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This approach was, however, known to be confronted with a number of difficulties: the values of <

In the conventional MO method, most of the GTF exponents are already determined to be the best for the lowest electronic structure of each atom and are generally fixed during the variational procedure. Though most of these basis sets are suitable for improvement of total energy in molecular systems, of course, there is no guarantee for energy-components. Also, although some attempts for optimizing the exponents [14, 15] and centers [16 - 18] in GTFs to the molecular systems are performed using various basis sets, many cases of such studies are focused on the improvement of total energies in the systems. To adopt the energy-component analysis for chemical reaction and phenomenon, the systematic recipe to seek the adequate basis sets for energy-components is indispensable.

We have proposed the fully variational molecular orbital (FVMO) method [19, 20], which is based on the optimization of all parameters under the variational principle. This FVMO method is already extended to the full-configuration interaction (CI) wave function [21], to improve the wave function of excited states or virtual orbitals. According to the FVMO scheme, the variational parameters such as the GTF exponents and centers are simultaneously optimized as well as the LCAO coefficients. It should be addressed that the quantum- mechanically Virial and Hellmann-Feynman theorems are always satisfied because of the optimization of the GTF exponents and centers, respectively. We note that these theorems are not always satisfied in the conventional MO method.

In this paper, we first discuss the efficiency and precision of total energy and energy-components, <

The Virial theorem for a polyatomic system of

where

holds good due to the optimization of orbital centers (floating) where the wave function force of Pulay term completely vanishes. Inserting Eq. (2) into Eq. (1), we obtain the following equation,

Satisfaction of Eq. (3) closely relates to the accuracy for the values of <

When we completely perform the optimization of both GTF exponents and centers, (that is, scaling and floating) the extended virial ratio of Eq. (4) must be exactly kept at 2.

Table 1 summarizes the optimized geometries of the H

We have analyzed the effect of the polarization functions for the H

Interestingly, the total energy of the [6s] GTFs with FVMO method is given as -1.1335197 hartree, which is really lower than that with [6s3p2d1f] set by the conventional MO method, -1.1335138 hartree in Table 2. In addition, the results of the FVMO calculation with [6s] GTFs give the deviation value of extended virial ratio as -1.160E-06, which is much closer to 2 than that with the [6s3p2d1f] set in Table 2, due to the simultaneous optimization of the GTF exponents and centers in the FVMO scheme.

We confirm that the FVMO method using only s-type GTFs gives more highly accurate values of energy-components than the conventional MO method adding further polarization functions.

In the case of the [6s] GTF basis set for H

GTF | [3s] | [4s] | [5s] | [6s] | [7s] | [8s] | [9s] | [10s] |
---|---|---|---|---|---|---|---|---|

R | 0.7285 | 0.7316 | 0.7320 | 0.7320 | 0.7323 | 0.7318 | 0.7318 | 0.7321 |

E | -1.1220680 | -1.1266829 | -1.1281712 | -1.1284869 | -1.1285256 | -1.1285347 | -1.1285868 | -1.1285840 |

<T> | 1.1473868 | 1.1369623 | 1.1289044 | 1.1286230 | 1.1279527 | 1.1285741 | 1.1286697 | 1.1283837 |

<V> | -2.2694548 | -2.263452 | -2.2570757 | -2.2571099 | -2.2564784 | -2.2571088 | -2.2572565 | -2.2569677 |

<V>/<T>+2 | 2.207E-02 | 9.041E-03 | 6.495E-04 | 1.205E-04 | -5.079E-04 | 3.489E-05 | 7.346E-05 | -1.775E-04 |

Extended virial+2 | 2.207E-02 | 9.041E-03 | 6.495E-04 | 1.215E-04 | -5.078E-04 | 3.514E-05 | 7.242E-05 | -1.765E-04 |

Exponents | 6.48055 | 19.2406 | 48.4837 | 98.1842 | 213.513 | 422.742 | 855.810 | 1685.52 |

0.981039 | 2.89915 | 7.31161 | 14.7041 | 31.9309 | 63.2595 | 125.855 | 249.958 | |

0.219797 | 0.653410 | 1.63318 | 3.42863 | 7.15706 | 14.2433 | 28.4045 | 55.6583 | |

0.177576 | 0.485632 | 0.949838 | 1.97352 | 3.98186 | 7.88875 | 15.2744 | ||

0.139434 | 0.329347 | 0.628792 | 1.27675 | 2.46923 | 4.86669 | |||

0.115846 | 0.224436 | 0.454000 | 0.890395 | 1.73163 | ||||

0.0874627 | 0.178063 | 0.152857 | 0.274373 | |||||

0.0756609 | 0.152857 | 0.116987 | ||||||

0.0635913 | 0.116987 | |||||||

0.0411335 |

GTF | [6s] | [6s1p] | [6s2p] | [6s3p] | [6s3p1d] | [6s3p2d] | [6s3p2d1f] |
---|---|---|---|---|---|---|---|

R | 0.7320 | 0.7353 | 0.7338 | 0.7335 | 0.7336 | 0.7336 | 0.7336 |

E | -1.1284869 | -1.1328392 | -1.1333967 | -1.1334585 | -1.1334925 | -1.1335134 | -1.1335138 |

<T> | 1.1286230 | 1.1308238 | 1.1334282 | 1.1338931 | 1.1338612 | 1.1339759 | 1.1339860 |

<V> | -2.2558857 | -2.2636630 | -2.2667249 | -2.2673516 | -2.2673537 | -2.2674893 | -2.2674997 |

Extended virial+2 | 1.205E-03 | -1.782E-03 | 2.784E-05 | 3.833E-04 | 3.252E-04 | 4.078E-04 | 4.164E-04 |

GTF | [3s] | [4s] | [5s] | [6s] | [7s] | [8s] | [9s] | [10s] |
---|---|---|---|---|---|---|---|---|

R | 0.7371 | 0.7340 | 0.7334 | 0.7334 | 0.7334 | 0.7334 | 0.7334 | 0.7334 |

E | -1.1275713 | -1.1321747 | -1.1332726 | -1.1335197 | -1.1336029 | -1.1336282 | -1.1336330 | -1.1336371 |

<T> | 1.1275714 | 1.1321742 | 1.1332726 | 1.1335197 | 1.1336025 | 1.1336282 | 1.1336330 | 1.1336371 |

<V> | -2.2551427 | -2.2643489 | -2.2665453 | -2.2670387 | -2.2672054 | -2.2672560 | -2.2672670 | -2.2672650 |

<V>/<T>+2 | 4.330E-08 | -3.763E-07 | 7.840E-08 | -7.146E-07 | -2.958E-07 | -3.741E-07 | 9.741E-07 | -8.130E-06 |

Extended virial+2 | 1.740E-08 | -3.916E-07 | -1.326E-07 | -1.160E-06 | -2.954E-07 | -3.744E-07 | 3.056E-06 | -8.560E-06 |

Optimized | 6.12609 | 16.3510 | 44.7276 | 112.879 | 209.167 | 561.102 | 577.198 | 1670.99 |

exponents | 0.899218 | 2.45153 | 6.71696 | 16.9653 | 31.4140 | 84.0410 | 88.1165 | 244.342 |

0.189920 | 0.559855 | 1.50317 | 3.90929 | 7.14731 | 19.1420 | 20.3148 | 54.7297 | |

0.149436 | 0.413733 | 1.11315 | 2.02165 | 5.40309 | 5.89829 | 15.4974 | ||

0.126556 | 0.342411 | 0.686733 | 1.70108 | 2.08384 | 5.07536 | |||

0.113462 | 0.249398 | 0.601118 | 0.908447 | 1.78849 | ||||

0.0946336 | 0.227547 | 0.399890 | 0.698974 | |||||

0.0898279 | 0.173315 | 0.287213 | ||||||

0.0747885 | 0.121171 | |||||||

0.0483265 | ||||||||

Optimized | 0.3618 | 0.3649 | 0.3657 | 0.3662 | 0.3666 | 0.3666 | 0.3665 | 0.3667 |

centers ^{a} | 0.3014 | 0.3363 | 0.3605 | 0.3653 | 0.3648 | 0.3664 | 0.3667 | 0.3664 |

0.3259 | 0.3196 | 0.3272 | 0.3514 | 0.3619 | 0.3638 | 0.3630 | 0.3665 | |

0.2903 | 0.3161 | 0.3219 | 0.3315 | 0.3597 | 0.3631 | 0.3623 | ||

0.2838 | 0.3146 | 0.3215 | 0.3266 | 0.3266 | 0.3609 | |||

0.2766 | 0.3025 | 0.3223 | 0.3294 | 0.3249 | ||||

0.2918 | 0.2966 | 0.3074 | 0.3259 | |||||

0.3038 | 0.3043 | 0.3004 | ||||||

0.2596 | 0.3000 | |||||||

0.3513 |

LiH | BH_{3} | CH_{4} | NH_{3} | H_{2}O | HF | |
---|---|---|---|---|---|---|

Conventional MO ^{a} | ||||||

E | -7.8633821 | -26.0707035 | -39.7268637 | -55.4554198 | -74.9659012 | -98.5728474 |

<T> | -7.9782692 | -26.3260663 | -39.4663855 | -55.0753995 | -74.5187885 | -97.7226603 |

<V> | -15.8416514 | -52.3967698 | -79.1932492 | -110.5308193 | -149.4846897 | -196.2955077 |

<V>/<T>+2 | 1.440E-02 | 9.700E-03 | -6.600E-03 | -6.900E-03 | -6.000E-03 | -8.700E-03 |

FVMO | ||||||

E | -7.9604716 | -26.3786774 | -40.1798208 | -56.1205610 | -75.9416137 | -99.7410064 |

<T> | 7.9604716 | 26.3786605 | 40.1798177 | 56.1205450 | 75.9416137 | 99.7410067 |

<V> | -15.9209432 | -52.7573378 | -80.3596386 | -112.2411060 | -151.8832221 | -199.4820131 |

<V>/<T>+2 | 1.900E-09 | -6.405E-07 | -9.470E-08 | -2.853E-07 | -6.950E-08 | 2.500E-09 |

Extended virial+2 | 1.870E-08 | -9.318E-07 | -7.360E-08 | -2.416E-07 | -8.590E-08 | 3.600E-09 |

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 Dr. Mayumi Ishida for her 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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[13] R. Poirier, P. Kari, and I. G. Csizmadia,

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[21] M. Tachikawa and Y. Osamura,

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[23] O. Matsuoka and K. Okamura,

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[27] H. Partridge,

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[31] T. H. Dunning Jr.,

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