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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 sp

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

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 sp

where

The energy gradient with respect to W is expressed as follows,

where

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.

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.

Table 1. Population for methane, ethane, acetylene, and benzene molecules using G-OPT.

orbital | exponent | CH_{4} | C_{2}H_{6} | C_{2}H_{4} | C_{2}H_{2} | C_{6}H_{6} |
---|---|---|---|---|---|---|

Carbon (9s GTF) | ||||||

c_{1s} | 6617.03 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

c_{2s} | 997.376 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 |

c_{3s} | 227.874 | 0.019 | 0.019 | 0.019 | 0.019 | 0.019 |

c_{4s} | 64.6899 | 0.132 | 0.132 | 0.132 | 0.132 | 0.132 |

c_{5s} | 21.0376 | 0.501 | 0.501 | 0.502 | 0.501 | 0.502 |

c_{6s} | 7.48015 | 0.883 | 0.883 | 0.883 | 0.883 | 0.883 |

c_{7s} | 2.79094 | 0.434 | 0.434 | 0.432 | 0.434 | 0.432 |

sum(c_{1s}-c_{7s}) | 1.971 | 1.971 | 1.971 | 1.974 | 1.970 | |

c_{8s} | 0.521367 | 0.703 | 0.695 | 0.746 | 0.727 | 0.742 |

c_{9s} | 0.159534 | 0.750 | 0.729 | 0.554 | 0.448 | 0.549 |

sum(c_{8s}-c_{9s}) | 1.454 | 1.424 | 1.300 | 1.175 | 1.291 | |

Carbon (5p GTF) | ||||||

c_{1p} | 18.6649 | 0.005 | 0.005 | 0.005 | 0.005 | 0.005 |

c_{2p} | 4.12264 | 0.104 | 0.103 | 0.101 | 0.103 | 0.102 |

c_{3p} | 1.19749 | 0.722 | 0.708 | 0.712 | 0.716 | 0.708 |

c_{4p} | 0.382713 | 1.780 | 1.754 | 1.647 | 1.641 | 1.678 |

c_{5p} | 0.121110 | 0.700 | 0.545 | 0.624 | 0.698 | 0.469 |

sum(c_{1p}-c_{5p}) | 3.311 | 3.115 | 3.089 | 3.163 | 2.963 | |

Hydrogen (4s GTF) | ||||||

c_{1s} | 19.2406 | 0.003 | 0.004 | 0.004 | 0.003 | 0.004 |

c_{2s} | 2.89915 | 0.072 | 0.073 | 0.072 | 0.070 | 0.072 |

c_{3s} | 0.653410 | 0.448 | 0.451 | 0.456 | 0.421 | 0.457 |

c_{4s} | 0.177576 | 0.293 | 0.303 | 0.289 | 0.195 | 0.243 |

sum(c_{1s}-c_{4s}) | 0.816 | 0.830 | 0.820 | 0.689 | 0.776 |

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 sp

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 sp^{3}-, and sp^{2}- (or sp-) hybrid characters, we have optimized the exponent values of p_{x}, p_{y}, and p_{z}, 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 p_{x}, p_{y}, and p_{z} for ethane molecule are about 1.2 and the molecular exponents of GTFs for s bond, e.g., p_{x} and p_{y} GTFs on ethylene and p_{x} 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., p_{z} GTFs on ethylene and p_{y} and p_{z} GTFs on acetylene, are almost the same or smaller than atomic ones. The difference of exponent values of p-type GTFs among sp^{3}-hybrid characters, and sp^{2}- 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 sp^{3}-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 c_{1s} and c_{4s} on methane and c_{4s} 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.

Molecule | parameter | G-OPT | EG-OPT | EG-OPT(U) | 6-311G | 6-311G(d) | 6-311G(3df,3pd) | cc-pVDZ | cc-pVTZ |
---|---|---|---|---|---|---|---|---|---|

CH_{4} | Energy^{ a} | -40.1880331 | -40.1898661 | - | -40.1881957 | -40.2026372 | -40.2125873 | -40.1987120 | -40.2134659 |

<V>/<T>+2 | 3.69E-05 | 1.27E-08 | - | ||||||

C-H^{ b} | 2.0461 | 2.0443 | - | 2.0418 | 2.0468 | 2.0443 | 2.0613 | 2.0451 | |

H-C-H^{ c} | 109.5 | 109.5 | - | 109.5 | 109.5 | 109.5 | 109.5 | 109.5 | |

C_{2}H_{6} | Energy^{ a} | -79.116984 | -79.2155970 | -79.2158979 | -79.2118020 | -79.2429298 | -79.2584048 | -79.2349446 | -79.2600347 |

<V>/<T>+2 | 4.10E-05 | 4.90E-09 | 2.80E-09 | ||||||

C-C^{ b} | 2.8986 | 2.8868 | 2.8878 | 2.8874 | 2.8837 | 2.8799 | 2.8818 | 2.8803 | |

C-H^{ b} | 2.0496 | 2.0489 | 2.0484 | 2.0462 | 2.0511 | 2.0481 | 2.0653 | 2.0486 | |

C-C-H^{ c} | 111.1 | 111.3 | 111.3 | 111.2 | 111.3 | 111.2 | 111.3 | 111.2 | |

C_{2}H_{4} | Energy^{ a} | -78.0202632 | -78.0227573 | -78.0251653 | -78.019444 | -78.0474750 | -78.0620018 | -78.0401652 | -78.0644200 |

<V>/<T>+2 | 4.60E-05 | 0.00 | 6.20E-08 | ||||||

C-C^{ b} | 2.4944 | 2.4917 | 2.4944 | 2.4945 | 2.4877 | 2.4798 | 2.4961 | 2.4829 | |

C-H^{ b} | 2.0358 | 2.0275 | 2.0284 | 2.0269 | 2.0340 | 2.0301 | 2.0483 | 2.0301 | |

C-C-H^{ c} | 122.0 | 122.0 | 121.9 | 121.9 | 121.8 | 121.7 | 121.6 | 121.6 | |

C_{2}H_{2} | Energy^{ a} | -76.8142737 | -76.8156792 | -76.8165721 | -76.8114344 | -76.8351440 | -76.8481958 | -76.8260431 | -76.8506239 |

<V>/<T>+2 | 2.10E-04 | 4.46E-08 | -1.40E-09 | ||||||

C-C^{ b} | 2.2441 | 2.2424 | 2.2446 | 2.2433 | 2.2354 | 2.2275 | 2.2524 | 2.2302 | |

C_{6}H_{6} | Energy^{ a} | -230.66457 | -230.674343 | - | -230.663035 | -230.743142 | -230.774218 | -230.722350 | -230.780511 |

<V>/<T>+2 | 2.36E-05 | 1.40E-09 | - | ||||||

C-C^{ b} | 2.6236 | 2.6227 | - | 2.6225 | 2.6183 | 2.6115 | 2.624 | 2.6128 | |

C-H^{ b} | 2.0279 | 2.0272 | - | 2.0236 | 2.0324 | 2.0284 | 2.0451 | 2.0284 | |

C-C-H^{ c} | 120.0 | 120.0 | - | 120.0 | 120.0 | 120.0 | 120.0 | 120.0 |

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 c_{4s} 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.

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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[ 3] F. Jensen,

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[10] S. Huzinaga,

[11] R. Krishnan, J. S. Binkey, R. Seeger, and J. A. Pople,

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