Theoretical Analysis of Weak Adjacent Substituent Effect on the

Article pubs.acs.org/JPCA

Theoretical Analysis of Weak Adjacent Substituent Effect on the Overtone Intensities of XH (X = C, O) Stretching Vibrations Hirokazu Takahashi and Satoshi Yabushita* Department of Chemistry, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan ABSTRACT: It is known that the overtone intensities of some set of OH and CH stretching vibrations show only a weak dependence on the adjacent substituent, in sharp contrast to the much stronger dependence of their fundamental intensities. To understand this characteristic, we calculated the fundamental and overtone intensities of the Δv = 1−6 transitions for the OH stretching of alcohols and acids and the CH stretching of hydrocarbons with different types of hybridization. Based on the local-mode model, from the three components of the dipole moment function (DMF) of each molecule, a onecomponent effective DMF that recovered about 95% of the total intensity for the Δv = 1−6 transitions was constructed and expressed as a sixth-order polynomial of the bond displacement ΔR, with the leading expansion coefficients M1, M2, and M3 for the linear, quadratic, and cubic terms, respectively. When these coefficients for each molecule were represented as points in the coordinate system O-M1 M2 M3, the points for some set of molecules were found to lie on a straight line. Interestingly, the line had a direction cosine such that the resultant transition moments exhibited a small substituent dependence of the overtone intensities. Moreover, the slope of the line could be well approximated by the Morse exponential parameter and the bond distance. These characteristics of the DMFs can be rationalized by using the calculated transition moments and the wave function expansion method with the eigenfunction of the Morse potential. It was also verified by the quasiclassical method of Medvedev that these characteristics of the DMFs are the intrinsic reason for the weak substituent dependence of the overtone intensities. It is emphasized that the graphical representation of the DMF parameters provides a comprehensive tool for discussing various aspects of vibrational intensities.

1. INTRODUCTION The infrared absorption intensity of a molecular vibration represents the change in electron density induced by the molecular motion. A harmonic oscillator model with a linear dipole moment function (DMF) can express only a fundamental absorption in the mid-infrared region. However, overtone transitions of the XH (X = C, O) stretching vibration, which has a large anharmonicity, are weakly observed in the nearinfrared and visible regions. In treating these overtone transitions, the local-mode model, in which each XH bond is represented as an independent1 or coupled2−4 anharmonic oscillator, has been applied successfully instead of the normalmode model. The infrared absorption intensity is calculated from the vibrational wave functions ϕv and the DMF μ, which express the mechanical and electrical properties of molecule, respectively. Whereas the potential energy surface (PES) and its eigenfunctions ϕv for the XH bond can be approximated very well by the Morse potential and its analytic eigenfunctions, such a model function for the DMF does not exist. Mecke’s empirical one-dimensional DMF model5 has been used to understand experimental absorption intensities,2,6−8 but it is generally not suitable for fitting the calculated DMF. Therefore, even though the overtone intensities contain much information about the DMF, the nature of the DMF has not been investigated thoroughly as compared to that of the PES. © XXXX American Chemical Society

The fundamental intensity of the XH (X = C, O) stretching vibration shows a strong substituent dependence, because the transition moment is dominated by the linear displacement term of DMF in most cases and is, therefore, approximated by only the first-derivative term of the DMF at the equilibrium bond length. For the overtone intensities, there is well-known characteristic called the normal intensity distribution law (NIDL), which means that the overtone intensities fall off in an exponential manner with increasing vibrational quantum number v.9 Although this NIDL has been recognized as very common characteristic of overtone intensities, it has been explained only by the nature of PES, and it does not provide any information concerning DMFs. Another interesting feature of overtone absorption intensities is that the intensities of OH or CH bonds with different substituents show only a little variance, which is called the “universal intensity concept” or “bond transferability”, in sharp contrast to the large differences observed for fundamental intensities. This small variance in the overtone intensities (SVOI) was first reported experimentally for the CH stretching overtones of hydrocarbon molecules.8,10−14 It was also reported for OH stretching overtones of alcohols and acids both Received: December 22, 2012 Revised: June 2, 2013

A

dx.doi.org/10.1021/jp312674r | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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vibrational wave function, and μ⃗(R) (D) is the DMF as a function of the XH bond length R. 2.2. DMF and PES. We calculated the PES and the DMF by the hybrid density functional theory method using the B3LYP functional with the 6-311++G(3d,3p) basis set within the Gaussian 03 program,24 as before.17 The basis set was changed from that used in our previous study17 to unify the results with other works on electron density distributions by the Atoms in Molecules (AIM)25 method. In such works, to increase the numerical accuracy of AIM calculations, the use of f-type basis functions is not allowed, so the 6-311++G(3d,3p) basis set is used. The calculated one-dimensional potential energy curves were fitted to Morse potentials, and the DMFs were fitted to sixth-order polynomials by the least-squares method. The number of sample points was 16, as before.17 We investigated the OH vibrations of alcohols and acids containing groups with different electron-attracting abilities and CH vibrations of hydrocarbons with different hybridizations. In particular, we studied tert-butyl alcohol (TB), methanol (ME), acetic acid (AA), and nitric acid (NA) for OH groups (ROH) and ethane (C2H6), benzene (C6H6), ethylene (C2H4), and acetylene (C2H2) for CH groups (RCH). 2.3. Effective One-Dimensional DMF. The transition moment vector, μ⃗ v0, in eq 1 is given by

experimentally6,15 and theoretically.15−17 This behavior might be a consequence of the substituent dependence of the DMF, that is, different degrees of nonlinearity, which is sometimes called electrical anharmonicity, because the second- and thirdderivative terms of the DMF make large contributions to the overtone intensities. Therefore, the analysis of SVOI should provide general but previously unknown characteristics of the molecular electronic structure through the substituent dependence of DMFs. It has also been reported that the overtone intensities are less sensitive than the fundamental intensities to the inclusion of electron correlations.18−20 Providing a physical explanation for these phenomena would be useful not only to satisfy scientific interest but also to judge the appropriate computational method for intensity calculations. In previous work,17 we first constructed DMFs by polynomial expansions in the directions parallel and perpendicular to the OH bond and found that the substituent dependencies of the linear- and quadratic-term contributions cancel each other at the overtone transitions. Second, we constructed effective one-dimensional DMFs derived from ab initio calculations and analyzed them with the vibrational wave function expansion method,21,22 finding that the apparent substituent dependence of the DMFs is caused essentially by the different amounts of fundamental components in the DMFs. This wave function expansion method represents a DMF as the sum of each vibrational-level component, which is equal to the product of the transition moment with the so-called quotient function, to be explained later in section 2.5. This method therefore explains the exponential falloff of the overtone intensities, because the quotient function becomes increasingly oscillatory as v increases, so the magnitudes of the overtone components (i.e., the transition moments) must exhibit an exponential falloff. In this article, we continue our previous investigation to show the conditions for the DMFs to satisfy the SVOI for XH (X = C, O) stretching vibrations with an emphasis on the adjacent substituent dependence. Through the polynomial expansion of one-dimensional DMFs, we found that, when the coefficients of the linear, quadratic, and cubic terms (M1−M3, respectively) for individual molecules are considered as coordinate points in three-dimensional space, these points lie on a straight line for each type of bond. The slope of the line can be expressed by the Morse parameter and the XH bond distance. The characteristics that we identify in this article will be essential in the interpretation of the overtone intensities of other types of XH bonds as well.

μv⃗0 = μvx0 ex⃗ + μvy0 ey⃗ + μvz0 ez⃗

(2)

μav0

where ea⃗ and are the a-axis unit vector and the a-axis component of the transition moment vector, respectively. Therefore, the absorption intensity is given as a sum of the three components A v0 = CABS[(μvx0 )2 + (μvy0 )2 + (μvz0 )2 ]vṽ 0 = A vx0 + A vy0 + A vz0

(3)

All of the molecules considered in this article have Cs symmetry, and the DMF in the direction perpendicular to the RXH plane is zero. Thus, we treat only two components of absorption intensity, namely, those in the directions parallel to the XH bond and perpendicular to it in the Cs plane A v0(v) = CABS[(μv0 )2 + (μv⊥0 )2 ] × vṽ 0 = A v0 + A v⊥0

(4)

The effective one-dimensional DMF, μ*, was constructed by taking the projection of the two-component DMF onto the unit vector e*⃗ , which is rotated from the XH bond axis by an angle θ, as shown in Figure 1. This angle θ is considered to be the effective

2. THEORY AND COMPUTATIONAL METHOD 2.1. Infrared Absorption Intensity. In the present calculations, we apply the local-mode model to the stretching motion of the XH bond. We calculate the integrated absorption coefficient Av0 (km mol−1)23 of each XH stretching transition as A v0 = ln 10

∫ ε(v)̃ dv ̃ Figure 1. Schematic diagram of the unit vectors e⃗∥, e⃗⊥, e⃗*, and e⃗*⊥ along with the rotation angle θ.

= CABS |μv⃗0 |2 vṽ 0 = CABSvṽ 0 |⟨ϕv|μ ⃗ (R )|ϕ0⟩|2

(1)

direction of the two-component transition moments and was determined by maximizing the sum of all of the overtone intensities, as described in the previous article.17 The optimized angles θ for ROH were essentially the same as before,17 and those

where CABS ≡ 8NAπ3/(300000hc) = 2.5066 km mol−1 cm D−2, ε(ṽ) is the molar extinction coefficient, ṽv0 (cm−1) is the transition energy, μ⃗ v0 (D) is the transition moment vector, ϕv is the B



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for C2H6, C6H6, C2H4, and C2H2 were −5.0°, 0.0°, −18.1°, and 0.0°, respectively. These one-dimensional DMFs recover about 95% of the total intensity for the Δv = 1−6 transitions, including the fundamental intensity. In the following sections, we present the results derived from the effective direction components of the absorption intensities, transition moments, and DMFs. 2.4. Morse Potential. The Morse potential26 is expressed as a function of the displacement of the XH bond length, ΔR, as follows V (ΔR ) = De(1 − e−αΔR )2

Table 1. Calculated Harmonic and Anharmonic Terms (cm−1) and Morse Parameters for ROH and RCH

(5)

where De is the dissociation energy and α is the scaling factor with units of inverse length. De and α are related to the frequency as ωe = α(2De/m)1/2, and the anharmonicity is χe = ℏωe/4De, where m is the reduced mass. The eigenfunction ϕv can be expressed analytically as ϕv(y) =

α(k − 2v − 1)v! s −y /2 2s ye Lv (y), Γ(k − v)

e y y − 2s d v − y v + 2s (e y ) v! dy v

(6)

where v is the vibrational quantum number; Γ is the gamma function; L2s v is the associated Laguerre polynomial, with realvalued index 2s; and k is the inverse of the anharmonicity coefficient. 2.5. Wave Function Expansion Method. For the analysis of the observed vibrational spectra of diatomic molecules, both Trischka and Salwen21 and Cashion22 reported the so-called wave function expansion method in obtaining the DMF from the observed vibrational absorption intensities. First, with the completeness of the wave functions, the function μ*(R) ϕ0(R) is expanded as

∑ ⟨ϕv|μ*|ϕ0⟩ϕv(R) v=0

μ*(R ) =

∑ ⟨ϕv|μ*|ϕ0⟩ v=0

ϕv(R )

(7)

ϕ0(R )

∑ dv0Fv0(R) v=0

De (au)

TB ME AA NA C2H6 C6H6 C2H4 C2H2

3708 3747 3835 3808 3066 3107 3166 3406

90.5 91.1 96.5 94.9 63.3 60.3 62.4 60.4

1.302 1.302 1.286 1.277 1.044 1.037 1.036 1.021

0.173 0.176 0.174 0.174 0.169 0.182 0.183 0.219

(9)

Table 2. Zeroth- to Third-Order Coefficients of the DMF, M0−M3, for ROH and RCH

∞

≡

α (a0−1)

As mentioned in the Theory and Computational Method section, the coefficients were determined by a least-squares fit and are presented in Table 2. It was found that the DMFs are

Then, dividing both sides by the ground-state wave function ϕ0(R), we obtain ∞

ωeχe (cm−1)

μ*(ΔR ) = M 0 + M1ΔR + M 2ΔR2 + M3ΔR3 + ···

∞

μ*(R ) ϕ0(R ) =

ωe (cm−1)

molecules. Because the substituent dependencies of all of the parameters are relatively small except for acetylene, the differences in their vibrational wave functions ϕv are very small, and the differences in the absorption intensities are essentially determined by the differences in the DMFs, as is evident from eq 1. Figure 2 shows the absorption intensity values calculated from the one-dimensional DMF of RCH. The results for ROH were almost the same as in our previous work (Figure 2 of ref 17). In the present article, we use the expression “small variance” when the overtone intensity values agree within a factor of 2. Therefore, the overtone intensities for the ROH group (TB, ME, AA, and NA) are said to exhibit a small variance for Δv ≥ 2. For the RCH group, the overtone intensities of C2H6, C6H6, and C2H4 exhibit a small variance for Δv ≥ 3, but acetylene does not belong to this molecular group. 3.2. Polynomial Expansion of DMF. In Figure 3, we present the effective one-dimensional DMFs with the equilibrium values offset to zero. To analyze their behavior, the DMF was expanded into polynomials in terms of the displacement ΔR, as follows

y = k e−αΔR , 2s = k − 2v − 1, k = χe−1 , Lv2s(y) =

molecule

(8)

Therefore, one can understand the transition moment as the expansion coefficient, dv0, of the DMF taking the quotient function Fv0(R) ≡ ϕv(R)/ϕ0(R) as the basis function in the region where ϕ0(R) ≠ 0. Along with eq 6, the wave function expansion method with the Morse oscillator can be considered as a least-squares fitting with the weight function [ϕ0(y)]2. From this point of view, we showed previously that, for ROH molecules, the overtone components of the DMFs are similar and only their fundamental components exhibit a significant substituent dependence.17

molecule

M0 (D)

M1 (D/a0)

M2 (D/a02)

M3 (D/a03)


−0.067 0.309 −0.320 2.289 0.000 0.000 0.000 0.000

0.239 0.387 0.539 0.721 −0.447 −0.260 −0.241 0.531

−0.335 −0.335 −0.216 −0.122 −0.345 −0.325 −0.294 0.000

−0.112 −0.125 −0.091 −0.045 0.009 −0.031 −0.006 −0.020

dominated by the linear and quadratic terms for most molecules. From this table, it is clear that, as the adjacent substituent becomes more electronegative, M1 and M2 become more positive whereas M3 is less sensitive than M1 and M2. Although a simple interpretation of these coefficients was provided in our previous article,17 we explain more in this article by using a point-charge model. Our preliminary study using AIM charge calculations showed that the variation in electron density upon XH stretching occurs mainly between the H and X atoms, whereas the electron density on the substituent part remains the

3. RESULTS AND DISCUSSION 3.1. Vibrational Calculation. In Table 1, we list the calculated values of the harmonic term ωe (cm−1), the anharmonic term ωeχe (cm−1), and the Morse parameters α (bohr−1, denoted as a0−1) and De (au) for ROH and RCH in the investigated C



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Figure 2. Absorption intensity values, Av (km/mol), calculated from the effective dipole moment function of RCH.

Figure 3. Effective direction dipole moment functions of (a) the ROH group [tert-butyl alcohol (TB, blue), methanol (ME, red), acetic acid (AA, yellow) and nitric acid (NA, green)] and (b) the RCH group [ethanol (C2H6, blue), benzene (C6H6, red), ethylene (C2H4, yellow) and acetylene (C2H2, green)].

same. Therefore, we consider only that the point charge q(R) on the stretching H atom depends on the XH bond length, R = Re + ΔR, and the point charge on the X atom accordingly. Defining the position of the X atom as the coordinate origin, as shown in Figure 1, the molecular dipole moment vector is expressed as μ ⃗ (R ) = q(R )Re ⃗ + μresidual ⃗

that is · e *⃗ ) + M1ΔR + M 2ΔR2 q(R )R cos θ = (M 0 − μresidual ⃗ + M3ΔR3 + ···

Equating the first, second, and third derivatives of the two sides of eq 12 at R = Re, it follows that

(10)

where the constant vector μ⃗residual represents the contribution of substituent R to the molecular dipole moment. Using the unit vector e*⃗ in the effective direction, μ*(R) can be expressed as μ*(R ) = μ ⃗ (R ) · e ⃗* = q(R )R cos θ + μresidual · e ⃗* ⃗ = M 0 + M1ΔR + M 2ΔR2 + M3ΔR3 + ···

(12)

[q(R e) + q′(R e)R e] cos θ = M1

(13)

[2q′(R e) + q″(R e)R e] cos θ = 2M 2

(14)

[3q″(R e) + q‴(R e)R e] cos θ ≈ 3q″(R e) cos θ = 6M3

(11)

(15) D



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sign of M1 for RCH depends on the molecule, as previously discussed in terms of atomic polar tensors and electro-optical parameters.29−31 Table 3 shows that the signs of the predicted q(Re) and q′(Re) values are almost the same as for ROH, but there are some differences in magnitude. Comparison of the q(Re) and q′(Re)Re values of each molecule shows that, in contrast to ROH, the magnitude of q′(Re)Re is larger and M1 = [q(Re) + q′(Re)Re] cos θ becomes negative except for acetylene. That is, for C2H6, C6H6, and C2H4, the negative sign of M1 suggests the dominance of the q′(Re)Re term and a smaller contribution from q(Re). Focusing on the substituent dependence, similarly to ROH, more electronegative substituents generally have larger q(Re) values and smaller |q′(Re)| and | q″(Re)| values. Therefore, the relation between Table 3 and the behavior in Figure 3b can be understood. As mentioned previously, q(Re), q′(Re), and q″(Re) are closely correlated with the electronegativity of the substituents. A molecule with a strongly polarized H atom shows a small electron-density migration from the X (X = C, O) atom to the H atom upon bond stretching. This characteristic causes the close correlation between coef f icients M1 and M2, which becomes essential for the interpretation of the substituent dependence of the shape of the DMF, moreover the substituent independence of the overtone intensities. 3.3. Linear to Quartic Term Contributions to the Transition Dipole Moment. In this section, to clarify the relation between Mn and the absorption intensity, we considered the following polynomial expansion of the DMF in the transition moment formula, which is a well-known technique in the analysis of overtone intensities16,17,22,32−35

where q‴(Re) in eq 15 is set to zero, because its contribution is considered to be small. Moreover, from eqs 13−15, M1−M3 can be related by the expression M2 =

q(R e) cos θ M1 − + R eM3 Re Re

(16)

The interpretation of eq 16 is provided later in section 3.5, which discusses the relation between Mn and the vibrational wave function. Table 3 lists the q(Re), q′(Re), and q″(Re) values calculated from eqs 13−15 for ROH. In these cases, note the relations Table 3. Charge q(Re) and Charge Derivatives q′(Re) and q″(Re) of H Atoms and cos θ for ROH and RCH molecule

q(Re) (e)

q′(Re) (e/a0)

q″(Re) (e/a02)

q′(Re)Re (e)

q″(Re)Re (e/a0)

cos θ


0.553 0.643 0.731 1.006 0.304 0.274 0.351 0.453

−0.152 −0.119 −0.057 −0.050 −0.365 −0.261 −0.295 0.039

−0.131 −0.138 −0.106 −0.057 0.009 −0.031 −0.007 −0.020

−0.276 −0.217 −0.104 −0.092 −0.752 −0.535 −0.604 0.078

−0.237 −0.250 −0.194 −0.105 0.019 −0.063 −0.014 −0.039

0.861 0.907 0.860 0.788 0.998 1.000 0.951 1.000

q(Re) > 0, q′(Re) < 0, q″(Re) < 0, M1 > 0, and M2 < 0. Here, q(Re) > 0 means that the H atom is positively charged at Re. The relation q′(Re) < 0 reflects the fact that, as the bond length increases, the H atom finally becomes neutral by receiving electron population from the O atom. Moreover, q″(Re) < 0 reflects that the rate at which the H atom receives electron population increases as the bond length increases around R = Re. Equations 13 and 14 are essential to understand the substituent dependencies of the slopes (M1) and the curvatures (nonlinearities, M2) of the DMFs. According to Table 2, NA has the largest M1 value and the smallest M2 and M3 values of the four ROH molecules, yielding the largest q(Re) value and the smallest |q′(Re)| value; thus, M1 ≈ q(Re) cos θ, and the M1 value reflects the almost constant charge on the H atom. However, the larger negative M2 values of TB and ME suggest smaller charges [q(Re)] and larger decreasing rates [q′(Re)] that make significant contributions to both M1 in eq 13 and M2 in eq 14 . For these reasons, compared with NA, TB and ME have reduced slopes and larger nonlinearities of their DMFs, as seen in Figure 3a. It should be emphasized here that the substituent dependencies of the q′(Re) and q″(Re) terms in eqs 13−15 cause a linear relation among the M1 and M2 values of the different molecules, which is the main subject of this study. Concerning the physical interpretation of the substituent dependence of q(Re) in Table 3, it is well-known that more electronegative substituents increase q(Re) and strongly polarize the OH bond. For the substituent dependence of q′(Re), from the present results, one can see the characteristics of a resistance; that is, as the substituent becomes more electronegative, electron-density migration between the O and H atoms becomes more difficult. Our understanding of the electron-density migration during XH stretching can be further improved by the concept of electron-cloud incomplete following, which has been used to explain internal rotation27 and out-of-plane bending motion.28,29 Next, we consider the q(Re), q′(Re), and q″(Re) values for RCH listed in Table 3. As can be seen in Table 2 and Figure 3, the

⟨v|μ*|0⟩ = ⟨v| ∑ MnΔRn|0⟩ = n=1

∑ Mn⟨v|ΔRn|0⟩ n=1

(17)

In this expression, the transition moment can be understood as the sum of the products of the series coefficients Mn and the matrix elements of ΔRn. Then, the substituent dependence of the transition moment is determined by the coefficients Mn, because the factors ⟨v|ΔRn|0⟩ are determined only by the PES and have a very small substituent dependence. For example, the relative differences in ⟨v|ΔRn|0⟩ for ROH for each v are up to 5%. Those among C2H6, C6H6, and C2H4 are 2−15%; however, those between C2H6 and acetylene are 5−50%. It is also important that the factors ⟨v|ΔRn|0⟩ are strongly dependent on the vibrational quantum number v. Based on the structure of eq 17, where each term exhibits a large substituent dependence only through the Mn coef ficients, a small variance in the overtone intensities can be explained only by the existence of a mechanism that cancels the substituent dependencies of dif ferent-order terms. To verify this hypothesis, contributions of the linear through quartic terms of the DMF to the transition moment were separated and are shown, including their signs, in Figures 4 and 5 for ROH and RCH, respectively. The sum of each contribution, which is almost equal to the transition moment calculated by the DMF represented by the sixth-order polynomial, is shown to the right of the decomposed bars for each molecule (dark blue bars). As is well-known, the intensity of the fundamental transition is dominated by the linear term, because ⟨1|ΔR|0⟩ is much larger than ⟨1|ΔRn|0⟩ (n ≥ 2). For overtone transitions, the contribution of the quadratic term becomes as large as that of the linear term, thereby weakening the substituent dependence of the linear term. The specific values of the matrix elements are almost the same as the results for ethanol in Table 8 of ref 16. For Δv ≥ 3, the contributions of the cubic and quartic terms become E



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Figure 4. Decomposition of transition moment dv0 for v = (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, and (f) 6 of ROH into the linear- (blue), quadratic- (red), cubic(yellow), and quartic- (sky blue) term contributions and sum of the linear- to quartic-term contributions (dark blue), including their signs.

where κv0 is called the Landau−Lifshitz tunneling exponent and can be expressed as

larger and further weaken the substituent dependence of the quadratic term. As explained in the preceding discussion, we have found that, for the satisfaction of SVOI, the following conditions are necessary: (1) there is a linear correlation between M1 and M2, and (2) the substituent dependencies of the PESs are small. More detailed expressions for these and additional conditions in a graphical form are provided in the next section. 3.4. Quasiclassical Expression of Transition Moment. In this section, we demonstrate that the characteristics of the DMFs explained in the preceding section are also an essential reason for the similar overtone intensities in the quasiclassical expression of transition moment. Medvedev studied overtone intensities using a quasiclassical approximation and found that they are more sensitive to the difference in the inner wall of the PES than to the DMF.36,37 In his quasiclassial approximation, the transition moment is given as the product of exponential factor exp(−κv0) and pre-exponential factor Q

dv0 = Q exp( −κv0)

2m lim [ ℏ R→R 0

∫R

κv 0 = −

∫R

R − v

V (R ) − εv dR

R − 0

V (R ) − ε0 dR ]

(19)

The integration for the first (second) integral is carried out over the classically forbidden region V(R) > εv (V(R) > ε0), that is, from the inner-wall turning points R−v (R−0 ) in states v (0) to a point R0 where the stretching potential tends to infinity. Whereas κv0 depends on the PES alone, the pre-exponential factor Q in eq 18 depends on both the PES and the DMF and can be expressed as Q = CvC0(ωvω0)1/2 π −1 Re{Z exp[i Im Φ(R 0)]}

(20)

Cv = (2π )1/4 (v !)−1/2 exp{0.5(v + 0.5)[ln(v + 0.5) − 1]}

(18)

(21) F



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Figure 5. Decomposition of transition moment dv0 for v = (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, and (f) 6 of RCH into the linear- (blue), quadratic- (red), cubic(yellow), and quartic- (sky blue) term contributions and sum of the linear- to quartic-term contributions (dark blue), including their signs.

Z=

∫L

μ(R )(rvr0)−1/2 exp[Φ(R ) − Φ(R 0)] dR

Φ(R ) =

i ℏ

∫R

R

p dR − + v v

i ℏ

∫R

the quantum values in Figure 4 within relative differences of less than 3%. Medvedev concluded that the exp(−κv0) term, which depends on the PES only, mainly contributes to overtone intensities because Q is a weakly dependent function of the vibrational quantum number v. However, we would emphasize that the characteristic of the DMF is also essential for the overtone intensity values, because we are focusing on very small variance as defined in section 3.1. To see the substituent dependence of Q explicitly, we considered the polynomial expansion of Q by ΔR around Re. By using the coefficients Mn in eq 17, Q can be expressed as

(22)

R

R + 0

p0 dR

(23)

where ωv (ω0), rv (r0), and R+v (R+0) are the frequencies, velocities of classical motion, and outer turning points, respectively, in the states with energies εv (ε0), and LR is the integration path displaced from the real axis into the upper half-plane. The quasiclassical transition moments for ROH were also calculated and are listed in Table 4. They are in agreement with

Q = M1Q(1) + M 2Q(2) + M3Q(3) + ···

Table 4. Quasiclassical Transition Moment dv0 (D) of ROH v

TB

ME

AA

NA

1 2 3 4 5 6

2.55 × 10−2 1.09 × 10−2 2.42 × 10−3 5.74 × 10−4 1.56 × 10−4 4.97 × 10−5

4.47 × 10−2 1.31 × 10−2 2.82 × 10−3 6.57 × 10−4 1.76 × 10−4 5.45 × 10−5

6.63 × 10−2 1.29 × 10−2 2.61 × 10−3 6.09 × 10−4 1.67 × 10−4 5.40 × 10−5

9.17 × 10−2 1.35 × 10−2 2.67 × 10−3 6.56 × 10−4 1.92 × 10−4 6.53 × 10−5

(24)

where Q(n) is the nth-order derivative of Q (Q(0) = 0). Note that Q(n) depend on the PES only, and thus, these derivatives have only a small substituent dependence. Multiplying eq 24 by exp(−κv0), the quasiclassical transition moment can be decomposed as ⟨v|μ*|0⟩quasiclassical = Q exp( −κv0) =

∑ MnQ(n) exp(−κv0) n=1

(25) G



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Figure 6. Plots of (M1, M2, M3) as coordinate points (blue circles ), the line obtained by least-squares fitting of these points (solid black line), the line perpendicular to the solid black line (dashed line), and the line directing the IAv vector (A = ME, C2H6) for v = 1−6 (blue, light blue, light green, orange, red, and purple lines, respectively) for (a) ROH and (b) RCH groups projected on the M1−M2 plane and (c) ROH and (d) RCH groups in M1M2M3 space. For v = 1, the angles between the Mn line and Iv vectors are 15° and 20° for ROH and RCH groups, respectively. For v = 2−6, the angles are 82−88° and 81−89° for ROH and RCH groups, respectively. These angles are defined in M1M2M3 space.

The plotting of individual components of eq 25 for ROH gave a figure almost identical to Figure 4 for the quantum components with indistinguishable differences. Therefore, quasiclassical calculations also support the conclusion that the substituent dependence of the M2-term contribution cancels out that of the M1-term contribution, because of the previously explained correlation between these terms. It is also noted that, if eq 24 included only the linear-term contribution, the substituent dependence of Q would become much larger for the overtone transitions. 3.5. Inner Product of Vectors M and Iv. In this section, to explain the conditions under which SVOI is satisfied from another point of view, we examine the case in which two molecules A and B have exactly the same overtone intensities. First, suppose the DMF of molecule A is approximated by the cubic polynomial μ A (ΔR ) = M1A ΔR + M 2A ΔR2 + M3A ΔR3

Because the transition moment in eq 17 is expressed as the inner product of these two vectors, the condition that the intensities for transitions to vibrational level v for molecules A and B become identical can be written as (MB − MA) ·IvA = 0

under the assumption that their PESs are exactly the same. This equation states that, when the M1−M3 values of different molecules are represented as coordinate points in threedimensional space, the line drawn from MA to MB, called the Mn line, should be orthogonal to the IAv vector, called the Iv line. In Figure 6, we present the calculated results for M1−M3, the line obtained by least-squares fitting of these points, and the lines directing the IAv vectors (v = 1−6). The reference molecule A for Figure 6a,c is ME, and that for Figure 6b,d is C2H6. For v = 1, the direction of Iv is almost parallel to the Mn line; therefore, eq 28 is not fulfilled, and a large substituent dependence is implied. On the other hand, for v = 2−6, the directions of Iv are almost perpendicular to the Mn line, and their v dependence is relatively small. Therefore, the M and Iv vectors approximately fulfill eq 28, which suggests a small substituent dependence for overtone transitions v = 2−6. A graphical representation of the M and Iv vectors provides a comprehensive tool for understanding various aspects concerning overtone intensities. We next present three other examples. First, it is clear that M1−M3 for the RCH group including acetylene lie on a straight line, even though it does not satisfy SVOI; see Figures 2 and 5. Then, the reason that acetylene is an

(26)

As shown in Figure 6, we define a position vector MA in the three-dimensional space (two-dimensional space for quadratic expansion) whose vector components are the Mn coefficients of the DMF, as well as a position vector IAv whose three components are the matrix elements, as follows MA = (M1A , M 2A , M3A ), IvA = (⟨ϕvA |ΔR |ϕ0A ⟩, ⟨ϕvA |ΔR2|ϕ0A ⟩, ⟨ϕvA |ΔR3|ϕ0A ⟩)

(28)

(27) H



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exception of SVOI is clearly because its PES is significantly different from those of the other RCHs. To confirm this statement, we calculated the transition moment by using the DMF of acetylene and the wave functions of C2H6 and obtained a result within 30% relative difference of the other molecules. Second, for a certain molecule A with an XH bond, one can similarly obtain the MA vectors with and without including electron correlation effects, which are denoted as MA,correlation and MA,HF, respectively. The Hartree−Fock method exposes two well-known shortcomings of the DMF of XH stretching vibrations. One is the overestimation of the ionic character, namely, the q(Req) value in eq 13, which causes the slope of the ) to be too positive. The other is incorrect DMF (MA,HF 1 dissociation limits, which causes the DMF to be less bent at longer bond distances. These problems can be solved by using electron correlation methods in such a manner that the difference vector ΔMA = MA,correlation − MA,HF has a dominant negative M1 component and a smaller negative M2 component in most molecules. Because the Iv vectors are not expected to have such a large correlation effect, one can expect ΔMA·I1 ≠ 0 and ΔMA·Iv ≈ 0 (v ≥ 2) and naturally understand the correlation effects studied before.18−20 Third, because of the accidental cancellation between the mechanical and electrical anharmonicities,33 or the so-called intensity anomaly of the NIDL,37 the intensity of a higher overtone can be exceptionally larger than that of a lower one. This occurs when the position vector MA and the Iv vector for a particular overtone level v are almost orthogonal, in which case the intensity becomes accidentally small, because MA·IAv ≈ 0. A simple relation among M1−M3 was derived in eq 16. The equilibrium bond length Re and cos θ are almost the same for ROH and RCH groups; therefore, the difference of eq 16 for molecules A and B leads to ΔM 2 = ΔM1/R e − Δq(R e) cos θ /R e + R eΔM3

μ A (ΔR ) =

μB (ΔR ) =

∞

∑ dvA0Fv0(ΔR) =

∑ MnA ΔRn

v=0

n=0

∞

∞

(30)

∑ dvB0Fv0(ΔR) = ∑ MnBΔRn v=0

(31)

n=0

From the calculation results, we know that the significant substituent dependence of dv0 appears only for v = 1 of ROH and only for v = 1 and 2 of RCH. Then, if the transition moments of molecules A and B satisfy the condition dAv0 = dBv0 (v ≥ 3), the difference between the DMFs is expressed as μB (ΔR ) − μ A (ΔR ) 2

=

∑ (dvB0 − dvA0)Fv0(ΔR) v=0 ∞

=

∑ (MnB − MnA)ΔRn

(32)

n=0

Taylor expansion of Fv0 leads to the identity for each of the ΔRn terms ∞

2

∑ ∑ (dvB0 − dvA0) n=0 v=0

Fv(0n)(0) n ΔR = n!

∞

∑ (MnB − MnA)ΔRn n=0

(33)

Explicit expressions for the Taylor expansion coefficients of Fv0 are given in Appendix. Comparing the coefficients of each term, we have (M1B − M1A ): (M 2B − M 2A ): (M3B − M3A ) 1 1 = 1: α(1 + β): α 2(1 + 3β) 2 6

(34)

where the constant β is defined as

(29)

B A ⎤ ⎡ d 20 − d 20 ⎥ β = (k − 4)⎢ B B A ⎥ ⎢⎣ (d10 − d10A)k (k − 2)/2(k − 3)(k − 5) − 4(d 20 − d 20 )⎦

suggesting that the slope of the M1−M2 line is 1/Re. The mean value of 1/Re is 0.55 a0−1 for ROH and 0.49 a0−1 for RCH. These values are in accord with the slopes of the lines obtained by least-squares fitting of M1 and M2, that is, 0.48 a0−1 for ROH in Figure 6a and 0.37 a0−1 for RCH in Figure 6b. Inspection of the parameters given in Tables 2 and 3 suggests that the deviations of the least-squares-fitted values from the 1/Re values are attributable to the substituent dependencies of the second and third terms of eq 29, −Δq(Re) cos θ/Re + ReΔM3, that is, the intercepts of the M1−M 2 lines in Figure 6a,b. Because eq 29 was derived by using only the characteristics of the DMF, in the following section, we present a more physically sensible expression for the slope of the Mn line. 3.6. Expressions for M1−M3 by the Wave Function Expansion Method. In the previous section, we showed that the Mn line and Iv vector are orthogonal to each other when the SVOI concept is satisfied. In this section, the DMF is expressed by the wave function expansion method21,22 and then by polynomial expansion. In this manner, M1−M3 can be expressed in terms of the Morse parameters. Moreover, we present the orthogonal condition of the Mn line and Iv vectors using the Morse parameters. Suppose the PES has no substituent dependence, in which case the DMFs of both molecules A and B are expressed by the same quotient functions and are also represented by the polynomial expansions, as follows

(35)

Equation 34 is the formula for the line with direction vector (1, α(1 + β)/2, α2(1 + 3β)/6). For the ROH group, we know that the transition moments of A and B also satisfy the relation dAv0 = dBv0 (v ≥ 2), that is, β ≈ 0. Then, eq 34 becomes (M1B − M1A ): (M 2B − M 2A ): (M3B − M3A ) = 1:

α α2 : 2 6 (36)

The direction vector of the Mn line determined by least-squares fitting of M1−M3 was (1, 0.48, 0.16), which is in accord with the (1, 0.65, 0.28) direction vector derived from the mean values of α/2 and α2/6 for the ROH group. For RCH, the direction vector of the Mn line determined by least-squares fitting was (1, 0.37, −0.02). Although the practical values of α(1 + β)/2 and α2(1 + 3β)/6 vary in the ranges 0.17 ∼ 0.42 and −0.18 ∼ 0.07, respectively, depending on the combination of molecules A and B, their mean values of 0.32 for α(1 + β)/2 and −0.02 for α2(1 + 3β)/6 are in accord with the values determined by the least-squares fitting. With the two theoretical models, we have shown that the slope of the M1−M2 line is approximately expressed as α/2 or 1/Re, implying that α/2 ≈ 1/Re. This can be understood by using an I



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empirical correlation between ωe and Re.38 We suppose the relationship R e nωe = C

approximation recently reviewed by Medvedev, this nature is also an essential reason for the similar overtone intensities. With a graphical representation of the M and Iv vectors, one can readily understand various aspects of overtone intensities, including the sensivity (insensitivity) of fundamental (overtone) intensities to the inclusion of electron correlation effects18−20 and the exceptional deviations from NIDL caused by accidental cancellation between mechanical and electrical anharmonicities,33 or the so-called intensity anomaly of the NIDL.37 The overtone intensity of acetylene was found to be an anomaly to the small substituent dependence. The Mn values of acetylene lie on the same line as those of the other RCH molecules, so the difference of the PES is responsible for the anomaly. There could be other anomalies similar to acetylene. By applying a categorization similar to that presented in this article to other sets of molecules, one could simply understand and categorize the substituent dependence of XH stretching overtone intensities.

(37)

where n and C are some constants depending on the set of molecules. Assuming n = 1, the value of C determined by leastsquares fitting is 0.0329 au for ROH and 0.0308 au for RCH with less than 6% relative differences. By substituting eq 37 with n = 1 into the relation α = ωe[m/(2De)]1/2, we obtain ω α = e 2 2

m C = 2De 2R e

m 2De

(38) 1/2

where the mean value of C/2[m/(2De)] is 1.18 for ROH and 1.06 for RCH. Therefore, we found that α/2 ≈ 1/Re. Next, we present the condition for the orthogonality between the Mn line and the Iv vector explained in eqs 26−28. The components of the Iv vector can be analytically expressed using the Morse parameters and the vibrational quantum number v.39 A full expression of the matrix elements is given in the Appendix. By the inner product of eqs 36 and A12, the condition for the orthogonality of the Mn line and Iv vector is obtained as m12 m π2 − m1 + 2 + 1 − =0 2 2 6

■

APPENDIX Expressions for the quotient functions F10 and F20 of a Morse oscillator are easily derived following refs 21 and 22 as F10(ΔR ) =

(39)

where m1 and m2 are functions that depend on the Morse parameters α and k and the vibrational quantum number v. See the Appendix for further details. For ME, the calculated values on the left-hand side of eq 39 are 0.019, −0.040, 0.064, 0.206, and 0.352 for v = 2−6, respectively. Although the values for v = 5 and 6 seem a little large, these values are insensitive to transition moment. In fact, the angles between the Mn line and the Iv vectors are 81−89° depending on v, and the relative differences of the transition moments are 2−26%. (The difference was taken relative to ME.) From the eigenfunction of the Morse oscillator and the calculation results for dv0, we found that the slope of the Mn line cannot vary independently from the PES, but should be expressed with the Morse parameter α. In other words, when SVOI is satisfied, there exists a significant relation between the electrical and mechanical properties of the XH bond.

F20(ΔR ) =

⎛ k − 2 αΔR ⎞ k − 3⎜ e − 1⎟ ⎝ k ⎠

(A1)

⎡ (k − 3)(k − 4) 2αΔR 2(k − 2)(k − 5) ⎢ e ⎣ 2k 2 k − 3 αΔR 1⎤ − + ⎥ e k 2⎦ (A2)

Taylor expansions of these functions give F10(ΔR ) = +

⎧k − 2⎡ (αΔR )2 k − 3⎨ ⎢1 + αΔR + 2! ⎩ k ⎣ ⎪

⎪

⎫ ⎤ (αΔR )3 + ···⎥ − 1⎬ = 3! ⎦ ⎭ ⎪

⎪

k−3

⎡ 2 (k − 2)α (k − 2)α 2 2 ×⎢ − + ΔR + ΔR 2k k ⎣ k

4. CONCLUSIONS We analyzed the one-dimensional effective DMFs, which are responsible for the total sum of the overtone intensities for OH stretching of alcohols and acids and CH stretching of hydrocarbons with different hybridizations of the C atom. We found that the M1−M3 coefficients of the one-dimensional DMFs lie on a straight line for each type of bond with similar XH stretching overtone intensities and similar PESs. For these sets of molecules, we showed that the Iv (v ≥ 2) lines, which contain information on the PES alone, are approximately orthogonal to the Mn line. With the help of a simple point-charge model, the linear relationship between M1 and M2 was interpreted as follows: An RXH molecule with a strong electronegative substituent R has the characteristics that (1) the XH bond is strongly polarized as Xδ−Hδ+ at the equilibrium structure and (2) a strong resistance to electron-density migration from atom X to atom H upon bond stretching. The slope of the Mn line can be expressed by the Morse parameter. The expression can be derived from the calculated results for the transition moment and analytical eigenfunction of Morse oscillator. We also found that, in the quasiclassical

+

⎤ (k − 2)α 3 3 ΔR + ···⎥ 6k ⎦

(A3)

⎧ (k − 3)(k − 4) 2(k − 2)(k − 5) ⎨ 2k 2 ⎩ ⎡ ⎤ (2αΔR )2 (2αΔR )3 ×⎢1 + 2αΔR + + + ···⎥ ! ! 2 3 ⎣ ⎦

F20(ΔR ) =

− = +

k− k

⎪

⎪

⎤ 1⎫ (αΔR )2 (αΔR )3 3⎡ + + ···⎥ + ⎬ ⎢1 + αΔR + 2! 3! 2⎭ ⎣ ⎦

⎪

⎪

⎡ k − 12 4(k − 3)α − ΔR 2(k − 2)(k − 5) ⎢ − 2 ⎣ k2 2k ⎤ (k − 8)(k − 3)α 2 2 (3k − 16)(k − 3)α 3 3 ΔR + ΔR + ···⎥ 2 2 ⎦ 2k 6k (A4)

Using these series coefficients, eqs 34 and 35 in the main text are obtained. J



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Specific expressions for the matrix elements of ΔR, ΔR2, and ΔR3 without binomial coefficients were reported by Gallas39 ⟨v|ΔR |0⟩ = −m0 /α

(A5)

⟨v|ΔR2|0⟩ = 2m0m1/α 2

(A6)

⟨v|ΔR3|0⟩ = −m0(3m12 + 3m2 − π 2)/α 3

(A7)

Vapor-Phase Alcohols and Acids. J. Phys. Chem. A 2001, 105, 3481− 3486. (7) Scheck, I.; Jortner, J.; Sage, M. L. Intensities of High-Energy Molecular C−H Vibrational Overtones. Chem. Phys. Lett. 1979, 64, 209−212. (8) Amrein, A.; Dübal, H. R.; Lewerenz, M.; Quack, M. Group Additivity and Overtone Intensities for the Isolated CH Chromophore. Chem. Phys. Lett. 1984, 112, 387−392. (9) Medvedev, E. S. Intensity Distributions in Molecular Overtone Vibrational Spectra. Chem. Phys. Lett. 1985, 120, 173−177. (10) Burberry, M. S.; Albrecht, A. C.; Swofford, R. L. Local Mode Overtone Intensities of C−H Stretching Modes in Alkanes and Methyl Substituted Benzenes. J. Chem. Phys. 1979, 70, 5522−5526. (11) Burberry, M. S.; Albrecht, A. C. Comments on the Concept of Universal Intensities in Local Mode Theory. J. Chem. Phys. 1979, 71, 4768−4769. (12) Lewerenz, M.; Quack, M. Vibrational Overtone Intensities of the Isolated CH and CD Chromophores in Fluoroform and Chloroform. Chem. Phys. Lett. 1986, 123, 197−202. (13) Ahmed, M. K.; Henry, B. R. A Local-Mode Analysis of the Overtone Spectra of Some Monosubstituted Cyclopropanes. J. Phys. Chem. 1987, 91, 5194−5202. (14) Lehmann, K. K. The Absolute Intensity of Visible Overtone Bands of Acetylene. J. Chem. Phys. 1989, 91, 2759−2760. (15) Vaida, V.; Feierabend, K. J.; Rontu, N.; Takahashi, K. SunlightInitiated Photochemistry: Excited Vibrational States of Atmospheric Chromophores. Int. J. Photoenergy 2008, 138091. (16) Takahashi, K.; Sugawara, M.; Yabushita, S. Theoretical Analysis on the Fundamental and Overtone OH Stretching Spectra of Several Simple Acids and Alcohols. J. Phys. Chem. A 2003, 107, 11092−11101. (17) Takahashi, K.; Sugawara, M.; Yabushita, S. Effective OneDimensional Dipole Moment Function for the OH Stretching Overtone Spectra of Simple Acids and Alcohols. J. Phys. Chem. A 2005, 109, 4242− 4251. (18) Kjaergaard, H. G.; Henry, B. R. Ab Initio Calculation of Dipole Moment Functions: Application to Vibrational Band Intensities of H2O. Mol. Phys. 1994, 83, 1099−1116. (19) Kjaergaard, H. G.; Daub, C. D.; Henry, B. R. The Role of Electron Correlation on Calculated XH-Stretching Vibrational Band Intensities. Mol. Phys. 1997, 90, 201−213. (20) Daub, C. D.; Henry, B. R.; Sage, M. L.; Kjaergaard, H. G. Modeling and Calculation of Dipole Moment Functions for XH Bonds. Can. J. Chem. 1999, 77, 1775−1781. (21) Trischka, J.; Salwen, H. Dipole Moment Function of Diatomic Molecules. J. Chem. Phys. 1959, 31, 218−225. (22) Cashion, K. A Method for Calculating Vibrational Transition Probabilities. J. Mol. Spectrosc. 1963, 10, 182−231. (23) Atkins, P. W. Molecular Quantum Mechanics, 3rd ed.; Oxford University Press: Oxford, U.K., 1997. (24) 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.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, revision C.02; Gaussian, Inc.: Wallingford, CT, 2004. (25) Bader, R. F. W. Atoms in Molecules. A Quantum Theory; Oxford University Press: New York, 1990.

where m0 = (− 1)v (k − 2v − 1)Γ(k − v − 1) Γ(v)/v(k − v − 1) Γ(k − 1)

(A8)

m1 = ψ (0)(k − v − 1) + ψ (0)(v) − ψ (0)(1) − ln k

m2 = ψ (1)(k − v − 1) + ψ (1)(v) + ψ (1)(1)

(A9) (A10)

and ψ is the polygamma function defined as the logarithmic derivative ψ (n)(z) = (d/dz)n + 1 ln Γ(z)

(A11)

From eqs A5−A7, the ratio of the ΔRn matrix element, that is, the direction vector component of the Iv line, is expressed as (⟨v|ΔR |0⟩, ⟨v|ΔR2|0⟩, ⟨v|ΔR3|0⟩) = (1, −2m1/α , (3m12 + 3m2 − π 2)/α 2)

(A12)

Because the inner product of eqs 36 and A12 gives m12 m π2 − m1 + 2 + 1 − 2 2 6

(A13)

the orthogonal condition, that is, eq A13 = 0, leads to eq 39 in the main text.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors acknowledge Mr. Kazutoshi Matsushita and Dr. Kaito Takahashi for their preliminary works. This work was supported by Grants-in-Aid for Scientific Research funded by MEXT in Japan. This work was also carried out within the MEXT-Supported Program for the Strategic Research Foundation at Private Universities, 2009−2013. The computations were partly carried out using the computer facilities at the Research Center for Computational Science, Okazaki National Research Institutes, Okazaki, Japan.

■

REFERENCES

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Theoretical Analysis of Weak Adjacent Substituent Effect on the

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