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An approximate method for calculating mean amplitudes of vibration from spectroscopic data—II
Spectrochlmicn
Act& 1050, pp. 341 to 348. Pergmon
Press Ltd.
Printed
in Northern
Irelmd
An approximate method for calculating mean amplitudes of vibration from spectroscopic data-II Application to simple polyatomic molecules S. J. CYVIN Institute
of Theoretical
Chemistry, Trondheim,
(Received
Technical Norway
21 December
University
of Norway,
1958)
Abstract-The approximate method for calculating mean-square amplitudes of vibration which was described in the first article of this series, has been applied to some special cases. The meansquare amplitudes have been expressed in terms of the atomic maasea and force constants for the following molecular structures: (i) linear eymmetrical XY,, ,(ii) linear symmetrical X,Y,, (iii) tetrahedral X, and (iv) plane symmetrical XY, molecules. IIltrodllction
IN the first article
of this series [l] (hereafter referred to as Paper I) an approximate method for computing the mean-square amplitudes of vibration in polyatomic molecules has been developed. It is an advantage of the method, that in several cases the mean-square amplitudes for the considered molecular model may be expressed explicitly in terms of the atomic masses and the force constants of the molecule. In the present work, this sort of expressions are evaluated for some symmetrical molecules including the plane symmetrical XY, molecular model as the most complicated case. According to equations (16) and (20) of Paper I, the matrices G, GFG, F-1 and. (FG F)-l are to be evaluated. However, the first term of the mean-square amplitude expression is always equal to a(,ux + ,a~), where px and ,uv denote the inverse masses of the adjacent atoms attached to the considered interatomic distance [2]. This fact may serve as a good check if the evaluation is performed by means of the G matrix. The present treatments of the special molecular models are in correspondence with the following scheme: (i) setting up the G and F matrices; (ii) determination of the corresponding inverse matrices; (iii) performing the matrix multiplications GFG and F-lG-lF-l; (’iv ) ex p ressing the interatomic displacements in terms of the chosen internal co-ordinates; and (v) determination of the terms of the meansquare amplitude formulae.
Linear symmetrical XY2 molecules . The simple case of linear symmetrical XY, molecules is taken as the first illustration of the method, although for this case the rigorous method for small harmonic vibrations [3] may also lead to explicit expressions [4]. [l] S. J. CYVIN, Spectrochim. Acta. 66 (1959). [2] E. B. WILSON, JR., J. Chena. Phys. 7, 1047 (1939); 9, 76 (1941); Y. MORINO, K. KUCHI~SU, A. TAKAEASHI and K. MAEDA, J. Chem. Phvs. 21, 1927 (1953). [3] Y. MORINO, K. KUCHITSU and T. SHINIANOUCHI, J. Client. Phys. 20, 726 (1952). [4] S. J. CYVIN, J. Chem. Phys. 30, 337 (1959).
341
S. J. CYVIN
Since we are interested in the interatomic displacements to a first approximation, only the non-degenerate vibrational modes need to be taken into account. The evalustion will be worked out in terms of the valence force co-ordinates rr, (r2, In these co-ordii.e. the deviations from equilibrium for the two X-Y distances. nates, the G and F matrices are reading*
k is the force constant for the X-Y constant. The corresponding inverse
stretching matrices
and k’ the bond-bond sre easily found to be
interaction
1 F-l=iqk +k')(lc - k')--k'/(k + Ic')(k k') k/(k +k')(lc - k') 1 [
G-l
-_
(Px + PYw+ux
PXIPPX
+ PY)PP
+ PY)PY
(Ps + PYWPUS
[
The G FG and F-l G-l F-l matrices GFG=
a
are specified b
[ 1
as follows,
F-lG-lF-1
;
+ PY)PY
=
[ I A
B
A
a
a = Q(pus+ pY)2 + px21 - 2k'CPx + PY)PX b = --BWps + /-+)ps + JWPS + PY)’ + ,d’l A = [(k2 + k'2)(px + PY) - 'L~Qu,ll(~ + k'12(h- W2(?4c + IUP)/+ B = [(k2 + V2)ps
- 21ck’(,ux + ,uJl/(k
+ ~%‘)~(k - 1~‘)~(2/4~ + ,uy),uy
The expression for the mean-square amplitude of the X-Y distance is obtained immediately from the diagonal elements of the evaluated matrices. The Y . . . Y interatomic distance deviations are expressed by TY...Y
=
1'1 +
r2
Hence the matrix Ui which enters into equation (20) of Paper I is the column matrix (1, l} in the present case, and the expressions which enter into the meansquare amplitude formula are consequently of the form 2(z + y). Here z denotes the diagonal and 2~ the off-diagonal element of one of the matrices G, G FG, F-l and (FG F)-I. The resulting formulae for the mean-square amplitudes (~2) are as follows : G&Y u;...y=
= dlux + yk(k 2apy
+ IUY) + Id)-l(k
hmk + W/JY2 + (k - wq-4~ + pyJ21 - l/)-l - $3[(k + ii’)-2/.+-l + (k - Id)-2(2px
- 2/9(k + k’)py2
* Throughout this paper omitted. ,ux and ,uy denote
+ 2y(k
+ k/)-l
- 2d(k + Ic’)-2py-1
the elements beneath the main diagonals the inverse masses of the X and Y atoms,
342
+ py)-l]
of symmetrical respectively.
matrices
will
be
in
approximate
method
for
calculating
mean
amplitudes
of vibration
from
Linear symmetrical X,Y, molecules
spectroscopic
data-II
_
It is profitable to utilize the molecular symmetry when working out the expressions in this case, since it is somewhat more complicated than the previous one. As before, only the non-degenerate vibrations will be taken into account. The inverse kinetic energy matrix and the potential energy matrix in terms of
Fig.
1. Notation used for the linear symmetrical X,Y, molecule. displacements from the equdibrium distances. Symmetry s, = 2-qr, + r*) Syrnm. species A,,: i s, = cl! $Symm. species A,,: s, = 2-q?., - T*)
The symbols co-ordinates:
denote
the
symmetry co-ordinates shall be denoted by theGermanletters 8 and $j,respectively. The applied symmetry co-ordinates are specified by Fig. 1 with the adherent text. In terms of these co-ordinates the energy matrices are reading Px +
PY
--2$is
0
211s
0
6= i
Ps f
k, + k,’ ; 5= PYI
2&k,’
0
k2
0
1
k, - k,’ I
1
The potential energy matrix, has been expressed by means of the force constants attached to the valence force co-ordinates, not the symmetry co-ordinates. k, is the force constant for the X-Y stretching and k’, for the interaction between the two X-Y bond stretchings. k, is the force constant for the X-X stretching and k’, for the X-Y and X-X bond interactions. By inversion one obtains
L
UPS + PY)l
L
KQJ
where
~1 = b/[(k, + k,'P, - 2(kz')21 K2 = (k, + W/W& + k,‘P, - 2(b,‘)21 K’ = -2%‘/[(kl + k,‘P, - 2(k2’121 K2
=
By matrix
l/(k,
- k,‘)
multiplication
@go
f
it is found
b”
81;
8-16-l8-1=
[^’
c
0
B
0 A2
343
1
s. J. h-VIN
a1
=
b
= 2(k, + k,’ + 2k, - 4k2’),ux2
(k,
+
c
= --2Vc,
a2
=
4
= C(kz - k,‘)2ps
B C
= !d(Jc, + h’ - 2bY2px = 2-*{[(k, + k,‘)k, - ,&
A2
=
(k,
h')(px
+
+
-
pd2
k,')(p,
k,')(px
+
+ +
2k2/4s2
py)px
--4k,'(px
-
+
PUY)PX
2%px2
+
%Vps
+ k,‘F,
-
W4)21-2(pxp~)-1
+
-
/-da
+ W)2,411k~
+ (k, + k~‘)2/4Hk~ 4 k,‘)k, - 2(k,‘)21-2(/w~)-1 + k,’ + 2k, - %‘)k,‘lpx - (k, + Wchs} x [(k,
(k,
-
A set of
k,')-2(px
+
+ h’)k,
-
2(k,‘)21-2(luxruy)-1
PI-)-~
representative
four
PY)PX
interatomic
distance
TX-Y
= 2-ybs,
1’x-x
= s,
TX . ..p
=
Let one of the matrices Q, $g%, matrices are of the typical form
is
given
by
+ S,)
2-y31
+
rp.. . p = 2q
deviations
8,)
+
82
f s,
3-1 and (gSi’j)-l
z
0
Y
0
be denoted
by
8.
All these
i 1 Xl
8=
52
For computing the expressions
the mean-square vl@Vl, F,SjV,,
amplitudes according to equation (20) of Paper v&V2 and i&V4 are needed, where
v, = (2-1, 0, 2-t);
v,
= (0,
v, = (2-5, 1, 2-i};
v, = (26, 1, O}
I,
1, O}
It is found = *(xl
V&V2
= $(x1 + x2) + y + 2%; V&V4
The mean-square &-Y
= a(px + SyVMk, -
t.&-=
+x21;
= y = 2x, + y + 2%
amplitudes
may now be evaluated.
+ PUP) - f&h(cLx
+ PY)~ + ~SPX~ -
+ k,‘)k,
%$,‘pxlW, = Zaps -
V2~V2
w4
- W2’)23-1
+ k,‘P,
-
+ (k, - W-l) - 3Wh2px + (k2’)2(clx + PI-) 2(k,‘)21-2(px/+r)-1 + (k, - k,‘r2(px + prF1}
2B(k, + k,’ + 2k2 -
WlXk, + k,’ -
~~‘CPU, + PYJPUXI
4k2’),ux2 + y(k, + k,‘)[(k,
2k,‘)21ux + (k, + k~‘)2puyl[(k~ 344
+ k,‘W,
+ k,‘)k, -
- 2(k2’)2]-1
2(k,‘)21-2(pxpy)-’
An
approximate
&...Y
=
4~s
method
for
+
-
1~1-1
calculating PP~(Ps~
+ h4?(k, + k’,) +
mean +
amplitudes
py2)
of vibration
2k’,~xpu-k-
-
from
spectroscopic
dat+II
+ k,,+i2 - 2k’dps - py)~xl
- 4k’d& + Ic’,P, - 2Wz)21-1 + (k, - k’,Y} - MUk, + k’J2(,e + PY) + kz2/+z+ W, + k’,Vws + W2Ppx + PY) - -Yk, + k’,P’,Plus + PY) - 6k,k’,pxl[@, + k’,)k, - 2W,)21-2(pxpyY + (k, - k’,F2(ps + ruyW k,
= 2wY - Wkl + k’dpY2 + y& + k’, + 2k2 - 4k’,)[(k, + k’,P, - 2W,)21-1 - -kY& + k’A2(px + puy)+ 4bi2px + 4(k, + k’,)k,px + 4(k’A2(4,e + PY) - 4(k, + k’,)k’,(2ps + puy)- l%k’,pu,lW, + k’,)k, - W’,)21-2(/wy)-1
~"Y...Y
Tetrahedral X, molecules The development for this type of molecules turns out to be less complicated than the previous case due to the high molecular symmetry. In fact, the treatment of tetrahedral X, molecules may also be carried through by the rigorous method.
Fig.
2. Notation
symm. symm.
species
used for the tetrahedrnl from equilibrium. A, :
species
E:
symm.species The symmetry be written
P, :
co-ordinates
4 s,, 1&, s,, s,, s,,
X, molecule. Symmetry = 6-+1 = 124(2r, = f(f.2 = 2-qr, = 2-q?, = 2-k(Ta
are given
The symbols denote displacements co-ordinates: + rz + ~3 + ~4 + rs + ~6) - T2 - 9.3 + 25-4 - 7.5 - r,) r3 + rs - To) - +J - TJ - ?-J
by Fig. 2 and the energy
matrices
may
8 = diag (4~~~ psj ius, 2px, 2px, 2~~) 5 = diag (&, g2, K,, K,, K,, K3) Let US assume that the potential field is given by the following force constants: k for the bond stretching, k’ for the interaction between a pair of adjacent bonds and k” for a pair of opposite bonds. Then we have K, = k + 4k’ + k”, K, = k .- 2k’ + k” and K, = k - k”. All the matrices to be evaluated are diagonal and obtained straightforward. 8-l 3-l OgS 5-16-1g-1 3
= diag (&+-l, = diag (K,-l,
,ux-i, ~~-1, +ps-l, +,us-l, &,~~-l) K2-l, K2-l, KS-l, KS-l, KS-l)
= diag (16RlpUX2, R2px2, . . . 4K,pX2, . . , . . ) = diag ( $K,-2px-1, K2-2px-1, . . , -&K3-2ps-1, ..,..) 345
S. J. CYVIN
The X-X
displacements
may be represented ‘)‘z= 6-*s,
yielding
the following
result
t&a = 2u/& - /?(jK, + iK,
+ 3-f&,
by
$ 2-Q5,,
for th’e mean-square
+ 2K,)pg+
amplitude.
y(&-l+
iKz-l
--6(&K,-a
+ iK,l)
+ iK,-2
+ $K,-2)/&
Fig. 3. Notation used for the plane symmetrical XY 3 molecule. The symbols denote the displacements from the equilibrium distances and angles. The equilibrium X-Y &stance is denoted by R. Symmetry co-ordinates: Symm. species A,‘: 4 = 3-)(71 -I- ?a + pa) St,, = 6-4(27, - r2 - T-J S,, = 6-?R(2a, - a, - as) Symm. species E’: = 2-q, - r3) 23i = 2-6R(a, - a$)
Plane symmetrical XY3 molecules In this case only the in-plane vibrations need to be considered. For. the introduction of symmetry co-ordinates, see Fig. 3. In terms of these co-ordinates, the energy matrices may be written
‘PY
0
%3/b
+
2PY)
0
0
0
$3"pu,
0
0
0
0
%3/b
6=
+
2clY)
3(3/b
+
2clY)
:3$% %(3Px f2PY)
L -K,
0
0
K,l? iv=
CD
0
O-
0
0
0
0
K,
r CD
346
--
An
approximate
If the 2f’R2
method
potential 2
aiaj
+
for
c&lculating
fun&ion 2gR
2
is given
riai
i-zj k + 2k’, K, = k - k’, @ = brevity be given by specifying way, s,:
= l//k;
6,; 3,: g,',
= (3~x + 2,+-)/6(3clx = l/R,; = K,/(K,aJ - l?);
By matrix
a2
=
multiplication
+
iK2(3f-4~
b =WbX2 c =
23”@2
mean
+
2g’R
f
amplitudes
of vibration
from
spectroscopic
d&-II
as 2 V = k C ri2 + 2k’ .z.ri~, +fR2 .x1. (riaj
q-,)2
+
z CC>+
(i, j = 1,‘;: 3), one ha; X, =
- f’ izd I’ = g - g’. The inverse matrices will for the different non-vanishing elements in the following
+ PY)PY;
6: @Ii 3,;
= (3k + 2h.)/2(3k = --3%&(3~u, + = Q/(K',@ - I?)
3;;
= --~/(K,Q
+.P~)P~ PUP)PY
- r2)
one obtains
+ YQx2 + 2d2
21~~)~
+8@(3~, + 3@)(3px
+ Wv~x + :3*r(3px
+ 2py)px
+ 2py)px + 2puy)px
+ uwpx
+ 2py)2 + 9/4x2]
A, = K1-2,uy-1
A2 = M3Q2
+ r2)(3px
+ 2py) +~3Wpx](~,o
- ry(3px
+ pu,)-lpp-~
B = E(K22 + 3rwpx + 2~~) + 3*~c2rpxw2~ - r2)-y3pu, C = -WW2Q + r2bx+ WC, + 3wy3px + 2py)] x (K,@ - ry(3px The interatomic
If one of the matrices
distance
deviations
are represented
by
rxmy = 3-*Sl + 6-*2S,, )'y...y =s, - 2-*S2, + S-5!& 8, 636, 3-l and (565)-1 is denoted x1
0 22
xi=
0
0
o-
2
0
0
Y
0
0
x2
2 Y-
347
by
+ E"p)-lpy-l + py)-lpy-l
S. J. Cm
it is found that the expressions +x1 + $x2 and x1 + $x2 + +y - 3-&z are to be respectively. used for computing the X-Y and Y . . . Y mean-square amplitudes, The resulting formulae are given below : U2X-Y
= 4pu~ +
PY)
+
--B&=&~Y~
W2(3p~
+
+ %%)pxl + M&-l - 4wiI-2~~-1 + w3P + rv3~ +
x U"Y...Y
=
2aruY
J72)-2(3Px
-
BUG
+
+ y[lil-l
+*(I<,
+
+
&iW2
+ 3-*W2
WC2
+
+
+
t&x2
21.4~) + 3*.w7~~1
PY)-lPY-l)
34)
+ 36rlpu,2
+ 34 + 3*2r)pi,+
3412clx
+ 34
+
+
+ 24w,+ - r3-11
3’u3pu,
(K24
-5%~)~
9(Jff22
+ 3*r)r(%
+
- ry-11
- 6{1i,-2~~-1
!V2)pu,
+ P~)I(K~+ - JJ2r2(3px + P~)-~P~-~>
Conclusion As a conclusion it may be stated that the general formulae turn out to be rather complicated when the considered molecular models are not quite simple. But individual molecules of higher complexity may be treated ?tumericaEly by the described method.
348
Act& 1050, pp. 341 to 348. Pergmon
Press Ltd.
Printed
in Northern
Irelmd
An approximate method for calculating mean amplitudes of vibration from spectroscopic data-II Application to simple polyatomic molecules S. J. CYVIN Institute
of Theoretical
Chemistry, Trondheim,
(Received
Technical Norway
21 December
University
of Norway,
1958)
Abstract-The approximate method for calculating mean-square amplitudes of vibration which was described in the first article of this series, has been applied to some special cases. The meansquare amplitudes have been expressed in terms of the atomic maasea and force constants for the following molecular structures: (i) linear eymmetrical XY,, ,(ii) linear symmetrical X,Y,, (iii) tetrahedral X, and (iv) plane symmetrical XY, molecules. IIltrodllction
IN the first article
of this series [l] (hereafter referred to as Paper I) an approximate method for computing the mean-square amplitudes of vibration in polyatomic molecules has been developed. It is an advantage of the method, that in several cases the mean-square amplitudes for the considered molecular model may be expressed explicitly in terms of the atomic masses and the force constants of the molecule. In the present work, this sort of expressions are evaluated for some symmetrical molecules including the plane symmetrical XY, molecular model as the most complicated case. According to equations (16) and (20) of Paper I, the matrices G, GFG, F-1 and. (FG F)-l are to be evaluated. However, the first term of the mean-square amplitude expression is always equal to a(,ux + ,a~), where px and ,uv denote the inverse masses of the adjacent atoms attached to the considered interatomic distance [2]. This fact may serve as a good check if the evaluation is performed by means of the G matrix. The present treatments of the special molecular models are in correspondence with the following scheme: (i) setting up the G and F matrices; (ii) determination of the corresponding inverse matrices; (iii) performing the matrix multiplications GFG and F-lG-lF-l; (’iv ) ex p ressing the interatomic displacements in terms of the chosen internal co-ordinates; and (v) determination of the terms of the meansquare amplitude formulae.
Linear symmetrical XY2 molecules . The simple case of linear symmetrical XY, molecules is taken as the first illustration of the method, although for this case the rigorous method for small harmonic vibrations [3] may also lead to explicit expressions [4]. [l] S. J. CYVIN, Spectrochim. Acta. 66 (1959). [2] E. B. WILSON, JR., J. Chena. Phys. 7, 1047 (1939); 9, 76 (1941); Y. MORINO, K. KUCHI~SU, A. TAKAEASHI and K. MAEDA, J. Chem. Phvs. 21, 1927 (1953). [3] Y. MORINO, K. KUCHITSU and T. SHINIANOUCHI, J. Client. Phys. 20, 726 (1952). [4] S. J. CYVIN, J. Chem. Phys. 30, 337 (1959).
341
S. J. CYVIN
Since we are interested in the interatomic displacements to a first approximation, only the non-degenerate vibrational modes need to be taken into account. The evalustion will be worked out in terms of the valence force co-ordinates rr, (r2, In these co-ordii.e. the deviations from equilibrium for the two X-Y distances. nates, the G and F matrices are reading*
k is the force constant for the X-Y constant. The corresponding inverse
stretching matrices
and k’ the bond-bond sre easily found to be
interaction
1 F-l=iqk +k')(lc - k')--k'/(k + Ic')(k k') k/(k +k')(lc - k') 1 [
G-l
-_
(Px + PYw+ux
PXIPPX
+ PY)PP
+ PY)PY
(Ps + PYWPUS
[
The G FG and F-l G-l F-l matrices GFG=
a
are specified b
[ 1
as follows,
F-lG-lF-1
;
+ PY)PY
=
[ I A
B
A
a
a = Q(pus+ pY)2 + px21 - 2k'CPx + PY)PX b = --BWps + /-+)ps + JWPS + PY)’ + ,d’l A = [(k2 + k'2)(px + PY) - 'L~Qu,ll(~ + k'12(h- W2(?4c + IUP)/+ B = [(k2 + V2)ps
- 21ck’(,ux + ,uJl/(k
+ ~%‘)~(k - 1~‘)~(2/4~ + ,uy),uy
The expression for the mean-square amplitude of the X-Y distance is obtained immediately from the diagonal elements of the evaluated matrices. The Y . . . Y interatomic distance deviations are expressed by TY...Y
=
1'1 +
r2
Hence the matrix Ui which enters into equation (20) of Paper I is the column matrix (1, l} in the present case, and the expressions which enter into the meansquare amplitude formula are consequently of the form 2(z + y). Here z denotes the diagonal and 2~ the off-diagonal element of one of the matrices G, G FG, F-l and (FG F)-I. The resulting formulae for the mean-square amplitudes (~2) are as follows : G&Y u;...y=
= dlux + yk(k 2apy
+ IUY) + Id)-l(k
hmk + W/JY2 + (k - wq-4~ + pyJ21 - l/)-l - $3[(k + ii’)-2/.+-l + (k - Id)-2(2px
- 2/9(k + k’)py2
* Throughout this paper omitted. ,ux and ,uy denote
+ 2y(k
+ k/)-l
- 2d(k + Ic’)-2py-1
the elements beneath the main diagonals the inverse masses of the X and Y atoms,
342
+ py)-l]
of symmetrical respectively.
matrices
will
be
in
approximate
method
for
calculating
mean
amplitudes
of vibration
from
Linear symmetrical X,Y, molecules
spectroscopic
data-II
_
It is profitable to utilize the molecular symmetry when working out the expressions in this case, since it is somewhat more complicated than the previous one. As before, only the non-degenerate vibrations will be taken into account. The inverse kinetic energy matrix and the potential energy matrix in terms of
Fig.
1. Notation used for the linear symmetrical X,Y, molecule. displacements from the equdibrium distances. Symmetry s, = 2-qr, + r*) Syrnm. species A,,: i s, = cl! $Symm. species A,,: s, = 2-q?., - T*)
The symbols co-ordinates:
denote
the
symmetry co-ordinates shall be denoted by theGermanletters 8 and $j,respectively. The applied symmetry co-ordinates are specified by Fig. 1 with the adherent text. In terms of these co-ordinates the energy matrices are reading Px +
PY
--2$is
0
211s
0
6= i
Ps f
k, + k,’ ; 5= PYI
2&k,’
0
k2
0
1
k, - k,’ I
1
The potential energy matrix, has been expressed by means of the force constants attached to the valence force co-ordinates, not the symmetry co-ordinates. k, is the force constant for the X-Y stretching and k’, for the interaction between the two X-Y bond stretchings. k, is the force constant for the X-X stretching and k’, for the X-Y and X-X bond interactions. By inversion one obtains
L
UPS + PY)l
L
KQJ
where
~1 = b/[(k, + k,'P, - 2(kz')21 K2 = (k, + W/W& + k,‘P, - 2(b,‘)21 K’ = -2%‘/[(kl + k,‘P, - 2(k2’121 K2
=
By matrix
l/(k,
- k,‘)
multiplication
@go
f
it is found
b”
81;
8-16-l8-1=
[^’
c
0
B
0 A2
343
1
s. J. h-VIN
a1
=
b
= 2(k, + k,’ + 2k, - 4k2’),ux2
(k,
+
c
= --2Vc,
a2
=
4
= C(kz - k,‘)2ps
B C
= !d(Jc, + h’ - 2bY2px = 2-*{[(k, + k,‘)k, - ,&
A2
=
(k,
h')(px
+
+
-
pd2
k,')(p,
k,')(px
+
+ +
2k2/4s2
py)px
--4k,'(px
-
+
PUY)PX
2%px2
+
%Vps
+ k,‘F,
-
W4)21-2(pxp~)-1
+
-
/-da
+ W)2,411k~
+ (k, + k~‘)2/4Hk~ 4 k,‘)k, - 2(k,‘)21-2(/w~)-1 + k,’ + 2k, - %‘)k,‘lpx - (k, + Wchs} x [(k,
(k,
-
A set of
k,')-2(px
+
+ h’)k,
-
2(k,‘)21-2(luxruy)-1
PI-)-~
representative
four
PY)PX
interatomic
distance
TX-Y
= 2-ybs,
1’x-x
= s,
TX . ..p
=
Let one of the matrices Q, $g%, matrices are of the typical form
is
given
by
+ S,)
2-y31
+
rp.. . p = 2q
deviations
8,)
+
82
f s,
3-1 and (gSi’j)-l
z
0
Y
0
be denoted
by
8.
All these
i 1 Xl
8=
52
For computing the expressions
the mean-square vl@Vl, F,SjV,,
amplitudes according to equation (20) of Paper v&V2 and i&V4 are needed, where
v, = (2-1, 0, 2-t);
v,
= (0,
v, = (2-5, 1, 2-i};
v, = (26, 1, O}
I,
1, O}
It is found = *(xl
V&V2
= $(x1 + x2) + y + 2%; V&V4
The mean-square &-Y
= a(px + SyVMk, -
t.&-=
+x21;
= y = 2x, + y + 2%
amplitudes
may now be evaluated.
+ PUP) - f&h(cLx
+ PY)~ + ~SPX~ -
+ k,‘)k,
%$,‘pxlW, = Zaps -
V2~V2
w4
- W2’)23-1
+ k,‘P,
-
+ (k, - W-l) - 3Wh2px + (k2’)2(clx + PI-) 2(k,‘)21-2(px/+r)-1 + (k, - k,‘r2(px + prF1}
2B(k, + k,’ + 2k2 -
WlXk, + k,’ -
~~‘CPU, + PYJPUXI
4k2’),ux2 + y(k, + k,‘)[(k,
2k,‘)21ux + (k, + k~‘)2puyl[(k~ 344
+ k,‘W,
+ k,‘)k, -
- 2(k2’)2]-1
2(k,‘)21-2(pxpy)-’
An
approximate
&...Y
=
4~s
method
for
+
-
1~1-1
calculating PP~(Ps~
+ h4?(k, + k’,) +
mean +
amplitudes
py2)
of vibration
2k’,~xpu-k-
-
from
spectroscopic
dat+II
+ k,,+i2 - 2k’dps - py)~xl
- 4k’d& + Ic’,P, - 2Wz)21-1 + (k, - k’,Y} - MUk, + k’J2(,e + PY) + kz2/+z+ W, + k’,Vws + W2Ppx + PY) - -Yk, + k’,P’,Plus + PY) - 6k,k’,pxl[@, + k’,)k, - 2W,)21-2(pxpyY + (k, - k’,F2(ps + ruyW k,
= 2wY - Wkl + k’dpY2 + y& + k’, + 2k2 - 4k’,)[(k, + k’,P, - 2W,)21-1 - -kY& + k’A2(px + puy)+ 4bi2px + 4(k, + k’,)k,px + 4(k’A2(4,e + PY) - 4(k, + k’,)k’,(2ps + puy)- l%k’,pu,lW, + k’,)k, - W’,)21-2(/wy)-1
~"Y...Y
Tetrahedral X, molecules The development for this type of molecules turns out to be less complicated than the previous case due to the high molecular symmetry. In fact, the treatment of tetrahedral X, molecules may also be carried through by the rigorous method.
Fig.
2. Notation
symm. symm.
species
used for the tetrahedrnl from equilibrium. A, :
species
E:
symm.species The symmetry be written
P, :
co-ordinates
4 s,, 1&, s,, s,, s,,
X, molecule. Symmetry = 6-+1 = 124(2r, = f(f.2 = 2-qr, = 2-q?, = 2-k(Ta
are given
The symbols denote displacements co-ordinates: + rz + ~3 + ~4 + rs + ~6) - T2 - 9.3 + 25-4 - 7.5 - r,) r3 + rs - To) - +J - TJ - ?-J
by Fig. 2 and the energy
matrices
may
8 = diag (4~~~ psj ius, 2px, 2px, 2~~) 5 = diag (&, g2, K,, K,, K,, K3) Let US assume that the potential field is given by the following force constants: k for the bond stretching, k’ for the interaction between a pair of adjacent bonds and k” for a pair of opposite bonds. Then we have K, = k + 4k’ + k”, K, = k .- 2k’ + k” and K, = k - k”. All the matrices to be evaluated are diagonal and obtained straightforward. 8-l 3-l OgS 5-16-1g-1 3
= diag (&+-l, = diag (K,-l,
,ux-i, ~~-1, +ps-l, +,us-l, &,~~-l) K2-l, K2-l, KS-l, KS-l, KS-l)
= diag (16RlpUX2, R2px2, . . . 4K,pX2, . . , . . ) = diag ( $K,-2px-1, K2-2px-1, . . , -&K3-2ps-1, ..,..) 345
S. J. CYVIN
The X-X
displacements
may be represented ‘)‘z= 6-*s,
yielding
the following
result
t&a = 2u/& - /?(jK, + iK,
+ 3-f&,
by
$ 2-Q5,,
for th’e mean-square
+ 2K,)pg+
amplitude.
y(&-l+
iKz-l
--6(&K,-a
+ iK,l)
+ iK,-2
+ $K,-2)/&
Fig. 3. Notation used for the plane symmetrical XY 3 molecule. The symbols denote the displacements from the equilibrium distances and angles. The equilibrium X-Y &stance is denoted by R. Symmetry co-ordinates: Symm. species A,‘: 4 = 3-)(71 -I- ?a + pa) St,, = 6-4(27, - r2 - T-J S,, = 6-?R(2a, - a, - as) Symm. species E’: = 2-q, - r3) 23i = 2-6R(a, - a$)
Plane symmetrical XY3 molecules In this case only the in-plane vibrations need to be considered. For. the introduction of symmetry co-ordinates, see Fig. 3. In terms of these co-ordinates, the energy matrices may be written
‘PY
0
%3/b
+
2PY)
0
0
0
$3"pu,
0
0
0
0
%3/b
6=
+
2clY)
3(3/b
+
2clY)
:3$% %(3Px f2PY)
L -K,
0
0
K,l? iv=
CD
0
O-
0
0
0
0
K,
r CD
346
--
An
approximate
If the 2f’R2
method
potential 2
aiaj
+
for
c&lculating
fun&ion 2gR
2
is given
riai
i-zj k + 2k’, K, = k - k’, @ = brevity be given by specifying way, s,:
= l//k;
6,; 3,: g,',
= (3~x + 2,+-)/6(3clx = l/R,; = K,/(K,aJ - l?);
By matrix
a2
=
multiplication
+
iK2(3f-4~
b =WbX2 c =
23”@2
mean
+
2g’R
f
amplitudes
of vibration
from
spectroscopic
d&-II
as 2 V = k C ri2 + 2k’ .z.ri~, +fR2 .x1. (riaj
q-,)2
+
z CC>+
(i, j = 1,‘;: 3), one ha; X, =
- f’ izd I’ = g - g’. The inverse matrices will for the different non-vanishing elements in the following
+ PY)PY;
6: @Ii 3,;
= (3k + 2h.)/2(3k = --3%&(3~u, + = Q/(K',@ - I?)
3;;
= --~/(K,Q
+.P~)P~ PUP)PY
- r2)
one obtains
+ YQx2 + 2d2
21~~)~
+8@(3~, + 3@)(3px
+ Wv~x + :3*r(3px
+ 2py)px
+ 2py)px + 2puy)px
+ uwpx
+ 2py)2 + 9/4x2]
A, = K1-2,uy-1
A2 = M3Q2
+ r2)(3px
+ 2py) +~3Wpx](~,o
- ry(3px
+ pu,)-lpp-~
B = E(K22 + 3rwpx + 2~~) + 3*~c2rpxw2~ - r2)-y3pu, C = -WW2Q + r2bx+ WC, + 3wy3px + 2py)] x (K,@ - ry(3px The interatomic
If one of the matrices
distance
deviations
are represented
by
rxmy = 3-*Sl + 6-*2S,, )'y...y =s, - 2-*S2, + S-5!& 8, 636, 3-l and (565)-1 is denoted x1
0 22
xi=
0
0
o-
2
0
0
Y
0
0
x2
2 Y-
347
by
+ E"p)-lpy-l + py)-lpy-l
S. J. Cm
it is found that the expressions +x1 + $x2 and x1 + $x2 + +y - 3-&z are to be respectively. used for computing the X-Y and Y . . . Y mean-square amplitudes, The resulting formulae are given below : U2X-Y
= 4pu~ +
PY)
+
--B&=&~Y~
W2(3p~
+
+ %%)pxl + M&-l - 4wiI-2~~-1 + w3P + rv3~ +
x U"Y...Y
=
2aruY
J72)-2(3Px
-
BUG
+
+ y[lil-l
+*(I<,
+
+
&iW2
+ 3-*W2
WC2
+
+
+
t&x2
21.4~) + 3*.w7~~1
PY)-lPY-l)
34)
+ 36rlpu,2
+ 34 + 3*2r)pi,+
3412clx
+ 34
+
+
+ 24w,+ - r3-11
3’u3pu,
(K24
-5%~)~
9(Jff22
+ 3*r)r(%
+
- ry-11
- 6{1i,-2~~-1
!V2)pu,
+ P~)I(K~+ - JJ2r2(3px + P~)-~P~-~>
Conclusion As a conclusion it may be stated that the general formulae turn out to be rather complicated when the considered molecular models are not quite simple. But individual molecules of higher complexity may be treated ?tumericaEly by the described method.
348