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Vibrational frequency shifts in IR absorption and emission spectra of liquid carbon monoxide. Monte-Carlo simulation
Journal of Molecular Liquids, 46 (1990) 129-139 Elsevier Science Publishers B.V., Amsterdam
129
VIBRATIONAL FREQUENCY SHIFTS IN IR ABSORPTION AND EMISSION SPECTRA OF LIQUID CARBON MONOXIDE. MONTE-CARLO SIMULATION"
S. KH. AKOPYAN and S.I. LUKYANOV Institute of Chemistry, Leningrad University,
199034, Leningrad (USSR)
(Received 28 November 1989) SUMMARY
Solvent induced frequency shifts in the IR absorption and emission spectra were simulated by Monte-Carlo procedure. The pair potential with repulsive, dispersive and electrostatic parts was used to study liquid carbon monoxide at T = 80K and p = 0.7982 g/om 3. It was shown that if single molecule of the system undergoes vibrational transition, the "red" frequency shift in emission spectrum is greater than the one in the absorption spectrum. The computer simulation results predict that if all molecules of the system are excited, the frequency shift of the emitting molecule turns "blue". This means frequency shift in the IR emission spectrum depends on the pumping intensity. The increasin E number of excited molecules in the system leads to the increase of the influence of repulsive intermolecular interactions on frequency shift.
INTRODUCTION Computer
simulation
spectroscopic, systems,
methods
thermodynamic
which
the liquid
(refs.
1-18). The molecular dynamics method is used in for this purpose, (refs.
spectrum
range
The influence interactions
of of
was
investigated
temperature
and
the components
upon spectroscopic,
Luck
the Monte-Carlo
In particular,
method
the behaviour
frequency in the carbon monoxide
by means density
a pair
thermodynamic
liquid CO, was considered.
"Dedicated $o Professor W.A.P.
liquid of
whereas
12-18).
of the shift of the fundamental vibrational IR absorption
characteristics
simultaneous
spectra of
was applied only in a few papers
the wide
determine of
most of the investigations
structural
to
have been recently applied for interpreting vibrational
liquids and solutions
and
enable
of Monte-Carlo variations
potential
of
and structural
method
(refs.
in
16-18).
intermolecular parameters
of
130 As
It can be observed from the results of experimental
investigation
of
liquid carbon monoxide vibrational relaxation rates (ref. 19), the lifetime of the exited exceeds
vibrational
the relaxation
state time
of
the molecules
of
the liquid.
forming
This
it,
means
substantially
that
equilibrium
conflguratlon of liquid molecules has time to realize around excited molecules while molecules are in the excited vibrational state. That is why it may seem interesting
to
estimate
the vibrational
frequency
shifts
in
the emission
spectrum of liquid carbon monoxide and to compare them with the corresponding values
of
the absorption
spectrum.
information about peculiarities and, particularly,
Considering
this,
of intermolecular
we
can get
interactions
additional
in the system
information about the analytical form of pair potentials,
the potential parameters, etc. The fundamental carbon
monoxide,
band,
observed
responses
to
in
the IR
absorption
the transition
of
spectrum
molecules
of
from
liquid
a ground
equilibrium level to an excited unequlllbrlum vlbratlonal level (see Fig. 1).
Q'Lt~
liquid
gas
liquid
Fig. I. Scheme of vlbratlonal levels of molecules in gas and liquid phases. In
accordance
an excited
with
vlbratlonal
the above-made state
of
the transition from an equilibrium
remark
concerning
CO
molecules
excited
level
in
the duration the liquid
to an unequlllbrlum
of
phase, ground
vibrational level will realize in the case of the emission spectrum of llquld carbon
monoxide.
Thus,
according
to
the scheme
in Fig.
1,
the vibrational
frequency shift in the emlsslon spectrum of liquid carbon monoxide should be greater, comparing with the frequency shift in the absorption spectrum. This paper shows the results of the simulation by the Monte-Carlo method of the vibrational frequency shift in the IR absorption and emission spectra of liquid cases
carbon occur:
monoxide 1) one of
at
such
pumping
the conslderated
intensities molecules
when
the following
is always
vibrational state; 2) all molecules of the system are excited.
two
in an excited
131 RESULTS AND DISCUSSION The Schr~kilnger equation for
a liquid consisting
of dlatomlc
molecules
reads:
Ho(Q) + T(X) + V(Q,X)} Cn(Q,X) = En@(Q,X)
(1)
where: N
(2)
Ho(Q) = Z h°l (ql) I=1
Is the sum of hamlltonlans of free anharmonlc osclllators by which dlatomlc molecules are approxlmated, T(X) is the kinetic energy of translational and rotational movements, interactions,
Q
is
V(Q,X)
is the potential energy of the intermolecular
the vibrational
coordinate,
X
is
the rotational
and
translational coordinates, M is the number of molecules in the system. Using the Born-0ppenhelmer approximation:
~(Q,x) = ~ ( Q , x ) ~ ( x )
(3)
one can r e p l a c e e q u a t i o n (1) by the two e q u a t i o n s :
~
oca) + vca, x)],,ca, x) = EvCX),vCa, X)
C4)
[T(X' + Ev(X)]~.CX' = Ev=~ (X'
(5)
Equation C4) describes the vibrational movement of molecules in the field of
intermolecular
forces,
equation
(5)
describes
their
rotational
and
translational movements. If
rotational
and
translational
movements
are
regarded
processes, the probability density of the system beln E in X exp [-i/KT)Ev (X])]
~
v
-(1ZKT)Ev(X 1) dx
J
as
classical
co~IEuratlon is:
132
P(Xj)dX
is
the probability
the configuration According
of
the system
space of classical
in
X]-dX,
Xj+dX
region
of
coordinates.
to the first approximation
of the M molecules
being
of the perturbation
theory,
the energy
in the quantum state V I V2...V M is:
M
Evcx ~---E v
v cx~ : ~ E° ÷ <~°cQ~nvcQ, x,~i~°cQ~ >
1 2
M
c~
VI
1=1
Here: M
0
#~(Q) = U ~v Cql) 1=1
(s)
t
E° o v and @v(ql) are eigenvalue and ! i anharmonic oscillator hol(ql). Usin E
eqn.(7),
vibrational vibrational
one
levels
and,
frequency
interactions.
can
eigenfunction
estimate in
a hamiltonian
the stabilization
this
way,
of molecules
under
In particular,
of
one
can
determine
the influence
N
a(e)
<¢1(qk)~ °
%Cq,.~lvcQ, x J
S
o(
1=1
t ~k
i ~:k
M
of
a free
combining
the shifts
of
the intermolecular frequency shift in
is excited,
)l~,~cq?n % %
i=l
M
of
in the case of the vibrational
IR absorption and emission spectra when one molecule
Au(X~ ¢e)) = h - "
energy
of
we obtain:
)>_
(9)
)
~,,~) Ill ¢0(qi o )> ¢o°( q~)IvcQ'x~ i=l 1=1
<11
where
a and
e
indexes
correspond
k is the number of a molecule, When
all
molecules
of
to absorption
and
emission,
respectively,
for which optical transition occurs.
the system
shift in an emission spectrum is:
are
excited,
the vibrational
frequency
133
M
: h
e
M
o
% IvcQ, x In,
-
(IO)
%
o
"
"
1=1
1=1
i q~k
1 ~k
The result observed
of the averaging
in a spectroscopic
I
/
)
over all configuration
experiment
states,
the shift
i.e.
is:
Au = IAv(X)P(X)dX
(11)
v
Integral
(eqn.
Metropolis
a NVT-ensemble. cell.
II)
algorithm
performed.
Molecules
of
used,
by
the Monte-Carlo
the simulation
the simulated
between
potentials
depending
potentials
were
16, 17).
to describe
on
collisions.
dipole-dipole
dispersive
coefficient
a semi-empirlcal has
into a Monte-Carlo
in cubic
boundary
conditions
were
simulated
coordinates
of
potential
by
pair
molecules. (potential
additive
Two I)
types
of
exp
-6
is
of reactions The attractive
interactions
of transfer of vibrational part
of
in the systems.
the potential The value
of
energy
describes isotropic
C is taken from (ref. 21), where it was obtained in 6 The anisotropic dispersive coefficient C (2), for CO 6 estimated in our work as well as in (ref. 22), usin E
way.
been
of
an isoelectronic mechanical
out
in the cubic cell was 61. were
type
Standard
Its repulsive part was taken from (ref. 20), where it was used
dispersive
the values
The first
method. carried
the number of molecules and
Periodic
the molecules
the rate constants
molecules
molecule,
state.
on vibrational
used.
was
of the system were placed
The total number of molecules
Interactions
of
computed
was
The cell size was chosen in accordance with
the density
(refs.
was
isotropic molecule
calculations.
and
anisotropic
N 2 obtained
in
dispersive
(ref. 23),
on
coefficients the basis
for
of quantum
134
~z = A-exp(-~R)x[e
11 1 1 + e-~e:21clT1
e -~8¢zlc2":z
+ e
m m m +m c21=E:22 m +m ; c o c o and p o l a r i z a b i l i t y a n t s o t r o p y o f m o l e c u l e s , o
-
where Cl=COSb~I, and ~ are
(O¢/Oq)
C2=COS(d2, eft--S12"
polarizability
and
(8~/8q)
o
(12)
are
c
-
,
their
_-
derivatives,
m
with
respect
to
vibrational
o
c o o r d i n a t e s . D i s t a n c e s and a n g l e s a r e g i v e n i n Fig. 2a.
×
\ Flg. 2. Coordinates determining molecular orlentatlon.
The second type potential describing electrostatic
interactions
a dipole
moment
studied,
both
mechanical
(potential II), was observed by adding the terms
dlpole-dlpole,
of
dlpole-quadruple in
the system
CO molecules
experimentally
calculations
and,
upon
and
to
and
intermolecular
theoretically,
at
present,
quadruple-quadruple
potential
on
the first
I.
Dependence
distance the basis four
was of
of
actively quantum
coefficients
of
the series are known (refs. 24, 25)
~(T)
= ~0+~I(T--Te)+~2(T--Te)2+~3(T--Te
)3
(13)
135 Vibrational
dependence of a quadruple moment of CO molecules
(14)
QCT] = Qo+Q, CT-T]+Q2(T-T] 2
was determined
only In quantum mechanical
-Px Px -Py Py J
[zp. Pz
r_
(ref. 26).
(15)
R3
f^ (2)-,1)- (2)_,1)3 (2)[^(
_ I,.")IIQ")I
calculations
(1)~ 2
.1
,,,^,2,3 ,,,[_r,,,~21
.,_.,_
, [^lr ,,n2r ,:.n 2 r ,,n2r ~2,~21,._r ,,,'~'r ,.,~2 _rr ,,,~2 r ,.,~21
+ ~ l"[t"x J t"x J +t", J L", J J ~'t"-Jr"-J-"t
L'- J÷t"-JJ
,1}}
Angles are glven In Flg. 2b.
Table 1 shows the values of potentlal parameters and standard deviations Uslng agreement
the above (within
experlmental
of their determlnatlon
values
of
parameters,
the llmlt of 0.6~),
values of internal energy
As the results
of calculatlons,
vary wlthln
between
can
obtain
the results
a
rather
of
accurate
simulation
(Table 2), by means of potentlal
in the case of potential
energy of the system can be descrlbed parameters
one
taken over from literature
(the first column).
II,
and
I.
the internal
accurately enough only if the values of
the llmlts of accuracy
of their determination.
We did
not select parameter values in detall and stopped at those values of potential parameters
at which the defference between simulation and experimental
does not exceed 1,6~ (the third column in Table I).
results
136
TABLE 1 Values of parameters in pair potentials. Parameter
Parameter value
A(kJ/mole) ~(A) 6 C (kJ.A /mole) 6
Potential I
49889,67 3,6 5093,93 t 556,67
C (2) (kJ.A6/mole) 6 (A3)
488,6
± 53,4
(ref. 20) (ref. 20) (ref. 21)
49889,67 3,6 5093,93
(refs. 21, 22)
488,6
Potential II 49889,67 3,6 4804,50 460,8
1,94 + 1,98
Cref. 27)
1.94
1,94
0,53
Cref.
0,53
0,53
(a~/aq) o (A2)
1,55 ± 0,08
(ref. 27)
1,50
1,63
(8~/8q) ° CA2)
2,82 ± 0,31
(ref. 27)
2,82
2,96
(A 3 )
#o(D)
27)
-0,1126 ± 0,002 [refs. 24, 25)
-0,113
#t(D/A)
3,11 ± 0,15
(refs. 24, 25)
2,96
#2(D/A2)
-0,15 t 0,28
(refs. 24, 25)
-0,15
#3(D/A3)
-2,36 ± 0,8 (refs. 24, 25) -2,241 -2,50 (ref. 26) -1,93 ± 0,04 0,935 (ref. 26)
-2,36
Qo(B) QI(B/A ) Q2(B/A2)
TABLE 2 Calculated
]
1,814
and experimental
internal
-1,90 0,94
(ref. 26)
energy
1,81
values
(in kJ/mole)
of
liquid
carbon monoxide. Calculated values
Experimental value
Potential I
Potential II
-5,48 ± 0,05
-5,53 + 0,04
The potentials
described
-5,45
above
were
used
for
the simulation
of
a vibrational frequency shift in IR absorption and emission spectra of liquid carbon
monoxide
at
T=80K
and
p=0,7982
g/cm 3.
In averaging
according the vibrational states of interacting molecules,
the potentials
the internal energy
of molecules was described by Morse potential. Considering
the absorption spectra,
the interaction energy of the shifted
k-th molecule with other molecules was calculated for the two cases:
137 I) v =u =...=v =0 1 0 H
v2=O...V =I...vM=O
2) v1=O,
Energy value, for the first case, both energy values means
of
eqn.
are used
(9).
For
is used to generate the Markov chain, and
to calculate
the emission
the vibrational spectra,
frequency
the interaction
shift by energy
is
calculated for the following cases:
v2=O...VK=I...vM=O
1) v1=O,
2) v =v =...=v =0 (one molecule is excited),
1
2
and
N
1) v =v =...=v =1 12 M 2)
vi=1, v2=l...vK=O...vx=l(all Determining
the vibrational
molecules are excited).
frequency shifts in IR absorption and emission
in the system with electrostatic potential values, As
I into
the frequency
interactions, shifts
by
we derived the contribution of
calculating
the interaction
energy
using both potentials. it
can
be
sufficiently absorption
seen
reliable and
from
Table
difference
emission
1,
using
potential
in vibrational
spectra
when
one
1,
one
frequency
molecule
cannot
shifts
is
observe
in
the IR
excited,
because
the mean-square error is great enough because of a comparatively small number of generated chains (from 6 to 12). Inclusion
of
components,
the system,
into
vibrational
frequency
greater.
This
characterizing
the potential
result
shifts may
be
results for
electrostatic
in
the fact
absorption
accounted
for
and by
that
emission
the fact
interactions
in
the difference
in
spectra
that
becomes
electrostatic
interactions depend essentially on molecular orientation. Consideration molecules
results
the vibrational -1 +3 cm Table excited
of the data given
4 gives
in greater frequency
the values
vibrational
the absorption
and
levels
in Table
changes
shift
3,
emission
that
in the properties
changes
the sign
of stabilization of
shows
CO molecules spectra.
As
of
and
excitation the system. reaches
energy of ground in
the liquid
one
can
see
from
Thus,
the value
and
state
of all
the first
relating Table
to 4,
138
TABLE 3 Vibrational
frequency
carbon monoxide
shifts
in IR absorption
and emission
spectra
of liquid
(in cm-1}.
Calculated values
Potential
Potential
II
-4,7 + 0,2 -5,2 + 0,2
-2,0 _+ 0, 1 -2,8 + 0,4 +3,0 -+ 0,6
+2,6 ± 0, 1 +7,1 + 0,3 -5,8 -+ 0,8
the stabilization
energy
of
the ground
unequlllbrlum
the stabilization
energy
of
the ground
equilibrium
molecules.
This
molecular
energy
discrepancy Fig. while
result
on
the values as
the basic
not
in
is
by
of
of
energy
of
level
shown that
in
level
used
than of
scheme
CO of
I.
The above
the scheme
of
levels
levels
therefore,
greater
28)
FiE.
in
the system
given
NVT-ensemble.
and,
is
vibrational
interaction
of
-4 (ref.
the usually
the fact
energy
simulation
the free
with state
energy
stabilization
a result
value
for,
the potential
of
agree
the liquid
can be accounted
1 is based
obtained
does
levels
value
Contribution of potential I
Total
Absorption Emission I Emission II
Experimental
I
in
In
Table
the latter
changes
and
in
only, 4
are
case,
entropy
of
the system are also taken into account. Thus,
the obtained
of vibrational
results give evidence
frequencies
realize for liquid carbon monoxide. molecules goes
in the IR emission
from
molecules
the "red"
spectrum
forces
is determined
in forming
with
state,
spectroscopic
that various
and emission
And the vibrational
the "blue",
in the excited vibrational
of repulsive monoxide.
into
for the fact
in the IR absorption
spectra
frequency
by pumping
the increase
of
shifts should
shift of CO
intensity
and
the number
of
which is due to the strengthening characteristics
of liquid carbon
139 TABLE 4 Stablllzatlon
energy values
(In kJ/mole) of vlbratlonal
levels v=O and v=l of
CO molecules in the llquld phase.
Absorption spectrum
v=O (equlllbr.)
v=l (unequlllbr.)
Emlsslon spectrum
v=O (unequlllbr.)
v=l (equlllbr.) -11,373
Potential Potential
I II
-10,783 -11,221
-10,839 -11,247
-11,313 -11,546
-11,579
REFERENCES 1 D.W. Oxtoby, D. Levesque, J . J . Wets, J. Chem. Phys., 68 (1978) 5528. 2 D.W. Oxtoby, Adv. Chem. Phys., 40 (1979) 1. 3 R.W. Impey, P.A. Madden, I.R. McDonald, Mol. Phys., 46 (1982) 513. 4 D. Levesque, J . J . Wets, D.W. Oxtoby, J. Chem. Phys., 79 (1983) 917. 5 A. De S a n t t s , M. Sampolt, R. V a l l a u r t , Mol. Phys., 53 (1984) 695. 6 P.A. Madden, M o l e c u l a r Dynamlcs S o l u t i o n o f S t a t i s t i c a l Mechanical Systems, ln: G. Clccotl, W.G. Hoover (Ed.),Proc. Int. School of Physlcs "Enrlco Ferml", Course XCVIII, North Holland, 1986, p. 371. 7 R. Banasle, E. Berger, E. Toukan, M.A. Rlecl, S.H. Chen, Chem. Phys. Lett., 132 (1986) 1 6 5 . 8 P.O. Westlund, R.M. L y n d e n - B e l l , Mol, Phys., 60 (1987) 1189. 9 M.W. Ewans, K. Refson, K.N. Swamy, G.S. L i e , E. K l e m e n t l , Phys. Rev., A36 (1987) 3935. 10 G. H e l n J e , W.A.P. Luck, K. H e l n z l n g e r , J. Phys. Chem., 91 (1987) 331. 11 G. H e l n J e , W.A.P. Luck, P. Bopp, Chem. Phys. L e t t . , 152 (1988) 358. 12 M.F. Herman, B . J . Berne, Chem. Phys. L e t t . , 77 (1981) 163. 13 M.F. Herman, B.J. Berne, J. Chem. P h y s . , 78 (1983) 4103. 14 T.W. Zerda, J. Zerda, J. Phys. Chem., 87 (1983) 149. 15 M.F. Herman, J. Chem. P h y s . , 87 ( 1 9 8 7 ) 4 7 9 4 . 16 S. Kh. Akopyan, S . I . Lukyanov, S.V. Shevkunov, J. Mol. S t r u c t . , 115 (1984) 307. 17 S. Kh. Akopyan, S , I . Lukyanov, S.V. Shevkunov, Chem. Phys. (USSR), 4 (1985) 1451. 18 S. Kh. Akopyan, S . I . Lukyanov, S.V. Shevkunov, J. Phys. Chem. (USSR), 62 (1988) 2021. 19 N. Legay-Sommalre, F. Legay, Chem. Phys. Lett., 52 (1977) 213. 20 M. Cacciatore, G.D. B111Ing, Chem. Phys. 58 (1981) 359. 21 G.A. Parker, R.T. Pack, J, Chem. Phys., 64 (1976) 2010. 22 G.A. Parker, R.T. Pack, J. Chem. Phys., 69 (1978) 3268. 23 R.M. Berns, A. van der Avolrd, J. Chem. Phys., 72 (1980) 6107. 24 R.A. Toth, R.H. Hunt, E.K. Plyler, J. Mol. Spectrosc., 32 (1969) 85. 25 C. Chackerlan, R.H. Tipping, J. Mol. Spectrosc., 99 {1983) 431. 26 W.L. Meerts, F.H. de Leeuw, A. Dynamus, Chem. Phys., 22 (1977) 319. 27 J. Oddershede, E.N. Svendsen, Chem. Phys., 64 (1982)359. 28 G.E. Ewlng, J. Chem. Phys., 37 (1962) 2250.
129
VIBRATIONAL FREQUENCY SHIFTS IN IR ABSORPTION AND EMISSION SPECTRA OF LIQUID CARBON MONOXIDE. MONTE-CARLO SIMULATION"
S. KH. AKOPYAN and S.I. LUKYANOV Institute of Chemistry, Leningrad University,
199034, Leningrad (USSR)
(Received 28 November 1989) SUMMARY
Solvent induced frequency shifts in the IR absorption and emission spectra were simulated by Monte-Carlo procedure. The pair potential with repulsive, dispersive and electrostatic parts was used to study liquid carbon monoxide at T = 80K and p = 0.7982 g/om 3. It was shown that if single molecule of the system undergoes vibrational transition, the "red" frequency shift in emission spectrum is greater than the one in the absorption spectrum. The computer simulation results predict that if all molecules of the system are excited, the frequency shift of the emitting molecule turns "blue". This means frequency shift in the IR emission spectrum depends on the pumping intensity. The increasin E number of excited molecules in the system leads to the increase of the influence of repulsive intermolecular interactions on frequency shift.
INTRODUCTION Computer
simulation
spectroscopic, systems,
methods
thermodynamic
which
the liquid
(refs.
1-18). The molecular dynamics method is used in for this purpose, (refs.
spectrum
range
The influence interactions
of of
was
investigated
temperature
and
the components
upon spectroscopic,
Luck
the Monte-Carlo
In particular,
method
the behaviour
frequency in the carbon monoxide
by means density
a pair
thermodynamic
liquid CO, was considered.
"Dedicated $o Professor W.A.P.
liquid of
whereas
12-18).
of the shift of the fundamental vibrational IR absorption
characteristics
simultaneous
spectra of
was applied only in a few papers
the wide
determine of
most of the investigations
structural
to
have been recently applied for interpreting vibrational
liquids and solutions
and
enable
of Monte-Carlo variations
potential
of
and structural
method
(refs.
in
16-18).
intermolecular parameters
of
130 As
It can be observed from the results of experimental
investigation
of
liquid carbon monoxide vibrational relaxation rates (ref. 19), the lifetime of the exited exceeds
vibrational
the relaxation
state time
of
the molecules
of
the liquid.
forming
This
it,
means
substantially
that
equilibrium
conflguratlon of liquid molecules has time to realize around excited molecules while molecules are in the excited vibrational state. That is why it may seem interesting
to
estimate
the vibrational
frequency
shifts
in
the emission
spectrum of liquid carbon monoxide and to compare them with the corresponding values
of
the absorption
spectrum.
information about peculiarities and, particularly,
Considering
this,
of intermolecular
we
can get
interactions
additional
in the system
information about the analytical form of pair potentials,
the potential parameters, etc. The fundamental carbon
monoxide,
band,
observed
responses
to
in
the IR
absorption
the transition
of
spectrum
molecules
of
from
liquid
a ground
equilibrium level to an excited unequlllbrlum vlbratlonal level (see Fig. 1).
Q'Lt~
liquid
gas
liquid
Fig. I. Scheme of vlbratlonal levels of molecules in gas and liquid phases. In
accordance
an excited
with
vlbratlonal
the above-made state
of
the transition from an equilibrium
remark
concerning
CO
molecules
excited
level
in
the duration the liquid
to an unequlllbrlum
of
phase, ground
vibrational level will realize in the case of the emission spectrum of llquld carbon
monoxide.
Thus,
according
to
the scheme
in Fig.
1,
the vibrational
frequency shift in the emlsslon spectrum of liquid carbon monoxide should be greater, comparing with the frequency shift in the absorption spectrum. This paper shows the results of the simulation by the Monte-Carlo method of the vibrational frequency shift in the IR absorption and emission spectra of liquid cases
carbon occur:
monoxide 1) one of
at
such
pumping
the conslderated
intensities molecules
when
the following
is always
vibrational state; 2) all molecules of the system are excited.
two
in an excited
131 RESULTS AND DISCUSSION The Schr~kilnger equation for
a liquid consisting
of dlatomlc
molecules
reads:
Ho(Q) + T(X) + V(Q,X)} Cn(Q,X) = En@(Q,X)
(1)
where: N
(2)
Ho(Q) = Z h°l (ql) I=1
Is the sum of hamlltonlans of free anharmonlc osclllators by which dlatomlc molecules are approxlmated, T(X) is the kinetic energy of translational and rotational movements, interactions,
Q
is
V(Q,X)
is the potential energy of the intermolecular
the vibrational
coordinate,
X
is
the rotational
and
translational coordinates, M is the number of molecules in the system. Using the Born-0ppenhelmer approximation:
~(Q,x) = ~ ( Q , x ) ~ ( x )
(3)
one can r e p l a c e e q u a t i o n (1) by the two e q u a t i o n s :
~
oca) + vca, x)],,ca, x) = EvCX),vCa, X)
C4)
[T(X' + Ev(X)]~.CX' = Ev=~ (X'
(5)
Equation C4) describes the vibrational movement of molecules in the field of
intermolecular
forces,
equation
(5)
describes
their
rotational
and
translational movements. If
rotational
and
translational
movements
are
regarded
processes, the probability density of the system beln E in X exp [-i/KT)Ev (X])]
~
v
-(1ZKT)Ev(X 1) dx
J
as
classical
co~IEuratlon is:
132
P(Xj)dX
is
the probability
the configuration According
of
the system
space of classical
in
X]-dX,
Xj+dX
region
of
coordinates.
to the first approximation
of the M molecules
being
of the perturbation
theory,
the energy
in the quantum state V I V2...V M is:
M
Evcx ~---E v
v cx~ : ~ E° ÷ <~°cQ~nvcQ, x,~i~°cQ~ >
1 2
M
c~
VI
1=1
Here: M
0
#~(Q) = U ~v Cql) 1=1
(s)
t
E° o v and @v(ql) are eigenvalue and ! i anharmonic oscillator hol(ql). Usin E
eqn.(7),
vibrational vibrational
one
levels
and,
frequency
interactions.
can
eigenfunction
estimate in
a hamiltonian
the stabilization
this
way,
of molecules
under
In particular,
of
one
can
determine
the influence
N
a(e)
<¢1(qk)~ °
%Cq,.~lvcQ, x J
S
o(
1=1
t ~k
i ~:k
M
of
a free
combining
the shifts
of
the intermolecular frequency shift in
is excited,
)l~,~cq?n % %
i=l
M
of
in the case of the vibrational
IR absorption and emission spectra when one molecule
Au(X~ ¢e)) = h - "
energy
of
we obtain:
)>_
(9)
)
~,,~) Ill ¢0(qi o )> ¢o°( q~)IvcQ'x~ i=l 1=1
<11
where
a and
e
indexes
correspond
k is the number of a molecule, When
all
molecules
of
to absorption
and
emission,
respectively,
for which optical transition occurs.
the system
shift in an emission spectrum is:
are
excited,
the vibrational
frequency
133
M
: h
e
M
o
% IvcQ, x In,
-
(IO)
%
o
"
"
1=1
1=1
i q~k
1 ~k
The result observed
of the averaging
in a spectroscopic
I
/
)
over all configuration
experiment
states,
the shift
i.e.
is:
Au = IAv(X)P(X)dX
(11)
v
Integral
(eqn.
Metropolis
a NVT-ensemble. cell.
II)
algorithm
performed.
Molecules
of
used,
by
the Monte-Carlo
the simulation
the simulated
between
potentials
depending
potentials
were
16, 17).
to describe
on
collisions.
dipole-dipole
dispersive
coefficient
a semi-empirlcal has
into a Monte-Carlo
in cubic
boundary
conditions
were
simulated
coordinates
of
potential
by
pair
molecules. (potential
additive
Two I)
types
of
exp
-6
is
of reactions The attractive
interactions
of transfer of vibrational part
of
in the systems.
the potential The value
of
energy
describes isotropic
C is taken from (ref. 21), where it was obtained in 6 The anisotropic dispersive coefficient C (2), for CO 6 estimated in our work as well as in (ref. 22), usin E
way.
been
of
an isoelectronic mechanical
out
in the cubic cell was 61. were
type
Standard
Its repulsive part was taken from (ref. 20), where it was used
dispersive
the values
The first
method. carried
the number of molecules and
Periodic
the molecules
the rate constants
molecules
molecule,
state.
on vibrational
used.
was
of the system were placed
The total number of molecules
Interactions
of
computed
was
The cell size was chosen in accordance with
the density
(refs.
was
isotropic molecule
calculations.
and
anisotropic
N 2 obtained
in
dispersive
(ref. 23),
on
coefficients the basis
for
of quantum
134
~z = A-exp(-~R)x[e
11 1 1 + e-~e:21clT1
e -~8¢zlc2":z
+ e
m m m +m c21=E:22 m +m ; c o c o and p o l a r i z a b i l i t y a n t s o t r o p y o f m o l e c u l e s , o
-
where Cl=COSb~I, and ~ are
(O¢/Oq)
C2=COS(d2, eft--S12"
polarizability
and
(8~/8q)
o
(12)
are
c
-
,
their
_-
derivatives,
m
with
respect
to
vibrational
o
c o o r d i n a t e s . D i s t a n c e s and a n g l e s a r e g i v e n i n Fig. 2a.
×
\ Flg. 2. Coordinates determining molecular orlentatlon.
The second type potential describing electrostatic
interactions
a dipole
moment
studied,
both
mechanical
(potential II), was observed by adding the terms
dlpole-dlpole,
of
dlpole-quadruple in
the system
CO molecules
experimentally
calculations
and,
upon
and
to
and
intermolecular
theoretically,
at
present,
quadruple-quadruple
potential
on
the first
I.
Dependence
distance the basis four
was of
of
actively quantum
coefficients
of
the series are known (refs. 24, 25)
~(T)
= ~0+~I(T--Te)+~2(T--Te)2+~3(T--Te
)3
(13)
135 Vibrational
dependence of a quadruple moment of CO molecules
(14)
QCT] = Qo+Q, CT-T]+Q2(T-T] 2
was determined
only In quantum mechanical
-Px Px -Py Py J
[zp. Pz
r_
(ref. 26).
(15)
R3
f^ (2)-,1)- (2)_,1)3 (2)[^(
_ I,.")IIQ")I
calculations
(1)~ 2
.1
,,,^,2,3 ,,,[_r,,,~21
.,_.,_
, [^lr ,,n2r ,:.n 2 r ,,n2r ~2,~21,._r ,,,'~'r ,.,~2 _rr ,,,~2 r ,.,~21
+ ~ l"[t"x J t"x J +t", J L", J J ~'t"-Jr"-J-"t
L'- J÷t"-JJ
,1}}
Angles are glven In Flg. 2b.
Table 1 shows the values of potentlal parameters and standard deviations Uslng agreement
the above (within
experlmental
of their determlnatlon
values
of
parameters,
the llmlt of 0.6~),
values of internal energy
As the results
of calculatlons,
vary wlthln
between
can
obtain
the results
a
rather
of
accurate
simulation
(Table 2), by means of potentlal
in the case of potential
energy of the system can be descrlbed parameters
one
taken over from literature
(the first column).
II,
and
I.
the internal
accurately enough only if the values of
the llmlts of accuracy
of their determination.
We did
not select parameter values in detall and stopped at those values of potential parameters
at which the defference between simulation and experimental
does not exceed 1,6~ (the third column in Table I).
results
136
TABLE 1 Values of parameters in pair potentials. Parameter
Parameter value
A(kJ/mole) ~(A) 6 C (kJ.A /mole) 6
Potential I
49889,67 3,6 5093,93 t 556,67
C (2) (kJ.A6/mole) 6 (A3)
488,6
± 53,4
(ref. 20) (ref. 20) (ref. 21)
49889,67 3,6 5093,93
(refs. 21, 22)
488,6
Potential II 49889,67 3,6 4804,50 460,8
1,94 + 1,98
Cref. 27)
1.94
1,94
0,53
Cref.
0,53
0,53
(a~/aq) o (A2)
1,55 ± 0,08
(ref. 27)
1,50
1,63
(8~/8q) ° CA2)
2,82 ± 0,31
(ref. 27)
2,82
2,96
(A 3 )
#o(D)
27)
-0,1126 ± 0,002 [refs. 24, 25)
-0,113
#t(D/A)
3,11 ± 0,15
(refs. 24, 25)
2,96
#2(D/A2)
-0,15 t 0,28
(refs. 24, 25)
-0,15
#3(D/A3)
-2,36 ± 0,8 (refs. 24, 25) -2,241 -2,50 (ref. 26) -1,93 ± 0,04 0,935 (ref. 26)
-2,36
Qo(B) QI(B/A ) Q2(B/A2)
TABLE 2 Calculated
]
1,814
and experimental
internal
-1,90 0,94
(ref. 26)
energy
1,81
values
(in kJ/mole)
of
liquid
carbon monoxide. Calculated values
Experimental value
Potential I
Potential II
-5,48 ± 0,05
-5,53 + 0,04
The potentials
described
-5,45
above
were
used
for
the simulation
of
a vibrational frequency shift in IR absorption and emission spectra of liquid carbon
monoxide
at
T=80K
and
p=0,7982
g/cm 3.
In averaging
according the vibrational states of interacting molecules,
the potentials
the internal energy
of molecules was described by Morse potential. Considering
the absorption spectra,
the interaction energy of the shifted
k-th molecule with other molecules was calculated for the two cases:
137 I) v =u =...=v =0 1 0 H
v2=O...V =I...vM=O
2) v1=O,
Energy value, for the first case, both energy values means
of
eqn.
are used
(9).
For
is used to generate the Markov chain, and
to calculate
the emission
the vibrational spectra,
frequency
the interaction
shift by energy
is
calculated for the following cases:
v2=O...VK=I...vM=O
1) v1=O,
2) v =v =...=v =0 (one molecule is excited),
1
2
and
N
1) v =v =...=v =1 12 M 2)
vi=1, v2=l...vK=O...vx=l(all Determining
the vibrational
molecules are excited).
frequency shifts in IR absorption and emission
in the system with electrostatic potential values, As
I into
the frequency
interactions, shifts
by
we derived the contribution of
calculating
the interaction
energy
using both potentials. it
can
be
sufficiently absorption
seen
reliable and
from
Table
difference
emission
1,
using
potential
in vibrational
spectra
when
one
1,
one
frequency
molecule
cannot
shifts
is
observe
in
the IR
excited,
because
the mean-square error is great enough because of a comparatively small number of generated chains (from 6 to 12). Inclusion
of
components,
the system,
into
vibrational
frequency
greater.
This
characterizing
the potential
result
shifts may
be
results for
electrostatic
in
the fact
absorption
accounted
for
and by
that
emission
the fact
interactions
in
the difference
in
spectra
that
becomes
electrostatic
interactions depend essentially on molecular orientation. Consideration molecules
results
the vibrational -1 +3 cm Table excited
of the data given
4 gives
in greater frequency
the values
vibrational
the absorption
and
levels
in Table
changes
shift
3,
emission
that
in the properties
changes
the sign
of stabilization of
shows
CO molecules spectra.
As
of
and
excitation the system. reaches
energy of ground in
the liquid
one
can
see
from
Thus,
the value
and
state
of all
the first
relating Table
to 4,
138
TABLE 3 Vibrational
frequency
carbon monoxide
shifts
in IR absorption
and emission
spectra
of liquid
(in cm-1}.
Calculated values
Potential
Potential
II
-4,7 + 0,2 -5,2 + 0,2
-2,0 _+ 0, 1 -2,8 + 0,4 +3,0 -+ 0,6
+2,6 ± 0, 1 +7,1 + 0,3 -5,8 -+ 0,8
the stabilization
energy
of
the ground
unequlllbrlum
the stabilization
energy
of
the ground
equilibrium
molecules.
This
molecular
energy
discrepancy Fig. while
result
on
the values as
the basic
not
in
is
by
of
of
energy
of
level
shown that
in
level
used
than of
scheme
CO of
I.
The above
the scheme
of
levels
levels
therefore,
greater
28)
FiE.
in
the system
given
NVT-ensemble.
and,
is
vibrational
interaction
of
-4 (ref.
the usually
the fact
energy
simulation
the free
with state
energy
stabilization
a result
value
for,
the potential
of
agree
the liquid
can be accounted
1 is based
obtained
does
levels
value
Contribution of potential I
Total
Absorption Emission I Emission II
Experimental
I
in
In
Table
the latter
changes
and
in
only, 4
are
case,
entropy
of
the system are also taken into account. Thus,
the obtained
of vibrational
results give evidence
frequencies
realize for liquid carbon monoxide. molecules goes
in the IR emission
from
molecules
the "red"
spectrum
forces
is determined
in forming
with
state,
spectroscopic
that various
and emission
And the vibrational
the "blue",
in the excited vibrational
of repulsive monoxide.
into
for the fact
in the IR absorption
spectra
frequency
by pumping
the increase
of
shifts should
shift of CO
intensity
and
the number
of
which is due to the strengthening characteristics
of liquid carbon
139 TABLE 4 Stablllzatlon
energy values
(In kJ/mole) of vlbratlonal
levels v=O and v=l of
CO molecules in the llquld phase.
Absorption spectrum
v=O (equlllbr.)
v=l (unequlllbr.)
Emlsslon spectrum
v=O (unequlllbr.)
v=l (equlllbr.) -11,373
Potential Potential
I II
-10,783 -11,221
-10,839 -11,247
-11,313 -11,546
-11,579
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