© Copyright 1998 by the American Chemical Society
VOLUME 102, NUMBER 14, APRIL 2, 1998
ARTICLES Molecular and Atomic Polarizabilities: Thole’s Model Revisited Piet Th. van Duijnen* and Marcel Swart Department of Chemistry (OMAC), UniVersity of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands ReceiVed: NoVember 14, 1997; In Final Form: January 19, 1998
Thole’s modified dipole interaction model for constructing molecular polarizabilities from effective, isotropic atomic polarizabilities is reviewed and extended. We report effective atomic polarizabilities for H, C, N, O, S, and the halogen atoms, independent of their chemical environment. They are obtained by fitting the model both to experimental and calculated molecular polarizabilities, the latter to enable one to model ab initio polarizabilities for various basis sets.
Introduction Atomic and molecular dipole polarizabilities are becoming of increasing importance, e.g., as parameters of “polarizable” force fields, as they are developed presently in the field of mixed quantum/classicalsor QM/MMscalculations on extended systems.1-5 In general, in microscopic descriptions of matter, polarizabilities should be used instead of “effective dielectric constants”, because in an arbitrary microscopic collection of point charges and polarizabilities the energy and the forces depend on the nature of the charges, and on the spatial arrangement of the charges and polarizabilities. As a consequence, interactions between like and unlike charges do in general not show the uniform scaling implied in using a (macroscopic!) dielectric “constant”.6,7 In particular, in classical force fields depending on atomic pair contributions it is common to model the inductive and dispersion interactions by means of isotropic atomic polarizabilities, which in general depend on the “chemical environment”. Usually they are then considered as adjustable parameters in fitting procedures, in which they lose their precise * Corresponding author. E-mail address:
[email protected]. Telephone: (+31)(0)50-3634373. Fax: (+31)(0)50-3634296
physical meaning. For example, in one of the first polarizable force fields, CHARMM,8 one finds for an aliphatic carbon atom a polarizability of 0.98 Å3, and for an aliphatic hydrogen 0.1 Å3. In a simple-minded way, i.e., summing the atomic contributions, this would give for CH4 a polarizability of 1.4 Å3 while the experimental value is 2.65 Å3. Similarly, for CH3OH, using the same polarizability for C, and defining O and H as “hydroxyl”, the model polarizability would be 2.16 Å3 versus an experimental value of 3.33 Å3. Likewise, in recent work of Thompson,3 the polarizabilities of H and O (0.170 and 0.528 Å3, respectively) lead to a polarizability for water of 0.70 Å3 instead of the experimental value, ∼1.5 Å3. In the force field for QM/MM or classical calculations, developed in our laboratory,4,5 we consider molecular polarizabilities not as adjustable parameters but rather as welldefined physical properties of the (sub)system(s) of interest. In these cases, we want polarizabilities as close as possible to their (experimental or calculated) vacuum values. Thole devised a model to calculate molecular polarizabilities9 from (model) isotropic atomic polarizabilities, although formally no relation exists between these entities. The atomic polarizabilities, without need to define atomic “types” or their “environment”, were adjustable parameters, obtained by fitting
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2400 J. Phys. Chem. A, Vol. 102, No. 14, 1998
van Duijnen and Swart
TABLE 1: Experimental and Fitted Molecular Polarizabilities (au) from Fit to 16 Moleculesa molecule
exp
acetone acetonitrile carbon monoxide cyclohexane cyclopentane dimethyl ether p-dioxane ethane ethanol formaldehyde hydrogen methanol nitrogen oxygen propane tert-butyl cyanide
43.12 30.23 13.16 74.23 61.75 35.36 58.04 30.23 34.28 16.53 5.33 22.40 11.88 10.80 43.05 64.72
deviation, %
16 mol
acetamide ethylene oxide formamide methane propanol water deviation, % a
linear model 42.57 28.80 13.08 73.45 61.38 35.27 59.27 29.90 35.22 17.46 5.29 22.58 11.98 10.54 42.30 65.83
dev, %
expon model Learning Set 42.47 29.22 13.06 72.41 60.81 35.17 58.57 30.12 35.07 18.00 5.29 22.76 11.95 10.48 42.18 65.15
1.27 4.74 0.64 1.05 0.61 0.26 2.13 1.09 2.74 5.66 0.74 0.80 0.83 2.43 1.74 1.72 1.78 ( 1.47
38.26 29.89 27.53 17.68 47.04 10.06 6 mol
38.51 29.30 25.86 17.46 47.40 10.06
dev, % 1.51 3.34 0.77 2.46 1.52 0.53 0.90 0.37 2.29 8.88 0.85 1.62 0.62 3.00 2.02 0.67
Thole 1 42.44 28.53 12.28 73.96 61.45 35.41 59.68 29.79 35.22 16.91 5.06 22.37 11.57 10.02 42.40 66.07
0.67 1.97 6.05 1.26 1.91 0.00 1.98 ( 1.95
0.43 0.73 4.91 1.10 0.49 1.50 1.53 ( 1.56
Thole 2
1.57 5.63 6.71 0.37 0.49 0.15 2.82 1.46 2.75 2.30 5.08 0.11 2.63 7.23 1.52 2.09
1.96 ( 1.99 Control Set 38.43 30.11 26.18 17.88 47.27 10.21
dev, %
42.65 28.61 12.28 73.89 61.55 35.36 59.66 30.10 35.50 17.17 5.13 22.54 11.61 10.06 42.45 66.34
2.68 ( 2.22 38.27 29.63 25.43 17.12 48.16 9.61
0.02 0.89 7.63 3.20 2.38 4.45 3.10 ( 2.49
dev, % 1.10 5.38 6.67 0.45 0.33 0.00 2.79 0.45 3.54 3.67 3.80 0.60 2.27 6.88 1.41 2.50 2.62 ( 2.17
38.53 29.63 25.58 17.21 48.39 9.72
0.71 0.90 7.11 2.67 2.87 3.36 2.94 ( 2.11
Thole 1: Thole’s atomic polarizabilities in present geometries. Thole 2: Taken from Thole’s article.
TABLE 2: Atomic Polarizabilties (au) Fitted to the Original Learning Set present
point p is given by: N
Thole
model atom
linear
expon
linear
expon
H C N O
3.5308 10.0533 7.6430 6.2955
3.0588 8.7939 6.6704 5.6480
3.4686 9.4614 7.4557 5.8171
2.8815 8.6716 6.5256 5.3042
a
1.7823
1.9088
1.662
2.089
µp ) rp[Fp -
acetamide ethylene oxide formamide methane propanol water
exp
STO
N31
DZV
DZP
TZV
TZP
38.26 5.12 25.90 27.46 28.95 29.89 11.81 21.24 21.93 22.85 27.53 9.98 16.66 17.64 18.82 17.68 5.54 11.59 11.96 12.60 47.04 18.43 33.61 34.56 36.24 10.06 2.39 3.90 4.20 5.09
30.26 23.56 19.96 12.79 37.46 5.01
31.08 24.17 20.59 13.40 38.60 5.56
the model to experimental polarizabilities of a small “learning set” of molecules, and proved in many unrelated cases to give molecular polarizabilities in most cases with experimental accuracy. In the repeated process of migrating to other computer platforms, Thole’s original programs and data got lost. Furthermore, we were in need of an extended “learning set” of both experimental and computed polarizabilities, the latter to compare fully quantum chemical with QM/MM and fully MM calculations. To this end we recoded Thole’s method, and we reparameterized it. In the following sections we review briefly the method and present results of its application on the polarizabilities of up to 70 different molecules chosen from varying classes. Thole’s Modified Dipole Interactions9 For a collection of N point dipole polarizabilities placed in a homogeneous electric field F, the induced dipole moment at
(1)
with rp the atomic polarizability tensor of p and Tpq the dipole field tensor:
[ ]
x2 xy xz 1 3 Tpq ) 3 - 5 yx y2 yz rpq rpq zx zy z2
TABLE 3: CPHF Polarizabilities (au) of the (Six) Control Set in Various Basis Sets molecule
∑Tpqµq]
q*p
(2)
where 1 is the 3 × 3 unit tensor, rpq is the distance between atoms p and q, and x, y, and z are the Cartesian components of the vector connecting the two. Equation 1 can be rearranged into the single matrix equation RM ) F, where M is a 3N vector containing the induced moments, and R, the relay matrix, is a 3N × 3N matrix:
[
r1-1 0 R) · · · 0
0 r2-1 ··· 0
0 ··· ·· · 0
][
0 0 T21 ··· + · · ··· · TN1 rN-1
T12 0 · · · ···
· · · T1N ··· ··· ·· · · · · ··· 0
]
(3)
containing the inverse of the (isotropic) atom polarizability tensors in 3 × 3 diagonal blocks which are coupled by the dipole field tensors Tpq. Inversion of R yields:
R-1 ) [r-1 + T]-1 ≡ A
(4)
A maps the linear response of the system to a (uniformly applied) electric field. It is just a polarizability, as is obvious from M ) AF, albeit in a 3N × 3N “atomic” representation. A can be reduced to a “normal” (3 × 3) molecular polarizability tensor:
Molecular and Atomic Polarizabilities
J. Phys. Chem. A, Vol. 102, No. 14, 1998 2401
TABLE 4: Atomic Polarizabilities from Fit to 16 ab Initio Molecular (CPHF) Polarizabilities STO
N31
DZV
DZP
TZV
TZP
basis set model
linear
expon
linear
expon
linear
expon
linear
expon
linear
expon
linear
expon
H C N O
0.5278 3.7757 2.1716 1.3761
0.5410 3.8570 2.1945 1.4326
1.2388 7.2605 4.6255 2.7017
1.1178 6.8721 3.9125 2.7382
1.2453 8.2030 4.9165 2.9236
1.0989 7.2352 4.1303 2.8681
.5388 8.0484 5.4064 3.5749
1.3333 7.2545 4.5343 3.3791
.2931 9.0813 5.3701 3.2613
1.1550 8.0552 4.4997 3.1986
1.6106 9.0015 5.6740 3.6476
1.3899 7.8995 4.7383 3.4939
a
1.9331
2.4055
1.7020
2.4966
1.7213
2.4864
1.6898
2.4450
1.6920
2.4297
1.6766
2.4290
dev
3.42
5.61
3.61
4.05
4.77
3.80
3.82
3.41
3.67
3.21
3.37
2.85
TABLE 5: Average Percentage of Experimental Polarizability
TABLE 7: Effective Atomic Polarizabilities (au) from Fit to 52 Molecules
basis set
STO
N31
DZV
DZP
TZV
TZP
SOP SOP (Unso¨ld) CPHF
32.2 65.3 35.8
55.9 79.3 66.2
56.1 82.5 68.7
62.8 70.5 73.0
62.4 84.4 74.7
67.0 74.2 77.8
TABLE 6: Empirical Atomic Polarizabilities for Some Halogen Atoms
a
atom
linear
expon
F Cl Bra a
3.1685 16.5421 22.8464 1.7823
3.0228 16.8649 23.2597 1.9088
Fitted to one experimental value. N
RGmn )
∑ (Aij)mn i,j)1
m,n ∈ {x,y,z}
(5)
empirical atom
linear
expon
ab initio (mean) linear
expon
2.7927 1.5350 1.3368 8.6959 8.7622 7.9421 6.5565 5.2663 4.5428 5.7494 2.6175 2.5694 3.0013 0.5819 1.1109 16.6984 15.0931 13.9500 16.1979 7.0741 9.0475 23.5714 12.1193 15.7347 36.9880 19.6286 23.4488
ab initio components linear
expon
0.5974 7.7764 4.7575 2.6442 0.8149 11.0208 7.5682 12.7339 20.2867
1.3849 6.5780 2.9855 2.3184 1.4906 10.8651 6.7129 12.3384 20.0081
H C N O F S Cl Br I
3.5020 10.1756 7.6048 6.3940 2.9413 19.7422 16.1145 22.6671 34.2434
a
1.7278
2.1304
1.6767
2.4380
1.6623
2.8990
dev, %
3.64
3.76
6.10
5.90
7.62
9.98
In eq 6 we have:
Tpq ) (RpRq)1/2t(x/(RpRq)1/6)
or, for that matter, to any partition of N in subgroups by summing corresponding 3 × 3 blocks. A well-known property of this (so far nonmodified) point dipole model is that it may lead to infinite polarization by the cooperative (head to tail) interaction between two induced dipoles in the direction of the line connecting the two. From the above, one may derive the resulting polarizability components:
with t a shape function, which does not depend on p and q, but is related to a well-behaved (model) charge distribution F(u). Thole originally investigated various forms for the charge distribution of which two survived:
{
3 (a - u) u
