Systematic Improvement of Energy-Components by Simultaneous Optimization of Exponents and Centers of Gaussian-Type Function Basis Sets for Molecular Self-Consistent-Field Wave Functions

Takayoshi ISHIMOTO and Masanori TACHIKAWA


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1 Introduction

The total energy (E) of a molecular system consists of the energy-components of kinetic energy (<T>) of electron and potential energy (<V>) in the Born-Oppenheimer approximation. The energy-component analysis [1 - 3] is one of the fascinating methods to elucidate what happens inside the system for chemical reactions and phenomenon. The first attempt along this strategy has been demonstrated 40 years ago by Ruedenberg et al. [4]. As a matter of fact, a number of papers have been published using this analysis; the nature of the Jahn-Teller effect [5, 6] and the Hund's rule [7, 8].
This approach was, however, known to be confronted with a number of difficulties: the values of <T> and <V> depend much more sharply on the accuracy of wave function than that of the E, since energy component values are estimated as not "eigenvalues" but expectation values of wave function [9]. In the case of the conventional linear combination of atomic orbital (LCAO) molecular orbital (MO) scheme where only LCAO coefficients of Gaussian-type functions (GTFs) [9 - 13] basis sets are optimized, large basis sets and multiple polarization functions should be required to improve energy-component values [9].
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, <T> and <V>, determined by the FVMO method in comparison with the results of the conventional MO method. We have applied the FVMO method to several molecules, to demonstrate the systematic improvement of the values of energy-components. The FVMO method should be expected as a very effective tool to estimate the energy component values for the first time in the current work.

2 Theory

The E obtained by the MO theory consists of the <T> and <V> under the adiabatic approximation. The energy- components closely relate to the quantum-mechanical theorems, such as the Virial and Hellmann-Feynman theorems which are not always satisfied in the conventional MO method.
The Virial theorem for a polyatomic system of

where Ra is a nuclear coordinate, is always satisfied in the FVMO method due to the optimization of exponents in GTFs (scaling). Additionally, the Hellmann- Feynman theorem,

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 <T> and <V>. To check both the Virial and Hellmann-Feynman theorems, we define the extended virial ratio as

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.

3 Results and Discussion

3. 1 Dependence on various numbers of the conventional GTFs for energy-components

We first discuss the H2 as the simplest molecule. Several basis functions for the hydrogen atom were reported by the variational [22, 23] or numerical optimization [24 - 27], or by fitting to the Slater-type function [28]. First, we have adopted the Huzinaga's basis sets from [3s] to [10s] [24], where all the GTF exponents were determined to be optimized for the hydrogen atom. Since the hydrogen atomic orbital shrinks by the kinetic balance when the hydrogen atom forms a chemical bonding, GTF exponents are multiplied by the square of a commonly used scale factor 1.2 [29, 30].
Table 1 summarizes the optimized geometries of the H2 molecule with total energies, energy-components, and the Huzinaga's GTF exponent values from [3s] to [10s]. We fixed these GTFs on each nucleus according to the conventional MO procedure. Table 1 demonstrates that the total energy is really improved as the number of GTFs increases from the [3s] to [9s]. The total energy with [10s] is, however, a little higher than that with [9s], which might be due to the redundancy of the basis set. The absolute values of <T> and <V> are also found to be small, as the number of s-type GTFs increases. Table 1 also shows the extended virial ratio, which must be exactly kept at 2 if complete basis sets are used as shown in the above section. Though one might expect that the virial ratio comes closer to 2 as the number of GTFs increases, the deviation of the ratio from 2 is of the order of 10-4, in spite of employing the large numbers of basis sets of [10s]. These results clearly show that the values of energy-components are not always improved systematically when the numbers of s-type GTFs increase with the conventional procedure.
We have analyzed the effect of the polarization functions for the H2 molecule with [6s] GTFs fixed on each nucleus at the optimized geometry. We adopted the [6s] GTFs by Huzinaga with Dunning [3p2d1f] polarization set [31]. Table 2 shows the optimized geometries, the total energies, energy-components, and extended virial ratio, respectively, with [6s], [6s1p], [6s2p], [6s3p], [6s3p1d], [6s3p2d], and [6s3p2d1f] basis sets. As the adopted basis sets are increased from the minimal to the extended basis, the total energy actually decreases due to the effect of polarization basis functions. Concerning the energy-components, contrary to the case of using only s-type GTFs, the absolute values of <T> and <V> become large due to the polarization functions. In addition, the deviation of virial ratio from 2 with [6s3p2d1f] set becomes 4.164E-04, which really comes close to 2 rather than that with the [6s] basis set of 1.205E-03. These results indicate that polarization functions should be required to obtain the more accurate values of energy-components within the scheme of the conventional MO method. The deviation of virial ratio from 2 is, however, greater than 1.E-05.

3. 2 Improvement of energy-components by the FVMO method

Second, we have applied the FVMO method to the H2 molecule. The optimized geometries, the total energies, energy-components, and extended virial ratios of H2 molecule with [3s] ~ [10s] GTFs are listed in Table 3. The GTF exponents optimized by the FVMO method shrink in comparison with the initial exponent values (see Table 1), while the optimized centers shift to the chemical bond region. We have also found the tendency that the centers of delocalized GTFs largely move from each nucleus. The total energies obtained by the FVMO calculations are actually lower than the values of the conventional MO calculations in Table 1. Increasing the number of s-type GTFs, the absolute values of <T> and <V> become large. This result is also obtained in the case of increasing the polarization functions (see Table 2). Optimizing the GTF exponents and centers simultaneously, the effect of polarization function is fully expected even though only s-type GTFs are used.
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.

3. 3 FVMO calculation for several hydride molecular systems

We finally performed the FVMO calculations for several hydride molecular systems with the optimization of the geometry as well as the exponents and centers. Table 4 summarizes the total energies, energy-components, and extended virial ratios for LiH, BH3, CH4, NH3, H2O, and HF molecules, respectively. The [3s] GTFs for H atoms and the [6s3p] GTFs for other heavy atoms were employed. For comparison, the conventional MO results are also listed in Table 4. In the conventional MO method, values of the virial ratios are found to deviate 10-2 ~ 10-3 order from 2 in all the molecules. Using the FVMO method, the deviation of virial ratio is drastically improved; in particular, the deviation of the extended virial ratio is of the order of 10-7 in the H2O molecule. For LiH, BH3, CH4, NH3, and HF molecules, the satisfactory extended virial ratios are also demonstrated as well as that of H2O. The total energies of FVMO calculation are lower than those of conventional MO because of the expansion of variational space. We confirm that the FVMO method should be the effective method to obtain the highly accurate energy-component values, and can be applied to the energy-component analysis for the polyatomic molecular systems.

4 Conclusions

We have demonstrated the systematic and drastic improvement of <T> and <V> by the FVMO method. Accurate values of <T> and <V> are important for the energy-component analysis. In the FVMO method, the Virial and Hellmann-Feynman theorems, which provide the relationship between <T> and <V>, are always satisfied because the GTF exponents and centers are completely optimized as well as the LCAO coefficients, simultaneously.
In the case of the [6s] GTF basis set for H2 molecule, we have confirmed that the total energy becomes lower than that of the conventional MO calculation with multiple polarized [6s3p2d1f] GTFs. The values of energy-components were drastically improved due to the optimizations of the GTF exponents and centers in the FVMO calculations. We also performed the FVMO calculations for the several hydride molecular systems. The energy-component analysis using the FVMO method may provide a promising method to analyze chemical problems such as steric effects, conjugation, and aromaticity emerging in polyatomic molecules.

Table 1. The optimized geometries with the total energies, energy-components, virial ratios, extended virial ratios, and the exponent values of s-type GTFs from [3s] to [10s]. The energy and energy-components are shown in hartree. The bond distances are shown in angstrom.
GTF[3s][4s][5s][6s][7s][8s][9s][10s]
R0.72850.73160.73200.73200.73230.73180.73180.7321
E-1.1220680-1.1266829-1.1281712-1.1284869-1.1285256-1.1285347-1.1285868-1.1285840
<T>1.14738681.13696231.12890441.12862301.12795271.12857411.12866971.1283837
<V>-2.2694548-2.263452-2.2570757-2.2571099-2.2564784-2.2571088-2.2572565-2.2569677
<V>/<T>+22.207E-029.041E-036.495E-041.205E-04-5.079E-043.489E-057.346E-05-1.775E-04
Extended virial+22.207E-029.041E-036.495E-041.215E-04-5.078E-043.514E-057.242E-05-1.765E-04
Exponents6.4805519.240648.483798.1842213.513422.742855.8101685.52
0.9810392.899157.3116114.704131.930963.2595125.855249.958
0.2197970.6534101.633183.428637.1570614.243328.404555.6583
0.1775760.4856320.9498381.973523.981867.8887515.2744
0.1394340.3293470.6287921.276752.469234.86669
0.1158460.2244360.4540000.8903951.73163
0.08746270.1780630.1528570.274373
0.07566090.1528570.116987
0.06359130.116987
0.0411335

Table 2. The optimized geometries, total energies, energy-components, and virial ratios of [6s] GTFs with various multiple polarization functions fixed on each nucleus. The energies and energy-components are shown in hartree. The bond distances are shown in angstrom.
GTF[6s][6s1p][6s2p][6s3p][6s3p1d][6s3p2d][6s3p2d1f]
R0.73200.73530.73380.73350.73360.73360.7336
E-1.1284869-1.1328392-1.1333967-1.1334585-1.1334925-1.1335134-1.1335138
<T>1.12862301.13082381.13342821.13389311.13386121.13397591.1339860
<V>-2.2558857-2.2636630-2.2667249-2.2673516-2.2673537-2.2674893-2.2674997
Extended virial+21.205E-03-1.782E-032.784E-053.833E-043.252E-044.078E-044.164E-04

Table 3. The optimized geometries with the total energies, energy-components, virial ratios, extended virial ratios, optimized exponents, and centers with several GTFs by the FVMO method. The energy and energy-components are shown in hartree. The bond distances are shown in angstrom.
GTF[3s][4s][5s][6s][7s][8s][9s][10s]
R0.73710.73400.73340.73340.73340.73340.73340.7334
E-1.1275713-1.1321747-1.1332726-1.1335197-1.1336029-1.1336282-1.1336330-1.1336371
<T>1.12757141.13217421.13327261.13351971.13360251.13362821.13363301.1336371
<V>-2.2551427-2.2643489-2.2665453-2.2670387-2.2672054-2.2672560-2.2672670-2.2672650
<V>/<T>+24.330E-08-3.763E-077.840E-08-7.146E-07-2.958E-07-3.741E-079.741E-07-8.130E-06
Extended virial+21.740E-08-3.916E-07-1.326E-07-1.160E-06-2.954E-07-3.744E-073.056E-06-8.560E-06
Optimized6.1260916.351044.7276112.879209.167561.102577.1981670.99
exponents0.8992182.451536.7169616.965331.414084.041088.1165244.342
0.1899200.5598551.503173.909297.1473119.142020.314854.7297
0.1494360.4137331.113152.021655.403095.8982915.4974
0.1265560.3424110.6867331.701082.083845.07536
0.1134620.2493980.6011180.9084471.78849
0.09463360.2275470.3998900.698974
0.08982790.1733150.287213
0.07478850.121171
0.0483265
Optimized0.36180.36490.36570.36620.36660.36660.36650.3667
centers a0.30140.33630.36050.36530.36480.36640.36670.3664
0.32590.31960.32720.35140.36190.36380.36300.3665
0.29030.31610.32190.33150.35970.36310.3623
0.28380.31460.32150.32660.32660.3609
0.27660.30250.32230.32940.3249
0.29180.29660.30740.3259
0.30380.30430.3004
0.25960.3000
0.3513
a Only the positive X coordinates (in angstrom) of GTF centers are shown.

Table 4. The total energies, energy-components, virial ratios, and extended virial ratios by FVMO calculation for the various simple molecules. The energies and energy-components are shown in hartree.
LiHBH3CH4NH3H2OHF
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>+21.440E-029.700E-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>7.960471626.378660540.179817756.120545075.941613799.7410067
<V>-15.9209432-52.7573378-80.3596386-112.2411060-151.8832221-199.4820131
<V>/<T>+21.900E-09-6.405E-07-9.470E-08-2.853E-07-6.950E-082.500E-09
Extended virial+21.870E-08-9.318E-07-7.360E-08-2.416E-07-8.590E-083.600E-09
a All calculations are performed with the standard STO-3G basis set [32, 33] using the GAUUSIAN 98 program [34].

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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