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Electromagnetic moments and fields induced by nuclear vibrational motion in molecules
Volume 179. number3
CHEMICAL PHYSICS LETTERS
19Aptil 1991
Electromagnetic moments and fields induced by nuclear vibrational motion in molecules Paolo Lazzeretti, Massimo Malagoli and Riccardo Zanasi Dipartimentodi Chimica dell’Universitddegli Studi di Modena. Via Campi 183. 41100 Modena, Italy Received 18 December 1990;in final form 4 February 1991
A simple model Hamiltonian is analyzed within the framework of time-dependent perturbation theory to show how nuclear vibrations induce instantaneous electromagnetic moments in the electron cloud of a molecule. These moments can be rationalized in terms of nuclear electromagnetic shielding tensors. The vibrational motion of a nucleus also gives rise to induced electric and magnetic fields at the other nuclei. This effect is discussed introducing nuclear electric and magnetic coupling tensors, which are related to nuclear shielding and other molecular properties. Nuclear electromagnetic couplings have been calculated for water, ammonia and methane molecules using the random-phase approximation.
1. Introduction Let us consider a closed-shell molecule with n electrons and N nuclei in its singlet ground state. It has been shown [ 1] that, in the presence of electromagnetic radiation, which, for the sake of simplicity, is assumed to be represented by a monochromatic plane wave of frequency w, oscillating electric and magnetic moments are induced in the electron cloud. These moments can be discussed in terms of electric polarizabilities and magnetic susceptibilities [ I 1. In addition, the perturbed electrons induce electric and magnetic fields at the nuclei, which can be rationalized introducing the idea of nuclear electromagnetic shieldings [ 2-5 ] and nuclear electromagnetic couplings [ 6 1. The present note is aimed at showing that these response tensors can also be used to discuss the instantaneous moments and fields induced within the electron cloud by nuclear motion.
2. Vibrationally induced
electronicmoments
Let us assume that the molecule lies in its equilibrium geometry specified by the nuclear positions RjO’, I= 1,2, .... N. The coordinate X1=&- Rf”l de-
scribes the displacement of nucleus I from equilibrium. In the absence of external perturbation, the static electronic Hamiltonian can be expanded in powers of Xf, H,=HiO’ +H6” +... ,
(1)
, = -F&
_FN-J fa
9
denoting by Fy-’ the force on nucleus I due to the other nuclei. The first-order function corresponding to Hamiltonian (2) can be obtained using the Hellmann-Feynman theorem [ 71. We can generalii this result via a heuristic procedure to discuss the frequency-dependent case. In the following example we will assume that the nuclei are classical particles, moving in phase as linear oscillators of frequency w. Accordingly we put in (2) X,(t) =X,,,, COS(~~).
(3)
The origin of time is chosen so that, corresponding to t = 0 the nuclear displacements are maximum, that is X,(O) =X1,,,. We consider the average values, correct to first or-
0009-2614/91/o 03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
297
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CHEMICAL PHYSICS LElTERS
der in the perturbed Hamiltonian (2), of an electronic operator T. Using time-dependent perturbation theory within the framework of our model, the contribution induced by the perturbation (e.g. by the nuclear motion) to the expectation value of Tin the “a” electronic state is given by
for the magnetic dipole operator
A(T),
for the linear momentum operator
=2Re
1 (alTY> exp( -ic+t)cj,,(l)
c #II
>
,
CXp(iUjJ)
z,[029~,(w)x,~+r~sn(o~~~l
1
(14)
where the notation and the definitions for the various quantities are the same as in previous papers [ 2-5 1. In particular, the nuclear shielding tensors are defined:
zJ
(6) where El; =F;/Z, is the operator for the electric field at nucleus J due to the electrons. From this equation we obtain the electronic multipoles induced by the harmonic vibration of the nuclei. Thus, for the electric dipole operator Pa=-e
(13)
=m,E,
fi(u;a-w2)
eN &
(12)
I
WIy)
Using eqs. (4) and (5 ) one finds the formula for the contribution to the observable induced by the inphase harmonic motion of the nuclei in our model system, MT),=
A{m,)=e,f, ZdSfd~F~~-~$d~)~~l,
(4)
where the coefficients are [ 21 Cja(t)=-
(11)
i ria, r=l
(7)
xJm((4E&lj)
(il4la>)
7
(17)
xRe((WLlj)
CiW,d~>).
(18)
Eqs. (S), (lo), ( 12) and (14) are also valid in the limit for w going to zero analyzed in refs. [ 2-5,8 1. In a change of origin r”=r’+d,
for the quadrupole operator
the electromagnetic shieldings transform [ 21 C&iW)=&(W;r’)+
&&%(w),
(19) (20)
298
19April 1991
CHEMICAL PHYSICS LETTERS
Volume 179, number 3
field of the electrons on nucleus I
consistently with A(m,(r”)>=.Um,(r’)>
(22) (21)
We also observe that from eqs. ( 7 )- ( 14) one gets definitions for derivatives of expectation values with respect to nuclear positions and nuclear velocities valid for any 6.1.Thus, for instance, a(k) ax,
=-z,e&,
-a(m,) ax,#
=-Z,e&,
where all the quantities depend on o and the partial derivatives are taken at constant values of all the other variables. These formulae are more general than those previously reported for the static case [ 2-5,8101. Eq. (8) provides a model to interpret some essential features of infrared spectroscopy, as it shows how molecular motion can induce molecular electric dipole moments which, in turn, interact with the electric field of IR radiation. In much the same way, eq. ( 12) accounts for a mechanism whereby molecular vibrations induce electron circulation, i.e. angular momentum (and, consequently, a magnetic moment) within the electron cloud. In vibrational circular dichroism (VCD), that is differential absorption of left- and right-circularly polarized IR radiation, the induced magnetic dipole interacts with the z/2 out-of-phase magnetic field [6] of the external radiation. One can notice that these results have been obtained within the Born-Oppenheimer approximation.
the electric field induced on nucleus Z by molecular vibration becomes
-&wxJpl
>
(23)
where
(24)
xW(alELIj)
(il-%la>),
(25)
can be callid nuclear electric coupling tensors [ 6 1. It can be shown that they are related to the electron relaxation term contributing to IR force constants
161. Similarly, from the operator for the magnetic field of electrons on nucleus Z, Bya=-
n MRI)
&ML,
ML= C i=l
Iri-R,13 ’
(26)
the contribution induced by the nuclear motion is A(BL}=-e -&_&$tJfil
f Z, [R&(w)X,, J=I ,
(27)
where 3. Vibrationally induced electric and magnetic fields at the nuclei The idea of nuclear electric and electromagnetic: couplings naturally arises when external radiation impinging on a molecule is described in terms of Hertz-Righi vectors [6]. We show hereafter that these molecular tensors are alsd useful to discuss effects due to nuclear vibration. From eq. (6), using the operator for the electric
(28)
(29)
299
Volume 179. number 3
CHEMICAL PHYSICS LETTERS
19 April 1991
are called nuclear electromagnetic coupling tensors
161. The oscillating fields (23) and (27) would average to zero over the vibration, that is‘integratingover a period, so that only instantaneous electromagnetic fields are induced by nuclear vibration. On the other hand, in the limit of zero frequency,&$(O) is not zero, as in similar cases analyzed in ref [ 9 1. In the absence of an external magnetic field the wavefunctions 1a) and Ij} may be chosen to be real if )a> is non-degenerate, then tensors (25) and (28) vanish for diamagnetic molecules. The electromagnetic couplings are related to the magnetoelectric shielding of nucleus I,
)-Km)
1*
(37)
The electric shieldings (15) and (16) and the electromagnetic shieldings (17) and (18) are related to the electric couplings
r&3(w) =-m-10-2,;, j&(w)=
z.&%&+c$9(0)1
,
-m-h-2 ,i, ZJ&$s(w) ,
(38) (39)
(30) &g(w)=-(2?nc)-‘w-2
xRe((4~h I.0 (jlR~l4 1,
(31)
&( W) =
via the equations
z--w
e2 m
-2 i
Z,[JgJw) -"igay
(32)
I=1
(33) The paramagnetic nuclear shieldings
c wz JO-w2
2c2m2ii J+a
(34)
(35) are also related to the nuclear electromagnetic couplings,
300
(2mC)
-I
,g,
z~~sysR~&i%W)
-
(41)
4. Calculation of electromagnetic couplings In a first attempt, tensors (29) have been calculated for some simple molecules using a computational scheme based upon the random-phase approximation (RPA) implemented within the SYSMO program [ 3-5,101. Basis sets of intermediate quality have been first considered for water, ammonia and methane molecule, taken from previous calculations [ 1I 1. Extended basis sets have also been adopted, known to yield near Hartree-Fock results for a number of second-order properties ]3,4,101. Two values of the frequency have been considered for each molecule, w = 0, that is the static case, w = 0.3 au (corresponding to a wavelength d of 1518.8 A) for water and ammonia and o= 0.2 au (corresponding to A= 2278.2 A) for methane, the same as in previous papers [ 3,4,10]. These values of w are inappropriate to discuss molecular vibration, but are large
19Aprill991
CHEMICAL PHYSICS LETTERS
Volume 179, number 3
Table 1 Nuclear electromagnetic coupling in the water molecule (in au) ‘)
Table 2 Nuclear electromagnetic coupling in the ammonia molecule (in au) w
eko.3
W=O
WC0
basis 101
basis 39
-4.910 2.381
- 1.694 1.306
- 17.865 3.809
-2.112 2.142
-1.106 - 1.370 1.160 1.485
-1.114 - 1.407 1.123 1.449
-0.924 -0.734 1.160 1.485
- 1.536 - 2.080 1.020 2.079
- 1.575 -2.174 1.078 2.141
- 0.094 0.850 0.641 - 0.649 6.932 10.021 -5.180 -7.143
basis 45
basis 95
-2.540
-0.056
- 5.868
-0.024
-1.179 -1.473 1.150 1.509
-0.463 0.615 1.362 - 1.258
- 0.605 0.620 1.513 - 1.493
-0.371 0.756 1.402 - 3.603
-0.638 0.688 1.595 -1.278
- 2.465 -3.996 0.714 2.604
-2.910 -5.251 0.785 2.711
-0.328 0.658 I .97&I -2.223
-0.371 0.699 2.144 -2.381
- 12.208 0.532 2.222 4.477
-2.021 0.529 2.422 -1.691
-0.101 0.867 0.638 -0.732
-0.316 2.464 1.271 - 1.050
-0.71 I 3.615 1.269 -1.160
1.558 -2.361 - 5.909 6.044
1.226 -2.617 -6.126 6.448
-6.195 - 7.487 - 8.005 17.729
- 12.848 -8.077 -8.213 19.826
6.074 9.801 -5.197 -6.632
39.044 21.604 - 5.537 - 12.274
59.187 21.316 -5.674 - 11.477
-0.271 -0.331 -0.645 0.144 0.229 -0.388 0.420 0.639 -0.062
-0.243 -0.330 -0.650 0.144 0.237 -0.420 9.432 0.680 - 0.049
10.497 5.506 - 0.905 0.253 0.482 - 0.430 - 6.243 -2.708 -0.127
2.034 0.436 -0.892 0.253 0.519 -0.473 -1.146 0.350 -0.106
‘) Cartesian coordinates (in au) are 0: (0,0,0.124144), H,: (0, 1.431530, -0.985266).
Table 3 Nuclear electromagnetic coupling in the methane molecule (in au) &ab)
w=o
wzo.2
basis 5 1
basis 110
basis 5 1
basis 110
-0.986 0.697
- 0.768 I .086
- 0.709 1.003
-1.113 0.787
$,cx1
-2.794 3.952
-2.930 4.143
- 3.223 4.558
-3.397 4.804
J4HI.“l XY #r,.n, xz
0.913 - 1.291
1.156 - 1.635
0.947 - 1.339
1.212 - 1.714
j&1
-0.161 0.623
-0.210 0.681
-0.060 0.832
$ik J&m zx
-0.491 -0.517
- 0.024 0.760 -0.597 0.407
-0.576 -0.664
-0.681 -0.550
&HI UiyFH! 11
U;KHI.C
JfHI.HZ
w=o.3
basis 10I
basis 39
a) Cartesian coordinates (in au) are C: (0, 0, O,), HI: (0, 1.683396, l.190341),HZ: (0, -1.683396, 1.190341). b) &nl= _,iuC$_H’ _ic$C = _,&$c >l;$‘b’“’= _&$Hl
basis 45
basis 95
‘) Cartesian coordinates (in au) are N: (0, 0, 0.127799), H,: (1.77100, 0, -0.591964), H,: (-0.885499, 1.533729, -0.591964). b) 2,“;” = _2”,” W.
enough to check the validity of eqs. (32), (36) and (40) against previously published nuclear shieldings [ 3,4, lo]. In any event, if w is small compared to the natural frequencies w,, one can predict from definition (29) that &$(w) x...@“(O). Accordingly, the zero-frequency limit might be a good approximation for w within the IR range. The results of the calculation are reported in tables l-3. This property seems to be largely affected by basis quality, especially as far as the heavy nuclei are concerned and in the case of non-zero frequency.
Acknowledgement The authors wish to thank Dr. P.W. Fowler for 301
Volume 179,number 3
CHEMICAL PHYSICS LElTERS
useful discussions. Financial support from the CICAIA of the Modena University, from the Italian MPI and from the Italian CNR (Progetto Finalizzato Sistemi Informatici e Calcolo ParalIelo) is also gratefully acknowledged.
References [ 11A.D. Buckingham, Advan. Chem. Phys. 12 ( 1967) 107. [2] P. Lazzerctti, Advan. Chem. Phys. 75 (1987) 507.
302
19April1991
[3] P. Lazzeretti and R. Zanasi, Phys. Rev. A 33 (1986) 3727. [4] P. Iazzerettiand R, Zanasi, J. Chem. Php. 87 (1987) 472. [S] P. Lazzeretti and P. Bertocchi, Chem. Phys. Letters 138 (1987) 465. [6] P. Lazzeretti, Chem. Phys. 134 (1989) 269. [ 71 P. Lamretti and R. Zanasi, Chem. Phys. Letters 118 (1985) 217. [8] P. Lazzeretti, Chem. Phys. Letters 160 (1989) 49. [9] A.D. Buckingham, P.W. Fowler and P.A. Galwas, Chem. Phys. 112 (1987) 1. [ 101P. Lazzeretti, R. Zanasi and R. Bursi, J. Chem. Phys. 89 (1988) 987. [ 111P. Lazzeretti and R. Zanasi. J. Chem. Phys. 77 (1982) 2448.
CHEMICAL PHYSICS LETTERS
19Aptil 1991
Electromagnetic moments and fields induced by nuclear vibrational motion in molecules Paolo Lazzeretti, Massimo Malagoli and Riccardo Zanasi Dipartimentodi Chimica dell’Universitddegli Studi di Modena. Via Campi 183. 41100 Modena, Italy Received 18 December 1990;in final form 4 February 1991
A simple model Hamiltonian is analyzed within the framework of time-dependent perturbation theory to show how nuclear vibrations induce instantaneous electromagnetic moments in the electron cloud of a molecule. These moments can be rationalized in terms of nuclear electromagnetic shielding tensors. The vibrational motion of a nucleus also gives rise to induced electric and magnetic fields at the other nuclei. This effect is discussed introducing nuclear electric and magnetic coupling tensors, which are related to nuclear shielding and other molecular properties. Nuclear electromagnetic couplings have been calculated for water, ammonia and methane molecules using the random-phase approximation.
1. Introduction Let us consider a closed-shell molecule with n electrons and N nuclei in its singlet ground state. It has been shown [ 1] that, in the presence of electromagnetic radiation, which, for the sake of simplicity, is assumed to be represented by a monochromatic plane wave of frequency w, oscillating electric and magnetic moments are induced in the electron cloud. These moments can be discussed in terms of electric polarizabilities and magnetic susceptibilities [ I 1. In addition, the perturbed electrons induce electric and magnetic fields at the nuclei, which can be rationalized introducing the idea of nuclear electromagnetic shieldings [ 2-5 ] and nuclear electromagnetic couplings [ 6 1. The present note is aimed at showing that these response tensors can also be used to discuss the instantaneous moments and fields induced within the electron cloud by nuclear motion.
2. Vibrationally induced
electronicmoments
Let us assume that the molecule lies in its equilibrium geometry specified by the nuclear positions RjO’, I= 1,2, .... N. The coordinate X1=&- Rf”l de-
scribes the displacement of nucleus I from equilibrium. In the absence of external perturbation, the static electronic Hamiltonian can be expanded in powers of Xf, H,=HiO’ +H6” +... ,
(1)
, = -F&
_FN-J fa
9
denoting by Fy-’ the force on nucleus I due to the other nuclei. The first-order function corresponding to Hamiltonian (2) can be obtained using the Hellmann-Feynman theorem [ 71. We can generalii this result via a heuristic procedure to discuss the frequency-dependent case. In the following example we will assume that the nuclei are classical particles, moving in phase as linear oscillators of frequency w. Accordingly we put in (2) X,(t) =X,,,, COS(~~).
(3)
The origin of time is chosen so that, corresponding to t = 0 the nuclear displacements are maximum, that is X,(O) =X1,,,. We consider the average values, correct to first or-
0009-2614/91/o 03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)
297
Volume 179, number 3
19April 1991
CHEMICAL PHYSICS LElTERS
der in the perturbed Hamiltonian (2), of an electronic operator T. Using time-dependent perturbation theory within the framework of our model, the contribution induced by the perturbation (e.g. by the nuclear motion) to the expectation value of Tin the “a” electronic state is given by
for the magnetic dipole operator
A(T),
for the linear momentum operator
=2Re
1 (alTY> exp( -ic+t)cj,,(l)
c #II
>
,
CXp(iUjJ)
z,[029~,(w)x,~+r~sn(o~~~l
1
(14)
where the notation and the definitions for the various quantities are the same as in previous papers [ 2-5 1. In particular, the nuclear shielding tensors are defined:
zJ
(6) where El; =F;/Z, is the operator for the electric field at nucleus J due to the electrons. From this equation we obtain the electronic multipoles induced by the harmonic vibration of the nuclei. Thus, for the electric dipole operator Pa=-e
(13)
=m,E,
fi(u;a-w2)
eN &
(12)
I
WIy)
Using eqs. (4) and (5 ) one finds the formula for the contribution to the observable induced by the inphase harmonic motion of the nuclei in our model system, MT),=
A{m,)=e,f, ZdSfd~F~~-~$d~)~~l,
(4)
where the coefficients are [ 21 Cja(t)=-
(11)
i ria, r=l
(7)
xJm((4E&lj)
(il4la>)
7
(17)
xRe((WLlj)
CiW,d~>).
(18)
Eqs. (S), (lo), ( 12) and (14) are also valid in the limit for w going to zero analyzed in refs. [ 2-5,8 1. In a change of origin r”=r’+d,
for the quadrupole operator
the electromagnetic shieldings transform [ 21 C&iW)=&(W;r’)+
&&%(w),
(19) (20)
298
19April 1991
CHEMICAL PHYSICS LETTERS
Volume 179, number 3
field of the electrons on nucleus I
consistently with A(m,(r”)>=.Um,(r’)>
(22) (21)
We also observe that from eqs. ( 7 )- ( 14) one gets definitions for derivatives of expectation values with respect to nuclear positions and nuclear velocities valid for any 6.1.Thus, for instance, a(k) ax,
=-z,e&,
-a(m,) ax,#
=-Z,e&,
where all the quantities depend on o and the partial derivatives are taken at constant values of all the other variables. These formulae are more general than those previously reported for the static case [ 2-5,8101. Eq. (8) provides a model to interpret some essential features of infrared spectroscopy, as it shows how molecular motion can induce molecular electric dipole moments which, in turn, interact with the electric field of IR radiation. In much the same way, eq. ( 12) accounts for a mechanism whereby molecular vibrations induce electron circulation, i.e. angular momentum (and, consequently, a magnetic moment) within the electron cloud. In vibrational circular dichroism (VCD), that is differential absorption of left- and right-circularly polarized IR radiation, the induced magnetic dipole interacts with the z/2 out-of-phase magnetic field [6] of the external radiation. One can notice that these results have been obtained within the Born-Oppenheimer approximation.
the electric field induced on nucleus Z by molecular vibration becomes
-&wxJpl
>
(23)
where
(24)
xW(alELIj)
(il-%la>),
(25)
can be callid nuclear electric coupling tensors [ 6 1. It can be shown that they are related to the electron relaxation term contributing to IR force constants
161. Similarly, from the operator for the magnetic field of electrons on nucleus Z, Bya=-
n MRI)
&ML,
ML= C i=l
Iri-R,13 ’
(26)
the contribution induced by the nuclear motion is A(BL}=-e -&_&$tJfil
f Z, [R&(w)X,, J=I ,
(27)
where 3. Vibrationally induced electric and magnetic fields at the nuclei The idea of nuclear electric and electromagnetic: couplings naturally arises when external radiation impinging on a molecule is described in terms of Hertz-Righi vectors [6]. We show hereafter that these molecular tensors are alsd useful to discuss effects due to nuclear vibration. From eq. (6), using the operator for the electric
(28)
(29)
299
Volume 179. number 3
CHEMICAL PHYSICS LETTERS
19 April 1991
are called nuclear electromagnetic coupling tensors
161. The oscillating fields (23) and (27) would average to zero over the vibration, that is‘integratingover a period, so that only instantaneous electromagnetic fields are induced by nuclear vibration. On the other hand, in the limit of zero frequency,&$(O) is not zero, as in similar cases analyzed in ref [ 9 1. In the absence of an external magnetic field the wavefunctions 1a) and Ij} may be chosen to be real if )a> is non-degenerate, then tensors (25) and (28) vanish for diamagnetic molecules. The electromagnetic couplings are related to the magnetoelectric shielding of nucleus I,
)-Km)
1*
(37)
The electric shieldings (15) and (16) and the electromagnetic shieldings (17) and (18) are related to the electric couplings
r&3(w) =-m-10-2,;, j&(w)=
z.&%&+c$9(0)1
,
-m-h-2 ,i, ZJ&$s(w) ,
(38) (39)
(30) &g(w)=-(2?nc)-‘w-2
xRe((4~h I.0 (jlR~l4 1,
(31)
&( W) =
via the equations
z--w
e2 m
-2 i
Z,[JgJw) -"igay
(32)
I=1
(33) The paramagnetic nuclear shieldings
c wz JO-w2
2c2m2ii J+a
(34)
(35) are also related to the nuclear electromagnetic couplings,
300
(2mC)
-I
,g,
z~~sysR~&i%W)
-
(41)
4. Calculation of electromagnetic couplings In a first attempt, tensors (29) have been calculated for some simple molecules using a computational scheme based upon the random-phase approximation (RPA) implemented within the SYSMO program [ 3-5,101. Basis sets of intermediate quality have been first considered for water, ammonia and methane molecule, taken from previous calculations [ 1I 1. Extended basis sets have also been adopted, known to yield near Hartree-Fock results for a number of second-order properties ]3,4,101. Two values of the frequency have been considered for each molecule, w = 0, that is the static case, w = 0.3 au (corresponding to a wavelength d of 1518.8 A) for water and ammonia and o= 0.2 au (corresponding to A= 2278.2 A) for methane, the same as in previous papers [ 3,4,10]. These values of w are inappropriate to discuss molecular vibration, but are large
19Aprill991
CHEMICAL PHYSICS LETTERS
Volume 179, number 3
Table 1 Nuclear electromagnetic coupling in the water molecule (in au) ‘)
Table 2 Nuclear electromagnetic coupling in the ammonia molecule (in au) w
eko.3
W=O
WC0
basis 101
basis 39
-4.910 2.381
- 1.694 1.306
- 17.865 3.809
-2.112 2.142
-1.106 - 1.370 1.160 1.485
-1.114 - 1.407 1.123 1.449
-0.924 -0.734 1.160 1.485
- 1.536 - 2.080 1.020 2.079
- 1.575 -2.174 1.078 2.141
- 0.094 0.850 0.641 - 0.649 6.932 10.021 -5.180 -7.143
basis 45
basis 95
-2.540
-0.056
- 5.868
-0.024
-1.179 -1.473 1.150 1.509
-0.463 0.615 1.362 - 1.258
- 0.605 0.620 1.513 - 1.493
-0.371 0.756 1.402 - 3.603
-0.638 0.688 1.595 -1.278
- 2.465 -3.996 0.714 2.604
-2.910 -5.251 0.785 2.711
-0.328 0.658 I .97&I -2.223
-0.371 0.699 2.144 -2.381
- 12.208 0.532 2.222 4.477
-2.021 0.529 2.422 -1.691
-0.101 0.867 0.638 -0.732
-0.316 2.464 1.271 - 1.050
-0.71 I 3.615 1.269 -1.160
1.558 -2.361 - 5.909 6.044
1.226 -2.617 -6.126 6.448
-6.195 - 7.487 - 8.005 17.729
- 12.848 -8.077 -8.213 19.826
6.074 9.801 -5.197 -6.632
39.044 21.604 - 5.537 - 12.274
59.187 21.316 -5.674 - 11.477
-0.271 -0.331 -0.645 0.144 0.229 -0.388 0.420 0.639 -0.062
-0.243 -0.330 -0.650 0.144 0.237 -0.420 9.432 0.680 - 0.049
10.497 5.506 - 0.905 0.253 0.482 - 0.430 - 6.243 -2.708 -0.127
2.034 0.436 -0.892 0.253 0.519 -0.473 -1.146 0.350 -0.106
‘) Cartesian coordinates (in au) are 0: (0,0,0.124144), H,: (0, 1.431530, -0.985266).
Table 3 Nuclear electromagnetic coupling in the methane molecule (in au) &ab)
w=o
wzo.2
basis 5 1
basis 110
basis 5 1
basis 110
-0.986 0.697
- 0.768 I .086
- 0.709 1.003
-1.113 0.787
$,cx1
-2.794 3.952
-2.930 4.143
- 3.223 4.558
-3.397 4.804
J4HI.“l XY #r,.n, xz
0.913 - 1.291
1.156 - 1.635
0.947 - 1.339
1.212 - 1.714
j&1
-0.161 0.623
-0.210 0.681
-0.060 0.832
$ik J&m zx
-0.491 -0.517
- 0.024 0.760 -0.597 0.407
-0.576 -0.664
-0.681 -0.550
&HI UiyFH! 11
U;KHI.C
JfHI.HZ
w=o.3
basis 10I
basis 39
a) Cartesian coordinates (in au) are C: (0, 0, O,), HI: (0, 1.683396, l.190341),HZ: (0, -1.683396, 1.190341). b) &nl= _,iuC$_H’ _ic$C = _,&$c >l;$‘b’“’= _&$Hl
basis 45
basis 95
‘) Cartesian coordinates (in au) are N: (0, 0, 0.127799), H,: (1.77100, 0, -0.591964), H,: (-0.885499, 1.533729, -0.591964). b) 2,“;” = _2”,” W.
enough to check the validity of eqs. (32), (36) and (40) against previously published nuclear shieldings [ 3,4, lo]. In any event, if w is small compared to the natural frequencies w,, one can predict from definition (29) that &$(w) x...@“(O). Accordingly, the zero-frequency limit might be a good approximation for w within the IR range. The results of the calculation are reported in tables l-3. This property seems to be largely affected by basis quality, especially as far as the heavy nuclei are concerned and in the case of non-zero frequency.
Acknowledgement The authors wish to thank Dr. P.W. Fowler for 301
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CHEMICAL PHYSICS LElTERS
useful discussions. Financial support from the CICAIA of the Modena University, from the Italian MPI and from the Italian CNR (Progetto Finalizzato Sistemi Informatici e Calcolo ParalIelo) is also gratefully acknowledged.
References [ 11A.D. Buckingham, Advan. Chem. Phys. 12 ( 1967) 107. [2] P. Lazzerctti, Advan. Chem. Phys. 75 (1987) 507.
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