1) The document discusses finding simple patterns between molecular properties and whether correlations reported in literature stand up to scrutiny. 2) It examines correlations between polarizability (α) and other properties like electronegativity, ionization energy, and finds poor correlations with RMSPE around 40%. 3) Better correlations are found between α and measures of molecular size like the square root of mean square distance of electrons from center (<r^2>)^(1/2) with RMSPE below 20%, and molecular volume with RMSPE around 15%. 4) The best correlation found is between α and the composite variable of volume over hardness cubed with three-quarters power, achieving a RMSPE of 7.3%.

1. Finding simple patterns is not always simple
Ajit Thakkar
Department of Chemistry
University of New Brunswick
Fredericton, New Brunswick E3B 5A3, Canada

2. Primitive patterns of understanding
• Find new, simple relationships among molecular properties, preferably
observable ones
• As emphasized by Charles Coulson, finding primitive patterns of understanding
is the crux of chemical research!
• Are some of the simple patterns in the literature really reliable?
• Can new simple chemical patterns be found in the 21st century?
• To answer the two questions above, we need “good” data . . .
Ajit Thakkar — Winter School, February 2015 3

3. What constitutes “good” data?
• Size: The database must be large enough
• Breadth and Balance: The database must be representative of a wide enough
class of molecules
• Consistency: The database must be consistent enough
• Reliability: The database must be accurate enough
Ajit Thakkar — Winter School, February 2015 4

4. TABS: A database of molecular structures and properties
• 1641 neutral, closed-shell, ground-state organic molecules (at least one C atom
and one or more H, N, O, F, S, Cl, and Br atoms)
• Average: 13 atoms, 56 electrons. Maximum: 34 atoms, 246 electrons
• At least 25 (often many more) instances of each of 24 functional categories
• Consistency is provided by using a (Pople) model chemistry: all data computed
at the same level
• Used B3LYP/aug-cc-pVTZ which should be reliable enough for correlations
among properties
• But double check reliability later
• Details: Blair & Thakkar, Comput. Theor. Chem. 1043, 13 (2014).
Ajit Thakkar — Winter School, February 2015 5

5. Correlations between α and other molecular properties
• There is a huge literature reporting simple correlations between the mean
dipole polarizability (α) and other molecular properties.
• Many references can be found listed in J. Chem. Phys., 141, 074306 (2014).
Apologies to those not mentioned.
• How many of these stand up to close scrutiny using the “good” TABS data?
Ajit Thakkar — Winter School, February 2015 6

6. Correlations between α and χ, η
• Polarizability ought to be inversely related to both electronegativity χ and
hardness η. For example,
– Brown (1961),
– Vela and G´azquez (1990),
– Nagle (1990),
– Ghanty and Ghosh (1993),
– Sim´on-Manso and Fuentealba (1998),
– Ayers (2007)
• Do such correlations really work well?
Ajit Thakkar — Winter School, February 2015 7

7. Polarizability correlates poorly with χ & η
• Using α vs. 1/ηo, where ηo = ǫHOMO − ǫLUMO, gives an RMSPE of 43%.
Equally weak correlations are obtained when either η = I − A or
χ2
= (I + A)2
/4 are used instead of ηo. Different powers of χ and η are worse.
0
40
80
120
160
0 5 10
α
1/ηo
Ajit Thakkar — Winter School, February 2015 8

8. Correlations between α and ionization energy
• Inverse correlations have been reported between α and ionization energy I and
sometimes I2 − I1 as well. For example,
– Dimitrieva and Plindov (1983),
– Fricke (1986),
– Rosseinsky (1994).
• Do such correlations really work well?
Ajit Thakkar — Winter School, February 2015 9

9. Polarizability correlates poorly with the ionization energy
• The best correlation with the vertical ionization energy I involves 1/I2
rather
than 1/I3
leading to an RMSPE of 36%.
0
40
80
120
160
0 6 12 18
α
1/I
2
Ajit Thakkar — Winter School, February 2015 10

10. Correlations between α and molecular size
• Correlation between α and Rk
(R is a characteristic length). For example,
– Wasastjerna (1922),
– Le Fevre (1950’s and 60’s).
Unexpectedly, k > 3 usually found.
• Does this correlation really work well?
• How do we measure molecular size?
Ajit Thakkar — Winter School, February 2015 11

11. Measures of molecular size can be extracted from ρ(r)
• (Bader) Volume V enclosed by 0.001 e/a3
0 isodensity surface of ρ(r)
• (Robb) Size: r2
= r2
ρ(r) dr (related to diamagnetic susceptibility)
• V ≈ 29.07 r2 1/2
has a MAPE of 11%
0
500
1000
1500
2000
0 25 50 75 100
V
<r2
>1/2
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12. The polarizability correlates better with molecular size
• If R = r2 1/2
is taken as the size or characteristic length parameter, then the
best correlation is α versus R [RMSPE = 18%], not versus R3
or R4
.
0
40
80
120
160
0 25 50 75 100
α
<r2
>1/2
• The egregious outliers have a very different spatial extent in one direction.
Ajit Thakkar — Winter School, February 2015 13

13. Correlations between α and molecular volume
• Volume should correlate with α. For example,
– Debye (1929),
– Cohen (1979),
– Gough (1989),
– Laidig and Bader (1990),
– Brinck, Murray and Politzer (1993).
• Does this correlation really work well?
Ajit Thakkar — Winter School, February 2015 14

14. The polarizability correlates fairly well with molecular volume
• V is the volume contained by the 0.001 e/a3
0 electron isodensity surface of
ρ(r). Using α vs. V and V 4/3
give RMSPE’s of 14.8% and 12%, respectively.
0
40
80
120
160
0 500 1000 1500 2000
α
V
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15. What have we found so far?
• The dipole polarizability α correlates only very roughly [RMSPE ≈ 40%] with
χ, η, I.
• Somewhat better correlations with an RMSPE under 20% are obtained with
R = r2 1/2
• Using the (Bader) volume brings the RMSPE down to 12%–15%
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16. What next?
• Can better correlations between α and other molecular properties be obtained?
• Try composite variables, even those for which no physical interpretation may be
apparent.
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17. One composite variable that works better is V/I
• A correlation between α and V/I has an RMSPE of 9.7%.
0
40
80
120
160
0 2000 4000 6000
α
V/I
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18. The best polarizability correlation we found
• Our best correlation vs. V/η
3/4
1 has an RMSPE=7.3%. It involves the hardness
η1 = 2(I − χo) where the electronegativity χo = −(ǫHOMO + ǫLUMO)/2. η1
uses Tozer and deProft’s approximation A = 2χo − I.
0
40
80
120
160
0 1000 2000 3000 4000
α
V/η1
3/4
• Details: Blair & Thakkar, J. Chem. Phys., 141, 074306 (2014).
Ajit Thakkar — Winter School, February 2015 19

19. Do the correlations depend on the model chemistry?
• B3LYP does not scale well with molecular size and overestimates polarizabilities
• Do the correlations change if the model chemistry is changed?
• Double check everything with CAM-B3LYP and ωB97XD
• Numbers change but nothing significant about the correlations changes
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20. Can we really find new, good, simple patterns?
• Connect properties of the electron density ρ(r) with properties of the electron
momentum density Π(p)
Ajit Thakkar — Winter School, February 2015 21

21. Notes on the electron momentum density
• Squaring does not commute with Fourier transformation. The electron
momentum density Π(p) is NOT the Fourier transform of ρ(r)
• Π(p) is the diagonal element of the Fourier transform of the r-space density
matrix
• All the nuclei have p = 0 in the Born-Oppenheimer approximation
• Moments of Π(p), pk
= pk
Π(p) dp, are finite for −2 ≤ k ≤ 4
• All the above pk
can be obtained from experiment
• Details: Thakkar, Adv. Chem. Phys., 128, 303 (2004).
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22. The Compton profile is an observable momentum property
• An observable directly related to Π(p) is the Compton profile J(q). It is the
intensity of inelastic (Compton) scattering at wavelengths shifted, by a
“Doppler broadening” mechanism, from the wavelength at which inelastic
scattering by a motionless electron would be predicted.
• In the impulse approximation, the gas-phase Compton profile for momentum
transfer q is related to the radial momentum density I(p) by
J(q) = 1
2
∞
|q|
I(p)/p dp
• The peak height of J is just J(0) = 1
2 1/p
• A sum rule tells us that
1/p2
= 2
∞
0
q−2
[J(0) − J(q)] dq
• Details: Thakkar, Adv. Chem. Phys., 128, 303 (2004).
Ajit Thakkar — Winter School, February 2015 23

23. How do we obtain molecular size from Π(p)?
• Molecular size tells us how far the electrons are spread out — that is, size is
sensitive to ρ(r) at larger values of |r|
• Reciprocity of position and momentum suggests that molecular size is most
likely to be correlated with properties sensitive to small values of |p|
• How about 1/p = 2J(0) and 1/p2
?
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24. The Compton profile peak height measures molecular volume
• V ≈ 40.81 J(0) has a MAPE of 5.5%
0
500
1000
1500
2000
0 50 100
V
<1/p> = 2J(0)
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25. Molecular volume is predicted rather well by 1/p2
• V ≈ 13.71 1/p2
has a MAPE of 3.1%.
V ≈ 12.70 1/p2
+ 72.11 has a MAPE of 2.1%
0
500
1000
1500
2000
0 50 100 150
V
<1/p
2
>
• Details: Blair & Thakkar, Chem. Phys. Lett., 609, 113 (2014)
Ajit Thakkar — Winter School, February 2015 26

26. The take-home message
• Finding simple (primitive) patterns is important . . .
• but finding them is not simple.
• Don’t believe all the patterns that are reported in the literature.
– For example, α versus other molecular properties.
• Yes, Virginia, it is possible to find good simple patterns even today.
– For example, 1/p2
versus Bader volume.
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