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Vibrationa-rotational structure in the angular distribution and intensity of photoelectrons from diatomic molecules. II. H2
Che$caI Physics 30 (1978) 109-317 0 North-Holland Publishing Company -_ -_; y _-
VIBRATIONAL-ROTATIONAL STRUCTURE IN THE ANGULAR DISTRIBUTION AND INTENSITY OF PHOTOELECXRONS FROM DIATOMIC MOLECULES. II. H2 Yukikazu ITIKAWA Institute of Space and Aeronautical Science. University of Tokyo Komaba. Meguroku. Tokyo 153, Japan
Received 22 August 1977
Total intensity and &g&r distribution of photoelectrons ejected from Hz are calculated for individual viirationalrotational transitions accompanied by photoionization The relevant transition moment is evaluated in the two-center spheroidal coordinates with varying internuclear distance. The calculations for 584 A and 736 A lines are compared with the observed spectra. The theoretical result clearly shows the dependence of the angular distribution on the initial and fml viirational states and the dominance of the rotational transition with AJ= 0 over that with IAJI = 2.
1. Introduction In molecular photoelectron spectroscopy, vibrational-rotational structure in intensity and angular distribution of ejected electrons can provide detailed information on the structure of the molecule concerned. In part I [l] of this series of papers (referred to as 178), a general theory has been developed to take into account vibrational and rotational motions in the formulation of photoionization cross section of molecules_ An application of the theory to a hydrogen molecule is made in the present paper. In a previous paper [2] (referred to as 173), the present author has made a theoretical study on the vibrational effect in the photoionization of H2and D,. He has shown that the Franck-Condon factor approximation is inadequate to provide the vibrational structure in the photoionization cross section. In his calculation, rotational motions have noi been considered explicitly so that his result corresponds to a cross section averaged over the molecular rotation. Furthermore the previous paper (173) gives only the total intensity of photoelectrons. The-present calculation is regarded as an extension of the previous one in that (1) rotational structure is to be considered explicitly; (2) angular distribution is calculated both for the rotational and for the vibrational transitions. To this end, correct information is needed on the phase
factor of the wavefunction of the ejected electron. Except for this additional requirement, the actual calculation follows the same way as in 173. The rigorous form of the electrostatic interaction between the departing electron and the residual ion is employed in the present calculation (the case C in 173). The hydrogen molecule is only the example for which detailed rotational spectra have been obtained experimentally. Dii [3] has already calculated the rotational dependence of the shape parameter of the angular distribution for H,. His calculation, however, is applicable only to the threshold region of the photoionization and he has given no result for the absolute intensity of the photoelectrons. It is quite interesting to see if a certain type of rotational transition is more favorable to occur than others. For that purpose, the total intensity, as well as the angular distribution, of photoelectrons is calculated for each vibrational-rotational transition in the present paper. To describe electronic structure of a diatomic molecule, two-center spheroidal coordinates are more natural than one-center spherical ones. Due to the apparent complexity, the two-center formalism has
been rarely used in the study of molecular photoionization. So far only a few authors have applied it to the calculation of the total intensity [2,4,5]. One of the objectives of the present paper is to show the feasibility ofthe two-center calculation for the angular distribution of photoelectrons.
110
Y. ItikawalPhotoionizatioion of diatomic molecules. II
2. Cross section formulas Here we consider the photoionization
The F-functions in eq. (6) imply that the summation overj is taken only when b&h I +j + 1 and r+ j + 1 are even. The quantity ((d(X)>)Iin eq. (5) is defined by
process
hn + Hz(X ‘“2, Ui,Ji) + H$(X 2Z, nf, Jf) + e , where (ufii) and (VfJf) denote the initial and final vibrotational states, respectively. The formulas obtained in 178 for Z - Z transitions can be applied to this case. When the moiecules are randomly oriented in the space, the differential cross section for the ejection of an electron with a wave vector k is given by do -&uiJi
-+u&)=$[1
+/lP2(COSe~)] ,
(1)
for polarized light and ~(“i~i
-+ VfJf) =$I
- :&(cos
O,)]
,
(2)
for unpolarized hght. The an&es 6, and @k specify the direction of the ejected electron with respect to the polarization and the incoming-direction of the incident light, respectively. The total cross section etot and the shape parameter p of the angular distribution are expressed as (see section 3 of ‘E78) utot =--
8n3e”v 4rr -f?(L=O), 3c 3112
(3)
fi = -1O1/2 B(L = 2)/B(L = 0) ,
(4)
({&})l = (_i>’ eirl!@’ (7)
M(ZXIR)= sd;l
jir;
Jl&(r;
r;) d’$To~ r;)
(8)
is the so-called transition moment. Here ri (s = 1, 2) denote the positions of the molecular electrons in terms of the molecule-fuced frame; ~1 is a factor arising from the Coulomb phase shift (T#) = arg r(r + 1 - iolk); 4 = m,e2fi2k); d 0) is the dipole operator (dcA) = TZsrs Y,,(<)); ai and xuf are the vibrational wavefunctions of the initial and fmal molecules; rpand 11 are the electronic part of the molecular wavefunctions of the initial and final (free electron + molecular ion) states. Note that the electronic wavefunctions p and $ depend on the internuclear distance R. According to 178, the final state function is composed of the functions for the molecular ion, &I, and the ejected electron, GErk, in such a way that $ Hk=2-
1’2 k&r;)
* cp(+)(r;) G.&;l)l
G&r;) .
(9)
Here we have omitted the (singlet) spin function. It should be noted that the free-electron function G has the asymptotic form
where
Gmx(r’) z
0-l
{exp[it?(k, l, r)] Ylk(?)
f incoming wave] ,
(10)
where Tfi@U
; JfJi)
= CS(l+j + 1 =even)&(I+it
1 =even)
C = (m e j2nfi2 k)‘12
(11)
B(k, Z,r) = kr - $rZ + L-QIn(27cr) + q(k) .
WI
As in 173, we use for (p(‘r the exact ground-state wavefunction of H$ derived by Bates et al. [6]. The determination of the function G is the subject of the next section. As for the initia! wavefunction, we employ the Weinbaum function for the ground state of Hz:
(6)
Ip(r;, 4) =%
]PJt;>P&~)
f4v%r(~&r(r~)l
43)
The detailed form of the gerade (I@ and tingeradc
Y. ItikawaiPItotoionization
(I& orbital functions and the numerical constants N, and Bw are shown in 173 and Flannery’s paper 173. Substituting eqs. (9) and (13) into (8) and considering the parity conservation and the symmetry properties of the wavefunctions, we have M(IOIR) = Q/2Tr)CV, M(I1 IR) = -(3/@
[IgJ,p(O +B,I,uJ”(O] Nw PP(r, x
, (14)
of &atomic molecules. II
eq. (21) is rewritten in terms of the spheroidal coordinates. The resulting equation has a linearly-independent solution of the foml G&n i) = g,,(E) C&&q) ei4” with q = 0, *I, *2, __. . The
(23)
function Q is the eigenfunc-
tion of the equation
WI
’
111
-p2)&-A,,
-h2q2
-L]n
where 1s = Jdi &V(r’) cp,(r’) , 1,” = fdr’
(7))=0,
l-7?*
z’ pC+)*(rr) Ip,(r’)
061
,
pq
(24)
and normalized to unity such that
(17) (25)
J:(O =sdr’z’ G&1
~a@‘) ,
(18)
Jgl
@r’) ,
(19)
=_jp~rxfG;I&r)
J”Ol =&G;&h(r’)
-
(20)
The transition moment M vanishes unless I= 1,3,5, ....
3. Calculations of the wavefunction for the ejected de&on and the transition moment
In eq. (24), A,, is the corresponding eigenvahre and (k being the wave number of the ejected electron). It should be noted that QPp4and A,, depend on k and R through their dependence on h. The quantum number p (p = 1&141+ 1, ...) distinguishes each eigenfunction of eq. (24). (Note that QP,_4 = Qpq). The function Q is called the (normalized) prolate spheroidal angular function f . The radial part of the function (23) satisfies the equation
h = kR/2
The wavefunction for the ejected electron, GEIl\, is determined by solving the equation (atomic units being used hereafter) 1-i ( V’)2 f V(r’) - E] G,,,(r’) = 0 .
(21)
Here V denotes the electrostatic interaction between the electron and the residual ion (Hz) and the effect of electron exchange has been neglected. Now we adopt the (two-center) prolate spheriodal coordinates. The position of the electron is designated -1
by E=CrA+rg)lR.
rI=CrA-rg)/R,
cp’,
where rA and ru are the distances of the electron from the two nuclei of the molecule and yl’is the azimuthal angle around the molecular axis. With the use of the relation $p(V’)2
+aqa0
Q211; s21q2 aE C
=-
‘1 -n2)&
1 +($2_
l)(l_$)
(27) In the present case, the potential L’is independent of the azimuthal angle cp’so that eq. (26) can be soIved separately for each ~7andgp,_q =gP4. The electrostatic potential of Hz is calculated from JT=_
4L-+
R 524
I
I q(‘)(r”)12 [r’ - r”l
dr”
’
with the use of the bound-state wavefunction LO(+). We
Aa(&’
(22)
i See,for example,Flammer [8] ; the present calculation followsthe method in ref. [9] _
112
-
Y. Itika~a/Photoionirntion
expand B in terms of the Legendre function PI(~) to have L
and retain only the first two nonvanishin terms (u,, i? is given and II& Then the interaction matrix iPPP, by $$
= ;R2i&(U 6,.
+ $R2 t?$? ij2(g) ,
of diatomic moIec@es. II underthe initialcondition f,(&1)=0_
~-
:
~~
_.
-. .--
;- _.i-
(37)
The electrostatic potential of @ behaves like that,ofa two-center Coulomb field with increasing ,$ Although no analytical expression is known for the spheroidal Coulomb phase shift, it c& be shown by means of the WKEImethod that a solution of eq. (36) has an asymptotic form (see the appendix)
(30) fP&) ‘%
where (31) (32)
Cw h-II2 cos(hE+a&+$,J.
(38)
hr atomic units, ark = l/k. Now we construct the appropriate solution GEl, in the form of a linear combination of the function (23):
(33) Details of the calculation of ui are shown in 173. In the following we ignore the coupling among the terms with different p in eq. (26). Furthermore we consider only one partial wave @ = 1) in the actual calculation. This restriction may be justified, at least ’ for the calculation of total intensity [4], in the case of photoelectrons with rather lower kinetic energy. Moreover, since some part of the anisotropy of the molecular field is automatically taken considered in the two-center formalism, the effect of higher partial waves may be less significant here than in the onecenter approach. More elaborate calculation with the p-coupling considered is now planned and will be reported in the future. With neglecting the coupling term, we solve
The coefficients dP\ are determined such that the function G,T~~thus obtained satisfies the proper asymptotic condition (10). In the asymptotic region, we have g~2r/R,
Q~CCOS~‘,
(40)
where 0’ is the angle between the directions of the vector r’ and the molecular axis. Considering eq. (38), we get a$ = Is,, 7r-1 (2/R) l/2 C;cl i(-i)’ X efql e-‘54
B$
,
(41)
where KP4 = gPq + :7r - ffk ln(z2)
(42)
and use has been made of the relation &&I) (34) It is covenient to introduce a function f w ($) = (E2 - 1P2 gP4(E) ’
(35)
and solve, instead of eq. (34) the equation
+
h2g2 + Apq - %f(Q f (Q =-J tp l--q2 $2 - 1 (E2 - 1)2I p4 (3;)
etiq’ = (279 1’2 &
~tYt4(cos-rtl,
cp’)(43)
The coefficient B$ can be obtained from the expansion of the function QP4 in terms of the Legendre function. It follows from the charact&tic properties of QP4 [8,9 ] that BP4depends on h (and hence on k and R), vanishes unless p + t = even, and satisfies an orthonormal condition c’B$B;,, t
= 6,t
.
(W
113
Y. ItikawafPhotoionization of diatomic molecules. ZZ
4. Vibrational structure
Let a new function GEph be defined by 2 GEPk = ?r-.-‘(2/R)‘/-z c;j ($2 - I)_“2 X_&(E)
ln this section, a rotationally averaged cross section
Qpi1,(71)eihp' -
Then we have GEZ~= i(-i)’ er’l F
e-‘sph B$&,h
.
(46)
Substituting (46) into (9), we can rewrite (7) into ((d%)Z = (-i).JdR
x=f(R) x,,(R)
X Ck$“B$i@(phlR),
(47)
P
where k(pXIR) is defined by the formula for_l4(filR) [eqs. (14) and (15)] with replacing GEIi by GEph in eqs. (lS)_C20). In general Eph and Z3$ are dependent on the internuclear distance R. The remaining problem is to solve eq. (36) under the boundary conditions (37) and (38). We need only the solution with 4 = 0,l and p = 1_ This is exactly the same problem as has been solved in the previous paper (173). At that t&ue we needed no information about the phase shift Sp4. (There the dependence of the phase shift on R has been implicitly ignored, see section 4.) The integration of the differential equation (36) is repeated here to determine the phase shift as well as the amplitude. Finally the integration over R [i eq. (47)] is carried out in the following way. Consider an integraI of the form Z=./dR
x:,(R)xvJR)F(R) .
(48)
First we fit the function F to a quadratic form F(R) = F(O)+ (R -It3
F(l) + (R - R,)*F(*) ,
(49)
where R, is the equilibrium internuclear distance of Hz. Then the integral (48) is evaluated with the formula 2
Z =n2
f
11
(
L
ir
x a -h--x N x x
(50)
- Re)“iui) =JdR
)I
L
-
-x-x
(54)
In particular the total cross section becomes !Jtet(fJi+ Uf) E (327r4e2v/9c) x 7
(l((d’o’NZl*+ 2lKd’%Zl2) .
WI
When we assume only the contribution of the partial wave with p = 1, the R-averaged transition moment M(A)NZis given by [see eq. (47)J Kdi’%Z = (-i) JdR x:&R) xvi(R) X eiE’Xsiifi(l(lhlR)
bfl(R - R,)nJOi>F(n) .
The coefficients $I@
is considered. In eq. (52) the initial population of the rotational states is denoted by gJ. Under the assumption that the kinetic energy of the ejected electron is independent of Ji and Jf and any vibration-rot&ion coupling is ignored, the rotational averaging of eq. (5) results in (see section 4 of 178)
.
(56)
If we assume further that the phase shift E and the coefficient B do not depend on R, we have ((&I)), = (-i) ei’rh R&
x:f(R) (R - Re)’ x~(R),(SI)
have been calculated in 173.
X j-a
xt,@)xU:W~(W 1 ‘
-
(57)
114
Y. ItikawajfJhotoionization
Table 1 Dependeneeofthe reduced total intensity (lin au) and the arlguh parameter (8) on the vibrational state of the ion (uf)The values wrrespkd to the ejection of photoelectrons from H, in its ground vibrational state (i.e., ui = 0) by the incidence of 584 A line nf
0 :l 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18
KUf) cahb)
cd. a)
0.1850 0.3543 0.4160 0.3914 0.3265 0.2543 0.1894 0.1380 0.0993 0.0712 0.0510 0.0367 0.0264 0.0189 0.0135 0.0094 0.0061 0.0033 0.0010
0.1859 0.3556 0.4169 0.3918 0.3264 0.2536 0.1891 0.1377 0.0991 0.0710 0.0509
1.675 1.726 1.766 1.798 1.824 1.847 1.863 1.878 1.891 1.902 1.911 1.919 1.925 1.931 1.935 1.938 1942 1.943 1.944
0.0263 0.0134 0.0060 0.0010
Eip.
1.75 r 0.05 c)
a) Calculated using formula (56). b) The intensity calculatedin ref.
121using the approximation (57). ’ C)Experimentalvaiue reported in ref. [IO j. Substituting this into eq. (55) and using the orthonormalization relation (44), we get Otot(Ui + Uf) = (3LH - 4 e 2 V/ 9c)
(@)I’
f 2lD’l)l2) , (58)
where
DC”)= j-dRx;&Rjx,$Rj~WZ)
.
(59)
It is obvious from the defntition of i that formula is equivalent to that used in the previous study (173) of the vibrational structure for H,. In 173, numerical results for otct(Ui + uf) have been obtained by using formula (58) and compared with experimental data. To see the validity of the approximation (57), the values obtained with the use of the more accurate formuIa (56) are compared in table I with the previous ones (for case Cj_ There are shown the reduced total intensity (58)
I(u~-+ ufj = $T 7 (,~dcO,&~ + 21~d%~l~j I‘ for ui = 0 and the incidence cross section is given by C,t(Ui
POJfI
Cal. a)
of dintomic molecules. II
+
Uf)
=
of 584 a photons.
(87r3e2v/3c) I(Ui
.-
+
Uf) _
(60) The total
(61)
It is seen from table 1 that the R-dependence of the phase shift and Bh has a very small effect on utct in this case. Thus the conclusion obtained in 173 is unchanged even if we evaluate more correctIy the Rdependence of the transition moment. After the publication of 173, Gardner and Samson [15] have reported another experimental determination of $0 + uf) for 584 A line. Although their values for uf =Z8 show reasonable agreement with the calculation in 173 (and the present paper), there is a significant discrepancy for higher levels (uf > 9). One of the possible causes of the difference is the ignorance of the higher partiai waves in the calculation. Chapman [16] showed in his study of $ that higher partial waves become more important for longer bond lengths. Also, for large uf, we need to carefully consider the vibrational-rotational coupling_ The shape parameter fl of the angular distribution of photoelectrons is also shown in table 1. In the calculation of 0, the correct form of ((d@)j)l [eq. (56)] is used. In principle we have to calculate ((&jQ for an Infinite number of 1. The actual value &, however, decreases rapidly with increasing difference II - 1I, unless h is too large [8,9] . In the present problem, therefore, only a few terms (up to I = 3 or 5) have a non-negligible contnIution to the summation over I and Tin eq. (53). The p thus calculated clearly shows the dependence on the molecular vibrational states, which is never accounted for by the Franck-Condon factor approximation. There is only one measurement of the vibrational dependence of /I for 584 A line [lo]. The absohrte value obtained for uf = 2 is shown in table 1. The agreement between theory and experiment is quite good. Carlson and Jonas [lo] measured also the relative change of the ratio of the photoelectron intensity detected at 20” to that at 90”, do/d/%@ = 20”) (62) R(” -+ ‘f) = do/& (0, = 900) ’
Y. Itikawa/Photoionizatioion
of diatomic molecules. II
115
Table 2 The same as table 1, but for the incidence of 736 A he. “f
Au,)
P(WfI
=k. a)
-k. a)
esp.
0
0.4413 6.8405 0.9793 0.9122 0.7536 9.5812
1.694 1.748 1.783 1.815 1.835 1.853
2.0 c 0.2”) 1.93 f 0.03 c)
1 2 3 4 5
a) Calculated using fommla (56). b) Experimental value obtained in ref. [lo]. C)Experimental value obtained in ref. [ 1l] .
0.8 I
.
.
.
.
.
.
very valuable to measure the vibrational
012345678
of I and II with continuously incident light _ Fig. 1. Relative dependence on the final vibrational state (uf) of the ratio of the differential photoionization cross sections, R = (do/dd(20°))/(dafd~(900)). for the impact of A 584 photons. Calculated values are compared with the measured ones [lo], both being normalized at uf = 2.
for various vf- They used unpolarized light. We can calculate the ratio by using the formula
R(0+ uf)= [1 - &(cos 20”)] /[ 1 - ;&(cos = (I - 0.41227 /3)/(1 + 0.25 0) ,
go”)] (63)
and the values of S in table 1. Fig. 1 shows the comparison between the measured and the calculated values. Both are normalized at I+ = 2. Although the absolute amount of the relative change of the ratio is in the same order of magnitude, the trend of its qdependence is very different. There could be many factors plausible to cause this discrepancy, but the definite answer is yet clear. Another experimental determination of fl(ui + vf)
was made for Hz by using Ne resonance line (736 A) [lo,1 l] _To compare with that, we calculate p (and I) also for 736 A line. The result is given in table 2. The calculated value of 0 is in rather poor agreement with the measured one. In this region of photon energies, there may be a contribution of autoionization process [12] _In fact the relative dependence on uf of the total intensity calculated (table 2) apparently differs from the measurement [12,13]. In this respect, it would be
5. Vibrational-rotational
dependence
varying wave length of
structure
Once we obtain the R-averaged transition moment (SO), it is ver:’ easy to calculate also the vibrationalrotational structure. In tables 3 and 4, we show the total intensity [I(vi, Ji + uf, Jf) defined analogously to (61)] for 584 A and 736 A lines, respectively. Tables 5 and 6 give the corresponding values of p_ In the actual calculation, we have neglected the slight difference of the electron kinetic energy due to the difference of the rotational states involved. Also we do not consider any vibration-rotation interaction. Table 3 Intensity (au) of the photoelectrons ejected in the process (Vi= O,Jr) + (uf,Jf) on the impact of A 584 photons Ji
Jf
Z(Ui
q=o
Uf’2
Uf’5
vf=8
0
0
0.1510
0.3611
0.2324
0.0933
2
0.0340
0.0549
0.0218
0.0061
1 3 0 2 4 1 3 5
0.1646 0.0204 0.0068 0.1607 0.0175 0.0087 OS600 0.0162
0.3831 0.0329 0.0110 0.3768 0.0282 0.0141 0.3758 0.0261
0.2412 0.0131 0.0044 0.2387 0.0112 0.0056 0.2383 0.0104
0.0957 0.0036 0.0012 0.0950 0.0031 0.0016 0.0949 0.0029
1
2
3
= 0) Ji ~
V~ Jf)
116
.Y. &ikwafPfzdtoionization_of
Table4
--
The smieastable 3, &it for ?, 736 photons J-1
Jf
I(Ui=0, Ji + y=o
Uf’l
Uf’5
0
0 2 1 3 0
0.3659 0.0753 0.3961 0.0452
0.7226 0.1179 0.7698 0.0707
0.5337 0.0475 0.5527 0.0285
12
3
Vf, Jf)
0.0151
0.0236
0.0095
2 4
0.3875
0.7563
0.5473
0.0387
0.0606
0.0244
1 3 5
0.0194 0.3860 0.0359
0.0303 0.7541
0.0122 0.5464 0.0226
0.0561
Table 5 Shape parameter of the angular distribution of photoelectrons ejected in the process (vi = 0, Ji) -) (uf, Jf) on the impact of A584 photons J-1
0 1 2 3
P(ui=0sJi + ufxJf)
Jf 0 1 2 3
AJ=rZ
0 1 2 3
Jf
Vf’O
Vf’2
Uf’5
of’s
2.0 1.854 1.893 1900
2.0 1.898 1.926 1.931
2.0 1.935 1.953 1.956
2.0 1.955 1.967 1.970
0.2338
0.2254
0.2153
0.2086
0 1 2 3
AJ=*2
p(Ui =0, Ji +
molecides. ZI
nates in the summation over I in (5). Thus ii for the-. transition I AJI = 2 has a constant value (= i), regardfess of the initial rotationaistate-(see section 5 in 178) The photoelectron intensity for the transition [ ATI > 2 is very weak in the present calculation (paitiy due to the neglect of the higher partial Waves) and is not shown here. In the photoelectron spectroscopy of Hz, rotational spectra were clearly observed for 736 A line [l l-131. &brink [13] reported the spectrum corresponding to the transition Vi= 0, Ji + Uf = 5;Jf. His spectrum indicates large intensities for the. processes(J;, Jf) = (l,l), (O,O),(2,2), (33) (in decreasing order of magnitude) and much less for (1,3), (0,2) and (2,4). The experiment was done at room temperature. Taking into account the thermal distribution of the initial rotational states, we calculate from the numbers in table 4 the relative intensity to be
I(O+O):1(2+2):1(3+3):1(1
Table 6 The same as table 5, but for A736 photons Ji
+ztomic
of; Jf)
Uf’O
Uf’l
Vf’5
2.0 1.864 1.900 1.907
2.0 1.890 1.920 1.925
2.0 1.938 1.955 1.958
0.2069
0.2047
0.2003
Thus we have a relation (see section 5
of 178)
B(uJ+ U’J’) = 4(ur’ -* V’J) . As is shown in 178, fi for the transition Jt = 0 + Jf = 0 has a constant value of 2. Under the present restriction of the partial wave top = 1, the term with I = 1 dorni-
+3):1(0+2):1(2+4)
= 1.0: 0.192: 0.174: 0.130: 0.052 : 0.017: 0.008. The theoretical result, therefore, can explain the trend in the measured spectrum. Niehaus and Ruf [l l] experimentally determined p for the transition Ui= 0, Ji + uf = 1, Jf at the impact of X736 photons. They could not resolve each member (0 + 0, 1 + 1,2 -+ 2) of the Q-branch transition and obtained /3(AJ= 0) = 1.95 f 0.03. This is fairly close to the theoretical one (see table 6). From the measurement of the angular dependence of the intensity ratio of the Q and S branches, they also obtained @J= 2) = 0.85 f 0.14. This is too large compared with the calculated value. This discrepancy could be ascribed to the various assumptions adopted in the present calculation. In particular the restriction of the electron partial wave top = 1 may affect considerably the angular dependence. Another possibility ofthe disagreement might come from the rather incomplete resolution of the experimental photoelectron spectra. If the photoelectrons for AJ= 0 transition contribute partly to the AJ = 2 iine, the apparent value of /3(AJ f 2) is larger than the correct one. Furthermore, as has been mentioned before, an autoionization process might contribute to the ejection of photoelectrons for 736 A photons. No more detailed test of the present calculation
-.
Y. ItikawafPhotoionization
c& be~rhade until more elaborate calculation (espe~cially considering the effect: of higher partial waves) + be performed or-rotational spectra with the higher resolution @ll be obtained for many other wave lengths of incident light.
of diatomic molecules. II
117
F
I @I’2(<‘)dt’ -
n/4
=& + (R/2h)lng
- h[(h2 +R,)/2h2
2 R2/8h43t;-l
+6 ($-2) f constant _
(A-7)
Thus, neglecting the term of the order $-I, we have the asymptotic form of the solution of eq. (A.l) as
Appendix: Asymptotic form of the radial wave function of the ejected electron
f
P4
5-m (t) Cpqh-“2
cos(hE + k-l
In r;+ 5,)
_(A.8)
Consider an asymptotic solution of the equation
h2E2 +Apq - F”;;‘(~) _q2 + fpq@l= 0 * 1 p-1 (E2 - 1)2 (A-l) 1
When ,$+ m, the electrostatic potential of H’; becomes V- -2/RE +6 (g-3) and
(A.3
hence we have
W$$ - -Rf; + O(.$-‘) _
(A-3)
In the asymptotic region, therefore, eq. (A-1) has the form (A.41
@((u=h2+R.$-l
+(h2 +Apq)E-2
_
(A.51
Applying the WKB method [ 141 to eq. (A.4), we get a solution of the form
1. .
f P4 (.$) =c P4 ~-I’4(~)cos &2($)d$’ - 7~14{A 6) [IFrom eq. (AS), the phase factor is calculated to be
[l] Y. Itlkawa,Chem.
Phys. 28 (1978)461.
[2] Y. Itikawa,J. Electron Spectry. 2 (1973) 125. [31 D. Dti, P&s. Rev.A6 (1973) 160. 141M.R. FIannery and U. Opik. Proc. Phys. Sot. 86 (1965) 491. [S] H.C. TuckweIl, I. Phys. B3 (1970) 293. [6] D.R. Bares, K. Ledsham and A.L. Stewart, Phil. Trans. Roy. Sot. A246 (1953) 215. [7] M.R. Flannery, Proc. Phys. Sot. 85 (1965) 1318. [8] C. Flammer, Spheroidal wave functions (Stanford Univ. Press, 1957). [9] Y. Itikawa and H. Hattori, Tables of Prolate.Spheroidal Functions, Contriiutions from the Research Group on Atomsand Molecules, No. 2 (1968). [lo] T.A. Carlson and A.E. Jonas, J. Chem. Phys. 55 (1971) 4913. [ 111 A. Niehaus and M.W. Ruf, Chem. Phys. Letters 11 (1971) [12] %R. Samson, Phys. Rept. 28 (1976) 303. [13] L. Asbrink, Chem. Phys. Letters 7 (1970) 549. [14] H.A. Bethe and R. Jackiw, Intermediate quantum mechanics, 2nd Ed., (Benjamin, New York, 1968)_ch. 1. [15] J.L. Gardner and J.A.R. Samson, J. Electron Spectry. 8 (1976) 123. [16] FM. Chapman Jr., J. Chem. Phys. 63 (1975) 2101.
VIBRATIONAL-ROTATIONAL STRUCTURE IN THE ANGULAR DISTRIBUTION AND INTENSITY OF PHOTOELECXRONS FROM DIATOMIC MOLECULES. II. H2 Yukikazu ITIKAWA Institute of Space and Aeronautical Science. University of Tokyo Komaba. Meguroku. Tokyo 153, Japan
Received 22 August 1977
Total intensity and &g&r distribution of photoelectrons ejected from Hz are calculated for individual viirationalrotational transitions accompanied by photoionization The relevant transition moment is evaluated in the two-center spheroidal coordinates with varying internuclear distance. The calculations for 584 A and 736 A lines are compared with the observed spectra. The theoretical result clearly shows the dependence of the angular distribution on the initial and fml viirational states and the dominance of the rotational transition with AJ= 0 over that with IAJI = 2.
1. Introduction In molecular photoelectron spectroscopy, vibrational-rotational structure in intensity and angular distribution of ejected electrons can provide detailed information on the structure of the molecule concerned. In part I [l] of this series of papers (referred to as 178), a general theory has been developed to take into account vibrational and rotational motions in the formulation of photoionization cross section of molecules_ An application of the theory to a hydrogen molecule is made in the present paper. In a previous paper [2] (referred to as 173), the present author has made a theoretical study on the vibrational effect in the photoionization of H2and D,. He has shown that the Franck-Condon factor approximation is inadequate to provide the vibrational structure in the photoionization cross section. In his calculation, rotational motions have noi been considered explicitly so that his result corresponds to a cross section averaged over the molecular rotation. Furthermore the previous paper (173) gives only the total intensity of photoelectrons. The-present calculation is regarded as an extension of the previous one in that (1) rotational structure is to be considered explicitly; (2) angular distribution is calculated both for the rotational and for the vibrational transitions. To this end, correct information is needed on the phase
factor of the wavefunction of the ejected electron. Except for this additional requirement, the actual calculation follows the same way as in 173. The rigorous form of the electrostatic interaction between the departing electron and the residual ion is employed in the present calculation (the case C in 173). The hydrogen molecule is only the example for which detailed rotational spectra have been obtained experimentally. Dii [3] has already calculated the rotational dependence of the shape parameter of the angular distribution for H,. His calculation, however, is applicable only to the threshold region of the photoionization and he has given no result for the absolute intensity of the photoelectrons. It is quite interesting to see if a certain type of rotational transition is more favorable to occur than others. For that purpose, the total intensity, as well as the angular distribution, of photoelectrons is calculated for each vibrational-rotational transition in the present paper. To describe electronic structure of a diatomic molecule, two-center spheroidal coordinates are more natural than one-center spherical ones. Due to the apparent complexity, the two-center formalism has
been rarely used in the study of molecular photoionization. So far only a few authors have applied it to the calculation of the total intensity [2,4,5]. One of the objectives of the present paper is to show the feasibility ofthe two-center calculation for the angular distribution of photoelectrons.
110
Y. ItikawalPhotoionizatioion of diatomic molecules. II
2. Cross section formulas Here we consider the photoionization
The F-functions in eq. (6) imply that the summation overj is taken only when b&h I +j + 1 and r+ j + 1 are even. The quantity ((d(X)>)Iin eq. (5) is defined by
process
hn + Hz(X ‘“2, Ui,Ji) + H$(X 2Z, nf, Jf) + e , where (ufii) and (VfJf) denote the initial and final vibrotational states, respectively. The formulas obtained in 178 for Z - Z transitions can be applied to this case. When the moiecules are randomly oriented in the space, the differential cross section for the ejection of an electron with a wave vector k is given by do -&uiJi
-+u&)=$[1
+/lP2(COSe~)] ,
(1)
for polarized light and ~(“i~i
-+ VfJf) =$I
- :&(cos
O,)]
,
(2)
for unpolarized hght. The an&es 6, and @k specify the direction of the ejected electron with respect to the polarization and the incoming-direction of the incident light, respectively. The total cross section etot and the shape parameter p of the angular distribution are expressed as (see section 3 of ‘E78) utot =--
8n3e”v 4rr -f?(L=O), 3c 3112
(3)
fi = -1O1/2 B(L = 2)/B(L = 0) ,
(4)
({&})l = (_i>’ eirl!@’ (7)
M(ZXIR)= sd;l
jir;
Jl&(r;
r;) d’$To~ r;)
(8)
is the so-called transition moment. Here ri (s = 1, 2) denote the positions of the molecular electrons in terms of the molecule-fuced frame; ~1 is a factor arising from the Coulomb phase shift (T#) = arg r(r + 1 - iolk); 4 = m,e2fi2k); d 0) is the dipole operator (dcA) = TZsrs Y,,(<)); ai and xuf are the vibrational wavefunctions of the initial and fmal molecules; rpand 11 are the electronic part of the molecular wavefunctions of the initial and final (free electron + molecular ion) states. Note that the electronic wavefunctions p and $ depend on the internuclear distance R. According to 178, the final state function is composed of the functions for the molecular ion, &I, and the ejected electron, GErk, in such a way that $ Hk=2-
1’2 k&r;)
* cp(+)(r;) G.&;l)l
G&r;) .
(9)
Here we have omitted the (singlet) spin function. It should be noted that the free-electron function G has the asymptotic form
where
Gmx(r’) z
0-l
{exp[it?(k, l, r)] Ylk(?)
f incoming wave] ,
(10)
where Tfi@U
; JfJi)
= CS(l+j + 1 =even)&(I+it
1 =even)
C = (m e j2nfi2 k)‘12
(11)
B(k, Z,r) = kr - $rZ + L-QIn(27cr) + q(k) .
WI
As in 173, we use for (p(‘r the exact ground-state wavefunction of H$ derived by Bates et al. [6]. The determination of the function G is the subject of the next section. As for the initia! wavefunction, we employ the Weinbaum function for the ground state of Hz:
(6)
Ip(r;, 4) =%
]PJt;>P&~)
f4v%r(~&r(r~)l
43)
The detailed form of the gerade (I@ and tingeradc
Y. ItikawaiPItotoionization
(I& orbital functions and the numerical constants N, and Bw are shown in 173 and Flannery’s paper 173. Substituting eqs. (9) and (13) into (8) and considering the parity conservation and the symmetry properties of the wavefunctions, we have M(IOIR) = Q/2Tr)CV, M(I1 IR) = -(3/@
[IgJ,p(O +B,I,uJ”(O] Nw PP(r, x
, (14)
of &atomic molecules. II
eq. (21) is rewritten in terms of the spheroidal coordinates. The resulting equation has a linearly-independent solution of the foml G&n i) = g,,(E) C&&q) ei4” with q = 0, *I, *2, __. . The
(23)
function Q is the eigenfunc-
tion of the equation
WI
’
111
-p2)&-A,,
-h2q2
-L]n
where 1s = Jdi &V(r’) cp,(r’) , 1,” = fdr’
(7))=0,
l-7?*
z’ pC+)*(rr) Ip,(r’)
061
,
pq
(24)
and normalized to unity such that
(17) (25)
J:(O =sdr’z’ G&1
~a@‘) ,
(18)
Jgl
@r’) ,
(19)
=_jp~rxfG;I&r)
J”Ol =&G;&h(r’)
-
(20)
The transition moment M vanishes unless I= 1,3,5, ....
3. Calculations of the wavefunction for the ejected de&on and the transition moment
In eq. (24), A,, is the corresponding eigenvahre and (k being the wave number of the ejected electron). It should be noted that QPp4and A,, depend on k and R through their dependence on h. The quantum number p (p = 1&141+ 1, ...) distinguishes each eigenfunction of eq. (24). (Note that QP,_4 = Qpq). The function Q is called the (normalized) prolate spheroidal angular function f . The radial part of the function (23) satisfies the equation
h = kR/2
The wavefunction for the ejected electron, GEIl\, is determined by solving the equation (atomic units being used hereafter) 1-i ( V’)2 f V(r’) - E] G,,,(r’) = 0 .
(21)
Here V denotes the electrostatic interaction between the electron and the residual ion (Hz) and the effect of electron exchange has been neglected. Now we adopt the (two-center) prolate spheriodal coordinates. The position of the electron is designated -1
by E=CrA+rg)lR.
rI=CrA-rg)/R,
cp’,
where rA and ru are the distances of the electron from the two nuclei of the molecule and yl’is the azimuthal angle around the molecular axis. With the use of the relation $p(V’)2
+aqa0
Q211; s21q2 aE C
=-
‘1 -n2)&
1 +($2_
l)(l_$)
(27) In the present case, the potential L’is independent of the azimuthal angle cp’so that eq. (26) can be soIved separately for each ~7andgp,_q =gP4. The electrostatic potential of Hz is calculated from JT=_
4L-+
R 524
I
I q(‘)(r”)12 [r’ - r”l
dr”
’
with the use of the bound-state wavefunction LO(+). We
Aa(&’
(22)
i See,for example,Flammer [8] ; the present calculation followsthe method in ref. [9] _
112
-
Y. Itika~a/Photoionirntion
expand B in terms of the Legendre function PI(~) to have L
and retain only the first two nonvanishin terms (u,, i? is given and II& Then the interaction matrix iPPP, by $$
= ;R2i&(U 6,.
+ $R2 t?$? ij2(g) ,
of diatomic moIec@es. II underthe initialcondition f,(&1)=0_
~-
:
~~
_.
-. .--
;- _.i-
(37)
The electrostatic potential of @ behaves like that,ofa two-center Coulomb field with increasing ,$ Although no analytical expression is known for the spheroidal Coulomb phase shift, it c& be shown by means of the WKEImethod that a solution of eq. (36) has an asymptotic form (see the appendix)
(30) fP&) ‘%
where (31) (32)
Cw h-II2 cos(hE+a&+$,J.
(38)
hr atomic units, ark = l/k. Now we construct the appropriate solution GEl, in the form of a linear combination of the function (23):
(33) Details of the calculation of ui are shown in 173. In the following we ignore the coupling among the terms with different p in eq. (26). Furthermore we consider only one partial wave @ = 1) in the actual calculation. This restriction may be justified, at least ’ for the calculation of total intensity [4], in the case of photoelectrons with rather lower kinetic energy. Moreover, since some part of the anisotropy of the molecular field is automatically taken considered in the two-center formalism, the effect of higher partial waves may be less significant here than in the onecenter approach. More elaborate calculation with the p-coupling considered is now planned and will be reported in the future. With neglecting the coupling term, we solve
The coefficients dP\ are determined such that the function G,T~~thus obtained satisfies the proper asymptotic condition (10). In the asymptotic region, we have g~2r/R,
Q~CCOS~‘,
(40)
where 0’ is the angle between the directions of the vector r’ and the molecular axis. Considering eq. (38), we get a$ = Is,, 7r-1 (2/R) l/2 C;cl i(-i)’ X efql e-‘54
B$
,
(41)
where KP4 = gPq + :7r - ffk ln(z2)
(42)
and use has been made of the relation &&I) (34) It is covenient to introduce a function f w ($) = (E2 - 1P2 gP4(E) ’
(35)
and solve, instead of eq. (34) the equation
+
h2g2 + Apq - %f(Q f (Q =-J tp l--q2 $2 - 1 (E2 - 1)2I p4 (3;)
etiq’ = (279 1’2 &
~tYt4(cos-rtl,
cp’)(43)
The coefficient B$ can be obtained from the expansion of the function QP4 in terms of the Legendre function. It follows from the charact&tic properties of QP4 [8,9 ] that BP4depends on h (and hence on k and R), vanishes unless p + t = even, and satisfies an orthonormal condition c’B$B;,, t
= 6,t
.
(W
113
Y. ItikawafPhotoionization of diatomic molecules. ZZ
4. Vibrational structure
Let a new function GEph be defined by 2 GEPk = ?r-.-‘(2/R)‘/-z c;j ($2 - I)_“2 X_&(E)
ln this section, a rotationally averaged cross section
Qpi1,(71)eihp' -
Then we have GEZ~= i(-i)’ er’l F
e-‘sph B$&,h
.
(46)
Substituting (46) into (9), we can rewrite (7) into ((d%)Z = (-i).JdR
x=f(R) x,,(R)
X Ck$“B$i@(phlR),
(47)
P
where k(pXIR) is defined by the formula for_l4(filR) [eqs. (14) and (15)] with replacing GEIi by GEph in eqs. (lS)_C20). In general Eph and Z3$ are dependent on the internuclear distance R. The remaining problem is to solve eq. (36) under the boundary conditions (37) and (38). We need only the solution with 4 = 0,l and p = 1_ This is exactly the same problem as has been solved in the previous paper (173). At that t&ue we needed no information about the phase shift Sp4. (There the dependence of the phase shift on R has been implicitly ignored, see section 4.) The integration of the differential equation (36) is repeated here to determine the phase shift as well as the amplitude. Finally the integration over R [i eq. (47)] is carried out in the following way. Consider an integraI of the form Z=./dR
x:,(R)xvJR)F(R) .
(48)
First we fit the function F to a quadratic form F(R) = F(O)+ (R -It3
F(l) + (R - R,)*F(*) ,
(49)
where R, is the equilibrium internuclear distance of Hz. Then the integral (48) is evaluated with the formula 2
Z =n2
f
11
(
L
ir
x a -h--x N x x
(50)
- Re)“iui) =JdR
)I
L
-
-x-x
(54)
In particular the total cross section becomes !Jtet(fJi+ Uf) E (327r4e2v/9c) x 7
(l((d’o’NZl*+ 2lKd’%Zl2) .
WI
When we assume only the contribution of the partial wave with p = 1, the R-averaged transition moment M(A)NZis given by [see eq. (47)J Kdi’%Z = (-i) JdR x:&R) xvi(R) X eiE’Xsiifi(l(lhlR)
bfl(R - R,)nJOi>F(n) .
The coefficients $I@
is considered. In eq. (52) the initial population of the rotational states is denoted by gJ. Under the assumption that the kinetic energy of the ejected electron is independent of Ji and Jf and any vibration-rot&ion coupling is ignored, the rotational averaging of eq. (5) results in (see section 4 of 178)
.
(56)
If we assume further that the phase shift E and the coefficient B do not depend on R, we have ((&I)), = (-i) ei’rh R&
x:f(R) (R - Re)’ x~(R),(SI)
have been calculated in 173.
X j-a
xt,@)xU:W~(W 1 ‘
-
(57)
114
Y. ItikawajfJhotoionization
Table 1 Dependeneeofthe reduced total intensity (lin au) and the arlguh parameter (8) on the vibrational state of the ion (uf)The values wrrespkd to the ejection of photoelectrons from H, in its ground vibrational state (i.e., ui = 0) by the incidence of 584 A line nf
0 :l 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18
KUf) cahb)
cd. a)
0.1850 0.3543 0.4160 0.3914 0.3265 0.2543 0.1894 0.1380 0.0993 0.0712 0.0510 0.0367 0.0264 0.0189 0.0135 0.0094 0.0061 0.0033 0.0010
0.1859 0.3556 0.4169 0.3918 0.3264 0.2536 0.1891 0.1377 0.0991 0.0710 0.0509
1.675 1.726 1.766 1.798 1.824 1.847 1.863 1.878 1.891 1.902 1.911 1.919 1.925 1.931 1.935 1.938 1942 1.943 1.944
0.0263 0.0134 0.0060 0.0010
Eip.
1.75 r 0.05 c)
a) Calculated using formula (56). b) The intensity calculatedin ref.
121using the approximation (57). ’ C)Experimentalvaiue reported in ref. [IO j. Substituting this into eq. (55) and using the orthonormalization relation (44), we get Otot(Ui + Uf) = (3LH - 4 e 2 V/ 9c)
(@)I’
f 2lD’l)l2) , (58)
where
DC”)= j-dRx;&Rjx,$Rj~WZ)
.
(59)
It is obvious from the defntition of i that formula is equivalent to that used in the previous study (173) of the vibrational structure for H,. In 173, numerical results for otct(Ui + uf) have been obtained by using formula (58) and compared with experimental data. To see the validity of the approximation (57), the values obtained with the use of the more accurate formuIa (56) are compared in table I with the previous ones (for case Cj_ There are shown the reduced total intensity (58)
I(u~-+ ufj = $T 7 (,~dcO,&~ + 21~d%~l~j I‘ for ui = 0 and the incidence cross section is given by C,t(Ui
POJfI
Cal. a)
of dintomic molecules. II
+
Uf)
=
of 584 a photons.
(87r3e2v/3c) I(Ui
.-
+
Uf) _
(60) The total
(61)
It is seen from table 1 that the R-dependence of the phase shift and Bh has a very small effect on utct in this case. Thus the conclusion obtained in 173 is unchanged even if we evaluate more correctIy the Rdependence of the transition moment. After the publication of 173, Gardner and Samson [15] have reported another experimental determination of $0 + uf) for 584 A line. Although their values for uf =Z8 show reasonable agreement with the calculation in 173 (and the present paper), there is a significant discrepancy for higher levels (uf > 9). One of the possible causes of the difference is the ignorance of the higher partiai waves in the calculation. Chapman [16] showed in his study of $ that higher partial waves become more important for longer bond lengths. Also, for large uf, we need to carefully consider the vibrational-rotational coupling_ The shape parameter fl of the angular distribution of photoelectrons is also shown in table 1. In the calculation of 0, the correct form of ((d@)j)l [eq. (56)] is used. In principle we have to calculate ((&jQ for an Infinite number of 1. The actual value &, however, decreases rapidly with increasing difference II - 1I, unless h is too large [8,9] . In the present problem, therefore, only a few terms (up to I = 3 or 5) have a non-negligible contnIution to the summation over I and Tin eq. (53). The p thus calculated clearly shows the dependence on the molecular vibrational states, which is never accounted for by the Franck-Condon factor approximation. There is only one measurement of the vibrational dependence of /I for 584 A line [lo]. The absohrte value obtained for uf = 2 is shown in table 1. The agreement between theory and experiment is quite good. Carlson and Jonas [lo] measured also the relative change of the ratio of the photoelectron intensity detected at 20” to that at 90”, do/d/%@ = 20”) (62) R(” -+ ‘f) = do/& (0, = 900) ’
Y. Itikawa/Photoionizatioion
of diatomic molecules. II
115
Table 2 The same as table 1, but for the incidence of 736 A he. “f
Au,)
P(WfI
=k. a)
-k. a)
esp.
0
0.4413 6.8405 0.9793 0.9122 0.7536 9.5812
1.694 1.748 1.783 1.815 1.835 1.853
2.0 c 0.2”) 1.93 f 0.03 c)
1 2 3 4 5
a) Calculated using fommla (56). b) Experimental value obtained in ref. [lo]. C)Experimental value obtained in ref. [ 1l] .
0.8 I
.
.
.
.
.
.
very valuable to measure the vibrational
012345678
of I and II with continuously incident light _ Fig. 1. Relative dependence on the final vibrational state (uf) of the ratio of the differential photoionization cross sections, R = (do/dd(20°))/(dafd~(900)). for the impact of A 584 photons. Calculated values are compared with the measured ones [lo], both being normalized at uf = 2.
for various vf- They used unpolarized light. We can calculate the ratio by using the formula
R(0+ uf)= [1 - &(cos 20”)] /[ 1 - ;&(cos = (I - 0.41227 /3)/(1 + 0.25 0) ,
go”)] (63)
and the values of S in table 1. Fig. 1 shows the comparison between the measured and the calculated values. Both are normalized at I+ = 2. Although the absolute amount of the relative change of the ratio is in the same order of magnitude, the trend of its qdependence is very different. There could be many factors plausible to cause this discrepancy, but the definite answer is yet clear. Another experimental determination of fl(ui + vf)
was made for Hz by using Ne resonance line (736 A) [lo,1 l] _To compare with that, we calculate p (and I) also for 736 A line. The result is given in table 2. The calculated value of 0 is in rather poor agreement with the measured one. In this region of photon energies, there may be a contribution of autoionization process [12] _In fact the relative dependence on uf of the total intensity calculated (table 2) apparently differs from the measurement [12,13]. In this respect, it would be
5. Vibrational-rotational
dependence
varying wave length of
structure
Once we obtain the R-averaged transition moment (SO), it is ver:’ easy to calculate also the vibrationalrotational structure. In tables 3 and 4, we show the total intensity [I(vi, Ji + uf, Jf) defined analogously to (61)] for 584 A and 736 A lines, respectively. Tables 5 and 6 give the corresponding values of p_ In the actual calculation, we have neglected the slight difference of the electron kinetic energy due to the difference of the rotational states involved. Also we do not consider any vibration-rotation interaction. Table 3 Intensity (au) of the photoelectrons ejected in the process (Vi= O,Jr) + (uf,Jf) on the impact of A 584 photons Ji
Jf
Z(Ui
q=o
Uf’2
Uf’5
vf=8
0
0
0.1510
0.3611
0.2324
0.0933
2
0.0340
0.0549
0.0218
0.0061
1 3 0 2 4 1 3 5
0.1646 0.0204 0.0068 0.1607 0.0175 0.0087 OS600 0.0162
0.3831 0.0329 0.0110 0.3768 0.0282 0.0141 0.3758 0.0261
0.2412 0.0131 0.0044 0.2387 0.0112 0.0056 0.2383 0.0104
0.0957 0.0036 0.0012 0.0950 0.0031 0.0016 0.0949 0.0029
1
2
3
= 0) Ji ~
V~ Jf)
116
.Y. &ikwafPfzdtoionization_of
Table4
--
The smieastable 3, &it for ?, 736 photons J-1
Jf
I(Ui=0, Ji + y=o
Uf’l
Uf’5
0
0 2 1 3 0
0.3659 0.0753 0.3961 0.0452
0.7226 0.1179 0.7698 0.0707
0.5337 0.0475 0.5527 0.0285
12
3
Vf, Jf)
0.0151
0.0236
0.0095
2 4
0.3875
0.7563
0.5473
0.0387
0.0606
0.0244
1 3 5
0.0194 0.3860 0.0359
0.0303 0.7541
0.0122 0.5464 0.0226
0.0561
Table 5 Shape parameter of the angular distribution of photoelectrons ejected in the process (vi = 0, Ji) -) (uf, Jf) on the impact of A584 photons J-1
0 1 2 3
P(ui=0sJi + ufxJf)
Jf 0 1 2 3
AJ=rZ
0 1 2 3
Jf
Vf’O
Vf’2
Uf’5
of’s
2.0 1.854 1.893 1900
2.0 1.898 1.926 1.931
2.0 1.935 1.953 1.956
2.0 1.955 1.967 1.970
0.2338
0.2254
0.2153
0.2086
0 1 2 3
AJ=*2
p(Ui =0, Ji +
molecides. ZI
nates in the summation over I in (5). Thus ii for the-. transition I AJI = 2 has a constant value (= i), regardfess of the initial rotationaistate-(see section 5 in 178) The photoelectron intensity for the transition [ ATI > 2 is very weak in the present calculation (paitiy due to the neglect of the higher partial Waves) and is not shown here. In the photoelectron spectroscopy of Hz, rotational spectra were clearly observed for 736 A line [l l-131. &brink [13] reported the spectrum corresponding to the transition Vi= 0, Ji + Uf = 5;Jf. His spectrum indicates large intensities for the. processes(J;, Jf) = (l,l), (O,O),(2,2), (33) (in decreasing order of magnitude) and much less for (1,3), (0,2) and (2,4). The experiment was done at room temperature. Taking into account the thermal distribution of the initial rotational states, we calculate from the numbers in table 4 the relative intensity to be
I(O+O):1(2+2):1(3+3):1(1
Table 6 The same as table 5, but for A736 photons Ji
+ztomic
of; Jf)
Uf’O
Uf’l
Vf’5
2.0 1.864 1.900 1.907
2.0 1.890 1.920 1.925
2.0 1.938 1.955 1.958
0.2069
0.2047
0.2003
Thus we have a relation (see section 5
of 178)
B(uJ+ U’J’) = 4(ur’ -* V’J) . As is shown in 178, fi for the transition Jt = 0 + Jf = 0 has a constant value of 2. Under the present restriction of the partial wave top = 1, the term with I = 1 dorni-
+3):1(0+2):1(2+4)
= 1.0: 0.192: 0.174: 0.130: 0.052 : 0.017: 0.008. The theoretical result, therefore, can explain the trend in the measured spectrum. Niehaus and Ruf [l l] experimentally determined p for the transition Ui= 0, Ji + uf = 1, Jf at the impact of X736 photons. They could not resolve each member (0 + 0, 1 + 1,2 -+ 2) of the Q-branch transition and obtained /3(AJ= 0) = 1.95 f 0.03. This is fairly close to the theoretical one (see table 6). From the measurement of the angular dependence of the intensity ratio of the Q and S branches, they also obtained @J= 2) = 0.85 f 0.14. This is too large compared with the calculated value. This discrepancy could be ascribed to the various assumptions adopted in the present calculation. In particular the restriction of the electron partial wave top = 1 may affect considerably the angular dependence. Another possibility ofthe disagreement might come from the rather incomplete resolution of the experimental photoelectron spectra. If the photoelectrons for AJ= 0 transition contribute partly to the AJ = 2 iine, the apparent value of /3(AJ f 2) is larger than the correct one. Furthermore, as has been mentioned before, an autoionization process might contribute to the ejection of photoelectrons for 736 A photons. No more detailed test of the present calculation
-.
Y. ItikawafPhotoionization
c& be~rhade until more elaborate calculation (espe~cially considering the effect: of higher partial waves) + be performed or-rotational spectra with the higher resolution @ll be obtained for many other wave lengths of incident light.
of diatomic molecules. II
117
F
I @I’2(<‘)dt’ -
n/4
=& + (R/2h)lng
- h[(h2 +R,)/2h2
2 R2/8h43t;-l
+6 ($-2) f constant _
(A-7)
Thus, neglecting the term of the order $-I, we have the asymptotic form of the solution of eq. (A.l) as
Appendix: Asymptotic form of the radial wave function of the ejected electron
f
P4
5-m (t) Cpqh-“2
cos(hE + k-l
In r;+ 5,)
_(A.8)
Consider an asymptotic solution of the equation
h2E2 +Apq - F”;;‘(~) _q2 + fpq@l= 0 * 1 p-1 (E2 - 1)2 (A-l) 1
When ,$+ m, the electrostatic potential of H’; becomes V- -2/RE +6 (g-3) and
(A.3
hence we have
W$$ - -Rf; + O(.$-‘) _
(A-3)
In the asymptotic region, therefore, eq. (A-1) has the form (A.41
@((u=h2+R.$-l
+(h2 +Apq)E-2
_
(A.51
Applying the WKB method [ 141 to eq. (A.4), we get a solution of the form
1. .
f P4 (.$) =c P4 ~-I’4(~)cos &2($)d$’ - 7~14{A 6) [IFrom eq. (AS), the phase factor is calculated to be
[l] Y. Itlkawa,Chem.
Phys. 28 (1978)461.
[2] Y. Itikawa,J. Electron Spectry. 2 (1973) 125. [31 D. Dti, P&s. Rev.A6 (1973) 160. 141M.R. FIannery and U. Opik. Proc. Phys. Sot. 86 (1965) 491. [S] H.C. TuckweIl, I. Phys. B3 (1970) 293. [6] D.R. Bares, K. Ledsham and A.L. Stewart, Phil. Trans. Roy. Sot. A246 (1953) 215. [7] M.R. Flannery, Proc. Phys. Sot. 85 (1965) 1318. [8] C. Flammer, Spheroidal wave functions (Stanford Univ. Press, 1957). [9] Y. Itikawa and H. Hattori, Tables of Prolate.Spheroidal Functions, Contriiutions from the Research Group on Atomsand Molecules, No. 2 (1968). [lo] T.A. Carlson and A.E. Jonas, J. Chem. Phys. 55 (1971) 4913. [ 111 A. Niehaus and M.W. Ruf, Chem. Phys. Letters 11 (1971) [12] %R. Samson, Phys. Rept. 28 (1976) 303. [13] L. Asbrink, Chem. Phys. Letters 7 (1970) 549. [14] H.A. Bethe and R. Jackiw, Intermediate quantum mechanics, 2nd Ed., (Benjamin, New York, 1968)_ch. 1. [15] J.L. Gardner and J.A.R. Samson, J. Electron Spectry. 8 (1976) 123. [16] FM. Chapman Jr., J. Chem. Phys. 63 (1975) 2101.