Electronic Structural Basis for the Atomic Partial Charges of Planar

Telephone and Fax: +81-54-238-4624. ... For planar molecular systems, atomic partial charges may be derived from out-of-plane dipole ...... Nakatsuji ...
0 downloads 0 Views 1MB Size
Download PDF
Article pubs.acs.org/JPCA

Electronic Structural Basis for the Atomic Partial Charges of Planar Molecular Systems Derived from Out-of-Plane Dipole Derivatives Hajime Torii* Department of Chemistry, Faculty of Education, and Department of Optoelectronics and Nanostructure Science, Graduate School of Science and Technology, Shizuoka University, 836 Ohya, Shizuoka 422-8529, Japan S Supporting Information *

ABSTRACT: The method of deriving atomic partial charges from out-of-plane dipole derivatives has gained renewed interest as a useful method for identifying the changes in charge distribution upon molecular motion. To understand the nature of those atomic partial charges and to explore possible wide applicability of this method, the modulations in the electron density induced by out-of-plane atomic displacements are analyzed by calculating twodimensional electronic polarization derivatives. It is shown that their amplitudes are substantially delocalized to the spatial regions of the adjacent atoms, suggesting the size of the planar region required for applying this method to locally planar systems, such as the peptide group capped by alkyl groups. It is also shown that, in molecular complexes, the amplitudes of electronic polarization derivatives are affected rather strongly by the polarization effect, indicating the way to use this method to estimate the extent of intermolecular charge transfer in hydrogen-bonded systems.

1. INTRODUCTION Electronic distributions in molecular systems are fundamental for understanding their various properties. Quite often, we employ the model of atomic partial charges, where the electron densities around individual atoms are lumped and assigned to the atomic sites. This model is useful and efficient in many cases, especially for examining intermolecular electrostatic interactions. However, since atomic partial charges are not observables, there are various ways to estimate them,1−11 resulting in varying results. Sometimes the difference is critical; for water clusters, the atomic partial charges derived from simple use of Mulliken population analysis indicate the wrong sign of intermolecular charge transfer (CT) occurring upon hydrogen-bond formation, and correction for the basis set superposition error is needed to obtain reasonable results.12,13 For planar molecular systems, atomic partial charges may be derived from out-of-plane dipole derivatives;10,11 within the framework of the atomic partial charge model, a dipole derivative has two terms, the equilibrium charge (EC) term (simple displacements of atoms with partial charges) and the charge flux (CF) term (changes in the atomic partial charges upon displacements of atoms),14,15 but for the out-of-plane displacements there is no CF term because of the symmetry. In some sense these atomic partial charges are well founded, since they are derived from dipole derivatives (related to the infrared (IR) intensity, an observable) and, by definition,16 reproduce the molecular dipole moment (an observable). Changes in these atomic partial charges upon in-plane molecular vibrations (calculated as dipole second derivatives) often give us important insight into the electronic origin of the strongly IR active modes, as clearly shown in some recent studies.17−19 However, it is also known that the electrons do not exactly © 2015 American Chemical Society

follow the atomic nuclei even upon the out-of-plane atomic displacements in planar systems;20−22 for example, in the case of an out-of-plane atomic displacement from the stable planar structure, the electrons follow the atomic nuclei just incompletely, and such an incomplete following is the origin of the restoring force acting on the displaced atom.21 Then, it will be desirable to examine how we can rationalize the atomic partial charges derived from out-of-plane dipole derivatives from the viewpoint of the modulations in the electron density. Such an analysis will also be useful for assessing the applicability of this method to locally planar systems, such as the peptide group capped by methyl or other alkyl groups, where discussion based on the group theory cannot verify the absence or negligibility of the CF term for the (locally) out-of-plane displacements. It will also be interesting to see how this method can be used for estimating the extent of intermolecular CT occurring upon hydrogen-bond formation, since this is an important property for interpreting the spectral intensities, such as those of the O−H stretching mode at ∼3400 cm−1 and the molecular translation (O···H stretching) mode at ∼200 cm−1 (∼6 THz) of liquid water, and even the nature of the molecular dipole moments of hydrogen-bonded systems.12,13,23−29 For these purposes, in the present study, theoretical analyses are carried out on the modulations in the electron density induced by out-of-plane atomic displacements in some molecular systems. The two-dimensional (2D) electronic polarization derivatives are calculated for quantitative analyses on the spatial distributions of those electron density Received: December 26, 2014 Revised: March 8, 2015 Published: March 11, 2015 3277

DOI: 10.1021/jp512884g J. Phys. Chem. A 2015, 119, 3277−3284

Article

The Journal of Physical Chemistry A

B. Effect of Intramolecular Polarization Induced by Intermolecular Electrostatic Field. As discussed below in section 3B, (∂pz(el)(x,y)/∂Zn)H_bond contains a significant contribution of intramolecular polarization induced by intermolecular electrostatic field, according to the usual classification of intermolecular interactions. The latter, denoted as (∂pz(el)(x,y)/∂Zn)polar, was approximately evaluated as

modulations. We take the cases of benzene, formamide−water 1:1 complex, and planar water dimer as example cases of planar molecular systems. Then, the case of the N-methylacetamide− water 1:1 complex is examined as an interesting locally planar (around the peptide group and the water molecule) system.

2. COMPUTATIONAL PROCEDURE The 2D electronic polarization derivative ∂pz(el)(x,y)/∂Zn was calculated from the electron density derivative ∂ρ(el)(x,y,z)/∂Zn as ∂pz(el) (x , y)/∂Zn ≡ −e

∫ dz z(∂ρ(el)(x , y , z)/∂Zn)

(∂pz(el) (x , y)/∂Zn)polar 2

=

− (∂pz(el) (x , y)/∂Zn)molecJ ]

= (∂pz(el) (x , y)/∂Zn)complex 2 J=1

(∂pz(el) (x , y)/∂Zn)CT

(2)

≡ (∂pz(el) (x , y)/∂Zn)H ‐ bond − (∂pz(el) (x , y)/∂Zn)polar

where (∂pz(el)(x,y)/∂Zn)molecJ in the second term was calculated for molecule J in the complex. In calculating this quantity, the basis sets of all the atoms in the complex were retained as in our previous studies12,13,27,28,31,32 to avoid the basis set superposition error, similarly to the calculations of intermolecular interaction energies based on the Counterpoise method, and the atomic locations (including those of the ghost atoms) were kept unchanged from those in the complex. Note that, if atom n does not belong to molecule J, Zn in (∂pz(el)(x,y)/ ∂Zn)molecJ represents the displacement of ghost atom n. The hydrogen bond induced component of the Mulliken charge was calculated in a similar way,12,13 as

(5)

The polarization-induced component of the Mulliken charge was calculated in a similar way. Assuming that atom n belongs to molecule M, it is expressed as (qn)polar =

∑ (qn)molecJ J=1

∑

[(qn)molecM + charges(K ) − (qn)molecM ]

K (≠ M )

(6)

where only the basis sets of the atoms of molecule M were used. Contrary to the case of the integral of (∂pz(el)(x,y)/ ∂Zn)polar, the sum of (qn)polar over atoms n in molecule M is equal to zero by definition. Therefore, the sum of (qn)H_bond over atoms n in any molecule provides an estimate of the extent of intermolecular CT. However, for the purpose of atom-wise discussion, (qn)CT was also defined as follows:

2

(qn)H ‐ bond = (qn)complex −

(4)

where the first term in the parentheses stands for ∂pz(el)(x,y)/ ∂Zn of molecule J in the electrostatic field generated by the atomic partial charges of molecule K, which were derived from the (separately evaluated) out-of-plane dipole derivatives of isolated molecule K (with the locations of all the atoms being kept unchanged from those in the complex). Contrary to the case of (∂pz(el)(x,y)/∂Zn)H_bond, only the basis sets of the atoms of molecule J were used in calculating each term in the summation of this equation. The sum of (∂pz(el)(x,y)/∂Zn)polar over atoms n in any molecule M, which is denoted as Σn∈molecM (∂pz(el)(x,y)/ ∂Zn)polar and represents the polarization of electrostatic origin induced by an out-of-plane displacement of molecule M, generally gives a nonzero value after integration on the xy plane. This is naturally understood by considering that an outof-plane displacement of either molecule J or K in the summation on the right-hand side of eq 4 gives rise to an out-of-plane component of intermolecular electrostatic field, so that the electron density of molecule J is modulated in total in the out-of-plane direction. Therefore, Σn∈molecM (∂pz(el)(x,y)/ ∂Zn)polar counts the electron density that electrostatically obeys the out-of-plane displacement of molecule M, and in this sense conforms to the nature of the atomic partial charges derived from out-of-plane dipole derivatives (as discussed below in section 3), but should be subtracted from Σ n∈molecM (∂pz(el)(x,y)/∂Zn)H_bond to estimate the extent of genuine intermolecular CT through the hydrogen bond. For this purpose, (∂pz(el)(x,y)/∂Zn)CT was defined as

(∂pz(el) (x , y)/∂Zn)H ‐ bond

∑ (∂pz(el) (x , y)/∂Zn)molecJ

[(∂pz(el) (x , y)/∂Zn)molecJ + charges(K )

j = 1 K (≠ J )

(1)

where Zn stands for a displacement of atom n along the z axis. Here, the atomic centers (of the original structure) are assumed to be located on the xy plane (z = 0). This quantity, after integration on the xy plane, represents the electronic contribution to the atomic partial charge of atom n. To obtain ∂ρ(el)(x,y,z)/∂Zn, the structure was first optimized (for the planar water dimer, the optimization was done with all the atoms being constrained on the xy plane), and the electron density ρ(el)(x,y,z) was calculated in a rectangular box of about 15 × 13 × 10 Å3 (the exact dimension depends on the system, and is determined so that each boundary of the box is at least 5 Å from any atom in the system) with the interval of 0.02 Å, for the equilibrium (Zn = 0) and displaced (Zn = 0.01 Å) structures, and a numerical differentiation was carried out. The calculations of ρ(el)(x,y,z) were done at the B3LYP/6-31+G(2df,p) level for benzene and at the MP2/6-31+G(2df,p) level for the other three systems, by using the Gaussian 03 program.30 The theoretical treatments after those calculations were carried out with our original programs. A. Effect of Hydrogen Bond Formation. For the formamide−water and N-methylacetamide−water 1:1 complexes and for the planar water dimer, the effect of hydrogen bond formation on ∂pz(el)(x,y)/∂Zn, denoted as (∂pz(el)(x,y)/ ∂Zn)H_bond, was calculated as

−

∑ ∑

(3)

where the basis sets of all the atoms in the complex were retained in calculating (qn)molecJ in the second term.

(qn)CT ≡ (qn)H ‐ bond − (qn)polar 3278

(7) DOI: 10.1021/jp512884g J. Phys. Chem. A 2015, 119, 3277−3284

Article

The Journal of Physical Chemistry A

lines in Figure 1), and ∂pz(el)(x,y)/∂Zn is integrated in each of these polyhedra. The results are shown in Table 1. In the case

C. 1D Plots and Their Running Integrals. To improve the understanding of the spatial distributions of ∂pz(el)(x,y)/ ∂Zn, 1D plots and their running integrals were also calculated. They are expressed as (∂pz(el,1D) (x)/∂Zn)kind =

∫ dy(∂pz(el)(x , y)/∂Zn)kind

(∂pz(el,1D,run+) (x)/∂Zn)kind =

∫

x

Table 1. Integrated Electronic Polarization Derivatives [Integration of ∂p(el)(x,y)/∂Zn] of Atomic Out-of-Plane Displacements (Zn) and the Corresponding Atomic Partial Charges Calculated for Benzenea

(8)

dξ(∂pz(el,1D) (ξ)/∂Zn)kind (9)

(∂pz

(el,1D,run−)

(x)/∂Zn)kind =

∫x dξ(∂pz

(el,1D)

displaced atom

integration regionb

integrated electronic polarization derivative/e

atomic partial chargec/e

C1

C1 H7 C2 , C 6 H8, H12 C3 , C 5 H9, H11 C4 H10 H7 C1 C2 , C 6 H8, H12 C3 , C 5 H9, H11 C4 H10

−4.666 −0.707 −0.360 −0.027 −0.013 0.015 0.009 −0.003 −0.757 −0.070 −0.012 −0.008 0.003 0.001 −0.002 −0.001

1.334 0.628 −0.092 −0.145 −0.171 −0.142 −0.132 −0.136d 0.243 0.173 0.149 0.132 0.138 0.139 0.137 0.136d

(ξ)/∂Zn)kind (10)

where x and y are taken as the horizontal and vertical axes, respectively, and “kind” is equal to H_bond, polar, or CT. With regard to the running integrals, eq 9 is used for the molecule on the −x side and eq 10 for the molecule on the +x side.

H7

3. RESULTS AND DISCUSSION A. Benzene. The 2D maps of ∂pz(el)(x,y)/∂Zn calculated for the out-of-plane displacements of one carbon and one hydrogen atom (denoted hereafter as C1 and H7, the other atoms being numbered in a usual way) of benzene are shown in Figure 1. It is seen that, although the main contribution to ∂pz(el)(x,y)/∂Zn arises (naturally) from the spatial region around the displaced atom, significant amplitudes of ∂pz(el)(x,y)/∂Zn are also calculated in the neighboring spatial regions. To see this more clearly, the whole space is divided into the Voronoi polyhedra of all the atoms (shown with dotted

a

Calculated at the B3LYP/6-31+G(2df,p) level. bDefined as the Voronoi polyhedron of each atom. cAccumulated value. For example, the value of 1.334 e on the first line is calculated as 6 e − 4.666 e, where 6 e is the electric charge of the carbon nucleus, and the value of 0.628 e on the second line is calculated as 1.334 e− 0.707 e (consider the truncation error in each value). dThese are the atomic partial charges that can be derived from the out-of-plane dipole derivatives.

of the displacement of C1, integration of ∂pz(el)(x,y)/∂Z1 only in the spatial region around C1 results in the integrated value of −4.666 e, which is only about 76% of the fully integrated value (−6.136 e) and is equivalent to the atomic partial charge of 1.334 e (= 6 e − 4.666 e). This means that the rest, which amounts to −1.470 e, arises from the polarization in the neighboring spatial regions. If we include the spatial regions around C2, C6, and H7, which are directly bonded to C1, we obtain a much closer value (−0.092 e) for the atomic partial charge. An even closer value (−0.145 e) is obtained if we also include the spatial regions around H8 and H12. In the case of the displacement of H7, integration of ∂pz(el)(x,y)/∂Z7 only in the spatial region around H7 results in the integrated value of −0.757 e, which is about 88% of the fully integrated value (−0.864 e) and is equivalent to the atomic partial charge of 0.243 e. As in the case of the displacement of C1, a much closer value is obtained if we include the spatial region of the directly bonded atom (0.173 e) or even a wider region extended to H8 and H12 (0.132 e). This delocalized nature of ∂pz(el)(x,y)/∂Zn is also related to the out-of-plane vibrational coupling constants, which are expressed as −∂Fz,m/∂Zn, where Fz,m is the force in the z direction acting on atom m. For example, an analysis on the origin of −∂Fz,m/∂Zn indicates that, among the electronic contribution to −∂Fz,m/∂Z1 with m = 2 and 7 (−2.026 and −0.732 Eh a0−2, respectively), 26.6% (−0.539 Eh a0−2) and 40.8% (−0.299 Eh a0−2) arise from ∂ρ(el)(x,y,z)/∂Z1 in the spatial region around atom m. The results shown up to this point indicate that the amplitudes of ∂pz(el)(x,y)/∂Zn [and ∂ρ(el)(x,y,z)/∂Zn giving rise

Figure 1. 2D maps of ∂pz(el)(x,y)/∂Zn for the out-of-plane displacements (Zn) of (a) one carbon and (b) one hydrogen atom (located on the right-hand side) of benzene calculated at the B3LYP/631+G(2df,p) level. The black open circles show the locations of the carbon and hydrogen atoms, and the dotted lines represent the boundaries of the Voronoi polyhedra of the atoms. The contours are drawn with the interval of 0.044 e a0−2 in the range from −1.1 to 1.1 e a0−2 for (a), and with the interval of 0.015 e a0−2 in the range from −0.375 to 0.375 e a0−2 for (b), with the color code shown on the lefthand side (red color is meant for an increase of electron density in the +z region). 3279

DOI: 10.1021/jp512884g J. Phys. Chem. A 2015, 119, 3277−3284

Article

The Journal of Physical Chemistry A to it] in the spatial regions directly adjacent to that of the displaced atom are sufficiently large and contribute significantly to the atomic partial charge and the vibrational coupling constants. They also suggest that the method of estimating atomic partial charges from out-of-plane dipole derivatives may be applicable to the cases where the system is locally planar in a spatial region wide enough in this sense. This latter point will be discussed in more detail in section 3C. B. Formamide−Water 1:1 Complex and Planar Water Dimer. How can this method be used for estimating the extent of intermolecular CT through hydrogen bonds,12,13,23−29,33−44 and are there any terms that should be carefully taken into account? To answer this question, as an example, we take the case of the formamide−water 1:1 complex, for which the planar structure is calculated as stable. The effect of hydrogen-bond formation is calculated as (∂pz(el)(x,y)/∂Zn)H_bond defined in eq 2, and the sum over the atoms in each molecule Σn∈molecM (∂pz(el)(x,y)/∂Zn)H_bond is calculated [J (in eq 2) and M are equal to 1 and 2 for formamide and water, respectively, in the present case]. The 2D maps of this quantity are shown in Figure 2a and f. [The 2D maps of (∂pz(el)(x,y)/∂Zn)complex and (∂pz(el)(x,y)/∂Zn)H_bond of individual atoms are shown in Figure S1 and S2 in the Supporting Information.] It is seen that Σn∈molecM (∂pz(el)(x,y)/∂Zn)H_bond is partially delocalized and extends to the spatial region of the counterpart molecule. The atomic partial charges derived from out-of-plane dipole derivatives count the electron density that obeys the out-ofplane displacements of the moving atoms, even if the electron density itself resides in the spatial region of the counterpart molecule. Then, the result shown in Figure 2a means that part of the electron density in the spatial region of water loses the control of the atomic displacements of formamide upon complex formation. This phenomenon occurs because the electron density is somewhat delocalized even in the case of an isolated molecule, and upon complex formation, part of that delocalized electron density becomes controlled by the counterpart molecule of the complex. In this sense, this 2D map of Σn∈molecM (∂pz(el)(x,y)/∂Zn)H_bond elucidates a different aspect of the behavior of electrons in complex formation from that recognized by just inspecting the electron density itself. The full integration of Σn∈molecM (∂pz(el)(x,y)/∂Zn)H_bond results in the value of ±0.0601 e, which is essentially equal to the sum of the atomic partial charges shown on the third column in Table 2 (part A). Considering that the atomic partial charges derived from outof-plane dipole derivatives reproduce the system’s dipole moment, it may seem as if the sum over the atoms in each molecule is a good estimate of the extent of intermolecular CT through the hydrogen bond. In fact, it contains significant contribution of intramolecular polarization induced by intermolecular electrostatic field, according to the usual classification of intermolecular interactions. This effect is approximately evaluated as (∂pz(el)(x,y)/∂Zn)polar defined in eq 4. The difference, calculated as (∂pz(el)(x,y)/∂Zn)CT ≡ (∂pz(el)(x,y)/∂Zn)H_bond − (∂pz(el)(x,y)/∂Zn)polar (eq 5), is considered to represent the genuine effect of intermolecular CT through the hydrogen bond. The results are also shown in Figure 2. It is seen that, although many prominent features of (∂pz(el)(x,y)/∂Zn)H_bond arise from those of (∂pz(el)(x,y)/ ∂Zn)polar, a significant extent of intermolecular CT is calculated for (∂pz(el)(x,y)/∂Zn)CT. Taking the full integration of the sum over the atoms in each molecule, it is estimated as 0.0321 e, as shown on the sixth column in Table 2. Note that, in contrast to

Figure 2. 2D maps of (a) Σn∈molecM (∂pz(el)(x,y)/∂Zn)H_bond, (b) Σn∈molecM (∂pz(el)(x,y)/∂Zn)polar, and (c) Σn∈molecM (∂pz(el)(x,y)/∂Zn)CT, and (d) the corresponding 1D plots (after integration along the y axis) and (e) their running integrals for M = 1 (formamide), and (f−j) the corresponding plots for M = 2 (water), of the formamide−water 1:1 complex calculated at the MP2/6-31+G(2df,p) level. The O3···H7 hydrogen bond is taken along the x (horizontal) axis. In the 2D maps, the black open circles show the locations of the atoms, and the contours are drawn with the interval of 0.5 × 10−3 e a0−2 in the range from −22.5 × 10−3 to 22.5 × 10−3 e a0−2, with the color code shown on the left-hand side. In the 1D plots, the red, green, and blue curves are for the H-bond, polar, and CT components, respectively, and the purple and pink vertical lines indicate the locations of the O3 and H7 atoms and the hydrogen-bond center.

the usual terminology of intramolecular polarization, the sum of the atomic partial charges derived from (∂pz(el)(x,y)/∂Zn)CT is different from that derived from (∂pz(el)(x,y)/∂Zn)H_bond. The 2D maps of Σn∈molecM (∂pz(el)(x,y)/∂Zn)CT and the corresponding 1D plots indicate that the electron density modulation is essentially localized within the molecule for M = 2 (water, hydrogen-bond donor) as shown in Figure 2h−j, but extends to the spatial region of the counterpart molecule for M = 1 (formamide, hydrogen-bond acceptor) as shown in Figure 2c− e. As another example, the same type of analysis is carried out for the planar water dimer, whose structure is optimized with all the atoms being constrained on the same plane. In this case, the bare Mulliken charges give the wrong sign of intermolecular 3280

DOI: 10.1021/jp512884g J. Phys. Chem. A 2015, 119, 3277−3284

Article

The Journal of Physical Chemistry A

Table 2. Atomic Partial Charges (In Units of e) of the Formamide−Water 1:1 Complex, the Planar Water Dimer, and the NMethylacetamide−Water 1:1 Complex Obtained from the Out-of-Plane Dipole Derivatives and from Mulliken Populationsa out-of-plane dipole derivatives molecule

atom

chargeb

H-bond induced

formamide

C1 H2 O3 N4 H5 H6 total H7 O8 H9 total

0.2118 0.1159 −0.3574 −0.5912 0.3499 0.3286 0.0576 0.2898 −0.7045 0.3571 −0.0576

0.0146 0.0064 0.0293 0.0150 −0.0049 −0.0002 0.0601 −0.0698 0.0140 −0.0043 −0.0601

O1 H2 H3 total H4 O5 H6 total

−0.6820 0.3679 0.3584 0.0443 0.3178 −0.7170 0.3549 −0.0443

water

acceptor

donor

NMA

water

C1 O2 N3 H4 C5 H6 H7,8 C9 H10 H11,12 total H13 O14 H15 total

0.2603 −0.6855 0.3519 −0.0733

polarization

Mulliken intermol. CTc

charged

Formamide−Water 1:1 Complex (A) 0.0064 0.0082 0.0646 0.0047 0.0017 0.1861 0.0068 0.0225 −0.3719 0.0163 −0.0013 −0.5509 −0.0045 −0.0004 0.3354 −0.0017 0.0015 0.3450 0.0280 0.0321 0.0083 −0.0455 −0.0242 0.4222 0.0342 −0.0202 −0.7980 −0.0167 0.0124 0.3675 −0.0280 −0.0321 −0.0083 Planar Water Dimer (B) 0.0428 0.0457 −0.0029 −0.7726 0.0076 −0.0041 0.0117 0.3851 −0.0018 −0.0075 0.0057 0.3793 0.0486 0.0341 0.0145 −0.0082 −0.0467 −0.0265 −0.0203 0.4402 0.0054 0.0051 0.0003 −0.7926 −0.0073 −0.0127 0.0054 0.3606 −0.0486 −0.0341 −0.0145 0.0082 N-Methylacetamide (NMA)−Water 1:1 Complex (C)f 0.3008 −0.4565 −0.3106 0.3290 −0.6216 0.1686 0.2297 −0.4451 0.1783 0.2008 0.0042 −0.0970 −0.0662 −0.0308 0.4506 0.0302 0.0589 −0.0287 −0.8202 −0.0080 −0.0293 0.0213 0.3654 −0.0748 −0.0366 −0.0382 −0.0042

H-bond induced

polarization

intermol. CTe

0.0112 0.0338 −0.0282 0.0013 0.0059 0.0011 0.0250 0.0198 −0.0383 −0.0066 −0.0250

0.0278 0.0336 −0.0672 0.0015 0.0049 −0.0006 0.0000 0.0448 −0.0417 −0.0031 0.0000

−0.0167 0.0003 0.0389 −0.0002 0.0010 0.0017 0.0250 −0.0250 0.0034 −0.0035 −0.0250

0.0032 0.0062 0.0115 0.0209 0.0149 −0.0270 −0.0087 −0.0209

−0.0161 0.0057 0.0103 0.0000 0.0488 −0.0412 −0.0076 0.0000

0.0193 0.0005 0.0011 0.0209 −0.0339 0.0142 −0.0011 −0.0209

−0.0051 −0.0195 0.0080 0.0049 −0.0006 0.0001 0.0176 −0.0004 0.0038 0.0017 0.0296 0.0192 −0.0387 −0.0101 −0.0296

0.0476 −0.0714 −0.0016 0.0045 0.0067 −0.0035 0.0096 −0.0053 0.0026 0.0006 0.0000 0.0598 −0.0540 −0.0058 0.0000

−0.0528 0.0519 0.0096 0.0005 −0.0073 0.0035 0.0079 0.0049 0.0012 0.0011 0.0296 −0.0406 0.0152 −0.0043 −0.0296

Calculated at the MP2/6-31+G(2df,p) level. bDipole moments are calculated as (μx, μy) = (−1.1199, 0.7208) ea0 and (−1.2852, −0.5300) ea0 for A and B, respectively, which are equal to the directly calculated values. The x axis is taken along the O···H hydrogen bond (O3···H7 for A and O1···H4 for B). cThis component gives rise to a dipole moment of (μx, μy) = (−0.1179, −0.0017) ea0 and (−0.0612, −0.0099) ea0 for A and B, respectively. d Dipole moments are calculated as (μx, μy) = (−1.0616, 0.6554) ea0, (−1.2442, −0.5523) ea0, and (−1.5784, 0.5140) ea0 for A, B, and C, respectively. eThis component gives rise to a dipole moment of (μx, μy) = (−0.0981, −0.0242) ea0, (−0.0588, 0.0085) ea0, and (−0.0963, −0.0147) ea0 for A, B, and C, respectively. fThe atomic partial charges obtained from the out-of-plane dipole derivatives are shown only for the water molecule. The directly calculated value of the dipole moment is (μx, μy) = (−1.5600, 0.6197) ea0, where the x axis is taken along the O···H hydrogen bond (O2···H13). a

contain significant contribution of the polarization effect. As shown in Figure 3, many prominent features of (∂pz(el)(x,y)/ ∂Zn)H_bond arise from those of (∂pz(el)(x,y)/∂Zn)polar, so that the latter must be subtracted from the former [to derive (∂pz(el)(x,y)/∂Zn)CT] to see the genuine effect of intermolecular CT through the hydrogen bond. By taking the sum over the atoms in each molecule, the extent of intermolecular CT is calculated as 0.0145 e, as shown in the sixth column in Table 2, which supports some recent estimates (0.01−0.02 e)12,33,36,43,44 of this quantity. The intermolecular CT component of the dipole moment, calculated with the atomic partial charges derived from

CT upon hydrogen-bond formation, as shown on the seventh column in Table 2 (part B), which is corrected12,13 by removing the basis set superposition error as shown on the eighth column. The atomic partial charges derived from out-of-plane dipole derivatives give the correct sign in this regard, both before and after the effect of hydrogen-bond formation is extracted (the third and fourth columns, respectively). In other words, the present result underscores the validity of the treatment on the Mulliken charges proposed in refs 12 and 13 for estimating the extent of intermolecular CT. However, as in the case of the formamide−water 1:1 complex, the atomic partial charges derived from out-of-plane dipole derivatives 3281

DOI: 10.1021/jp512884g J. Phys. Chem. A 2015, 119, 3277−3284

Article

The Journal of Physical Chemistry A

the extent of intermolecular CT (0.0145 e) multiplied by the O1···H4 hydrogen bond distance (2.011 Å or 3.801 a0) provides a good estimate (0.0553 ea0) of the intermolecular CT component of the dipole moment, the modulation of electron density involved in it is not that simple. C. N-Methylacetamide−Water 1:1 Complex. The results shown in Figure 2 f−j obtained for the out-of-plane displacement of water in the formamide−water 1:1 complex indicate that the electron density is modulated in this case mostly in the spatial regions of the water molecule and the hydrogen-bond accepting oxygen atom in the formamide molecule. The latter corresponds to the directly bonded atom to the displaced atom discussed in section 3A. Therefore, it is expected that the method of deriving atomic partial charges from out-of-plane dipole derivatives is applicable to the cases where the system is locally planar in such a spatial region for estimating the molecular partial charge of the water molecule and, hence, the extent of intermolecular CT upon hydrogen bond formation. To see the situation more concretely, calculations are done for the N-methylacetamide−water 1:1 complex as an example case. The 2D maps of Σn∈molecM (∂pz(el)(x,y)/∂Zn)H_bond, Σn∈molecM (∂pz(el)(x,y)/∂Zn)polar, and Σn∈molecM (∂pz(el)(x,y)/ ∂Zn)CT for M = 2 (water) are shown in Figure 4a−c, and the corresponding 1D plots and their running integrals in Figure 4d−e. Here, the atomic partial charges of an isolated Nmethylacetamide molecule appearing in eq 4 (molecule K) for calculating (∂pz(el)(x,y)/∂Zn)polar were estimated by the CHelpG method.7 It is clearly seen that, qualitatively, the electron density modulations occur quite similarly to the case of the formamide−water 1:1 complex, in accord with the expectation described above. In fact, by inspecting the 2D maps of (∂pz(el)(x,y)/∂Zn)H_bond, (∂pz(el)(x,y)/∂Zn)polar, and (∂pz(el)(x,y)/∂Zn)CT calculated for individual atoms n in the water molecule (n = 13−15) shown in Figure S5 in the Supporting Information, it is found that somewhat large amplitudes of (∂pz(el)(x,y)/∂Zn)H_bond are calculated for n = 13 and 14 (parts d and e) in the spatial region of the C-methyl hydrogen atoms, which are located out of plane. However, most of these arise from the polarization term (shown in parts g and h), so that they are mostly canceled when calculating (∂p z (el) (x,y)/∂Z n ) CT (parts j and k). Since only the intermolecular electrostatic interactions are involved in the polarization term, it is considered that the existence of the amplitudes of (∂pz(el)(x,y)/∂Zn)polar in the spatial region of the C-methyl hydrogen atoms does not diminish the applicability of the method of deriving atomic partial charges from the outof-plane dipole derivatives. The extent of intermolecular CT estimated from the full integration of Σn∈molecM (∂pz(el)(x,y)/∂Zn)CT for M = 2 (water) is 0.0382 e (in magnitude), as shown on the sixth column in Table 2 (part C), which is close to (but slightly larger than) the value estimated for the formamide−water 1:1 complex (0.0321 e) and much larger than that for the planar water dimer (0.0145 e). Multiplying the value (0.0382 e) by the O2···H13 hydrogen bond distance (1.869 Å or 3.533 a0), we obtain an estimate of the intermolecular CT component of the dipole moment as (μx, μy)CT = (−0.1349, 0.0000) ea0. The closeness of this value to the directly calculated value [(μx, μy)CT = (−0.1348, 0.0172) ea0, calculated in the same way as eq 7, substituting qn by μx or μy] supports the validity of the view that we can regard (∂pz(el)(x,y)/∂Zn)CT as an estimate of an atomic partial charge (intermolecular CT component) also for this locally planar

Figure 3. 2D maps of (a) Σn∈molecM (∂pz(el)(x,y)/∂Zn)H_bond, (b) Σn∈molecM (∂pz(el)(x,y)/∂Zn)polar, and (c) Σn∈molecM (∂pz(el)(x,y)/∂Zn)CT, and (d) the corresponding 1D plots (after integration along the y axis) and (e) their running integrals for M = 1 (hydrogen-bond acceptor), and (f−j) the corresponding plots for M = 2 (hydrogen-bond donor), of the planar water dimer calculated at the MP2/6-31+G(2df,p) level. The O1···H4 hydrogen bond is taken along the x (horizontal) axis. In the 2D maps, the black open circles show the locations of the atoms, and the contours are drawn with the interval of 0.36 × 10−3 e a0−2 in the range from −16.2 × 10−3 to 16.2 × 10−3 e a0−2, with the color code shown on the left-hand side. In the 1D plots, the red, green, and blue curves are for the H-bond, polar, and CT components, respectively, and the purple and pink vertical lines indicate the locations of the O1 and H4 atoms and the hydrogen-bond center.

(∂pz(el)(x,y)/∂Zn)CT shown on the sixth column in Table 2, is 0.0620 ea0 (0.158 D) in magnitude and is nearly parallel to the O1···H4 hydrogen bond. Those atomic partial charges indicate that the atom H4 is mainly responsible for the gain of electron density on the hydrogen bond donor molecule through this intermolecular CT, and this view is also supported by the same type of terms of the Mulliken charges (shown on the tenth column), but by inspecting Figure 3h, it is clear that the electron density is also significantly modulated in the spatial region around O5. With regard to the hydrogen bond acceptor molecule, the atom H2 is mainly responsible for the loss of electron density when it is lumped atom-wise, but it is clearly seen in Figure 3c that the electron density is also significantly modulated in the spatial region around O1. Therefore, although 3282

DOI: 10.1021/jp512884g J. Phys. Chem. A 2015, 119, 3277−3284

Article

The Journal of Physical Chemistry A

and 0.0600 e Å −1 for the formamide−water and Nmethylacetamide−water 1:1 complexes, which is larger than that obtained for the planar water dimer (0.0276 e Å−1).45 This result suggests that the modulations in the hydrogen bond distances between peptide and water in aqueous solutions of proteins are important for the intensity generation in the THz spectra, similarly to the case of neat liquid water.13,28 This point will be examined further in future studies.

4. CONCLUDING REMARKS In the present study, the modulations in the electron density induced by out-of-plane atomic displacements have been analyzed theoretically for benzene, formamide−water, and Nmethylacetamide−water 1:1 complexes, and planar water dimer by calculating the 2D electronic polarization derivative ∂pz(el)(x,y)/∂Zn. The results have clearly indicated how the electrons behave in relation to the out-of-plane dipole derivatives of planar and locally planar molecular systems, providing us insight into the nature of the atomic partial charges derived from them. Those atomic partial charges are well founded in some sense, because they are directly related to observables (molecular dipole moment and IR intensity). However, according to the usual classification of intermolecular interactions, they contain a significant contribution of intramolecular polarization induced by intermolecular electrostatic field, as shown in Figures 2−4 and Table 2, so that this factor should be carefully taken into account for estimating the extent of intermolecular CT through hydrogen bonds in molecular complexes. The extent of intermolecular CT of the Nmethylacetamide−water 1:1 complex calculated in this way (0.0382 e) multiplied by the O···H hydrogen bond distance (3.533 a0) gives a good estimate of the intermolecular CT component of the dipole moment (very close to the directly calculated value), suggesting the validity of the application of this method of deriving atomic partial charges to this locally planar system (planar in the CO···HOH part). This opens a way to estimate the extent of modulations in intermolecular CT upon molecular motion (molecular translations and rotations) in such systems with this method.

Figure 4. 2D maps of (a) Σn∈molecM (∂pz(el)(x,y)/∂Zn)H_bond, (b) Σn∈molecM (∂pz(el)(x,y)/∂Zn)polar, and (c) Σn∈molecM (∂pz(el)(x,y)/∂Zn)CT, and (d) the corresponding 1D plots (after integration along the y axis) and (e) their running integrals for M = 2 (water) of the Nmethylacetamide−water 1:1 complex calculated at the MP2/631+G(2df,p) level. The O2···H13 hydrogen bond is taken along the x (horizontal) axis. In the 2D maps, the black open circles show the locations of the atoms, and the contours are drawn with the interval of 0.5 × 10−3 e a0−2 in the range from −22.5 × 10−3 to 22.5 × 10−3 e a0−2, with the color code shown on the left-hand side. In the 1D plots, the red, green, and blue curves are for the H-bond, polar, and CT components, respectively, and the purple and pink vertical lines indicate the locations of the O2 and H13 atoms and the hydrogen-bond center.

■

ASSOCIATED CONTENT

* Supporting Information S

2D maps of the electronic polarization derivatives calculated for out-of-plane displacements of individual atoms in the formamide−water and N-methylacetamide−water 1:1 complexes and the planar water dimer. This material is available free of charge via the Internet at http://pubs.acs.org.

■

system. A similar result is obtained for the formamide−water 1:1 complex (a strictly planar system); the extent of intermolecular CT (0.0321 e) multiplied by the O3···H7 hydrogen bond distance (1.930 Å or 3.647 a0) gives an estimate of the intermolecular CT component of the dipole moment as (μx, μy)CT = (−0.1171, 0.0000) ea0, which is very close to the directly calculated value [(μx, μy)CT = (−0.1179, −0.0017) ea0]. Having obtained a reasonable way to estimate the extent of intermolecular CT upon hydrogen-bond formation, we can go one step further to calculate its derivative upon molecular translation along the O···H direction. It is calculated as 0.0585

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Telephone and Fax: +8154-238-4624. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This study was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology. 3283

DOI: 10.1021/jp512884g J. Phys. Chem. A 2015, 119, 3277−3284

Article

The Journal of Physical Chemistry A

■

(25) Heyden, M.; Sun, J.; Funkner, S.; Mathias, G.; Forbert, H.; Havenith, M.; Marx, D. Dissecting the THz Spectrum of Liquid Water from First Principles via Correlations in Time and Space. Proc. Natl. Acad. Sci. U.S.A. 2010, 107, 12068−12073. (26) Wan, Q.; Spanu, L.; Galli, G. A.; Gygi, F. Raman Spectra of Liquid Water from Ab Initio Molecular Dynamics: Vibrational Signatures of Charge Fluctuations in the Hydrogen Bond Network. J. Chem. Theory Comput. 2013, 9, 4124−4130. (27) Torii, H. Intra- and Intermolecular Charge Fluxes Induced by the OH Stretching Mode of Water and Their Effects on the Infrared Intensities and Intermolecular Vibrational Coupling. J. Phys. Chem. B 2010, 114, 13403−13409. (28) Torii, H. Intermolecular Electron Density Modulations in Water and Their Effects on the Far-Infrared Spectral Profiles at 6 THz. J. Phys. Chem. B 2011, 115, 6636−6643. (29) Hamm, P.; Savolainen, J. Two-Dimensional-Raman-Terahertz Spectroscopy of Water: Theory. J. Chem. Phys. 2012, 136, 094516. (30) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C. et al. Gaussian 03, revision D.01; Gaussian, Inc., Wallingford CT, 2004. (31) Torii, H. Intermolecular Charge Flux as the Origin of Infrared Intensity Enhancement upon Halogen-Bond Formation of the Peptide Group. J. Chem. Phys. 2010, 133, 034504. (32) Torii, H. Mechanism of the Secondary Structure Dependence of the Infrared Intensity of the Amide II Mode of Peptide Chains. J. Phys. Chem. Lett. 2012, 3, 112−116. (33) Lee, A. J.; Rick, S. W. The Effects of Charge Transfer on the Properties of Liquid Water. J. Chem. Phys. 2011, 134, 184507. (34) Soniat, M.; Rick, S. W. Charge Transfer Effects of Ions at the Liquid Water/Vapor Interface. J. Chem. Phys. 2014, 140, 184703. (35) Yao, Y.; Kanai, Y.; Berkowitz, M. L. Role of Charge Transfer in Water Diffusivity in Aqueous Ionic Solutions. J. Phys. Chem. Lett. 2014, 5, 2711−2716. (36) Vácha, R.; Marsalek, O.; Willard, A. P.; Jan Bonthuis, D.; Netz, R. R.; Jungwirth, P. Charge Transfer between Water Molecules as the Possible Origin of the Observed Charging at the Surface of Pure Water. J. Phys. Chem. Lett. 2012, 3, 107−111. (37) Ben-Amotz, D. Unveiling Electron Promiscuity. J. Phys. Chem. Lett. 2011, 2, 1216−1222. (38) Ufimtsev, I. S.; Luehr, N.; Martinez, T. J. Charge Transfer and Polarization in Solvated Proteins from Ab Initio Molecular Dynamics. J. Phys. Chem. Lett. 2011, 2, 1789−1793. (39) Zhao, Z.; Rogers, D. M.; Beck, T. L. Polarization and Charge Transfer in the Hydration of Chloride Ions. J. Chem. Phys. 2010, 132, 014502. (40) Khaliullin, R. Z.; Bell, A. T.; Head-Gordon, M. Electron Donation in the Water−Water Hydrogen Bond. Chem.Eur. J. 2009, 15, 851−855. (41) Marenich, A. V.; Olson, R. M.; Chamberlin, A. C.; Cramer, C. J.; Truhlar, D. G. Polarization Effects in Aqueous and Nonaqueous Solutions. J. Chem. Theory Comput. 2007, 3, 2055−2067. (42) Dal Peraro, M.; Raugei, S.; Carloni, P.; Klein, M. L. Solute− Solvent Charge Transfer in Aqueous Solution. ChemPhysChem 2005, 6, 1715−1718. (43) Glendening, E. D. Natural Energy Decomposition Analysis: Extension to Density Functional Methods and Analysis of Cooperative Effects in Water Clusters. J. Phys. Chem. A 2005, 109, 11936−11940. (44) Ronca, E.; Belpassi, L.; Tarantelli, F. A Quantitative View of Charge Transfer in the Hydrogen Bond: The Water Dimer Case. ChemPhysChem 2014, 15, 2682−2687. (45) This value is smaller than that shown in ref 13, partly because the latter is obtained from the Mulliken charges (thus overestimated by a factor of about 2), and partly because the water molecules in the planar water dimer are not tetrahedrally hydrogen bonded.

REFERENCES

(1) Mulliken, R. S. Electronic Population Analysis on LCAO−MO Molecular Wave Functions. I. J. Chem. Phys. 1955, 23, 1833−1840. (2) Hirshfeld, F. L. Bonded-Atom Fragments for Describing Molecular Charge Densities. Theor. Chim. Acta 1977, 44, 129−138. (3) Reed, A. E.; Weinstock, R. B.; Weinhold, F. Natural Population Analysis. J. Chem. Phys. 1985, 83, 735−746. (4) Singh, U. C.; Kollman, P. A. An Approach to Computing Electrostatic Charges for Molecules. J. Comput. Chem. 1984, 5, 129− 145. (5) Besler, B. H.; Merz, K. M., Jr.; Kollman, P. A. Atomic Charges Derived from Semiempirical Methods. J. Comput. Chem. 1990, 11, 431−439. (6) Chirlian, L. E.; Francl, M. M. Atomic Charges Derived from Electrostatic Potentials: A Detailed Study. J. Comput. Chem. 1987, 8, 894−905. (7) Breneman, C. M.; Wiberg, K. B. Determining Atom-Centered Monopoles from Molecular Electrostatic Potentials. The Need for High Sampling Density in Formamide Conformational Analysis. J. Comput. Chem. 1990, 11, 361−373. (8) Hu, H.; Lu, Z.; Yang, W. Fitting Molecular Electrostatic Potentials from Quantum Mechanical Calculations. J. Chem. Theory Comput. 2007, 3, 1004−1013. (9) Cioslowski, J. A New Population Analysis Based on Atomic Polar Tensors. J. Am. Chem. Soc. 1989, 111, 8333−8336. (10) Dinur, U.; Hagler, A. T. Determination of Atomic Point Charges and Point Dipoles from the Cartesian Derivatives of the Molecular Dipole Moment and Second Moments, and from Energy Second Derivatives of Planar Dimers. I. Theory. J. Chem. Phys. 1989, 91, 2949−2958. (11) Dinur, U. Charge Flux and Electrostatic Forces in Planar Molecules. J. Phys. Chem. 1991, 95, 6201−6211. (12) Torii, H. Extended Nature of the Molecular Dipole of Hydrogen-Bonded Water. J. Phys. Chem. A 2013, 117, 2044−2051. (13) Torii, H. Cooperative Contributions of the Intermolecular Charge Fluxes and Intramolecular Polarizations in the Far-Infrared Spectral Intensities of Liquid Water. J. Chem. Theory Comput. 2014, 10, 1219−1227. (14) Decius, J. C. An Effective Atomic Charge Model for Infrared Intensities. J. Mol. Spectrosc. 1975, 57, 348−362. (15) van Straten, A. J.; Smit, W. M. A. Bond Charge Parameters from Integrated Infrared Intensities. J. Mol. Spectrosc. 1976, 62, 297−312. (16) Note that each in-plane element of molecular dipole moment is equal to the product of the dipole derivatives and the (out-of-plane) displacements of the molecular rotations around the in-plane perpendicular axis. (17) Galimberti, D.; Milani, A.; Castiglioni, C. Charge Mobility in Molecules: Charge Fluxes from Second Derivatives of the Molecular Dipole. J. Chem. Phys. 2013, 138, 164115. (18) Galimberti, D.; Milani, A.; Castiglioni, C. Infrared Intensities and Charge Mobility in Hydrogen Bonded Complexes. J. Chem. Phys. 2013, 139, 074304. (19) Torii, H. Delocalized Electrons in Infrared Intensities. J. Mol. Struct. 2014, 1056/1057, 84−96. (20) Nakatsuji, H. Common Natures of the Electron Cloud of the System Undergoing Change in Nuclear Configuration. J. Am. Chem. Soc. 1974, 96, 24−30. (21) Nakatsuji, H. Electron-Cloud Following and Preceding and the Shapes of Molecules. J. Am. Chem. Soc. 1974, 96, 30−37. (22) This does not mean the breakdown of the Born−Oppenheimer approximation. Even within this approximation, the centroids of the local electron densities do not exactly follow the atomic nuclei. (23) Sharma, M.; Resta, R.; Car, R. Intermolecular Dynamical Charge Fluctuations in Water: A Signature of the H-Bond Network. Phys. Rev. Lett. 2005, 95, 187401. (24) Chen, W.; Sharma, M.; Resta, R.; Galli, G.; Car, R. Role of Dipolar Correlations in the Infrared Spectra of Water and Ice. Phys. Rev. B 2008, 77, 245114. 3284

DOI: 10.1021/jp512884g J. Phys. Chem. A 2015, 119, 3277−3284