Anharmonicity in the Vibrational Spectra of ... - ACS Publications

Subscriber access provided by University of Otago Library

Article

Anharmonicity in the Vibrational Spectra of Naphthalene and Naphthalene-D8: Experiment and Theory Shubhadip Chakraborty, Subrata Banik, and Puspendu Kumar Das J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b09034 • Publication Date (Web): 17 Nov 2016 Downloaded from http://pubs.acs.org on November 19, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry A is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Anharmonicity in the Vibrational Spectra of Naphthalene and Naphthalene-d8: Experiment and Theory Shubhadip Chakraborty*†, Subrata Banik*§, and Puspendu K. Das† †

Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore, India.

§

Advanced Centre for Research in High Energy Materials and School of Chemistry, University

of Hyderabad, Hyderabad, India.

1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract In this paper, we report the gas phase Infrared (IR) spectra of naphthalene and naphthalene-d8 recorded in the mid-infrared region (3200-500 cm-1) using a heated multi-pass long path gas cell. Several combination bands appear as shoulders and satellite peaks in the 3200-2600 cm-1region along with the C-H stretch fundamental bands. Experimental IR spectra of these molecules were systematically analyzed with vibrational self-consistent field (VSCF) theory, vibrational second order perturbation theory (VPT2) and vibrational couple cluster method (VCCM) with two different potential energy surfaces obtained using B3LYP and MP2 methods. A comparative study between these two PESs was made to match the observed spectra. Final assignment of the IR spectra of naphthalene and naphthalene-d8 was done using the VCCM with MP2 potential which provided the best match.


Page 2 of 45

Page 3 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1. Introduction Polycyclic Aromatic Hydrocarbons (PAHs) are one of the major environmental pollutants originating from various natural and anthropogenic sources.1 Especially, airborne PAHs cause severe health disorders.2 Naphthalene is one of the PAHs (vapor pressure ~ 10-2torr at 250C) that is widely present in the atmosphere from combustion of fossil fuels, tobacco burning, the use of mothballs, fumigants, deodorizers etc. It is used as an intermediate in the production of phthalic anhydride, pesticides etc.3 Infrared spectra of naphthalene and its deuterated derivative have been reported and assignment of bands has been done within the harmonic approximation.4-7 In these studies, the harmonic frequencies were scaled with suitable factors to match the experimental spectra. Such scaling technique was found to give reasonable agreement with the fundamental vibrations. However, the harmonic approximation was unable to account for the broad band structures in the spectra, especially in the region of 3200-2600 cm-1. This region of the spectrum is primarily dominated by C-H stretches. The frequencies of the C-H stretching modes are typically double the frequency of C-C stretching or C-C-H in-plane bending modes. Thus, the CH stretch fundamentals are coupled with two or several higher quanta excitations of low frequency fundamentals by Fermi or other kind of interactions. Such interaction affects the frequencies and intensities of the C-H fundamentals. Several combination and overtone bands borrow significant intensities from the C-H stretching fundamentals and appear as satellite peaks or shoulders along with C-H stretching vibrations. Anharmonic frequency calculations include such interactions between fundamental and overtone or combination bands and reproduce the band structures to a certain extent. However, such calculation on a molecule as large as naphthalene (48 modes of vibrations spanning over 8 irreducible representations in the D2h point group) is a challenging task. It requires extensive electronic structure calculation to generate the



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

potential energy surface (PES) and a sophisticated ab initio method to solve for the vibrational Schrödinger equation. A quartic expansion of the potential energy around equilibrium in massweighted normal coordinates is commonly used for the generation of PES for polyatomic molecules. This requires third and fourth order derivatives of potential energy with respect to the mass weighted normal coordinates which are computationally time consuming for large molecules. Moreover, one needs a similar expansion of dipole moment surface to calculate the anharmonic intensities. Various packages are available commercially to compute anharmonic spectra of polyatomic molecules. Barone8 first implemented the module for anharmonic calculation in the Gaussian 03 Rev C.01 version9. It was restricted up to the calculation of cubic and quartic force constants until recently when G09 Rev D.0110 was released. In this version Barone and Bloino11 have implemented a methodology to compute the anharmonic intensity. Cané and co-workers12 made an extensive calculation on the cubic and semidiagonal quartic PES in normal coordinates and applied vibrational second order perturbation theory (VPT2) to describe the IR spectra of naphthalene. Although their calculations yielded accurate description of the fundamental vibrations of naphthalene and the results matched well with the experimental IR spectrum of naphthalene6,7,12-16, the nonfundamental bands appearing in the C-H stretching region were not assigned. Very recently, Mackie and co-workers17 and Bloino18 made systematic studies of the anharmonic vibrational spectra of naphthalene and its deuterated derivative based on VPT2 calculation. To overcome the problem of singularities inherent in the VPT2 method, they used a variational calculation on the so called polyad of vibrational modes that are involved in Fermi resonances. From these studies, it appears that higher quanta states play a significant role to obtain the accurate band positions of the C-H stretches affected by Fermi coupling. Apart from VPT2 method one can also compute the anharmonic spectra of polyatomic molecules using


Page 4 of 45

Page 5 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


vibrational self-consistent field19-30 (VSCF) or vibrational coupled cluster31-38 method (VCCM). In the VSCF method, the vibrational Hamiltonian is treated variationally, whereas the VCCM is neither variational nor perturbative. Since these three approaches (VCCM, VSCF and VPT2) are based on three different philosophies, they treat the anharmonicity in different ways. As mentioned earlier, the IR spectra of naphthalene and its isotopic analog are affected by several strong resonance effects especially in the spectral region dominated by C-H(D) fundamental transitions. The accuracy of the theoretical description depends on how well the vibrational method describes the resonances and the anharmonic effects. Irrespective of the method used to solve the vibrational Schrödinger equation, the accuracy of resultant spectra also depends on the electronic structure method used to generate the quartic PES. The calculations of quartic PES of a molecule of the size of naphthalene is computationally costly. To the best of our knowledge, the calculations of anharmonic spectra of naphthalene have been limited to the use of density functional approaches.12-13, 17, 39 In this work, we revisit the assignment of the gas phase IR spectra of naphthalene and its deuterated derivative with two objectives: (1) To assign the spectral transitions of naphthalene and naphthalene-d8 in 3200-800 cm-1 and 2400-600 cm-1, respectively and (2) To see how different electronic structure calculation affects the PES and thus the band positions and intensities. We used three different methods, VCCM, VSCF and VPT2 in our calculations. A systematic analysis

of

coupling

strengths

between

the

C-H(D)

stretching

fundamentals

with

nonfundamentals was performed. To calculate the quartic PES, we used the density functional theory (B3LYP) and second order Møller Plesset perturbation theory (MP2).



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 45

2. Experiment Infrared spectra of naphthalene and its deuterated isomer were recorded from 4000-400 cm-1 in the gas phase using a multi pass long path gas cell of optical path length of 6.0 meter. Naphthalene (99%) and its deuterated isomer (99%) were purchased from Sigma Aldrich and Cambridge isotope laboratories Inc., respectively and were used without further purification. All spectra were recorded with an infrared spectrometer (Bruker optics, Vertex 70) equipped with KBr broad band beam splitter and liquid nitrogen cooled mercury-cadmium-telluride (LN2MCT) detector. The spectral resolution was kept at 0.2 cm-1 and all spectra were averaged over 4096 scans. Both naphthalene and its deuterated isomer have vapor pressures of ~ 10-2torr at room temperature. The compound was heated in the sample chamber and the vapor was transferred inside the gas cell using ultra high purity (UHP) argon gas. Atmospheric moisture and carbon dioxide interferes with the experimental spectrum. Therefore, to remove the moisture and carbon dioxide we have purged the spectrometer with UHP nitrogen during the entire duration of experiment. The collected interferrogram was converted to the spectrum with the aid of Blackman Haris 3 Term apodization function. 3. Theoretical Methodology To describe anharmonic vibrations in a polyatomic molecule, the Watson Hamiltonian is used = ∑

+ + + ,

(1)

where, , and are the mass weighted normal coordinates and their conjugate momenta. and are the Corriolis coupling and Watson mass terms, respectively. The potential energy functional is taken as a quartic polynomial in the Taylor series expansion,


Page 7 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


= ∑ + ∑ + ∑ .

(2)

where, are the harmonic vibrational frequencies of the ith normal mode, and are the third and fourth derivatives of the potential energy with respect to the mass weighted normal coordinates around the equilibrium geometry. Note that such quartic expansion of potential energy surface in the Taylor series may be problematic for floppy molecules, or, molecules with hindered rotational modes etc. In such cases, the higher order terms are significantly large, and, thus the Taylor series expansion may not even converge. However, for a semi-rigid molecule like naphthalene the anharmonic cubic and quartic terms are significantly smaller than the harmonic frequencies and convergence was not a problem. The use of quartic PES in mass weighted normal coordinates is very common for the vibrational study of semi-rigid molecules. The intensity of a vibrational transition from an initial state i, to a final state f, is =

∆"#$% #&' #%( )# .

(3)

Here, ∆" is the transition energy, and&' is the dipole moment surface (DMS) in mass

weighted normal coordinates. #$% #&' #%( )# is the square of dipole transition matrix element between two states. % and %( are the corresponding wave functions for the initial and final state, obtained by solving the Schrödinger equation associated with the vibrational Hamiltonian in eq. 1. A cubic expansion of DMS is usually considered to be consistent with a quartic PES. ' ' &' = *+' + ∑ *' + ∑ * + ∑ *

(4)

' ' Here, *' , * , * , etc are the first, second and third order derivatives of the dipole moment

vector with respect to the normal coordinates along , = -, /, 0 directions. At present, only the linear dipole terms are implemented in our VSCF and VCCM code. Thus, in this work, we 7 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 45

considered only the linear expansion of DMS to calculate the spectral intensities with VCCM and VSCF. From our earlier work40,41, we note that such linear approximation for the DMS give reasonable accuracy of the intensities of fundamental transitions. The nonfundamental transitions that are strongly coupled with the fundamental states are also described well with the linear approximation of the DMS. We excluded Coriolis coupling and Watson terms in the VCCM and VSCF calculations. Naphthalene, being a large molecule, has a large moment of inertia and thus, the Coriolis coupling will have negligible effect on its vibrations. 3.1. Vibrational coupled cluster method: The vibrational coupled cluster method has emerged as one of the most accurate methods to calculate the vibrational spectra of semi-rigid molecules. Two different representations are used in the VCCM method. One is the basis set representation developed and used by Christiansen and co-workers,31-33 and other is the bosonic representation developed by Prasad and co-workers.34-38 Here, we have used the bosonic representation. The details of the method can be found elsewhere.34-38 Here we give a brief description of the VCCM in the bosonic representation. The VCCM consists of three steps. In the first step, an effective harmonic oscillator (EHO) approximation42 is invoked to optimize the ground state wave function. Here, a product of N Gaussian functions 9

7 /

Φ2 = 3 4 ∑:; 5 6 46

,

(5)

is posited as the ground state wave function for N vibrational degrees of freedom. Such reference function is then variationally optimized with respect to the and 2 to obtain the optimized ground state energy in the EHO approximation. In the second step, the coupled cluster wave function for the ground vibrational state is parametrized as


Page 9 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


%< = 3 = 3 4> |Φ2 @

(6)

The cluster operators, S and σ are, respectively, excitation and de-excitation operators from the EHO ground state (vacuum), consisting of connected one-boson, two-boson, etc. terms. These cluster operators are written in terms of harmonic oscillator ladder operators A and AB , C = ∑ D AB + ∑ D AB AB + ∑ D AB AB AB + ⋯

(7a)

F = ∑ F A + ∑ F A A + ∑ F A A A + ⋯

(7b)

We consider these expansions up to at most four boson terms. We invoked subsystem embedding condition43 to derive the working equations for S and F cluster matrix elements. Under this condition, the cluster matrix elements S are decoupled from the cluster matrix elements F. Thus, the working equations for the cluster matrix elements S and the ground state energy are given by $Φ+ #+(( #Φ2 ) = 0

(8a)

$Φ2 #+(( #Φ2 ) = "
|Φ2 〉.

(10)

Here, N N N ΩN = ∑ Ω AB + ∑ Ω AB AB + ∑ Ω AB AB AB + … … … ….

(11)

The working equation for the excitation energy is given by P+(( , ΩN Q = Δ"N ΩN .

(12)

Solving this equation is equivalent to the diagonalization of +(( matrix in the space of the

excited states of the EHO. Eq. 12 is projected in the manifold of zeroth order excited states determined by the truncation in Ω. The vibrational excitation energies are then obtained directly as the eigenvalues of +(( . Here we used four boson operators for the expansion in eq. 11. This

approach to the calculation of the transition energies, within the framework of coupled cluster approach, is alternately called the coupled cluster linear response theory44,

45

(CCLRT) or the

equation of motion coupled cluster46, 47 (EOMCC) method. We used the effective operator approach37,40,48 to evaluate the square of dipole transition matrix elements for the calculation of spectral intensity. In this approach, the working equation for the square of the dipole transition matrix element between initial state i and final state f is

' ' #$% #&' #%( )# = $S #&+(( #T( )$S( #&+(( #T ),


(13)

Page 11 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


' where, S and T are the left and right eigenvectors of +(( and &+(( is the effective dipole

operator ' &+(( = 3 > 3 4= &' 3 = 3 4> .

(14)

Note that the VCCM calculation requires the construction and subsequent diagonalization of effective Hamiltonian matrix to get the excited states. The dimension of such effective Hamiltonian matrix increases exponentially with the number of degrees of freedom. Thus, it demands a huge memory space to store the effective Hamiltonian matrix and the left and right eigenvectors. In addition, the diagonalization of such matrix requires large CPU time. The effective Hamiltonian matrix is block diagonalized for each symmetry irreducible representation. Even with such block structure, the size of matrix turns out to be large for naphthalene. For naphthalene the dimension of effective Hamiltonian matrix for ag irreducible representation is about 35000×35000. Diagonalization of such large matrix and subsequent storage of this matrix and its eigenvectors are beyond our computational resource. In this work we invoke an efficient approximation to make the VCCM calculation feasible for naphthalene. Our approximation is based on the separation of low and high frequency vibrational modes. In the first step, the EHO approximation is invoked with all the 48 vibrational modes. The VCCM calculation is then carried out with 35 modes, excluding 13 low frequency modes from naphthalene. Since we are mainly interested in the spectral C-H(D) stretching region, we anticipate that such approximation will have a minimum effect on the accuracy of our calculations due to two reasons. One, due to variational EHO approximation for the ground reference state, the zeroth order frequencies are now variationally optimized effective harmonic frequencies. The anharmonic effects due to the low frequency modes are taken care to some extent in the renormalized frequencies of the



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 45

EHO. Second, since naphthalene is a semi-rigid molecule, the coupling between such low frequency modes and CH stretching is small. Thus, we expect that removing the low frequency vibrations will have negligible effect on the spectral region dominated by CH stretching fundamentals. 3.2. Vibrational self-consistent field (VSCF) theory: The VSCF method has been developed and used extensively by several authors.19-30 Here, a direct product of N one mode functions is posited as the vibrational wave function. Each one mode function is expanded in an orthonormal basis set, usually denoted as modals. By invoking the variation principle, one obtains the optimized modals as the eigenfunctions of an effective single mode Hamiltonian. The potential of this single particle Hamiltonian is generated by averaging the non-separable many body Hamiltonian over the other modes. One needs to solve the equations for the modal functions in self-consistent manner. The VSCF energy for a particular vibrational state is then obtained as the expectation value of the Hamiltonian in eq. 1 with the optimized wave functions of the vibrational states. Once the optimized vibrational wave functions are obtained, the intensity for a vibrational transition is calculated by using eq. 3. Note that, the wave functions for the initial and the final states from VSCF calculations are not orthogonal. However, as Gerber and co-workers23-29 showed that the overlap between these two VSCF wave functions is very small. Thus, we neglect the error in the intensity values due to such non-orthogonality of the VSCF wave functions. We note that Christiansen and co-workers30 proposed an alternate approach to calculate the spectral intensity within VSCF framework using linear response theory. Although the linear response theory based VSCF formalism to calculate the spectral intensities does not suffer from such orthogonality problem, it allows calculation of the transition moments if the states involve excitation to at most one mode at a time30. 12 ACS Paragon Plus Environment

Page 13 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.3. Second order vibrational perturbation theory: The second order vibrational perturbation theory (VPT2) is another way to calculate the anharmonic vibrational spectra of polyatomic molecules.49-54 Handy and co-workers48 first implemented the VPT2 in the SPECTRO package51,52 to calculate the vibrational spectra of polyatomic molecules. A modified version of VPT2 has been implemented in Gaussian 09 by Barone and others.8-11 In this method, the zeroth order harmonic oscillator part of the Hamiltonian is taken as zeroth order Hamiltonian and the cubic, quartic and the Coriolis terms are considered as perturbation. The working equation for the vibrational energies takes the form

" U = V2 + ∑ WU + X + ∑ V WU + X U +

(15)

Here, are the harmonic frequencies, U , the vibrational states, and, V2 and V are the zero point contribution and anharmonic constants derived from the cubic, quartic and Corriolis coupling terms of the PES. The VPT2 method encounters singularities in the calculations of the anharmonic constants if two vibrational states have strong Fermi resonances. To overcome the problem of singularities, Barone used variational calculations with the states that are involved in the resonances in his implementation8. Once the transition frequencies are obtained, the intensities of the transitions can be calculated using the Formula derived by Bloino and Barone11. Note that the calculation of intensity also suffers from singularity due to strong Fermi resonance. To overcome this problem, the deperturbed VPT211 approach is used. Here, a threshold value is set to identify the resonance terms and the corresponding resonance terms are dropped from calculation (see reference 11 for detail). 4. Computational Details



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 45

All the electronic structure calculations for the PES and DMS were performed by using GAUSSIAN 09 package. The Gaussian package calculates the analytic hessian for the harmonic frequencies and then numerically differentiates the analytic hessian to obtain the cubic and quartic terms of the PES in dimensionless normal coordinates. For B3LYP calculations the geometry was optimized tightly and ultrafine keyword was used as recommended by Barone8. We used 6-311G (2d, 2p) basis set in our calculations to generate the quartic potential. The VSCF and VCCM calculations were carried out with our in-house FORTRAN code. The VPT2 results were obtained with Gaussian09 D.01 package10. We used the default threshold values implemented in Gaussian 09 to detect resonances in vibrational frequencies as well as in intensities (10 cm-1 for Fermi resonances, 10 cm-1 for Darling-Dennison resonances). For VSCF calculations, we used 8 harmonic oscillator basis functions for each vibrational mode. For VCCM calculations, we exclude the modes that have experimental frequencies below 750 cm-1 for naphthalene and below 600 cm-1 for naphthalene-d8. 5. Results and Discussion All the 48 normal modes of vibration of naphthalene and naphthalene-d8 are classified into 8 one dimensional irreducible representation, ag, au, b1g, b1u, b2g, b2u, b3g and b3u under the D2h point group. Among them only b1u, b2u and b3umodes are IR active and the remaining is Raman active. 5.1.C-H(D) stretching region In Figures 1 and 2, we report the experimental IR spectra of naphthalene and naphthalene-d8 in the C-H and C-D stretching region along with the stick spectra calculated from vibrational couple cluster method with MP2 PES, hereafter, VCCM-MP2 PES. The eight C-H


Page 15 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and C-D stretching modes of naphthalene and naphthalene-d8 belong to ag, b3g, b1u and b2u representations. In Table 1 and Table S1 (given in the supporting informations), the mode descriptions of all the fundamentals with their harmonic intensities are listed for naphthalene and naphthalene-d8, respectively. The anharmonic frequencies of naphthalene and naphthalene-d8, calculated with VSCF, VPT2 and VCCM with both MP2 and B3LYP PES along with the experimental frequencies are given in Table 2 and Table S2. The intensities for the fundamental IR transitions from all the calculation are given in Table S3 (supporting informations). The VPT2 calculation with Gaussian 09 uses semi-diagonal cubic DMS, whereas, we used linear DMS for our VCCM and VSCF calculation. Thus, the magnitude of VPT2 intensities should be compared with VSCF or VCCM values. However, the relative values in each calculation help us to reduce the uncertainties of the assignments, especially for the combination bands. We assign only those transitions that are bright in both VPT2 and VCCM calculation. From the VCCM-MP2 PES calculation we assign four bands at 3058.2 (ν17), 3005.7 (ν18), 3067.9 (ν29) and 3032.3 (ν30) cm-1 to C-H stretch fundamentals. In the polarized infrared spectrum of naphthalene crystal, Broude and Umarov14 observed only 3 bands in this region and assigned 3004.0 (ν18), 3068.0 (ν29) and 3048.0 (ν30) cm-1 bands to b1u and b2u symmetry C-H stretches of naphthalene. They did not observe the ν17 band of naphthalene. While our ν18 and ν29 assignments are same as theirs, our ν30 band is off by 16 cm-1 from the frequency assigned by them. Cané and co-workers12 also assigned three bands at 3014.0, 3078.0 and 3042.0 cm-1 in the gas phase IR spectrum of naphthalene as ν18, ν29 and ν30 fundamentals. Their assignment of those three bands agrees closely to our results. These authors did not mention anything about the assignment of the ν17 mode. They “firmly” assigned two C-H stretch fundamentals at 3065.0 (ν17,b1u) and 3078.0 cm-1(ν29,b2u) for naphthalene, whereas, the assignment for the other two



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 45

fundamentals, ν18 and ν30 was uncertain because of low intensity and probably overlapped with the nearby high intensity fundamental bands. Very recently, Mackie and co-workers17 performed high resolution gas phase IR absorption study of naphthalene in molecular beam. They assigned the observed bands using a modified version of the SPECTRO program.52 They observed all four fundamentals of naphthalene, but could not assign all of them using their formulation. In our experiment we have seen and assigned all four C-H stretch fundamentals using VCCM methodology. Moreover, the VCCM calculation gives an idea about various types of anharmonic couplings among the C-H stretches with nearby several two or higher quanta states. This kind of detailed analysis of the coupling behavior in the C-H and C-D stretching region for large aromatic molecule is not reported so far in the literature. In this context it is important to mention that the Fermi and various other kind of anharmonic resonances play a significant role for an in depth analysis of the observed bands (both in terms of position and intensity) in this region. Notably, the Raman transitions due to the C-H stretching fundamentals, reported by Cané and co-workers12 at 3057.0 (ν1), 3051.0 (ν2) and 3057.0 (ν37), 3018.0 (ν38), are well reproduced in our VCCM-MP2 PES calculation. In 2009, Pirali and co-workers39 recorded the high resolution IR spectra of thermally excited naphthalene. Their calculated anharmonic frequency for the C-H stretching region was found to be downshifted than the experimental frequencies; especially the ν30 mode was underestimated by 52 cm-1. For naphthalene-d8 four IR active modes were observed at 2275.1 (ν17), 2253.1 (ν18), 2300.8 (ν29) and 2260.1 (ν30) cm-1. With our VCCM calculation we assigned all of them as C-D stretching fundamental band. Our results match very well (within 2 cm-1) with the earlier assignment of naphthalene-d8.12 5.2. Comparison among the theoretical methods and PESs:


Page 17 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Next, we turn our attention to compare the VCCM, VPT2 and VSCF results. In Table 2 and Table S2 (supporting informations), we present anharmonic frequencies calculated with VSCF, VPT2 and VCCM in MP2 and B3LYP PES along with the observed frequencies for naphthalene and naphthalene-d8, respectively. In Figure 3a and 3b, we plot the errors in the fundamental frequencies with respect to the experimental values in all the three vibrational methods with two different PESs for both molecules. We note that for naphthalene, both the VCCM and VPT2 methods provide reasonable match between the experiment and theory in the spectral region 2000-600 cm-1. The errors in the fundamental transitions by these two methods with both MP2 and B3LYP PES lie within 20 cm-1. However, the VSCF with both the PES gives poor description for the fundamental frequencies in this region of spectra. For example, the error in the VSCF value of ν46 mode fundamental frequency is as high as -60 cm-1 with both MP2 and B3LYP PES. We note that the VSCF transition energies are overestimated compared to the experimental values for the both PES. This highlights the importance of correlation effects for the accurate description for these transition energies. Similar pattern was found in the lower region of spectrum of naphthalene-d8. For the most of the fundamentals in the region of 1700500 cm-1, both VCCM and VPT2 gives reasonable results. The C-H(D) stretching fundamentals deserve special attention. These frequencies differ significantly with different PES and different vibrational methods. In the VCCM calculations of naphthalene, with B3LYP PES maximum deviation of 80 cm-1 was observed between the experiment and theory, whereas the use of MP2 PES gives a much better agreement. This is probably because the C-H stretch harmonic frequencies in MP2 method are higher about 25-30 cm-1 compared to B3LYP frequencies. Therefore, the final anharmonic frequencies in the VCCM calculations are much closer to the experiment. The standard deviation in VCCM-MP2 PES for



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 45

the frequencies of all the experimentally observed fundamental transitions is 11 cm-1 compared to 30 cm-1 with B3LYP PES. Likewise, the VCCM calculation of naphthalene-d8 with B3LYP PES results the standard deviation in the C-D stretching region of 37 cm-1, whereas the MP2 PES gives a standard deviation of 7 cm-1. On the other hand, in the VPT2 calculations the MP2 PES overestimates most of C-H stretching frequencies. The VPT2 along with B3LYP PES gives closer results to the experimental values. In the VSCF calculations, the MP2 PES overestimates the C-H frequencies significantly. The maximum error in VSCF with MP2 PES is as high as about -80 cm-1. However, the VSCF results are consistent with its descriptions for the lower energy fundamentals. Being a single particle approximation, the VSCF yields excitation energies for the fundamental systematically higher than the experimental values. Surprisingly, the VSCF provides good descriptions for some of the C-H stretching fundamentals with B3LYP PES. The accuracy of VSCF results is probably due to cancellation of errors in the B3LYP PES by the errors due to absence of correlation effects in the VSCF descriptions. For naphthalene-d8, both MP2 and B3LYP PES encounter similar magnitude of absolute error in the VPT2 calculations. Both VSCF and VPT2 give better results with B3LYP PES than MP2 PES. The standard deviation in VPT2 for all the observed fundamental transition frequencies is 16 cm-1 with B3LYP PES against 21 cm-1 with MP2 PES. The standard deviation in VSCF is 26 cm-1 with B3LYP PES, whereas, is 37 cm-1 with MP2 PES. Thus, the best description for the fundamental frequencies is obtained with VCCM-MP2 PES results. This pattern is also consistent for naphthalene-d8. Here, the VCCM gives better results with MP2 PES compared to B3LYP PES. The standard deviation with MP2 PES is 13.6 cm-1 as compared to 21 cm-1 with B3LYP PES in the VCCM calculations. In this context, we note that several authors used a hybrid PES, where, the harmonic frequencies are calculated with a more accurate method (e.g. CCSD(T) etc.) and


Page 19 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


cubic and quartic terms with computationally less costly method e.g. B3LYP to improve accuracy in the frequency calculation54,55. 5.3. Nonfundamental bands near the C-H(D) stretching region: The broad nature of the IR spectra near to C-H(D) stretching region is due to several nonfundamental transitions. Using VCCM-MP2 PES with linear dipole surface we are able to assign five prominent shoulder or satellite bands for naphthalene and two for naphthalene-d8. In Table 3 we report the VCCM frequencies of these combination bands along with their experimental values. It is well known that the spectra around 3000 cm-1are dominated by Fermi and several higher order couplings with two or higher quanta states with the fundamentals. These two or higher quanta excitations borrow significant intensity from the C-H(D) stretching fundamentals. The VCCM due to its VCI like description for the excited states gives accurate descriptions for these couplings. In Table 4, we report some intense states in this region with their two most important contributions for VCCM-MP2 PES calculations. The VCI coefficient56 values for these two or higher quanta states are significantly high in these C-H(D) fundamental states. For example, for naphthalene, the state assigned to be ν17 fundamental has contribution 0.1419 from the combination band (ν32 + ν39), whereas its own contribution is only 0.5971. As a consequence, (ν32 + ν39) transition borrows major chunk of intensity from ν17 fundamental and appear as shoulder peak in this region. We assign the shoulder peak at 3077.5 cm-1 as the ν32 + ν39 combination band. The calculated frequency with VCCM-MP2 PES is 3078 cm-1. We note that with the consideration of relative intensities, the assignments with the VCCM are consistent with the VPT2 results from Gaussian 09 calculation. For example, the ν32 + ν39 band has intensity value 13.1 km/mol with VCCM calculations compared to 7.0 km/mole with VPT2. Since the VCCM and VPT2 describe the coupling strengths between the states in different way, their 19 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 45

description of band positions varies significantly. Similarly, the use of different PES with a same method deviate the frequencies. For example, the calculated frequency for above mentioned ν32 + ν39 combination band in VPT2 calculations is 3059 cm-1. The VCCM-B3LYP PES yields frequency value for this transition as 3000 cm-1. Such deviations in the band positions with different method and different PES make the assignment of the non-fundamental bands in this region tedious and uncertain. However, when the coupling between the two quanta states and the fundamental state is very strong, with both PES and both VCCM and VPT2, the combination band ought to be bright. From Table 3, we can see that the ν32 + ν39 combination is bright in all the calculations (> 5 km/mol). Its relative frequency with respect to experimental fundamental band at 3068 cm-1 is best described at 3078 cm-1 by VCCM-MP2 PES. Thus the use of two different PES and different methods makes the assignment of non-fundamental bands in this region more certain. We still encounter difficulties in assignment of the combination bands when the calculated numbers for two different combination states are close to each other. For example, in the experimental spectrum of naphthalene we observe a shoulder band at 3017.5 cm-1. In the VCCM-MP2 PES we get two bright transitions corresponds to ν5 + ν19 and ν3 + ν32 with frequencies 2998 and 2997 cm-1 respectively. The pattern of the results is consistent with all the other methods and PES. These two transitions are unresolvable within the experimental resolutions. Similarly, the resultant frequencies of ν4 + ν20 and ν20 + ν40 transitions in all the theoretical methods are close to each other. These states are together assigned for the experimental band at 2879.6 cm-1. Note that, in our calculations we find several transitions with intensity higher than 1.0 km/mol, but no such bands were observed in the experimental spectrum. These bands are probably overlapped with the nearby broad bands and cannot be resolved in our experimental resolution. As an example, Mackie and co-workers17 in their high resolution


Page 21 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


molecular beam experiment of naphthalene has observed a band of b1u symmetry at 2963.8 cm-1. In our experiment, a broad band was observed in that region so we could not make any conclusive statement about that band. Nonetheless, from our VCCM-MP2 calculation we can see a band at 2968.6 cm-1. Similarly for the naphthalene-d8, we find that there are several two quanta combination bands strongly coupled with different C-D stretching fundamentals. The bottom panel of Table 4 lists some of these states that are strongly coupled with the C-D stretching fundamentals. These two quanta transitions appear as shoulder or satellite peaks in this region. The fundamental ν17 strongly coupled with the combination state ν3 + ν23. This state borrows intensity for the ν17 transition. This ν3 + ν23 state is in turn strongly coupled with the ν31 + ν43 state. In VCCM-MP2 PES we get a bright state with major contribution from the ν31 + ν43 state. Thus, the experimental band at 2285 cm-1 is assigned as the ν31 + ν43 combination band. The VCCM-MP2 PES value close to it is at 2286 cm-1. Note that we could not assign some moderately intense peaks in naphthalene-d8. We do not find any bright combination transitions for the corresponding experimental bands. In our calculation, the intensity of the overtone bands comes only from the mechanical anharmonicity. The higher order DMS terms in VCCM are probably needed to be included in order to assign these transitions. Next, we turn to the comparison of experimental and theoretical the frequencies of combination bands in the C-H(D) stretching region. From Table 3, we find that VCCM along with the MP2 PES gives better description than the VPT2 calculations with the same MP2 PES. For naphthalene, in the experimental spectra (Figure 1), we observe two shoulders at 3101 cm-1 and 3079 cm-1 with frequencies higher than the fundamental C-H frequencies. In the VCCMMP2 PES, the combination bands are simulated at 3095 cm-1 and 3078 cm-1. In the VPT2 21 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 45

calculation with same MP2 PES, these two transitions appear at frequencies 3072 and 3051 cm-1 which are lower than the highest energy fundamental band at 3095 cm-1. The spectral transitions in this region are better represented by the VCCM than the VPT2 method. In case of VSCF calculation, we find that the frequencies for the combination transitions with MP2 PES are higher than the experimental values, as in case of fundamental transitions. However, for B3LYP PES, the results do not follow any trend. For some cases, e.g., the VSCF frequency is lower than the experimental value. The error in the B3LYP PES is more prominent here. Between the two PES used here, we find that like the fundamentals, the MP2 PES gives better description for the combination frequencies than the B3LYP method. The combination frequencies are highly underestimated by the B3LYP PES. 6. Conclusion: In this work, we studied the experimental room temperature IR spectra of naphthalene and naphthalene-d8 with vibrational coupled cluster method. The 15 low energy modes were omitted to make the implementation of VCCM feasible for these molecules. However, we used a variational EHO approximation for the ground state to consider the anharmonic effects of these low energy vibrations in our descriptions. We anticipate that such approximation will have minimum effect on the accuracy of the transition energies in the C-H stretching region. We find good agreement between the experimental spectrum and VCCM results with MP2 PES. Note that we used only linear expansion of the DMS to simulate the theoretical spectra. The VCCM results are compared with the results of VPT2 and VSCF calculations in terms of the accuracy in transition energies. Along with it, we compared the results obtained with two different PESs based on MP2 and B3LYP electronic structure calculations. The results differ significantly with different vibrational methods and different PES. The differences are more 22 ACS Paragon Plus Environment

Page 23 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


prominent in the higher energy region (C-H(D) stretching). In this region, with MP2 PES, the VCCM gives better description than the VPT2 for the fundamental frequencies. Moreover, the band positions of the combination excitations are not reproducible compared to the experimental spectra with VPT2 calculations. The two shoulder peaks with higher frequencies than the fundamentals do not appear in the VPT2 calculations. These two bands are well reproduced in the VCCM calculation. In the VCCM, one uses variationally optimized reference function for the ground state. In contrast, in the VPT2, we use harmonic oscillator zeroth order Hamiltonian. Moreover, the VCI like diagonalization for the excitation energies in the VCCM with the state space defined by as many as four quanta excitation makes the VCCM results more accurate. In comparison with the VSCF method, we find that the transition frequencies are overestimated with this MP2 PES in the VSCF calculations than VCCM or VPT2 results. This reflects that one needs to account the correlation effects for the proper description of the IR spectra of these two molecules. With the analysis of effects of electronic structure calculations to generate the quartic PES using two different methods MP2 and B3LYP, we find that the MP2 gives more reliable descriptions than the B3LYP. The B3LYP frequencies are underestimated for the C-H(D) region, and it produces large error especially for the C-H(D) stretching region in the VCCM calculations. Although the MP2 takes more CPU time to generate the quartic PES than the B3LYP method, the use of MP2 is recommended over B3LYP to get the accurate results especially for the spectral region dominated by C-H(D) stretching fundamentals. The accidental matches of C-H fundamental frequencies of naphthalene with VSCF and B3LYP is misguiding. The lower frequency fundamentals and the combination bands in the C-H stretching region are poorly describe by VSCF withB3LYP PES.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Supporting information Description of the normal modes of naphthalene-d8 and anharmonic frequencies of naphthalened8 calculated in VSCF, VPT2 and VCCM method are listed in Tables S1 and S2, respectively. The intensities for the fundamental IR transitions from all the calculation are given in Table S3. Experimental and theoretical stick spectra of naphthalene and naphthalene-d8 from 2000-500 cm-1 are shown in Figure S3 and S4. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHORS INFORMATION Corresponding authors *E-mail: [email protected] Phone: 91-80-2293-2582, Fax: 91-80-2360-1552 *E-mail: [email protected] Acknowledgement: We thank M. Durga Prasad for his critical reading of the manuscript and many helpful comments. We are grateful to CSIR, Govt. of India for generous funding of this work. SB thanks ACRHEM funded by DRDO, India, for a Research Associateship.


Page 24 of 45

Page 25 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


References (1) Ravindra, K.; Sokhi, R.; Grieken, R. V. Atomospheric polycyclic aromatic hydrocarbons: Source attribution, emission factors and regulation. Atmos. Environ. 2008, 42, 2895-2921. (2) Jia, C; Batterman, S. A critical review of naphthalene sources and exposures relevant to indoor and outdoor air. Int. J. Environ. Res. Public Health 2010, 7, 2903-2939. (3) CEH. Chemical Economics Handbook, V. 27.; SRI International: Menlo Park, CA, USA, 2000. (4) Sellers, H.; Pulay, P.; Boggs, J. E. Theoretical prediction of vibrational spectra. 2. Force field, spectroscopically refined geometry, and reassignment of the vibrational spectrum of naphthalene. J. Am. Chem. Soc. 1985, 107, 6487-6494. (5) Langhoff, S. R. Theoretical infrared spectra for polycyclic aromatic hydrocarbon neutrals, cations, and anions. J. Phys. Chem. 1996, 100, 2819-2841. (6) Bauschlicher, C. W. Jr.; Langhoff, S. R.; Sandford, S. A. Infrared spectra of perdeuterated naphthalene, phenanthrene, chrysene, and pyrene. J. Phys. Chem. A 1997, 101, 2414-2422. (7) Hudgins, D. H.; Sandford, S. A. Infrared spectroscopy of matrix isolated polycyclic aromatic hydrocarbons. 1. PAHs containing two to four rings. J. Phys. Chem. A 1998, 102, 329-343. (8) Barone, V. Anharmonic vibrational properties by a fully automated second-order perturbative approach. J. Chem. Phys. 2005, 122, 014108. (9) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, Jr., J. A.; Vreven, T.; Kudin, K. N.; Burant, J. C.; et. al. Gaussian 03, Revision C.02, Gaussian, Inc., Wallingford CT, 2004. 25 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(10) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M.A.; Cheeseman, J. R.; Scalmani,G.; Barone, V.; Mennucci, B.; Petersson, G. A. et. al. Gaussian 09, Revision D.01, Gaussian, Inc., Wallingford CT, 2009. (11) Bloino, J.; Barone, V. A second-order perturbation theory route to vibrational averages and transition properties of molecules: general formulation and application to infrared and vibrational circular dichroismspectroscopies. J. Chem. Phys. 2012, 136, 124108. (12) Cané, E.; Miani, A.; Trombetti, A. Anharmonic force fields of naphthalene-h8 and naphthalene-d8. J. Phys. Chem. A 2007, 111, 8218-8222. (13) Cané, E.; Palmieri, P.; Tarroni, R.; Trombetti, A.; Handy, N. C. The high-resolution infrared spectra of naphthalene-h8 and naphthalene-d8: comparison of scaled SCF and density functional force fields. Gazz.Chim. Ital. 1996, 126, 289-295. (14) Broude, V. L.; Umarov, L. M. Interpretation of the ir spectrum of a naphthalene crystal in the 3000-3100 cm-1 region of valence vibrations of carbon-hydrogen bonds. Optikai Spektroskopiya 1975, 39, 68-74. (15) Stenman, F. Raman scattering from molecular crystals. I. powdered naphthalene. J. Chem. Phys. 1971, 54, 4217-4222. (16) Hanson, D. M.; Gee, A. R. Raman scattering tensors for single crystals of naphthalene. J. Chem. Phys. 1969, 51, 5052-5062. (17) Mackie, C. J.; Candian, A.; Huang, X.; Maltseva, E.; Petrignani, A.; Oomens, J.; Buma, W. J.; Lee, T. J.; Tielens, A. G. G. M. The anharmonic quartic force field infrared spectra of three


Page 26 of 45

Page 27 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


polycyclic aromatic hydrocarbons: Naphthalene, anthracene, and tetracene. J. Chem. Phys. 2015, 143, 224314. (18) Bloino, J. A VPT2 route to Near-Infrared spectroscopy: The role of mechanical and electrical anharmonicity. J. Phys. Chem. A 2015, 119, 5269. (19) Bowman, J. M. Self consisted field energies and wave functions for coupled oscillators. J. Chem. Phys. 1978, 68, 608. (20) Carney, D. G.; Sprandel, L. L.; Kern, C. W. Variational approach to vibration-rotation spectroscopy for polyatomic molecules. Adv. Chem. Phys. 1978, 37, 305-381. (21) Bowman, J. M. The Self-Consistent-Field approach to polyatomic vibrations. Acc. Chem. Res. 1986, 19, 202-208. (22) Christoffel, K. M.; Bowman, J. M. Investigations of self-consistent field, scf ci and virtual state configuration interaction vibrational energies for a model three-mode system. Chem. Phys. Lett. 1982, 85, 220-224. (23) Gerber, R. B.; Chaban, G. M.;Brauer, B.; Miller, Y. in: C.E. Dykstra, G. Frenking, K. Kim, G. Suseria, (Eds.), Theory and Applications of computational chemistry: the first 40 years, Elsevier, 2005, Chapter 9, p165. (24) Matsunaga, N.; and Chaban, G. M. and Gerber, R. B. Degenerate perturbation theory corrections for the vibrational self-consistent field approximation: Method and applications. J. Chem. Phys. 2002, 117, 3541.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 45

(25) Jung, J. O.; Gerber, R. B. Vibrational wave functions and spectroscopy of (H2O)n, n=2,3,4,5: Vibrational self‐consistent field with correlation corrections. J. Chem. Phys. 1996, 105, 10332. (26) Chaban, G.; Jung, J. O.; Gerber, R. B. Degenerate perturbation theory corrections for the vibrational self-consistent field approximation: Method and applications. J. Chem. Phys. 2002, 117, 3541. (27) Chaban, G. M.; Jung, J. O.; Gerber, R. B. Ab initio calculation of anharmonic vibrational states of polyatomic systems: Electronic structure combined with vibrational self-consistent field. J. Chem. Phys. 1999, 111, 1823. (28) Bowman, J. M.; Carter, S.; Huang. X. Multimode: A code to calculate rovibrational energies of polyatomic molecules. Int. Rev. Chem. Phys. 2003, 22, 533-549. (29) Pele, L.; Brauer, B.; Gerber, R. B. An efficient parallelization scheme for molecular dynamics simulations with many-body, flexible, polarizable empirical potentials: application to water. Theor. Chem. Acc. 2007, 117, 69-72. (30) Seidler, P.; Kongsted, P.; Christiansen, O. Calculation of vibrational infrared intensities and raman activities using explicit anharmonic wave functions. J. Phys. Chem. A. 2007, 111, 1120511213. (31) Christiansen, O. Vibrational couple cluster theory. J. Chem. Phys. 2004, 120, 2149. (32) Christiansen, O. Response theory for vibrational wave functions. J. Chem. Phys. 2005, 122, 194105.


Page 29 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(33) Seidler, P.; Christiansen, O. Vibrational excitation energies from vibrational response theory. J. Chem. Phys. 2007, 126, 204101. (34) Nagalakshmi, V.; Lakshminarayana, V.; Sumithra, G.; Durga Prasad, M. Couple cluster description of anharmonic molecular vibrations. Application to O3 and SO2 Chem. Phys. Lett. 1994, 217, 279-282. (35) Durga Prasad, M. Calculation of vibrational spectra by the coupled cluster method Applications to H2S. Indian J. Chem. 2000, 39, 196. (36) Banik, S.; Pal, S.; Durga Prasad, M. Calculation of vibrational energy of molecule using coupled cluster linear response theory in bosonic representation: Convergence studies. J. Chem. Phys. 2008, 129, 134111. (37) Banik, S.; Pal, S.; Durga Prasad, M. Calculation of dipole transition matrix elements and expectation values by vibrational coupled cluster method. J. Chem. Theor. Comput. 2010, 6, 3198-3204. (38) Banik, S.; Pal, S.; Durga Prasad, M. Vibrational multi-reference coupled cluster theory in bosonic representation. J. Chem. Phys. 2012, 137, 114108. (39) Pirali, O.; Vervloet, M.; Mulas, G.; Malloci G.; Joblin, C. High-resolution infrared absorption spectroscopy of thermally excited naphthalene: Measurements and calculations of anharmonic parameters and vibrational interactions. Phys. Chem. Chem. Phys. 2009, 11, 34433454.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 45

(40) Banik, S.; Durga Prasad, M. On the spectral intensities of vibrational transitions in polyatomic molecules: Role of electrical and mechanical anharmonicities. Theor. Chem. Acc. 2012, 131, 1282. (41) Banik, S. On the choice of electronic structure method to calculate the quartic potential energy surface for the vibrational calculations of polyatomic molecules. Theor. Chem. Acc. 2016, 135, 203. (42) Roy, T. K; Durga Prasad, M. Effective harmonic oscillator description of anharmonic molecular vibrations. J. Chem. Sci. 2009, 121, 805-810. (43) Mukherjee, D. On the heirarcy equations of the wave operator for open-shell systems. Pramana, 1979, 12, 203-225. (44) Monkhorst, H. Calculation of properties with the coupled-cluster method. Int. J. Quantum Chem. 1977, 11, 421-432. (45) Mukherjee, D.; Mukherjee, P. K.A response-function approach to the direct calculation of the transition-energy in multiple-cluster expansion formalism. Chem Phys. 1979, 39, 325-335. (46) Comeau, D. C.; Bartlett, R. J. The equation-of-motion coupled-cluster method. Applications to open- and closed-shell reference states. Chem. Phys. Lett. 1993, 207, 414-423. (47) Emrich, K. An extension of the coupled cluster formalism to excited states (I). Nucl. Phys. A 1981, 351, 379-396. (48) Durga Prasad, M. On the calculation of expectation values and transition matrix elements by coupled cluster method. Theor.Chim. Acta. 1994, 88, 383-388.


Page 31 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(49) Nielson, H. H. The vibration-rotation energies of molecules. Rev. Mod. Phys.1951, 23, 90136. (50) Amos. R. D.; Handy. N. C.; Green. W. H.; Jayatilaka. D.; Willets.A; Palmeiri. P. Anharmonic vibrational properties of CH2F2: A comparison of theory and experiment. J. Chem. Phys. 1991, 95, 8323-8336. (51) Willettes, A.; Gaw, J. F.; Green, J. F.; Handy, N. C. “SPECTRO a theoretical spectroscopy package,” version 2.0, Cambridge, UK, 1990. (52) Gaw, J. F.; Willettes, A.;Green, J. F.; Handy, N. C. “SPECTRO a theoretical spectroscopy package,” version 3.0, Cambridge, UK, 1996. (53) Bloino, J.; Baiardi, A.; Biczysko, Z. Aiming an accurate prediction of vibrational and electronic spectra of medium-to-large molecules: An overview. Int. J. Quantum Chem. 2016, 116, 1543-1574. (54) Barone, V.; Biczysko, M.; Bloino, J.; Puzzarini, C. Accurate molecular structures and infrared spectra of trans-2,3-dideuterooxirane, methyloxirane and trans-2,3-dimethyloxirane. J. Chem. Phys. 2014, 141, 034107. (55) Rauhut, G. Efficient calculation of potential energy surfaces for the generation of vibrational wave functions. J. Chem. Phys. 2004, 121, 9313-9322. (56) As mentioned earlier, the EOM-CC method for the description of the excited states can be treated as the vibrational configuration interaction (VCI) method that diagonalizes the effective Hamiltonian in the space of excited states of EHO. In other words, each excited states in the EOM-CC description is written as the linear combination of all possible excited states of EHO.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 45

The co-efficient (VCI co-efficient) such EHO excited states directly gives the account of how the vibrational the modes mixes with each other for a given excited states.


Page 33 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 1. Normal modes of Naphthalene Mode ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 ν10 ν11 ν12 ν13 ν14 ν15 ν16 ν17 ν18

ω (Harm.) (MP2) 3221.6 (0.0) 3191.2 (0.0) 1605.7 (0.0) 1488.3 (0.0) 1436.1 (0.0) 1179.1 (0.0) 1042.6 (0.0) 767.5 (0.0) 516.6 (0.0) 924.1 (0.0) 825.9 (0.0) 596.7 (0.0) 183.5 (0.0) 918.5 (0.0) 717.5 (0.0) 387.1 (0.0) 3206.5 (42.8) 3185.6 (9.9)

Mode description

Mode

Aromatic C-H str.

ν19

Aromatic C-H str.

ν20

C-C str (major)

ν21

H-C-C i-p* bend (major)

ν22

C-C str. (major)

ν23

H-C-C i-p bend (major)

ν24

C-C str. + H-C-C i-p bend C-C str. + H-C-C i-p bend + C-C-C bend C-C-C ring bend

ν25

H-C-C o-o-p# bend

ν28

H-C-C o-o-p bend

ν29

ring torsion

ν30

ring torsion

ν31

H-C-C o-o-p bend

ν32

H-C-C o-o-p bend

ν33

ring torsion

ν34

Aromatic C-H str.

ν35

Aromatic C-H str.

ν36

ν26 ν27

ω (Harm.) (MP2) 1622.7 (3.5) 1413.1 (6.6) 1284.2 (7.8) 1143.3 (2.3) 808.2 (0.0) 358.2 (0.9) 938.2 (0.0) 846.7 (0.0) 677.3 (0.0) 462.5 (0.0) 3220.7 (37.5) 3188.7 (0.04) 1551.4 (7.9) 1471.5 (1.0) 1251.2 (1.5) 1177.5 (0.2) 1034.2 (5.6) 624.9 (3.4)


Mode description C-C str. + H-C-C i-p bend H-C-C i-p bend H-C-C i-p bend H-C-C i-p bend C-C-C ring bend + H-CC i-p bend C-C-C ring bend H-C-C o-o-p bend H-C-C o-o-p bend ring torsion ring torsion Aromatic C-H str. Aromatic C-H str. C-C str. + H-C-C i-p bend H-C-C i-p bend H-C-C i-p bend H-C-C i-p bend C-C str. + H-C-C i-p bend ring torsion


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 45

Table 1 (continued..). Normal modes of Naphthalene Mode

ω (Harm) Mode description MP2 ν37 3206.0 Aromatic C-H str. (0.0) ν38 3183.8 Aromatic C-H str. (0.0) ν39 1670.1 C-C str. + H-C-C i-p (0.0) bend ν40 1484.4 H-C-C i-p bend (0.0) ν41 1262.8 H-C-C i-p bend (0.0) 1163.4 C-C str. + H-C-C i-p ν42 (0.0) bend * # i-p: in-plane, o-o-p: out-of-plane

Mode ν43 ν44 ν45 ν46 ν47 ν48

ω (Harm) MP2 945.3 (0.0) 510.3 (0.0) 926.3 (1.3) 782.5 (105.0) 474.5 (16.5) 169.7 (2.1)

Mode description C-C-C ring bend + HC-C i-p bend C-C-C ring bend H-C-C o-o-p bend H-C-C o-o-p bend ring torsion ring torsion

Harmonic intensities in km/mol of each normal mode are reported within parentheses.


Page 35 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2. Comparison between VSCF, VCCM and VPT2 results with experimental frequencies of naphthalene Mode ag ν1 ν2 ν3 ν4 ν5 ν6 ν7 au ν10 ν11 b1g ν14 b1uν17

MP2/6-311G(2d,2p) ν(VSCF) ν(VPT2) ν(VCCM) 3115.3 3086.5 3062.0 3085.5 3068.7 3047.5 1583.9 1567.3 1568.1 1475.2 1461.5 1460.0 1418.9 1404.6 1411.0 1182.2 1167.9 1166.1 1038.7 1028.0 1027.3 1039.3 1009.2 1008.6 892.8 843.1 851.4 993.7 952.9 957.9 3101.9 3080.9 3046.7

B3LYP/6-311G(2d,2p) ν(VSCF) ν(VPT2) ν(VCCM) 3073.3 3050.6 3002.6 3046.1 3032.7 2971.2 1589.6 1574.2 1573.7 1480.5 1471.0 1465.9 1371.5 1358.8 1363.1 1187.6 1174.6 1171.8 1041.2 1032.6 1030.2 1026.5 981.1 994.9 887.5 832.1 849.2 993.7 945.9 959.9 3059.6 3041.3 3020.1

Expt. 3057* 3051* 1576* 1464* 1383* 1168* 1025* 952* 3058.2

ν18

3083.0

3080.4

3005.0

3042.0

3004.9

2946.9

3005.7

ν19 ν20 ν21 ν22 ν23 b2gν25 ν26 b2uν29

1603. 1407.8 1277.2 1144.0 807.1 1029.5 925.8 3103.4

1588.2 1397.7 1269.1 1134.2 802.8 996.7 884.7 3094.9

1587.2 1394.5 1268.2 1132.7 802.6 998.7 887.8 3066.3

1619.8 1409.5 1278.5 1148.0 805.9 1030.5 923.7 3060.2

1605.2 1398.2 1276.9 1138.1 801.6 986.7 885.8 3048.6

1603.5 1395.8 1270.0 1136.7 801.6 998.7 895.7 3017.4

1603.1 1389.5 1267.4 1129.4 979* 880* 3067.9

ν30

3074.3

3071.9

3053.2

3033.9

3022.0

2964.4

3032.3

ν31 ν32 ν33 ν34 ν35

1532.2 1456.8 1242.7 1178.3 1027.6

1517.6 1439.7 1231.0 1163.0 1016.9

1517.1 1437.7 1231.4 1162.6 1017.4

1529.8 1376.6 1222.3 1164.9 1027.3

1514.7 1364.7 1214.6 1149.9 1017.8

1514.4 1358.1 1211.3 1150.8 1017.9

1514.5 1011.6



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 45

Table 2 (continued..) Comparison between VSCF, VCCM and VPT2 results with experimental frequencies of naphthalene Mode

MP2/6-311G(2d,2p) ν(VSCF) ν(VPT2 ) ν (VCCM)

B3LYP/6-311G(2d,2p) ν(VSCF) ν(VPT2) ν(VCCM)

Expt.

3099.9 b3g ν37 3087.7 3062.8 3056.3 3038.7 3008.8 3057* 3080.0 ν38 3037.2 3002.3 3038.9 3009.0 2967.6 3018* 1646.7 ν39 1628.2 1629.3 1646.3 1627.6 1628.2 1629* 1472.1 1461.9 1458.8 1477.5 1468.7 1463.9 1458* ν40 ν41 1261.7 1251.3 1249.1 1264.4 1250.7 1252.4 1240* ν42 1162.4 1149.7 1148.8 1167.3 1156.3 1153.7 1145* 944.0 940.9 941.1 947.2 943.6 944.2 936* ν43 b3u ν45 1010.9 974.5 977.6 1009.2 966.5 978.0 957.8 ν46 842.8 792.6 802.3 841.5 789.5 804.9 781.9 1 SD 37.0 21.3 11.4 26.2 16.6 31.9 1 SD is the standard deviation. The modes included in the VCCM calculations are reported here. *

The values are taken from the Raman spectra of reference 12.


0.75

Expt. 3067.9 (ν29)

0.50

3077.4 (ν39+ν32)

0.25

3058.3(ν17) 3043.8 3032.3 (ν30)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Relative Intensity

Page 37 of 45

3101.8 (ν31+ν3)

0.00

3005.7 (ν18) 2978.2 (ν31+ν4) 2849.5 (ν4+ν20)

VCCM

0.75 0.50 0.25 0.00 3300

3200

3100

3000

2900

2800

2700

2600

Wavenumber (cm-1)

Figure 1. Experimental (Blue) and Theoretical (Red) stick spectra of naphthalene. Y axis left hand side relative intensity (experiment) and right hand side (theory). Theoretical intensities and the transition energies were calculated using VCCM. See Table 4 for the details description of VCCM transitions.

1.00

2275.1 (ν17)

2283.1 (ν43+ν31)

Expt.

0.75

2260.1 (ν30)

2300.8 (ν29)

0.50

2253.1 (ν18)

0.25 0.00

VCCM

0.50 0.25 0.00 2500

2450

2400

2350

2300

2250

2200

Wavenumber (cm-1)

Figure 2. Experimental (Blue) and Theoretical (VCCM, Red) stick spectra of naphthalene-d8. Y axis left hand side relative intensity (experiment) and right hand side (theory). Theoretical intensities and the transition energies were calculated using VCCM. See Table 4 for the detailed description of the VCCM transitions.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 45

Table 3. VCCM, VPT2, VSCF calculated frequencies, experimental frequencies and state descriptions of naphthalene and naphthalene-d8 in MP2 (top) and B3LYP (bottom) methods. Calculated IR intensities (km/mol) by VCCM and VPT2 method are given in parentheses.

No

Sym.

Theo. Freq.

Expt.

VCCM

VPT2

VSCF

3094.8(6.7) 3091.0(1.2) 3078(13.1) 3000.1(1.0) 3035.7(3.0) 3070.9(1.5) 2997.6(2.6) 2972.8(4.4) 2996.6(3.1) 2924.1(2.4) 2969.9(2.1) a* 2968.6(1.8) 2964.4(3.7) 2850.9(1.3) 2854.9(3.3) 2846.4(1.2) 2851.4(1.3)

3071.8(2.9) 3098.8(0.01) 3051.0(7.0) 2980.4(2.5) 3042.6(0.6) 3069.3(4.2) 2989.4(0.1) 2956.4(0.4) 3001.4(0.5) 2933.8(0.4) 2974.0(0.3) 2974.8(1.1) 2975.5(0.2) 2980.5(0.6) 2856(0.05) 2865.4(0.3) 2856.9(0.02) 2861.1(0.2)

3115.3 3119 3102.6 3023 3075.1 3097 3021.9 2991 3041.4 2966 3004.3 3007 3007.9 3010 2885.0 2892 2882.7 2889

State description

C10H8 1

b2u

2

b1u

3

b2u

4.

b1u

5.

b2u

6.

b1u

7.

b2u

8.

b1u

9.

b2u

3101.8

(ν 3 + ν 31 )

3077.4

(ν 32 + ν 39 )

?

(ν 19 + ν 40 )

3017.5(?)

(ν 5 + ν 19 )

?

(ν 3 + ν 32 )

?

(ν 31 + ν 40 )

2978.2

(ν 4 + ν 31 )

2849.5

(ν 4 + ν 20 )

?

(ν 20 + ν 40 )

C10D8 1.

b1u

2286.3(2.4) 2286.5(0.2) 2305 2283.1 2283.0(0.3) 2280.6(0.2) 2301 2. b1u 2239.6(4.8) 2296.4(0.5) 2268 2241.8 2141.2(0.7) 2144.3(0.1) 2164 3. b2u 2276.0(2.3) 2282.3(0.04) 2291 ? 2289.1(0.3) 2287.9(0.7) 2299 4. b2u 2287.2(2.9) 2285.7(0.6) 2310 ? 2294.0(1.3) 2287.1(1.2) 2310 a* We do not find any state that has dominating contribution from zeroth order (ν 31 with this B3LYP PES in our VCCM calculation.


(ν 31 + ν 43 ) (ν 32 + ν 43 ) (ν 20 + ν 41 ) (ν 31 + ν 7 ) + ν 40 ) state

Page 39 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 4. List of some bright states in the VCCM calculations with MP2 PES in the region dominated by C-H or C-D fundamentals for naphthalene and naphthalene-d8.Two most significant contributions are printed.

Sym. b1u

ν 17 ← 0 . 419 (ν 17 ) − 0 . 167 (ν 32 + ν 39 )

VCCM Transition energy 3046.7

b1u

ν 32 + ν 39 ← 0 . 204 (ν 17 ) + 0 . 627 (ν 32 + ν 39 )

3078.1

b1u

ν 18 ← 0 . 396 (ν 18 ) − 0 . 221 (ν 4 + ν 19 )

3005.0

b1u

ν 4 + ν 19 ← 0 . 238 (ν 18 ) + 0 . 632 (ν 4 + ν 19 )

3063.5

b2u

State description (C10H8)

ν 3 + ν 32 ← 0 . 626 (ν 3 + ν 32 ) − 0 . 088 (ν 29 )

2996.6

b2u

ν 4 + ν 31 ← 0 . 822 (ν 4 + ν 31 ) − 0 . 044 (ν 30 )

2968.6

b2u

ν 29 ← 0 . 309 (ν 29 ) + 0 . 255 (ν 19 + ν 40 )

3066.3

b2u

ν 3 + ν 31 ← 0 . 632 (ν 3 + ν 31 ) − 0 . 178 (ν 29 )

3094.8

b2u

ν 30 ← 0 . 285 (ν 30 ) + 0 . 268 (ν 11 + ν 20 + ν 46 )

3053.2

Sym.

State description (C10D8)

VCCM Transition energy

b1u

ν 17 ← 0 . 597 (ν 17 ) + 0 . 142 (ν 23 + ν 3 )

2274.2

b1u

ν 31 + ν 43 ← 0 . 446 (ν 31 + ν 43 ) − 0 . 373 (ν 23 + ν 3 )

2286.3

b1u

ν 18 ← 0 . 642 (ν 18 ) − 0 . 110 (ν 32 + ν 43 )

2258.4

b1u

ν 32 + ν 43 ← 0 . 734 (ν 32 + ν 43 ) + 0 . 134 (ν 18 )

2239.6

b2u

ν 29 ← 0 . 440 (ν 31 + ν 7 ) − 0 . 356 (ν 29 )

2301.5

b2u

ν 31 + ν 7 ← 0 . 299 (ν 31 + ν 7 ) + 0 .258 (ν 20 + ν 41 )

2287.2

b2u

ν 30 ← 0 . 404 (ν 30 ) − 0 . 099 (ν 35 + ν 4 )

2259.9



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(a)

(b)

Figure 3 (a) and (b): Comparison among VSCF, VPT2 and VCCM calculations with MP2 and B3LYP PESs of naphthalene and naphthalene-d8


Page 40 of 45

Page 41 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table of Contents



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Comparison among VSCF, VPT2 and VCCM level of theory with MP2 and B3LYP PES of naphthalene and naphthalene-d8 254x190mm (300 x 300 DPI)

ACS Paragon Plus Environment

Page 42 of 45

Page 43 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Comparison among VSCF, VPT2 and VCCM level of theory with MP2 and B3LYP PES of naphthalene and naphthalene-d8 254x190mm (300 x 300 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 1. Experimental (Blue) and Theoretical (Red) stick spectra of naphthalene. Y axis left hand side relative intensity (experiment) and right hand side (theory). Theoretical intensities and the transition energies were calculated using VCCM. See Table 4 for the details description of VCCM transitions. 289x202mm (300 x 300 DPI)


Page 44 of 45

Page 45 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 2. Experimental (Blue) and Theoretical (VCCM, Red) stick spectra of naphthalene-d8. Y axis left hand side relative intensity (experiment) and right hand side (theory). Theoretical intensities and the transition energies were calculated using VCCM. See Table 4 for the detailed description of the VCCM transitions. 289x202mm (300 x 300 DPI)


Anharmonicity in the Vibrational Spectra of ... - ACS Publications

Recommend Documents