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Connection between the nuclear electric shielding tensor and the infrared intensity
Volume 112, number 2
CONNECTION
BETWEEN THE NUCLEAR
AND THE INFRARED P. LAZZERETTI
30 November
CHEMICAL PHYSICS LETTERS
ELECTRIC
SHIELDING
1981
TENSOR
INTENSITY
and R. ZANASI
Istifuto di Chimica Organicae Cenrro di CMcoIo Elettronico delI’Unil~ersit& Via Campi 183. III 100 hfodcna, Iralj
Received 19 September 1984; in final form 9 October 1964
The integrated IR intensity can be directly related to the electric shielding. tensor of the nuclei in a molecule. This allows us to measure, by IR spectroscopy, the effective electric field at the nuclei of a molecule immersed in an esternal electric field. Relations chit with tlte dynamic polarizability and the magnetizability, which implies the possibilit_v of a unified treatment of electric and magnetic second-order properties
For many years the integrated infrared intensity has been considered a parameter of relatively less importance than the absorption frequency_ This was due to an overall failure to reduce intensity data of a molecule to some physically meaningful quantity and to the lack of adequate theoretical interpretation of the experimental results. In particular, attempts to relate this this spectroscopic observable to the molecular electronic structure were not fruitful. Renewed attention is presently being paid to the rationalization of IR intensities: a promising approach has been independently suggested by two groups [l--4] in terms of atomic polar tensors. For the Ith nucleus, it is defined as the derivative of the dipole moment m with respect to the nuclear Cartesian coordinate R, PI=
amliJR,_
(1)
In practice, it is expedient to refer the polar tensor to normal coordinates in discussing the IR intensity [14]_ For the ith fundamental band the integrated absorption coefficient can be written Ai =E; lam/ag,I'
,
“have some exceptionally interesting properties”. They can be used to define atomic effective charges and satisfy the sum rule $Z=*.
(3)
In this note we show that the polar tensor can be related to the nuclear electric shielding [5-S], which permits important conclusions and reveals that information on the electron distribution can be obtained from IR intensity. The nuclear electric shielding y’ of atom 1, with nuclear charge 2,. immersed in an external electric field E,,,,was introduced by Stemheimer [9],taking into account an earlier discussion by Feynman [IO], Accurate calculations on atoms [ 11,13] are available_ The definition of yz can be extended to the case of a molecule in a spatially unifomi field, periodic with frequency w. In dipole-length formalism and adopting atomic units [6-81
(2)
where K isa collection of constants, and the derivative with respect to the normal coordinate Qi can be expressed in terms of PI after coordinate transformation [ I-
(4) where
41.
It has been recognized
[2] that the polar tensors
0 009-2614/84/S 03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
(5) B-V.
103
Volume 112, number 2
These definitions hold for a molecule with 12electrons (IV nuclei) with coordinates rj (RI)_ In order to show the connections with the polar tensor, consider the molecular Born-Oppenheimer presence of a static field E:
Hamiltonian
[S] in the
Z,&(O)
= amjaR, = Pr _
(14)
Eq. (14) shows the connection shielding [7,8]
between
the electric
(15)
&3=Q+$3
N H=Ho+R-E-&IRl-E.
30 November 1984
CHEMICAL PHYSICS LETTERS
(6)
I
The (real) molecular wavefunction is expanded in powers of E:
\k=\koiE,\kEff+___
(7)
(sum over repeated indices)_ The equilibrium for the forces on the nuclei reads
condition
and the polar tensor P$_ The shielding tensor y1 conveys a large amount of information. According to its definition [5-131, it enables one to determine the effective electric field and the actual Lorentz force at the nucleus I in the presence of the external perturbation. Assuming linear response [7,8], -%&I,
0) = &J)E,,,GJ)
-
It is also related to the dynamic
(16) polarizability
[7,8]
A’ (8) where the force on the fth nucleus is fF;Y-‘_
Fr=-VIH=ZIE+F;’
(9)
(10) Summing over 1, the last term in (9), expressing the force due to the other nuclei, cancels out since the action equals the reaction_ At first order in Ewe get from (8)
(11) where E;’ = E,{/ZI and
a(w)
= w-2 F
A’
satisfying A’
X=Xa+KP,
104
_
(17)
(18)
where the paramagnetic
contribution
can be written
=(1/4c”)(K$),
(19)
cLY~(OILliXjlLIO).
(20)
the Hellmann-Feynrnan @Z-Q)=2
c’ i
(L@)=3i
~‘~,~~(OlK~l~)(~lLIO). i
IV
the dipole moment m = -
-y’(O)]
This is an interesting resolution of the molecular polarizability into atomic terms: eq. (17) rigorously defines the polarizability af of atom I within the molecule [8], and can be used to rationalize the atomic additivity schemes empirically found to hold [14]_ It can be proven that (11) is a restatement of the Thomas_Reiche-Kuhn sum rule written in mixed length-acceleration formalism [7,8]_ Note that it is equivalent to eq. (3): rediscovered by spectroscopists [l-4] in a different context. As a matter of fact, eq. (11) is a deep relation. Using the torque formalism for the magnetizability and the velocity definition [7,8] for the electric shielding, it can be proven that (11) is also a gauge-invariance condition: The magnetizability is
XP =(1/4c2)(E,P) For wavefunctions rheoreni [ 131
Z#(o)
(21)
denoting by L the electronic angular momentum and byKt=1[Ho, L] the torque exerted by the nuclei on the electrons_ Introducing the velocity definition [7,8 ]
Volume 112, number 2
30 November 1984
CHEMICAL PHYSICS LETTERS
Table 1 Comparison bet\veentheoretical and experimental shielding tensors in the water moleculea)
theory
0 yxx
Y1.Y
yZ=
YH’ xx
Hl yYY
Hl Y,%!
Hl rzy
Hl -r=Z
1.063 1.082
1.049 1.057
1.034 1.037
0.639 0.671
0.717 0.770
-0.062 -0.077
-0.057 -0.062
0.776 0.851
[ 161
c1p. [17]
0
0
3) The results arc for the coordinate system of ref. [ll]: -1.431530, -0.85266). in bohr.
O(o.0,
0.0, 0.124144),
Hl (0.0,
1.431530,
-0.985266),
H2 (0.0,
I
y'(0)
=
-(L!i/ZI)
C
(OIE~ljXjlPIO)o$
,
(22)
i using (15), and imposing the translational invariance [7,8,15] for the total magnetizability (IS), eq. (11) is rediscovered in the form
quite encouraging, which supports the reliability of the present approach_
References
N 2i
C'~~'(O~~,~~~)(~~PIO)=CZI~~(O)=,I~, J
-
J
(23) where P = Z_fpi is the electronic linear momentum and Fiy = i [Ho, P] is the operator expressing the force
of the nuclei on the electrons. Eqs. (14), (16), (17) and (23) reveal the deep physical meaning of the yf tensor. In particular, they suggest that electric and mametic second-order properties are connected and can be reduced to a unitary and synthetic theoretical framework, where y1 plays a central role. Eqs. (I), (2) and (14) relate IR intensity, nuclear electric shielding and polar tensors, so that (4) and (23) can be used to evaluate [6-81 IR intensities a priori. This may serve as a substantial help to spectroscopists, as intensity measurements are rather difficult [4]. On the other hand, the same relations explicitly show the far-reaching nature of the concept of nuclear electric shielding and state that important _ information on electronic structure is available from- IR _ intensity data. In table 1 we compare accurate theoretical values [ 161 and experimental data from a recent
intensity analysis [17] of the water molecule: the coordinate transformations linking Q and RI, PI and PQ (the latter is accessible from actual IR intensity measurements) are reported in ref. [17], within the framework of L-matrix approximation. The results are
[I ] J.F. Biage, J. Hcrranz and J. hlorcillo, An. R. Sot. Esp. Fis. Quim A57 (1961) 81. [2] W.B. Person and J-H. Newton, J. Chem. Phys. 61 (1974)
1040. [3] 1V.B. Person and J.H. Newton, J. Chem. Phys. 64 (1976) 3036. [4] W-B. Person and G. Zerbi. eds., Vibrational
intensities in infrared and Raman spectroscopy (Elsevicr, Amsterdam, 1982).
[S 1 H. Snmbe. J. Chem. Phys. 58 (1973) 4779. [6] P. Laueretti and R. Zanasi, Phys. Rev. A24 (1981) 1696. [7] P. Lazzeretti, E. Rossi and R. Zanasi. Phys. Rev. A27 (1983) 1301; J. Chem. Phys. 79 (1983) 889. [S] P. Lazzeretti and R. Zanasi, Chem. Phys. Letters 109 (1984) 89. [9] R.hl. Sternheimer, Phys. Rev. 96 (1954) 951. [lo ] R.P. Feynman, Phys. Rev. 56 (1939) 340. [ 111 H-P_ Kelly. Advan. Theor_ Phys 2 (1967) 75, and references therein. [13] A. Dalzamo. Advan. Phys. 11 (1962) 281, and references therein. [13] S.T. Epstein, The variation method in quantum chemistry (Academic Press, New York, 1974). [14] KG. Denbigh. Trans. Faraday Sot. 36 (1940) 936. [15] G.P. A.r$hini, hl. hirtcstro and R. Moccia. J. Cbem. Phys. 49 (1968) 882. [ 161 P. Laucretti and R. Zanasi, Chem. Phys. Letters 71 (1980) 529. [17] B.A. Zilles and W-B. Person, J. Chem. Phys. 79 (1983) 65.
105
CONNECTION
BETWEEN THE NUCLEAR
AND THE INFRARED P. LAZZERETTI
30 November
CHEMICAL PHYSICS LETTERS
ELECTRIC
SHIELDING
1981
TENSOR
INTENSITY
and R. ZANASI
Istifuto di Chimica Organicae Cenrro di CMcoIo Elettronico delI’Unil~ersit& Via Campi 183. III 100 hfodcna, Iralj
Received 19 September 1984; in final form 9 October 1964
The integrated IR intensity can be directly related to the electric shielding. tensor of the nuclei in a molecule. This allows us to measure, by IR spectroscopy, the effective electric field at the nuclei of a molecule immersed in an esternal electric field. Relations chit with tlte dynamic polarizability and the magnetizability, which implies the possibilit_v of a unified treatment of electric and magnetic second-order properties
For many years the integrated infrared intensity has been considered a parameter of relatively less importance than the absorption frequency_ This was due to an overall failure to reduce intensity data of a molecule to some physically meaningful quantity and to the lack of adequate theoretical interpretation of the experimental results. In particular, attempts to relate this this spectroscopic observable to the molecular electronic structure were not fruitful. Renewed attention is presently being paid to the rationalization of IR intensities: a promising approach has been independently suggested by two groups [l--4] in terms of atomic polar tensors. For the Ith nucleus, it is defined as the derivative of the dipole moment m with respect to the nuclear Cartesian coordinate R, PI=
amliJR,_
(1)
In practice, it is expedient to refer the polar tensor to normal coordinates in discussing the IR intensity [14]_ For the ith fundamental band the integrated absorption coefficient can be written Ai =E; lam/ag,I'
,
“have some exceptionally interesting properties”. They can be used to define atomic effective charges and satisfy the sum rule $Z=*.
(3)
In this note we show that the polar tensor can be related to the nuclear electric shielding [5-S], which permits important conclusions and reveals that information on the electron distribution can be obtained from IR intensity. The nuclear electric shielding y’ of atom 1, with nuclear charge 2,. immersed in an external electric field E,,,,was introduced by Stemheimer [9],taking into account an earlier discussion by Feynman [IO], Accurate calculations on atoms [ 11,13] are available_ The definition of yz can be extended to the case of a molecule in a spatially unifomi field, periodic with frequency w. In dipole-length formalism and adopting atomic units [6-81
(2)
where K isa collection of constants, and the derivative with respect to the normal coordinate Qi can be expressed in terms of PI after coordinate transformation [ I-
(4) where
41.
It has been recognized
[2] that the polar tensors
0 009-2614/84/S 03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
(5) B-V.
103
Volume 112, number 2
These definitions hold for a molecule with 12electrons (IV nuclei) with coordinates rj (RI)_ In order to show the connections with the polar tensor, consider the molecular Born-Oppenheimer presence of a static field E:
Hamiltonian
[S] in the
Z,&(O)
= amjaR, = Pr _
(14)
Eq. (14) shows the connection shielding [7,8]
between
the electric
(15)
&3=Q+$3
N H=Ho+R-E-&IRl-E.
30 November 1984
CHEMICAL PHYSICS LETTERS
(6)
I
The (real) molecular wavefunction is expanded in powers of E:
\k=\koiE,\kEff+___
(7)
(sum over repeated indices)_ The equilibrium for the forces on the nuclei reads
condition
and the polar tensor P$_ The shielding tensor y1 conveys a large amount of information. According to its definition [5-131, it enables one to determine the effective electric field and the actual Lorentz force at the nucleus I in the presence of the external perturbation. Assuming linear response [7,8], -%&I,
0) = &J)E,,,GJ)
-
It is also related to the dynamic
(16) polarizability
[7,8]
A’ (8) where the force on the fth nucleus is fF;Y-‘_
Fr=-VIH=ZIE+F;’
(9)
(10) Summing over 1, the last term in (9), expressing the force due to the other nuclei, cancels out since the action equals the reaction_ At first order in Ewe get from (8)
(11) where E;’ = E,{/ZI and
a(w)
= w-2 F
A’
satisfying A’
X=Xa+KP,
104
_
(17)
(18)
where the paramagnetic
contribution
can be written
=(1/4c”)(K$),
(19)
cLY~(OILliXjlLIO).
(20)
the Hellmann-Feynrnan @Z-Q)=2
c’ i
(L@)=3i
~‘~,~~(OlK~l~)(~lLIO). i
IV
the dipole moment m = -
-y’(O)]
This is an interesting resolution of the molecular polarizability into atomic terms: eq. (17) rigorously defines the polarizability af of atom I within the molecule [8], and can be used to rationalize the atomic additivity schemes empirically found to hold [14]_ It can be proven that (11) is a restatement of the Thomas_Reiche-Kuhn sum rule written in mixed length-acceleration formalism [7,8]_ Note that it is equivalent to eq. (3): rediscovered by spectroscopists [l-4] in a different context. As a matter of fact, eq. (11) is a deep relation. Using the torque formalism for the magnetizability and the velocity definition [7,8] for the electric shielding, it can be proven that (11) is also a gauge-invariance condition: The magnetizability is
XP =(1/4c2)(E,P) For wavefunctions rheoreni [ 131
Z#(o)
(21)
denoting by L the electronic angular momentum and byKt=1[Ho, L] the torque exerted by the nuclei on the electrons_ Introducing the velocity definition [7,8 ]
Volume 112, number 2
30 November 1984
CHEMICAL PHYSICS LETTERS
Table 1 Comparison bet\veentheoretical and experimental shielding tensors in the water moleculea)
theory
0 yxx
Y1.Y
yZ=
YH’ xx
Hl yYY
Hl Y,%!
Hl rzy
Hl -r=Z
1.063 1.082
1.049 1.057
1.034 1.037
0.639 0.671
0.717 0.770
-0.062 -0.077
-0.057 -0.062
0.776 0.851
[ 161
c1p. [17]
0
0
3) The results arc for the coordinate system of ref. [ll]: -1.431530, -0.85266). in bohr.
O(o.0,
0.0, 0.124144),
Hl (0.0,
1.431530,
-0.985266),
H2 (0.0,
I
y'(0)
=
-(L!i/ZI)
C
(OIE~ljXjlPIO)o$
,
(22)
i using (15), and imposing the translational invariance [7,8,15] for the total magnetizability (IS), eq. (11) is rediscovered in the form
quite encouraging, which supports the reliability of the present approach_
References
N 2i
C'~~'(O~~,~~~)(~~PIO)=CZI~~(O)=,I~, J
-
J
(23) where P = Z_fpi is the electronic linear momentum and Fiy = i [Ho, P] is the operator expressing the force
of the nuclei on the electrons. Eqs. (14), (16), (17) and (23) reveal the deep physical meaning of the yf tensor. In particular, they suggest that electric and mametic second-order properties are connected and can be reduced to a unitary and synthetic theoretical framework, where y1 plays a central role. Eqs. (I), (2) and (14) relate IR intensity, nuclear electric shielding and polar tensors, so that (4) and (23) can be used to evaluate [6-81 IR intensities a priori. This may serve as a substantial help to spectroscopists, as intensity measurements are rather difficult [4]. On the other hand, the same relations explicitly show the far-reaching nature of the concept of nuclear electric shielding and state that important _ information on electronic structure is available from- IR _ intensity data. In table 1 we compare accurate theoretical values [ 161 and experimental data from a recent
intensity analysis [17] of the water molecule: the coordinate transformations linking Q and RI, PI and PQ (the latter is accessible from actual IR intensity measurements) are reported in ref. [17], within the framework of L-matrix approximation. The results are
[I ] J.F. Biage, J. Hcrranz and J. hlorcillo, An. R. Sot. Esp. Fis. Quim A57 (1961) 81. [2] W.B. Person and J-H. Newton, J. Chem. Phys. 61 (1974)
1040. [3] 1V.B. Person and J.H. Newton, J. Chem. Phys. 64 (1976) 3036. [4] W-B. Person and G. Zerbi. eds., Vibrational
intensities in infrared and Raman spectroscopy (Elsevicr, Amsterdam, 1982).
[S 1 H. Snmbe. J. Chem. Phys. 58 (1973) 4779. [6] P. Laueretti and R. Zanasi, Phys. Rev. A24 (1981) 1696. [7] P. Lazzeretti, E. Rossi and R. Zanasi. Phys. Rev. A27 (1983) 1301; J. Chem. Phys. 79 (1983) 889. [S] P. Lazzeretti and R. Zanasi, Chem. Phys. Letters 109 (1984) 89. [9] R.hl. Sternheimer, Phys. Rev. 96 (1954) 951. [lo ] R.P. Feynman, Phys. Rev. 56 (1939) 340. [ 111 H-P_ Kelly. Advan. Theor_ Phys 2 (1967) 75, and references therein. [13] A. Dalzamo. Advan. Phys. 11 (1962) 281, and references therein. [13] S.T. Epstein, The variation method in quantum chemistry (Academic Press, New York, 1974). [14] KG. Denbigh. Trans. Faraday Sot. 36 (1940) 936. [15] G.P. A.r$hini, hl. hirtcstro and R. Moccia. J. Cbem. Phys. 49 (1968) 882. [ 161 P. Laucretti and R. Zanasi, Chem. Phys. Letters 71 (1980) 529. [17] B.A. Zilles and W-B. Person, J. Chem. Phys. 79 (1983) 65.
105