Hyperpolarizability is a function of frequency. This is called dispersion. Because of the Kramers-Kronig relations, researchers expect that a material that is dispersing light is also absorbing it. Where there is both dispersion and absorption, the molecular polarizabilities are complex functions of the frequency. This led researchers to consider atomic additivity in both the real and imaginary parts of the ordinary and hyperpolarizabilities. This effort is desirable not only from a theoretical point of view, but also because of the existence of a large body of complex refractive index data, which may be used to test the additivity principle with the complex valued ordinary dipole polarizability

Baird, James K.

English

NASA Technical Reports Server

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Center: Consortium for Materials Development in Space 
The University of Alabama in Hunstville (UAH) 
Project Name: "Highly Non-Linear Optical Organic Crystals and Films" 
Subproject Name: "Theory of Atomic Additivity in Molecular Hyperpolarizabilities" 
The University of Alabama in Huntsville 
James K. Baird 
Annual Report: September 15, 1986 to September 14, 1987 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  
Introduction 
The non-linear optical properties of molecular crystals depend upon the 
hyperpolarizabilities of the molecules they contain. Consider an electric field 
with components Ej, j = 1,2,3, measured in a Cartesian system of coordinates. 
The i-th component, mi, of the electric dipole induced in this molecule by the 
field is given by 
mi = aijEj + OijkEjEk f * * *  
where on the right hand side, L e  double occurrence of a subscript implies a 
summation over all values ( 1 ,  2 ,  and 3) which the index may take on. The second 
rank tensor coefficient, Uij, is called the ordinary dipole polarizability, 
while the third rank tensor coefficient, Bijk, is  called the first dipole hyper- 
polarizability. 
If the applied electric field oscillates at frequency, w .  the induced dipole 
moment, mi, oscillates at frequency o and also at frequency 20. The oscillation 
at 2w is the origin of the non-linear optical phenomenon known as frequency 
doubling. Using the time dependent perturbation theory of quantum mechanics, the 
part of Bijk associated with frequency doubling is 
https://ntrs.nasa.gov/search.jsp?R=19880002485 2020-03-20T08:42:23+00:00Z
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uno) - & 1 
(2) J bkonpinlpj 10 (0n2-d (0~2-d 
where the on are resonance frequencies for excitation of the molecule from the 
ground electronic state to the n-th excited state.l 
dipole transition moments connecting the p-th and q-th electronic states of the 
molecule and are given by 
The pPpq are electric 
(3) 
where r represents all of the electronic coordinates, $p(r) is the molecular 
orbital for the p-th state, and is the electric dipole moment operator. The 
subscript, zero, in Eq. (2) refers to the ground state. It is clear from Eq. 
(3) that, if a molecule is t o  have a non-zero value for @ijk (20), it cannot 
have a center of symmetry. 
In the laboratory, @ijk (20) may be determined by laser frequency doubling 
experiments or indirectly by combining measurements of absorption spectral line 
strengths with the difference in the absorption frequency and the fluorescence 
frequency found f o r  the molecule.2 
This structure contains 3 carbon atoms, 5 hydrogen atoms, one carbonyl oxygen 
(=O), and one OH. By Landolt's Rule, the molar refraction of propionic acid is 
given by a linear combination of atoms and bond refractions with coefficients 
specified by the stoichiometry. Using the table, we write 
R = 3(2.591) + 5(1.028) + 2.122 + 2.553 = 17.588 
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(7) 
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The molecular weight and density of propionic acid are M = 74.08 gm/mole and d 
= 1.0030 gm/cm3, respectively. Solving Eq. ( 5 )  for n and substituting the mole- 
cular weight, density, and the value of R given by Eq. (7), we obtain a pre- 
dicted value for the refractive index, 
+ 2(d/M)R % 
n = [I ] = 1.3877 (8) 
1 - (d/M)R 
This is to be compared with the measured value n = 1.3869 reported in the 
Handbook of Chemistry and Physics. 
The Additivity Property for Absorbinp Compounds 
According to Eq. (2), the hyperpolarizability is a function of frequency. 
This is called dispersion. Because of the Kramers-Kronig relations, we can 
expect that a material which is dispersing light is also absorbing it.7 Where 
there is dispersion and absorption, the molecular polarizabilities are complex 
functions of the frequency. This has led us to consider atomic additivity in 
both the real and imaginary parts of the ordinary and hyperpolarizabilities. 
This effort is desirable not only from a theoretical point of  view, but also 
because of the existence of a large body of complex refractive index data, which 
may be used to test additivity principle with the complex valued ordinary dipole 
polarizability. 
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To date, we have shown using Maxwell's equations, that if the wavelength of 
the light is is long compared with the molecular size, the Lorentz-Lorenz equa- 
tion becomes 
112 - 1 
f12 + 2 
= [j d (9) 
where the complex refractive index, fi, has real and imaginary parts n and k, 
respectively, and where the complex valued molar refraction, f i ,  has real and 
imaginary parts, R', and R". Eq. (9) may be used to test the additivity con- 
cept on the case of dispersion/absorption 
In the vacuum ultraviolet, both n and k have been measured for a number of 
compounds , 8  among them being cyclohexane (no carbon-carbon double bonds), cyclo- 
hexene (one double bond), and cyclohexadiene (two double bonds).9 Substituting 
n and k, molecular weight, and density into Eq. (9), we have solved for l? f o r  
each compound. Our hypothesis of additivity in the case of these three 
dispersing/absorbing compounds can be tested in the following way: We let Bo, 
81,  and fi2 be the complex valued molar refractions of the compounds with none, 
one, and two double bonds. Letting f i ,  El, and be the complex valued refrac- 
tions of the hydrogen atom, the carbon-carbon single bond, and the carbon-carbon 
double bond, we have assuming the additivity principle, 
These three equations are n o t  linearly independent, and we find 
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which serves as a check on the additivity principle. We have found Eq. (13) to 
obeyed within experimental error throughout the entire wavelength region (775 to 
1770 Angstroms) where complex refractive index data are available for the cyclic 
hexanes. This result gives us confidence that the additivity principle extends 
to dispersive systems in general. 
The Quantum Theory of Additivity 
A theoretical proof of the additivity idea, nevertheless, would be useful. 
For non-dispersive wavelengths (e.g., for most organic molecules at the Na D 
line), the electric field of the light wave may be assumed to be static. Since 
there are in this case no quantum transitions between states, the variation 
method of quantum mechanics is applicable and has been used with molecular orbi- 
tal theory of the dipole moment to demonstrate that the additivity principle 
applies to the ordinary dipole polarizability in the limit of zero frequency.10 
We are searching for a similar proof for dispersing systems. Using time 
dependent perturbation theory, we have found that the real part, aij'(w), and 
the imaginary part, aij'(o), of the ordinary dipole polarizability satisfy the 
equations, 
The quantity, yj, is the rate of  decay of the 1-th(excited) state of the mole- 
cule due to the joint effect of collisions and spontaneous emission. In the 
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limit yp -. 0, aij"(w) becomes the difference of delta functions in accord with 
Fermi's Golden Rule.11 We maintain Yp finite, in both Eq. (14) and (15) to 
avoid this singularity in the latter. 
Eq. (2) is the real part of the first hyperpolarizability and is analogous 
to Eq. (14) with yp = 0. One notes that Eqs. ( 2 ) ,  (14), and (15) all contain 
the same set of transition dipole matrix elements, so that if additivity applies 
to one, it should apply to all. We propose to demonstrate this by following 
closely the proof already available in the static case. 
Applications 
We envisage three important applications of the additivity principle as gen- 
eralized to dispersing/absorbing systems: 
1. We have already mentioned the concept of predicting molecular dipole 
hyperpolarizabilities from bond and atomic hyperpolarizabilities. Using 
the concepts demonstrated above f o r  propionic acid and for the cyclic 
hexanes, we propose t o  construct a table of bond hyperpolarizability 
contributions found by solving appropriate linear systems of equations 
involving measured molecular hyperpolarizabilities. These tables will 
be used to predict the hyperpolarizabilities of molecules not y e t  
synthesized. 
2. The rate of energy loss of a fast charge particle (e.g. proton or alpha) 
moving through matter is determined by the optical properties of the 
material.12 
particle is the source of an electric field, and because the particle is 
moving, the field i s  time varying. However, by Fourier's theorem, a 
To understand this, one should note that the charge of the 
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time varying field may be decomposed into sinusoidal frequency com- 
ponents. Individually, these have the same effect on the material as 
light. By demonstrating the additivity principle, we shall show that 
fast charged particles lose energy in covalently bonded materials by 
depositing it in the individual chemical bonds. For a complex material 
such as a living cell, there are many types of molecules but a far fewer 
number of bond types. By showing that chemical bonds are the fundamen- 
tal structure in energy loss, we will greatly simplify the study of the 
structure of ionization tracks, which is so important in radiation 
biology and chemistry. 
3. In a dilute gas, a"(@) determines the intensities of spectral lines. In 
condensed matter, we have shown that both a'(@) and a"(@) are involved, 
combined in a non-linear way. By demonstrating the additivity prin- 
ciple, we shall show that the intensities of spectral lines are deter- 
mined by the number and types of bonds in the molecules. This should 
permit us to examine regularities in spectra with new insight. 
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References 
1. . I .  L. Oudar and J. Xyss, Phys. Rev. 26, 2016 (1982). 
2. N. G. Bakhshiev, M. I. Knyazhanskii, V. I. Minkim, 0. A. Osipov, and G. V. 
Saidov, Russian Chemical Reviews 36, 704 (1969). 
3. Reference 1, page 2019. 
4. A. D. Buckingham and B. J.Orr, Quart. Revs. 21, 195 (1967). 
5. K. G. Denbigh, Trans. Far. SOC. 36, 936 (1940). 
6. Handbook of Chemistry and Physics, 59th Edition, 1978/79, page E-356. 
7. A. S. Davydov, Quantum Mechanics, Pergamon Press, Oxford, U.K., 1965, 
451. 
8. M. W. Williams, D. W. Young, J .  C. Ashley, and E. T. Arakawa, J. Appl 
Phys. 58, 4360 (1985) and references therein. 
9. B. T .  Sowers, E. T. Arakawa, and R. D. Birkhoff, J. Chem. Phys. 54, 871 
(1971). 
10. K. J. Miller and J. A. Savchik, J. Am. Chem. SOC. 79, 7206 (1979). 
11. R. G. Gordon, "Correlation Functions for Molecular Motion" in Advances in 
Mametic Resonance, J. S .  Waugh, editor, Vol. 3, 1968, page 4. 
12. H. Frohlich and R. L. Platzman, Phys. Rev. 92, 1152 (1953). 
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Theory of atomic additivity in molecular hyperpolizabilities

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