Study of the transferability of some molecular properties

Journal of Molecular Structure (Theochem), 181 (1988) 237-243 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

STUDY OF THE TRANSFERABILITY PROPERTIES

237

OF SOME MOLECULAR

E. KAPUY Quantum Theory Group, Physics Institute, Technical University of Budapest and Department of Theoretical Physics, Jdzsef Attila University of Szeged (Hungary) R. DAUDEL Laboratoire de Chimie Physique, Universitk Pierre et Marie Curie, Paris (France) C. KOZMUTZA Quantum Theory Group, Physics Institute, Technical University of Budapest, Budapest (Hungary) (Received 25 April 1988)

ABSTRACT An ab initio SCF method has been applied to normal saturated hydrocarbons C2n+lH4n+4 (n=0,1,2,3,4), by using a standard STO-3G .basis set. The localized orbital5 of the molecules obtained by Boys’ procedure are investigated and compared. It is shown that, in spite of the “tails” the contributions of the localized orbit& (orbital energy, charge distribution, effective volume, etc.), by increasing the length of the system, converge to definite values depending only upon the position of the corresponding localized orbital in the molecule. This offres a possibility of explaining the transferability of certain molecular properties in a natural way.

INTRODUCTION

The aim of this paper is to show why certain properties of some related systems (molecules, polymers and solids) are transferable. The transferability of various molecular properties (orbital energies, bond energies, orbital electric moments, etc. ) is considered as a long established fact in chemistry. For systems of certain types, e.g. aliphatic hydrocarbons, the bond energies are apparently transferable to an accuracy exceeding that of the results obtained by the Hartree-Fock (HF) method by an order of magnitude

[Il.

In the framework of the independent particle model the one-electron properties (including the total energy at equilibrium geometry) can be written as the sum of contributions from the individual orbitals. This means that the transferability of the one-electron properties is implied by the transferability

236 TABLE 1 Total energies Molecule

Total energy (a.u.)

CI-L C& W&z CJ%, G&w

- 39.72029 - 116.87134 - 194.02255 -271.17381 - 346.32505

of the orbitals. As is well known, however, the canonical orbitals are non-transferable and even the localized molecular orbitals (LMOs) penetrating each other considerably [ 2 ] and having delocalized “tails” [ 3 ] cannot be considered transferable in the strict sense. The transferability of LMOs has been investigated in various papers [ 4-91. The efforts in this direction have been reviewed by O’Leary et al. [lo]. Transferability deserves careful examination because the transferability of the LMOs in related systems would mean that the wavefunctions of larger systems could be constructed from the LMOs of smaller ones without further optimization. CALCULATIONS

The canonical HF equations have been solved for Czn+1H4n+4(n = 0,1,2,3,4), by applying the program system SYCETY utilizing efficiently the CpVsymmetry of the systems [ 11-141. A standard STO-3G basis set has been used with a model geometry RCH= 1.094 A, Rcc = 1.526 A and tetrahedral valence angles. The total energies are shown in Table 1. The dipole moments of these molecules are given in Table 2. The dipole moment vector coincides with the intersection of the two mirror planes of these molecules. The positive direction points from the central C atom at the midpoint of the line connecting the two H atoms. As can be seen the dipole moments are alternating. In the case of C3H8 the calculated value is only 20% of the measured one [ 151. This is probTABLE 2 Dipole momenta (in D ) Molecule

Calculated

Exp.

C&e

0.017 -0.019 0.020 - 0.020

0.083

‘XI,, W-I,, CJ-&o

1151

239 TABLE 3 Orbital energies (in a.u. ) of LMOs

C,H, = CIH?

CH,

W-L

-0.611105 -0.611105

-0.608965 -0.610092

Extrapolated (n=r;o) -0.607716 -0.608886

-0.607276 -0.608433

-0.607089 -0.608238

-0.606951 -0.608091

ably due to the poor basis set and/or to the model geometry chosen. The direction is, however, likely to be correct [ 161. The occupied and the virtual orbitals have been localized with Boys’ procedure [17]. In the following sections we discuss the transferability of certain properties. ORBITAL ENERGIES

We investigate first the elements of the Fock matrix in the localized representation. We take as examples the diagonal elements (“orbital energies” of LMOs) of two CH bonds both at the end of the systems studied: CIH1= denotes an LMO lying in the plane of the C atoms and CiHy an LMO pointing out of that plane (see Table 3). It is apparent that the entries of Table 3 converge when the length of the molecule increases, showing that the localized “orbital energies” are practically insensitive to changes at the other end of the molecule. Similar conclusions can be drawn for the “orbital energies” of the other orbitals and for the off-diagonal Fock matrix elements of orbital pairs. ORBITAL DELOCALIZATION

’

As is well known, the localized orbitals are not strictly localized, i.e. they extend over the whole molecule. We characterize the “delocalization” of the LMOs by giving the net atomic populations n(i,a) = C]c6,1” r where Ci, is the coefficient of the contracted r-th basis function of atom a in LMO i (Only valence shell basis functions are included). In Table 4 we show the net atomic populations of CIH1= in the molecules studied. As can be seen the main contributions (from C, and H,=) are large, while those of the first and second neighbours (H?, Hk, C,, H? , Hi, C3 are still important (l-0.01% ). After the second neighbours however, the net atomic populations rapidly decrease. With increasing the length of the molecule the important contributions are apparently converging to definite values, which means that they are

240 TABLE 4 Net atomic populations of LMO CIHI= Neighbourhood 0th

Atom

CH,

Cl

0.35102

0.34917

0.34877

0.34866

0.34862

H,=

0.28673

0.29045

0.29083

0.29094

0.29098

HDHL 1 1

0.33253-2

0.35178-2

0.35254-2

0.35275-2

0.35282-2

Cz U-L-)

0.33253-2

0.40996-2

0.39825-2

0.39791-2

0.39783-2

Hi’ 3;

0.37481-3

0.37508-3

0.37462-3

0.37450-3

G

0.25567-2

0.25175-2

0.25123-2

0.25109-2

0.13015-4

0.12958-4

0.12941-4

1st

2nd

Hi’&? 3rd

C,W=)

0.14672-4

0.30690-3

0.30426-3

0.30389-3

H? 34

0.28036-3

0.16702-5

0.16837-5

0.16747-5

C5

0.39287-4

0.38367-4

0.38231-4

J% ,Hk

0.86881-6

0.77300-6

0.76888-6

GUS=)

0.89647-5

0.10080-4

0.99769-5

H; 5%

0.95370-7

0.95888-7

G

0.19877-5

0.19439-5

Hi’ J-G

0.27403-7

0.30019-7

C,u-G=)

0.37067-6

0.41493-6

4th

5th

6th

7th

Hf3-I:

0.59722-g

G

0.10071-6

H:,Hk

0.14346-g

8th

9th 0.20282-7

241

transferable if an environment of adequate extension is taken into account. Characteristic of the charge distribution of CIH1= is that there is a displacement of charge from the centre, i.e. the C, and Hi= atoms, to atoms CZC3. (The CIH1= bond is trans coplanar to C&C,.) This has already been noted by England et al. [ 3 ] when analyzing the results of an INDO calculation carried out for some hydrocarbons. Similar conclusions can be drawn for the charge distribution of C, Hy (see Table 5). There is a displacement of charge from the C, and HF atoms to atom Hi. (C, HF is tram coplanar to C2Hk .) The important net atomic populations (the main, first and second neighbours) apparently converge to definite values with increase in length of the molecule. After the second neighbours the contributions decrease even more rapidly than in the case of CIH1=. It should be noted, however, that there is a “disturbance” propagating towards the end of the molecules and as a result, for example, the contribution of C6 is larger than that of C&. FIRST AND SECOND ORDER ELECTRIC MOMENTS AND “EFFECTIVE” VOLUME OF LMOs

The first electric moment of LMO @i is defined as Ei = (#i 1r I&) and the length of centroid of #i as (EP) 1’2 [ 91. In Table 6 the centroid lengths of LMO CIHl = (first row) and of LMO C,HF (second row) are given. The origin is taken as the position of nucleus C1. As can be seen by increasing the molecule size the centroid lengths converge to definite values. The second electric moments are defined in the following way. Calculating the expectation values (%I)

=

(9il

(u--“i)

tvwvi)

Ik)

u,u

=

xJf,z

we obtain a symmetric tensor which can be diagonalized. The square roots of the diagonal elements of the diagonal second moment tensor are the so-called dispersions (Z ) iI2 (9 ) iI2 (2 ) ii2 Daudel et al. [ 17,181 has shown that the product Vi = (3 ):I2 (4 ) f’” (2 ) f’” can be taken as the measure of an “effective volume” of LMO i. In Table 6 we show the effective volume of CIH1= (third row) and of C!,HF (fourth row). We see that with increasing the length of the molecule the ViSof the LMOs converge to definite values, i.e. they are completely insensitive to changes in distant parts of the systems considered. CONCLUSION

The above investigations unambiguously prove that, in spite of the tails, the parts of the LMOs which determine the most important properties are localized to the centre and to the first and second neighbour atoms. These parts of

242 TABLE 5 Net atomic populations of LMO Cl Hy Neighbourhood 0th

Atom

CH,

CSHB

CsHxz

GH,,

GHzo

Cl

0.35102

0.34975

0.34943

0.34937

0.34935

H?

0.28673 0.33253-2

0.29034 0.35253-2

0.29065

0135316-2

29071 0.35348-2

0.29073 0.35360-2

0.33253-2 0.33253-2

0.35475-2 0.40865-2 0.39717-3

0.35552-2 0.40605-2 0.39752-3

0.35559-2 0.40565-2 0.39719-3

0.35561-2 0.40551-2 0.39708-3

0.19712-2 0.75986-3 0.18829-6

0.19662-2 0.75877-3 0.67281-6

0.19657-2 0.75820-3 0.69027-6

0.19655-2 0.75798-3 0.69547-6

0.20116-6 0.12743-4

0.36888-7 0.14620-4 0.11012-5

0.34442-7 0.14589-4 0.11056-5

0.33946-7 0.14582-4 0.11042-5

0.14827-5 0.18357-6 0.33513-6

0.14781-5 0.18937-6 0.30092-6

0.14804-5 0.18846-6 0.30041-6

0.15111-9 0.26470-6

0.38402-g 0.67041-g

0.39629-g 0.32741-6 0.66633-g

0.12895-g 0.18778-7 0.70588-11

0.11471-g 0.18422-7 0.11545-10

0.10455-B 0.20119-B

0.91856-g 0.23630-B 0.52497-10

HI= 1st

2nd

3rd

4th

H4 C5

HF 5th Hk C,U-G=) H:

0.32983-6

6th

7th

Hk C,(W)

I-C? 8th

Hk C9

0.15461-g 0.10784-B

the LMOs are transferable as a whole. The shape of the LMOs however, depends upon the basis. Larger bases having more diffuse functions may result in an increase of the effective volume [ 171, But significant changes cannot be expected. For the virtual orbitals the situation is different. Their localizability

243 TABLE 6 Centroid length (ri) and effective volume ( Vi) of LMOs (in a.~. ) Parameter

Orbital

Molecule CHI

“i

V,

&Hi = C,Hf CIHI= ClH?

1.4047 1.4047 0.6234 0.6234

1.4093 1.4099 0.6294 0.6266

1.4096 1.4103 0.6303 0.6265

1.4098 1.4104 0.6303 0.6265

1.4098 1.4104 0.6303 0.6265

depends strongly on the basis. For STO-3G basis the localizability is similar to that of the occupied orbitals [ 191. By using larger basis sets with more diffuse functions and polarization functions we obtain virtual orbitals which are not all well localized. This may affect the calculation of the correlation energy. ACKNOWLEDGEMENT

The authors are much indebted to Dr. F. Bartha and Zs. Ozoroczy for carrying out parts of the calculation.

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