Interpretation of Semiclassical Transition Moments through Wave

Article pubs.acs.org/JPCA

Interpretation of Semiclassical Transition Moments through Wave Function Expansion of Dipole Moment Functions with Applications to the OH Stretching Spectra of Simple Acids and Alcohols Hirokazu Takahashi,† Kaito Takahashi,‡ and Satoshi Yabushita*,† †

Department of Chemistry, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522 Japan ‡ Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei, Taiwan ABSTRACT: Semiclassical description of molecular vibrations has provided us with various computational approximations and enhanced our conceptual understanding of quantum mechanics. In this study, the transition moments of the OH stretching fundamental and overtone intensities (Δv = 1−6) of some alcohols and acids are calculated by three kinds of semiclassical methods, correspondence-principle (CP) approximation, quasiclassical approximation, and uniform WKB approximation, and their respective transition moments are compared to those by the quantum theory. On the basis of the local mode picture, the one-dimensional potential energy curves and the dipole moment functions (DMFs) were obtained by density functional theory calculations and then fitted to Morse functions and sixth-order polynomials, respectively. It was shown that both the transition energies and the absorption intensities derived in the semiclassical methods reproduced their respective quantum values. In particular, the CP approximation reproduces the quantum transition moments if the formula given by Naccache is used for the action integral value. On the basis of these semiclassical results, we present a picture to understand the small variance in the overtone intensities of these acids and alcohols. Another important result is the ratios of semiclassical-to-quantum transition moment are almost independent of the applied molecules even with a great molecular variance of the DMFs, and they depend only on the nature of the semiclassical approximations and the quantum number. The difference between the semiclassical and quantum transition moments was analyzed in terms of a hitherto unrecognized concept that the Fourier expansion of the time dependent DMF in the CP treatment is a kind of the wave function expansion method using trigonometric functions as the quotient functions. For a Morse oscillator, we derive the analytic and approximate expressions of the quotient functions in terms of the bond displace coordinate in both the CP and the quantum mechanical frameworks and discuss the methodological dependence of the calculated transition moments. As a byproduct, we have found a simple derivation of the DMF expression first derived by Timm and Mecke long time ago.

1. INTRODUCTION The local mode model has shown a great success in treating XH bond stretching vibration of molecules, where X = C, O, and so on.1−4 Under this model, the vibrational wave function is described as a product of anharmonic oscillators using the internal coordinates. The fundamental and overtone intensities of the XH bonds in polyatomic molecules provide us with detailed information concerning the interplay between the socalled mechanical anharmonicity in the potential functions and the electrical anharmonicity reflected in the nonlinear dipole moment functions (DMFs). For example, we know a linear DMF gives only a fundamental transition for a harmonic oscillator. Then, what is the shape of a DMF that has only a f undamental transition and no overtone intensities for a Morse oscillator? The present paper will give the quantum and semiclassical answers to this question. A well-known characteristic of the XH stretching overtone transitions is the so-called normal intensity distributions low (NIDL), which states that the 0 → v overtone intensities show © 2015 American Chemical Society

a falloff in an exponential manner with increasing v, as has been explained by Medvedev5 by applying the quasiclassical approximation due to Landau−Lifshitz.6 Another well-known characteristic is that their overtone intensities show much smaller molecular variance than the fundamental intensity.7−10 For example, upon examining the CH vibrational spectra of hydrocarbons, Burberry and Albrecht suggested a “universal intensity concept for the local mode model”: the intensities of CH bond with different substituents show only a small variance for Δv ≥ 3 transitions.7,8 This feature led to the expectation that there might be some universal anharmonic DMFs that would be similar among these different molecules. However, our constructed effective one-dimensional dipole moment functions (1-D DMFs) of some acids and alcohols have shown a great variance even though their overtone Received: March 2, 2015 Revised: April 27, 2015 Published: April 28, 2015 4834

DOI: 10.1021/acs.jpca.5b02050 J. Phys. Chem. A 2015, 119, 4834−4845

Article

The Journal of Physical Chemistry A transition moments have similar values.11,12 This puzzling problem was solved recently.13 For a set of molecules RXH (X = C, O) exhibiting a small variance of overtone intensities (SVOI), their substituent dependence appears only in the DMFs; their XH stretching potential curves, thus their vibrational wave functions ϕv in the local mode model have a negligibly small difference. If the DMF of molecule A is expressed by a polynomial of the bond displacement ΔR as μA(ΔR) = ΣMnAΔRn, and also that of molecule B as μB(ΔR) = ΣMnBΔRn, their respective 0 → v transition moments are given by the inner products of MA·Iv and MB·Iv. Here, we have defined the DMF coefficient vectors MA = (MA1 , MA2 , ...) and MB = (MBB, MB2 , ...) for molecules A and B, respectively, and what we call the Iv vector, Iv = (⟨ϕv|ΔR|ϕ0⟩, ⟨ϕv|ΔR2|ϕ0⟩, ...), which is common to both molecules. Then, the condition for these two molecules to have the same overtone intensities is not MA = MB but (MA − MB)·Iv =0 only for v ≥ 2. In fact, if the coefficient vectors M = (M1, M2, ...) for individual molecules were plotted as coordinate points in the two- or three-dimensional space, they were found to lie on a straight line, called the Mn line, which was approximately orthogonal to the Iv vectors for v ≥ 2.13 In the present paper, we attempt to present another picture to understand the SVOI concept for OH stretching spectra of acids and alcohols based on various semiclassical methods. One of the motivations of the current work is to compare the accuracy of those methods by applying to the realistic nonlinear DMFs and Morse potentials. These parameters were obtained from ab initio calculations for the local mode OH stretching vibration of the previously studied four molecules, nitric acid (O2NOH), acetic acid (CH3COOH), methanol (CH3OH), and tert-butyl alcohol ((CH3)3COH). The approximate methods examined in this paper are as follows. First, in section 3.1(a), we study a semiclassical method, called the correspondence principle (CP) method, that evaluates the transition moment from the Fourier amplitudes of the time dependent DMF (TDDMF) with the classical mechanics.14−22 The basic idea behind this comes from the CP given by Heisenberg in the early days of quantum mechanics.23 With a linear DMF, thus without electrical anharmonicity, it has been shown that these classical mechanical quantities reproduce the quantum mechanical ones for several analytical potentials, including harmonic potentials and anharmonic Morse potentials.16 However, the DMFs of the above acids and alcohols showed a great molecular variance even though their overtone transition moments have similar values in the previous quantum treatment. Our interest is to assess the accuracy of these semiclassical methods in the overtone transition moments calculated with realistic nonlinear DMFs and also to give a semiclassical interpretation for the SVOI concept through the behavior of their Fourier components of the DMFs. Second, in section 3.1(b), we examine another semiclassical method, which is called quasiclassical (QC) method and has been developed by Medvedev5 based on the Landau−Lifshitz approximate method.6 In this method, a transition moment is expressed as the product of the exponential factor exp(−κv0) and the pre-exponential factor Q. It was generally believed that the exp(−κv0) term has dominant contributions to overtone intensities because Q is a function weakly dependent on v. However, we have previously clarified that the characteristic of DMFs, especially the linear dependence between the M1 and M2 parameters on the substituents also play an essential role for SVOI through the Q factor in the QC method. In this paper,

the accuracy of this method is further examined and compared with other methods. Third, in section 3.1(c), we calculate the transition moments numerically with the so-called uniform WKB approximation, which was developed to solve the well-known problem that primitive WKB wave functions diverge at the classical turning points.22 Furthermore, to provide a new perspective of the vibrational overtone intensities, the SVOI concept is analyzed by treating the classical TDDMFs within the CP treatment. In section 3.2, to explain the difference between the CP and quantum transition moments, we emphasize a hitherto unfamiliar concept that the Fourier series expansion of TDDMF for the CP treatment is a kind of the wave function expansion method24,25 for quantum theory with the basis functions being the quotient functions of vibrational wave functions. In section 3.3, for a Morse problem, we present the analytical expressions and plots of the quotient functions derived from both CP and quantum methods, and explain their difference in the transition moments. These results are discussed in relation to the approximate DMF expression derived by Timm and Mecke.26

2. THEORY 2.1. Quantum Theory. Here, only a minimum description is given because more details have been explained in a previous paper.13 In quantum mechanics, we solve the Schrödinger equation for the 1-D molecular vibration ⎡ ℏ2 d2 ⎤ + V (ΔR )⎥ϕv(ΔR ) = Evϕv(ΔR ) ⎢− 2 ⎣ 2m dΔR ⎦

(1)

where v, ΔR, m, V(ΔR), and ϕv are the vibrational quantum number, the displacement of OH internuclear distance, the reduced mass, the potential energy curves (PEC), and the vibrational wave function, respectively. We calculate the PEC and the DMF by the B3LYP functional27,28 with the 6-311+ +G(3df,3pd) basis set29−32 with the Gaussian 03 program33 as before.11−13 First, the potential energies and dipole moments of each molecule are calculated by changing only ΔR with keeping the remaining structure parameters based on the local mode picture, then they are fitted to a Morse function34 V(ΔR) = De(1 − e−αΔR)2 and a sixth-order polynomial of ΔR. Here De is the dissociation energy and α is the scaling factor; they are related to the harmonic frequency for infinitesimal amplitude as ωe = α(2De/m)1/2 and to the anharmonicity coefficient as χe = (ℏωe/4De). Then, the eigenvalues are given as Ev = ℏωe(v + 1/ 2) − ℏωeχe(v + 1/2)2, and the corresponding eigenfunctions ϕv can be expressed analytically as ϕv(y) =

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

y = k e−αΔR ,

2s = k − 2v − 1, v

Lv2s(y) = Γ(k − v)

∑ m=0

k = χe−1

( −1)m ym m ! (v − m) !Γ(k − 2v + m) (2)

where Γ is the gamma function, is the associated Laguerre polynomial of the variable y defined above, with 2s being a real valued index and k is the inverse of the anharmonicity coefficient and is about 40 for the OH bonds studied in this work. We have previously shown that the absorption intensities L2s v

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The Journal of Physical Chemistry A

the quotient f unctions derived f rom approximate wave f unctions, the expansion coef f icients, namely the transition moments derived f rom this approximate wave f unctions, are also approximate ones. Then we can discuss the accuracy of the transition moments from the functional form of the quotient functions. It is interesting to point out that the orthonormal relations of eq 5 with the appropriate range of the vibrational coordinate and the weight function define the specific orthogonal polynomials. For example, as shown in 2.1, each eigenfunction of a Morse oscillator is a product of the associated Laguerre polynomial and ϕ0(y) except for the normalization coefficient. Then, considering y = ke−αq as the variable, {ϕ0(y)}2 is the weight function for the orthogonal expansion with the quotient functions Fv0. In this context, the wave function expansion method with a Morse oscillator is considered as a least-squares fit with the weight function {ϕ0(y)}2. In a similar view, the functions {(1/(2π)1/2)eivθ} of the angle variable θ defined in (0, 2π) are orthonormal and, as will be explained later, their real part {cos vθ} is related to the Tchebycheff polynomial of the variable cos θ; thus the theoretical framework is numerically equivalent to a Fourier analysis. We note that this wave function expansion method can be used for any forms of wave functions and has already been considered by us for numerically obtained vibrational wave functions.12 In the present paper, this method is applied to analytic forms of wave functions mainly to analyze the relationship between quantum mechanical and semiclassical vibrational wave functions. 2.3. Correspondence Principle Method. Classical and semiclassical descriptions of periodic motions such as molecular vibration can be carried out using the angle variable θ and the action variable J as the canonical variables instead of the bond length R and its conjugate momentum p. Under the new coordinate, the Hamiltonian is cyclic and contains only the action variable J. As discussed by Marcus et al.,15,17,18 their simultaneous CP eigenfunctions are

of the aforementioned four molecules are essentially represented by the effective 1-D DMF μ*(ΔR).12 We continue the analyses by using μ*(ΔR) denoting it as μ(ΔR) for simplicity. 2.2. Wave Function Expansion Method. In this work, the transition moments for overtone transitions with the exact as well as various approximate wave functions are calculated. The usual way is to calculate the matrix elements of DMF with the initial- and final-state wave functions and compare their values to discuss the accuracy. This is an indirect method in that the relationship between the accuracy of the wave functions and the transition moment cannot be predicted before the actual calculation. However, if we use the wave function expansion method24,25 to represent the DMF and consider its least-squares definition,12 which is satisfied by any orthogonal polynomials as discussed later with eq 6, we can understand a more direct relation between the wave functions and the transition moments. Suppose we have a set of approximate or exact vibrational wave functions {φv(q)} of vibrational coordinate q, defined in the region (a, b). In the usual quantum treatment, q is the displacement ΔR, and (a, b) = (−∞, +∞), but in the classical case, the region is limited between the left- and right-turning points (a, b) = (ΔR), or with the angle variable (a, b) = (0, 2π). First, with the completeness of the wave functions{φv(q)}, the function μ(q)φ0(q) is expanded as ∞

μ(q)φ0(q) =

∑ ⟨φv|μ|φ0⟩φv(q)

(3)

v=0

Dividing both sides by the ground-state wave function φ0 (q), we obtain ∞

μ(q) =

∑ ⟨φv|μ|φ0⟩ v=0

φv(q) φ0(q)

∞

≡

∑ dv0Fv0(q) v=0

(4)

Therefore, the transition moment can be recognized as the expansion coefficient dv0 of the DMF with taking the quotient function Fv0(q) ≡ φv(q)/φ0(q) as the basis function in the region that φ0(q) ≠ 0. Note that the expansion of the DMF in eq 4 is meaningful only within the 0 → v transition manifold, and cannot be applied to a transition outside it, say, 1 → 2 transition. As is readily recognized, the quotient functions Fv0(q) satisfy the following orthonormal relations with respect to the weight function of the squared ground-state wave function, (Fm0 , Fn0) ≡

∫a

b

Fm0(q) Fn0(q) φ0(q)2 dq = δm , n

I (c) ≡ (μ −

(7)

The action integral for this particular semiclassical state is usually quantized with Jv =

1 2π

⎛

(5)

⎜

⟨ϕvcp|μ|ϕ0cp⟩ = ⟨ϕ0cp|μ|ϕvcp⟩ =

N

∑ cmFm0 , μ − ∑ cnFn0) m=0

= (μ , μ) − 2 ∑ cn(μ , Fn0) + n=0

n=0

(8)

1 2π

∫0

2π

μ(J ,θ ) exp[ivθ ] dθ

This means that the 0 → v transition moment is the vth Fourier component of the DMF, which is treated as a classical function of the action and angle variables. The most popular averaged action value J employed for the transition is the symmetrized semiclassical expression named by Nikitin et al., as follows,

N

∑ cn

⎟

(9)

n=0 N

⎞

∮ p dR = ℏ⎝v + 12 ⎠

However, to compute 0 → v transition properties, the earlier studies by Marcus et al. and Nikitin et al.20,21 show that the transition moment and the frequency can be extracted from classical quantities with appropriately averaged action variable J with the initial and final semiclassical states. For example, the transition moment of the 1-D DMF μ(R) is approximated by the classical quantity as follows,

Next, with the above-defined inner-product, let us consider the square norm of the difference function with the first N-term expansion, μ(q) − ΣNn=0cnFn0(q), N

1 ivθ e 2π

ϕvcp(θ ) =

2

(6)

Here, cn is an arbitrary expansion coefficient. Because the minimum condition for the least-squares, ∂I/∂cn = 0, yields cn = (μ, Fn0) ≡ ⟨φn|μ|φ0⟩, we understand that the least-squares fit of μ(q) = ΣNn=0cnFn0(q) with the weight function φ0(q)2 gives the expansion coefficients equal to the transition moments. If we use

J sym ≡ 4836

Jv + J0 2

=

ℏ (v + 1) 2

(10) DOI: 10.1021/acs.jpca.5b02050 J. Phys. Chem. A 2015, 119, 4834−4845

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The Journal of Physical Chemistry A Table 1. Mean Values of the Acids and Alcohols for the Approximate-to-Quantum Transition Energy Ratiosa v 1 sym

J Jimp uniform WKB uniform WKB (orthogonalized) a

1.000 1.000 0.954 0.960

(0.0) (0.0) (4.1 × 10−5) (3.5 × 10−5)

2 1.000 1.005 0.974 0.976

3

(0.0) (7.2 × 10−5) (3.3 × 10−5) (7.5 × 10−6)

1.000 1.010 0.981 0.983

4

(0.0) (1.6 × 10−4) (3.2 × 10−5) (1.3 × 10−5)

1.000 1.016 0.984 0.987

5

(0.0) (2.7 × 10−4) (3.4 × 10−5) (2.3 × 10−5)

1.000 1.023 0.987 0.989

6

(0.0) (3.9 × 10−4) (3.4 × 10−5) (2.7 × 10−5)

1.000 1.030 0.989 0.990

(0.0) (5.3 × 10−4) (3.6 × 10−5) (3.8 × 10−5)

The standard deviations are given in parentheses.

where each of Jv and J0 is defined by eq 8. In addition to the Jsym above, we have also calculated the transition moment using an improved action integral suggested by Naccache16

⎛ n! ⎞1/(n − m) J imp = ℏ⎜ ⎟ ⎝ m! ⎠

under the initial condition mentioned above, the semiclassical wave function for v ≠ 0 contains both Fourier components;18 thus the semiclassical quotient function for the physical state 1/2 ivθ with v > 0 is written as Fcp + e−ivθ)]/ϕcp v0 = [(1/(2π) )(e 0 = 2 cos vθ. Conversely, we arrived at a physical interpretation that the quantum Fv0 plays a role corresponding to 2 cos vθ, a basis function of the Fourier series expansion of TDDMF in the CP treatment. In our actual calculation, the 1-D DMF was expanded into sixth-order polynomials in terms of the displacement of XH bond length ΔR, as follows,

(11)

for the transition between the states m and n. This formula was originally devised to satisfy the condition that the semiclassical matrix elements of momentum operators of a harmonic oscillator agree with the quantum ones. Although Naccache did not explain why eq 11 gives very accurate values for the matrix element of ΔR, Shirts explained later that for a Morse oscillator the matrix element of ΔR with semiclassical method using the above condition equals to that of quantum mechanics for k (inverse of the anharmonicity, eq 2) sufficiently larger than m and n.19 Then the semiclassical values derived with this Jimp are expected to reproduce the quantum values more precisely than using Jsym. In this paper, we consider only v = 0 for the initial state, then Jimp can be expressed as J

imp

1/ v

= ℏ(v ! )

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

In this expression, the so-called electrical anharmonicity is expressed by the Mn (n ≥ 2) terms. For the Morse oscillator satisfying the initial condition, the displacement ΔR is given by the angle variable θ as follows,17,19,35 ΔR =

imp

Along a classical trajectory with either J = J or J , the DMF μ(J,θ) in eq 9 oscillates periodically with the angle variable. With the frequency ω of the classical vibrational motion on a PEC, the angle variable is θ = ωt, where, for the anharmonic motion, the angular frequency ω is dependent on the amplitude, namely, the action integral; thus, in the above equation ω = ω(J). The initial conditions of the trajectory were taken as ΔR(t=0) = ΔR>, where ΔR> is the right turning point at J and ΔṘ (t=0) = 0, under which the Fourier transform in eq 9 has only cosine terms and results in the following semiclassical transition moment expression, ⟨ϕvcp|μ|ϕ0cp⟩ =

1 2π

∫0

2π

μ(J ,θ ) cos(vθ ) dθ ≡

P=

(16)

E /De

(17)

and the classical energy E is expressed by J as follows,

E = Jωe − J 2 ωeχe /ℏ

(18)

Therefore, the angular frequency ω(J) of the classical vibration is given by ω(J ) = ∂E /∂J = ωe − 2Jωeχe /ℏ

(19)

Then, the classical overtone angular frequency for a 0 → v transition is given by

μv 2

1 1 + P cos θ ln α 1 − P2

where P is related to the vibrational energy E by

(12) sym

(15)

vω(J ) = v(ωe − 2Jωeχe /ℏ)

(13)

(20)

To obtain the Fourier amplitudes of μ(ΔR), the ones for powers of the coordinate (ΔR(t)n)v, which correspond to the quantum mechanical matrix elements of ΔRn were evaluated by the following integrals17,19 by using the Mathematica program.

Here the usual convention to define the Fourier amplitudes has been employed for μv.19 The above eq 13 states that each Fourier coefficient μv is twice the semiclassical transition moment; thus the 1-D DMF can be expressed as a function of the angle variable as follows,

(ΔR(t )n )v =

μ(J ,θ ) = μ0 /2 + μ1 cos θ + μ2 cos 2θ + μ3 cos 3θ + ⋯

1 πα n

∫0

2π

⎛ 1 + P cos θ ⎞n ⎜ln ⎟ cos(vθ ) dθ ⎝ 1 − P2 ⎠ (21)

= ⟨ϕ0cp|μ|ϕ0cp⟩ + ⟨ϕ1cp|μ|ϕ0cp⟩·2 cos θ + ⟨ϕ2cp|μ|ϕ0cp⟩·2 cos 2θ + ⋯

3. RESULTS AND DISCUSSION 3.1. Transition Energy and Transition Dipole Moment. Both the CP and QC results are compared with those with the quantum calculation reported in our earlier papers.11−13 3.1(a). Correspondence Principle Method. We present the ratios of the approximately calculated transition energy to the quantum one in Table 1 for transitions to v = 1−6. Because

(14)

As explained in section 2.2, we emphasize here that 2 cos vθ corresponds to the quotient function for the semiclassical method. From eq 7, the v = 0 wave function is simply ϕcp 0 (θ) = 1/(2π)1/2 and if μ(J,θ) is interpreted to have both positive (v) and negative (−v) Fourier components of {[1/(2π)1/2]eivθ}, 4837


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The Journal of Physical Chemistry A Table 2. Mean Values of the Acids and Alcohols for the Approximate-to-Quantum Transition Moment Ratiosa v 1 sym

J Jimp quasiclassical uniform WKB uniform WKB (orthogonalized) a

1.001 1.001 1.027 1.016 1.009

(1.1 (1.1 (4.1 (2.3 (1.2

2 × × × × ×

−4

10 ) 10−4) 10−4) 10−3) 10−3)

1.062 1.001 1.017 0.959 0.981

(1.2 (8.4 (4.2 (2.7 (1.4

3 × × × × ×

−4

10 ) 10−4) 10−5) 10−2) 10−2)

1.159 0.994 1.012 1.074 1.055

(1.1 (5.7 (1.0 (4.4 (3.4

4 × × × × ×

−4

10 ) 10−4) 10−5) 10−2) 10−2)

1.283 0.983 1.009 1.643 1.347

(1.7 (2.6 (9.7 (2.2 (1.2

5 × × × × ×

−4

10 ) 10−3) 10−7) 10−1) 10−1)

1.436 0.969 1.008 2.546 1.723

(3.4 (5.1 (5.4 (3.5 (1.3

6 × × × × ×

−4

10 ) 10−3) 10−6) 10−1) 10−1)

1.619 0.956 1.006 2.545 1.483

(6.7 (7.2 (7.9 (1.2 (1.3

× × × × ×

10−4) 10−3) 10−6) 10−1) 10−1)

The standard deviations are given in parentheses.

ΔRn should have almost the same value, irrespective of the power n,

these ratios have shown only a small substituent dependence, their average value over four molecules was given here along with the standard deviation in parentheses. The transition energies calculated by Jsym are in complete agreement with quantum value, because their expressions are known to be identical for a Morse potential.17 Those by Jimp show slight differences from the quantum values; however, their discrepancy is still less than 3%. Thus, the semiclassical values using either Jsym or Jimp reproduce the quantum transition energies very well. The corresponding transition moments calculated with the ab initio nonlinear DMF are summarized in Table 2, again in the form of the mean value and the standard deviation for the ratios of the approximate transition moment to the exact quantum one. Although the improved action variable Jimp gives results much closer to the quantum results than Jsym, as in the previous work with the matrix element of ΔR,18,19 the overall trends of the transition moments are given by the Fourier amplitudes calculated using Jsym as well as Jimp. With Jsym, very similar average ratios of 1.00, 1.06, 1.16, 1.28, 1.42, and 1.60 for Δv = 1−6 were also obtained36 by using the grid variational method37 with the numerically tabulated form of onedimensional potential energies and dipole moment values, which were provided directly from quantum chemical calculations; thus the results should be free from the fitting procedure. This fact suggests that the SVOI can be reproduced by the CP method as well. Furthermore, the difference between the quantum and the CP (both with Jsym and Jimp) transition moments increases with increasing v. The detailed analysis using TDDMF and the quotient function will be given later in sections 3.2 and 3.3. Interestingly, the ratios of the transition moments derived with Jsym to the quantum method are almost identical to those given by Nikitin et al., who used a Morse potential and a linear DMF and explained that the deviation from unity originates from the use of an average action variable Jsym in the calculation of the Fourier amplitude.20,21 Because the ratios of the transition moments shown in Table 2 have a very small substituent dependence, even though their DMFs have a great molecular variance and their matrix elements of ΔR2 contribute largely to their overtone transition moments (see Figures 3 and 4 of ref 13), these interesting results need further examination. For a particular 0 → v transition of molecule A, the ratio of the CP transition moment to the quantum one is given as follows, μvCP (A) 0 μv0 (A)

=

⟨ϕvcp|ΔR |ϕ0cp⟩ ⟨ϕv|ΔR |ϕ0⟩

⟨ϕv|ΔR2|ϕ0⟩

··· =

⟨ϕvcp|ΔRn|ϕ0cp⟩ ⟨ϕv|ΔRn|ϕ0⟩ (23)

which also suggests that the following relation holds for each 0 → v transition, ⟨ϕv|ΔR |ϕ0⟩:⟨ϕv|ΔR2|ϕ0⟩:⟨ϕv|ΔR3|ϕ0⟩⋯ = ⟨ϕvcp|ΔR |ϕ0cp⟩:⟨ϕvcp|ΔR2|ϕ0cp⟩:⟨ϕvcp|ΔR3|ϕ0cp⟩⋯

(24)

From this analysis, it becomes evident that the ratios shown in Table 2 are specific for the 0 → v transition for the Morse oscillator and independent of the molecular DMF parameters used. This interesting phenomenon seems to hold as long as their potentials, and therefore, vibrational wave functions are essentially the same. In fact, these matrix elements have been given analytically, and their ratio of the matrix element of ΔR2 to that of ΔR in quantum theory is given as follows38 ⟨ϕv|ΔR2|ϕ0⟩ ⟨ϕv|ΔR |ϕ0⟩

2 = − (ψ (0)(v) − ψ (0)(1) + ψ (0)(k − v − 1) α − ln k)

(25)

where ψ (z) is the polygamma function defined as the logarithmic derivative of the gamma function. For the semiclassical method, the corresponding ratio was given as19 (n)

⎡ 2 ⎢ (0) = − ψ (v) − ψ (0)(1) ⟨ϕvcp|ΔR |ϕ0cp⟩ α ⎢⎣

⟨ϕvcp|ΔR2|ϕ0cp⟩ ∞

+ v∑ l=1

l ⎤ k−J 1 ⎛ J ⎞ ⎥ − ln k ⎜ ⎟ − ln ⎥⎦ l(l + v) ⎝ k − J ⎠ (k − 2J )2

(26)

Here and in what follows, we have redefined J as the action variable divided by Planck’s constant ℏ. By comparing eqs 25 and 26, we find that ψ(0)(k − v − 1) in the quantum method corresponds to the third and fourth terms of eq 26. The relative difference between these terms for Jsym and Jimp were 0.006− 0.02% and 0.006−0.9%, respectively, and Jsym gave a slightly closer value than Jimp for this ratio. The ratios in eqs 25 and 26 are nothing but the slope of what we called the Iv line.13 We have previously clarified that, to satisfy the SVOI concept, the slope of the Iv line should be approximately −2/α and have a small v dependence for v ≥ 2; then it becomes perpendicular to the Mn line, because the slope of the M1−M2 line in quantum treatment for these molecules

⟨ϕvcp|M1A ΔR1 + M 2A ΔR2 + ···|ϕ0cp⟩ ⟨ϕv|M1A ΔR1 + M 2A ΔR2 + ···|ϕ0⟩

=

⟨ϕvcp|ΔR2|ϕ0cp⟩

(22)

Because the ratio M1:M2:M3:... critically depends on the molecules, a very small substituent dependence of the above ratio implies that the following ratios of the matrix elements of 4838


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The Journal of Physical Chemistry A can be approximately expressed as α/2. We have derived these results by combining the wave function expansion method and the Taylor expansion of μ(R) with the condition that the molecules showing SVOI have exactly the same PEC and overtone intensities for v ≥ 2.13 As will be discussed later in section 3.3, the semiclassical expression of the quotient function for v = 1 is essentially identical to the quantum one (see eqs 33 and 31, respectively), from the above-mentioned procedure, we can find that the slope of the M1−M2 line for the CP method is also α/2. We now state that the CP method with both Jsym and Jimp gives extremely good value for the slope of the Iv line and then it also satisfies the SVOI concept. We can give here another mathematical condition that describes the SVOI. Suppose molecules A and B have the same PEC and vibrational wave functions, and their DMFs are approximated by cubic polynomials. Under those conditions, their overtone intensities are identical if 3

= (− 1)v − 1(ωvω0)1/2 (ωey)−1 ⎡ 2π(v + 1/2)2v + 1(k − v − 1/2)k − v − 1/2 ⎤1/2 ⎥ ×⎢ ⎣ ⎦ e v + 1v ! (k − 1/2)k − 1/2 (29)

where ϕqc v is the QC vibrational wave function, Cv is the normalization constant, and ωv is the classical frequency. The ratio of eq 29 to the quantum matrix element should depend on the k value. However, their k dependence was found to be almost independent for k = 20−80. In particular, the relative differences of the ratios are less than 0.1% for v = 1−6. For the quantum method using the wavefunction defined in eq 2, by using Stirling’s approximation (Γ(z + 1) ≈ (2π z)1/2(z/e)z) to the factorial and the gamma functions appearing in the quantum expression, almost the same expression as the QC method above can be derived as follows.5

3

⟨ϕv| ∑ MnA ΔRn|ϕ0⟩ = ⟨ϕv| ∑ MnBΔRn|ϕ0⟩ n=1

⟨ϕvqc|ΔR |ϕ0qc⟩ = (− 1)v − 1CvC0(ωvω0)1/2 (ωey)−1 exp[− κv0]

n=1

⎡ v !Γ(k − v) ⎤1/2 ⟨ϕv|ΔR |ϕ0⟩ = ( −1)v − 1(ωvω0)1/2 (ωey)−1⎢ ⎥ ⎣ Γ(k) ⎦

(27)

≈ ( −1)v − 1(ωvω0)1/2 (ωey)−1

As discussed in the Introduction, eq 27 can be expressed as (MA − MB)·Iv = 0, and this is the condition that the M1, M2, and M3 of molecules A and B should lie on the same “plane” perpendicular to the Iv line. Thus, for the overtones that have a large contribution from cubic term of the DMF, we should consider the M1−M2−M3 plane instead of the M1−M2 line. However, as mentioned before, M1, M2, and M3 for the previously mentioned four molecules lie on a straight line and SVOI is satisfied even by excluding the cubic terms. Thereby, to discuss the difference between quantum and semiclassical methods concisely, we only consider linear and quadratic terms in what follows. 3.1(b). Quasiclassical Method. Recognizing vibrational overtone transitions as dynamical tunneling, Medvedev has developed another semiclassical method, the so-called quasiclassical (QC) approximate method, and has applied the method to XH overtone intensities.5,39−41 The QC transition moments are expressed as the product of exponential factor exp(−κv0) and pre-exponential factor Q as follows, μvQC = Q exp( −κv0) 0

⎡ 2π v v + 1/2(k − v − 1)k − v − 1/2 ⎤1/2 ⎥ ×⎢ ⎣ ⎦ (k − 1)k − 1/2 (30) 13

In a previous work, we have shown the results of the QC transition moments by using sixth order 1-D DMF μ(ΔR) to represent various degrees of electrical anharmonicity. We also present their ratios to the quantum transition moments in Table 2. Because we have used the 6-311++G(3df,3pd) basis set, which is slightly different from the previously used 6-311+ +G(3d,3p) basis set, the absolute value of the transition moments becomes 1−20% smaller than the previous values for v = 1−6. However, the ratio of the QC to quantum transition moments is almost the same for both basis sets. Surprisingly, the ratios of the transition moment calculated by using sixthorder DMF given in Table 2 are almost the same as those calculated with a linear DMF.39 Because this result is analogous to the previously mentioned CP case, it is considered that the corresponding relations written in eqs 23 and 24 are also satisfied for the QC method. Actually, we have calculated the slope of the Iv line for the QC method using Medvedev’s most accurate numerical integration method along the complex trajectory given in eq 31 of ref 39, and found its relative differences from the quantum one are less than 0.02%. Though the analytical expression corresponding to eq 26 for the QC method cannot be derived, from eq 25 of our previous paper,13 the slope of the Iv line depends not on the exponential factor but on the pre-exponential factor Q. 3.1(c). Uniform WKB Method. In this section, we present the results with the uniform WKB method that represents the wave function using the Airy function.22,42,43 Although the uniform WKB wave function satisfies the continuity around the classical turning points, the two functions continued from the left and right turning points do not necessarily have the same logarithmic derivative value in the classically allowed region; moreover, different eigenstates lack the orthogonality in general. These problems have caused serious numerical errors, because the overtone intensities are very small and sensitive quantities. Therefore, we followed the procedure by Miller44 to

(28)

where κv0, called Landau−Lifshitz tunneling exponent, depends on the PEC alone, and the pre-exponential factor Q depends on both the PEC and the DMF. Note this formula is independent of the value of the DMF at ΔR = 0, because it uses orthogonal wave functions.5 Medvedev has also reported that, under a certain condition, his QC formula is reduced to the semiclassical Fourier transform formula shown in eq 9.39,40 An improved semiclassical formula for the matrix element of ΔR presented by Nikitin et al.20,21 corresponds to this case. The QC method requires a sophisticated mathematical method and shows in general excellent agreement with quantum values. For example, the ratio between the matrix element of the linear DMF ⟨v|ΔR|0⟩ for the QC and the quantum method has been reported.39 Furthermore, we can also recognize the accuracy of the QC method by comparing its analytical expression of ⟨v|ΔR|0⟩ to the corresponding quantum one. For the QC method, the matrix element can be written as 4839


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Figure 1. Time dependence of the effective one-dimensional dipole moment functions μ(ΔR(t)) and 0.2 D × cos vθ for (a) v = 1, (b) v = 2, (c) v = 3, and (d) v = 4. The dipole moment functions for nitric acid (green line), acetic acid (yellow line), methanol (red line), and tert-butyl alcohol (blue line) are given. 0.2 D × cos vθ is given in dashed lines.

are very similar. Therefore, the substituent dependence of the TDDMFs for the CP method is essentially determined by that of Mn. One will notice that for the fundamental transition, both cos θ and the TDDMFs are slowly varying functions in the same phase. On the contrary, for the overtones, cos vθ is a fast oscillating function, whereas the TDDMF is still a slowly varying function. From these behaviors, it is obvious that the transition moment of the overtones will be much smaller than the fundamental one, and show an exponential decrease with increasing v. The main reason for this is that the DMF is a smooth function of ΔR, dominated by the linear term, M1ΔR in eq 15; therefore, the TDDMF shows a similar behavior to the motion of ΔR(t), a simple oscillation along the PEC with a frequency of ω(J). Furthermore, because the anharmonic term in the Hamiltonian is much smaller than the harmonic term, the classical frequency ω(J) and thus the motion of R(t) show a small v dependence, as eq 19 shows. Looking at the TDDMFs of the overtones shown in Figure 1b−d, we still see the differences between the molecules, raising questions concerning SVOI. However, for the calculation of the transition moments we take the vth Fourier component (eq 13); thus, subtracting the Fourier component of the fundamental v = 1 from the TDDMF does not affect the values of the overtone transition moments. In Figure 2a−c, we show the TDDMFs containing only the information on overtones, obtained by subtracting their respective v = 1 components 2μCP 10 cos θ from the TDDMF in Figure 1. Looking at the similarity of these functions, one can easily understand SVOI among the four molecules. We note that subtracting the v = 1 Fourier component from TDDMF can be related to

obtain the uniform WKB wave functions and the eigenvalues by imposing for the two wave functions to have the same logarithmic derivative at ΔR = 0. After having found the proper eigenfunctions for v = 0−7, we have further orthogonalized them with the symmetric orthogonalization scheme.45,46 The eigenvalues and transition moments calculated by using these uniform WKB wave functions are also given in Tables 1 and 2, respectively, as the ratios to the quantum values. Both eigenvalues and transition moments are improved by the orthogonalization. Though the orthogonalized uniform WKB eigenvalues for v = 1−3 show less accuracy than the semiclassical values, it shows comparable values to those with Jimp for v = 4−6. Even though the uniform WKB transition moments show the accuracy comparable to or better than those with Jimp, their standard deviations for the four molecules are the worst for all v. 3.2. Time Dependent Dipole Moment Function. From the results so far, it becomes clear that the CP transition moments show similar trends as the quantum results, namely, the exponential decrease in the absolute value with increasing the quantum number v and similar overtone intensities for the alcohols and acids. In the following, we examine the TDDMF μ(ΔR(t)) and obtain a detailed understanding on the traits of the CP transition moments. Figure 1a−d shows the TDDMFs of eq 14 as functions of time. Note that these TDDMFs contain the information on not only the electrical anharmonicity described by Mn parameters in eq 15 but also the mechanical anharmonicity through the time dependence of the displacement in eq 16. Due to the similarity in the PEC, the cosine functions, the basis of Fourier expansion, of the four molecules 4840


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accuracy of the CP transition moments can be understood by comparing the expressions of their quotient functions. Using eq 2, the fundamental component of the quotient function for quantum theory is derived as follows, quantum F10 (ΔR ) =

⎛ k − 2 αΔR ⎞ k − 3⎜ e − 1⎟ ⎝ k ⎠

(31)

For the CP method, as mentioned before, the quotient function is expressed as 2 cos vθ. From eqs 16−18, cos θ can be expressed with ΔR and the action variable J as follows, cos θ =

⎧⎛ k − 2J ⎞2 ⎫ ⎟ e αΔR − 1⎬ ⎨⎜ 2 J(k − J ) ⎩⎝ k ⎠ ⎭ k

⎪

⎪

⎪

⎪

(32) sym

The detailed derivation is given in Appendix. When v = 1, J = Jimp = 1, then the quotient functions have the same expression for both Jsym and Jimp as follows, sym imp F10 (ΔR ) = F10 (ΔR )

= 2 cos θ ⎧ (k − 2)2 αΔR ⎫ k ⎨ − 1⎬ e 2 ⎭ k−1⎩ k

=

(33)

Because of the complication of their expressions, we next consider their asymptotic expressions. With a sufficiently large k value, eqs 31 and 33 are reduced to the same expression as follows quantum F10 (ΔR ) ≈

k (e αΔR − 1)

sym imp F10 (ΔR ) = F10 (ΔR ) ≈

(34)

k (e αΔR − 1)

(35)

Thus, it is clear that the transition moment calculated by these CP methods for v = 1 show a good agreement with the quantum theory. Next, the complete expressions of the quotient functions for v = 2 are written as follows,

Figure 2. Time dependence of overtone components in the effective one-dimensional DMFs, which were obtained by subtracting their respective v = 1 component 2μCP 10 cos θ from the dipole moment functions in Figure 1 for (a) v = 2, (b) v = 3, and (c) v = 4. The functions for nitric acid (green line), acetic acid (yellow line), methanol (red line), and tert-butyl alcohol (blue line) are given.

quantum F20 (ΔR ) =

2(k − 2)(k − 5)

2 ⎧ (k − 3)(k − 4) 2αΔR ⎫ 2(k − 3) αΔR ×⎨ − + 1⎬ e e 2 ⎭ ⎩ k k (36)

subtracting the fundamental contribution of μ10F10(ΔR) from the DMF as carried out in our previous paper.12 This correspondence can also be understandable from the fact that each Fourier component in the CP method corresponds to the respective quotient function in quantum mechanics, as explained in section 2.3. To confirm this interesting aspect, in the next section, we express the CP quotient functions as a function of ΔR and compare to their quantum ones. 3.3. Comparison of the Quotient Functions. In this section, we compare the quantum and CP quotient functions for a Morse oscillator analytically and graphically to discuss the difference of the transition moments calculated with Jsym and Jimp. Although the formula of vibrational wave function for the CP expression eq 7 looks very different from that given in eq 2, their quotient functions will be shown to be fairly similar. Moreover, for v ≥ 2, we also recognize that the quotient functions for Jimp are closer to quantum ones than that for Jsym in the classically allowed region. From the least-squares meaning of the transition moments as discussed in 2.2, the

sym F20 (ΔR ) =

4(k − 3)2 ⎧ (k − 3)2 2αΔR ⎨ − 2e αΔR + 1 e 3(2k − 3) ⎩ k 2 3(2k − 3) ⎫ ⎬ 2(k − 3)2 ⎭

+

imp F20 (ΔR ) =

(37)

2 (k − 2 2 )2 ⎧ (k − 2 2 )2 2αΔR ⎨ e 2(k − 2 ) ⎩ k2 − 2e αΔR + 1 +

⎫ 2 ⎬ 2 (k − 2 2 ) ⎭

(38)

For a large k value, these expressions can be approximated as quantum F20 (ΔR ) ≈

= 4841

2 k 2αΔR (e − 2e αΔR + 1) 2 2 k αΔR (e − 1)2 2

(39)


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Figure 3. Quotient functions of methanol determined by quantum (blue line) and correspondence principle methods with Jsym (red line) and Jimp (yellow line) for (a) v = 1, (b) v = 2, (c) v = 3, and (d) v = 4. Dashed lines show the coordinates of the classical turning points determined by Jsym. sym F20 (ΔR ) ≈

this form of the DMF expression was first derived by Timm and Mecke almost 80 years ago in a completely different manner.26 As explained above and in the Appendix, although the coefficient to the factor (eαΔR − 1)v with Jsym is slightly different from that of the quantum theory, those with Jimp are completely identical to the quantum coefficients, implying Jimp gives more accurate transition moments than Jsym does. In Figure 3a−d, we have drawn the actual quotient functions (not the asymptotic approximate forms given in eqs 39−41) of methanol for the quantum and CP methods for v = 1−4, respectively. We first point out their similarity to the quotient functions calculated previously for numerically obtained anharmonic OH vibrational wave functions of nitric acid shown in Figure 7 of ref 12. If TDDMF and CP vibrational wave functions are to be integrated as a function of ΔR, the integration range is not (−∞, +∞), but limited within the classically allowed region (ΔR), where αΔR> = −ln(1 − P) and αΔR< = −ln(1 + P) from eq 16. The turning points determined by Jsym are also shown in Figure 3, and those by Jimp for v = 2−4 cover slightly narrower regions than those by Jsym. Similarity found in their expressions in eqs 31−38 for the quantum and CP quotient functions can be also seen graphically, especially in Figure 3a for the v = 1 case, and the similarities for v = 1 and 2 are extended even to their tunneling regions. The quotient functions for CP case are the Tchebycheff polynomials Fcp v0 (cos θ) = 2 cos vθ = 2Tv(cos θ), where T1(x) = x, T2(x) = 2x2 − 1, ..., and are continued smoothly beyond the boundary, i.e., at x = cos θ = ±1, where

2k 2αΔR 2k αΔR (e − 2e αΔR + 1) = (e − 1)2 3 3 (40)

imp F20 (ΔR ) ≈

=

2 k 2αΔR (e − 2e αΔR + 1) 2 2 k αΔR (e − 1)2 2

(41)

Equations 39−41 contain the same factor of (eαΔR − 1)2; therefore, the difference of their proportional coefficient determines the accuracy of the transition moment for v = 2. As shown in Table 2, Jsym produces about 6% larger whereas Jimp produces almost the same transition moment as quantum theory. In Appendix, we have derived general expressions of these quantum and approximate quotient functions for higher v. It is clear that these expressions have a common form of Jdependent proportional coefficient times (eαΔR − 1)v. Thus, as long as 0 ≤ v ≪ k, we can approximate the DMF μ (ΔR) as follows, μ(ΔR ) =

∑ ⟨v|μ|0⟩Fv0 ≈ ∑ ⟨v|μ|0⟩ v=0

v=0

kv αΔR (e − 1)v v! (42)

with the transition moments being the expansion coefficients in the least-squares sense. It is very important to point out that 4842


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The Journal of Physical Chemistry A v Fcp v0 takes the value of (±1) ·2. Because the classical trajectory in the CP treatment for 0 → v excitation has an average energy between E0 and Ev, their turning points exist outside those for the v = 0 level. Considering this point and the least-squares meaning of the wave function expansion method, it is clear that the 0 → v transition moment with the CP method contains the information on the DMF in the tunneling region of the v = 0 level. Note that the quantum and CP quotient functions are normalized in different ways, because their weight functions and the integration regions are different. In spite of these differences, it is interesting to point out their similarities. We can now give a clear answer to the question asked in the Introduction. If a DMF is proportional to eq 31 for the quantum case and to eq 33 for the CP case, the Morse oscillator has a nonzero transition dipole moment only for the fundamental transition and no overtone intensities. Figure 3a shows that such a DMF increases exponentially as the bond stretches, which is quite unusual behavior. This behavior is easily understood because having no overtone intensities is expressed as μv0 = M·Iv = 0 for all v ≥ 2, which necessitates M1 > 0 and M2 > 0 in the case of the Morse oscillator. (See Figure 6 of ref 13.) In fact, following the discussion given in section 3.5 of ref 12, the fundamental component of the quotient functions in Figure 3a represents the difference of the DMFs of molecules satisfying the ideal SVOI concept. Combining Figures 3a−d suggests that a linear DMF with anharmonic PEC has fundamental and overtone intensities satisfying the NIDL explained in Introduction.

derived by Timm and Mecke, who used a complicated algebra.26 In this paper, we have discussed quotient functions in a simple one-dimensional local mode model for the analysis of dipole moment functions and the transition moments, but quotient functions can be applied to various studies because they are just the ratios of the excited-state to the ground-state wave functions. For example, the theoretical method in ref 47 can be considered such an application. We have recently seen increasing activities in more sophisticated vibrational intensity calculations with multidimensional DMFs.48−56 The concept explained in this paper can be a useful tool to analyze the results and to consider the mathematical expressions of the DMFs.

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APPENDIX

The quotient functions for the quantum theory are expressed as22,23 Fv0 ≡ =

ϕv ϕ0

(k − 2v − 1)v !

=

Γ(k − 1) −v 2s y Lv (y) Γ(k − v)

(k − 2v − 1) Γ(k − 1) v! Γ(k − v)

v

∑ (mv )(−1)m

m=0

v−m Γ(k − v) ⎛ e αΔR ⎞ × ⎜ ⎟ Γ(k − 2v + m) ⎝ k ⎠

4. CONCLUSIONS We calculated the fundamental and overtone transition energies and transition moments for the OH stretching vibration in simple acids and alcohols using various approximate methods for the anharmonic PECs and nonlinear DMFs obtained from ab initio calculations. It was shown that these approximate methods can reproduce respectively the quantum values for both the transition energies and the transition moments. The most accurate transition moment values are given by the correspondence principle methods with Jimp for v = 1−3 and the quasiclassical method for v = 4−6. It was also found that the accuracy of the correspondence principle and quasiclassical transition moments does not depend on the nonlinearity of the DMFs but depends weakly on the quantum level v, as long as their potentials and therefore vibrational wave functions are essentially the same. This result suggests that the approximateto-quantum ratio of ⟨v|ΔR2|0⟩ is almost identical to that of ⟨v|ΔR|0⟩, which leads to the fact that the slope of the Iv line can be calculated with extremely high accuracy for both the correspondence principle and quasiclassical methods. The physical reason for these aspects can be explained on the basis of the mathematical analyses made by Medvedev.5,39−41 From a new viewpoint that 2 cos vθ, the basis of the Fourier expansion of TDDMF, corresponds to the quotient function of the vibrational wave function, the difference between the semiclassical and quantum transition moments can be reinterpreted as the difference of their quotient functions as a function of ΔR. To confirm this, we have derived approximate analytical expressions of the correspondence principle quotient functions and have shown that those for Jimp have the same leading term as the quantum expressions. Furthermore, for the limit of 0 ≤ v ≪ k, we have shown that the wave function expansion method simply yields the same expression of DMF

(A1)

The value of k for the OH bonds studied in this paper is about 40, sufficiently large compared to v = 1−6; thereby, Fv0 can be approximated as follows, Fv0 ≅ =

k v−1 k v! kv v!

⎛

v

∑ (mv )(−1)m kv − m⎜ e

αΔR ⎞v − m

⎟ ⎝ k ⎠

m=0

v

∑ (mv )(−1)m e(v − m)αΔR

m=0

v

=

k αΔR (e − 1)v v!

(A2)

As mentioned in section 2.3, for the semiclassical CP method, the vth quotient function corresponds to 2 cos vθ, and therefore to the Tchebycheff polynomial Tv of the variable cos θ, as follows,57 Fvcp0 (cos θ) = 2 cos vθ = 2Tv(cos θ)

(A3)

To explain the relation between the quantum and semiclassical expressions, we compare eq A2 with only the highest order term of Tv(x), which is derived from the three-term recurrence formula and expressed as 2v−1xv. Therefore, Fvcp0 (cos θ) ≅ 2v cosv θ

(A4)

From eqs 16−18, cos θ can be written by ΔR and J. 4843


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The Journal of Physical Chemistry A 1 {(1 − P 2)e αΔR − 1} P De ⎧⎛ E⎞ ⎨⎜1 − ⎟e αΔR − = E ⎩⎝ De ⎠

(7) 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. (8) Burberry, M. S.; Albrecht, A. C. Comments on the Concept of Universal Intensities in Local Mode Theory. J. Chem. Phys. 1979, 71, 4768−4769. (9) 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. (10) Lehmann, K. K.; Smith, A. M. Where Does Overtone Intensity Come From? J. Chem. Phys. 1990, 93, 6140−6147. (11) 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. (12) 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. (13) Takahashi, H.; Yabushita, S. Theoretical Analysis of Weak Adjacent Substituent Effect on the Overtone Intensities for the XH (X=C, O) Stretching Vibrations. J. Phys. Chem. A 2013, 117, 5491. (14) Percival, I. C.; Richards, D. A Correspondence Principle for Strongly Coupled States. J. Phys. B 1970, 3, 1035−1046. (15) Marcus, R. A. Extension of the WKB Method to Wave Functions and Transition Probability Amplitudes (S-Matrix) for Inelastic or Reactive Collisions. Chem. Phys. Lett. 1970, 7, 525−532. (16) Naccache, P. F. Matrix Elements and Correspondence Principles. J. Phys. B 1972, 5, 1308−1319. (17) Koszykowski, M. L.; Noid, D. W.; Marcus, R. A. Semiclassical Theory of Intensities of Vibrational Fundamentals, Overtones, and Combination Bands. J. Phys. Chem. 1982, 86, 2113−2117. (18) Wardlaw, D. M.; Noid, D. W.; Marcus, R. A. Semiclassical and Quantum Vibrational Intensities. J. Phys. Chem. 1984, 88, 536−547. (19) Shirts, R. B. Use of Classical Fourier Amplitudes as Quantum Matrix Elements: A Comparison of Morse Oscillator Fourier Coefficients with Quantum Matrix Elements. J. Phys. Chem. 1987, 91, 2258−2267. (20) Nikitin, E. E.; Noda, C.; Zare, R. N. On the Quasiclassical Calculation of Fundamental and Overtone Intensities. J. Chem. Phys. 1993, 98, 46−59. (21) Nikitin, E. E.; Pitaevskii, L. P. Calculation of the Landau Quasiclassical Exponent from the Fourier Components of Classical Functions. Phys. Rev. A 1994, 49, 695−703. (22) Child, M. S. Semiclassical Mechanics with Molecular Applications; Oxford: Clarendon, U.K., 1991). (23) Heisenberg, W. The Physical Principles of the Quantum Theory; Dover: New York, 1930. (24) Trischka, J.; Salwen, H. Dipole Moment Function of Diatomic Molecules. J. Chem. Phys. 1959, 31, 218−225. (25) Cashion, K. A Method for Calculating Vibrational Transition Probabilities. J. Mol. Spectrosc. 1963, 10, 182−231. (26) Timm, B.; Mecke, R. Quantitative Absorption Measurements on the C-H Overtones of Simple Hydrocarbons. Z. Phys. 1936, 98, 363− 381. (27) Becke, A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648−5652. (28) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev. B 1988, 37, 785−789. (29) McLean, A. D.; Chandler, G. S. Contracted Gaussian Basis Sets for Molecular Calculations. I. Second Row Atoms, Z = 11−18. J. Chem. Phys. 1980, 72, 5639−5648. (30) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. SelfConsistent Molecular Orbital Methods. XX. A Basis Set for Correlated Wave Functions. J. Chem. Phys. 1980, 72, 650−654. (31) Clark, T.; Chandrasekhar, J.; Spitznagel, G. W.; Schleyer, P. v. R. Efficient Diffuse Function-Augmented Basis Sets for Anion Calculations. III. The 3-21+G Basis Set for First-Row Elements, Li−F. J. Comput. Chem. 1983, 4, 294−301.

cos θ =

⎪ ⎪

ωe 1 4χe Jωe − J 2 ωeχe

=

=

⎫ 1⎬ ⎭ ⎪ ⎪

⎧ ⎫ ⎛ 4χ (Jω − J 2 ωeχe ) ⎞ αΔR ⎪ ⎪ ⎟⎟e ⎨⎜⎜1 − e e − 1⎬ ⎪ ⎪ ω ⎝ ⎠ e ⎩ ⎭

⎫ ⎧⎛ k − 2J ⎞2 ⎟ e αΔR − 1⎬ ⎨⎜ ⎝ ⎠ k 2 J(k − J ) ⎩ ⎭ k

⎪

⎪

⎪

⎪

(A5)

With sufficiently large k, cos θ can be approximated as cos θ ≅

1 2

k αΔR (e − 1) J

(A6)

From eqs A3 and A4, the quotient function can be approximated as 2Tv(cos θ ) ≅

⎛ k ⎞v αΔR − 1)v ⎜ ⎟ (e ⎝J⎠

(A7)

The approximate semiclassical quotient functions have the same factor (eαΔR − 1)v as the quantum expression in eq A2. Then, with Jsym, the coefficient of (eαΔR − 1)v becomes [(2k/(v + 1))v]1/2, slightly different from that in eq A2. On the contrary, with Jimp, the coefficient becomes (kv/v!)1/2 and agrees with that in eq A2. Conversely, as Shirts has shown for the matrix element of ΔR,19 the expression of Jimp can be derived by the equality (A2) = (A7). Thereby, Jimp gives a closer expression of the quotient function to the quantum one and more accurate transition moments as shown in Table 2.

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AUTHOR INFORMATION

Corresponding Author

*S. Yabushita. E-mail: [email protected]. Tel: +81-45566-1715. Fax: +81-45-566-1697. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI Grant Number 24550032 and 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 Institutes. K.T. thanks Ministry of Science and Technology of Taiwan (Grant: NSC 102-2113-M-001-012MY3) for support.

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REFERENCES

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Interpretation of Semiclassical Transition Moments through Wave

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