A numerical study of molecular information entropies

4 March 1994

CHEWCAL PHYSICS LETTERS Chemical Physics Letters 2 19 ( 1994) 15-20

A numerical study of molecular information entropies Minhhuy H6 a, Robin P. Sagar a, JosC M. PCrez-JordA a, Vedene H. Smith Jr. ‘, Rodolfo 0. Esquivel b aDepartment of Chemistry, Queens University, Kingston, Ontario, Canada K7L 3N6 b Departamento de Quimica, Vniversidad Autdnoma Metropolitana Apartado Postal 55-534, Iztapalapa, 09340 Mkxico DF, h4Pxico Received 20 October 1993; in tinal form 10 December 1993

AbStEWt Molecular information entropies are computed by means of a three-dimensional numerical integration from wavefunctions expanded in a variety of Gaussian basis sets at different levels. The results substantiate the use of the entropy sum as a measure of basis set quality. This sum is also shown to be sensitive to electron correlation. The previously observed trends for atomic systems computed from Slater-type orbitals are seen to be present in the results from wavefunctions expanded in Gaussian-type orbitals.

1. Introduction Recently, there has appeared numerous successful applications of information theoretical concepts for the analysis of a large variety of physical problems. Our efforts lie in the direction of attempting to elucidate chemical and physical properties of atomic and molecular systems from within the information theory framework. The foundations of information theory rests with Shannon [ I] who attempted to measure the uncertainty of information of a system with a continuous probability distribution p(x), by the relationship

putation of these information entropies in terms of certain expectation values have also been introduced [61. Most importantly, Gadre et al. [ 41 introduced the concept of the sum of information entropies in position and momentum space as being an important quantity in atomic structure theory. These quantities are respectively defined as m S,=-

dx.

(2)

n(p)lnn(p)&,

(3)

co

(1)

There have been attempts to study atomic information entropies at both the Thomas-Fermi [2], Thomas-Fermi-Amaldi [ 3 ] and the Hartree-Fock [ 41 levels. Most recently, these frameworks have been extended beyond Hartree-Fock to include correlation effects [ 5 1. Extremely tight bounds for the com-

p(r)lnp(r)dr,

0

&=-

S= - I P(X) ln&)

s

I 0

where p(r) is the electronic charge density and z(p) is the corresponding momentum density. Studies at the near-Hartree-Fock (NHF ) level for atoms indicated that the sum S,=S,,+S, is a maximum for the ground state. Also this sum increases with an amelioration of the basis set and thus is po-

0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)00029-P

16

hf. H6 et al. /Chemical Physics Letters 219 (1994) 15-20

tentially a good indicator of basis set quality [ 41. Such an indicator would be of vital importance in judging the quality of the charge density generated from an approximate wavefunction. Also, the entropy sum is invariant to scaling in contrast to the individual entropies which are not [ 41. Studies at the correlated level for the lithium atom isoelectronic series substantiated the observation at the NHF level although small discrepancies were observed for some members [5]. In this Letter, we will undertake an examination of the trends present at the molecular level for some diatomic species. We hope to establish the ground work herein upon which subsequent studies of chemical structure, properties and reactivity, as analyzed from the information theory perspective, may be achieved. Previous atomic studies [ 4,5 ] have utilized basis sets consisting of Slater-type orbital (ST0 ) expansions. We shall employ wavefunctions expanded in Gaussian-type orbitals (GTOs) since these are more readily available for molecular systems in the well-known quantum chemistry packages such as GAUSSIAN [ 7 ] or MOTECC [ 8 1. These programs also permit access to a large variety of different basis sets thereby providing a larger data set which would lend more weight in substantiating observed trends. Before investigating diatomic species, in order to provide a consistent study, we present the entropies for the first and second row atoms computed from a variety of different basis sets. From this we hope to determine whether trends previously seen from the ST0 calculations are maintained in the results from the GTO calculations. In this study we do not employ spherically averaged densities as previously done at the atomic level but rather perform the full three-dimensional integration in order to access the total information content of the three-dimensional distributions.

2. Results and discussion Atomic and molecular position and momentum space entropies, S, and S,, are computed from wavefunctions obtained from the HONDO program in the MOTECC package [ 81. All wavefunctions consist of basis set expansion of Gaussian-type orbitals. We employedtheSTO-3G [9,10],3-21G [11,12] and63 1G [ 13,141 basis for all atoms and diatomic spe-

ties. In addition, we used the polarized 3-2 1G* basis set [ 15 ] for the second-row atoms and 6-3 1G** basis set [ 16 ] for all atoms. The electronic configurations and geometries of the ground states of the diatomic species were taken from Huber and Herzberg [ 17 1. In addition, the correlated wavefunctions for all atoms and diatomic species are obtained at the CISD level using the 6-3 1G basis set. The respective entropies were computed from the charge densities via Eqs. (2) and (3) with the use of an automatic three-dimensional quadrature scheme [ 18-201 which has been shown to yield good results in terms of efficacy and accuracy on integrations concerning charge densities. Note that this scheme differs from previous atomic calculations [ 4,5 ] where a one-dimensional integration was performed utilizing the spherically averaged densities. Also in this work, we employed unit-normalized densities,

I pdx) dx= 1

9

(4)

in contrast to the previous works [ 4,5 ] where the densities pi were normalized to the number of electrons in the system. At any rate, the corresponding unit-normalized entropies, So, are related to the Nnormalized ones, S,, by So=SN/N+lnN,

(5)

where N is the number of electrons in the system. We present in Tables 1 and 2 the normalized values for the first and second row atoms for a variety of basis sets. All values reported in Tables l-3 are in atomic units. Since the criterion conventionally used to judge the quality of the basis set is the energy, we include it in Tables 1 and 2. It has been observed at the correlated level that no simple relationships exist between the entropies and the energies [ 5 1. Immediately apparent from Tables 1 and 2 is the fact that the entropy sum does increase in going from the standard STO-3G basis to the better quality basis sets for all the atoms studied in accordance with the energies. For the atoms N through 0, the use of the extended 6-3 1G basis set does result in a greater entropy sum than the case of the 3-21G basis set. However, this is not the case for the remainder of the atoms. The influence of the inclusion of polarization functions on the 6-3 1G basis set increases the entropy sum for lith-

M. Ht? et al. /Chemical

Table 1 U-normalized information ues are in atomic units

entropies

Physics Letters 219 (1994) 15-20

for first row atoms. All val-

Table 2 U-normalized information values are in atomic units

STG-3G 3-21G 6-31G 6-31G” CISD

-7.3155 -7.3815 -7.4312 -7.4313 -7.4314

3.4529 3.7212 3.6652 3.6679 3.6654

4.1922 3.975 1 4.0266 4.0247 4.0265

7.645 1 7.6963 7.6918 7.6926 7.6919

Be

SIG-3G 3-21G 6-31G 6-31G” CISD

-14.3518 - 14.4868 - 14.5667 - 14.5669 - 14.6122

3.3179 3.6059 3.5501 3.5510 3.5620

4.4699 4.2036 4.2554 4.2550 4.3144

7.7879 7.8095 7.8055 7.8060 7.8764

SIG-3G 3-21G 6-31G 6-31G* CISD

-24.1489 - 24.3897 -24.5194 - 24.5220 -24.5623

2.9917 3.3699 3.3145 3.3113 3.3118

5.0207 4.6806 4.7259 4.7297 4.7668

8.0124 8.0506 8.0404 8.0410 8.0786

STO-3G 3-21G 6-31G 6-31G# CISD

-37.1983 -37.4810 - 37.6778 - 37.6808 -37.7156

2.8306 3.0711 3.0537 3.0537 3.0610

5.3497 5.1426 5.1588 5.1597 5.1701

8.1803 8.2137 8.2125 8.2134 8.2311

STO-3G 3-21G 6-31G 6-31G” CISD

-53.7190 -54.1053 - 54.3850 - 54.3854 -54.4091

2.6488 2.7772 2.7963 2.7976 2.8079

5.6534 5.5549 5.5516 5.5503 5.5461

8.3023 8.3321 8.3479 8.3479 8.3541

0

STG-3G 3-21G 6-31G 6-31G” CISD

-73.8041 -74.3936 - 74.7803 - 74.7839 -74.8381

2.3654 2.5053 2.5342 2.5327 2.5511

5.9639 5.8679 5.8615 5.8618 5.8541

8.3293 8.3732 8.3957 8.3945 8.405 1

F

STO-3G 3-21G 6-31G 6-31G-’ CISD

-97.9865 -98.8450 -99.3608 -99.3649 -99.4378

2.1064 2.2454 2.2718 2.2689 2.2861

6.2509 6.1670 6.1627 6.1633 6.1563

8.3573 8.4124 8.4344 8.4323 8.4425

Ne

STG-3G 3-21G 6-31G 6-3lG” CISD

- 126.6045

1.8413 2.0014 2.0318 2.0311 2.0472

6.5402 6.4477 6.4424 6.4423 6.4352

8.3816 8.4491 8.4742 8.4734 8.4823

B

C

N

-

127.8038 128.4738 128.4744 128.5863

entropies

for second row atoms.

All

Energy

S(P)

S(n)

ST

STO-3G 3-21G 3-21G* 6-31G 6-3 1G” CISD

- 159.6684 - 160.8540 - 160.8550 - 161.8414 - 161.8414 -161.8426

2.1008 2.3305 2.3304 2.3352 2.3350 2.3368

6.5832 6.4788 6.4792 6.4793 6.4794 6.4782

8.6840 8.8093 8.8096 8.8144 8.8144 8.8151

STO-3G 3-21G 3-21G’ 6-31G 6-31G” CISD

-

2.1356 2.3929 2.3976 2.3924 2.3921 2.3877

6.685 1 6.5190 6.5158 6.5175 6.5174 6.5393

8.8207 8.9119 8.9134 8.9099 8.9096 8.9270

Al

STO-3G 3-21G 3-21G* 6-31G 6-31G” CISD

-238.8583 -240.5510 -240.5869 -241.8541 -241.8569 -241.8857

2.1613 2.4299 2.4350 2.4303 2.4278 2.4337

6.7988 6.6131 6.6085 6.6123 6.6144 6.6216

8.9601 9.0430 9.0435 9.0426 9.0422 9.0552

Si

STO-3G 3-21G 3-2lG’ 6-31G 6-31Ga CISD

-285.4662 -287.3444 -287.3944 -288.8284 -288.8317 -288.8519

2.1921 2.4086 2.4141 2.4072 2.4059 2.4155

6.8815 6.7278 6.7226 6.7288 6.7301 6.7293

9.0736 9.1364 9.1367 9.1360 9.1360 9.1448

P

STG-3G 3-21G 3-21G’ 6-31G 6-31G” CISD

- 336.8687 - 339.0000 -339.0595 - 340.6890 - 340.6902 - 340.6996

2.1700 2.3586 2.3644 2.3559 2.3561 2.3609

6.9833 6.8476 6.8417 6.8501 6.8496 6.8477

9.1533 9.2062 9.2061 9.2060 9.2057 9.2086

S

STO-3G 3-21G 3-2 1G* 6-31G 6-31G” CISD

-393.1302 -395.5513 -395.6312 -397.4714 - 397.4759 - 397.4866

2.1285 2.2957 2.3006 2.2924 2.2918 2.2973

7.0629 6.9440 6.9387 6.9470 6.9475 6.9442

9.1914 9.2397 9.2393 9.2395 9.2393 9.2415

Cl

STO-3G 3-21G 3-21G’ 6-31G 6-31G” CISD

-454.5421 -457.2765 -457.3710 -459.4429 -459.4479 -459.4659

2.1503 2.2201 2.2246 2.2155 2.2152 2.2206

7.0831 7.0467 7.0416 7.0502 7.0505 7.0472

9.2334 9.2668 9.2662 9.2657 9.2656 9.2678

Energy Li

17

Na

197.0073 198.4681 198.4852 199.5952 199.5956 199.6301

M. H6 et al. /Chemical Physics Letters 219 (1994) 15-20

18

Table 3 U-normalized information entropies for the diatomic series. All values are in atomic units Basis set

Energy

H-H

STO-3G 3-21G 6-31G CISD

-1.1166 - 1.1229 -1.1267 -1.1516

3.8221 3.9297 3.9492 3.9354

2.7100 2.6013 2.5916 2.6436

6.5321 6.5309 6.5408 6.5790

H-O

STO-3G 3-21G 6-31G CISD

-74.3626 -74.9700 -75.3631 - 75.4593

2.6970 2.7912 2.8229 2.8355

5.7108 5.6347 5.6253 5.6258

8.4079 8.4258 8.4482 8.4614

H-F

STO-3G 3-21G 6-31G CISD

-98.5707 -99.4597 -99.9834 -100.1103

2.4243 2.5122 2.5424 2.5551

6.0265 5.9573 5.9483 5.9474

8.4508 8.4695 8.4907 8.5026

H-Cl

STO-3G 3-21G 6-31G CISD

-455.1348 -457.8692 -460.0369 -460.0952

2.3797 2.4440 2.4399 2.4462

6.8928 6.8595 6.8633 6.8620

9.2726 9.3036 9.3032 9.3082

C-H

STO-3G 3-21G 6-31G CISD

-37.7698 -38.0519 -38.2512 -38.3150

3.2021 3.3763 3.3663 3.3696

5.0369 4.8850 4.8977 4.9167

8.2390 8.2614 8.2640 8.2863

c-c

STO-3G 3-21G 6-31G CISD

- 74.4220 - 74.9539 -75.3485 - 75.5842

3.4224 3.5745 3.5576 3.5295

5.2936 5.1818 5.1960 5.2645

8.7160 8.7563 8.7536 8.7941

c-o

STO-3G 3-21G 6-31G CISD

-

111.2245 112.0932 112.6672 112.8635

3.1275 3.2285 3.2304 3.2321

5.7144 5.6546 5.6572 5.6687

8.8419 8.8831 8.8876 8.9008

C-F

STO-3G 3-21G 6-31G CISD

- 135.3042 - 136.4246 -137.1220 - 137.2999

2.9897 3.1172 3.1183 3.1308

5.9350 5.8549 5.8562 5.8555

8.9247 8.9721 8.9745 8.9864

c-s

STO-3G 3-21G 6-31G CISD

-430.4713 -433.1220 -435.2391 -435.3864

2.8521 2.9848 2.9786 2.9743

6.6099 6.5150 6.5203 6.5317

9.4620 9.4999 9.4989 9.5060

C-Cl

STO-3G 3-21G 6-31G CISD

-491.8116 -494.7829 -497.1523 -497.2652

2.8514 2.9307 2.9218 2.9275

6.6603 6.6128 6.6201 6.6218

9.5117 9.5435 9.5419 9.5493

M. Hb et al. /Chemical PhysicsLetters219 (I 994) 15-20

60

-20

' 0

I

I' 4

I

I

1'1'

12 a Atomic Number

I 16

2

Fig. 1. N-normalized information entropies of the fint and second row atoms versus atomic number; ( 0 ) ST, ( A ) S,,, and ( 0 ) SW

ium through carbon, while for oxygen through silicon polarization functions on the 3-2 1G basis set serve to increase the entropy sum. With the exception of the lithium atom, the entropy sum is seen to enhance with the inclusion of electron correlation. In Fig. 1 we present a plot of the information entropies versus atomic number for the atoms at the HF/6-31GQ level using the N-normalized values. There is a great similarity in trends to the analogous plot presented by Gadre et al. in ref. [4]. Note that in that work STOs and spherically averaged densities were employed. Using the neon atom as an example, our value of S, at the HF/6-31GQ level is 38.6819 compared to 38.87 17 from Gadre et al. [ 41. A comparison of values obtained using three-dimensional integration and one-dimensional integration with spherically averaged densities shows little difference for the atoms in S state. However, for the other states there were significant differences in the total entropies usually occurring in the first decimal place for the N-normalized values. The largest difference was the boron atom (‘P state) which yields a difference of about 1% at the HF/6-3 1G” level. The entropy sum increases for the ST0 calculations [ 4,6] in comparison with the calculations done at the split valence

19

GTO (6-3 1G”) level, with the exception of the Na atom, for which both calculations employ spherically averaged densities. In Table 3 we present the entropies for some diatomic species. Note that there is an enhancement of the entropy sum on going from the STO-3G to higher levels for all the species with the exception of the hydrogen molecule. Note however that the difference between the STO-3G and 3-21G level is small in comparison to the other species. Furthermore, the entropy sum is also seen to increase on going from the 3-21G level to the 6-31G level for all members except the H-Cl, C-S and C-Cl species. As in the case of the atoms, the S, value is smallest going from STO3G to 6-3 1G basis sets and the reverse trend is noted for the values of S,. Similarly, it is observed that the entropy sum increases with the inclusion of electron correlation in the wavefunctions for the diatomic species. This trend was also observed for nested basis sets. Hh et al. [ 5 ] studied the atomic information entropies calculated from configuration interaction wavefunctions for members of the lithium isoelectronic series. The CI wavefunctions were built in a cumulatively added manner. Starting with NHF wavefunctions by Clementi and Roetti [ 211 new functions were formed by adding new orbitals for each I-symmetry until the final CI wavefunctions, which satisfy a density convergence criterion, are obtained. Except for small discrepancies, the information entropies were enhanced in general with basis set size. These atomic results together with those of the diatomic species suggest that the entropy sum is indeed a good measure of basis set quality when differences in terms of the energy are relatively large. However, when energetic differences are small, the observed trend varies from system to system. Furthermore, if we assume that the entropy sum is a good estimate of the quality of the density produced then our results would suggest that at least for basis sets close in an energetic sense that the densities do not behave in the same manner as the energies, i.e. a lower energy does not necessarily imply a better density. The effect of the addition of polarization functions for the atomic cases is observed to be small, in most cases of the order of a change in the energy. For the individual entropies, S,, and S,, there does not seem to be any consistent trend although S, increases in going from the STO-3G basis while S, decreases.

20

M. H6 et al. /Chemical Physics Letters 219 (1994) IS-20

3. Conclusions

References

We have computed the atomic and molecular information entropies from wavefunctions expanded in Gaussian basis sets employing three-dimensional numerical integration. The results show that the gross features for the atomic cases previously observed in wavefunctions expanded in STOs are preserved. Also, the use of the entropy sum as a measure of basis set quality is substantiated by the observation that this sum enhances in going from STO-3G to better basis sets, at both the atomic and diatomic level. This is also to be true on going from the GTO (6-3 lG**) basis set to the ST0 basis sets. Furthermore, the result also holds true in going from Hartree-Fock to the correlated level. For the atoms, the influence of polarization functions and more complete basis sets on the entropy sum varies from system to system. On the other hand, the entropy sum for most of the studied diatomic species enhances on going from the STO-3G to the 6-3 1G level with a few exceptions. These results raise questions as to the nature of a criterion for the goodness of a density generated from an approximate wavefunction. Further studies of relationships between the information entropies and energies of molecular species are planned in this laboratory.

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Acknowledgement

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERCC) and by the Mexican Research Council (CONACyt ). JMPJ would like to thank the Spanish Ministerio de Education y Ciencia for a postdoctoral grant.