Analysis on Solvated Molecules with a New Energy Partitioning

Yasuaki Kikuchi , Motoki Ishii , Kin-ya Akiba , Hiromi Nakai. Chemical Physics ... Hiromi Nakai , Yuji Kurabayashi , Michio Katouda , Teruo Atsumi. Ch...
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12714

J. Phys. Chem. B 2006, 110, 12714-12720

Analysis on Solvated Molecules with a New Energy Partitioning Scheme for Intra- and Intermolecular Interactions Hirofumi Sato* and Shigeyoshi Sakaki Department of Molecular Engineering, Kyoto UniVersity, Kyoto, 615-8510, Japan ReceiVed: February 17, 2006; In Final Form: April 22, 2006

A new partitioning scheme for the total energy of molecules is presented. In the scheme, the Hartree-Fock total energy of a molecular system is represented as the sum of one- and two-center terms exactly. The present method provides physically reasonable behavior for a wide range of interactions, and intermolecular interaction is treated equivalently with intramolecular interaction. The method is applied to analysis on the inter- and intramolecular interactions of molecular complexes both in gas phase and in aqueous solution. The results strongly indicate that the present method is a powerful tool to understand not only the bonding nature of molecules but also interaction between molecules.

Introduction It is well-known that the electronic structure of a molecule considerably changes when the molecule is dissolved in solvent. On this occasion, how can we understand this event in a traditional chemical sense? Even when obtaining the wave function of the molecule, it is numerical description of the molecule and is often far from chemical intuition. In other words, the issues that we are interested in are whether the chemical bond is strengthened by solvent and why it occurs. A key to answering these questions is to partition the total energy of a molecule as follows,

E)

EIJ ∑I EI + ∑ J>I

(1)

where EI is the energy contribution from atom I in the molecule (one-center part) and EIJ represents the energy contribution from the interaction between two atoms, I and J (two-center part). The quantity, EIJ, is expected to correlate with the strength of the chemical bond. It should be noted, however, as pointed out previously, that EIJ is different from the bond dissociation energy.1 The bond dissociation energy is a well-defined quantity and can be determined uniquely by both experimental and quantum-chemical computational studies. The process of bond dissociation involves bond breaking and electronic relaxation in all the atoms of the molecule. On the other hand, EIJ describes the interaction energy between atoms as the bond is still kept. It is different from the dissociation energy, while it is likely that a correlation exists between EIJ and the bond dissociation energy. Energy partitioning has been well-recognized in semiempirical methods such as CNDO.2 But, This is not the case for ab initio molecular theory,3 since the treatment of the many-body contribution is not trivial. The way to partition ab initio total energy has an unlimited number of possibilities, and there is no unique way to define EI and EIJ. Ichikawa et al. proposed a partitioning scheme which describes the bond-dissociation limit correctly.4 Mayer reported a somewhat different scheme.1 His method was extended to be combined with density functional * Corresponding author.

theory.5 Nakai has proposed another type of partitioning scheme called bond-EDA.6 Other partitioning techniques have been proposed by Mayer et al, but they do not reproduce the total energy of the system exactly.7-9,12 Nakai also reported another type of interesting energy-partitioning scheme called EDA.10 Their method was combined with the quantum mechanics/ molecular mechanics (QM/MM) Hamiltonian.11 A molecule is regarded as an assembly of atoms; thus, the interaction between molecules is the sum of interactions between these atoms. In this sense, intra- and intermolecular interactions are considered to be equivalent and should be treated in the same manner. An energy partitioning scheme fulfilling this requirement is highly desired for the deep understanding of chemical phenomena. Furthermore, it should be addressed that potential functions for intermolecular interaction widely used in the field of molecular simulations are generally described by the sum of interactions between atoms and are usually developed through trial and error processes. A systematic way, with which we can relate the quantum chemical interaction energy to the pairwise one, could be very helpful to construct new potential functions for molecular simulations. In this study, a new partitioning scheme is presented, which provides very reasonable description of intra- and intermolecular interaction energies. Solvation effects on the electronic structure of the solute are also described. Method In the present study, we limits ourselves to the Hartree-Fock case, but its extension to correlation level may be achieved using a similar approach by Vyboishchikov et al.5 The total energy of a molecular system is given by

E)

1

Dabhba + ∑ (ab|cd) ∑ 2abcd ab

{

}

1 DabDcd - DacDbd + 2 ZIZJ

∑ II)

(13)

comes down to the energy defined in EDA.10 All the computational procedures were carried out with program code GAMESS13 modified by us. The results of Ichikawa’s and Mayer’s methods presented here were computed by our own program, whose consistency was checked by comparison with their original programs.14 Results and Discussions Acetylene, Ethylene, and Ethane. Table 1 shows the calculated energy components of acetylene, which is a illustrative result of the present partitioning method. The four diagonal elements represent energy components assigned to each atom (EI), and the six off-diagonal elements are the interaction between two atoms (EIJ). C1-C2 and C1-H1 (C2-H2) respectively form chemical bonds, with large contributions in the two-center terms (-293.3 and -93.4 kcal/mol). On the other hand, vicinal H-H contribution is reasonably small (2.9 kcal/ mol), showing that the interaction between the two hydrogen atoms is negligible. Table 2 lists the series of hydrocarbon molecules involving single, double, and triple bonds. Results are very reasonable: As the bond order increases, the C-C interaction energy becomes greater. The energy in acetylene is about three times as strong as that in ethane. On the other hand, the C-H (bonding) interaction energy is about 100 kcal/mol in all molecules, not depending on the type of molecule. Compared

12716 J. Phys. Chem. B, Vol. 110, No. 25, 2006

Sato and Sakaki

TABLE 1: All the Energy Components (EIJ) of Acetylenea C1 C2 H1 H2

C1

C2

H1

H2

(-37.5694) -293.3 -93.4 -9.4

(-37.5694) -9.4 -93.4

(-0.4462) 2.9

(-0.4462)

a

Calculated by the 6-31G** basis set and given in kcal/mol. Values in parentheses are in atomic units.

TABLE 2: Energy Components of Ethane, Ethylene, and Acetylenea components b

C Hb CsC CsH (bonding) H‚‚‚H (geminal) H‚‚‚ H (vicinal) C‚‚‚ H (vicinal)

ethane

ethylene

acetylene

-4.1 6.5 -109.9 -108.1 14.1 3.4 -1.8 11.1

15.0 11.0 -156.5 -111.3 16.5 4.7 -1.1 13.3

67.6 19.3 -293.3 -93.4 2.9 -9.4

a

Figure 1. “Interaction energy” (Eint) between ammonia and water molecules as a function of the N-O distance. This plot was computed from Hartree-Fock total energy (∆E) and the three partitioning scheme. 6-31G** basis sets were used.

Calculated by the 6-31G** basis set and given in kcal/mol. Difference from the ROHF atomic energy, -37.6771 au (C) and -0.4982 au (H).

b

to the energy components calculated by Mayer’s method (hereafter, we call it “method M”), the present method shows moderate values: the two-center components of C-C are smaller by ca. 10 kcal/mol, and those of C-H are also slightly smaller in absolute value.1 The deviation from the energy of a free atom in the one-center component is much less than that estimated by Mayer’s method. It should be noted that this onecenter component is related to the electronic distortion (relaxation)15 of the atom, as well as to the number of electrons belonging to the atom. If the number of the electron is conserved, the distortion always raises the energy positively. If the number of the electron is not conserved, it can contribute to either increase or decrease of the energy. Very recently, Nakai et al. reported similar analysis using cc-pVDZ basis set.6 The two-center components of CsH were reported as -117 and -121 kcal/mol, respectively for ethane and ethylene, which are close to our results. Our estimations of the two-center components of CsC are higher by 40-70 kcal/ mol compared to their results (-149 and -222 kcal/mol, respectively for CsC and CdC). The one-center terms are also largely different. Since it is not easy to judge which method is most suitable to describe the bonding nature, further careful investigation is highly desired. Inter- and Intramolecular Interactions on the NH3-H2O Complex. The method is also useful to analyze interaction between molecules. Here, we have studied the interaction energy between water and ammonia as a function of R (O-N distance). On the basis of the present partitioning procedure, the interaction between ammonia (V) and water (W) is defined, as follows:

Eint )

∑

EIJ

(14)

I∈V,J∈W

The remains of the total energy express the locally defined energy of each molecule.

EV )

EI + ∑ ∑ I∈V

EIJ

∑ EI + ∑

EIJ

J>I(I,J∈V)

EW )

I∈W

(15)

J>I(I,J∈W)

The change in the energy of a molecule along R is evaluated

Figure 2. Locally defined energy of ammonia molecule computed by the three partitioning scheme (∆EV), as a function of the N-O distance.

by subtracting the energy of each isolated molecule.

∆EV(R) ) EV(R) - EV(isolated) ∆EW(R) ) EW(R) - EW(isolated)

(16)

The interaction energy computed by the standard Hartree-Fock procedure (∆E) is given by

∆E(R) ) EV+W(R) - {EV(isolated) + EW(isolated)} ) Eint(R) + ∆EV(R) + ∆EW(R) (17) Eint values along R calculated by the three different partitioning methods are plotted in Figure 1. The dashed line represents the exact interaction energy, ∆E. At the region with weak interaction (greater R), all three methods correctly provide the same energy. But when approaching two of the molecules, the Eint curve by method M gradually shifts more negative from the ∆E value. On the other hand, the Eint curve by Ichikawa’s method (we call it “method I”) gradually shifts more positive, as it approaches the two molecules. The Eint curve by the present method behaves nicely and is close to ∆E in all the region of R. Figures 2 and 3 respectively represent the change of the locally defined energy of ammonia and water with respect to the energy of the isolated molecule. Both ∆EV and ∆EW evaluated by method I show negatively divergent behavior, which relates to the strong repulsive interaction appearing in

Molecular Interaction Energy Partitioning Scheme

J. Phys. Chem. B, Vol. 110, No. 25, 2006 12717

Figure 3. Locally defined energy of a water molecule computed by the three partitioning scheme (∆EW), as a function of the N-O distance.

TABLE 3: All the Energy Components (∆EIJ) of the NH3-H2O Complex in Gas Phasea H1 H1 O2 H3 N4 H5 H6 H7

-5.0 -1.5 2.8 -26.7 7.1 7.5 7.5

O2

H3

N4

H5

H6

H7

∆EW ) -1.9 0.8 -11.5 12.5 Eint ) -6.6 78.1 -65.3 -19.2 14.2 -18.9 14.0 -18.9 14.0

TABLE 4: Lennard-Jones Potential Parameters for Solute-Water Interaction atom

∆EV ) 2.2 1.6 -5.6 -3.6 -3.6

Figure 4. Components of interaction energy represented by the sum of atom-atom contributions.

4.6 0.7 0.7

3.4 0.6

 (kcal/mol) Solute

3.4

Calculated by the 6-31G** basis set at RN-O ) 3.0 Å and given in kcal/mol. a

Eint (Figure 1). Both ∆EV and ∆EW calculated by method M present positively divergent behavior, which relates to the strong attractive interaction appearing in Eint. The energy computed by the present method shows a much more moderate change than that by methods I or M. This tendency has been already pointed out in the aforementioned hydrocarbon calculations. We would like to emphasize here that the Eint represents the interaction energy in the supermolecule and does not contain the distortion (relaxation) energy, which is included in ∆E (see eq 17). Thus, ∆E must be different from Eint evaluated by any of three partitioning methods as long as the contribution from the electronic distortion in the molecule is not negligible. The change in individual atomic contribution is defined as follows,

∆EI ) EI(R) - EI(isolated) ∆EIJ ) EIJ(R) - EIJ(isolated)

σ (Å)

(18)

Table 3 lists all the energy components at R ) 3.0 Å in the gas phase. As expected, the greatest attractive contribution is found in the hydrogen bonding between H3 and N4 (-65 kcal/mol). At the same time, the interaction between electron-rich atoms, O2-N4, shows strong repulsion (+78 kcal/mol). Other intermolecular components such as O2-H5 and N4-H1 are presumably attributed to Coulomb attraction between these atoms. By forming the complex, all the chemical bonds in the molecules, namely, H1-O2, H3-O2, H5-N4, H6-N4, and H7-N4, are strengthened. It is noted that the sum of all these values is exactly the same with Hartree-Fock interaction energy (∆E). Solvation Effect on Molecular Interactions on the NH3H2O Complex. It is of great interest to apply these partitioning

O N H B F

3.15 3.42 1.00 3.71 2.72

0.152 0.170 0.056 0.136 0.348

O H

3.17 1.00

Solvent Water 0.155 0.056

q -0.834a -1.02a b

-0.82 0.41

a,b The values are used only for the evaluation of ∆E b classical. The value 0.417 is used for H2O, and 0.34, for NH3.

methods to analysis of solvation effects on the solute molecule. In the classical and empirical type of pairwise models, such as OPLS16 and TIP3P,17 interaction between molecules is generally represented by the sum of interactions between individually constituting atoms. These models are used to investigate intermolecular interaction in solvent. Figure 4 illustrates the components of the molecular interactions of the NH3-H2O system (R ) 3.0 Å) computed by the three different methods, in comparison with the classical model computed by OPLS and TIP3P (∆Eclassical). The label “gas” denotes the interaction energy computed by the standard ab initio molecular orbital (MO) method, and “RISM-SCF” indicates the interaction energy evaluated in an aqueous environment (298.15 K, 1.0 g/cm3) by using the RISM-SCF method.18,19 The parameters for the solute molecule were taken from the literature,16,21,22 as summarized in Table 4. The sum of all these components for the classical model (∆Eclassical) represents the total interaction energy between water and ammonia (-5.5 kcal/mol). In any case, the intermolecular energy is determined by a delicate balance among a great amount of components. For example, the positively greatest contribution comes from N-O interaction, which is 94.2 kcal/mol in the classical model and 75-80 kcal/mol in the ab initio models. As expected, the negatively greatest contribution is from the hydrogen bonding N-H pair (-70 to -65 kcal/mol) both in the classical and ab initio models. It is noted that the total interaction energy and details of each term of the classical model are very similar to those evaluated by the RISM-SCF method but are considerably different from those evaluated in gas-phase computations.

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Sato and Sakaki TABLE 5: Solvation Effect on the NH3-H2O Complexa H1

O2

H3

N4

H5

H6

H7

EW,dist ) 3.5 H1 O2 H3

16.6 -25.7 5.0

N4 H5 H6 H7

-8.6 2.7 2.8 2.8

17.1 -14.7 5.1 Edist int ) -3.9 18.7 -12.8 -5.2 2.0 -4.6 1.6 -4.6 1.6

EV,dist ) 4.1 12.1 -10.0 -12.7 -12.7

7.6 2.2 2.2

7.0 1.4

7.0

Calculated by the 6-31G** basis set at RN-O ) 3.0 Å and given in kcal/mol. a

Figure 5. Components of interaction energy between N-O and N-H along R. Dashed lines are by the classical model, lines with cross marks are by gas-phase wave functions, and others are by the RISM-SCF method. Figure 7. Energy change in two-center contributions computed by the present partitioning scheme. All the values are given in kcal/mol. It is noted that only greater contributions are shown. See Table 5 for other values.

Figure 6. Change in Mulliken gross population (upper) and EDA (lower, given in kcal/mol) assigned at each atom upon transferring from gas to solution phase.

Figure 5 shows energy profiles of these components along R. The dashed lines are the classical interactions, and the lines with cross marks (+) are evaluated with the standard gas-phase computations. As seen in the figure, the curves of the classical interactions are close to those computed in aqueous solution (denoted as “sol.”) but considerably different from these components in the gas phase. These results relate to the fact that the classical models were constructed to reproduce the intermolecular interaction in solvent well. The three partitioning schemes give virtually identical profiles in the longer R region (R g 3.5 Å). However, the present method provides much better agreement with the classical profiles than methods I and M in the shorter R region. In other words, the energy profiles computed by the empirical potential and by the present partitioning scheme give very similar energy behavior in the wide range of R. It should be remembered that the components of interaction in the gas phase are very different from those in aqueous solution, as is clearly shown in Figures 4 and 5, indicating that the wave function in aqueous solution is very different from that in the gas phase. The components of interaction in aqueous solution resemble well those by OPLS and TIP3P. The present results strongly suggest that interaction potential derived by the standard ab initio MO calculation carried out in the gas phase might not be applicable for molecular simulation in the solution phase. Charge population has been a typical tool to analyze the difference in wave functions between isolated and solvated molecules. Figure 6 summarizes the changes in Mulliken gross population upon transferring from the gas phase to the aqueous solution phase. The results suggest that the center of the complex, around the N-O bond, attracts more electrons but that hydrogen atoms opening onto the solvent lose electrons. The effective number of transferring electrons from ammonia to water, which is the sum of the charge over the molecule, is 0.025 and 0.047 in the gas and solution phases, respectively.

These results clearly indicate that the solvent induces larger changes in the electronic structure to stabilize the interaction. In the framework of the present partitioning scheme, solvation effects on wave function can be understood from a different point of view. The energy difference between the isolated and solvated molecules is due to distortion and given by sol gas Edist IJ ) EIJ - EIJ

(19)

where superscripts “sol” and “gas” denote that energies are evaluated in the solution phase and gas, respectively. Table 5 shows Edist IJ computed by the present method, and noticeable contributions are illustrated in Figure 7. One can see that all the bonds between hydrogen and heteroatoms are strengthened by the solvation, but the interaction between N and O becomes more repulsive, presumably because of the increase of the negative charge in these atoms. All the H-H interactions also become repulsive due to the decrease of electrons on the hydrogen atoms. It is likely that the ionic character is enhanced in the chemical bonds of the solvated molecule. In fact, Mayer’s covalent bond order indices20 in these bonds decrease upon the transfer. The same conclusions can be drawn from the results of methods I and M. Stated another way, the electronic structures of ammonia and water molecules are polarized and Coulombic intermolecular interaction between them is enhanced in aqueous solution. It is noted that sum of Edist IJ represents the distortion energy of the system. The sums over water, ammonia, and their interaction are 3.5, 4.1, and -3.9 kcal/mol, respectively, as listed in Table 5, showing that the interaction between the two molecules becomes more attractive due to the electronic structure change in the individual molecules. Solvation Effect on Molecular Interactions on the ChargeTransfer Complex of NH3-BF3. Another example is NH3BF3, which is well-known as a charge-transfer complex.23 The sum of Mulliken charges in the NH3 moiety at the gas-phase optimized geometry is +0.23|e|, exhibiting its strong chargetransfer character in this complex. Because of this remarkable change in electronic structure in each moiety, the contribution from electronic distortion (relaxation) in the total interaction energy (∆E) is not negligible, which is different from the case of NH3-H2O. Table 6 shows the energy components of the

Molecular Interaction Energy Partitioning Scheme

J. Phys. Chem. B, Vol. 110, No. 25, 2006 12719

TABLE 6: All the Energy Components (∆EIJ) of the NH3-BF3 Complex in the Gas Phasea B1

F2

F3

F4

N5

H6

H7

TABLE 8: Basis-Set Dependency of the MPA Results for the NH3 Moiety in the Two Complexes NH3-H2O

H8

∆EB ) - 32.5

B1 F2 F3 F4 N5 H6 H7 H8

-12.1 -4.0 -9.1 -4.0 6.3 -9.1 -4.0 6.3 6.3 Eint ) -21.4 -237.7 67.9 67.9 55.1 -15.3 -17.8 55.1 -17.8 -15.3 55.1 -17.8 -17.8

-9.1 67.9 -17.8 -17.8 -15.3

∆EV ) 9.5 -16.0 -11.8 17.5 -11.8 2.8 -11.8 2.8

17.5 2.8

17.5

a

Calculated by the 6-31G** basis set at the optimized geometry. All the values are given in kcal/mol.

TABLE 7: Solvation Effect of the NH3-BF3 Complexa B1

F2

F3 EB,dist

B1 F2 F3 F4 N5 H6 H7 H8

-6.3 -0.3 0.7 -0.3 1.6 0.7 -0.2 1.6 1.6 Eint ) -9.3 -10.5 4.7 4.7 2.9 -1.9 -2.6 2.9 -2.6 -1.9 2.8 -2.6 -2.6

F4

N5

H6

H7

H8

EV,dist ) 13.1 3.6 -9.0 9.9 -9.0 2.4 -9.2 2.3

9.9 2.3

9.9

) -0.2

0.7 4.7 -2.7 -2.7 -1.9

a

Calculated by the 6-31G** basis set at the optimized geometry. All the values are given in kcal/mol.

complex. Since the number of electrons in the BF3 moiety increases, the locally defined energy (see eq 15) significantly lowers. On the other hand, the locally defined energy of the NH3 moiety, which loses electrons, positively increases. As expected, the greatest contribution in the molecular interaction comes from that between the boron and nitrogen atoms, which are directly connected. At the same time, the interactions between the boron and hydrogen atoms and that between the nitrogen and fluorine atoms become strongly repulsive. All these interactions are essentially electrostatic, since Mulliken charges on the boron, fluorine, nitrogen, and hydrogen atoms are 1.02|e|, -0.42|e|, -0.79|e|, and 0.34|e|, respectively. It is noted that the sum of the three energies (Eint, ∆EB, and ∆EV) is exactly the same as the Hartee-Fock interaction energy (∆E). The intermolecular interaction in an aqueous solution is strengthened by 9 kcal/mol compared to the complex in the gas phase. The Mulliken charge in the NH3 moiety increases to +0.31|e|, indicating that the charge-transfer character is enhanced in the condensed phase. EV,dist is a positive, large value (13 kcal/mol; see Table 7), attributed to the electron decrease in the NH3 moiety. Similar discussion to that in the NH3-H2O complex is presented: The bonds between the hydrogen and nitrogen atoms are strengthened by the solvation, and all the H-H interaction becomes more repulsive. In other words, the ionic character in the chemical bonds is enhanced in the solvated molecule. Conclusions It is natural to understand that atoms constitute a molecule in a chemical sense. We have proposed a new partitioning scheme to distribute the total energy of a molecule into atomic components and their interactions. Needless to say, there is innumerable possibility to introduce energy partitioning schemes for the electronic energy. The present method provides reliable understanding of the interaction between atoms, not only in an intermolecular sense but in an intramolecular sense.

4-31G 6-31G 6-31G(d) 6-31G(d,p) 6-31G(2d) 6-31G(2d,p) 6-31G(3d) 6-31G(3df) 6-31+G 6-31+G(d) 6-31+G(d,p) 6-311G 6-311G(d) 6-311G(d,p) 6-311G(2d) 6-311G(2df) 6-311+G(1d) 6-311+G(d,p) 6-311+G(2d) 6-311+G(2df) 6-311+G(3d) 6-311+G(3df) DZV DZP TZV TZP average minimum maximum standard deviation

NH3-BF3

population

ratio (%)

population

ratio (%)

0.029 0.020 0.022 0.025 0.033 0.034 0.043 0.042 -0.033 -0.026 -0.021 0.025 0.027 0.037 0.038 0.039 -0.055 -0.030 -0.018 -0.017 -0.008 -0.003 -0.002 0.004 -0.011 -0.007 0.007

0.289 0.199 0.218 0.250 0.331 0.343 0.430 0.420 -0.327 -0.256 -0.206 0.255 0.268 0.365 0.379 0.390 -0.546 -0.297 -0.182 -0.171 -0.078 -0.028 -0.022 0.036 -0.106 -0.073

0.161 0.189 0.228 0.228 0.266 0.264 0.144 0.203 0.063 0.136 0.183 0.201 0.275 0.339 0.320 0.345 0.113 0.183 0.314 0.348 0.147 0.169 0.356 0.367 0.195 0.289 0.232

1.610 1.887 2.283 2.278 2.664 2.640 1.435 2.029 0.629 1.357 1.826 2.007 2.746 3.390 3.195 3.453 1.129 1.827 3.136 3.481 1.474 1.687 3.560 3.665 1.954 2.894

-0.546 0.430 0.279

0.629 3.665 0.846

The method is applied to analysis on the inter- and intramolecular interactions of a molecular complex in aqueous solution. The method provides an intuitive interpretation of the solvation effect on a molecular system. We found that the ionic bonding character becomes prominent in aqueous solution. Acknowledgment. We would like to express our sincere thanks to Prof. Ichikawa at Hoshi College of Pharmacy, for providing us his program code and useful discussion. We acknowledge financial support by the Grant-in Aid for Scientific Research on Priority Areas “Water and biomolecules” h(43016041223 and 430-18031019) and by the Grant-in Aid for Encouragement of Young Scientists (17750012), both from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) Japan. Appendix: Basis-Set Dependency It is well-known that the results of the Mulliken charge population analysis (MPA) are significantly dependent on the adopted basis functions. Since the present method is related to MPA, numerical assessments of the present method are examined in this appendix. The procedure employed here follows the analysis by Nakai et al.24 The MPA results obtained by the Hartree-Fock calculations with various basis functions including Pople’s 6-31G series25 and Dunning-Huzinaga’s double/triple-ζ series (DZV, etc., denoted the build-in basis sets in the GAMESS program package13,26) are shown in Table 8. In the table, the sum of the gross population in the NH3 moiety is shown together with its ratio with respect to the total number of electrons in the moiety. In the NH3-H2O system, charge transfer between the molecules does not occur and the total Mulliken charge in the NH3 moiety is close to zero. The situation is different in the NH3-BF3

12720 J. Phys. Chem. B, Vol. 110, No. 25, 2006

Sato and Sakaki

TABLE 9: Basis-Set Dependency of the Locally Defined Energy for the NH3 Moiety in the Two Complexes NH3-H2O 4-31G 6-31G 6-31G(d) 6-31G(d,p) 6-31G(2d) 6-31G(2d,p) 6-31G(3d) 6-31G(3df) 6-31+G 6-31+G(d) 6-31+G(d,p) 6-311G 6-311G(d) 6-311G(d,p) 6-311G(2d) 6-311G(2df) 6-311+G(1d) 6-311+G(d,p) 6-311+G(2d) 6-311+G(2df) 6-311+G(3d) 6-311+G(3df) DZV DZP TZV TZP minimum maximum standard deviation

NH3-BF3

EV (au)

ratio (%)

EV (au)

ratio (%)

-56.094 -56.156 -56.178 -56.192 -56.179 -56.193 -56.178 -56.185 -56.177 -56.199 -56.208 -56.176 -56.198 -56.204 -56.192 -56.195 -56.218 -56.221 -56.203 -56.207 -56.200 -56.200 -56.177 -56.197 -56.191 -56.212

99.98 99.99 99.99 99.99 99.99 99.99 99.98 99.98 100.02 100.02 100.01 100.00 99.99 99.99 99.98 99.98 100.02 100.01 99.99 99.99 99.98 99.98 100.00 100.00 100.01 100.01 99.98 100.02 0.014

-56.070 -56.147 -56.147 -56.181 -56.153 -56.180 -56.074 -56.122 -56.212 -56.222 -56.202 -56.188 -56.181 -56.164 -56.129 -56.150 -56.230 -56.218 -56.101 -56.122 -56.060 -56.131 -56.118 -56.153 -56.172 -56.163

99.94 99.98 99.93 99.97 99.94 99.97 99.79 99.87 100.08 100.06 100.00 100.02 99.96 99.92 99.87 99.90 100.05 100.01 99.81 99.84 99.73 99.85 99.90 99.92 99.98 99.92 99.73 100.08 0.084

complex. The averaged value of the total Mulliken charge, 0.23|e|, is thirty times greater than that in the NH3-H2O system, indicating the charge-transfer character in NH3-BF3. The standard deviations are not small for both of the complexes, suggesting that the results of MPA, more or less, depend on the choice of the basis set. Table 9 lists the results of the locally defined energy of the NH3 moiety (EV) computed by the present method. The ratio is calculated with respect to the total energy of the isolated molecule.

Ratio (%) ) EV/EV(isolated) × 100 The difference in the ratio is attributed to the distortion of the

electronic structure as well as to the change in the total number of electrons belonging to the NH3 moiety. For this cause, the ratios become both higher and lower than 100%. The standard deviation of the ratio is significantly small, suggesting that the basis-set dependency of the present method is small in comparison with that of MPA. References and Notes (1) Mayer, I. Chem. Phys. Lett. 2003, 382, 265. (2) Fischer, H.; Kollmar, H. Theor. Chim. Acta 1970, 16, 163. (3) Kollmar, H. Theor. Chim. Acta 1978, 50, 235. (4) Ichikawa, H.; Yoshida, A. Int. J. Quan. Chem. 1999, 71, 35. (5) Vyboishchikov, S. F.; Salvador, P.; Duran, M. J. Chem. Phys. 2005, 122, 244110. (6) Nakai, H.; Kikuchi, Y. J. Theor. Comput. Chem. 2005, 4, 317. (7) Salvador, P.; Duran, M.; Mayer, I. J. Chem. Phys. 2001, 115, 1153. (8) Mayer, I.; Hamza, A. Theor. Chim. Acc. 2001, 105, 360. (9) Hamza, A.; Mayer, I. Theor. Chim. Acc. 2003, 109, 109. (10) Nakai, H. Chem. Phys. Lett. 2002, 363, 73. (11) (a) Kawamura, Y.; Nakai, H. Chem. Phys. Lett. 2005, 410, 64. (b) Sato, H.; Hirata, F.; Sakaki, S. J. Phys. Chem. A 2004, 108, 2097. (c) Sato, H.; Sakaki, S. J. Phys. Chem. A 2004, 108, 1629. (12) Mayer, I. Chem. Phys. Lett. 2000, 332, 381. (13) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S. J.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. J. Comput. Chem. 1993, 14, 1347. (14) Mayer, I.; Hamza, A. available from http://occam.chemres.hu/ programs/. (15) Note that the term “distortion” used in the present work is the deviation of electronic distribution in individual molecules from their isolated situation and does not include contribution from the interaction between them after this electronic change. If the number of electrons in each molecule is conserved and the molecular geometry is fixed, the energy contribution from the distortion is always a positive value, due to the variational principle for the wave function. (16) Rizzo, R. C.; Jorgensen, W. L. J. Am. Chem. Soc. 1999, 121, 4827. (17) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. J. Chem. Phys. 1983, 79, 926. (18) Ten-no, S.; Hirata, F.; Kato, S. Chem. Phys. Lett. 1993, 214, 391. Ten-no, S.; Hirata, F.; Kato, S. J. Chem. Phys. 1994, 100, 7443. (19) Sato, H.; Hirata, F.; Kato, S. J. Chem. Phys. 1996, 105, 1546. (20) Mayer, I. Chem. Phys. Lett. 1983, 97, 270. A Ä ngya´n, J. G.; Loos, M.; Mayer, I. J. Phys. Chem. 1994, 98, 5244. (21) Chen, X.; Bartolotti, L.; Ishaq, K.; Tropsha, A. J. Chem. Phys. 1994, 15, 333. (22) (a) Berendsen, H. J. C.; Postma, J. P. M.; van Gunstern, W. F.; Hermans, J. In Intermolecular Forces; Pullman, B., Ed.; Reidel Publ.: Cy, Dordrecht, 1981. (b) Sato, H.; Hirata, F. J. Am. Chem. Soc. 1999, 121, 3460. (23) Umeyama, H.; Morokuma, K. J. Am. Chem. Soc. 1976, 98, 7208. (24) Yamauchi, Y.; Nakai, H. J. Chem. Phys. 2005, 123, 034101. (25) (a) Hariharan, P. C.; Pople, J. A. Theor. Chem. Acta 1973, 28, 213. (b) Francl, M. M.; Petro, W. J.; Hehre, W. J.; Binkley, J. S.; Gordon, M. S.; De Frees, D. J.; Pople, J. A. J. Chem. Phys. 1982, 77, 3654. (26) Dunning, T. H., Jr.; Hay, P. J. In Methods of Electronic Structure Theory; Shaefer, H. F., III, Ed.; Plenum Press: New York, 1977; pp 1-27. Dunning, T. H. J. Chem. Phys. 1971, 55, 716. Huzinaga, S. J. Chem. Phys. 1965, 42, 1293.