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Frequency-dependence of the nuclear electric shielding in the HF molecule via equation-of-motion-approaches
207
Journal o f Molecular Structure, 93 (1983) 207--212 THEOCHEM Elsevier Science Publishers B.V., A m s t e r d a m -- Printed in The Netherlands
FREQUENCY-DEPENDENCE
OF THE NUCLEAR ELECTRIC
SHIELDING
IN THE
HF MOLECULE VIA E Q U A T I O N - 0 F - M O T I O N - A P P R O A C H E S
P. LAZZERETTI, E. ROSSI
Istituto
di Chimica Organica,
In the presence dependent molecule
and R. ZANASI
external
via Campi
of a spatially uniform, electric
is p o l a r i s e d
field E(=,t),
ponse,
alternating
(refs.l,2).
functions
For instance,
discussed
the m o l e c u l a r
can be ratio-
and generalized
(=)E (=,t) can be aT
in terms of dynamic polarizability.
induced by the external p e r t u r b a t i o n
cloud of a
assuming a linear res-
the induced dipole moment m (=,t)=a O
(Italy)
time-
the electronic
and the relative p h e n o m e n o l o g y
nalized w i t h i n the theory of response susceptibilities
183, 41100 Modena
T
The electric
field
at an arbitrary point r in
space is
(1)
F (~,=,t)=-YaB(r,=)Es(=,t)
where
the dynamic electric
uliar m o l e c u l a r property, to the p e r t u r b i n g
external
(dipole)
shielding tensor ~, a pec-
is introduced to describe field (ref.3).
ordinate RI, the n u c l e a r electric
the response
For a nucleus
I, with co-
shielding tensor can be defined
(ref.4)
V
(RI,=)~VI~x (=)=-2E'<01Zn j (r.-rjo )o In> x
(=2 _=2)-1= on
(2)
on
within the dipole length formalism.
Alternative
finition can be given in dipole velocity
0166-1280/83/$03.00
equivalent
formalism:
© 1983 Elsevier Science Publishers B.V.
de-
208 and in a c c e l e r a t i o n formalism: I (w)=-2z~
In>
x [(W~n - w 2 )mort ] -I
I0 >
(2'')
This is an important m o l e c u l a r quantity,
since its k n o w l e d g e w o u l d
permit to evaluate the effective electric field at an a r b i t r a r y point, nuclei
in particular,
n u c l e a r b o n d relaxation, easily shown (refs.3,4)
the Lorentz forces causing inter -
geometrical that,
changes and so on. It can be
in the static case,
the q u a n t i t i e s
defined via eq.2 fulfill the r e l a t i o n
zIZI~I
(O)=N6
(3)
w h i c h is a T h o m a s - R e i c h e - K u h n sum rule a c c e l e r a t i o n formalism.
frequency,
in m i x e d
length-
It can be shown (ref.4) that eq.3 is a
restatement of the h y p e r v i r i a l Invariance c o n d i t i o n
(ref.5)
(ref.7).
theorem (ref.6) and of gaugeFor the general case of an a r b i t r a r y
the dynamic p o l a r i z a b i l i t y is related (ref.4) to the
n u c l e a r electric shielding:
a ~ (~)=~-2zjZj[y~, (m)-TJ~, (O)1
In the exat calculations, Approximation
(RPA)
(4)
and also w i t h i n the exact R a n d o m - P h a s e -
(ref.8),
the numerical
estimates p r o v i d e d by
eqs. 2, 2', 2'' are equal, eqs. 3, 4 are exactly obeyed. calculations,
based on a truncated space,
e q u a t i o n s indicate the quality of a
In actual
deviations from these
chosen set of expansion.
The p r e v i o u s theory has b e e n applied to investigate the linear response of the h y d r o g e n fluoride m o l e c u l e in the w range from zero up to the lower electronic
resonance frequencies. An e x t e n d e d
basis set of 72 c o n t r a c t e d G a u s s i a n f u n c t i o n s has b e e n p r e p a r e d to build up L C A O - S C F m o l e c u l a r orbitals forming the q u a s i v a c u u m IYSCF > ground state. The c o n t r a c t i o n scheme is (14s8p4d/8s3pld)-->
209
[9s6p4d/6s3pld],
w h i c h gives
a n e a r HF energy,
The RPA e q u a t i o n s
have b e e n s o l v e d w i t h i n
sation p r o p a g a t o r
approach
(TDA)
(ref.8).
and the S i n g l e - T r a n s i t i o n
been
also employed.
with
the C o u p l e d H a r t r e e - F o c k
Fock
(TDHF)
approach.
method
corresponding
the c o r r e s p o n d i n g
dipole m o m e n t
(ref.lO).
and its g r a d i e n t
reliable
were
predictions,
celeration tical
nuclear
have b e e n range
a further
shown
orbitals,
indication
0<=<0.3
a.u.
nuclear
to each other.
length-acceleration of the r e l i a b i l i t y dependence
in the ac-
of the theoreobtained
O t h e r nice computed
in length eq.3
in values,
set of c a n o n i c a l
average
up to 9.88
less
results values
formalisms.
is 9.70,
which
is
of the t h e o r e t i c a l
of the n u c l e a r Cauchy moments
shielding
is
in the range
x (2) = 0 .725,
assuming
expansion
(0) +
Also,
a numerical
2 (
(ref.ll).
and a c c e l e r a t i o n
for a c o m p l e t e
I and 2. The average
• (=)=X
proximate
velocity
are x (0)=1.014 , x (1) = 0 •406,
the t r u n c a t e d
close
length,
are close
The f r e q u e n c y
in Figs.
from the e x p e r i m e n t a l
Sambe e q u a t i o n s
for the TRK sum rule:
from m i x e d
shieldings.
obtained
in Table
when compared
is g i v e n by the RPA results
from 9.60 in a c c e l e r a t i o n
The result
are c o l l e c t e d
A check on the good quality
formalisms:
obtained
of the cal-
and 9 4 - 9 5 % of h y d r o g e n
should be the same a l l o w i n g
Hartree-Fock
w i t h the R e b a n e
The STA and TDA seem to f u r n i s h
shielding
have
Hartree-
w h i c h are too large p a r t i c u l a r l y
formalism.
the d i f f e r e n t which
obtained.
(ref.9)
The results
case
through
In fact up to 9 7 - 9 8 % of fluorine shielding
(STA)
seem to be accurate,
estimates,
Approximation
or T i m e - D e p e n d e n t
to the static
I: the length RPA p r e d i c t i o n s with
(CHF)
polari-
that the R P A is e q u i v a l e n t
The TDA is e q u i v a l e n t
variation-perturbation culations
The T a m m - D a n c o f f
we note
hartree.
the f i r s t - o r d e r
Approximation
Incidentally
-I00.069661
1)+w4X(2)
dynamic
test
(see Table
polarizability
to the actual
RPA s tensor
(5)
II) has shown that obtained in length
through
eq.
formalism.
the ap4 is very
210
TABLE I
-
Static
nuclear
electric
shielding.
F
H
~! (0)
~1 I (01
XAv(0)
X! (0)
x l I (0)
XAv(0)
STA
(1) (v) (a)
1.2952 1.1572 2.9095
1.4202 1.2235 2.4637
1.3369 1.1793 2.?609
0.6332 0.5432 0.6830
0.5610 0.3863 0.3469
0.6091 0.4909 0.5710
TDA
(1) (v) (a)
1.1430 0.9294 1.8246
1.2142 0.9889 1.4030
1.1667 0.9492 1.6841
0.6319 0.5548 0.2243
0.6368 0.5075 0.3051
0.6335 0.5390 0.2512
RPA
(1) (v) (a)
1.0141 1.0079 1.0055
1.0146 1.0137 1.0090
1.0143 1.0057 1.0067
0.5555 0.5517 0.5184
0.5907 0.5905 0.5965
0.5672 0.5646 0.5443
1.0453
1.04
1.04
0.5922
0.62
0.60
a
Exp
aExperimental values r e t a i n e d here: R ( H - F ) = I . 7 3 2 8 bohr, m = 0 . 7 0 6 6 a.u., ~m/~r=0.38 a.u., see refs. 3c,12. The formulas relating the n u c l e a r electric s h i e l d i n g to these quantities are given by Sambe (ref.ll): ~ll(F)=I/9(9+am/~r); x!(F)=i/9(9+m/r); ~ll(H)=l-~m/~r; ~ ! ( H ) = l - m / r
a
TABLE II - Dynamic p o l a r i s a b i l i t y in HF in a.u..
~!(=) RPA 0.37 0.40 0.42 0.434 0.44 0.46 0.48 0.52 0.54
6.837 8.942 16.742 -20.298 -5.038 2.406 4.328 7.626 17.031
Approx.
~11 (=) b
6.110 7.630 13.050 -12.336 -1.830 3.384 4.848 7.830 17.152
RPA 7.756 8.350 8.862 9.299 9.511 10.366 11.552 16.431 23.447
Approx.
b
7.634 8.218 8.722 9.151 9.360 10.200 11.366 16.156 23.042
aIn the range O
frequency
dependent
nuclear
211
o I
/ o
o
£
IO.O0
'
o.~o
'
o.~o
'
o.~o
'
0.40 '
'
o.~o
' 0.60 ' f.O(a.u.)
Fig.l. Frequency dependence of RPA electric shielding of F nucleus. Solid (dashed) lines are relative to perpendicular (parallel) components.
0.
&
o .
= 0
oo
o o
'o:oo 'o.1o
'o.~o
'o~o
' 0.40'
'o.~o 031a ' °Jio
Fig.2. Frequency dependence of RPA electric shielding of H nucleus. Solid (dashed) lines are relative to perpendicular (parallel) components.
212
REFERENCES 1
P. C. Martin, Measurement and Correlation Functions, p. 37 of "Many-Body Physics", ed. C. de Witt and R. Balian, Summer School in Theoretical Physics, Les Houches 1967, Gordon and Breach, New York 1968. 2 D. N. Zubarev, Nonequilibrium Statistical Thermodynamics, Plenum, New York, 1974. 3 P. Lazzeretti and R. Zanasi, Phys. Rev. A24 (1981) 1696; 2_55 (1982) 1790 and references therein; J. Phys. B, At. Molec. Phys. 15 (1982) 521. 4 P. Lazzeretti, E. Rossi and R. Zanasi, Phys. Rev. A, submitted for publication. 5 See, for instance, J. O. Hirschfelder, W. Byers-Brown and S. T. Epstein, Adv. Quantum Chemistry, ed. P. O. Lowdin, Academic, 1964, vo]. i, p.267. 6 See, for instance, S. T. Epstein, The Variation Method in Quantum Chemistry, Academic, New York, 1974. 7 G. P. Arrighini, M. Maestro and R. Moccia, J. Chem. Phys. 44 (1968) 882. 8 See, for instance, J. 0ddershede, Adv. Quantum Chemistry, ed. P. O. Lowdin, Academic, New York, 1978, vol. II, p. 275. 9 T. H. Dunning and V. McKoy, J. Chem. Phys. 47 (1973) 1735. I0 J. K. Rebane, Opt. Spectr. 8 (1960) 242. II H. Sambe, J. Chem. Phys. 58 (1973) 4779. 12 D. Steele and W.B. Person, Molecular Spectroscopy,Chem. Soc. Specialist Periodical Report, London, 1974, vol. 2, p. 357.
Journal o f Molecular Structure, 93 (1983) 207--212 THEOCHEM Elsevier Science Publishers B.V., A m s t e r d a m -- Printed in The Netherlands
FREQUENCY-DEPENDENCE
OF THE NUCLEAR ELECTRIC
SHIELDING
IN THE
HF MOLECULE VIA E Q U A T I O N - 0 F - M O T I O N - A P P R O A C H E S
P. LAZZERETTI, E. ROSSI
Istituto
di Chimica Organica,
In the presence dependent molecule
and R. ZANASI
external
via Campi
of a spatially uniform, electric
is p o l a r i s e d
field E(=,t),
ponse,
alternating
(refs.l,2).
functions
For instance,
discussed
the m o l e c u l a r
can be ratio-
and generalized
(=)E (=,t) can be aT
in terms of dynamic polarizability.
induced by the external p e r t u r b a t i o n
cloud of a
assuming a linear res-
the induced dipole moment m (=,t)=a O
(Italy)
time-
the electronic
and the relative p h e n o m e n o l o g y
nalized w i t h i n the theory of response susceptibilities
183, 41100 Modena
T
The electric
field
at an arbitrary point r in
space is
(1)
F (~,=,t)=-YaB(r,=)Es(=,t)
where
the dynamic electric
uliar m o l e c u l a r property, to the p e r t u r b i n g
external
(dipole)
shielding tensor ~, a pec-
is introduced to describe field (ref.3).
ordinate RI, the n u c l e a r electric
the response
For a nucleus
I, with co-
shielding tensor can be defined
(ref.4)
V
(RI,=)~VI~x (=)=-2E'<01Zn j (r.-rjo )o In>
(=2 _=2)-1= on
(2)
on
within the dipole length formalism.
Alternative
finition can be given in dipole velocity
0166-1280/83/$03.00
equivalent
formalism:
© 1983 Elsevier Science Publishers B.V.
de-
208 and in a c c e l e r a t i o n formalism: I (w)=-2z~
In>
x [(W~n - w 2 )mort ] -I
I0 >
(2'')
This is an important m o l e c u l a r quantity,
since its k n o w l e d g e w o u l d
permit to evaluate the effective electric field at an a r b i t r a r y point, nuclei
in particular,
n u c l e a r b o n d relaxation, easily shown (refs.3,4)
the Lorentz forces causing inter -
geometrical that,
changes and so on. It can be
in the static case,
the q u a n t i t i e s
defined via eq.2 fulfill the r e l a t i o n
zIZI~I
(O)=N6
(3)
w h i c h is a T h o m a s - R e i c h e - K u h n sum rule a c c e l e r a t i o n formalism.
frequency,
in m i x e d
length-
It can be shown (ref.4) that eq.3 is a
restatement of the h y p e r v i r i a l Invariance c o n d i t i o n
(ref.5)
(ref.7).
theorem (ref.6) and of gaugeFor the general case of an a r b i t r a r y
the dynamic p o l a r i z a b i l i t y is related (ref.4) to the
n u c l e a r electric shielding:
a ~ (~)=~-2zjZj[y~, (m)-TJ~, (O)1
In the exat calculations, Approximation
(RPA)
(4)
and also w i t h i n the exact R a n d o m - P h a s e -
(ref.8),
the numerical
estimates p r o v i d e d by
eqs. 2, 2', 2'' are equal, eqs. 3, 4 are exactly obeyed. calculations,
based on a truncated space,
e q u a t i o n s indicate the quality of a
In actual
deviations from these
chosen set of expansion.
The p r e v i o u s theory has b e e n applied to investigate the linear response of the h y d r o g e n fluoride m o l e c u l e in the w range from zero up to the lower electronic
resonance frequencies. An e x t e n d e d
basis set of 72 c o n t r a c t e d G a u s s i a n f u n c t i o n s has b e e n p r e p a r e d to build up L C A O - S C F m o l e c u l a r orbitals forming the q u a s i v a c u u m IYSCF > ground state. The c o n t r a c t i o n scheme is (14s8p4d/8s3pld)-->
209
[9s6p4d/6s3pld],
w h i c h gives
a n e a r HF energy,
The RPA e q u a t i o n s
have b e e n s o l v e d w i t h i n
sation p r o p a g a t o r
approach
(TDA)
(ref.8).
and the S i n g l e - T r a n s i t i o n
been
also employed.
with
the C o u p l e d H a r t r e e - F o c k
Fock
(TDHF)
approach.
method
corresponding
the c o r r e s p o n d i n g
dipole m o m e n t
(ref.lO).
and its g r a d i e n t
reliable
were
predictions,
celeration tical
nuclear
have b e e n range
a further
shown
orbitals,
indication
0<=<0.3
a.u.
nuclear
to each other.
length-acceleration of the r e l i a b i l i t y dependence
in the ac-
of the theoreobtained
O t h e r nice computed
in length eq.3
in values,
set of c a n o n i c a l
average
up to 9.88
less
results values
formalisms.
is 9.70,
which
is
of the t h e o r e t i c a l
of the n u c l e a r Cauchy moments
shielding
is
in the range
x (2) = 0 .725,
assuming
expansion
(0) +
Also,
a numerical
2 (
(ref.ll).
and a c c e l e r a t i o n
for a c o m p l e t e
I and 2. The average
• (=)=X
proximate
velocity
are x (0)=1.014 , x (1) = 0 •406,
the t r u n c a t e d
close
length,
are close
The f r e q u e n c y
in Figs.
from the e x p e r i m e n t a l
Sambe e q u a t i o n s
for the TRK sum rule:
from m i x e d
shieldings.
obtained
in Table
when compared
is g i v e n by the RPA results
from 9.60 in a c c e l e r a t i o n
The result
are c o l l e c t e d
A check on the good quality
formalisms:
obtained
of the cal-
and 9 4 - 9 5 % of h y d r o g e n
should be the same a l l o w i n g
Hartree-Fock
w i t h the R e b a n e
The STA and TDA seem to f u r n i s h
shielding
have
Hartree-
w h i c h are too large p a r t i c u l a r l y
formalism.
the d i f f e r e n t which
obtained.
(ref.9)
The results
case
through
In fact up to 9 7 - 9 8 % of fluorine shielding
(STA)
seem to be accurate,
estimates,
Approximation
or T i m e - D e p e n d e n t
to the static
I: the length RPA p r e d i c t i o n s with
(CHF)
polari-
that the R P A is e q u i v a l e n t
The TDA is e q u i v a l e n t
variation-perturbation culations
The T a m m - D a n c o f f
we note
hartree.
the f i r s t - o r d e r
Approximation
Incidentally
-I00.069661
1)+w4X(2)
dynamic
test
(see Table
polarizability
to the actual
RPA s tensor
(5)
II) has shown that obtained in length
through
eq.
formalism.
the ap4 is very
210
TABLE I
-
Static
nuclear
electric
shielding.
F
H
~! (0)
~1 I (01
XAv(0)
X! (0)
x l I (0)
XAv(0)
STA
(1) (v) (a)
1.2952 1.1572 2.9095
1.4202 1.2235 2.4637
1.3369 1.1793 2.?609
0.6332 0.5432 0.6830
0.5610 0.3863 0.3469
0.6091 0.4909 0.5710
TDA
(1) (v) (a)
1.1430 0.9294 1.8246
1.2142 0.9889 1.4030
1.1667 0.9492 1.6841
0.6319 0.5548 0.2243
0.6368 0.5075 0.3051
0.6335 0.5390 0.2512
RPA
(1) (v) (a)
1.0141 1.0079 1.0055
1.0146 1.0137 1.0090
1.0143 1.0057 1.0067
0.5555 0.5517 0.5184
0.5907 0.5905 0.5965
0.5672 0.5646 0.5443
1.0453
1.04
1.04
0.5922
0.62
0.60
a
Exp
aExperimental values r e t a i n e d here: R ( H - F ) = I . 7 3 2 8 bohr, m = 0 . 7 0 6 6 a.u., ~m/~r=0.38 a.u., see refs. 3c,12. The formulas relating the n u c l e a r electric s h i e l d i n g to these quantities are given by Sambe (ref.ll): ~ll(F)=I/9(9+am/~r); x!(F)=i/9(9+m/r); ~ll(H)=l-~m/~r; ~ ! ( H ) = l - m / r
a
TABLE II - Dynamic p o l a r i s a b i l i t y in HF in a.u..
~!(=) RPA 0.37 0.40 0.42 0.434 0.44 0.46 0.48 0.52 0.54
6.837 8.942 16.742 -20.298 -5.038 2.406 4.328 7.626 17.031
Approx.
~11 (=) b
6.110 7.630 13.050 -12.336 -1.830 3.384 4.848 7.830 17.152
RPA 7.756 8.350 8.862 9.299 9.511 10.366 11.552 16.431 23.447
Approx.
b
7.634 8.218 8.722 9.151 9.360 10.200 11.366 16.156 23.042
aIn the range O
frequency
dependent
nuclear
211
o I
/ o
o
£
IO.O0
'
o.~o
'
o.~o
'
o.~o
'
0.40 '
'
o.~o
' 0.60 ' f.O(a.u.)
Fig.l. Frequency dependence of RPA electric shielding of F nucleus. Solid (dashed) lines are relative to perpendicular (parallel) components.
0.
&
o .
= 0
oo
o o
'o:oo 'o.1o
'o.~o
'o~o
' 0.40'
'o.~o 031a ' °Jio
Fig.2. Frequency dependence of RPA electric shielding of H nucleus. Solid (dashed) lines are relative to perpendicular (parallel) components.
212
REFERENCES 1
P. C. Martin, Measurement and Correlation Functions, p. 37 of "Many-Body Physics", ed. C. de Witt and R. Balian, Summer School in Theoretical Physics, Les Houches 1967, Gordon and Breach, New York 1968. 2 D. N. Zubarev, Nonequilibrium Statistical Thermodynamics, Plenum, New York, 1974. 3 P. Lazzeretti and R. Zanasi, Phys. Rev. A24 (1981) 1696; 2_55 (1982) 1790 and references therein; J. Phys. B, At. Molec. Phys. 15 (1982) 521. 4 P. Lazzeretti, E. Rossi and R. Zanasi, Phys. Rev. A, submitted for publication. 5 See, for instance, J. O. Hirschfelder, W. Byers-Brown and S. T. Epstein, Adv. Quantum Chemistry, ed. P. O. Lowdin, Academic, 1964, vo]. i, p.267. 6 See, for instance, S. T. Epstein, The Variation Method in Quantum Chemistry, Academic, New York, 1974. 7 G. P. Arrighini, M. Maestro and R. Moccia, J. Chem. Phys. 44 (1968) 882. 8 See, for instance, J. 0ddershede, Adv. Quantum Chemistry, ed. P. O. Lowdin, Academic, New York, 1978, vol. II, p. 275. 9 T. H. Dunning and V. McKoy, J. Chem. Phys. 47 (1973) 1735. I0 J. K. Rebane, Opt. Spectr. 8 (1960) 242. II H. Sambe, J. Chem. Phys. 58 (1973) 4779. 12 D. Steele and W.B. Person, Molecular Spectroscopy,Chem. Soc. Specialist Periodical Report, London, 1974, vol. 2, p. 357.