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The r-centroid approximation and generalized functions
Volume 17, number 2
CHEMICAL
15 November 1972
PHYSICS LETTERS
THE r-CENTROID APPROXIMATION AND GENERALISED
FUNCTIONS
K.W. NICHOLLS Centre for Researcil iti Experimentcl Space Science, York Utrivetsity, Toronto, Otltmio. Cannda Received 12 Seprember 1972
The r-centroid approximation implies that molecular vibrational to relatively high sequence members of generaked functions.
In the spectroscopy of diatomic molecules, two of the most commonly used derived quantities of molecular vibrational wavefunctions q,, and qII,.la, are the Frannck-Condon factor ~v~v~~= 11 \I.,,~9,,~fdrlz and the r-centroid rvlv,, f I] which is a characteristic internuclear separation associated with the (v’, Y”) transition. It is defined as &
= I\Tly.~ky.,rdl.!l\lly~\Ily,.
It has been demonstrated centroid approximation
dr .
numerically
E ,I = I\zly~\k ),11P cIr//\k,,.~,,~! vv
dr=
(1) [I!] that the r-
(T,‘lylJ’
(2)
(where I? is an integer less than about 10) may be applied to most bands of many band systems through such equations
as
where&-) is a poiynomial function of r. Eq. (3) is commonly used to interpret intensity measurements on molecular spectra in terms of transition probability parameters. In such applicationsg(r) is often the electronic transition moment of the band system in question.
The adequacy of the r-centroid approximation can be tested for any band by studying how Y:l,c. defined as
252
wavefunction
products
should behave similarly
departs from unity for Y > 2. e-v,, and Yi*,f* are both unity by definition. Drake and Nicholls [3] made a preliminary numerical study of the behaviour of Y>vr(O-(S)] and the properties of generalised functions and distributions [S] which were being developed in the mathematical literature at about the same time that r-centroids were being applied to molecular problems. In this way the r-centroid approximation may be placed on a more satisfying mathemat-
Volume
17, number 2
ical basis than has hitherto
been possible,
and the con-
ditions under which the approximation holds and breaks down may be more clearly understood. In the theory of general&d functions and weak convergence [S], 2 sequence of generalised functions b,(r - ro) is defined relative to a test functionf(r) such that
The sequence member Qn can take many acceptable forms which converge to specific genelaliscd function, of which the Dirac de1 ta function 6(r), the Heaviside step function H(r) and the step function sgn(r) are most commonly used in applications. The pre-limit form of eq. (5) becomes an increasingly better approximation tof(ro) as Iz increases. The structures of the pre-limit form of eq. (5) and of eqs. (I), (2) and (3) are very similar. @,,(r - r,-,) can be identified with Q,$r)$,z$r)/J ~Q,,f(r)!l?,~~(r) dr, and f(r) can be respectively identified with r, r”, or g(r). Provided that the wavefunction product is sharply peaked in one region of r (and also probably oscillatory), the pre-limit form of eq. (5) may be expected to be a good representation of eqs. (1), (2) and (3). When the overlap integral is very small, no part of the wavefunction product is likely to be of large magnitude relative to the rest. Eqs. (1), (2) and (3) will then not be well represented by eq. (5), and the r-centroid approximation will break down. When the overlap integral is non-zero, part of the wavefunction product is usually locally peaked in one region ofr, and tends to be more and more similar in
shape to a particular Q,, of eq. (5). Indeed, at the limit, as has been suggested elsewhere [6] a sufficient criterion for operation of the r-centroid approximation, is a b-function behaviour of the wavefunction product *,,\k,,v. While this limit (which corresponds to II-+=) in eq. (5) is never completely application, an approximate
IS November 1972
CHEMICAL PHYSICS LETTERS
reached in the r-centroid genenlised function se-
quence member representation
of the waveftinction
product illuminates the nature of the r-centroid approximation rather well. Of the various forms for sequences of general&d functions $,,(r- rO) which are commonly used to describe the h-function [5], the form sin n(r - rO)/n(r - ro) is particularly useful here. r of course is expressed in appropriate dimensionless un,its
which can later be interpreted in specific length units (atomic units, Angstroms, etc.) in any particular application. This sequence function is peaked at r. and, depending upon the sequence member )I, oscillates with rapidly decreasing amplitude on either side of ‘0. It is very reminiscent of the shape of vibrational wavefunction products encountered in numerical studies of molecular overlap integrals. Some insight can then be gained about the genera1 nature of the r-centroid approximation by examining the properties of the right-hand member of eq. (3) when the normal&d wavefunction product is represented by sin n(r - ro)/n(r - ro), and the polynomial g(r) is represented by CQ,, rnr. Eq. (3) may then be rewritten sin n(r - r, j G,, = c am $I-“’ ---dr = c CI,,~f,n,, , 7i(T - ‘0) I?1 nl
(6a)
where I
rrn sin )z(r - rO) dr. ??l)i _In(r-- To) =
(6b)
If z = r - ro, eq. (6b) becomes a
TtI ,11,1=
s
,,1
(z+‘O)
sin tlz ,-dz
(74
.
-a
The range of integration, though formally --a~ + m, in practice Is limited to a finite segment r. -a
1 +z/To)“* sin(nz)/z ,
which if it is sufticicntly cant range of integation rr [sin(nz)/z
peaked that over the signifiz/r0 < 1, may be written as
+ (m/r) sin(rzz)]
.
The contribution from the second term to the integral is zero. The contribution of the first term is 30” Si(a) where Si is the sine integral. When (I is large this is
nr?, which is not a bad approximation
when a is
smaller. In fact, the properties of sin(rlz)/z suggests that if a is set at one of its zeros, such that tza = p, p should not have to exceed three or four and could be smaller. Eq. (7a) may be written
L7
+
... +
s
z‘72-l sinlIz dz .
C7b)
-u
253
Volume
17, number
CHEMICAL
2.
Eq. (4) may also be rewritten
Ymn xl.
a
Yrnn=
-u
y[tin~nz~,zl dz
(rgp
-a
The term J’T, [sin(nz)/z] dz which occurs in eqs. (75) and (8a) is twice the sine integral. Even for relatively small values of II, and finite limits, it oscillates with decreasing amplitude (Sa) then becomes
about
a
nfo”Y,nn = J(z +rJrn
rr with increase
sy
&
= ‘Tlrn”
(8d)
Under these circumstances eqs. (6a) and (7d) imply that
1(z +r())‘” [sin(rzz)/zl dz
YT".. =
15 November 1972
PHYSICS LETTERS
in a. Eq.
@b)
.
-a
Eqs. (7b) and (8b) becoine
Gn=C=,nlmn~Cnm~~=g(ro).
(10)
m
III
On the other hand, when Am,, is not small ccmpared to 1, which occurs when rz is small and the wavefunction product is not strongly peaked, the approximation will not hold. These points are well iliustrated in table 1 in which the dependence of AmI, upon rtz, II and p is displayed for values of m between zero and five. Illustrative numerical values of A,*” are also displayed for p = 1,2,3, y. = 1 for the two extreme values of 12. 1 and 10. In practical applications of the r-centroid method through equations such as (3), (6a) and (10) in which
molecular Froperties are expressed as empirical power series of r, the polynomial in r is seldom of degee higher than three or four.
??I
nk=l ‘;n,1= 1tlC
0k 4 III
Ak- l,n
=( 1 + Am,,1
3
(SC)
where
A
=O;
zx,!,rr
s is an integer,
Pb)
Thus the natcre of the r-centroid approximation which arises from thk locally peaked Eature of the oscillatory wavefunction product has been demonstrated to have close similarity in behaviour to the S-sequence finction sin n(r - ro)/n(r - ko)_ Whether or not well developed peaks form, depends in a very sensitive manner upon the relative phases of the two wavefunctions through the Franck-Condon principle. Under conditions of significant constructive overlap between the two wave functions a”,, and air,!, , eq. (5) is closely approximated and the Y:*vft wifi not differ significantly from unity, as noted in a very large number of cases
[41However, in cases where there is no significant structive
PC) The nature
and limitations of the r-centroid approximation are evident from eqs. (7~) and (8~). When A ,yrn is much less th.an 1, which occurs for large values of 11 and implies a strongly peaked wavefunction product,
where ‘0 is. the r-centroid
rYaYI, and
interference
between
conthe two wavefunctions,
eq. (5) is not necessarily a good representation
of eqs.
(I), (2) and (3) which are themselves inexact. The wavefunction product is then not similar in shape to the 6-sequence function sin n(r- ro)/n(r - r00) and the Y,“*,,. may be expected to diverge signific*antly from unity, as has been observed in a number of cases, particularly for bands of hydrides and deuterides. This work has been supported by research grants from the National Research Council of Canada and from the Defence Research Board of Canada.
Volume
17. number
2
CHEhIlCAL
15 November
PHYSICS LETTERS
1972
Table 1 of A,, upon m, r~and p
Dependence
A nw @cl = 1)
p=2
p=l m
n= 1
A rnn
n= 10
I7 =
10
p=3 -/I=1
77=10
I
77=
0 0
0 0
0 0
0
-0.04
6
0.06
-0.007
1
0.01
-
0 1
0
2
i-W+&
0
0 0
0 0
2
0.02
-4
0.3
0.003
-
0.00016
-1.57
-0.015
532
0.002
-
-0.0028
107
0
3
(-l)p+l
4
(-1)F+1&
m
p
(-HP+1
5
19.3
1-l. 24(pf;;;6)) 0
*
3.93
‘1+,,,,;;-6)
R.W. Nicholls, in: Physical chemistry, an advanced treaand W’. Jost tise, Vol. 3, eds. M. Eyring, D. Henderson (Academic Press, New York, 1969) ch. 3; B.H. Armstrong and R.W. Nicholls, Emission, absorption and rnnsfcr of radiation in heated atmospheres (Pcrgamon Press, London, 1972). P.A. Fraser, Can. J. Phys. 32 (1954) 515. J. Drake and R.W. Nicholls, Chem. Phys. Letters 3 (1969)
457. McCnIlum,
Space
Science,
W.R.
Jnrmain
and R.W. Nicholls,
1, Centre for Rcsexch York
University,
31.5
-0.0 1
>
References
scopic Report
0.05
(
0 t
J.C.
0.7
March
Spectro-
in Experimenti 1970;
J.C. McCallum and R.W. Nicholls, Spectroscopic
Report
2, Centre for Reenrch in Esperimental Space Science. York Univerity, October 1971; J.C. McCallum, W.R. Jarmain and R.W. Nicholls, Spectro-
scopic Report 3, Ccntre for Resexch in Esperimcntal Space Science, York University, March 1972; Spectroscopic Report 4, Centrc for Research in Experimental May 1972. Vols. 1, 2 (Herman, 151 L. Schwartz, Thborie des distributions, Paris, 1950.1951); G. Temple, J. Lond. hlath. Sot. 28 (1953) 134; Proc. Roy.
Sparr Science, York University,
Sot. A228 (19.55) 125: hf. Lkhthill, !ntroduction
to Fourier analysis and generUniv. Press, London, 1959). [cl R.W. Nicholls and A.L. Stewart, in: Atomic and molecular al&d
functions
(Cambridge
processes, ed. D.R. Bates (Audemic 1962) ch. 2
Press,
New York,
255
CHEMICAL
15 November 1972
PHYSICS LETTERS
THE r-CENTROID APPROXIMATION AND GENERALISED
FUNCTIONS
K.W. NICHOLLS Centre for Researcil iti Experimentcl Space Science, York Utrivetsity, Toronto, Otltmio. Cannda Received 12 Seprember 1972
The r-centroid approximation implies that molecular vibrational to relatively high sequence members of generaked functions.
In the spectroscopy of diatomic molecules, two of the most commonly used derived quantities of molecular vibrational wavefunctions q,, and qII,.la, are the Frannck-Condon factor ~v~v~~= 11 \I.,,~9,,~fdrlz and the r-centroid rvlv,, f I] which is a characteristic internuclear separation associated with the (v’, Y”) transition. It is defined as &
= I\Tly.~ky.,rdl.!l\lly~\Ily,.
It has been demonstrated centroid approximation
dr .
numerically
E ,I = I\zly~\k ),11P cIr//\k,,.~,,~! vv
dr=
(1) [I!] that the r-
(T,‘lylJ’
(2)
(where I? is an integer less than about 10) may be applied to most bands of many band systems through such equations
as
where&-) is a poiynomial function of r. Eq. (3) is commonly used to interpret intensity measurements on molecular spectra in terms of transition probability parameters. In such applicationsg(r) is often the electronic transition moment of the band system in question.
The adequacy of the r-centroid approximation can be tested for any band by studying how Y:l,c. defined as
252
wavefunction
products
should behave similarly
departs from unity for Y > 2. e-v,, and Yi*,f* are both unity by definition. Drake and Nicholls [3] made a preliminary numerical study of the behaviour of Y>vr(O
Volume
17, number 2
ical basis than has hitherto
been possible,
and the con-
ditions under which the approximation holds and breaks down may be more clearly understood. In the theory of general&d functions and weak convergence [S], 2 sequence of generalised functions b,(r - ro) is defined relative to a test functionf(r) such that
The sequence member Qn can take many acceptable forms which converge to specific genelaliscd function, of which the Dirac de1 ta function 6(r), the Heaviside step function H(r) and the step function sgn(r) are most commonly used in applications. The pre-limit form of eq. (5) becomes an increasingly better approximation tof(ro) as Iz increases. The structures of the pre-limit form of eq. (5) and of eqs. (I), (2) and (3) are very similar. @,,(r - r,-,) can be identified with Q,$r)$,z$r)/J ~Q,,f(r)!l?,~~(r) dr, and f(r) can be respectively identified with r, r”, or g(r). Provided that the wavefunction product is sharply peaked in one region of r (and also probably oscillatory), the pre-limit form of eq. (5) may be expected to be a good representation of eqs. (1), (2) and (3). When the overlap integral is very small, no part of the wavefunction product is likely to be of large magnitude relative to the rest. Eqs. (1), (2) and (3) will then not be well represented by eq. (5), and the r-centroid approximation will break down. When the overlap integral is non-zero, part of the wavefunction product is usually locally peaked in one region ofr, and tends to be more and more similar in
shape to a particular Q,, of eq. (5). Indeed, at the limit, as has been suggested elsewhere [6] a sufficient criterion for operation of the r-centroid approximation, is a b-function behaviour of the wavefunction product *,,\k,,v. While this limit (which corresponds to II-+=) in eq. (5) is never completely application, an approximate
IS November 1972
CHEMICAL PHYSICS LETTERS
reached in the r-centroid genenlised function se-
quence member representation
of the waveftinction
product illuminates the nature of the r-centroid approximation rather well. Of the various forms for sequences of general&d functions $,,(r- rO) which are commonly used to describe the h-function [5], the form sin n(r - rO)/n(r - ro) is particularly useful here. r of course is expressed in appropriate dimensionless un,its
which can later be interpreted in specific length units (atomic units, Angstroms, etc.) in any particular application. This sequence function is peaked at r. and, depending upon the sequence member )I, oscillates with rapidly decreasing amplitude on either side of ‘0. It is very reminiscent of the shape of vibrational wavefunction products encountered in numerical studies of molecular overlap integrals. Some insight can then be gained about the genera1 nature of the r-centroid approximation by examining the properties of the right-hand member of eq. (3) when the normal&d wavefunction product is represented by sin n(r - ro)/n(r - ro), and the polynomial g(r) is represented by CQ,, rnr. Eq. (3) may then be rewritten sin n(r - r, j G,, = c am $I-“’ ---dr = c CI,,~f,n,, , 7i(T - ‘0) I?1 nl
(6a)
where I
rrn sin )z(r - rO) dr. ??l)i _In(r-- To) =
(6b)
If z = r - ro, eq. (6b) becomes a
TtI ,11,1=
s
,,1
(z+‘O)
sin tlz ,-dz
(74
.
-a
The range of integration, though formally --a~ + m, in practice Is limited to a finite segment r. -a
1 +z/To)“* sin(nz)/z ,
which if it is sufticicntly cant range of integation rr [sin(nz)/z
peaked that over the signifiz/r0 < 1, may be written as
+ (m/r) sin(rzz)]
.
The contribution from the second term to the integral is zero. The contribution of the first term is 30” Si(a) where Si is the sine integral. When (I is large this is
nr?, which is not a bad approximation
when a is
smaller. In fact, the properties of sin(rlz)/z suggests that if a is set at one of its zeros, such that tza = p, p should not have to exceed three or four and could be smaller. Eq. (7a) may be written
L7
+
... +
s
z‘72-l sinlIz dz .
C7b)
-u
253
Volume
17, number
CHEMICAL
2.
Eq. (4) may also be rewritten
Ymn xl.
a
Yrnn=
-u
y[tin~nz~,zl dz
(rgp
-a
The term J’T, [sin(nz)/z] dz which occurs in eqs. (75) and (8a) is twice the sine integral. Even for relatively small values of II, and finite limits, it oscillates with decreasing amplitude (Sa) then becomes
about
a
nfo”Y,nn = J(z +rJrn
rr with increase
sy
&
= ‘Tlrn”
(8d)
Under these circumstances eqs. (6a) and (7d) imply that
1(z +r())‘” [sin(rzz)/zl dz
YT".. =
15 November 1972
PHYSICS LETTERS
in a. Eq.
@b)
.
-a
Eqs. (7b) and (8b) becoine
Gn=C=,nlmn~Cnm~~=g(ro).
(10)
m
III
On the other hand, when Am,, is not small ccmpared to 1, which occurs when rz is small and the wavefunction product is not strongly peaked, the approximation will not hold. These points are well iliustrated in table 1 in which the dependence of AmI, upon rtz, II and p is displayed for values of m between zero and five. Illustrative numerical values of A,*” are also displayed for p = 1,2,3, y. = 1 for the two extreme values of 12. 1 and 10. In practical applications of the r-centroid method through equations such as (3), (6a) and (10) in which
molecular Froperties are expressed as empirical power series of r, the polynomial in r is seldom of degee higher than three or four.
??I
nk=l ‘;n,1= 1tlC
0k 4 III
Ak- l,n
=( 1 + Am,,1
3
(SC)
where
A
=O;
zx,!,rr
s is an integer,
Pb)
Thus the natcre of the r-centroid approximation which arises from thk locally peaked Eature of the oscillatory wavefunction product has been demonstrated to have close similarity in behaviour to the S-sequence finction sin n(r - ro)/n(r - ko)_ Whether or not well developed peaks form, depends in a very sensitive manner upon the relative phases of the two wavefunctions through the Franck-Condon principle. Under conditions of significant constructive overlap between the two wave functions a”,, and air,!, , eq. (5) is closely approximated and the Y:*vft wifi not differ significantly from unity, as noted in a very large number of cases
[41However, in cases where there is no significant structive
PC) The nature
and limitations of the r-centroid approximation are evident from eqs. (7~) and (8~). When A ,yrn is much less th.an 1, which occurs for large values of 11 and implies a strongly peaked wavefunction product,
where ‘0 is. the r-centroid
rYaYI, and
interference
between
conthe two wavefunctions,
eq. (5) is not necessarily a good representation
of eqs.
(I), (2) and (3) which are themselves inexact. The wavefunction product is then not similar in shape to the 6-sequence function sin n(r- ro)/n(r - r00) and the Y,“*,,. may be expected to diverge signific*antly from unity, as has been observed in a number of cases, particularly for bands of hydrides and deuterides. This work has been supported by research grants from the National Research Council of Canada and from the Defence Research Board of Canada.
Volume
17. number
2
CHEhIlCAL
15 November
PHYSICS LETTERS
1972
Table 1 of A,, upon m, r~and p
Dependence
A nw @cl = 1)
p=2
p=l m
n= 1
A rnn
n= 10
I7 =
10
p=3 -/I=1
77=10
I
77=
0 0
0 0
0 0
0
-0.04
6
0.06
-0.007
1
0.01
-
0 1
0
2
i-W+&
0
0 0
0 0
2
0.02
-4
0.3
0.003
-
0.00016
-1.57
-0.015
532
0.002
-
-0.0028
107
0
3
(-l)p+l
4
(-1)F+1&
m
p
(-HP+1
5
19.3
1-l. 24(pf;;;6)) 0
*
3.93
‘1+,,,,;;-6)
R.W. Nicholls, in: Physical chemistry, an advanced treaand W’. Jost tise, Vol. 3, eds. M. Eyring, D. Henderson (Academic Press, New York, 1969) ch. 3; B.H. Armstrong and R.W. Nicholls, Emission, absorption and rnnsfcr of radiation in heated atmospheres (Pcrgamon Press, London, 1972). P.A. Fraser, Can. J. Phys. 32 (1954) 515. J. Drake and R.W. Nicholls, Chem. Phys. Letters 3 (1969)
457. McCnIlum,
Space
Science,
W.R.
Jnrmain
and R.W. Nicholls,
1, Centre for Rcsexch York
University,
31.5
-0.0 1
>
References
scopic Report
0.05
(
0 t
J.C.
0.7
March
Spectro-
in Experimenti 1970;
J.C. McCallum and R.W. Nicholls, Spectroscopic
Report
2, Centre for Reenrch in Esperimental Space Science. York Univerity, October 1971; J.C. McCallum, W.R. Jarmain and R.W. Nicholls, Spectro-
scopic Report 3, Ccntre for Resexch in Esperimcntal Space Science, York University, March 1972; Spectroscopic Report 4, Centrc for Research in Experimental May 1972. Vols. 1, 2 (Herman, 151 L. Schwartz, Thborie des distributions, Paris, 1950.1951); G. Temple, J. Lond. hlath. Sot. 28 (1953) 134; Proc. Roy.
Sparr Science, York University,
Sot. A228 (19.55) 125: hf. Lkhthill, !ntroduction
to Fourier analysis and generUniv. Press, London, 1959). [cl R.W. Nicholls and A.L. Stewart, in: Atomic and molecular al&d
functions
(Cambridge
processes, ed. D.R. Bates (Audemic 1962) ch. 2
Press,
New York,
255