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Angular correlation functions of spherical-top carbonyl complexes in gaseous N2 influenced by coriolis coupling
15
CHEhIlCAL PHYSICS LETTERS
Volume 23, number 4
December 1973
ANGULAR CORRELATlON FUNCTIONS OF SPHERLCAL-TOP CARBONYL COMPLEXES 1N GASEOUS N, INFLUENCED BY CORlOLlS COUPLLNG K. MijLLER, Solid Srore Physics
Laboratory.
P. ETIQUE
F. KNEUBtiHL
and
ETH Hhggerberg,
Received
17 Scpkmber
CH-8049
Zurich, Swirrerhtld
1973
The Fourier transform of measured line shapes of the triply degencratc vibrations of carbonyl complexes in gaseous N2 shows srrong devialions from rhe angular correlation functions calcuhted by Steele for the rigid sphericzdtop molecules. These deviations are explained theoretically with rhe aid of Coriolis coupling and vibrational relaxation. For molecules in the gas phase it is possible LO separate the Coriolis coupling from the vibrational relaxation.
Spherical-top carbonyls of heavy metals, e.g., Cr(CO), , Mo(CO), , W(CO), , Ni(CO), , show Coriolis coupling for the triply degenerate IR-active vibrations [l ,2] . This is connected with the narrowing of some specific vibration-rotation bands and drastic changes of the corresponding angular correlation functions. Therefore we have measured in detail the band shapes of the vibrations v5 of Ni(CO), and ve, v7, us of the three other carbonyls mentioned above, all dissolved in gaseous
N2 at room
temperature
and 1 atm.
Sub-
sequently we determined the autocorrelation functions (ACF’S) of the transition dipole moments [3, 41 . The observed band shapes differ considerably from those calculated by Steele [5] for the rigid spherical top. The deviations are due to Coriolis coupling and vibrational relaxation. Including Coriolis coupling and negiecting higher order terms, tie write the hamiltonian of a triply degenerate vibration of a spherical-top molecule H =I&
+ (J -
The Coriolis coupling is represented I is defined in the molecule-fixed tem. Its equation of motion is aK(t)jat
by the last term. coordinate sys-
= N(f).
indicates the internal torque caused by the Coriolis coupling. The potential errergy and the torque N(t) of w(t) relative to J are determined by <:
N
E
pot = -_(
N(c) = -(t/o
I
[J x K(t)] .
By a change of the frame of reference to a space-. fixed system, the equation of motion is transformed into: dw(t)/dr = aK(r)/& + [o X r(r)] ,
where the vector o denotes the angular frequency of the rotating molecule. Since the molecule is sphericd, o corresponds to J/I. Thus the change of r(t) can be described by
w.)~/Z,
dK(r)/dt = [(1 - <)/I] [J x K(f)].
with the eigenvalues
Before we discuss the solutions of this equation, we pay attention to the ACF of the transition dipole moment m, which can be calculated from the Fourier
+ (fi*/U)[J(Jtl) + K(K+l)] - (&(J-K), where J = total angular momentum, II = vibrational angular momentum, and < = Coriolis coupling constant.
.:
.,, .-;. : .,,
‘,‘, ..., : -. .’ :’
band shape. :
,,.. ,,489,: .‘: .’ : ., .. . ‘. .:.., ,. ,, :: :: .: .,‘,... : ., . ._,;. : : ‘. 1~.:_ ,‘:-. .:: .’ : ,. : -, -.< : _. .. ‘. ..L: ._,. .‘.) : .: _. ._ .‘;:,_, ,.,. :/ .‘_.:.,_..I: ,.,.,,,. ,_-. .,-_ :.. ,,.: :.1:.‘:.,:,_,...; .,- ‘., I i,. ‘.I-::_.‘j.;. .: .,_.1:.:: 1 .._ ._‘..J’._,.; .‘I: : I i’. .:: :; :.,, :.1_ ..- _:: :,,_,, _,.,,.; .t:,.. : L_Fy,:.:. .-:..,;;“’ ,,.
:,
transform of the measured absorption This ACF is defmed by:
^ _:
..,
,‘,
:
:
Volume 23, number 4
15 December 1973
CHEMlCAL PHYSICS LETTERS t G*‘(tl
Fig. 1. Influence off on the rotational rei&tion
function Grot(l) of molecules in the gas phase,
.-' :
Fig. 2. Vibrational r’&xatiq
functions G V’kj of the vibration .. v8 for: (1) C&O)&
.-4q: _-
: _’ .,.” ,- .: : ., ,: -. ,;.. . . .,.‘. : ., :. 2.. 2 .., . . _. _: ;:, .. ._- .,‘-. .: -_’ : :i.
(2),MokO)&, (3) W(CXX6 in gaseous Nz.
‘.
_
,‘.
,: _: i, -:
(1,.
., : ,’ ,,: :.
.,” ‘. ‘,
‘,
‘.
_’ .’ .. .;‘-,, ‘. ,‘,‘,:.. 1,. .__. ., --.: . . . . ,,.: _:...: .- ., , ‘. .I _;, ,, ._ ,._ ‘..._, ‘...’ .-,.y,.._“ .~,‘. ., .,’ ;_ _’ -,,, ” ..,, ‘i’
Volume 23, number 4
hr(f)*m(O)/
[m(O)]
CHEMICAL PHYSICS LETTERS
*>
obtained
= c[m(o)??2(r)lm(o)~~(o)lu(O)~u(t)~, where U(C) indicates the direction of the transition dipole and m(t) its magnitude. u(t) describes the rotation of the transition dipole and m(f) denotes its vibrational relaxation. Assuming weak coupling between the vibrational and the rotational-translational modes of the molecules, Bratos [6] and Morawitz [7] fmd a separation of the above ACF into a vibrational and a rotational relaxation function:
(m(O)-m(r)/m(O)m(O)) = ~nz(O)m(t)/m(O)m(O)~~u(O)~u(t)~
= G’*((r)Gro’(r).
We now assume the rotational relaxation of the tnnsition dipole m(r) to correspond to the relaxation of the vibrational angular momentum K(T): G”‘(r)
= (K(O)-K(T)/K(O)K(O)).
the equation of reference. It can be solved exactly for: J(t) = J = constant.
We wish to thank Dr. B. Keller for valuable discussions.
References
tions of K and the Boltzmann distribution of the Ss, we find the following ACF, also shown in fig. 1:
For I= 0, this rotational
[2] [3] [4]
By taking the average over all possible initial direc-
-
<)*f*]
- 5)*r*]. relaxation
equals the ACF
top.
model by Kubo [8] and Anderson [9] based on a stochastic variation of the vibration frequency assuming a Gauss-Gauss process.
[I] L.H. Jones, R.S. McDowell and M. Goldbbtt.
to the problem of the solution of of motion of K(t) in a space-fixed frame
X exp[-(k7’/2f)(l
[5] for the rigid spherical
Experimentally we observe the product G’O’(r)Gqf). S’rnce we are able to calculate Grol(t) at least for dilute gases, the experiment allows us to determine Cd(r). As an example we show in fig. 2 the functions Gvrr’((t) of the vibration q for Cr(CO),, MOM and W(CO)6 with < = 0.3,028, and 0.18, respectively. All Gti((t) can be approximated by a gaussian function. This procedure is justified by a theoretical
We now return
G;pt(r) =+ + f [I - (kT/l)(l
by Steele
15 December 1973
[S ] [6] 17) [8 [9]
]
Inorg.
Chcm. 8 (1969) 2349. L.H. Jones, R.S. fvlcDowell and hl. Goldblatt, J. Chcm. Phys. 48 (1968) 2663. R.G. Gordon, in: Advances in magnetic resortancz, Vol. 3, cd. J.S.Waugh (Academic Press, New York, 1968). B. Keller and F. Kneubiihl, Helv. Phys. Acta 45 (1972) 1127. W.A. Steele, J. Chcm. Phys. 38 (1963) 2411. S. Bratos. J. Rios and Y. Guisuni, J. Chem. Phys. 52 (1970) 439. H. hlorawitz and K.B. Eianthal, J. Chem. Phys. 55 (1971) 887. R. Kubo, in: Stochastic processes in chemical physics, ed. K.E. Shuler (Interscience, New York, 1969). P.W. Anderson, J. Phys. Sot. Japan 9 (1954) 316.
CHEhIlCAL PHYSICS LETTERS
Volume 23, number 4
December 1973
ANGULAR CORRELATlON FUNCTIONS OF SPHERLCAL-TOP CARBONYL COMPLEXES 1N GASEOUS N, INFLUENCED BY CORlOLlS COUPLLNG K. MijLLER, Solid Srore Physics
Laboratory.
P. ETIQUE
F. KNEUBtiHL
and
ETH Hhggerberg,
Received
17 Scpkmber
CH-8049
Zurich, Swirrerhtld
1973
The Fourier transform of measured line shapes of the triply degencratc vibrations of carbonyl complexes in gaseous N2 shows srrong devialions from rhe angular correlation functions calcuhted by Steele for the rigid sphericzdtop molecules. These deviations are explained theoretically with rhe aid of Coriolis coupling and vibrational relaxation. For molecules in the gas phase it is possible LO separate the Coriolis coupling from the vibrational relaxation.
Spherical-top carbonyls of heavy metals, e.g., Cr(CO), , Mo(CO), , W(CO), , Ni(CO), , show Coriolis coupling for the triply degenerate IR-active vibrations [l ,2] . This is connected with the narrowing of some specific vibration-rotation bands and drastic changes of the corresponding angular correlation functions. Therefore we have measured in detail the band shapes of the vibrations v5 of Ni(CO), and ve, v7, us of the three other carbonyls mentioned above, all dissolved in gaseous
N2 at room
temperature
and 1 atm.
Sub-
sequently we determined the autocorrelation functions (ACF’S) of the transition dipole moments [3, 41 . The observed band shapes differ considerably from those calculated by Steele [5] for the rigid spherical top. The deviations are due to Coriolis coupling and vibrational relaxation. Including Coriolis coupling and negiecting higher order terms, tie write the hamiltonian of a triply degenerate vibration of a spherical-top molecule H =I&
+ (J -
The Coriolis coupling is represented I is defined in the molecule-fixed tem. Its equation of motion is aK(t)jat
by the last term. coordinate sys-
= N(f).
indicates the internal torque caused by the Coriolis coupling. The potential errergy and the torque N(t) of w(t) relative to J are determined by <:
N
E
pot = -_(
N(c) = -(t/o
I
[J x K(t)] .
By a change of the frame of reference to a space-. fixed system, the equation of motion is transformed into: dw(t)/dr = aK(r)/& + [o X r(r)] ,
where the vector o denotes the angular frequency of the rotating molecule. Since the molecule is sphericd, o corresponds to J/I. Thus the change of r(t) can be described by
w.)~/Z,
dK(r)/dt = [(1 - <)/I] [J x K(f)].
with the eigenvalues
Before we discuss the solutions of this equation, we pay attention to the ACF of the transition dipole moment m, which can be calculated from the Fourier
+ (fi*/U)[J(Jtl) + K(K+l)] - (&(J-K), where J = total angular momentum, II = vibrational angular momentum, and < = Coriolis coupling constant.
.:
.,, .-;. : .,,
‘,‘, ..., : -. .’ :’
band shape. :
,,.. ,,489,: .‘: .’ : ., .. . ‘. .:.., ,. ,, :: :: .: .,‘,... : ., . ._,;. : : ‘. 1~.:_ ,‘:-. .:: .’ : ,. : -, -.< : _. .. ‘. ..L: ._,. .‘.) : .: _. ._ .‘;:,_, ,.,. :/ .‘_.:.,_..I: ,.,.,,,. ,_-. .,-_ :.. ,,.: :.1:.‘:.,:,_,...; .,- ‘., I i,. ‘.I-::_.‘j.;. .: .,_.1:.:: 1 .._ ._‘..J’._,.; .‘I: : I i’. .:: :; :.,, :.1_ ..- _:: :,,_,, _,.,,.; .t:,.. : L_Fy,:.:. .-:..,;;“’ ,,.
:,
transform of the measured absorption This ACF is defmed by:
^ _:
..,
,‘,
:
:
Volume 23, number 4
15 December 1973
CHEMlCAL PHYSICS LETTERS t G*‘(tl
Fig. 1. Influence off on the rotational rei&tion
function Grot(l) of molecules in the gas phase,
.-' :
Fig. 2. Vibrational r’&xatiq
functions G V’kj of the vibration .. v8 for: (1) C&O)&
.-4q: _-
: _’ .,.” ,- .: : ., ,: -. ,;.. . . .,.‘. : ., :. 2.. 2 .., . . _. _: ;:, .. ._- .,‘-. .: -_’ : :i.
(2),MokO)&, (3) W(CXX6 in gaseous Nz.
‘.
_
,‘.
,: _: i, -:
(1,.
., : ,’ ,,: :.
.,” ‘. ‘,
‘,
‘.
_’ .’ .. .;‘-,, ‘. ,‘,‘,:.. 1,. .__. ., --.: . . . . ,,.: _:...: .- ., , ‘. .I _;, ,, ._ ,._ ‘..._, ‘...’ .-,.y,.._“ .~,‘. ., .,’ ;_ _’ -,,, ” ..,, ‘i’
Volume 23, number 4
hr(f)*m(O)/
[m(O)]
CHEMICAL PHYSICS LETTERS
*>
obtained
= c[m(o)??2(r)lm(o)~~(o)lu(O)~u(t)~, where U(C) indicates the direction of the transition dipole and m(t) its magnitude. u(t) describes the rotation of the transition dipole and m(f) denotes its vibrational relaxation. Assuming weak coupling between the vibrational and the rotational-translational modes of the molecules, Bratos [6] and Morawitz [7] fmd a separation of the above ACF into a vibrational and a rotational relaxation function:
(m(O)-m(r)/m(O)m(O)) = ~nz(O)m(t)/m(O)m(O)~~u(O)~u(t)~
= G’*((r)Gro’(r).
We now assume the rotational relaxation of the tnnsition dipole m(r) to correspond to the relaxation of the vibrational angular momentum K(T): G”‘(r)
= (K(O)-K(T)/K(O)K(O)).
the equation of reference. It can be solved exactly for: J(t) = J = constant.
We wish to thank Dr. B. Keller for valuable discussions.
References
tions of K and the Boltzmann distribution of the Ss, we find the following ACF, also shown in fig. 1:
For I= 0, this rotational
[2] [3] [4]
By taking the average over all possible initial direc-
-
<)*f*]
- 5)*r*]. relaxation
equals the ACF
top.
model by Kubo [8] and Anderson [9] based on a stochastic variation of the vibration frequency assuming a Gauss-Gauss process.
[I] L.H. Jones, R.S. McDowell and M. Goldbbtt.
to the problem of the solution of of motion of K(t) in a space-fixed frame
X exp[-(k7’/2f)(l
[5] for the rigid spherical
Experimentally we observe the product G’O’(r)Gqf). S’rnce we are able to calculate Grol(t) at least for dilute gases, the experiment allows us to determine Cd(r). As an example we show in fig. 2 the functions Gvrr’((t) of the vibration q for Cr(CO),, MOM and W(CO)6 with < = 0.3,028, and 0.18, respectively. All Gti((t) can be approximated by a gaussian function. This procedure is justified by a theoretical
We now return
G;pt(r) =+ + f [I - (kT/l)(l
by Steele
15 December 1973
[S ] [6] 17) [8 [9]
]
Inorg.
Chcm. 8 (1969) 2349. L.H. Jones, R.S. fvlcDowell and hl. Goldblatt, J. Chcm. Phys. 48 (1968) 2663. R.G. Gordon, in: Advances in magnetic resortancz, Vol. 3, cd. J.S.Waugh (Academic Press, New York, 1968). B. Keller and F. Kneubiihl, Helv. Phys. Acta 45 (1972) 1127. W.A. Steele, J. Chcm. Phys. 38 (1963) 2411. S. Bratos. J. Rios and Y. Guisuni, J. Chem. Phys. 52 (1970) 439. H. hlorawitz and K.B. Eianthal, J. Chem. Phys. 55 (1971) 887. R. Kubo, in: Stochastic processes in chemical physics, ed. K.E. Shuler (Interscience, New York, 1969). P.W. Anderson, J. Phys. Sot. Japan 9 (1954) 316.