d e m Energieunterschied der Niveaus ab, ähnlich wie
w i r d nicht die Planck-Verteilung sein)
die G r ö ß e d e r induzierten E m i s s i o n v o n d e r Strah-
so w i r d auch bei den induzierten S t o ß p r o z e s s e n eine
lungsdichte, ihrer Frequenzverteilung u n d d e r A n -
spezielle
regungsenergie d e s Ü b e r g a n g e s a b h ä n g t .
F e r m i - V e r t e i l u n g sein) vorteilhaft sein.
Genauso
w i e f ü r die technische A n w e n d u n g der induzierten
Emission
1
2
eine
spezielle
Strahlungsverteilung
(es
A . EINSTEIN, Berichte der Deutschen Physikalischen Gesellschaft Nr. 13/14, S. 318 [1916].
A . EINSTEIN, Physik. Z . 1 8 , 1 2 1 [ 1 9 1 7 ] .
Elektronenverteilung
Herrn Dr. M.
Hinweise.
8
SCHINDLER
(es
günstig ist,
wird
nicht
die
danken wir für wertvolle
D. J. BLOCHINZEW, Grundlagen der Quantenmechanik, VEB
Deutscher Verlag der Wissenschaften. Berlin 1961, S. 462 ff.
The Dipole Moment Function of H 7 9 Br Molecule
D . N . URQUHART, T . D . CLARK, and B . S. R A O
Department of Physics, University of North Dakota
(Z. Naturforsch. 27 a, 1563—1565 [1972] ; received 2 August 1972)
The radial Schrödinger wave equation with Morse potential function is solved for H79Br molecule. The resulting vibration-rotation eigenfunctions are then used to compute the matrix elements
of (r—r e ) w . These are combined with the experimental values of the electric dipole matrix elements
to calculate the dipole moment coefficients, M 1 and M 2 .
Introduction
m e r can h e o b t a i n e d f r o m the experimentally measured
In the case of the v i b r a t i o n - r o t a t i o n
o c c u r r i n g in a given electronic state of
transitions
a hetero-
M"'v"
= /
V:>,j> ( r ) M(r)
where
Wv">r{r)
r2dr
(1)
electric
d i p o l e m o m e n t f u n c t i o n w h i c h is c o n v e n i e n t l y
ex-
p a n d e d about the e q u i l i b r i u m internuclear distance,
r e , and is written as
M(r)
=M0
+ M1(r-re)
+ M2(r-re)2 + ...
where M0 is the permanent d i p o l e m o m e n t a n d
M2,
(2)
whereas
the
latter
V(r)
which
adequately
represents
the
true potential of the d i a t o m i c m o l e c u l e . It is g i v e n
by
f u n c t i o n s of the u p p e r and l o w e r v i b r a t i o n - r o t a t i o n
is the m o l e c u l a r
strengths
In the present w o r k we have chosen the M o r s e p o tential f u n c t i o n 1
and W v " t j " ( r ) are the r a d i a l eigen-
states respectively. M(r)
line
used to represent the d i a t o m i c m o l e c u l e .
nuclear d i a t o m i c m o l e c u l e , the electric d i p o l e m a trix element M l > ' S j " is g i v e n b y
spectral
d e p e n d o n the t y p e of the potential f u n c t i o n
V\r) = D e [ l - e x p { - / S ( r - r e ) } ]
(4)
2
w h e r e De is the dissociation energy of the m o l e c u l e
expressed in c m - 1 units and ß is a constant t o b e
determined.
T h e radial S c h r ö d i n g e r wave equation f o r a dia t o m i c m o l e c u l e c a n b e written as
Mx,
etc. are the d i p o l e m o m e n t c o e f f i c i e n t s . Substi-
tuting E q . ( 2 ) in E q . ( 1 ) w e get
Ml'^j" = MJ Kr, (r) (r - r e ) 2 V . y " (r) r 2 dr
+ M2 f S ^ V ( r ) ( r - r e ) 2 S V , y r 2 d r + . . . .
(3)
T h e coefficients Mx , M2,
the values of
etc. c a n b e d e t e r m i n e d if
the d i p o l e m a t r i x
elements f o r
the
various vibration-rotation transitions and the eigenfunctions of the states i n v o l v e d are k n o w n . T h e f o r Reprint requests to Dr. B. SESH R A O , Department of
Physics, University of North Dakota, Grand Forks, N.D.
58201, USA.
0
(5)
w h e r e ju is the r e d u c e d mass of the m o l e c u l e a n d
EViJ
is the e n e r g y
of
the vibration-rotation
state
charaterized b y the vibrational q u a n t u m n u m b e r v
and the rotational
quantum n u m b e r J. W h e n
M o r s e potential [ E q . ( 4 ) ] is substituted f o r
the
V(r),
the a b o v e equation c a n b e solved r i g o r o u s l y f o r the
case / = 0 to yield the p u r e vibrational
tios,
rFv.
U s i n g these e i g e n f u n c t i o n s
eigenfuncHEAPS
and
HERZBERG 2 evaluated the two integrals in E q .
(3)
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T h e eigenvalues E were o b t a i n e n d in the f o l l o w -
f o r the case / = 0 in terms of the usual m o l e c u l a r
used their expressions and the ex-
ing w a y . A value of Ev> j was arbitrarily chosen and
perimentally d e t e r m i n e d vibrational matrix elements
the radial w a v e e q u a t i o n w a s s o l v e d outward f r o m
to determine the d i p o l e m o m e n t coefficients Mx
r = 1 0 - 8 a 0 and i n w a r d f r o m r = 7 a 0 up to the equi-
constants. R A O
Ml"
3
and M2 f o r the h y d r o g e n b r o m i d e molecule,
Thus,
in
his calculations,
RAO
3
ignored
H79Br.
the
in-
l i b r i u m internuclear
arithmic
distance, r e .
A t re,
the log-
derivatives f o r these two solutions
were
fluence of rotation o n the d i p o l e m o m e n t f u n c t i o n .
evaluated and c o m p a r e d . T h e initial estimate of EVt j
In this p a p e r we have extended R a o ' s w o r k
was then changed and the c o m p a r i s o n of the loga-
on
H 7 9 B r m o l e c u l e to i n c l u d e this rotational contribu-
rithmic derivatives was repeated. T h i s p r o c e s s was
tion. T o calculate the e i g e n f u n c t i o n s of the v a r i o u s
c o n t i n u e d until the two were equal at r e . T h e value
vibration-rotation states of the H B r m o l e c u l e ,
of Ev> j satisfying this c o n d i t i o n w a s taken as the
the
radial S c h r ö d i n g e r w a v e equation using the M o r s e
a p p r o p r i a t e eigenvalue.
potential f u n c t i o n w a s solved numerically with the
A s a check o n the a c c u r a c y of o u r c o m p u t e r p r o -
aid o f I B M 3 6 0 - 4 0 c o m p u t e r . T h e s e e i g e n f u n c t i o n s
g r a m , w e ran it with the M o r s e parameters f o r HCl
were then used to
in
m o l e c u l e used b y CASHION 6 . T h e vibrational eigen-
E q u a t i o n ( 3 ) . Finally the d i p o l e m o m e n t c o e f f i c i e n t s
values o b t a i n e d f r o m o u r p r o g r a m were then c o m -
were evaluated b y using these integrals and the ex-
p a r e d with those f o u n d b y C a s h i o n w h o used the
perimental values of the matrix elements of the 1 — 0
c o m p u t e r techniques d e v e l o p e d b y C o o l e y a n d Nu-
and
m e r o v to solve the radial w a v e equation. T h e s e are
2—0
calculate the t w o
vibration-rotation
bands
integrals
of
the
H79Br
molecule.
s h o w n in T a b l e 1. T h e excellent agreement between
the two sets of the eigenvalues gives us the confidence
Calculations
A l l the m o l e c u l a r constants f o r the H B r m o l e c u l e
Table 1. The vibrational eigenvalues calculated using Cashion's program and our program for HCl molecule.
used in the present calculations were obtained f r o m
the data of RANK et al. 4 .
In o r d e r to find an initial estimate of De
and ß
to b e used in the calculations, the vibrational term
values G(v)
Ev,
Cashion's
V
A) Morse Parameters, De and ß
were first calculated f r o m the m o l e c u l a r
constants f o r the vibrational states with v = 0 , 1, 2 ,
in the m e t h o d s w e have used.
0
in Rydbergs
0.013491005
0.039651547
0.064716808
0.088686793
0.111561503
0.133340931
0
1
2
3
4
5
Ours
Difference
0.013490990
0.039651519
0.064716808
0.088686793
0.111561503
0.133340940
1.5 x IO" 8
2.8 x IO" 8
0
0
0
0.9 x IO" 8
and 3. T h e s e term values are related to M o r s e paraF r o m the eigenvalues calculated using the initial
meters t h r o u g h the e x p r e s s i o n
M o r s e parameters Dc
De2h
G(v)=ß]/1
\ 2 TI1 c
/U
(v +
8 7tz c
/U
(v
+
v
h2.
(6)
T h e values of De and ß were then obtained b y using
a least squares fit to the a b o v e equation.
and ß, the vibration-rotation
transition f r e q u e n c i e s were f o u n d . These were then
c o m p a r e d to the e x p e r i m e n t a l l y measured transition
f r e q u e n c i e s . T h e parameters w e r e then systematically
v a r i e d until a g o o d agreement between the calculated and
experimental
transition
frequencies
were
found.
B) Eigenvalues and Eigenfunctions
T h e e i g e n f u n c t i o n s c o r r e s p o n d i n g to these eigenequation
values
were
n u m e r i c a l l y , we used a c o m p u t e r p r o g r a m similar
chosen
was such that the e i g e n f u n c t i o n s were
to that used b y LouCKS 5 . This p r o g r a m uses the
m a d e p o s i t i v e f o r the increasing values of the inter-
Runge-Kutta and M i l n e techniques. In the p r o g r a m ,
nuclear distance, r. T h e e i g e n f u n c t i o n s were then
a g r i d of 3 8 8 points was chosen f o r the internuclear
n o r m a l i z e d and checked f o r the o r t h o g o n a l i t y
distance r within a r a n g e of 1 X 1 0 - 8 a 0
dition and the values of the o r t h o g o n a l i t y integrals
To
solve the radial Schrödinger w a v e
where a 0 is the B o h r unit of length.
r <i 7 a 0
then
obtained.
The
sign
were f o u n d to b e less than 1 0 ~ ° .
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convention
all
con-
Results and Discussion
values of the integrals in E q .
( 3 ) to calculate the
values of the d i p o l e m o m e n t coefficients, M x
T h e values of the M o r s e parameters De
and
ß
and
M2 . T h e y are f o u n d to b e Mx = 0 . 3 1 5 D e b y e / Ä and
were f o u n d to be 0 . 3 5 5 9 3 7 R y d b e r g s and 0 . 8 6 1 5 4 9
Mo = 0 . 5 7 5
a 0 _ 1 respectively. Using these values w e have eval-
the least squares m e t h o d and is confined to quadra-
D e b y e / Ä 2 . T h e a b o v e calculation
uses
uated the eigenvalues and the eigenfunctions of the
tic fit o n l y . W e have tried and rejected the c u b i c fit
first eleven states in the 1 — 0 and 2 — 0 vibration-
since it g a v e a larger standard error. T a k i n g M0 =
rotation b a n d s . F o r the 1 — 0 band, the sign o f the
0 . 7 8 8 D e b y e f o r the p e r m a n e n t dipole m o m e n t as
values of the first integral in E q . ( 3 ) turned out to
listed b y WESSON 9 , the d i p o l e m o m e n t f u n c t i o n of
b e negative. T h i s m a d e us choose a negative sign
the H 7 9 B r m o l e c u l e can b e expressed b y the f o l l o w -
f o r the value of MQJ"
ing e q u a t i o n :
cient Mt
in order to keep the c o e f f i -
to b e positive. T o b e consistent with the
rotational distribution of the d i p o l e matrix elements
M(r)
and a positive sign f o r
M o j " . T h e values of the d i p o l e matrix elements, o b -
(7)
T h e values obtained b y RAO 3 , w h o i g n o r e d the
of b o t h the 1 — 0 and 2 — 0 bands, we had to choose
a negative sign f o r MQJ"
= 0 . 7 8 8 + 0 . 3 1 5 (r - r e ) + 0 . 5 7 5 (r - r e ) 2 .
rotational
contribution
Mx = 0 . 3 1 6 D e b y e / Ä
in
and
his
calculations,
are
M2 = 0 . 5 1 8 D e b y e / Ä 2 .
tained f r o m the papers published b y RAO and his
These are very close to the values presented in this
students 7 ' 8 , were then c o m b i n e d with the c o m p u t e d
paper.
1
2
3
4
5
P. M. MORSE, Phys. Rev. 34, 57 [1929].
S. HEAPS and G. HERZBERG, Z. Phys. 133, 48 [1952].
H.
6
7
B. S. RAO, J. Phys. B 4, 791 [1971].
D. H. R A N K , U . F I N K , and T. A. W I G G I N S , J . Mol. Spectry.
18, 170 [1965],
T. LOUCKS, Augmented Plane Wave Method, W. A. Benjamin, Inc., New York 1967, Chapter 3.
8
9
J. K. CASHION, J. Chem. Phys. 39, 1872 [1963].
B. S. R A O and L . H. L I N D Q U I S T , Can. J. Phys. 46, 2739
[1968].
B. P. GUSTAFSON and B. S. R A O , Can. J. Phys. 48, 330
[1970].
L. G. WESSON, Tables of Electric Dipole Moments, M.I.T.
Press, Cambridge, Mass., 1948, p. 3.
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