Optimized Molecular Exponents on Gaussian Basis Sets for Hybrid Orbitals of Hydrocarbon Molecules

Takayoshi ISHIMOTO and Masanori TACHIKAWA


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1 Introduction

The linear combination of atomic orbitals (LCAO) expansion is widely used in molecular orbital (MO) calculations. According to the concept of LCAO approximation, many basis sets, such as the correlation consistent [1, 2] and polarization consistent basis sets [3, 4], have been proposed. In these basis sets, the Gaussian-type function (GTF) is often used due to the convenience of molecular integral calculations. Since most of the GTF exponents are already determined to obtain the best for the lowest electronic structure of each atom (atomic exponents) which are generally fixed during the variational procedure, only LCAO coefficients are optimized in the conventional MO calculations.
There are, however, some questions as to swhether atomic exponents are appropriate for the molecular calculation or the exponent values should be modified for various bonding characters. Some attempts for these questions have been employed by directly optimizing the exponents in molecules (molecular exponents) including first-row elements [5, 6]. Recently, a detailed description of the exponent values has been reported for the excited states of hydrogen molecule with full-CI wave function [7]. However, the molecular exponents of different bonding characters such as sp3-, sp2-, and sp-type hybrids have not been systematically determined, yet. It is expected that the calculation using adequate exponents for each hybrid state will enable us to obtain precise molecular properties.
Nowadays, due to the rapid progress of computational techniques and new methodology, theoretical analysis for the function of biological molecules is one of the most important subjects, via ab initio MO calculation. For example, the fragment molecular orbital (FMO) method [8] has theoretically made it possible to calculate large molecules, such as DNA or protein. In the FMO method a large molecule is divided into small fragments, where the total energy and total physical properties are estimated from the fragments' and fragment pairs' MOs. For the improvement of physical properties, the optimum molecular exponents with a few basis functions for each fragment or fragment pair are expected to show the power for larger molecules. Before the optimization of molecular exponents for the large molecular systems, we should determine the characteristics and optimum molecular exponents for small molecules as the first step of the various hybrid states based on the comparison with the atomic exponents.
In this paper, we optimized the exponent values in GTF for several hybrid states of hydrocarbon molecules in order to elucidate the adequate molecular exponents. Methane and ethane, ethylene and benzene, and acetylene are used to express sp3-, sp2, and sp-hybrid states, respectively. The populations of GTFs and geometrical parameters induced by the difference of hybrid-states are also analyzed in these molecules. In order to show the efficiency of our method, we also compare the results with some high quality basis sets at Hartree-Fock level of calculation.

2 Method

The mth Cartesian GTF (CGTF) is represented as follows:

where N(am) is a normalization factor. am and Rm={Xm, Ym, Zm} are GTF exponent and coordinate of GTF center, respectively. We denote all optimized parameters of am and Rm as W. All GTF centers were fixed on each nucleus through this paper.
The energy gradient with respect to W is expressed as follows,

where fi is the occupation number of the ith molecular orbital, SijΩ, hijΩ, and (ij|kl)Ω are each derivative of the overlap, the one-electron, and the two-electron integral, eij is the Lagrangian matrix, and aij and bij are the coupling constants of Coulomb (ii|jj)Ω and exchange (ij|ji)Ω integrals, respectively.
The gradient calculus of CGTF with respect to exponent is expressed as follows,

We note that the first term of equation (3) has completely vanished under the Hartree-Fock condition.

3 Computational Procedure

We have employed the partial and fully optimization for the GTF exponents and the geometrical parameters, i.e., nuclear position, at Hartree-Fock level. We abbreviate the calculations of geometry optimization as G-OPT and both geometry and GTF exponents optimizations as EG-OPT, respectively. We have performed G-OPT and EG-OPT for methane, ethane, ethylene, acetylene, and benzene molecules, under Td, D3h, D2h, D¥h, and D6h symmetry restrictions, respectively. The initial values of the exponents are taken from [9s5p] GTFs for carbon [9] and [4s] for hydrogen [10], where the basis set of hydrogen is scaled by the conventional universal factor of 1.2.
Since the optimization of variational parameters is carried out in a multidimensional hyperenergy surface, we encounter the problem of multiple energy minima. We have checked by using different initial parameters for the GTF exponents, and have obtained the same result in most cases.

4 Results and Discussion

4. 1 Populations by G-OPT

First, the conventional atomic basis sets are used for various hydrocarbon molecules. Table 1 summarizes the initial exponent values and electronic population for methane, ethane, ethylene, acetylene, and benzene molecules by G-OPT. In order to discuss the change of population, the sum of populations for s-type GTFs between c1s and c7s (sum(c1s-c7s)), s-type GTFs c8s and c9s (sum(c8s-c9s)), p-type GTFs (sum(c1p-c5p)) on carbon, and s-type GTFs (sum(c1s-c4s)) on hydrogen are also summarized in Table 1. The sum(c1s-c7s) on carbon has almost the same values among each molecule, since these GTFs are used to express the 1s core orbital. Contrary to them, sum(c8s-c9s) and the populations of two small exponents (c4p and c5p) of the p-type GTFs are different to express the different hybrid orbitals. We have also obtained a different population for the smallest exponent (c4s) of the s-type GTFs on hydrogen. We note that the conventional MO method describes the molecular orbital by optimizing only LCAO coefficients. It is expected to express more flexible molecular orbital by optimizing the exponent values, as well as LCAO coefficients.

Table 1. Population for methane, ethane, acetylene, and benzene molecules using G-OPT.
orbitalexponentCH4C2H6C2H4C2H2C6H6
Carbon (9s GTF)
c1s6617.030.0000.0000.0000.0000.000
c2s997.3760.0020.0020.0020.0020.002
c3s227.8740.0190.0190.0190.0190.019
c4s64.68990.1320.1320.1320.1320.132
c5s21.03760.5010.5010.5020.5010.502
c6s7.480150.8830.8830.8830.8830.883
c7s2.790940.4340.4340.4320.4340.432
sum(c1s-c7s)1.9711.9711.9711.9741.970
c8s0.5213670.7030.6950.7460.7270.742
c9s0.1595340.7500.7290.5540.4480.549
sum(c8s-c9s)1.4541.4241.3001.1751.291
Carbon (5p GTF)
c1p18.66490.0050.0050.0050.0050.005
c2p4.122640.1040.1030.1010.1030.102
c3p1.197490.7220.7080.7120.7160.708
c4p0.3827131.7801.7541.6471.6411.678
c5p0.1211100.7000.5450.6240.6980.469
sum(c1p-c5p)3.3113.1153.0893.1632.963
Hydrogen (4s GTF)
c1s19.24060.0030.0040.0040.0030.004
c2s2.899150.0720.0730.0720.0700.072
c3s0.6534100.4480.4510.4560.4210.457
c4s0.1775760.2930.3030.2890.1950.243
sum(c1s-c4s)0.8160.8300.8200.6890.776

4. 2 Optimized molecular exponents by EG-OPT

In order to determine the adequate molecular exponents for each hybrid character, we have performed EG-OPT for methane, ethane, ethylene, acetylene, and benzene molecules, respectively. Figure 1 shows the scale factor (lower panel) for basis functions of each molecule with corresponding electronic population (upper panel). Large scale factors indicate that the optimized molecular exponents become more localized than the atomic ones. Comparing the values of molecular and atomic exponents, we have found that most of the molecular exponents of s-type GTFs are slightly larger than the corresponding atomic ones. The first seven s-type core GTFs (from c1s and c7s) among each molecule are almost the same, while the last two s-type GTFs of c8s and c9s, and p-type GTFs are much different among molecules. This is because these GTFs contribute to express the occupied molecular orbitals for different hybrid character of valence electrons.
It is interesting to see that the scale factors for p-type GTFs on methane and ethane which have no p orbitals are more than 1.2. That is, the molecular exponents of p-type GTFs on sp3-hybrid character are larger than the atomic ones. Contrary to the sp3-hybrid character case, the molecular exponents of p-type GTFs in ethylene and acetylene are smaller than the atomic ones, while those of benzene are a little larger than the atomic ones. The electronic population of s-type GTFs of carbon and hydrogen are almost the same among each molecule. The p-type GTFs of carbon (especially c3p and c5p), however, show drastic difference. For example, the populations of the c3p for ethylene and acetylene having small scale factor are large, while those for methane, ethane, and benzene become small.


Figure 1. The scale factor (lower panel) and population (upper panel) of each orbital on carbon (9s5p) and hydrogen (4s) for methane, ethane, ethylene, acetylene, and benzene using EG-OPT. The scale factor is defined as (optimized molecular exponent value/atomic exponent value)1/2.

To clearly explore the difference of molecular exponent for p-type GTFs between sp3-, and sp2- (or sp-) hybrid characters, we have optimized the exponent values of px, py, and pz, individually, using "uncontracted" p-type GTF basis sets, abbreviated by EG-OPT(U). The scale factor (lower panel) and population (upper panel) of ethane, ethylene, and acetylene calculated by EG-OPT(U) is shown in Figure 2. Here, we fix C atoms on x-axis and H atoms of ethylene on xy-plane. Figure 2 indicates that the scale factors of px, py, and pz for ethane molecule are about 1.2 and the molecular exponents of GTFs for s bond, e.g., px and py GTFs on ethylene and px GTFs on acetylene, are larger than the corresponding atomic ones. Contrary to s bond case, the molecular exponents of GTFs for p bond, e.g., pz GTFs on ethylene and py and pz GTFs on acetylene, are almost the same or smaller than atomic ones. The difference of exponent values of p-type GTFs among sp3-hybrid characters, and sp2- and sp-ones for EG-OPT is owing to the difference of the existences of s and p bonds. Our results clearly show the importance of scaling factor of 1.2 for carbon p-type GTF in sp3-hybrid as well as for hydrogen, although the conventional basis sets are scaled 1.2 for hydrogen only.


Figure 2. The scale factor (lower panel) and population (upper panel) of each orbital of carbon (5p) for methane, ethane, ethylene, acetylene, and benzene using EG-OPT(U). The scale factor is defined as (optimized molecular exponent value/atomic exponent value)1/2.

Taking notice of the scale factor on hydrogen (see Figure 1), the molecular exponents of s-type GTFs between c1s and c4s on methane and c4s on ethane and benzene are slightly smaller than the corresponding ones on ethylene and acetylene. As the molecular exponent of p-type GTFs on carbon becomes larger, the molecular exponent of s-type GTFs on hydrogen tends to be smaller. The exponent optimization of GTFs shows the importance of s-type GTFs on hydrogen in the expression of the molecular orbitals.

4. 3 Comparison of total energies, deviation of virial ratios, and geometrical parameters by G-OPT and EG-OPT

Table 2 shows the optimized geometrical parameters, total energies, and deviation of virial ratios by G-OPT and EG-OPT. Here, we note that the virial ratio consists of kinetic energy (<T>) of electrons and potential energy (<V>). If the GTF exponents are optimized as well as the LCAO coefficients, such a quantum- mechanically Virial theorem is always satisfied. The deviations of virial ratio of EG-OPT and EG-OPT(U) are also close to 0. In addition, the total energies of EG-OPT and EG-OPT(U) are improved compared to G-OPT due to the expansion of variational space. We performed the conventional MO calculation using the various high quality basis sets in order to show the efficiency of results of EG-OPT. The total energies and optimized geometrical parameters of 6-311G [11], 6-311G(d) [11], 6-311G(3df,3pd) [11], cc-pVDZ [1], and cc-pVTZ [1] are also shown in Table 2. The total energies of EG-OPT are slightly smaller than those of 6-311G. Almost all geometrical parameters of EG-OPT are close to the results using the cc-pVTZ. The basis set using the optimized molecular exponent enables us to describe the molecular energy and structures as well as the conventional high quality basis set.

Table 2. Total energy, deviation of virial ratio, and geometry for methane, ethane, ethylene, acetylene, and benzene molecules by G-OPT, EG-OPT, EG-OPT(U), and conventional MO with various basis sets.
MoleculeparameterG-OPTEG-OPTEG-OPT(U)6-311G6-311G(d)6-311G(3df,3pd)cc-pVDZcc-pVTZ
CH4Energy a-40.1880331-40.1898661--40.1881957-40.2026372-40.2125873-40.1987120-40.2134659
<V>/<T>+23.69E-051.27E-08-
C-H b2.04612.0443-2.04182.04682.04432.06132.0451
H-C-H c109.5109.5-109.5109.5109.5109.5109.5
C2H6Energy a-79.116984-79.2155970-79.2158979-79.2118020-79.2429298-79.2584048-79.2349446-79.2600347
<V>/<T>+24.10E-054.90E-092.80E-09
C-C b2.89862.88682.88782.88742.88372.87992.88182.8803
C-H b2.04962.04892.04842.04622.05112.04812.06532.0486
C-C-H c111.1111.3111.3111.2111.3111.2111.3111.2
C2H4Energy a-78.0202632-78.0227573-78.0251653-78.019444-78.0474750-78.0620018-78.0401652-78.0644200
<V>/<T>+24.60E-050.006.20E-08
C-C b2.49442.49172.49442.49452.48772.47982.49612.4829
C-H b2.03582.02752.02842.02692.03402.03012.04832.0301
C-C-H c122.0122.0121.9121.9121.8121.7121.6121.6
C2H2Energy a-76.8142737-76.8156792-76.8165721-76.8114344-76.8351440-76.8481958-76.8260431-76.8506239
<V>/<T>+22.10E-044.46E-08-1.40E-09
C-C b2.24412.24242.24462.24332.23542.22752.25242.2302
C6H6Energy a-230.66457-230.674343--230.663035-230.743142-230.774218-230.722350-230.780511
<V>/<T>+22.36E-051.40E-09-
C-C b2.62362.6227-2.62252.61832.61152.6242.6128
C-H b2.02792.0272-2.02362.03242.02842.04512.0284
C-C-H c120.0120.0-120.0120.0120.0120.0120.0
a Units are in hartree.b Units are in bohr.c Units are in degree.

Taking notice of the C-H bond length by the EG-OPT, as the C-H bond lengths become long, the molecular exponent values of c4s on hydrogen tend to decrease, in order of acetylene, ethylene, benzene, methane, and ethane (see Figure 1). We have found the relationship between the molecular exponent and C-H bond length because the short C-H bond length requires the large exponent.

5 Conclusions

We have simultaneously optimized the molecular exponent and geometry of hydrocarbon molecules, methane, ethane, ethylene, acetylene, and benzene, in order to elucidate the adequate basis set for sp3-, sp2-, and sp-hybrid characters. We have found the different optimum molecular exponent of carbon on each hybrid character, sp3, sp2, and sp. Our results suggest that it is better to scale 1.2 not only for hydrogen s-type GTFs but also carbon p-type GTFs in sp3-hybrid. The difference of s and p bonds in C-C enables us to describe flexibly due to the optimization of uncontracted p-type GTFs. We also found that detailed analysis concerning the chemical bond region is possible by the change of molecular exponents. The larger population of s-type GTFs on hydrogen of all molecules obtained by EG-OPT than that by G-OPT shows the importance of these GTFs for the expression of the molecular orbitals.
The basis set based on the optimized molecular exponents clearly shows reasonable pictures about the total energy and geometrical parameters in comparison with the results using the high quality basis sets. This study is a first step to extend the optimizations of geometry and GTF exponents for large molecules.

Part of this work is supported by Grant-in-Aid for Scientific Research and for the priority area by Ministry of Education, Culture, Sports, Science and Technology, Japan, and the grand for 2009 Strategic Research Project (No.K2107) of YCU, Japan. We thank Mr. Nobuyuki Ikeda for his calculation. We would like to dedicate this article to the memory of Dr. Kazuhide Mori of Waseda Computational Science Consortium. At all times he had encouraged us and given us many helpful discussions. We pray his soul may rest in peace.

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