrix elements in (38) according @an’-ru

n PU(J)JM

:(,,E’

= zB:zJ(J+l)

287

to (39a) and (39b) we get for J + 0:

J( J + 1) - 3M2 (2J--1)(2’+3)

L’

+

(;;$;J”-‘+

(~+i)~--bf* (25+2)(25+3f’”

(414

and forJ=O

It is the first term in (41a) that is usually exclusively quoted in textbooks for the additronal energy which a rotational level of a ‘Z state acquires in a uniform external electric field according to second-order Stark effect. This confinement not only implies the approximatrons (40a) and (40b) and c,’ = 0 in (41a) and (41b), but according to (21) also the total neglect of any interaction with other levels except for the very next rotational ones. This neglect may be justified as far as the electronic polartzabilities are concerned whenever perturbing electronic states are far away and the perturbed level does not extend into the region of an avoided curve crossing (see section 3). For the vibrational blanch polarizabilities, however, we should be aware of the fact that, depending on the degree of anharmonicity that is important for the investigated level and on the course of the dipole moment function, there may be a substantial non-zero contribution from the vibrational branch polarizabilities that should be considered_ Unfortunately even advanced textbooks do not account for this contnbution, and so we suspect some of the dipole moments given in the literature of being erroneously determined_

3. The A’Z+,

d = 5 dipole moment of ‘LiH

The first electronically excited state A’Z+ of the lithium hydrides and deuterides is endowed with some prominent features: like the other alkali hydrides it has an anomalously shaped potential energy curve which originates from avoided curve crossings and leads to an anomalous dependence of the anharmonicity and of the rotational constant on vibration. Also common to all alkali hydrides is the strongly R dependent dipole moment function which makes the dipole moments vary strongly with the vibrational quantum numbers. For lithium hydride and deuteride, however, the dipole moment function has a umque course: at small internuclear separattons it is negative -down to a minimum of -2.05 au at 2.50 a, making the molecule behave like Li-H+ in this region, whereas for large distances between the nuclei it is positive with a maximum of 4.2 au at 9.00 CZ~and the ionic composition of the molecule is Li*H[5-71. This behaviour is due to a drastic change in the composition of the electronic wavefunction in the region of the avoided crossing which makes the molecule experience different ionic compositions in the course of each vibrational period. For low vibrational quantum numbers the molecule preferentially probes the negative branch of the dipole moment function and the observable dipole moment is negative. On the other hand, for higher vibrational excitations the larger probability of finding the molecule in the vicinity of the outer turning point favours a positive dipole moment. For 7LiH the zero-crossover point, where the contributions of the positive and negative branch of the dipole moment function cancel on vibrational averaging, is almost reached in the vibrational level u’ = 5. For this level the dipole moment is theoretically predicted to be negative with the smallest absolute value which also is strongly J dependent, because of the reasons mentioned in section 2: J = 0: - 0.030au, J = 1: -0.028 au, J = 2: -0.025 au f7]. Exposed to an external

.288

. _

AL Bneger / Stark eff&ct m ‘B states of dratomic mifecules

electric field a ‘LiH molecule in this level therefore will closely resemble a homonuclear molecule regarding its interaction with the field, i.e. the polarizability terms will be very important. Recently [2] we succeeded in obtaining Stark quantum beats [S] from the A’Z+, u’ = 5, J’ = l level of ‘LiH at electric field strengths of the order of 100 kV/cm. The field-reduced splitting was determined to be AW/E’ = l-157(70) kHz/(kV/cm)‘_ This splitting turned out to be two times larger than it could be expected from the data available for all known states. (Due to a mistake in the definition of a, in ref_ [2] the contribution of a:-” to the splitting was too large by a factor of 2. This led to the larger deviation factor of 2.64 in ref. [2]_) Possible reasons for this discrepancy have already been mentioned in ref. [2] and we will now discuss them in more detail_ We start this discussion by recalling the analysis we pursued when trying to explain the splitting. This analysis was based on all parts of expression (21). Due to the limited knowledge about the electronic states of even this simplest heteronuclear molecule we were not able to take account of the interactions of the A’-?’ - . d = 5, J’ = 1 level with all allowed rovibronic levels of actually all other electronic ‘Z states [part III in (21)] but had to restrict the contributions to this part to the X ‘Z+ ground state and. similarly, instead of collecting the interactions with the levels of all ‘II states [part IV in (21)] we had to be satisfied with the one for the very next B ‘II state. These are the only states where experimental data are available for the mutual interaction kvith the perturbed state. No account at all was taken of interactions with unbound states or levels. Except for the B state. this can be justified by almost complete Franck-Condon overlaps between the bound states and levels. Speaking in terms of polarizabilities. this means that for the parallel electronic polarizability of the perturbed level only the contribution of the X ‘Z+ ground state and for the perpendicular electronic polarizability only the contnbution of the B’H state has been taken into account. Contrarily to the foregoing_ the values for the the vibrational branch polarizabilities [part II in (21)] are based on contributions from the first 15 vibrational levels where the most reliable data are available for. These data comprise the dipole moment function of Partridge et al. [7] who provided the most accurate ab initio MC SCF calculation in the Born-Oppenheimer approximation. and vibrational wavefunctions that have been obtained by numerically solving the Schrbdinger equation for the nuclear motion on the adiabatically corrected “effective” (i.e. including the centrifugal barrier) A’Z+ potential energy curve as given by Vidal and Stwalley [15]_ Although theoretical calculations [16] with large term CI wavefunctions in an elliptical orbital basis showed that at least for the X state there is only a qualitative. but not a quantitative agreement with the adiabatic corrections reported by ref. [15], it IS our conviction that these data still are the best and most reliable ones avatlable for the A’Z’ state of ‘LiH at present. These data also have been used for calculating the interaction with the adjacent rotational levels J’ = J + 1 [part I in (21)]. In this context the data revealed that approximating this contribution by expression (41a) and omitting the ef terms. as is usually done for evaluating the dipole moment of a vibrational level in a ‘Z state. in this case would mtroduce an additional error in the level splitting contributed by part I of (21) of = 6.54%;.1 he source for this comparattvely Ltrge error is the almost complete cancellation of large positive and negative contributions to the vibrational average of the dtpole moment functton in tlus vibrational level. which makes the matrix elements be very sensitive even to small alterations in any of their constituents. Concerning the interactions with other electronic states the quality of the data does not seem to be as consistent as for the interactions within the A’Z‘ state. These data have been given by Stwalley and co-workers [17-191. They are based on the transition moment functions of the ab initio MC SCF calculations of Docken and Hinze [5] and use hybrid potential energy curves for the states X ‘8, AIZf and B’H. These hybrid potential energy curves are composed from experimentally based Rydberg-Klein-Rees (RKR) parts describing the bottom of the well and theoretical long-range parts that were obtained by slightly scaling the ab initio results of Docken and Hinze [4] in the region of interpolation for joining both. The upper repulsive part of the potential energy curve \\as also taken from the same ab initio results and joined to the RKR part by interpolation. Since all transition matrix elements are given for J = 0 only, the

M

Brieger / Stark ejfect m ‘P stales of dratomrcmolecules

289

data used for parts III and IV exactly correspond to case J = 0 contributions to the parallel .md perpendicular electronic polarizabilities, respectively, as defined in (22a)-(22c). By the interactions quoted above, the main contributions to the total scalar and tensor-polarizability certainly have-been taken into account. Collecting all available data according (26) we arrive at the following expressions which have been used for deducing the field-reduced level shifts and splittings in the A’Z+ v’ = 5 state of ‘LiH for the special cases J = 1, M = 0, f 1: &:O,Ez

= -- [j&$(1,2) = - &&1.2)

WI:;1 *,/E2 AW;:;z/E'=

-[&&(1,2)

+ $&l,O)] -

- [&2$(1,2)

&2$(1,2)

-

+&$(l,O)l

$jCxy(o)

+ &;f(l,O)] -

- [+f(l,2)

-$af-X(O)

-$&&B(O),

$&fyyo),

++;(l,O)]

(42a) (42b)

-fa;-x(0)

+$r^,-B(O)_

(42c)

From the analysis described above we obtain (in units of D*/cm-‘) &(1,2)

= 1.111 x 10-3.

z$‘(1.2) = 6.253 x 10-3, at-x(O)

= -6.671

x 10-3,

p;(l,O)

= -2.618

$(l,O)

= 9.658 x 10-3,

o;-“(o)

x 10-3.

= 8.197 x 10-s.

Using these numbers the following numerical shifts and splittings contributed by parts I through IV of expression (21) are obtained and listed in the same order as in (42) above (in units of kHz/(kV/cm)‘): H(&,/E’ W$i *,/E2 AFV&/E'=

= 2.436 - 20.654 + 16.918 - 0.139 = - 1.439, = -0.939

- 5.286 + 5.639 - 0.277 = -0.863,

3.375 - 15.369~11.278+0.139=

-0.577.

This situation is illustrated in fig. 1 which depicts the contributions of the leading terms in parts I-IV of expression (21) to the field-reduced level shifts Wnf/E2, when using the data quoted above. Fig. 1 demonstrates the counteraction and almost total cancellation of the parts I and II by III, representing the combined action of the parallel rotational and vibrational branch polarizabilities pt and z$ and of the parallel electronic polarizability a,fMX. respectively_ The influence of the part labelled IV which is equivalent to aqeB is almost negligible_ The surprisingly large contribution of the X1x+ ground state to the parallel electronic polarizability is directly related to the occurrence of an avoided curve crossing in this region: for u’ = 5 in the A state the cIassical outer turning point of the vibrational motion does not only mark the R value for the nearest approach between the potential energy curves of the A and the X state, but it also indicates the region where the maximum in the A-X transition moment function occurs. This maximum originates from the rearrangement of the electronic configurations in the crossing region [20]_ A third reason for the enhancement of at-x is inferred from the shapes of both the A and X hybrid potential energy curves which in both cases have the same slope at the common position of the outer turning points [18]. The identical sIope ensures an optimum overlap of the upper and lower vibrational wavefunction in this region, i.e. large Franck-Condon factors for the participating vibrational levels v” = 17 and v” = 18 of the ground state. This makes these two levels contribute 65% to cytmx! After having combined all known contributions to parts I-IV according to (21) the resulting theoretical field-reduced level splitting is W/E2 = 0.577 kHz/(kV/cm)’ and thus turns out to be smaller than the experimentally determined one by a factor of two. At this point we should mention that approximating part I of (21) in the usual way by only the dipole moment term of expression (41a) we would have run into severe consequences regarding the further interpretation of the splitting. This approximation diminishes the splitting contributed by part I by 6.5% as pointed out above. The resulting theoretical overall splitting,

Af- Bneger

290

/ Srurk

efjecr in

‘Z

slates

of

dmtomlc mok&s

however, wouid have been enlarged by as much as 35.3% by this manipulation. A straightforward explanation for the discrepancy discovered above. certainly could be that due to the lack of comprehensive information we were not able to account for all contributions to the pohuizability terms. Whether or not all of the missing splitting has to be attributed to contributions from unknown electronic states to the electronic polarizabilities of the investigated level strongly depends on the accuracy of the known contributions. If these ones have the correct magnitude. then, in fact, the missing portion originates from interactions with unknown electronic states. An agreement with the measured splitting, however could also be achieved by ahering the influences of the main contributors. i.e. either by reducing part Ii1 (aii;L-“) by 5.1% or by strengthening the combined action of parts I and II by 4.8%. The latter manipulation would involve a correspondingly slight modification of the dipole moment function. This has been discussed in some detail in ref. [2], where it has been shown that a slight slnft of the dipole moment function of = 0.007 au is sufficient for totally explaining the missing splittin g. In this case. the correction is practically totally generated by part I alone. Being convinced. however. that the most reliable data entered into the first and second part of (21). we concentrate our discussion on the electronic polarizabilities. Each contnbution to the electronic polarizabilities will be inspected separately for posstble shortcomings_ In order to do so we start by commenting on their properties_ Inspecting the definitions (22a)-(22c) for the parallel and perpendicular electronic polarizabilities we realize that an> perturbing state lying above (below) the perturbed one will contribute a positive (negative) portion to any of these polarizabilities. Therefore. cx;i’-x is negative and n$- B is positive. The smallness of a:-” is astonishing at first glance. since the B ‘II state is the next electronic state lying only 6800 cm-’ above the investigated level. and also the A-B transition moment function is comparable in magnitude with the A-X transition moment function [5.6]. Furthermore, since the B state shares the sctme dissociation limit wtth the A state. it is embedded into the upmost vibrational levels of the A state well. The range of the B state potential energy cume. however. is only very limited compared with the extension of the A state well. and so the overlap of the vibrational wavefunctions is only very poor. Ii

I

WMIE2

[A,v=S)t[A,v%5)

3

l

III

IV

(A-X)

f (A-e)

M=O

i

o

/r

v’;S,J’d

’

I I t I .,

_

I

-lS-

Fig 1. Theoreucal conrnbutions to the Stark sphttinp of the Irxcl At\‘+, u’ = 5. J’ = 1 of ‘LtH. The contnbutions ansmg from the rotational (I) .md vtbrattonal (II) branch polarizabditics .trc almost completely compensated essentially by the contnbution of the X ground state (III) to the parallel electronic polariibility. (IV) is thr contribution of the B’lX stat2 to the perpendtcular polarizabthty. (III) and (IV) are the only known portions of the polarizabdity tensor of the mvestlgated level.

hf. Bneger / Stark effect tn ‘P states of dtafomic molecules

29i

Furthermore, being very-narrow and not deep the B state well can accommodate on!y three vibrationallevels. The poor overlap of the-vibrational wavefunctionof the investigated level with the bound ones of’the B state- is expressed by the fact -that the sum.of the Eranck-Condon factors_covers -only .6.75_ [19]. Therefore, it is possible that there is a large contribution to a,A-B due to interactions. with unbound levels of the B state which has not been accounted for. If these bound-free interactions had to cover the missing splitting, how laro,e should. their contributions be? Since in our kind of -quantum-beat experiments no information about the level order can be gained, the actual level order could equally well be inverted as opposed to the one suggested by fig. 1. This. would happen if the bound-free contribution to (r:-B wouid amount to 1.87 kHz/(kV/cm)’ which is 13.5 times the contribution of the three bound levels. Since almost the same factor is missing for a complete Franck-Condon overlap. these bound-free interactiors seem to be some of the main candidates for explaining the missing splitting. Before looking for other electronic ‘r states we first note that since a2-B enters with the negative sign into expression (36) for the tensor polarizability which is one of the parameters determining the level splitting, a2-B is supporting the action of a,pMx and so will any contribution to the perpendicular polarizabihty of unknown higher ‘II states. Each ‘IT state. however, shares the same dissociation limit with a ‘Z state, whose contribution to the parallel polarizability in the tensor polarizability counteracts the contribution of the accompanying ‘II state to the perpendicular polarizability. Besides of these, there are further ‘Z states that dissociate to ‘!&_ states of the lithium atom. The next ‘Z state above the A state is one of this kind. Using the asymptotic configurations Li(2s) + H(ls), Li(2p) + H(ls), Li(3s) + H(ls), Li(3p) + H(ls), Li(3d) + H(ls) and Li(4s) + H(ls), respectively, and mixing one or two appropriate ones with the ionic configuration Li’(ls’) + H-(ls’), Adelman and Herschbach [21] have calculated approximate potential energy curves for the first six ‘Z states, the A and the X state included_ Their results indicate that due to the interaction with the ionic curve the next four ‘Z states above the A state closely follow the ionic curve. These ‘Z states dissociate to the 3s, 3p, 3d and 4s states of the lithium atom, respectively. Due to the interaction with the ionic configuration ‘all potential energy curves for higher ‘Z states are shifted outward along the ionic curve and, besides of the next ‘X state, probably are totally outside of the range of the A’Z+ state well. This probably also holds for associated ‘TI states. An interaction, if any, of the A state with these higher states can possibly only occur by the very limited overlap of the vibrational wavefunctions in the common region of outer and inner turning points, respectively. Therefore, this behaviour probably renders these higher ‘Z and ‘II states inactive in the electronic polarizability terms of the A state, as far as bound 1eveIs are concerned. After all, the only further candidate possibly left for a sufficiently strong interaction with the A state, according to these calculations, seems to be the next ‘Z state, from which the molecule dissociates into Li(3s) -t H(ls). In the inner region, however, the potential energy curve of this eIectronic state is represented by one of the roots obtained from the diagonalization of the hamiltonian for three configurations two of which are the covalent ones Li(2s) + H(ls) and Li(2p) + H(ls). The third configuration is the ionic one Li(2s’) + H(ls’). The two further roots of the 3 x 3 secular equation are the potential energy curves for the A and the X state. The agreement with the ab initio calculations of Docken and Hinze [4] is very good for the ground state and fairly good (within 25% atmost) for the excited A state. The potential energy curve for the next ‘Z state obtained from this simple model is rather shallow, widely extended and shifted out and upward along the ionic curve. The minimum is located = 12200 cm-’ above the investigated level in the long-range region of the A state well. Due to the very hmited number of interacting configurations the position and the depth of the well for the next ‘Z state strongly depend on the choice and on the number of configurations, whether one or two, interacting with the ionic one. This, of course, strongly influences also the interaction with the investigated level of the A state. If the potential energy curve obtained from the asymptotic approximation for the next ‘2 state refers to reality as much as the ones for the A and the X state do, it certainly can be regarded as an estimate for the position and the shape of the actual well for the next ‘23 state. In any case, this one will probably be rather deep, widely extended and due to its shallowness is likely to be able to accommodate a large number

292

M. Brieger / Stark effect in ‘Z= states of dratomic moleczdes

of vibrational levels. This state also is likely to possess huge positive dipole moments. Lying above the perturbed A’Z+ state all unknown higher ‘Z states will contribute positively to a,[, thus reducing the Influence of a,,A-x _ If these interactions are not negligible there either must be a stronger influence of the perpendicular polarizability in order to achieve the correct splitting for the inverted level scheme (relative to the one shown in fig. 1). or these contributions to the parailel polarizabiiity must be larger than necessary for cancelling the perpendicular polarizability in order to establish the correct splitting with the level order given in fig. 1. and investigate whether there is any hint that this contribution possibly is too Next, we turn to $‘-x large. We have already mentioned that reducing the value given above for a,?-” by 5.1% is sufficient for totally explaining the splitting_ The amount of this reduction certainly also depends on to what accuracy this contribution has been determined. From preliminary results of lifetime measurements in the vibrational levels u’ = 0 through u’ = 7 of the A’Z f state of ‘LiH [22] we deduce lifetimes that uniformly are 10% longer than those_calculated on a semitheoretical basis by Zen-&e et al. [23]. Compared with other experimental results ]24,25] the deviations are 11% [24] for u’ = 2. 15.4% [24] and 6.6% [25], respectively, for o’ = 5 and - 2.2% [24] for o’ = 7. All values have been determined for J’ = 3 except for u’ = 0 where J’ = 4. To make sure that these results are not degraded by an unrecognized systematic error they are subjected to reexamination and partial redetermination. Provtded the former findings are affirmed, these results indicate that for a wide range of vibrational levels the A-X transition probability is smaller by 10% than theoretically [23] predictable when using the scLme data. which were used for calculating a,pmX. This behaviour shows the desired tendency and. since the potentials turn out to be correctly understood (see below), it could imply the need for a correction of the transition moment function of ref. [5] which was used in these calculations_ Due to the very different energy dependences of the terms in the sums. however, both quantities are hardly comparable: each that enter into the transition probability and into a,,A-y vibrational level u” of the X ‘2+ ground state contributes the portion ]O’e:,‘,“]’ (EC?_, - E”z)’ to the transition probability of the investigated level A’Z+. u’ = 5 (D,?::L’. denotes the transition matrix element). The contribution of the same level X. u” to ai>-“, however, IS given by ]O’~:~,“]’ (E’4-s - ELF)-‘, and thus totally different transitions and interactions, respectively, are important for both quantities. This is illustrated in fig. 2. The top part of fig. 2 shows the dependences of the squares of the rotationless transition matrix elements on the transition energy. as taken from ref. [18]. The heavy dots mark the magnitudes of the matrix elements_ The numbers attached to them refer to the quantum numbers of the vibrational levels in the X state. The dashed tine merely has been drawn to guide the eye. It represents a sketch of the vibrational wavefunction of the upper level, as it would approximately look like in an energy representation, if the vibrational quantization in the lower X ‘Z + potential energy curve could be switched off. What immediately attracts attention are the large matrix elements for the transitions to the vibrational levels u” = 17 and 18 of the X ground state. Reasons for this behaviour have been outlined above. The center of fig. 2 is occupied by the terms of the sum that represents a,,A-x_ Due to the hyperbohc energy dependence. the influence of the interaction with the high vibrational levels u” = 17 and 18 is further emphasized and makes them contribute 65% as mentioned above. This behaviour is in contrast to the one for the terms that contribute to the transition probability which is depicted on the bottom of fig. 2. Because of the large matrix elements, the partial decays to the levels u” = 17 and 18 together still contribute 21% to the width of the upper level, but in this case decay channels are important that interact only negligibly with the upper level in the sense of a,,A-X _ Therefore, both quantities may show the same tendency, but are hardly comparable. For ‘LiH the vibrational levels u” = 17 and 18 have not yet been investigated experimentally. Reliable data for the shape of the Xix+ potential energy curve of ‘LiH only exist for vibrational levels up to t.” = 12 [15]. Recently. however, investigating the fluorescence emitted from the Arf laser excited A’X+, u’ = 12, J’ = 10 level of 6LiH V erma and Stwalley [26] were able to observe all partial decays into vibrational -levels of the X ground state up to u” = 21. Based upon the results of the subsequent analysis

M. krieger / Starkefffecrn ‘X states of dtatomic molecules _

293

they-constructed a RKR potential energy curve covering_more _th+n:99% of the % state well. Comparing their new pqtentid energy cur& with ihose of ab i&o calculations, they f@und-the- old. DC&ken--I-&e calculatioqs [4] to-be in pest agreement, and also the hybrid potential energy curve [17] fits amazingly well. Since it can be expected that the electronic tensor polarizability is only weakly dependent on f, more clearness about the actual contribution of the polarizabilities could be achieved according to eq. (37) by independently determining the electronic tensor polarizability from the Stark splitting for another angular momentum, preferentially J = 2. Keeping the experimental arrangement unchanged, much higher electric field strengths would be necessary for obtaining a sufficiently developed quantum beat. This beat then also would be composed from the superposition of two beat frequencies which have a ratio of 1: 3. Changing the geometry in excitation and observation for an optimum detection of Am = _+2 beats not only would 5 103

1105

;? ia:\ L I t .I

f

ISlO~

2104

~S104

3104 I ~3.10"

117

I

I

’

:

-4

10-l

: ’, : : : t

--3104

: I I

--210-l

i '

410“

Fig. 2. Plot of the squared dipole matrix elements [18] for AtX+ . u’ = 5 to X *2+, d’ transitions (top). of the terms that sum up in at-X (center) and of the individual A’Z+, J = 5 to X IZ+, U” transition probabditxs [18] pottom) versus traxwtion energy E$_s - E$

M. Zheger / Stark effect in ‘2 srat& of dratomic molecules

294

suppress the second beat frequency, but also wouId help for par&thy overcoming the need for much higher field strengths_ This is expressed by the following reIations that hold in the J = 2 c&se for the field-reduced splitting between the sublevels M = 0, M’ = & 2; M = 0. M’ f 1 and M = + 1, M’ = +2, respectively: W4 (a=) Ar#trcz) -. o-2-* x--7 I = 3AWc2) 4

W)

If in expression (42~) the infhrence of the electronic polarizabilities is sufficiently covered by a,tbx(0) and a:-B(0). then for J = 2 their contribution to the splitting should not change by much more than 5% Besides of the difficuIties with the eIectric field strength in this special case. quantum-beat experiments in motecular levels with J > 1 generally have the disadvantage of a rapidly decreasing degree of modulation as a function of J.

Acknowledgement The author wishes to express his gratitude to Professor A. Hese and Dr. A. Renn for stimulating discussions and critically reading the manuscript_ This work was supported by the Deutsche Forschungsgemeinschaft. Sonderforschungsbereich 161.

Appendix To obtain relation (31) we start with a special form of the expression for the product of two D-functions WI

Making use of the closure properties of the rotational wavefunctions C C CIJ’M:~‘)(J’M~~‘~=

1,

(-4-2)

J’ M’ _I’ me obtain with these functions for the diagonal matrix element

=~~~(-1)Q’-Q(2K+1) h- Q Q’

;

,’

_E,)(;

;

_;)(JMA,Bh?+!,JMA)

(A-3)

=CCC(-1)Q’-Q-nf--~(2Jt1)(2K~l) K Q Q’ Kll -0’

J

)( p

S

-:

N -M

K

J

Q‘

M

(A-3’)

._ -- _ -

.

.-

M. Briegei;/

;

_

Stark effect TV ‘2 srates of-@atomic mo&uIes _ -_- _ . _ _

1

_

-

295 1

HeiT (12) hti be&i applied for_bbtaining (A;3’). -In (A-3’) t& last two 3j-$mbols -&ff& f&m zero only if _ Q’ = Q = O_ This implies for the first ttio.on_es that .q = r and p = s. Using (9) Ihe left-hand side of (A.3Pj.LC then can be written as the sum_over_the squared absolute values‘for the matrix elements. Apbl$ng (12) to these matrix elements we obtain the left-&and side of (31), the right-hand side being given by‘ (A.3’) tith .. Q’=&Oodq=r,p ‘S, -

References [I] [2] [3] [4] [S] [6]

M. Brieger, A. Hese, A. Renn and A. Sodetk. Chem. Phys. Lelten 76 (1980) 465. M. Brieger. A. Hese, A. Renn and A. Sodetk, Chem. Phys. 75 (1983) 1. P.J. Dagdigian, J. Chem. Phys. 73 (1980) 2049. K.K. Docken and J. Hinze, J_ Chem. Phys. 57 (1972) 4928. K-K. Docken and J. Hinze, J. Chem. Phys. 57 (1972) 4936. H. Partridge and S.R. Langhoff. J. Chem. Phys. 74 (1981) 2361.

[7] H. Partridge. S.R. Langhoff, WC. Stwalley and W.T. Zemke, J. Chem. Phys. 75 (1981) 2299. [S] S. Haroche. in: Topics in applied physics, Vol. 13. ed. K. Shimoda (Springer, Berhn, 1976). and references therein. [9] [lo] Ill] [12] [13]

A R.W. McKellar, in: Proceedings of the 8th Colloquium on High Resolution Molecular Spectroscopy, Tours. France (1983). M. Miiushima. The theory of rotating diatomic molecules (Wiley. New York, 1975) A.R. Edmonds. Angular momentum in quantum mechanics (Prmceton Univ. Press, Princeton, 1960). J.R.P. Angel and P.G.H. Sandars. Proc. Roy. Sot. A305 (1968) 125. A D. Bu~ki~@t%n, in: MTP international reviews of science, Vol. 3. Spectroscopy (Butterworths, London. 1972). and references therein. [14] W Liptay. in: Excited states, Vol. 1. ed. EC. Lii (Acaderruc Press, New York. 1974). and references therern. [15] CR Vidal and WC. Stwailey. J. Chem. Phys. 77 (1982) 883. 1161 D M. Bishop and L_M. Cheung. J. Chem. Phys. 78 (1983) 1396. [17] W.C. Stwalley. W-T. Zemke. K.R Way, K.C. Li and T.R Proctor. J. Chem. Phys 66 (1977 ) 5412. [18] W-T. Zen&e and WC. Stwalley. J. Chem. Phys. 68 (1978) 4619. [19] W-T. Zen&e. K.R. Way and W.C. Stwalley, J. Chem. Phys. 69 (1978) 402. [20] M. Oppenheimer and K.K. Docken. Chem. Phys. Letters 29 (1974) 349 [21] S.A. Adelman and D R. Herschbach. Mol. Phys. 33 (1977) 793. 1221 M. Brieger, A. Hese, A. Renn and A. Sodeik, to be published. (231 [24] [25] (261

W T Zemke, J B. Crooks and W-C. Stwalley, J. Chem. Phys. 68 (1978) 4628. P J Dagdigian. J. Chem. Phys. 64 (1976) 2609. P-H. Wine and LA Melton, J. Chem. Phys 64 (1976) 2692. K.K. Verma and WC. Stwalley. J Chem. Phys. 77 (1982) 2350.